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[ [ "Photon-assisted confinement-induced resonances for ultracold atoms" ], [ "Abstract We solve the two-particle s-wave scattering for an ultracold atom gas confined in a quasi-one-dimensional trapping potential which is periodically modulated.", "The interaction between the atoms is included in terms of Fermi's pseudopotential.", "For a modulated isotropic transverse harmonic confinement, the atomic center of mass and relative degrees of freedom decouple and an exact solution is possible.", "We use the Floquet approach to show that additional photon-assisted resonant scattering channels open up due to the harmonic modulation.", "Applying the Bethe-Peierls boundary condition, we obtain the general scattering solution of the time-dependent Schr\\\"odinger equation which is universal at low energies.", "The binding energies and the effective one-dimensional scattering length can be controlled by the external driving." ], [ "Photon-assisted confinement-induced resonances for ultracold atoms Vicente Leyton$^1$ Maryam Roghani$^1$ Vittorio Peano$^2$ Michael Thorwart$^1$ $^1$ I. Institut für Theoretische Physik, Universität Hamburg, Jungiusstraße 9, 20355 Hamburg, Germany $^2$ Institute for Theoretical Physics II, Friedrich-Alexander-Universität Erlangen-Nürnberg, Staudtstraße 7, 91058 Erlangen, Germany We solve the two-particle s-wave scattering for an ultracold atom gas confined in a quasi-one-dimensional trapping potential which is periodically modulated.", "The interaction between the atoms is included in terms of Fermi's pseudopotential.", "For a modulated isotropic transverse harmonic confinement, the atomic center of mass and relative degrees of freedom decouple and an exact solution is possible.", "We use the Floquet approach to show that additional photon-assisted resonant scattering channels open up due to the harmonic modulation.", "Applying the Bethe-Peierls boundary condition, we obtain the general scattering solution of the time-dependent Schrödinger equation which is universal at low energies.", "The binding energies and the effective one-dimensional scattering length can be controlled by the external driving.", "34.50.-s, 03.65.Nk, 05.30.Jp, 37.10.Jk The ability to accurately control the effective atomic interactions in ultracold atom gases has opened the doorway to novel exciting physics in the recent years.", "The currently available experimental tools allow for a powerful implementation of analog quantum simulators realized by cold-atom assemblies [1], [2].", "In the regime of strong atomic interactions, the quantum gas becomes scale-invariant and shows universal physical aspects quantified in terms of a few dimensionless coefficients.", "For instance, the three-dimensional (3D) $s$ -wave scattering length $a$ characterizes the interatomic interactions and has to be compared to the mean interatomic distance which is of the order of the particles' inverse momenta $k^{-1}$ .", "It can be controlled over several orders of magnitude via the help of a magnetic field tuned accross a Feshbach resonance [3].", "In 3D, the tunability of the scattering length, for instance, reveals the cross-over from the weakly interacting Bardeen-Cooper-Schrieffer superfluid state, where $1/(ka)\\rightarrow -\\infty $ , to the strongly interacting Bose-Einstein condensate of dimer molecules, where $1/(ka)\\rightarrow +\\infty $ [1].", "In quasi-one-dimensional (1D) gases, another relevant length scale appears in form of the transverse confinement length $a_\\perp $ .", "Then, the scattering of two tightly confined quantum particles is known to induce universal low-energy features in form of confinement-induced resonances (CIRs) [4], [5], [6].", "At low energies, only the transverse ground state of the confining potential is significantly populated whereas the higher transverse states can be only virtually populated during the elastic collisions.", "In this regime, the remaining scattering processes in the longitudinal direction can be characterized by the effective 1D interaction strength $g_{1D}$ .", "It is governed by a single parameter, being the ratio of $a$ and the zero-point-fluctuation length scale $a_\\perp $ , irrespective of the details of the confinement.", "By tuning the confinement strength (or the $3D$ scattering length via a Feshbach resonance [3]) across a CIR, it is possible to cross over from strongly repulsive to strongly attractive interactions.", "CIRs have been observed in a strongly confined 1D gas of fermionic K atoms [7] and of bosonic Cs atoms [8].", "This has allowed to investigate the cross-over from a strongly repulsive Tonks-Girardeau gas to a strongly attractive Super-Tonks-Girardeau gas [8].", "CIRs have also been observed in a strongly interacting 2D Fermi gas [9] and in mixed dimensions [10].", "In close analogy to a Feshbach resonance, the CIR occurs when the continuum threshold for the lowest transverse state (the open channel) has the same energy as a bound state formed by two particles being in some transverse excited states (the closed channels) [5].", "Put differently, the transverse orbital degrees of freedom of the confined atoms play the same role as the internal atomic spin degrees of freedom for a Feshbach resonance [3].", "When the transverse confinement of two equal atom species is purely harmonic, only one such bound state exists, leading to a single universal CIR [4], [5], [6].", "This feature can be traced back to the separability of the center of mass and relative coordinates [11].", "In turn, a multitude of CIRs appears when these degrees of freedom are no longer separable, i.e., for a mixture of different species [11], anisotropic [12], [13], [14] and anharmonic confinement [11], [15], [14], [12], [13], [16], and in mixed dimensions [17], [10].", "Dipolar CIRs have been predicted to occur also in presence of long-range anisotropic interactions between different atomic angular momentum states [18].", "Coupled CIRs have also been predicted at higher energies when all partial scattering waves are taken into account [19].", "As an alternative to the “orbital” Feshbach resonance to control the atomic scattering, a time-dependent modulation of internal atomic states generates an optical Feshbach resonance [20].", "It occurs when the optical radiation resonantly couples two atoms in their respective electronic ground state to a molecular state formed by electronically excited states, i.e., the relevant closed channels in this case.", "This mechanism has been also experimentally demonstrated [21].", "In this work, we propose a novel mechanism to coherently manipulate the scattering properties of cold atoms in quasi-1D traps via a time-dependent RF modulation of the trapping potential.", "We consider a tight trap where the atoms have been initially cooled to their transverse ground state.", "The trap eigenfrequency $\\omega _0$ is modulated by a periodic field such that $\\omega ^2(t)=\\omega ^2_0-F\\cos \\omega _{\\rm ex}t$ .", "The modulation is switched on adiabatically in order to keep all atoms in the same transverse Floquet quantum state.", "We show that a new type of CIR is induced by the virtual molecular recombination of the atoms (in their electronic ground state) during the collision.", "This process is mediated by the emission of $m$ virtual photons by the scattering atoms at the continuum threshold $\\hbar \\omega _0$ , see the level scheme in Fig.", "REF .", "Thus, the virtual transition to the bound state with energy $E_B$ becomes resonant when $\\hbar (\\omega _0- m\\omega _{\\rm ex})=E_B$ , leading to a series of photon-assisted CIRs.", "This process is fundamentally different from any type of Feshbach resonances as it does not involve a bound state formed by the closed channels.", "By tuning the modulation field parameters, the photon-assisted CIRs can easily be tuned.", "Figure: Sketch of the energy level scheme.", "A standard CIR occurswhenthe energy of the scattering atoms at the continuum threshold (horizontal line,blue online) matches the energy E B * E^*_B of a bound state formed by the closedchannels (dashed line, light green online).", "The energy E B E_B of the realbound state is shown as a continuum line (dark green online).", "A photon-assistedCIR occurs when the binding energy ℏω 0 -E B \\hbar \\omega _0-E_B matches the energy ofmm photons.", "The photon energy ℏω ex \\hbar \\omega _{\\rm ex} is indicated by wavy linesat the photon-assisted CIRs for the single-photon and two-photon resonances.The quasienergy is the energy folded on an energy interval ℏω ex \\hbar \\omega _{\\rm ex}In a frame of reference where the centre of mass of the two atoms is at rest, the two-body problem can be mapped to the scattering of a trapped single particle, with the reduced mass $\\mu $ , by a central potential $U(r)$ .", "The centre of mass and relative degrees of freedom decouple for a harmonic trap also in presence of a parametric time-dependent modulation.", "The Hamiltonian can then be written as $ H(r,t) = H_0(t) +U(r), \\quad H_0(r,t) = \\frac{p^2 }{2\\mu }+ \\frac{1}{2}\\mu \\, \\omega ^2(t) r^2_\\perp .$ We set the $z$ -direction as the direction of free evolution.", "The confinement is defined over the $x-y$ -plane, implying that $r_{\\perp }=(x,y)$ .", "Furthermore, we choose the frequency $\\omega (t)$ such that the solutions of the classical equation of motion $\\ddot{x}+\\omega ^2(t)x=0$ are stable.", "We consider the limit where the interaction range is much shorter than the amplitude of zero point fluctuations $a_\\perp =(\\mu \\omega _0)^{-1/2}$ in the static trap ($\\hbar =1$ ).", "Thus, the scattering dynamics is governed by a single parameter, the 3D scattering length $a$ .", "In this limit, the low-energy interparticle interaction $U(r)$ can be described by Fermi's pseudopotential [22] $U({r}) = \\frac{2\\pi a}{\\mu } \\delta ({r})\\frac{\\partial }{\\partial r} r .$ We consider the atoms initially in the adiabatic transverse ground state and having a longitudinal momentum $k$ .", "They are described by the incoming wave function $\\psi _{\\rm in}(t)=\\exp [-i\\varepsilon t]\\exp [ikz] u_0(x,t)u_0(y,t),$ where $\\varepsilon $ is the quasienergy $\\varepsilon =k^2/2\\mu +\\nu $ of the atoms.", "Here, the time-periodic functions $u_n(x,t)$ are the Floquet eigenstates of the parametrically driven harmonic oscillator with quasienergy $(n+1/2)\\nu $ [23], [24], [25].", "Notice that, when the driving is switched off adiabatically, $\\nu \\rightarrow \\omega _0$ and the $u_n(x)$ become the eigenstates of the harmonic oscillator.", "Our goal is to compute the full solution $\\psi (t)$ which includes the scattered wave $\\psi _{\\rm out}(t)$ such that $\\psi (t)=\\psi _{\\rm in}(t)+\\psi _{\\rm out}(t)$ .", "It is most convenient to introduce the time-periodic Floquet state $\\phi (t)=\\exp [i\\varepsilon t]\\psi (t)$ as the solution of the eigenvalue problem ${\\cal H}(r,t)\\, \\phi (r,t) = \\varepsilon \\,\\phi (r,t)$ , where ${\\cal H}\\equiv H - i\\partial _t$ is the Floquet Hamiltonian [24].", "With $T=2\\pi /\\omega _{\\rm ex}$ being the external modulation period, we can formally write $\\phi (r,t) = \\phi _{\\rm in}(r,t) + \\int _0^T \\frac{dt^{\\prime }}{T}{\\cal G}_\\varepsilon (r,0;t,t^{\\prime }) \\frac{f (t^{\\prime })}{2 \\mu },$ where $\\phi _{\\rm in}(t)=\\exp [i\\varepsilon t]\\psi _{\\rm in}(t)$ and the second term on the r.h.s.", "represents the outgoing scattered wave function $\\phi _{\\rm out}(t)$ .", "The intergral kernel ${\\cal G}_\\varepsilon =({\\cal H}_0 - \\varepsilon - i0)^{-1}$ is the retarded Floquet-Green's function with ${\\cal H}_0 = H_0 -i \\partial _t$ .", "This wave function has to fullfill the Bethe-Peierls boundary condition $\\phi (r\\rightarrow 0,t) \\simeq \\left(1-\\frac{r}{a}\\right) \\frac{f(t)}{4\\pi r}\\, ,$ with the Bethe-Peierls amplitude $f(t)$ yet to be determined.", "For large $z$ , the asymptotic scattered wave can be decomposed into partial waves as $\\phi _{\\rm out}({\\bf {r}})\\approx \\sum _{\\begin{array}{c}\\mathbf {n} \\\\m= {\\rm open}\\end{array}}S^m_{\\mathbf {n}}\\left(\\frac{k}{k_{nm}}\\right)^{1/2}e^{ik_{nm}|z|}u_{n_x}(x,t)u_{n_y}(y,t)\\,.$ Here, $|S^m_{\\mathbf {n}}|^2$ is the probability of the atoms to be excited into the transverse state with quantum number ${\\bf n}=(n_x,n_y)$ after absorbing $m>0$ photons from the field and thereby acquiring the momentum $k_{nm}$ , which is determined via $\\frac{k_{nm}^2}{2\\mu }+(n+1)\\nu =\\varepsilon +m\\omega _{\\rm ex},\\qquad n=n_x+n_y\\, .$ The number of available open channels depends on $m$ .", "The S-matrix elements $S^m_{\\mathbf {n}}$ are determined by the Bethe-Peierls boundary condition Eq.", "(REF ) (for technical detials, see Ref.", "[25]) as $S^m_{\\mathbf {n}} =\\frac{i}{2\\sqrt{kk_{nm}}T}\\int _0^{T}\\!\\!", "dt^\\prime e^{im\\omega _{\\rm ex}t}u^*_{n_x}(0,t^{\\prime })u^*_{n_y}(0,t^{\\prime })f(t^{\\prime })\\,.$ Next, we insert Eq.", "(REF ) into Eq.", "(REF ) and define a scalar product $\\langle f|g\\rangle =T^{-1}\\int _0^T dt \\, f^*(t)g(t)$ on the Hilbert space of time-periodic functions.", "Then, we can derive an inhomogeneous linear equation for the Bethe-Peierls amplitude $f(t)$ as $\\left(\\zeta _\\varepsilon +\\frac{a_\\perp }{a}\\right) |f \\rangle =-4\\pi {\\cal N}^{-1/2}|{\\rm in}\\rangle \\ .$ Here, we have introduced the regularized integral kernel $\\langle t|\\zeta _\\varepsilon |t^{\\prime }\\rangle =\\frac{2\\pi a_\\perp }{ \\mu }\\left[ {\\cal G}_\\varepsilon (r,0;t,t^{\\prime }) -\\delta (t-t^{\\prime })\\frac{T\\mu }{2\\pi r}\\right]_{r\\rightarrow 0}$ and the normalized vector $|{\\rm in}\\rangle $ via $\\langle t|{\\rm in}\\rangle ={\\cal N}^{1/2}a_\\perp \\phi _{\\rm in}(0,t)$ .", "For small initial momenta, $k\\ll k_{nm}$ , the scattering is dominated by elastic collisions.", "In this regime, it is convenient to divide the kernel $\\zeta _\\varepsilon $ into a smooth part $\\tilde{\\zeta }_\\varepsilon $ , that can be evaluated for $k=0$ , and the contribution of the channel of the incoming atoms, where the $k$ -dependence is retained.", "This yields [25] $|f\\rangle = -4\\pi {\\cal N}^{-1/2}\\frac{ ka_{\\rm 1D}}{ ka_{\\rm 1D}-i}\\left(\\tilde{\\zeta }_\\nu + \\frac{a_\\perp }{ a}\\right)^{-1}|{\\rm in}\\rangle \\, .$ Here, it is convenient to introduce the effective 1D scattering length $a_{1D}$ for the longitudinal scattering according to $\\frac{a_\\perp }{a_{\\rm 1D}} &=& -2\\pi {\\cal N}^{-1} \\langle {\\rm in}| \\left[\\tilde{\\zeta }_\\nu +\\frac{a_\\perp }{a} \\right]^{-1}|{\\rm in} \\rangle \\, .$ Since the operator $\\tilde{\\zeta }_\\nu $ is not Hermitian, it acquires an imaginary part whose meaning is discussed further below.", "Using Eq.", "(REF ) in Eq.", "(REF ), we obtain the $S$ -matrix element $S^0_{00}=-\\frac{i}{k a_{\\rm 1D}-i}\\,.$ This is the probability amplitude for the reflection of a 1D particle due to the effective scattering potential $U_{\\rm 1D}=g_{\\rm 1D}\\delta (z)$ with complex interaction strength $g_{1D}=-1/(\\mu a_{\\rm 1D})$ .", "Its imaginary part refers to the loss of atoms in the excited transverse states which occurs due to inelastic scattering processes into other channels provided by the modulation.", "From this, we obtain the elastic cross section, which is the probability of an elastic scattering event, as $\\sigma _\\ell =|S^0_{00}|^2$ and its inelastic counterpart as $\\sigma _r=1-\\sigma _\\ell -|1+S^0_{00}|^2$ , respectively.", "From Eq.", "(REF ), we find $&&\\sigma _\\ell =\\left( 1+k^2|a_{\\rm 1D}|^2-2k\\, {\\rm Im}\\, a_{\\rm 1D}\\right)^{-1},\\nonumber \\\\&&\\sigma _r=-2\\sigma _\\ell k\\, {\\rm Im}\\, a_{\\rm 1D},$ (${\\rm Im}\\, a_{\\rm 1D}<0$ ).", "The effective quasi-1D scattering cross sections (for the case $|{\\rm Im }\\, a_{\\rm 1D}|/|a_{\\rm 1D}|=0.15$ ) are shown in Fig.", "REF .", "The probability of an elastic scattering event tends to one for small realtive momenta of the scattering atoms, $k\\ll 1/|a_{\\rm 1D}|$ .", "On the other hand, the scattering is dominated by inelastic scattering events for comparatively larger momenta, $k\\gg |{\\rm Im}\\,a_{\\rm 1D}|^{-1}$ .", "Hence, when $|a_{\\rm 1D}|$ becomes smaller than the typical longitudinal de Broglie wavelength of the atoms a scattering resonance results.", "Figure: Effective scattering cross sections in quasi-1D: total(σ\\sigma ) andelastic (σ ℓ \\sigma _\\ell ) cross section as a function of the relative longitudinalmomentum kk for | Im a 1D |/|a 1D |=0.15|{\\rm Im }\\, a_{\\rm 1D}|/|a_{\\rm 1D}|=0.15.Formally, the 1D scattering length $a_{\\rm 1D}$ can be expressed in terms of the spectrum $\\lbrace \\lambda _m\\rbrace $ and the right and left eigenvalues $|v_m^R\\rangle $ and $|v_m^L\\rangle $ of the kernel $\\tilde{\\zeta }_\\varepsilon $ , $\\frac{a_\\perp }{ a_{\\rm 1D}} =-2\\pi {\\cal N}^{-1}\\sum _m\\frac{\\langle {\\rm in}|v^R_m\\rangle \\langle v^L_m|{\\rm in}\\rangle }{\\lambda _m+a_\\perp / a} \\, .$ Hence, if several eigenvectors significantly overlap with the vector $|{\\rm in}\\rangle $ of the incoming wave, more than one scattering resonances may occur.", "The resonances in $|a_\\perp /a_{\\rm 1D}|$ are well resolved Lorentzian peaks with their center determined by the resonance condition $a_\\perp /a={\\rm Re}\\, \\lambda _m$ and with their line width given by ${\\rm Im} \\, \\lambda _m$ , if their mutual distance exceeds the corresponding widths.", "Figure: Photon-assisted broadening of the standard CIR.", "Theimaginary and real parts ofa ⊥ /a 1D a_\\perp /a_{\\rm 1D} are shown in (a) and (b), respectively, for ω ex =1.2ω 0 \\omega _{\\rm ex}=1.2 \\omega _0 and F=10 -4 ω 0 2 F=10^{-4}\\omega _0^2.", "The dashed vertical line indicatesthe resonance position in absence of driving,a ⊥ /a=-ζ(1/2,1)a_\\perp /a=-\\zeta (1/2,1).", "The resonance is broadened by driving-inducedinelastictransitions to excited transverse channels.", "(c) The width of the CIR has aquadratic dependence on the driving amplitude,Im λ 0 ∝F 2 \\lambda _0 \\propto F^2.The kernel $\\tilde{\\zeta }_\\varepsilon $ can be computed fully analytically only for the trivial case $F=0$ , see Supplemental Information [25].", "In this case, the left and right eigenvectors, $|v_m^L\\rangle $ and $|v_m^R\\rangle $ , are plane waves, $\\langle t|v_m^R\\rangle ^*=\\langle v^L_R|t\\rangle =\\exp [im\\omega _{\\rm ex}t]$ , and the incoming wave is given by $|{\\rm in}\\rangle =|v^R_0\\rangle $ .", "Thus, the overlap is zero for $m\\ne 0$ and we recover the standard result: there is only one CIR for $a_\\perp /a=-\\lambda _0=-\\zeta (1/2,1)$ ($\\zeta (x,y)$ is the Hurwitz zeta function) when the energy of the virtual bound state formed by the closed channels, coincides with the continuum threshold [4].", "If the driving is weak and far away from any parametric resonance $F\\ll \\omega _{\\rm ex}^2,\\qquad |2\\omega _0-m\\omega _d|\\gg \\omega _d (4F/\\omega _d^2)^m,$ we expect the eigenvalues $\\lbrace \\lambda _m\\rbrace $ of the kernel $\\zeta _\\varepsilon $ to only smoothly deviate from their values for $F=0$ .", "To obtain a specific result, we evaluate and diagonalize the kernel $\\tilde{\\zeta }_\\varepsilon $ with a semi-analytical procedure outlined in the Supplementary Material [25].", "The standard CIR for a weak non-resonant driving is shown in Figs.", "REF (a) and (b).", "The zero-photon resonance is clearly broadened by the inelastic collisions.", "In Fig.", "REF (c) we show that the width ${\\rm Im}\\lambda _0\\propto F^2$ .", "Figure: Photon-assisted confinement-induced resonances (imaginarypart ofa ⊥ /a 1D a_\\perp /a_{\\rm 1D} in (a) and real part in (b)) due to a one-photonabsorption from and subsequent emission into the driving field (m=-1m=-1) forω ex =1.2ω 0 \\omega _{\\rm ex}=1.2 \\omega _0 and for varying modulation amplitudes FF.For a finite driving there is no selection rule preventing the remaining eigenvectors of the kernel $\\tilde{\\zeta }_\\varepsilon $ to yield a finite contribution in the r.h.s of Eq.", "(REF ) to $a_{1D}$ .", "For weak non-resonant driving, we can label the eigenvalues $\\lambda _m$ with the number $m$ of radiofrequency photons which are virtually absorbed (emitted for $m<0$ ) and later re-emitted (re-absorbed) during an elastic collision.", "From Eq.", "(REF ), we see that the contribution to $a_{1D}$ from the processes where $m$ photons are virtually absorbed is largest for $a_\\perp /a=-{\\rm Re}\\lambda _{m}$ .", "In the Supplemental material, we show that, for the case $m>0$ , this occurs when the $m$ -photon transition from the continuum threshold $\\hbar \\omega _0$ to the virtual bound state with energy $E_B^*(m)$ formed by the transverse channels which are still closed after the absorption of $m$ photons [with transverse energy $E>E_M={\\rm Int} [m\\omega _{\\rm ex}+\\omega _0]$ ] is resonant, $E^*_B (m)=\\omega _0+m\\omega _{\\rm ex}$ .", "We do not expect that these processes lead to a scattering resonance because the corresponding bound states leak very quickly into the open channels.", "In fact, the imaginary part ${\\rm Im}\\lambda _m$ can be shown to be finite even for $F\\rightarrow 0$ , ${\\rm Im}\\lambda _m\\gtrsim 1$ [25].", "On the other hand,the contribution to $a_{1D}$ from processes where the photons are first emitted ($m<0$ ) is largest for [25] $\\frac{a_\\perp }{a}=-{\\rm Re}\\lambda _{m}\\approx -\\zeta (1/2, |m|\\omega _{\\rm ex}/2\\omega _0)$ Notice that the energy $E_B$ of the molecular bound state (for $F=0$ ) is given by $a_\\perp /a=-\\zeta (1/2, (E_B-1)/2\\omega _0)$ [5].", "Hence, the processes where $|m|$ -photons are virtually emitted leads to the largest enhancement of scattering when the molecular recombination accompanied by the emission of $m$ photons is resonant, $\\omega _0-|m|\\omega _{\\rm ex}\\approx E_B$ .", "Since $\\lim _{F\\rightarrow 0}{\\rm Im} \\lambda _m=0$ [25] (the molecular bound state can only dissociate because of the driving), these processes lead to sharp CIRs.", "The resonance in the case of resonant emission of a single photon ($m=-1$ ) is shown in Fig.", "REF for different values of $F$ .", "Notice that, the scattering resonances investigated here involve the true molecular bound state and not a virtual bound state formed by the closed channels.", "Therefore, they are not Feshbach-type resonances but rather the dynamical equivalent of a shape resonance (a scattering resonance that occurs when a generic potential has a bound state closed to the continuum threshold) [26].", "Conclusions - We have shown that the s-wave scattering of two atoms confined in a tight quasi-1D trap can be coherently controlled by a RF modulation of the transverse confinement.", "The scattering in the atom cloud in the adiabatic transverse ground state can be efficiently described as atoms in a 1D waveguide interacting via a short range interaction.", "In this case, the coupling constant $g_{1D}$ acquires an imaginary part and incorporates inelastic scattering into the transverse excited states.", "This mechanism generates a new kind of photon-assisted or dynamical confinement-induced resonances.", "The results are universally valid in the regime of low energies.", "The dynamical confinement-induced resonances are another example in which photon-assisted processes carry signatures of atomic interparticle interactions which have also been seen in the photon-assisted tunneling in a Bose-Einstein condensate [27].", "Such processes could be used for avoided-level-crossing spectroscoy in strongly interacting quantum gases [28].", "Acknowledgements - We acknowledge support from the DFG Sonderforschungsbereich 925 “Light induced dynamics and control of correlated quantum systems” (project C8)." ] ]
1403.0348
[ [ "Impact of Secondary Acceleration in Gamma-Ray Bursts" ], [ "Abstract We discuss the acceleration of secondary muons, pions, and kaons in gamma-ray bursts within the internal shock scenario, and their impact on the neutrino fluxes.", "We introduce a two-zone model consisting of an acceleration zone (the shocks) and a radiation zone (the plasma downstream the shocks).", "The acceleration in the shocks, which is an unavoidable consequence of the efficient proton acceleration, requires efficient transport from the radiation back to the acceleration zone.", "On the other hand, stochastic acceleration in the radiation zone can enhance the secondary spectra of muons and kaons significantly if there is a sufficiently large turbulent region.", "Overall, it is plausible that neutrino spectra can be enhanced by up to a factor of two at the peak by stochastic acceleration, that an additional spectral peaks appears from shock acceleration of the secondary muons and pions, and that the neutrino production from kaon decays is enhanced.", "Depending on the GRB parameters, the general conclusions concerning the limits to the internal shock scenario obtained by recent IceCube and ANTARES analyses may be affected by up to a factor of two by secondary acceleration.", "Most of the changes occur at energies above 10^7 GeV, so the effects for next-generation radio-detection experiments will be more pronounced.", "In the future, however, if GRBs are detected as high-energy neutrino sources, the detection of one or several pronounced peaks around 10^6 GeV or higher energies could help to derive the basic properties of the magnetic field strength in the GRB." ], [ "Introduction", "Gamma-ray bursts (GRBs) are a candidate class for the origin of the ultra-high energy cosmic rays (UHECRs).", "A popular scenario is the internal shock model, where the prompt $\\gamma $ -ray emission originates from the radiation of particles accelerated by internal shocks in the ejected material  [45], [47] (see [46], [40] for reviews).", "If a significant baryon flux is accelerated, the GRBs may be a plausible source for the UHECRs.", "In this case, substantial production of secondary pions, muons, and also kaons are expected from photohadronic interactions between the baryons and the radiation field; these will decay into neutrinos and other decay products [50], [12].", "Gamma-rays at TeV energies and above are co-produced in the photohadronic process, but are subject to interactions with the internal photon field from the radiation processes, including synchrotron radiation and inverse Compton scattering, in the GRB.", "The photons are expected to cascade down via pair production cascades so that they can be detected at $\\sim $ GeV energies.", "Several of such GRBs have been detected be Fermi-LAT [6], but the associated neutrino production per burst is generally expected to be rather low, see e.g.", "[19].", "In general, neutrino detection from GRBs with IceCube therefore needs to be done via the stacking of a larger number of bursts, see e.g.", "[3], [4].", "Very stringent neutrino flux limits for the internal shock scenario have been recently obtained by the IceCube collaboration using the stacking approach [3], [4].", "Using timing, energy, and directional information for the individual bursts, new limits have been obtained, which are basically background-free and which are significantly below earlier predictions based on gamma-ray observations [50], [25], [18], [2].", "These predictions have been recently revised from the theoretical perspective [29], [37], [26], yielding about a factor of ten lower expected flux [29] depending on the analytical method compared to; see also [7] for an analysis by the ANTARES collaboration using this method.", "This discrepancy comes mainly from the energy dependence of the mean free path of the protons, the integration over the full photon target spectrum (instead of using the break energy for the pion production efficiency), and several other corrections adding up in the same direction; see Fig.", "1 (left) in [29].", "It should be noted at this point, that the absolute normalization of the neutrino flux scales linearly with the ratio of the luminosity in protons to electrons (baryonic loading).", "In typical models [50], [25], [18], [2], this ratio is usually assumed to be 10, while theoretical considerations suggest a value of 100 [48] if GRBs are to be the sources of the UHECRs.", "In a recent study [15], this value is self-consistently derived from the combined UHECR source and propagation model, including the fit of the UHECR data.", "For an injection index of two, it is demonstrated that this value depends on the burst parameters, and that values between 10 and 100 are plausible.A baryonic loading of 10 requires, however, “typical” source parameters $\\Gamma \\sim 400$ and $L_{\\gamma ,\\mathrm {iso}} \\sim 10^{53} \\, \\mathrm {erg \\, s^{-1}}$ , whereas $\\Gamma \\sim 300$ and $L_{\\gamma ,\\mathrm {iso}} \\sim 10^{52} \\, \\mathrm {erg \\, s^{-1}}$ more point towards a baryonic loading of order 100; see Fig.", "7 in [15].", "Note that these numbers depend strongly on the proton and electron/photon input spectral shapes and energy ranges, and according to basic theory of stochastic acceleration, it can easily vary between 1000 and $0.1$  [39].", "As further demonstrated in [15], the improved modeling of the GRB spectra can be used to constrain the central parameters of the calculation, i.e., the ratio of protons to electrons and the boost factor.", "The effects discussed in our study could then contribute to determining another basic property of the GRB, namely the magnetic field strength.", "Another argument can be used when relating the neutrino and UHECR fluxes directly if the cosmic rays escape as neutrons produced in the same interactions as the neutrinos [8].", "Since this possibility is strongly disfavored [4] it is conceivable that other escape mechanisms dominate for UHECR escape from GRBs [14].", "For instance, if the Larmor radius can reach the shell width at the highest energies, it is plausible that a fraction of the cosmic rays can directly escape.", "Other possible mechanisms include diffusion out of the shells.", "In [15], it was demonstrated that even current IceCube data already imply that these alternative escape mechanisms must dominate if GRBs ought to be the sources of the UHECR, and that future IceCube data will exert pressure on these alternative options as well.", "The secondary pions, muons, and kaons produced by photohadronic interactions will typically either decay (at low energies) or lose energy by synchrotron radiation (at high energies).", "At the point where decay and synchrotron timescales are equal, a spectral break in the secondary, and therefore also in the neutrino spectrum is expected.", "This is the so-called “cooling break”, see e.g.", "[50].", "Additional processes, which potentially affect the secondary spectra, are: adiabatic losses, interactions with the radiation field [31], and acceleration of the secondaries in the shocks or by stochastic acceleration [36], [42], [34].", "In this study, we focus on the quantitative impact of the secondary acceleration on the neutrino fluxes, and the conditions for a significant contribution of this effect.", "A substantial enhancement of the secondary spectra would increase the tension between the recent IceCube observations and the predictions, and would therefore be critical for the interpretation of the recent IceCube results." ], [ "Model description", "The effect of linear acceleration on the secondaries has been discussed in [34], where it was demonstrated that significant acceleration effects can be expected if the secondaries can pile up over a large enough energy range.", "In GRBs, one has shock (Fermi 1st order) acceleration and, possibly, stochastic (Fermi 2nd order) acceleration in the plasma downstream the shock if turbulent magnetic fields are present; see [42], where qualitative estimates for the secondary acceleration are made.", "An important effect is that the secondaries will be mostly produced by photohadronic interactions downstream of the shock, where high photon and proton densities are available over the dynamical timescale of the collision $t^{\\prime }_{\\mathrm {dyn}}$ (as usual, we refer to quantities in the shock rest frame, SRF, by primed quantities).", "We therefore propose a two-zone model, including: Acceleration zone (I): Forward and reverse shocks.", "Radiation zone (II): Plasma downstream the shocks.", "The collision of the shells in the internal shock model is illustrated in Fig.", "REF , where the different zones are shown as well.", "Although we do not consider the explicit time dependence, it is a good approximation to use a steady state model with constant effective densities over the dynamical timescale $t^{\\prime }_{\\mathrm {dyn}}$ .", "This dynamical timescale is typically related to the width of the shells $\\Delta r^{\\prime }$ by $c t^{\\prime }_{\\mathrm {dyn}} \\simeq \\Delta r^{\\prime } \\simeq \\Gamma \\, c \\, \\frac{t_v}{1+z} \\, ,$ where $t_v$ is the (observed) variability timescale and $\\Gamma $ is the appropriately averaged Lorentz boost of the shells, see [35].", "We henceforth assume that the adiabatic cooling timescale is of the same order of magnitude $t^{\\prime }_{\\mathrm {ad}} \\sim t^{\\prime }_{\\mathrm {dyn}}$ .", "The description of the secondary acceleration in GRBs faces several challenges.", "First of all, all species (pions, muons, and kaons) may be accelerated.", "Second, muons are produced by pion decays, which may be accelerated themselves.", "Third, it is expected that an efficient secondary acceleration is a consequence of the primary (proton, electron) acceleration.", "The amount of accelerated secondaries depends on the transport between radiation and acceleration zones.", "And fourth, the spectra of the secondaries, which originate from photohadronic interactions between protons and radiation, are no trivial power laws.", "It is therefore a priori not clear over what energy range the secondaries could pile up, and what the impact of spectral effects is.", "Our model aims to address these issues in a way as self-consistent as technically feasible, using state-of-the-art technology.", "For the description of the target photon spectrum, we chose a framework relying on gamma-ray observations: it is assumed that the observed gamma-ray spectrum is representative for the spectrum within the source.", "Based on energy partition arguments, the baryonic and magnetic field densities can be obtained from the gamma-ray spectrum, and the secondary and neutrino fluxes can be computed from the photohadronic interactions between matter and radiation fields, see [25], [18], [2].", "This framework is slightly different from completely self-consistent (theoretical) approaches generating the target photon spectrum from the radiation processes such as synchrotron radiation and inverse Compton scattering of co-accelerated electrons, or photon production from $\\pi ^0$ decays, see e.g.", "[10], [11].", "The advantage of this approach is that it matches the gamma-ray observations by construction, but the drawback is that it cannot explain them.", "The overall parameters (photon density, $B^{\\prime }$ ) are assumed to be similar for both zones.", "We include additional pion and kaon production modes using the methods in [30], based on the physics of SOPHIA [41]; see also [43] for their impact.", "We also include flavor mixing [20], magnetic field effects on the secondaries, see [32], [38], [16], the kinematics of the weak decays [38], and the recent revisions of the normalization in [29].", "For details on the underlying model, see [17]." ], [ "Acceleration of protons (zone I)", "We assume that protons are accelerated by Fermi shock acceleration in the acceleration zone to obtain a power law spectrum with spectral index $\\alpha _p \\simeq 2$ .", "The acceleration rate is empirically described, as usual, by (see, e.g., [27]) $t^{\\prime -1}_{\\mathrm {acc},I} = \\eta _I \\frac{c}{R_L^{\\prime }} = \\eta _I \\frac{c^2 e B^{\\prime }}{E^{\\prime }} = 9 \\cdot 10^3 \\, \\eta _I \\, \\frac{B^{\\prime } \\, [\\text{G}]}{E^{\\prime } \\, [\\text{GeV}]}$ with the acceleration efficiency $0.1 \\lesssim \\eta _I \\lesssim 1$ in that definition.", "Here, the energy gain is $t^{\\prime -1} \\equiv E^{\\prime -1} |dE^{\\prime }/dt^{\\prime }|$ , $\\eta _I$ corresponds to the fractional energy gain per cycle, and $R_L^{\\prime }$ to the cycle time.", "The acceleration can only be efficient up to the maximal (or critical) energy, where escape or energy losses start to dominate over the acceleration efficiency.", "We neglect photohadronic losses since the considered bursts are optically thin to neutron escape.", "We assume that synchrotron or adiabatic losses ($t^{\\prime -1}_{\\text{ad}} \\simeq c/\\Delta r^{\\prime }$ ) limit the maximal energy, whatever loss rate is larger, or the dynamical timescale.", "We obtain the maximal particle energies from $t^{-1}_{\\mathrm {acc}}=t^{-1}_{\\mathrm {loss}}$ as $&& E^{\\prime }_{c} \\, [\\text{GeV}] = \\nonumber \\\\&&\\min \\left( 2.3 \\cdot 10^{11} \\, (m \\, [\\text{GeV}])^2 \\sqrt{\\frac{\\eta _I}{B^{\\prime } \\, [\\text{G}]}} \\, ,\\right.\\nonumber \\\\&& \\, \\left.9 \\cdot 10^3 \\eta _I \\, B^{\\prime } \\, [\\text{G}] \\, \\frac{\\Gamma \\, t_v \\, [\\text{s}]}{1+z} \\right)\\,.$ The first entry corresponds to the synchrotron-limited case, and the second entry to the adiabatic loss or dynamical timescale-limited case.", "Note that Eq.", "(REF ) can be applied to the protons as well as the secondaries, at least if they are accelerated by shock acceleration.", "The magnetic field can be estimated by energy partition arguments from the observables as [17] $B^{\\prime } \\simeq 130 \\, \\left( \\frac{\\epsilon _B}{\\epsilon _e} \\right)^{\\frac{1}{2}} \\, \\left( \\frac{L_{\\text{iso}}}{10^{52} \\, \\mathrm {erg \\, s^{-1}}}\\right)^{\\frac{1}{2}} \\, \\left( \\frac{\\Gamma }{10^{2.5}} \\right)^{-3} \\, \\left( \\frac{t_v}{0.01 \\, } \\right)^{-1} \\, \\left( \\frac{1+\\textit {z}}{3} \\right) \\, \\text{G} \\, ,$ where $\\epsilon _B/\\epsilon _e \\simeq 1$ describes equipartition between magnetic field energy and kinetic energy of the electrons.", "Note that the variability timescale $t_v$ is given in the observer's frame at Earth, not in the source (engine) frame.", "The protons are injected from the acceleration into the radiation zone with a spectrum $Q^{\\prime }_{II,p} \\propto (E^{\\prime }_p)^{-\\alpha _p}$ , where $Q^{\\prime }$ carries units of $[\\mathrm {GeV^{-1} \\, cm^{-3} \\, s^{-1}}]$ .", "In that zone, the protons may interact with photons to produce secondaries, which undergo synchrotron losses, decay, escape (over the dynamical timescale), and adiabatic losses, see [54] for details.", "In addition, we consider stochastic acceleration (second order Fermi acceleration) in the spirit of [42], following [53], [52], [42].", "If there is substantial turbulence, this stochastic (Fermi 2nd order) acceleration may be important.", "For the sake of simplicity, we assume that the turbulent region covers the whole zone II; if only a fraction is affected, only a fraction of particles will be accelerated and one can trivially obtain the results from our figures.", "The steady state kinetic equation for the secondary muons, pions, and kaons in zone II is given by $Q^{\\prime }_{II,i} = \\frac{N^{\\prime }_{II,i}}{t^{\\prime }_{\\mathrm {esc}}} - \\frac{\\partial }{\\partial E^{\\prime }} \\left( \\frac{E^{\\prime } N^{\\prime }_{II,i}}{t^{\\prime }_{\\mathrm {loss}}} \\right) + \\frac{\\partial }{\\partial E^{\\prime }} \\left( \\frac{E^{\\prime } N^{\\prime }_{II,i}}{t^{\\prime }_{\\mathrm {acc},II}} \\right) - \\frac{\\partial }{\\partial E^{\\prime }} \\left( \\frac{E^{\\prime 2}}{2 t^{\\prime }_{\\mathrm {acc},I}} \\frac{\\partial N^{\\prime }_{II,i}}{\\partial E^{\\prime }} \\right)\\, ,$ where $Q_i^{\\prime }$ the injection of species $i$ from photohadronic processes or parent decays and $N_i^{\\prime }$ is the steady state density (units $[\\mathrm {GeV^{-1} \\, cm^{-3}}]$ ).", "The first term on the r.h.s.", "describes escape by decay or escape over the dynamical timescale, i.e., $t^{\\prime -1}_{\\mathrm {esc}} = t^{\\prime -1}_{\\mathrm {decay}}+t^{\\prime -1}_{\\mathrm {dyn}}$ .", "The second term describes energy losses, i.e., $t^{\\prime -1}_{\\mathrm {loss}} = t^{\\prime -1}_{\\mathrm {synchr}}+t^{\\prime -1}_{\\mathrm {ad}}$ .", "Without acceleration, decay typically dominates at low energies and synchrotron losses at high energies, and for $t^{\\prime -1}_{\\mathrm {synchr}} \\simeq t^{\\prime -1}_{\\mathrm {decay}}$ , a spectral break by two powers is expected.", "The last two terms in Eq.", "(REF ) are characteristic for stochastic acceleration and always come together with a fixed relative magnitude, see e.g.", "[53], [52].", "We assume that the acceleration timescale $t^{\\prime }_{\\mathrm {acc},II}$ is given by $t^{\\prime }_{\\mathrm {acc},II} \\equiv \\frac{E^{\\prime 2}}{2 D^{\\prime }_{EE}} = \\tilde{\\eta }_{II}^{-1} \\frac{l^{\\prime }_{\\mathrm {tur}}}{c} \\left( \\frac{R^{\\prime }_L}{l^{\\prime }_{\\mathrm {tur}}} \\right)^{2-q} \\simeq \\eta _{II}^{-1} t^{\\prime }_{\\mathrm {dyn}} = \\eta _{II}^{-1} \\, \\Gamma \\, \\frac{t_v}{1+z} \\, ,$ following [42]; see discussion therein.", "The energy diffusion coefficient is assumed to be $D^{\\prime }_{EE} \\propto E^{\\prime q}$ , and $l^{\\prime }_{\\mathrm {tur}}$ is the length scale of the turbulence – which can be estimated from the typical lifetime of the turbulence.", "In the third step, we have chosen $q \\simeq 2$  [42], and we have re-parametrized the acceleration timescale in terms of the shell width and turbulence length scale as $ \\eta _{II} = \\tilde{\\eta }_{II} \\Delta R^{\\prime } / l^{\\prime }_{\\mathrm {tur}}$ .", "We expect significant effects of stochastic acceleration if $\\eta _{II} > 1$ , since then the acceleration exceeds the escape in a certain energy window.", "Let's consider the most extreme case, the kaons, which have the highest energies at their cooling break.", "If these are to be accelerated and confined, the condition $R^{\\prime }_L < c t^{\\prime }_{\\mathrm {acc},II} \\lesssim l^{\\prime }_{\\mathrm {tur}} < \\Delta R^{\\prime } \\simeq c t^{\\prime }_{\\mathrm {dyn}}$ implies that the stochastic acceleration timescale is longer than the shock acceleration timescale but shorter than the hydrodynamical timescale.", "One finds $1 < \\eta _{II} \\lesssim 10$ as a reasonable parameter range; cf., Eq.", "(REF ).In the most extreme case, if $\\eta _{I}=1$ and the maximal energy is dominated by adiabatic losses, the kaons will take about 35% of the maximal proton energy.", "Thus, their Larmor radius will be of the order of one tenth of the size of the region.", "In other cases and for other species, it will be smaller.", "We neglect acceleration of the primary protons in zone II, since one can show analytically that the Fermi 2nd order acceleration only changes the overall normalization of a simple power law.", "The proton spectrum normalization is however determined by energy partition arguments, so the impact of Fermi 2nd order re-acceleration can be absorbed in a re-definition of the baryonic loading.", "In addition, the target photon spectrum is based on observation, which means that we do not need to consider the acceleration of electrons in the spirit of [42] to describe the prompt emission spectrum.", "As in [42], we assume that the secondaries in the relevant energy range cannot escape, since $R^{\\prime }_L < \\Delta R^{\\prime }$ .", "Apart from stochastic acceleration in the radiation zone, it is conceivable that a substantial fraction of secondaries is transported back to the acceleration zone I by diffusion.", "We characterize this fraction as $f_{\\text{diff}} \\simeq \\lambda ^{\\prime }/\\Delta R^{\\prime }$ , where $\\lambda ^{\\prime }$ is the diffusion length over the dynamical timescale.", "Our description closely follows [14] in that aspect, and we assume that the secondaries are produced uniformly over the radiation zone II.", "The fraction $f_{\\text{diff}}$ of particles which can diffuse back to the shock front within the dynamical timescale can be estimated from the diffusion length $\\lambda ^{\\prime } \\simeq \\sqrt{D^{\\prime }_{xx} \\, t^{\\prime }_{\\mathrm {dyn}}}$ as $f_{\\text{diff}} = \\min \\left( \\frac{\\lambda ^{\\prime }}{c t^{\\prime }_{\\mathrm {dyn}}} , 1 \\right) \\, ,$ where $D^{\\prime }_{xx}$ is the spatial diffusion coefficient.", "This definition ensures that $f_{\\text{diff}} \\le 1$ .", "For example, for Bohm-like diffusion, one has $D^{\\prime }_{xx} \\propto E^{\\prime }$ and for Kolmogorov-like diffusion, one has $D^{\\prime }_{xx} \\propto E^{\\prime 1/3}$ [49], [48], and as a consequence, $f_{\\text{diff}} \\propto \\sqrt{E^{\\prime }}$ and $f_{\\text{diff}} \\propto E^{\\prime 1/6}$ , respectively.", "As a lower limit, it can be shown that a fraction $f_{\\text{dir}}=R^{\\prime }_L/(c t^{\\prime }_{\\mathrm {dyn}}) \\propto E^{\\prime }$ of the secondaries can directly escape from the radiation zone in the same way as cosmic rays from the shells [14].", "That is, when the Larmor radius becomes comparable to the shell width, all particles will reach back to the shocks.", "It is therefore reasonable to normalize the transport back to zone I in the way that for $R^{\\prime }_L=\\Delta R^{\\prime }$ all particles are efficiently transported.That is, we choose $f_{\\text{diff}}=(R^{\\prime }_L/(c t^{\\prime }_{\\mathrm {dyn}}))^\\gamma $ with $D^{\\prime }_{xx} \\propto E^{\\prime 2 \\gamma }$ .", "Muons, pions, and kaons will typically not reach these high energies, since synchrotron losses lead to a spectral break.", "As a consequence, only the fraction $f_{\\text{diff}} \\simeq \\sqrt{E^{\\prime }_{\\mathrm {break}}/E^{\\prime }_{\\mathrm {c}}}$ will diffuse back, at the most, where $E^{\\prime }_{\\mathrm {c}}$ is the maximal energy in Eq.", "(REF ).", "For pions and kaons the break energies are typically higher than for muons, which means that a larger fraction of pions and kaons should be transported back to zone I.", "Note that stochastic acceleration and the transport by diffusion are connected via transport theory.", "Specifically, it is shown in e.g.", "[48], that in the relativistic limit of $E\\approx p\\cdot c$ , the product of the spatial and momentum diffusion coefficients is given as $D^{\\prime }_{EE}\\cdot D^{\\prime }_{xx}=\\frac{4E^2\\,v_{A}^{2}}{3\\,a\\,(4-a^2)\\,(4-a)\\,w}\\,.$ where $w$ is a constant parameter defining the turbulence scale, which is often included in the definition of the Alfvén velocity $v_{A}$ (see e.g.", "[21] for a summary).", "The wave spectrum follows a power law $k^{a}$ with the index $a$ connected to the spatial diffusion coefficient as $D_{xx}\\propto E^{\\prime 2-a}$ .", "For Kolmogorov-type diffusion, $a=5/3$ , while in the Bohm-case, $a=1$ .", "This means that efficient stochastic acceleration $t^{\\prime }_{\\mathrm {acc},II} \\propto {D^{\\prime }_{EE}}^{-1}$ in zone II, see Eq.", "(REF ), implies inefficient spatial transport, and vice versa.", "In particular, if $D^{\\prime }_{EE} \\propto {E^{\\prime }}^{q}$ , as we assumed above, $D^{\\prime }_{xx} \\propto {E^{\\prime }}^{2-q}$ .", "Therefore, $q \\sim 2$ is roughly consistent with Kolmogorov diffusion, which we use as a standard in the following.", "It is conceivable from this discussion that the acceleration of the secondaries dominates either in zone I or zone II as a function of energy, depending on the efficiency of transport back to the acceleration zone versus stochastic acceleration.", "The injection from the radiation back into the shock zone is given by $Q^{\\prime }_{I,i}=N^{\\prime }_{II,i} \\, t^{\\prime -1}_{\\mathrm {dyn}} \\, f_{\\text{diff}}(E^{\\prime })=N^{\\prime }_{II,i} \\, t^{\\prime -1}_{\\mathrm {eff,diff}} \\, ,$ where one can define the effective diffusion timescale $t^{\\prime -1}_{\\mathrm {eff,diff}} \\equiv t^{\\prime -1}_{\\mathrm {dyn}} f_{\\text{diff}}$ .", "Note that $N^{\\prime }_{II,i} t^{\\prime -1}_{\\mathrm {dyn}}$ is the ejected spectrum if all particles can escape from zone II over the dynamical timescale, whereas $f_{\\text{diff}} \\le 1$ characterizes the energy-dependent fraction obtained from Eq.", "(REF ).", "In addition, note that $t^{\\prime -1}_{\\mathrm {eff,diff}} \\le t^{\\prime -1}_{\\mathrm {dyn}} \\simeq t^{\\prime -1}_{\\mathrm {ad}}$ , so diffusion is always less efficient than the adiabatic cooling or escape over the dynamical timescale in zone II, and is therefore not included in Eq.", "(REF ).", "The corresponding kinetic equation for the secondaries in zone I is given in the steady state by $Q^{\\prime }_{I,i} = \\frac{N^{\\prime }_{I,i}}{t^{\\prime }_{\\mathrm {esc}}} - \\frac{\\partial }{\\partial E^{\\prime }} \\left( \\frac{E^{\\prime } N^{\\prime }_{I,i}}{t^{\\prime }_{\\mathrm {synchr}}} \\right) + \\frac{\\partial }{\\partial E^{\\prime }} \\left( \\frac{E^{\\prime } N^{\\prime }_{I,i}}{t^{\\prime }_{\\mathrm {acc},I}} \\right) \\,$ with the same acceleration efficiency as for the protons Eq.", "(REF ).", "That is, we assume that the secondaries undergo acceleration similar to the protons, suffer from synchrotron losses, and escape via decay and escape from the acceleration zone over the dynamical timescale, i.e., $t^{\\prime -1}_{\\mathrm {esc}} = t^{\\prime -1}_{\\mathrm {dyn}}+t^{\\prime -1}_{\\mathrm {decay}}$ .", "Since $t^{-1}_{\\mathrm {acc},I} > t^{-1}_{\\mathrm {decay}}$ in order to have significant secondary acceleration (both have the same energy dependence), the particles at the highest energies can typically escape over the dynamical timescale from zone I before they decay.", "Therefore, we assume that accelerated secondaries decay in the radiation zone.That is only relevant for accelerated pions which may decay in the acceleration zone, such that the resulting muons are guaranteed to be in the shock from the beginning.", "One may ask if this approach is consistent with the textbook version of Fermi shock acceleration.", "In that version, the proton index is given by $\\alpha _p = P_{\\mathrm {esc}}/\\eta _I+1$ , where $P_{\\mathrm {esc}}$ is the (constant) escape probability per cycle and $\\eta $ is the (constant) fractional energy gain per cycle.", "The ratio $P_{\\mathrm {esc}}/\\eta _I = 3/(\\chi -1) \\simeq 1$ depends on the compression ratio $\\chi $ only, where $\\chi \\simeq 4$ for a strong shock.", "As a consequence, a “intrinsic” escape term $t^{-1}_{\\mathrm {esc,shock}}=t^{-1}_{\\mathrm {acc}}$ is needed for a self-consistent kinetic simulation.", "In our approach, we checked analytically and numerically that such an additional escape term $t^{\\prime -1}_{\\mathrm {esc,shock}} \\simeq P_{\\mathrm {esc}}/T^{\\prime }_{\\mathrm {cycle}} \\propto E^{\\prime -1}$ with $P_{\\mathrm {esc}}=\\eta _I$ and $T^{\\prime }_{\\mathrm {cycle}} \\simeq R_L^{\\prime }/c$ produces an $E^{\\prime -2}$ ejection spectrum for the protons if a narrow-energetic particle distribution is injected.", "Here, it is crucial that acceleration and escape terms carry the same energy dependence (which is implied by the constant energy gain and escape probability per cycle), and that $Q^{\\prime }_{\\mathrm {esc}}=N^{\\prime }/t^{\\prime }_{\\mathrm {esc,shock}}$ , which means that $Q^{\\prime }_{\\mathrm {esc}}$ and $N^{\\prime }$ have different energy dependencies.", "For the secondaries, such an escape term will suppress the spectra somewhat, depending on the spectral index of the injection (determined by the ratio $P_{\\mathrm {esc}}/\\eta _I$ ).", "For the sake of simplicity, we assume that the secondaries will escape via decay or over the dynamical timescale only.", "This is in a way the most aggressive assumption one can make, which will however support our conclusions.", "It may also apply if the escape properties change over time, the acceleration site of the secondaries is different from the one of the primaries, or if the secondaries, which have lower energies than the protons, are trapped in magnetic fields, whereas the protons are injected into the shock at relatively high energies with a larger Larmor radius." ], [ "Impact of acceleration effects on the secondaries", "In order to illustrate the impact of the acceleration on the secondaries, we choose the GRB parameters listed in the second column of Table REF for a burst chosen to reproduce the properties of the Waxman-Bahcall burst [50], [51] (plateau between $10^5$ and $10^7 \\, \\mathrm {GeV}$ in $E_\\nu ^2 F_\\nu $ ), see [16].", "The corresponding inverse timescales are shown in Fig.", "REF for the secondary muons, pions, and kaons (in the different panels, in SRF).", "We can use this figure to discuss the expected behavior in the different zones.", "In the acceleration zone (I), acceleration or synchrotron losses dominate for all species.", "The maximal (critical) energy can be obtained from Eq.", "(REF ) as for protons, where adiabatic losses are also included as possibility to limit the maximal energy.", "From the figure it is clear that it is close to each other for muons and pions, whereas it is significantly higher for kaons.", "Here, all secondary species can be, in principle, efficiently accelerated in the shock, since $t^{\\prime -1}_{\\mathrm {decay}} \\ll t^{\\prime -1}_{\\mathrm {acc},I}$ .", "The largest difference between decay and acceleration, which have the same energy dependence, is obtained for muons, the smallest for kaons.", "Therefore, one may expect that muons are most efficiently accelerated, see also [34].", "The pile-up depends on the energy efficiency range of the acceleration.", "For GRBs, that is non-trivial to determine, since the secondary spectrum has a spectral break coming from the the gamma-ray spectrum; hence the potential pile-up range is given by the interval between that break and the critical energy.", "Another break, the synchrotron cooling break, can be obtained from $t^{\\prime -1}_{\\mathrm {decay}}=t^{\\prime -1}_{\\mathrm {synchr}}$ , and is lowest for muons and highest for kaons.", "It shows up in all cases at lower energies than $E_c$ .", "Even more complicated, the critical energy is above the cooling break in all cases, which means that its energy is beyond the peak energy of the spectrum, and that it is not guaranteed that the peak flux of the spectrum will be increased at the absolute maximum.", "Note that it is not simply possible to lower the magnetic field to reduce the cooling and enhance the effect of the acceleration, since the acceleration efficiency will be reduced, whereas the cooling break will persist as adiabatic cooling break even if synchrotron losses are suppressed (where $t^{\\prime -1}_{\\mathrm {decay}}=t^{\\prime -1}_{\\mathrm {ad}}$ ).", "We will however discuss the conditions for the possibly largest acceleration effects in the next section.", "As far as the transport between radiation zone, where the secondaries are mostly produced, and the acceleration zone is concerned, we show the effective diffusion rates $t^{\\prime -1}_{\\mathrm {eff,diff}}$ (cf., Eq.", "(REF )) for the Kolmogorov and Bohm cases in the figure as upper and lower dashed curves, respectively.", "It is clear that the higher the critical energy, the more particles will be transported back to the shock.", "Therefore, the transport is expected to be most efficient for kaons, which somewhat compensates for the less efficient acceleration – depending on the transport type.", "The Kolmogorov and Bohm cases give the range of plausible transport scenarios.", "Perfect transport (all particles transported back to the shock over the dynamical timescale) would correspond to $t^{\\prime -1}_{\\mathrm {eff,diff}}=t^{\\prime -1}_{\\mathrm {dyn}}$ .", "As most conservative assumption, only the particles not scattering at all may reach back to the shock, corresponding to the direct escape in [14].", "In that case, the transport is only efficient if the Larmor radius reaches the size of the region.", "We checked that the results for the Kolmogorov case are already quite similar to the perfect transport case, whereas the Bohm case and steeper energy dependencies lead to very small amounts of secondary acceleration, see below.", "For the stochastic acceleration in zone II, the largest effects are expected if $t^{-1}_{\\mathrm {acc},II}$ dominates over the synchrotron and decay timescales in the radiation zone.", "Because of the shallow dependence on energy, a small window (about one order of magnitude in energy) can be found for muons and kaons in Fig.", "REF , whereas pions are hardly affected for the chosen acceleration efficiency.", "In summary, we expect the most interesting results for muons, which may be efficiently accelerated in both zones, and kaons, which may be efficiently transported back to the shock and efficiently accelerated in the radiation zone.", "Figure: Effect of shock acceleration on steady state densities N ' N^{\\prime } of muons, pions, and kaons for two different transport mechanisms between radiation and acceleration zone (different panels, in SRF).", "The plots use the burst SB from Table  and η I =0.1\\eta _I=0.1.", "The dashed curves are shown without acceleration for comparison.These qualitative considerations are quantitatively supported by numerical simulations.", "Let us focus on Fig.", "REF first, where the effect of shock acceleration only is shown on the secondary muons, pions, and kaons.", "This figure shows the steady state spectra $N^{\\prime }$ , which differ in shape from the neutrino ejection spectra; see also App.", "A in [17].", "Solving Eq.", "(REF ) for decay only, one obtains $Q^{\\prime }=(t^{\\prime }_{\\mathrm {decay}})^{-1} N^{\\prime }$ below the peak of the spectra.", "Therefore, $Q^{\\prime } \\propto (E^{\\prime })^{-2}$ for $N^{\\prime } \\propto (E^{\\prime })^{-1}$ , which can be compared to the neutrino ejection spectra in terms of shape.", "The left panel in Fig.", "REF uses Kolmogorov diffusion from the radiation to the acceleration zone, the right panel Bohm diffusion, where these may be regarded as the optimistic and conservative cases for the transport.", "The shock acceleration leads to the pile-up spikes at the critical energies, marked in Fig.", "REF , which are also observed in [34].", "The acceleration components are most prominent for Kolmogorov diffusion, where many secondaries are transported back to the shock, and least prominent for the Bohm diffusion.", "As we discussed above, because of the balance between transport and acceleration efficiency, muons and kaons are mostly accelerated, whereas the effect on the pions is smaller.", "In either case, the spikes in the muon or pion spectra are washed out in the neutrino spectrum, because the kinematics of the weak decays re-distributes the parents' energies.", "The spikes lead to shoulders in the neutrino spectra, as can be seen in Fig.", "REF .", "Most importantly, the effect of muon acceleration may be shadowed by the regular pion spectrum, as it is evident from the right panel.", "Therefore, in the Bohm case, the effect of acceleration on the neutrinos is hardly visible.", "In the Kolmogorov case, on the other hand, two distinctive peaks (from pion/muon acceleration and from kaon acceleration) should be visible, where the one from pion/muon acceleration is closest to the overall peak of the spectrum and therefore perhaps easiest to detect.", "In the following, we will only discuss the transport by Kolmogorov diffusion, which may be optimistic but is the minimal requirement to observe significant effects on the neutrino spectra.", "As a minor detail, in Fig.", "REF , left panel, a small enhancement above the critical energy for muons, which comes from the fact that some of the muons are injected above the critical energy from accelerated pions.", "Figure: Effect of shock acceleration, stochastic acceleration, and both accelerations combined (in different panels) on steady state densities N ' N^{\\prime } for muons, pions, and kaons.", "Here, the burst SB from Table , η I =0.1\\eta _I=0.1, and η II =2.5\\eta _{II}=2.5 have been used.", "The different panels are for acceleration in the shock (left), in the radiation zone (middle), and both (right).", "Here, Kolmogorov diffusion has been used as transport mechanism between the two zones.The dashed curves are shown without acceleration for comparison.Apart from shock acceleration of secondaries transported back to the shock, stochastic acceleration in a turbulent radiation zone could be relevant.", "In order to maximize the effect, we assume that the whole radiation zone is turbulent, and show the impact of shock acceleration (left panel), stochastic acceleration (middle panel), and acceleration in both zones (right panel) in Fig.", "REF .", "As discussed above, stochastic acceleration can lead to a significant enhancement of the muon and kaon spectra at their peaks, where stochastic acceleration is efficient over about one order of magnitude in energy for the chosen acceleration efficiency.", "The combined effect of acceleration in both zones is shown in the right panel, and is (to a first approximation) an addition of the two effects.", "One could in principle assume even somewhat more extreme acceleration efficiencies in zone II, which leads to a much stronger enhancement.", "However, current neutrino data [4] already puts constraints on scenarios with more optimistic secondary acceleration." ], [ "Impact on neutrino fluences", "The upper left panel of Fig.", "REF shows the neutrino spectra for the standard burst from Table REF .", "In the muon neutrino fluence, the enhancement of the peaks from muon and kaon decays in the case of stochastic acceleration can be clearly seen.", "For the shock acceleration, the spikes in Fig.", "REF translate into peaks at energies higher by a factor of $\\Gamma /(1+z)$ .", "The combined effect enhances the neutrino spectrum by about 50% in that case, leading to an additional peak at about $10^8 \\, \\mathrm {GeV}$ , and increases the neutrino peak from kaon decays significantly.", "The spiky secondary particle spectra lead to $E^{-1}$ spectra for the neutrinos, since the kinematics of weak decays cannot exceed this spectral index.", "It is, of course, an interesting question how much these observations depend on the parameter values.", "We therefore choose three different, recently observed (by Fermi) GRBs as examples: GRB 080916C, GRB 090902B, and GRB 091024.", "GRB 080916C is one of the brightest bursts ever seen, although at a large redshift, and one of the best studied Fermi-LAT bursts.", "The gamma-ray spectrum of GRB 090902B has a relatively steep cutoff, and might therefore be representative for a class of bursts for which the gamma-ray spectrum can be fit with a single power law with exponential cutoff as well.", "GRB 091024 can be regarded as a typical example representative for many Fermi-GBM bursts [44], except for the long duration.", "Note that GRB 080916C and GRB 090902B have an exceptionally large $\\Gamma \\gtrsim 1000$ , whereas $\\Gamma \\simeq 200$ for the last burst.", "All three observed bursts have in common that the required parameters for the neutrino flux computation can be taken from the literature; see the Table REF and its caption for the references.", "Note that these bursts have been also studied in the context of neutrino decays [13] and the normalization question [28], [54].", "We show in Fig.", "REF the neutrino spectra for the four representative GRBs listed in Table REF , The effects of the secondary acceleration on the neutrino spectra are depicted in Fig.", "REF .", "To a first approximation, the effects are dominated by the strength of the magnetic field strength, which can be estimated with Eq.", "(REF ).", "GRB 091024 has a similar magnetic field (about 60 kG) to our Standard Burst (about 290 kG), whereas the magnetic fields for GRB 080916C (4 kG) and GRB 090902B (6 kG) are significantly lower because of their large Lorentz boosts.", "Consequently, the spectral shapes of the neutrino spectra are very different, dominated by an adiabatic cooling break which changes the spectrum only by one power.", "In these cases, it is possible that the stochastic and shock acceleration effects add up.", "One may ask the question when these largest effects can be expected and if they can be enhanced.", "The peak of the secondary spectrum in $E^2 N^{\\prime }$ is given by the cooling break $t^{\\prime -1}_{\\mathrm {synchr}}=t^{\\prime -1}_{\\mathrm {decay}}$ .", "The critical energy for the shock acceleration is typically given by $t^{\\prime -1}_{\\mathrm {synchr}}=t^{\\prime -1}_{\\mathrm {acc},I}$ .", "The critical energy for the stochastic acceleration is determined by $t^{\\prime -1}_{\\mathrm {synchr}}=t^{\\prime -1}_{\\mathrm {acc},II}$ , which means that stochastic acceleration can be efficient up to relatively large energies for small $B^{\\prime }$ (as one can see in the figure).", "One expects the maximal enhancement effect at the peak for $t^{\\prime -1}_{\\mathrm {decay}}=t^{\\prime -1}_{\\mathrm {acc,I}} \\simeq t^{\\prime -1}_{\\mathrm {synchr}}$ , where the cooling break and critical energy for shock acceleration coincide.", "This condition translates into a critical magnetic field $B_c^{\\prime } \\simeq 10^{-4} \\, \\eta _I^{-1} \\frac{m \\, [\\mathrm {GeV}]}{\\tau _0 \\, [\\mathrm {s}]} \\, \\mathrm {G} \\, ,$ where $m$ is the mass of the secondary and $\\tau _0$ its rest-frame lifetime.", "The ratio $m/\\tau _0 \\simeq 4.8 \\cdot 10^{4} \\, \\mathrm {GeV \\, s^{-1}}$ is smallest for muons, where $B_c^{\\prime } \\simeq 50 \\, \\mathrm {G}$ (for $\\eta _I=0.1$ ).", "Since for muons, the acceleration is most efficient, the pions and kaons will rather decay than being accelerated in that case.", "The closest parameters can be found for the GRBs with high Lorentz factors (to achieve high energies) and relatively low magnetic fields GRB080916C and GRB090902B, best seen in the lower left panel of Fig.", "REF as additional peak only a factor of few above the energy of the absolute maximum.", "In principle, one can also find such a critical magnetic field for kaons, for which $m/\\tau _0 \\simeq 4 \\cdot 10^{7} \\, \\mathrm {GeV \\, s^{-1}}$ so $B_c^{\\prime } \\simeq 40 \\, \\mathrm {kG}$ (for $\\eta _I=0.1$ ).", "This case is not so far away from what is shown in Fig.", "REF ($B^{\\prime } \\simeq 290 \\, \\mathrm {kG}$ ).", "However, the kaon peak is far away from the absolute peak of the spectrum.", "For pions, the spectral peak is closer to that of the muons and the acceleration is less efficient.", "Again, one cannot arbitrary reduce the magnetic field, since low $B^{\\prime }$ mean low proton acceleration efficiencies and low maximal energies.", "In the shown cases, there is a balance between a low $B^{\\prime }$ and high energies obtained by high $\\Gamma $ , which cannot be separated independently.", "Finally, there is some impact of the acceleration efficiencies $\\eta _I$ and $\\eta _{II}$ on the neutrino result, which shift the peaks as qualitatively expected." ], [ "Summary and conclusions", "The aim of this study has been to address the quantitative importance of the acceleration of secondary muons, pions, and kaons for the neutrino fluxes.", "We have therefore extended the model by [29], which predicts the neutrino fluxes from gamma-ray observations in the internal shock model and which the current state-of-the-art GRB stacking analyses in neutrino telescopes are based on, by the acceleration effects of the secondaries – as discussed in a more general sense in [34].", "One of the key issues has been a separate description of the acceleration zone (the shocks) and the radiation zone (the plasma downstream the shocks) in a two-zone model, since it is plausible that the shock acceleration and the photohadronic processes, leading to the secondary production, happen dominantly in different regions.", "Two classes of acceleration have been implemented for the secondaries: shock acceleration in the acceleration zone and stochastic acceleration in the (possibly turbulent) plasma in the radiation zone.", "An important component of the model has been the transport of the secondaries from the radiation zone back to the acceleration zone, which we describe by Kolmogorov diffusion (optimistic) or Bohm diffusion (conservative) – assuming that at the highest energies, where the Larmor radius reaches the size of the region, all secondaries are efficiently transported.", "The shock acceleration of the secondaries is then just a consequence of the efficient proton acceleration if they can be transported back to the shocks, whereas the stochastic acceleration depends on the size of the turbulent region.", "In both cases, some uncertainty arises from the acceleration efficiencies, which may vary within reasonable limits.", "We have shown that both the muon and kaon spectra can be significantly modified by shock acceleration: the muon spectrum, because muons have a long lifetime over which they can be accelerated, and the kaon spectrum, because kaons are most efficiently transported back to the acceleration zone at their highest energies (they have the highest synchrotron cooling break).", "The shock acceleration leads to additional peaks determined by the critical energy, where acceleration and energy loss or escape rates are equal.", "These peaks translate into corresponding peaks of the neutrino spectra, smeared out by the kinematics of the weak decays.", "The most significant enhancement at the peak is expected from the muon spectrum if the magnetic field is low and the Lorentz boost is high, since then the critical energy may coincide with the peak energy.", "Too low magnetic fields, on the other hand, mean that the protons cannot be efficiently accelerated.", "We note that our model is fully self-consistent in the sense that it is taken into account that muons are produced by pion decays, which may be accelerated themselves.", "The amount of shock acceleration depends critically on the transport between radiation and acceleration zone.", "For Bohm diffusion or even slower transport processes, hardly any modification of the neutrino spectra is observed, since the enhancement of the muon spectrum is completely shadowed by the regular pion spectrum present at higher energies.", "On the other hand, the results for Kolmogorov diffusion are already close to the perfect transport case (all particles efficiently transported over the dynamical timescale).", "The stochastic acceleration can be very efficient for muons and kaons, since their cooling breaks occur at a smaller (decay and synchrotron loss) rates than the one for pions, which means that the stochastic acceleration can be dominant at these breaks.", "The consequence is an enhancement at the cooling break (if the break comes from synchrotron losses) or beyond (if it comes from adiabatic losses).", "In the latter case, the shock and stochastic acceleration effects can add up and lead to an additional peak in the neutrino spectrum with a significant enhancement.", "It is conceivable that efficient stochastic acceleration means inefficient transport, i.e., the two acceleration effects are mutually exclusive in terms of their energy ranges, and that such an effect can be only observed for a flat enough energy dependence of the diffusion coefficient (such as Kolmogorov diffusion).", "Depending on the specific GRB parameters, secondary particle acceleration can enhance the neutrino flux by up to an overall factor of two.", "The enhancement is typically largest at higher energies, around $10^{8}$ GeV or above.", "This enhancement is relevant for extremely high-energy searches with IceCube at energies around $10^{8}$  GeV and above.", "In particular, southern hemisphere searches are sensitive at these energies, as the background of atmospheric muons is sufficiently small at those high energies, see [1] for the latest point source sensitivity of IceCube.", "Even northern hemisphere searches might already be sensitive to the enhancement.", "Next generation instruments like KM3NeT and a high-energy extension of IceCube will be able to constrain the parameter space for secondary particle acceleration effects even further.", "Other future experiments which have good sensitivity above $10^{8}$  GeV concern the radio emission from neutrino-induced showers, such as ARA [9] and ARIANNA [22], [33].", "These conclusions will somewhat depend on the choices of the acceleration rates, which means that we cannot exclude larger effects for individual bursts.", "Note that some of our choices (such as for transport and size of the turbulent region) are already on the optimistic side.", "WW acknowledges support from DFG grants WI 2639/3-1 and WI 2639/4-1, the FP7 Invisibles network (Marie Curie Actions, PITN-GA-2011-289442), and the “Helmholtz Alliance for Astroparticle Physics HAP”, funded by the Initiative and Networking fund of the Helmholtz association.", "JBT acknowledges support from the Research Department of Plasmas with Complex Interactions (Bochum) and from the MERCUR Project Pr-2012-0008.", "The work of SK was supported in part by U.S. National Science Foundation under grant PHY-1307472 and the U.S. Department of Energy under contract number DE-AC-76SF00098.", "We are grateful to P. Baerwald, R. Tarkeshian, and E. Waxman for useful discussions.", "We thank members of the IceCube collaboration for useful discussions." ] ]
1403.0574
[ [ "Some properties of the group of birational maps generated by the\n automorphisms of $\\mathbb{P}^n_\\mathbb{C}$ and the standard involution" ], [ "Abstract We give some properties of the subgroup $G_n(\\mathbb{C})$ of the group of birational self-maps of $\\mathbb{P}^n_\\mathbb{C}$ generated by the standard involution and the group of automorphisms of $\\mathbb{P}^n_\\mathbb{C}$.", "We prove that there is no nontrivial finite-dimensional linear representation of $G_n(\\mathbb{C})$.", "We also establish that $G_n(\\mathbb{C})$ is perfect, and that $G_n(\\mathbb{C})$ equipped with the Zariski topology is simple.", "Furthermore if $\\varphi$ is an automorphism of $\\mathrm{Bir}(\\mathbb{P}^n_\\mathbb{C})$, then up to birational conjugacy, and up to the action of a field automorphism $\\varphi_{\\vert G_n(\\mathbb{C})}$ is trivial." ], [ "Introduction", "The group $\\mathrm {Bir}(\\mathbb {P}^2_\\mathbb {C})$ of birational self-maps of $\\mathbb {P}^2_\\mathbb {C}$ , also called the Cremona group of rank 2, has been the object of a lot of studies.", "For finite subgroups let us mention for example [3], [27], [9] ; other subgroups have been dealt with ([21], [23]), and some group properties have been established ([21], [22], [19], [14], [10], [12], [5], [6], [4]).", "One can also find a lot of properties between algebraic geometry and dynamics ([26], [13], [7]).", "The Cremona group in higher dimension is far less well known ; let us mention some references about finite subgroups ([42], [43], [41], [40], [33]), about algebraic subgroups of maximal rank ([18], [51], [47], [49], [48]), about other subgroups ([38], [39]), about (abstract) homomorphisms from $\\mathrm {PGL}(r+1;\\mathbb {C})$ to the group $\\mathrm {Bir}(M)$ where $M$ denotes a complex projective variety ([11]), and about maps of small bidegree ([35], [36], [34], [29], [24]).", "In this article we consider the subgroup of birational self-maps of $\\mathbb {P}^n_\\mathbb {C}$ introduced by Coble in [15] $G_n(\\mathbb {C})=\\langle \\sigma _n,\\,\\mathrm {Aut}(\\mathbb {P}^n_\\mathbb {C})\\rangle $ where $\\sigma _n$ denotes the involution $(z_0:z_1:\\ldots :z_n)\\dashrightarrow \\left(\\prod _{\\stackrel{i=0}{i\\ne 0}}^nz_i:\\prod _{\\stackrel{i=0}{i\\ne 1}}^nz_i:\\ldots :\\prod _{\\stackrel{i=0}{i\\ne n}}^nz_i\\right).$ Hudson also deals with this group ([29]) : \"For a general space transformation, there is nothing to answer either to a plane characteristic or Noether theorem.", "There is however a group of transformations, called punctual because each is determined by a set of points, which are defined to satisfy an analogue of Noether theorem, and possess characteristics, and for which we can set up parallels to a good deal of the plane theory.\"", "Note that the maps of $G_3(\\mathbb {C})$ are in fact not so \"punctual\" ([8]).", "It follows from Noether theorem ([1], [44]) that $G_2(\\mathbb {C})$ coincides with $\\mathrm {Bir}(\\mathbb {P}^2_\\mathbb {C})$  ; it is not the case in higher dimension where $G_n(\\mathbb {C})$ is a strict subgroup of $\\mathrm {Bir}(\\mathbb {P}^n_\\mathbb {C})$ (see [29], [35]).", "However the following theorems show that $G_n(\\mathbb {C})$ shares good properties with $G_2(\\mathbb {C})=\\mathrm {Bir}(\\mathbb {P}^2_\\mathbb {C})$ .", "In [14] we proved that for any integer $n\\ge 2$ the group $\\mathrm {Bir}(\\mathbb {P}^n_\\mathbb {k})$ , where $\\mathbb {k}$ denotes an algebraically closed field, is not linear ; we obtain a similar statement for $G_n(\\mathbb {k})$ , $n\\ge 2$  : Theorem A If $\\mathbb {k}$ is an algebraically closed field, there is no nontrivial finite-dimensional linear representation of $G_n(\\mathbb {k})$ over any field.", "The group $G_n(\\mathbb {C})$ contains some \"big\" subgroups : Proposition B The group of polynomial automorphisms of $\\mathbb {C}^n$ generated by the affine automorphisms and the Jonquières ones is a subgroup of $G_n(\\mathbb {C})$ .", "If $\\mathfrak {g}_0$ , $\\mathfrak {g}_1$ , $\\ldots $ , $\\mathfrak {g}_k$ are some generic automorphisms of $\\mathbb {P}^n_\\mathbb {C}$ , then $\\langle \\mathfrak {g}_0\\sigma _n,\\,\\mathfrak {g}_1\\sigma _n,\\,\\ldots ,\\,\\mathfrak {g}_k\\sigma _n\\rangle \\subset G_n(\\mathbb {C})$ is a free subgroup of $G_n(\\mathbb {C})$ .", "Remark 1.1 For the meaning of \"generic\" see the proof of Proposition REF .", "In [14] we establish that $G_2(\\mathbb {C})=\\mathrm {Bir}(\\mathbb {P}^2_\\mathbb {C})$ is perfect, i.e.", "$[G_2(\\mathbb {C}),G_2(\\mathbb {C})]=G_2(\\mathbb {C})$  ; the same holds for any $n$  : Theorem C If $\\mathbb {k}$ is an algebraically closed field, $G_n(\\mathbb {k})$ is perfect.", "In [21] we determine the automorphisms group of $G_2(\\mathbb {C})=\\mathrm {Bir}(\\mathbb {P}^2_\\mathbb {C})$  ; in higher dimensions we have a similar description.", "Before giving a precise result, let us introduce some notation : the group of the field automorphisms acts on $\\mathrm {Bir}(\\mathbb {P}^n_\\mathbb {C})$  : if $f$ is an element of $\\mathrm {Bir}(\\mathbb {P}^n_\\mathbb {C})$ , and $\\kappa $ is a field automorphism we denote by ${}^{\\kappa }\\!\\, f$ the element obtained by letting $\\kappa $ acting on $f$ .", "Theorem D Let $\\varphi $ be an automorphism of $\\mathrm {Bir}(\\mathbb {P}^n_\\mathbb {C})$ .", "There exist $\\kappa $ an automorphism of the field $\\mathbb {C}$ , and $\\psi $ a birational map of $\\mathbb {P}^n_\\mathbb {C}$ such that $\\varphi (f)={}^{\\kappa }\\!\\,(\\psi f\\psi ^{-1}) \\qquad \\forall \\,f\\in G_n(\\mathbb {C}).$ The question \"is the Cremona group simple ?\"", "is a very old one ; Cantat and Lamy recently gave a negative answer in dimension 2 (see [12]).", "One can consider the same question when $G_2(\\mathbb {k})$ is equipped with the Zariski topology ($\\mathbb {k}$ denotes here an algebraically closed field) ; Blanc looked at it, and obtained a positive answer ([5]).", "What about $G_n(\\mathbb {k})$   ?", "Proposition E If $\\mathbb {k}$ is an algebraically closed field, the group $G_n(\\mathbb {k})$ , equipped with the Zariski topology, is simple." ], [ "Organisation of the article", "We first recall a result of Pan about the set of group generators of $\\mathrm {Bir}(\\mathbb {P}^n_\\mathbb {C})$ , $n\\ge 3$ (see §) ; we then note that as soon as $n\\ge 3$ , there are birational maps of degree $n=\\deg \\sigma _n$ that do not belong to $G_n(\\mathbb {C})$ .", "In § we prove Theorem REF , and in § Proposition REF .", "Let us remark that the fact that the group of tame automorphisms is contained in $G_n(\\mathbb {C})$ implies that $G_n(\\mathbb {C})$ contains maps of any degree, it was not obvious a priori.", "In § we study the normal subgroup in $G_n(\\mathbb {C})$ generated by $\\sigma _n$ (resp.", "by an automorphism of $\\mathbb {P}^n_\\mathbb {C}$ ) ; it allows us to establish Theorem REF .", "We finish § with the proofs of Theorem REF , and Proposition REF .", "I would like to thank D. Cerveau for his helpful and continuous listening.", "Thanks to the referee that helps me to improve the exposition.", "Thanks to I. Dolgachev for pointing out me that Coble introduced the group $G_n(\\mathbb {C})$ in [15], and to J. Blanc, J. Diller, F. Han, M. Jonsson, J.-L. Lin for their remarks and comments." ], [ "Some definitions", "A polynomial automorphism $\\varphi $ of $\\mathbb {C}^n$ is a map $\\mathbb {C}^n\\rightarrow \\mathbb {C}^n$ of the type $(z_0,z_1,\\ldots ,z_{n-1})\\mapsto \\big (\\varphi _0(z_0,z_1,\\ldots ,z_{n-1}),\\varphi _1(z_0,z_1,\\ldots ,z_{n-1}),\\ldots ,\\varphi _{n-1}(z_0,z_1,\\ldots ,z_{n-1})\\big ),$ with $\\varphi _i\\in \\mathbb {C}[z_0,z_1,\\ldots ,z_{n-1}]$ , that is bijective ; we denote $\\varphi $ by $\\varphi =(\\varphi _0,\\varphi _1,\\ldots ,\\varphi _{n-1})$ .", "A rational self-map $\\phi \\colon \\mathbb {P}^n_\\mathbb {C}\\dashrightarrow \\mathbb {P}^n_\\mathbb {C}$ is given by $(z_0:z_1:\\ldots :z_n)\\dashrightarrow \\big (\\phi _0(z_0,z_1,\\ldots ,z_n):\\phi _1(z_0,z_1,\\ldots ,z_n):\\ldots :\\phi _n(z_0,z_1,\\ldots ,z_n)\\big )$ where the $\\phi _i$ are homogeneous polynomials of the same positive degree, and without common factor of positive degree.", "Let us denote by $\\mathbb {C}[z_0,z_1,\\ldots ,z_n]_d$ the set of homogeneous polynomials in $z_0$ , $z_1$ , $\\ldots $ , $z_n$ of degree $d$ .", "The degree of $\\phi $ is by definition the degree of the $\\phi _i$ .", "A birational self-map of $\\mathbb {P}^n_\\mathbb {C}$ is a rational self-map that admits a rational inverse.", "The set of polynomial automorphisms of $\\mathbb {C}^n$ (resp.", "birational self-maps of $\\mathbb {P}^n_\\mathbb {C}$ ) form a group denoted $\\mathrm {Aut}(\\mathbb {C}^n)$ (resp.", "$\\mathrm {Bir}(\\mathbb {P}^n_\\mathbb {C})$ )." ], [ "A result of Pan", "Let us recall a construction of Pan ([35]) which, given a birational self-map of $\\mathbb {P}^n_\\mathbb {C}$ , allows one to construct a birational self-map of $\\mathbb {P}^{n+1}_\\mathbb {C}$ .", "Let $P\\in \\mathbb {C}[z_0,z_1,\\ldots ,z_n]_d$ , $Q\\in \\mathbb {C}[z_0,z_1,\\ldots ,z_n]_\\ell $ , and let $R_0$ , $R_1$ , $\\ldots $ , $R_{n-1}\\in \\mathbb {C}[z_0,z_1,\\ldots ,z_{n-1}]_{d-\\ell }$ be some homogeneous polynomials.", "Denote by $\\Psi _{P,Q,R}\\colon \\mathbb {P}^n_\\mathbb {C}\\dashrightarrow \\mathbb {P}^n_\\mathbb {C}$ and $\\widetilde{\\Psi }\\colon \\mathbb {P}^{n-1}_\\mathbb {C}\\dashrightarrow \\mathbb {P}^{n-1}_\\mathbb {C}$ the rational maps defined by $\\Psi _{P,Q,R}=\\big (QR_0:QR_1:\\ldots :QR_{n-1}:P\\big )\\qquad \\&\\qquad \\widetilde{\\Psi }_R=\\big (R_0:R_1:\\ldots :R_{n-1}\\big ).$ Lemma 2.1 ([35]) Let $d$ , $\\ell $ be some integers such that $d\\ge \\ell +1\\ge 2$ .", "Take $Q$ in $\\mathbb {C}[z_0,z_1,\\ldots ,z_n]_\\ell $ , and $P$ in $\\mathbb {C}[z_0,z_1,\\ldots ,z_n]_d$ without common factors.", "Let $R_1$ , $\\ldots $ , $R_n$ be some elements of $\\mathbb {C}[z_0,z_1,\\ldots ,z_{n-1}]_{d-\\ell }$ .", "Assume that $P=z_nP_{d-1}+P_d \\qquad \\qquad Q=z_nQ_{\\ell -1}+Q_\\ell $ with $P_{d-1}$ , $P_d$ , $Q_{\\ell -1}$ , $Q_\\ell \\in \\mathbb {C}[z_0,z_1,\\ldots ,z_{n-1}]$ of degree $d-1$ , resp.", "$d$ , resp.", "$\\ell -1$ , resp.", "$\\ell $ and such that $(P_{d-1},Q_{\\ell -1})\\ne (0,0)$ .", "The map $\\Psi _{P,Q,R}$ is birational if and only if $\\widetilde{\\Psi }_R$ is birational.", "Let us give the motivation of this construction : Theorem 2.2 ([29], [35]) Any set of group generators of $\\mathrm {Bir}(\\mathbb {P}^n_\\mathbb {C})$ , $n\\ge 3$ , contains uncountably many non-linear maps.", "We will give an idea of the proof of this statement.", "Lemma 2.3 ([35]) Let $n\\ge 3$ .", "Let $\\mathcal {S}$ be an hypersurface of $\\mathbb {P}^n_\\mathbb {C}$ of degree $\\ell \\ge 1$ having a point $p$ of multiplicity $\\ge \\ell -1$ .", "Then there exists a birational self-map of $\\mathbb {P}^n_\\mathbb {C}$ of degree $d\\ge \\ell +1$ that blows down $\\mathcal {S}$ onto a point.", "One can assume without loss of generality that $p=(0:0:\\ldots :0:1)$ .", "Denote by $q^{\\prime }=0$ the equation of $\\mathcal {S}$ , and take a generic plane passing through $p$ given by the equation $h=0$ .", "Finally choose $P=z_nP_{d-1}+P_d$ such that $\\bullet $ $P_{d-1}\\ne 0$  ; $\\bullet $ $\\textrm {pgcd}\\,(P,hq^{\\prime })=1$ .", "Now set $Q=h^{d-\\ell -1}q^{\\prime }$ , $R_i=z_i$ , and conclude with Lemma REF .", "[Proof of Theorem REF ] Let us consider the family of hypersurfaces given by $q(z_1,z_2,z_3)=0$ where $q=0$ defines a smooth curve $\\mathcal {C}_q$ of degree $\\ell $ on $\\lbrace z_0=z_4=z_5=\\ldots =z_n=0\\rbrace $ .", "Let us note that $q=0$ is birationally equivalent to $\\mathbb {P}^{n-2}_\\mathbb {C}\\times \\mathcal {C}_q$ .", "Furthermore $q=0$ and $q^{\\prime }=0$ are birationally equivalent if and only if $\\mathcal {C}_q$ and $\\mathcal {C}_{q^{\\prime }}$ are isomorphic.", "Note that for $\\ell =2$ the set of isomorphism classes of smooth cubics is a 1-parameter family, and that according to Lemma REF for any $\\mathcal {C}_q$ there exists a birational self-map of $\\mathbb {P}^n_\\mathbb {C}$ that blows down $\\mathcal {C}_q$ onto a point.", "Hence any set of group generators of $\\mathrm {Bir}(\\mathbb {P}^n_\\mathbb {C})$ , $n\\ge 3$ , has to contain uncountably many non-linear maps.", "One can take $d=\\ell +1$ in Lemma REF .", "In particular Corollary 2.4 As soon as $n\\ge 3$ , there are birational maps of degree $n=\\deg \\sigma _n$ that do not belong to $G_n(\\mathbb {C})$ .", "Remark 2.5 The maps $\\Psi _{P,Q,R}$ that are birational form a subgroup of $\\mathrm {Bir}(\\mathbb {P}^n_\\mathbb {C})$ denoted by $\\mathrm {J}_0(1;\\mathbb {P}^n_\\mathbb {C})$ , and studied in [37] : in particular $\\mathrm {J}_0(1;\\mathbb {P}^3_\\mathbb {C})$ inherits the property of Theorem REF ." ], [ "A first remark", "Let $\\phi $ be a birational map of $\\mathbb {P}^3_\\mathbb {C}$ .", "A regular resolution of $\\phi $ is a morphism $\\pi \\colon Z\\rightarrow \\mathbb {P}^3_\\mathbb {C}$ which is a sequence of blow-ups $\\pi =\\pi _1\\circ \\ldots \\circ \\pi _r$ along smooth irreducible centers, such that $\\phi \\circ \\pi \\colon Z\\rightarrow \\mathbb {P}^3_\\mathbb {C}$ is a birational morphism, and each center $B_i$ of the blow-up $\\pi _i\\colon Z_i\\rightarrow Z_{i-1}$ is contained in the base locus of the induced map $Z_{i-1}\\dashrightarrow \\mathbb {P}^3_\\mathbb {C}$ .", "It follows from Hironaka that such a resolution always exists.", "If $B$ is a smooth irreducible center of a blow-up in a smooth projective complex variety of dimension 3, then $B$ is either a point, or a smooth curve.", "We define the genus of $B$ as follows : it is 0 if $B$ is a point, the usual genus otherwise.", "Frumkin defines the genus of $\\phi $ to be the maximum of the genera of the centers of the blow-ups in the resolution of $\\phi $ (see [28]), and shows that this definition does not depend on the choice of the regular resolution.", "In [32] an other definition of the genus of a birational map is given.", "Let us recall that if $E$ is an irreducible divisor contracted by a birational map between smooth projective complex varieties of dimension 3, then $E$ is birational to $\\mathbb {P}^1_\\mathbb {C}\\times \\mathcal {C}$ , where $\\mathcal {C}$ denotes a smooth curve ([32]).", "The genus of a birational map $\\phi $ of $\\mathbb {P}^3_\\mathbb {C}$ is the maximum of the genera of the irreducible divisors in $\\mathbb {P}^3_\\mathbb {C}$ contracted by $\\phi $ .", "Lamy proves that these two definitions of genus agree ([32]).", "Let $\\phi $ be in $\\mathrm {Bir}(\\mathbb {P}^3_\\mathbb {C})$ , and let $\\mathcal {H}$ be an irreducible hypersurface of $\\mathbb {P}^3_\\mathbb {C}$ .", "We say that $\\mathcal {H}$ is $\\phi $ -exceptional if $\\phi $ is not injective on any open subset of $\\mathcal {H}$ (or equivalently if there is an open subset of $\\mathcal {H}$ which is mapped into a subset of codimension $\\ge 2$ by $\\phi $ ).", "Let $\\phi _1$ , $\\ldots $ , $\\phi _k$ be in $\\mathrm {Bir}(\\mathbb {P}^3_\\mathbb {C})$ , and let $\\phi =\\phi _k\\circ \\ldots \\circ \\phi _1$ .", "Let $\\mathcal {H}$ be an irreducible hypersurface of $\\mathbb {P}^3_\\mathbb {C}$ .", "If $\\mathcal {H}$ is $\\phi $ -exceptional, then there exists $1\\le i\\le k$ and a $\\phi _i$ -exceptional hypersurface $\\mathcal {H}_i$ such that $\\phi _{i-1}\\circ \\ldots \\circ \\phi _1$ realizes a birational isomorphism from $\\mathcal {H}$ to $\\mathcal {H}_i$  ; $\\phi _i$ contracts $\\mathcal {H}_i$ .", "In particular one has the following statement.", "Proposition 2.6 The group $G_3(\\mathbb {C})$ is contained in the subgroup of birational self-maps of $\\mathbb {P}^3_\\mathbb {C}$ of genus 0.", "If $V$ is a finite dimensional vector space over $\\mathbb {C}$ there is no faithful linear representation $\\mathrm {Bir}(\\mathbb {P}^n_\\mathbb {C})\\rightarrow \\mathrm {GL}(V)$ (see [14]).", "The proof of this statement is based on the following Lemma due to Birkhoff ([2]) : if $\\mathfrak {a}$ , $\\mathfrak {b}$ and $\\mathfrak {c}$ are three elements of $\\mathrm {GL}(n;\\mathbb {C})$ such that $[\\mathfrak {a},\\mathfrak {b}]=\\mathfrak {c},\\quad [\\mathfrak {a},\\mathfrak {c}]=[\\mathfrak {b},\\mathfrak {c}]=\\mathrm {id},\\quad \\mathfrak {c}^p=\\mathrm {id} \\text{ for some $p$ prime}$ then $p\\le n$ .", "Assume that there exists an injective homomorphism $\\rho $ from $\\mathrm {Bir}(\\mathbb {P}^2_\\mathbb {C})$ to $\\mathrm {GL}(n;\\mathbb {C})$ .", "For any $p>n$ prime consider in the affine chart $z_2=1$ the maps $(\\exp (2\\mathbf {i}\\pi /p)z_0,z_1),\\qquad (z_0,z_0z_1),\\qquad (z_0,\\exp (-2\\mathbf {i}\\pi /p)z_1).$ The image by $\\rho $ of these maps satisfy Birkhoff Lemma so $p\\le n$  : contradiction.", "In any dimension we have the same property : $G_n(\\mathbb {C})$ is not linear, i.e.", "if $V$ is a finite dimensional vector space over $\\mathbb {C}$ there is no faithful linear representation $G_n(\\mathbb {C})\\rightarrow \\mathrm {GL}(V)$ .", "Actually $G_n(\\mathbb {C})$ satisfies a more precise property due to Cornulier in dimension 2 (see [16]) : Proposition 3.1 The group $G_n(\\mathbb {C})$ has no non-trivial finite dimensional representation.", "Lemma 3.2 The map $\\varsigma =\\big (z_0z_{n-1}:z_1z_{n-1}:\\ldots :z_{n-2}z_{n-1}: z_{n-1}z_n:z_n^2\\big )$ belongs to $G_n(\\mathbb {C})$ .", "We have $\\varsigma =\\mathfrak {a}_1\\sigma _n\\mathfrak {a}_2\\sigma _n\\mathfrak {a}_3$ where $\\mathfrak {a}_1=\\big (z_2-z_1:z_3-z_1:\\ldots :z_n-z_1:z_1:z_1-z_0\\big ),$ $\\mathfrak {a}_2=\\big (z_{n-1}+z_n:z_n:z_0:z_1:\\ldots :z_{n-2}\\big ),$ $\\mathfrak {a}_3=\\big (z_0+z_n:z_1+z_n:\\ldots :z_{n-2}+z_n:z_{n-1}-z_n:z_n\\big ).$ [Proof of Proposition REF ] We adapt the proof of [16].", "Let us now work in the affine chart $z_n=1$ .", "By Lemma REF in $G_n(\\mathbb {C})$ there is a natural copy of $H=(\\mathbb {C}^*)^n\\rtimes \\mathbb {Z}$  ; indeed $\\langle \\varsigma =\\big (z_0z_{n-1},z_1z_{n-1},\\ldots ,z_{n-2}z_{n-1}, z_{n-1}\\big )\\rangle \\simeq \\mathbb {Z}$ acts on $\\big \\lbrace (\\alpha _0z_0,\\alpha _1z_1,\\ldots ,\\alpha _{n-1}z_{n-1})\\,\\vert \\,\\alpha _i\\in \\mathbb {C}^*\\big \\rbrace \\simeq (\\mathbb {C}^*)^n$ and $H$ is the group of maps $\\big \\lbrace (\\alpha _0z_0z_{n-1}^k,\\alpha _1z_1z_{n-1}^k,\\ldots ,\\alpha _{n-2}z_{n-2}z_{n-1}^k,\\alpha _{n-1}z_{n-1})\\,\\vert \\,\\alpha _i\\in \\mathbb {C}^*,\\,k\\in \\mathbb {Z}\\big \\rbrace .$ Consider any linear representation $\\rho \\colon H\\rightarrow \\mathrm {GL}(k;\\mathbb {C})$ .", "If $p$ is prime, and if $\\xi _p$ is a primitive $p$ -root of unity, set $\\mathfrak {g}_p=(\\xi _pz_0,\\xi _pz_1,\\ldots ,\\xi _pz_{n-1}), \\quad \\mathfrak {h}_p=(\\xi _p z_0,\\xi _p z_1,\\ldots ,\\xi _p z_{n-2},z_{n-1}).$ Then $\\mathfrak {h}_p=[\\varsigma ,\\mathfrak {g}_p]$ commutes with both $\\phi $ and $\\mathfrak {g}_p$ .", "By [2] if $\\rho (\\mathfrak {g}_p)\\ne 1$ , then $k\\ge p$ .", "Picking $p$ to be greater than $k$ , this shows that if we have an arbitrary representation $f\\colon G_n(\\mathbb {C})\\rightarrow \\mathrm {GL}(k;\\mathbb {C})$ , the restriction $f_{\\vert \\mathrm {PGL}(n+1;\\mathbb {C})}$ is not faithful.", "Since $\\mathrm {PGL}(n+1;\\mathbb {C})$ is simple, this implies that $f$ is trivial on $\\mathrm {PGL}(n+1;\\mathbb {C})$ .", "We conclude by using the fact that the two involutions $-\\mathrm {id}$ and $\\sigma _n$ are conjugate via the map $\\psi $ given by $\\left(\\frac{z_0+1}{z_0-1},\\frac{z_1+1}{z_1-1},\\ldots ,\\frac{z_{n-1}+1}{z_{n-1}-1}\\right)$ and $\\psi =\\mathfrak {a}_1\\sigma _n\\mathfrak {a}_2$ where $\\mathfrak {a}_1$ and $\\mathfrak {a}_2$ denote the two following automorphisms of $\\mathbb {P}^n_\\mathbb {C}$ $\\mathfrak {a}_1=\\big (z_0+1,z_1+1,\\ldots ,z_{n-1}+1\\big ),$ $\\mathfrak {a}_2=\\Big (\\frac{z_0-1}{2},\\frac{z_1-1}{2},\\ldots ,\\frac{z_{n-1}-1}{2}\\Big ).$ Remark 3.3 Proposition REF is also true for $G_n(\\mathbb {k})$ where $\\mathbb {k}$ is an algebraically closed field." ], [ "The tame automorphisms", "The automorphisms of $\\mathbb {C}^n$ written in the form $(\\phi _0,\\phi _1,\\ldots ,\\phi _{n-1})$ where $\\phi _i=\\phi _i(z_i,z_{i+1},\\ldots ,z_{n-1})$ depends only on $z_i$ , $z_{i+1}$ , $\\ldots $ , $z_{n-1}$ form the Jonquières subgroup $\\mathrm {J}_n\\subset \\mathrm {Aut}(\\mathbb {C}^n)$ .", "A polynomial automorphism $(\\phi _0,\\phi _1,\\ldots ,\\phi _{n-1})$ where all the $\\phi _i$ are linear is an affine transformation.", "Denote by $\\mathrm {Aff}_n$ the group of affine transformations ; $\\mathrm {Aff}_n$ is the semi-direct product of $\\mathrm {GL}(n;\\mathbb {C})$ with the commutative unipotent subgroup of translations.", "We have the following inclusions $\\mathrm {GL}(n;\\mathbb {C})\\subset \\mathrm {Aff}_n\\subset \\mathrm {Aut}(\\mathbb {C}^n).$ The subgroup $\\mathrm {Tame}_n\\subset \\mathrm {Aut}(\\mathbb {C}^n)$ generated by $\\mathrm {J}_n$ and $\\mathrm {Aff}_n$ is called the group of tame automorphisms.", "For $n=2$ one has $\\mathrm {Tame}_2=\\mathrm {Aut}(\\mathbb {C}^2)$ , this follows from the fact that $\\mathrm {Aut}(\\mathbb {C}^2)=\\mathrm {J}_2\\ast _{\\mathrm {J}_2\\cap \\mathrm {Aff}_2}\\mathrm {Aff}_2$ (see [30]).", "The group $\\mathrm {Tame}_3$ does not coincide with $\\mathrm {Aut}(\\mathbb {C}^3)$  : the Nagata automorphism is not tame ([46]).", "Derksen gives a set of generators of $\\mathrm {Tame}_n$ (see [50] for a proof) : Theorem 4.1 Let $n\\ge 3$ be a natural integer.", "The group $\\mathrm {Tame}_n$ is generated by $\\mathrm {Aff}_n$ , and the Jonquières map $\\big (z_0+z_1^2,z_1,z_2,\\ldots ,z_{n-1}\\big )$ .", "Proposition 4.2 The group $G_n(\\mathbb {C})$ contains the group of tame polynomial automorphisms of $\\mathbb {C}^n$ .", "The inclusion $\\mathrm {Aff}_n\\subset \\mathrm {Aut}(\\mathbb {P}^n_\\mathbb {C})$ is obvious ; according to Theorem REF we thus just have to prove that $\\big (z_0+z_1^2,z_1,z_2,\\ldots ,z_{n-1}\\big )$ belongs to $G_n(\\mathbb {C})$ .", "But $\\big (z_0z_n+z_1^2:z_1z_n:z_2z_n:\\ldots :z_{n-1}z_n:z_n^2\\big )=\\mathfrak {g}_1\\sigma _n \\mathfrak {g}_2\\sigma _n \\mathfrak {g}_3\\sigma _n \\mathfrak {g}_2\\sigma _n\\mathfrak {g}_4$ where $& \\mathfrak {g}_1=\\big (z_2-z_1+z_0:2z_1-z_0:z_3:z_4:\\ldots :z_n:z_1-z_0\\big ), \\\\& \\mathfrak {g}_2=\\big (z_0+z_2:z_0:z_1:z_3:z_4:\\ldots :z_n\\big ), \\\\& \\mathfrak {g}_3=\\big (-z_1:z_0+z_2-3z_1:z_0:z_3:z_4:\\ldots :z_n\\big ), \\\\& \\mathfrak {g}_4=\\big (z_1-z_n:-2z_n-z_0:2z_n-z_1:-z_2:-z_3:\\ldots :-z_{n-1}\\big ).$" ], [ "Free groups and $G_n(\\mathbb {C})$", "Following the idea of [14] we prove that : Proposition 4.3 Let $\\mathfrak {g}_0$ , $\\mathfrak {g}_1$ , $\\ldots $ , $\\mathfrak {g}_k$ be some generic elements of $\\mathrm {Aut}(\\mathbb {P}^n_\\mathbb {C})$ .", "The group generated by $\\mathfrak {g}_0$ , $\\mathfrak {g}_1$ , $\\ldots $ , $\\mathfrak {g}_k$ , and $\\sigma _n$ is the free product $\\overbrace{\\mathbb {Z}\\ast \\ldots \\ast \\mathbb {Z}}^{k+1}\\,\\ast \\,(\\mathbb {Z}/2\\mathbb {Z}),$ the $\\mathfrak {g}_i$ 's and $\\sigma _n$ being the generators for the factors of this free product.", "In particular the subgroup $\\langle \\mathfrak {g}_0\\sigma _n,\\,\\mathfrak {g}_1\\sigma _n,\\,\\ldots ,\\,\\mathfrak {g}_k\\sigma _n\\rangle $ of $G_n(\\mathbb {C})$ is a free group.", "Remark 4.4 The meaning of \"generic\" is explained in the proof below.", "Let us show the statement for $k=0$ (in the general case it is sufficient to replace the free product $\\mathbb {Z}\\ast \\mathbb {Z}/2\\mathbb {Z}$ by $\\mathbb {Z}\\ast \\mathbb {Z}\\ast \\ldots \\ast \\mathbb {Z}\\ast \\mathbb {Z}/2\\mathbb {Z}$ ).", "If $\\langle \\mathfrak {g},\\sigma _n\\rangle $ is not isomorphic to $\\mathbb {Z}\\ast \\mathbb {Z}/2\\mathbb {Z}$ , then there exists a word $M_\\mathfrak {g}$ in $\\mathbb {Z}\\ast \\mathbb {Z}/2\\mathbb {Z}$ such that $M_\\mathfrak {g}(\\mathfrak {g},\\sigma _n)=\\mathrm {id}$ .", "Note that the set of words $M_\\mathfrak {g}$ is countable, and that for a given word $M$ the set $R_M=\\big \\lbrace \\mathfrak {g}\\,\\big \\vert \\, M(\\mathfrak {g},\\sigma _n)=\\mathrm {id}\\big \\rbrace $ is algebraic in $\\mathrm {Aut}(\\mathbb {P}^n_\\mathbb {C})$ .", "Consider an automorphism $\\mathfrak {g}$ written in the following form $\\big (\\alpha z_0+\\beta z_1:\\gamma z_0+\\delta z_1:z_2:z_3:\\ldots :z_n\\big )$ where $\\left[\\begin{array}{cc}\\alpha & \\beta \\\\\\gamma & \\delta \\end{array}\\right]\\in \\mathrm {PGL}(2;\\mathbb {C})$ .", "Since the pencil $z_0=tz_1$ is invariant by both $\\sigma _n$ and $\\mathfrak {g}$ , one inherits a linear representation $\\langle \\mathfrak {g},\\,\\sigma _n\\rangle \\rightarrow \\mathrm {PGL}(2;\\mathbb {C})$ defined by $\\mathfrak {g}\\colon t\\mapsto \\frac{\\alpha t+\\beta }{\\gamma t+\\delta },\\qquad \\sigma _n\\colon t\\mapsto \\frac{1}{t}.$ But the group generated by $\\left[\\begin{array}{cc} \\alpha & \\beta \\\\ \\gamma & \\delta \\end{array}\\right]$ and $\\left[\\begin{array}{cc} 0 & 1 \\\\ 1 & 0 \\end{array}\\right]$ is generically isomorphic to $\\mathbb {Z}\\ast \\mathbb {Z}/2\\mathbb {Z}$ (see [17]).", "Hence the complements $R_M^C$ are dense open subsets, and their intersection is dense by Baire property." ], [ "The group $G_n(\\mathbb {C})$ is perfect", "If $\\mathrm {G}$ is a group, and if $g$ is an element of $\\mathrm {G}$ , we denote by $\\mathrm {N}(g;\\mathrm {G})=\\langle fgf^{-1}\\,\\vert \\,f\\in \\mathrm {G}\\rangle .$ the normal subgroup generated by $g$ in $\\mathrm {G}$ .", "Proposition 5.1 The following assertions hold : $\\mathrm {N}(\\mathfrak {g};\\mathrm {PGL}(n+1;\\mathbb {C}))=\\mathrm {PGL}(n+1;\\mathbb {C})$ for any $\\mathfrak {g}\\in \\mathrm {PGL}(n+1;\\mathbb {C})\\setminus \\lbrace \\mathrm {id}\\rbrace $  ; $\\mathrm {N}(\\sigma _n;G_n(\\mathbb {C}))=G_n(\\mathbb {C})$  ; $\\mathrm {N}(\\mathfrak {g};G_n(\\mathbb {C}))=G_n(\\mathbb {C})$ for any $\\mathfrak {g}\\in \\mathrm {PGL}(n+1;\\mathbb {C})\\setminus \\lbrace \\mathrm {id}\\rbrace $ .", "Let us work in the affine chart $z_n=1$ .", "Since $\\mathrm {PGL}(n+1;\\mathbb {C})$ is simple one has the first assertion.", "Let $\\phi $ be in $G_n(\\mathbb {C})$  ; there exist $\\mathfrak {g}_0$ , $\\mathfrak {g}_1$ , $\\ldots $ , $\\mathfrak {g}_k$ in $\\mathrm {Aut}(\\mathbb {P}^n_\\mathbb {C})$ such that $\\phi =(\\mathfrak {g}_0)\\,\\sigma _n\\, \\mathfrak {g}_1\\,\\sigma _n\\,\\ldots \\sigma _n\\,\\mathfrak {g}_k\\,(\\sigma _n).$ As $\\mathrm {PGL}(n+1;\\mathbb {C})$ is simple $\\mathrm {N}(-\\mathrm {id};\\mathrm {PGL}(n+1;\\mathbb {C}))=\\mathrm {PGL}(n+1;\\mathbb {C}),$ and for any $0\\le i\\le k$ there exist $\\mathfrak {f}_{i,0}$ , $\\mathfrak {f}_{i,1}$ , $\\ldots $ , $\\mathfrak {f}_{i,\\ell _i}$ in $\\mathrm {PGL}(n+1;\\mathbb {C})$ such that $\\mathfrak {g}_i=\\mathfrak {f}_{i,0}\\big (-\\mathrm {id}\\big )\\mathfrak {f}_{i,0}^{-1}\\,\\mathfrak {f}_{i,1}\\big (-\\mathrm {id}\\big )\\mathfrak {f}_{i,1}^{-1}\\,\\ldots \\,\\mathfrak {f}_{i,\\ell _i}\\big (-\\mathrm {id}\\big )\\mathfrak {f}_{i,\\ell _i}^{-1}.$ We conclude by using the fact that $-\\mathrm {id}$ and $\\sigma _n$ are conjugate via an element of $G_n(\\mathbb {C})$ (see the proof of Proposition REF ).", "Fix $\\mathfrak {g}$ in $\\mathrm {PGL}(n+1;\\mathbb {C})\\setminus \\lbrace \\mathrm {id}\\rbrace $ .", "Since $\\mathrm {N}(\\mathfrak {g};\\mathrm {PGL}(n+1;\\mathbb {C}))=\\mathrm {PGL}(n+1;\\mathbb {C})$ , the involution $-\\mathrm {id}$ can be written as a composition of some conjugates of $\\mathfrak {g}$ .", "The maps $-\\mathrm {id}$ and $\\sigma _n$ being conjugate one has $\\sigma _n=(f_0 \\mathfrak {g}f_0^{-1})\\,(f_1 \\mathfrak {g}f_1^{-1})\\,\\ldots \\,(f_\\ell \\mathfrak {g}f_\\ell ^{-1})$ for some $f_i$ in $G_n(\\mathbb {C})$ .", "So $\\mathrm {N}(\\sigma _n;G_n(\\mathbb {C}))\\subset \\mathrm {N}(\\mathfrak {g};G_n(\\mathbb {C}))$ , and one concludes with the second assertion.", "Corollary 5.2 The group $G_n(\\mathbb {C})$ satisfies the following properties : $G_n(\\mathbb {C})$ is perfect, i.e.", "$[G_n(\\mathbb {C}),G_n(\\mathbb {C})]=G_n(\\mathbb {C})$  ; for any $\\phi $ in $G_n(\\mathbb {C})$ there exist $\\mathfrak {g}_0$ , $\\mathfrak {g}_1$ , $\\ldots $ , $\\mathfrak {g}_k$ automorphisms of $\\mathbb {P}^n_\\mathbb {C}$ such that $\\phi =(\\mathfrak {g}_0\\sigma _n\\mathfrak {g}_0^{-1})(\\mathfrak {g}_1\\sigma _n\\mathfrak {g}_1^{-1})\\ldots (\\mathfrak {g}_k\\sigma _n\\mathfrak {g}_k^{-1})$ The third assertion of Proposition REF implies that any element of $G_n(\\mathbb {C})$ can be written as a composition of some conjugates of $\\mathfrak {t}=\\big (z_0:z_1+z_n:z_2+z_n:\\ldots :z_{n-1}+z_n:z_n\\big ).$ As $\\mathfrak {t}=\\Big [\\big (z_0:3z_1:3z_2:\\ldots :3z_{n-1}:z_n\\big ),\\left(2z_0:z_1+z_n:z_2+z_n:\\ldots :z_{n-1}+z_n:2z_n\\right)\\Big ],$ the group $G_n(\\mathbb {C})$ is perfect.", "For any $\\alpha _0$ , $\\alpha _1$ , $\\ldots $ , $\\alpha _{n}$ in $\\mathbb {C}^*$ set $\\mathfrak {d}(\\alpha _0,\\alpha _1,\\ldots ,\\alpha _{n})=(\\alpha _0z_0:\\alpha _1z_1:\\ldots :\\alpha _{n}z_{n})$ , and let us define $\\mathrm {H}$ as follows : $\\mathrm {H}=\\Big \\lbrace \\mathfrak {g}_0\\sigma _n\\mathfrak {g}_0^{-1}\\,\\mathfrak {g}_1\\sigma _n\\mathfrak {g}_1^{-1}\\,\\ldots \\,\\mathfrak {g}_\\ell \\sigma _n\\mathfrak {g}_\\ell ^{-1}\\,\\vert \\,\\mathfrak {g}_i\\in \\mathrm {PGL}(n+1;\\mathbb {C}),\\,\\ell \\in \\mathbb {N}\\Big \\rbrace .$ The second assertion of the Corollary is then equivalent to $\\mathrm {H}=G_n(\\mathbb {C})$ .", "Let us remark that $\\mathrm {H}$ is a group that contains $\\sigma _n$ , and that $\\mathrm {PGL}(n+1;\\mathbb {C})$ acts by conjugacy on it.", "One can check that $\\mathfrak {d}_\\alpha \\,\\sigma _n\\,\\mathfrak {d}_\\alpha ^{-1}=\\mathfrak {d}_\\alpha ^2\\,\\sigma _n=\\sigma _n\\,\\mathfrak {d}_\\alpha ^{-2}.$ Hence for each $\\mathfrak {g}$ in $\\mathrm {PGL}(n+1;\\mathbb {C})$ we have $\\mathfrak {g}\\mathfrak {d}_\\alpha \\,\\sigma _n\\,\\mathfrak {d}_\\alpha ^{-1}\\mathfrak {g}^{-1}=(\\mathfrak {g}\\mathfrak {d}_\\alpha ^2\\mathfrak {g}^{-1})(\\mathfrak {g}\\sigma _n\\mathfrak {g}^{-1})$ , so $\\mathfrak {g}\\mathfrak {d}_\\alpha ^2\\mathfrak {g}^{-1}$ belongs to $\\mathrm {H}$ .", "Since any automorphism of $\\mathbb {P}^n_\\mathbb {C}$ can be written as a product of diagonalizable matrices, $\\mathrm {PGL}(n+1;\\mathbb {C})\\subset \\mathrm {H}$ ." ], [ "On the restriction of automorphisms of the group birational maps to $G_n(\\mathbb {C})$", "If $M$ is a projective variety defined over a field $\\mathbb {k}\\subset \\mathbb {C}$ the group $\\mathrm {Aut}_\\mathbb {k}(\\mathbb {C})$ of automorphisms of the field extension $\\mathbb {C}/\\mathbb {k}$ acts on $M(\\mathbb {C})$ , and on both $\\mathrm {Aut}(M)$ and $\\mathrm {Bir}(M)$ as follows ${}^{\\kappa }\\!\\,\\psi (p)=(\\kappa \\psi \\kappa ^{-1})(p)$ for any $\\kappa $ in $\\mathrm {Aut}_\\mathbb {k}(\\mathbb {C})$ , any $\\psi $ in $\\mathrm {Bir}(M)$ , and any point $p$ in $M(\\mathbb {C})$ for with both sides of $(\\ref {eq:fieldaut})$ are well defined.", "Hence $\\mathrm {Aut}_\\mathbb {k}(\\mathbb {C})$ acts by automorphisms on $\\mathrm {Bir}(M)$ .", "If $\\kappa \\colon \\mathbb {C}\\rightarrow \\mathbb {C}$ is a morphism field, this contruction gives an injective morphism $\\mathrm {Aut}(\\mathbb {P}^n_\\mathbb {C})\\rightarrow \\mathrm {Aut}(\\mathbb {P}^n_\\mathbb {C})\\qquad \\mathfrak {g}\\mapsto \\mathfrak {g}^\\vee .$ Indeed, write $\\mathbb {C}$ as the algebraic closure of a purely transcendental extension $\\mathbb {Q}(x_i,i\\in I)$ of $\\mathbb {Q}$  ; if $f\\colon I\\rightarrow I$ is an injective map, then there exists a field morphism $\\kappa \\colon \\mathbb {C}\\rightarrow \\mathbb {C}\\qquad x_i\\mapsto x_{f(i)}.$ Note that such a morphism is surjective if and only if $f$ is onto.", "In 2006, using the structure of amalgamated product of $\\mathrm {Aut}(\\mathbb {C}^2)$ , the automorphisms of this group have been described : Theorem 5.3 ([20]) Let $\\varphi $ be an automorphism of $\\mathrm {Aut}(\\mathbb {C}^2)$ .", "There exist a polynomial automorphism $\\psi $ of $\\mathbb {C}^2$ , and a field automorphism $\\kappa $ such that $\\varphi (f)={}^{\\kappa }\\!\\,(\\psi f\\psi ^{-1})\\qquad \\forall \\,f\\in \\mathrm {Aut}(\\mathbb {C}^2).$ Then, in 2011, Kraft and Stampfli show that every automorphism of $\\mathrm {Aut}(\\mathbb {C}^n)$ is inner up to field automorphisms when restricted to the group $\\mathrm {Tame}_n$  : Theorem 5.4 ([31]) Let $\\varphi $ be an automorphism of $\\mathrm {Aut}(\\mathbb {C}^n)$ .", "There exist a polynomial automorphism $\\psi $ of $\\mathbb {C}^n$ , and a field automorphism $\\kappa $ such that $\\varphi (f)={}^{\\kappa }\\!\\,(\\psi f\\psi ^{-1})\\qquad \\forall \\,f\\in \\mathrm {Tame}_n.$ Even if $\\mathrm {Bir}(\\mathbb {P}^2_\\mathbb {C})$ hasn't the same structure as $\\mathrm {Aut}(\\mathbb {C}^2)$ (see Appendix of [12]) the automorphisms group of $\\mathrm {Bir}(\\mathbb {P}^2_\\mathbb {C})$ can be described, and a similar result as Theorem REF is obtained ([21]).", "There is no such result in higher dimension ; nevertheless in [11] Cantat classifies all (abstract) homomorphisms from $\\mathrm {PGL}(k+1;\\mathbb {C})$ to the group $\\mathrm {Bir}(M)$ of birational maps of a complex projective variety $M$ , provided $k\\ge \\dim _\\mathbb {C}M$ .", "Before recalling his statement let us introduce some notation.", "Given $\\mathfrak {g}$ in $\\mathrm {Aut}(\\mathbb {P}^n_\\mathbb {C})=\\mathrm {PGL}(n+1;\\mathbb {C})$ we denote by ${{}^{\\mathrm {t}}\\!\\, } \\mathfrak {g}$ the linear transpose of $\\mathfrak {g}$ .", "The involution $\\mathfrak {g}\\mapsto \\mathfrak {g}^{\\vee }=({}^{\\mathrm {t}}\\!\\, \\mathfrak {g})^{-1}$ determines an exterior and algebraic automorphism of the group $\\mathrm {Aut}(\\mathbb {P}^n_\\mathbb {C})$ (see [25]).", "Theorem 5.5 ([11]) Let $M$ be a smooth, connected, complex projective variety, and let $n$ be its dimension.", "Let $k$ be a positive integer, and let $\\rho \\colon \\mathrm {Aut}(\\mathbb {P}^k_\\mathbb {C})\\rightarrow \\mathrm {Bir}(M)$ be an injective morphism of groups.", "Then $n\\ge k$ , and if $n=k$ there exists a field morphism $\\kappa \\colon \\mathbb {C}\\rightarrow \\mathbb {C}$ , and a birational map $\\psi \\colon M\\dashrightarrow \\mathbb {P}^n_\\mathbb {C}$ such that either $\\psi \\,\\rho (\\mathfrak {g})\\,\\psi ^{-1}={}^{\\kappa }\\!\\, \\mathfrak {g}\\qquad \\forall \\,\\mathfrak {g}\\in \\mathrm {Aut}(\\mathbb {P}^n_\\mathbb {C})$ or $\\psi \\,\\rho (\\mathfrak {g})\\,\\psi ^{-1}=({}^{\\kappa }\\!\\, \\mathfrak {g})^{\\vee }\\qquad \\forall \\,\\mathfrak {g}\\in \\mathrm {Aut}(\\mathbb {P}^n_\\mathbb {C});$ in particular $M$ is rational.", "Moreover, $\\kappa $ is an automorphism of $\\mathbb {C}$ if $\\rho $ is an isomorphism.", "Let us give the proof of Theorem REF  : Theorem 5.6 Let $\\varphi $ be an automorphism of $\\mathrm {Bir}(\\mathbb {P}^n_\\mathbb {C})$ .", "There exists a birational map $\\psi $ of $\\mathbb {P}^n_\\mathbb {C}$ , and a field automorphism $\\kappa $ such that $\\varphi (g)={}^{\\kappa }\\!\\, (\\psi g\\psi ^{-1})\\qquad \\forall \\,g\\in G_n(\\mathbb {C}).$ Let us consider $\\varphi \\in \\mathrm {Aut}(\\mathrm {Bir}(\\mathbb {P}^n_\\mathbb {C}))$ .", "Theorem REF implies that up to birational conjugacy and up the action of a field automorphism $\\left\\lbrace \\begin{array}{ll}\\text{ either $\\varphi (\\mathfrak {g})=\\mathfrak {g}\\qquad \\forall \\,\\mathfrak {g}\\in \\mathrm {Aut}(\\mathbb {P}^n_\\mathbb {C})$}\\\\\\text{or $\\varphi (\\mathfrak {g})=\\mathfrak {g}^{\\vee }\\qquad \\forall \\,\\mathfrak {g}\\in \\mathrm {Aut}(\\mathbb {P}^n_\\mathbb {C}).$}\\end{array}\\right.$ In other words up to birational conjugacy and up to the action of a field automorphism one cas assume that either $\\varphi _{\\vert \\mathrm {Aut}(\\mathbb {P}^n_\\mathbb {C})}\\colon \\mathfrak {g}\\mapsto \\mathfrak {g}$ , or $\\varphi _{\\vert \\mathrm {Aut}(\\mathbb {P}^n_\\mathbb {C})}\\colon \\mathfrak {g}\\mapsto \\mathfrak {g}^\\vee $ .", "Now determine $\\varphi (\\sigma _n)$ .", "Let us work in the affine chart $z_n=1$ .", "For $0\\le i\\le n-2$ denote by $\\tau _i$ the automorphism of $\\mathbb {P}^n_\\mathbb {C}$ that permutes $z_i$ and $z_{n-1}$ $\\tau _i=\\big (z_0,z_1,\\ldots ,z_{i-1},z_{n-1},z_{i+1},z_{i+2},\\ldots ,z_{n-2},z_i\\big ).$ Let $\\eta $ be given by $\\eta =\\left(z_0,z_1,\\ldots ,z_{n-2},\\frac{1}{z_{n-1}}\\right).$ One has $\\sigma _n=\\big (\\tau _0\\eta \\tau _0\\big )\\,\\big (\\tau _1\\eta \\tau _1\\big )\\,\\ldots \\,\\big (\\tau _{n-2}\\eta \\tau _{n-2}\\big )\\,\\eta $ so $\\varphi (\\sigma _n)=\\big (\\varphi (\\tau _0)\\varphi (\\eta )\\varphi (\\tau _0)\\big )\\big (\\varphi (\\tau _1)\\varphi (\\eta )\\varphi (\\tau _1)\\big )\\ldots \\big (\\varphi (\\tau _{n-2})\\varphi (\\eta )\\varphi (\\tau _{n-2})\\big )\\varphi (\\eta ).$ Since any $\\tau _i$ belongs to $\\mathrm {Aut}(\\mathbb {P}^n_\\mathbb {C})$ one can, thanks to (REF ), compute $\\varphi (\\tau _i)$ , and one gets : $\\varphi (\\tau _i)=\\tau _i$ .", "Let us now focus on $\\varphi (\\eta )$ .", "We will distinguish the two cases of (REF ).", "Assume that $\\varphi _{\\vert \\mathrm {PGL}(n+1;\\mathbb {C})}=\\mathrm {id}$ .", "For any $\\alpha =(\\alpha _0,\\alpha _1,\\ldots ,\\alpha _{n-1})$ in $(\\mathbb {C}^*)^n$ set $\\mathfrak {d}_\\alpha =(\\alpha _0z_0,\\alpha _1z_1,\\ldots ,\\alpha _{n-1}z_{n-1});$ the involution $\\eta $ satisfies for any $\\alpha =(\\alpha _0,\\alpha _1,\\ldots ,\\alpha _{n-1})\\in (\\mathbb {C}^*)^n$ $\\mathfrak {d}_\\beta \\eta =\\eta \\mathfrak {d}_\\alpha $ where $\\beta =(\\alpha _0,\\alpha _1,\\ldots ,\\alpha _{n-1}^{-1})$ .", "Hence $\\varphi (\\eta )=\\left(\\pm z_0,\\pm z_1,\\ldots ,\\pm z_{n-2},\\frac{\\alpha }{z_{n-1}}\\right)$ for $\\alpha \\in \\mathbb {C}^*$ .", "As $\\eta $ commutes with $\\mathfrak {t}=\\big (z_0+1,z_1+1,\\ldots ,z_{n-2}+1,z_{n-1}\\big ),$ the image $\\varphi (\\eta )$ of $\\eta $ commutes to $\\varphi (\\mathfrak {t})=\\mathfrak {t}$ .", "Therefore $\\varphi (\\eta )=\\left(z_0,z_1,\\ldots ,z_{n-2},\\frac{\\alpha }{z_{n-1}}\\right).$ If $\\mathfrak {h}_n=\\left(\\frac{z_0}{z_0-1},\\frac{z_0-z_1}{z_0-1},\\frac{z_0-z_2}{z_0-1},\\ldots ,\\frac{z_0-z_{n-1}}{z_0-1}\\right)$ then $\\varphi (\\mathfrak {h}_n)=\\mathfrak {h}_n$ , and $(\\mathfrak {h}_n\\sigma _n)^3=\\mathrm {id}$ implies that $\\varphi (\\sigma _n)=~\\sigma _n$ .", "If $\\varphi _{\\vert \\mathrm {PGL}(n+1;\\mathbb {C})}$ coincides with $\\mathfrak {g}\\mapsto \\mathfrak {g}^{\\vee }$ , a similar argument yields $\\big (\\varphi (\\mathfrak {h}_n)\\varphi (\\sigma _n)\\big )^3\\ne \\mathrm {id}$ ." ], [ "Simplicity of $G_n(\\mathbb {C})$", "An algebraic family of $\\mathrm {Bir}(\\mathbb {P}^n_\\mathbb {C})$ is the data of a rational map $\\phi \\colon M\\times \\mathbb {P}^n_\\mathbb {C}\\dashrightarrow \\mathbb {P}^n_\\mathbb {C},$ where $M$ is a $\\mathbb {C}$ -variety, defined on a dense open subset $\\mathcal {U}$ such that for any $m\\in M$ the intersection $\\mathcal {U}_m=\\mathcal {U}\\cap (\\lbrace m\\rbrace \\times \\mathbb {P}^n_\\mathbb {C})$ is a dense open subset of $\\lbrace m\\rbrace \\times \\mathbb {P}^n_\\mathbb {C}$ , and the restriction of $\\mathrm {id}\\times \\phi $ to $\\mathcal {U}$ is an isomorphism of $\\mathcal {U}$ on a dense open subset of $M\\times \\mathbb {P}^n_\\mathbb {C}$ .", "For any $m\\in M$ the birational map $z\\dashrightarrow \\phi (m,z)$ represents an element $\\phi _m$ in $\\mathrm {Bir}(\\mathbb {P}^n_\\mathbb {C})$  ; the map $M\\rightarrow \\mathrm {Bir}(\\mathbb {P}^n_\\mathbb {C}),\\qquad m\\mapsto \\phi _m$ is called morphism from $M$ to $\\mathrm {Bir}(\\mathbb {P}^n_\\mathbb {C})$ .", "These notions yield the natural Zariski topology on $\\mathrm {Bir}(\\mathbb {P}^n_\\mathbb {C})$ , introduced by Demazure ([18]) and Serre ([45]) : the subset $\\Omega $ of $\\mathrm {Bir}(\\mathbb {P}^n_\\mathbb {C})$ is closed if for any $\\mathbb {C}$ -variety $M$ , and any morphism $M\\rightarrow \\mathrm {Bir}(\\mathbb {P}^n_\\mathbb {C})$ the preimage of $\\Omega $ in $M$ is closed.", "Note that in restriction to $\\mathrm {Aut}(\\mathbb {P}^n_\\mathbb {C})$ one obtains the usual Zariski topology of the algebraic group $\\mathrm {Aut}(\\mathbb {P}^n_\\mathbb {C})=\\mathrm {PGL}(n+1;\\mathbb {C})$ .", "Let us recall the following statement : Proposition 5.7 ([5]) Let $n\\ge 2$ .", "Let $\\mathrm {H}$ be a non-trivial, normal, and closed subgroup of $\\mathrm {Bir}(\\mathbb {P}^n_\\mathbb {C})$ .", "Then $\\mathrm {H}$ contains $\\mathrm {Aut}(\\mathbb {P}^n_\\mathbb {C})$ and $\\mathrm {PSL}\\big (2;\\mathbb {C}(z_0,z_1,\\ldots ,z_{n-2})\\big )$ .", "In our context we have a similar statement : Proposition 5.8 Let $n\\ge 2$ .", "Let $\\mathrm {H}$ be a non-trivial, normal, and closed subgroup of $G_n(\\mathbb {C})$ .", "Then $\\mathrm {H}$ contains $\\mathrm {Aut}(\\mathbb {P}^n_\\mathbb {C})$ and $\\sigma _n$ .", "A similar argument as in [5] allows us to prove that $\\mathrm {Aut}(\\mathbb {P}^n_\\mathbb {C})$ is contained in $\\mathrm {H}$ .", "The fact $-\\mathrm {id}$ and $\\sigma _n$ are conjugate in $G_n(\\mathbb {C})$ (see Proof of Proposition REF ) yields the conclusion.", "The proof of Proposition REF follows from Proposition REF and Corollary REF ." ] ]
1403.0346
[ [ "Quantum steering ellipsoids, extremal physical states and monogamy" ], [ "Abstract Any two-qubit state can be faithfully represented by a steering ellipsoid inside the Bloch sphere, but not every ellipsoid inside the Bloch sphere corresponds to a two-qubit state.", "We give necessary and sufficient conditions for when the geometric data describe a physical state and investigate maximal volume ellipsoids lying on the physical-unphysical boundary.", "We derive monogamy relations for steering that are strictly stronger than the Coffman-Kundu-Wootters (CKW) inequality for monogamy of concurrence.", "The CKW result is thus found to follow from the simple perspective of steering ellipsoid geometry.", "Remarkably, we can also use steering ellipsoids to derive non-trivial results in classical Euclidean geometry, extending Euler's inequality for the circumradius and inradius of a triangle." ], [ "Introduction", "The Bloch vector representation of a single qubit is an invaluable visualisation tool for the complete state of a two-level quantum system.", "Properties of the system such as mixedness, coherence and even dynamics are readily encoded into geometric properties of the Bloch vector.", "The extraordinary effort expended in the last 20 years on better understanding quantum correlations has led to several proposals for an analogous geometric picture of the state of two qubits , , .", "One such means is provided by the quantum steering ellipsoid , , , , which is the set of all Bloch vectors to which one party's qubit could be `steered' (remotely collapsed) if another party were able to perform all possible measurements on the other qubit.", "It was shown recently  that the steering ellipsoid formalism provides a faithful representation of all two-qubit states and that many much-studied properties, such as entanglement and discord, could be obtained directly from the ellipsoid.", "Moreover steering ellipsoids revealed entirely new features of two-qubit systems, namely the notions of complete and incomplete steering, and a purely geometric condition for entanglement in terms of nested convex solids within the Bloch sphere.", "However, one may well wonder if there is much more to be said about two-qubit states and whether the intuitions obtained from yet another representation could be useful beyond the simplest bipartite case.", "We emphatically answer this in the affirmative.", "Consider a scenario with three parties, Alice, Bob and Charlie, each possessing a qubit.", "Bob performs measurements on his system to steer Alice and Charlie.", "We show that the volumes $V_{A|B}$ and $V_{C|B}$ of the two resulting steering ellipsoids obey a tight inequality that we call the monogamy of steering (Theorem REF ): $\\sqrt{V_{A|B}}+\\sqrt{V_{C|B}}\\le \\sqrt{\\frac{4\\pi }{3}}.$ We also prove an upper bound for the concurrence of a state in terms of the volume of its steering ellipsoid (Theorem REF ).", "Using this we show that the well-known CKW inequality for the monogamy of concurrence  can be derived from the monogamy of steering.", "The monogamy of steering is therefore strictly stronger than the CKW result, as well as being more geometrically intuitive.", "The picture that emerges, which was hinted at in Ref.", "by the nested tetrahedron condition for separability, is that the volume of a steering ellipsoid is a fundamental property capturing much of the non-trivial quantum correlations.", "But how large can a steering ellipsoid be?", "Clearly the steering ellipsoid cannot puncture the Bloch sphere.", "However, not all ellipsoids contained in the Bloch sphere correspond to physical states.", "We begin our analysis by giving necessary and sufficient conditions for a steering ellipsoid to represent a valid quantum state (Theorem REF ).", "The conditions relate the ellipsoid's centre, semiaxes and orientation in a highly non-trivial manner.", "We subsequently clarify these geometric constraints on physical states by considering the limits they impose on steering ellipsoid volume for a fixed ellipsoid centre.", "This gives rise to a family of extremal volume states (Figure REF ) which, in Theorem REF , allows us to place bounds on how large an ellipsoid may be before it becomes first entangled and then unphysical.", "The maximal volume states that we give in equation (REF ) are found to be very special.", "In addition to being Choi-isomorphic to the amplitude-damping channel, these states maximise concurrence over the set of all states that have steering ellipsoids with a given centre (Theorem REF ).", "This endows steering ellipsoid volume with a clear operational meaning.", "A curious aside of the steering ellipsoid formalism is its connection with classical Euclidean geometry.", "By investigating the geometry of separable steering ellipsoids, in Section REF we arrive at a novel derivation of a famous inequality of Euler's in two and three dimensions.", "On a plane, it relates a triangle's circumradius and inradius; in three dimensions, the result extends to tetrahedra and spheres.", "Furthermore, we give a generalisation of Euler's result to ellipsoids, a full discussion of which appears in Ref. .", "The term `steering' was originally used by Schrödinger  in the context of his study into the complete set of states/ensembles that a remote system could be collapsed to, given some (pure) initial entangled state.", "The steering ellipsoid we study is the natural extension of that work to mixed states (of qubits).", "Schrödinger was motivated to perform such a characterisation by the EPR paper .", "The question of whether the ensembles one steers to are consistent with a local quantum model has been recently formalised  into a criterion for `EPR steerability' that provides a distinct notion of nonlocality to that of entanglement: the EPR-steerable states are a strict subset of the entangled states.", "We note that the existence of a steering ellipsoid with nonzero volume is necessary, but not sufficient, for a demonstration of EPR-steering.", "It is an open question whether the quantum steering ellipsoid can provide a geometric intuition for EPR-steerable states as it can for separable, entangled and discordant states, although progress has recently been made .", "Figure: References" ] ]
1403.0418
[ [ "A unified proof of the Howe-Moore property" ], [ "Abstract We provide a unified proof of all known examples of locally compact groups that enjoy the Howe-Moore property, namely, the vanishing at infinity of all matrix coefficients of the group unitary representations that are without non-zero invariant vectors.", "These examples are: connected, non-compact, simple real Lie groups with finite center, isotropic simple algebraic groups over non Archimedean local fields and closed, topologically simple subgroups of Aut(T) that act 2-transitively on the boundary at infinity of T, where T is a bi-regular tree with valence > 2 at every vertex." ], [ "Introduction", "For a locally compact group $G$ , the Howe–Moore property asserts that all matrix coefficients of its unitary representations, that are without non-zero $G$ –invariant vectors, vanish at infinity.", "This property was first established by Howe and Moore  and Zimmer , around 1977, for connected, non-compact, simple real Lie groups that are with finite center.", "It plays an important role in the Mostow rigidity theorem, as well as in various other rigidity results.", "The original proof of the Howe–Moore property uses the geometry of semi-simple real Lie groups.", "Nowadays the proof is more algebraic (see for example Bekka and Mayer  or Morris  for the case of $\\operatorname{SL}(2,\\mathbb {R})$ ).", "In the same article, Howe and Moore also treat the case of isotropic simple algebraic groups over non Archimedean local fields (see Howe and Moore  and Definition REF ).", "By Bruhat and Tits , to such a group one associates a locally finite thick Euclidean building $\\Delta $ where the group acts continuously and properly, by type-preserving automorphisms and strongly transitively (for strong transitivity see Definition REF ).", "For other totally disconnected analogs of semi-simple real Lie groups, namely, closed, strongly transitive subgroups of $\\mathrm {Aut}(\\Delta )$ , where $\\Delta $ is a locally finite thick Euclidean building, the study of the Howe–Moore property was initiated by Lubotzky and Mozes  in the particular case of $\\Delta $ being a $d$ –regular tree $T_d$ , with $d \\ge 3$ .", "Using the geometry of horocycles and horoballs, they prove that the index-two subgroup of $\\mathrm {Aut}(T_{d})$ , which preserves the 2–coloring of the tree $T_{d}$ , enjoys this property.", "A more general result, in the context of $d$ –regular trees, was obtained by Burger and Mozes .", "There they show that every closed, topologically simple subgroup of $\\mathrm {Aut}(T_{d})$ which is 2–transitive on the boundary has the Howe–Moore property.", "We mention that the latter class of subgroups of $\\mathrm {Aut}(T_d)$ contains also non linear examples.", "For completeness, we add the known fact that in the case of a thick tree $T$ , the strongly transitive action of a subgroup of $\\mathrm {Aut}(T)$ on $T$ is equivalent to the 2–transitive action of that group on the boundary of $T$ (see for example Caprace and Ciobotaru ); in particular, the latter mentioned equivalence implies that $T$ is a bi-regular tree.", "Regarding all examples presented above, this article proposes to give a unified proof of the Howe–Moore property: Theorem 1.1 (See Theorem REF and Section REF ) Let $G$ be a connected, non-compact, simple real Lie group, with finite center, or an isotropic simple algebraic group over a non Archimedean local field, or a closed, topologically simple subgroup of $\\mathrm {Aut}(T)$ that acts 2–transitively on the boundary $\\partial T$ , where $T$ is a bi-regular tree with valence $ \\ge 3$ at every vertex.", "Then $G$ admits the Howe–Moore property.", "The main ingredients of the unified proof, are: i)  the $K_1A^{+}K_2$ decomposition of these groups, where $K_1,K_2$ are compact subsets and $A^{+}$ is the `maximal abelian' sub semi-group part (see Section ); ii)  the `contraction' subgroups $U^{\\pm }_{\\alpha }$ (see Definition REF ) with respect to hyperbolic elements, combined with basic facts regarding normal operators (see Sections REF , REF ).", "More precisely, the above mentioned ingredients are assembled to obtain the following criterion, used to verify the Howe–Moore property: Theorem 1.2 (See Theorem REF ) Let $G$ be a second countable, locally compact group having the following properties: i)  $G$ admits a decomposition $K_1A^{+}K_2$ , where $K_1,K_2 \\subset G$ are compact subsets and $A^{+}$ is an abelian sub semi-group of $G$ ; ii)  every sequence $\\alpha =\\lbrace g_{n}\\rbrace _{n>0} \\subset A^{+}$ , with $g_{n} \\rightarrow \\infty $ , admits a subsequence $\\beta =\\lbrace g_{n_k}\\rbrace _{n_k>0}$ such that $G= \\overline{ \\left\\langle U^{+}_{\\beta },U^{-}_{\\beta }\\right\\rangle }$ .", "Then $G$ has the Howe–Moore property.", "The idea behind this criterion is inspired by a proof given for $\\operatorname{SL}(2, \\mathbb {R})$ by Dave Witte Morris .", "To the best of our knowledge, there are no other known examples of locally compact or even discrete groups enjoying the Howe–Moore property, beside those covered by the unified proof given in this article.", "Some necessary algebraic conditions for a locally compact group admitting the Howe–Moore property are described in Cluckers–de Cornulier–Louvet–Tessera–Valette .", "Those conditions are quite elementary, but probably far from being sufficient.", "Even for the two classes of totally disconnected locally compact groups of Theorem REF , we still do not know to what extent the Howe–Moore property relates with the $KA^{+}K$ decomposition and with the strong transitivity on the corresponding Euclidean building; in this direction, only partial results have been obtained so far." ], [ "Basic definitions", "In what follows all Hilbert spaces are assumed to be complex, not necessarily separable, and their inner product is denoted by $\\left\\langle \\cdot , \\cdot \\right\\rangle $ .", "Let $B(\\mathcal {H})$ be the space of all bounded linear operators $ F: \\mathcal {H} \\rightarrow \\mathcal {H}$ .", "Recall that the closed unit ball of a Hilbert space $\\mathcal {H}$ is compact with respect to the weak topology and moreover, by the Eberlein–Šmulian Theorem, compactness is equivalent to sequential compactness.", "Hence, any sequence of vectors $\\lbrace v_n\\rbrace _{n\\ge 0} \\subset \\mathcal {H}$ , with their norms being bounded above, contains a weakly convergent subsequence $ \\lbrace v_{n_{k}}\\rbrace _{k>0}$ in $\\mathcal {H}$ .", "Namely, there exists a vector $v \\in \\mathcal {H}$ such that for any $w \\in \\mathcal {H}$ one has that $\\lim \\limits _{n_k \\rightarrow \\infty } \\left\\langle v_{n_{k}} , w \\right\\rangle =\\left\\langle v , w \\right\\rangle $ .", "In the same spirit, if $\\mathcal {H}$ is a separable Hilbert space, it is well-known that the closed unit ball in $B(\\mathcal {H})$ is compact and metrizable with respect to the weak operator topology.", "Thus, any sequence of operators $\\lbrace F_n\\rbrace _{n\\ge 0} \\subset B(\\mathcal {H})$ , with their operator norms being bounded above, contains a weakly convergent subsequence $ \\lbrace F_{n_{k}}\\rbrace _{k>0}$ in $B(\\mathcal {H})$ .", "Namely, there exists $F$ in $B(\\mathcal {H})$ such that for all $v,w \\in \\mathcal {H}$ one has that $\\lim \\limits _{n_{k} \\rightarrow \\infty } \\left\\langle F_{n_{k}}v , w \\right\\rangle =\\left\\langle Fv , w \\right\\rangle $ Definition 2.1 Let $\\mathcal {H}$ be a Hilbert space.", "We say that $E \\in B(\\mathcal {H})$ is normal if $E^{*}E=EE^{*}$ .", "It is well-known that this is equivalent to $\\left\\langle EE^{*}v, v \\right\\rangle =\\left\\langle E^{*}Ev, v \\right\\rangle $ , for every $v \\in \\mathcal {H}$ .", "Moreover, $U \\in B(\\mathcal {H})$ is called unitary if $UU^{*}=U^{*}U=I$ or, equivalently, if $\\left\\langle U \\xi ,U \\eta \\right\\rangle =\\left\\langle \\xi , \\eta \\right\\rangle $ for all $\\xi , \\eta \\in \\mathcal {H}$ and $U$ is onto.", "We denote by $\\mathcal {U}(\\mathcal {H})$ the group of all unitary operators of $\\mathcal {H}$ .", "Definition 2.2 A unitary representation of a topological group $G$ into a Hilbert space $\\mathcal {H}$ is a group homomorphism $\\pi : G \\rightarrow \\mathcal {U}(\\mathcal {H})$ , which is moreover strongly continuous: the map $g \\in G \\mapsto \\pi (g) v \\in \\mathcal {H}$ is continuous for every $v \\in \\mathcal {H}$ .", "Instead of $\\pi : G \\rightarrow \\mathcal {U}(\\mathcal {H})$ we often write $(\\pi , \\mathcal {H})$ for a unitary representation of $G$ .", "Furthermore, any two vectors $v, w \\in \\mathcal {H}$ define a continuous function $c_{v, w} : G \\rightarrow \\mathbb {C}$ , given by $c_{v, w}(g): = \\left\\langle \\pi (g)v , w \\right\\rangle $ and we call it the associated $( v, w)$ –matrix coefficient.", "Moreover, we say that $c_{v, w}$ is $\\mathbf {\\mathrm {C}_0}$ if for every $\\epsilon >0$ , the subset $\\lbrace g \\in G \\; : \\; |c_{v, w}(g)| \\ge \\epsilon \\rbrace $ is compact in $G$ .", "Remark 2.3 (See Bekka, de la Harpe and Valette ) We mention that for a unitary representation $(\\pi , \\mathcal {H})$ of a topological group $G$ , the Hilbert space $\\mathcal {H}$ can be decomposed as a direct sum $\\mathcal {H}=\\bigoplus _{i} \\mathcal {H}_{i}$ of mutually orthogonal, closed and $G$ –invariant subspaces $\\mathcal {H}_{i}$ , such that the restriction of $\\pi $ to $\\mathcal {H}_{i}$ is cyclic for every $i$ (i.e., for every $i$ there exists a non-zero vector $v \\in \\mathcal {H}_{i}$ such that $\\pi (G)v$ is dense in $\\mathcal {H}_{i}$ ).", "Moreover, the following well-known lemma asserts that for `nice' topological groups, only separable Hilbert spaces can be considered when working with unitary representations.", "For the convenience of the reader, we include its proof.", "Lemma 2.4 Let $G$ be a separable locally compact group.", "Then for every cyclic unitary representation $(\\pi , \\mathcal {H})$ of $G$ , the corresponding Hilbert space $ \\mathcal {H}$ is separable.", "In particular, all irreducible unitary representations of $G$ are over separable Hilbert spaces.", "Let $(\\pi , \\mathcal {H})$ be a cyclic unitary representation of $G$ and denote by $v$ a cyclic vector of it.", "Then, we have that $f: G \\rightarrow \\mathcal {H}$ , defined by $f(g):= \\pi (g)v$ , is a continuous function.", "Moreover, by the definition of a cyclic vector, we know that the linear span of $\\pi (G)v$ is dense in $\\mathcal {H}$ .", "As $G$ is separable, let $Q$ be a countable dense subset of it.", "Thus $f(Q)$ is dense in $\\pi (G)v$ and therefore the linear span of $\\pi (Q)v$ is also dense in $\\mathcal {H}$ .", "The conclusion thus follows.", "Remark 2.5 The class of separable locally compact groups contains non-compact, simple real Lie groups with finite center and also closed, non-compact subgroups of $\\mathrm {Aut}(\\Delta )$ , where $\\Delta $ is a locally finite thick Euclidean building.", "In particular, all non-compact, simple $p$ –adic Lie groups are separable." ], [ "The Howe–Moore property", "Definition 2.6 Let $G$ be a locally compact group and let $G \\bigcup \\lbrace \\infty \\rbrace $ be the one point compactification of $G$ .", "We say that $G$ has the Howe–Moore property if the set of all unitary representations of $G$ is the union of the ones having non-zero $G$ –invariant vectors and the ones for which all matrix coefficients are $\\mathrm {C}_0$ .", "When a matrix coefficient is $\\mathrm {C}_0$ , we say also that it vanishes at $\\infty $.", "Throughout this article we assume that all locally compact groups are second countable; this is to simplify our notation, by using sequences instead of nets.", "The second reason is given by the next remark.", "Remark 2.7 By , to prove the Howe–Moore property for a second countable, locally compact group $G$ it is enough to verify the $C_0$ condition only for all irreducible, non-trivial, unitary representations of $G$ , which, by Lemma REF , are over separable Hilbert spaces.", "Moreover, the following well-known lemma shows that matrix coefficients not vanishing at $\\infty $ give rise to particular matrix coefficients with the same property.", "Lemma 2.8 Let $G$ be a second countable, locally compact group and $(\\pi , \\mathcal {H})$ be a unitary representation of $G$ .", "Suppose there exist two non-zero vectors $v, w \\in \\mathcal {H}$ such that the $( v, w )$ –matrix coefficient does not vanish at $\\infty $ .", "Then the $( v, v)$ –matrix coefficient does not vanish at $\\infty $ .", "By hypothesis, there exists a sequence $\\lbrace g_n\\rbrace _{n\\ge 0} \\subset G$ and $\\epsilon >0$ such that $g_n \\rightarrow \\infty $ and $\\vert \\left\\langle \\pi (g_{n})v, w \\right\\rangle \\vert \\ge \\epsilon $ .", "Thus $w$ is not orthogonal to the Hilbert subspace generated by $\\pi (G)v$ .", "Denote this Hilbert subspace by $\\overline{ \\langle \\pi (G)v \\rangle }$ and remark this is $G$ –invariant.", "Moreover, $\\lbrace \\pi (g_n) v\\rbrace _{n\\ge 0}$ being bounded in the norm of $\\mathcal {H}$ , there exist $v_0 \\in \\mathcal {H}$ and a subsequence $\\lbrace n_k\\rbrace _{k\\ge 0}$ such that $\\lbrace \\pi (g_{n_k}) v\\rbrace _{k\\ge 0}$ weakly converges to $v_0$ .", "Thus $\\lim \\limits _{n_k \\rightarrow \\infty } \\left\\langle \\pi (g_{n_k}) v , w^{\\prime } \\right\\rangle =\\left\\langle v_0 , w^{\\prime } \\right\\rangle $ , for every $w^{\\prime } \\in \\mathcal {H}$ .", "From here we conclude that $v_0$ is a non-zero vector.", "Furthermore, we claim that $v_0$ is a vector in $\\overline{ \\langle \\pi (G)v \\rangle }$ .", "Indeed, suppose the converse that $v_0=w_1+w_2$ , with $w_1 \\in \\overline{ \\langle \\pi (G)v \\rangle }$ and $w_2$ a non-zero vector orthogonal to $\\overline{ \\langle \\pi (G)v \\rangle }$ .", "Taking $w^{\\prime }=w_2$ , we obtain that $\\lim \\limits _{n_k \\rightarrow \\infty } \\left\\langle \\pi (g_{n_k}) v , w_2 \\right\\rangle =0=\\left\\langle v_0 , w_2 \\right\\rangle $ , which is impossible.", "The claim follows and we conclude that there exists $g \\in G$ such that $\\left\\langle v_0 , \\pi (g)v \\right\\rangle \\ne 0$ .", "For this $g$ we obtain that $\\lim \\limits _{n_k \\rightarrow \\infty } \\left\\langle \\pi (g_{n_k}) v , \\pi (g)v \\right\\rangle =\\left\\langle v_0 , \\pi (g)v \\right\\rangle $ .", "The conclusion follows by only remarking that $\\vert \\left\\langle \\pi (g^{-1}g_{n_k})v, v \\right\\rangle \\vert \\nrightarrow 0$ .", "In addition, the following easy lemma asserts that compact subsets do not count for the Howe–Moore property, given a `polar decomposition' of the locally compact group.", "Lemma 2.9 Let $G$ be a second countable, locally compact group admitting a decomposition $G=K_{1}AK_{2}$ , where $K_{1},K_{2}$ are compact subset and $A$ is any subset of $G$ .", "Let $(\\pi , \\mathcal {H})$ be a unitary representation of $G$ .", "If for every sequence $\\lbrace a_{n}\\rbrace _{n>0} \\subset A$ , with $\\lbrace a_{n}\\rbrace _{n>0} \\rightarrow \\infty $ , the corresponding matrix coefficients are $C_0$ , then all matrix coefficients of $G$ , with respect to $(\\pi , \\mathcal {H})$ , vanish at $\\infty $ .", "[Proof] Suppose there exist a sequence $\\lbrace g_{n}\\rbrace _{n>0} \\subset G $ , with $g_{n} \\rightarrow \\infty $ , and two non-zero vectors $v,w \\in \\mathcal {H}$ such that the corresponding matrix coefficient $c_{v,w}(g_{n})$ does not vanish at $\\infty $ .", "As $G=K_{1}AK_{2}$ one has $g_{n}=k_{n}a_{n}h_{n}$ , where $k_{n} \\in K_1, h_{n} \\in K_2$ and $a_{n} \\in A$ , for every $n>0$ .", "We have thus $a_{n} \\rightarrow \\infty $ .", "As $K_{1},K_{2}$ are compact sets and by passing to a subsequence, we can assume that $\\lbrace \\pi (k_{n})^{-1}w \\rbrace _{n>0}$ and $\\lbrace \\pi (h_{n})v \\rbrace _{n>0}$ converge, in the norm of $\\mathcal {H}$ , to the vectors $w_{0}$ , respectively $v_{0}$ .", "One has: $\\begin{split}\\vert \\left\\langle \\pi (k_{n}a_{n}h_{n})v,w \\right\\rangle \\vert &= \\vert \\left\\langle \\pi (a_{n}h_{n})v,\\pi (k_{n})^{-1}w \\right\\rangle -\\left\\langle \\pi (a_{n}h_{n})v,w_{0} \\right\\rangle \\\\&+ \\left\\langle \\pi (h_{n})v,\\pi (a_{n})^{-1} w_{0} \\right\\rangle - \\left\\langle v_{0},\\pi (a_{n})^{-1} w_{0} \\right\\rangle + \\left\\langle \\pi (a_{n}) v_{0}, w_{0} \\right\\rangle \\vert \\\\& \\le \\Vert \\pi (k_{n})^{-1}w-w_0 \\Vert \\cdot \\Vert v \\Vert + \\Vert (h_{n})v -v_0\\Vert \\cdot \\Vert w_0\\Vert + \\vert \\left\\langle \\pi (a_{n}) v_{0}, w_{0} \\right\\rangle \\vert .", "\\\\\\end{split}$ As the first two terms tend to zero as $n \\rightarrow \\infty $ , we have that the matrix coefficient $c_{v_{0},w_{0}}(a_{n})$ does not vanish at $\\infty $ .", "We obtain a contradiction and the conclusion follows.", "Let us add here some necessary algebraic conditions for a non-compact locally compact group to admit the Howe–Moore property; however, those conditions are far from being sufficient.", "Proposition 2.10 (See Prop.", "3.2 in ) Let $G$ be a non-compact locally compact group with the Howe–Moore property.", "Then any proper open subgroup of $G$ is compact and thus of infinite index in $G$ ." ], [ "Mautner's phenomenon", "One of the main ideas to verify the Howe–Moore property is to search for `big' subgroups $H \\le G$ admitting at least one non-zero $H$ –invariant vector in $\\mathcal {H}$ .", "Mautner's phenomenon gives us a method to construct such subgroups.", "First, let us introduce the following: Definition 2.11 Let $G$ be a locally compact group.", "Let $\\alpha =\\lbrace g_{n}\\rbrace _{n>0}$ be a sequence in $G$ .", "This sequence $\\alpha $ defines the set: $U^{+}_{\\alpha }:= \\lbrace g \\in G \\; | \\; \\lim \\limits _{n \\rightarrow \\infty } g_{n}^{-1}gg_{n}=e \\rbrace .$ Notice that $U^{+}_{\\alpha }$ is a subgroup of $G$ .", "It is called the contraction group corresponding to $\\alpha $ .", "Because it doesn't need to be closed in general, we denote by $N^{+}_{\\alpha }$ the closed subgroup $\\overline{U^{+}_{\\alpha }}$ .", "In the same way, but using $g_{n}gg_{n}^{-1}$ we define $U^{-}_{\\alpha }$ and $N^{-}_{\\alpha }$ .", "When $\\alpha =\\lbrace a^{n}\\rbrace _{n>0}$ , for some $a \\in G$ , we simply use the notation $U^{\\pm }_{a}:=U^{\\pm }_{\\alpha }$ and $N^{\\pm }_{a}:=N^{\\pm }_{\\alpha }$ .", "We record the following lemma, known as Mautner's phenomenon (see Bekka–Mayer ): Lemma 2.12 Let $G$ be a second countable, locally compact group and $(\\pi , \\mathcal {H})$ be a unitary representation.", "Let $\\lbrace g_{n}\\rbrace _{n>0}$ be a sequence in $G$ and take $v \\in \\mathcal {H}$ a non-zero vector.", "Suppose moreover that $\\lbrace \\pi (g_{n})v\\rbrace _{n>0}$ weakly converges to $v_{0}$ .", "Then $v_{0}$ is $N^{+}_{\\alpha }$ –invariant.", "Let $w \\in \\mathcal {H}$ be an arbitrary vector and take $g \\in U^{+}_{\\alpha }$ .", "We have: $\\begin{split}|\\left\\langle \\pi (g)v_{0}-v_{0}, w \\right\\rangle |&=\\lim _{n \\rightarrow \\infty }|\\left\\langle \\pi (g g_{n})v-\\pi (g_{n})v, w \\right\\rangle |\\\\&=\\lim _{n \\rightarrow \\infty }|\\left\\langle \\pi (g_{n}^{-1}g g_{n})v-v, \\pi (g_{n}^{-1})w \\right\\rangle |\\\\&\\le \\lim _{n \\rightarrow \\infty }\\Vert (\\pi (g_{n}^{-1}g g_{n})-Id_{\\mathcal {H}})v \\Vert \\cdot \\Vert w\\Vert =0.\\\\\\end{split}$ Since $U^{+}_{\\alpha }$ is dense in $N^{+}_{\\alpha }$ and $\\pi $ is continuous, it follows that $N^{+}_{\\alpha }$ fixes $v_{0}$ ." ], [ "Normal operators as weak limits", "This elementary lemma is a key step used in Section .", "Lemma 2.13 Let $\\mathcal {H}$ be a separable Hilbert space and consider a sequence $\\lbrace U_{n} \\rbrace _{n>0} \\subset \\mathcal {U}(\\mathcal {H})$ of pairwise commuting unitary operators.", "Then there exists a subsequence $\\lbrace U_{n_{k}} \\rbrace _{n_{k}}$ which weakly converges to an operator $E \\in B(\\mathcal {H})$ .", "Moreover, any such weak limit is normal.", "[First proof] As $\\lbrace U_{n} \\rbrace _{n>0}$ are unitary operators, they have norm one in $B(\\mathcal {H})$ .", "Hence, there is a subsequence $\\lbrace U_{n_{k}} \\rbrace _{n_{k}}$ which weakly converges to an operator $E \\in B(\\mathcal {H})$ .", "It remains to prove that $E$ is normal.", "In what follows we replace the subsequence $\\lbrace U_{n_{k}} \\rbrace _{n_{k}}$ by the sequence $\\lbrace U_{n} \\rbrace _{n>0}$ .", "Notice that, as the sequence $\\lbrace U_{n} \\rbrace _{n>0}$ converges weakly to $E$ , the sequence $\\lbrace U_{n}^{*} \\rbrace _{n>0}$ converges weakly to $E^{*}$ .", "By hypothesis, the unitary operators $\\lbrace U_{n} \\rbrace _{n>0}$ commute pairwise, thus $U_{n}$ and $U_{k}^{*}$ commute for all $n,k>0$ .", "Therefore for all $v,w \\in \\mathcal {H}$ one has: $\\begin{split}\\left\\langle EE^{*}v, w \\right\\rangle &= \\lim _{n \\rightarrow \\infty }\\left\\langle U_{n}(E^{*} v), w \\right\\rangle = \\lim _{n \\rightarrow \\infty } \\left\\langle E^{*} v, U_{n}^{*} w \\right\\rangle \\\\&= \\lim _{n \\rightarrow \\infty }( \\lim _{k \\rightarrow \\infty } \\left\\langle U_{k}^{*} v , U_{n}^{*} w \\right\\rangle )= \\lim _{n \\rightarrow \\infty }( \\lim _{k \\rightarrow \\infty } \\left\\langle U_{n} U_{k}^{*} v, w \\right\\rangle )\\\\&= \\lim _{n \\rightarrow \\infty }( \\lim _{k \\rightarrow \\infty } \\left\\langle U_{k}^{*} U_{n} v, w \\right\\rangle )= \\lim _{n \\rightarrow \\infty } \\left\\langle U_{n} v , E w \\right\\rangle =\\left\\langle E v, E w \\right\\rangle \\\\&= \\left\\langle E^{*} E v, w \\right\\rangle .\\\\\\end{split}$ [Second proof] Another way to see this is using von Neumann algebras.", "More precisely, as the unitary operators $\\lbrace U_{n}\\rbrace _{n>0}$ commute pairwise, they generate a commutative von Neumann algebra which is closed, by definition, in the weak topology.", "Thus the weak limit $E$ is also contained in this von Neumann algebra.", "We conclude that $E$ is normal as operator.", "The pairwise commutativity condition in Lemma REF is essential to obtain a normal weak limit.", "The following is an example in that sense.", "Example 2.14 Consider the Hilbert space $ \\mathcal {H}:=\\ell ^{2}(\\mathbb {Z})\\bigoplus \\ell ^{2}(\\mathbb {Z})$ and for every $n >0$ , let $U_{n}:=\\left( \\begin{array}{cc} 0 & \\lambda _{n} \\\\ 1 & 0 \\end{array} \\right) $ , where $\\lambda _n$ is the unitary operator of translation by $n$ of the Hilbert space $\\ell ^{2}(\\mathbb {Z})$ .", "Notice that the sequence of unitary operators $\\lbrace \\lambda _n\\rbrace _{n>0}$ weakly converges to 0.", "Therefore, $\\lbrace U_{n} \\rbrace _{n>0}$ is a sequence of unitary operators of $\\mathcal {H}$ which do not commute pairwise and whose weak limit is the operator $E=\\left( \\begin{array}{cc} 0 & 0 \\\\ 1 & 0 \\end{array} \\right)$ .", "Equally, the weak limit of $\\lbrace U_{n}^{*} \\rbrace _{n>0}$ is the operator $E^{*}=\\left( \\begin{array}{cc} 0 & 1 \\\\ 0& 0 \\end{array} \\right)$ and one can notice that $EE^{*}=\\left( \\begin{array}{cc} 0 & 0 \\\\ 0 & 1 \\end{array} \\right)$ and $E^{*}E=\\left( \\begin{array}{cc} 1 & 0 \\\\ 0 & 0 \\end{array} \\right)$ .", "Therefore, $E$ is not normal." ], [ "An algebraic criterion", "This section proposes a criterion, given by Theorem REF , to verify the Howe–Moore property, over separable Hilbert spaces, for locally compact groups admitting a special decomposition.", "Its proof uses the idea of Dave Witte Morris , which relies on a more elaborate version of Mautner's phenomenon; for further references this idea is recorded in a separate lemma.", "Lemma 3.1 Let $G$ be a second countable, locally compact group and $(\\pi , \\mathcal {H})$ be a unitary representation of $G$ , without $G$ –invariant vectors.", "Assume there exists a sequence $\\alpha =\\lbrace g_{n}\\rbrace _{n>0}$ of pairwise commuting elements of $G$ such that $g_n \\rightarrow \\infty $ and $G= \\overline{ \\left\\langle U^{+}_{\\alpha },U^{-}_{\\alpha }\\right\\rangle }$ .", "Then all matrix coefficients of $(\\pi , \\mathcal {H})$ corresponding to $\\lbrace g_{n}\\rbrace _{n>0}$ are $C_0$ .", "As $G$ is second countable and using Lemma REF , we can assume, without loss of generality, that the Hilbert space $\\mathcal {H}$ is separable.", "In particular, by restricting to a subsequence, we can consider that the sequence $\\lbrace \\pi (g_n)\\rbrace _{n>0}$ weakly converges to an operator $E$ .", "By contraposition, we assume there exist two non-zero vectors $v,w \\in \\mathcal {H}$ (considered fixed for what follows) such that $ \\vert \\left\\langle \\pi (g_{n})v, w \\right\\rangle \\vert \\nrightarrow 0$ .", "We want to prove that $(\\pi , \\mathcal {H})$ has a non-zero $G$ –invariant vector.", "Because $\\lbrace \\pi (g_n)\\rbrace _{n>0}$ weakly converge to an operator $E$ , by Lemma REF we have moreover that $E$ is normal and $E \\ne 0$ .", "We now claim that the following important property of $E$ holds: $\\pi (u_{+})E\\pi (u_{-})=E$ , for every $u_{+} \\in U^{+}_{\\alpha }$ and every $u_{-} \\in U^{-}_{\\alpha }$ .", "Proof of the claim.", "Let $u_{-} \\in U^{-}_{\\alpha }$ .", "For every $\\phi , \\psi \\in \\mathcal {H}$ one has: $\\begin{split}|\\left\\langle E\\pi (u_{-})\\phi , \\psi \\right\\rangle - \\left\\langle E\\phi , \\psi \\right\\rangle |&= |\\lim _{n \\rightarrow \\infty } \\left\\langle \\pi (g_{n} u_{-})\\phi -\\pi (g_{n})\\phi , \\psi \\right\\rangle |\\\\&= |\\lim _{n \\rightarrow \\infty } \\left\\langle \\pi (g_{n} u_{-} g_{n}^{-1}) \\pi (g_{n})\\phi - \\pi (g_{n})\\phi , \\psi \\right\\rangle |\\\\&=|\\lim _{n \\rightarrow \\infty } \\left\\langle \\pi (g_{n})\\phi , \\pi (g_{n} u_{-}^{-1} g_{n}^{-1}) \\psi -\\psi \\right\\rangle | \\\\&\\le \\lim _{n \\rightarrow \\infty } \\Vert \\pi (g_{n})\\phi \\Vert \\cdot \\Vert (\\pi (g_{n} u_{-}^{-1} g_{n}^{-1}-e)\\psi \\Vert =0, \\\\\\end{split}$ as $\\lim \\limits _{n \\rightarrow \\infty } g_{n}u_{-}g_{n}^{-1}=e$ .", "We conclude that $E\\pi (u_{-})=E$ , for every $u_{-} \\in U^{-}_{\\alpha }$ .", "In the same way one proves that $\\pi (u_{+})E=E$ , for every $u_{+} \\in U^{+}_{\\alpha }$ .", "This proves the claim.", "By the claim we obtain that, for every $\\phi \\in \\mathcal {H}$ , the vector $E(\\phi )$ (respectively, $E^{*}(\\phi )$ ) is $U^{+}_{\\alpha }$ (respectively, $U^{-}_{\\alpha }$ ) invariant.", "In particular, this is the case for the vectors $E(v)$ and respectively, $E^{*}(w)$ .", "Recall that $E$ is normal and different from zero.", "Thus, we have that $E^{*}E=EE^{*}\\ne 0$ (otherwise $0=\\left\\langle E^{*}E (\\phi ), \\phi \\right\\rangle =\\Vert E (\\phi ) \\Vert ^{2}$ for every $\\phi \\in \\mathcal {H}$ contradicting $E \\ne 0$ ).", "In this way we are able to find a non-zero vector $\\psi \\in E(\\mathcal {H})\\bigcap E^{*}(\\mathcal {H})\\ne \\lbrace 0\\rbrace \\subset \\mathcal {H}$ which is $\\left\\langle N^{+}_{\\alpha },N^{-}_{\\alpha } \\right\\rangle $ –invariant, so also $G$ –invariant.", "The contradiction follows and the lemma stands proven.", "Theorem 3.2 Let $G$ be a second countable, locally compact group having the following properties: i)  $G$ admits a decomposition $K_1A^{+}K_2$ , where $K_1,K_2 \\subset G$ are compact subsets and $A^{+}$ is an abelian sub semi-group of $G$ ; ii)  every sequence $\\alpha =\\lbrace g_{n}\\rbrace _{n>0} \\subset A^{+}$ , with $g_{n} \\rightarrow \\infty $ , admits a subsequence $\\beta =\\lbrace g_{n_k}\\rbrace _{n_k>0}$ such that $G= \\overline{ \\left\\langle U^{+}_{\\beta },U^{-}_{\\beta }\\right\\rangle }$ .", "Then $G$ has the Howe–Moore property.", "Notice that by Lemma REF and the fact that $G$ is also separable, it is enough to prove the Howe–Moore property only for unitary representations that are over separable Hilbert spaces.", "Therefore, let $(\\pi , \\mathcal {H})$ be a unitary representation of $G$ , without non-zero $G$ –invariant vectors and where $ \\mathcal {H}$ a separable Hilbert space.", "Suppose that not all matrix coefficients of $(\\pi , \\mathcal {H})$ vanish at $\\infty $ .", "Using Lemma REF there exists a sequence $\\lbrace g_{n} \\rbrace _{n>0} \\subset A^{+}$ and two non-zero vectors $v,w \\in \\mathcal {H}$ such that $g_{n} \\rightarrow \\infty $ and $ \\vert \\left\\langle \\pi (g_{n})v, w \\right\\rangle \\vert \\nrightarrow 0$ .", "By passing to a subsequence $\\lbrace g_{n_{k}}\\rbrace _{k>0}$ and using the separability of $\\mathcal {H}$ (see Lemma REF ), assume that $\\lbrace \\pi (g_{n_{k}}) \\rbrace _{n_{k}}$ weakly converges to an operator $E$ and that $\\vert \\left\\langle \\pi (g_{n_k})v, w \\right\\rangle \\vert >C>0$ , for every $n_k$ and where $C$ is a constant.", "Applying Lemma REF to a corresponding subsequence of $\\alpha =\\lbrace \\pi (g_{n_{k}}) \\rbrace _{n_{k}} $ given by hypothesis ii), we obtain a contradiction.", "Our criterion stands proven." ], [ "Unified proof of known examples", "As announced, in this section we apply the criterion provided by Theorem REF to the classes of groups mentioned in the introduction.", "Thereby we deduce, in a unified fashion, that the Howe–Moore property holds for all those classes." ], [ "Real Lie groups", "For the class of connected, non-compact, simple real Lie groups with finite center the proof is the same as the one given in Bekka–Mayer ; still we outline the ideas.", "It is a well-known fact that a connected, non-compact, semi-simple real Lie group $G$ , with finite center, has a $KA^{+}K$ decomposition (called the polar decomposition) where $K$ is a maximal compact subgroup, $A < G$ is the closed subgroup corresponding to the maximal abelian sub-algebra of the Lie algebra of $G$ and $A^{+} \\subset A$ is the corresponding positive sub semi-group part (see  for more details).", "Example 4.1 Let $G=\\operatorname{SL}(n, \\mathbf {R})$ .", "Its maximal compact subgroup $K$ , up to conjugation, is $\\operatorname{SO}(n, \\mathbf {R})$ .", "The subgroup $A$ is the group of all diagonal matrices in $\\operatorname{SL}(n, \\mathbf {R})$ with positive entries and $A^{+}$ is the subset of $A$ formed by the elements $a \\in A$ having the diagonal coefficients $a_{ii}$ verifying $a_{ii} \\le a_{i+1, i+1}$ , for every $1\\le i \\le n-1$ .", "Therefore, to use our criterion Theorem REF for connected, non-compact, simple real Lie groups, with finite center, it is enough to prove that for every sequence $\\alpha = \\left( g_{n}\\right)_{n \\ge 1} \\subset A^{+}$ with $g_{n} \\rightarrow \\infty $ one has that $\\overline{ \\left\\langle U^{+}_{\\alpha },U^{-}_{\\alpha }\\right\\rangle } =G$ .", "More exactly, this is given by the following general lemma (see ): Lemma 4.2 Let $G$ be a connected, non-compact, semi-simple real Lie group, with finite center.", "Using the above notation, we have: i)  For every $a \\in A^{+}$ , the subgroup $\\overline{ \\left\\langle U^{+}_{a},U^{-}_{a}\\right\\rangle }$ is normal and non-discrete in $G$ .", "ii)  For every $\\alpha =\\left( g_{n}\\right)_{n \\ge 1} \\subset A^{+}$ , with $g_{n} \\rightarrow \\infty $ , there exist $b \\in A^{+}$ and a subsequence $\\beta =\\lbrace g_{n_k}\\rbrace _{n_k>0}$ of $\\alpha $ such that $\\overline{ \\left\\langle U^{+}_{\\beta },U^{-}_{\\beta }\\right\\rangle } =\\overline{ \\left\\langle U^{+}_{b},U^{-}_{b}\\right\\rangle } $ ; moreover $N^{\\pm }_{b}= N^{\\pm }_{\\beta }$ .", "The proof of this lemma uses the Lie algebra of $G$ and the root system." ], [ "Euclidean building groups", "Let us now turn our attention to the totally disconnected analogs of semi-simple real Lie groups; namely locally compact groups acting continuously and properly by automorphisms and strongly transitively on locally finite thick Euclidean buildings.", "For the definition of a building and the related concepts the reader can freely consult the book of Abramenko and Brown .", "Notation 4.3 To fix the notation, let us denote a locally finite thick Euclidean building by $\\Delta $ .", "Fix for what follows an apartment $\\mathcal {A}$ in $\\Delta $ , a chamber $C$ in $\\mathcal {A}$ and a special vertex $x_0$ of $C$ .", "By $\\mathrm {Ch}(\\mathcal {B})$ we denote the set of all chambers of an apartment $\\mathcal {B}$ in $\\Delta $ and by $\\partial \\mathcal {B}$ its corresponding apartment in the spherical building at infinity $\\partial \\Delta $ .", "We make use also of the following: Definition 4.4 Let $\\Delta $ be a Euclidean or a spherical building and $G$ be a subgroup of the full automorphisms group $\\mathrm {Aut}(\\Delta )$ of $\\Delta $ .", "We say that $G$ acts strongly transitively on $\\Delta $ if for any two pairs $(A_1,c_{1})$ and $(A_2,c_{2})$ consisting of an apartment $A_i$ and a chamber $c_i \\in \\mathrm {Ch}(A_i)$ , there exists $g \\in G$ such that $g(A_1)=A_2$ and $g(c_{1})=c_{2}$ .", "Moreover, since buildings are colorable (i.e., the vertices of any chamber are colored differently and any chamber uses the same set of colors), we say that $g \\in \\mathrm {Aut}(\\Delta )$ is type-preserving if $g$ preserves the coloration of the building.", "We say that $G \\le \\mathrm {Aut}(\\Delta )$ is type-preserving if all elements of $G$ are type-preserving.", "Remark 4.5 Recall that the subgroup of all type-preserving automorphisms of $\\Delta $ is of finite index in $\\mathrm {Aut}(\\Delta )$ (see Abramenko–Brown ).", "Therefore, if a subgroup of $\\mathrm {Aut}(\\Delta )$ enjoys the Howe–Moore property then, by Proposition REF , that group must be type-preserving.", "For the rest of this article, by Remark REF , we design $G$ to be a closed, strongly transitive and type-preserving subgroup of $\\mathrm {Aut}(\\Delta )$ .", "For most of such groups $G$ we want to verify the two conditions given in Theorem REF .", "The first one is the well-known polar decomposition whose proof is recalled in the next subsection." ], [ "Polar decomposition", "Let $\\mathrm {Stab}_{G}(\\mathcal {A})$ , respectively $\\mathrm {Fix}_{G}(\\mathcal {A})$ , be the stabilizer subgroup, respectively the pointwise stabilizer subgroup, in $G$ of an apartment $\\mathcal {A}$ of $\\Delta $ .", "Recall that $\\mathrm {Stab}_{G}(\\mathcal {A}) / \\mathrm {Fix}_{G}(\\mathcal {A})$ is the Weyl group $W$ , corresponding to the Euclidean building $\\Delta $ and that $W$ is generated by the reflexions through the walls of a chamber $C \\in \\mathrm {Ch}(\\mathcal {A})$ .", "Another well-known fact is that $W$ contains a maximal abelian normal subgroup isomorphic to $\\mathbb {Z}^{m}$ , whose elements are Euclidean translation automorphisms of the apartment $\\mathcal {A}$ and $m$ is the Euclidean dimension of $\\mathcal {A}$ .", "Denote this maximal abelian normal subgroup by $A$ and remark that its elements are induced by hyperbolic automorphisms of $\\Delta $ .", "In particular, every element of $A$ can be lifted (not in a unique way) to a hyperbolic element of $G$ .", "Remark 4.6 Notice that, for a general closed, strongly transitive and type-preserving subgroup $G$ of $\\mathrm {Aut}(\\Delta )$ , its corresponding abelian subgroup $A < W$ does not necessarily lift to an abelian subgroup of $G$ .", "Still, this is the case if we consider that $G$ is a semi-simple algebraic group over a non Archimedean local field.", "By the theory of Bruhat–Tits, such a group gives rise to a Euclidean building where the group $G$ acts by automorphisms and strongly transitively.", "The abelian group $A <W$ is in fact the restriction to $\\mathcal {A}$ of a maximal split torus of $G$ , which is abelian.", "Let us now construct a basis of $A$ and define the sub semi-group $A^+$ .", "For this, let $x_0 \\in \\mathcal {A}$ be a special vertex and let $Q$ be a sector in $\\mathcal {A}$ based at $x_0$ .", "Recall that $W_{x_0}$ , the stabilizer subgroup in $W$ of the vertex $x_0$ , is transitive on the set of sectors of $\\mathcal {A}$ that are based at $x_0$ .", "This implies that for every $a \\in A$ , there exists $k \\in W_{x_0}$ such that $ka k^{-1}$ maps $x_0$ in $Q$ .", "Moreover, as the Euclidean dimension of $\\mathcal {A}$ is $m$ and $A$ is isomorphic to $\\mathbb {Z}^{m}$ , one can choose a basis $\\lbrace \\gamma _1,\\cdots , \\gamma _m\\rbrace $ of $A$ such that $\\gamma _i$ maps $x_0$ in $Q$ , for every $i \\in \\lbrace 1,\\cdots , m\\rbrace $ .", "Define now $A^{+} \\subset A$ to be the sub semi-group consisting of translations mapping $x_0$ in $Q$ .", "Remark also that every element of $A^{+}$ is a product of positive powers of elements from the basis $\\lbrace \\gamma _1,\\cdots , \\gamma _m\\rbrace $ .", "The subset $A^{+}$ is the desired `maximal abelian' and `positive' sub semi-group part of $A$ .", "Using the above notation we recall below the proof of the polar decomposition.", "Lemma 4.7 Let $G$ be a closed, strongly transitive and type-preserving subgroup of $\\mathrm {Aut}(\\Delta )$ .", "Take $\\mathcal {A}$ an apartment in $\\Delta $ and $x_0$ be a special vertex in $\\mathcal {A}$ .", "Then $G=\\mathrm {Stab}_{G}(x_0) A^{+} \\mathrm {Stab}_{G}(x_0)$ .", "Remark 4.8 We mention that the equality $G=\\mathrm {Stab}_{G}(x_0) A^{+} \\mathrm {Stab}_{G}(x_0)$ is an abuse of notation, as $A$ is not a subgroup of $G$ .", "In fact, the polar decomposition is coming from the Bruhat decomposition of $G$ with respect to the affine BN-pair.", "The reader can consult the book of Abramenko and Brown .", "[Proof of Lemma REF ] Take $g \\in G$ and let $\\mathcal {B}$ be an apartment in $\\Delta $ such that $C, g(C) \\in \\mathrm {Ch}(\\mathcal {B})$ .", "As $G$ is strongly transitive, there exists $k \\in G$ such that $k(\\mathcal {B})=\\mathcal {A}$ and $k(C)=C$ pointwise.", "Therefore $k \\in \\mathrm {Stab}_{G}(x_0)$ .", "As $ G$ is type-preserving, $kg(x_0)$ is a special vertex of the same type as $x_0$ .", "Therefore, using the fact that $W$ is simply transitive on special vertices of the same type, there exists a translation automorphism $a$ of the apartment $\\mathcal {A}$ sending $kg(x_0)$ into $x_0$ .", "We conclude that $akg(x_0)=x_0$ and thus $akg \\in \\mathrm {Stab}_{G}(x_0)$ .", "By writing the element $a$ using elements from $A^{+}$ and $ \\mathrm {Stab}_{G}(x_0)$ , we have that $g \\in \\mathrm {Stab}_{G}(x_0) A^{+} \\mathrm {Stab}_{G}(x_0)$ .", "Therefore the polar decomposition of the group $G$ is proven, as $\\mathrm {Stab}_{G}(x_0)$ , being the stabilizer of the vertex $x_0$ , is compact and open in $G$ .", "Example 4.9 (See Platonov–Rapinchuk  page 151) For $G=\\operatorname{SL}(n, \\mathbf {Q}_p)$ the `good' maximal compact subgroup is $\\operatorname{SL}(n, \\mathbf {Z}_{p})$ and $A$ is the group of diagonal matrices in $\\operatorname{SL}(n, \\mathbf {Q}_p)$ of the form $\\operatorname{diag}(p^{a_1}, \\cdots , p^{a_n})$ , with the condition that $(a_1,\\cdots , a_n) \\in \\mathbf {Z}^{n}$ and $\\sum \\limits _{i=1}^{n} a_i=0$ .", "In this case, the sub semi-group $A^{+}$ is the subset of $A$ such that $a_1 \\le a_2 \\le \\cdots \\le a_n$ .", "Example 4.10 (See Burger–Mozes ) When the Euclidean building is a bi-regular tree $T_{bireg}$ , with valence $\\ge 3$ at every vertex, and $G$ is a closed, type-preserving subgroup of $\\mathrm {Aut}(T_{bireg})$ , with 2–transitive action on the boundary of $T_{bireg}$ , the polar decomposition goes as follows.", "Notice that $G$ acts without fixed point on the boundary of $T_{bireg}$ and from , we have that $G$ is not compact.", "By Tits , $G$ must contain a hyperbolic element.", "Moreover, by using the 2–transitivity of $G$ on the boundary of $T_{bireg}$ , we can construct a hyperbolic element $a \\in G$ whose translation length is 2.", "Denote by $\\ell \\subset T_{bireg}$ the translation axis of $a$ , which is a bi-infinite geodesic line of the tree $T_{bireg}$ .", "Let $x_0$ be a vertex of $\\ell $ .", "The maximal compact subgroup of the polar decomposition of $G$ is the stabilizer subgroup in $G$ of $x_0$ , the subgroup $A =\\langle a\\rangle $ and $A^{+}=\\lbrace a^{n} \\; \\vert \\; n \\ge 1\\rbrace $ .", "We also mention the known fact that in the case of a thick tree $T$ , the strongly transitive action of a subgroup of $\\mathrm {Aut}(T)$ on $T$ is equivalent to the 2–transitive action of that group on the boundary of $T$ (see for example Caprace–Ciobotaru ).", "Related to the polar decomposition from Lemma REF , we recall another well-known decomposition of the group $G$ .", "Let $c$ be an ideal chamber in $\\partial \\mathcal {A}$ .", "Denote by $G_c^{0}: =\\lbrace g \\in G \\; \\vert \\; g(c)=c \\text{ and } g \\text{ fixes at least one vertex of } \\Delta \\rbrace $ the `pointwise' stabilizer subgroup in $G$ corresponding to $c \\in \\mathrm {Ch}(\\partial \\mathcal {A})$ .", "Remark that $G_c^{0}$ is a closed subgroup of $\\mathrm {Stab}_{G}(c)$ and it does not contain any hyperbolic automorphisms of $\\Delta $ .", "Indeed, let $g \\in G_c^{0}$ and let $x_g \\in \\Delta $ be a vertex fixed by $g$ .", "Consider an apartment $\\mathcal {B}$ in $\\Delta $ such that $x_g \\in \\mathcal {B}$ and $c \\in \\mathrm {Ch}(\\partial \\mathcal {B})$ .", "Then $\\mathcal {A}$ and $\\mathcal {B}$ share a common sector pointing to the chamber at infinity $c$ .", "As $g(c)=c$ and $g$ fixes the vertex $x_g$ , $g$ must fix a sector in $\\mathcal {A} \\cap \\mathcal {B}$ that points to the chamber $c$ .", "Using this observation, it is easy to see that $G_c^{0}$ is a group.", "To prove that $G_c^{0}$ is closed, let $\\lbrace g_n\\rbrace _{n>0} \\subset G_c^{0}$ such that $g_n \\rightarrow g$ , for some $g \\in G$ .", "We have that $g(c)=c$ .", "Moreover, as $\\lbrace g_n\\rbrace _{n>0}$ are elliptic elements in $G$ , $g$ cannot be hyperbolic.", "We have that $g \\in G_c^{0}$ as desired.", "Take $g \\in G$ and let $\\mathcal {B}$ be an apartment in $\\Delta $ such that $g^{-1}(C) \\in \\mathrm {Ch}(\\mathcal {B})$ and $c \\in \\mathrm {Ch}(\\partial \\mathcal {B})$ .", "Remark that $\\mathcal {A}$ and $\\mathcal {B}$ have a sector in common, corresponding to the chamber at infinity $c$ .", "Thus, by the strongly transitive action, there exists $n \\in G_c^{0}$ such that $n(\\mathcal {B})=\\mathcal {A}$ .", "We obtain that $ng^{-1}(C) \\in \\mathrm {Ch}(\\mathcal {A})$ .", "As $G$ is type-preserving, there exists a translation automorphism $a \\in A $ of the apartment $\\mathcal {A}$ such that $ang^{-1}(x_0)=x_0$ .", "From here we conclude that $ang^{-1} \\in \\mathrm {Stab}_{G}(x_0)$ and thus $g \\in \\mathrm {Stab}_{G}(x_0) A G_c^{0}$ .", "Therefore $G=\\mathrm {Stab}_{G}(x_0) A G_c^{0}.$ Let us now start verifying the second condition given by Theorem REF , for a closed, strongly transitive and type-preserving subgroup $G$ of $\\mathrm {Aut}(\\Delta )$ .", "Its first step is given by the following well-known proposition; for the convenience of the reader we include its proof.", "Regarding the closed subgroup $G_c^{0}$ defined in (REF ), by the strongly transitive action of $G$ on $\\Delta $ , we have that $G_c^{0}$ acts transitively on the set $\\mathrm {Opp}(c)$ of all chambers opposite $c$ of $\\mathrm {Ch}(\\partial \\Delta )$ .", "Indeed, let $c_1 \\ne c_2 \\in \\mathrm {Opp}(c)$ .", "Denote by $A_1$ and respectively, by $A_2$ , the unique apartments of $\\Delta $ such that $c_1,c \\in \\mathrm {Ch}(\\partial A_1)$ and respectively, $c_{2},c \\in \\mathrm {Ch}(\\partial A_2)$ .", "Let $Q \\subset A_1\\cap A_2 $ be a sector pointing to the chamber at infinity $c$ and choose a chamber $C$ of $\\mathrm {Ch}(\\Delta )$ contained in the interior of $Q$ .", "Then, by the strong transitive action of $G$ , there exists $g \\in G$ such that $g(C)=C$ and $g(A_1)=A_2$ .", "In addition, we have that $g(Q)=Q$ pointwise; therefore $g \\in G_c^{0}$ and $g(c_1)=c_2$ .", "Proposition 4.11 Let $G$ be a closed, strongly transitive and type-preserving subgroup of $\\mathrm {Aut}(\\Delta )$ .", "Let $\\mathcal {A}$ be an apartment of $\\Delta $ and take $c _+\\in \\mathrm {Ch}(\\partial \\mathcal {A})$ .", "Denote by $c_-$ the chamber opposite $c_+$ in $\\partial \\mathcal {A}$ .", "Then $G=\\langle G_{c_+}^{0}, G_{c_-}^{0} \\rangle $ .", "Before starting the proof, we state the next useful lemma.", "Lemma 4.12 Let $G$ be a closed, strongly transitive and type-preserving subgroup of $\\mathrm {Aut}(\\Delta )$ .", "Let $\\mathcal {A}$ be an apartment of $\\Delta $ and take $c _+\\in \\mathrm {Ch}(\\partial \\mathcal {A})$ .", "Denote by $c_-$ the chamber opposite $c_+$ in $\\partial \\mathcal {A}$ .", "Then $\\mathrm {Stab}_{G}(\\mathcal {A})=\\mathrm {Stab}_{G}( \\partial \\mathcal {A}) \\le \\langle G_{c_+}^{0}, G_{c_-}^{0} \\rangle $ .", "The equality $\\mathrm {Stab}_{G}(\\mathcal {A})=\\mathrm {Stab}_{G}( \\partial \\mathcal {A})$ is straightforward.", "Following Caprace–Ciobotaru  we start with a basic observation.", "Let $\\mathcal {A}^{\\prime }$ be an apartment of $\\Delta $ such that the intersection $\\mathcal {A}\\cap \\mathcal {A}^{\\prime }$ is a half-apartment.", "We claim that there is some $u \\in \\langle G_{c_+}^{0}, G_{c_-}^{0} \\rangle $ mapping $\\mathcal {A}^{\\prime }$ to $\\mathcal {A}$ and fixing $\\mathcal {A} \\cap \\mathcal {A}^{\\prime }$ pointwise.", "Indeed, because $c_+,c_-$ are opposite in $\\partial \\mathcal {A}$ , they are separated by any wall of $\\mathcal {A}$ .", "Thus, $\\mathcal {A} \\cap \\mathcal {A}^{\\prime }$ , being a half-apartment, contains a sector $Q$ corresponding either to $c_+$ , or to $c_-$ .", "We apply the strong transitivity of $G$ ; there exists an element $g \\in G$ fixing pointwise a chamber in the interior of the sector $Q$ and sending $\\mathcal {A}^{\\prime }$ to $\\mathcal {A}$ .", "Then $g \\in \\langle G_{c_+}^{0}, G_{c_-}^{0} \\rangle $ .", "Let now $M$ be a wall of $\\mathcal {A}$ .", "Let $H$ and $H^{\\prime }$ be the two half-apartments of $\\mathcal {A}$ determined by $M$ .", "Since $\\Delta $ is thick, there exists a half-apartment $H^{\\prime \\prime }$ such that $H \\cup H^{\\prime \\prime }$ and $H^{\\prime } \\cup H^{\\prime \\prime }$ are both apartments of $\\Delta $ .", "By the above claim, we can find an element $u \\in \\langle G_{c_+}^{0}, G_{c_-}^{0} \\rangle $ fixing $H$ pointwise and mapping $H^{\\prime }$ to $H^{\\prime \\prime }$ .", "Similarly, there are elements $v, w \\in \\langle G_{c_+}^{0}, G_{c_-}^{0} \\rangle $ fixing $H^{\\prime }$ pointwise and such that $v(H^{\\prime \\prime }) = H$ and $w(H) = u^{-1}(H^{\\prime })$ .", "Now we set $r := vuw$ .", "By construction $r$ fixes pointwise the wall $M$ and $r \\in \\langle G_{c_+}^{0}, G_{c_-}^{0} \\rangle $ .", "Moreover we have $r(H) = vu u^{-1}(H^{\\prime }) = v(H^{\\prime }) = H^{\\prime }$ and $r(H^{\\prime }) = vu(H^{\\prime }) = v(H^{\\prime \\prime }) = H,$ so that $r$ swaps $H$ and $H^{\\prime }$ .", "It follows that $r$ stabilizes the apartment $\\mathcal {A}$ and acts on $\\mathcal {A}$ as the reflection through the wall $M$ .", "Since this holds for an arbitrary wall of $\\mathcal {A}$ and $\\mathrm {Stab}_{G}(\\mathcal {A}) /\\mathrm {Fix}_{G}(\\mathcal {A})$ , being the Weyl group $W$ , is generated by reflexions through the walls of a chamber $C \\in \\mathrm {Ch}(\\mathcal {A})$ , our conclusion follows.", "[Proof of Proposition REF ] Take $g \\in G$ and let $\\mathcal {B}$ an apartment of $\\Delta $ such that $g(c_+), c_+ \\in \\mathrm {Ch}(\\partial \\mathcal {B})$ .", "Then $\\mathcal {A}$ and $\\mathcal {B}$ have a sector in common, corresponding to the chamber at infinity $c_+$ .", "By the strong transitivity of $G$ , take $h_1 \\in G_{c_+}^{0}$ such that $h_1(\\mathcal {B})=\\mathcal {A}$ .", "Therefore $h_1g(c_+) \\in \\mathrm {Ch}(\\partial \\mathcal {A})$ .", "Recall that the strong transitivity of $G$ on $\\Delta $ implies that $G$ acts strongly transitively on $\\partial \\Delta $ (see Garrett ).", "Applying Lemma REF , there exists $h_2 \\in \\mathrm {Stab}_{G}( \\partial \\mathcal {A}) \\le \\langle G_{c_+}^{0}, G_{c_-}^{0} \\rangle $ such that $h_2h_1g(c_+)=c_+ $ .", "Thus $h_2h_1g(\\mathcal {A})$ and $\\mathcal {A}$ have a sector in common corresponding to the chamber at infinity $c_+$ .", "Take now $h_3 \\in G_{c_+}^{0}$ with $h_3h_2h_1g(\\mathcal {A})= \\mathcal {A}$ .", "Therefore $h_3h_2h_1g \\in \\mathrm {Stab}_{G}(\\mathcal {A})$ and by Lemma REF the conclusion follows.", "Recall from Baumgartner and Willis  the following general theory.", "Let $G $ be a totally disconnected locally compact group and take $a \\in G$ .", "Let $P^{+}_{a}:=\\lbrace g \\in G \\; \\vert \\; \\lbrace a^{-n}g a^{n}\\rbrace _{n \\in \\mathbb {N}} \\text{ is bounded}\\rbrace .$ In the same way, but using $a^{n}g a^{-n}$ , we define $P^{-}_{a}$ .", "Following , $P^{+}_{a}$ is a closed subgroup of $G$ and denote $U^{+}_{a}:=\\lbrace g \\in G \\; \\vert \\; \\lim _{n \\rightarrow \\infty }a^{-n}g a^{n}=e\\rbrace .$ By , $P^{+}_{a}$ and $U^{+}_{a}$ are called the parabolic, respectively the contraction, subgroups associated to $a$ , where in general $U^{+}_{a}$ is not closed.", "Moreover, from the algebraic point of view, we have the following Levi decomposition: Theorem 4.13 (See Baumgartner and Willis ) Let $G$ be a totally disconnected locally compact group which is also metrizable.", "Let $a \\in G$ .", "Then $P^{+}_{a}= U^{+}_{a} M_a$ , where $M_{a}:= P^{+}_{a} \\cap P^{-}_{a}$ .", "Moreover, $U^{+}_{a}$ is normal in $P^{+}_{a}$ .", "Notice that Theorem REF applies to our context of $G$ being a closed, strongly transitive and type-preserving subgroup of $\\mathrm {Aut}(\\Delta )$ , where $\\Delta $ is a locally finite Euclidean building.", "Remark 4.14 In the setting of a locally finite Euclidean building $\\Delta $ , a subset $F \\subset \\mathrm {Aut}(\\Delta )$ is called bounded if there exists a point $y \\in \\Delta $ such that the set $F \\cdot y$ is a bounded subset of $\\Delta $ .", "For equivalent definitions see Abramenko–Brown .", "Those characterizations are useful and used in what follows when working with elements of $P^{+}_{a}$ .", "This subsection proposes to link the `algebraic' Levi decomposition from Theorem REF with the subgroup $G_c^{0}$ defined in (REF ), a key ingredient being given by the following: Proposition 4.15 Let $G$ be a closed, non-compact and type-preserving subgroup of $\\mathrm {Aut}(\\Delta )$ .", "Let $\\mathcal {A}$ be an apartment in $\\Delta $ and assume there exists a hyperbolic automorphism $a \\in \\mathrm {Stab}_{G}(\\mathcal {A})$ .", "Denote its attracting endpoint by $\\xi _+ \\in \\partial \\mathcal {A}$ and let $c_+ \\in \\mathrm {Ch}(\\partial \\mathcal {A})$ , with $\\xi _+ \\in c_+$ .", "Let $G_{c_+}:= \\lbrace g \\in G \\; \\vert \\; g(c_+)=c_+\\rbrace $ and $G_{\\xi _+}:= \\lbrace g \\in G \\; \\vert \\; g(\\xi _+)=\\xi _+\\rbrace $ .", "Then $G_{c_+} \\le P^{+}_{a} = G_{\\xi _+}= G_{\\sigma }$ , where $\\sigma $ is the unique simplex in $\\partial \\mathcal {A}$ such that $\\xi _+$ is contained in the interior of $\\sigma $ ($\\sigma =\\xi _+$ if and only if $\\xi _+$ is a vertex).", "In particular, we obtain that $P^{+}_{a} \\cap P^{-}_{a} = G_{\\lbrace \\xi _-, \\xi _+\\rbrace }$ , where $G_{\\lbrace \\xi _-, \\xi _+\\rbrace }:=\\lbrace g \\in G \\; \\vert \\; g(\\xi _+)=\\xi _+ \\text{ and } g(\\xi _-)=\\xi _-\\rbrace $ .", "The equality $ G_{\\xi _+}= G_{\\sigma }$ is immediate as $G$ is type-preserving.", "Let us first prove that $P^{+}_{a} \\le G_{\\xi _+}$ .", "Take $g \\in P^{+}_{a}$ and let $x_0$ be a special vertex of $\\mathcal {A}$ .", "By the definition of $P^{+}_{a}$ , the set $\\lbrace a^{-n}g a^{n}(x_0) \\; \\vert \\; n \\in \\mathbb {N}\\rbrace $ is a bounded subset of $\\Delta $ and thus is finite.", "Extract a subsequence $n_k \\rightarrow \\infty $ such that $a^{-n_k}g a^{n_k}(x_0)=y$ , for every $n_k$ .", "As $G$ is type-preserving, the point $y$ is also a special vertex of $\\Delta $ .", "Now, because $a^{n_k}(x_0) \\rightarrow \\xi _+$ and $a^{n_k}(y) \\rightarrow \\xi _+$ (in the cone topology of $\\Delta \\cup \\partial \\Delta $ ), one can conclude that $g(\\xi _+)=\\xi _+$ .", "Therefore $g \\in G_{\\xi _+}$ .", "Let us show that $G_{\\xi _+} \\le P^{+}_{a}$ .", "Let $g \\in G_{\\xi _+}$ .", "As $g(\\xi _+)=\\xi _+$ there exists a geodesic ray in $\\mathcal {A}$ , say $[y, \\xi _+)$ , such that $g([y, \\xi _+)) \\subset \\mathcal {A}$ is parallel to $[y, \\xi _+)$ .", "Moreover, we have that $\\operatorname{dist}_{\\mathcal {A}}(g(t),t) = \\operatorname{dist}_{\\mathcal {A}}(g(y),y)$ , for every $t \\in [y, \\xi _+)$ .", "To prove that $g \\in P^{+}_{a}$ it is enough to show that the set $\\lbrace a^{-n}g a^{n}(z) \\; \\vert \\; n \\in \\mathbb {N}\\rbrace $ is a bounded subset of $\\Delta $ , for some point $z \\in \\mathcal {A}$ .", "Take $z=y$ .", "Because $a$ is a hyperbolic element in $\\mathrm {Stab}_{G}(\\mathcal {A})$ , $a^{n}(y) \\in [y, \\xi _+)$ .", "From here, we immediately conclude that $\\lbrace a^{-n}g a^{n}(z) \\; \\vert \\; n \\in \\mathbb {N}\\rbrace $ is indeed a bounded set.", "The fact that $G_{c_+} \\le P^{+}_{a} $ is an immediate consequence of the equality $P^{+}_{a} = G_{\\xi _+}$ , as $G$ is type-preserving and $G_{c_+} \\le G_{\\xi _+}$ .", "Remark 4.16 For a general hyperbolic element $a \\in \\mathrm {Stab}_{G}(\\mathcal {A})$ we do not necessarily have the equality $P^{+}_{a} =G_{c_+}$ in Proposition REF .", "However, if $a$ is a strongly regular hyperbolic element of $\\mathrm {Stab}_{G}(\\mathcal {A})$ (see Caprace–Ciobotaru ), the equality $P^{+}_{a} =G_{c_+}$ holds, as in this case $G_{c_+}=G_{\\xi _+}$ .", "In addition, if we take $\\Delta $ to be a locally finite tree, the equality $P^{+}_{a} =G_{c_+}$ is always verified as the chambers at infinity are just the ends of the tree, which are points.", "By combining Proposition REF and Theorem REF we obtain the geometric Levi decomposition: Corollary 4.17 Let $G$ be a closed and type-preserving subgroup of $\\mathrm {Aut}(\\Delta )$ .", "Let $\\mathcal {A}$ be an apartment in $\\Delta $ and assume there exists a hyperbolic automorphism $a \\in \\mathrm {Stab}_{G}(\\mathcal {A})$ .", "Denote its attracting endpoint by $ \\xi _+ \\in \\partial \\mathcal {A}$ and let $\\sigma $ be a simplex in $\\partial \\mathcal {A}$ such that $\\xi _+ \\in \\sigma $ .", "Let also $c_+ \\in \\mathrm {Ch}(\\partial \\mathcal {A})$ with $\\xi _+ \\in c_+$ .", "Then $G_{\\sigma }= U^{+}_{a} (M_a \\cap G_{\\sigma })$ , where $M_{a}:= P^{+}_{a} \\cap P^{-}_{a}$ .", "In particular, $G_{c_+}^{0}= U^{+}_{a} (M_a \\cap G_{c_+}^{0})$ .", "In addition $U^{+}_{a} $ is normal in $G_{c_+}^{0}$ , respectively, $G_{\\sigma }$ .", "By the definition of $U^{+}_{a}$ one can easily verify that $U^{+}_{a} \\le G_{c_+}^{0} \\le G_{\\sigma } \\le P^{+}_{a}$ .", "The conclusion follows by applying Theorem REF .", "Furthermore, by Proposition REF and Corollary REF we obtain: Corollary 4.18 Let $G$ be a closed, strongly transitive and type-preserving subgroup of $\\mathrm {Aut}(\\Delta )$ .", "Let $\\mathcal {A}$ be an apartment of $\\Delta $ and let $a \\in \\mathrm {Stab}_{G}(\\mathcal {A})$ be a hyperbolic automorphism.", "Then $\\overline{\\langle U^{+}_{a},U^{-}_{a} \\rangle }$ is normal in $G$ .", "Let $c_-,c_+ \\in \\mathrm {Ch}(\\partial \\mathcal {A})$ be two opposite ideal chambers containing the repelling point and respectively the attracting point of the hyperbolic element $a$ .", "By Proposition REF we have that $G=\\langle G_{c_-}^{0}, G_{c_+}^{0} \\rangle $ .", "Therefore, to prove that $\\overline{\\langle U^{+}_{a},U^{-}_{a} \\rangle }$ is normal in $G$ , it is enough to verify that, for every $g \\in G_{c_{\\pm }}^{0}$ , we have that $g U^{\\pm }_{a}g^{-1} \\in \\langle U^{+}_{a},U^{-}_{a} \\rangle $ .", "Indeed, using the decomposition of $G_{c_{\\pm }}^{0}$ given by Corollary REF , and the fact that $U^{\\pm }_{a} $ is normal in $G_{c_\\pm }^{0}$ , the conclusion follows.", "Using all the ingredients presented in the above subsections, the goal here is to provide the proof of the following theorem: Theorem 4.19 Let $G$ be an isotropic simple algebraic group over a non Archimedean local field or a closed, topologically simple subgroup of $\\mathrm {Aut}(T)$ that acts 2–transitively on the boundary $\\partial T$ , where $T$ is a bi-regular tree with valence $\\ge 3$ at every vertex.", "Then $G$ admits the Howe–Moore property.", "To prove Theorem REF , in the case of algebraic groups, it remains to verify the second condition given by Theorem REF , namely, the analogue of Lemma REF .", "As in the case of connected simple real Lie groups with finite center, the analogue lemma uses the theory of root group datum related to Euclidean buildings; therefore, we briefly recall some definitions and state some known properties of this theory from Weiss , Abramenko–Brown   and Ronan .", "We mention that the proof of the analogue of Lemma REF , stated in Proposition REF for the case of Euclidean buildings, works in the same spirit as for real Lie groups.", "Definition 4.20 Let $X$ be a Euclidean or a spherical building.", "A half-apartment $h$ of $X$ is called a root of $X$.", "By abuse of notation, we use $\\partial h$ to denote the boundary wall in $X$ which is determined by $h$ .", "Moreover, if $X$ is a spherical building, the root group $U_{h}$ corresponding to a root $h$ of $X$ is defined to be the set of all $g \\in \\mathrm {Aut}(X)$ such that $g$ fixes pointwise $h$ and also the star of every panel contained in $h \\setminus \\partial h$ .", "We denote $U_{h}^{*}:=U_{h} \\setminus \\lbrace \\operatorname{id}\\rbrace $ .", "Example 4.21 If $X$ is a locally finite tree without vertex of valence one, then a root $h$ of $X$ corresponds to an infinite ray in the tree and its boundary $\\partial h$ is just the base-vertex in $X$ of that infinite ray.", "Denote by $\\partial X$ the boundary at infinity of the tree $X$ .", "For a point $\\xi \\in \\partial X$ , its corresponding root group is the contraction group $U^{+}_{a} \\le \\mathrm {Aut}(X)$ defined in (REF ) above and where $a$ is a hyperbolic element of $\\mathrm {Aut}(X)$ with $\\xi $ its attracting endpoint (see Caprace–De Medts ).", "Definition 4.22 Let $X$ be a thick spherical building.", "For a root $h$ of $X$ , we denote by $\\mathcal {A}(h)$ the set of all apartments in $X$ that contain $h$ .", "We say that $X$ is Moufang if, for every root $h$ of $X$ , the root group $U_{h} < \\mathrm {Aut}(X)$ is transitive on $\\mathcal {A}(h)$ .", "It is immediate that the boundary at infinity of a locally finite tree without vertex of valence one is Moufang.", "Recall that for a semi-simple algebraic group $\\mathbb {G}$ over a non Archimedean local field, by Bruhat and Tits  one associates a locally finite thick Euclidean building $\\Delta $ where $\\mathbb {G}$ acts by automorphisms and strongly transitively.", "More precisely: Definition 4.23 Let $\\mathbb {G}$ be a semi-simple algebraic group over a non Archimedean local field $k$ , of rank $\\ge 1$ .", "Let $G=\\mathbb {G}(k)$ be the group of all $k$ –rational points of $\\mathbb {G}$ .", "By , to the group $G$ one associates a locally finite thick Euclidean building $\\Delta $ where $G$ acts by type-preserving automorphisms and strongly transitively.", "Moreover, by , the spherical building $\\partial \\Delta $ at infinity of $\\Delta $ is Moufang.", "Let $G^{+}:= \\langle U_h \\le G \\; \\vert \\; h \\text{ is a root of } \\partial \\Delta \\rangle $ .", "It is known that $G^{+}$ is a closed normal subgroup of $G$ and that the factor group $G/G^{+}$ is compact (see Margulis ).", "We call $G^{+}$ an isotropic simple algebraic group over the non Archimedean local field $k$.", "From its definition, notice that $G^{+}$ acts by type-preserving automorphisms and also strongly transitively on $\\Delta $ .", "Proposition 4.24 (See Weiss ) Let $\\Delta $ be a Euclidean building such that its corresponding building at infinity $\\partial \\Delta $ is Moufang.", "Let $\\mathcal {A}$ be an apartment of $\\Delta $ and let $h$ be a root of $\\partial \\mathcal {A}$ .", "For $u \\in U_{h}^{*}$ let $h_u := \\mathcal {A} \\cap u(\\mathcal {A})$ .", "Then $h_u$ is a root of $\\mathcal {A}$ such that its boundary at infinity $h_u(\\infty )$ is $h$ .", "Moreover, $u$ acts trivially on $h_u$ .", "Conversely, let $\\mathcal {H}$ be a root of $\\mathcal {A}$ such that its boundary at infinity $\\mathcal {H}(\\infty )$ is $h$ .", "Then there exists $u \\in U_{h}^{*}$ such that $\\mathcal {H}=h_u$ .", "Definition 4.25 Let $\\Delta $ be a Euclidean building such that its corresponding building at infinity $\\partial \\Delta $ is Moufang.", "Let $\\mathcal {H}$ be a root of an apartment $\\mathcal {A}$ of $\\Delta $ and denote $h := \\mathcal {H}(\\infty )$ the corresponding root in $\\partial \\mathcal {A}$ .", "With respect to $U_{h}$ , the affine root group corresponding to $\\mathcal {H}$ is the set $U_{\\mathcal {H}}:= \\lbrace u \\in U_{h}^{*}\\;\\vert \\; h_u=\\mathcal {H}\\rbrace \\cup \\operatorname{id}$ , where $h_u := \\mathcal {A} \\cap u(\\mathcal {A})$ , for $u \\in U_{h}^{*}$ .", "Example 4.26 Let $X$ is a locally finite tree without vertex of valence one as in Example REF .", "Let $\\mathcal {H}$ be an infinite ray of $X$ and denote by $h \\in \\partial X$ the endpoint at infinity of $\\mathcal {H}$ .", "Then $U_{\\mathcal {H}}:= \\lbrace g \\in U_{a}^{+} \\; \\vert \\; h_g=\\mathcal {H} \\rbrace \\cup \\operatorname{id}$ , where $U_{a}^{+}$ is defined like in Example REF .", "Proposition 4.27 (See Bruhat–Tits ) Let $\\mathbb {G}$ be a semi-simple algebraic group over a non Archimedean local field $k$ and let $G=\\mathbb {G}(k)$ be the group of all $k$ –rational points of $\\mathbb {G}$ .", "Denote by $\\Delta $ the corresponding locally finite thick Euclidean building on which $G$ acts by type-preserving automorphisms and strongly transitively.", "Let $\\mathcal {H}$ be a root of $\\Delta $ and let $x \\in \\mathcal {H}$ .", "Then the radius of the ball of $\\Delta $ around $x$ which is fixed pointwise by $U_{\\mathcal {H}} < G$ goes to infinity as the distance from $x$ to $\\partial \\mathcal {H}$ goes to infinity.", "From Propositions REF and REF we immediately obtain: Corollary 4.28 Let $\\mathbb {G}$ be a semi-simple algebraic group over a non Archimedean local field $k$ and let $G=\\mathbb {G}(k)$ be the group of all $k$ –rational points of $\\mathbb {G}$ .", "Denote by $\\Delta $ the corresponding locally finite thick Euclidean building on which $G$ acts by type-preserving automorphisms and strongly transitively.", "Let $\\mathcal {A}$ be an apartment of $\\Delta $ and let $h$ be a root of $\\partial \\mathcal {A}$ .", "Let $\\alpha =\\lbrace a_n\\rbrace _{n>0} \\subset \\mathrm {Stab}_{G}(\\mathcal {A})$ with $a_n \\rightarrow \\infty $ .", "Assume that $a_n(x_0) \\rightarrow \\xi \\in h \\setminus \\partial h$ , in the cone topology, where $x_0 \\in \\mathcal {A}$ is some special vertex.", "Then $U_{h} \\le U_{\\alpha }^{+}$ .", "In particular, we obtain that $U_{h} \\le U_{a}^{+}$ , for every hyperbolic element $a \\in \\mathrm {Stab}_{G}(\\mathcal {A})$ whose attracting endpoint $\\xi _+ \\in h \\setminus \\partial h$ .", "As $U_h$ is the union of its corresponding affine root groups $U_{\\mathcal {H}}$ , it is enough to prove that $U_{\\mathcal {H}} \\le U_{\\alpha }^{+}$ .", "Indeed, let $u \\in U_{\\mathcal {H}}$ .", "As $a_n(x_0) \\rightarrow \\xi \\in h \\setminus \\partial h$ , in the cone topology, where $x_0$ is a special vertex of $\\mathcal {A}$ , we have that the intersection of the geodesic ray $[x_0, \\xi )$ with the root $\\mathcal {H}$ is a geodesic ray with endpoint $\\xi $ .", "Therefore, for every standard open neighborhood $V \\subset \\mathcal {H}$ of $\\xi $ with respect to the cone topology of $\\mathcal {A}$ , there exists $N$ such that $a_n(x_0) \\in V \\subset \\mathcal {H}$ , for every $n \\ge N$ .", "By Proposition REF , we immediately obtain that $u \\in U_{\\alpha }^{+}$ .", "Another important property is recorded by the following corollary.", "Corollary 4.29 Let $\\mathbb {G}$ be a semi-simple algebraic group over a non Archimedean local field $k$ and let $G=\\mathbb {G}(k)$ be the group of all $k$ –rational points of $\\mathbb {G}$ .", "Denote by $\\Delta $ the corresponding locally finite thick Euclidean building on which $G$ acts by type-preserving automorphisms and strongly transitively.", "Let $\\mathcal {A}$ be an apartment of $\\Delta $ and let $a \\in \\mathrm {Stab}_{G}(\\mathcal {A})$ be a hyperbolic element whose attracting endpoint in $\\partial \\mathcal {A}$ is denoted by $\\xi _+$ .", "Let $\\sigma $ be the unique simplex in $\\partial \\mathcal {A}$ such that $\\xi _{+}$ is contained the interior of $\\sigma $ and denote by $\\mathrm {St}(\\sigma )$ the star of $\\sigma $ in $\\partial \\mathcal {A}$ .", "Then $U_{a}^{+}= \\langle U_h\\; \\vert \\; h \\text{ is a root of } \\partial \\mathcal {A} \\text{ with } \\mathrm {St}(\\sigma ) \\subset h\\rangle $ .", "The conclusion follows by combining Corollary REF with the theory of root group datum from Ronan .", "The analogue of Lemma REF is given by the following proposition.", "Proposition 4.30 Let $\\mathbb {G}$ be a semi-simple algebraic group over a non Archimedean local field $k$ and let $G=\\mathbb {G}(k)$ be the group of all $k$ –rational points of $\\mathbb {G}$ .", "Denote by $\\Delta $ the corresponding locally finite thick Euclidean building on which $G$ acts by type-preserving automorphisms and strongly transitively.", "Let $\\mathcal {A}$ be the apartment that corresponds to the abelian sub semi-group $A^{+}$ of the Weyl group $W$ of $G$ .", "Let $\\alpha =\\lbrace a_n\\rbrace _{n>0} \\subset A^{+}$ with $a_n \\rightarrow \\infty $ and such that $a_{n}(x_0) \\rightarrow \\xi \\in \\partial \\mathcal {A}$ , for some special vertex $x_0 \\in \\mathcal {A}$ .", "Then there exists a hyperbolic element $b \\in \\mathrm {Stab}_{G}(\\mathcal {A})$ such that $U^{\\pm }_{b} \\le U^{\\pm }_{\\alpha }$ .", "Let $x_0$ be a fixed special vertex in the apartment $\\mathcal {A}$ and let $\\lbrace \\gamma _1,\\cdots , \\gamma _m\\rbrace $ be the basis of $A^{+}$ described in the beginning of Section REF .", "Moreover, we can choose the basis $\\lbrace \\gamma _1,\\cdots , \\gamma _m\\rbrace $ such that the attracting endpoint of $\\gamma _j$ , for every $j \\in \\lbrace 1, \\cdots , m\\rbrace $ , is a vertex in $\\partial \\mathcal {A}$ (i.e., it is a vertex of the chamber at infinity determined by the sector that defines $A^{+}$ ).", "Let $\\sigma $ be the unique simplex of $\\partial \\mathcal {A}$ that contains $\\xi $ in its interior ($\\xi = \\sigma $ if and only if $\\xi $ is a vertex in $\\partial \\mathcal {A}$ ).", "Let $\\lbrace \\gamma _{i_1},\\cdots , \\gamma _{i_l}\\rbrace $ be the set of the elements of $\\lbrace \\gamma _1,\\cdots , \\gamma _m\\rbrace $ whose attracting endpoints determine the simplex $\\sigma $ .", "Define $b:= \\gamma _{i_1}\\cdot ... \\cdot \\gamma _{i_l}$ and notice that $b$ is a hyperbolic element in $\\mathrm {Stab}_{G}(\\mathcal {A})$ .", "Moreover, the attracting endpoint $\\xi _{+}$ of $b$ and the point $\\xi $ are contained in the interior of the simplex $\\sigma $ of $\\partial \\mathcal {A}$ .", "Apply Corollary REF to $b$ and $\\sigma $ .", "Then, by Corollary REF we obtain that $U^{\\pm }_{b} \\le U^{\\pm }_{\\alpha }$ .", "[Proof of Theorem REF ] By Lemma REF and Remark REF it is enough to verify the Howe–Moore property only for separable Hilbert spaces.", "Let $G$ be an isotropic simple algebraic group over a non Archimedean local field or a closed, topologically simple subgroup of $\\mathrm {Aut}(T)$ that acts 2–transitively on the boundary $\\partial T$ , where $T$ is a bi-regular tree with valence $ \\ge 3$ at every vertex.", "For such $G$ , we verify the two conditions of Theorem REF .", "In both cases, the first condition of Theorem REF is verified by applying Lemma REF to $G$ .", "Let us verify the second condition of Theorem REF .", "Notice that, in the case of a bi-regular tree, the corresponding group $A^{+}$ of $G$ , given by the polar decomposition of $G$ , is generated by a single hyperbolic element $a \\in G$ (see Example REF ).", "Therefore, the second condition follows immediately from Corollary REF applied to $G$ .", "When $G$ is an isotropic simple algebraic group over a non Archimedean local field, to obtain the Howe–Moore property, it is enough to verify the second condition of Theorem REF only for a sequence $\\alpha =\\lbrace a_n\\rbrace _{n>0} \\subset A^{+}$ with $a_n \\rightarrow \\infty $ and such that $a_{n}(x_0) \\rightarrow \\xi \\in \\partial \\mathcal {A}$ , for some special vertex $x_0 \\in \\mathcal {A}$ .", "The latter assertion follows from the fact that $\\partial \\Delta \\cup \\Delta $ is compact with respect to the cone topology.", "Then, apply to $\\alpha =\\lbrace a_n\\rbrace _{n>0}$ Proposition REF and Corollary REF .", "Notice that, for every hyperbolic element $b \\in \\mathrm {Stab}_G(\\mathcal {A})$ , the group $\\overline{\\langle U^{+}_{b},U^{-}_{b} \\rangle }$ , which is normal in $G$ , acts non-trivially on $\\Delta $ .", "By Tits , we obtain that for every hyperbolic element $b \\in \\mathrm {Stab}_G(\\mathcal {A})$ , the group $\\overline{\\langle U^{+}_{b},U^{-}_{b} \\rangle }$ equals $G$ .", "The theorem stands proven.", "Acknowledgements.", "This article is part of author's PhD project conducted under the supervision of Pierre-Emmanuel Caprace.", "I would like to thank him for valuable conversations and support during this project.", "My thanks go also to Mihai Berbec and Steven Deprez for interesting discussions, in the early stage of this work, on von Neumann algebras, as well as providing the elegant proof of Lemma REF and the Example REF .", "I also thank the referee for his/her useful comments.", "ABbook author=Abramenko, Peter, author=Brown, Kenneth S., title=Buildings, series=Graduate Texts in Mathematics, volume=248, note=Theory and applications, publisher=Springer, place=New York, date=2008, BW04article author=Baumgartner, Udo , author=Willis, George A., title=Contraction groups and scales of automorphisms of totally disconnected locally compact groups, journal=Israel Journal of Mathematics, year=2004, issn=0021-2172, volume=142, number=1, doi=10.1007/BF02771534, url=http://dx.doi.org/10.1007/BF02771534, publisher=Springer-Verlag, pages=221-248, language=English BHVbook author = Bekka, Bachir, author=de la Harpe, Pierre, author=Valette, Alain, title = Kazhdan's Property (T), publisher = New Mathematical Monographs, Cambridge University Press, volume=11 year = 2008, BMbook author=Bekka, M. 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1403.0223
[ [ "On the Behavioral Interpretation of System-Environment Fit and\n Auto-Resilience" ], [ "Abstract Already 71 years ago Rosenblueth, Wiener, and Bigelow introduced the concept of the \"behavioristic study of natural events\" and proposed a classification of systems according to the quality of the behaviors they are able to exercise.", "In this paper we consider the problem of the resilience of a system when deployed in a changing environment, which we tackle by considering the behaviors both the system organs and the environment mutually exercise.", "We then introduce a partial order and a metric space for those behaviors, and we use them to define a behavioral interpretation of the concept of system-environment fit.", "Moreover we suggest that behaviors based on the extrapolation of future environmental requirements would allow systems to proactively improve their own system-environment fit and optimally evolve their resilience.", "Finally we describe how we plan to express a complex optimization strategy in terms of the concepts introduced in this paper." ], [ "Introduction", "Let us consider a familiar case of “systems”: the human beings.", "Human beings are generally considered as the highest peak of biological evolution.", "Their behavioral and teleological characteristics [1] set them apart from other system classes [2] and make them appear to be more “gifted” than other beings, e.g., dogs.", "But how do the superior qualities of mankind translate in terms of resilience?", "Under stressful or turbulent conditions we know that often a man will result “better” than a dog: superior awareness, consciousness, manual and technical dexterity, and reasoning; advanced ability to reuse experience, learn, develop science, as well as other factors, they all lead to the apparently “obvious” conclusion that mankind has a greater ability to tolerate adverse conditions.", "And though, it is also quite easy to find counterexamples.", "If a threat, e.g., comes with ultrasonic noise, a dog may perceive the threat and react—for instance by running away—while a man may stay unaware until too late.", "Or consider the case of miners: inability to perceive toxic gases makes them vulnerable to, e.g., carbon monoxide and dioxide, methane, and other lethal gases [3].", "A simpler system able to perceive the threat and flee would have more chances to survive.", "Perception of course is but one of a number of “systemic features” that need to be available in order to counterbalance a threat.", "So how do we tell whether a system is fit to stand the new conditions characterizing a changing environment?", "How do we reason about the quality of resilience?", "And, even more importantly, how do we make sure that a system “stays fit” if the environment changes?", "The above questions are discussed and, to some extent, addressed in this paper.", "Our starting point here is the conjecture that resilience is no absolute figure; rather, it is the result of a match with a deployment environment.", "Whatever its structure, organization, architecture, capabilities, and resources, a system is only robust as long as its “provisions” (its system characteristics, including the ability to develop knowledge and “wisdom”) match the current environmental conditions.", "A second cornerstone of the present discussion is given by the assumption that the interactions between systems and environments can be expressed and reasoned upon by considering the behaviors expressed during those interactions.", "In other words, a system-environment fit is the result of the match between the behaviors exercised by a system and those exercised by its environment (including other systems, the users, etc.)", "A third and final assumption is that reasoning about a system's resilience is facilitated by considering the behaviors of those system “organs” (namely, sub-systems) responsible for the following abilities: the ability to perceive change; the ability to ascertain the consequences of change; the ability to plan a line of defense against threats deriving from change; the ability to enact the defense plan being conceived in step 3; and, finally, the ability to treasure up past experience and continuously improve, to some extent, abilities 1–4.", "As can be clearly seen, the above abilities correspond to the components of the so-called MAPE-K loop of autonomic computing [4].", "We shall refer to those abilities as well as the organs that embed them as to the “systemic features.” In what follows we first focus in Sect.", "on the concept of behavior and recall the five major classes of behaviors according to Rosenblueth, Wiener, and Bigelow [1] and Boulding [2].", "We then introduce a system's cybernetic class by associating each of the systemic features with its own behavior class.", "After this, in Sect.", ", we introduce a behavioral formulation of the concepts of supply and system-environment fit as measures of the optimality of a given design with respect to the current environmental conditions.", "Section  then suggests how proactive and/or social behaviors that would be able to track supply and system-environment fit would pave the way to systems able to self-tune their systemic features in function of the experienced or predicted environmental conditions.", "An application of the concepts presented in this work is briefly described in Sect. .", "Our conclusions are finally stated in Sect.", "." ], [ "Systemic Features", "As mentioned above, an important attribute towards achieving robustness is given by what we called in Sect.", "as the “systemic features”, or the behaviors typical of the system under scrutiny.", "Such behaviors are the subject of the present section.", "In what follows we first recall in Sect.", "REF what are the main behavioral classes.", "The main sources here are the classic works by Rosenblueth, Wiener, and Bigelow [1] and Boulding [2].", "In the first work, classes were identified by the Authors by considering the system in isolation.", "In the second one Boulding introduced an additional class considering the social dimension.", "After this, in Sect.", "REF , we consider an exemplary system; we identify in it the main system organs responsible for resilience; and associate behavioral classes to those organs.", "By doing so we characterize $C$ , namely the “cybernetic class” of the system under consideration." ], [ "Behavioral Classes", "Already 71 years ago Rosenblueth, Wiener, and Bigelow [1] introduced the concept of the “behavioristic study of natural events”, namely “the examination of the output of the object and of the relations of this output to the input”.", "The term “object” in the cited paper corresponds to that of “system”.", "In that renowned text the Authors purposely “omit the specific structure and the intrinsic organization” of the systems under scrutiny and classify them exclusively on the basis of the quality of the “change produced in the surroundings by the object”, namely the system's behavior.", "The Authors identify in particular four major classes of behaviorsFor the sake of brevity we will not discuss here passive behavior.", ": ${\\beta _{\\scriptsize \\hbox{ran}}}$  : Random behavior.", "This is an active form of behavior that does not appear to serve a specific purpose or reach a specific state.", "A source of electro-magnetic interference exercises random behavior.", "${\\beta _{\\scriptsize \\hbox{pur}}}$  : Purposeful behavior.", "This is behavior that serves a purpose and is directed towards a specific goal.", "Quoting the Authors, in purposeful behavior we can observe a “final condition toward which the movement [of the object] strives”.", "Servo-mechanisms are examples of purposeful behavior.", "${\\beta _{\\scriptsize \\hbox{rea}}}$  : Reactive behavior.", "This is behavior that “involve[s] a continuous feed-back from the goal that modifies and guides the behaving object”.", "Examples of this behavior include phototropism, namely the tendency we observe, e.g., in certain plants, to grow towards the light, and gravitropism, viz.", "the tendency of plant roots to grow downward.", "Reactive behaviors require the system to be open [5] (able that is to continuously perceive, communicate, and interact with external systems and the environment) and to embody some form of feedback loop.", "${\\beta _{\\scriptsize \\hbox{pro}}}$  : Proactive behavior.", "This is behavior directed towards the extrapolated future state of the goal.", "The Authors in [1] classify proactive behavior according to its “order”, namely the amount of context variables taken into account in the extrapolation.", "Kenneth Boulding in his classic paper [2] introduces an additional class: ${\\beta _{\\scriptsize \\hbox{soc}}}$  : Social behaviors.", "This class is based on the concept of social organization.", "Quoting the Author, in such systems “the unit is not perhaps the person—the individual human as such—but the `role'—that part of the person which is concerned with the organization or situation in question, and it is tempting to define social organizations, or almost any social system, as a set of role tied together with channels of communication.” Social behaviors may take different forms and be, e.g., mutualistic, commensalistic, co-evolutive, or co-opetitive [6], [7], [8].", "For more information we refer the Reader to [9].", "We shall define $\\pi $ as a projection map returning, for each of the above behavior classes, an integer in $\\lbrace 1,\\ldots ,5\\rbrace $ ($\\pi (\\hbox{${\\beta _{\\scriptsize \\hbox{ran}}}$})=1$ , ..., $\\pi (\\hbox{${\\beta _{\\scriptsize \\hbox{soc}}}$})=5$ ).", "For any behavior ${\\beta _{\\scriptsize x}}$ and any set of context figures $F$ , notation ${\\beta ^{\\scriptsize F}_{\\scriptsize x}}$ will be used to denote that ${\\beta _{\\scriptsize x}}$ is exercised by considering the context figures in $F$ .", "Thus if, for instance, $F=(\\hbox{speed},\\hbox{luminosity})$ , then ${\\beta ^{\\scriptsize F}_{\\scriptsize \\hbox{rea}}}$ refers to a reactive behavior that responds to changes in speed and light.", "For any behavior ${\\beta _{\\scriptsize x}}$ and any integer $n>0$ , notation ${\\beta ^{\\scriptsize n}_{\\scriptsize x}}$ will be used to denote that ${\\beta _{\\scriptsize x}}$ is exercised by considering $n$ context figures, without specifying which ones.", "As an example, behaviour ${\\beta ^{\\scriptsize |F|}_{\\scriptsize \\hbox{pro}}}$ , with $F$ defined as above, identifies an order-2 proactive behavior while ${\\beta ^{\\scriptsize F}_{\\scriptsize \\hbox{pro}}}$ says in addition that that behavior considers both speed and luminosity to extrapolate the future position of the goal.", "We now introduce the concept of partial order among behaviors.", "Definition 1 (Partial order of behaviors) Given any two behaviors $\\beta _1$ and $\\beta _2$ we shall say that $\\beta _1 \\prec \\beta _2$ if and only if either of the following conditions holds: $\\pi (\\beta _1) < \\pi (\\beta _2)$ .", "$\\left(\\pi (\\beta _1) = \\pi (\\beta _2)\\right) \\wedge $ $\\left(\\exists (F, G): \\beta _1=\\hbox{${\\beta ^{\\scriptsize F}_{\\scriptsize 1}}$} \\wedge \\beta _2=\\hbox{${\\beta ^{\\scriptsize G}_{\\scriptsize 2}}$} \\wedge F \\subsetneq G\\right)$ .", "$\\left(\\pi (\\beta _1) = \\pi (\\beta _2) = \\hbox{${\\beta _{\\scriptsize \\hbox{pro}}}$}\\right) \\wedge $ $\\left(\\exists (n,m): \\beta _1=\\hbox{${\\beta ^{\\scriptsize n}_{\\scriptsize 1}}$} \\wedge \\beta _2=\\hbox{${\\beta ^{\\scriptsize m}_{\\scriptsize 2}}$} \\wedge n<m\\right)$ .", "Whenever two behaviors $\\beta _1$ and $\\beta _2$ are such that $\\beta _1 \\prec \\beta _2$ , it is possible to define some notion of distance between the two behaviors by considering an arithmetizationA classic example of arithmetization may be found in the renowned work [10] by Kurt Gödel.", "based on, e.g., the following factors used as exponents of three different prime numbers: $\\pi (\\beta _2) - \\pi (\\beta _1)$ .", "$|G\\setminus F|$ .", "$m-n$ .", "In what follows we shall assume that some metric function, $\\mathbf {dist}$ , has been defined." ], [ "Cybernetic Class", "The behavioral classes recalled in REF may be applied to the five “systemic features” introduced in Sect. .", "For any system $s$ we shall refer to the systemic features of $s$ through the following 5-tuple: $\\left(C_M(s), C_A(s), C_P(s), C_E(s), C_K(s)\\right),$ whose components orderly correspond to the abilities introduced in Sect.", "as well as to the stages of MAPE-K loops [4].", "System $s$ will me omitted when it can be implicitly identified without introducing ambiguity.", "Definition 2 (Cybernetic Class) For any given system $s$ we define as cybernetic class the 5-tuple $C(s) = \\left(\\beta _{C_M(s)}, \\beta _{C_A(s)}, \\beta _{C_P(s)}, \\beta _{C_E(s)}, \\beta _{C_K(s)}\\right),$ where, for any $x\\in \\lbrace M,A,P,E,K\\rbrace $ , $\\beta _{C_x(s)}$ represents the behavior class assigned to systemic feature $C_x$ of $s$ , or $\\varnothing $ if $s$ does not include $C_x$ altogether.", "As can be clearly understood, a system's cybernetic class is a qualitative metric that does not provide a full coverage of the systemic characteristics of the system.", "As such it should be complemented with quantitative assessments of the quality of service of its system organs—namely the sub-systems responsible for hosting its systemic features (REF ).", "In particular for $C_M(s)$ and $C_E(s)$ —namely, the features corresponding to the abilities of perception and actuation—it is useful to complement the notion of behavior with a characterization of the set of context variables that are under the “sphere of action” of the corresponding organs.", "For $C_M(s)$ this means specifying the set of context figures that may be timely perceived by $s$  [11], [3].", "Interestingly enough, this concept closely corresponds to that of the powers of representation in Leibniz [12].", "When considering $C_E(s)$ , the sphere of action could be represented by the set of the context figures that may be controlled—to a certain extent—through system behaviors.", "We observe that features $C_M$ and $C_E$ are intrinsically purposeful.", "We believe that notation ${\\beta ^{\\scriptsize F}_{\\scriptsize \\hbox{pur}}}$ provides a convenient and homogeneous way to express the behavior class and the spheres of action of both $M$ and $E$ organs.", "It is now possible to characterize a system's cybernetic class through notation (REF ).", "As an example, by following the assessments proposed in [13], the adaptively redundant data structures described in [14] have the following cybernetic class $ C_1 = (\\hbox{${\\beta _{\\scriptsize \\hbox{pur}}}$}, \\hbox{${\\beta ^{\\scriptsize 1}_{\\scriptsize \\hbox{pro}}}$}, \\hbox{${\\beta _{\\scriptsize \\hbox{pur}}}$}, \\hbox{${\\beta _{\\scriptsize \\hbox{pur}}}$}, \\varnothing ), $ while the adaptive $N$ -version programming system introduced in [15], [16] is $ C_2 = (\\hbox{${\\beta _{\\scriptsize \\hbox{pur}}}$}, \\hbox{${\\beta ^{\\scriptsize 2}_{\\scriptsize \\hbox{pro}}}$}, \\hbox{${\\beta _{\\scriptsize \\hbox{pur}}}$}, \\hbox{${\\beta _{\\scriptsize \\hbox{pur}}}$}, \\hbox{${\\beta _{\\scriptsize \\hbox{pur}}}$}).", "$ We believe the notion and notation of cybernetic class provide a convenient way to compare qualitatively the systemic features of any two systems with reference to their robustness.", "As an example, by comparing the above 5-tuples $C_1$ and $C_2$ one may easily realize how the major strength of those two systems lies in their analytic organs, both of which are capable of proactive behaviors (${\\beta _{\\scriptsize \\hbox{pro}}}$ )—though in a simpler fashion in $C_1$ .", "Another noteworthy difference is the presence of a knowledge organ in $C_2$ , which indicates that the second system is able to accrue and make use of the past experience in order to improve its action—to some extent and exclusively through ${\\beta _{\\scriptsize \\hbox{pur}}}$ behaviors.", "We conjecture that the action of the knowledge organ in this case corresponds to so-called antifragility [17], [18], namely the ability to “treasure up” the past experience so as to improve one's system-environment fit." ], [ "System-Environment Fit", "What presented in Sect.", "allows for a system to be characterized—to some extent—in terms of its “systemic features”—the provisions that is that play a role when responding to change.", "As a way to identify the “quality” of those provisions in that section we made use of the different behavioral classes as defined in [1], [2], and introduced $C(s)$ as well as its components.", "Here we move our attention to a second aspect that, we conjecture, needs to be considered when assessing a system's resilience.", "This second aspect tells us how the cybernetic class matches the requirements of dynamically changing environmental conditions.", "As already anticipated in Sect.", ", in what follows we assume that the evolution of an environment may also be expressed as a behavior.", "Said behavior may be of any of the types listed in Sect.", "REF and as such it may result in the dynamic variation of a number of “firing context figures”.", "In fact those figures characterize and, in a sense, set the boundaries of an ecoregion, namely “an area defined by its environmental conditions” [19].", "An environment may be the result of the action of, e.g., a human being (a “user”), or a software managing an ambient, or for instance it may be the result of purposeless (random) behavior—such as a source of electro-magnetic interference.", "As a consequence, an environment may behave randomly or exhibit a recognizable trend; in the latter case the variation of its context figures may be such that it allows for tracking or speculation (extrapolation of future states).", "Moreover, an environment may exhibit the same behavior for a relatively long period of time or it may vary dynamically its character.", "We shall refer in what follows to the dynamic evolution of environmental behavior as to an environment's turbulence.", "Diagrams such as the one in Fig.", "REF may be used to represent the dynamic evolution of environments.", "Figure: Exemplification of turbulence, namely the dynamic evolution of environmental behavior (shown hereas adotted line).", "Abscissas are time, “now” beingthe current time.", "Ordinates are the behavior classes exercised by the environment.It is now possible to propose a definition of two indicators for the quality of resilience: the system supply relative to an environment and the system-environment fit.", "Definition 3 (System supply) Given a system $S$ deployed in an environment $E$ , characterized respectively by behaviors $\\beta ^S(t)$ and $\\beta ^E(t)$ ; and given a metric function $\\mathbf {dist}$ ; we define as supply at time $t$ with respect to $\\beta ^E(t)$ the following value: $\\mathbf {supply}(S,E,t) = \\nonumber \\\\= {\\left\\lbrace \\begin{array}{ll}\\mathbf {dist}(\\beta ^S(t),\\beta ^E(t)) & \\mbox{if } \\beta ^E(t)\\prec \\beta ^S(t)\\\\-\\mathbf {dist}(\\beta ^S(t),\\beta ^E(t)) & \\mbox{if } \\beta ^S(t)\\prec \\beta ^E(t)\\\\0 & {0.5}{\\mbox{if $\\beta ^E(t)$ and $\\beta ^S(t)$ ex-}\\\\\\mbox{press the same behaviors.}}\\end{array}\\right.", "}$ Supply can be positive (oversupply), negative (undersupply), or zero (perfect supply).", "Definition 4 (System-environment fit) Given the same conditions as in Definition REF , we define as the system-environment fit at time $t$ the function $\\mathbf {fit}(S,E,t) = \\\\ \\nonumber \\\\= {\\left\\lbrace \\begin{array}{ll}1 / (1 + \\mathbf {supply}(S,E,t) ) & \\mbox{if } \\mathbf {supply}(S,E,t) \\ge 0\\\\-\\infty & \\mbox{otherwise.}\\\\\\end{array}\\right.", "}$ The above definition expresses system-environment fit as a function returning 1 in the case of best fit; slowly scaling down with oversupply; and returning $-\\infty $ in case of undersupply.", "It is not the only possible such definition of course: an alternative one is given, for instance, by having $\\mathbf {supply}^2$ instead of $\\mathbf {supply}$ .", "Figure REF exemplifies a system-environment fit in the case of two behaviors $\\beta ^S$ and $\\beta ^E$ with $S\\subsetneq E$ .", "$E$ consists of five context figures identified by integers $1,\\dots ,5$ while $S$ consists of context figures $1,\\dots ,4$ .", "The system behavior is assumed to be constant; if $S=C(M)$ this means that the system's perception organ constantly monitors the four figures $1,\\dots ,4$ .", "On the contrary $\\beta ^E$ varies with time.", "Five time segments are exemplified ($s_1,\\dots ,s_5$ ) during which the following context figures are affected: $s_1$ : Figures $1,\\dots ,4$ .", "$s_2$ : Figure 1 and figure 4.", "$s_3$ : Figure 4.", "$s_4$ : Figures $1,\\dots ,4$ .", "$s_5$ : Figures $1,\\dots ,5$ .", "Figures are represented as boxed integers, with an empty box meaning that the figure is not affected by the environment and a filled box meaning the figure is affected.", "The behaviour of the environment is constant within a time segment and changes at the next one.", "This is shown through the sets at the bottom of Fig.", "REF : for each segment $t_s\\in \\lbrace s_1,\\dots ,s_5\\rbrace $ the superset is $E(t_s)$ while the subset is $S(s_t)$ , namely $E(s_t)\\cap S$ .", "The relative supply and the system-environment fit also change with the time segments.", "During $s_1$ and $s_4$ there is perfect supply and best fit: the behavior exercised by the environment is evenly matched by the features of the system.", "During $s_2$ and $s_3$ the systemic features are more than enough to match the current environmental conditions—a case of what we referred to as “oversupply”.", "Correspondingly, fit is rather low.", "In $s_5$ we have the opposite situation: the systemic features—for instance, pertaining to a perception organ—are insufficient to become aware of all the changes produced by the environment.", "In particular here changes connected with figure 5 go undetected.", "This is a case of “undersupply”, corresponding to the “worst possible” system-environment fit.", "Figure: Exemplification of supply and system-environment fit." ], [ "Optimal Resilience", "The two functions introduced in Sect.", ", $\\mathbf {supply}$ and $\\mathbf {fit}$ , may be interpreted as measures of the optimality of a given design with respect to the current environmental conditions.", "Whenever those conditions allow it and a partial order “$\\prec $ ” exists for the behaviors at play, then it is possible to consider system behaviors of the following forms: $\\hbox{${\\beta ^{\\scriptsize F}_{\\scriptsize \\hbox{pro}}}$}$ , with $F$ including figures $\\mathbf {supply}$ and $\\mathbf {fit}$ .", "Such behavior, when exercised by system organs for analysis, planning, and knowledge management, translates in the possibility to become aware and speculate on the possible future robustness requirements.", "If this is coupled with the possibility to revise one's system organs by enabling or disabling, e.g., the ability to perceive certain context figures depending on the extrapolated future environmental conditions, then a system could proactively improve its own system-environment fit.", "$\\hbox{${\\beta ^{\\scriptsize F}_{\\scriptsize \\hbox{soc}}}$}$ , with $F$ including figures $\\mathbf {supply}$ and $\\mathbf {fit}$ .", "Analysis, planning, and knowledge management behaviors of this type aim at artificially augmenting or reducing the system features by establishing / disestablishing collaborative relationships as exemplified in the “canary-in-the-coal-mine” scenario of [3].", "As we did in the paper just cited we propose to call behaviors such as REF ) and REF ) as auto-resilient.", "Finally, we remark how the formulation of system-environment fit presented in this work may also be tailored so as to include overheads and costs." ], [ "An Application: Project LittleSister", "LittleSister [20] is an ICON project financed by the iMinds research institute and the Flemish Government Agency for Innovation by Science and Technology (IWT).", "The project aims to deliver a low-cost telemonitoring [21] solution for home care and is to run until the end of year 2014.", "LittleSister adopts a connectionist approach in which the collective action of an interconnected network of simple units [22] (battery-powered mouse sensors) replaces the adoption of more powerful and expensive complex devices (smart cameras).", "In order for this approach to be effective the mentioned collective action is to guarantee that an optimal trade-off between energy efficiency, performance, and safety is dynamically sustained.", "We plan to express this optimal trade-off in terms of a system-environment fit.", "Obviously the formulation of the LittleSister system-environment fit will be considerably more complex than the one introduced in the present work.", "A key role will be played in particular by the LittleSister awareness organ, which will be used to determine the level of criticality of the current situation and set an operative mode ranging from “energy-saving-first” to “safety-first”.", "This operative mode will be included in the set of context figures of the social behavior $\\hbox{${\\beta ^{\\scriptsize F}_{\\scriptsize \\hbox{soc}}}$}$ of LittleSister's sensors.", "Depending on the requirements expressed by the current operative mode and other context figures, the system-environment fit will vary, which will translate in a variable selection and number of sensors to be activated.", "The goal we aim to reach is being able to sustain at the same time both maximum safety and minimum energy expenditure." ], [ "Conclusions", "The questions we have posed in Sect.", "have been answered, to some extent, by defining a conceptual framework for their discussion.", "The nature of our framework is behavioral and “sits on the shoulders” of the work carried out in the first half of last Century by “giants” such as Bogdanov, Wiener, von Bertalanffy, Boulding, and several others—in turn based on the intriguingly modern ideas of “elder giants” such as Leibniz [12] and Aristotle [23].", "Within our framework we have introduced a behavioral formulation of the concepts of supply and system-environment fit as measures of the optimality of a system with respect to the current conditions of the environment in which the system is deployed.", "Moreover, we have suggested how complex abilities such as auto-resilience and antifragility may be expressed in terms of behaviors able to track supply and fit measures and evolve the systemic features in function of the hypothesized future environmental conditions.", "Practical application of the concepts in this article has been briefly discussed by considering a strategy for optimizing the collective behavior of the mouse sensors used in project LittleSister.", "As can be clearly understood, our work is far from being exhaustive or complete.", "In particular discussing context figures without referring to a “range”, or sphere of action, makes it difficult to compare behaviors such as auditory perception in animals.", "Our future work will include extending our conceptual framework accordingly.", "Another direction we intend to take is the application of our concepts towards the design of antifragile computing systems; the Reader may refer to [18] for a few preliminary ideas about this." ], [ "Acknowledgments", "I would like to express my gratitude to Alan Carter for helping me with the pictures in this paper.", "Many thanks to Dr. Tom Leckrone (https://twitter.com/SemprePhi) for introducing me to the work of Alexander A. Malinovsky (A.", "A. Bogdanov).", "This work was partially supported by iMinds—Interdisciplinary institute for Technology, a research institute funded by the Flemish Government—as well as by the Flemish Government Agency for Innovation by Science and Technology (IWT).", "The iMinds LittleSister project is a project co-funded by iMinds with project support of IWT.", "Partners involved in the project are Universiteit Antwerpen, Universiteit Gent, Vrije Universiteit Brussel, Xetal, Christelijke Mutualiteit vzw, Niko Projects, JF Oceans BVBA, and SBD NV.", "16" ] ]
1403.0339
[ [ "Asymptotic estimate for the number of Gaussian packets on three\n decorated graphs" ], [ "Abstract We study a topological space obtained from a graph via replacing vertices with smooth Riemannian manifolds, i.e.", "a decorated graph.", "We construct a semiclassical asymptotics of the solutions of Cauchy problem for a time-dependent Schr\\\"{o}dinger equation on a decorated graph with a localized initial function.", "The main term of our asymptotic solution at an arbitrary finite time is the sum of Gaussian packets and generalized Gaussian packets.", "We study the number of such packets as time goes to infinity.", "We prove asymptotic estimations for this number for the following decorated graphs: cylinder with a segment, two dimensional torus with a segment, three dimensional torus with a segment.", "Also we prove general theorem about a manifold with a segment and apply it to the case of a uniformly secure manifold." ], [ "Introduction", "Differential equations and differential operators on decorated graphs have been intensively studied over the past thirty years (see, for example, [1], [2], [3], [4] and references therein).", "The study of the motion of Gaussian packets arises when considering a Cauchy problem for a time-dependent Schrödinger equation.", "Let us consider the Cauchy problem for a decorated graph, which is a singular space obtained by gluing the ends of segments to the surfaces of dimension two or three.", "The initial conditions are a Gaussian packet with support on one of the edges.", "We look for a semiclassical solution (see, for example, [7], [8] and references therein).", "Upon reaching the end of the segment, the packet forms an expanding wavefront on the surface.", "If the front reaches another point of gluing then a new Gaussian packet starts to move along the corresponding edge and so on.", "We are interested in finding the asymptotic behavior of the number of supports of the Gaussian packets on the edges of a decorated graph at time $T$ .", "A detailed description of the formulation of this problem is given in [10].", "Wavefront propagation on surfaces is associated with the properties of geodesics on the surface.", "There are two variations.", "In the first the number of geodesics connecting two points on the surface is finite.", "As an example we can take a standard sphere with an edge.", "In this situation we can construct from one-dimensional edges and geodesics on the surfaces an equivalent metric graph and describe the statistics of the distribution of Gaussian packets using our results obtained before (see, e.g., [5], [6]).", "Theorems on the asymptotic behavior of the number of packets $\\mathop {\\hbox{}\\mathrm {N}}\\nolimits (T)$ and the uniformity of their distribution for this case were proven in [10].", "The second variation is more general, where the equivalent geometric graph is infinite.", "The main question here is how the number of geodesics joining two given points increases as time goes to infinity.", "The time and the maximum length are synonymous in this context.", "Some results in this area can be found in [13].", "Let $h$ be a topological entropy for a compact Riemannian manifold $M$ .", "R. Mañé has shown in [16] (see references therein) that for quite general situations $h=\\lim \\limits _{T\\rightarrow \\infty }\\frac{1}{T}\\int _{M{\\times }M}\\mathop {\\mathrm {log}}\\nolimits CF_T(x,y)\\mathrm {d}\\mu (x)\\mathrm {d}\\mu (y)$ , where $CF_T(x, y)$ is the number of geodesics joining $x$ to $y$ of maximum length $T$ .", "But this equation may fail if $M$ has conjugate points.", "There are examples where the growth of $CF_T (x, x)$ is arbitrarily large for some exceptional points $x \\in M$ (see [17]).", "There are also examples in which the limit in the equation is smaller than the topological entropy for an open set of configurations (see [15]).", "A Riemannian manifold is said to be uniformly secure (see [17]) if there is a finite number $k$ such that all geodesics connecting an arbitrary pair of points in the manifold can be blocked by $k$ point obstacles.", "It is proven in [17] that the number of geodesics with length $\\le T$ between every pair of points in a uniformly secure manifold grows polynomially as $T \\rightarrow \\infty $ .", "According to the results of M. Gromov and R. Mañé (see [16]), the fundamental group of such a manifold is virtually nilpotent, and the topological entropy of its geodesic flow is zero.", "Furthermore, if a uniformly secure manifold has no conjugate points, then it is flat.", "This follows from the virtual nilpotency of its fundamental group either via the theorems of Croke-Schroeder and Burago-Ivanov, or by a more recent work of Lebedeva [18].", "In the present article we consider the case where the number of geodesics grows polynomially, for example a compact Riemannian manifold that is uniformly secure.", "We study in detail two examples: a standard cylinder and a flat torus with one edge.", "In the situation where the equivalent lengths of the edges of the graph are linearly independent over the field of rational numbers, there is a one-to-one correspondence between the times in which the birth of new Gaussian packets occurs and nonnegative solutions of infinite linear inequality where the right hand side equals $T$ .", "Such inequalities arise in number theory, namely in the analysis of the asymptotic behavior of the number of partitions of natural numbers.", "We can recall the results of G. H. Hardy and S. Ramanujan, J. V. Uspensky, G. Rademacher, P. Erdös and others.", "In our case, the length of the geodesics will not be a sequence of natural numbers.", "Therefore, at the first stage of our research we obtain only an upper bound from the classical results.", "On the other hand, a lower bound was proven in [10], namely for infinite number of edges $\\mathop {\\hbox{}\\mathrm {N}}\\nolimits (T)$ grows faster than any polynomial.", "Later we found a connection between the problem we had studied and questions arising in the study of Bose-Maslov gas entropy, which in recent years was carried out by V. P. Maslov and V. E. Nazaikinskii.", "We used a 2013 result of V. E. Nazaikinskii (see [9]) to obtain asymptotic formulas for the logarithm of the number of Gaussian packets on the edge.", "If linear independence over $\\mathbb {Q}$ does not hold (such situation is certainly possible because we can even have many geodesics with the same length, see e.g.", "[12]), the resulting asymptotic formula becomes an upper bound for $\\mathop {\\hbox{}\\mathrm {N}}\\nolimits (T)$ .", "It should be noted that the same results can be obtained using Additive Abstract Prime Number Theorem from [11].", "A computer experiment conducted jointly with O. V. Sobolev has confirmed the correctness of the estimates.", "The simulation results for a decorated graph obtained by attaching a length of the cylinder are also given in [10]." ], [ "Preliminary remarks and definitions", "A decorated graph is a topological space, obtained from a metric graph by replacing vertices by smooth manifolds of dimensions two or three.", "Consider a finite number of smooth complete Riemannian manifolds $M_j$ , and a number of segments $\\gamma _i$ , endowed with regular parametrization.", "For each endpoint $y$ of an arbitrary segment $\\gamma _j$ fix a point $\\tilde{y}$ on one of the manifolds $M_k$ ; we assume all points $\\tilde{y}$ to be distinct.", "A decorated graph $\\Gamma _d$ is a quotient space of the disjoint sum $\\bigsqcup _kM_k\\bigsqcup _j\\gamma _j$ by the equivalence $y \\sim \\tilde{y}$ .", "The Schrödinger equation on a decorated graph is defined as follows (see [2] and [10] for detailed explanation; the original ideas were presented in [3],[4]).", "Let $V$ be a real valued continuous function on $ \\Gamma _d$ , smooth on the edges.", "Let $V_j$ and $V_k$ be restrictions of $V$ to $\\gamma _j$ and to $M_k$ respectively.", "Consider a direct sum $\\widehat{H}_0=\\bigoplus \\limits _{j=1}^{E}\\left(-\\frac{h^2}{2}\\frac{\\mathrm {d}^2}{\\mathrm {d}z_j^2}+V_j\\right)\\bigoplus \\limits _{k=1}^{V}\\left(-\\frac{h^2}{2}\\Delta _k+V_k\\right) $ with the domain $H^2(\\Gamma )=\\bigoplus \\limits _{j=1}^{E}H^2(\\gamma _j)\\bigoplus \\limits _{k=1}^{V} H^2(M_k).$ Here $\\frac{\\mathrm {d}^2}{\\mathrm {d}z_j^2}$ is an operator of the second derivative on $\\gamma _j$ with respect to a fixed parametrization with Neumann boundary conditions, $\\Delta _k$ is the Laplace–Beltrami operator on $M_k$ .", "Definition.", "The Schrödinger operator $\\widehat{H}$ is a self-adjoint extension of the restriction $\\widehat{H_0}|_{L}$ , where $L=\\lbrace \\psi \\in H^2(\\Gamma ),\\quad \\psi (y_s)=0\\rbrace .$ Domain of the operator $\\widehat{H}$ contains functions with singularities in the points $y_j$ .", "Namely, let $G(x,y,\\lambda )$ be the Green function on $M_k$ (integral kernel of the resolvent) of $\\Delta $ , corresponding to the spectral parameter $\\lambda $ .", "This function has the following asymptotics as $x\\rightarrow y$ : $G(x,y,\\lambda )=F_0(x,y)+F_1$ , where $F_1$ is a continuous function and $F_0$ is independent of $\\lambda $ and has the form $F_0=-\\frac{c_2}{2\\pi }\\mathop {\\mathrm {ln}}\\nolimits \\rho $ if $\\mathop {\\mathrm {dim}}\\nolimits M=2$ and $F_0=\\frac{c_3}{4\\pi \\rho }$ if $\\mathop {\\mathrm {dim}}\\nolimits M=3$ .", "Here $c_j(x,y)$ is continuous, $c_j(y,y)=1$ , $\\rho $ is the distance between $x$ and $y$ .", "The function $\\psi $ from the domain of the operator $\\widehat{H}$ has the following asymptotics as $x\\rightarrow y_j$ : $ \\psi =\\alpha _jF_0(x)+b_j+o(1)$ .", "Now for each endpoint of the segment consider a pair $\\psi (y)$ , $h\\psi ^\\prime (y)$ and a vector $\\xi =(u,v)$ ,$u=(h\\psi ^\\prime (y_1),\\dots ,h\\psi ^\\prime (y_{2E}),\\alpha _1,\\dots ,\\alpha _{2E})$ ,$v=(\\psi (y_1),\\dots ,\\psi (y_{2E}),hb_1,\\dots ,hb_{2E})$ .Consider a standard skew-Hermitian form $[\\xi ^1,\\xi ^2]=\\sum _{j=1}^{4E}(u^1_j\\bar{v}^2_j-v_j^1\\bar{u}_j^2)$ in $\\mathbb {C}^{4E}\\oplus \\mathbb {C}^{4E}$ .", "Let us fix the Lagrangian plane $\\Lambda \\subset \\mathbb {C}^{4E}\\oplus \\mathbb {C}^{4E}$ .", "A self-adjoint extension $\\widehat{H}$ is defined by the coupling conditions $\\xi \\in \\Lambda $ or equivalently $\\quad -i(I+U)u+(I-U)v=0,$ where $U$ is a unitary matrix defining $\\Lambda $ and $I$ is an identity matrix.", "We will consider only local coupling conditions, i,e.", "$\\Lambda =\\bigoplus \\limits _y{\\Lambda }_y$ , where ${\\Lambda }_y\\subset \\mathbb {C}^4$ is defined for each point $y$ separately.", "In Theorem 3.1 of the article [10] it is shown how scattering of a Gaussian packet at the point of gluing of the edge to the surface takes place.", "Further analysis of the number of packets on a decorated graph is as follows.", "Since wavefront propagation occurs along the geodesic, we can construct a matching metric graph for a given decorated graph.", "Vertices for this new graph are the points of gluing and edges are all geodesics on the surfaces connecting points of gluing and the old edges.", "We are interested in the number of old Gaussian packets, i.e.", "those whose support will lie on the edges of the original graph.", "The number of packets could change only in those moments of time that have the form of linear combinations of edge travel time.", "So the number of packets is equal to the number of sets $\\lbrace n_j\\rbrace $ satisfying some inequations of this kind: $n_1t_{l_1}+\\ldots +n_mt_{l_m}{\\le }T,$ where $t_j$ is a travel time of the $j$ -th edge of equivalent graph and all $t_j$ are linearly independent over $\\mathbb {Q}$ .", "In the case where an equivalent graph contains an infinite number of edges we obtain an infinite inequality.", "In the work [9] of V. E.Nazaikinskii the formula for $\\mathop {\\mathrm {ln}}\\nolimits \\mathop {\\hbox{}\\mathrm {N}}\\nolimits (E)$ i.e.", "entropy of a gas with full energy less than $E$ was obtained.", "$\\mathop {\\hbox{}\\mathrm {N}}\\nolimits (E)$ is defined via the number of solutions of an infinite linear inequality, where $E$ is on the right hand side of the inequality.", "Let us state this useful theorem.", "Theorem 1.1 Let $\\mathop {\\hbox{}\\mathrm {N}}\\nolimits (T)$ be the number of non-negative integer solutions of inequality $\\sum _{i=1}^\\infty \\lambda _i N_i \\le T$ and for sequence $\\lambda _j$ a counting function $\\rho (\\lambda ) = \\#\\lbrace j | \\lambda _j \\le \\lambda \\rbrace $ has asymptotics $\\rho (\\lambda ) = c_0 \\lambda ^{1+\\gamma } (1 + O(\\lambda ^{-\\varepsilon })), \\varepsilon > 0.$ Then $\\mathop {\\mathrm {ln}}\\nolimits \\mathop {\\hbox{}\\mathrm {N}}\\nolimits (T) =(\\gamma + 2) \\left( \\frac{c_0 \\Gamma (\\gamma + 2) \\zeta (\\gamma +2)}{(\\gamma + 1)^{\\gamma + 1}} \\right)^{\\frac{1}{\\gamma + 2}}T^{\\frac{\\gamma + 1}{\\gamma + 2}} (1+o(1))$ as $T$ goes to infinity.", "In next two sections we will consider two examples which show how to construct the asymptotics for $\\mathop {\\hbox{}\\mathrm {N}}\\nolimits (T)$ for a given decorated graph." ], [ "Decorated graph, obtained by gluing a segment to a cylinder", "Let us consider a circular cylinder with a length of a circle equaling $b$ .", "Points $A$ and $B$ lie on a ruling of the cylinder at the distance $a$ from each other.", "The wave front that begins to spread from the point $A$ will reach the point $A$ again in the time of the form $nb$ .", "Point of the front which is the first to return to the point $A$ will pass the cycle $e^{\\prime }$ with the passage time $t^{\\prime }=b$ .", "Similarly, we define a cycle $e^{\\prime \\prime }$ from $B$ to $B$ with the passage time $t^{\\prime \\prime }=b$ .", "Let us assume that wavefront propagates from the point $A$ reaches the point $B$ at time $\\sqrt{(kb)^2+a^2} (k \\ge 0)$ .", "One of points on wave front that reaches $B$ will passed way $e_k$ during the time $t_k =\\sqrt{(kb)^2+a^2}$ .", "Also, let $e^{\\prime \\prime \\prime }$ be a segment that is glued to the cylinder with the travel time $t^{\\prime \\prime \\prime }$ .", "We can compose an infinite graph from the edges obtained earlier (see Pic. 1).", "[draw, color = red!75!black, line width = 1pt] (-70pt,0pt) .. controls (-30pt,25pt) and (-10pt,25pt) .. ( 0pt,25pt) .. controls (10pt,25pt) and (30pt,25pt) .. ( 70pt,0pt) (-70pt,0pt) – ( 70pt,0pt) (-70pt,0pt) .. controls (-30pt,-25pt) and (-10pt,-25pt) .. ( 0pt,-25pt) .. controls (10pt,-25pt) and (30pt,-25pt) .. ( 70pt,0pt) (-70pt,0pt) .. controls (-30pt,-50pt) and (-10pt,-50pt) .. ( 0pt,-50pt) .. controls (10pt,-50pt) and (30pt,-50pt) .. ( 70pt,0pt) (0pt,-58pt) node [color = black] $\\vdots $ (-70pt,0pt) .. controls (-30pt,-70pt) and (-10pt,-70pt) .. ( 0pt,-70pt) .. controls (10pt,-70pt) and (30pt,-70pt) .. ( 70pt,0pt) (0pt,-76pt) node [color = black] $\\vdots $ (-70pt,0pt) .. controls (-115pt,45pt) and (-115pt,-45pt) .. (-70pt,0pt) ( 70pt,0pt) .. controls ( 115pt,45pt) and ( 115pt,-45pt) .. ( 70pt,0pt) ; [draw, color = blue!75!black, line width = 1pt] (0pt,25pt) node [anchor = south] $e^{\\prime \\prime \\prime }$ (0pt,0pt) node [anchor = south] $e_0$ (0pt,-25pt) node [anchor = south] $e_1$ (0pt,-50pt) node [anchor = south] $e_2$ (-70pt,5pt) node [anchor = south] $A$ ( 70pt,5pt) node [anchor = south] $B$ (-105pt,0pt) node [anchor = base east] $e^{\\prime }$ ( 105pt,0pt) node [anchor = base west] $e^{\\prime \\prime }$ ; [draw, fill = green!75!black, line width = 0pt] (-70pt,0pt) circle (2pt) ( 70pt,0pt) circle (2pt) ; Pic. 1.", "Equivalent graph.", "Theorem 2.1 For decorated graph obtained by attaching a segment to a flat cylinder, the following asymptotic estimate holds, as $T$ goes to infinity: $\\mathop {\\mathrm {ln}}\\nolimits \\mathop {\\hbox{}\\mathrm {N}}\\nolimits (T) \\le \\sqrt{\\frac{}{}}{2}{3b} \\ \\pi \\ T^{\\frac{1}{2}} (1+o(1))$ If times $\\lbrace t^{\\prime }\\rbrace \\cup \\lbrace t_i \\rbrace _{i=0}^\\infty $ are linearly independent over $\\mathbb {Q}$ then inequality turns into equality.", "For almost all real $a$ and $b$ times $\\lbrace t^{\\prime }\\rbrace \\cup \\lbrace t_i \\rbrace _{i=0}^\\infty $ are linearly independent over $\\mathbb {Q}$ .", "Proof We list all times at which $\\mathop {\\hbox{}\\mathrm {N}}\\nolimits (T)$ grows by one.", "It happens when: 1) A packet arrives at point $A$ .", "I.e.", "at time of a kind: $T = t^{\\prime } n^{\\prime } + t^{\\prime \\prime } n^{\\prime \\prime } + t^{\\prime \\prime \\prime } n^{\\prime \\prime \\prime } + \\sum _{i=0}^k t_i n_i$ for some $k$ , and non-negative integer $n^{\\prime }, n^{\\prime \\prime }, n_0, \\ldots n_k$ satisfy the following conditions:a) $\\sum _{i=0}^k n_i + n^{\\prime \\prime \\prime }$ is evenb) if $\\sum _{i=0}^k n_i + n^{\\prime \\prime \\prime } = 0$ , then $n^{\\prime \\prime } = 0$ ,c) if $\\sum _{i=0}^k n_i + n^{\\prime \\prime \\prime } > 0$ , then $n^{\\prime \\prime } \\ge 0$ .", "A set of paths $\\gamma _T$ from $A$ to $A$ corresponds to every moment of time $T$ .", "Function $\\mathop {\\hbox{}\\mathrm {N}}\\nolimits (T)$ does not increase at time $T$ if the following condition is fulfilled: there is a path in $\\gamma _T$ , by which we return to point $A$ and we arrive by $e^{\\prime \\prime \\prime }$ at final moment of time.", "Obviously, there is no such path if and only if $n^{\\prime \\prime \\prime }=0$ .", "Thus a new packet is born in point $A$ at times of a kind $t^{\\prime }n+\\sum _{i=0}^k t_i n_i$ , where $n_i\\ge 0$ , $n\\ge 0$ , $\\sum _{i=0}^k n_i$ is even.", "2) A packet arrives at point $B$ .", "Similarly, a new packet is born in point $B$ at times of a kind $t^{\\prime } n+\\sum _{i=0}^k t_i n_i$ , where $n_i\\ge 0$ , $n\\ge 0$ , and $\\sum _{i=0}^k n_i$ is odd.", "Thus $\\mathop {\\hbox{}\\mathrm {N}}\\nolimits (T)$ equals the number of times of a kind $t^{\\prime } n+\\sum _{i=0}^k t_in_i, n_i\\ge 0, n\\ge 0$ , and times less than $T$ .", "If $\\lbrace t^{\\prime }\\rbrace \\cup \\lbrace t_i\\rbrace _{i=0}^\\infty $ are linearly independent over $\\mathbb {Q}$ , then there exists one-to-one correspondence between such times and sets $(n,n_0,n_1,...)$ .", "Hence, $\\mathop {\\hbox{}\\mathrm {N}}\\nolimits (T)$ is equal to the number of solutions of inequality $t^{\\prime } n+\\sum _{i=0}^k t_i n_i \\le T$ .", "Let us use the theorem REF : $\\rho (\\lambda ) = [\\frac{1}{b} \\sqrt{\\lambda ^2 - a^2}]$ , thus $c_0=\\frac{1}{b}, \\gamma =0$ and we obtain $\\mathop {\\mathrm {ln}}\\nolimits \\mathop {\\hbox{}\\mathrm {N}}\\nolimits (T) = \\sqrt{\\frac{}{}}{2}{3b} \\pi T^{\\frac{1}{2}} (1+o(1)).$ If $\\lbrace t^{\\prime }\\rbrace \\cup \\lbrace t_i \\rbrace _{i=0}^\\infty $ are linearly dependent over $\\mathbb {Q}$ , then different sets $(n,n_0,..)$ correspond to one moment of time and then $\\mathop {\\mathrm {ln}}\\nolimits \\mathop {\\hbox{}\\mathrm {N}}\\nolimits (T) \\le \\sqrt{\\frac{}{}}{2}{3b} \\pi T^{\\frac{1}{2}} (1+o(1)).$ It remains only to explain because of why for almost all real $a$ and $b$ times $\\lbrace t^{\\prime }\\rbrace \\cup \\lbrace t_k \\rbrace _{k=0}^\\infty $ are linearly independent over $\\mathbb {Q}$ .", "Let us take $b=1$ without loss of generality.", "It is suffcient to prove that a set $TS=\\cup _{k=0}^\\infty \\lbrace t_k=\\sqrt{k^2+a^2} \\rbrace $ is linearly independent over $\\mathbb {Q}$ .", "Suppose that this is not true.", "This means that there is a finite list of rational numbers $\\alpha _j, j=1,\\ldots ,m$ such that finite linear combination of elements of $TS$ with coefficients $\\alpha _j$ equals zero.", "Hence the set of $a$ values such that $TS$ is $\\mathbb {Q}$ -linearly dependent is smaller than the set of finite sequences of rational numbers.", "Since the latter set is countable, so is the set of such $a$ .", "Hence the set $TS$ is linearly independent over $\\mathbb {Q}$ for almost all real $a$ .", "Remark.", "One can prove that times $t_k$ are $\\mathbb {Q}$ -linearly independent if $a$ is a transcendent number (for example one can take $a=\\pi $ ) in the following manner.", "Since $a$ is transcendent, hence $\\mathbb {Q}(a)$ is isomorphic to the field $\\mathbb {Q}(x)$ of rational functions over $\\mathbb {Q}$ .", "To show that the numbers $\\sqrt{n^2+a^2}$ are $\\mathbb {Q}$ -linearly independent is therefore equivalent to the statement that the functions $\\sqrt{n^2+x^2}$ are linearly independent over $\\mathbb {Q}$ .", "Suppose that there exists a nontrivial linear combination of such functions that equals zero: $\\sum \\limits _{j=1}^n\\alpha _j\\sqrt{j^2+x^2}=0$ .", "We will prove that all $\\alpha _j=0$ in the following way.", "We take one summand $\\alpha _j\\sqrt{j^2+x^2}$ and place it on the right hand side.", "In the neighborhood of the point $j$ on the complex plane the rest of the sum is holomorphic function, so $\\alpha _j$ should be zero.", "We can repeat this procedure for all $j$ ." ], [ "Decorated graph obtained by gluing segment to flat torus", "Let us take a flat torus with fundamental cycles of lengthes $a$ and $b$ .", "Let us consider a fundamental rectangle with sides $a, b$ and take points $A=(0,0)$ , $B=(c,d)$ in it.", "We glue a segment $e^{\\prime \\prime \\prime }$ with travel time $t^{\\prime \\prime \\prime }$ to points $A, B$ .", "[draw, color = red!75!black, line width = 1pt] (-80pt,0pt) ++(0pt, 0pt) – +(160pt,0pt) ++(0pt,-25pt) – +(160pt,0pt) ++(0pt,-25pt) – +(160pt,0pt) ++(0pt,-25pt) – +(160pt,0pt) (-75pt,5pt) ++(0pt, 0pt) – +(0pt,-85pt) ++(50pt, 0pt) – +(0pt,-85pt) ++(50pt, 0pt) – +(0pt,-85pt) ++(50pt, 0pt) – +(0pt,-85pt) ; [draw, color = blue!75!black, line width = 0pt, fill = green!75!black] (-25pt,0pt) circle (2pt) node [anchor = south east] $A$ ++(35pt,-20pt) circle (2pt) node [anchor = base west] $B$ ; [color = black, line width = .5pt, arrows = latex'-latex'] [draw] (-25pt,3pt) – +(50pt,0pt) ; [draw] (-28pt,0pt) – +(0pt,-25pt) ; [draw] (-25pt,-20pt) – +(35pt,0pt) ; [draw] (10pt,0pt) – +(0pt,-20pt) ; [draw, color = black] (-25pt,3pt) +(25pt,0pt) node [anchor = south] $b$ (-28pt,0pt) +(0pt,-12.5pt) node [anchor = east] $a$ (-13pt,-20pt) node [anchor = south] $c$ (10pt,-10pt) node [anchor = east] $d$ ; Pic. 2.", "Gluing of a segment to the flat torus.", "Wave packet begins its propagation from point $A$ and reaches point $A$ again at times of the following kind $\\lbrace \\sqrt{(na)^2+(mb)^2}| n\\ge 0,m\\ge 0, n^2+m^2\\ne 0 \\rbrace $ .", "Let us arrange them in ascending order: $t^{\\prime }_0,t^{\\prime }_1, \\ldots $ .", "One of the two front points reaches $A$ having pass the way $e^{\\prime }_k$ in time $t^{\\prime }_k$ (the other point passes this way in the other direction).", "In a similar way, we consider front propagating from point $B$ and we obtain times $\\lbrace t^{\\prime \\prime }_i\\rbrace _{i=0}^\\infty $ (where $t^{\\prime \\prime }_i=t^{\\prime }_i$ ) and paths $\\lbrace e^{\\prime \\prime }_i\\rbrace _{i=0}^\\infty $ .", "Front propagating from point $A$ reaches point $B$ in times of the following kind $\\lbrace \\sqrt{(c+na)^2+(d+mb)^2}| n,m\\in \\mathbb {Z} \\rbrace $ .", "Let us arrange them in ascending order: $t_0, t_1, \\ldots $ .", "Let front point reach $B$ in time $t_k$ having pass way $e_k$ .", "We can construct an infinite graph from paths obtained before (see Pic. 3).", "[draw, color = red!75!black, line width = 1pt] (-70pt,0pt) .. controls (-30pt,25pt) and (-10pt,25pt) .. ( 0pt,25pt) .. controls (10pt,25pt) and (30pt,25pt) .. ( 70pt,0pt) (-70pt,0pt) – ( 70pt,0pt) (-70pt,0pt) .. controls (-30pt,-25pt) and (-10pt,-25pt) .. ( 0pt,-25pt) .. controls (10pt,-25pt) and (30pt,-25pt) .. ( 70pt,0pt) (-70pt,0pt) .. controls (-30pt,-50pt) and (-10pt,-50pt) .. ( 0pt,-50pt) .. controls (10pt,-50pt) and (30pt,-50pt) .. ( 70pt,0pt) (0pt,-58pt) node [color = black] $\\vdots $ (-70pt,0pt) .. controls (-30pt,-70pt) and (-10pt,-70pt) .. ( 0pt,-70pt) .. controls (10pt,-70pt) and (30pt,-70pt) .. ( 70pt,0pt) (0pt,-76pt) node [color = black] $\\vdots $ (-70pt,0pt) .. controls (-100pt,25pt) and (-100pt,-25pt) .. (-70pt,0pt) (-70pt,0pt) .. controls (-120pt,45pt) and (-120pt,-45pt) .. (-70pt,0pt) (-70pt,0pt) .. controls (-150pt,75pt) and (-150pt,-75pt) .. (-70pt,0pt) ( 70pt,0pt) .. controls ( 100pt,25pt) and ( 100pt,-25pt) .. ( 70pt,0pt) ( 70pt,0pt) .. controls ( 120pt,45pt) and ( 120pt,-45pt) .. ( 70pt,0pt) ( 70pt,0pt) .. controls ( 150pt,75pt) and ( 150pt,-75pt) .. ( 70pt,0pt) ; [draw, color = blue!75!black, line width = 1pt] (0pt,25pt) node [anchor = south] $e^{\\prime \\prime \\prime }$ (0pt,0pt) node [anchor = south] $e_0$ (0pt,-25pt) node [anchor = south] $e_1$ (0pt,-50pt) node [anchor = south] $e_2$ (-70pt,5pt) node [anchor = south] $A$ ( 70pt,5pt) node [anchor = south] $B$ (-90pt,0pt) node [anchor = base east] $e_0^{\\prime }$ (-105pt,0pt) node [anchor = base east] $e_1^{\\prime }$ (-117pt,0pt) node [anchor = base east, color = black] ... ( 90pt,0pt) node [anchor = base west] $e_0^{\\prime \\prime }$ ( 105pt,0pt) node [anchor = base west] $e_1^{\\prime \\prime }$ ( 117pt,0pt) node [anchor = base west, color = black] ... ; [draw, fill = green!75!black, line width = 0pt] (-70pt,0pt) circle (2pt) ( 70pt,0pt) circle (2pt) ; Pic. 3.", "Equivalent graph for torus.", "Theorem 3.1 For a decorated graph obtained by attaching a segment to a flat 2-dimensional torus, the following asymptotic estimate holds, as $T$ goes to infinity: $\\mathop {\\mathrm {ln}}\\nolimits \\mathop {\\hbox{}\\mathrm {N}}\\nolimits (T) \\le 3 \\left(\\frac{5\\pi }{8ab} \\zeta (3)\\right)^{\\frac{1}{3}}T^\\frac{2}{3} (1+o(1))$ If $\\lbrace t^{\\prime }_i\\rbrace _{i=0}^\\infty \\cup \\lbrace t_i\\rbrace _{i=0}^\\infty $ are linearly independent over $\\mathbb {Q}$ , then inequality turns into equality.", "Proof.", "1) Packets reach point $A$ at times of the form $W =\\Big \\lbrace t^{\\prime \\prime \\prime } n^{\\prime \\prime \\prime } +\\sum _{i=0}n^{\\prime }_i t^{\\prime }_i +\\sum _{i=0} n_i t_i \\ \\Big |\\ n^{\\prime \\prime \\prime } + \\sum _{i=0}n_i \\hbox{ is even} \\Big \\rbrace .$ But it may happen that at such time another packet comes to the point $A$ by edge $e^{\\prime \\prime \\prime }$ All such moments are defined by condition $n^{\\prime \\prime \\prime } > 0$ , and $\\sum _{i=0}n_i $ is even.", "We exclude from $W$ all such times and obtain $W^{\\prime } =\\Big \\lbrace \\sum _{i=0}n^{\\prime }_i t^{\\prime }_i + \\sum _{i=0} n_i t_i \\ \\Big |\\ \\sum _{i=0}n_i \\hbox{ is even} \\Big \\rbrace .$ At each time from $W^{\\prime }$ a new packet starts from point $A$ by edge $e^{\\prime \\prime \\prime }$ .", "2) Similarly we obtain that at each time from $W^{\\prime \\prime } =\\Big \\lbrace \\sum _{i=0}n^{\\prime }_i t^{\\prime }_i + \\sum _{i=0} n_i t_i \\ \\Big |\\ \\sum _{i=0}n_i \\hbox{ is odd} \\Big \\rbrace $ a new packet starts from point $B$ by edge $e^{\\prime \\prime \\prime }$ .", "We join times from cases 1 and 2: $Q = W^{\\prime } \\cup W^{\\prime \\prime }$ $Q = \\Big \\lbrace \\sum _{i=0}n^{\\prime }_i t^{\\prime }_i + \\sum _{i=0} n_i t_i \\ \\Big \\rbrace $ Thus $\\mathop {\\hbox{}\\mathrm {N}}\\nolimits (T) = \\lbrace t\\in Q| t \\le T \\rbrace $ .", "If $\\lbrace t^{\\prime }_i\\rbrace _{i=0}^\\infty \\cup \\lbrace t_i\\rbrace _{i=0}^\\infty $ are linearly independent over $\\mathbb {Q}$ , then there is a bijection between $Q$ and sets $\\lbrace n_i\\rbrace _{i=0}^\\infty \\cup \\lbrace n^{\\prime }_i\\rbrace _{i=0}^\\infty $ .", "Thus $\\mathop {\\hbox{}\\mathrm {N}}\\nolimits (T)$ is equal to the number of inequality solutions $\\sum _{i=0}n^{\\prime }_i t^{\\prime }_i +\\sum _{i=0} n_i t_i \\le T$ .", "Counting function in our situation is given by the following formula $\\rho (\\lambda ) = \\# \\lbrace \\sqrt{(c+na)^2+(d+mb)^2} \\le \\lambda | n,m\\in \\mathbb {Z} \\rbrace +$ $\\# \\lbrace \\sqrt{(na)^2+(mb)^2} \\le \\lambda | n,m\\ge 0,n,m\\in \\mathbb {Z} \\rbrace = \\frac{5\\pi \\lambda ^2}{4ab} (1+O(\\lambda ^{-\\varepsilon }))$ We apply REF with $c_0 = \\frac{5\\pi }{4ab}, \\gamma =1$ and finish the proof.", "If $\\lbrace t^{\\prime }_i\\rbrace _{i=0}^\\infty \\cup \\lbrace t_i\\rbrace _{i=0}^\\infty $ are linearly dependent over $\\mathbb {Q}$ then different sets of integers $\\lbrace n_i\\rbrace _{i=0}^\\infty \\cup \\lbrace n^{\\prime }_i\\rbrace _{i=0}^\\infty $ can correspond to one time of packets birth.", "It means that Theorem REF gives an upper bound for $\\mathop {\\mathrm {ln}}\\nolimits \\mathop {\\hbox{}\\mathrm {N}}\\nolimits (T)$ .", "Let us consider a decorated graph obtained by gluing an edge $e^{\\prime \\prime \\prime }$ to a 3-dimensional flat torus with fundamental cycle lengths equal $a, b, c$ .", "Suppose that edge is glued at points $A=(0,0,0), B=(d,e,f)$ .", "In this case $\\lbrace t_i\\rbrace _{i=0}^\\infty = \\lbrace \\sqrt{(d+na)^2+(e+bm)^2+(f+cl)^2}|n,m,l \\in \\mathbb {Z} \\rbrace $ $\\lbrace t^{\\prime }_i\\rbrace _{i=0}^\\infty = \\lbrace \\sqrt{(na)^2+(bm)^2+(cl)^2}|n,m,l \\ge 0,n,m,l \\in \\mathbb {Z} \\rbrace .$ Similarly to the previous theorem we can obtain Theorem 3.2 For a decorated graph obtained by attaching a segment to a flat 3-dimensional torus, the following asymptotic estimate holds, as $T$ goes to infinity: $\\mathop {\\mathrm {ln}}\\nolimits \\mathop {\\hbox{}\\mathrm {N}}\\nolimits (T) \\le 4 \\left(\\frac{\\pi }{3abc} \\zeta (4)\\right)^{\\frac{1}{4}} T^\\frac{3}{4} (1+o(1))$ If $\\lbrace t^{\\prime }_i\\rbrace _{i=0}^\\infty \\cup \\lbrace t_i\\rbrace _{i=0}^\\infty $ are linearly independent over $\\mathbb {Q}$ then inequality turns into equality." ], [ "Uniformly secure manifolds", "Definition.", "A Riemannian manifold is said to be uniformly secure if there is a finite number $s$ such that all geodesics connecting an arbitrary pair of points in the manifold can be blocked by $s$ point obstacles.", "Let us remind a theorem that describes uniformly secure manifolds.", "Theorem 4.1 (K. Burns, E. Gutkin).", "Let $M$ be a compact Riemannian manifold that is uniformly secure.", "Then the topological entropy of the geodesic flow for $M$ is zero, and the fundamental group of $M$ is virtually nilpotent.", "If, in addition, $M$ has no conjugate points, then $M$ is flat.", "We prove a theorem for decorated graphs constructed from uniformly secure manifolds.", "Theorem 4.2 Let a segment with the travel time $L$ be glued at two points $A$ and $B$ on the surface $M$ .", "Suppose that for geodesics connecting $A$ with $A$ , $B$ with $B$ , $A$ with $B$ the following condition holds: the number $g(\\lambda )$ of geodesics whose length is equal or less than $\\lambda $ equals $g(\\lambda ) = c_0\\lambda ^{1+ \\gamma }(1 + O(\\lambda ^{-\\varepsilon })), \\varepsilon > 0.$ Then $ln N(T) \\le (\\gamma + 2) \\left(\\frac{3 c_0 \\Gamma (\\gamma + 2)\\zeta (\\gamma + 2)}{(\\gamma + 1)^{\\gamma + 1}}\\right)^{\\frac{1}{\\gamma + 2}}T^{\\frac{\\gamma + 1}{\\gamma + 2}}(1 + o (1)).$ Proof We denote by $t^{\\prime }_i (i \\ge 0)$ the length (i.e.", "propagation time) of geodesics joining $A$ and $A$ , by $t_i (i \\ge 0)$ the length of the geodesic connecting $A$ and $B$ , by $t^{\\prime \\prime }_i (i \\ge 0)$ the length of the geodesic connecting $B$ and $B$ .", "Let the packet came from point A.", "$N (T)$ can be increased only in the time of the form (each of these times corresponds to a path from $A$ to $A$ or $A$ to $B$ ) $T = \\left\\lbrace \\sum t^{\\prime }_i n^{\\prime }_i + \\sum t^{\\prime \\prime } _ i n^{\\prime \\prime } _ i + \\sum t_i n_i + Ln^{\\prime \\prime \\prime } \\right\\rbrace $ where $n^{\\prime \\prime \\prime }, n_i, n^{\\prime }_i, n^{\\prime \\prime }_ i$ are non-negative integers such that if $\\forall i: n_i = 0$ and $ n^{\\prime \\prime \\prime }= 0$ , then $\\forall i: n^{\\prime \\prime }_ i = 0$ .", "Let $T_A$ be the times in which packets arrive at the point $A$ .", "Elements of $T_A$ are characterized by the condition: $n^{\\prime \\prime } + \\sum n_i$ is even.", "Similarly, let $T_B$ be times in which packets arrive at the point $B$ .", "Elements of $T_B$ characterized by the condition: $n ^{\\prime \\prime } + \\sum n_i$ is odd.", "Then $T = T_A \\cup T_B$ .", "Now we decompose $T_A = T_ {A_1} \\cup T_ {A_2}$ , where $T_ {A_1}$ are times satisfying $n^{\\prime \\prime \\prime }\\ne 0$ , and $T_ {A_2}$ are times satisfying condition $n ^{\\prime \\prime \\prime }= 0$ .", "Similarly, we decompose $T_B = T_ {B_1} \\cup T_ {B_2}$ , where $T_ {B_1}$ are times satisfying $n^{\\prime \\prime \\prime }\\ne 0$ , and $T_ {B_2}$ are times satisfying condition $n^{\\prime \\prime \\prime }= 0$ .", "Note that the number of packets on the edge $e^{\\prime \\prime \\prime }$ will not increase in the time belongs to $T_ {A_1}$ or $T_ {B_1}$ because always there exists packets that come from the edge $e^{\\prime \\prime \\prime }$ at this time.", "Therefore, $N (T)$ does not exceed the number of solutions of the inequality $\\sum t^{\\prime }_i n^{\\prime }_i + \\sum t^{\\prime \\prime } _ i n^{\\prime \\prime } _ i + \\sum t_i n_i \\le T$ .", "Analogically, $N(T)$ does not exceed the number of solutions of the inequality $\\sum s_i n_i \\le T$ .", "Consequence 4.1 Let a segment with the travel time $L$ be glued at two points $A$ and $B$ on the surface $M$ .", "Suppose that $M$ is uniformly secure.", "Then $\\ln N(T) \\le (q + 2) \\left( \\frac{3C_g \\Gamma (q + 2) \\zeta (q +2)}{(q + 1)^{q + 1}} \\right)^{\\frac{1}{q + 2}}T^{\\frac{q + 1}{q + 2}} (1+o(1)),$ where $C_g$ and $q$ depends only on $M$ .", "Proof We use the following theorem (see [17]): Lemma 4.1 (K. Burns, E. Gutkin).", "Let $M$ be a compact Riemannian manifold.", "If $M$ is uniformly secure, then there are positive constants $C$ and $q$ such that for any pair $x, y \\in M$ we have $CF_{(x, y)}(T){\\le }C_gT^q$ .", "So the number of geodesics grows polynomially.", "Then we apply our previous theorem and get the result." ], [ "Further questions", "We have obtained estimates for $\\mathop {\\mathrm {ln}}\\nolimits \\mathop {\\hbox{}\\mathrm {N}}\\nolimits (T)$ for the situation where the number of geodesics grows polynomially.", "Such surfaces are not only the cylinder and flat tori.", "For all manifolds with polynomial growth it is possible to carry out similar calculations and obtain an upper bound for the number of Gaussian packets.", "But, as already mentioned above, for a large class of Riemannian manifolds the number $CF_T(x,y)$ grows as $e^{hT}$ .", "In this case Theorem REF is not applicable and the question of the asymptotic behavior of $\\mathop {\\hbox{}\\mathrm {N}}\\nolimits (T)$ remains open.", "Moreover, the arithmetic properties of geodesic lengths affect the accuracy of the estimate.", "Therefore, it would be interesting to consider results related to the linear independence over $\\mathbb {Q}$ of the lengths of geodesics joining two given points on a surface." ], [ "Acknowledgments", "The authors are grateful to A. I. Shafarevich, N. S. Gusev and O. V. Sobolev, V. E. Nazaikinskii and N. G. Moschevitin, for useful discussions and attention to their work.", "The work was supported by the grant “The National Research University Higher School of Economics' Academic Fund Program in 2013-2014, research grant No.12-01-0164”.", "Authors thank Joseph H. Silverman and Jan-Christoph Schlage-Puchta for their very useful comments." ] ]
1403.0263
[ [ "A Stochastic Geometry Analysis of Inter-cell Interference Coordination\n and Intra-cell Diversity" ], [ "Abstract Inter-cell interference coordination (ICIC) and intra-cell diversity (ICD) play important roles in improving cellular downlink coverage.", "Modeling cellular base stations (BSs) as a homogeneous Poisson point process (PPP), this paper provides explicit finite-integral expressions for the coverage probability with ICIC and ICD, taking into account the temporal/spectral correlation of the signal and interference.", "In addition, we show that in the high-reliability regime, where the user outage probability goes to zero, ICIC and ICD affect the network coverage in drastically different ways: ICD can provide order gain while ICIC only offers linear gain.", "In the high-spectral efficiency regime where the SIR threshold goes to infinity, the order difference in the coverage probability does not exist, however the linear difference makes ICIC a better scheme than ICD for realistic path loss exponents.", "Consequently, depending on the SIR requirements, different combinations of ICIC and ICD optimize the coverage probability." ], [ "Motivation and Main Contributions", "Recently, the Poisson point process (PPP) has been shown to be a tractable and realistic model of cellular networks [2].", "However, the baseline PPP model predicts the coverage probability of the typical user to be less than 60% if the signal-to-interference-plus-noise ratio (SINR) is set to 0 dB—even if noise is neglected.", "This is clearly insufficient to provide reasonable user experiences in the network.", "To improve the user experiences, in cellular systems, the importance of inter-cell interference coordination (ICIC) and intra-cell diversity (ICD) have long been recognized[3], [4].", "Yet, so far, most of the PPP-based cellular analyses lack a careful treatment of these two important aspects of the network, partly due to the lack of a well-established approach to deal with the resulting temporal or spectral correlation [5].", "Modeling the cellular network as a homogeneous PPP, this paper explicitly takes into account the temporal/spectral correlation and analyzes the benefits of ICIC and ICD in cellular downlink under idealized assumptions.", "Consider the case where a user is always served by the BS that provides the strongest signal averaged over small-scale fading but not shadowingWithout shadowing, this is the nearest BS association policy as used, for example, in [2].. For ICD, we consider the case where the serving BS always transmit to the user in $M$ resource blocks (RBs) simultaneously and the user always decodes from the RB with the best SIR (selection combining).", "For ICIC, we assume under $K$ -BS coordination, the RBs that the user is assigned are silenced at the next $K-1$ strongest BSs.", "Note that both of the schemes create extra load (reserved RBs) in the network: ICIC at the adjacent cells and ICD at the serving cell.", "Therefore, it is important to quantify the benefits of ICIC and ICD in order to design efficient systems.", "The main contribution of this paper is to provide explicit expressions for the coverage probability with $K$ -BS coordination and $M$ -RB selection combining.", "Notably, we show that, in the high-reliability regime, where the outage probability goes to zero, the coverage gains due to ICIC and ICD are qualitatively different: ICD provides order gain while ICIC only offers linear gain.", "In contrast, in the high-spectral efficiency regime, where the SIR threshold goes to infinity, such order difference does not exist and ICIC usually offers larger (linear) gain than ICD in terms of coverage probability.", "The techniques presented in this paper have the potential to lead to a better understanding of the performance of more complex cooperation schemes in wireless networks, which inevitably involve temporal or spectral correlation." ], [ "ICIC, ICD and Related Works", "Generally speaking, inter-cell interference coordination (ICIC) assigns different time/frequency/spatial dimensions to users from different cells and thus reduces the inter-cell interference.", "Conventional ICIC schemes are mostly based on the idea of frequency reuse.", "The resource allocation under cell-centric ICIC is designed offline and does not depend on the user deployment.", "While such schemes are advantageous due to their simplicity and small signaling overhead, they are clearly suboptimal since the pre-designed frequency reuse pattern cannot cope well with the dynamics of user distribution and channel variation.", "Therefore, there have been significant efforts in facilitating ICIC schemes, where the interference coordination (channel assignment) is based on real user locations and channel conditions and enabled by multi-cell coordination.", "Different user-centric (coordination-based) ICIC schemes in OFDMA-based networks are well summarized in the recent survey papers [6], [7], [8].", "Conventionally, most of the performance analyses of ICIC are based on network-level simulation, and the hexagonal-grid model is frequently used [7].", "Since real cellular deployments are subject to many practical constraints, recently more and more analyses are based on randomly distributed BSs, mostly using the PPP as the model.", "These stochastic geometry-based models not only provide alternatives to the classic grid models but also come with extra mathematical tractability [2], [9], [10].", "In terms of the treatment of ICIC, the most relevant papers to this one are [11], [12], [13], where the authors analyzed partial frequency reuse schemes using independent thinning.", "The authors in [14], [15] considered BS coordination based on clusters grouped by tessellations.", "Different from these papers, this paper focuses on user-centric ICIC schemes where the spatial correlation of the coordinated cells is explicitly accounted for.", "It is worth noting that ICIC is closely related to multi-cell processing (MCP) and coordinated multipoint (CoMP) transmission, see [16], [17], [14], [15] and the references therein.", "MCP/CoMP emphasizes the multi-antenna aspects of the cell coordination, while the form of ICIC considered in this paper does not take into account the use of MIMO (joint transmission) techniques and thus is not subject to the considerable signaling and processing overheads of typical MCP/CoMP schemes, which include symbol-level synchronization and joint precoder design [8].", "Thus it can be considered as a simple form of MCP/CoMP that is light on overhead.", "Intra-cell diversity (ICD) describes the diversity gain achieved by having the serving BS opportunistically assigns users with their best channels.", "In cellular systems, diversity exists in space, time, frequency and among users[4].", "It is well acknowledged that diversity can significantly boost the network coverage.", "However, conventional analyses of diversity usually do not include the treatment of interference, e.g., [18], [19].", "In order to analytically characterize diversity in wireless networks with interference, a careful treatment of interference correlation is necessary, otherwise the results may be misleading.", "Therefore, there have been a few recent efforts in understanding this correlation[20], [21], [22], [23], [24], [25].", "Notably, [20] shows that in an ad hoc type network, simple retransmission schemes do not result in diversity gain if interference correlation is consideredDifferent from conventional SNR-based diversity analysis, [20] calculates the diversity gain by considering the case where signal to interference ratio (SIR) goes to infinity, which is an analog of the classic (interference-less) notional of diversity.", "This paper follows the same analogy.. Analyzing the intra-cell diversity (ICD) under interference correlation, this paper shows that a diversity gain can be obtained in a cellular setting where the receiver is always connected to the strongest BS, in sharp contrast with the conclusion drawn from ad hoc type networks in [20]." ], [ "Paper Organization", "The rest of the paper is organized as follows: Section  presents the system model and discusses the comparability of ICIC and ICD.", "Sections  and  derive the coverage probability for the case with ICIC or ICD only, respectively, and provide results on the asymptotic behavior of the coverage probability in the high-reliability as well as high-spectral efficiency regimes.", "The case with both ICIC and ICD is analyzed in Section .", "We validate our model and discuss fundamental trade-offs between ICIC and ICD in Section .", "The paper is concluded in Section ." ], [ "System Model", "Considering the typical user at the origin $o$ , we use a homogeneous Poisson point process (PPP) $\\Phi \\subset \\mathbb {R}^2$ with intensity $\\lambda $ to model the locations of BSs on the plane.", "To each element of the ground process $x\\in \\Phi $ , we add independent marksFor analytical tractability, the spatial shadowing correlation due to common obstacles is not considered in this model.", "$S_x\\in \\mathbb {R}^+$ and $h^m_x\\in \\mathbb {R}^+$ , where $m\\in [M]$ and $M\\in \\mathbb {N}$ ,We use $[n]$ , to denote the set $\\lbrace 1,2,\\cdots ,n\\rbrace $ .", "to denote the (large-scale) shadowing and (power) fading effect on the link from $x$ to $o$ at the $m$ -th resource block (RB), and the combined (marked) PPP is denoted as $\\hat{\\Phi }= \\lbrace (x_i, S_{x_i}, (h^m_{x_i})_{m=1}^M)\\rbrace $ .", "In particular, under power law path loss, the received power at the typical user $o$ at the $m$ -th RB from a BS at $x\\in \\Phi $ is $P_x = S_x h^m_x \\Vert x\\Vert ^{-\\alpha },$ where $\\alpha $ is the path loss exponent.", "In this paper, we focus on Rayleigh fading, i.e., $h_x$ is exponentially distributed with unit mean but allow the shadowing distribution to be (almost) arbitrary.", "Fig.", "REF shows a realization of a PPP-modeled cellular network under $K$ -BS coordination with lognormal shadowing.", "Due to the shadowing effect, the $K$ strongest BSs under coordination are not necessarily the $K$ nearest BSs.", "Figure: A realization of the cellular network modeled by a homogeneous PPP Φ\\Phi .The network is under KK-BS (K=5K=5) coordination with lognormal shadowing.The typical user is denoted by ∘\\circ ,the BSs by ×\\times , the serving BS by ⋄\\Diamond and the coordinatednon-serving BS by □\\Box .The base station locations (ground process $\\Phi $ ) and the shadowing random variables $S_x$ are static over time and frequency (i.e., over all RBs), which is the main reason of the spectral/temporal correlation of signal and interference.", "In comparison, the (small-scale) fading $h_x^m$ is iid over RBs.", "Both $S_x$ and $h_x^m$ are iid over space (over $x$ ).", "The user is assumed to be associated with the strongest (without fading) BS and is called covered (without ICIC) at the $m$ -th RB iff $\\textnormal {SIR}_m = \\frac{S_{x_0} h^m_{x_0} \\Vert x_0\\Vert ^{-\\alpha }}{\\sum _{y\\in \\Phi \\setminus \\lbrace x_0\\rbrace } S_y h^m_y \\Vert y\\Vert ^{-\\alpha }} > \\theta ,$ where $x_0 = \\operatornamewithlimits{arg\\,max}_{x\\in \\Phi } S_x \\Vert x\\Vert ^{-\\alpha }$ is the serving BS." ], [ "The Path Loss Process with Shadowing (PLPS)", "Definition 1 (The path loss process with shadowing) The path loss process with shadowing (PLPS) $\\Xi $ is the point process on $\\mathbb {R}^+$ mapped from $\\hat{\\Phi }$ , where $\\Xi = \\lbrace \\xi _i = \\frac{\\Vert x\\Vert ^\\alpha }{S_x}, x\\in \\Phi \\rbrace $ and the indices $i\\in \\mathbb {N}$ are introduced such that $\\xi _k<\\xi _j$ for all $k<j$ .", "Note that the PLPS is an ordered process.", "It captures the effect of shadowing and spatial node distribution of the network at the same time, and consequently, determines the BS association.", "Lemma 1 The PLPS $\\Xi $ is a one-dimensional PPP with intensity measure $\\Lambda ((0,r])=\\lambda \\pi r^\\delta \\mathbb {E}[S^\\delta ]$ , where $\\delta =2/\\alpha $ , $S\\stackrel{\\textnormal {d}}{=}S_x$ and $\\stackrel{\\textnormal {d}}{=}$ means equality in distribution.", "The proof of Lemma REF is analogous to that of [26] and is omitted from the paper.", "The intensity measure of the PLPS demonstrates the necessity of the $\\delta $ -th moment constraint on the shadowing random variable $S_x$ .", "Without this constraint, the aggregate received power (with or without fading) is unbounded almost surely." ], [ "The Coverage Probability and Effective Load Model", "Similar to the construction of $\\hat{\\Phi }$ , We construct a marked PLPS $\\hat{\\Xi }= \\lbrace (\\xi _i, (h^m_{\\xi _i})_{m=1}^M,\\chi _{\\xi _i})\\rbrace $ , where we put two marks on each element of the PLPS $\\Xi $ : $h^m_\\xi = h^m_x,\\; m\\in [M],\\; x\\in \\Phi $ , are the iid fading random variables directly mapped from $\\hat{\\Phi }$ ; $\\chi _{\\xi }\\in \\lbrace 0,1\\rbrace $ indicates whether a BS represented by $\\xi $ is transmitting at the RB(s) assigned to the typical userIt is assumed that the RBs are grouped into chunks of size $M$ , i.e., each BS either transmits at all the $M$ RBs or does not transmit at any of these RBs..", "In the case where no ambiguity is introduced, we will use $h^m_i$ as an abbreviation for $h^m_{\\xi _i}$ and $\\chi _i$ as a short of $\\chi _{\\xi _i}$ .", "For example, if no ICIC is considered, we have $\\chi _i=1,\\;\\forall i$ ,We assume all the BSs are fully loaded, i.e., each RB is either used in downlink transmission or silenced due to coordination.", "and the coverage condition in (REF ) can be written in terms of the marked PLPS as $\\textnormal {SIR}_m = \\frac{h^m_1 \\xi _1^{-1}}{\\sum _{i=2}^\\infty h^m_i \\xi _i^{-1}} >\\theta .$ With ICIC, the value of $\\chi _i$ is determined by the scheduling policy.", "Given $\\chi _i$ , the coverage condition at the $m$ -th RB under $K$ -BS coordination can be expressed in terms of the marked PLPS as $\\textnormal {SIR}_{K,m} = \\frac{h^m_1 \\xi _1^{-1}}{\\sum _{i=2}^\\infty \\chi _i h^m_i \\xi _i^{-1}} >\\theta .$ By $K$ -BS coordination (ICIC), we assume the $K-1$ strongest non-serving BSs of the typical user do not transmit at the RBs to which the user is assignedThis can be implemented by letting the UE to identify the $K$ strongest BSs and then reserve the RBs at all of them..", "Thus, we have $\\chi _i = 0,\\;\\forall i\\in [K]\\setminus \\lbrace 1\\rbrace $ .By default $\\chi _1 = 1$ .", "For $i>K$ , the exact value of $\\chi _i$ is hard to model since the BSs can either transmit to its own users in the RB(s) assigned to the typical user or reserve these RB(s) for users in nearby cells, and the muted BSs can effectively “coordinate\" with multiple serving BSs at the same time.", "Therefore, the resulting density of the active BSs outside the $K$ coordinating BSs is a complex function of the user distribution, (joint) scheduling algorithms and shadowing distribution.", "In order to maintain tractability, we assume $\\chi _i, \\; i>K$ are iid Bernoulli random variables with (transmitting) probability $1/\\kappa $ .", "Such modeling is justified by the random distribution of the users and the shadowing effect [27].", "Here, $\\kappa \\in [1,K]$ is called the effective load of ICIC.", "$\\kappa = K$ implies all the coordinating BS clusters do not overlap while $\\kappa = 1$ represents the scenario where all the users assigned to the same RB(s) in the network share the same $K-1$ muted BSs.", "The actual value of $\\kappa $ lies between these two extremesThe statement is true under the full-load assumption.", "In the case where some cells may contain no users, it is possible that $\\kappa >1$ while $K=1$ .", "But this does not have a large influence on the accuracy of the analyses as is shown in detail in Section .", "and is determined by the scheduling procedure which this paper does not explicitly study.", "However, we assume that $\\kappa $ is known.", "The accuracy of this model will be validated in Section .", "Let ${\\bf S}_{K,m}\\triangleq \\lbrace \\textnormal {SIR}_{K,m}>\\theta \\rbrace $ be the event of coverage at the $m$ -th RB.", "We consider the coverage probability with inter-cell interference coordination (ICIC) and intra-cell diversity (ICD) formally defined as follows.", "Definition 2 The coverage probability with $K$ -BS coordination and $M$ -RB selection combining is $\\mathsf {P}^{\\textnormal {c}}_{K,M} = \\mathsf {P}^{\\cup \\textnormal {c}}_{K,M} \\triangleq \\mathbb {P}(\\cup _{m=1}^M {\\bf S}_{K,m}).$ In other words, the typical user is covered iff the received SIR at any of the $M$ RBs is greater than $\\theta $ .", "The superscript c denotes coverage and $\\cup $ stresses that $\\mathsf {P}^{\\cup \\textnormal {c}}_{K,M}$ is the probability of being covered in at least one of the M RBs.", "(If there is no possibility of confusion, we will use $\\mathsf {P}^{\\cup \\textnormal {c}}_{K,M}$ and $\\mathsf {P}^{\\textnormal {c}}_{K,M}$ interchangeably.)" ], [ "System Load and Comparability", "In the baseline case without ICIC and ICD, each user occupies a single RB at the serving BS.", "With (only) $M$ -RB selection combining, each user occupies $M$ RBs at the serving BS.", "Thus, the system load is increased by a factor of $M$ .", "The load effect of ICIC can be described by the effective load $\\kappa $ since, as discussed above, in a network with $K$ -BS coordination there are $1/\\kappa $ of the BSs actively serving the users in a single RB whereas each BS serves one user in every RB in the baseline case, i.e., the load is increased by a factor of $\\kappa $ due to ICIC.", "The fundamental comparability of ICIC and ICD comes from the similarity in introducing extra load in the system and will be explored in more detail in Section ." ], [ "Intercell Interference Coordination (ICIC)", "This section focuses on the effect of ICIC on the coverage probability.", "Since no ICD is considered, we will omit the superscript $m$ on the fading random variable $h^m_\\xi ,\\;\\xi \\in \\Xi $ , for simplicity." ], [ "Integral Form of Coverage Probability", "Our analysis will be relying on a statistical property of the marked PLPS $\\hat{\\Xi }$ stated in the following lemma.", "Lemma 2 For $\\hat{\\Xi }= \\lbrace (\\xi _i, h_i, \\chi _i)\\rbrace $ , let $X_k = \\xi _1/\\xi _k$ and $Y_k = \\xi _k^{-1}/I_k$ , where $I_k \\triangleq \\sum _{i=k+1}^{\\infty } \\chi _i h_i \\xi _i^{-1}$ .", "For all $k\\in \\mathbb {N}$ , the two random variables $X_k$ and $Y_k$ are independent.", "If $k=1$ , the lemma is trivially true, since $X_1\\equiv 1$ while $Y_1$ has some non-degenerate distribution.", "For $k\\ge 2$ , $x_1\\in [0,1]$ and $x_2\\in \\mathbb {R}^+$ , the joint ccdf of $\\xi _1/\\xi _k$ and $\\xi _k/I_k$ can be expressed as $\\hspace{20.0pt}&\\hspace{-20.0pt}\\mathbb {P}(X_k > x_1, Y_1 > x_2) \\\\&= \\mathbb {E}_{\\xi _k}\\left[\\mathbb {P}\\left(\\frac{\\xi _1}{\\xi _k} > x_1, \\frac{\\xi _k^{-1}}{I_k} > x_2\\right) \\mid \\xi _k\\right] \\\\&\\stackrel{\\text{\\scriptsize (a)}}{=} \\mathbb {E}_{\\xi _k}\\left[\\mathbb {P}\\left(\\frac{\\xi _1}{\\xi _k }> x_1\\right)\\mathbb {P}\\left(\\frac{\\xi _k^{-1}}{I_k} > x_2\\right) \\mid \\xi _k\\right] \\\\&\\stackrel{\\text{\\scriptsize (b)}}{=} \\mathbb {P}\\left(\\frac{\\xi _1}{\\xi _k} > x_1\\right) \\mathbb {E}_{\\xi _k}\\left[\\mathbb {P}\\left(\\frac{\\xi _k^{-1}}{I_k} > x_2\\right) \\mid \\xi _k\\right] \\\\&= \\mathbb {P}\\left(\\frac{\\xi _1}{\\xi _k} > x_1\\right)\\mathbb {P}\\left(\\frac{\\xi _k^{-1}}{I_k} > x_2\\right),$ where (a) is due to the fact that $\\lbrace \\xi _i, i < k\\rbrace $ and $\\lbrace \\xi _i, i > k\\rbrace $ are conditionally independent given $\\xi _k$ by the Poisson property and $\\lbrace h_i\\rbrace $ , $\\lbrace \\chi _i\\rbrace $ are iid and independent from $\\Xi $ .", "(b) holds since, conditioning on $\\xi _k$ implies that there are $k-1$ points on $[0,\\xi _k)$ .", "Thus, thanks to the Poisson property, it can be shown that given $\\xi _k$ , $\\xi _1/\\xi _k$ follows the same distribution as that of the minimum of $k-1$ iid random variables with cdf $\\min \\lbrace x^\\delta ,1\\rbrace \\mathsf {1}_{\\mathbb {R}^+}(x)$ .In fact, for general inhomogeneous PPP on $\\mathbb {R}^+$ of intensity measure $\\Lambda (\\cdot )$ , given there are $N$ points on $[0,x_0)$ the joint distribution of the locations of the $N$ points is the same as that of $N$ iid random variables with cdf $\\Lambda ([0,x))/\\Lambda ([0,x_0))$ [10].", "Since the resulting conditional distribution of $\\xi _1/\\xi _k$ does not depend on $\\xi _k$ , this distribution is also the marginal distribution of $\\xi _1/\\xi _k$ as is stated in the lemma.", "Furthermore, due to Lemma REF , it is straightforward to obtain the ccdf of $\\xi _1/\\xi _k,\\;\\forall k\\ge 2$ , which is formalized in the following lemma.", "Lemma 3 For all $k\\in \\mathbb {N}\\setminus \\lbrace 1\\rbrace $ , The ccdf of $\\xi _1/\\xi _k$ is $\\mathbb {P}\\left(\\frac{\\xi _1}{\\xi _k} > x\\right) = (1-x^{\\delta })^{k-1}, \\; x\\in [0, 1].$ As discussed in the proof of Lemma REF , by the Poisson property and the intensity measure of $\\Xi $ given in Lemma REF , conditioned on $\\xi _k, k > 1$ , $\\xi _i/\\xi _k\\stackrel{\\textnormal {d}}{=}X_{i:k-1}$ , where $X_{i:k-1}$ denotes the $i$ -th order statistics of $k-1$ iid random variables with cdf $\\mathbb {P}(X<x) = \\min \\lbrace x^\\delta ,1\\rbrace \\mathsf {1}_{\\mathbb {R}^+}(x)$ .", "Thus, $\\mathbb {P}(\\frac{\\xi _1}{\\xi _k} > x)$ is the probability that all the $k-1$ iid random variables are larger than $x$ .", "Lemma 4 For $\\hat{\\Xi }= \\lbrace (\\xi _i, h_i)\\rbrace $ , let $I_\\rho = \\sum _{\\xi \\in \\Xi \\cap (\\rho ,\\infty )} \\chi _\\xi h_{\\xi } \\xi ^{-1}$ for $\\rho >0$ .", "The Laplace transform of $\\rho I_\\rho $ is $\\mathcal {L}_{\\rho I_\\rho } (s) = \\exp \\left( -\\frac{\\lambda }{\\kappa }\\pi \\mathbb {E}[S^\\delta ] C(s) \\rho ^\\delta \\right),$ where $C(s) = \\frac{s\\delta }{1-\\delta } {_2 F_1} (1,1-\\delta ;2-\\delta ;-s) $ and $_2 F_1 (a,b;c;z)$ is the Gauss hypergeometric function.", "First, we can calculate the Laplace transform of $I_\\rho $ using the probability generating functional (PGFL) of PPP [10], i.e., $\\mathcal {L}_{I_\\rho } (s) &= \\mathbb {E}[\\exp (-s\\sum _{\\xi \\in \\Xi \\cap (\\rho ,\\infty )} \\chi _\\xi h_\\xi \\xi ^{-1})] \\\\&= \\mathbb {E}_{\\Xi }\\prod _{\\xi \\in \\Xi \\cap (\\rho ,\\infty )} \\mathbb {E}_{h,\\chi }[\\exp (-s\\chi h\\xi ^{-1})] \\\\&= \\exp \\left( - \\mathbb {E}_{h,\\chi } \\left[\\int _\\rho ^\\infty (1-e^{-s\\chi h/x}) \\Lambda (\\textnormal {d}x)\\right] \\right),$ where $\\chi $ is a Bernoulli random variable with mean $\\frac{1}{\\kappa }$ , $\\Lambda (\\cdot )$ is the intensity measure of $\\Xi $ and by Lemma REF , $\\Lambda (\\textnormal {d}x)=\\lambda \\pi \\mathbb {E}[S^\\delta ]\\delta x^{\\delta -1} \\textnormal {d}x$ .", "Then, straightforward algebraic manipulation yields $\\mathcal {L}_{I_\\rho } (s) =\\\\ \\exp \\left( - \\frac{\\lambda }{\\kappa } \\pi \\mathbb {E}[S^\\delta ]\\mathbb {E}_h\\left[ (sh)^\\delta \\gamma (1-\\delta ,\\frac{sh}{\\rho })- \\rho ^\\delta (1-e^{-\\frac{sh}{\\rho }}) \\right] \\right).$ Since, for an arbitrary random variable $X$ and constant $u$ , $\\mathcal {L}_{uX}(s) \\equiv \\mathcal {L}_X(us)$ , we have $\\mathcal {L}_{\\rho I_\\rho } (s) = \\mathcal {L}_{I_\\rho } ({s\\rho }) = \\exp \\left( -\\frac{\\lambda }{\\kappa }\\pi \\mathbb {E}[S^\\delta ] C(s) \\rho ^\\delta \\right),$ where $C(s) = \\mathbb {E}_h[(sh)^\\delta \\gamma (1-\\delta ,sh)+e^{-sh}-1]$ .", "The proof is completed by considering the exponential distribution of $h$ .", "Here, $I_\\rho $ can be understood as the interference from BSs having a (non-fading) received power weaker than $\\rho ^{-1}$ .", "In the case without shadowing, i.e., $S_x\\equiv 1$ , it can also be understood as the interference coming from outside a disk centered at the typical user with radius $\\rho ^{\\frac{1}{\\alpha }}$ .", "Lemma 5 The Laplace transform of $\\xi _k I_k$ is $\\mathcal {L}_{\\xi _k I_k}(s) = \\frac{1}{\\left(C_\\kappa (s,1)\\right)^k},$ where $C_\\kappa (s,m) = \\frac{\\kappa -1}{\\kappa }+\\frac{1}{\\kappa } {_2F_1}(m,-\\delta ;1-\\delta ;-s)$ .", "First, the pdf of $\\xi _k$ can be derived analogously to the derivation of [26] as $f_{\\xi _k}(x) = (\\lambda \\pi \\mathbb {E}[S^\\delta ])^k\\frac{\\delta x^{k\\delta -1}}{\\Gamma (k)}\\exp \\left(-\\lambda \\pi \\mathbb {E}[S^\\delta ] x^\\delta \\right).$ Then, thanks to Lemma REF , the Laplace transform of $\\xi _k I_{\\xi _k}$ can be obtained by deconditioning $\\mathcal {L}_{\\rho I_\\rho }(s)$ (given $\\rho $ ) over the distribution of $\\xi _k$ .", "This leads to $\\mathcal {L}_{\\xi _k I_k}(s) = \\frac{1}{\\left(1+\\frac{1}{\\kappa }C(s)\\right)^k},$ where $1+\\frac{1}{\\kappa } C(s) = C_\\kappa (s,1)$ .", "Note that although the path loss exponent $\\alpha $ is not explicitly taken as a parameter of $C_\\kappa (\\cdot ,\\cdot )$ , $C_\\kappa (\\cdot ,\\cdot )$ depends on $\\alpha $ by definition.", "Thus, the value of $\\alpha $ affects all the results.", "Since we consider Rayleigh fading, the coverage probability without ICIC is just the Laplace transform of $\\xi _1 I_1$ .", "The special case of $k=\\kappa =1$ of Lemma REF corresponds to the well-known coverage probability in cellular networks (under the PPP model) without ICIC or ICD [2], $\\mathsf {P}^\\textnormal {c}_{1,1} = \\frac{1}{C_1(\\theta ,1)}.", "$ Note that since we consider the full load case and $\\kappa \\in [1,K]$ , $K=1$ implies $\\kappa =1$ .", "In the more general case $K>1$ , $\\kappa $ depends on the user distribution and the scheduling policy and thus is hard to determine.", "However, treating $\\kappa $ as a parameter we obtain the following theorem addressing the case with non-trivial coordination.", "Theorem 1 ($K$ -BS coordination) The coverage probability for the typical user under $K$ -cell coordination ($K>1$ ) is $\\mathsf {P}^\\textnormal {c}_{K,1} = (K-1) \\int _0^1 \\frac{(1-x^\\delta )^{K-2} \\delta x^{\\delta -1}}{\\left(C_\\kappa (\\theta x, 1)\\right)^{K}} \\textnormal {d}x,$ where $C_\\kappa (s,m) = \\frac{\\kappa -1}{\\kappa }+\\frac{1}{\\kappa } {_2F_1}(m,-\\delta ;1-\\delta ;-s)$ .", "The coverage probability can be written in terms of the PLPS as $\\mathsf {P}^\\textnormal {c}_{K,1} = \\mathbb {P}(h_1 \\xi _1^{-1} > \\theta I_K) = \\mathbb {P}\\left(\\frac{h_1 \\xi _K^{-1}}{I_K} > \\theta \\frac{\\xi _1}{\\xi _K}\\right),$ where $h_1$ is exponentially distributed with mean 1, and thus $\\mathbb {P}(\\frac{h_1 \\xi _K^{-1}}{I_K} > x) = \\mathcal {L}_{\\xi _K I_K}(x)$ .", "Since ${h_1 \\xi _K^{-1}}/{I_K}$ and ${\\xi _1}/{\\xi _K}$ are statistically independent (Lemma REF ), we can calculate the coverage probability by $\\mathsf {P}^\\textnormal {c}_{K,1} = \\int _0^1 \\mathcal {L}_{\\xi _K I_K}(\\theta x) \\textnormal {d}F_{\\xi _1/\\xi _K}( x),$ where $F_{\\xi _1/\\xi _K} (x) = 1-(1-x^{\\delta })^{K-1}$ is the cdf of $\\xi _1/\\xi _K$ given by Lemma REF .", "The theorem is thus proved by change of variables.", "The finite integral in (REF ) can be straightforwardly evaluated numerically.", "Remark 1 The Gauss hypergeometric function can be cumbersome to evaluate numerically, especially when embedded in an integral, as in (REF ).", "Alternatively, $C_1$ can be expressed as $C_1(s,m)=\\frac{1}{(s+1)^m}+s^\\delta m\\textnormal {B}^\\textnormal {u}_{\\frac{1}{s+1}}(m+\\delta ,1-\\delta ),$ and $C_\\kappa (s,m) = \\frac{\\kappa -1}{\\kappa }+\\frac{1}{\\kappa }C_1(s,m)$ .", "Here, $\\textnormal {B}^\\textnormal {u}_x(a,b) = \\int _x^{1} y^{a-1} (1-y)^{b-1} \\textnormal {d}y$ , $x\\in [0,1]$ , is the upper incomplete beta function, which can be calculated much more efficiently in many cases.", "In Matlab, the speed-up compared with the hypergeometric function is at least a factor of 30.", "Fig.", "REF demonstrates the effect of ICIC on the coverage probability for $\\kappa = 1$ and $\\kappa = K$ .", "The former case may be interpreted as a lower bound and the latter case an upper bound.", "As expected, the larger $K$ , the higher the coverage probability for all $\\theta $ .", "On the other hand, the marginal gain of cell coordination decreases with increasing $K$ since the interference, if any, from far away BSs is attenuated by the long link distance and affects the SIR less.", "Fig.", "REF also shows that larger $\\kappa $ results in larger coordination gain in terms of SIR.", "This is due to the fact that coordination not only mutes the strongest $K-1$ interferers but also thins the interfering BSs outside the coordinating cluster.", "However, this does not mean that the system will be better-off by implementing a larger $\\kappa $ .", "Instead, from the load perspective, the (SIR) gain is accompanied with the loss in bandwidth (increased load) since fewer BSs are actively serving users.", "The SIR-load trade-off will be further discussed along with the model validation in Section .", "Figure: The coverage probability under KK-BS coordination, 𝖯 K,1 c \\mathsf {P}^\\textnormal {c}_{K,1}, for K=1,2,3,4,5K=1,2,3,4,5(lower to upper) and κ=K\\kappa =K and κ=1\\kappa = 1.", "The path loss exponent α=4\\alpha = 4.When K=1K=1, the dashed line and solid line overlap." ], [ "ICIC in the High-Reliability Regime", "While the finite integral expression given in Theorem REF is easy to evaluate numerically, it is also desirable to find a simpler estimate that lends itself to a more direct interpretation of the benefit of ICIC.", "This subsection investigates the asymptotic behavior of ICIC when $\\theta \\rightarrow 0$ .", "Note that $\\theta \\rightarrow 0$ refers to the high-reliability regime since in this limit the typical user is covered almost surely.", "In practice, the high-reliability regime ($\\theta \\rightarrow 0$ ) is usually where the control channels operate.", "In the LTE system (narrowband), the lowest MCS mode for downlink transmission supports an SINR about -7 dB and thus may also be suitable for the high-reliability analysis.", "In wide-band systems (e.g., CDMA, UWB), the system is more robust against interference and noise.", "Thus, $\\theta $ is much smaller and the high-reliability analysis is more applicable.", "Proposition 1 Let $\\mathsf {P}^\\textnormal {o}_{K,1} = 1-\\mathsf {P}^\\textnormal {c}_{K,1}$ be the outage probability of the typical user for $K\\in \\mathbb {N}$ .", "Then, $\\mathsf {P}^\\textnormal {o}_{K,1} \\sim a_K \\theta , \\text{ as } \\theta \\rightarrow 0,$ where $a_K = \\frac{1}{\\kappa }\\frac{K!", "}{(1+\\delta ^{-1})_{K-1}}\\frac{\\delta }{1-\\delta } $ and $(x)_n = \\prod _{i=0}^{n-1} (x+i)$ is the (Pochhammer) rising factorial.", "See Appendix .", "Proposition REF shows that for pure ICIC schemes, the number of coordinating BSs only linearly affects the outage probability in the high-reliability regime.", "However, depending on the value of $\\theta $ , even the linear effect may be significant.", "In Fig.", "REF , we plot the coefficient $a_K$ for $K = 1,2,3,4,5$ as a function of the path loss exponent $\\alpha $ , assuming $\\kappa = 1$ .", "The difference (in ratio) between $a_K$ for different $K$ indicates the usefulness of ICIC, and this figure shows that ICIC is more useful when the path loss exponent $\\alpha $ is large.", "This is consistent with intuition, since the smaller the path loss exponent, the more the interference depends on the far-away interferers and thus the less useful the local interference coordination is.", "For other values of $\\kappa $ , the same trend can be observed.", "Figure: The asymptotic coverage probability coefficient a K a_K from Proposition as a function of the path loss exponent α\\alpha under KK-cell coordination (for K=1,2,3,4,5K=1,2,3,4,5, upper to lower)." ], [ "ICIC in High Spectral Efficiency Regime", "The other asymptotic regime is when $\\theta \\rightarrow \\infty $ .", "In this regime, the coverage probability goes to zero while the spectral efficiency goes to infinity.", "Thus, it is of interest to study how the coverage probability decays with $\\theta $ .", "Proposition 2 The coverage probability of the typical user for $K$ -BS ($K>1$ ) coordination satisfies $\\mathsf {P}^\\textnormal {c}_{K,1} \\sim b_K \\theta ^{-\\delta }, \\text{ as } \\theta \\rightarrow \\infty ,$ where $b_K = (K-1)\\int _0^\\infty \\frac{\\delta x^{\\delta -1}}{\\left(C_\\kappa (x,1)\\right)^K}\\textnormal {d}x$ .", "We prove the proposition by studying the asymptotic behavior of $\\theta ^\\delta \\mathsf {P}^\\textnormal {c}_{K,1}$ .", "Using Theorem REF and a change of variable, we have $\\theta ^\\delta \\mathsf {P}^\\textnormal {c}_{K,1}= (K-1) \\int _0^\\theta \\frac{\\delta x^{\\delta -1}(1- x^\\delta /\\theta ^\\delta )^{K-2}}{\\left(C_\\kappa (x,1)\\right)^K} \\textnormal {d}x.$ Considering the sequence of functions (indexed by $\\theta $ ) $f_\\theta (x) \\triangleq \\frac{x^{\\delta -1}(1- x^\\delta /\\theta ^\\delta )^{K-2}}{\\left(C_\\kappa (x,1)\\right)^K},$ we have $\\theta ^{\\prime }>\\theta \\Rightarrow f_{\\theta ^{\\prime }}(x) > f_{\\theta }(x), \\;\\forall x $ , and $f_\\theta (x)$ converges to $f(x)\\triangleq {x^{\\delta -1}}/{\\left(C_\\kappa (x,1)\\right)^K}$ as $\\theta \\rightarrow \\infty $ .", "Therefore, $\\lim _{\\theta \\rightarrow \\infty } \\theta ^\\delta \\mathsf {P}^\\textnormal {c}_{K,1}&= \\lim _{\\theta \\rightarrow \\infty } (K-1)\\delta \\int _0^\\theta f_\\theta (x) \\textnormal {d}x \\\\&\\stackrel{\\text{\\scriptsize (a)}}{\\le } (K-1)\\delta \\lim _{\\theta \\rightarrow \\infty } \\int _0^\\infty f_\\theta (x) \\textnormal {d}x \\\\&\\stackrel{\\text{\\scriptsize (b)}}{=} (K-1)\\delta \\int _0^\\infty f (x) \\textnormal {d}x,$ where (b) is due to the monotone convergence theorem.", "Further, since $f_\\theta (x) \\le f(x)$ and $\\lim _{x\\rightarrow \\infty } f(x) = 0$ , we have $\\lim _{\\theta \\rightarrow \\infty }\\int _{\\theta }^\\infty f_\\theta (x) \\textnormal {d}x = 0$ .", "This allows replacing the inequality (a) with equality and completes the proof.", "Proposition REF shows, just like in the high-reliability regime, that $\\mathsf {P}^\\textnormal {c}_{K,1} = \\Theta (\\theta ^\\delta )$ is not affected by the particular choice of $K$ and $\\kappa $ .", "Since $\\delta = 2/\\alpha $ , the coverage probability decays faster when $\\alpha $ is smaller in the high spectral efficiency regime, consistent with intuition." ], [ "Intra-cell Diversity (ICD) ", "ICIC creates additional load to the neighboring cells by reserving the RBs at the coordinated BSs.", "The extra load improves the coverage probability as it reduces the inter-cell interference.", "In contrast, with selection combining (SC), the serving BS transmits to the typical user at $M$ RBs simultaneously, and the user is covered if the maximum SIR (over the $M$ RBs) exceeds $\\theta $ .", "Like ICIC, SC can also improve the network coverage at the cost of introducing extra load to the BSs.", "Different from ICIC, SC takes advantage of the intra-cell diversity (ICD) by reserving RBs at the serving cell.", "This section provides a baseline analysis on the coverage with ICD (but without ICIC)." ], [ "General Coverage Expression", "Theorem 2 The joint success probability of transmission over $M$ RBs (without ICIC) is $\\mathsf {P}^{\\cap \\textnormal {c}}_{1,M} = \\mathbb {P}(\\bigcap _{m=1}^M {\\bf S}_{1,m})= \\frac{1}{C_1(\\theta ,M)}.$ Since the proof of Theorem REF is a degenerate version of that of a more general result stated in Theorem REF , we defer the discussion of the proof to Section .", "A similar result was obtained in [24] where a slightly different framework was used and the shadowing effect not explicitly modeled.", "Due to the inclusion-exclusion principle, we have the coverage probability with selection combining over $M$ RBs: Corollary 1 ($M$ -RB selection combining) The coverage probability over $M$ RBs without BS-coordination is $\\mathsf {P}^{\\cup \\textnormal {c}}_{1,M} = \\sum _{m=1}^M (-1)^{m+1} {M\\atopwithdelims ()m} \\mathsf {P}^{\\cap \\textnormal {c}}_{1,m},$ where $\\mathsf {P}^{\\cap \\textnormal {c}}_{1,m}$ is given by Theorem REF .", "Fig.", "REF compares the coverage probability under $M$ -RB selection combining, $\\mathsf {P}^{\\cup \\textnormal {c}}_{1,M}$ for $M=1,\\cdots ,5$ .", "As expected, the more RBs assigned to the users, the higher the coverage probability.", "Also, similar to the ICIC case, the marginal gain in coverage probability due to ICD diminishes with $M$ .", "However, comparing Figs.", "REF  and REF , we can already observe dramatic difference: with the same overhead, the coverage gain of ICD looks more evident than that of ICIC in the high-reliability regime, i.e., when $\\theta \\rightarrow 0$ .", "This observation will be formalized in the following subsection.", "Figure: The coverage probability with selection combining overMM RBs without ICIC for M=1,2,3,4,5M=1,2,3,4,5 (lower to upper).", "Here, α=4\\alpha =4." ], [ "ICD in the High-Reliability Regime", "Proposition 3 Let $\\mathsf {P}^{\\cap \\textnormal {o}}_{1,M} = 1-\\mathsf {P}^{\\cup \\textnormal {c}}_{1,M}$ be the outage probability of the typical user under $M$ -RB selection combining.", "We have $\\mathsf {P}^{\\cap \\textnormal {o}}_{1,M}\\sim a_M \\theta ^{M}, \\text{ as } \\theta \\rightarrow 0,$ where $a_M=\\frac{\\partial ^M}{\\partial x^M}\\left({{_1 F_1} (-\\delta ;1-\\delta ; x)}\\right)^{-1}\\!\\big |_{x=0}$ and ${_1 F_1}(a;b;z)$ is the confluent hypergeometric function of the first kind.", "The proof of Proposition REF can be found in Appendix .", "Remark 2 Although Proposition REF provides a neat expression for the constant in front of $\\theta ^M$ in the expansion of the outage probability, numerically evaluating the $M$ -th derivative of the reciprocal of confluent hypergeometric function may not be straightforward.", "A relatively simple approach is to resort to Faà di Bruno's formula.", "Alternatively, one can directly consider (REF ) and simplify it by introducing the Bell polynomial [28]: $a_M = \\sum _{i=1}^M (-1)^i i!", "\\,\\textnormal {Bell}_{M,i}(\\bar{\\tau }(1),\\bar{\\tau }(2),\\cdots ,\\bar{\\tau }(M-i+1)),$ where $\\bar{\\tau }(j) \\triangleq j!", "\\tau (j) = \\frac{(-\\delta )_j}{(1-\\delta )_j},$ and $\\textnormal {Bell}_{m,i}(x_1,\\cdots ,x_{m-i+1}) = \\\\ \\frac{1}{i!", "}\\sum _{j_i\\ge 1}^{\\sum _{i=1}^k j_i=m} {m \\atopwithdelims ()j_1,\\cdots ,j_i} x_{j_1}\\cdots x_{j_i},$ which can be efficiently evaluated numericallyThere was a typo in the version published in the December issue of IEEE Transactions on Wireless Communication where $\\bar{\\tau }$ was mistaken as $\\tau $ .", "The typo this corrected in this manuscript.. To better understand Proposition REF we introduce the following definition of the diversity gain in interference-limited networks, which is consistent with the diversity gain defined in [20] and is analogous to the conventional diversity defined (only) for interference-less cases, see e.g., [4].", "Definition 3 (Diversity (order) gain in interference-limited networks) The diversity (order) gain, or simply diversity, of interference-limited networks is $d \\triangleq \\lim _{\\theta \\rightarrow 0} \\frac{\\log \\mathbb {P}(\\textnormal {SIR}<\\theta )}{\\log \\theta }.$ Clearly, Proposition REF shows that a diversity gain can be obtained by selection combining—in sharp contrast with the results presented in [20], where the authors show that there is no such gain in retransmission.", "The reason of the difference lies in the different association assumptions.", "[20] considers the case where the desired transmitter is at a fixed distance to the receiver which is independent from the interferer distribution.", "However, this paper assumes that the user is associated with the strongest BS (on average).", "In other words, the signal strength from the desired transmitter and the interference are correlated.", "Proposition REF together with [20] demonstrates that this correlation is critical in terms of the time/spectral diversity.", "Figure: Asymptotic behavior (and approximation) of the outage probability 𝖯 1,M ∩o \\mathsf {P}^{\\cap \\textnormal {o}}_{1,M} with MM-RB joint transmission for M=1,2,3,4,5M = 1,2,3,4,5 (upper to lower).", "Here, α=4\\alpha =4.Propositions REF  and REF quantitatively explain the visual contrast between Figs.", "REF  and REF in the high-reliability regime ($\\theta \\rightarrow 0$ ).", "While ICIC reduces the interference by muting nearby interferers, the number of coordinated BSs only affects the outage probability by the coefficient and does not change the fact that $\\mathsf {P}^\\textnormal {o}_{K,1}=\\Theta (\\theta )$ as $\\theta \\rightarrow 0$ .", "In contrast, ICD affects the outage probability by both the coefficient and the exponent.", "Fig.", "REF compares the asymptotic approximation, i.e., $a_M \\theta ^M$ , with the exact expression provided in Corollary REF .", "A reasonably accurate match can be found for small $\\theta $ , and the range where the approximation is accurate is larger when $M$ is smaller.", "Thus, despite the fact that the main purpose of Proposition REF was to indicate the qualitative behavior of ICD, the analytical tractability of $a_M$ also provides useful approximations in applications with small coding rate, e.g., spread spectrum/UWB communication, node discovery, etc." ], [ "ICD in High Spectral Efficiency Regime", "For completeness, we also consider the high spectral efficiency regime where $\\theta \\rightarrow \\infty $ .", "Proposition 4 The coverage probability of the typical user under $M$ -RB selection combining satisfies $\\mathsf {P}^\\textnormal {c}_{1,M} \\sim b_M \\theta ^{-\\delta }, \\text{ as } \\theta \\rightarrow \\infty ,$ where $b_M = \\sum _{m=1}^M (-1)^{m+1} {M \\atopwithdelims ()m} \\frac{\\Gamma (m)}{\\Gamma (1-\\delta )\\Gamma (m+\\delta )} $ .", "We proceed by (first) considering $\\theta ^\\delta \\mathsf {P}^{\\cap \\textnormal {c}}_{1,M}$ .", "By Theorem REF , we have $\\theta ^\\delta \\mathsf {P}^{\\cap \\textnormal {c}}_{1,M} &= \\frac{\\theta ^\\delta }{_2 F_1(m,-\\delta ;1-\\delta ;-\\theta )} \\\\&\\stackrel{\\text{\\scriptsize (a)}}{=} \\left( \\frac{\\theta }{1+\\theta }\\right)^\\delta \\frac{1}{_2 F_1(-\\delta ,1-\\delta -m;1-\\delta ;\\frac{\\theta }{1+\\theta })} $ where (a) comes from [29].", "Since $_2 F_1(-\\delta ,1-\\delta -m;1-\\delta ;1) = {\\Gamma (1-\\delta )\\Gamma (m+\\delta )}/{\\Gamma (m)}$ , we have $\\lim _{\\theta \\rightarrow \\infty } \\theta ^\\delta \\mathsf {P}^{\\cap \\textnormal {c}}_{1,M} = \\frac{\\Gamma (m)}{\\Gamma (1-\\delta )\\Gamma (m+\\delta )}$ , which leads to the proposition thanks to Corollary REF .", "Comparing Propositions REF  and REF , we see that unlike the high-reliability regime, the coverage probabilities of ICIC and ICD do not have order difference in the high spectral efficiency regime.", "However, the difference in coefficients ($b_K$ and $b_M$ ) can also incur significant difference in the coverage probability.", "Fig.", "REF compares the coefficients for different path loss exponent $\\alpha $ assuming $\\kappa = 1$ .", "Note that $\\kappa = 1$ corresponds to the smallest possible $b_K$ .", "Yet, even so, $b_K$ still dominates $b_M$ for most realistic $\\alpha $ .", "This implies that ICIC is often more effective than ICD in the high spectral efficiency regime.", "Figure: b K b_K (K=1,2,3,4,5K=1,2,3,4,5) in Proposition and b M b_M (M=1,2,3,4,5M=1,2,3,4,5) in Proposition  for different values of α\\alpha ,Here, κ=1\\kappa = 1." ], [ "ICIC and ICD", "Sections  and  provided the coverage analysis in cellular networks with ICIC and ICD separately.", "This section considers the scenario where the network takes advantage of ICIC and ICD at the same time.", "In particular, we will evaluate the coverage probability $\\mathsf {P}^{\\cup \\textnormal {c}}_{K,M}$ when the typical user is assigned with $M$ RBs with independent fading at the serving BS and all the $M$ RBs are also reserved at the $K-1$ strongest non-serving BSs." ], [ "The General Coverage Expression", "In order to derive the coverage probability, we first generalize Lemma REF beyond Rayleigh fading.", "In particular, for a generic fading random variable $H$ , we introduce the following definition.", "Definition 4 For a PLPS $\\Xi = \\lbrace \\xi _i\\rbrace $ , let ${I}_k^{H}$ be the interference from the BSs weaker (without fading) than the $k$ -th strongest BS, i.e., ${I}_k^{H} = \\sum _{i>k} H_i \\xi _i^{-1}$ , where $H_i\\stackrel{\\textnormal {\\scriptsize d}}{=} H,\\;\\forall i\\in \\mathbb {N},$ are iid.", "Similarly, we define ${I}_\\rho ^{H} = \\sum _{\\xi \\in \\Xi \\cap (\\rho ,\\infty )} H_\\xi \\xi _\\rho ^{-1}$ to be the interference from BSs with average (over fading) received power less than $\\rho ^{-1}$ .", "Then, we obtain a more general version of Lemma REF as follows.", "Lemma 6 For general fading random variables $H\\ge 0$ and $\\mathbb {E}[H^\\delta ]<\\infty $ , the Laplace transform of ${\\xi _k I^H_k}$ is $\\mathcal {L}_{\\xi _k I^H_k}(s) = \\frac{1}{\\left(1-\\frac{1}{\\kappa }+\\frac{1}{\\kappa }\\mathbb {E}_H\\left[e^{-sH}+s^\\delta H^\\delta \\gamma (1-\\delta ,sH)\\right]\\right)^k}.$ The proof of the lemma follows exactly that of Lemmas REF  and REF .", "The only difference is that we do not factor in the distribution of the fading random variable $H$ .", "More precisely, we can first show $\\mathcal {L}_{\\xi _\\rho I^H_\\rho }(s) = \\\\\\exp \\left(-\\frac{\\lambda }{\\kappa }\\pi \\mathbb {E}[S^\\delta ]\\rho ^\\delta \\mathbb {E}_H[e^{-sH}+s^\\delta H^\\delta \\gamma (1-\\delta ,sH)-1]\\right).$ Then, integrating $\\rho $ over the distribution of $\\xi _k$ gives the desired result.", "Note that the condition $\\mathbb {E}[H^\\delta ]<\\infty $ in Lemma REF is sufficient (but not necessary) to guarantee the existence of the Laplace transform.", "As will become clear shortly, for the purpose of this section, the most important case of $H$ is when $H$ is a gamma random variable with pdf $f_H(x) = \\frac{1}{\\Gamma (m)}x^{m-1} e^{-x}$ , where $m\\in \\mathbb {N}$ .", "For this case, we have the following lemma.", "Lemma 7 For $m\\in \\mathbb {N}$ , if $H$ is a gamma random variable with pdf $f_H(x) = \\frac{1}{\\Gamma (m)}x^{m-1} e^{-x}$ , $\\mathcal {L}_{\\xi _k I^H_k}(s) = \\frac{1}{\\left(C_\\kappa (s,m)\\right)^k}.$ Almost trivially based on Lemma REF , Lemma REF helps to show the following theorem.", "Theorem 3 For all $M\\in \\mathbb {N}$ and $K>1$ , the joint coverage probability over $M$ -RBs under $K$ -cell coordination is $\\mathsf {P}^{\\cap \\textnormal {c}}_{K,M} = \\mathbb {P}(\\bigcap _{m=1}^M {\\bf S}_{K,m}) = (K-1) \\int _0^1 \\frac{(1-x^\\delta )^{K-2} \\delta x^{\\delta -1}}{\\left(C_\\kappa (\\theta x,M)\\right)^{K}} \\textnormal {d}x.$ Let $h_i^m$ be the fading coefficient from the $i$ -th strongest (on average) BS at RB $m$ for $m\\in [M]$ .", "By definition, we have $\\mathsf {P}^{\\cap \\textnormal {c}}_{K,M}=\\mathbb {E}_{\\Xi } \\mathbb {P}\\left(h_1^m \\xi _1^{-1} >\\theta \\sum _{i>K} \\chi _i h_i^m \\xi _i^{-1},\\; \\forall m\\in [M]\\right)$ Due to the conditional independence (given $\\Xi $ ) across $m$ , (REF ) can be further simplified as $\\mathsf {P}^{\\cap \\textnormal {c}}_{K,M}&= \\mathbb {E}_\\Xi \\prod _{m=1}^M \\mathbb {P}\\left(h_1^m>\\theta \\xi _1\\sum _{i>K} \\chi _i h_i^m \\xi _i^{-1}\\right) \\\\&= \\mathbb {E}_\\Xi \\mathbb {E}\\prod _{m=1}^M \\exp \\left(-\\theta \\xi _1\\sum _{i>K} \\chi _i h_i^m \\xi _i^{-1}\\right), \\\\&= \\mathbb {E}_\\Xi \\mathbb {E}\\exp \\left(-\\theta \\xi _1\\sum _{i>K} \\chi _i \\underbrace{\\sum _{m=1}^M h_i^m}_{H_i} \\xi _i^{-1}\\right), $ where the inner expectation in (REF ) is taken over $h_i^m$ for $m\\in [M]$ and $i\\in \\mathbb {N}$ , and due to the independence (across $m$ and $i$ ) and (exponential) distribution of $h_i^m$ , $H_i$ are iid gamma distributed with pdf $f(x)=\\frac{1}{\\Gamma (M)}x^{M-1} e^{-x}$ .", "Further, writing $\\xi _1$ as $\\frac{\\xi _1}{\\xi _K} \\xi _K$ and letting $\\Xi _K = \\lbrace \\xi _i\\rbrace _{i=K+1}^{\\infty }$ , we obtain the following expression by taking advantage of the statistical independence shown in Lemma REF : $\\mathsf {P}^{\\cap \\textnormal {c}}_{K,M} = \\mathbb {E}_{\\frac{\\xi _1}{\\xi _K}} \\mathcal {L}_{\\xi _k I^H_K} \\left(\\theta \\frac{\\xi _1}{\\xi _K}\\right),$ where $\\mathcal {L}_{\\xi _K I^H_K}(\\cdot )$ is given in Lemma REF .", "The proof is completed by plugging in Lemma REF .", "Note that although Theorem REF does not explicitly address the case $K=1$ , the same proof technique applies to this (easier) case, where the treatment of the random variable $\\xi _1/\\xi _K$ is unnecessary since it has a degenerate distribution ($\\equiv 1$ ).", "Thus, the proof of Theorem REF is evident and omitted from the paper.", "Due to the inclusion-exclusion principle, we immediately obtain the following corollary.", "Corollary 2 ($K$ -BS coordination and $M$ -RB selection combining) The coverage probability over $M$ RBs with $K$ BS-coordination is $\\mathsf {P}^{\\cup \\textnormal {c}}_{K,M} = \\sum _{m=1}^M (-1)^{m+1} {M\\atopwithdelims ()m} \\mathsf {P}^{\\cap \\textnormal {c}}_{K,m},$ where $\\mathsf {P}^{\\cap \\textnormal {c}}_{K,m}$ is given by Theorem REF .", "Figure: The outage probability 𝖯 K,M ∩o \\mathsf {P}^{\\cap \\textnormal {o}}_{K,M} under KK-BS coordination over MM RBsfor K=1,2,3,4,5K=1,2,3,4,5 (upper to lower) and M=1,2M=1,2.", "Here, κ=1\\kappa = 1." ], [ "The High-Reliability Regime", "Proposition 5 Let $\\mathsf {P}^{\\cap \\textnormal {o}}_{K,M} = 1-\\mathsf {P}^{\\cup \\textnormal {c}}_{K,M}$ be the outage probability of the typical user under $M$ -RB selection combining and $K$ -BS coordination.", "We have $\\mathsf {P}^{\\cap \\textnormal {o}}_{K,M}\\sim a(K,M) \\theta ^{M}, \\text{ as } \\theta \\rightarrow 0,$ where $a(K,M)>0\\; ,\\forall K,M\\in \\mathbb {N}$ .", "Proposition REF combines Proposition REF  and REF .", "Its proof is analogous to that of Proposition REF (but more tedious) and is thus omitted from the paper.", "Proposition REF gives quantitative evidence on why a pure ICD scheme maximizes the coverage probability in the high-reliability regime.", "In Fig.", "REF , we plot the outage probability for difference number of coordinated cell $K=1,2,3,4,5$ and in-cell diversity $M=1,2$ assuming $\\kappa =1$ and observe the consistency with Proposition REF ." ], [ "The Effective Load Model", "In Section , we introduced the effective load $\\kappa $ and modeled the impact of the out-of-cluster coordination on the interference by independent thinning of the interferer field with retaining probability $1/\\kappa $ .", "Although, remarkably, $\\kappa $ disappears when considering the diversity order of the network, it is still of interest to evaluate the accuracy of such modeling in the non-asymptotic regime.", "To this end, we set up the following ICIC simulation to validate the effective load model.", "We consider the users are distributed as a homogeneous PPP $\\Phi _\\textnormal {u}$ with density $\\lambda _\\textnormal {u}$ independent from the BS process $\\Phi $ .", "We assume a single channel and a random scheduling policy where we pick every user exactly once at a random order.", "A picked user is scheduled iff its strongest BSs are not (already) serving another user or coordinating (i.e., being muted) with user(s) in other cell(s) and its second to $K$ -th strongest BSs are not transmitting (serving other users).", "Thus, after the scheduling phase, there are at most $\\Phi (B)$ users scheduled where $B\\subset \\mathbb {R}^2$ is the simulation region since there are at most $\\Phi (B)$ serving BSs.", "In reality, the number of scheduled users is often much less than $\\Phi (B)$ since 1) there is always a positive probability that there are empty cells due to the randomness in BS and user locationsThis also implies that the full-load assumption does not hold (exactly) in the simulation.", "; 2) when $K>1$ some BSs are muted due to coordination.", "The ratio between the number of BSs $\\Phi (B)$ and the number of scheduled users (which equals the number of serving BSs) is consistent with the definition of the effective load $\\kappa $ and thus is a natural estimate.", "Under lognormal shadowing with standard deviation $\\sigma $ , we empirically measured $\\kappa $ as in Table REF .", "It is observed that our simulation results in the estimates $\\hat{\\kappa }$ that can be well approximated by an affine function of $K$ and the function depends on the shadowing variance.", "The fact that more severe shadowing results in smaller $\\kappa $ can be explained in the case $K=1$ .", "In this case, the only reason that $\\kappa >1$ is the existence of empty cells and the larger $\\kappa $ is the more empty cells there are.", "Independent shadowing reduces the spatial correlation of the sizes of nearby Poisson Voronoi cells and thus naturally reduces the variance of the number of users in each cell, resulting in a smaller number of empty cells.", "Table: Estimated κ\\kappa Fig.", "REF compares the coverage probability under $K$ -BS coordination predicted by Theorem REF using the estimated $\\kappa $ from Table REF with the simulation results.", "We picked the case where $\\sigma = 0$  dB since this is the worst case in terms of matching analytical results with the simulation due to the size correlation of Poisson Voronoi cells.", "To see this more clearly, consider the case $K=1$ , where the simulation and analysis match almost completely in the figure.", "The match is expected but not entirely trivial since the process of transmitting BSs is no longer a PPP.", "More specifically, a BS is transmitting iff there is at least one user in its Voronoi cell, i.e., the ground process $\\Phi $ is thinned by the user process $\\Phi _\\textnormal {u}$ .", "However, the thinning events are spatially dependent due to the dependence in the sizes of the Voronoi cells.", "As a result, clustered BSs are less likely to be serving users at the same time and thus the resulting transmitting BS process is more regular than a PPP.", "Yet, Fig.", "REF shows that the deviation from a PPP is small when $\\lambda _\\textnormal {u}=10\\lambda $ .In fact, the match for $K=1$ is still quite good for smaller user densities, say $\\lambda _\\textnormal {u} = 5\\lambda $ .", "Shadowing breaks the spatial dependence of the interfering field (transmitting BSs) and consequently improves the accuracy of the analysis.", "When $\\sigma =10$  dB, the difference between the simulated coverage probability and the one predicted in Theorem REF are almost visually indistinguishable (Fig.", "REF ).", "These results validate the effective load model for analyzing ICIC in the non-asymptotic regime.", "Figure: The coverage probability comparison between the analytical coverage probabilityderived in Theorem and the simulation results with K=1,2,3,4,5K=1,2,3,4,5.Here, the BS density λ=1\\lambda =1, user density λ u =10\\lambda _\\textnormal {u} = 10.Lognormal shadowing with variance σ 2 \\sigma ^2 (in dB 2 ^2) is considered." ], [ "ICIC-ICD Trade-off", "The analyses in Sections ,  and  shows the significantly different behavior of ICIC and ICD schemes despite the fact that both the schemes improves the coverage probability through generating extra load in the system.", "In particular, Propositions REF , REF  and REF show that when $\\theta \\rightarrow 0$ , ICD will have a larger impact on the coverage probability due to the diversity gain.", "In contrast, Propositions REF  and REF suggest that when $\\theta \\rightarrow \\infty $ , ICIC will be more effective since the linear gain is typically larger (Fig.", "REF ).", "Intuitively, a ICIC-ICD combined scheme should present a trade-off between the performance in these two regimes.", "To make a fair comparison between different ICIC-ICD combined schemes, we need to control their load on the system in terms of RBs used.", "By the construction of the model, we observe that the load introduced by ICIC is the effective load $\\kappa $ times the load without ICIC since $1/\\kappa $ is the fraction of active transmitting BSs, which, in the single-channel case, is proportional to the number (or, density) of users being served.", "Similarly, the load introduced by ICD is $M$ times the load without ICD since $M$ RBs are grouped to serve a single user (while without ICD they could be used to serve $M$ users).", "Thus, under both $K$ -BS coordination and $M$ -RB selection combining, the system load is proportional to $\\kappa M$ which we term ICIC-ICD load factor.", "Fig.", "REF plots the coverage probability of three ICIC-ICD combined schemes with different but similar ICIC-ICD load factor $\\kappa M$ using both the analytical result and simulation.", "As is shown in the figure, a hybrid ICIC-ICD scheme (i.e., with $K,M>1$ ) provides a trade-off between the good performance of ICIC and ICD in the two asymptotic regimes.", "In general, a hybrid scheme could provide the highest coverage probability for intermediate $\\theta $ , and the crossing point depends on all the system parameters.", "Figure: The coverage probability under KK-BS coordinationand MM-RB selection combining 𝖯 K,M ∪c \\mathsf {P}^{\\cup \\textnormal {c}}_{K,M} withdifferent combinations (K,M)(K,M).Here, α=4\\alpha =4, σ=0\\sigma =0 dB.", "The left figure shows the part for θ∈[-20dB,-5dB]\\theta \\in [-20~\\textnormal {dB},-5~\\textnormal {dB}]and the right figure for θ∈[-5dB,20dB]\\theta \\in [-5~\\textnormal {dB},20~\\textnormal {dB}]." ], [ "ICIC-ICD-Load Trade-off", "Another more fundamental trade-off is between the load and the ICIC-ICD combined schemes.", "In other words, how to find the optimal combination $(K,M)$ that takes the load into account.", "While the complexity of this problem prohibits a detailed exploration in this paper, we give a simple example to explain the trade-off.", "Assume all the users in the network are transmitting at the same rate $\\log (1+\\theta )$ and the network employs the random scheduling procedure as described in Section REF .", "Then the (average) throughput of the typical scheduled user is $\\log (1+\\theta ) \\mathsf {P}^{\\cup \\textnormal {c}}_{K,M}$ in the interference-limited network.", "Under the ICIC and ICD schemes, the number of user being served per RB is (on average) $1/\\kappa M$ times those who can be served in the baseline case without ICIC and ICD.", "Therefore, for fixed $\\theta $ , the spatially averaged (per user) throughput is proportional to $\\mathsf {P}^{\\cup \\textnormal {c}}_{K,M}/\\kappa M$ .", "Intuitively, it is the product of the probability of a random chosen user being scheduled ($\\propto 1/\\kappa M$ ) and the probability of successful transmission ($\\mathsf {P}^{\\cup \\textnormal {c}}_{K,M}$ ).", "Then we can find optimal combination $(K^*,M^*) = \\operatornamewithlimits{arg\\,max}_{(K,M)\\in \\mathbb {N}^2} \\frac{\\mathsf {P}^{\\cup \\textnormal {c}}_{K,M}}{\\kappa M}$ using exhaustive search.", "Alternatively, we can enforce an outage constraint and find the optimal $(K,M)$ combination such that $(K^*,M^*) = \\operatornamewithlimits{arg\\,max}_{(K,M)\\in \\mathbb {N}^2} \\frac{\\mathsf {P}^{\\cup \\textnormal {c}}_{K,M}}{\\kappa M} \\sf {1}_{[1-\\epsilon ,1]}(\\mathsf {P}^{\\cup \\textnormal {c}}_{K,M})$ Fig.", "REF plots the exhaustive search result for $(K^*,M^*)$ defined in (REF ) and (REF ).", "In the simulation, we limit our search space for both $K$ and $M$ to $\\lbrace 1,2,\\cdots ,20\\rbrace $ and we use the affine function $\\kappa = \\eta _0 + \\eta _1 K$ to approximate $\\kappa $ , which turns out to be an accurate fit in our simulation (see the data in Table REF ).", "Fig.", "REF shows that as $\\theta $ increases, it is beneficial to increase $K$ .", "This is consistent with the result derived in Propositions REF  and REF and Fig.", "REF , which show that ICIC is more effective in improving coverage probability for large $\\theta $ .", "If there is no outage constraint, it is more desirable to keep both $K$ and $M$ (and thus the load factor) small.", "This is true especially for small $\\theta $ since the impact of $(K,M)$ on $\\mathsf {P}^{\\cup \\textnormal {c}}_{K,M}$ is small ($\\mathsf {P}^{\\cup \\textnormal {c}}_{K,M}\\approx 1$ ) but both $\\kappa $ (a function of $K$ ) and $M$ linearly affect the load factor and thus the average throughput.", "The incentive to increase $(K,M)$ is higher if an outage constraint is imposed.", "Although it is still more desirable to increase $K$ (both due to its usefulness in the high-spectral efficiency regime and its smaller impact on the load factor), the increase in $M$ also has a non-trivial impact: a slight increase in $M$ could significantly reduce the optimal value of $K$ .", "This is an observation of practical importance, since the cost of increasing $K$ is usually much higher than that of increasing $M$ due to the signaling overhead that ICIC requires.", "Figure: The optimal (K * ,M * )(K^*,M^*) as a function of θ\\theta .The top subfigure is optimized for average throughput, see ().The bottom subfigure is optimized for average throughput under an outage constraint, see ()with ϵ=0.05\\epsilon = 0.05.10 dB shadowing is considered." ], [ "Conclusions ", "This paper provides explicit expressions for the coverage probability of inter-cell interference coordination (ICIC) and intra-cell diversity (ICD) in cellular networks modeled by a homogeneous Poisson point process (PPP).", "Examining the high-reliability regime, we demonstrate a drastically different behavior of ICIC and ICD despite their similarity in creating extra load in the network.", "In particular, ICD, under the form of selection combining (SC), provides diversity gain while ICIC can only linearly affect the outage probability in the high-reliability regime.", "In contrast, in the high-spectral efficiency regime, ICIC provides higher coverage probability for realistic path loss exponents.", "All the analytical results derived in the paper are invariant to the network density and the shadowing distribution.", "The fact that ICD under selection combining provides diversity gain in cellular networks even with temporal/spectral interference correlation contrasts with the corresponding results in ad hoc networks, where [20] shows no such gain exists.", "This shows that the spatial dependence between the desired transmitter and the interferers is critical in harnessing the diversity gain.", "In the non-asymptotic regime, we propose an effective load model to analyze the effect of ICIC.", "The model is validated with simulations and proven to be very accurate.", "Using these analytical results, we explore the fundamental trade-off between ICIC-ICD and system load in cellular systems." ], [ "Proof of Proposition ", "We prove the theorem by calculating $\\lim _{\\theta \\rightarrow 0}\\frac{\\mathsf {P}^\\textnormal {o}_{K,1}}{\\theta }$ .", "First, consider the case where $K=1$ .", "Since $\\mathsf {P}^\\textnormal {o}_{K,1} = 1-\\mathsf {P}^\\textnormal {c}_{K,1}$ , we have $\\lim _{\\theta \\rightarrow 0}\\frac{\\mathsf {P}^\\textnormal {o}_{K,1}}{\\theta }= \\lim _{\\theta \\rightarrow 0} \\frac{C_\\kappa (\\theta ,1)-1}{\\theta C_\\kappa (\\theta ,1)}= \\lim _{\\theta \\rightarrow 0} C^{\\prime }_\\kappa (\\theta ,1),$ where $C^{\\prime }_\\kappa (x,1) = \\frac{\\textnormal {d}}{\\textnormal {d}x}C_\\kappa (x,1)$ , and the last equality is due to L'Hospital's rule and the fact that $C_1(0,1) = 1$ .", "Moreover, we have $\\lim _{\\theta \\rightarrow 0} C^{\\prime }_\\kappa (\\theta ,1) = \\frac{1}{\\kappa } C^{\\prime }_1(0,1)= \\frac{1}{\\kappa } \\frac{\\delta }{1-\\delta }$ due to the series expansion of the Gauss hypergeometric function ${_2 F_1} (a,b;c;z) = \\sum _{n=0}^{\\infty }\\frac{(a)_n (b)_n}{(c)_n} \\frac{z^n}{n!", "}$ .", "Thus, we proved (REF ) is true for $K=1$ .", "For $K\\ge 2$ , by Theorem REF , we have $\\mathsf {P}^\\textnormal {o}_{K,1} = \\\\(K-1)\\delta \\int _0^1 \\left(1-\\frac{1}{(C_\\kappa (\\theta x,1))^K}\\right)(1-x^\\delta )^{K-2}x^{\\delta -1}\\textnormal {d}x,$ where the integral, by change of variable $y = \\theta x$ , can be written as $\\frac{1}{\\theta ^\\delta }\\int _0^\\theta \\Delta _K(y)\\left(1-\\frac{y^\\delta }{\\theta ^\\delta }\\right)^{K-2}y^{\\delta -1}\\textnormal {d}y,$ where $\\Delta _K(y) = 1-(C_\\kappa (y,1))^{-K}$ .", "Therefore, we have $\\lim _{\\theta \\rightarrow 0}\\frac{\\mathsf {P}^\\textnormal {o}_{K,1}}{\\theta }= \\lim _{\\theta \\rightarrow 0} \\frac{(K-1)\\delta }{\\theta ^{\\delta +1}}\\int _0^\\theta \\Delta _K(y)\\left(1-\\frac{y^\\delta }{\\theta ^\\delta }\\right)^{K-2}y^{\\delta -1}\\textnormal {d}y,$ where the RHS can be simplified by (repetitively) applying L'Hospital's rule as follows: $&\\phantom{={}} \\lim _{\\theta \\rightarrow 0} \\frac{(K-1)\\delta }{\\theta ^{\\delta +1}}\\int _0^\\theta \\Delta _K(y)\\left(1-\\frac{y^\\delta }{\\theta ^\\delta }\\right)^{K-2}y^{\\delta -1}\\textnormal {d}y \\\\&= \\lim _{\\theta \\rightarrow 0} \\frac{(K-2)_2 \\delta ^2}{(\\delta +1)\\theta ^{2\\delta +1}} \\int _0^\\theta \\Delta _K(y)\\left(1-\\frac{y^\\delta }{\\theta ^\\delta }\\right)^{K-3}y^{2\\delta -1}\\textnormal {d}y \\\\&= \\cdots \\\\&= \\frac{(K-1)!", "\\delta ^{K-1}}{\\prod _{k=1}^{K-2}(k\\delta +1)} \\lim _{\\theta \\rightarrow 0} \\frac{1}{\\theta ^{(K-1)\\delta +1}}\\int _0^\\theta \\Delta _K(y) y^{(K-1)\\delta -1}\\textnormal {d}y \\\\&= \\frac{(K-1)!", "\\delta ^{K-1}}{\\prod _{k=1}^{K-1}(k\\delta +1)} \\lim _{\\theta \\rightarrow 0} \\frac{\\Delta _K(\\theta )}{\\theta },$ where $\\lim _{\\theta \\rightarrow 0} \\frac{\\Delta _K(\\theta )}{\\theta } = \\lim _{\\theta \\rightarrow 0} \\frac{(C_\\kappa (\\theta ,1))^K-1}{\\theta (C_\\kappa (\\theta ,1))^K}.$ Note that $\\lim _{\\theta \\rightarrow 0} C_\\kappa (\\theta ,1) = 1$ and thus $\\lim _{\\theta \\rightarrow 0} \\frac{\\Delta _K(\\theta )}{\\theta } = \\lim _{\\theta \\rightarrow 0} \\frac{(C_\\kappa (\\theta ,1))^K-1}{\\theta }$ , which by L'Hospital's rule can be further simplified as $K\\lim _{\\theta \\rightarrow 0} (C_\\kappa (\\theta ,1))^{K-1} C^{\\prime }_\\kappa (\\theta ,1)$ .", "Therefore, thanks to (REF ), we have $\\lim _{\\theta \\rightarrow 0} \\frac{\\Delta _K(\\theta )}{\\theta } = K C^{\\prime }_\\kappa (0,1) = \\frac{K}{\\kappa } \\frac{\\delta }{1-\\delta }.", "$ Combining (REF ) and (REF ) completes the proof." ], [ "Proof of Proposition ", "In order the prove Proposition REF , we first introduce two useful lemmas.", "Letting $\\mathsf {D}^k_u = \\frac{\\partial ^k}{\\partial u^k}$ , the following lemma states a simple algebraic fact which will turn out to be useful in the asymptotic analysis.", "Lemma 8 For any $c\\in \\mathbb {R}$ , we have $\\mathsf {D}^k_u\\left.\\left(1-\\frac{1}{c(1+u)}\\right)^M\\right|_{u=0} = \\\\\\sum _{j=1}^k {M \\atopwithdelims ()j}{k \\atopwithdelims ()j} j!", "(M)_{k-j} (-1)^{k-j} \\left(1-\\frac{1}{c}\\right)^{M-j}.", "$ First, expressing $\\left(1-\\frac{1}{c(1+u)}\\right)^M$ as $(cu+c-1)^M(\\frac{1}{c(1+u)})^M$ , by the Leibniz rule, we can expand the $k$ -th order derivative as $\\mathsf {D}^k_u\\left(1-\\frac{1}{c(1+u)}\\right)^M = \\\\\\sum _{j=0}^k {k \\atopwithdelims ()j} \\mathsf {D}^j_u(cu+c-1)^M \\mathsf {D}^{k-j}_u\\frac{1}{c^M (1+u)^M},$ where $\\left.", "\\mathsf {D}^j_u(cu+c-1)^M\\right|_{u=0}&= \\mathsf {D}^j_u\\left.", "\\sum _{m=1}^M {M \\atopwithdelims ()m} (cu)^m (c-1)^{M-m} \\right|_{u=0} \\\\&= {M \\atopwithdelims ()j} j!", "c^j (c-1)^{M-j},$ and $\\mathsf {D}^{k-j}_u\\left.\\frac{1}{c^M (1+u)^M} \\right|_{u=0} = \\frac{(-1)^{k-j}}{c^M} (M)_{k-j}.$ This gives the desired expansion.", "Thanks to Lemma REF , we have the following result.", "Lemma 9 Given $n$ arbitrary nonnegative integers $k_1,k_2,\\cdots ,k_n\\in \\mathbb {N}\\cup \\lbrace 0\\rbrace $ and $A_n\\triangleq \\sum ^n_{i=1} k_i \\ge 1$ , we have $\\sum _{m=1}^{M} {M \\atopwithdelims ()m} (-1)^{m+A_n} \\prod _{i=1}^n (m)_{k_i} ={\\left\\lbrace \\begin{array}{ll}0, &\\text{if } A_n < M \\\\M!, &\\text{if } A_n = M\\end{array}\\right.", "}$ for all $M\\in \\mathbb {N}$ .", "We prove the lemma by induction.", "First, consider the case where $n=1$ .", "Then for all $k_1>0$ , we have $\\hspace{20.0pt}&\\hspace{-20.0pt}\\sum _{m=1}^M (-1)^{k_1+m} {M\\atopwithdelims ()m} (m)_{k_1} \\\\&= \\sum _{m=1}^M (-1)^{m} {M\\atopwithdelims ()m} \\mathsf {D}^{k_1}_u\\left.\\frac{1}{(1+u)^m}\\right|_{u=0} \\\\&= \\mathsf {D}^{k_1}_u \\left.\\left( \\left(1-\\frac{1}{1+u}\\right)^M -1 \\right)\\right|_{u=0} \\\\&= \\mathsf {D}^{k_1}_u \\left.\\frac{u^M}{(1+u)^M}\\right|_{u=0},$ where the $k_1$ -th derivative can be expanded by the Leibniz rule, i.e., $\\mathsf {D}^{k_1}_u\\frac{u^M}{(1+u)^M} = \\sum _{i=0}^{k_1} {k_1\\atopwithdelims ()i}\\left(\\mathsf {D}^i_u u^M \\right)\\left(\\mathsf {D}^{k_1-i}_u \\frac{1}{(1+u)^M} \\right),$ which is 0 when $u=0$ if $k_1<M$ .", "When $k_1=M$ , the only non-zero term in the sum is the one with $i=k_1=M$ and thus is $\\mathsf {D}^{k_1}_u\\frac{u^M}{(1+u)^M}|_{u=0}=M!$ .", "Therefore, the lemma is true for $n=1$ .", "Second, we prove it for the case $0<A_n < M$ with general $n$ by induction.", "Assume $\\sum _{m=1}^{M} {M \\atopwithdelims ()m} (-1)^{m+A_{n-1}} \\prod _{i=1}^{n-1} (m)_{k_i} = \\\\\\sum _{m=1}^{M} {M \\atopwithdelims ()m} (-1)^{m} \\prod _{i=1}^{n-1} \\left.\\mathsf {D}^{k_i}_{u_i} \\left(\\frac{1}{1+u_i}\\right)^m \\right|_{\\begin{array}{c}u_i=0 \\\\ i\\in [n]\\end{array}} = 0, $ for all $n-1$ nonnegative integers $\\lbrace k_i\\rbrace _{i=1}^{n-1}$ with $0<A_{n-1}=\\sum _{i=1}^{n-1} k_i < M$ .", "Then, we consider the case for $n$ , move all the $\\mathsf {D}^{k_i}_{u_i}$ to the front and, analogous to the $n=1$ case, obtain $\\sum _{m=1}^{M} {M \\atopwithdelims ()m} (-1)^{m+\\sum \\limits ^{n}_{i=1} k_i} \\prod _{i=1}^{n} (m)_{k_i} = \\\\\\left.\\left(\\prod _{i=1}^{n} \\mathsf {D}^{k_i}_{u_i}\\right) \\left(1-\\frac{1}{\\prod _{i=1}^{n}(1+u_i)}\\right)^M\\right|_{\\begin{array}{c}u_i=0 \\\\ i\\in [n]\\end{array}}.$ Expanding only the $k_n$ -th order derivative using Lemma REF , we can express (REF ) as $\\left(\\prod _{i=1}^{n-1} \\mathsf {D}^{k_i}_{u_i}\\right)\\sum _{j=1}^{k_n} a_{k_n,j,M} \\left(1-\\frac{1}{\\prod _{i=1}^{n-1}(1+u_i)}\\right)^{M-j},$ where $a_{k_n,j,M}={M \\atopwithdelims ()j}{k_n \\atopwithdelims ()j} j!", "(M)_{k_n-j} (-1)^{k_n-j}$ is independent from $k_i$ and $u_i$ for all $i\\in [n-1]$ .", "We then can move the derivative operators inside the summation.", "Further, since $j\\le k_n < M-\\sum _{i=1}^{n-1} k_i$ , we have $\\sum _{i=1}^{n-1} k_i < M-j$ for all $j$ in the summation, which leads to the observation that $\\left.\\left(\\prod _{i=1}^{n-1} \\mathsf {D}^{k_i}_{u_i}\\right)\\left(1-\\frac{1}{\\prod _{i=1}^{n-1}(1+u_i)}\\right)^{M-j}\\right|_{\\begin{array}{c}u_i=0 \\\\ i\\in [n]\\end{array}} =0,\\; \\forall j\\in [k_n]$ by our assumption on the $n-1$ case.", "Thus the lemma is proved for the case $0<A_n < M$ for all $n\\in \\mathbb {N}$ .", "For the case $A_n = M$ , we see, by the first part of the proof, that $\\left.\\left(\\prod _{i=1}^{n-1} \\mathsf {D}^{k_i}_{u_i}\\right)\\left(1-\\frac{1}{\\prod _{i=1}^{n-1}(1+u_i)}\\right)^{M-j}\\right|_{\\begin{array}{c}u_i=0 \\\\ i\\in [n]\\end{array}}$ can be non-zero only if $j=k_n$ .", "Thus (REF ) can be simplified to $\\left.", "{M \\atopwithdelims ()k_n} k_n !", "\\left(\\prod _{i=1}^{n-1} \\mathsf {D}^{k_i}_{u_i}\\right) \\left(1-\\frac{1}{\\prod _{i=1}^{n-1}(1+u_i)}\\right)^{M-k_n}\\right|_{\\begin{array}{c}u_i=0 \\\\ i\\in [n]\\end{array}},$ which is $M!$ if we assume the lemma is true for $n-1$ .", "Since $n$ is arbitrarily chosen, the lemma is proved for all $n\\in \\mathbb {N}$ .", "With Lemmas REF  and REF , we are able to proceed with the proof of Proposition REF as follows.", "[Proof (of Proposition REF )] By Corollary REF , we have $\\mathsf {P}^{\\cap \\textnormal {o}}_{1,M}&= 1-\\sum _{m=1}^M (-1)^{m+1} {M\\atopwithdelims ()m} \\frac{1}{C_1(\\theta ,m)} \\\\&= \\sum _{m=1}^M (-1)^{m+1} {M\\atopwithdelims ()m} \\left(1-\\frac{1}{C_1(\\theta ,m)}\\right).", "$ We then proceed the proof by considering the Taylor expansion of ${1}/{C_1(x,m)}$ at $x = 0$ .", "To find the $n$ -th derivative of ${1}/{C_1(x,m)}$ we treat $\\frac{1}{C_1(x,m)}$ as a composite of $f(x)=x^{-1}$ and $C_1(x,m)$ , where the derivatives of $C_1(x,m)$ is available by the series expansion of hypergeometric function mentioned before.", "Then, by Faà di Bruno's formula [28], we have $\\left.\\mathsf {D}^n_x\\left(\\frac{1}{C_1(x,m)}\\right)\\right|_{x=0}= \\\\\\sum _{{\\bf b}\\in {\\cal B}_n}\\frac{n!", "(\\sum _{i=1}^{n} b_i)!", "}{\\prod _{i=1}^n (b_i!)}", "\\prod _{i=1}^n\\left( \\frac{(m)_i (-\\delta )_i}{(1-\\delta )_i i!}", "\\right)^{b_i},$ where ${\\cal B}_n$ is the set of $n$ -tuples of non-negative integers $(b_i)_{i=1}^n$ with $\\sum _{i=1}^n{i b_i}=n$ , and ${\\bf b} = (b_i)_{i=1}^n$ .", "(REF ) directly leads to the Taylor expansion of ${1}/{C_1(\\theta ,m)}$ , which combined with (REF ) leads to a series expansion of $\\mathsf {P}^{\\cap \\textnormal {o}}_{1,M}$ as function of $\\theta $ , $\\mathsf {P}^{\\cap \\textnormal {o}}_{1,M}=\\\\\\sum _{m=1}^M (-1)^{m} {M\\atopwithdelims ()m}\\sum _{n=1}^\\infty \\theta ^n\\sum _{{\\bf b}\\in {\\cal B}_n}\\frac{ (\\sum \\limits _{i=1}^{n} b_i)!", "}{\\prod \\limits _{i=1}^n (b_i!)}", "\\prod _{i=1}^n\\big ( (m)_i\\tau (i) \\big )^{b_i},$ where $\\tau (i) \\triangleq \\frac{ (-\\delta )_i}{(1-\\delta )_i i!", "}$ .", "Rearranging the sums and products in expression above yields $\\mathsf {P}^{\\cap \\textnormal {o}}_{1,M} = \\sum _{n=1}^\\infty a_n \\theta ^n$ , where $a_n = \\sum _{{\\bf b}\\in {\\cal B}_n}\\frac{ (\\sum \\limits _{i=1}^{n} b_i)!", "}{\\prod \\limits _{i=1}^n (b_i!)}", "\\prod _{i=1}^n \\big (\\tau (i)\\big )^{b_i}\\sum _{m=1}^M (-1)^{m} {M\\atopwithdelims ()m}\\prod _{i=1}^n \\big ((m)_i\\big )^{b_i}.$ Recall that ${\\bf b}\\in {\\cal B}_n$ indicates $\\sum _{i=1}^n{i b_i}=n$ .", "By Lemma REF , we have $a_n = 0$ for all $n<M$ , i.e., $\\mathsf {P}^{\\cap \\textnormal {o}}_{1,M} = O(\\theta ^M)$ as $\\theta \\rightarrow 0$ .", "Further, Lemma REF helps us to obtain the coefficient in front of $\\theta ^M$ , i.e., $a_M = \\sum _{{\\bf b}\\in {\\cal B}_M}\\frac{ M!", "(-1)^{\\sum _{i=1}^{M} b_i} (\\sum _{i=1}^{M} b_i)!", "}{\\prod _{i=1}^M (b_i!)}", "\\prod _{i=1}^M \\big (\\tau (i)\\big )^{b_i},$ which leads to the concise expression in the proposition by reusing Faà di Bruno's formula." ] ]
1403.0012
[ [ "On particle production in jets with quark-like and gluon-like\n fragmentation" ], [ "Abstract We study the $p/\\pi$ ratio in jets produced in simulated proton-proton collisions at $\\sqrt{s_{NN}}=7~\\mathrm{TeV}$ using Pythia $6.4$ Monte-Carlo generator.", "We compare the $p/\\pi$ ratio in the selected quark-like and gluon-like jets to a reference samples of quark- and gluon-jets tagged at Monte-Carlo level.", "We observe that the contamination in the selected jets significantly influences the observed ratios.", "This suggests, that the origin of the jet fixes the value of the $p/\\pi$ ratio within the model that we use." ], [ "Introduction", "At LHC, partons inside hadrons colliding at high energies may experience hard scatterings.", "This kind of interaction produces correlated showers of particles, that are experimentally identified as jets.", "We identify two type of partons as origins of such showers: quarks and gluons.", "The basic $2\\rightarrow 2$ hard scattering processes are summarized in Tab.", "REF[1]: Table: Basic QCD jets processes with their IDs in PYTHIAThe properties of the jet we observe is determined by how the original parton fragments along its way.", "There are differences between quarks and gluons.", "These differences are theoretically embraced in the QCD Casimir factors (also known as color factors), which are proportional to the probability that a parton radiates a soft gluon.", "Gluon's color factor ($\\mathrm {C_{A}}$ ) is more than twice larger than that of a quark ($\\mathrm {C_{F}}$ ) [2]: $\\frac{C_{A}}{C_{F}}\\ =\\ \\frac{3}{4/3}\\ =\\ 2.25,$ which means that gluons are expected to form higher multiplicity jets with softer fragments distributed in a larger jet-cone.", "The differences between quark- and gluon-jets were tested extensively at LEP in $e^{+}e^{-}$ collisions[3] and later at Tevatron in $p\\bar{p}$ collisions [4].", "In both experiments the above expectations have been fulfilled.", "Furthermore, at LEP the $C_{A},\\ C_{F}$ factors have been measured to be $C_{A} =2.89+-0.01(stat.", ")+- 0.21(syst.", ")\\ \\mathrm {and}\\ C_{F}=1.30+-0.01(stat.)+-0.09(syst.", ")$ .", "These are consistent with the QCD predictions [5].", "At LEP it was also observed that gluon-jets on average produce more protons than quark-jets [6].", "Of course, this finding has been included into the parameters of the fragmentation functions we use in Monte-Carlo (MC) simulations [7].", "On the other hand, this effect was not explained theoretically which opens possibilities for an investigation of this subject.", "In our previous work[8] we studied particle production in gluon- and quark-enriched events.", "In this work, we designed an exercise, to see, what is the crucial factor in a MC model, that determines the $p/\\pi $ ratio in a jet.", "Is it the nature of the leading parton, or is it the way that a jet object fills the phase space with particles?", "First step to be taken to answering this question, is to obtain samples of quark- and gluon-jets.", "There are efforts put into identifying jets as quarks and gluons at LHC.", "A thorough theoretical approach was taken in this matter in [9].", "The cited work aims at obtaining high-purity samples.", "We decided to take a rather different approach.", "In order to understand where the excess of protons in gluon-jets comes from and to answer the above question, we allow for contamination on purpose and observe how the final ratios change with respect to reference samples of MC quark- and gluon-jets.", "To summarize, the aim of our study is to see, how the observed differences between quark- and gluon-jets determine the final identified particle spectra within a widely used MC model.", "Further we want to motivate a study on data, that divides the jets into two samples with either quark-like or gluon-like fragmentation.", "On comparison with reference samples of quark- and gluon-jets, we can conclude whether the resulting $p/\\pi $ ratio is determined by the way the jet fragments or its origin.", "Our aim is not to compare different MC models.", "Instead we want to introduce a way to look at jets, that can shed more light on why gluon-jets produce more protons relative to quark-jets.", "If it is the origin that matters, the contamination will significantly affect the final particle spectra.", "If it is the fragmentation that is important, we should not observe difference within a certain fragmentation class, no matter the contamination." ], [ "The separation method", "In this section we introduce a method to separate the jets into two fragmentation classes.", "We do this by looking at the energy distribution inside a jet and the respective number of tracks.", "To obtain the cuts we use 3–jet and $\\gamma $ –jet events.", "We chose to work with these, since they provide samples of quark- and gluon-jets as they are their experimental sources." ], [ "Data sample and event selection", "To separate our sample we use the sets of 3–jet and $\\gamma $ –jet events obtained from simulated proton-proton(pp) collisions at $\\sqrt{s_{NN}} = 7~\\mathrm {TeV}$ .", "This way we calibrate the cuts used to distinguish between quark-like and gluon-like jets.", "In order to obtain these sets, we generated $100~\\textrm {milion}$ events containing QCD hard processes and $60~\\textrm {milion}$ events containing direct-photon production.", "For our study we use $\\mathrm {Pythia}~6.4$ MC generator[1], Perugia 0 tune[10].", "To reconstruct jets we use the $\\mathrm {anti-k_{T}}$ jet-finding algorithm [11] with the following parameters: $R=0.4$ , $p_{T}^{jet}>5~\\mathrm {GeV/c}$ , $|\\eta _{jet}|<0.5$ and $|\\eta _{particle}|<0.8$ .", "We have used the same set of kinematic cuts that are used by the ALICE collaboration at LHC.", "We have made such a decision, since we are interested to study identified particles inside jets and ALICE provides the necessary particle-identification capabilities to do so.", "Further, we divided these samples into a sample containing 2 leading jets from 2– and 3–jet events, a sample containing the least energetic jet from 3–jet events and finally a sample of jets from $\\gamma $ –jet pairs.", "In order to obtain these samples, we imposed specific event-selection criteria.", "What we are looking for in a 2–jet event is a pair of jets, which are well balanced and produced in plane.", "In order to do so, we selected events with two reconstructed jets and aplanarity[1] less than $0.01$ ." ], [ "Selection of 3–jet events", "What we are looking for in a 3–jet event is a triplet of jets, which are produced out of event-plane and take a topology, which is close to a 'Mercedes-Star like' one.", "In order to do so, we require aplanarity to be greater than $0.05$ in events, where 3 jets were reconstructed.", "We consider the least energetic jet to be a gluon." ], [ "Selection of $\\gamma $ –jet events", "For a $\\gamma $ –jet event we are looking for an event with a pair of direct photon and a jet with the following properties: $\\textrm {aplanarity}\\-<\\-0.006$ , $p_{T}^{imbalance}\\-<\\-0.1$$p_{T}^{imbalance}=(p_{T}^{\\gamma }-p_{T}^{jet})/p_{T}^{\\gamma }$ and $\\Delta \\varphi _{\\gamma -jet} \\in (2.8;3.4)$ ." ], [ "By these selections we acquired a sample of mixed quark– and gluon–jets (leading jets from the 2– and 3–jet events, further QG(MC)), that will serve as a pool for our selection, sample of gluon-jets (least energetic jet from the 3–jet event sample, further G(3J)) and finally, a sample of quark-jets from the $\\gamma $ –jet events (the single jet from $\\gamma $ –jet pair, further Q($\\gamma $ J)).", "A table with abbreviations of different jet-samples to be used further in the text can be found in Tab.", "REF on Page REF .", "Table: Table explaining the naming convention in the text and legends of figures" ], [ "Selection of quark- and gluon-like jets", "In this subsection we introduce a set of cuts that will be used on the QG(MC) to select quark-like and gluon-like jets.", "These set of cuts are designed to select jets with similar fragmentation to that of G(3J) and Q($\\gamma $ J)." ], [ "$R(90)$ and {{formula:72878823-d7e0-40c7-9967-b891be8eb12f}} variables", "At first we looked at how the energy is distributed in different jets based on which sample they belong to.", "This we did using a jet-shape-like variable, which we call $R(90)$ .", "It is the sub-cone size, which contains $90\\%$ of jet's momentum.", "The distribution of $\\langle R(90)\\rangle \\-vs.\\-p_{T}^{jet}$ is shown in Fig.", "REF .", "We see that the value for G(3J) is higher than for Q($\\gamma $ J), as expected from the quark and gluon difference, and further the value for QG(MC) lies between these two, suggesting, that it is a mixture of quarks and gluons.", "However, to be able to use the G(3J) and Q($\\gamma $ J) samples to introduce a selection cut, we need to show that indeed, on combination of the G(3J) and Q($\\gamma $ J), we obtain the same distributions of $R(90)$ as with QG(MC).", "This comparison is shown in Fig.", "REF in the bottom of left panel, where the average values and widths of $R(90)$ are compared for QG(MC) and Q($\\gamma $ J)+G(3J).", "On combination, the average values and widths are comparable.", "Figure: Left: R(90)R(90) as a function of jet's momentum for different jet-selections(top).", "In the two bottom panels the sum of G(3J) and Q(γ\\gamma J) is compared to QG(MC) in terms of average values and widths.", "Right: ΔR(90)\\Delta R(90) as a function of jet's momentum for different jet selections(top).", "In the two bottom panels the sum of G(3J) and Q(γ\\gamma J) is compared to QG(MC) in terms of average values and widths.We see, that the $R(90)$ depends on jet's momentum.", "To reduce this dependence we introduce $\\Delta R(90)_{\\lbrace G(3J);Q(\\gamma J)\\rbrace }=R(90)_{\\lbrace G(3J);Q(\\gamma J)\\rbrace }-\\langle R(90)\\rangle _{QG(MC)}$ at a given $p_{T}^{jet}$ momentum bin.", "This way we obtain the distribution in Fig.", "REF , right panel.", "We see that we have reduced the momentum dependence and can thus work in a wider range of momenta.", "We chose to work with the jets $p_{T}^{jet}\\in (16;56)~\\mathrm {GeV/c}$ .", "The distribution of $\\Delta R(90)$ for G(3J) and Q($\\gamma $ J) in given momentum range is shown in Fig.", "REF .", "We can distinguish $\\Delta R(90)$ intervals in which either the G(3J) or Q($\\gamma $ J) dominate, although they are overlapping.", "We select the following cuts for quark-like and gluon-like jets: $\\mathrm {G(sel)}:\\-\\Delta R(90)\\in (0.02, 0.04)$ , $\\mathrm {Q(sel)}:\\-\\Delta R(90)\\in (0.02, 0.04)$ .", "Figure: ΔR(90)\\Delta R(90) distribution for G(3J) and Q(γ\\gamma J)." ], [ "Number of tracks", "We want to obtain 2 samples of jets so, that each of them contains jets with similar, quark– or gluon–like fragmentation.", "The first step is to select jets with similar energy distribution inside a jet–cone as described in previous paragraph.", "The second step is to look at the number of tracks inside each jet in the chosen $\\Delta R(90)$ sub-intervals based on whether this jet comes from G(3J) or Q($\\gamma $ J).", "The distributions of the number of tracks inside jets are shown in Fig.", "REF .", "We see, that based on which sample the jet came from, the distribution of the number of tracks varies according to expectations, even in a narrow $\\Delta R(90)$ bin.", "This means, that Q($\\gamma $ J) have smaller number of tracks than G(3J).", "To enhance the separation of the jets into two samples of different fragmentation, we apply an additional cut on the number of tracks in a jet: $\\mathrm {G(sel)}:\\-N_{tracks}=8$ , $\\mathrm {Q(sel)}:\\-N_{tracks}=3$ .", "The selection we have made, allows for contamination of the selected samples.", "As mentioned earlier, we do not want to get rid off this contamination, rather we want to see, how the contamination will influence particle spectra inside jets.", "Figure: Number of tracks distribution for G(3J) and Q(γ\\gamma J) events in selected ΔR(90)\\Delta R(90) intervals." ], [ "Comparison of selected quark– and gluon–like jets", "In this section we compare the $p/\\pi $ ratios of the selected jets to the 3–jet and $\\gamma $ –jet samples and MC quark–jets (Q(MC)) and MC gluon–jets (G(MC)).", "First we compare the Q($\\gamma $ J) and G(3J) samples to Q(MC) and G(MC).", "The MC jet–samples were obtained by simulating events with hard scatterings producing either quarks or gluons in the final state.", "These processes and their respective process IDs can be found in Tab.", "REF on page REF .", "Subsequently, in these events we ran the jet–finding algorithm and selected 2–jet events as described in the previous section.", "The comparison is shown in Fig.", "REF , in the very left panel.", "Figure: p/πp/\\pi ratio compared for different jet selections.", "Left: p/πp/\\pi ratio for γ\\gamma –jet, 3–jet and Q(MC) and G(MC).", "Middle: p/πp/\\pi ratio for a subset of Q(MC) and G(MC), passing the criteria selecting quark– and gluon–like jets, respectively, compared to Q(MC) and G(MC).", "Right: p/πp/\\pi ratio for Q(sel) and G(sel) compared to the one for Q(MC) and G(MC).As can be seen, the $p/\\pi $ ratios between the respective samples of quark– and gluon–jets are comparable and our assumption to use G(3J) and Q($\\gamma $ J) for studying the influence of quarks and gluons on the $p/\\pi $ ratios is justified within the used MC model.", "Since we are applying cuts to the mixed QG(MC) sample, before we compare it to the G(MC) and Q(MC), we need to see if the cuts themselves distort the studied ratio.", "In order to see this, from each G(MC) and Q(MC) we select a subset of jets passing the selection criteria either for quark– or gluon–like jets.", "From G(MC) we select a subset of jets passing gluon–like criteria (G(sel) $\\in $ G(MC)) and from Q(MC) we select a subset of jets passing quark–like criteria (Q(sel) $\\in $ Q(MC)).", "The comparison of these subsets with the inclusive G(MC) and Q(MC) is shown in Fig.", "REF , in the middle.", "We see that, even though we did select a very narrow fragmentation–class of jets, the ratios are comparable with the MC samples.", "This already suggests that in Pythia it is rather the origin than the fragmentation property of jet itself which determines the identified particle spectra inside a jet.", "Finally, we proceed with the comparison of the Q(sel) and G(sel) samples with the Q(MC) and G(MC).", "The comparison is shown in the very right panel of Fig.", "REF .", "We observe that the gluon–like selection works fine, however, the selected quark–like jets have the ratio closer to G(MC) than to Q(MC).", "The reason lies in the high contamination of the Q(sel) with the real gluons.", "We are at energies where the processes producing two gluons in the final state of a hard-scattering have highest probability to occur [12] and this gives them a high probability to pass our quark selection criteria, thus contaminating the quark–like jets.", "This supports our statement from previous paragraph, where we say that in the particular MC model which we use it is rather the origin of the jet than its fragmentation that determines the final particle spectra." ], [ "Discussion and conclusions", "In this work we have motivated a novel study of particle production in jets, based on their fragmentation, in order to complement other studies focusing on the origin of jets.", "We have concluded that despite the different fragmentation of jets originating from quarks or gluons, it has no influence on the final particle spectra inside these jets – within the used MC model.", "To see whether this statement holds, we suggest to proceed with similar study on experimental data, investigating the dependence of the $p/\\pi $ ration inside jets on their fragmentation." ], [ "acknowledgements", "This work was funded by the OTKA Grant NK77816 and NK106119.", "I would like to personally thank Tamás Sándor Bíró and Péter Lévai for valuable consultations." ] ]
1403.0412
[ [ "Measuring the spin of black holes in binary systems using gravitational\n waves" ], [ "Abstract Compact binary coalescences are the most promising sources of gravitational waves (GWs) for ground based detectors.", "Binary systems containing one or two spinning black holes are particularly interesting due to spin-orbit (and eventual spin-spin) interactions, and the opportunity of measuring spins directly through GW observations.", "In this letter we analyze simulated signals emitted by spinning binaries with several values of masses, spins, orientation, and signal-to-noise ratio.", "We find that spin magnitudes and tilt angles can be estimated to accuracy of a few percent for neutron star--black hole systems and $\\sim$ 5-30% for black hole binaries.", "In contrast, the difference in the azimuth angles of the spins, which may be used to check if spins are locked into resonant configurations, cannot be constrained.", "We observe that the best performances are obtained when the line of sight is perpendicular to the system's total angular momentum, and that a sudden change of behavior occurs when a system is observed from angles such that the plane of the orbit can be seen both from above and below during the time the signal is in band.", "This study suggests that the measurement of black hole spin by means of GWs can be as precise as what can be obtained from X-ray binaries." ], [ "Introduction", "Advanced LIGO [1] and Advanced Virgo [2] will start collecting data in 2015-2016 [3].", "KAGRA [4] and LIGO India [5] will join the network later in the decade.", "Ground based detectors are expected to make the first direct detection of gravitational radiation and to start gravitational-wave (GW) astronomy.", "The most promising sources are compact binary coalescences (CBC) of two neutron stars (BNS), two black holes (BBH) or a neutron star and a black hole (NSBH), which could be detected at a rate of 40, 20, and 10 per year respectively [6]These are the realistic detection rates of [6].", "The possible values span two orders of magnitude..", "Analysis of such signals promises to shed light on several open problems in astrophysics.", "Accurate estimation of neutron star and black hole masses will help in checking the existence of a gap between the maximum mass of a neutron star and the minimum black hole mass [7].", "The observed distribution of spin magnitudes and tilts will help in understanding binary formation and evolution, including issues such as supernovae kicks and common envelope phases.", "Measurement of the difference in the azimuth angles could reveal if spin vectors in CBC are locked into resonant configurations [8], [9].", "Parameter estimation of detected signals should thus have a central role in the next years, with a large impact in astrophysics.", "Inside the LIGO-Virgo collaboration, reliable Bayesian parameter estimation algorithms [10], [11], [12], [13], [14] have been written and extensively used.", "Much work has focused on spinless CBC sources (e.g.", "[15] and references therein), which can be a good approximation when both objects are neutron stars [17].", "Systems with spins aligned with the orbital angular momentum, and the resulting large mass-spin degeneracy, have been extensively studied [18], [19], [20], [21].", "Several papers have analyzed NSBH systems and the best way to parametrize the signals they emit [23], [24], [34], also assessing spin measurement [10], [11], [22].", "Fewer studies have focused on systems with two precessing spins [25], [26], [27], [28], usually analyzing only a few signals.", "In this letter we consider a larger set of NSBH and BBH, where both objects have precessing spins.", "We perform parameter estimation using an advanced LIGO-Virgo network [3], and find that the black hole spin in NSBH and BBH systems can be estimated with a precision comparable to spin measurements from X-ray binaries [29] using the continuum-fitting [30] or “Fe-line” [31], [32] methods.", "Both of them are indirect measurements which rely on assumptions about the disk physics, which may bias the measurements.", "This is why having an independent and direct way to estimate the spin of black holes, GWs, is of great importance.", "Furthermore, we analyze the dependence of parameter estimation capabilities on the orientation of the CBC.", "We find that errors are smallest when the line of sight is perpendicular to the total angular momentum, which is expected [23], [33].", "We show how there is a clear change of behavior in the parameter estimation capability if the plane of the orbit can be observed both from above and below, due to precession." ], [ "Method", "Signals emitted by quasi-circular CBC with generic spins depend on 15 unknown parameters [25] with non-trivial correlations [20], [21], [22].", "In this work we are primarily interested in how parameter estimation performances depend on the different possible spin configurations.", "We have therefore chosen a set of simulations which explore the spin parameter space, using only a small subset of the other parameters, in order to study the phenomenology of the results.", "In particular we assigned fixed values of masses to our simulated systems: NSBH were chosen to have mass $(1.4,10) M_\\odot $ , while we considered two possible kinds of BBH, $(7.5,7.5) M_\\odot $ and $(10,5) M_\\odot $ .", "For the NSBH, the reduced spin magnitude ($a\\equiv |\\vec{S}|/m^2$ ) of the black hole was 0.9 while the neutron star had a spin of 0.1.", "For the BBHs, all pair-wise combinations of 0.9 and 0.1 were used.", "For each of these systems, we considered two possible orientations of the spin vectors $\\vec{S}_1$ and $\\vec{S}_2$ : both spins forming a tilt angle $\\tau $ of $60^\\circ $ with respect to the orbital angular momentum and parallel to each other Due to precession, tilt angles evolve with time or, equivalently, frequency.", "We quote their values at 100Hz [34]., or $\\vec{S}_1$ forming an angle of $45^\\circ $ degrees and $\\vec{S}_2$ an angle of $135^\\circ $ .", "In both cases the orbital angular momentum $\\vec{L}$ and the spins lie on the same plane at the reference frequency.", "The first configuration is such that it maximizes the scalar product of the spins while the second maximizes the cross product.", "Thus we explore the cases that give large values of the spin interaction terms in the post-Newtonian expansion [35], [36], with a stronger precession in the first case, because the resulting total spin will be more misaligned with respect to $\\vec{L}$ .", "Each system was analyzed with three possible orientations, i.e.", "the angle $\\theta _{\\vec{J}\\vec{N}}$ between the total angular momentum and the line of sight, as shown by the colorbars in Fig.", "REF and Fig.", "REF below.", "To study the dependence of parameter estimation capabilities on the loudness of the event we have analyzed all systems at 3 network signal-to-noise ratios (SNR [25]): threshold for detection (12), moderate (17), or high (30).", "These values correspond to distances in the range [68 - 970] Mpc, the exact value depending on the mass, spin, and orientation.", "Waveforms were generated using the SpinTaylorT4 (STT4) approximant [35], [36], working at the 3.5 Post-Newtonian (PN) phase order, while neglecting amplitude corrections (which have negligible effects, at least for NSBH [37]).", "STT4 waveforms can only describe the inspiral part of a waveform, and one expects the merger and ringdown to become more significant for more massive binaries.", "Our choice was forced by the lack of reliable IMR [38] waveforms with precessing spins at the time of the analysis.", "Furthermore, it has been shown that merger and ringdown do not play significant role for systems with masses below $\\sim 20 M_\\odot $  [39].", "Because this study is not about sky localization accuracy, and to better appreciate the effect of the intrinsic and orientation parameters on parameter estimation, we have put all sources in the same sky positionWe have verified this sky position is not “special”, and that nearly all sky positions would lead to very similar results., which is considered unknown.", "For the same reason, even though we performed the analysis using the design strain sensitivity of LIGO and Virgo [3], we have assumed the actual realization of the noise was zero.", "The uncertainties we quote are equal to the frequentistic average over several noise realizations at the 1/SNR$^3$ level [15], [16].", "The analysis was carried out using the Nested Sampling [40], [12] version of LALInference, the Bayesian parameter estimation tool developed by the LIGO-Virgo Collaboration  [25], [14], [41], using spins' magnitude, tilt and azimuth difference to parametrize spins [24]." ], [ "Results", "For equal mass systems the angle between the spins and the total angular momentum, J, is almost constant, whereas in the unequal mass case it oscillates inducing more waveform modulations.", "Moreover, in the unequal mass and unequal spin case the spin-dependent terms in the waveform phase are larger for our configuration [42], which should aid parameter estimation.", "We thus expect spin parameters to be best estimated for NSBH, while the worst case scenario should be found for equal mass and equal spin BBH.", "On the top panel of Fig.", "REF we show the percent error in the estimation of the BH dimensionless spin magnitude $a_1$ for the NSBH systems.", "We notice that even at moderate SNRs it can be as small as 5%, and become of the order of $1-2$ % for loud signals.", "These accuracies are comparable to what can be obtained with X-ray binaries [29], and thus GWs should provide a reliable and independent way of measuring the spins of black holesX-ray binaries will not produce GWs measurable with LIGO and Virgo, we are thus not suggesting all methods can be used on the same systems..", "Figure: 1-sigma error in the estimation of the spin magnitude (top, percent) and tilt angle (bottom, radians) of the black hole for NSBH systems.", "The color represents θ J →N → \\theta _{\\vec{J}\\vec{N}} in radians.", "Small symbols are systems for which both objects have a tilt angle of 60 ∘ ^\\circ ; large symbols have τ 1 =45 ∘ \\tau _1=45^\\circ and τ 2 =135 ∘ \\tau _2=135^\\circ .We also notice how, generally speaking, configurations in which the systems are seen from angles close to $\\pi /2$ lead to better parameter estimation.", "This is to be expected because the most favorable lines of sight are those for which, during their precession around J, $\\vec{L}$ and the spins vectors show the largest variation along the line of sight.", "This occurs when $\\vec{N} \\perp \\vec{J}$ , i.e.", "the amount of modulation in the waveform increases for decreasing $|\\theta _{\\vec{J}\\vec{N}} -\\pi /2|$  [33], [23].", "Although this is the most probable configuration for isotropically distributed orientations, systems with $\\theta _{\\vec{J}\\vec{N}}$ $\\sim \\pi /2$ are harder to detect because the emission pattern is minimum on the plane of the orbit [25].", "Finally, we notice that the two tilt configurations lead to similar results, especially for well measured events.", "On the bottom panel of Fig.", "REF we show the error of the BH tilt angle.", "Here the errors can be as good as $\\sim 0.04$  Rad for loud signals and can be below $0.1$  Rad even at moderate SNR.", "At threshold SNR we see a large spread of possible outcomes, including instances of multimodal and highly correlated posteriors that make it difficult to uniquely resolve the systems' parameters.", "As expected, the NS spin is not well estimated, due to the small spin of the NS.", "We find that the relative error in the estimation of the component masses is $\\sim 2-5$ % at high SNR (the range encompasses all events), $\\sim 3-5\\%$ at medium SNR, and can be $\\sim 10-30\\%$ at threshold.", "Errors for the chirp mass $\\mathcal {M}\\equiv \\left[\\frac{m_1^3 m_2^3}{m_1+m_2}\\right]^\\frac{1}{5}$ are always smaller than 1% (and of the order of 0.1% for SNR=30 events for all spin orientations and $\\theta _{\\vec{J}\\vec{N}}$ ).", "Precessing spins induce modulation of both the phase and the amplitude of the waveform: in particular, amplitude modulation should help break the well known degeneracy between distance and inclination [10], [25], [15].", "We find that this is indeed the case, and that luminosity distance can be estimated as well as $4-8$ % for loud events and $\\lesssim 10\\%$ for SNR 17 events.", "This is much better than what can be obtained for spinless signals [44] and is comparable to calibration induced errors [45].", "This suggests GWs from spinning NSBH may be optimal standard sirens [46].", "Finally, we remark that accurately measured events were usually estimated quite precisely, with a single posterior mode well peaked at the true value.", "We will report more about precision in a forthcoming publication.", "We now move to binary black holes, for which we expect worse spin measurements as a result of the smaller component mass difference.", "In what follows we will quote numbers for the $(10,5)M_\\odot $ systems.", "For equal mass $(7.5,7.5)$  $M_\\odot $ BBHs, errors for systems with spin 0.9-0.1 are a factor of several higher than the corresponding $(10,5) M_\\odot $ systems.", "When both spins are 0.9 the errors are only slightly worse than for equal mass.", "This is because when the spins are similar the waveform is less sensitive to the mass ratio (see e.g.", "eq 3.21 of [42]).", "In Fig.", "REF we show the relative error of the spin magnitude of the more massive black hole.", "Comparison against the top panel of Fig.", "REF reveals that errors for BBH are indeed larger by a factor of a few.", "To be more precise, relative errors of $5\\%$ are possible for loud events, but errors of $10-30$ % are otherwise more typical.", "When both spins are small (not shown in the plot), it is hard to estimate the spin magnitude, and the posterior distribution only shows some hint that small values may be preferred.", "Figure: 1-sigma percent error in the estimation of the most massive black hole spin magnitude in (10-5)(10-5)M ⊙ M_\\odot BBHs.The color represents θ J →N → \\theta _{\\vec{J}\\vec{N}} in radians.", "Small symbols are systems for which both objects have a tilt angle of 60 ∘ ^\\circ , large symbols have τ 1 =45 ∘ \\tau _1=45^\\circ and τ 2 =135 ∘ \\tau _2=135^\\circ .", "Circles are systems where both objects have spin magnitude of 0.9; squares have a spin of 0.9 in the 10M ⊙ M_\\odot black hole, and 0.1 in lighter one.", "Results for s 1 =s 2 =0.1s_1=s_2=0.1 systems are not shown, as the errors in that case are above 100%.Similar conclusions apply for the tilt angles, which cannot be estimated with errors smaller than $\\sim 0.1$  Rad, and for which more typical errors are above $0.2$  Rad.", "We observe strong correlations between the accuracies of the measurement of $a_1$ and $\\tau _1$ .", "This is not unexpected since, at the lowest order, what matters is $\\vec{L}\\cdot \\vec{S}$ .", "A small variation in the spin magnitude can thus be compensated by a variation of the tilt angle.", "These are summarized in Fig.", "REF .", "The upper branch is made of systems where both objects have spin magnitude of 0.1, which makes it hard to estimate the tilt angle.", "In the lower branch we find all systems for which $a_1=0.9$ .", "The best configurations are those with the largest spin-magnitude ratio (i.e.", "$a_1=0.9$ and $a_2=0.1$ ) and inclination angles $\\theta _{\\vec{J} \\vec{N}} \\sim \\pi /2$ .", "Figure: Correlation between 1-σ\\sigma errors for the spin magnitude and the tilt angle of the 10M ⊙ 10 M_\\odot black hole.", "Triangles are systems with reduced spin magnitude of 0.9 for both black holes.", "Stars are 0.9-0.1 systems, and circles are 0.1-0.1 systems.", "The size is proportional to the SNR (12,17, or 30).Similarly to the NSBH systems, the errors associated with the smaller of the two bodies are larger.", "We find that the magnitude of $\\vec{S}_2$ can be estimated with relative errors of 15% (25%) for loud (moderate) SNRs when the systems are seen from $\\theta _{\\vec{J}\\vec{N}}$ not too close to 0 or $\\pi $ and $a_2=0.9$ .", "Furthermore, the errors in the estimation of $a_2$ for $a_1=a_2=0.9$ systems with small $\\theta _{\\vec{J}\\vec{N}}$ show barely any variation with the SNR, which is due to the near total lack of modulation and the small size of the secondary black hole.", "When $a_2=0.1$ the spin magnitude of the lighter black hole cannot be constrained.", "The chirp masses of $(10,5)$ $M_\\odot $ BBH are estimated with relative errors between 1% (weak signals) and $\\sim 0.3\\%$ (loud events), and these errors do not show large variations with spin magnitude or orientation.", "This is to be expected because the chirp mass is already estimated quite well at the 0 PN order [47], and thus is not affected much by higher order spin terms.", "The error for the component masses shows some dependence on the spin configuration, being slightly better for strongly spinning systems, and being of the order of $\\sim 7-15\\%$ ($\\sim 10-20\\%$ ) for high (medium-low) SNR events.", "The distance is estimated slightly worse than for NSBH, which is not surprising given the higher mass of the BBH (more massive systems generate shorter waveforms, which usually leads to larger errors [47], [48]).", "Finally, we observe that the only configurations for which the posterior distribution of the spins' azimuthal angle difference is significantly different from the prior are $\\theta _{\\vec{J}\\vec{N}}$ $\\sim \\pi /2$ , SNR 30, events with both spins equal to 0.9.", "However the associated errors are still large, $\\sim 30-40$ degs.", "While discussing the NSBH results, we mentioned that the smallest errors are obtained when $\\theta _{\\vec{J}\\vec{N}}$ is close to $\\pi /2$ , and we see the same for BBH systems.", "We thus complemented our study by taking one of the $(10,5)~M_\\odot $ BBHs (the one with $a_1$ =0.9, $a_2$ =0.1, $\\cos \\tau _1 =\\cos \\tau _2 =0.5$ and SNR=17) and analyzing it with different $\\theta _{\\vec{J}\\vec{N}}$ .", "In Fig.", "REF we show the 1-$\\sigma $ errors for the magnitude (circles) and tilt (pentagons) of the 10$M_\\odot $ BH spin against the value of $\\theta _{\\vec{J}\\vec{N}}$ .", "The SNR is kept fixed by varying the distance.", "As expected the errors reach their minimum and look symmetrical around $\\theta _{\\vec{J} \\vec{N}} \\sim \\pi /2$ rad, while the maximum is reached when $\\vec{J} || \\vec{N}$  [33], [23].", "Figure: 1-σ\\sigma errors for the spin magnitude (circles), tilt angle (pentagons, radians) and θ J →N → \\theta _{\\vec{J}\\vec{N}} (triangles, radians) against the injected value of θ J →N → \\theta _{\\vec{J}\\vec{N}} (in units of π\\pi ).", "A change of behavior in the region of θ J →N → \\theta _{\\vec{J}\\vec{N}} [1.2-1.8][1.2-1.8] rad is clearly visible.Fig.", "REF also shows the variation of the errors in the estimation of $\\theta _{\\vec{J}\\vec{N}}$ (triangles).", "We see that the error reaches a minimum when $\\theta _{\\vec{J}\\vec{N}}$ $\\sim \\pi /2$ , but now there appears to be a change of behavior in the region $\\theta _{\\vec{J}\\vec{N}}$ $ \\in [1.2-1.8]$ rad.", "The boundary angles of this region correspond to lines of sight such that the plane of the orbit will be observed both from above and below during its precession (“tropical region”).", "The width of this region is twice the precession angle, i.e.", "the angle between Jand $\\vec{L}$ .", "We also notice that in the tropical region there is a strong reduction in the mass-spin and distance-iota correlations, with a corresponding, often dramatic, reduction of the errors of these quantities." ], [ "Conclusions", "We report an initial parameter estimation analysis of the gravitational radiation emitted by spinning compact binary systems consisting of two black holes or a neutron star and a black hole.", "We simulated systems with different values of masses, spins, signal-to-noise ratio, and orbital inclination.", "We find that the magnitude of black hole spins in a $(1.4,10)$ $M_\\odot $ NSBH can be estimated to an accuracy of a few percent.", "This is comparable to what can be obtained with X-ray binaries, and is not affected by the same systematics.", "The tilt angle of the black hole spin can be pinned down with an error of $\\lesssim 0.1$  Rad.", "accuracy.", "Our analysis of solar mass BBH shows that the errors in spin magnitude ($5-30\\%$ ) and angles ($\\gtrsim 0.1$  Rad) will be larger than for NSBH, mainly due to the smaller mass ratio.", "We considered both equal and unequal (ratio 2:1) mass BBHs.", "The errors are slightly larger for equal mass systems if the spins are similar, and noticeably larger when the spins are different.", "We show that the errors in the estimation of the spins are at their minimum when the line of sight is perpendicular to the total angular momentum, and we observe that correlations and errors of other parameters are sensibly smaller when systems are seen from their tropical regions.", "We find that the difference in the spins' azimuthal angles cannot be constrained unless both objects have large spins, the SNR is high, and the system is observed from its tropical region, in which case the errors are still $\\sim 30$ degs.", "A forthcoming work will focus on the aspects we had to neglect in this letter and will expand in several directions." ], [ "Acknowledgments", "SV, RL and VR acknowledge the support of the National Science Foundation and the LIGO Laboratory.", "LIGO was constructed by the California Institute of Technology and Massachusetts Institute of Technology with funding from the National Science Foundation and operates under cooperative agreement PHY-0757058.", "JV was supported by the research program of the Foundation for Fundamental Research on Matter (FOM), which is partially supported by the Netherlands Organisation for Scientific Research (NWO), and by STFC grant ST/K005014/1.", "VR is supported by a Richard Chase Tolman fellowship at the California Institute of Technology.", "RS is supported by the FAPESP grant 2013/04538-5.", "The authors would like to acknowledge the LIGO Data Grid clusters, without which the simulations could not have been performed.", "Specifically, these include the Syracuse University Gravitation and Relativity cluster, which is supported by NSF awards PHY-1040231 and PHY-1104371.", "Also, we thank the Albert Einstein Institute in Hannover, supported by the Max-Planck-Gesellschaft, for use of the Atlas high-performance computing cluster.", "We would also like to thank D. Chakrabarty, W. Del Pozzo, R. Essick, B. Farr, W. Farr, V. Grinberg, F. Harrison, S. Hughes, V. Kalogera, E. Katsavounidis, A. Lundgren, E. Ochsner, R. O'Shaughnessy, R. Penna, and A. Weinstein for useful comments and suggestions.", "This is LIGO document number P1400024." ] ]
1403.0129
[ [ "Langevin dynamics, large deviations and instantons for the\n quasi-geostrophic model and two-dimensional Euler equations" ], [ "Abstract We investigate a class of simple models for Langevin dynamics of turbulent flows, including the one-layer quasi-geostrophic equation and the two-dimensional Euler equations.", "Starting from a path integral representation of the transition probability, we compute the most probable fluctuation paths from one attractor to any state within its basin of attraction.", "We prove that such fluctuation paths are the time reversed trajectories of the relaxation paths for a corresponding dual dynamics, which are also within the framework of quasi-geostrophic Langevin dynamics.", "Cases with or without detailed balance are studied.", "We discuss a specific example for which the stationary measure displays either a second order (continuous) or a first order (discontinuous) phase transition and a tricritical point.", "In situations where a first order phase transition is observed, the dynamics are bistable.", "Then, the transition paths between two coexisting attractors are instantons (fluctuation paths from an attractor to a saddle), which are related to the relaxation paths of the corresponding dual dynamics.", "For this example, we show how one can analytically determine the instantons and compute the transition probabilities for rare transitions between two attractors." ], [ "Introduction", "Many natural and experimental turbulent flows display bistable behavior, in which one observes rare and abrupt dynamical transitions between two attractors that correspond to very different subregions of the phase space.", "The most prominent natural examples are probably the Earth magnetic field reversals (over geological timescales), or the Dansgaard-Oeschger events that have affected the Earth climate during the last glacial period, and are probably due to several attractors of the turbulent ocean dynamics [53].", "Experimental studies include examples in two-dimensional turbulence [59], [40], [14], [26], rotating tank experiments [66], [58] related to the quasi-geostrophic dynamics of oceans (Kuroshio current bistability [58], [52]) and atmospheres (weather regime blockings), three dimensional turbulent flows in a Von Kármán geometry [54], the magnetic field reversal in MHD experiments [4], [26], Rayleigh-Bénard convection cells [48], [18], [60], [16].", "The theoretical understanding of these transitions is an extremely difficult problem due to the large number of degrees of freedoms, the broad spectrum of timescales and the non-equilibrium nature of these flows.", "Up to now there have been an extremely limited number of theoretical results, the analysis being mostly limited to analogies with models with few degrees of freedom.", "One example with an interesting phenomenological approach results in the clever use of symmetry arguments in order to describe effectively the largest scales of MHD experiments [50].", "This strategy has been fruitful in several examples in regimes close to deterministic bifurcations, where the hypothesis of describing the turbulent flow by few dominant modes, even if based only up to now on empirical arguments, is likely to be relevant.", "In fact it has led to the prediction of non-trivial qualitative features of the rare transitions.", "The main problem is in how to develop a general theory for these phenomena?", "When a complex turbulent flow switches at random times from one subregion of the phase space to another, the first theoretical aim is to characterize and predict the observed attractors.", "This is already a non-trivial task, as no picture based on a potential landscape is available.", "Indeed, this is especially tricky when the transition is not related to any symmetry breaking.", "An additional theoretical challenge is in being able to compute the transition rates between attractors.", "It is also often the case that most transition paths from one attractor to another concentrate close to a single unique path, therefore a natural objective is to compute this most probable transition path.", "In order to achieve these goals, it is convenient to think about the framework of large deviation theory, in order to describe either the stationary distribution of the system, or in computing the transition probabilities of the stochastic process.", "In principle, we could argue that from a path integral representation of the transition probabilities [68], and the study of its semi-classical limit in an asymptotic expansion with a well chosen small parameter, we could derive a large deviation rate function that would characterize the attractors and various other properties of the system.", "When this semi-classical approach is relevant, one expects a large deviation result, similar to the one obtained through the Freidlin-Wentzell theory [24].", "If this notion is correct, then this would explain why these rare transitions share many analogies with phase transitions in statistical mechanics and stochastic dynamics with few degrees of freedom.", "The theoretical issues in order to assess the validity of such a broad approach are however numerous: what is the natural asymptotic large deviation parameter?", "Why and when should the finite dimensional picture be valid?", "How does one actually compute the large deviation rate function and characterize its minima?", "Should one expect that the dynamics of the rare transition be well described by few degrees of freedom?", "And so on.", "Up to now, these questions have no clear or precise answers for any meaningful turbulence problems.", "The aim of this paper is to make small steps in this direction.", "We will study the class of models that describe two-dimensional and quasi-geostrophic dynamics.", "Those are arguably the simplest class of turbulence models for which phase transitions and bistability phenomena exist.", "For simplicity, we will consider forces which are stochastic, white in time, Gaussian noises.", "In previous papers, we have given partial answers to the theoretical challenges discussed above.", "For instance, for the two-dimensional stochastic Navier-Stokes equations, we have argued [14] that in the inertial limit (weak noise and dissipation), one should expect the invariant measure to be concentrated close to the attractors of the inertial dynamics (the two-dimensional Euler equations).", "This partially answers the issue of characterizing the attractors, and helps us to empirically find the bistable regimes, based on bifurcation diagrams for the inertial dynamics.", "Indeed, numerical simulations showed that the Navier-Stokes dynamics actually concentrates close to the set of attractors of the two-dimensional Euler equations [14], and display bistable behavior in some parameter range.", "In order to develop further the theoretical understanding, we have used stochastic averaging techniques to describe the long time dynamics of the barotropic quasi-geostrophic model in a regime where the main attractors are simple parallel flows (zonal jets) [13].", "Moreover, this model also displays multiple attractors [13], which can be studied using large deviation theory.", "Furthermore, we have also developed a similar theoretical approach for the stochastic Vlasov equations where bistability was also discussed [44], [45].", "However these works only partially address the theoretical questions: mainly in predicting the set of attractors and in determining the phase transitions and bistability regimes.", "However, up to now it has not been possible to explicitly compute the transition rates and transition probabilities for these systems.", "For turbulent dynamics, in inertial limits, the attractors are expected to be subclasses of the attractors of the inertial dynamics, as we discussed above.", "The natural attractors of the inertial dynamics are those derived from the microcanonical measures, namely the macroscopic equilibria of the Miller-Robert-Sommeria theory [55], [43], [56], [57] (please note the many recent contributions to the application of this theory [63], [65], [29], [30], [31], [62], [47], [46], [67], [11], [22], [51]).", "In essence, these microcanonical measures are characterized by an entropy functional that is actually a large deviation rate functional (see for instance see [42], [8]).", "As explained in [9], the related entropy maximization is closely related to energy-Casimir variational problems.", "This link highlights the possibility that energy-Casimir functionals are natural potentials for the effective description of the largest scales in these turbulent flows.", "We address this point further in the conclusion.", "The goal of this paper is to define and to study a class of Langevin dynamics associated to energy-Casimir potentials and in the investigation of the related stochastic process.", "We show that this stochastic process is an equilibrium one, in the sense that either it verifies detailed balance, or a generalization of the detailed balance property.", "In the latter, the time reversed stochastic process is not simply the initial process but belongs to the same class of physical model (for instance in Langevin dynamics of particles in magnetic fields).", "From this time reversal symmetry, identified at the level of the action, we can show that the quasi-potential related to the action minimization can be explicitly computed, and is actually the energy-Casimir functional.", "Moreover, we can also explain why fluctuation trajectories (the most probable paths to get a rare fluctuation) are time reversed relaxation trajectories of the dual dynamics, as in classical Langevin dynamics.", "In situations with bistability (when the quasi-potential has two or more local minima), we recover the classical picture: an Arrhenius law for the transition rate and a typical transition trajectory that follows an instanton trajectory (the time reversed trajectory of the relaxation path of the dual dynamics from the lowest saddle point).", "All these properties are derived from the orthogonality of the Hamiltonian part of the dynamics to the potential part, which is a consequence of the fact that the potential is conserved under the Hamiltonian dynamics.", "We discuss a specific example where the energy-Casimir functional leads to bistable regimes, and describe a bifurcation diagram that includes a tricritical point (a bifurcation from a first order phase transition to a second order phase transition).", "Close to the critical point, the turbulent dynamics can be reduced to the effective dynamics involving only a few degrees of freedom related to the null space of the potential at the transition point, by analogy with the phenomenology of bifurcations in deterministic systems.", "However, far away from the tricritical point such a reduction does not seem to be relevant.", "These Langevin dynamics are very interesting examples of turbulent dynamics, that fit within the classical framework of equilibrium stochastic thermodynamics.", "All the recent results related to stochastic thermodynamics: Gallavotti-Cohen fluctuation relations, relations between the entropy production and the probability of paths, and so on, could be easily generalized for these Langevin dynamics.", "Together with genuine turbulence dynamics, they also display fascinating dynamical behavior including phase transitions.", "The relevance of these dynamics for real physical phenomena should however be questioned.", "As discussed in the paper and in the conclusion, several examples of these Langevin dynamics actually relate to physical microscopic dissipation mechanisms (linear friction and/or viscosity), but is not true in general.", "When this analogy is incorrect, these dynamics should be understood, at best, as effective models for the largest scales of the flows.", "All these aspects and the resulting limitations and benefits of these model to real flows are further discussed in the conclusion.", "This Langevin dynamics approach also opens up a new set of very interesting theoretical and mathematical issues.", "For instance, dynamics that involve white in space noise, or colored noise but with vanishing related frictions: under which conditions are the stochastic dynamics well-posed?", "Would dynamics with regularized noise lead to qualitative similar behavior?", "What are the necessary and sufficient conditions for the formal computations performed in this work to be mathematically founded?", "Some of these questions are related to recent advances in the mathematics of stochastic partial differential equations [38], [37], [33], [34], [6], [7], [27].", "Again, these aspects are further discussed in the conclusion.", "In Section we discuss a general framework for Langevin dynamics.", "Starting from a few hypotheses (Liouville theorem, transversality condition, and relation between friction and noise amplitude), we derives the time reversal symmetry properties of the stochastic process.", "Section applies this framework to two-dimensional and quasi-geostrophic turbulence models.", "Section discusses a specific case where a tricritical point is a situation for bistability, and finally Section concludes by emphasizing the interest and limitations of these Langevin models and outlining the perspectives." ], [ "Langevin dynamics and equilibrium instantons", "The aim of this section is to describe the general framework for Langevin dynamics.", "We first define Langevin dynamics in subsection REF , as stochastic, ordinary or partial, differential equations, for which the deterministic part is composed of a vector field with a Liouville property (conservation of phase space volume, Eq.", "(REF )) plus a potential force with potential $\\mathcal {G}$ .", "The conservative part of the dynamics are assumed to be transverse to the gradient of the potential (REF ).", "The stochastic force is defined as the derivative of a Brownian process, with a correlation function identical to that of the kernel of the potential force.", "We derive the main properties of Langevin dynamics: its invariant measure is a Gibbs measure with potential $\\mathcal {G}$ .", "As the Langevin dynamics is a Markov process, we can define the time reversed Markov process, which also satisfies Langevin dynamics which is usually related to the original dynamics.", "We call this process the reversed, or dual Langevin dynamics.", "We study this time-reversal symmetry through the symmetry of the action, describing transition path probabilities.", "Based on this symmetry, we describe the relation between relaxation paths (most probable paths for a relaxation from any initial state to an attractor of the system) and fluctuation paths (most probable paths to observe a fluctuation starting from an attractor and ending at any point of the system).", "As we explain, for Langevin dynamics, fluctuation paths are time reversed trajectories of relaxation paths of the dual dynamics.", "These properties, for instance the relation between fluctuation and relaxation paths, can be considered as a generalization of Onsager reciprocal relations.", "However, they are valid for fluctuations arbitrarily far from the main attractor, and for relaxation dynamics that do not necessarily need to be linear.", "Such a symmetry between the fluctuation and relaxation paths is somehow a classical remark in statistical mechanics.", "For instance, the relation between the action symmetry and detailed balance can be found in [35], discussion of these properties can also be found in [39], and additionally we have been told that this symmetry may be traced back to Onsager and Machlup [49].", "Even if the basic ideas seem classical, we do not yet know of any references where the general structure of Langevin dynamics, and its relation with the symmetries of relaxation and fluctuation paths are discussed.", "We also note an interesting discussion of this symmetry in [61].", "This symmetry is also clearly related to the Gallavotti-Cohen fluctuation relations [23], [25].", "The fact that large deviation functionals can be computed explicitly when the dynamics can be decomposed into the sum of a gradient and a transverse part is explained in the book of Freidlin-Wentzell [24].", "In our problem, this transversality comes from the Hamiltonian structure and the fact that the potential is a conserved quantity of the Hamiltonian dynamics.", "As explained very clearly in [5], for non-equilibrium systems, the deterministic vector field can also be decomposed into the sum of the gradient of the quasi-potential plus a transverse part, the transversality condition being then equivalent to the Hamilton-Jacobi equation.", "Similar ideas can also be found in works of Graham in the 1980s and 1990s (see for instance [28])." ], [ "Langevin dynamics with potential $\\mathcal {G}$ ", "We call the Langevin dynamics for the potential $\\mathcal {G}$ the stochastic dynamics given by $\\frac{\\partial q}{\\partial t}=\\mathcal {F}\\left[q\\right]({\\bf r})-\\alpha \\int _{\\mathcal {D}}\\, C({\\bf r},{\\bf r}^{\\prime })\\frac{\\delta \\mathcal {G}}{\\delta q({\\bf r}^{\\prime })}\\left[q\\right] \\,{\\rm d}{\\bf r}^{\\prime }+\\sqrt{2\\alpha \\gamma }\\, \\eta ,$ where $\\mathcal {F}$ satisfies a Liouville property (defined below, Eq.", "(REF )), $\\mathcal {G}$ is a conserved quantity of the dynamics defined by $\\mathcal {F}$ (see Eq.", "(REF )), and the stochastic force $\\eta $ is a Gaussian process, white in time, with correlation function $\\mathbb {E}\\left[\\eta ({\\bf r},t)\\eta ({\\bf r}^{\\prime },t^{\\prime })\\right]=C({\\bf r},{\\bf r}^{\\prime })\\delta (t-t^{\\prime })$ .", "As it is a correlation function, $C$ is a symmetric positive function, i.e.", "for any function $\\phi $ over $\\mathcal {D}$ $\\int _{\\mathcal {D}}\\int _{\\mathcal {D}}\\,\\phi \\left({\\bf r}\\right)\\, C({\\bf r},{\\bf r}^{\\prime })\\, \\phi \\left({\\bf r}^{\\prime }\\right)\\, {\\rm d}{\\bf r}\\, {\\rm d}{\\bf r}^{\\prime } \\ge 0 ,$ and $C({\\bf r},{\\bf r}^{\\prime })=C({\\bf r}^{\\prime },{\\bf r})$ .", "For simplicity, we assume in the following that $C$ is positive definite and has an inverse $C^{-1}$ such that $\\int _{\\mathcal {D}}\\, C({\\bf r},{\\bf r}_{1})\\, C^{-1}({\\bf r}_{1},{\\bf r}^{\\prime })\\, {\\rm d}{\\bf r}_{1}=\\delta \\left({\\bf r}-{\\bf r}^{\\prime }\\right).$ The variable $q$ is either finite dimensional (for instance $q\\in \\mathbb {R}^{N}$ ), or a field (for instance a two-dimensional field for solution of the two-dimensional Euler equations).", "If $q\\in \\mathbb {R}^{N}$ , we assume that the deterministic dynamical system $\\frac{\\partial q}{\\partial t}=\\mathcal {F}\\left[q\\right],$ conserves the Lebesgue measure $\\prod _{i=1}^{N}\\, {\\rm d}q_{i}$ , or equivalently that the divergence of the vector field $\\mathcal {F}$ is zero: $\\nabla \\cdot \\mathcal {F}\\equiv \\sum _{i=1}^{N}\\frac{\\partial \\mathcal {F}}{\\partial q_{i}}=0.$ We call this property a Liouville property.", "If $q$ is a field (for instance a two-dimensional vorticity or potential vorticity field, for the two-dimensional Euler or quasi-geostrophic equations) defined over a domain $\\mathcal {D}$ , we assume that a Liouville property holds, in the sense that the formal generalization of the finite dimensional Liouville property, $\\nabla \\cdot \\mathcal {F}\\equiv \\int _{\\mathcal {D}}\\,\\frac{\\delta \\mathcal {F}}{\\delta q({\\bf r})}\\,{\\rm d}{\\bf r}=0,$ is verified.", "We further assume that the deterministic dynamical system (REF ) has $\\mathcal {G}$ as a conserved quantity.", "Then for any $q$ : $\\int _{\\mathcal {D}}\\,\\mathcal {F}\\left[q\\right](\\mathbf {r})\\frac{\\delta \\mathcal {G}}{\\delta q({\\bf r})}\\left[q\\right]\\,{\\rm d}{\\bf r}=0.$ This equation is a transversality property between the the vector field $\\mathcal {F}$ and the gradient of the potential $\\mathcal {G}$ .", "These two hypotheses, Liouville (REF ) and the conservation of the potential (REF ), are verified if the dynamical system is Hamiltonian: $\\mathcal {F}[q]=\\left\\lbrace q,\\mathcal {H}\\right\\rbrace ,$ with $\\mathcal {G}$ being one of its conserved quantity, for instance $\\mathcal {G}=\\mathcal {H}$ .", "We stress however that $\\mathcal {G}$ does not need to be $\\mathcal {H}$ in general.", "The major property of Langevin dynamics is that the stationary probability density functional is known a-priori and is given by $P_{s}[q]=\\frac{1}{Z}\\exp \\left(-\\frac{\\mathcal {G}[q]}{\\gamma }\\right),$ where $Z$ is a normalization constant.", "At a formal level, this can be easily checked by writing the Fokker-Planck equation for the evolution of the probability functional.", "Then the property that $P_{s}$ is stationary readily follows from the Liouville property and the fact that $\\mathcal {G}$ is a conserved quantity for the deterministic dynamics." ], [ "Reversed Langevin dynamics", "We consider the linear operator $I$ to be a linear involution on the space of fields $q$ ($I^{2}={\\rm Id}$ ).", "Therefore, we define the reversed Langevin dynamics, with respect to $I$ , as $\\frac{\\partial q}{\\partial t}=\\mathcal {F}_{r}\\left[q\\right](\\mathbf {r})-\\alpha \\int _{\\mathcal {D}}\\, C_{r}({\\bf r},{\\bf r}^{\\prime })\\frac{\\delta \\mathcal {G}_{r}}{\\delta q({\\bf r}^{\\prime })}\\left[q\\right]\\,{\\rm d}{\\bf r}^{\\prime }+\\sqrt{2\\alpha \\gamma }\\eta ,$ where $\\mathcal {F}_{r}=-I\\circ \\mathcal {F}\\circ I,$ $C_{r}=I^{+}CI,$ here $I^{+}$ is the adjoint of $I$ for the $L^{2}$ scalar product, and $\\mathcal {G}_{r}\\left[q\\right]=\\mathcal {G}\\left[I\\left[q\\right]\\right].$ From the properties of $\\mathcal {F}$ , $C$ and $\\mathcal {G}$ , one can demonstrate that a Liouville property holds for $\\mathcal {F}_{r}$ , that $C_{r}$ is positive definite, and that $\\mathcal {G}_{r}$ is a conserved quantity for the dynamics $\\frac{\\partial q}{\\partial t}=\\mathcal {F}_{r}\\left[q\\right]$ for any $q$ : $\\int _{\\mathcal {D}}\\,\\mathcal {F}_{r}\\left[q\\right](\\mathbf {r})\\frac{\\delta \\mathcal {G}_{r}}{\\delta q({\\bf r})}\\left[q\\right]\\,{\\rm d}{\\bf r}=0.$ As a consequence, the reversed Langevin dynamics (REF ) is also Langevin.", "A very interesting case is when the deterministic dynamics is symmetric with respect to time reversal.", "Then there exists a linear involution $I$ such that $\\mathcal {F}=\\mathcal {F}_{r}=-I\\circ \\mathcal {F}\\circ I.$ Moreover, if $C$ and $\\mathcal {G}$ are symmetric with respect to the involution: $C_{r}=C$ and $\\mathcal {G}_{r}=\\mathcal {G},$ then the reversed Langevin dynamics are nothing else than the original Langevin dynamics.", "In this case, we say that the Langevin dynamics are time-reversible.", "Simple examples of time-reversible Langevin dynamics are the overdamped processes: $\\dot{q}=-\\int _{\\mathcal {D}}\\, C({\\bf r},{\\bf r}^{\\prime })\\frac{\\delta \\mathcal {G}}{\\delta q({\\bf r}^{\\prime })}\\left[q\\right]\\, {\\rm d}{\\bf r}^{\\prime }+\\sqrt{2\\gamma }\\eta ,$ which can be proved to be time-reversible with the involution $I={\\rm Id}$ , the canonical Langevin dynamics ${\\left\\lbrace \\begin{array}{ll}\\dot{x} = p,\\nonumber \\\\\\dot{p} = -\\frac{{\\rm d}V}{{\\rm d}x}-\\alpha p+\\sqrt{2\\alpha k_{B}T}\\eta ,\\nonumber \\end{array}\\right.", "}$ with $I\\left(x,p\\right)^T=\\left(x,-p\\right)^T$ , or the two-dimensional stochastic Euler equations: $\\frac{\\partial \\omega }{\\partial t}+{\\bf v}\\cdot \\nabla \\omega =-\\alpha \\int _{\\mathcal {D}}\\, C({\\bf r},{\\bf r}^{\\prime })\\frac{\\delta \\mathcal {G}}{\\delta \\omega ({\\bf r}^{\\prime })}\\, {\\rm d}{\\bf r}^{\\prime }+\\sqrt{2\\alpha \\gamma }\\eta ,\\quad {\\rm with}\\quad {\\bf v}=\\mathbf {e}_{z}\\times \\nabla \\psi ,\\nonumber $ under the assumption that $\\mathcal {G}$ is conserved by the Euler dynamics, and is an even functional ($\\mathcal {G}\\left[-\\omega \\right]=\\mathcal {G}\\left[\\omega \\right]$ ).", "For the two-dimensional Euler equations, the natural involution corresponding to time-reversal symmetry is $I\\left[\\omega \\right]=-\\omega $ .", "In the following, we will also consider cases when the Langevin dynamics are not time-reversible, for instance the two-dimensional stochastic Euler equations when $\\mathcal {G}$ is not even, or the quasi-geostrophic equations with topography $h(y)\\ne 0$ ." ], [ "Path integrals, action, and time-reversal symmetry", "The Lagrangian $\\mathcal {L}$ associated to the Langevin dynamics (REF ) is defined as $\\mathcal {L}\\left[q,\\frac{\\partial q}{\\partial t}\\right]&=&\\frac{1}{4\\alpha }\\int _{\\mathcal {D}}\\int _{\\mathcal {D}}\\,\\left(\\frac{\\partial q}{\\partial t}-\\mathcal {F}\\left[q\\right](\\mathbf {r})+\\alpha \\int _{\\mathcal {D}}C({\\bf r},{\\bf r}_{1})\\frac{\\delta \\mathcal {G}}{\\delta q({\\bf r}_{1})}\\left[q\\right]\\mbox{d}{\\bf r}_{1}\\right)\\nonumber \\\\&\\times &C^{-1}({\\bf r},{\\bf r}^{\\prime })\\left(\\frac{\\partial q}{\\partial t}-\\mathcal {F}\\left[q\\right](\\mathbf {r}^{\\prime })+\\alpha \\int _{\\mathcal {D}}\\, C({\\bf r}^{\\prime },{\\bf r}_{2})\\frac{\\delta \\mathcal {G}}{\\delta q({\\bf r}_{2})}\\left[q\\right]\\mbox{d}{\\bf r}_{2}\\right)\\,{\\rm d}{\\bf r}\\,{\\rm d}{\\bf r}^{\\prime },$ and the action functional as $\\mathcal {A}_{(0,T)}\\left[q\\right]=\\int _{0}^{T}\\mathcal {L}\\left[q(t),\\frac{\\partial q}{\\partial t}(t)\\right]\\,{\\rm d}t.$ Consequently, the Lagrangian of the reverse process is defined as $\\mathcal {L}_{r}\\left[q,\\frac{\\partial q}{\\partial t}\\right]&=&\\frac{1}{4\\alpha }\\int _{\\mathcal {D}}\\int _{\\mathcal {D}}\\,\\left(\\frac{\\partial q}{\\partial t}-\\mathcal {F}_{r}\\left[q\\right](\\mathbf {r})+\\alpha \\int _{\\mathcal {D}}\\, C_{r}({\\bf r},{\\bf r}_{1})\\frac{\\delta \\mathcal {G}_{r}}{\\delta q({\\bf r}_{1})}\\left[q\\right]\\mbox{d}{\\bf r}_{1}\\right)\\nonumber \\\\&\\times &C_{r}^{-1}({\\bf r},{\\bf r}^{\\prime })\\left(\\frac{\\partial q}{\\partial t}-\\mathcal {F}_{r}\\left[q\\right](\\mathbf {r}^{\\prime })+\\alpha \\int _{\\mathcal {D}}\\, C_{r}({\\bf r}^{\\prime },{\\bf r}_{2})\\frac{\\delta \\mathcal {G}_{r}}{\\delta q({\\bf r}_{2})}\\left[q\\right]\\mbox{d}{\\bf r}_{2}\\right)\\,{\\rm d}{\\bf r}\\, {\\rm d}{\\bf r}^{\\prime },$ with the time-reversed action functional $\\mathcal {A}_{r}$ defined accordingly.", "Using the Onsager-Machlup formalism, we know that $P\\left[q_{T},T;q_{0},0\\right]$ , the transition probability to go from the state $q_{0}$ at time 0 to the state $q_{T}$ at time $T$ , can be expressed as $P\\left[q_{T},T;q_{0},0\\right]=\\int ^{q(T)=q_T}_{q(0)=q_0}\\,\\mathcal {D}\\left[q\\right]\\,{\\exp }\\left(-\\frac{\\mathcal {\\mathcal {A}}}{\\gamma }\\right),$ where we have used the fact that the Jacobian $J\\left[q\\right]=\\left|\\det \\left[\\frac{\\delta }{\\delta q({\\bf r}^{\\prime })}\\left(\\dot{q}-\\mathcal {F}[q]+\\alpha \\int _{\\mathcal {D}}\\, C({\\bf r},{\\bf r}_{1})\\frac{\\delta \\mathcal {G}}{\\delta q({\\bf r}_{1})}\\left[q\\right]\\, {\\rm d}{\\bf r}_{1}\\right)\\right]\\right|,$ is formally equal to a $q$ -independent constant when we interpret our stochastic partial differential equation using Ito's convention [68].", "This constant can be included in the definition of the functional integration measure.", "For a given path $\\left\\lbrace q(t)\\right\\rbrace _{0\\le t\\le T}$ , we define the reversed path by $q_{r}(t)=I\\left[q(T-t)\\right]$ .", "The main interest of the reversed process stems from the study of temporal symmetries of the stochastic process and the remark that $\\mathcal {A}\\left[q_{r},T\\right]=\\mathcal {A}_{r}\\left[q,T\\right]-\\left(\\mathcal {G}\\left[q(T)\\right]-\\mathcal {G}\\left[q(0)\\right]\\right),$ or equivalently, using (REF ), $\\mathcal {A}\\left[q,T\\right]=\\mathcal {A}_{r}\\left[q_{r},T\\right]+\\left(\\mathcal {G}\\left[q(T)\\right]-\\mathcal {G}\\left[q(0)\\right]\\right).$ Let us prove this equality.", "Using the definition of $\\mathcal {F}_{r}$ , $\\mathcal {G}_{r}$ and $C_{r}$ , (Eqs.", "(REF -REF )), and using that $\\frac{\\delta \\mathcal {G}_{r}}{\\delta q(\\mathbf {r})}\\left[q\\right]=I\\frac{\\delta \\mathcal {G}}{\\delta q(\\mathbf {r})}\\left[I\\left[q\\right]\\right],\\nonumber $ with $I^{2}={\\rm Id}$ , we have $\\mathcal {L}\\left[I\\left[q\\right],-\\frac{\\partial }{\\partial t}I\\left[q\\right]\\right]&=&\\frac{1}{4\\alpha }\\int _{\\mathcal {D}}\\int _{\\mathcal {D}}\\,\\left(\\frac{\\partial q}{\\partial t}-\\mathcal {F}_{r}\\left[q\\right](\\mathbf {r})-\\alpha \\int _{\\mathcal {D}}\\, C_{r}({\\bf r},{\\bf r}_{1})\\frac{\\delta \\mathcal {G}_{r}}{\\delta q({\\bf r}_{1})}\\left[q\\right]\\mbox{d}{\\bf r}_{1}\\right)\\nonumber \\\\&\\times &C_{r}^{-1}({\\bf r},{\\bf r}^{\\prime })\\left(\\frac{\\partial q}{\\partial t}-\\mathcal {F}_{r}\\left[q\\right](\\mathbf {r}^{\\prime })-\\alpha \\int _{\\mathcal {D}}\\, C_{r}({\\bf r}^{\\prime },{\\bf r}_{2})\\frac{\\delta \\mathcal {G}_{r}}{\\delta q({\\bf r}_{2})}\\left[q\\right]{\\rm d}{\\bf r}_{2}\\right)\\,{\\rm d}{\\bf r}\\, {\\rm d}{\\bf r}^{\\prime }.\\nonumber $ Then, by expanding and using the conservation of $\\mathcal {G}_{r}$ we arrive to $\\mathcal {L}\\left[I\\left[q\\right],-\\frac{\\partial I\\left[q\\right]}{\\partial t}\\right]=\\mathcal {L}_{r}\\left[q,\\frac{\\partial q}{\\partial t}\\right]-\\int _{\\mathcal {D}}\\,\\frac{\\partial q}{\\partial t}\\frac{\\delta \\mathcal {G}}{\\delta q({\\bf r})}\\,{\\rm d}{\\bf r},$ or equivalently, $\\mathcal {L}\\left[I\\left[q\\right],-\\frac{\\partial I\\left[q\\right]}{\\partial t}\\right]=\\mathcal {L}_{r}\\left[q,\\frac{\\partial q}{\\partial t}\\right]-\\frac{{\\rm d}}{{\\rm d}t}\\mathcal {G}\\left[q\\right].$ Using the above formula and (REF ) in order to compute $\\mathcal {A}\\left[q_{r},T\\right]$ , we obtain (REF ).", "Performing the change of variable $q_{r}(t)=I\\left[q(T-t)\\right]$ in the path integral representation (REF ), and using the action duality formula (REF ), we obtain $P\\left[q_{T},T;q_{0},0\\right]\\exp \\left(-\\frac{\\mathcal {G}\\left[q_{0}\\right]}{\\gamma }\\right)=P_{r}\\left[I\\left[q_{0}\\right],T;I\\left[q_{T}\\right],0\\right]\\exp \\left(-\\frac{\\mathcal {G}_{r}\\left[I\\left[q_{T}\\right]\\right]}{\\gamma }\\right),$ where $P_{r}$ is a transition probability for the reversed process.", "We have thus obtain a relation between the transition probability of the direct, forward, process and that of the reversed one." ], [ "Detailed balance for reversible processes", "If we assume that the Langevin equation is time-reversible, then the direct and the reverse processes are the same, and the duality relation for the transition probability implies $P\\left[q_{T},T;q_{0},0\\right]\\exp \\left(-\\frac{\\mathcal {G}\\left[q_{0}\\right]}{\\gamma }\\right)=P\\left[I\\left[q_{0}\\right],T;I\\left[q_{T}\\right],0\\right]\\exp \\left(-\\frac{\\mathcal {G}\\left[I\\left[q_{T}\\right]\\right]}{\\gamma }\\right),$ where it is also true that $\\exp \\left(-\\mathcal {G}\\left[I\\left[q_{T}\\right]\\right]/\\gamma \\right)=\\exp \\left(-\\mathcal {G}\\left[q_{T}\\right]/\\gamma \\right)$ .", "This result is the detailed balance property for the stochastic process.", "When the reverse process is different from the direct process, then in general, detailed balance should not be verified." ], [ "Steady states and critical points of the potential $\\mathcal {G}$", "Let us prove that any non-degenerate critical point of the potential is also a steady state of the deterministic dynamics.", "This is a classical result in mechanics, i.e.", "any non-degenerate critical point of the energy is a steady state.", "Extrema of the stationary PDF are critical points of the potential $\\mathcal {G}$ .", "Such a critical point $q_{c}$ verifies $\\frac{\\delta \\mathcal {G}}{\\delta q({\\bf r})}\\left[q_{c}\\right]=0.$ We assume that the critical point is non-degenerate, that the second variations of $\\mathcal {G}$ has no null eigenvalues.", "More explicitly, the relation $\\int _{\\mathcal {D}}\\,\\frac{\\delta ^{2}\\mathcal {G}}{\\delta q({\\bf r})\\delta q({\\bf r}^{\\prime })}\\left[q_{c}\\right]\\,\\phi (\\mathbf {r}^{\\prime })=0,$ implies that $\\phi =0$ .", "Now, we can prove that $q_{c}$ is also a steady state of the Hamiltonian dynamics.", "We use the property that $\\mathcal {G}$ is conserved.", "By taking the variational derivative $\\delta /\\delta q({\\bf r})$ of Eq.", "(REF ) we obtain that for any $q$ $\\int _{\\mathcal {D}}\\,\\frac{\\delta ^{2}\\mathcal {G}}{\\delta q({\\bf r}_{2})\\delta q({\\bf r})}\\left[q\\right]\\mathcal {F}\\left[q\\right](\\mathbf {r}_{2})\\,{\\rm d}{\\bf r_{2}}+\\int _{\\mathcal {D}}\\,\\frac{\\delta \\mathcal {G}}{\\delta q({\\bf r}_{2})}\\left[q\\right]\\frac{\\delta \\mathcal {F}}{\\delta q({\\bf r})}\\left[q\\right](\\mathbf {r}_{2})\\,{\\rm d}{\\bf r_{2}}=0.$ If we apply this formula at the critical point $q_{c}$ , we can conclude that $\\int _{\\mathcal {D}}\\,\\frac{\\delta ^{2}\\mathcal {G}}{\\delta q({\\bf r}_{2})\\delta q({\\bf r})}\\left[q_{c}\\right]\\,\\mathcal {F}\\left[q_{c}\\right](\\mathbf {r}_{2})\\, {\\rm d}{\\bf r_{2}}=0.$ Moreover, using that $\\mathcal {G}$ is non-degenerate we observe that for all $\\mathbf {r}$ $\\mathcal {F}\\left[q_{c}\\right](\\mathbf {r})=0,$ and thus $q_{c}$ is a steady state of the deterministic dynamics.", "The remark that non-degenerate critical points of conserved quantity are steady states also extends to their stability properties.", "Any stable and non-degenerate minima or maxima of a conserved quantity is a stable fixed point of the deterministic dynamics (again, think of the energy or angular momentum in classical mechanics).", "These points are probably about as old as classical mechanics.", "For infinite-dimensional problems, like the two-dimensional Euler equations or other fluid mechanics problems, the issue may be more subtle.", "Indeed, one should be careful of possible norm inequivalence (an infinite number of small scales can do a lot).", "But proofs about stability of critical points of conserved quantities can still be obtained on a case by case basis.", "For instance, we refer to the two Arnold stability theorems for the two-dimensional Euler equations [1], or their generalization to many other fluid mechanics problems [32].", "Another important point is that from relations (REF ) and (REF ), it is clear that if $q_{s}$ is a steady state of the deterministic dynamics, then $I\\left[q_{s}\\right]$ is a steady state of the reversed dynamics, and vice-versa.", "Moreover, if $q_{c}$ is a critical point of the potential $\\mathcal {G}$ , then $I\\left[q_{c}\\right]$ will be a critical point of $\\mathcal {G}_{r}$ .", "The stability properties (minima, global minima, local minima, number of unstable directions, and so on) of $q_{c}$ , with respect to the minimization of $\\mathcal {G}$ , will agree with the stability properties of $I\\left[q_{c}\\right]$ with respect to the minimization of $\\mathcal {G}_{r}$ ." ], [ "Relaxation dynamics and Lyapunov functionals", "We define a relaxation path to be a solution of the relaxation dynamics: $\\frac{\\partial q}{\\partial t}=\\mathcal {F}\\left[q\\right](\\mathbf {r})-\\alpha \\int _{\\mathcal {D}}\\, C({\\bf r},{\\bf r}^{\\prime })\\frac{\\delta \\mathcal {G}}{\\delta q({\\bf r}^{\\prime })}\\left[q\\right]\\,{\\rm d}{\\bf r}^{\\prime }.$ For any relaxation path $q(t)$ , using the property that $\\mathcal {G}$ is conserved by the inertial dynamics we can easily prove that $\\frac{\\rm d}{{\\rm d}t}\\mathcal {G}\\left[q(t)\\right]=-\\alpha \\int _{\\mathcal {D}}\\, C({\\bf r},{\\bf r}^{\\prime })\\frac{\\delta \\mathcal {G}}{\\delta q({\\bf r}^{\\prime })}\\frac{\\delta \\mathcal {G}}{\\delta q({\\bf r})}\\,{\\rm d}{\\bf r}\\,{\\rm d}{\\bf r}^{\\prime }\\le 0,$ where we have used the positive definiteness of $C$ for establishing the inequality.", "Thus, we can conclude that $\\mathcal {G}$ is a Lyapunov functional for the relaxation dynamics.", "From this, we state that any minima of the potential is stable for the relaxation dynamics." ], [ "Action minima, relaxation paths of the dual dynamics and instantons", "We consider action minima, subjected to fixed boundary conditions $A_{(0,T)}\\left[q_{0},q_{T}\\right]=\\min _{\\left\\lbrace q\\,\\left|\\,q(0)=q_{0},\\ q(T)=q_{T}\\right\\rbrace \\right.", "}\\mathcal {A}_{(0,T)}\\left[q\\right].$ This variational problem important for many questions.", "For instance, it describes the most probable path to go from state $q_{0}$ to state $q_{T}$ .", "Moreover, as will be discussed in the next section, it will be useful in order to describe large deviation results.", "From the definition of the action (REF -REF ), and as $C$ is positive definite, it is clear that $A_{(0,T)}\\left[q_{0},q_{T}\\right]\\ge 0.$ Furthermore, using the action duality relation given by Eq.", "(REF ), we also conclude that $A_{(0,T)}\\left[q_{0},q_{T}\\right]\\ge \\mathcal {G}\\left[q_{T}\\right]-\\mathcal {G}\\left[q_{0}\\right].$ It is self-evident from the definition of the relaxation paths (REF ), and from the structure of the action functional (REF -REF ) that a relaxation path has zero action.", "This should be physically intuiative, as no noise is needed for the system to follow such a path.", "Then, if there exists a relaxation path between $q_{0}$ and $q_{T}$ taking time $T$ , ($\\left\\lbrace q(t)\\right\\rbrace _{0\\le t\\le T}$ such that $q(0)=q_{0}$ and $q(T)=q_{T}$ ), we deduce that $A_{(0,T)}\\left[q_{0},q_{T}\\right]=0.$ Similarly, using the duality relation (REF ), if there exists a relaxation path for the reversed dynamics between $I\\left[q_{T}\\right]$ and $I\\left[q_{0}\\right]$ , we surmise that $A_{(0,T)}\\left[q_{0},q_{T}\\right]=\\mathcal {G}\\left[q_{T}\\right]-\\mathcal {G}\\left[q_{0}\\right].$ This is an important statement.", "Indeed, the reversed dynamics has properties very similar to that of the original dynamics (it has the same fixed points, the same attractors, and the same saddles up to the application of the involution $I$ ), but in the argument above, we see that the final and end-points of the relaxation paths have been exchanged from $q_{0}$ and $q_{T}$ to $I\\left[q_{T}\\right]$ and $I\\left[q_{0}\\right]$ respectively.", "This will be especially useful when the starting point is one of the local minima of the potential $\\mathcal {G}$ , and thus one of the attractors of the reversed dynamics.", "Consider now the case when $q_{0}$ is a local minima of $\\mathcal {G}$ .", "Then as it is also an attractor of the relaxation dynamics, no non-trivial relaxation path will start at $q_{0}$ .", "But, for all $q_{T}$ inside the basin of attraction of $q_{0}$ , there exists a relaxation path from $q_{T}$ to $q_{0}$ .", "Generically, this path will take an infinite amount of time $T=\\infty $ , e.g.", "if there is an exponential relaxation.", "Consequently, there will also be a relaxation path for the dual dynamics from $I\\left[q_{T}\\right]$ to $I\\left[q_{0}\\right]$ taking infinite time.", "Therefore, for relaxation dynamics, we have that for all $q_{T}$ in the basin of attraction of an local minima of $q_{0}$ $A_{(-\\infty ,0)}\\left[q_{0},q_{T}\\right]=\\mathcal {G}\\left[q_{T}\\right]-\\mathcal {G}\\left[q_{0}\\right].$ For many problems, e.g.", "when one considers the stationary distribution, the action minima $A_{(-\\infty ,0)}\\left[q_{0},q_{T}\\right]$ becomes an essential quantity.", "If $q_{T}$ is in basin of attraction of $q_{1}\\ne q_{0}$ , then as there exists a relaxation path from $q_{1}$ to $q_{T}$ , we can infer that $A_{(-\\infty ,0)}\\left[q_{0},q_{T}\\right]=A_{(-\\infty ,0)}\\left[q_{0},q_{1}\\right].$ Moreover, it is easily understood that the action minima will correspond to the relaxation trajectory, in the reversed dynamics, from the saddle $q_{s}(q_{0},q_{1})$ that belongs to the closure of the basin of attractions of both $q_{0}$ and $q_{1}$ , with the smallest possible value of the potential $\\mathcal {G}\\left[q_{s}(q_{0},q_{1})\\right]$ .", "Hence, if $q_{T}$ is within the basin of attraction of $q_{1}$ we have $A_{(-\\infty ,0)}\\left[q_{0},q_{T}\\right]=A_{(-\\infty ,0)}\\left[q_{0},q_{1}\\right]=A_{(-\\infty ,\\infty )}\\left[q_{0},q_{s}(q_{_{0},}q_{1})\\right]=\\mathcal {G}\\left[q_{s}(q_{0},q_{1})\\right]-\\mathcal {G}\\left[q_{0}\\right].$ Ultimately, the minimizers of the action, between local minima of the potential and saddles, of infinite time, are immensely important.", "These trajectories are called instantons.", "As it should be obvious from the previous discussion, instantons for Langevin dynamics are the reversed time trajectories of relaxation paths of the reversed dynamics.", "Instantons are thus fluctuation paths for the Langevin dynamics.", "More explicitly, if $\\left\\lbrace q_{r}(t)\\right\\rbrace _{-\\infty \\le t\\le \\infty }$ is a relaxation path for the reversed dynamics between a saddle $I\\left[q_{s}\\right]$ and the attractor $I\\left[q_{0}\\right]$ , then the instanton between $q_{0}$ and $q_{s}$ is given by $\\left\\lbrace I\\left[q_{r}(-t)\\right]\\right\\rbrace _{-\\infty \\le t\\le \\infty }$ .", "As instantons are the most probable fluctuation paths between attractors and saddles, they require an infinite amount of time to leave the attractor and an infinite amount of time to converge to the saddle.", "Moreover, they are degenerate, in the sense that if $\\left\\lbrace q_{r}(t)\\right\\rbrace _{-\\infty \\le t\\le \\infty }$ is an instanton, then for any $\\tau $ , $\\left\\lbrace q_{r}(t+\\tau )\\right\\rbrace _{-\\infty \\le t\\le \\infty }$ is also an instanton." ], [ "Large deviations, Freidlin-Wentzell theory and entropic effects", "Up to now, we have discussed only the symmetry properties of the action functional (REF ) and of the action minima (REF ).", "In the limit of small noise, $\\gamma \\rightarrow 0$ (see subsection (REF )), one directly observes, from the path integral representation of the transition probability (REF ) that the minima of the action will play a crucial role.", "Indeed, the path integral will then be seen as a Laplace integral, and a Laplace principle will be used in order to derive a large deviation result $P\\left[q_{T},T;q_{0},0\\right]\\underset{\\gamma \\rightarrow 0}{=}\\exp \\left(-\\frac{A_{(0,T)}\\left[q_{0},q_{T}\\right]}{\\gamma }+{\\rm o}\\left(\\frac{1}{\\gamma }\\right)\\right),$ where $A_{(0,T)}\\left[q_{0},q_{T}\\right]=\\min _{\\left\\lbrace q\\,\\left|\\,q(0)=q_{0},\\ q(T)=q_{T}\\right.\\right\\rbrace }\\mathcal {A}_{(0,T)}\\left[q\\right]$ , and where ${\\rm o}\\left(1/\\gamma \\right)$ are subdominant contributions.", "Physicist, through explicit computations, have discussed many examples where this Laplace principle may or may not be correct for small $\\gamma $ .", "In quantum mechanics, evaluations of path integrals in the limit of small $\\hbar $ , or in the WKB approximation, which also involves the evaluation of path integrals through a saddle point approximation.", "On the mathematical side, the study of sufficient hypotheses in order to rigorously prove such large deviation results (REF ) is one of the main aspects of Freidlin-Wentzell theory [24].", "Roughly speaking, Freidlin and Wentzell proved that for finite dimensional stochastic dynamics, under generic hypotheses, a large deviation result actually holds.", "However, we draw the attention of the reader to the fact that for infinite dimensional field equations, e.g turbulence models, a large deviation result (REF ) is far from obvious in the limit of small $\\gamma $ .", "It may be expected to be true if, for instance, if the degrees of freedom at the smallest scales can be proven to have a negligible effect upon the dynamics, such that it is qualitatively similar to an effective finite dimensional system.", "For the turbulence model we present here, such a property is not obvious at all.", "Studying this issue in general is an extremely difficult task.", "The path integral taken over Gaussian fluctuations around the critical point is given by the determinant of the second variation of the action functional and this determinant is typically infinite for infinitely many degrees of freedom.", "Therefore it requires a regularization which can either lead to a renormalization of constants in (REF ) or to a completely different answer.", "This problem goes beyond the scope of this paper, however, we will return to this discussion for a specific case in the conclusion." ], [ "The two-dimensional Euler and quasi-geostrophic equilibrium dynamics", "In this section, we apply the formalism outlined previously to turbulence models.", "We explain why the two main hypotheses of Langevin dynamics (Liouville property and conservation of the potential related to the transversality condition) are verified.", "We assume that the kernel in front of the gradient part and the noise autocorrelation are identical.", "Then all of the time-reversal properties and the Lyapunov properties discussed in the previous section apply to these turbulence models.", "An interesting aspect, explained below, is that depending on the properties of the potential $\\mathcal {G}$ (even or not), and of the model (with or without topography), the Langevin dynamics can be either symmetric under time reversal or not.", "We consider the Langevin dynamics associated to the quasigeostrophic equations in a periodic domain $\\mathcal {D}=[0,2\\pi l_{x})\\times [0,2\\pi )$ with aspect ratio $l_{x}$ to be given as $\\frac{\\partial q}{\\partial t}+\\mathbf {v}\\left[q-h\\right]\\cdot \\mathbf {\\nabla }q & = & -\\alpha \\int _{\\mathcal {D}}\\, C({\\bf r},{\\bf r}^{\\prime })\\frac{\\delta \\mathcal {G}}{\\delta q({\\bf r}^{\\prime })}\\,{\\rm d}{\\bf r}^{\\prime }+\\sqrt{2\\alpha \\gamma }\\eta ,\\\\\\mathbf {v}=\\mathbf {e}_{z}\\times \\mathbf {\\nabla }\\psi ,\\quad \\omega &=&\\Delta \\psi , \\quad q=\\omega +h(\\mathbf {r}),$ with potential $\\mathcal {G}.$ The stochastic force $\\eta $ is a Gaussian process, white in time, with correlation function $\\mathbb {E}\\left[\\eta (\\mathbf {r},t)\\eta (\\mathbf {r}^{\\prime },t^{\\prime })\\right]=C(\\mathbf {r},\\mathbf {r}^{\\prime })\\delta (t-t^{\\prime })$ .", "The potential $\\mathcal {G}$ and the assumption of Langevin dynamics are discussed in section REF .", "Moreover, the topography $h(\\mathbf {r})$ is such that $\\int _{\\mathcal {D}}\\, h\\left(\\mathbf {r}\\right)\\, {\\rm d}{\\bf r}=0$ .", "We consider $G$ to be the Green's function of the Laplacian operator ($G=\\Delta ^{-1}$ ) for doubly periodic functions with zero averages.", "Then, the equations relating the potential vorticity $q$ , the stream function $\\psi $ , and the velocity are inverted as $\\psi (\\mathbf {r})=\\int _{\\mathcal {D}}\\, G\\left(\\mathbf {r},{\\bf r}^{\\prime }\\right)\\left[q({\\bf r}^{\\prime })-h({\\bf r}^{\\prime })\\right]\\, {\\rm d}{\\bf r}^{\\prime },$ and $\\mathbf {v}\\left[\\omega \\right](\\mathbf {r})=\\int _{\\mathcal {D}}\\,\\mathbf {e}_{z}\\times \\nabla _{\\mathbf {r}^{\\prime }}G\\left(\\mathbf {r},{\\bf r}^{\\prime }\\right)\\omega (\\mathbf {r}^{\\prime })\\, {\\rm d}{\\bf r}^{\\prime },$ respectively.", "Here, $\\mathbf {v}\\left[\\omega \\right]$ is the operator that allows us to compute the velocity from the vorticity.", "When $h=0$ , these dynamics correspond to the two-dimensional Euler equilibrium dynamics." ], [ "Conserved quantity and Liouville property", "From the velocity-vorticity relationship, it is easily checked that the kinetic energy can be expressed as $\\mathcal {E}=-\\frac{1}{2}\\int _{\\mathcal {D}}\\,\\left[q-h\\left(\\mathbf {r}\\right)\\right]\\psi \\, {\\rm d}\\mathbf {r}=\\frac{1}{2}\\int _{\\mathcal {D}}\\,\\left(\\nabla \\psi \\right)^{2}\\, {\\rm d}\\mathbf {r},$ and, for any sufficiently smooth real function $s$ , the Casimir functionals are defined as $\\mathcal {C}_{s}=\\int _{\\mathcal {D}}\\, s(q)\\, {\\rm d}\\mathbf {r},$ which are all conserved quantities of the deterministic quasi-geostrophic dynamics (Eqs.", "(REF ) for $\\alpha =0$ ).", "For any $s$ , and any $\\beta $ the functional $\\mathcal {G}=\\mathcal {C}_{s}+\\beta \\mathcal {E},$ will be the correct potential for Langevin dynamics.", "Moreover, as the deterministic equations (Eqs.", "(REF ) for $\\alpha $ =0) essentially correspond to a transport equation by a divergence-less velocity field, the Liouville property (REF ) is formally verified $\\nabla \\cdot \\mathcal {F}\\equiv -\\int _{\\mathcal {D}}\\,\\mathbf {v}\\left[q-h\\right]\\cdot \\mathbf {\\nabla }q\\,{\\rm d}\\mathbf {r}=-\\int _{\\mathcal {D}}\\,\\nabla \\cdot \\left(\\mathbf {v}\\left[q-h\\right]q\\right)\\,{\\rm d}\\mathbf {r}=0.$ Then the formalism of section applies with $\\mathcal {F}\\left[q\\right]=-\\mathbf {v}\\left[q-h\\right]\\cdot \\nabla q$ ." ], [ "Reversed dynamics and detailed balance", "For the two-dimensional Euler or quasi-geostrophic equations, the relevant involution corresponding to a time reversal is $I\\left[q\\right]=-q.$ Using (REF -REF ) we conclude that $\\mathcal {F}_{r}\\left[q\\right]=\\mathbf {v}\\left[q+h\\right]\\cdot \\nabla q,$ $C_{r}=C$ and $\\mathcal {G}_{r}\\left[q\\right]=\\mathcal {G}\\left[-q\\right].$ From these equations, we observe that for the two-dimensional Euler equations ($h=0$ ), $\\mathcal {F}_{r}=\\mathcal {F}$ , and thus we conclude that the dynamics are time-reversible (see Eq.", "(REF )).", "The time reversibility condition on $\\mathcal {G}$ (see Eq.", "(REF )) imposes that the potential $\\mathcal {G}$ must be even.", "There we have two cases: For the two-dimensional Euler equations with an even potential $\\mathcal {G}$ , the Langevin dynamics are time-reversible and detailed balance is verified.", "When either $h\\ne 0$ (quasi-geostrophic) or when $\\mathcal {G}$ is not even, then the Langevin dynamics are not time-reversible.", "The original dynamics are conjugated to another Langevin dynamics where $h$ has to be replaced by $-h$ and $\\mathcal {G}$ by $\\mathcal {G}_{r}\\left[q\\right]=\\mathcal {G}\\left[-q\\right]$ .", "In this case, detailed balance is not verified." ], [ "Instanton equation", "As discussed in section , the instantons from one attractor to a saddle are given by the reverse of the relaxation paths of the corresponding reversed dynamics.", "From (REF ) applied to the case where $\\mathcal {F}_{r}\\left[q\\right]=\\mathbf {v}\\left[q+h\\right]\\cdot \\nabla q$ , and $\\mathcal {G}_{r}\\left[q\\right]=\\mathcal {G}\\left[-q\\right]$ , we determine that the equation of these relaxation paths is $\\frac{\\partial q}{\\partial t}+\\mathbf {v}\\left[q+h\\right]\\cdot \\nabla q=-\\alpha \\int _{\\mathcal {D}}\\, C({\\bf r},{\\bf r}^{\\prime })\\frac{\\delta \\mathcal {G}}{\\delta q({\\bf r}^{\\prime })}\\left[-q\\right] {\\rm d}{\\bf r}^{\\prime }.$" ], [ "Energy, enstrophy, and energy-enstrophy ensembles and physical dissipation", "In this subsection, we consider the special case when the potential is given in the following form $\\mathcal {G}=\\int _{\\mathcal {D}}\\,\\frac{q^{2}}{2}\\,{\\rm d}\\mathbf {r}+\\beta \\mathcal {E}.$ This structure is referred to as the potential enstrophy ensemble (when $\\beta =0$ ), the enstrophy ensemble (when $\\beta =0$ and $h=0$ ), or generally as the energy-enstrophy ensemble.", "The properties of the corresponding invariant measures have been discussed on a number of occasions, starting with the works of Kraichnan [36] in the case of Galerkin truncations of the dynamics, and for some cases without discretization, see for instance [11] and references therein.", "For specific choices of the potential $\\mathcal {G}$ and of the kernel $C$ , the friction term can also be identified with a classical physical dissipation mechanism.", "For instance, if $C({\\bf r},{\\bf r}^{\\prime })=\\Delta \\delta (\\mathbf {r}-\\mathbf {r}^{\\prime })$ , and the potential takes the form of (REF ), then the dissipative term on the right hand side of (REF ) is $-\\alpha \\int _{\\mathcal {D}}\\, C({\\bf r},{\\bf r}^{\\prime })\\frac{\\delta \\mathcal {G}}{\\delta q({\\bf r}^{\\prime })}\\left[q\\right]\\,{\\rm d}{\\bf r}^{\\prime }=\\alpha \\Delta q-\\alpha \\beta q,$ which leads to a diffusion type dissipation with viscosity $\\alpha $ and a linear friction with friction parameter $\\alpha \\beta $ .", "Such a linear friction can model the effects of three-dimensional boundary layers on the quasi two-dimensional bulk vorticity, that appear in experiments with a very large aspect ratio, rotating tank experiments, or soap film experiments.", "The fact that for the enstrophy ensemble, the quasi-potential is simply the enstrophy, the relaxation and fluctuation paths can be easily computed explicitly in many scenarios, as is discussed in [12].", "For the majority of the other cases, the dissipative term on the right hand side of (REF ) cannot be identified as a microscopic dissipation mechanism nor as a physical mechanism.", "There is however another possible interpretation of this kind of friction term.", "As explained in [9], entropy maxima subjected to constraints related to the conservation of energy and the distribution of vorticity, are also extrema of energy-Casimir functionals.", "By analogy with the Allen-Cahn equation in statistical mechanics, that uses the free energy as a potential, it seems reasonable to describe the largest scales of turbulent flows as evolving through a gradient term of the energy-Casimir functional.", "Such models have been considered in the past (see, for example [19], [20] and references therein).", "At this stage, this should be considered as a phenomenological approach, as no clear theoretical results exist to support this view.", "In order to fully determine the quasi-geostrophic Langevin dynamics (REF ), we need to specify the topography function and the potential $\\mathcal {G}$ .", "Given the infinite number of conserved quantities for the quasi-geostrophic dynamics, there are many possible choices.", "We are interested in the description of the phenomenology of phase transitions and instanton theory in situations of first order transitions.", "Therefore, we will illustrate such a phenomenology through two examples.", "For the first example, we choose a topography give by $h\\left(\\mathbf {r}\\right)=H\\cos \\left(2y\\right)$ , such that $q=\\Delta \\psi +H\\cos \\left(2y\\right),$ and consider the potential $\\mathcal {G}=\\mathcal {C}+\\beta \\mathcal {E},$ with energy (REF ), $\\beta $ the inverse temperature, and where $\\mathcal {C}$ is the Casimir functional $\\mathcal {C}=\\int _{\\mathcal {D}}\\,\\frac{q^{2}}{2}-a_{4}\\frac{q^{4}}{4}+a_{6}\\frac{q^{6}}{6}\\, {\\rm d}\\mathbf {r},$ where we assume that $a_{6}>0$ ." ], [ "Zonal phase transitions", "We first consider the structure of the minima of the potential $\\mathcal {G}$ (REF ), and then their bifurcations when the parameters $\\epsilon $ and $a_{4}$ are changed, where $\\epsilon $ is defined by $\\beta =-1+\\epsilon .$ At low positive temperature ($\\beta \\rightarrow \\infty $ ), we expect to observe energy minima, which correspond to $\\psi =0$ and $q=H\\cos \\left(2y\\right)$ .", "As the energy is convex, for positive $\\beta $ and small enough $a_{4}$ , both $\\mathcal {C}$ and $\\beta \\mathcal {E}$ will also be convex.", "Henceforth, we expect that $\\mathcal {G}$ will contain an unique global minimum and no local minima.", "For large enough $\\beta $ , this equilibrium state will be dominated by the topographic effect.", "For small negative $\\beta $ , the change of convexity of $\\beta \\mathcal {E}$ from convex to concave will not change this picture.", "However, for smaller $\\beta $ (more negative and higher absolute value), we expect a phase transition to occur as the potential $\\mathcal {G}$ will become locally concave.", "If $a_{4}>0$ , with sufficiently large values, this will be a first order phase transition.", "If $a_{4}<0$ with sufficiently large values, this will be a second order phase transition.", "When $H=0$ , a bifurcation occurs for $\\beta =-1$ ($\\epsilon =0$ ) and $a_{4}=0$ , as can be easily checked (see [22]).", "This bifurcation is due to the vanishing of the Hessian at $\\beta =-1$ ($\\epsilon =0$ ) and $a_{4}=0$ .", "As discussed in many papers [21], [64], [15], [22], for the quadratic Casimir functional $\\mathcal {C}_{2}=\\int _{\\mathcal {D}}\\,q^{2}/2\\, {\\rm d}{\\bf r}$ , the first bifurcation involves the eigenfunction of $-\\Delta $ with the lowest eigenvalue.", "If we assume that the aspect ratio $l_{x}$ (defined just before equation (REF )) satisfies $l_{x}<1$ , then the smallest eigenvalue is the one corresponding to the zonal mode proportional to $\\cos \\left(y\\right)$ .", "Because we are interested by transitions between two zonal states, we assume from now on that $l_{x}<1$ .", "For non-zero, but sufficiently small, $H$ there will still be a bifurcation for $\\epsilon $ and $a_{4}$ close to zero.", "This is the regime that we wish to consider.", "The null space of the Hessian is spanned by eigenfunctions $\\cos \\left(y\\right)$ and $\\sin \\left(y\\right)$ , therefore as a consequence, for small enough $\\epsilon $ , $a_{4}$ and $H$ , we expect that the bifurcation can be described by a normal form involving only the projection of the field $q$ onto the null space.", "Hence, we decompose the fields into a contribution arising through its projection onto this null space and its orthogonal complement: $\\psi =A\\cos \\left(y\\right)+B\\sin \\left(y\\right)+\\psi ^{\\prime }$ where $\\int _{\\mathcal {D}}\\exp \\left(iy\\right)\\, \\psi ^{\\prime }(\\mathbf {r})\\, {\\rm d}{\\bf r} =0.$ Then $q=-A\\cos \\left(y\\right)-B\\sin \\left(y\\right)+q^{\\prime },$ with $\\int _{\\mathcal {D}}\\,\\exp \\left(iy\\right)\\, q^{\\prime }(\\mathbf {r}) \\,{\\rm d}{\\bf r}=0$ .", "The fact that the bifurcation can be described by a normal form over the null space of the Hessian can be expected on a general basis.", "It can actually be justified by using Lyapunov-Schmidt reduction, as performed and explained in [22] for a number of examples for the two-dimensional Euler and quasi-geostrophic equations.", "Then all other degrees of freedoms describing the minima $q_{c}$ of $\\mathcal {G}$ are slaved to $A$ and $B$ , in the sense that they can be simply expressed as functions of $A$ and $B$ themselves.", "Even though the following example is not treated in the paper [22], it would not be difficult.", "Therefore, we omit the details of the Lyapunov-Schmidt reduction here for simplicity.", "Instead, we rather propose a more heuristic discussion.", "Our strategy, will be in treating the problem perturbatively by assuming that $\\epsilon \\ll 1$ , $\\epsilon a_{6}\\ll a_{4}^{2}$ , and $a_{4}H^{2}\\ll \\epsilon $ (note that it implies that $a_{6}H^{4}\\ll \\epsilon $ ).", "We make these assumptions in order to get an explicit description of the phase transition.", "However, it is important to understand that the theory that predicts the transition rates and the instantons does not depend on these assumptions, and that the same phenomenology will remain valid beyond the perturbative regime.", "We will assume that $\\psi ^{\\prime }$ and $q^{\\prime }$ are first order corrections in all of the three perturbation parameters.", "By rewriting the potential $\\mathcal {G}$ , taking into account only the leading order contributions, and using Eqs.", "(REF ), (REF ) and (REF -REF ), we get after some straightforward computations that $\\mathcal {E}=\\pi ^{2}l_{x}\\left(A^{2}+B^{2}\\right)+\\frac{1}{2}\\int _{\\mathcal {D}}\\,\\left[H\\cos (2y)-q^{\\prime }\\right]\\psi ^{\\prime } \\, {\\rm d}\\mathbf {r},$ and $\\mathcal {G}=\\pi ^{2}l_{x}\\mathcal {G}_{0}(A,B)+\\mathcal {G}_{1}(A,B)\\left[q^{\\prime }\\right]+{\\rm lower~order~terms},$ with $\\mathcal {G}_{0}(A,B)=\\epsilon \\left(A^{2}+B^{2}\\right)-\\frac{3a_{4}}{8}\\left(A^{2}+B^{2}\\right)^{2}+\\frac{5a_{6}}{24}\\left(A^{2}+B^{2}\\right)^{3}+\\mathcal {O}\\left(\\epsilon a_{4}\\right),$ and $\\mathcal {G}_{1}(A,B)\\left[q^{\\prime }\\right]&=&\\frac{\\epsilon -1}{2}\\int _{\\mathcal {D}}\\,\\left[H\\cos \\left(2y\\right)-q^{\\prime }\\right]\\psi ^{\\prime }\\,{\\rm d}\\mathbf {r}\\nonumber \\\\&&+\\frac{1}{2}\\int _{\\mathcal {D}}\\, q^{\\prime 2}\\left\\lbrace 1-3a_{4}\\left[A\\cos (y)+B\\sin (y)\\right]^{2}+5a_{6}\\left[A\\cos (y)+B\\sin (y)\\right]^{4}\\right\\rbrace \\,{\\rm d}\\mathbf {r}.$ We further assume that $a_{4}A^{2}\\ll \\epsilon $ , $a_{6}A^{4}\\ll \\epsilon $ and $\\epsilon \\ll 1$ .", "Then.", "the leading order terms are obtained from the minimization of the first integral and $\\psi ^{\\prime }=\\left[\\frac{H}{3}\\cos (2y)\\right]\\left[1+\\mathcal {O}\\left(\\epsilon \\right)+\\mathcal {O}\\left(a_{4}A^{2}\\right)+\\mathcal {O}\\left(a_{6}A^{4}\\right)\\right],$ or equivalently $q^{\\prime }=-\\frac{H}{3}\\cos (2y)\\left[1+\\mathcal {O}\\left(\\epsilon \\right)+\\mathcal {O}\\left(a_{4}A^{2}\\right)+\\mathcal {O}\\left(a_{6}A^{4}\\right)\\right].$ We use this expression in order to compute the leading order contributions to $G_{1}(A,B)=\\min _{q^{\\prime }}\\mathcal {\\, G}_{1}(A,B)\\left[q^{\\prime }\\right]$ .", "After lengthy but straightforward computations, we get the leading order contribution to be $G_{1}=\\min _{q^{\\prime }}\\mathcal {G}_{1}=-\\frac{{H^{2}}}{3}-\\frac{\\pi ^{2}l_{x}a_{4}H^{2}}{6}\\left(A^{2}+B^{2}\\right)+\\frac{5\\pi ^{2}l_{x}a_{6}H^{2}}{144}\\left[5\\left(A^{2}+B^{2}\\right)^{2}+2\\left(A^{2}-B^{2}\\right)^{2}\\right],$ and subsequently we obtain $\\min _{q}\\mathcal {G}=\\min _{(A,B)}\\,\\pi ^{2}l_{x}G(A,B)$ with $G$ given at leading order by $G(A,B)&=&-\\frac{{H^{2}}}{3}+\\left(\\epsilon -\\frac{a_{4}H^{2}}{6}+\\frac{5a_{6}H^{4}}{216}\\right)\\left(A^{2}+B^{2}\\right)\\nonumber \\\\&&+\\left(-\\frac{3a_{4}}{8}+\\frac{25a_{6}H^{2}}{144}\\right)\\left(A^{2}+B^{2}\\right)^{2}+\\frac{5a_{6}}{24}\\left(A^{2}+B^{2}\\right)^{3}+\\frac{5a_{6}H^{2}}{72}\\left(A^{2}-B^{2}\\right)^{2}.$ $G(A,B)$ is the normal form that describes the phase transition in the limit $a_{4}A^{2}\\ll 1$ , and $a_{6}A^{4}\\ll 1$ and $\\epsilon \\ll 1$ .", "The fact that $G$ is a normal form for small enough $a_{4}$ , $a_{6}$ , and $H$ , implies that the gradient of $\\mathcal {G}$ in the directions transverse to $q=A\\cos \\left(y\\right)+B\\sin \\left(y\\right)$ are much steeper than the gradient of $G$ .", "A more complete derivation could easily be performed along the lines discussed in [22].", "Figure: Contour plot (left) and surface plot (right) of the reduced potential surface G(A,B)G(A,B) (see Eq.", "()) for parameters: ϵ=1.6×10 -2 \\epsilon =1.6\\times 10^{-2}, H=4H=4,a 4 =6×10 -4 a_{4}=6\\times 10^{-4}, a 6 =3.6×10 -6 a_{6}=3.6\\times 10^{-6}.", "For these parameter, GG has four global minima withA=B\\left|A\\right|=\\left|B\\right| and one local minima at A=B=0A=B=0.", "This structure with four non-trivial attractorsis due to symmetry breaking imposed by the topography h(y)=Hcos2yh(y)=H\\cos \\left(2y\\right).We observe that the term proportional to $\\left(A^{2}-B^{2}\\right)^{2}$ breaks the symmetry between $A$ and $B$ .", "Its minimization imposes that $A^{2}=B^{2}$ .", "Then either $A=B$ , or $A=-B$ .", "If we take into account that minimizing with respect to $A^{2}+B^{2}$ will give only the absolute value of $A$ , we can surmise that we will have four equivalent non-trivial solutions: $q_{i}=-\\frac{H}{3}\\cos \\left(2y\\right)+\\sqrt{2}\\left|A\\right|(\\epsilon ,a_{4},a_{6})\\cos (y+\\phi _{i}),$ with $\\phi _{i}$ taking one of the four value $\\left\\lbrace -\\frac{3\\pi }{4},-\\frac{\\pi }{4},\\frac{\\pi }{4},\\frac{3\\pi }{4}\\right\\rbrace $ , with $\\left|A\\right|$ minimizing $\\tilde{G}(\\left|A\\right|)=-\\frac{H^{2}}{3}+2\\left(\\epsilon -\\frac{{a_{4}H^{2}}}{6}+\\frac{5a_{6}H^{4}}{216}\\right)\\left|A\\right|^{2}+4\\left(\\frac{3a_{4}}{8}+\\frac{25a_{6}H^{2}}{144}\\right)\\left|A\\right|^{4}+\\frac{5a_{6}}{3}\\left|A\\right|^{6}.$ The reduced potential $G$ is plotted in figure REF for the case $\\epsilon >0$ and $a_{4}>0$ .", "The structure has four non-trivial attractors due to a breaking of the symmetry imposed by the topography $h(y)=H\\cos \\left(2y\\right)$ .", "For $\\epsilon <0$ , the minima of $G$ have the symmetries of $h$ (potential vorticity profile have a reflexion symmetry with respect to both $y=0$ or $y=\\pi $ and an anti-reflection symmetry with respect to both $y=\\pi /2$ and $y=3\\pi /2$ ).", "For $\\epsilon >0$ this symmetry is broken leading to four different attractors.", "In figure REF , we show the potential vorticity of two of the attractors, the corresponding saddle and the topography.", "Figure: The plot depicts the topography(h(y)=Hcos2yh(y)=H\\cos \\left(2y\\right), symmetric red curve) and two non-trivial attractors of thepotential vorticity qq (black solid lines) corresponding to two minima of the effective potential GG (see Eq.", "(),and figure ) for parameter values ϵ>0\\epsilon >0 anda 4 >0a_{4}>0.", "Additionally, we show the saddle between the two attractorsof the effect potential GG (dashed black curve).Figure: We show the phase diagram fora tricritical point corresponding to the maximization of the normalform s(m)=-m 6 -3b 2m 4 -3am 2 s(m)=-m^{6}-\\frac{3b}{2}m^{4}-3am^{2} (taken from ).The inset show the qualitative shape of the potential ss when theparameters aa and bb are changed.", "The black solid line corresponds to a line of firstorder (discontinuous) phase transition.", "The black dashed line is a second order phase transition line.", "At the tricritical point (a=b=0a=b=0), the first order phase transition change to a second order phase transition.Considering the reduced potential $\\tilde{G}$ (Eq.", "(REF )), we recognize that the structure contains a tricritical point: a point at which a first order transition line switches to a second order transition line.", "Figure REF shows a normal form for a tricritical point.", "The reduced potential $\\tilde{G}$ (Eq.", "(REF )) has the same normal form structure with $a=\\frac{2}{5a_{6}}\\left(\\epsilon -\\frac{a_{4}H^{2}}{6}+\\frac{5a_{6}H^{4}}{216}\\right)$ and $b=\\frac{8}{5a_{6}}\\left(\\frac{3a_{4}}{8}+\\frac{25a_{6}H^{2}}{144}\\right)$ .", "From this last equation, we can conclude that for $a_{4}<25a_{6}H^{2}/54$ ($a_{4}<0$ at leading order), we have a continuous phase transition for $\\epsilon ={35a_{6}H^{4}}/{648}$ (zero at leading order).", "For $a_{4}={25a_{6}H^{2}}/{54}$ ($a_{4}=0$ at leading order), we have a tricritical point.", "Therefore, the transition is between a state given, at leading order, by $q=-\\frac{H}{3}\\cos \\left(2y\\right)$ to one of the four states given by $q_{i}=-\\frac{H}{3}\\cos \\left(2y\\right)+\\sqrt{2}\\left|A\\right|(\\epsilon ,a_{4},a_{6})\\cos \\left(y+\\phi _{i}\\right),$ where $\\phi _{i}\\in \\left\\lbrace -\\frac{3\\pi }{4},-\\frac{\\pi }{4},\\frac{\\pi }{4},\\frac{3\\pi }{4}\\right\\rbrace $ , and $\\left|A\\right|(\\epsilon ,a_{4},a_{6})$ being the non-zero minimizer of (REF ).", "For $a_{4}>0$ and $\\epsilon $ close to zero, we have the coexistence of both ot these states, and thus the transition when $\\epsilon $ is increased is of first order.", "For $a_{4}<0$ and $\\epsilon $ close to zero, the transition when $\\epsilon $ is increased is a second order (continuous) transition." ], [ "Instantons for the topography phase transition", "To summarize, we know how to describe and compute the instantons corresponding to the phase transitions between zonal flows.", "In section we have derived the general theory for Langevin dynamics for field problems with potential $\\mathcal {G}$ , and have concluded in section REF that instantons are the time reversed trajectories of relaxation paths for the reversed dynamics.", "The corresponding equation of motion for the relaxation paths for the reversed dynamics for the quasi-geostrophic dynamics has then been derived in section REF .", "The general theory and Eq.", "(REF ) show that for the quasi-geostrophic dynamics, the reversed dynamics is simply the quasi-geostrophic dynamics where $h$ has been replaced by $-h$ and $\\mathcal {G}$ by $\\mathcal {G}_{r}$ , with $\\mathcal {G}_{r}\\left[q\\right]=\\mathcal {G}\\left[-q\\right]$ .", "In the example we discussed now, $\\mathcal {G}$ is even (see Eq.", "(REF )) such that $\\mathcal {G}_{r}=\\mathcal {G}$ .", "We remark, that over the set of zonal flows $\\mathbf {v}=U(y)\\mathbf {e}_{x}$ , the nonlinear term of the quasi-geostrophic equation vanishes: $\\mathbf {v}\\left[q+h\\right]\\cdot \\nabla q=0$ .", "As a consequence, when the instanton remains a zonal flow, the fact that $h$ has to be replaced by $-h$ has no consequence.", "Let us now argue that the instanton is actually generically a zonal flow.", "We assume for simplicity that the stochastic forces are homogeneous (invariant by translation in both directions).", "Then $C\\left(\\mathbf {r},\\mathbf {r^{\\prime }}\\right)=C\\left(\\mathbf {r}-\\mathbf {r^{\\prime }}\\right)=C_{z}(y-y^{\\prime })+C_{m}(y-y^{\\prime },x-x^{\\prime })$ where $C_{z}(y)=\\frac{1}{2\\pi l_{x}}\\int _{_{0}}^{2\\pi l_{x}}\\, C(x,y)\\,{\\rm d}x$ is the zonal part of the correlation function, and $C_{m}=C-C_{z}$ the non-zonal or meridional part.", "As the nonlinear term of the two-dimensional Euler equations identically vanishes, the relaxation dynamics has a solution among the set of zonal flows.", "If $C_{z}$ is non-degenerate (positive definite as a correlation function), then relaxation paths will exist through the gradient dynamics $\\frac{\\partial q}{\\partial t}=-2\\pi \\alpha l_{x}\\int _{_{0}}^{2\\pi }\\, C_{z}(y-y^{\\prime })\\frac{\\delta \\mathcal {G}}{\\delta q(y^{\\prime })}\\, {\\rm d}y^{\\prime },$ where $q=q(y)$ is the zonal potential vorticity field.", "Moreover, as argued in section REF , the fact that $G$ (REF ) is a normal form for small enough $a_{4}$ , $a_{6}$ , and $H$ , implies that the gradient of $\\mathcal {G}$ in directions transverse to $q=A\\cos \\left(y\\right)+B\\cos \\left(y\\right)$ are much steeper than the gradient of $G$ .", "As a consequence, at leading order the relaxation paths will be given by the relaxation paths for the effective two-degrees of freedom $G$ .", "Then, from (REF ), (REF ), and (REF ) we obtain that, at leading order, for the relaxation path given by (REF -REF ), the dynamics of $A$ and $B$ are given by $\\frac{{\\rm d}A}{{\\rm d}t}=-c\\frac{\\partial G}{\\partial A}\\quad {\\rm and}\\quad \\frac{{\\rm d}B}{{\\rm d}t}=-c\\frac{\\partial G}{\\partial B},$ with $c=-\\alpha l_{x}\\int _{0}^{2\\pi }\\, C_{z}(y)\\cos \\left(y\\right)\\, {\\rm d} y$ , where we recall that $G$ is given by Eq.", "(REF ).", "From this result the relaxation paths are easily computed.", "Using the fact that fluctuation paths are time reversed trajectories of relaxation paths, instanton are also easily obtained.", "One of the resulting relaxation paths (blue curve) and one of the instantons (red curve) are depicted in figure REF overlapped on the contours of the potential $G$ in the $(A,B)$ -plane.", "The corresponding two attractor involved, together with the saddle point and examples of two intermediate states are shown in figure REF .", "Figure: Contour plot of the reduced potentialsurface G(A,B)G(A,B) (same as figure ) with the superimposedtransition path between two attractors denoted by • via a saddle ▪\\blacksquare .", "The instanton (most probable fluctuation path from one attractorto a saddle) is show by the solid red line, while the corresponding relaxation path from the saddle to the second attractor is given by the solid blue line.In this case, the instanton and the relaxation paths are actually the reverse of one another.Figure: The potential vorticityq(y)q(y) for two of the non-trivial attractors (solid black curves), the corresponding saddle between the attractors (dashed black curve), and two intermediateprofiles along the instanton path (solid red curve) and the relaxationpath (solid blue curve)." ], [ "Dimensional analysis ", "In this section, we briefly discuss dimensional analysis for the dynamics (REF ), with topography $h=H\\cos \\left(2y\\right)$ , and potential $\\mathcal {G}$ given by Eqs.", "(REF -REF ).", "We recall these equations for clarity: $\\frac{\\partial q}{\\partial t}+\\mathbf {v}\\left[q-H\\cos \\left(2y\\right)\\right]\\cdot \\nabla q & = & -\\alpha \\int _{\\mathcal {D}}\\, C({\\bf r}-{\\bf r}^{\\prime })\\frac{\\delta \\mathcal {G}}{\\delta q({\\bf r}^{\\prime })}\\,{\\rm d}{\\bf r}^{\\prime }+\\sqrt{2\\alpha \\gamma }\\eta ,\\\\\\mathbf {v}=\\mathbf {e}_{z}\\times \\nabla \\psi ,\\quad \\omega &=&\\Delta \\psi , \\quad q=\\omega +H\\cos \\left(2y\\right),$ with $\\mathcal {G}=\\int _{\\mathcal {D}}\\frac{q^{2}}{2}-a_{4}\\frac{q^{4}}{4}+a_{6}\\frac{q^{6}}{4}\\, {\\rm d}{\\bf r}-\\left(1-\\epsilon \\right)\\mathcal {E}.$ First, let us discuss a set of convenient non-dimensional units for our problem.", "We express length in units of the domain size.", "The dynamics involve the following parameters $\\alpha $ ($s^{-1}$ ), $\\gamma $ ($s^{2}$ ), $H$ ($s^{-1}$ ), $a_{4}$ ($s^{2}$ ), $a_{6}$ ($s^{4}$ ), $\\beta $ or $\\epsilon $ (no dimension), the aspect ratio $l_{x}$ (no dimension), and the force spectrum $C$ (no dimension), energy $\\mathcal {E}$ ($s^{-2}$ ), and Casimirs $\\mathcal {C}$ ($s^{-2}$ ).", "We are interested mainly in the range of parameter for which the dynamics is bistable.", "Moreover, it will be especially useful to consider the perturbative regime close to the bifurcation described in section REF .", "As a consequence, we choose $\\epsilon \\ll 1$ , $a_{4}>0$ and $a_{4}$ sufficiently small (as discussed below), and $H$ sufficiently small ($a_{4}H^{2}\\ll \\epsilon $ and $a_{6}H^{2}\\ll a_{4}$ ) such that the phase transition is close to the one occurring for $H=0$ .", "We recall that these assumptions are made in order to get an explicit description of the phase transitions, however it is important to understand that the theory that predicts the transition rates and the instantons does not depend on these assumptions and that the same phenomenology will remain valid beyond this perturbative regime.", "As discussed in section REF , with these hypotheses, the lower values of $\\mathcal {G}$ are approximated by the normal form $G$ (REF ).", "From (REF ), we conclude that if we assume $\\epsilon a_{6}\\ll a_{4}^{2}$ , then the order of magnitude of $A$ , the amplitude of the large scale mode, is $\\left(\\epsilon /a_{4}\\right)^{1/2}$ .", "As we have chosen $a_{4}H^{2}\\ll \\epsilon $ , the correction due to the topography is of sub-leading order (see Eq.", "(REF )).", "The kinetic energy of the largest scale mode is then of the order $\\epsilon /a_{4}$ .", "Subsequently, we choose $\\left(a_{4}/\\epsilon \\right)^{1/2}$ as a time unit.", "We denote $H^{\\prime }=\\left(a_{4}/\\epsilon \\right)^{1/2}H$ , $\\gamma ^{\\prime }=\\left(a_{4}/\\epsilon \\right)^{3/2}\\gamma $ , $\\alpha ^{\\prime }=\\left(a_{4}/\\epsilon \\right)^{1/2}\\alpha $ , $a^{\\prime }_{6}=(\\epsilon /a_{4})^{2}a_{6}$ , and $q^{\\prime }=\\left({a_{4}/\\epsilon }\\right)^{1/2}q$ to be the dimensionless variables in this time unit.", "Therefore, we can write the non-dimensional equations, dropping the prime variables as $\\frac{\\partial q}{\\partial t}+\\mathbf {v}\\left[q-H\\cos \\left(2y\\right)\\right]\\cdot \\nabla q & = & -\\alpha \\int _{\\mathcal {D}}\\, C({\\bf r}-{\\bf r}^{\\prime })\\frac{\\delta \\mathcal {G}}{\\delta q({\\bf r}^{\\prime })}\\,{\\rm d}{\\bf r}^{\\prime }+\\sqrt{2\\alpha \\gamma }\\eta ,\\\\\\mathbf {v}=\\mathbf {e}_{z}\\times \\nabla \\psi ,\\quad \\omega &=&\\Delta \\psi , \\quad q=\\omega +H\\cos \\left(2y\\right),$ with $\\mathcal {G}=\\int _{\\mathcal {D}}\\,\\frac{q^{2}}{2}-\\epsilon \\frac{q^{4}}{4}+a_{6}\\frac{q^{6}}{6}\\, {\\rm d}{\\bf r}-\\left(1-\\epsilon \\right)\\mathcal {E}.$ Within these non-dimensional variables, $\\epsilon $ controls the distance to the bifurcation.", "The approximation of the large scale dynamics by a few number of modes will then be valid for $\\epsilon \\ll 1$ , and the approximation that the topography is a second order effect is controlled by $H^{2}\\ll \\epsilon $ and $a_{6}H^{2}\\ll 1$ (this also implies $a_{6}H^{4}\\ll \\epsilon $ ).", "We now give a qualitative picture of the dynamics.", "Recall that the stationary distribution of the stochastic process is given by $P_{s}=Z^{-1}\\exp \\left(-\\mathcal {G}/\\gamma \\right)$ .", "The gradient of $\\mathcal {G}$ in the directions which are transverse with respect to the modes $A\\cos \\left(y\\right)+B\\sin \\left(y\\right)$ is of order one, whereas the stochastic force is multiplied by $\\gamma ^{1/2}$ .", "As a consequence, typical values of fluctuations for the stationary measure in these transverse directions are of order $\\gamma ^{1/2}$ .", "Finally, the non-dimensional parameter $\\alpha $ controls the relative order of magnitude of the inertial (or Hamiltonian) part of the dynamics, compared to the dissipative gradient terms in (REF )." ], [ "Conclusions and perspectives", "We have defined Langevin dynamics for two-dimensional and quasi-geostrophic turbulent flows.", "These dynamics have an energy-Casimir invariant measure.", "The dissipative part of the dynamics derives from a potential that is transverse to the Hamiltonian part of the dynamics.", "Moreover, the noise autocorrelation function is the same as the kernel defining the dissipative part.", "Under these hypotheses, the action is modified in a simple manner under time reversal.", "It is either symmetric leading to detailed balance, or leads to a dual action which describes dynamics that belong to the same family of physical model.", "These symmetries put these Langevin dynamics in the framework of classical Langevin dynamics.", "For instance, fluctuation paths are time reversed trajectories of relaxation paths of the dual dynamics.", "This gives a very simple characterization of fluctuation paths, of large deviations, and of large deviation paths, when they exist.", "We have proposed and analyzed cases with phase transitions, both continuous and discontinuous, and of a tricritical point.", "This opens the study to a rich phenomenology of processes, including bistable situations.", "These Langevin dynamics with exact theoretical prediction will be very useful benchmarks for future tests of numerical algorithms aimed at computing large deviations in turbulence problems.", "Several interesting concepts could be developed in the future.", "These Langevin dynamics, give examples of turbulence problems for which the recent results of stochastic thermodynamics could be extended, e.g.", "it would be very interesting to study Gallavotti-Cohen fluctuation relations [25], or entropy production [3], [34] in this setup.", "The temporal response of the system to external driving or change of parameters could also be studied in relation to recently studied non-equilibrium linear response for Markovian dynamics [2], [41].", "Let us come back to two important and related issues not discussed in this paper.", "Firstly, is it possible to give a clear mathematical meaning to the Langevin dynamics (REF ), given that it may involve very rough forces through the noise term?", "Or of smooth noise combined with very weak friction?", "Secondly, for the dynamics (REF ), will large deviation results (REF ) be valid?", "In order to discuss these two questions, let us consider a special case of Langevin dynamics (REF ), with $C({\\bf r},{\\bf r}^{\\prime })=\\Delta \\delta (\\mathbf {r}-\\mathbf {r}^{\\prime })$ , corresponding to the enstrophy ensemble (see section REF ).", "From a physical point of view, it has been identified for a long time that the dynamics can not be given a simple physical interpretation.", "Indeed, for the enstrophy measure, the expectations of both the energy and enstrophy are infinite.", "Even the expectation for the velocity field is not defined, and most of the realization do not lead to a physical velocity field.", "This is related to some of the mathematical results in [6].", "These remarks give a negative answer to the first question.", "Still, it has been observed [12] that, at a formal level, the minimization of the action can be computed explicitly and leads to a quasi-potential which is indeed the enstrophy as may have been expected.", "A natural physical question is then to understand what happens if the noise is regularized at a scale $\\delta $ , much smaller than the domain size.", "Recently, we have been aware of the work by Brzezniak, Cerrai and Freidlin [17], that actually considers this problem.", "Their mathematical result, is that for any finite $\\delta $ the dynamics are well defined.", "Moreover, that for any finite $\\delta $ , a large deviation principle for exit times from a bounded domain holds when the noise amplitude goes to zero (when $\\gamma $ goes to zero in our notation, see Eq.", "(REF )).", "These large deviations are actually described by the minimization of the action functional (REF -REF ), with a kernel $C_{\\delta }$ taking into account the noise regularization.", "When $\\delta $ goes to zero, the large deviation functional and the minimizers of the actions actually converge to the one corresponding to the enstrophy ensemble [17].", "These results justify the formal computation in [12], and equivalent results would justify the formal computations presented in this current work.", "However, we stress that for these results to hold, the order of the limits ($\\gamma \\rightarrow 0$ and $\\delta \\rightarrow 0$ afterwards) is crucial.", "As discussed above, for the enstrophy ensemble, it is necessary to regularize the noise first in order to obtain meaningful dynamics.", "However, it is not yet clear which are the relavent cases, depending on the kernel $C$ or the potential $\\mathcal {G}$ , when such a regularization is necessary or not?", "For instance, when $a_{4}<0$ or $a_{6}>0$ , see Eq.", "(REF ), such a regularization may be unnecessary, or with a potential controlling the extremal values of the vorticity field, such a regularization would be unnecessary.", "This question could be the subject of further studies.", "The dynamics could also be regularized at the level of the dissipation, for instance by adding small scale dissipation in the form of hyperviscosity with small coefficient.", "In order to conclude, we stress once more that, for applications it would be desirable to go beyond the Langevin dynamics considered in this paper.", "A first step could be for the derivation of the slow dynamics of zonal jets in quasi-geostrophic models [13], followed by large deviation computations.", "We consider progresses in this direction and in other directions in future works." ] ]
1403.0216
[ [ "Recurrent events of synchrony in complex networks of pulse-coupled\n oscillators" ], [ "Abstract We present and analyze deterministic complex networks of pulse-coupled oscillators that exhibits recurrent events comprised of an increase and a decline in synchrony.", "Events emerging from the networks may form an oscillatory behavior or may be separated by periods of asynchrony with varying duration.", "The phenomenon is specific to spatial networks with both short- and long-ranged connections and requires delayed interactions and refractoriness of oscillators." ], [ "Recurrent events of synchrony in complex networks of pulse-coupled oscillators A. Rothkegel1,2,3 K. Lehnertz1,2,3 A. Rothkegel & K. Lehnertz 1Department of Epileptology, University of Bonn, Germany 2Helmholtz Institute for Radiation and Nuclear Physics, University of Bonn, Germany 3Interdisciplinary Center for Complex Systems, University of Bonn, Germany 89.75.Kd 05.45.Xt 84.35.+i 87.10.-e We present and analyze deterministic complex networks of pulse-coupled oscillators that exhibits recurrent events comprised of an increase and a decline in synchrony.", "Events emerging from the networks may form an oscillatory behavior or may be separated by periods of asynchrony with varying duration.", "The phenomenon is specific to spatial networks with both short- and long-ranged connections and requires delayed interactions and refractoriness of oscillators.", "Nature possesses various examples of systems composed of many oscillatory elements in which, through mutual interactions, collective dynamics emerges [1], [2], [3].", "In many biological networks (like the heart or the brain) interactions are short-lasting and may be modeled as pulses, which are fired at a given oscillator phase [4], [5].", "Such pulse-coupled oscillators show, for large arrangements, a variety of collective dynamics with a convergence of global observables after transients.", "Examples range from stable synchronous [5], [6] and asynchronous states [7], phase-locking [8], partial synchrony [9], [10], [11], to transitions from asynchronous to synchronous states [12].", "The human brain with its 100 billion neurons, however, may show recurrent changes in global synchrony (as seen, e.g., on the electroencephalogram) associated with physiologic or pathophysiologic functioning [13], [14].", "In this Letter, we present a spatial network model of pulse-coupled oscillators (PCOs), which shows collective dynamics with no convergence to either synchrony, partial synchrony, or asynchrony.", "Instead, the network generates – even without noise influences or a change of parameters – recurrent events that are comprised of an increase and decline in synchrony.", "Events may occur at random looking times and be separated by prolonged periods of asynchrony.", "Events may also be initiated immediately after completion of the preceding event leading to an oscillatory behavior.", "The oscillators thus generate a rhythm, which is not directly linked to their intrinsic time-scales (the delay of interactions and the duration of the refractory period) but is an emerging property of the network.", "We study networks of oscillators $n \\in N$ with phases $\\phi _n(t) \\in [ 0,1 ]$ , $\\frac{d \\phi _n}{d t} = 1$ .", "If for some $t_f$ and some oscillator $n$ the phase reaches 1 ($\\phi _n(t_f) = 1$ ), it is reset to 0 ($\\phi _n(t_f^+) = 0$ ) and we introduce a phase jump in all oscillators $n^{\\prime }$ which are adjacent to $n$ according to some directed graph: $\\phi _{n^{\\prime }}(t_f^+) = \\phi _{n^{\\prime }}(t_f) + \\Delta \\left(\\phi _{n^{\\prime }}(t_f)\\right).$ $\\Delta $ denotes the phase response curve.", "A refractory period of length $\\vartheta $ can easily be incorporated into $\\Delta $ by setting $\\Delta (\\phi ) = 0$ for $\\phi < \\vartheta $ .", "Consider identical time delays $\\tau < \\vartheta $ on all outgoing connections of oscillator $n$ , such that all its excitations are received when $\\phi _n = \\tau $ .", "In this case, $n$ can be replaced equivalently with an undelayed oscillator with periodically shifted phase and phase response curve.", "We can thus incorporate both refractoriness and delay into an arbitrary phase response curve $\\Delta _\\epsilon $ (yielding an formally undelayed phase response curve $\\Delta $ ) by setting: $\\Delta (\\phi ) ={\\left\\lbrace \\begin{array}{ll}(1 - \\vartheta )\\Delta _\\epsilon (\\frac{\\phi - \\vartheta + \\tau }{1 - \\vartheta })& \\vartheta - \\tau < \\phi < 1 - \\tau ,\\\\0 & \\mbox{otherwise}.", "\\\\\\end{array}\\right.", "}$ Here $\\epsilon $ denotes some coupling strength.", "Our choice for $\\Delta _\\epsilon (\\phi )$ is the first order in $\\epsilon $ approximation of standard integrate-and-fire neurons with excitatory coupling, threshold and eigenfrequency normalized to 1, and exponential charging: $\\Delta _\\epsilon (\\phi ) =\\min \\left\\lbrace - \\epsilon \\frac{(1-\\alpha )}{\\ln (\\alpha )} \\alpha ^{ -\\phi } , 1 - \\phi \\right\\rbrace .$ $\\alpha $ determines the curvature of $\\Delta _\\epsilon $ and therefore the amount of leakage.", "The non-leaky case is obtained for $\\alpha \\rightarrow 1$ .", "In contrast to delayed excitatory couplings without refractory periods [15], [16], [12], [17] the inclusion of a refractory period allows for synchronous states that are stable even if the sums of incoming connection weights are non-identical.", "To demonstrate this, we consider the dynamical evolution of a perturbation $\\lbrace \\phi _n(0) | n \\in N\\rbrace $ with size $\\delta _0:= \\max \\limits _n \\phi _n(0) - \\min \\limits _n \\phi _n(0)$ .", "Let $n_>$ ($n_<$ ) denote an oscillator with maximal (minimal) phase.", "For small perturbations with $\\delta _0 < \\vartheta - \\tau $ , each oscillator will fire and will then stay refractory until every other oscillator has fired once.", "We can thus define the size of the perturbation $\\delta _1 = \\max \\limits _n \\phi _n(t_1) - \\min \\limits _n \\phi _n(t_1)$ for some time $t_1$ at which each oscillator has fired exactly once.", "$n_>$ will not get excited and will not be overtaken by any other oscillator.", "As we consider solely advances in phase (no inhibition), we can thus conclude $\\delta _1 \\le \\delta _0$ , or $\\delta _{i+1} \\le \\delta _{i}$ per induction.", "Moreover, we can give an upper bound for $\\delta _\\infty = \\lim \\limits _{i \\rightarrow \\infty } \\delta _i$ .", "Since for our choice of the phase response curve, phase-locking of two oscillators is always achieved in a finite number of oscillations, and as we consider only a finite number of oscillators, we can choose a time $t_i^{\\prime }$ after which all excitations are received refractorily.", "The length of the shortest path between $n_>$ to $n_<$ is bounded by the diameter $d$ of the graph (the longest shortest path).", "As excitations are received refractorily, the difference in phase for every pair of oscillators along such a path is smaller than or equal $\\tau $ , and therefore $\\delta _\\infty = \\phi _{n_>} - \\phi _{n_<} \\le d \\tau $ .", "In particular, for the case of vanishing time delay, complete synchrony is attracting for all strongly connected graphs (i.e., a graph in which there is a path in each direction for every pair of nodes).", "For non-vanishing time delay, we have almost synchronous states with phases distributed in an interval of size $d \\tau $ .", "These states are surrounded by an attracting region in state space (assuming $d \\tau < \\vartheta - \\tau $ ).", "Whether or not almost synchronous states are reached, when starting from homogeneously distributed phases, depends on the relative size of $\\tau $ and $\\vartheta $ .", "E.g., for a random network and $\\vartheta $ slightly larger than $\\tau $ (or for similar PCO networks with refractory periods smaller than the time delay) synchronous states are unstable and we observe asynchronous behavior, while for large $\\vartheta $ almost synchronous states are globally attracting.", "In the following, we choose $\\vartheta $ between these two extremes and investigate the collective dynamics of $\\left|N\\right|=100.000$ identical PCOs arranged on a spatial network with both short- and long-ranged connections.", "We connect each oscillator bidirectionally to its $k$ nearest neighbors in a one-dimensional chain with cyclic boundaries.", "We then remove every directed connection with probability $\\rho $ and introduce a directed connection between two randomly chosen, unconnected oscillators, thereby avoiding self-connections (cf.", "[18]).", "To characterize the collective dynamical behavior, we define an order parameter $r(t) = 1/\\left|N\\right| \\left|\\sum _{n \\in N} e^{2 \\pi i \\phi _n(t)}\\right|$ , which yields 1 for complete synchrony and 0 for some balanced distribution of the phases of oscillators.", "Figure: Examples of recurrent events of synchrony in PCO networks (ρ=0.5,k=50\\rho = 0.5, k = 50, τ=0.002,ϵ=0.01,α=0.9\\tau = 0.002, \\epsilon = 0.01, \\alpha = 0.9) as assessed by the order parameter r(t)r(t).", "For ϑ=0.027\\vartheta = 0.027 (a) events are separated by periods of asynchrony with varying duration.", "For ϑ=0.03\\vartheta = 0.03 (b) events form an (almost) periodic behavior.", "Events last for several collective firings of oscillators (for the chosen parameters the number of firings amounts to ≈80\\approx 80).Starting from homogeneously distributed phases, we observe recurrent events of (partial) synchrony in our PCO networks (Fig.", "REF ).", "Events are stereotypical and can be characterized by an increase of the order parameter $r$ up to some peak value ($r<1$ ) and a subsequent decline.", "For $\\vartheta $ a few times larger than $\\tau $ , events emerge at random looking times, and are separated by prolonged periods of asynchronous firing of oscillators (Fig.", "REF a).", "For larger $\\vartheta $ , events emerge almost immediately after the preceding event is completed (Fig.", "REF b).", "We observe time periods $T$ between peaks of successive events to decrease with increasing refractory period $\\vartheta $ (Fig.", "REF ).", "In contrast, the network dynamics during an event and in particular its duration are only marginally affected by the choice of $\\vartheta \\in [0.027, 0.03]$ .", "In the following, we thus investigate emergence and network dynamics of a single event (Fig.", "REF ).", "Figure: Distributions of time periods TT between peaks of successive events for different refractory periods ϑ\\vartheta (observation interval t∈[0,50.000]t \\in [0,50.000]).", "Inset shows mean (μ T \\mu _T; black dots) and standard deviation (σ T \\sigma _T; gray dots) of TT in dependence on ϑ\\vartheta .In order to characterize the asynchronous behavior prior to the event, we consider the evolution of nearby trajectories in the state space spanned by the phases $\\phi _n, n \\in N$ using a notion of distance that takes into account their cyclicity: $d(\\phi _n,\\phi _m)=\\min \\lbrace | \\phi _n - \\phi _m| , 1-|\\phi _n - \\phi _m|\\rbrace $ (cf.", "[12]).", "Trajectories are separated whenever an oscillator with phase $\\phi $ is excited such that $\\Delta ^{\\prime }(\\phi ) > 0$ , which is the case for the largest part of our phase response curve (i.e., for $\\phi \\in [\\vartheta - \\tau , 1 - \\tau ]$ ).", "Note that due to the jump in the phase response curve at $\\phi = \\vartheta - \\tau $ , two nearby trajectories may be separated by an amount which is independent on their distance.", "For small initial distances, however, this situation becomes increasingly unlikely and we observe an exponential divergence of nearby trajectories, in contrast to inhibitory pulse-coupled networks (investigated in Refs.", "[19], [20]).", "Spatial correlations of phases decay over 50 – 100 oscillators, and phases of oscillators which are linked by a long-ranged connection are uncorrelated.", "The network dynamics thus resembles spatiotemporal chaos as observed in excitable media [21].", "The chaotic behavior, however, is unstable [22] and as soon as there is a large enough concentration of phases from distributed oscillators, the network will start to synchronize.", "However, a stable orbit (as observed e.g.", "in Ref.", "[12]) is not reached in our network.", "Figure: Temporal evolution of phases φ\\phi of a part of the network (oscillator 1 to 2.000; φ=0\\phi = 0 black, φ=1\\phi = 1 white) (top) and of the order parameter rr calculated for the whole network (100.000 oscillators) (bottom).The data are from the first event shown in Fig.", "a.", "The instantaneous frequency ν r \\nu _r of the low-amplitude oscillation of rr is slightly higher than the eigenfrequency of oscillators and coincides during the event with the frequency ν c \\nu _c of collective firings.Figure: Order parameter rr versus normalized mean phase distance SS (a) and instantaneous frequency ν r \\nu _r (b) for data shown in Fig.", "bottom.", "We define SS as the mean distance of phases of spatially neighbored oscillators D loc :=1/N∑ n d(φ n ,φ n+1 )D_{\\rm loc}:=1/\\left|N\\right| \\sum _n d(\\phi _n, \\phi _{n+1}) divided by the mean distance between any two oscillators D:=1/N 2 ∑ n,m d(φ n ,φ m )D:=1/\\left|N\\right|^2 \\sum _{n,m} d( \\phi _n ,\\phi _m ).", "SS measures the scatteredness of the oscillators in the phase cluster.", "If the oscillators of the phase cluster are homogeneously scattered over the network D loc ≈DD_{\\rm loc} \\approx D and thus S≈1S \\approx 1.", "If they form connected segments in the network D loc <DD_{\\rm loc} < D and SS takes on smaller values.", "In contrast to SS, the frequency ν r \\nu _r takes on similar values on both the ascending and descending part of the event.During an event a part of the oscillators forms a single, slightly dispersed phase cluster.", "The size of the cluster, i. e., the number of contributing oscillators, is reflected in the order parameter $r$ (Fig.", "REF ).", "The chaotic dynamics of the remaining oscillators (asynchronous part) is disturbed by collective firings of the cluster.", "Oscillators with a phase slightly smaller than the phase of the cluster are more likely to be entrained to the cluster.", "These oscillators are distributed all over the network (considering the chaotic spatiotemporal dynamics of the asynchronous firing), and their entrainment thus leads to a cluster which is scattered over the network during the ascending part of the event (Fig.", "REF a).", "Intuitively, the possibility for entrainment depends on the number of excitations $N_E$ per collective firing received by the asynchronous part and on the frequency $\\nu _c$ of collective firings.", "$N_E$ increases with cluster size but also depends on the spatial distribution of the cluster.", "On the other hand, $\\nu _c$ decreases with cluster size (Fig.", "REF b).", "This is because excitations via connections which start and end in the cluster are received by refractory oscillators and thus do not increase $\\nu _c$ .", "Excitations from the asynchronous part to the cluster do increase $\\nu _c$ but since their number declines with an increasing cluster size, $\\nu _c$ decreases.", "The impact of the slowing of collective firings on the possibility for entrainment outweighs the impact of the increase of $N_E$ .", "Thus synchronization in our network is not complete.", "At the peak of the event, the collective firing is too slow to entrain any more oscillators from the asynchronous part despite the large number of excitations per collective firing.", "The oscillators now form a few, small asynchronous segments which are embedded into synchronous segments (belonging to the phase cluster).", "During the descending part of the event, asynchronous segments grow (Fig.", "REF ).", "Consider neighboring oscillators well inside a synchronous segment.", "These oscillators receive excitations via long-ranged connections from oscillators which are distributed homogeneously over the asynchronous part.", "The resulting phase shifts thus differ only marginally, and these oscillators remain in the cluster.", "In contrast, oscillators located near the boundary of a synchronous segment receive (in addition to the excitations via long-ranged connections) correlated excitations via short-ranged connections from the asynchronous part.", "The resulting phase shift decreases with the distance to the boundary, and these oscillators are torn out of the cluster, leading to a growing of asynchronous segments.", "The cluster is thus composed of a few connected segments, in contrast to the scattered spatial distribution during the ascending part (Fig.", "REF a).", "Note that the growing of asynchronous segments can only be observed for networks for which in the initial lattice nodes are connected to at least $k=10$ nearby nodes.", "For nearest-neighbor coupling, asynchrony spreads via both short- and long-ranged connections which does not lead to events of synchrony.", "With a decreasing cluster size, the frequency of collective firings $\\nu _c$ increases again (Fig.", "REF b).", "Nevertheless, entrainment to the cluster is not possible.", "This is because $N_E$ depends on the spatial distribution of the cluster.", "For a given cluster size, $N_E$ is large if the oscillators of the cluster are scattered as here excitations via both short- and long-ranged connections contribute.", "$N_E$ is small if the oscillators of the cluster forms connected segments as here mainly excitations via long-ranged connections contribute.", "Therefore the cluster is not able to entrain oscillators from the asynchronous part as it was the case during the ascending part of the event.", "The network thus returns to asynchronous firing, concluding the event.", "Spatial networks of spiking elements can exhibit a variety of dynamical behaviors [23], [24], [25], [26], [27].", "We here reported on a novel phenomenon, namely recurrent events of synchrony, emerging robustly from networks of identical, deterministic oscillators.", "The phenomenon relies on both refractoriness in the local dynamics and delayed excitatory coupling and is specific to networks with both short- and long-ranged connections.", "With adjusted parameters of oscillators it can be observed for replacement probabilities $ 0.35 \\lesssim \\rho \\lesssim 0.60$ .", "Furthermore, recurrent events of synchrony can be observed for two-dimensional spatial PCO networks and for similar phase response functions without a jump.", "Nevertheless, the precise dynamical origin of the events needs further investigation.", "Complex networks with short- and long-ranged connections have frequently been used to model brain functions and dysfunctions (see, e.g., [28], [29], [30], [31], [32], [33]) since these networks represent a simple approximation of the synaptic wiring in the brain [34], [35].", "In our network, events of synchrony may emerge at random looking times, with prolonged inter-event periods of asynchronous firing of oscillators.", "This collective phenomenon resembles the dynamics of epileptic brains with seizures as recurrent rare events of overly synchronous neuronal activity.", "In contrast to previous modeling approaches to the dynamics into and out of seizures that rely either on changes in some control parameter or on noise to generate transitions [36], our network is capable to transit spontaneously despite the deterministic setup.", "Our approach may thus significantly improve recent modeling approaches of the disease [37].", "Events of synchrony in our network may also occur (almost) periodically, where the oscillators generate a rhythm, which is not directly linked to their intrinsic time-scales but is an emerging property of the network.", "The described mechanism that leads to the oscillation may advance understanding of the generation of the respiratory rhythm, which persists even in small slices of mammalian brains after the attenuation of postsynaptic inhibition [38].", "We are grateful to Stephan Bialonski and Ulrike Feudel for helpful comments.", "This work was supported by the Deutsche Forschungsgemeinschaft (LE 660/4-1)." ] ]
1403.0101
[ [ "Semi-classical analysis of the inner product of Bethe states" ], [ "Abstract We study the inner product of two Bethe states, one of which is taken on-shell, in an inhomogeneous XXX chain in the Sutherland limit, where the number of magnons is comparable with the length L of the chain and the magnon rapidities arrange in a small number of macroscopically large Bethe strings.", "The leading order in the large L limit is known to be expressed through a contour integral of a dilogarithm.", "Here we derive the subleading term.", "Our analysis is based on a new contour-integral representation of the inner product in terms of a Fredholm determinant.", "We give two derivations of the sub-leading term.", "Besides a direct derivation by solving a Riemann-Hilbert problem, we give a less rigorous, but more intuitive derivation by field-theoretical methods.", "For that we represent the Fredholm determinant as an expectation value in a Fock space of chiral fermions and then bosonize.", "We construct a collective field for the bosonized theory, the short wave-length part of which may be evaluated exactly, while the long wave-length part is amenable to a $1/L$ expansion.", "Our treatment thus results in a systematic 1/L expansion of structure factors within the Sutherland limit." ], [ "Introduction and summary", "The computation of structure factors, matrix elements of operators between eigenstates, analytically in exactly integrable systems remains a challenging task.", "In very small systems one may obtain results by employing determinant formulas derived from the algebraic Bethe ansatz.", "The determinants are of matrices whose size increase with the number of particles, such that fully analytical computations go quickly out of hand.", "In the thermodynamical limit of large number of particles, the computation of such determinants becomes intractable, except in special limits, usually accompanied by a phenomenon which in physical terms may be viewed as a condensation of excitations.", "The most familiar cases are the condensation of magnons into bound complexes with large spin in the Heisenberg ferromagnet as discovered by Sutherland [1] (hence the limit is sometimes called the `Sutherland limit'), the condensation of solitons in the quantum Sine-Gordon model to quasi-periodic solutions of the KdV [2] equation, or the condensation of Cooper pairs in a superconductor [3], to form either single or multiple condensates, the latter being described by the Richardson model (a particular example of Gaudin magnets).", "More recently, bound complexes of magnons has been studied in the context of the integrability in gauge and string theories [4][5] (see also the review [6]).", "Some correlation functions in supersymmetric Yang-Mills theories can be expressed in terms of inner products of Bethe states in a chain of spins [7] and can be cast in the form of a determinant [8].", "The thermodynamical limit here is the limit of `heavy' fields in the Yang-Mills theory, which correspond, by the AdS/CFT duality, to classical solutions of the string-theory sigma model.", "The three-point function of heavy fields is exponentially small and can be thought of as a process of semi-classical tunelling [9].", "The leading order computations performed in [9], [10], [11] gave an explicit expression of the exponent as a contour integral of a dilogarithm.", "In the present paper we give a method to compute the higher orders of the semi-classical expansion and give an explicit formula for the pre-exponential factor.", "We focus on the XXX spin chain (the isotropic Heiseberg magnet), where the thermodynamical limit corresponding to long-wavelength excitations above the ferromagnetic vacuum.", "In view of the applications, we consider the more general case of an inhomogeneous spin chain with twisted periodic boundary condition.", "We will consider $M$ -magnon Bethe states in a chain of length $L$ in the thermodynamical limit where $M, L\\rightarrow \\infty $ and $M /L\\sim 1$ .", "Our goal is to propose a systematic method for computing the $1/L$ expansion for the (logarithm of the) inner product of a Bethe eigenstate and an off-shell Bethe state.", "In particular we obtained an explicit expression for the subleading term, given below.", "Let $| { {\\bf u} }\\rangle $ and $| { \\bf v }\\rangle $ be two $M$ -magnon Bethe states in a XXX spin chain of length $L$ , characterized by the rapidities $ { {\\bf u} }=\\lbrace u_1, \\dots , u_M\\rbrace $ and $ { \\bf v }=\\lbrace v_1, \\dots , v_M\\rbrace $ .", "One of the two states is required to be on-shell in the sense that its rapidities satisfy the Bethe equations.", "The two Bethe states are characterised by their pseudo-momenta $ p_ { {\\bf u} }(x) = \\sum _{j=1}^M {1\\over x- u_j} - {L\\over 2 x},\\qquad p_ { \\bf v }(x) = \\sum _{j=1}^M {1\\over x- v_j} - {L\\over 2 x}.", "$ In the semi-classical (thermodynamical) limit the root distributions are described by continuous densities along one or several line segments in the rapidity plane.", "We call these line segments arcs because of the typical form they take.", "Each arc represents a branched cut of the pseudo-momentum [4], [5].", "The inner product can be considered as the amplitude for semi-classical tunelling with $\\hbar = 1/L$ and as such is expected to have a $1/L$ expansion of the form $ \\langle { {\\bf u} }| { \\bf v }\\rangle = e^{{\\cal F}_0+ {\\cal F}_1 + \\dots },\\qquad {\\cal F}_n\\sim L^{1-n}.$ We obtained for the first two terms the following expressions in terms of contour integrals: $ {\\cal F}_0&=& \\oint \\limits _{ {\\mathcal {C}}} {dx\\over 2\\pi } \\ \\text{Li}_2[e^{ip_ { {\\bf u} }(x)+ip_ { \\bf v }(x)}]\\, , \\\\\\nonumber \\\\ {\\cal F}_1&=&- {\\textstyle {1\\over 2}} \\oint \\limits _{ {\\mathcal {C}}\\times {\\mathcal {C}}} {dx\\, dy \\over (2\\pi )^2}\\ { \\log \\left[ 1-e^{i p_ { {\\bf u} }(x)+i p_ { \\bf v }(x)} \\right] \\ \\log \\left[1-e^{i p_ { {\\bf u} }(y)+i p_ { \\bf v }(y)} \\right]\\over (x-y)^2} \\, ,$ where the contour of integration $ {\\mathcal {C}}$ encircles the roots $ { {\\bf u} }$ and $ { \\bf v }$ .", "This expression is valid, after redefinition of the quasimomenta, for an inhomogeneous twisted XXX spin chain.", "The first term, containing a contour integral of the dilogarithmic function, is of order $L$ , because the typical size of the cuts is of order $L$ .", "It was first derived in [9] for the special case when the rapidities $ { {\\bf u} }$ are sent to infinity, and for general $ { {\\bf u} }$ and $ { \\bf v }$ in [10], [11].", "The expression of the subleading term, which is of order $L^0$ , is the main result of this paper.", "Our method is an improvement of the semi-classical computations in [10], [11], which used a representation of the Slavnov's determinant [12] in terms of a simpler quantity, the ${A}$ -functionalThe ${A}$ -functional generalises a quantity defined in [7], whose thermodynamical limit was computed in [9]..", "The most symmetric form of such a representation was found in [13].", "The present computation is based on a new representation of the ${A}$ -functional as a Fredholm determinant, where the integration kernel is defined for a specific contour in the complex plane.", "We compute the semi-classical limit of this Fredholm determinant in two different ways.", "The first, rigorous, method consists in solving the Riemann-Hilbert problem for the Fredholm kernel.", "The second, less rigorous but more intuitive, method uses field-theoretical formulation of the Fredholm determinant in terms of free chiral fermions.", "After bosonization, we solve exactly the resulting field theory at small distances to obtain an effective infrared field theory.", "The semiclassical expansion of the effective infrared theory can be also thought of as Mayer expansion for a gas of dipole charges living on certain contour in the rapidity plane.", "The leading and the sub-leading order are encoded in a saddle-point equation, which resembles the `TBA-like' equations considered in [14].", "The paper is structured as follows.", "In Section we recall the basics of the Algebraic Bethe Ansatz for the XXX spin chain and the expression of the inner product in terms of the ${A}$ -functional.", "In this section we also derive the determinantal representation of the ${A}$ -functional, which is the starting point for our semi-classical analysis.", "In Section we develop the Riemann-Hilbert approach and find an explicit expression for the subleading term.", "In Section we derive the same result by field-theoretical methods." ], [ "Algebraic Bethe Ansatz", "We first recollect some well known facts about the (twisted) periodic XXX spin chain.", "The inhomogeneous XXX spin chain of length $L$ is defined by the monodromy matrix $ M_\\alpha (u)=\\prod _{k=1}^L R_{\\alpha k}(u-z_k ) $ where the auxiliary space is denoted by the index $\\alpha $ .", "The rational R-matrix can be taken in the form $ R_{\\alpha \\beta }(u)=\\frac{u}{u+\\varepsilon }\\,I_{\\alpha \\beta }+\\frac{\\varepsilon }{u+\\varepsilon }\\, P_{\\alpha \\beta } ,$ with the operator $P_{\\alpha \\beta }$ acting as a permutation of the spins in the spaces $\\alpha $ and $\\beta $ .", "The monodromy matrix depends of a set of $L$ variables $ { { \\bf z} }= \\lbrace z_1, \\dots , z_L\\rbrace $ called inhomogeneities, associated with the sites of the chain.", "Sometimes one uses the notation $ z_l = \\theta _l + \\varepsilon /2, \\quad l=1,\\dots , L. $ The isotropic Heisenberg Hamiltonian describes the homogeneous point $\\theta _k=0$ or $z_k = \\varepsilon /2$ .", "The standard normalization of the rapidity variable $u$ is such that $\\varepsilon = i$ , but we prefer to keep $\\varepsilon $ as a free parameter.", "The monodromy matrix obeys the Yang-Baxter equation $ R_{\\alpha \\alpha ^{\\prime }}(u-u^{\\prime }) M _\\alpha (u) M _{\\alpha ^{\\prime }}(u^{\\prime })= M _{\\alpha ^{\\prime }}(u^{\\prime })M_\\alpha (u)R_{\\alpha \\alpha ^{\\prime }}(u-u^{\\prime }).$ Its diagonal matrix elements are traditionally denoted by $ M _\\alpha (u)=\\left(\\begin{array}{cc}A(u) & B(u) \\\\ C(u) &D(u)\\end{array}\\right)_{\\!", "\\alpha }.", "$ The operators $A(u)$ and $D(u)$ act on the pseudo-vacuum $|\\Omega \\rangle =|\\uparrow \\uparrow \\ldots \\uparrow \\rangle $ as $ A(u) |\\Omega \\rangle = a(u) |\\Omega \\rangle , \\qquad D(u)|\\Omega \\rangle = d(u) |\\Omega \\rangle , $ where the eigenvalues $a(u)$ and $d(u)$ are given, in the normalization (REF ) of the R-matrix, by $ a(u)=1 \\; , & \\quad & d(u) = {Q_ { { \\bf z} }(u)\\over Q_ { { \\bf z} }(u+\\varepsilon )}.\\;$ Here and below we will systematically denote by $Q_ { \\bf w }$ the monic polynomial with roots $ { \\bf w }$ : $Q_{ { \\bf w }}(u) \\equiv \\prod _{i=1}^M (u-w_i) , \\qquad { \\bf w }\\equiv \\lbrace w_1, \\dots , w_M\\rbrace \\, .$ Besides the inhomogeneities $ { { \\bf z} }$ is convenient to introduce another deformation parameter $\\kappa $ by choosing twisted-periodic boundary condition at length $L$ .", "The transfer matrix for the twisted chain, $T(u)={\\rm tr} _a \\left[ ( ^{1\\ 0}_{0\\ \\kappa })M _a(u)\\right] =A(u)+\\kappa \\, D(u),$ commutes with itself for any value of the spectral parameter, and the algebra of the matrix elements is the same as for the homogeneous XXX model.", "The Hilbert space is a Fock space spanned by states obtained from the pseudo vacuum by acting with the `raising operators' $B(u)$ : $| { {\\bf u} }\\rangle =B(u_1)\\ldots B(u_M)|\\Omega \\rangle \\;.$ If the rapidities $ { {\\bf u} }=\\lbrace u_1,\\ldots ,u_M\\rbrace $ are generic, the state is called `off-shell', and the state is called `on-shell' if the rapidities obey the Bethe Ansatz equations.", "The Bethe equations for the twisted chain read $ {a(u_j)\\over d(u_j)} + \\kappa \\, {Q_ { {\\bf u} }( u_j+\\varepsilon )\\over Q_ { {\\bf u} }(u_j-\\varepsilon )}=1.$ The `on-shell' states are eigenstates of the transfer matrix $T(x)$ with the eigenvalue $t(x)=\\frac{Q_ { {\\bf u} }(x-\\varepsilon )}{Q_ { {\\bf u} }(x)}+\\kappa \\,\\frac{d(x)}{a(x)}\\frac{Q_ { {\\bf u} }(x+\\varepsilon )}{Q_ { {\\bf u} }(x)} \\;.$" ], [ "The inner product in terms of the ${A}$ -functional", "We consider the bilinear form, $ ( { \\bf v }, { {\\bf u} }) = \\langle \\Omega | \\prod _{j=1}^M {\\mathcal {C}}(v_j)\\ \\prod _{j=1}^M{\\cal B}(u_j) |\\Omega \\rangle $ which we will refer to as inner product and which is related to the scalar product byThis follows from the complex Hermitian conjugation convention $B(u)^\\dag = - C(u^*)$ .", "$ \\left( { {\\bf u} }, { \\bf v }\\right)= (-1)^M\\langle { {\\bf u} }^*| { \\bf v }\\rangle .", "$ The inner product of two Bethe vectors can be computed using the commutation relations (REF ) and the action of the diagonal elements of the monodromy matrices on the pseudo-vacuum (REF ).", "The result is written down by Korepin [15] as a double sum over partitions.", "It was shown by N. Slavnov [12] that if one of the two states is on-shell, the Korepin sum can be written as a determinant.", "The Slavnov determinant for a twisted periodic XXX chain can be expressed in terms of a simpler quantity ${A}_ { \\bf w }[f]$ , which we call ${A}$ -functional, and which depends on the set of rapidities $ { \\bf w }=\\lbrace w_j\\rbrace _{j=1}^N$ and the function $f(u)$ .", "The inner product is equal, up to a simple factor, to the ${A}$ -functional (REF ) with $ { \\bf w }= { {\\bf u} }\\cup { \\bf v }$ and $f (u) = \\kappa \\, d(u)/a(u)$ $ \\left( { {\\bf u} }, { \\bf v }\\right) = \\prod _{j =1}^M d(u_j )a(v_j )\\ {A}_{ { {\\bf u} }\\cup { \\bf v }}[\\kappa \\, d/a]\\, .", "$ The formula (REF ) was derived by Y. Matsuo and one of the authors [13] for purely periodic chain (no twist), but the proof given there works without change also in the case of a twist.A twisted version of the determinant formula was also discussed by Kazama, Komatsu and Nishimura [16].", "This form of the inner product is particularly useful due to its symmetry in the rapidities $ { {\\bf u} }$ and $ { \\bf v }$ .", "The ${A}$ -functional is defined as the ratio of $N\\times N$ determinants [11] $ {A}_ { \\bf w }[f]\\equiv \\det _{jk}\\left( w_j^{k-1} - f(w_j) \\ (w_j+\\varepsilon )^{k-1}\\right)/ \\det \\left( w_j^{k-1} \\right) \\, .", "$ Expanding the determinant, one obtains an alternative expression as a sum over the partitions of $ { \\bf w }= \\lbrace w_1,\\dots , w_N\\rbrace $ into two disjoint subsets $ { \\bf w }_\\alpha = \\lbrace w_j\\rbrace _{j\\in \\alpha }$ and $ { \\bf w }_{\\bar{\\alpha }}= \\lbrace w_j\\rbrace _{j\\in \\bar{\\alpha }}$ : $\\begin{aligned}{A}_{ { \\bf w }} [f] &= \\sum _{\\alpha }\\ \\ (-1)^{|\\alpha |} \\prod _{ j\\in \\alpha } f(w_j)\\prod _{j\\in \\alpha , k\\in \\bar{\\alpha }} {w_{j}-w_k+ \\varepsilon \\over w_{j} - w_k} .\\end{aligned}$ In this form the ${A}$ -functional appeared (with $f= \\kappa \\, d/a$ ) as one of the building blocks in the expression for the three-point function in a supersymmetric Yang-Mills theory [9].", "In this paper we will be interested in functional argument of the the form $ f(u) = \\kappa \\, {d(u)\\over a(u)}=\\kappa \\, {Q_ { { \\bf z} }(u)\\over Q_ { { \\bf z} }(u+\\varepsilon )}, $ which is relevant for the inhomogeneous twisted XXX chain.", "We will use a special notation for the ${A}$ -functional as a function of the magnon rapidities $ { \\bf w }= \\lbrace w_j\\rbrace _{j=1}^N$ , the inhomogeneities $ { { \\bf z} }=\\lbrace z_l\\rbrace _{l=1}^L$ , the twist $\\kappa $ and the shift parameter $\\varepsilon $ : $ {A}_{ { \\bf w }, { { \\bf z} }}^{[\\varepsilon , \\kappa ]} \\equiv {A}_ { \\bf w }[ \\kappa \\, d/a].", "$ In these notations the inner product reads $ \\left( { {\\bf u} }, { \\bf v }\\right) = \\prod _{j =1}^M d(u_j )a(v_j )\\ {A}^{[\\varepsilon , \\kappa ]} _{ { \\bf w }, { { \\bf z} }} \\qquad \\qquad \\qquad ( { \\bf w }= { {\\bf u} }\\cup { \\bf v }).", "$" ], [ " The ${A}$ -functional as an {{formula:4ea5a69e-f6e0-4c29-a82a-fde186a9b837}} determinent", "In this paper we will use another determinant representation, ${A}_{ { {\\bf u} }, { { \\bf z} }}^{[\\varepsilon , \\kappa ]} = \\det \\left({ 1} -\\kappa \\, K \\right)$ where ${ 1}$ is the the $N\\times N$ identity matrix and the matrix $K$ has matrix elements $ K_{jk} &=&\\frac{\\varepsilon E_j}{u_{j} -u_k+ \\varepsilon } \\qquad \\qquad (j,k=1,\\dots , N)\\, , \\\\E_j &\\equiv & {Q_{ { { \\bf z} }}(u_j)\\over Q_{ { { \\bf z} }}(u_j+\\varepsilon )}\\prod _{k(\\ne j)} {u_{j} -u_k+ \\varepsilon \\over u_{j}-u_k}\\,.$ To prove (REF ), we write the sum over the partitions in (REF ) as a double sum in one of the subsets: $\\begin{aligned}{A}_{ { {\\bf u} }, { { \\bf z} }} ^{[\\varepsilon , \\kappa ]} &= \\sum _{\\alpha }\\ (-\\kappa )^{|\\alpha |} \\prod _{j\\in \\alpha } { E_j } \\ \\prod _{ j, k\\in \\alpha \\, ; \\, j\\ne k} {u_{j}-u_k\\over u_{j}-u_k+ \\varepsilon }\\end{aligned}$ and apply the Cauchy identity $(j,k\\in \\alpha )$ $ \\prod _{j\\ne k} {u_{j}-u_k\\over u_{j}-u_k+ \\varepsilon } = \\det {\\varepsilon \\over u_{j}-u_k+ \\varepsilon } .$ The new determinant representation (REF ) has the advantage that it exponentiates in a simple way: $ \\log {A}_{ { {\\bf u} }, { { \\bf z} }}^{[\\varepsilon , \\kappa ]}=- \\sum _{n=1}^\\infty \\ {\\kappa ^n\\over n} \\sum _{j_1, \\dots , j_n=1}^N {\\varepsilon \\ E_{j_1}\\over u_{j_1}-u_{j_2}+\\varepsilon } \\ {\\varepsilon \\ E_{j_2}\\over u_{j_1}-u_{j_3}+\\varepsilon } \\ \\cdots \\ {\\varepsilon \\ E_{j_n}\\over u_{j_n}-u_{j_1}+\\varepsilon }.$" ], [ "Semiclassical limit: from discrete data to meromorphic functions", "We are going to study the semi-classical limit $L\\rightarrow \\infty ,N\\rightarrow \\infty $ with $\\alpha =N/L$ finite, when the roots $ { {\\bf u} }$ arrange in one or several arcs of macroscopic size.", "We can also choose an $L$ -dependent normalisation of the rapidity variable so that $\\varepsilon \\sim 1/L$ .", "Then the typical size of the arcs will be of order $L^0$ .", "For our task it is advantageous to replace the discrete data $ { {\\bf u} }$ and $ { { \\bf z} }$ by the external potential $ \\Phi (x) \\equiv \\log Q_ { {\\bf u} }(x) -\\log Q_ { { \\bf z} }(x).$ In the semi-classical limit the arcs condense in one or more cuts of the meromorphic function $p(u) \\equiv \\partial \\Phi (u)$ .", "The discontinuities across the cuts are approximated by continuous densities which change slowly at distances of order $\\varepsilon $ .", "The crucial observation which will allow to reformulate the problem in terms of the external potential $\\Phi $ is that factors $E_j$ defined in () are the residues of the same meromorphic function at $x=u_j$ : $ E_j= {1\\over \\varepsilon } \\ \\underset{x\\rightarrow u_j}{ \\text{Res}} {\\cal Q}_\\varepsilon (x)\\,\\qquad (j=1,\\dots , N) .$ The function $ {\\cal Q}_\\varepsilon $ is defined as $ {\\cal Q}_\\varepsilon (x) = {Q_ { {\\bf u} }(x+\\varepsilon )\\over Q_ { {\\bf u} }(x)} {Q_ { { \\bf z} }(x)\\over Q_ { { \\bf z} }(x+\\varepsilon )}= e^{ \\Phi (x_j+\\varepsilon )-\\Phi (x_j) }.", "$ With the help of (REF ) one can write the sum in the $n$ -th term of the series (REF ) by a multiple contour integral along a contour $ {\\mathcal {C}}_ { {\\bf u} }$ which encircles all the roots $ { {\\bf u} }$ .", "The weight function ${\\cal Q}_\\varepsilon (x) $ strongly fluctuates when $x$ approaches $ { {\\bf u} }$ or $ { { \\bf z} }$ , but if $x$ is far from both $ { { \\bf z} }$ and $ { {\\bf u} }$ , it changes slowly at distances $\\sim \\varepsilon $ .", "Our goal is to reformulate the inner product in terms of contour integrals where the contour of integration $ {\\mathcal {C}}$ is placed far from the singularities of the function ${\\cal Q}_\\varepsilon $ , unlike the original contour $ {\\mathcal {C}}_ { {\\bf u} }$ .", "Then the weight ${\\cal Q}_\\varepsilon $ can be replaced by $ {\\cal Q}(x)= \\lim _{\\varepsilon \\rightarrow 0} {\\cal Q}_\\varepsilon (x)= e^{ \\varepsilon \\partial \\Phi (x)}.", "$ We will achieve this goal by two different, and in a sense complementary approaches.", "The first one relies on the solution of a Rieman-Hilbert problem, while the second one uses field-theoretical concepts.", "In both approaches the general idea is the same as in the original computation in [9], namely to introduce a cutoff $\\Lambda $ such that $|\\varepsilon |\\ll \\Lambda \\ll L|\\varepsilon |$ and split the problem into a fast (short-distance) and slow (large-distance) parts.", "The final result does not depend on the precise value of the cutoff $\\Lambda $ ." ], [ "Riemann-Hilbert Approach", "We will represent the linear operator with matrix (REF ) as an integral operator acting in a space of functions with given analytic properties.", "Then the determinant (REF ) takes the form of a Fredholm determinant.", "In the semi-classical limit it is possible to split the resolvent for the Fredholm kernel into slow and fast pieces.", "The fast piece can be evaluated exactly, while the computation of the slow piece is done by solving a standard scalar Riemann-Hilbert problem." ], [ "The ${A}$ -functional as a Fredholm determinant ", "We represent an $N$ -dimensional vector $ { { \\bf f} }= \\lbrace f_1,\\dots , f_N\\rbrace $ as a meromorphic function $f(u) $ , which has poles at $u=u_j$ with residues $f_j$ and no other singularities: $f(x)\\equiv \\sum _j \\frac{f_j}{x-u_j}.$ The functions $ e_j(x)= {1\\over x-u_j} \\qquad (j=1,\\dots , N).$ form a canonical basis in the $N$ -dimensional space of meromorphic functions analytic everywhere except on $ { {\\bf u} }$ .", "The matrix (REF ) defines a linear operator in this basis.", "In order to be able to compute traces, we should also give a functional representation of the dual space.", "The elements $\\tilde{ { { \\bf f} }}$ of the dual space with respect to the scalar product $\\tilde{ { { \\bf f} }}\\cdot { { \\bf f} }= \\tilde{f}_1 f_1 +\\dots + \\tilde{f}_Nf_N$ can be mapped to the space of functions $\\tilde{f}(x) $ which are analytic in the vicinity of $ { {\\bf u} }$ .", "With such a function we associate a dual vector $\\tilde{ { { \\bf f} }} $ with coordinates $ \\tilde{f}_j\\equiv \\tilde{f}(u_j)$ .", "This function is of course not unique.", "The scalar product may then be represented by a contour integral $\\langle \\tilde{f} |f \\rangle = \\oint \\limits _{ {\\mathcal {C}}_ { {\\bf u} }} {dx\\over 2\\pi i} \\ \\tilde{f}(x) f(x),$ where $\\mathcal {C}_ { {\\bf u} }$ is a contour surrounding the $ { {\\bf u} }$ 's and is contained in the domain of analyticity of $\\tilde{f}$ .", "We will use Dirac notations $ f(x) = \\langle x|f\\rangle , \\tilde{f}(x)= \\langle \\tilde{f}|x\\rangle $ , $f_j = \\langle j|f\\rangle $ and $\\tilde{f}_j= \\langle \\tilde{f}|j\\rangle $ , so that $|f\\rangle =\\sum _{j=1}^N |j\\rangle \\langle j|f\\rangle ,\\quad \\langle \\tilde{f}| =\\sum _{j=1}^N\\ \\langle \\tilde{f}|j\\rangle \\langle j| .$ The functional representations of the basis vectors in the direct and the dual spaces are $ |j\\rangle \\rightarrow \\langle x|j\\rangle \\equiv {1\\over x-u_j}; \\qquad \\langle j|\\rightarrow \\langle j |x\\rangle : \\ \\ \\langle j | u_k\\rangle = \\delta _{jk} \\quad (k=1,\\dots , N).", "$ The functions corresponding to the elements of the dual basis are defined up to an arbitrary meromorphic function that vanish on $ { {\\bf u} }$ .", "The functional representation of the matrix $K$ is given by an integral operator ${ \\mathcal { K} }$ , which acts on the function $f(x)\\equiv \\langle x|f\\rangle $ as $\\langle x|\\mathcal {K}|f\\rangle =\\frac{1}{2\\pi i} \\oint _{ {\\mathcal {C}}_ { {\\bf u} }} {\\cal Q}_\\varepsilon (y) \\,f(y+\\varepsilon ) \\frac{dy}{x-y}\\, ,$ where the function $ {\\cal Q}_\\varepsilon $ is defined by (REF ).", "The contour of integration $ {\\mathcal {C}}$ in this formula is chosen to encircles all $u_j$ but leaves the points $z_l-\\varepsilon $ as well as the point $x$ outside.", "Note that there are no other poles of the integrand, as the poles of $f(y+\\varepsilon )$ are compensated by the zeros of ${\\cal Q}_\\varepsilon $ .", "Applying (REF ), we obtain the action in the canonical basis $ [{ \\mathcal { K} }f]_j =\\sum _k {\\varepsilon E_jf_{k}\\over u_j-u_k+\\varepsilon },$ which agrees with ().", "Now the $N\\times N$ determinant (REF ) takes the form of a Fredholm determinant ${A}_{ { {\\bf u} }, { { \\bf z} }}^{[\\varepsilon , \\kappa ]}=\\det \\left( 1 -\\kappa \\, { \\mathcal { K} }\\right) .$ One may recast the definition of the Fredholm Kernel, Eq.", "(REF ), in the following operator form, which will be very useful in extracting the semiclassical limit, $ { \\mathcal { K} }= {\\mathcal {P}}_ { {\\bf u} }\\ e^{-\\Phi } \\ {\\mathbb {D}}_\\varepsilon \\, e^{\\Phi },$ where ${\\mathbb {D}}_\\varepsilon =e^{\\varepsilon \\partial }$ is the shift operator, acting as ${\\mathbb {D}}_\\varepsilon f(x) = f( x+\\varepsilon )$ , and $ {\\mathcal {P}}_ { {\\bf u} }= {\\mathcal {P}}_ { {\\bf u} }^2$ is the operator projecting onto the space of functions (REF ): $ [ {\\mathcal {P}}_ { {\\bf u} }f](x) = \\oint _{ {\\mathcal {C}}} {du\\over 2\\pi i} {f(u)\\over x-u} \\ \\qquad (u\\ \\text{is outside }\\ {\\mathcal {C}}).", "$ Let us stress on the important fact that the contour $ {\\mathcal {C}}$ , which encircles the set $ { {\\bf u} }$ , can be placed at macroscopic distance from the roots in $ { {\\bf u} }$ .", "Indeed, the resolvent has no other poles than $x=u_j$ and $x=z_l-\\varepsilon $ .", "Along the deformed contour $ {\\mathcal {C}}$ the factor ${\\cal Q}_\\varepsilon $ in the Fredholm kernel changes slowly at distances of order $\\varepsilon $ .", "We will denote functions in the image of $\\mathcal {P}_ { {\\bf u} }$ with a `$+$ ' subscript and functions in the kernel of $\\mathcal {P}_ { {\\bf u} }$ with a `$-$ ' subscript.", "Thus it will be implied that $ \\mathcal {P}_ { {\\bf u} }\\, g_+ =g_+, \\quad \\mathcal {P}_ { {\\bf u} }\\,g_-=0.", "$ We will also use the `$+$ ' subscript to denote functions which are in the image of $\\mathcal {P}_ { {\\bf u} }$ up to a polynomial, namely $\\mathcal {P}_ { {\\bf u} }g_+ = g_+ + P$ , where $P$ is a polynomial." ], [ "Resolvent of the Fredholm kernel", "We proceed by writing the logarithm of the ${A}$ -functional as follows: $\\log {A}_{ { {\\bf u} }, { { \\bf z} }}^{[\\varepsilon ,\\kappa ]}=\\int _0^\\kappa \\frac{d\\alpha }{\\alpha } {\\rm tr} \\left[ {1}-\\left({1}-\\alpha K \\right)^{-1} \\right],$ which leaves us with the task of computing the trace of the resolvent $\\left({1}-\\alpha K \\right)^{-1}$ .", "Here $K$ is the $N\\times N$ matrix defined in (REF ).", "We wish to find a functional representation of the resolvent, which we denote by $\\mathcal {F}$ and define as $\\langle x|\\mathcal {F}|f\\rangle = \\sum _i \\frac{\\left[({1}-\\alpha K)^{-1} { { \\bf f} }\\right]_i}{x-u_i}.$ One can compute ${\\rm tr} ({1}-\\alpha K)^{-1}$ in terms of $\\mathcal {F}$ as follows: ${\\rm tr} ({1}-\\alpha K)^{-1} = \\sum _{i=1}^N \\ \\langle i|\\mathcal {F}| i\\rangle .$ The function $F(x,u_i)\\equiv \\langle x|\\mathcal {F}|i\\rangle $ will appear repeatedly in the following.", "We shall analytically continue $F(x,u_i)$ in the variable $u_i$ , so that $u_i$ can be thought of as a general complex variable rather than one of the roots from the set $ { {\\bf u} }$ .", "This analytical continuation is not unique, but the ambiguity is arguably exponentially small and will be neglected in the following.", "To compute $F(x,u)$ explicitly, we note the following identity: $&\\langle x|({1}-\\alpha \\mathcal {K})|f\\rangle =e^{-\\Phi (x)}\\left(1-\\alpha {\\mathbb {D}}_\\varepsilon \\right) e^{\\Phi (x)} f(x)+\\alpha \\sum _{l=1}^L \\frac{ Q_{ { {\\bf u} }}(z_l)}{Q^{\\prime }_{ { { \\bf z} }}(z_l)}\\frac{e^{-\\Phi (z_l-\\varepsilon )}f(z_l)}{x-z_l+\\varepsilon },$ which is obtained making use of (REF ) and taking the projection by removing the poles explicitly, the latter being located at the points $z_l-\\varepsilon $ .", "The function $F(x,u)$ satisfies by definition $& (1-\\alpha \\mathcal {K})F(x,u_j)= \\frac{1}{x-u_j }\\qquad (j=1,\\dots , N).$ Analytically continuing both sides away from the set $ { {\\bf u} }$ we obtain for the meromorphic function $F(x,u)$ the equation $& (1-\\alpha \\mathcal {K})F(x,u)= \\frac{1}{x-u }.$ Substituting the function $F(x,u) $ for $f(x)$ in Eq.", "(REF ) leads to $F(x,u) = e^{-\\Phi (x)} \\left(1-\\alpha {\\mathbb {D}}_\\varepsilon \\right)^{-1}e^{\\Phi (x)}\\left(\\frac{1}{x-u}- \\alpha \\sum _{l=1}^L e^{-\\Phi (z_l-\\varepsilon )}\\frac{Q_{ { {\\bf u} }}(z_l)}{Q^{\\prime }_{ { { \\bf z} }}(z_l)}\\frac{F(z_l,u)}{x-z_l+\\varepsilon }\\right).$ Then the equation may be solved self-consistently by treating $F(z_l,u)$ on the right hand side as external parameters, solving for $F(x,u)$ and then requiring that by evaluating $F(x,u)$ at $x=z_l$ we recover these same parameters.", "Indeed, setting $x$ to $z_l$ in Eq.", "(REF ) and representing $\\left(1-\\alpha {\\mathbb {D}}_\\varepsilon \\right)^{-1}$ as $\\sum _n\\alpha ^n {\\mathbb {D}}_\\varepsilon ^n $ , one realizes that only the $n=0$ term in this sum contributes, which makes the application of the self-consistency straightforward, leading to: $\\left(\\begin{array}{c}F(z_1,u) \\\\F(z_2,u) \\\\.", "\\\\.", "\\\\F(z_L,u)\\end{array} \\right)=({1} - \\tilde{K})^{-1}\\left( \\begin{array}{c}\\frac{1}{z_1-u} \\\\ \\frac{1}{z_2-u} \\\\ .", "\\\\ .", "\\\\ \\frac{1}{z_L-u}\\end{array}\\right),$ with the $L\\times L$ matrix $\\tilde{K}$ given by $\\tilde{K}_{ln} =- \\frac{Q_{ { { \\bf z} }}(z_{n}-\\varepsilon )Q_{ { {\\bf u} }}(z_n)}{Q^{\\prime }_{ { { \\bf z} }}(z_n)Q_{ { {\\bf u} }}(z_n-\\varepsilon )}\\ \\frac{1}{z_l-z_n+\\varepsilon }.$" ], [ "Separation into fast and slow pieces", "We compute ${\\rm tr} (1-\\alpha K)^{-1}$ by splitting the rhs of Eq.", "(REF ) into two parts, $F=F^{ \\text{ fast}}+F^{ \\text{slow}}$ , as follows: $&F^ \\text{ fast}(x,u) =e^{-\\Phi (x)} \\left(1-\\alpha {\\mathbb {D}}_\\varepsilon \\right)^{-1}e^{\\Phi (x)}\\frac{1}{x-u}, \\\\&F^{ \\text{slow}}(x,u) =- \\alpha \\, e^{-\\Phi (x)} \\left(1-\\alpha {\\mathbb {D}}_\\varepsilon \\right)^{-1}e^{\\Phi (x)}\\sum _{l=1}^L e^{-\\Phi (z_l-\\varepsilon )} \\frac{Q_{ { {\\bf u} }}(z_l)}{Q^{\\prime }_{ { { \\bf z} }}(z_l)}\\frac{F(z_l,u)}{x-z_l+\\varepsilon }.$ We start by computing the contribution of $F^ \\text{ fast}(x,u)$ to ${\\rm tr} \\left({1}-\\alpha K \\right)^{-1}$ : $&\\sum _{j=1}^N \\underset{x\\rightarrow u_j}{\\rm Res} F^{ \\text{ fast}}(x,u_j)=\\sum _{j=1}^N \\left(1+\\sum _{n=1}^\\infty \\frac{\\alpha ^n}{n\\varepsilon }e^{\\Phi (u_j+n\\varepsilon )} \\frac{Q_{ { { \\bf z} }}(u_j)}{Q^{\\prime }_{ { {\\bf u} }}(u_j)} \\right) =\\nonumber \\\\&=N- \\oint \\limits _ {\\mathcal {C}}{dx\\over 2\\pi i}\\ e^{-\\Phi (x)}\\log \\left(1-\\alpha \\,{\\mathbb {D}}_\\varepsilon \\right) e^{\\Phi (x)}.$ In order to find the contribution of $F^{ \\text{slow}}$ to ${\\rm tr} \\left({1}-\\alpha K \\right)^{-1}$ , we define an integral operator $\\mathcal {F}^{ \\text{slow}}$ with the following action: $\\langle x|\\mathcal {F}^{\\, \\text{slow}}|f\\rangle = \\oint \\limits _ {\\mathcal {C}}{du\\over 2\\pi i} \\ F^{ \\text{slow}}(x,u)f(u).$ We are interested in computing the trace $\\sum _{j=1}^N\\, \\langle j|\\mathcal {F}^{ \\text{slow}}|j\\rangle ,$ the contribution of $\\mathcal {F}^{\\, \\text{slow}}$ to (REF ).", "For that we introduce another complete set of states $| m\\rangle $ , represented by functions $ \\langle x|m\\rangle =f_m(x) $ analytic on and inside the contour $ {\\mathcal {C}}$ , and a dual set $\\langle m|$ , obeying $\\langle m^{\\prime }|m\\rangle =\\delta _{m^{\\prime },m},$ represented by functions $\\langle m|j\\rangle = \\tilde{f}_m(j),$ the domain of analyticity of which contains $ {\\mathcal {C}}$ .", "The quantum number $m$ is discrete if the contour $ {\\mathcal {C}}$ is compact and continuous otherwise.", "For example, if there exists a circle centered at the point $x_0$ such that the set $ { {\\bf u} }$ is inside the circle and the set $ { { \\bf z} }$ is outside the circle, then we let $\\mathcal {C}$ be this circle and choose $f_m(x) = (x-x_0)^{-m}$ , $\\tilde{f}_m(x) = (x-x_0)^{m-1}$ for $m\\ge 1$ .", "The definition of $|m\\rangle $ and $\\langle m|$ imply $ \\oint _ {\\mathcal {C}}{dx\\over 2\\pi i} \\langle m|x\\rangle \\langle x|m^{\\prime }\\rangle =\\delta _{m,m^{\\prime }}\\ , \\qquad \\sum _m \\langle x |m\\rangle \\langle m|x^{\\prime }\\rangle = {1\\over x-x^{\\prime }}\\, , $ which allows to write the trace as $\\sum _{j=1}^N\\langle j|\\mathcal {F}^{ \\text{slow}}|j\\rangle =& \\sum _{j=1}^N\\sum _m\\langle j|\\mathcal {F}^{ \\text{slow}}|m\\rangle \\langle m| j\\rangle = \\sum _{j,m}\\langle m|j\\rangle \\langle j|\\mathcal {F}^{ \\text{slow}}|m\\rangle \\nonumber \\\\=&\\sum _m \\langle m|\\mathcal {F}^{ \\text{slow}}|m\\rangle .$ The first equality follows from the relation $\\langle x|j\\rangle =\\sum _m\\langle x|m\\rangle \\langle m|j\\rangle $ for any $x$ , which is true by the definition of $|m\\rangle $ as a complete set.", "To prove the last equality, we will show that $\\langle x|\\mathcal {F}^{ \\text{slow}}|m\\rangle $ has only simple poles at the $u_i$ 's and has additional singularities only around the $z_i$ 's.", "For such functions, the sum of $|j\\rangle \\langle j|$ acts as the identity operator, and one can write $ \\sum _{j=1}^N\\langle m|j\\rangle \\langle j|\\mathcal {F}^{ \\text{slow}}|m\\rangle =\\oint \\limits _ {\\mathcal {C}}{dx\\over 2\\pi i} \\langle m|x\\rangle \\langle x| {\\cal F}^ \\text{slow}|m\\rangle .", "$ Indeed, the left hand side of (REF ) is the sum of the $N$ residues of the integrand inside the contour $ {\\mathcal {C}}$ .", "The last identity (REF ) of is then a consequence of (REF ) and (REF ).", "We are left with the task of showing the above-mentioned analytical properties of $\\langle x|\\mathcal {F}^{ \\text{slow}}|m\\rangle $ .", "Namely we must show that $\\langle x|\\mathcal {F}^{ \\text{slow}}|m\\rangle $ has only simple poles at the $u_i$ 's.", "Indeed, combining () and (REF ), we obtain $\\langle x|\\mathcal {F}^{ \\text{slow}}|m\\rangle &=\\alpha e^{-\\Phi (x)} \\left(1-\\alpha {\\mathbb {D}}_\\varepsilon \\right)^{-1}e^{\\Phi (x)} \\times \\nonumber \\\\\\nonumber &\\\\&\\times \\left(\\begin{array}{c} \\frac{ Q_{ { { \\bf z} }}(z_1-\\varepsilon )Q_{ { {\\bf u} }}(z_1)}{Q_{ { {\\bf u} }}(z_1-\\varepsilon )Q^{\\prime }_{ { { \\bf z} }}(z_1)}\\frac{1}{x-z_1+\\varepsilon } \\\\.", "\\\\.", "\\\\\\frac{ Q_{ { { \\bf z} }}(z_L-\\varepsilon )Q_{ { {\\bf u} }}(z_L)}{Q_{ { {\\bf u} }}(z_L-\\varepsilon )Q^{\\prime }_{ { { \\bf z} }}(z_L)}\\frac{1}{x-z_L+\\varepsilon } \\end{array}\\right)^t({1}-\\alpha \\tilde{K}^t)^{-1}\\left( \\begin{array}{c}f_m(z_1) \\\\ .", "\\\\ .\\\\ f_m(z_L)\\end{array}\\right),$ whereupon the required analytic properties become apparent.", "This concludes the proof of (REF ).", "Writing (REF ) in terms of $F^{ \\text{slow}}$ yields $\\sum _i\\langle j | {{\\cal F}}^{ \\text{slow}}|j\\rangle &= \\sum _m \\oint {dx \\over 2\\pi i}\\oint {du\\over 2\\pi i} \\langle m| x\\rangle F^{ \\text{slow}}(x,u) \\langle u| m\\rangle = \\nonumber \\\\&= \\oint {dx \\over 2\\pi i}\\oint {du\\over 2\\pi i} \\frac{F^{ \\text{slow}}(x,u)}{u-x}, $ where the contour for the the integral in $u$ encircles the contour for the integral in $x$ .", "Taking into account that $ \\oint \\frac{F^{{ \\text{ fast}}}(x,u)}{u-x} du=0$ , we can also write $\\sum _{j=1}^N\\langle j|\\mathcal {F}^{ \\text{slow}}|j\\rangle = \\oint {dx \\over 2\\pi i}\\oint {du\\over 2\\pi i} \\ \\frac{F(x,u)}{u-x} .$ Combining (REF ), (REF ) and (REF ), we obtain: $\\log {A}_{ { {\\bf u} }, { { \\bf z} }}^{[\\varepsilon ,\\kappa ]} =\\int _0^\\kappa \\frac{d\\alpha }{\\alpha } \\left[\\oint \\limits _ {\\mathcal {C}}\\frac{dx}{2\\pi i}e^{-\\Phi (x)}\\log \\left(1-\\alpha {\\mathbb {D}}_\\varepsilon \\right) e^{\\Phi (x)} +\\oint \\limits _ {\\mathcal {C}}\\frac{dx}{2\\pi i}\\oint \\limits _ {\\mathcal {C}}\\frac{du}{2\\pi i}\\ \\frac{F^{}(x,u)}{x-u} \\right].$ The representation (REF ) of $\\log {A}_{ { {\\bf u} }, { { \\bf z} }}$ is only useful if the $1/N$ expansion of $F$ is computable.", "This turns out to be the case, and we undertake the task of performing this expansion in the following." ], [ "Semi-classical expansion of the slow piece ", "The resolvent $F(x,u)$ satisfies the defining equation (REF ), which can be written, making use of (REF ), in the form $ {\\mathcal {P}}_ { {\\bf u} }\\left[F(x,u)-\\alpha {\\cal Q}_\\varepsilon (u) F(x+\\varepsilon ,u) \\right]=\\frac{1}{x-u}.$ We can treat $\\varepsilon $ in the argument of $F$ as a small parameter.", "Indeed, since the contour $ {\\mathcal {C}}$ is at macroscopic distance from the arcs formed by the roots $ { {\\bf u} }$ , the function $F(x, u)$ changes slowly at distances of order $\\varepsilon $ .", "We thus obtain the expansion $ {\\mathcal {P}}_ { {\\bf u} }\\left[F(x,u)-\\alpha {\\cal Q}_\\varepsilon (x)\\left( F(x,u) + \\varepsilon F^{\\prime }(x,u)+\\dots \\right)\\right]= \\frac{1}{x-u}.$ This equation is solved order by order in powers of $\\varepsilon $ , $F=F^{(0)}+F^{(1)}+\\dots $ , with the leading order satisfying $ {\\mathcal {P}}_ { {\\bf u} }\\left[\\left(1-\\alpha {\\cal Q}_\\varepsilon (x)\\right) F^{(0)} (x,u)\\right]=\\frac{1}{x-u} \\, ,$ while the next to leading order can be easily seen to be given by $F^{(1)} (x,u)= -\\alpha \\varepsilon \\oint \\limits _ {\\mathcal {C}}{dv\\over 2\\pi i}\\ F^{(0)}(x,v) {\\cal Q}(v) \\partial _{v}F^{(0)}(v,u)\\, ,$ where ${\\cal Q}(x)$ is the limit as $\\varepsilon \\rightarrow 0$ of $\\mathcal {Q}_\\varepsilon (x)$ , Eq.", "(REF ).", "Computing yet higher orders is likewise mechanical.", "The function $F^{(0)}$ satisfies a standard problem in the theory of integral equations, Eq.", "(REF ).", "There is a standard method of solution of such equations [17].", "Namely, we decompose $1-\\alpha {\\cal Q}_\\varepsilon (x)$ into two parts, $1-\\alpha {\\cal Q}(x)= U_-(x)U_+(x),$ where $U_+(x)$ is analytic away from the arcs formed by the roots $ { {\\bf u} }$ and behaves at $x\\rightarrow \\infty $ as $U_+(x)\\rightarrow x^n$ for some $n$ of order 1, while $U_-(x)$ is analytic around the arcs, having no zeros around the arcs.", "We give an explicit expression for $U_\\pm $ in the following.", "We can write Eq.", "(REF ) as $&\\left(1-\\alpha {\\cal Q}(x) \\right)F^{(0)}+g_- = \\frac{1}{x-u}$ for some function $g_-$ regular around the arcs.", "Using the decomposition (REF ), we write $&U_+F^{(0)}+\\frac{g_-}{U_-} = \\frac{1}{(x-u)U_-}.$ We now apply the projector $ {\\mathcal {P}}_ { {\\bf u} }$ to both sides of this equation.", "The second term on the left hand side drops out while the first term yields a polynomial of degree $n-1$ in $x$ for $n$ positive, and zero otherwise.", "We denote this polynomial as $P_{n-1}(x;u)$ as it is also a function of $u$ .", "We thus have $ {\\mathcal {P}}_ { {\\bf u} }[U_+(x)F^{(0)}(x,u)] =U_+(x)F^{(0)} (x,u)-P_{n-1}(x;u),$ from which one obtains: $ F^{(0)}(x,u)=\\frac{1}{U_+(x)} \\left\\lbrace {\\mathcal {P}}_ { {\\bf u} }\\left[\\frac{1}{(x-u)U_-(x)}\\right]+ P_{n-1}(x;u) \\right\\rbrace .$ Finally, using the analytical properties of $U_-$ and the definition of $ {\\mathcal {P}}_ { {\\bf u} }$ it is easy to see that $ {\\mathcal {P}}_ { {\\bf u} }\\left[\\frac{1}{U_-(x)(x-u)}\\right]=\\frac{1}{U_-(u)(x-u)},$ and Eq.", "(REF ) simplifies to $F^{(0)}(x,u)=\\frac{P_{n-1}(x;u)}{U_+(x)}+\\frac{1}{U_+(x)U_-(u)(x-u)}.$ The coefficients of the polynomial $P_{n-1}(x,u)$ can be found by solving (REF ) for $F^{(0)}$ around infinity to order $x^{-n}$ and using $P_{n-1}(x;u) =\\left[U_+(x)F^{(0)}(x,u)\\right]_>$ , where the subscript `$>$ ' denotes the positive (polynomial in $x$ ) part of the Laurent expansion around infinity.", "Note that $n$ is of order 1 and $U_+$ is known (to be computed below) such that the task of finding $P_{n-1}(x;u)$ is relatively simple.", "Eq.", "(REF ) represents a solution for $F^{(0)}$ given the decomposition (REF ).", "Fortunately, the functions $U_\\pm $ may be computed explicitly.", "Assume that the phase of the complex function $1-\\alpha {\\cal Q}(x)$ winds $n_a$ times as $x$ moves around the $a$ -th arc once (where we use the conventions of positive winding for counterclockwise rotation).", "Namely, we assume that the rational function $1-\\alpha {\\cal Q}(x)$ has $n_a$ more zeros than poles a microscopic distance around the $a$ -th arc.", "Below we shall call $n_a$ below simply 'the winding number'.", "Let $n=\\sum _an_a$ .", "For each $a$ we find a rational function $R_a(x)$ such that [$1-\\alpha {\\cal Q}_\\varepsilon (x)]R(x)$ has winding number 0 around all arcs, where $R(x)=\\prod _a R_a(x)$ .", "We choose the functions $R_a(x)$ as follows.", "If $n_a<0$ , we take $R_a(x) =\\prod _{i=1}^{|n_a|} (x-u_{j^{(a)}_i}), $ where $u_{j^{(a)}_i}$ , $i=1, \\dots , |n_a|$ , are arbitrary roots belonging to the $a$ -th arc.", "The final result at given order does not depend on this choice to the corresponding order.", "If $n_a>0$ , we choose $R_a(x) =\\prod ^{n_a}_{i=1}\\frac{1}{x-\\alpha _{j^{(a)}_i}}, $ where $\\alpha _{j_i}^{(a)}$ are a set of $n_a$ roots of $1-\\alpha {\\cal Q}(x)$ around the $a$ -th arc.", "The functions $U_+(x)$ and $U_-(x)$ are then given by $U_+ (x)&=\\frac{\\exp \\left\\lbrace {\\mathcal {P}}_ { {\\bf u} }\\left[\\log {\\left(1-\\alpha {\\cal Q}(x)\\right) R(x)}\\right]\\right\\rbrace }{R(x)} , \\\\U_-(x) &=\\frac{1-\\alpha {\\cal Q}(x)}{U_+(x)}.$" ], [ "Semi-classical expansion of the fast piece", "To obtain the semiclasical expansion of the first term in (REF ), we need to be able to expand the expression $\\oint e^{-\\Phi }\\log \\left(1-\\alpha {\\mathbb {D}}_\\varepsilon \\right)e^{\\Phi }.$ This is possible only if the contour of integration $ {\\mathcal {C}}$ is far from both $ { {\\bf u} }$ and $ { { \\bf z} }$ .", "At leading order $\\oint \\limits _ {\\mathcal {C}}{dx\\over 2\\pi }\\ e^{-\\Phi }\\log \\left(1-\\alpha {\\mathbb {D}}_\\varepsilon \\right) e^{\\Phi } =\\oint \\limits _ {\\mathcal {C}}{dx\\over 2\\pi }\\ \\log \\left(1-\\alpha {\\cal Q}(x) \\right) +{\\mathcal { O} }(1/L).$ The $O(L^0)$ correction is an integral of pure derivative and vanishes.", "Note that the integrand on the right hand side has logarithmic branch cuts emanating from the arcs formed by the points from the set $ { {\\bf u} }$ whenever the winding number $n_a$ of the function $1-\\alpha {\\cal Q}$ around the $a$ -th arc is non-zero.", "By construction, the function ${\\cal Q}(x)$ has no winding numbers, but only cuts along the arcs.", "The cuts appear as the result of merging of the poles and the zeros of $ {\\cal Q}_\\varepsilon (x)$ in the limit $\\varepsilon \\sim 1/L \\rightarrow 0$ .", "When $\\alpha $ is small, the function $1-\\alpha {\\cal Q}_\\varepsilon $ has no zeros near the cuts and the contour of integration may be drawn to simply encircle the cuts.", "As $\\alpha $ increases, a number of zeros of $1-\\alpha {\\cal Q}_\\varepsilon $ on the second sheet can move through the cuts to the first sheet.", "The contour of integration in (REF ) should be drawn to surround those zeros.", "This presents no problem as moving the contour away from the arc is consistent with the expansion in (REF ).", "If, on the other hand, an extra zero (one that was not there at small $\\alpha $ ) of $1-\\alpha {\\cal Q}_\\varepsilon $ approaches the $a$ -th arc, as $\\alpha $ increases, the contour of integration must be drawn between that zero and the arc.", "Eventually, that zero may approach the arc up to a microscopic distance, and the approximation leading to (REF ) will be invalidated.", "To deal with this scenario one must separate out he roots around the zero and compute their contribution to the fast piece by performing the sum in (REF ) for those roots more directly.", "We do not show how this is done explicitly in this paper, rather it will be the subject of future work." ], [ "The leading order result", "Only the fast piece contributes to the leading order ($\\sim L$ ) of the semiclassical expansion of $\\log {A}_{ { {\\bf u} }, { { \\bf z} }}^{[\\varepsilon ,\\kappa ]}$ , since the slow piece can be easily seen to be of order $L^0$ : $\\log {A}_{ { {\\bf u} }, { { \\bf z} }}^{[\\varepsilon ,\\kappa ]} =\\int _0^\\kappa \\frac{d\\alpha }{\\alpha } \\oint \\limits _ {\\mathcal {C}}{dx\\over 2\\pi i}\\ \\log \\left[1-\\alpha {\\cal Q}(x)\\right],$ where the branch cut of the logarithmic function is to be taken according to the prescription in the previous subsection.", "Sometimes the contour should be deformed so that part of it passes in the second sheet, as explained in [11].", "In any case, the integral over $\\alpha $ can be taken and the final result is $\\log {A}_{ { {\\bf u} }, { { \\bf z} }}^{[\\varepsilon ,\\kappa ]} =- {1\\over \\varepsilon }\\oint \\limits _{ {\\mathcal {C}}} {dx\\over 2\\pi i}\\ \\text{Li}_2\\left(\\kappa Q(x)\\right).$" ], [ "Subleading order for zero winding numbers ", "In this section we will assume that $n_a=0$ to avoid the complications that arise in the case of non-vanishing winding numbers.", "With this assumption we will write a compact expression for the leading and the subleading orders.", "Combining (REF ) and (REF ), with $P_{n-1}(u)=0$ (which is appropriate for $n_a=0$ ), we obtain at leading order $&\\sum _j \\langle j|\\mathcal {F}^{ \\text{slow}}|j\\rangle =-\\oint \\limits _ {\\mathcal {C}}\\frac{dx}{2\\pi i}\\oint \\limits _ {\\mathcal {C}}\\frac{du}{2\\pi i}\\frac{1}{U_+(x)U_-(u )(x-u )^2}= \\nonumber \\\\&=-\\oint \\limits _ {\\mathcal {C}}\\frac{du}{2 \\pi i } \\ \\frac{\\partial _uU_+(u)}{U^2_+(u)U_-(u)} =-\\oint \\limits _ {\\mathcal {C}}\\frac{du}{2 \\pi i } \\ \\frac{\\partial _u\\log \\left[U_+(u)\\right]}{\\left(1-\\alpha Q(u)\\right) } .$ Using the explicit definition of $U_+, $ Eq.", "(REF ), where we take $R(u)=1$ , this can be further written as $&\\sum _j\\langle j|\\mathcal {F}^{ \\text{slow}}|j\\rangle = \\frac{dxdu}{(2 \\pi i)^2} \\ \\frac{1}{1-\\alpha {\\cal Q}(x)} \\frac{1}{(x-u)^2} \\log \\left( 1-\\alpha {\\cal Q}(u)\\right) = \\nonumber \\\\&=- \\frac{\\alpha }{2}\\partial _\\alpha \\frac{dxdu}{(2 \\pi i)^2} \\ \\log \\left( 1-\\alpha {\\cal Q}(x) \\right) \\frac{1}{(x-u)^2} \\log \\left( 1-\\alpha {\\cal Q}(u) \\right).$ Adding the contributions from the fast and the slow pieces, we arrive at the following approximation for $\\log {A}_{ { {\\bf u} }, { { \\bf z} }}^{[\\varepsilon ,\\kappa ]}$ , correct to order $O(1):$ $ \\log {A}_{ { {\\bf u} }, { { \\bf z} }}^{[\\varepsilon ,\\kappa ]} &=- {1\\over \\varepsilon } \\oint \\limits _ {\\mathcal {C}}{dx\\over 2\\pi i} e^{-\\Phi (x)} \\, \\text{Li}_2\\left(\\kappa {\\mathbb {D}}_\\varepsilon \\right) e^{\\Phi (x)} + \\nonumber \\\\&+\\frac{1}{2} \\limits _{ {\\mathcal {C}}\\times {\\mathcal {C}}}\\frac{dx du}{(2 \\pi i )^2} \\log \\left[ 1-\\kappa {\\cal Q}(x) \\right]\\frac{1}{(x-u)^2} \\log \\left[ 1-\\kappa {\\cal Q}(u) \\right].$ The first term on the right hand side is easily expandable in $1/ L$ as the contour of integration can be taken to be well away from the arcs formed by the points of the set $ { {\\bf u} }$ .", "The second term is an ${\\mathcal { O} }(1)$ correction.", "Higher order correction are straightforward to compute by incorporating the contribution of $F^{(n)}$ for $n>0$ .", "As mentioned above, the case when some winding numbers are non-zero implies more complicated expressions, which we do not develop here." ], [ "Effective field theory for the semiclassical limit", "In this section we reformulate the ${A}$ -functional in terms of a chiral fermion or, after bosonization, in terms of a chiral boson with exponential interaction.", "The interaction is weak at large distances but becomes singular at distances of order $\\varepsilon $ , where the two-point function develops poles.", "Our goal is to formulate an effective field theory for the limit $\\varepsilon \\rightarrow 0$ .", "For that we split the theory into a fast and a slow component and integrate with respect to the fast component." ], [ "Free fermions", "This determinant (REF ) is a particular case of the $\\tau $ -functions considered in section 9 of [20] and can be expressed as a Fock-space expectation value for a Neveu-Schwarz chiral fermion living in the rapidity complex plane and having mode expansion $ \\psi (u)= \\sum _{r\\in {\\mathbb {Z}}+ {1\\over 2}}\\psi _{r}\\ u^{-r-{1\\over 2}}, \\ \\ \\ \\psi ^* (u)= \\sum _{r\\in {\\mathbb {Z}}+ {1\\over 2}} \\psi ^*_{r}\\ u^{r- {1\\over 2}} .", "$ The fermion modes are assumed to satisfy the anticommutation relations $ [\\psi _{r},\\psi ^*_{s}]_+=\\delta _{rs}\\, , $ and the left/right vacuum states are defined by $ \\langle 0 | \\psi _{-r}= \\langle 0 | \\psi ^*_{r} = 0\\ \\ \\text{and}\\ \\ \\ \\psi _{r}\\, | 0 \\rangle = \\psi ^*_{-r} | 0 \\rangle = 0,\\ \\ \\ \\text{for} \\ r> 0 .", "$ The operator $ \\psi ^*_r$ creates a particle (or annihilates a hole) with mode number $r$ and the operator $\\psi _r$ annihilates a particle (or creates a hole) with mode number $r$ .", "The particles carry charge 1, while the holes carry charge $-1$ .", "The charge zero vacuum states are obtained by filling the Dirac see up to level zero.", "Any correlation function of the operators (REF ) is a determinant of two-point correlators $ \\langle 0 | \\psi (u) \\psi ^*(v) | 0 \\rangle = \\langle 0 | \\psi ^*(u)\\psi (v) | 0 \\rangle = {1\\over u-v}\\, .", "$ The expectation value of several pairs of fermions is given by the determinant of the two-point functions.", "Obviously the determinant (REF ) is equal to the expectation value $ \\text{Det}(1-\\kappa K) = \\langle 0 | \\exp \\left(\\kappa \\varepsilon \\sum _{j=1}^NE_j \\, \\psi ^*(u_j ) \\psi (u_j+ \\varepsilon )\\right) | 0 \\rangle .", "$ The discrete sum of fermion bilinears in the exponent on the rhs of (REF ) can be written, with the help of (REF ), as an integral along the contour $ {\\mathcal {C}}_ { {\\bf u} }$ which encircles the points $u_1,\\dots , u_N$ , and the Fock space representation (REF ) takes the form $ {A}_{ { {\\bf u} }, { { \\bf z} }}^{[\\varepsilon , \\kappa ]} = \\langle 0 | \\exp \\left({\\kappa }\\oint _{ {\\mathcal {C}}_ { {\\bf u} }} {d x\\over 2\\pi i} {\\cal Q}_\\varepsilon (x) \\, \\psi ^* (x )\\psi (x+\\varepsilon )\\right) | 0 \\rangle , $ where the weight function ${\\cal Q}_\\varepsilon (x)$ is defined by Eq.", "(REF )." ], [ "Bosonic field with exponential interaction", "Alternatively, one can express the ${A}$ -function in term of a chiral boson $\\phi (x)$ with two-point function $ \\langle 0 | \\phi (x) \\phi (y) | 0 \\rangle = \\log (x-y).$ After bosonization $\\psi (x) \\rightarrow e^{\\phi (x)}$ and $\\psi ^*(x) \\rightarrow e^{-\\phi (x)}$ , where we assumed that the exponents of the gaussian field are normally ordered, the fermion bilinear $\\psi ^*(x) \\psi (x+\\varepsilon )$ becomes, up to a numerical factor, a chiral exponential fieldOur convention is that the exponential is normally ordered, $e^{\\phi (u)-\\phi (v)} \\equiv \\ :e^{\\phi (u)-\\phi (v)}:$   , so that $\\langle 0|e^{\\phi (u)-\\phi (v)} |0\\rangle =1$ .", "$ { \\cal V}_\\varepsilon (x) \\equiv e^{\\phi (x+\\varepsilon ) - \\phi (x)}.$ The numerical factor is determined by the OPE $ e^{-\\phi (x)} \\, e^{\\phi (u)} ={1\\over x-u} e^{\\phi (u)-\\phi (x)}\\qquad (u= x+\\varepsilon )\\, ,$ so that the fermion bilinear bosonizes as $ \\psi ^*(x) \\psi (x+ \\varepsilon )\\quad \\ \\rightarrow \\ \\quad e^{-\\phi (x)}e^{\\phi (x+\\varepsilon ) } = - {1\\over \\varepsilon }\\, { \\cal V}_\\varepsilon (x) .", "$ The resulting bosonic field theory is that of a two-dimensional gaussian field $\\phi (x,\\bar{x})$ perturbed by a chiral interaction term $\\kappa {\\cal Q}_\\varepsilon (x){ \\cal V}_\\varepsilon (x)$ .", "Expanding the exponential in series and using that the $n$ -point correlator of the exponential field is a product of all two-point correlators $ \\langle 0 | { \\cal V}_\\varepsilon (x) { \\cal V}_\\varepsilon (y) | 0 \\rangle = { (x-y)^2 \\over (x-y+\\varepsilon )(x-y-\\varepsilon )} \\, ,$ we obtain that the expectation value (REF ) is given by the grand-canonical Coulomb-gas partition function $\\begin{aligned}{A}_{ { {\\bf u} }, { { \\bf z} }}^{[\\varepsilon , \\kappa ]}&= \\sum _{n=0}^N {(-\\kappa )^n\\over n!", "}\\prod _{j=1}^n \\oint _{ {\\mathcal {C}}_ { {\\bf u} }} {dx_j \\over 2\\pi i} \\ \\ { {\\cal Q}_\\varepsilon (x_j)\\over \\varepsilon }\\ \\ \\prod _{k>j}^n { (x_{j}-x_k)^2 \\over (x_{j}-x_k)^2- \\varepsilon ^2 } \\, .\\end{aligned}$ In the contour integral representations (REF ) and (REF ) the integration contour $ {\\mathcal {C}}_ { {\\bf u} }$ is drawn close to the poles of $ {\\cal Q}_\\varepsilon $ .", "We would like to deform the contours away from these poles, where $ {\\cal Q}_\\varepsilon $ can be considered as a sufficiently smooth.", "In the $n$ -th term of the series we can deform sequentially the integration contours away from the set $ { {\\bf u} }$ , so that the $n$ contours form a nested configuration separating the $ { {\\bf u} }$ -poles and the $ { { \\bf z} }$ -poles of the function $ {\\cal Q}_\\varepsilon $ .", "If the subsequent contours are spaced by $\\varepsilon $ , then the poles at $x_j-x_k=\\pm \\varepsilon $ of the integrand do not contribute." ], [ "Integrating out the fast modes", "The multiple contour integral can be evaluated in the semiclassical limit by splitting the integrand into slow and fast parts.", "We thus introduce an intermediate scale $\\Lambda $ such that $ |\\varepsilon |\\ll \\Lambda \\ll N|\\varepsilon |\\,$ and split the bosonic field into a fast and a slow components, $ \\phi = \\phi _{ \\text{slow}} + \\phi _{ \\text{ fast}}.$ Up to exponential terms the two-point function of the bosonic field is approximated at small distances by that of the fast component and at large distances by that of the slow component.", "Below we will perform explicitly the integration with respect to $\\phi _ \\text{ fast}$ to obtain an effective interaction for $\\phi _ \\text{slow}$ .", "Since the two-point function of the exponential fields with $\\phi $ replaced by $\\phi _ \\text{slow}$ does not contain poles, the nested contours spaced by $\\varepsilon $ can be replaced by a single contour $ {\\mathcal {C}}$ placed sufficiently far from the sets $ { { \\bf z} }$ and $ { {\\bf u} }$ where the integrand has poles.The splitting into a fast and a slow components can be done explicitly if the contour $ {\\mathcal {C}}$ can be placed along the real axis.", "Introduce a cutoff $\\Lambda $ such that $|\\varepsilon |\\ll \\Lambda \\ll N|\\varepsilon |$ .", "Then the bosonic field has a continuum of Fourier modes $\\alpha _E$ and the slow and fast parts can be defined as $\\phi _{ \\text{slow}} (x)= \\int _{|E|<\\Lambda } dE\\, \\alpha _E \\ e^{i E x}, \\phi _{ \\text{ fast}}(x) = \\int _{ |E|>\\Lambda } dE \\, \\alpha _E\\ e^{iE x}$ .", "The propagators of the slow and the fast components are $ \\langle 0 | \\partial \\phi _{ \\text{slow}} (x) , \\phi _{ \\text{slow}} (y) | 0 \\rangle = (1- e^{i \\Lambda (x-y)})/(x-y),\\ \\langle 0 | \\partial \\phi _{ \\text{ fast}} (x) , \\phi _{ \\text{ fast}}(y) | 0 \\rangle = e^{i \\Lambda (x-y)}/(x-y).$ The propagator for the slow component contains a strongly oscillating term whose role is to kill the pole at $x=y$ and which can be neglected far from the diagonal, while the numerator in the propagator of the fast component can be replaced by 1 at small distances.", "The effects of the cutoff are thus exponentially small and do not influence the perturbative quasiclassical expansion.", "The effective interaction for the slow component is of the form $ S_{\\text{eff}}[\\phi _ \\text{slow}]=\\sum _{n\\ge 1}\\oint _ {\\mathcal {C}}{dx\\over 2\\pi i} \\ V_{\\text{eff}}^{(n)}(x)\\, ,$ where $n$ -th term is the contribution of the connected $n$ -point function of the exponential field ${ \\cal V}_\\varepsilon (x)$ with respect to the fast component, $ V_{\\text{eff}}^{(n)}(x)={\\left({-\\kappa / \\varepsilon } \\right)^n\\over n!}", "\\oint {dx_1\\over 2\\pi i} \\dots {dx_n\\over 2\\pi i}\\ \\delta ( x-x_1)\\Big \\langle \\!\\!\\Big \\langle \\prod _{j=1}^n \\ {\\cal Q}_\\varepsilon (x_j) \\, { \\cal V}_\\varepsilon (x_j)\\Big \\rangle \\!\\!\\Big \\rangle _{\\!\\!", "\\text{ fast}} .$ $ \\Xi _n(x)={\\left({-1/ i\\varepsilon } \\right)^n\\over n!}", "\\oint {dx_1\\over 2\\pi i} \\dots {dx_n\\over 2\\pi i}\\ \\delta ( x-x_1)\\Big \\langle \\!\\!\\Big \\langle \\prod _{j=1}^n \\ {\\cal Q}(x_j) \\, { \\cal V}(x_j)\\Big \\rangle \\!\\!\\Big \\rangle _{\\!\\!", "\\text{ fast}} .$ To compute the connected correlation function $\\langle \\!\\langle \\ \\rangle \\!\\rangle _{ \\text{ fast}}$ we represent the product of $n$ exponential fields in the form $ { \\cal V}_\\varepsilon (x_1)\\dots { \\cal V}_\\varepsilon (x_n)=\\prod _{j<k} {(x_j-x_k)^2\\over (x_j-x_k)^2- \\varepsilon ^2}\\ :{ \\cal V}_\\varepsilon (x_1)\\dots { \\cal V}_\\varepsilon (x_n):\\, ,$ where $:\\ :$ signifies normal product.", "By definition the normal product of exponential fields has vacuum expectation value 1.", "Assuming that all distances $|x_j-x_k|$ are small compared to the scale $ \\Lambda $ , we can interpret the operator product expansion (REF ) as $ \\big \\langle { \\cal V}_\\varepsilon (x_1)\\dots { \\cal V}_\\varepsilon (x_n)\\big \\rangle _{\\!", "\\text{ fast}}\\approx \\prod _{j<k} {(x_j-x_k)^2\\over (x_j-x_k)^2- \\varepsilon ^2}\\ : { \\cal V}_\\varepsilon ^ \\text{slow}(x_1)\\dots { \\cal V}_\\varepsilon (x_n)^ \\text{slow}:\\, .$ $ \\big \\langle { \\cal V}(x_1)\\dots { \\cal V}(x_n)\\big \\rangle _{\\!", "\\text{ fast}}\\approx \\prod _{j<k} {x_{jk}^2\\over x_{jk}^2+ \\varepsilon ^2}\\ : { \\cal V}^ \\text{slow}(x_1)\\dots { \\cal V}(x_n)^ \\text{slow}:\\, .$ To extract the connected component of the $n$ -point function we apply the Cauchy identity (REF ) and represent the Cauchy determinant as a sum over permutations.", "The result is the sum of the (identical) contributions of the $(n-1)!$ permutations representing maximal cycles of length $n$ .This is basically the calculation done in ref.", "[21].", "The difference is in the extra factor and in our convention to choose $x=x_1$ as collective coordinate, while in [21] $x= (x_1+\\dots +x_n)/n$ .", "Note that the contribution of the permutations with more than one cycle vanishes automatically.", "We find ($x_{jk} \\equiv x_j-x_k$ ) $ V_{\\text{eff}}^{(n)}(x_1)&=&{ (n-1)!\\over n!}", "\\oint { \\prod _{k=1}^n{\\cal Q}_\\varepsilon (x_k) \\, { \\cal V}(x_k)^ \\text{slow}\\ {dx_2\\over 2\\pi i}\\, \\dots {dx_n\\over 2\\pi i}\\,\\over (\\varepsilon -x_{12} )\\dots (\\varepsilon -x_{n-1,n})(\\varepsilon - x_{n,1})}\\nonumber \\\\\\nonumber \\\\&=&-{Q_{n\\varepsilon }(x_1) \\over n^2\\varepsilon } \\ { \\cal V}_{n\\varepsilon }(x)\\, ,$ with $ {\\cal Q}_{n\\varepsilon }(x) &\\equiv &Q_{\\varepsilon }(x) Q_\\varepsilon (x+\\varepsilon ) \\dots Q_\\varepsilon (x+ n\\varepsilon ) \\ = e^{-\\Phi (x)+\\Phi (x+n\\varepsilon )} \\, ,\\\\{ \\cal V}_{n\\varepsilon }(x) &\\equiv &: { \\cal V}_{\\varepsilon }(x) { \\cal V}_{\\varepsilon }(x+\\varepsilon )\\dots { \\cal V}_{\\varepsilon }(x+n\\varepsilon ):\\ =e^ {-\\phi (x) + \\phi (x+ n\\varepsilon )}.", "$ (Here and below $\\phi $ denotes $\\phi _ \\text{slow}$ .)", "The resulting expression for the ${A}$ -functional in terms of the effective infrared theory is $ {A}_{ { {\\bf u} }, { { \\bf z} }}^{[\\varepsilon ,\\kappa ]} = \\langle 0 | \\exp \\left(-{1\\over \\varepsilon } \\sum _{n=1}^ {\\Lambda /\\varepsilon } {\\kappa ^n\\over n^2}\\oint _{ {\\mathcal {C}}_ { {\\bf u} }} {d x\\over 2\\pi i} {\\cal Q}_{n\\varepsilon } (x) \\,{ \\cal V}_{n\\varepsilon }(x)\\right) | 0 \\rangle + \\text{non-perturbative}.", "$ By construction the spacing $n\\varepsilon $ should be smaller than the scale $\\Lambda $ , but if the sum over $n$ in the exponent is extended to infinity, this will introduce exponentially small terms and will not change the $1/L$ expansion.", "Introducing the shift operator ${\\mathbb {D}}_\\varepsilon = e^{\\varepsilon \\partial _x}$ , the series in the exponent can be formally summed up as $ \\begin{aligned}{A}_{ { {\\bf u} }, { { \\bf z} }}^{[\\varepsilon ,\\kappa ]}&= \\langle 0 | \\exp \\left(- {1\\over \\varepsilon } \\oint \\limits _{ {\\mathcal {C}}} {dx\\over 2\\pi i} \\ :e^{-\\Phi (x)-\\phi (x) } \\ \\text{Li}_2(\\kappa \\, {\\mathbb {D}}_\\varepsilon )\\ e^{ \\Phi (x) +\\phi (x)}: \\right) | 0 \\rangle \\,+ \\text{non-perturbative}.\\end{aligned}$ Another way to write this expression, without using the normal product and redefining $\\phi +\\Phi \\rightarrow \\phi $ , is $ \\begin{aligned}{A}_{ { {\\bf u} }, { { \\bf z} }}^{[\\varepsilon ,\\kappa ]}&= \\langle 0 | \\exp \\Big (- \\oint \\limits _{ {\\mathcal {C}}} {dx\\over 2\\pi i} \\, e^{-\\phi (x) } \\ \\log (1-\\kappa \\, {\\mathbb {D}}_\\varepsilon )\\ e^{ \\phi (x)}: \\Big ) | 0 \\rangle \\,+ \\text{non-perturbative},\\\\& \\langle 0 | \\phi (x) | 0 \\rangle =\\Phi (x), \\ \\ \\langle 0 | \\phi (x)\\phi (y) | 0 \\rangle = \\log (x-y).\\end{aligned}$" ], [ "One-dimensional effective theory in the semiclassical\nlimit", "In the semiclassical limit $ \\hbar \\equiv 1/ L\\rightarrow 0 , \\qquad \\ell \\equiv L\\varepsilon \\sim 1, \\qquad \\alpha = N/L\\sim 1 $ the classical field $\\Phi $ grows as $1/\\hbar $ , but $ \\Phi (x+n\\varepsilon )-\\Phi (x) = n \\varepsilon \\, \\partial \\Phi (x) + \\dots $ remains finite, as well as the range of integration and size of the contour $ {\\mathcal {C}}$ .", "Furthermore, the distribution of the roots $u_j$ is assumed to be of the form of the finite zone solutions of the Bethe equations [5], which are described by hyperlliptic curves.", "The roots $ { {\\bf u} }$ condense into one or several arcs, which become the cuts of the meromorphic function $ \\partial \\Phi (x) = \\sum _{j=1}^N {1\\over x-u_j}-\\sum _{l=1}^L {1\\over x- z_l} .", "$ We assume that the inhomogeneities $ { { \\bf z} }$ are centered around the origin of the rapidity plane, but we do non make any other assumptions about them.", "We will concentrate on the leading term (of order $1/\\varepsilon $ ) and the subleading term (of order 1), and will ignore the corrections that vanish in the limit $\\varepsilon \\rightarrow 0$ .", "Then, using the approximation (REF ), we expand the exponent in (REF ) as $ \\begin{aligned}{A}_{ { {\\bf u} }, { { \\bf z} }}^{[\\varepsilon ,\\kappa ]} & = \\langle 0 | \\exp \\oint _{ {\\mathcal {C}}}{dx\\over 2\\pi i}\\left( - {1\\over \\varepsilon }: \\text{Li}_2(\\kappa \\, {\\cal Q}\\, e^{ -\\varepsilon \\varphi }) : - :\\log (1- {\\cal Q}\\, e^{-\\varepsilon \\varphi })\\partial \\varphi : + \\dots \\right) | 0 \\rangle \\, ,\\end{aligned}$ where we introduced the derivative field $ \\varphi (x) = - \\partial \\phi (x)$ and used the notation (REF ).", "We can retain only the first term on the rhs of (REF ), since the second term is a full derivative and can be neglected.", "Now we can pass from Fock-space to path-integral formalism.", "For that we express the expectation value (REF ) as a path integral for the $(0+1)$ -dimensional field $\\varphi (x)$ defined on the contour $ {\\mathcal {C}}$ and having two-point function $ G(x,u)\\equiv \\langle \\varphi (x)\\varphi (u)\\rangle = \\ {1\\over (x-u)^2}\\, .$ Introducing a second field $\\rho $ linearly coupled to $\\phi $ we write the ${A}$ -functional as a path integral $ {A}_{ { {\\bf u} }, { { \\bf z} }}^{[\\varepsilon ,\\kappa ]} &=& \\int [D\\varphi \\, D\\rho ]\\ e^{- {\\cal Y}[\\varphi , \\rho ]}\\, ,$ with the action functional given by $ {\\cal Y}[\\varphi , \\rho ]= \\oint \\limits _ {\\mathcal {C}}{dx \\over 2\\pi i}\\left( {1\\over \\varepsilon } \\text{Li}_2(\\kappa {\\cal Q}(x) e^{-\\varepsilon \\varphi (x)})+\\varphi (x)\\rho (x)\\right) + {\\textstyle {1\\over 2}} \\oint \\limits _{ {\\mathcal {C}}\\times {\\mathcal {C}}} {dx\\, du \\over (2\\pi i)^2}\\ \\rho (x)G(x, u )\\rho (u).", "$ The double integral in the second term can be understood as a principal value.", "Indeed, the contribution $\\rho \\rho ^{\\prime }$ of the pole at $x=u$ is pure derivative and vanishes after being contour-integrated.", "In the approximation we are looking for, the ${A}$ -functional is given by the saddle-point action $ \\log {A}^{[\\varepsilon , \\kappa ]} _{ { {\\bf u} }, { { \\bf z} }}&= {{\\cal Y}_{c}} + {\\mathcal { O} }(\\varepsilon ), \\qquad {\\cal Y}_{c}= {\\cal Y}[\\varphi _c, \\rho _c], $ where the saddle point $\\varphi _c$ is given by a couple of TBA-like equations $ \\varphi _c(x) = - \\!\\!\\!\\!\\!\\!", "\\int \\limits _ {\\mathcal {C}}{dy\\over 2\\pi i} G(x-y) \\rho _c(y), \\quad \\rho _c(x) = - \\log \\left( 1-\\kappa {\\cal Q}(x)e^{-\\varepsilon \\varphi _c(x)}\\right).$ After solving for $\\rho _c$ , one obtains a non-linear integral equationSuch type of integral equations first appeared as alternative formulation of the Thermodinamic Bethe Ansatz without strings [22], [23], and most recently in supersymmetric gauge theories [24], [14].", "If the space-time variable $x$ scales as $\\varepsilon ^0$ , there is no need to solve the non-linear integral equation, because only the leading order in $\\varepsilon $ matters.", "We don't exclude that the above analysis can be carried on for weaker assumptions about the distribution of the roots $ { {\\bf u} }$ , such that $x$ scales as $\\varepsilon ^1$ , in which case the non-linear integral equation does not contain a small parameter.", "for the classical field $\\varphi _c$ : $ \\varphi _c(x) =- \\!\\!\\!\\!\\!\\!", "\\int \\limits _{ {\\mathcal {C}}}{dy\\over 2\\pi i}G(x-y) \\log \\left( 1-\\kappa {\\cal Q}(y) e^{-\\varepsilon \\varphi _c(y)}\\right) .$ Expanding $ {\\cal Y}_c &=& \\oint \\limits _{ {\\mathcal {C}}}{dx\\over 2\\pi i} \\ \\left[ -{1\\over \\varepsilon } \\text{Li}_2(\\kappa {\\cal Q}(x) e^{-\\varepsilon \\varphi _c(x)})- {\\textstyle {1\\over 2}} \\varphi _c(x) \\log \\left( 1-\\kappa {\\cal Q}(x) e^{-\\varepsilon \\varphi _c(x)}\\right)\\right]$ up to $O(\\varepsilon )$ , we obtain an explicit expression for the leading and the subleading terms: $ \\begin{aligned}\\hspace{-8.5359pt}\\log {A}^{[\\varepsilon , \\kappa ]} _{ { {\\bf u} }, { { \\bf z} }} &= - {1\\over \\varepsilon }\\oint \\limits _{ {\\mathcal {C}}} {dx\\over 2\\pi i} \\ \\text{Li}_2[\\kappa {\\cal Q}(x)] + {\\textstyle {1\\over 2}} \\oint \\limits _{ {\\mathcal {C}}\\times {\\mathcal {C}}} {dx\\, du \\over (2\\pi i)^2}\\ { \\log \\left[1-\\kappa {\\cal Q}(x) \\right] \\ \\log \\left[ 1-\\kappa {\\cal Q}(u) \\right]\\over (x-u)^2} \\\\&+{\\mathcal { O} }(\\varepsilon ),\\end{aligned}$ where the double integral is understood as a principal value.", "The expression (REF ) obtained by the field-theory method is identical to the result obtained by solving the Riemann-Hilbert problem, Eq.", "(REF ).", "Taking $\\varepsilon =i$ and ${\\cal Q}=\\exp (ip_ { {\\bf u} }+ip_ { \\bf v })$ , we obtain the expression for the leading and the sub-leading terms of the inner product, Eqs.", "(REF )–().", "Here we neglected the trivial factors in the expression (REF ) of the inner product through the ${A}$ -functional.", "The choice of the contour $ {\\mathcal {C}}$ is a subtle issue and depends on the analytic properties of the function ${\\cal Q}(x)$ , as discussed above in Section .", "In any particular case one can first find explicitly the function ${\\cal Q}(x)$ in the limit of a small filling fractions ($\\alpha \\equiv N_a/L\\ll 1$ , where $N_a$ is the number of roots that form the $a$ -th arc), then place the contour $ {\\mathcal {C}}$ so that it does not cross any cuts of $ \\text{Li}_2({\\cal Q}(x))$ .", "If the fillings are not too large, this choice of the contour will remain valid also for $N_a/L\\sim 1$ .", "However, it is possible that at some critical filling that one of the the zeros of $1-{\\cal Q}$ approaches the $a$ -th arc.", "Such a situation has been analysed in [25].", "If this is the case, the contour of integration should be deformed to avoid the logarithmic cut starting with this zero, possibly passing to the second sheet." ], [ "Relation to the Mayer expansion of non-ideal gas", "The semi-classical limit of the ${A}$ -functional resembles the so-called Nekrasov-Shatashvili limit of instanton partition functions of deformed ${ \\cal N}=2$ supersymmetric gauge theories [14].", "The methods developed to study this limit, outlined in [14] and recently worked out in great detail in [18], [19], are based on the iterated Mayer expansion for a non-ideal gas of particles confined along a contour $ {\\mathcal {C}}$ .", "Below we are going to explain the connection between our approach and the Mayer expansion.", "The exponential field (REF ) creates a pair of Coulomb charges with opposite signs spaced at distance $\\varepsilon $ .", "One can think of such a pair as a `fundamental' particle with zero electric charge but non-vanishing dipole and higher charges.", "The sum (REF ) is the grand partition function of such `fundamental' dipoles confined on the contour $ {\\mathcal {C}}_ { {\\bf u} }$ , or equivalently, on a sequence of nested contours surrounding the set $ { {\\bf u} }$ .", "The fundamental dipoles interact with the external potential $\\Phi (x)$ and pairwise among themselves.", "The pairwise interaction is determined by the two-point correlator (REF ).", "Subtracting the product of the one-point functions $ \\langle 0 | { \\cal V}_\\varepsilon | 0 \\rangle =1$ , one obtains for the connected correlator of two dipoles $ \\langle \\!\\langle { \\cal V}_\\varepsilon (x){ \\cal V}_\\varepsilon (y)\\rangle \\!\\rangle ={\\varepsilon ^2\\over (x-y)^2 - \\varepsilon ^2}.", "$ The interaction between two dipoles depends both on the distance and on the direction.", "If $\\varepsilon =|\\varepsilon |i$ , then the force between two dipoles is repulsive if they are spaced horizontally and attractive if they are spaced vertically.", "As the interaction rapidly decreases at large distances, one can compute the thermodynamics of the dipole gas by performing Mayer (cumulant) expansion.", "The poles of the pair-wise interaction potential at $x-y=\\pm \\varepsilon $ lead to a phenomenon called in [14] clustering of instanton particles.", "The fundamental dipole can form `bound states' of $n$ fundamental dipoles, whose field-theoretical counterpart are the exponential fields (REF ).", "A composite particle made of $n$ fundamental dipoles behaves as a pair of positive and negative electric charges spaced at distance $n\\varepsilon $ .", "By the operator representation (REF ), the ${A}$ -functional is the grand partition function of a non-ideal gas made of the fundamental particles and all kinds of composite particles.", "The particles of this gas interact with the external potential $\\Phi (x)$ and pairwise as $ \\langle \\!\\langle { \\cal V}_{m\\varepsilon }(x)\\ { \\cal V}_{n\\varepsilon } (y)\\rangle \\!\\rangle ={\\varepsilon ^2 \\, mn \\over (x-y+m\\varepsilon )(x-y - n\\varepsilon )}.", "$ The effective one-dimensional theory (REF ) describes the limit when only the dipole charge is taken into account, while the quadruple etc.", "charges, small by powers of $\\varepsilon $ , are neglected.", "The first term in our final formula (REF ) corresponds to the dilute gas approximation, in which the charges interact only with the external potential, while the sub-leading second term takes into account the pairwise interactions." ], [ "Acknowledgments", "I.K.", "thanks J.-E. Bourgine, D. Fioravanti, N. Gromov, S. Komatsu, Y. Matsuo and F. Ravanini for illuminating discussions.", "We are grateful for the hospitality at the Simons Center for Geometry and Physics, where this work has been initiated.", "EB is grateful for the hospitality at the University of Cologne.", "This work has been supported by European Programme IRSES UNIFY (Grant No 269217), the Israel Science Foundation (Grant No.", "852/11) and by the Binational Science Foundation (Grant No.", "2010345)." ] ]
1403.0358
[ [ "Counting solutions without zeros or repetitions of a linear congruence\n and rarefaction in b-multiplicative sequences" ], [ "Abstract Consider a strongly $b$-multiplicative sequence and a prime $p$.", "Studying its $p$-rarefaction consists in characterizing the asymptotic behaviour of the sums of the first terms indexed by the multiples of $p$.", "The integer values of the \"norm\" $3$-variate polynomial $\\mathcal N_{p,i_1,i_2}(Y_0,Y_1,Y_2)\\!", ":=\\!\\prod_{j=1}^{p-1}\\left(Y_0{+}\\zeta_p^{i_1j}Y_1{+}\\zeta_p^{i_2j}Y_2\\right),$ where $\\zeta_p$ is a primitive $p$-th root of unity, and $i_1,i_2{\\in}\\{1,2,\\dots,p{-}1\\},$ determine this asymptotic behaviour.", "It will be shown that a combinatorial method can be applied to $\\mathcal N_{p,i_1,i_2}(Y_0,Y_1,Y_2).$ The method enables deducing functional relations between the coefficients as well as various properties of the coefficients of $\\mathcal N_{p,i_1,i_2}(Y_0,Y_1,Y_2)$, in particular for $i_1{=}1,i_2{=}2,3.$ This method provides relations between binomial coefficients.", "It gives new proofs of the two identities $\\prod_{j=1}^{p-1}\\left(1{-}\\zeta_p^j\\right){=}p$ and $\\prod_{j=1}^{p-1}\\left(1{+}\\zeta_p^j{-}\\zeta_p^{2j}\\right){=}L_p$ (the $p$-th Lucas number).", "The sign and the residue modulo $p$ of the symmetric polynomials of $1{+}\\zeta_p{-}\\zeta_p^2$ can also be obtained.", "An algorithm for computation of coefficients of $\\mathcal N_{p,i_1,i_2}(Y_0,Y_1,Y_2)$ is developed." ], [ "Introduction", "This article deals with a combinatorial method adapted to the coefficients of homogeneous 3-variate “norm\" polynomials which determine the asymptotic behaviour of rarified sums of a strongly $b$ -multiplicative sequence.", "The general definition of a strongly $b$ -multiplicative sequence of complex numbers (see [1]) can be written as: Definition 1 Let $(t_n)_{n\\geqslant 0}$ be a sequence of complex numbers and $b\\geqslant 2$ an integer.", "The sequence $(t_n)_{n\\geqslant 0}$ is called strongly b-multiplicative if it satisfies, for each $n\\in \\mathbb {N}$ , the equation $t_n=\\prod _{i=0}^l t_{c_i},$ where $n=\\sum _{i=0}^l c_ib^i$ is the $b$ -ary expansion of a natural integer $n$ .", "Additionally, we ask that $t_0=1$ or $t_n$ is identically zero.", "This definition ensures that $t_n$ does not depend on the choice of the $b$ -ary expansion of $n$ (for $b$ -ary expansions which may or may not start with zeroes).", "If the values of a strongly $b$ -multiplicative sequence are either 0 or roots of unity, it is $b$ -automatic.", "An example of such sequence is the $\\lbrace 1,-1\\rbrace $ -valued Thue-Morse sequence defined by $b=2,t_1=-1$ (referred as $A106400$ in OEIS, cf [16]).", "Further in this text we are going to refer to this sequence as the Thue-Morse sequence.", "A survey on the strongly $b$ -multiplicative sequences with values in an arbitrary compact group can be found in [6].", "Rarified sums (or $p$ -rarified sums, the term is due to [7]) of a sequence $(t_n)_{n\\geqslant 0}$ are the sums of initial terms of the subsequence $(t_{pn})_{n\\geqslant 0}$ (the rarefaction step $p$ is supposed to be a prime number in this paper).", "The problem of estimating the speed of growth of these sums has been studied in [8],[5],[9],[10],[11].", "The following result has been proved in a special case.", "Proposition 1.1 (see [9], Theorem $5.1$ ) Let $(t_n)_{n\\geqslant 0}$ be the Thue-Morse sequence.", "Suppose that $b=2$ is a generator of the multiplicative group $\\mathbb {F}_p^\\times $ .", "Then, $\\sum _{n<N,p \\,\\mid \\, n}t_n=O\\left(N^\\frac{\\log p}{(p-1)\\log 2}\\right)$ and this exponent cannot be decreased.", "We are going to study this problem in a more general case.", "In Section we generalize Proposition REF to Proposition REF valid for a large subclass of strongly $b$ -multiplicative sequences (with different values of $b$ ).", "Proposition REF describes the speed of growth of $p$ -rarified sums of a strongly $b$ -multiplicative sequence in a form $\\sum _{n<N,p \\,\\mid \\, n}t_n=O\\left(N^{\\frac{1}{(p-1)\\log b}{\\log \\left(\\mathbf {N}_{\\mathbb {Q}(\\zeta _p)/\\mathbb {Q}}(\\sum _{c=0}^{b-1} t_c\\zeta _p^c)\\right)}}\\right)$ similar to (REF ).", "The more general formula (REF ) contains the quantity $\\xi \\left((t_n)_{n\\geqslant 0},p\\right):=\\mathbf {N}_{\\mathbb {Q}(\\zeta _p)/\\mathbb {Q}}\\left(\\sum _{j=0}^{b-1} t_j\\zeta _p^j\\right).$ The properties of this norm expression provide information about the speed of growth of rarified sums.", "In Sections , , we develop a method to study these norms.", "Denote by $d((t_n)_{n\\geqslant 0})$ the number of nonzero terms among $t_1,\\dots ,t_{b-1}$ .", "For technical reasons, the method described in this article concerns only the strongly $b$ -multiplicative sequences such that $d((t_n)_{n\\geqslant 0}){\\leqslant }2$ ; in general, if $d\\geqslant 3$ , it leads to too difficult computations.", "On the other hand, the case where $d((t_n)_{n\\geqslant 0}){=}1$ (which concerns, for example, the Thue-Morse sequence) is relatively easy.", "In the short description of the method, which follows, we assume that $d((t_n)_{n\\geqslant 0}){=}2$ .", "Our method consists in dealing with a strongly $b$ -multiplicative sequence of monomials instead of the initial strongly $b$ -multiplicative sequence of complex numbers.", "Let $i_1,i_2{\\in }\\lbrace 1,\\dots ,b{-}1\\rbrace $ be the two indices such that $t_{i_1}{\\ne }0$ and $t_{i_2}{\\ne }0$ .", "Then the strongly $b$ -multiplicative sequence of monomials associated with the sequence $(t_n)_{n\\geqslant 0}$ and the choice of the order of $i_1,i_2$ is defined by $T_0&=1,&\\\\T_{i_1}&=Y_1,&\\\\T_{i_2}&=Y_2,&\\\\T_c&=0\\text{ if }c\\in \\lbrace 1,\\dots ,b{-}1\\rbrace \\setminus \\lbrace i_1,i_2\\rbrace \\\\T_n&=\\prod _{i=0}^l T_{c_i}\\text{ otherwise,}$ where $n=\\sum _{i=0}^l c_ib^i$ is the $b$ -ary expansion of a natural integer $n$ .", "Clearly, $T_n{=}0$ if and only if $t_n{=}0$ .", "For example, if $b{=}3$ and $i_1{=}1,i_2{=}2$ then the sequence $(T_n)_{n\\geqslant 0}$ starts with $1,Y_1,Y_2,Y_1,Y_1^2,Y_1Y_2,Y_2,Y_1Y_2,Y_2^2,\\dots $ If $b{=}5$ and $i_1{=}1,i_2{=}2$ , it starts with $1,Y_1,Y_2,0,0,Y_1,Y_1^2,Y_1Y_2,0,0,Y_2,\\dots $ One can define the $p$ -rarefied sums of the sequence $(T_n)_{n\\geqslant 0}$ and the formal object $\\prod _{j=1}^{p-1}\\left(\\sum _{c=0}^{b-1} \\zeta _p^{jc}T_c\\right)$ which plays the role of $\\xi \\left((T_n)_{n\\geqslant 0},p\\right)$ .", "One can write (REF ) explicitly as $\\bar{\\mathcal {N}}_{p,i_1,i_2}(Y_1,Y_2)=\\prod _{j=1}^{p-1}\\left(1+\\zeta _p^{i_1j}Y_1+\\zeta _p^{i_2j}Y_2\\right),$ and homogenize this polynomial, which defines $\\mathcal {N}_{p,i_1,i_2}(Y_0,Y_1,Y_2)=\\prod _{j=1}^{p-1}\\left(Y_0+\\zeta _p^{i_1j}Y_1+\\zeta _p^{i_2j}Y_2\\right).$ The norm (REF ) is then recovered as the value $\\mathcal {N}_{p,i_1,i_2}(1,t_{i_1},t_{i_2}).$ By definition, $\\mathcal {N}_{p,i_1,i_2}(Y_0,Y_1,Y_2)$ is the norm of $(Y_0+\\zeta _p^{i_1}Y_1+\\zeta _p^{i_2}Y_2)$ as a polynomial in the 4 variables $Y_0,Y_1,Y_2,\\zeta _p$ relative to the extension of fields $\\mathbb {Q}(\\zeta _p)/\\mathbb {Q}$ in the sense of the extended definition of norm introduced in [18].", "The form (REF ) of “norm\" polynomial reveals to be common for the strongly $b$ -multiplicative sequences which satisfy $d((t_n)_{n\\geqslant 0}){=}2$ (as defined above).", "In order to retrieve a particular sequence from this form, one should set the formal variables $Y_0,Y_1,Y_2$ to special values, fix the two residue classes $i_1,i_2$ and take a base $b$ (bigger than the smallest positive representatives of $i_1,i_2,$ and such that the residue class of $b$ modulo $p$ is in $\\mathbb {F}_p^\\times $ , and it generates this multiplicative group).", "Since the form (REF ) inherits the properties of its coefficients, any functional relation between these coefficients can be considered as a key result.", "In this context, Sections and enunciate a combinatorial interpretation of the coefficients of $\\mathcal {N}_{p,i_1,i_2}$ in terms of the following counting problem.", "Problem 1 Let $p$ be a prime number, let $\\mathbf {f}$ be a vector of length $p{-}1$ , all elements of which are residue classes modulo $p$ among $0,i_1,i_2$ (i.e., $\\mathbf {f}{\\in }\\lbrace 0,i_1,i_2\\rbrace ^{p{-}1}{\\subset }\\mathbb {F}_p^{p-1}$ ).", "Let $i$ be an element of $\\mathbb {F}_p.$ Find the number of vectors $\\mathbf {x}{\\in }\\mathbb {F}_p^{p-1}$ which are permutations of $(1,2,\\dots ,p-1)$ and such that $\\mathbf {f}\\cdot \\mathbf {x}=i.$ The equivalence of Problem REF and the problem of determining the coefficients of $\\mathcal {N}_{p,i_1,i_2}$ is made explicit in Proposition REF .", "Our main result is the following.", "Theorem 1.1 (equivalent to Theorem REF ) Let $p$ be an odd prime, and $i_1,i_2\\in \\mathbb {F}_p^\\times $ such that $i_1\\ne i_2$ .", "Denote by $\\triangle ^{i_1,i_2}(n_1,n_2,p)$ the coefficient of the term $Y_0^{p-1-n_1-n_2}Y_1^{n_1}Y_2^{n_2}$ in $\\mathcal {N}_{p,i_1,i_2}(Y_0,Y_1,Y_2)$ .", "Fix exponents $n_1,n_2\\in \\lbrace 1,\\dots ,p-2\\rbrace $ such that $n_1+n_2<p.$ Then, $\\triangle ^{i_1,i_2}(n_1,n_2,p)\\equiv -\\triangle ^{i_1,i_2}(n_1-1,n_2,p)-\\triangle ^{i_1,i_2}(n_1,n_2-1,p)\\text{ \\rm mod }p$ and if $p\\nmid n_1i_1+n_2i_2$ , the equality $\\triangle ^{i_1,i_2}(n_1,n_2,p)=-\\triangle ^{i_1,i_2}(n_1-1,n_2,p)-\\triangle ^{i_1,i_2}(n_1,n_2-1,p)$ holds.", "The relation (REF ) (similar to the recurrence equation of the Pascal's triangle) can be used to find closed formulas for some classes of coefficients (for all of them in the case $i_1=1,i_2=2$ ) and to find the remaining coefficients in a fast algorithmic way.", "A closed formula for these coefficients is a final goal.", "In Section we describe an algorithm in $O(p^2)$ additions that calculates the coefficients of (REF ) using this relation.", "We study the case $i_1=1,i_2=2$ and re-prove the result $\\prod _{j=1}^{p-1}\\left(1{+}\\zeta _p^j{-}\\zeta _p^{2j}\\right){=}L_p,$ the $p$ -th Lucas number (i.e., the $p$ -th term of the sequence referred as $A000032$ by OEIS, cf [16]).", "We formulate two corollaries of the new proof.", "We also state some results about the case $i_1=1,i_2=3$ .", "Throughout the paper, $|X|$ and $\\#X$ will both refer to the size of a finite set $X$ ; the symbol $\\#$ followed by a system of equations, congruences or inequalities will denote the number of solutions; and $\\sum X$ , standing for $\\sum _{x\\in X}x$ , will refer to the sum of a finite subset $X$ of a commutative group with additive notation.", "The results of this article (except Theorem REF and Subsection REF ) are part of the Ph.D. thesis [2]." ], [ "Partial sums of a stronly $q$ -multiplicative sequence.", "We are going to prove an asymptotic result about partial sums of a strongly $q$ -multiplicative sequence used in the proof of Proposition REF .", "Lemma 2.1 Let $q{\\geqslant }2$ be an integer and consider a strongly $q$ -multiplicative sequence $(\\tau _n)_{n\\geqslant 0}$ of complex numbers of absolute value smaller than or equal to 1.", "Denote the partial sums of $(\\tau _n)_{n\\geqslant 0}$ by $\\psi (N):=\\sum _{n<N}\\tau _n\\ (N\\in \\mathbb {N}).$ Then we have the following.", "If $|\\psi (q)|\\leqslant 1$ , then $\\psi (N)=O(\\log N).$ If $|\\psi (q)|>1$ , then $\\psi (N)=O\\left(N^{\\frac{1}{\\log q}{\\left(\\log \\left|\\sum _{c=0}^{q-1}\\tau _c\\right|\\right)}}\\right).$ Denote, for any $N,Q\\in \\mathbb {N}$ ($Q>0$ ), $\\eta (N,Q):=Q\\left\\lfloor \\frac{N}{Q}\\right\\rfloor .$ If $Q=q^m$ , then the $q$ -ary expansion of $\\eta (N,Q)$ can be obtained from the $q$ -ary expansion of $N$ by replacing the last $m$ digits by zeroes.", "Suppose that $N$ is a natural integer with $q$ -ary expansion $N=\\sum _{i=0}^l c_iq^i$ .", "Then, $&\\psi (N){=} \\sum \\limits _{n=0}^{\\eta (N,q^l)-1}\\tau _n{+} \\sum \\limits _{n=\\eta (N,q^l)}^{\\eta (N,q^{l-1})-1}\\tau _n{+}\\dots {+}\\sum \\limits _{n=\\eta (N,q)}^{N-1}\\tau _n\\\\&{=}\\left(\\sum \\limits _{c=0}^{c_l-1}\\tau _c\\right)\\left(\\sum \\limits _{c=0}^{q-1}\\tau _c\\right)^l{+}\\tau _{c_l}\\left(\\sum \\limits _{c=0}^{c_{l-1}-1}\\tau _c\\right)\\left(\\sum \\limits _{c=0}^{q-1}\\tau _c\\right)^{\\makebox{[}2pt]{\\scriptsize \\,l{-}1}}{+}\\dots {+}\\prod _{k=1}^l\\tau _{c_k}{\\cdot }\\left(\\sum \\limits _{c=0}^{c_0-1}\\tau _c\\right)\\\\&=\\sum \\limits _{i=0}^l\\left(\\prod \\limits _{k=i+1}^l\\tau _{c_k}\\right){\\cdot } \\psi (c_i) {\\cdot } \\psi (q)^i.$ If $|\\psi (q)|\\leqslant 1$ then each term of the sum (REF ) is bounded (by the maximum of $|\\psi (c)|,c=1,\\dots ,q-1$ ), therefore $\\psi (N)=O(l)=O(\\log N).$ Suppose that $|\\psi (q)|>1$ .", "Then we are going to extend the definition of the function $\\psi (x)$ to all real $x\\geqslant 0$ using the right-hand side of the formula (REF ).", "This requires to check that the result does not depend on the choice of the $q$ -ary expansion of the argument.", "Take $x=q^{-m}X$ where $m\\in \\mathbb {Z},X\\in \\mathbb {N},$ and the $q$ -ary expansion of $X$ is $X=\\sum _{i=0}^{m+l}c_{i-m}q^i.$ Then the two $q$ -ary expansions of $x$ are $x=\\sum _{i=-m}^{l}c_{i}q^i=\\sum _{i=-m+1}^{l}c_{i}q^i+(c_{-m}-1)q^{-m}+\\sum _{i=-\\infty }^{-m-1}(q-1)q^i.$ We have to prove the identity $\\sum \\limits _{i=-m}^{l} \\left(\\prod _{k=i+1}^l\\tau _{c_k}\\right)\\cdot \\psi (c_i) d(q)^i=\\\\\\sum \\limits _{i=-m+1}^l \\left(\\prod _{k=i+1}^l\\tau _{c_k}\\right)\\cdot \\psi (c_i) d(q)^i+ \\left(\\prod \\limits _{k=-m+1}^l\\tau _{c_k}\\right)\\cdot \\psi (c_{-m}-1) \\psi (q)^{-m}\\\\+ \\psi (q-1)\\left(\\sum \\limits _{i=-\\infty }^{-m-1} \\left(\\prod \\limits _{k=i+1}^l\\tau _{c_k}\\right)\\cdot \\psi (q)^i \\right).$ where we denote $c_i=q-1$ for $i<-m$ .", "Indeed, some sub-expressions of the right-hand side of (REF ) can be simplified.", "The last summand can be factored with one factor being $\\psi (q-1)\\left(\\sum \\limits _{i=-\\infty }^{-m-1} \\left(\\prod \\limits _{k=i+1}^{-m-1}\\tau _{c_k}\\right)\\cdot \\psi (q)^i \\right)=\\psi (q-1)\\sum \\limits _{i=-\\infty }^{-m-1}\\tau _{q-1}^{-m-i-1}\\psi (q)^i\\\\=\\psi (q-1)\\tau _{q-1}^{-m-1}\\left(\\frac{\\tau _{q-1}}{\\psi (q)}\\right)^m\\frac{1}{\\frac{\\psi (q)}{\\tau _{q-1}}-1}=\\psi (q-1)\\frac{1}{\\psi (q)^m(\\psi (q)-\\tau -{q-1})}\\\\=\\psi (q)^{-m}.$ Next, the sum of the two last summands in (REF ) is $\\left(\\prod \\limits _{k=-m+1}^l\\tau _{c_k}\\right)\\cdot \\psi (c_{-m}-1) \\psi (q)^{-m}+\\left(\\prod _{k=-m}^l\\tau _{c_k}\\right)\\psi (q)^{-m}\\\\=\\left(\\prod \\limits _{k>-m}\\tau _{c_k}\\right)\\psi (c_{-m})\\psi (q^{-m}).$ These transformations reduce the right-hand side of (REF ) to the form of the left-hand side, proving the identity.", "Therefore, $\\psi (x)$ is a well-defined function of a real argument.", "This function is continuous.", "Indeed, consider a sequence $(x_n)_{n}$ of positive real numbers which converges to $x>0$ .", "Suppose that either $x_n>x$ for all $n$ or $x_n<x$ for all $n$ .", "Let $x=\\sum _{i=-\\infty }^{l}c_{i}q^i$ be the $q$ -ary expansion of $x$ which has a property chosen depending on the choice above: if $x_n>x,$ the expansion of $x$ does not end by $q-1^{\\prime }$ s, if $x_n<x,$ it does not end by zeroes.", "In both cases, for each $m>0$ there is a rang $\\tilde{n}$ such that $n>\\tilde{n}$ implies that any $q$ -ary expansion of $x_n$ (denote it by $x_n=\\sum _{i=-\\infty }^{l}{\\bar{c}}_{i}q^i$ ) has all digits before radix point and $m$ digits after radix point identical to those of $x$ .", "This property implies: $|\\psi (x)-\\psi (x_n)|=\\left|\\sum \\limits _{i=-\\infty }^{m-1}\\prod _{k>i}\\tau _{c_k}\\cdot d(c_i)d(b)^i -\\sum \\limits _{i=-\\infty }^{m-1}\\prod _{k>i}\\tau (\\bar{c}_k)\\cdot d(\\bar{c}_i)d(b)^i\\right|\\\\[5mm]\\leqslant 2\\max _{c\\in \\lbrace 0,\\dots ,b-1\\rbrace } \\sum _{i<-m}|d(b)|^i \\xrightarrow[m\\rightarrow \\infty ]{}0,$ which proves that the sequence $(\\psi (x_n))_n$ converges to $\\psi (x)$ .", "By the definition (REF ), for any $x\\geqslant 0,$ $\\psi (qx)=\\psi (q)\\psi (x).$ Therefore, $\\psi (x)&=\\left(\\frac{x}{q^l}\\right){\\psi (q)}^l\\text{ where }l=\\lfloor \\log _qx\\rfloor ,\\text{ therefore}\\\\|\\psi (x)|&\\leqslant \\left(\\max _{x\\in [1,q]} |\\psi (x)|\\right)|\\psi (q)|^l=O(|\\psi (q)|^l)=O(x^\\frac{\\log |\\psi (q)|}{\\log q}).$ Lemma is proved.", "Remark that the second part of this Theorem (REF ) has been proved in the article [10] (formula ($2.9$ )).", "The previous Lemma leads to the following asymptotic result about the $p$ -rarified sums.", "Proposition 2.1 Consider a strongly $b$ -multiplicative sequence $(t_n)_{n\\geqslant 0}$ with values in $\\lbrace -1,0,1\\rbrace $ and a prime number $p$ such that $b<p$ is a generator of the multiplicative group $\\mathbb {F}_p^\\times $ .", "Suppose that the following inequality holds: $\\left|\\mathbf {N}_{\\mathbb {Q}(\\zeta _p)/\\mathbb {Q}}\\left(\\sum _{c=0}^{b-1} t_c\\zeta _p^c\\right)\\right|>\\max \\left((\\sum _{c=0}^{b-1}t_c)^{p-1},1\\right)$ where $\\zeta _p$ denotes a primitive $p$ -th root of unity and $\\mathbf {N}_{L/K}$ denotes the norm.", "Then we have the following estimation: $\\sum _{n<N,p \\,\\mid \\, n}t_n=O\\left(N^{\\frac{1}{(p-1)\\log b}{\\log \\left(\\mathbf {N}_{\\mathbb {Q}(\\zeta _p)/\\mathbb {Q}}(\\sum _{c=0}^{b-1} t_c\\zeta _p^c)\\right)}}\\right)$ The norm in (REF ) is real and nonnegative because it is a product of $\\frac{p-1}{2}$ complex-conjugate pairs.", "Furthermore, it is bigger than 1 by the hypothesis (REF ).", "This proves that the right-hand side of (REF ) has a meaning.", "The $p$ -rarefied sum in the left-hand side can be expanded as $\\sum _{n<N}1_{p|n} t_n= \\sum _{n<N}\\frac{1}{p}\\left(1+\\zeta _p^n+\\zeta _p^{2n}+\\ldots +\\zeta _p^{(p-1)n}\\right)t_n\\\\=\\frac{1}{p}\\left(\\sum _{n<N}t_n+\\sum _{n<N}\\sum _{j\\in \\mathbb {F}_p^\\times }\\zeta _p^{jn}t_n\\right).$ Remark that the sequences $(t_n)_{n\\geqslant 0}$ and $(\\zeta _p^{jn}t_n)_{n\\geqslant 0}$ ($j\\in \\lbrace 1,\\ldots ,p{-}1\\rbrace $ ), which appear in the previous formula, are strongly $b^{p-1}$ -multiplicative.", "By Lemma REF applied to the sequence $(t_n)_{n\\geqslant 0}$ , one has one of the two estimations $\\sum _{n<N}t_n&=O\\left(N^{\\frac{1}{\\log b}\\log \\left|\\sum _{c=0}^{b-1}t_c\\right|}\\right)\\text{ or}\\\\\\sum _{n<N}t_n&=O(\\log N).$ In both cases we get (using the hypothesis (REF )), $\\sum _{n<N}t_n=O\\left(N^{\\frac{1}{(p-1)\\log b}{\\log \\left(\\mathbf {N}_{\\mathbb {Q}(\\zeta _p)/\\mathbb {Q}}(\\sum _{j=0}^{b-1} t_j\\zeta _p^j)\\right)}}\\right).$ Lemma REF applied to a sequence of the form $(\\zeta _p^{jn}t_n)_{n\\geqslant 0}$ states that $\\sum _{n<N}\\zeta _p^{jn}t_n&=O\\left(N^{\\frac{1}{(p-1)\\log b}\\log \\left|\\sum _{c=0}^{b^{p-1}-1}\\zeta _p^{j c}t_c\\right|}\\right)\\text{ or} \\\\\\sum _{n<N}\\zeta _p^{jn}t_n&=O(\\log N).$ On the other hand, one can expand the norm of $(\\sum _{c=0}^{b-1} t_c\\zeta _p^c)$ as $\\mathbf {N}_{\\mathbb {Q}(\\zeta _p)/\\mathbb {Q}}(\\sum _{c=0}^{b-1} t_c\\zeta _p^c)=\\prod _{i=0}^{p-2}\\left(\\sum _{c=0}^{b-1}\\zeta _p^{b^icj}t_c\\right)=\\\\\\sum _{c_0,\\dots ,c_{p-2}\\in \\lbrace 0,\\dots ,b-1\\rbrace } \\zeta _p^{j(c_0+b c_1+\\dots +b^{p-2}c_{p-2})}t_{c_0}\\dots t_{c_{p-2}}$ where the sequences $(c_0,\\dots ,c_{p-2})$ are nothing else than all possible choices of the index $c$ when the product is expanded.", "The change of variable $n:=c_0+bc_1+\\dots +b^{p-2}c_{p-2}$ leads to a new variable which goes through all integers from 0 to $b^{p-1}-1$ .", "Therefore, $\\mathbf {N}_{\\mathbb {Q}(\\zeta _p)/\\mathbb {Q}}(\\sum _{c=0}^{b-1} t_c\\zeta _p^c)=\\sum _{n=0}^{b^{p-1}-1}\\zeta _p^nt_n,$ which is the sum involved in (REF ).", "Therefore, $\\sum _{n<N}\\zeta _p^{jn}t_n=O\\left(N^{\\frac{1}{(p-1)\\log b}{\\log \\left(\\mathbf {N}_{\\mathbb {Q}(\\zeta _p)/\\mathbb {Q}}(\\sum _{c=0}^{b-1} t_c\\zeta _p^c)\\right)}}\\right).$ Equations (REF ), (REF ) and (REF ) lead to the conclusion of the Proposition.", "Proposition REF generalizes the first part of Proposition REF (the estimation (REF )) as the Thue-Morse sequence satisfies the conditions of validity of Propoition REF and we get the following: $\\mathbf {N}_{\\mathbb {Q}(\\zeta _p)/\\mathbb {Q}}\\left(\\sum _{j=0}^{b-1} t_j\\zeta _p^j\\right)=\\mathbf {N}_{\\mathbb {Q}(\\zeta _p)/\\mathbb {Q}}(1-\\zeta _p)=p.$ Another situation where the norms $\\mathbf {N}_{\\mathbb {Q}(\\zeta _p)/\\mathbb {Q}}\\left(\\sum _{j=0}^{b-1} t_j\\zeta _p^j\\right)$ can be calculated in a straightforward way is the situation where $b=3,t_0{=}t_1{=}1,t_2{=}{-}1$ .", "Using the resultant of the two polynomials $S(X)=X^{p-1}+\\dots +1$ and $R(X){=}X^2{-}X{-}1$ , one obtains $\\mathbf {N}_{\\mathbb {Q}(\\zeta _p)/\\mathbb {Q}}(1+\\zeta _p-\\zeta _p^2)=L_p$ the $p$ -th term of the Lucas sequence (referred as $A000032$ by OEIS, cf [16]) defined recursively by $L_0\\!=\\!2,{L_1\\!=\\!1},L_{n+2}=L_n{+}L_{n+1}$ .", "This result is proved in a different way in Section REF ." ], [ "Combinatorics of partitions of a set.", "In this section we are going to give an alternative proof of the formula $\\prod _{j=1}^{j=p-1}\\left(X-\\zeta ^j\\right)=1+X+\\dots +X^{p-1},$ and the methods of this proof will be re-used in the proof of the functional equation in Section .", "The new proof uses the properties of the partially ordered sets $\\Pi _n$ of partitions of a set of size $n$ (a good reference about the properties of those is the Chapter $3.10.4$ of [17]).", "We are going to prove the following statement, which is equivalent to (REF ).", "Lemma 3.1 Let $p$ be a prime number and $0\\leqslant n<p$ an integer.", "Define $A_0(n,p)$ as the number of subsets of $\\mathbb {F}_p^\\times $ of $n$ elements that sum up to 0 modulo $p$ and $A_1(n,p)$ the number of those subsets that sum up to 1.", "Then $A_0(n,p)-A_1(n,p)=(-1)^n.", "$ Let us begin the proof with an obvious observation: if we define similarly the numbers $A_2(n,p)$ , $A_3(n,p)$ , ..., $A_{p-1}(n,p)$ , they will all be equal to $A_1(n,p)$ , since multiplying a set that sums to 1 by a constant residue $c\\in \\mathbb {F}_p^\\times $ gives a set that sums to $c$ , and this correspondence is one-to-one.", "Let us deal with a simpler version of the Lemma that allows repetitions and counts sequences instead of subsets, which is formalized in the following.", "Definition 2 Denote by $E^{k_1,\\dots ,k_n}_x(n,p)$ (where $x\\in \\mathbb {F}_p$ and $k_1,\\dots ,k_n\\in \\mathbb {F}_p^\\times $ ) the number of sequences $(x_1,x_2,\\dots ,x_n)$ of elements of $\\mathbb {F}_p^\\times $ such that $\\sum _{i=1}^n k_i x_i=x.", "$ Then we get the following.", "Lemma 3.2 If $n$ is even, $E^{k_1,k_2,\\dots ,k_n}_0(n,p)=\\frac{(p{-}1)^n{+}p{-}1}{p}\\quad \\text{\\rm and }\\quad E^{k_1,k_2,\\dots ,k_n}_1(n,p)=\\frac{(p{-}1)^n-1}{p};$     if $n$ is odd, $E^{k_1,k_2,\\dots ,k_n}_0(n,p)=\\frac{(p{-}1)^n{-}p{+}1}{p}\\quad \\text{\\rm and }\\quad E^{k_1,k_2,\\dots ,k_n}_1(n,p)=\\frac{(p{-}1)^n+1}{p}.$ In both cases, $E^{k_1,k_2,\\dots ,k_n}_0(n,p)-E^{k_1,k_2,\\dots ,k_n}_1(n,p)=(-1)^n.", "$ By induction on n. For $n=0$ or $n=1$ the result is trivial.", "For bigger $n$ we always get: $E_0^{k_1,k_2,\\dots ,k_n}(n,p)=(n-1)E_1^{k_1,k_2,\\dots ,k_{n-1}}(n-1,p) $ and $E_1^{k_1,k_2,\\dots ,k_n}(n,p)=E_0^{k_1,k_2,\\dots ,k_{n-1}}\\textit {}(n-1,p)+(p-2)E_1^{k_1,k_2,\\dots ,k_{n-1}}(n-1,p), $ since the sequences of length $n$ of linear combination (with coefficients $k_i$ ) equal to $x$ are exactly expansions of sequences of length $n-1$ of linear combination different from $x$ , and this correspondence is one-to-one.", "Injecting formulas for $n-1$ concludes the induction.", "Now we are going to prove Lemma REF for small $n$ .", "If $n=0$ or $n=1$ , Lemma is clear.", "For $n=2$ , there is one more sequence $(x,y)\\in \\mathbb {F}_p^\\times {^2}$ that sums up to 0, but that counts the sequences of the form $(x,x)$ which should be removed.", "Since $p$ is prime, these sequences contribute once for every nonzero residue modulo $p$ , and removing them increases the zero's “advantage\" to 2.", "Now, we have to identify $(x,y)$ and $(y,x)$ to be the same, so we get the difference 1 back, establishing Lemma 1 for $n=2$ .", "For $n=3$ , counting all the sequences $(x,y,z)\\in \\mathbb {F}_p^\\times $ gives a difference $E_0-E_1=-1 $ .", "The sequences $(x,x,z)$ contribute one time more often to the sum equal to 0, so removing them adds $-1$ to the total difference.", "The same thing applies to sequences of the form $(x,y,y)$ and $(x,y,x)$ .", "After removing them, we get an intermediate difference of $-4$ , but the triples of the form $(x,x,x)$ have been removed 3 times, which is equivalent to saying they count $-2$ times.", "Therefore, they should be “reinjected\" with coefficient 2.", "As $p$ is prime and bigger than 3, the redundant triples contribute once for each nonzero residue; therefore we accumulate the difference of $-4-2=-6$ .", "We have then to identify permutations, that is to divide the score by 6 which gives the final result $-1$ .", "Here is the explicit calculation for the case $n=4$ : $\\begin{array}{ll}1 & \\text{\\rm (corresponds to $E_0(4,p)-E_1(4,p)$)}\\\\+6 & \\text{\\rm (for removing \\begin{tabular}{ccc} (x,x,y,z),& (x,y,x,z),&(x,y,z,x),\\\\ (x,y,y,z),&(x,y,z,y),&(x,y,z,z) \\end{tabular})} \\\\[2mm]+2\\times 4 & \\text{\\rm (for re-injecting \\begin{tabular}{cc}(x,x,x,y),& (x,x,y,x),\\\\ (x,y,x,x)& and (x,y,y,y)\\end{tabular})}\\\\[2mm]+1\\times 3 & \\text{\\rm (for re-injecting $(x,x,y,y),(x,y,x,y)$ and $(x,y,y,x)$)}\\\\[2mm]+6\\times 1 & \\text{\\rm (for removing $(x,x,x,x)$)}\\\\[2mm]=24 &\\end{array}$ which is $4!$ , therefore Lemma REF is proved for $n=4$ .", "For a general $n$ we can calculate the difference between the number of sequences that sum up to 0 and the number of those that sum up to 1 by assigning to all sequences in $\\mathbb {F}_p^\\times {^n}$ an intermediate coefficient equal to one, then by reducing it by one for each couple of equal terms, then increasing by 2 for each triple of equal terms, and so on, proceeding by successive adjustments of coefficients, each step corresponding to a “poker combination\" of $n$ cards.", "If after adding the contributions of all the steps and the initial $(-1)^n$ , we get $(-1)^nn!$ , Lemma REF is valid for $n$ independently from $p$ provided that $p>n$ is prime.", "Let us introduce a formalization of these concepts using the notions exposed in [12].", "Call a partition of the set $\\lbrace 1,2,\\dots ,n\\rbrace $ a choice of pairwise disjoint nonempty subsets $B_1,B_2,\\dots , B_c$ of $\\lbrace 1,2,\\dots ,n\\rbrace $ of non-increasing sizes $|B_i|$ , and such that $B_1\\cup B_2 \\cup \\dots \\cup B_c=\\lbrace 1,2,\\dots ,n\\rbrace $ .", "The set $\\Pi _n$ of all partitions of $\\lbrace 1,2,\\dots ,n\\rbrace $ is partially ordered by reverse refinement: for each two partitions $\\tau $ and $\\pi $ , we say that $\\tau \\geqslant \\pi $ if each block of $\\pi $ is included in a block of $\\tau $ .", "We define the Möbius function $\\mu (\\hat{0},x)$ on $\\Pi _n$ recursively by:    if $x=\\lbrace \\lbrace 1\\rbrace ,\\lbrace 2\\rbrace ,\\dots ,\\lbrace n\\rbrace \\rbrace =\\hat{0}$ , then $\\mu (\\hat{0},x)=1$ ;    if $x$ is bigger than $\\hat{0}$ , then $\\mu (\\hat{0},x)=-\\sum _{\\begin{array}{c}y\\in \\Pi _n\\\\y<x\\end{array}}\\mu (\\hat{0},y).", "$ By the Corollary to the Proposition 3 section 7 of [15] and the first Theorem from the section $5.2.1$ of [12], if $x$ is a partition of type $(\\lambda _1,\\dots ,\\lambda _n)$ , then $\\mu (\\hat{0},x)=\\prod _{i=1}^{n}(-1)^{\\lambda _i-1}(\\lambda _i-1)!$ This formula will be useful in Section .", "We are also going to use the following definition: let $x=(x_1,x_2,\\dots ,x_n)$ be a sequence of $n$ nonzero residues modulo $p$ seen as a function $x:\\lbrace 1,2,\\dots ,n\\rbrace \\rightarrow \\mathbb {F}_p^\\times .", "$ Then the coimage of $x$ is the partition of $\\lbrace 1,2,\\dots ,n\\rbrace $ , whose blocks are the nonempty preimages of elements of $\\mathbb {F}_p^\\times $ .", "Now we can prove the following proposition that puts together all the previous study.", "Lemma 3.3 The difference $A_0(n,p)-A_1(n,p) $ does not depend on $p$ provided that $p$ is a prime number bigger than $n$ .", "We are going to describe an algorithm that computes this difference (which is the one applied earlier for small values of the argument).", "For each partition $x\\in \\Pi _n$ , denote by $r_0(x,p)$ the number of sequences $(x_1,x_2,\\dots ,x_n)$ of elements of $\\mathbb {F}_p^\\times $ of coimage $x$ that sum up to 0, and denote by $r_1(x,p)$ the number of those sequences of coimage $x$ that sum up to 1 and denote $r(x,p)=r_0(x,p)-r_1(x,p)$ .", "Then, $n!(A_0(n,p)-A_1(n,p))=r(\\hat{0},p).", "$ Denote, for each partition $y$ of $\\lbrace 1,2,\\dots ,n\\rbrace $ , $s(y,p)=\\sum _{x\\geqslant y} r(x,p).", "$ Then, by Proposition REF , $s(y,p)=(-1)^{c(y)}$ where $c(y)$ is the number of blocks in the partition $y$ .", "By the Möbius inversion formula (see [12]), $r(\\hat{0},p)=\\sum _{y\\in \\Pi _n} \\mu (\\hat{0},y)s(y,p)=\\sum _{y\\in \\Pi _n}(-1)^{c(y)}\\mu (\\hat{0},y) .$ If we compute this sum, we get the value of $A_0(n,p)-A_1(n,p)$ in a way that does not depend on $p$ .", "The last move consists in proving that $\\sum _{y\\in \\Pi _n}(-1)^{c(y)}\\mu (\\hat{0},y)=(-1)^nn!$ in a way that uses the equivalence with Lemma REF .", "This proof may seem to be artificial because it is no longer used in the Section , and a purely combinatorial and more general proof exists: see the final formula of Chapter $3.10.4$ of [17].", "Remark that $A_0(n,p)=A_0(n,p-1-n)$ since saying that the sum of some subset of $\\mathbb {F}_p^\\times $ is 0 is equivalent to saying that the sum of its complement is 0.", "For the same kind of reason, $A_1(n,p)=A_{-1}(n,p-1-n)=A_1(n,p-1-n)$ .", "Now we can prove Lemma REF by induction on $n$ .", "It has already been proved for small values of $n$ .", "If $n>4$ , by Bertrand's postulate, there is a prime number $p^{\\prime }$ such that $n<p^{\\prime }<2n$ .", "Replace $p$ by $p^{\\prime }$ (by the proposition REF this leads to an equivalent statement), then $n$ by $p^{\\prime }-1-n$ (using the above remark).", "As $p^{\\prime }-1-n<n$ , the step of induction is done.", "This proof can be analysed from the following point of view: how fast does the number of steps of induction grow as function of $n$ ?", "Suppose that one step of induction reduces Lemma REF for $n$ to Lemma REF for the number $f(n)$ and denote by $R(n)$ the number of steps of induction needed to reach one of the numbers 0 or 1 (the formal definitions will follow).", "We can prove then the following upper bound on $R(n)$ .", "Theorem 3.1 Let $\\mathrm {nextprime}(n):=\\min \\lbrace p>n \\mid p\\text{ \\rm prime}\\rbrace $ and $f(n):=\\mathrm {nextprime}(n)-n-1$ for each $n\\in \\mathbb {N}$ .", "Further, denote $R(n):=\\min \\lbrace k \\mid f^k(n)\\in \\lbrace 0,1\\rbrace \\rbrace .$ This definition makes sense, for $f(n)<n$ for each $n>1$ by the Bertrand's postulate.", "The function $R(n)$ satisfies the estimation $R(n)=O(\\log \\log n).$ Denote $\\theta =0.525$ .", "By Theorem 1 of [3], there is a constant $N_0$ such that for all $n>N_0$ , the interval $[n-n^{\\theta },n]$ contains a prime number.", "We are going to deduce from this the following result: for each $\\bar{\\theta }\\in ]0.525,1[$ there exists a constant $N_1$ such that $n>N_1$ implies $f(n)<n^{\\bar{\\theta }}$ .", "Indeed, suppose $n>N_0$ and denote $\\bar{p}=\\mathrm {nextprime}(n)-1$ .", "Then, by the result cited above, $n\\geqslant \\bar{p}-\\bar{p}^\\theta .$ The function $\\begin{array}{rrcl}u:&[N_0,+\\infty [&\\rightarrow &[u(N_0),+\\infty [\\\\&x&\\mapsto &x-x^\\theta \\end{array}$ is strictly increasing, continuous and equivalent to $x$ .", "Therefore, the same is valid for its inverse $u^{-1}$ .", "By (REF ), $\\bar{p}\\leqslant u^{-1}(n)$ , therefore $\\forall n>N_1\\ f(n)=\\bar{p}-n\\leqslant \\bar{p}^\\theta \\leqslant (u^{-1}(n))^\\theta <n^{\\bar{\\theta }}$ for each $\\bar{\\theta }\\in ]\\theta ,1[$ and for a bound $N_1\\geqslant N_0$ that may depend on $\\bar{\\theta }$ .", "The end of the proof is analogous to that of Theorem $1.1$ of [13].", "Denote by $l$ the integer such that $f^{l+1}(n)<N_1\\leqslant f^{l}(n)$ .", "Then: $n^{{\\bar{\\theta }}^l}\\geqslant N_0$ therefore $l\\log {\\bar{\\theta }}+\\log \\log n\\geqslant \\log \\log N_1$ which implies $l\\leqslant -\\frac{\\log \\log n}{\\log {\\bar{\\theta }}}.$ Put $b=\\max _{1\\leqslant m\\leqslant N_0}R(m)$ , it is a constant.", "We get: $R(n)\\leqslant l+1+b\\leqslant -\\frac{\\log \\log n}{\\log {\\bar{\\theta }}}+1+b$ which proves our claim." ], [ "Pascal's equation.", "We are going to prove the functional equation satisfied by the coefficients of the polynomial $\\mathcal {N}_{p,i_1,i_2}(Y_0,Y_1,Y_2)$ (introduced in (REF )).", "To do this, we are going to describe a combinatorial interpretation of these numbers.", "Definition 3 Let $p,i_1,i_2$ be fixed as in Introduction and $n_1,n_2$ be nonnegative integers such that $n_1+n_2\\leqslant p-1$ .", "Define $C^{i_1,i_2}_i(n_1,n_2,p)&=\\\\&\\#\\left\\lbrace (x_1,\\dots ,x_{n_1+n_2})\\in {\\mathbb {F}_p^\\times }^{n_1+n_2}\\left|\\begin{array}{@{}c@{}}x_k\\ne x_l\\text{ if }k\\ne l,\\\\[2pt]\\displaystyle i_1\\sum _{k=1}^{n_1}x_k+i_2{\\sum _{k=n_1+1}^{n_1+n_2}}x_k=i\\end{array}\\right.\\right\\rbrace \\\\\\multicolumn{2}{l}{\\text{and}}\\\\A^{i_1,i_2}_i(n_1,n_2,p)&=\\#\\left\\lbrace (X_1,X_2)\\in \\mathcal {P}(\\mathbb {F}_p^\\times )^2\\ \\left|\\begin{array}{@{}c@{}}|X_1|=n_1,|X_2|=n_2,\\\\ X_1\\cap X_2=\\emptyset ,\\\\ i_1\\sum X_1+i_2\\sum X_2=i\\end{array} \\right.\\right\\rbrace .$ Definition REF matches with the notations from the previous section because of the identity $A^{i_1,i_2}_i(n,0,p)=A^{i_1,i_2}_i(0,n,p)=A_i(n,p)$ (independently from $i_1,i_2$ ).", "One can also see that the answer to Problem REF is $(p{-}1{-}n_1{-}n_2)!C^{i_1,i_2}_i(n_1,n_2,p)$ where $n_1$ (resp., $n_2$ ) is the number of coordinates of the vector $\\mathbf {f}$ equal to $i_1$ (resp., $i_2$ ).", "From this definition one can see that $C^{i_1,i_2}_1(n_1,n_2,p)=\\dots =C^{i_1,i_2}_{p-1}(n_1,n_2,p),\\\\\\sum _{i=0}^{p-1}C^{i_1,i_2}_i(n_1,n_2,p)=(p-1)\\dots (p-n_1-n_2),$ and for any $i$ , $A^{i_1,i_2}_i(n_1,n_2,p)=\\frac{C^{i_1,i_2}_i(n_1,n_2,p)}{n_1!n_2!", "}$ .", "Only one linear equation should be added to these in order to be able to determine all the numbers defined by (REF ) and ().", "Proposition REF below suggests to research the value of $\\triangle ^{i_1,i_2}(n_1,n_2,p)=A^{i_1,i_2}_0(n_1,n_2,p)-A^{i_1,i_2}_1(n_1,n_2,p)\\\\=\\sum _{\\begin{array}{c}X_1,X_2\\subset \\mathbb {F}_p^\\times \\\\|X_1|=n_1, |X_2|=n_2,\\\\X_1\\cap X_2=\\emptyset \\end{array}}\\zeta _p^{i_1\\sum X_1+i_2\\sum X_2}.$ We can express the symmetric polynomials of the quantities $(Y_0+\\zeta _p^{i_1j}Y_1+\\zeta _p^{i_2j}Y_2)$ in terms of the previously defined numbers via the following.", "Proposition 4.1 Let $i_1,i_2$ be two different elements of $\\mathbb {F}_p^\\times $ and denote by $\\sigma _{v,(j=1,\\dots ,p-1)}$ the elementary symmetric polynomial of degree $v$ in quantities that depend on an index $j$ varying from 1 to $p-1$ .", "Then we have the following formal expansion: $\\sigma _{p-1-\\delta ,(j=1,\\dots ,p-1)}\\left(Y_0+\\zeta _p^{i_1j}Y_1+\\zeta _p^{i_2j}Y_2\\right)=\\\\\\sum _{\\begin{array}{c}0\\leqslant n_0,n_1,n_2\\leqslant p-1 \\\\n_0+n_1+n_2=p-1\\\\n_0\\geqslant \\delta \\end{array}}\\genfrac(){0.0pt}{}{n_0}{\\delta }\\triangle ^{i_1,i_2}(n_1,n_2,p)Y_0^{n_0-\\delta }Y_1^{n_1}Y_2^{n_2}.$ In particular, $\\mathcal {N}_{p,i_1,i_2}(Y_0,Y_1,Y_2)=\\\\\\sum _{\\begin{array}{c}0\\leqslant n_0,n_1,n_2\\leqslant p-1 \\\\n_0+n_1+n_2=p-1 \\end{array}}\\triangle ^{i_1,i_2}(n_1,n_2,p)Y_0^{n_0}Y_1^{n_1}Y_2^{n_2}.$ The symmetric polynomial develops as: $\\sigma _{p-1-\\delta ,(j=1,\\dots ,p-1)}\\left(Y_0+\\zeta _p^{i_1j}Y_1+\\zeta _p^{i_2j}Y_2\\right)\\\\=\\sum _{\\begin{array}{c}X\\subset \\mathbb {F}_p^\\times ,\\\\|X|=p{-}1{-}\\delta \\end{array}}\\prod _{j\\in X}\\left(Y_0+\\zeta _p^{i_1j}Y_1+\\zeta _p^{i_2j}Y_2\\right)\\\\=\\sum _{\\begin{array}{c}X\\subset \\mathbb {F}_p^\\times ,\\\\|X|=p{-}1{-}\\delta \\end{array}}\\sum _{\\begin{array}{c}X_0,X_1,X_2,\\\\X_0\\cup X_1\\cup X_2=X,\\\\X_0,X_1,X_2\\text{ disjoint}\\end{array}}\\zeta _p^{i_1\\sum X_1+i_2\\sum X_2}Y_0^{|X_0|}Y_1^{|X_1|}Y_2^{|X_2|}\\\\=\\sum _{\\begin{array}{c}X^{\\prime }_0,X_1,X_2,\\\\X^{\\prime }_0\\cup X_1\\cup X_2=\\mathbb {F}_p^\\times \\\\X^{\\prime }_0,X_1,X_2\\text{ disjoint}\\end{array}}\\genfrac(){0.0pt}{}{|X^{\\prime }_0|}{\\delta }\\zeta _p^{i_1\\sum X_1+i_2\\sum X_2} Y_0^{|X^{\\prime }_0|-\\delta }Y_1^{X_1}Y_2^{X_2}.$ When we group the terms of this sum by sizes $n_0=|X^{\\prime }_0|,n_1=|X_1|,n_2=|X_2|$ we obtain (REF ).", "The method of proof of Lemma REF can be generalized into the following conditional closed formula for the coefficients $\\triangle ^{i_1,i_2}(n_1,n_2,p)$ : Proposition 4.2 Let $p$ be an odd prime, and $i_1,i_2\\in \\mathbb {F}_p^\\times $ , $n_1,n_2\\in \\lbrace 1,\\dots ,p-2\\rbrace $ such that $i_1\\ne i_2$ and $n_1+n_2<p$ .", "Suppose that the multiset consisting of $i_1$ with multiplicity $n_1$ and of $i_2$ with multiplicity $n_2$ should have no nonempty subset of sum multiple of $p$ .", "Then, $\\triangle ^{i_1,i_2}(n_1,n_2,p)=(-1)^{n_1+n_2}\\genfrac(){0.0pt}{}{n_1+n_2}{n_1}.$ Define $f_k=\\left\\lbrace \\begin{array}{cl}i_1&\\text{ if $k\\in \\lbrace 1,\\dots ,n_1\\rbrace $}\\\\i_2&\\text{ if $k\\in \\lbrace n_1+1,\\dots ,n_1+n_2\\rbrace $}\\end{array}\\right.$ and $n:=n_1+n_2 $ .", "Next, for each partition $x{\\in }\\Pi _n$ and each $i{\\in }\\mathbb {F}_p$ denote by $r^{i_1,i_2}_i(x,n_1,n_2,p)$ the number of sequences $(x_1,x_2,\\dots ,x_{n{=}n_1{+}n_2})$ of elements of $\\mathbb {F}_p^\\times $ of coimage $x$ such that $i_1\\sum _{k=1}^{n_1}x_k+i_2{\\sum _{k=n_1+1}^{n_1+n_2}}x_k=i.$ Let us also define $r^{i_1,i_2}(x,n_1,n_2,p)=r^{i_1,i_2}_0(x,n_1,n_2,p)-r^{i_1,i_2}_1(x,n_1,n_2,p)$ and $s^{i_1,i_2}_i(x,n_1,n_2,p)=\\sum _{x^{\\prime }\\geqslant x} r^{i_1,i_2}_i(x^{\\prime },n_1,n_2,p),\\\\s^{i_1,i_2}(x,n_1,n_2,p)=s^{i_1,i_2}_0(x,n_1,n_2,p)-s^{i_1,i_2}_1(x,n_1,n_2,p).$ By definition, $C^{i_1,i_2}_0(n_1,n_2,p)-C^{i_1,i_2}_1(n_1,n_2,p)=r^{i_1,i_2}(\\hat{0},n_1,n_2,p).$ Consider a partition $x\\in \\Pi _n.$ The number $s^{i_1,i_2}_i(x,n_1,n_2,p)$ (defined by the formula (REF )) admits an equivalent definition as the number of sequences $(x_1,x_2,\\dots ,x_n)$ of elements of $\\mathbb {F}_p^\\times $ of coimage greater than or equal to $x$ (in the sense of partitions) which satisfy (REF ).", "Denote by $B_1,\\dots ,B_{c(x)}$ the blocks of $x$ and $f_{B_j}=\\sum _{k\\in B_j}f_k (j=1,\\dots ,c(x)).", "$ Then, $s^{i_1,i_2}_i(x,n_1,n_2,p)$ is the number of sequences $(x_{B_1},\\dots ,x_{B_{c(x)}})$ of elements of $\\mathbb {F}_p^\\times $ (where the terms can be equal or distinct) such that $\\sum _{j=1}^{c(x)} f_{B_j} x_{B_j}=i.$ By the hypotheses of the Proposition, all $f_{B_j}$ are nonzero in $\\mathbb {F}_p$ .", "Therefore, by Proposition REF , $s^{i_1,i_2}(x,n_1,n_2,p)=(-1)^{c(x)}.$ By the Möbius inversion formula and the formula (REF ), $r(\\hat{0},,n_1,n_2,p)=\\sum _{y\\in \\Pi _n}(-1)^{c(y)}\\mu (\\hat{0},y)=(-1)^nn!.$ Therefore, $\\triangle ^{i_1,i_2}(n_1,n_2,p)=\\frac{1}{n_1!n_2!", "}(C^{i_1,i_2}_0(n_1,n_2,p)-C^{i_1,i_2}_1(n_1,n_2,p))$ which concludes the proof.", "The condition of Proposition REF holds, for example, if the smallest positive representatives of $i_1$ and $i_2$ verify $n_1i_1+n_2i_2<p$ .", "Without the condition formulated in Proposition REF , (REF ) becomes false: for example, $\\triangle ^{2,3}(1,1,5)=-3$ .", "For the general case, we are going to replace the closed formula by a recursive equation in which the parameters $i_1,i_2,p$ are fixed, and the recursion is on different values of $n_1,n_2$ .", "The equation is similar to the equation of the Pascal's triangle, and can be formulated as follows: Theorem 4.3 (“Colored\" Pascal's equation) Let $p$ be an odd prime, and $i_1,i_2\\in \\mathbb {F}_p^\\times $ , $n_1,n_2\\in \\lbrace 1,\\dots ,p-2\\rbrace $ such that $i_1\\ne i_2$ and $n_1+n_2<p$ .", "Then, $C^{i_1,i_2}_0(n_1,n_2,p)-C^{i_1,i_2}_1(n_1,n_2,p)\\equiv \\\\ n_1C^{i_1,i_2}_1(n_1-1,n_2,p)+n_2C^{i_1,i_2}_1(n_1,n_2-1,p)-\\\\-n_1C^{i_1,i_2}_0(n_1-1,n_2,p)-n_2C^{i_1,i_2}_0(n_1,n_2-1,p)\\text{ \\rm mod }p$ and if $p\\nmid n_1i_1+n_2i_2$ , the equality $C^{i_1,i_2}_0(n_1,n_2,p)-C^{i_1,i_2}_1(n_1,n_2,p)=\\\\n_1C^{i_1,i_2}_1(n_1-1,n_2,p)+n_2C^{i_1,i_2}_1(n_1,n_2-1,p)-\\\\-n_1C^{i_1,i_2}_0(n_1-1,n_2,p)-n_2C^{i_1,i_2}_0(n_1,n_2-1,p)$ holds.", "We are going to use the notations of the beginning of the previous proof until the formula (REF ).", "We are also going to call a hindrance a subset $X$ of $\\lbrace 1,\\dots ,n_1+n_2\\rbrace $ such that $\\sum _{m\\in X}f_m\\equiv 0\\text{ mod }p$ .", "Proposition REF corresponds to the case when there are no hindrances.", "Then $C^{i_1,i_2}_0(n_1,n_2,p)-C^{i_1,i_2}_1(n_1,n_2,p)=(-1)^{n_1+n_2}(n_1+n_2)!", "$ and this number is the opposite of $n$ times $(-1)^{n-1}(n-1)!$ .", "In general, the formula (REF ) should be replaced by: $s^{i_1,i_2}(y,n_1,n_2,p)=(1-p)^{d(y)}(-1)^{c(y)}$ if the partition $y$ of $\\lbrace 1,\\dots ,n_1+n_2\\rbrace $ contains $d(y)$ blocks that are hindrances.", "We should, indeed, count the solutions of the congruences (REF ) for $i=0,1$ (in nonzero residues modulo $p$ ) and evaluate the difference.", "Proposition REF states that if we pay no attention to the indices $j$ that correspond to hindrances (i.e., such that $f_{B_j}=0$ ), the difference between numbers of solutions of $\\sum _{j} f_{B_j}x_{B_j}=0$ and $\\sum _{j} f_{B_j}x_{B_j}=1$ is $(-1)^{c-d(y)}$ .", "Moreover, the values of $x_{B_j}$ where $B_j$ are hindrances can be chosen arbitrarily (from $p-1$ options each).", "The product of these contributions leads to (REF ).", "The formula (REF ) can be rewritten as $s^{i_1,i_2}(y,n_1,n_2,p)=\\sum _{l=0}^{d(y)}\\sum _{\\begin{array}{c}X_1,X_2,\\dots ,X_l \\\\\\text{hindrances contained in $y$}\\end{array}}(-1)^{c(y)-l}p^l $ where the order of $X_1,X_2,\\dots ,X_l$ is irrelevant in the sum.", "Then we get: $&C^{i_1,i_2}_0(n_1,n_2,p)-C^{i_1,i_2}_1(n_1,n_2,p)=\\sum _{y\\in \\Pi _n} \\mu (\\hat{0},y)s^{i_1,i_2}(y,n_1,n_2,p)\\\\&=\\sum _{\\begin{array}{c}X_1,X_2,\\dots ,X_l \\\\\\text{disjoint hindrances}\\end{array}}\\sum _{\\begin{array}{c}y\\in \\Pi _n \\\\\\text{$y$ contains $X_1,\\dots ,X_l$}\\\\\\text{as blocks}\\end{array}}(-1)^{c(y)-l}\\mu (\\hat{0},y)p^l\\\\&=\\sum _{X_1,\\dots ,X_l} (-1)^{|X_1|+|X_2|+\\dots +|X_l|-l}(|X_1|-1)!", "(|X_2|-1)!\\dots (|X_l|-1)!p^l\\\\&\\times [r]{\\sum _{\\begin{array}{c}y\\in \\Pi _n \\\\\\text{$y$ contains $X_1,\\dots ,X_l$}\\end{array}}}\\hspace{10.0pt}\\mu (\\hat{0},y{-}X_1{-}X_2{-}\\dots {-}X_l)(-1)^{c(y-X_1-X_2-\\dots -X_l)}$ by factoring $\\mu (\\hat{0},y)$ according to the formula (REF ).", "In the last sum, $(y-X_1-X_2-\\dots -X_l)$ denotes the partition $y$ , where the blocks $X_1,\\dots ,X_l$ are removed (which is a partition of $(n_1+n_2-|X_1|-\\dots -|X_l|)$ elements).", "By applying (REF ) to the last sum of (), we get $C^{i_1,i_2}_0(n_1,n_2,p)-C^{i_1,i_2}_1(n_1,n_2,p)=\\\\\\sum _{X_1,\\dots ,X_l} (|X_1|{-}1)!", "(|X_2|{-}1)!\\dots (|X_l|{-}1)!", "(-1)^{n_1+n_2-l}\\\\p^l(n_1{+}n_2{-}|X_1|{-}\\dots {-}|X_l|)!.$ From (REF ), $C^{i_1,i_2}_0(n_1,n_2,p)-C^{i_1,i_2}_1(n_1,n_2,p)\\equiv (-1)^{n_1+n_2}(n_1+n_2)!\\text{ \\rm mod }p,$ which implies (REF ).", "Suppose that $\\lbrace 1,\\dots ,n_1+n_2\\rbrace $ is not a hindrance.", "In order to prove (REF ), remark that the sum (REF ) can be split as $\\sum _{X_1,\\dots ,X_l} (|X_1|{-}1)!", "(|X_2|{-}1)!\\dots (|X_l|{-}1)!", "(-1)^{n_1+n_2-l}\\\\p^l(n_1{+}n_2{-}|X_1|{-}\\dots {-}|X_l|)!= \\\\\\multicolumn{1}{l}{-\\sum _{m=1}^{n_1+n_2} \\sum _{\\begin{array}{c}X_1,X_2,\\dots ,X_l \\\\\\text{disjoint hindrances}\\\\\\text{not containing $m$}\\end{array}} (|X_1|{-}1)!", "(|X_2|{-}1)!\\dots (|X_l|{-}1)!", "}\\\\(-1)^{n_1+n_2-l-1}p^l(n_1{+}n_2{-}|X_1|{-}\\dots {-}|X_l|{-}1)!$ then gathered into two parts according to the values of $f_m$ : $C^{i_1,i_2}_0(n_1,n_2,p)-C^{i_1,i_2}_1(n_1,n_2,p)= \\\\-n_1\\sum _{\\begin{array}{c}X_1,X_2,\\dots ,X_l \\\\\\text{disjoint hindrances}\\\\\\text{not containing $1$}\\end{array}} (|X_1|{-}1)!", "(|X_2|{-}1)!\\dots (|X_l|{-}1)!\\\\(-1)^{n_1+n_2-l-1}p^l(n_1{+}n_2{-}|X_1|{-}\\dots {-}|X_l|{-}1)!\\\\-n_2\\sum _{\\begin{array}{c}X_1,X_2,\\dots ,X_l \\\\\\text{disjoint hindrances}\\\\\\text{not containing $n_1+1$}\\end{array}} (|X_1|{-}1)!", "(|X_2|{-}1)!\\dots (|X_l|{-}1)!\\\\(-1)^{n_1+n_2-l-1}p^l(n_1{+}n_2{-}|X_1|{-}\\dots {-}|X_l|{-}1)!$ By identifying each sum in the last formula to the right-hand side of (REF ) with one of the arguments $n_1$ or $n_2$ decreased by 1, we get (REF ).", "The numbers $\\triangle ^{i_1,i_2}(n_1,n_2,p)$ satisfy a similar equation.", "Theorem 4.4 (“Uncolored\" Pascal's equation) Let $p$ be an odd prime, and $i_1,i_2\\in \\mathbb {F}_p^\\times $ , $n_1,n_2\\in \\lbrace 1,\\dots ,p-2\\rbrace $ such that $i_1\\ne i_2$ and $n_1+n_2<p$ .", "Then, $\\triangle ^{i_1,i_2}(n_1,n_2,p)\\equiv -\\triangle ^{i_1,i_2}(n_1-1,n_2,p)-\\triangle ^{i_1,i_2}(n_1,n_2-1,p)\\text{ \\rm mod }p$ and if $p\\nmid n_1i_1+n_2i_2$ , the equality $\\triangle ^{i_1,i_2}(n_1,n_2,p)=-\\triangle ^{i_1,i_2}(n_1-1,n_2,p)-\\triangle ^{i_1,i_2}(n_1,n_2-1,p)$ holds.", "Division of both sides of (REF ) by $n_1!n_2!$ (which is not multiple of $p$ ) gives (REF ) and division by the same number of (REF ) gives (REF ).", "Theorem REF follows directly from Proposition REF and Theorem REF ." ], [ "Algorithm.", "Let us define formally $\\triangle ^{i_1,i_2}(n_1,n_2)=0$ when one of $n_1$ , $n_2$ is negative or $n_1+n_2\\geqslant p$ .", "Then (REF ) is valid for any $n_1,n_2\\in \\mathbb {N}^2$ such that $p\\nmid n_1i_1+n_2i_2$ .", "Indeed: if $n_1=0$ or $n_2=0$ , the identification $A^{i_1,i_2}(n_1,n_2,p)=A(\\max (n_1,n_2),p)$ implies (REF ) via Lemma REF .", "When $n_1+n_2=p-1$ , one can use the hypothesis $X_1\\cup X_2=\\mathbb {F}_p^\\times $ and the identity $\\sum \\mathbb {F}_p^\\times =0$ to prove $i_1\\sum X_1+i_2\\sum X_2=(i_1-i_2)\\sum X_1, $ which implies $A^{i_1,i_2}_i(n_1,n_2,p)=A^{i_1-i_2,i_2}_i(n_1,0,p)$ , therefore $\\triangle ^{i_1,i_2}(n_1,n_2)=(-1)^{n_1}$ .", "The equation (REF ) is valid, therefore, when $n_1+n_2=p$ .", "We can now prove that the functional relation (REF ), together with these border values, characterizes the function $\\triangle ^{i_1,i_2}(\\cdot ,\\cdot ,p)$ as a function defined on $\\mathbb {Z}_{\\geqslant -1}^2$ , with values in $\\mathbb {Z}$ .", "Theorem 5.1 Let $p$ be an odd prime, let $i_1,i_2$ be two distinct elements of $\\lbrace 1,\\dots ,p-1\\rbrace $ , and let $d:\\mathbb {Z}_{\\geqslant -1}^2\\rightarrow \\mathbb {Z}$ be a function such that $&d(0,0)=1, \\\\&d(n_1,n_2)=0 \\text{ if } n_1{=}-1\\text{ or }n_2{=}-1\\text{ or }n_1+n_2\\geqslant p,\\\\&d(n_1,n_2)+d(n_1-1,n_2)+d(n_1,n_2-1)=0 \\text{ if } p\\nmid n_1i_1+n_2i_2.$ Then, $d(n_1,n_2)=\\triangle ^{i_1,i_2}(n_1,n_2,p)$ .", "Define $\\delta (n_1,n_2)=d(n_1,n_2)-\\triangle ^{i_1,i_2}(n_1,n_2,p)$ .", "Then the function $\\delta $ satisfies (), () and $\\delta (0,0)=0$ .", "In order to prove the theorem we should prove that $\\delta =0$ .", "By applying () successively to $n_2=0$ and $n_1=1,\\dots ,p-1$ one proves that $\\delta (0,0)=-\\delta (1,0)=\\delta (2,0)=\\dots =\\delta (p-1,0)$ .", "By applying it to $n_1=0$ and $n_2=1,\\dots ,p-1$ one proves that $\\delta (0,0)=-\\delta (0,1)=\\dots =\\delta (0,p-1)$ .", "Let us prove the identity $\\delta (n_1,n_2)=0$ by induction on $\\tilde{n}:=p-n_1-n_2\\in \\lbrace 0,\\dots ,p-2\\rbrace $ .", "If $\\tilde{n}=0$ , then $\\delta (n_1,n_2)=0$ as a part of the hypothesis ().", "Suppose that the Theorem is proved for $\\tilde{n}\\in \\lbrace 0,\\dots ,p-3\\rbrace $ , let us prove it for $\\tilde{n}+1$ .", "Denote $(n_1^S,n_2^S)$ the solution of $\\left\\lbrace \\begin{array}{l}i_1n_1^S+i_2n_2^S\\equiv 0\\text{ mod }p\\\\n_1^S+n_2^S=p-\\tilde{n}\\\\(n_1^S,n_2^S)\\in \\lbrace 1,\\dots ,p\\rbrace ^2.\\end{array}\\right.$ If one applies the functional relation () to a point where $n_2=p-\\tilde{n}-n_1$ (with the restriction $n_1\\ne n_1^S$ ), and uses the induction hypothesis, one gets $\\delta (n_1-1,p-\\tilde{n}-n_1)+\\delta (n_1,p-\\tilde{n}-n_1-1)=0.$ By applying (REF ) successively to $n_1=1,\\dots ,n_1^S-1$ , we prove $\\delta (n_1,p-\\tilde{n}-n_1-1)=0$ for $n_1$ in the same range $1,\\dots ,n_1^S-1$ .", "If $n_1^S\\geqslant p-\\tilde{n}-1$ , this concludes the step of induction.", "Otherwise, by applying (REF ) successively to $n_1=p-\\tilde{n}-1,\\dots ,n_1^S+1$ (in the decreasing order of values of $n_1$ ), we prove $\\delta (n_1,p-\\tilde{n}-n_1-1)=0$ for $n_1$ in the range $p-\\tilde{n},\\dots ,n_1^S$ .", "This concludes the induction and proves $\\delta (n_1,n_2)=0$ for all $(n_1,n_2)$ .", "The previous proof corresponds to the Algorithm REF , which computes the values of the function $\\triangle ^{a,b}(x,y,p)$ line by line.", "It executes one addition per number to compute, therefore its execution time is proportional to the size of the answer.", "Given an odd prime $p$ and two distinct elements $i_1,i_2$ of $\\mathbb {F}_p^\\times $ , we are going to call the array of all values of $\\triangle ^{i_1,i_2}(n_1,n_2,p)$ for $n_1,n_2\\geqslant 0, n_1+n_2<p$ a finite Pascal's triangle, and we will use geometrical terminology when it seems to make exposition simpler.", "We are going to call sources the points $(n_1,n_2)$ such that $p|i_1n_1+i_2n_2$ .", "Define $f^{i_1,i_2}(n_1,n_2,p)=\\triangle ^{i_1,i_2}(n_1,n_2,p)+\\triangle ^{i_1,i_2}(n_1-1,n_2,p)+\\triangle ^{i_1,i_2}(n_1,n_2-1,p).$ The value of $f^{i_1,i_2}(n_1,n_2,p)$ (which we will call force) is nonzero only at sources, where it can be computed using (REF ) combined with the end of the proof of Theorem REF : $n_1!n_2!f^{i_1,i_2}(n_1,n_2,p)=\\sum _{\\begin{array}{c}X_1,X_2,\\dots ,X_l\\\\\\text{partition of }\\lbrace 1,\\dots ,n_1+n_2\\rbrace ,\\\\\\forall j\\ p\\,\\mid \\,\\sum _{m\\in X_j}f_m\\end{array}}\\\\(|X_1|-1)!", "(|X_2|-1)!\\dots (|X_l|-1)!", "(-1)^{n_1+n_2-l}p^l(n_1+n_2-|X_1|-\\dots -|X_l|)!.$ This formula uses the notation (REF ) in order to describe the fact that summation goes through all partitions of $\\lbrace 1,\\dots ,n_1+n_2\\rbrace $ into hindrances.", "The definition (REF ) implies, by linearity of the Pascal's equation: $\\triangle ^{i_1,i_2}(n_1,n_2,p)=\\sum _{\\begin{array}{c}0{\\leqslant } k{\\leqslant } n_1\\\\ 0{\\leqslant } l{\\leqslant } n_2 \\\\ p\\,\\mid \\,i_1k{+}i_2l\\end{array}} f^{i_1,i_2}(k,l,p)(-1)^{n_1+n_2-k-l}\\genfrac(){0.0pt}{}{n_1{+}n_2{-}k{-}l}{n_1-k}.$ Calculate a finite Pascal's triangle.", "Arguments $p,a,b$ : $p$ prime, $0<a<b<p$ Allocate the integer array data[$0..p-1$ ][$0..p-1$ ] (values of $\\triangle ^{a,b}(x,y,p)$ ), the boolean array reg[$0..p{-}1$ ][$0..p{-}1$ ] (information about sources) $x=0,..,p-1,y=0,..,p-1$ reg$[x][y]= (a\\cdot x+b\\cdot y\\lnot \\equiv 0\\mbox{ mod }p)$ data$[0][0]=$ data$[p-1][0]=$ data$[0][p-1]=1$    resolution at the edges for $x=1,\\dots ,p{-}2$ do data$[x][p{-}1{-}x]=-$ data$[x{-}1][p{-}x]$ end for for $x=1,\\dots ,p{-}2$ do data$[x][0]=-$ data$[x{-}1][0]$ end for for $y=1,\\dots ,p{-}2$ do data$[0][y]=-$ data$[0][y{-}1]$ end for    resolution inside $n=p-2,..,1$ $x=1,..,n-1$ $y\\leftarrow n-x $ $\\textrm {reg}[x][y+1]$ $\\textrm {data}[x][y]=-\\textrm {data}[x-1][y+1]-\\textrm {data}[x][y+1]$ Stop the inner loop $y=1,..,n-1$ $x\\leftarrow n-y $ $\\textrm {reg}[x+1][y]$ $\\textrm {data}[x][y]=-\\textrm {data}[x+1][y-1]-\\textrm {data}[x+1][y]$ Stop the inner loop    Print the result $n=0,..,p-1$ $y=0,..,n$ Print data$[n-y][y]$ , reg$[n-y][y]$ Print newline" ], [ "The case $i_1=1,i_2=2$ .", "We can find a closed formula for the numbers $\\triangle ^{1,2}(n_1,n_2,p)$ using the identity $ \\triangle ^{1,2}(n_1,n_2,p)=\\triangle ^{1,2}(n_1,p-1-n_1-n_2,p).$ It follows indeed from the fact that for each disjoint couple $X_1,X_2\\subset \\mathbb {F}_p^\\times $ , as in the definition (), $\\sum X_1+2\\sum X_2=-\\left(\\sum X_1+2\\sum (\\mathbb {F}_p^\\times \\setminus X_1\\setminus X_2)\\right).$ Formula (REF ) applies to at least one side of (REF ) for each $(n_1,n_2)$ (and to both sides of (REF ) if $n_1+2n_2=p-1$ ), leading to $\\triangle ^{1,2}(n_1,n_2,p)=\\left\\lbrace \\begin{array}{cl}(-1)^{n_1+n_2}\\genfrac(){0.0pt}{}{n_1+n_2}{n_1}&\\text{ if }n_1+2n_2\\leqslant p-1\\\\(-1)^{n_2}\\genfrac(){0.0pt}{}{p-1-n_2}{n_1}&\\text{ if }n_1+2n_2\\geqslant p-1.\\end{array}\\right.$ Therefore, this Pascal's triangle is symmetric with respect to the axis $n_1+2n_2=p-1$ .", "Figure: Coefficients of ∏ j=1 10 X+ζ 11 j Y+ζ 11 2j Z\\prod _{j=1}^{10}\\left(X+\\zeta _{11}^{j}Y+\\zeta _{11}^{2j}Z\\right)One can deduce (REF ) from (REF ) in the following way: by (REF ), $\\prod _{j=1}^{j=p-1}\\left(1+\\zeta _p-\\zeta _p^2\\right)=\\sum _{n_1,n_2\\in \\mathbb {N}}(-1)^{n_2}\\triangle ^{1,2}(n_1,n_2,p)\\\\=\\sum _{n_1,n_2\\in \\mathbb {N}}(-1)^{n_2-1}(\\triangle ^{1,2}(n_1{-}1,n_2,p)+\\triangle ^{1,2}(n_1,n_2{-}1,p)-f^{1,2}(n_1,n_2,p))\\\\=\\sum _{n_1,n_2\\in \\mathbb {N}}((-1)^{n_2-1}\\triangle ^{1,2}(n_1,n_2-1,p) -(-1)^{n_2}\\triangle ^{1,2}(n_1-1,n_2,p)\\\\ + (-1)^{n_2}f^{1,2}(n_1,n_2,p))\\\\=\\sum _{n_1,n_2\\in \\mathbb {N}}(-1)^{n_2}f^{1,2}(n_1,n_2,p)$ because massive cancellation occurs in the sum of differences of values of the function $(-1)^y\\triangle ^{1,2}(x,y,p)$ .", "Suppose $n_1,n_2>0$ and $n_1+2n_2=p$ (therefore $n_1$ is odd).", "Then $f^{1,2}(n_1,n_2,p)=\\triangle ^{1,2}(n_1-1,n_2,p)+\\triangle ^{1,2}(n_1,n_2-1,p)+\\triangle ^{1,2}(n_1,n_2,p)\\\\=(-1)^{n_2}\\genfrac(){0.0pt}{}{n_1+n_2-1}{n_1-1}+2(-1)^{n_2}\\genfrac(){0.0pt}{}{n_1+n_2-1}{n_1}\\\\=(-1)^{n_2}\\left(\\genfrac(){0.0pt}{}{n_1+n_2}{n_1}+\\genfrac(){0.0pt}{}{n_1+n_2-1}{n_1} \\right)\\\\=(-1)^{n_2}\\left(\\genfrac(){0.0pt}{}{p-n_2}{n_2}+\\genfrac(){0.0pt}{}{p-n_2-1}{n_2-1} \\right).$ The absolute value of (REF ) can be interpreted as the number of ways to put $n_2$ identical disjoint dominoes on a discrete circle of length $p$ .", "Indeed (see also [4]), for any $k\\leqslant \\frac{p-1}{2}$ $\\#\\lbrace k\\text{ disjoint dominoes on a circle of length }p\\rbrace \\\\=\\#\\lbrace k\\text{ disjoint dominoes on a line segment of length }p\\rbrace \\\\+\\#\\lbrace k-1\\text{ disjoint dominoes on a line segment of length }p-2\\rbrace \\\\=\\genfrac(){0.0pt}{}{p-k}{k}+\\genfrac(){0.0pt}{}{p-k-1}{k-1}.$ The sum (REF ) contains three terms not covered by the hypotheses of (REF ): these correspond to $n_1{=}n_2{=}0$ , $n_1{=}p,n_2{=}0$ , $n_1{=}0,n_2{=}p$ and they equal respectively 1, 1 and $-1$ .", "The overall contribution of these terms can be identified to the number of ways to put 0 dominoes on a discrete circle of length $p$ .", "Therefore, the norm (REF ) equals to the number of ways to put any number of identical disjoint dominoes on a discrete circle of length $p$ , which is proved in [4] to be $L_p$ .", "For example, if $p=11$ , the numbers are those of Figure REF ($\\blacklozenge $ denotes a source)." ], [ "Application: an identity for binomial coefficients.", "The formulas (REF ) and (REF ) have another application.", "As $1-\\zeta _p+\\zeta _p^2=\\frac{1+\\zeta _p^3}{1+\\zeta _p}$ , we get in a similar way to (REF ): $1=\\prod _{j=1}^{j=p-1}\\left(1-\\zeta _p+\\zeta _p^2\\right)=\\sum _{n_1,n_2\\in \\mathbb {N}}(-1)^{n_1}\\triangle ^{1,2}(n_1,n_2,p)\\\\=\\sum _{n_1,n_2\\in \\mathbb {N}}(-1)^{n_1}f^{1,2}(n_1,n_2,p).$ We further get: $1=1+\\sum _{n_1,n_2\\in \\mathbb {N}^*}(-1)^{n_1}f^{1,2}(n_1,n_2,p)=1-\\sum _{n_1,n_2\\in \\mathbb {N}^*}f^{1,2}(n_1,n_2,p).$ The formula (REF ) leads to the following combinatorial identityThe previous proof implies (REF ) in the case of prime $p{\\geqslant }5$ .", "The Zeilberger's algorithm (implemented in Maple 17, see also Chapter 6 of the book [14]) generalizes it for any $p{\\geqslant }5$ congruent to 1 or 5 modulo 6: $\\sum _{k=1}^{\\frac{p-1}{2}}(-1)^k\\left(\\genfrac(){0.0pt}{}{p-k}{k}+\\genfrac(){0.0pt}{}{p-k-1}{k-1}\\right)=0.$" ], [ "Second application: expression for a symmetric polynomial.", "We can formulate an expression for an arbitrary symmetric polynomial of the numbers $(1+\\zeta _p^j-\\zeta _p^{2j})$ which is: Theorem 5.2 Let $p\\geqslant 5$ be prime and $\\delta \\in \\lbrace 0,\\dots ,p-2\\rbrace $ an integer.", "Then $\\sigma _{p-1-\\delta ,(j=1,\\dots ,p-1)}(1+\\zeta _p^j-\\zeta _p^{2j})$ (see the notation of Proposition REF ) equals $\\genfrac(){0.0pt}{}{p-1}{\\delta }$ plus the sum of “weights\" of ways of putting a number $n{>}0$ of disjoint dominoes on a discrete circle of length $p$ , the weights being $\\genfrac(){0.0pt}{}{n-1}{\\delta }$ .", "As a consequence, $\\sigma _{p-1-\\delta }(1+\\zeta -\\zeta ^2)>0$ and $\\sigma _{p-1-\\delta }(1+\\zeta -\\zeta ^2)\\equiv \\genfrac(){0.0pt}{}{p-1}{\\delta }\\text{ \\rm mod }p.$ By Proposition REF , we get a similar expression to (REF ) $\\sigma _{p-1-\\delta ,(j=1,\\dots ,p-1)}\\left(1+\\zeta _p^{j}+\\zeta _p^{2j}\\right)\\\\=\\sum _{n_1,n_2\\in \\mathbb {N}}(-1)^{n_2}\\genfrac(){0.0pt}{}{p-1-n_1-n_2}{\\delta }\\triangle ^{1,2}(n_1,n_2,p)\\\\=\\sum _{\\tilde{n}=1}^p\\sum _{\\begin{array}{c}n_1,n_2\\\\n_1+n_2=p-\\tilde{n}\\end{array}}(-1)^{n_2}\\genfrac(){0.0pt}{}{\\tilde{n}-1}{\\delta }\\\\(-\\triangle ^{1,2}(n_1-1,n_2,p)-\\triangle ^{1,2}(n_1,n_2-1,p)+f^{1,2}(n_1,n_2,p))\\\\=\\sum _{\\tilde{n}=1}^p\\sum _{\\begin{array}{c}n_1,n_2\\\\n_1+n_2=p-\\tilde{n}\\end{array}}(-1)^{n_2}\\genfrac(){0.0pt}{}{\\tilde{n}-1}{\\delta }f^{1,2}(n_1,n_2,p).$ The identity (REF ) leads to $\\sigma _{p-1-\\delta ,(j=1,\\dots ,p-1)}\\left(1+\\zeta _p^{j}+\\zeta _p^{2j}\\right)\\\\=\\genfrac(){0.0pt}{}{p-1}{\\delta }+\\sum _{n_2=1}^{\\frac{p-1}{2}}\\genfrac(){0.0pt}{}{n_2-1}{\\delta }\\left(\\genfrac(){0.0pt}{}{p-n_2}{n_2}+\\genfrac(){0.0pt}{}{p-n_2-1}{n_2-1}\\right)$ and the discussion that follows the formula (REF ) identifies each number $(-1)^{n_2}f^{1,2}(n_1,n_2,p)$ as the number of ways to put $n_2$ disjoint dominoes on a discrete circle of length $p$ ." ], [ "The case $i_1=1,i_2=3$ .", "In this case the formula $\\triangle ^{1,3}(n_1,n_2,p)=\\triangle ^{2,3}(n_1,p-1-n_1-n_2,p)$ is analogous to (REF ) and implies $\\triangle ^{1,3}(n_1,n_2,p)=\\left\\lbrace \\begin{array}{@{}c@{}l@{}l}(-1)^{n_1+n_2}\\genfrac(){0.0pt}{}{n_1+n_2}{n_1}&\\text{ if }&n_1+3n_2\\leqslant p-1\\\\(-1)^{n_2}\\genfrac(){0.0pt}{}{p-1-n_2}{n_1}&\\text{ if }&n_1+3n_2\\geqslant 2p-2\\text{ or }\\\\&&n_1+3n_2=2p-4,\\end{array}\\right.$ therefore, in two regions, the coefficients of the triangle are identical to the previous case.", "The coefficients in the middle region can be calculated using the general formula (REF ).", "Let us specify different quantities used there, namely the position of sources and the associated forces.", "The sources are the integer points situated on two lines: the upper line with equation $n_1+3n_2=p$ and the lower line with equation $n_1+3n_2=2p$ .", "One can see that the number of integer points on the upper line of sources is $\\#\\left\\lbrace \\begin{array}{c}0<n_1<p\\\\0<n_2<p\\\\n_1+3n_2=p \\end{array}\\right\\rbrace =\\left\\lfloor \\frac{p}{3}\\right\\rfloor $ and the number of integer points on the lower line is $\\#\\left\\lbrace \\begin{array}{c}0<n_1<p\\\\0<n_2<p\\\\n_1+3n_2=2p \\end{array}\\right\\rbrace =\\textrm {rnd}(\\frac{p}{6}),$ the closest integer to $\\frac{p}{6}$ .", "If $(n_1,n_2)$ is a point on the upper line of sources, the value of $f^{1,3}(n_1,n_2,p)$ has a simple expression given by (REF ): $f^{1,3}(n_1,n_2,p)=\\frac{(n_1+n_2-1)!p}{n_1!n_2!", "}$ because the sum consists of the single term associated to ${X{=}\\lbrace 1,\\dots ,n_1{+}n_2\\rbrace }$ .", "Under the same hypotheses, (REF ) implies $\\triangle ^{1,3}(n_1,n_2,p)=\\frac{(n_1+n_2-1)!p}{n_1!n_2!", "}-\\genfrac(){0.0pt}{}{n_1+n_2}{n_1}=2\\genfrac(){0.0pt}{}{n_1{+}n_2{-}1}{n_1}.$ In any point $(n_1,n_2)$ such that $p\\leqslant n_1+3n_2<2p$ , the formula (REF ) takes the following form: $\\triangle ^{1,3}(n_1,n_2,p)=(-1)^{n_1+n_2}\\genfrac(){0.0pt}{}{n_1+n_2}{n_1}\\\\+\\sum _{\\begin{array}{c}0{<} k{\\leqslant } n_1\\\\ 0{<}l{\\leqslant } n_2 \\\\ p=i_1k{+}i_2l\\end{array}} f^{1,3}(k,l,p)(-1)^{n_1+n_2-k-l}\\genfrac(){0.0pt}{}{n_1{+}n_2{-}k{-}l}{n_1-k}.$ We can also compute a simple expression for the forces of sources on the lower line.", "Suppose that $n_1,n_2>0$ and $n_1+3n_2=2p$ .", "Then, by (REF ) and (REF ), $f^{1,3}(n_1,n_2,p)=\\triangle ^{1,3}(n_1,n_2,p)+\\triangle ^{1,3}(n_1-1,n_2,p)+\\triangle ^{1,3}(n_1,n_2-1,p)\\\\=f^{2,3}(n_1,p-n_1-n_2,p).$ By (REF ) (the sum, once again, consists of a single term because $2n_1+3(p{-}n_1{-}n_2){=}p$ ), $f^{2,3}(n_1,p-n_1-n_2,p)=\\frac{(-1)^{n_2}(p-n_2-1)!p}{n_1!(p-n_1-n_2)!", "}.$ For example, if $p=11$ , the numbers are those of Figure REF ($\\blacklozenge $ denotes a source)." ] ]
1403.0542
[ [ "Comparing disease control policies for interacting wild populations" ], [ "Abstract We consider interacting population systems of predator-prey type, presenting four models of control strategies for epidemics among the prey.", "In particular to contain the transmissible disease, safety niches are considered, assuming they lessen the disease spread, but do not protect prey from predators.", "This represents a novelty with respect to standard ecosystems where the refuge prevents predators' attacks.", "The niche is assumed either to protect the healthy individuals, or to hinder the infected ones to get in contact with the susceptibles, or finally to reduce altogether contacts that might lead to new cases of the infection.", "In addition a standard culling procedure is also analysed.", "The effectiveness of the different strategies are compared.", "Probably the environments providing a place where disease carriers cannot come in contact with the healthy individuals, or where their contact rates are lowered, seem to preferable for disease containment." ], [ "Introduction", "In population models predator-prey and competition systems play a dominant role, since the blossoming of this discipline about a century ago.", "In more recent times, more refined models try to better describe reality.", "Since prey try to seek protection against attacks of their predators in the features of the environment, scientists have tried to incorporate this behavior into the interaction models.", "Early contributions in this respect can be found in [8], [4], [7].", "The introduction of refuges has lead to the observation that the Lotka-Volterra models gets stabilized [3] even to show global asymptotic stability, [1], [2].", "This shows the relevant role that spatial refuges exert in shaping the dynamics of predator-prey interplay.", "The refuge is expressed in the equations by reducing the amount of prey population available for hunting by the predators.", "In this classical setting, if $Y$ denotes the prey population that can take cover, by $Y_n$ we denote the number of individuals who find protection in the niches that are available for their safety.", "Thus there are only $Y-Y_n$ individuals that can interact with the predators.", "There could be several functional forms that can be chosen for $Y_n$ .", "The simplest one is a constant value, $Y_n=Y_0$ , with $Y_0\\in \\textbf {R}_+$ , or alternatively one could take a linear function of the prey population, $Y_n=Y_0 Y$ , [3] or also a linear function of the predators $X$ , $Y_n=Y_0 X$ [9].", "Ecoepidemiology investigates the influence of diseases in ecosystems, see Chapter 7 of [5].", "It appears therefore that the refuges for some of the populations involved can be introduced also in this context.", "However, instead of using the environmental niches as protection against the predators, i.e.", "as an ecological tool as described above, we employ them in order to investigate whether they can influence the disease spread, i.e.", "we give them an epidemiological meaning.", "Therefore, it is not against predators that prey are protected, but we rather consider the case in which the healthy prey for some reason due to the conformation of the environment can avoid to come in contact with disease-carriers of their own population and therefore be somewhat protected from the epidemics.", "This is achieved by reduced contact rates that they have with infected individuals.", "Of all the various possible types of niche, to keep things simple, we just take the constant case, $Y_n=Y_0$ .", "In the next Sections, we present three models for the refuges and one for another common disease-control method, namely culling, based on the ecoepidemic system presented in [10].", "The first three differ in the way the refuge is modeled.", "In Section 2, some of the susceptibles are prevented from interaction with infected individuals.", "In Section 3, it is part of the infected that are unable to become in contact with healthy individuals.", "In Section 4, we look at a reduced contact rate.", "Section 5 contains the analysis of the culling strategy.", "After a brief discussion of bistability of some equilibria, the final Section compares the findings." ], [ "The refuge for the healthy prey", "Consider at first the system in which the susceptibles are stronger and therefore able to reach places unattainable by the diseased individuals, because these indeed are weakened by the disease.", "Thus the infectious individuals cannot come in contact with the healthy remote individuals, and therefore cannot infect them.", "Let $s$ denote the fixed number of susceptibles that escape from the spread of the epidemics using the refuge.", "The model is formulated as follows.", "The healthy prey $R$ reproduce with net reproduction rate $a$ , are subject to intraspecific competition only with other sound individuals at rate $b$ and are hunted by predators at rate $c$ .", "Those that can be infected by the diseased prey individuals $U$ , as discussed above, leave their class at rate $\\lambda $ , to enter into the class of sick inviduals.", "The latter do not reproduce, are hunted at a rate $k\\ne c$ by the predators.", "Here $k>c$ means that they are weaker than sound ones, and therefore easier to capture, while $k<c$ instead takes into account the fact that they might be less palatable than the healthy ones.", "Finally, they can recover the disease at rate $\\omega $ and therefore reenter into the $R$ population.", "As mentioned above, infected are assumed not to contribute to intraspecific pressure, either of sound prey or among themselves; this again is grounded in the fact that their disease-related weakness prevents them to compete with the other individuals in the population.", "The predators are assumed to have also other food sources, for which they reproduce at rate $d$ , but clearly get a benefit from the interactions with the healthy prey expressed by the parameter $e<c$ .", "This constraint expresses the fact that the amount of food they get from the captured prey cannot exceed its mass.", "So far all the system parameters are nonnegative.", "For the predators hunting the infected prey, instead, we could model two different situations.", "For $h>0$ , the infected cause a damage to the predators, killing them.", "In this paper we concentrate only on this case.", "In the opposite case we could have the normal situation in which predators get a reward from capturing the diseased prey, so that in this situation we would have $0<-h<k$ .", "In summary, the ecoepidemic model with inclusion of a disease-safety niche for the susceptibles reads $\\frac{dR}{dt}&=&R[a-bR-cF] + \\omega U -\\lambda \\max \\lbrace 0,(R-s)\\rbrace U,\\\\ \\nonumber \\frac{dU}{dt}&=&\\lambda \\max \\lbrace 0,(R-s)\\rbrace U - U[ kF + \\omega ]- \\mu U , \\\\ \\nonumber \\frac{dF}{dt}&=&F[d+eR-fF-hU] .$ When $R<s$ , the last term in the first equation and the first one in the second equation vanish, the maximum function preventing them to provide positive and negative contributions to these equations respectively, which makes no sense biologically.", "It follows also that for $R<s$ the infected prey in the system disappear, since in the second equation the term on the right hand side is always negative.", "Thus the system settles to one of the equilibria of the classical disease-free predator-prey model, with logistic correction for the prey alternative food supply for the predators, see [10] for its brief analysis.", "For the benefit of the reader a short summary of its findings is presented also here at the top of Section ." ], [ "Equilibria", "The equilibria $P_k=\\left(R_k,U_k,R_k\\right)$ of (REF ) are $P_1=\\left(0,0,0\\right)$ , $P_2=\\left(0, 0,d f^{-1} \\right)$ , $P_3=\\left(a b^{-1}, 0, 0 \\right)$ , $P_4=\\left({af-cd \\over bf+ce}, 0, {ae+bd \\over bf+ce} \\right),\\quad P_5&=&\\left({\\lambda s+ \\omega + \\mu \\over \\lambda }, {(a-bR_5)R_5 \\over \\mu },0 \\right).$ The first three points are always feasible, $P_4$ is feasible for $af\\ge cd,$ and $P_5$ is whenever $a\\ge bR_5$ , i.e.", "for $b(\\lambda s+ \\omega + \\mu )\\le a\\lambda .$ Then there is coexistence $P_6=\\left(R_6, U_6, F_6\\right)$ .", "Its population values are obtained solving for $F$ and $U$ respectively the second and third equations in (REF ), taking obviously $R_6>s$ , thus giving $F_6=\\frac{1}{k} \\left[ \\lambda (R_6-s) - \\omega -\\mu \\right], \\quad U_6=\\frac{1}{h} \\left[ d+eR_6 - f F_6 \\right].$ Note that this makes sense only for $R_6>s$ , since otherwise the second equilibrium equation gives either $U_6=0$ or $F_6=-\\omega k^{-1}$ , but both results are in contrast with coexistence.", "Substituting into the first one, we obtain the quadratic equation $W(R)\\equiv \\sum _{k=0}^2 a_k R^k=0$ whose roots give the values of $R_6$ .", "Its coefficients have the following values $a_2 = \\frac{\\lambda }{h} \\left( \\frac{f}{k} \\lambda - e\\right) -b -\\frac{c}{k} \\lambda , \\quad a_0=\\frac{1}{hk} \\left( dk+fs\\lambda +f(\\omega +\\mu ) \\right) (s\\lambda +\\omega ),\\\\a_1=a+\\frac{c}{k} (s\\lambda +\\omega +\\mu ) +\\frac{1}{hk} [ (s\\lambda +\\omega ) (ek-f\\lambda ) - \\lambda (dk+fs\\lambda +f(\\omega + \\mu )) ].$ Now, since $a_0>0$ , if the parabola $W(R)$ is concave one positive root will exist.", "Thus a sufficient condition for the existence of $P_6$ is $a_2<0$ , i.e., explicitly, $f\\lambda ^2 < e\\lambda k+h[bk +c\\lambda ].$ For feasibility, we need also the other population values at a nonnegative level, a fact which is attained for $U_6$ if $ek>f\\lambda $ , else we must impose it $s < R_6 < \\frac{dk+fs\\lambda +f(\\omega +\\mu )}{f\\lambda -ek}$ as we do for $F_6$ to obtain $R_6 > s + \\frac{\\omega +\\mu }{\\lambda }.$" ], [ "Stability", "Denoting as usual by $H(x)$ the Heaviside function, $H(x)=1$ for $x>0$ , $H(x)=0$ for $x\\le 0$ , the Jacobian of (REF ) is $J=\\left[\\begin{array}{ccc}a-2bR-\\lambda H(R-s) U-cF & -\\lambda \\max \\lbrace 0,(R-s)\\rbrace +\\omega & -cR \\\\\\lambda H(R-s) U & \\lambda \\max \\lbrace 0,(R-s)\\rbrace -kF-\\omega - \\mu & -kU \\\\eF & -hF& J_{33}\\end{array}\\right],$ $J_{33}= d+eR-hU-2fF $ .", "The eigenvalues for $P_1$ are $-\\omega - \\mu $ , $d$ , $a$ , entailing its instability.", "Those for $P_2$ are $-(dk+f(\\omega + \\mu ))f^{-1}$ , $-d$ , $(af-cd)f^{-1}$ giving the stability condition $af<cd.$ Comparing this condition with (REF ), we observe that there is a transcritical bifurcation, for which $P_4$ emanates from $P_2$ when the latter becomes unstable.", "In other words, introducing the healthy prey invasion number $R^{(i)}\\equiv \\frac{af}{cd},$ we have that for $R^{(i)}>1$ the healthy prey establish themselves in the environment.", "For $P_3$ the eigenvalues are $(bd+ae)b^{-1}$ , $(\\lambda a-\\lambda s b-b(\\omega + \\mu ))b^{-1}$ , $-a$ , giving instability.", "Figure: The coexistence equilibrium is stably attained for the following choice of parameters:a=21a=21, b=0.3b=0.3, c=1c=1, d=1d=1, e=0.5e=0.5, f=0.9f=0.9, h=0.1h=0.1, k=10k=10, λ=10.2\\lambda =10.2, ω=0.8\\omega =0.8, μ=2.8\\mu =2.8, s=0.9s=0.9.At $P_4$ one eigenvalue is easily factored out, $\\frac{\\lambda (af-cd)-k(bd+ae)}{ce+bf}-\\lambda s-\\omega -\\mu ,$ while the remaining ones are roots of the quadratic equation $T(\\delta )=\\delta ^2+b_1\\delta +b_2=0,$ where letting $D=ce+bf$ , $b_1 & = & \\frac{t_1}{D},\\quad b_2=\\frac{t_3}{D},\\quad t_1=af(b+e)+bd(f-c)\\\\t_3 & = & (bd+ae)(af-cd),\\quad t_2=t_1^2+4t_3(bf+ce).$ Explicitly, $T_{1,2}=\\frac{-b_1\\pm \\sqrt{b_1^2-4b_2}}{2}=\\frac{t_1\\pm \\sqrt{t_2}}{2(ec+bf)}.$ By the feasibility condition (REF ), $t_3<0$ so that $t_2<t_1^2$ .", "Hence both roots of (REF ) have negative real part.", "Stability hinges then just on the first eigenvalue, i.e.", "$\\lambda R_4<kF_4+\\lambda s+\\omega +\\mu $ or explicitly the following condition $\\lambda \\frac{af-cd}{bf+ce}<k\\frac{ae+bd}{bf+ce}+\\lambda s+\\omega +\\mu .$ An eigenvalue of $ P_5$ is $d+e R_5-h U_5$ , the remaining ones are the roots of $T(\\theta ) = \\theta ^2-c_1 \\theta +c_2=0$ , with $c_1 = a-2b R_5-\\lambda U_5, \\quad c_2 = \\mu \\lambda U_5>0.$ Explicitly, $T_{1,2}=\\frac{c_1\\pm \\sqrt{c_1^2-4c_2}}{2}.$ By Descartes' rule of signs, both have negative real part if $a<2b R_5+\\lambda U_5$ .", "But this inequality always holds, since $&&\\frac{a-2b R_5}{\\lambda }=\\frac{a\\lambda - 2b (\\lambda s + \\omega +\\mu )}{\\lambda ^2} =\\\\&&=\\frac{\\mu U_5}{\\lambda s + \\omega +\\mu }- \\frac{b(\\lambda s + \\omega +\\mu )}{\\lambda ^2}< U_5-\\frac{b(\\lambda s + \\omega +\\mu )}{\\lambda ^2}<U_5.$ Stability then hinges only on the first eigenvalue $U_5> \\frac{d+e R_5 }{h}.$ For the coexistence equilibrium $P_6$ , we have run some simulations to show not only that it satisfies the feasibility conditions (REF ) and (REF ), but that it can be attained at a stable level.", "Figure REF shows one such instance, for the hypothetical parameter values $s=0.9$ and $a=21, \\quad b=0.3, \\quad c=1, \\quad d=1, \\quad e=0.5, \\quad f=0.9, \\\\ \\nonumber h=0.1, \\quad k=10, \\quad \\lambda =10.2, \\quad \\mu =2.8, \\quad \\omega =0.8.$ Here the $R_6$ equilibrium value is much higher than the number of individuals $s$ that can take cover in the safety niche.", "Observe also that the same inequality holds also for all the healthy prey population values before attaining the equilibrium level." ], [ "The cover for the infected", "Assume now that part of the infected are somehow confined in an environment in which healthy prey cannot enter.", "In this way the contagion risk is reduced.", "Let $p$ denote the fixed number of infected that inhabit the unreacheable territory.", "With the remaining notation similar to model (REF ), the system in our present case reads $\\frac{dR}{dt}&=&R[a-bR-cF-\\lambda \\max \\lbrace 0,(U-p)\\rbrace ] + \\omega U ,\\\\ \\nonumber \\frac{dU}{dt}&=&\\lambda \\max \\lbrace 0,(U-p)\\rbrace R - U[ kF + \\omega ]- \\mu U , \\\\ \\nonumber \\frac{dF}{dt}&=&F[d+eR-fF-hU] .$ Again, here we have to remark that for $U<p$ the contributions to the infected class is to be understood to drop to zero.", "In such case, once again, the infected prey in the system vanish, and the system settles to any equilibrium of the classical disease-free predator-prey model, [10]." ], [ "Equilibria", "For (REF ) the equilibria are again the origin $\\widetilde{P}_1\\equiv P_1 =\\left(0,0,0\\right)$ and the point $\\widetilde{P}_2\\equiv P_2$ , while the healthy prey thrives again at $\\widetilde{P}_3\\equiv P_3$ , coexistence of healthy prey and predators is attained at level $\\widetilde{P}_4\\equiv P_4$ and the predator-free point $\\widetilde{P}_5=\\left( \\widetilde{R}_5,{1\\over \\mu } \\widetilde{R}_5 (a-b\\widetilde{R}_5 ),0 \\right),$ where $\\widetilde{R}_5$ solves the quadratic equation $b\\lambda R^2 -R [a\\lambda +b(\\omega +\\mu )] +a (\\mu +\\omega ) -p\\lambda =0.$ In view of the convexity of this parabola, there is exactly one positive root if $a (\\mu +\\omega ) <p\\lambda ,$ while there are two such positive roots if $a (\\mu +\\omega ) >p\\lambda , \\quad [a\\lambda +b(\\omega +\\mu )]^2 > 4b\\lambda [a (\\mu +\\omega ) -p\\lambda ].$ In addition $\\widetilde{P}_5$ is feasible for the condition $\\widetilde{R}_5 \\le \\frac{a}{b}.$ The presence of the coexistence equilibrium $\\widetilde{P}_6=(\\widetilde{R}_6,\\widetilde{U}_6,\\widetilde{F}_6)$ can be discussed as follows.", "We take $U>p$ , else the second equilibrium equation of (REF ) cannot be solved for positive values of the populations.", "From the last equilibrium equation of (REF ) we solve for $F$ obtaining $\\widetilde{F}_6=\\frac{1}{f} (d+eR-hU)$ and substitute into the remaining equations to obtain two conic sections $\\Psi (R,U)\\equiv -\\left( b+\\frac{c}{f} e\\right) R^2 + \\left( \\frac{c}{f} h - \\lambda \\right) RU + \\left( p\\lambda -\\frac{c}{f} d+a\\right) R +\\omega U=0,\\\\\\Phi (R,U)\\equiv \\frac{k}{f} hU^2+ \\left(\\lambda - e\\frac{k}{f}\\right) RU - \\left(\\frac{k}{f}d +\\omega +\\mu \\right)U -p \\lambda R=0,$ of which we seek an intersection $(\\widetilde{R}_6,\\widetilde{U}_6)$ in the first quadrant.", "We study each one of them separately.", "The implicit function $\\Phi =0$ can be solved as a function $R = \\rho (U)$ , $\\rho (U) \\equiv U \\frac{kh U - \\left[ f(\\omega +\\mu )+dk \\right]}{fp\\lambda +(ek-f\\lambda )U} .$ The function has a zero at the origin and another one at $U^{0}=[f(\\mu +\\omega )+kd] (hk)^{-1}>0$ .", "It has also a vertical asymptote at $U^{\\infty }=fp\\lambda (f\\lambda -ek)^{-1}$ .", "Asymptotically, for large $U$ , we find $\\rho (U) \\sim \\alpha U \\equiv \\frac{hk}{ek-f\\lambda } U.$ We can rewrite $\\rho $ as follows, and then compute its second derivative: $\\rho (U) = \\alpha \\frac{U-U^0}{U^{\\infty }-U^0} U, \\quad \\rho ^{\\prime \\prime }(U) = -2\\alpha U^{\\infty }\\frac{U^0-U^{\\infty }}{(U-U^{\\infty })^3}.$ Observe that $\\alpha >0$ if and only if $U^{\\infty }<0$ .", "There are three possible situations that can arise, depending on the sign of $U^{\\infty }$ .", "(A) $U^{\\infty }<0<U^0$ ; in this case there is a feasible branch mapping $[U^0,+\\infty )$ surjectively onto $[0, \\infty )$ ; the feasible branch of $\\rho (U)$ is increasing; the function is convex for $U>U^{\\infty }$ and thus the whole feasible branch is.", "(B) $0<U^{\\infty }<U^0$ ; in this case there is a feasible branch mapping $(U^{\\infty },U^0 ]$ surjectively onto $[0, +\\infty )$ ; the feasible branch of $\\rho (U)$ is decreasing; the function is convex for $U>U^{\\infty }$ and thus the whole feasible branch is.", "(C) $0<U^0<U^{\\infty }$ ; in this case there is a feasible branch mapping $[U^0,U^{\\infty })$ surjectively onto $[0, +\\infty )$ ; the feasible branch of $\\rho (U)$ is increasing; the function is convex for $U<U^{\\infty }$ and thus the whole feasible branch is.", "The inverse function $U=\\rho ^{-1}(R)$ maps $[0,+\\infty )$ surjectively onto $[U^0,+\\infty )$ , $(U^{\\infty },U^0 ]$ and $[U^0,U^{\\infty })$ respectively in each case (A), (B), (C).", "We proceed similarly with the implicit function $\\Psi (R,U)=0$ , rewriting it as $U=\\xi (R)$ , $\\xi (R) \\equiv R \\frac{(bf+ce)R + cd-af-fp\\lambda }{\\omega f +(ch-f\\lambda )R} .$ It has a zero at $R^0=(af+fp\\lambda -cd)(bf+ce)^{-1}$ , a vertical asymptote at $R^{\\infty }=\\omega f (f\\lambda -ch)^{-1}$ and asymptotically it behaves like a straight line, $\\xi (R) \\sim \\gamma R \\equiv \\frac{bf+ce}{ch-f\\lambda } R.$ Rewrite it again in compact form, so that $\\xi (R) = \\gamma R \\frac{R-R^0}{R-R^{\\infty }}, \\quad \\xi ^{\\prime \\prime }(R) = -2\\gamma R^{\\infty } \\frac{R^0-R^{\\infty }}{(R-R^{\\infty })^3}.$ Here $\\gamma >0$ if and only if $R^{\\infty }<0$ .", "In this case, more alternatives arise, since here also $R^0$ can be negative.", "We list them as follows: (I) $R^{\\infty }<R^0<0$ ; there is an increasing feasible branch mapping $[0,+\\infty )$ surjectively onto $[0, \\infty )$ ; the feasible branch is convex.", "(II) $R^0<R^{\\infty }<0$ ; as for (I) there is an increasing feasible branch mapping $(0,+\\infty )$ surjectively onto $[0, +\\infty )$ ; the feasible branch is concave.", "(III) $R^0<0<R^{\\infty }$ ; the feasible branch is increasing and maps $[0,R^{\\infty })$ surjectively onto $[0, +\\infty )$ ; the feasible branch is convex.", "(IV) $R^{\\infty }<0<R^0$ ; the increasing feasible branch maps here $[R^0,+\\infty )$ surjectively onto $[0, \\infty )$ ; the feasible branch is convex.", "(V) $0<R^0<R^{\\infty }$ ; in this case there is an increasing feasible branch mapping $[R^0, R^{\\infty })$ surjectively onto $[0, +\\infty )$ ; the feasible branch is convex.", "(VI) $0<R^{\\infty }<R^0$ ; the is feasible branch decreases, mapping $(R^{\\infty },R^0]$ surjectively onto $[0, +\\infty )$ ; the feasible branch is convex.", "The coexistence equilibrium is represented by the intersections of $\\rho ^{-1}$ and $\\xi $ .", "Now in view of the surjectivity and the continuity of these functions, whenever one vertical asymptote, either $U^{\\infty }$ or $R^{\\infty }$ is feasible, the intersection is guaranteed.", "The only cases that are questionable are (A)-(I), (A)-(II) and (A)-(IV).", "In these cases we compare the asymptotic behaviors of the two functions.", "To guarantee an intersection, we need to have $\\alpha ^{-1} < \\gamma $ , comparing (REF ) and (REF ).", "This condition becomes $bhk+ek\\lambda + ch\\lambda > f \\lambda ^2.$ Now case (A)-(I) and (A)-(II) both correspond to $U^{\\infty }<0, \\quad R^0<0, \\quad R^{\\infty }<0,$ while (A)-(IV) gives the same situation with only the second above inequality reversed.", "Combining the two, we are left with the first and the third conditions, namely $ek>f\\lambda , \\quad f\\lambda <ch.$ Use of these into (REF ) shows that the inequality is always satisfied, $bhk+ek\\lambda + ch\\lambda - f \\lambda ^2>bhk+ek\\lambda >0.$ Hence a feasible intersection exists also in these cases.", "Uniqueness follows in view of the convexity properties of the feasible branches of the functions $\\rho ^{-1}$ and $\\xi $ .", "We have thus shown the following result.", "Theorem.", "The feasible coexistence equilibrium $\\widetilde{P}_6$ always exists and is unique." ], [ "Stability", "The Jacobian of (REF ) is $\\widetilde{J} =\\left[\\begin{array}{ccc}a-2bR-cF-\\lambda \\max \\lbrace 0,(U-p)\\rbrace & -\\lambda R H(U-p)+\\omega & -cR \\\\\\lambda \\max \\lbrace 0,(U-p)\\rbrace & \\lambda R H(U-p)-kF-\\omega -\\mu & -kU \\\\eF & -hF & \\widetilde{J}_{33}\\end{array}\\right] ,$ $\\widetilde{J}_{33}=d+eR-hU-2fF$ .", "$\\widetilde{P}_1$ is always unstable, since the eigenvalues are $a$ , $d$ and $-\\omega -\\mu $ .", "For $\\widetilde{P}_2$ we find the eigenvalues $-d$ , $-(dk+f\\omega +f\\mu )f^{-1}$ and $(af-cd)f^{-1}$ , giving again the stability condition (REF ).", "Figure: Left: The coexistence equilibrium P ˜ 5 \\widetilde{P}_5 is achieved when μ=0.28\\mu =0.28 and p=0.1p=0.1 and the remaining parameters are given by() as in Figure .Right: The disease-free equilibrium is attained for μ=0.28\\mu =0.28 and p=0.4p=0.4 with the remaining parameters given by() as in Figure .Note that the diseased population UU falls below the level pp very soon, and consequently both the healthy prey first and subsequentlythe predators pick up, and finally settle to the coexistence equilibrium of the underlying demographic model.The point $\\widetilde{P}_3$ is unstable, in view of the eigenvalues $-a$ , $-\\omega -\\mu $ , $(ae+bd)b^{-1}>0$ .", "For $\\widetilde{P}_4$ we find the eigenvalue $-k \\widetilde{F} -\\omega -\\mu <0$ ; the Routh-Hurwitz conditions on the remaining minor of $\\widetilde{J}$ are satisfied, the determinant being $ce \\widetilde{R}_4 \\widetilde{F}_4>0$ , the trace instead leading to $-(ae+bd)f(bf+ce)^{-1}<0$ .", "Thus, when feasible, $\\widetilde{P}_4$ is unconditionally stable.", "For the point $\\widetilde{P}_5$ the Jacobian factorizes to give one explicit eigenvalue, from which the first stability condition can be obtained, $d+e \\widetilde{R}_5 < h \\widetilde{U}_5,$ and a quadratic equation, for which the Routh-Hurwitz criterion provides the remaining stability conditions $b \\widetilde{R}_5^2 +(\\omega -\\lambda \\widetilde{R}_5) \\widetilde{R}_5 + \\omega \\widetilde{U}_5>0, \\quad (b \\widetilde{R}_5^2 +\\omega \\widetilde{U}_5) (\\omega -\\lambda \\widetilde{R}_5) + \\omega \\lambda ^2 \\widetilde{R}_5^2( \\widetilde{U}_5 -p) >0.$ With the help of some simulations we can show that the coexistence equilibrium can be stably achieved, Figure REF left.", "The refuge parameter used is $p=0.1$ while all the remaining ones are those (REF ) as in Figure REF .", "Note that in this case raising the niche level to $p=0.4$ causes the infected population at some point to fall below this threshold, so that they are wiped out, Figure REF right.", "So while we stated that the disease-free point is not an equilibrium of (REF ) per se, in suitable situations it would certainly occur.", "In fact when the infected population $U$ becomes smaller than the level $p$ , and this occurs pretty early in the simulation as observed in Figure REF right, the sound prey first and then also the predator populations suddenly surge to finally settle to the coexistence equilibrium of the underlying demographic model." ], [ "The reduced contacts", "We consider now another situation, in which we assume that it is the rate of contacts between infected and susceptibles that gets somewhat reduced, due to the effect of a protective niche.", "In this case then we introduce the fraction $0\\le q \\le 1$ of avoided contacts.", "The model, using again the very same previous notation, now becomes $\\frac{dR}{dt}&=&R[a-bR-cF-(1-q)\\lambda U] + \\omega U ,\\\\ \\nonumber \\frac{dU}{dt}&=&U[(1-q)\\lambda R - kF - \\omega -\\mu ], \\\\ \\nonumber \\frac{dF}{dt}&=&F[d+eR-fF-hU] .$ Clearly, by redefining $\\beta =(1-q)\\lambda $ for $\\omega =0$ we get the same model studied in [10].", "For the convenience of the reader we summarize the basic results on the equilibria in which at least one of the population vanishes and then extend the study for the coexistence, to encompass here the situation $\\omega \\ne 0$ not considered in [10] for this specific equilibrium." ], [ "Equilibria", "The equilibria are again all the equilibria of the system (REF ), namely the origin $\\widehat{P}_1\\equiv P_1\\equiv \\widetilde{P}_1$ , and $\\widehat{P}_2\\equiv P_2\\equiv \\widetilde{P}_2$ , $\\widehat{P}_3\\equiv P_3$ , $\\widehat{P}_4\\equiv P_4$ .", "For feasibility of $\\widehat{P}_4$ clearly we need again (REF ).", "Then we have $\\widehat{P}_5 = \\left({\\omega +\\mu \\over \\lambda (1-q)}, \\frac{(a-b\\widehat{R}_5)\\widehat{R}_5}{\\mu }, 0 \\right),$ which is feasible if $a\\lambda (1-q)\\ge b (\\omega +\\mu ).$ Coexistence $\\widehat{P}_6=(\\widehat{R}_6,\\widehat{U}_6,\\widehat{F}_6)$ is obtained by solving the second equation in (REF ) at equilibrium and substituting into the third equation of (REF ) to get $\\widehat{F}_6 = {(1-q) \\lambda \\widehat{R}_5 - \\omega -\\mu \\over k}, \\quad \\widehat{U}_6=\\left( {e \\over h} - {f \\over hk}(1-q) \\lambda \\right) \\widehat{R}_5 + {d \\over h} +{f \\over hk} (\\omega +\\mu ),$ and finally from the first equation in (REF ) we get the quadratic equation $\\sum _{k=0}^2 c_k R^k$ , whose roots determine the value of $\\widehat{R}_6$ , with $c_0= (d k \\omega + f \\omega (\\omega +\\mu )) (hk)^{-1}>0$ and $c_2=\\left( {c \\over k} - {e \\over h} \\right) (1-q)\\lambda + {f \\over hk}(1-q)^2 \\lambda ^2-b, \\\\c_1= a + {c \\over k} (\\omega +\\mu ) + {e \\over h} \\omega - (1-q)\\lambda \\left({d \\over h}+ 2{f \\over hk} (\\omega +\\mu ) \\right).$ Again we can apply Descartes' rule to have at least a positive root.", "This occurs for one root if we impose either one of the alternative conditions $c_2<0, \\quad c_1<0; \\qquad c_2<0, \\quad c_1>0,$ and we get two positive roots if $c_2>0, \\quad c_1<0.$ We do not write explicitly these conditions.", "For feasibility we must impose $\\widehat{R}_6 > \\frac{\\omega +\\mu }{(1-q) \\lambda k}$ and the condition $\\widehat{R}_6 > \\frac{dk+f (\\omega +\\mu )}{ek - f (1-q) \\lambda }, \\quad ek > f (1-q) \\lambda ,$ since the opposite one $ek < f (1-q) \\lambda $ would give a negative value for $\\widehat{R}_6$ ." ], [ "Stability", "The Jacobian in this case is $\\widehat{J}=\\left[\\begin{array}{ccc}\\widehat{J}_{11} & -(1-q)\\lambda R+\\omega & -cR \\\\(1-q)\\lambda U & \\widehat{J}_{22} & -kU \\\\eF & -hF & \\widehat{J}_{33}\\end{array}\\right],$ where $\\widehat{J}_{11}= a-2bR-(1-q)\\lambda U-cF$ , $\\widehat{J}_{22}=(1-q)\\lambda R-kF-\\omega -\\mu $ , $\\widehat{J}_{33}= d+eR-hU-2fF $ .", "For $\\widehat{P}_1$ the eigenvalues are $-\\omega -\\mu $ , $d$ , $a$ , showing its instability.", "The eigenvalues of $\\widehat{P}_2$ are $-(dk+f(\\omega +\\mu ))f^{-1}$ , $-d$ , $(af-cd)f^{-1}$ , for which the stability condition is (REF ).", "Here again comparing (REF ) with (REF ) we observe the existence of a transcritical bifurcation, for which the same conclusions, using the healthy prey invasion number (REF ) can be drawn as for the model with refuge for the healthy prey (REF ).", "The eigenvalues of $\\widehat{P}_3$ are $(bd+ae)b^{-1}$ , $[(1-q)\\lambda a-b(\\omega +\\mu )]b^{-1}$ , $-a$ , thus it is unstable.", "For $\\widehat{P}_4$ one eigenvalue can easily be factored out, while the other ones are the roots of the quadratic equation (REF ).", "Thus, as found formerly, by feasibility (REF ) both its roots have negative real part, and stability depends only on the first eigenvalue, namely it is given by $(1-q)\\lambda R_4<kF_4+\\omega +\\mu $ , a condition that can also be explicitly written as $(1-q)\\lambda \\frac{af-cd}{bf+ce}<k\\frac{ae+bd}{bf+ce}+\\omega +\\mu .$ An eigenvalue of $\\widehat{P}_5$ is $d+e \\widehat{R}_5-h \\widehat{U}_5$ .", "The other ones are the roots of $T(\\theta ) = \\theta ^2-c_1 \\theta +c_2=0$ , with $c_1 = a-2b \\widehat{R}_5-(1-q) \\lambda \\widehat{U}_5 = -\\omega \\frac{\\widehat{U}_5}{\\widehat{R}_5 } -b\\widehat{R}_5 <0,\\quad c_2 = \\mu (1-q) \\lambda \\widehat{U}_5>0,$ so that both roots have negative real parts.", "Stability is achieved for $h \\left( a-b {\\omega +\\mu \\over \\lambda (1-q)}\\right) {\\omega +\\mu \\over \\lambda (1-q)}> \\mu \\left( d+e {\\omega +\\mu \\over \\lambda (1-q)} \\right) .$ Figure REF shows the result of a simulation with the same parameter values (REF ) as for Figure REF , but for $q=0.1$ , assessing the stability of the coexistence equilibrium $\\widehat{P}_5$ .", "Figure: The coexistence equilibrium P ^ 5 \\widehat{P}_5is attained for the same parameters () as in Figure withq=0.1q=0.1." ], [ "Culling", "In order to eradicate the disease, another common method employed is the elimination of the infected individuals, once spotted.", "Let $u(U)$ denote the control policy exercised by the farmer or the veterinarians on the infected population.", "We assume it to be a linear function of the number of infected, $u(U)=\\delta U$ .", "This control measure is of course assumed to be alternative to the safety niches.", "Therefore the model (REF ), without safety niche, then modifies as follows $R^{\\prime }&=&R[a-bR-cF-\\lambda U] + \\omega U , \\\\ \\nonumber U^{\\prime }&=&U[\\lambda R - kF - \\omega ] -(\\delta +\\mu ) U,\\\\ \\nonumber F^{\\prime }&=&F[d+eR-fF-hU] .$ where all the parameters retain their meaning as in (REF )." ], [ "Equilibria", "Since only the infected equation in (REF ) is affected by this change, for $U=0$ , we easily find the very same points $\\bar{P}_1=P_1$ , $\\bar{P}_2=P_2$ , $\\bar{P}_3=P_3$ , $\\bar{P}_4=P_4$ of (REF ), the latter having clearly the same feasibility condition (REF ).", "For $U > 0$ we find the predator-free point $\\bar{P}_5=\\left(\\frac{\\omega +\\delta +\\mu }{\\lambda }, \\frac{a\\lambda (\\omega +\\delta +\\mu )-b(\\omega +\\delta +\\mu )^2}{(\\delta +\\mu )\\lambda ^2},0\\right),$ It is feasible for $a\\lambda \\ge b (\\omega + \\delta +\\mu ).$ We then have the coexistence equilibrium $\\bar{P}_6=[\\bar{R}_6,\\bar{U}_6,\\bar{F}_6]$ , whose population values are found by solving the last equilibrium equation in (REF ) to get $\\bar{F}_6 = { \\lambda \\bar{R}_6 - \\omega -\\delta -\\mu \\over k},$ and by substituting into the second one of (REF ) we find $\\bar{U}_6 =\\left( {e \\over h} - {f \\over hk}\\lambda \\right) \\bar{R}_5 + {d \\over h} + {f \\over hk}(\\omega + \\delta +\\mu ),$ and finally from the first one of (REF ) we get the quadratic equation $a_2 R^2+a_1R+a_0=0$ with $a_2=-bhk -\\left( ch - ek \\right) \\lambda + f \\lambda ^2, \\quad a_0= d k \\omega + f \\omega (\\omega +\\delta +\\mu ), \\\\a_1= (ak + c (\\omega + \\delta +\\mu ) - \\lambda k) \\left( d k + f (\\omega +\\delta +\\mu ) \\right)+ \\omega \\left( ek -f\\lambda \\right),$ whose positive roots give the value of $\\bar{R}_6$ .", "Since $a_0>0$ , imposing $a_2<0$ ensures that exactly one positive root exists.", "Therefore a sufficient condition for feasibility and uniqueness is $f \\lambda ^2< bhk +\\left( ch - ek \\right) \\lambda .$ Alternatively, there will be two positive roots if $a_1^2>4a_2a_0$ , $a_2>0$ and $a_1<0$ , a situation that we however do not explore any further.", "For feasibility, we need further to require $\\bar{R}_6 > \\frac{1}{\\lambda } ( \\omega +\\delta +\\mu ), \\quad (f\\lambda -ek) \\bar{R}_6 < kd+f ( \\omega +\\delta +\\mu ).$" ], [ "Stability", "The Jacobian of (REF ) is $\\bar{J}=\\left[\\begin{array}{ccc}a-2bR-\\lambda U-cF & -\\lambda R+\\omega & -cR \\\\\\lambda U & \\lambda R-kF-\\omega -\\delta -\\mu & -kU \\\\eF & -hF & d+eR-hU-2fF\\end{array}\\right].$ Minor modifications involve the eigenvalues at the equilibria that coincide with those of (REF ).", "$\\bar{P}_1$ and $\\bar{P}_3$ retain their instability here too, $\\bar{P}_2$ is stable when (REF ) holds, $\\bar{P}_4$ has two eigenvalues with negative real parts as for (REF ), but the first one now also contains the culling term, so that the stability condition (REF ) gets here replaced by the more general condition $\\lambda \\bar{R}_4<k \\bar{F}_4+\\omega +\\delta +\\mu $ or, explicitly, $\\lambda (af-cd) < k(bd+ae) + (ce+bf)(\\omega +\\delta +\\mu ).$ For $\\bar{P}_5$ one eigenvalue is $d+e \\bar{R}_5-h \\bar{U}_5$ .", "The other ones are the roots of $T(\\theta ) = \\theta ^2+c_1 \\theta +c_2=0$ , with $c_1 = a-2b\\bar{R}_5-\\lambda \\bar{U}_5, \\quad c_2 = \\delta \\lambda \\bar{U}_5>0.$ Explicitly, $T_{1,2}=\\frac{-c_1\\pm \\sqrt{c_1^2-4c_2}}{2}.$ By Descartes' rule of signs, both have negative real parts if we impose $a-2b\\bar{R}_5-\\lambda \\bar{U}_5<0,$ i.e.", "$\\bar{U}_5>\\frac{a-2b \\bar{R}_5}{\\lambda }.$ This inequality is always satisfied, since using the equilibrium values, the right hand side becomes $\\frac{a-2b \\bar{R}_5}{\\lambda }=\\frac{a}{\\lambda }-2\\frac{b}{\\lambda }\\frac{\\omega +\\delta +\\mu }{\\lambda }=\\frac{a\\lambda - 2b (\\omega + \\delta +\\mu )}{\\lambda ^2}\\\\=\\frac{\\delta \\bar{U}_5}{\\omega +\\delta +\\mu }- \\frac{b(\\omega +\\delta +\\mu )}{\\delta ^2}< \\bar{U}_5-\\frac{b(\\omega +\\delta +\\mu )}{\\delta ^2},$ and the last expression is always smaller than $\\bar{U}_5$ as required.", "Stability hinges on the first eigenvalue only, giving $\\bar{U}_5> \\frac{d+e \\bar{R}_5 }{h}.$" ], [ "Bifurcations", "In this short Section we highlight a few other features of the models.", "For the model (REF ), i.e.", "the refuge for the healthy prey, there is a transcritical bifurcation for which $P_4$ emanates from $P_2$ when the parameters satisfy the critical condition $af=cd,$ compare (REF ) and (REF ).", "Furthermore $P_2$ and $P_5$ are both simultaneously stable if both (REF ) and (REF ) hold.", "Rewriting extensively the latter, we find indeed that (REF ) is its consequence.", "Explicitly, we have $af<cd,\\quad \\frac{h(a\\lambda ( \\lambda s +\\omega +\\mu ) -b( \\lambda s +\\omega +\\mu )^2)}{\\lambda ^2 \\mu }>d+\\frac{e(\\lambda s +\\omega + \\mu )}{\\lambda }.$ Also $P_4$ and $P_5$ are stable simultaneously if $af>cd,\\quad \\frac{h(a\\lambda ( \\lambda s +\\omega +\\mu ) -b( \\lambda s +\\omega +\\mu )^2)}{\\lambda ^2 \\mu }>d+\\frac{e(\\lambda s +\\omega + \\mu )}{\\lambda }.$ In case of the reduced contacts, model (REF ), bistability occurs between the same two pairs of equilibria, with slightly different conditions, namely $af<cd,\\quad \\frac{h(a\\lambda (1-q)( \\omega +\\mu ) -b( \\omega +\\mu )^2)}{\\lambda ^2 (1-q)^2 \\mu }>d+\\frac{e(\\omega + \\mu )}{\\lambda (1-q)}.$ for $\\widehat{P}_2$ and $\\widehat{P}_5$ , while for $\\widehat{P}_4$ and $\\widehat{P}_5$ they become $af>cd,\\quad \\frac{h(a\\lambda (1-q)( \\omega +\\mu ) -b( \\omega +\\mu )^2)}{\\lambda ^2 (1-q)^2 \\mu }>d+\\frac{e(\\omega + \\mu )}{\\lambda (1-q)}.$ Finally, for the model with culling, (REF ), the very same pairs of points are providing bistability once again, with conditions for $\\bar{P}_2$ and $\\bar{P}_5$ given by $af<cd, \\quad \\frac{a\\lambda (\\omega +\\delta +\\mu )-b(\\omega +\\delta +\\mu )^2}{(\\delta +\\mu )\\lambda }> \\frac{d\\lambda + e (\\omega +\\delta +\\mu )}{h},$ while those for $\\bar{P}_4$ and $\\bar{P}_5$ are $af>cd, \\quad \\frac{a\\lambda (\\omega +\\delta +\\mu )-b(\\omega +\\delta +\\mu )^2}{(\\delta +\\mu )\\lambda }> \\frac{d\\lambda + e (\\omega +\\delta +\\mu )}{h}.$ This result is illustrated in Figure REF , for the parameter set $ a=10, b=2, c=1, d=0.1, e=0.2, f=1, h=0.2, k=3, \\lambda =0.75, \\omega =0.9, \\delta =0.6$ .", "The points $\\bar{P}_1$ , $\\bar{P}_2$ , $\\bar{P}_3$ , $\\bar{P}_4$ and $\\bar{P}_5$ are all feasible.", "The equilibria $\\bar{P}_4$ and $\\bar{P}_5$ are both stable, while $\\bar{P}_1$ , $\\bar{P}_2$ , $\\bar{P}_3$ are not.", "Figure: Bistability of P 4 P_4 and P ¯ 5 \\bar{P}_5.", "Left frame: trajectories fromdifferent initial conditions tend to the two equilibria; Right: separatrix surface.", "Bothplots are obtained for the following set of parameter valuesa=10,b=2,c=1,d=0.1,e=0.2,f=1,h=0.2,k=3,λ=0.75,ω=0.9,δ=0.6 a=10, b=2, c=1, d=0.1, e=0.2, f=1, h=0.2, k=3, \\lambda =0.75, \\omega =0.9, \\delta =0.6.Note that the figure has been rotated, the origin lies in the bottom right corner; it isan unstable equilibrium marked with the red dot.", "Also shown with a black dot on the axis is the saddlepoint P ¯ 3 \\bar{P}_3." ], [ "The underlying demographic model", "The classical quadratic predator-prey model underlying these ecoepidemic systems is obtained by eliminating the variable $U$ and its corresponding equation in (REF ).", "This differs from the classical Lotka-Volterra model in which no extra food source is available for predators, which therefore experience an exponential mortality in absence of the prey.", "A related model in which the predator's carrying capacity depends on the prey population size had been introduced in [6].", "The reduced system with no infected, which is the projection of (REF ) onto the disease-free $R-F$ phase plane, has the following equilibria: $Q_1=(0,0), \\quad Q_2=\\left( 0,\\frac{d}{f}\\right), \\quad Q_3=\\left( \\frac{a}{b},0\\right), \\quad Q_4=\\left( \\frac{af-cd}{bf+ce},\\frac{ae+bd}{bf+ce}\\right).$ The latter is feasible when (REF ) holds.", "$Q_1$ and $Q_3$ are both unstable, in view of their respective eigenvalues $a$ , $d$ and $-a$ , $(ae+bd)b^{-1}$ .", "For $Q_2$ we find $(af-cd)f^{-1}$ , $-d$ showing that it is stable exactly when (REF ) holds.", "The eigenvalues of $Q_4$ are complex conjugate, with negative real part, so that $Q_4$ is unconditionally stable.", "Being the only such equilibrium, local stability implies global stability.", "This fact could be shown also via a suitable Lyapunov function." ], [ "Models equilibria summary", "The demographic equilibria $P_1$ -$P_4$ (labeled with the notation of the model (REF )) are the same in the four systems.", "Of these, the first three are always feasible, and $P_1$ and $P_3$ are always unstable.", "The feasibility condition for $P_4$ is always (REF ).", "The predator-free equilibrium differs in each case, because of the prey levels $R_5$ attained in each model.", "The predators settle always at the level $\\mu ^{-1}R_5(a-bR_5)$ .", "Below, we summarize the feasibility conditions in each case.", "Table: NO_CAPTIONThe stability condition for $P_2$ is always (REF ).", "The stability conditions for each equilibrium, assuming feasibility, are instead Table: NO_CAPTION" ], [ "Attainable equilibria with vanishing populations", "The ecoepidemic system exhibits a similar range of behaviors as the demographic ecosystem: predator and prey coexistence is allowed, both with and without infected, compare $P_4$ and $P_6$ , and also the predators-only equilibrium $P_2$ ; this is biologically meaningful recalling that they have other food sources available.", "Comparing feasibility and stability conditions for $P_2$ and $P_4$ a transcritical bifurcation is seen to arise whenever (REF ) holds.", "This clearly stems from the purely demographic model underlying all these ecoepidemic models.", "Evidently, in the prey-free environment expressed by equilibrium $P_2$ , the role of the refuge for the prey is nonexistent.", "In fact the refuge-related parameters appear neither in its feasibility nor in its stability conditions.", "The same does not occur, not surprisingly either, for the disease-free equilibrium $P_4$ .", "In fact its feasibility and the population levels are not affected by the size of the refuges in any model, but the stability of this equilibrium does in fact depend on this parameter.", "The way in which the refuges' parameters $s$ , $q$ and $\\delta $ appear in the stability conditions differs.", "Considering (REF ), (REF ) and (REF ), we find that $s$ , $q$ and $\\delta $ have a stabilizing effect for the ecoepidemic system, a result which as mentioned agrees with former findings in the literature for predator-prey models, [3].", "In fact, in the case of the reduced contacts model, the refuge favors stability since, mathematically, the left hand side becomes smaller due to a positive $q$ , while in the case of a refuge for the healthy prey and of culling it is the right hand side that gets increased by the presence of $s$ and $\\delta $ respectively.", "However, since $q$ is a fraction, denoting the relative reduction in the frequency of contacts, while $s$ represents the number of refuges and $\\delta $ the culling rate, it is more likely that $s$ and $\\delta $ could be sensibly larger than $q$ and therefore have a more marked influence on the stability of this equilibrium.", "Comparison of (REF ) with (REF ) shows that $\\delta $ must be compared with $s\\lambda $ , to assess which model provides the less stringent stability conditions.", "Comparison of (REF ) with (REF ) instead reduces to comparing $\\delta $ with $q\\lambda (af-cd)(bf+ce)^{-1}$ .", "Similary we must compare $s$ with $q (af-cd)(bf+ce)^{-1}$ to assess the largest stability condition between (REF ) and (REF ).", "The feasibility conditions for the equilibrium $P_5$ in all models, namely (REF ), (REF ), (REF ), are always an explicit restatement of (REF ).", "Thus the predator-free equilibrium $P_5$ entails that the size of surviving healthy individuals drops below the level of equilibrium $P_3$ , when they would thrive alone in the disease-free environment, if the equilibrium were stable.", "This is at first sight a somewhat counterintuitive result.", "Indeed it is true that the niches help the infected not to get in contact with the susceptibles, but then one would expect also an advantage for the healthy individuals.", "On the other hand, we can explain it saying that they cannot exceed the carrying capacity of the environment, which is exactly achieved by the healthy prey when they would thrive alone in the predator-free environment, at $P_3$ .", "While the presence of the predators could contribute toward the eradication of the disease helping the system to settle at $P_4$ , their absence cannot improve the environment conditions so that the healthy prey grow beyond the level allowed by the available resources.", "Another way of looking at this situation is to observe that in this case the niche stabilizes the otherwise unstable predator-free equilibrium, at the price of making the disease endemic.", "When feasible, the predator-free equilibrium $P_5$ is stable if the conditions (REF ), (REF ), (REF ), (REF ) hold, all expressing the same relation, while the model with the cover for infected in addition needs also (REF ).", "To compare effects of the various types of refuge is not immediate.", "The refuge-related parameters appear in all models in both healthy and infected prey, and therefore on both sides of the stability conditions.", "The latter are reduced to the inequality $bh R_5^2 +(e\\mu -ah) R_5 +d\\mu <0,$ in which $R_5$ has the value provided by each model.", "Denoting by $R_5^{\\pm }$ the roots of the associated equation to the above inequality, which are real if $(ah-e\\mu )>4bdh\\mu ,$ a condition that we now assume, the effect of the cover in each case can be estimated via the inequalities $R_5^- \\le s + \\frac{\\omega +\\mu }{\\lambda } \\le R_5^+, \\quad R_5^- \\le \\frac{\\omega +\\mu }{(1-q)\\lambda } \\le R_5^+, \\quad R_5^- \\le \\frac{\\omega +\\mu +\\delta }{\\lambda } \\le R_5^+.$" ], [ "Models coexistence equilibria", "The numerical experiments with the coexistence equilibria of the three models show that using the set of demographic parameter values in (REF ), i.e.", "those given by the first row, the system settles to the demographic disease-free equilibrium $(23.2475, 0, 14.0261)$ , whose projection onto the $R-F$ phase plane corresponds of course to the equilibrium of the underlying classical predator-prey system, $(23.2475, 14.0261)$ .", "If we now introduce the disease, with the related parameter values found in the second row of (REF ), we find the ecoepidemic equilibrium $(2.7450, 1.7848, 2.4334)$ .", "As we can easily observe, the disease has a large impact on the system, reducing both its populations by an order of magnitude.", "Although the epidemics affects only the prey, its effect is felt also by the predators.", "This can easily be interpreted, because a reduced food supply, due to a lower prey population caused by the disease, must reduce also the predator population and, in addition, consumption of infected prey is harmful for the predators.", "In other words, diseases, as stated many times in ecoepidemiological research, affect the whole ecosystems, and therefore in environmental studies they cannot be easily neglected." ], [ "Effects of safety refuges on coexistence", "Coming back to the effects of our safety refuges, we have run simulations using the previous parameter values (REF ), with various sizes for the refuge coefficients $s$ , $p$ and $q$ .", "As remarked earlier the proviso holds, that in the models (REF ) and (REF ) a check is implemented, for which when $U<p$ and $R<s$ the next to last term in the first equation and the first one in the second equation are set to zero in both (REF ) and (REF ).", "The results are reported in Figure REF .", "Figure: Equilibrium population values of system () as function of the controls.", "Clockwise from theupper left corner: refuge size ss, refuge size pp, culling rate δ\\delta , refuge size qq.Comparison of the results indicates that for the healthy refuge, the healthy prey and the predators at equilibrium increase in a linear fashion their numbers as $s$ grows, while the infected appear to reach a plateau.", "When the infected prey have a cover, there is a threshold value of its size $p$ beyond which the disease disappears and the other populations suddenly jump to the level of the corresponding demographic, disease-free, classical model and stay there independently of the value of $p$ .", "A similar result holds also when it is the contact rate that gets reduced, i.e.", "for model (REF ).", "In this case the equilibria behavior before the threshold value of $q$ is reached appears to be smoother than in the previous case of system (REF ).", "For the culling policy instead, in this case at least, the healthy prey slightly increase their levels as the rate $\\delta $ grows, but the infected do not vary much and in particular the disease is not eradicated.", "We also discovered persistent oscillations triggered by the use of infected refuges, i.e.", "through the parameter $p$ , Figure REF .", "Figure: Limit cycles obtained when the control over the infected is exercised through a refuge." ], [ "Combined effects of refuge and epidemiological parameters", "Consider the ecoepidemic model without any disease control.", "In Figure REF center we show the infected level as a function of the epidemic parameters $\\lambda $ and $\\omega $ for a fixed choice of the demographic parameters, namely $a=50, \\quad b=0.3, \\quad d=30, \\quad e=0.5, \\quad c=0.6, \\quad f=0.9, \\quad h=0.23, \\quad k=0.3.$ Figure: No disease control: top healthy prey; center infected prey; bottom predators.Observing the level of infected, we choose as reference values for the epidemic parameters $\\lambda =0.8$ and $\\omega =5$ , corresponding to the peak in the infectives.", "When performing simulations with the various disease controls, we will show the simulations results versus the control parameter and one of the epidemic parameters at the time.", "When using as epidemic parameter $\\lambda $ for instance, we will have to compare the figure with the line in Figure REF given by the intersection of the surface with the plane $\\omega =5$ .", "This function raises up to a maximum and then decreases.", "This function has to be compared with the situation when some control is implemented.", "To make things clearer, consider introducing the protected areas for the healthy prey, i.e.", "let us give to $s$ nonzero values.", "In Figure REF we plot the population levels as functions of both $\\lambda $ and $s$ .", "Here the value of $\\omega $ as said is kept at level 5, and independently of the fixed value of $s$ chosen, we see that the equilibrium values of the infected, center frame, as a function of $\\lambda $ has a similar behavior as if no control were present, it raises up to a maximum and then decreases.", "As function of $s$ it is slightly decreasing.", "Note that the maximum of infected with no disease control for $\\omega =5$ and $\\lambda =2$ is about 35, Figure REF center.", "When refuges for the healthy prey are present the number of infected remains about the same for increasing values of $s$ , Figure REF center, probably indicating a scarce effect of this measure to contain the disease propagation.", "If we study the same situation as a function of the parameter $\\omega $ , we have Figure REF center, for $\\lambda =0.8$ , shown under a different angle to better indicate that for large values of the control $s$ and the recovery rate $\\omega $ the disease gets eradicated.", "Now in Figure REF center we need to restrict the surface to the plane $\\omega =5$ .", "The resulting function decreases with increasing $s$ , starting from a value that for $s$ close to zero is comparable to the reference one.", "If we compare the healthy prey equilibrium values, for $s \\approx 0$ and $\\omega =5$ , Figure REF top, the situation is similar to the case of no disease control, Figure REF top.", "Supplying the healthy prey refuges, has the benefit that the equilibrium level of the latter increase, e.g.", "for $s=10$ and $\\lambda =10$ we find $R=10$ , certainly higher than the level in Figure REF top for $\\omega =5$ and for the corresponding value of $\\lambda =10$ .", "Similarly we find that as a function of $\\omega $ the equilibrium level surface is almost always above the level $R=10$ , thus improving over the case of no control, Figure REF top.", "In particular in this latter case the increase in number of healthy prey is quite dramatic.", "Figure: Control with protected areas for the healthy prey, for fixed ω=5\\omega =5.Figure: Control with protected areas for the healthy prey, for fixed λ=2\\lambda =2.", "For the infected the plot is shown under a different angle,to show disease eradication for suitable values of the parameters." ], [ "Comparison of the four different controls", "We now consider the four different controls.", "Plotting in the same frame the infected as function of all of them and of $\\lambda $ , Figure REF , left to right and top to bottom the controls being $s$ , $p$ , $q$ and $\\delta $ , we observe some differences in the infected equilibrium levels.", "As already discussed above, the parameter $s$ seems to lead in general to a rather higher prevalence, uniformly and independently of the contact rate $\\lambda $ .", "A sufficiently high value of both $p$ and $q$ , for not too high values of $\\lambda $ , lead to disease eradication, e.g.", "$\\lambda =1$ and $p=7$ or $q=0.7$ .", "However, both seem to have drawbacks: the “inappropriate” use of $p$ or $q$ leads to a high peak in the prevalence, for $\\lambda \\approx 2$ and $q \\approx 0.7$ .", "This occurs throughout the possible ranges of the controls and of the disease transmission rate, following the peaks in the two frames.", "The difference however is that the peak is rather steady when the control $q$ is used, while it decreases slowly in case of $p$ .", "So among these two controls, the refuge for the infected prey is preferable.", "In fact, in this case a choice of a large $p$ when $\\lambda $ is also large leads to persistent oscillations, as remarked earlier, Figure REF , which correspond to the uneven portion of the surface in the upper right corner of Figure REF , frame for the control $p$ .", "Culling markedly decreases the peak of the prevalence when $\\lambda =0.8$ , but it gives a much smaller range for which the disease is eradicated compared to the use of $q$ and $p$ .", "The “zero level” surface has a larger area indeed in the frames for the reduced contacts and the refuges for infected prey controls than what we find in the frame for culling.", "For large values of the transmission rate and low levels of the controls $p$ , $q$ and $\\delta $ the number of infected at equilibrium settles to about the same value $U=10$ .", "For larger implementations of these controls however, there is a marked difference.", "For $q$ the prevalence shoots up and only for extreme values of the control it goes down and eventually disappears.", "When using culling, the infected equilibrium levels do not change much even if high rates of abatement are employed.", "For the refuges for infected prey strategy, prevalence remains about the same, then there is a regime of oscillatory behavior, and finally for larger values of the control the disease is eradicated.", "To better study the limit cycles, we plot in Figure REF the parameter space of the controls used versus the disease transmission rate.", "The curves in each plot separate the region in which the disease is eradicated, the one having as border the vertical axis, from the region where the disease is endemic, the one bordering the horizontal axis.", "The region of the limit cycles appears only when the $p$ control is used, at the interface of the two regimes, for large values of the contact rate $\\lambda $ .", "The largest area for the disease-free equilibrium is therefore observed in case control is exercised through the parameter $p$ .", "In Figure REF we also compare the loci of the equilibria in the various controls versus the disease recovery rate parameter space.", "Here the region containing the origin represents always the endemic equilibrium.", "The $s$ control exhibits the smallest disease-free equilibrium region, the very small triangle in the top right corner.", "Similarly to it behaves culling.", "The reduced contacts and the refuge for infected prey controls have much larger regions where the disease is eradicated, with the largest region apparently being provided by the former policy, recalling that $q$ is a fraction and cannot exceed 1.", "Comparing the healthy prey and predators levels, Figures REF and REF , similar conclusions can be drawn.", "Culling and refuge for the healthy prey seem to behave similarly to each other, certainly less effectively than the other two policies.", "Among these two, as far as the healthy prey are concerned, it seems to be preferable not to use culling, since for large transmission rate $\\lambda \\approx 10$ , for high values of the control $s$ , they have a value around 10, while they attain much smaller values independently of the culling rate used.", "The predators levels are instead about the same for both policies also for large $\\lambda $ .", "The policies of refuges for the infected prey and of reducing the contact rate instead, when heavily implemented, i.e.", "for large values of the parameters $p$ and $q$ , boost both healthy prey and predators populations levels, especially in presence of high transmission rates, see the left top corners of the corresponding figures.", "A clear advantage is obtained by providing refuges for the infected prey, where they are less able to transmit the disease, see the top right frames in both Figures REF and REF ." ], [ "Final considerations", "In summary it seems that no strategy is the best alone.", "A clear exception are the safety refuges for healthy prey, in that they do not seem to be effective in controlling the disease levels and therefore should not be used.", "Selective culling on infected prey has adverse effects on healthy prey and predators, but it is preferable to control through reduced contacts in terms of smaller disease prevalence.", "In presence of a high transmission rate the best policy is to use refuges for the infected individuals, taking into account however that an insufficient use of this control may trigger persistent oscillations in the system.", "Thus in this type of predator-prey ecoepidemic system with disease just in the prey, for an endemic disease, the ecosystem with a place where some of the healthy individuals can be segregated from coming in contact with disease carriers would exhibit the worst features to preserve the epidemics to spread.", "Probably the most indicated strategies are providing areas for the infected prey where they cannot come in contact with the healthy ones, Reducing the contact rate and culling seem instead to have mixed effects.", "This result could possibly give some hints to field ecologists as how to fight diseases in wild populations, in case some artificial refuges for the diseased individuals, unreachable by the healthy animals, can be provided in specific real-life situations.", "Figure: Infectives as function of the various controls; left to right and top to bottom ss, pp, qq and δ\\delta , for fixed ω=5\\omega =5.", "Note that the spikes in the top right plot correspond to the situations in which the equilibriumis ustable and the coexistence is attained through persistent oscillations.Figure: Loci of the equilibria in the various controls-λ\\lambda parameter space.Infectives levels are function of the various controls; left to right and top to bottom ss, pp, qq and δ\\delta , for fixed ω=5\\omega =5.", "In the top left and bottom right frames,the region to the left of the vertical line is the disease-free equilibrium, to its right we have the endemic equilibrium.Similarly in the other frames,above the curve there is disease eradication, below the disease is endemic.", "In the plot with the control pp also the oscillatoryregion is indicated, at the border of the previous two regions for high transmission rates.Figure: Loci of the equilibria in the various controls-ω\\omega parameter space.Infectives levels are function of the various controls; left to right and top to bottom ss, pp, qq and δ\\delta , for fixed ω=5\\omega =5.The region containing the origin represents the endemic equilibrium.For the ss control, the disease-free equilibrium region is a very small triangle in the top right corner.The spots that occasionally appear correspond to very tiny oscillations, that can be disregarded.The largest region in this parameter space providing disease eradication is given by the reduced contacts policy.Figure: Healthy prey as function of the various controls; left to right and top to bottom ss, pp, qq and δ\\delta , for fixed ω=5\\omega =5.Figure: Predators as function of the various controls; left to right and top to bottom ss, pp, qq and δ\\delta , for fixed ω=5\\omega =5.EV is indebted to Prof. Cristobal Vargas for a useful discussion upon this matter, leading to the analysis of the model with culling.", "This research was partially supported by the project “Metodi numerici in teoria delle popolazioni” of the Dipartimento di Matematica “Giuseppe Peano”." ] ]
1403.0472
[ [ "The spectroscopic signature of Kondo screening on single adatoms in\n Na(Fe0.96Co0.03Mn0.01)As" ], [ "Abstract The electronic states of surface adatoms in Na(Fe0.96Co0.03Mn0.01)As have been studied by low temperature scanning tunneling spectroscopy.", "The spectra recorded on the adatoms display both superconducting coherence peaks and an asymmetric resonance in a larger energy scale.", "The Fano-type line shape of the spectra points towards a possible Kondo effect at play.", "The apparent energy position of the resonance peak shifts about 5 meV to the Fermi level when measured across the critical temperature, supporting that the Bogoliubov quasiparticle is responsible for the Kondo screening in the superconducting state.", "The tunneling spectra do not show the subgap bound states, which is explained as the weak pair breaking effect given by the weak and broad scattering potential after the Kondo screening." ], [ "The spectroscopic signature of Kondo screening on single adatoms in Na(Fe$_{0.96}$ Co$_{0.03}$ Mn$_{0.01}$ )As Zhenyu Wang$^{1}$ , Delong Fang$^2$ , Qiang Deng$^2$ , Huan Yang$^{2,*}$ , Cong Ren$^1$ and Hai-Hu Wen$^{2,*}$ $^1$ National Laboratory for Superconductivity, Institute of Physics and National Laboratory for Condensed Matter Physics, Chinese Academy of Sciences, Beijing 100190, China $^2$ Center for Superconducting Physics and Materials, National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, China The electronic states of surface adatoms in Na(Fe$_{0.96}$ Co$_{0.03}$ Mn$_{0.01}$ )As have been studied by low temperature scanning tunneling spectroscopy.", "The spectra recorded on the adatoms display both superconducting coherence peaks and an asymmetric resonance in a larger energy scale.", "The Fano-type line shape of the spectra points towards a possible Kondo effect at play.", "The apparent energy position of the resonance peak shifts about $5\\;$ meV to the Fermi level when measured across the critical temperature, supporting that the Bogoliubov quasiparticle is responsible for the Kondo screening in the superconducting state.", "The tunneling spectra do not show the subgap bound states, which is explained as the weak pair breaking effect given by the weak and broad scattering potential after the Kondo screening.", "74.55.+v, 68.37.Ef, 72.10.Fk, 74.70.Xa According to the Bardeen-Cooper-Schrieffer theory, metals become superconducting when Cooper pairs are formed via electron-phonon interaction and condense below the critical temperature ($T_\\mathrm {c}$ ).", "The Cooper pairs with the $s$ -wave symmetry can survive in the presence of non-magnetic impurities, while magnetic impurities are expected to suppress superconductivity and induce bound states within the superconducting gap[1], [2].", "The energy of the in-gap state is dominated by the interaction strength between the magnetic impurities and the Cooper pairs[3].", "On the other hand, in normal metals containing magnetic impurities, the magnetic coupling between the local impurity spin and the itinerant electrons can lead to another singlet formation, namely to form a Kondo screening cloud, at temperature below the characteristic Kondo temperature ($T_\\mathrm {K}$ ) of the system[4], [5].", "The competition between the Kondo screening and the opening of superconducting gap can be described by the ratio of $k_\\mathrm {B}T_\\mathrm {K}$ to $\\Delta $[6], where $k_\\mathrm {B}$ is the Boltzmann constant and the order parameter $\\Delta $ governs the superconducting pairing strength.", "Low temperature scanning tunneling microscopy (STM) and spectroscopy (STS) provide a direct method to study the local density of states (LDOS) near the impurities.", "The in-gap states induced by magnetic impurities (Mn, Gd) adsorbed on the surface of Nb single crystal have been proved[7], in consistent with the pioneer calculation of Yu-Shiba-Rusinov[8], [9], [10].", "In the high temperature superconductors, besides magnetic impurities, non-magnetic Zn substitution in Bi$_2$ Sr$_2$ CaCu$_2$ O$_{8+x}$ and Cu substitution in the optimally doped Na(Fe$_{1-x}$ Co$_{x}$ )As also showed clear in-gap resonant states, providing strong evidence of the sign-reversal order parameters in those materials[11], [12].", "Meanwhile, STM experiments have probed various kinds of magnetic atoms absorbed on metallic surfaces, and the Kondo resonance-like structures have also been observed in the differential conductance near the Fermi energy[13], [14].", "The asymmetric line shapes resemble that of a Fano resonance[15].", "Moreover, in a system with variable localized magnetic coupling strengths, a balance between Kondo screening and superconducting pair-breaking interaction has been revealed[16].", "The result showed that the energy of the bound state is located close to the superconducting gap edge for $k_\\mathrm {B}T_\\mathrm {K} \\gg \\Delta $ and deeply in the gap for $k_\\mathrm {B}T_\\mathrm {K} \\sim \\Delta $ .", "In iron pnictides, the superconductivity is widely believed to have a magnetic origin[17].", "Therefore it deserves further investigation to identify the interplay of superconducting pairing and the Kondo screening when a magnetic impurity is presented in those compounds.", "Figure: (Color online) (a) Topographic STMimage of the Na(Fe 0.96 _{0.96}Co 0.03 _{0.03}Mn 0.01 _{0.01})As with three kinds of typical defects.Inset: Atomic-resolution topography in a defect free region.Each of the 2×12\\times 1 rectangular block corresponds to a Co-impurity.", "(b,c) Topographies in a 17.517.5 nm ×17.5\\times 17.5 nm region before andafter the “picking-up” process, respectively (see text).", "(d) The spatial dependenceof height zz measured along the red and dark lines marked in (b) and (c).The tunneling conditions were V s V_\\mathrm {s} = 130 mV and I t I_\\mathrm {t} = 50 pA for (a), (b) and (c),V s V_\\mathrm {s} = 50 mV and I t I_\\mathrm {t} = 100 pA for the inset of (a).High quality Na(Fe$_{0.96}$ Co$_{0.03}$ Mn$_{0.01}$ )As single crystals ($T_\\mathrm {c} \\approx 12.8\\;$ K) were synthesized by the flux method[18].", "All STM and STS tunneling measurements were carried out with an ultrahigh-vacuum low-temperature scanning probe microscope USM-1300 (Unisoku).", "The samples were cleaved in an ultra-high vacuum with a base pressure about $1.1 \\times 10^{-10}\\;$ Torr, then immediately inserted into the microscopy head, which was kept at a low temperature.", "In all measurements, Pt/Ir tips were used.", "The tunneling spectra were acquired using the lock-in technique with ac modulation of $0.8\\;$ mV at $987.5\\;$ Hz.", "Figure: (Color online) (a) Zoom in on a 4.3 nm ×\\times 4.3 nm area containing an adatom, which was takenwith V s V_\\mathrm {s} = 100 mV and I t I_\\mathrm {t} = 50 pA.(b) Typical tunneling spectra measured on and off the adatom in (a) at 1.71.7 K.(c) Spatially resolved spectra of dI/dVdI/dV versus bias voltage recorded along the trajectory (17 points) indicated in (a).The spectra have been shifted vertically for clarity.", "(d) Tunneling spectra with various tunneling current set-points.", "The datahave been normalized to the conductance value at V s =40V_\\mathrm {s} = 40 meV.Fig.", "REF (a) depicts a constant current topographic image with three types of defects on the cleaved surface of Na(Fe$_{0.96}$ Co$_{0.03}$ Mn$_{0.01}$ )As: round protrusions (labeled A), vacancy-like depressions (labeled B) and orthogonal dumbbell-like features (labeled C).", "The inset shows an atomic-resolution image we achieved far away from those defects, with atomic spacing $\\sim 3.9\\;$ Å.", "The weakest bond in this crystal sits between the two adjacent Na layers, therefore the topmost layer after cleaving is the one with Na atoms.", "Beside the observed square lattice, one can see some rectangular blocks, which have already been identified as the Co dopants on the Fe-layer[19].", "For the Mn substitutions, it can only be identified with lower bias voltage during the scanning.", "The type B defects, which may correspond to the missing atoms, were widely distributed on the ever-scanned surface, and no significant spectroscopic differences were detected spatially.", "For the type C defects, the electronic behavior needs further study due to the very low coverage.", "The coverage of type A defects varied dramatically in different places.", "In most area, the type A defects were absent.", "Intriguingly, this kind of defects can be removed by bringing the STM tip close enough to the surface.", "For example, Fig.", "REF (b) and (c) present images of the same area before and after making such action.", "We therefore argue that type A defects are adatoms sitting on the surface.", "Fig.", "REF (d) displays line profiles, in terms of height variation, obtained along the red line marked in Fig.", "REF (b) and the black line in Fig.", "REF (c), respectively.", "The apparent height of the adatom is about $0.7\\;$ Å, measured as the distance between peak and the average surrounding background level.", "We focus the rest of this paper on the electronic properties near these adatoms.", "In Fig.", "REF (b) the tunneling spectra recorded at 1.7 K on and off a single adatom are reported.", "The red solid curve shows $dI/dV$ spectrum measured with the tip held over defect-free surface.", "The spectrum exhibits two clear peaks at an energy level of about $\\pm 4.2\\;$ meV (marked with black arrow in the negative bias side), which are associated with the superconducting gap.", "An additional conductance feature appears at higher energy (marked with red arrow), reminiscent of the bosonic mode observed in the Na(Fe$_{0.975}$ Co$_{0.025}$ )As single crystal[20].", "In the sample we studied here, the mode energy of $\\Omega \\sim 5\\;$ meV $\\sim 4.5k_\\mathrm {B}T_\\mathrm {c}$ , closely agrees with our previous work.", "The black solid curve displays spectrum recorded with the tip held over the center of an adatom.", "The superconducting coherence peaks and the bosonic mode structures are clearly visible, or even get enhanced.", "Additionally, the spectrum shows a pronounced background peak well outside the superconducting gap (marked with blue arrow), and the LDOS of the unoccupied side is strongly suppressed.", "These features were observed on hundreds of adatoms at different locations, and the peak position varies with location, ranging from $-14$ to $-25\\;$ meV.", "This kind of adatoms has also been detected on the surface of Na(Fe$_{0.975}$ Co$_{0.025}$ )As with similar electronic behavior.", "In Fig.", "REF (c), we display a set of tunneling spectra with tip held at varying distance from the center of an adatom indicated in Fig.", "REF (a).", "The resonance-like background feature both decreases in amplitude and changes shape as measured outward.", "At the position of about $15\\;$ Å from the center of the adatom, the feature is mostly gone.", "We also measured a series of $dI/dV$ spectra at various tunneling current set-points, i.e., with junction resistances ranging from 200 to $3200\\;$ M$\\Omega $ .", "Fig.", "REF (d) shows the spectra which have been normalized to the conductance value at $40\\;$ meV.", "No significant change was observed over this resistance range, which can rule out some tip induced effects[21].", "Figure: (Color online) (a) The evolution of the tunneling conductance at an adatom site with temperatureincreased from 1.7 K to 19 K. (b) Spectrum taken at 16 K, when the sample is in normal state.", "The full lineis a Fano fit to the curve, and the parameter of the fit is given in the figure.", "(c) Typical tunneling differential conductance (gg) measured on and off the adatom,which have been normalized to the value at about -5-5 mV.", "(d) g on /g off g_\\mathrm {on}/g_\\mathrm {off}, which was successfullyfitted with the modified Fano function.Fig.", "REF (a) shows the evolution of the STS spectra with temperature up to $19\\;$ K, and it is clear that the pronounced asymmetric background lineshape tends to become more symmetric and the resonance peak moves slightly to the Fermi energy as the temperature is increased.", "Above $13\\;$ K, when the sample is in the normal state, the superconducting coherence peaks disappear and the curves show only the asymmetric resonance feature.", "This type of asymmetric spectra with dip or peak features near the Fermi energy have been observed for the individual magnetic impurity on metal surface and interpreted as the Kondo resonance[13], [14], [22].", "Below $T_\\mathrm {K}$ , the spin of impurity can be flipped by an itinerant electron while simultaneously a spin excitation state called the Kondo resonance is created close to the Fermi energy.", "This spin exchange can modify the energy spectrum of the system.", "In STS measurement, an electron tunneling from a tip to the Kondo resonance actually has two different paths with different probabilities (the orbital of impurity and the continuum), leading to a quantum interference term.", "This term gives rise to a so-called Fano lineshape and described as $\\rho (E) \\propto \\rho _\\mathrm {0} + \\frac{(q+\\epsilon )^2}{1+\\epsilon ^2}$ where $\\epsilon = (E - E_\\mathrm {K})/\\Gamma $ is the normalized energy, with $E_\\mathrm {K}$ as the energy position of the resonance from the Fermi level and $\\Gamma = k_\\mathrm {B}T_\\mathrm {K}$ as the half-width at half-maximum of the curve.", "The parameter $q$ is related to the interference of the two channels contributing to the Fano line shape[23], and the Fano resonance shape thus depends on $q$ .", "Fig.", "REF (b) shows the spectrum obtained at $16\\;$ K and the corresponding fit to Eq.", "REF , which yields a half-width of $\\Gamma \\sim 13.1\\;$ meV.", "To investigate how these adatoms modify the spectra in superconducting state, we normalized the spectra obtained on and off the adatoms at $1.7\\;$ K at energy of about $-5\\;$ meV, and then used Fano equation to fit $g_{on}/g_{off}$ ($g = dI/dV \\sim $ LDOS).", "The outcomes are depicted in Fig.", "REF (c) and Fig.", "REF (d), respectively.", "One excellent fit of the spectrum outside the superconducting feature is achieved by introducing a linear background term into Eq.", "REF[24], and yields a value of $k_\\mathrm {B}T_\\mathrm {K} \\sim 10.5\\;$ meV and thus $T_\\mathrm {K}=122\\;$ K. With the excellent agreement to the Fano fitting of the tunneling spectra, we argue that the adatoms are most likely to be excess magnetic impurities, such as Fe or Co, which induce a strong Kondo resonance in the vicinity.", "Fig.", "REF (a) allows us to semi-quantitatively determine the temperature dependence of the resonance characters.", "A close look into the variation of the resonance peak position reveals that it moves to Fermi energy slightly with elevating temperature.", "As a result, Fig.", "REF (a) shows the temperature dependence of the resonance peak energy directly extracted from the $dI/dV$ curves.", "The absolute value decreases from $21.5\\;$ meV to $16.4\\;$ meV, which cannot be simply explained by the thermal broadening.", "Interestingly, the resonance peak almost does not change its location at temperature below $9\\;$ K or above $T_\\mathrm {c}$ .", "This decrease can be rationalized qualitatively considering that the Bogoliubov quasiparticle is responsible for screening the local spin of the adatom, which needs an energy of $\\Delta (T)$ to overcome the superconducting pairing potential.", "Furthermore, the resonance peak energy changes slowly at low temperature but drop down very quickly near $T_\\mathrm {c}$ , giving further support to this picture.", "Similar temperature dependent behavior were observed on other adatoms in our measurements.", "Another intriguing observation is that, with elevating temperature, the asymmetric background of the lineshape is gradually washed out.", "One important characteristics of the Kondo effect is the broadening and reduction of the Kondo resonance with increasing temperature[25].", "In the Fermi liquid model, the Kondo peak height has a temperature dependent behavior and is predicted to decay slowly with $1 - c(T/T_\\mathrm {K})^2$ for $T \\ll T_\\mathrm {K}$[26].", "However, determining the absolute intensity of the conductance for different spectra in STS measurements especially with varying temperature is a challenge.", "Considering that the LDOS at the positive bias side is relatively featureless, we used a simplest analytical treatment to illustrate the temperature dependence of the resonance feature, i.e., we normalized the spectra to the conductance value at about $20\\;$ meV and then extracted the magnitude of the peak in the normalized curves as a description of the Kondo resonance peak height.", "The result arising from this process and a theoretical fit with $c/T_\\mathrm {K}^2=0.0020\\pm 0.0002$ are shown in Fig.", "REF (b).", "One can see that the decrease of the peak height follows the expected behavior of a Kondo resonance at low temperature.", "Figure: (Color online) (a) Temperature dependence of the resonance peak energy directly extracted from Fig.", "(a).The absolute value decreases from 21.521.5meV to 16.416.4meV.", "The blue short dot lines are to guide the eye.", "(b) Temperature dependence of the magnitude of resonance peak height,which can be understood with the Kondo effect model (blue line).No clear in-gap bound states were detected on these magnetic adatoms, in striking contrast with the pervious STS measurement on Mn impurities in this material[12] and the conventional superconductor Nb[7].", "This seems to create a dilemma since the magnetic impurities are always regarded as pair-breaker in superconductors.", "Nevertheless, the competition between Kondo screening and superconductivity should be taken into account.", "The absence of in-gap states was also detected for adsorbed Co adatom on the Cu (111) surface using a superconducting Nb tip[27].", "STS performed on manganese-phthalocyanine molecules on Pb(111) substrate revealed the evolution from a Kondo screened singlet ground state to a unscreened multiplet ground state, and the in-gap states was found to be induced very close to the gap edge when $k_\\mathrm {B}T_\\mathrm {K}/\\Delta \\gg 1$[16].", "The absence of the in-gap bound states in our present case may be understood as a consequence of the $s\\pm $ pairing together with the Kondo screening effect on the spin of the adatom.", "In iron pnictides, superconducting pairing is suggested to be established by exchanging the spin fluctuations leading to a gap with reversed signs on the hole and electron pockets[28], [29], [30], [31].", "STM measurements involving impurity effects have been successfully performed and many interesting features have been reported[32], [33], [34], [35], [36], [37].", "For the adatom we studied here, the larger energy scale of the Kondo resonance channel, with $k_\\mathrm {B}T_\\mathrm {K} \\sim 3\\Delta $ , may give rise to the strong screening of the local spin of the adatom.", "At the mean time, the scattering potential gets much broader due to this Kondo screening.", "For the inter-pocket $s\\pm $ pairing with large momentum transfer during the pair-scattering process, a broadened scattering potential can hardly act as the effective pair breaker.", "Both effects mentioned here will weaken the pair-breaking.", "This is quite similar to the case of Co dopants in Na(Fe$_{1-x}$ Co$_{x}$ )As[19].", "Therefore, although the adatoms discovered in our experiment can produce clear Kondo effect and modify the tunneling spectrum significantly, the absence of in-gap bound states may however gain an explanation showing the consistency with the $s\\pm $ pairing.", "In summary, we have presented spatial evolution and temperature dependence of the tunneling spectra associated with the adatoms on iron pnictide Na(Fe$_{0.96}$ Co$_{0.03}$ Mn$_{0.01}$ )As.", "At the adatom and below $T_c$ , the superconducting spectrum is significantly modified with a Kondo resonance like background.", "The temperature dependence of the resonance is in good agreement with the predicted Kondo behavior, giving further evidence that the Kondo effect is playing the role here.", "The spectra in the superconducting state reveal the absence of in-gap bound states predicted theoretically for magnetic impurities.", "This is understood as the consequence of the Kondo screening effect on the spin of the adatom, and the broadened scattering potential, both will weaken the pair breaking effect with the scenario of $s\\pm $ pairing.", "This work is supported by the Ministry of Science and Technology of China (973 projects: 2011CBA00102, 2012CB821403, 2010CB923002), NSF and PAPD of China.", "$^*$ [email protected], [email protected]" ] ]
1403.0288
[ [ "SMERED: A Bayesian Approach to Graphical Record Linkage and\n De-duplication" ], [ "Abstract We propose a novel unsupervised approach for linking records across arbitrarily many files, while simultaneously detecting duplicate records within files.", "Our key innovation is to represent the pattern of links between records as a {\\em bipartite} graph, in which records are directly linked to latent true individuals, and only indirectly linked to other records.", "This flexible new representation of the linkage structure naturally allows us to estimate the attributes of the unique observable people in the population, calculate $k$-way posterior probabilities of matches across records, and propagate the uncertainty of record linkage into later analyses.", "Our linkage structure lends itself to an efficient, linear-time, hybrid Markov chain Monte Carlo algorithm, which overcomes many obstacles encountered by previously proposed methods of record linkage, despite the high dimensional parameter space.", "We assess our results on real and simulated data." ], [ "Introduction", "When data about individuals comes from multiple sources, it is essential to match, or link, records from different files that correspond to the same individual.", "Other names associated with record linkage are entity disambiguation and coreference resolution, meaning that records which are linked or co-referent can be thought of as corresponding to the same underlying entity.", "Solving this problem is not just important as a preliminary to statistical analysis; the noise and distortions in typical data files make it a difficult, and intrinsically high-dimensional, problem [8], [11], [16], [17].", "We propose a Bayesian approach to the record linkage problem based on a parametric model that addresses matching $k$ files simultaneously and includes duplicate records within lists.", "We represent the pattern of matches and non-matches as a bipartite graph, in which records are directly linked to the true but latent individuals which they represent, and only indirectly linked to other records.", "Such linkage structures allow us to simultaneously solve three problems: record linkage, de-duplication, and estimation of unique observable population attributes.", "The Bayesian paradigm naturally handles uncertainty about linkage, which poses a difficult challenge to frequentist record linkage techniques.", "[13] review Bayesian contributions to record linkage.", "Doing so permits valid statistical inference regarding posterior matching probabilities of records and propagation of errors as discussed in §.", "To estimate our model, we develop a hybrid MCMC algorithm, in the spirit of [9], which runs in linear time in the number of records and the number of MCMC iterations, even in high-dimensional parameter spaces.", "Our algorithm permits duplication across and within lists but runs faster if there are known to be no duplicates within lists.", "We achieve further gains in speed using standard record linkage blocking techniques [2].", "We apply our method to data from the National Long Term Care Survey (NLTCS), which tracked and surveyed approximately 20,000 people at five-year intervals.", "At each wave of the survey, some individuals had died and were replaced by a new cohort, so the files contain overlapping but not identical sets of individuals, with no within-file duplicates." ], [ "Related Work", "The classical work of [5] considered linking two files in terms of Neyman-Pearson hypothesis testing.", "Compared to this baseline, our approach is distinctive in that it handles multiple files, models distortion explicitly, offers a Bayesian treatment of uncertainty and error propagation, and employs a sophisticated graphical data structure for inference to latent individuals.", "Fellegi-Sunter methods based upon [5] can extend to $k>2$ files [14], but they break down for even moderately large $k$ or complex data sets.", "Moreover, they give little information about uncertainty in matches, or about the true values of noise-distorted records.", "The idea of modeling the distortion process originates with the “Hit-Miss Model” by [3], which anticipates part of our model in §REF .", "The specific distortion model we use is however closer to that introduced in [7], as part of a nonparametric frequentist technique for matching $k=2$ files.", "We differ from [7] by introducing latent individuals and distortion through a Bayesian model.", "Within the Bayesian paradigm, most work has focused on specialized problems related to linking two files, which propagate uncertainty [6], [15], [12], [1].", "These contributions, while valuable, do not easily generalize to multiple files and duplicate detection.", "Two recent papers [4], [6] are most relevant to the novelty of our work, namely the linkage structure.", "To aid recovering information about the population from distorted records, [6] called for developing “more sophisticated network data structures.\"", "Our linkage graphs are such a data structure with the added benefit of permitting de-duplication and handling multiple files.", "Moreover, due to exact error propagation, our methods are also easily integrated with other analytic procedures.", "Algorithmically, the closest approach to our linkage structure is the graphical representation in [4], for de-duplication within one file.", "Their representation is a unaparatite graph, where records are linked to each other.", "Our use of a bipartite graph with latents individuals naturally fits in the Bayesian paradigm along with distortion.", "Our method is the first to handle record linkage and de-duplication, while also modeling distortion and running in linear time." ], [ "Notation, Assumptions, and Linkage Structure", "We begin by defining some notation, where we have $k$ files or lists.", "For simplicity, we assume that all files contain the same $p$ fields, which are all categorical, field $\\ell $ having $M_{\\ell }$ levels.", "We also assume that every record is complete.", "(Handling missing-at-random fields within records is a minor extension within the Bayesian framework.)", "Let $\\mathbf {x}_{ij}$ be the data for the $j$ th record in file $i$ , where $i=1,\\ldots ,k$ , $j=1,\\ldots ,n_i$ , and $n_i$ is the number of records in file $i$ ; $\\mathbf {x}_{ij}$ is a categorical vector of length $p$ .", "Let $\\mathbf {y}_{j^{\\prime }}$ be the latent vector of true field values for the $j^{\\prime }$ th individual in the population (or rather aggregate sample), where $j^{\\prime }=1,\\ldots ,N$ , $N$ being the total number of observed individuals from the population.", "$N$ could be as small as 1 if every record in every file refers to the same individual or as large as $N_{\\max } \\equiv \\sum _{i=1}^k{n_i}$ if no datasets share any individuals.", "Now define the linkage structure $\\mathbf {\\Lambda }=\\lbrace \\lambda _{ij}\\;;\\;i=1,\\ldots ,k\\;;\\;j=1,\\ldots ,n_i\\rbrace $ where $\\lambda _{ij}$ is an integer from 1 to $N_{\\max }$ indicating which latent individual the $j$ th record in file $i$ refers to, i.e., $\\mathbf {x}_{ij}$ is a possibly-distorted measurement of $\\mathbf {y}_{\\lambda _{ij}}$ .", "Finally, $z_{ij\\ell }$ is 1 or 0 according to whether or not a particular field $\\ell $ is distorted in $\\mathbf {x}_{ij}.$ As usual, we use $I$ for indicator functions (e.g., $I(x_{ij\\ell }=m)$ is 1 when the $\\ell $ th field in record $j$ in file $i$ has the value $m$ ), and $\\delta _a$ for the distribution of a point mass at $a$ (e.g., $\\delta _{y_{\\lambda _{ij}\\ell }}$ ).", "The vector $\\mathbf {\\theta }_{\\ell }$ of length $M_{\\ell }$ denotes the multinomial probabilities.", "For clarity, we always index as follows: ${i=1,\\ldots ,k;}$ ${j=1,\\ldots ,n_i;}$ ${j^{\\prime }=1,\\ldots ,N;}$ ${\\ell =1,\\ldots ,p;}$ ${m=1, \\ldots , M_{\\ell }.", "}$" ], [ "Independent Fields Model", "We assume that the files are conditionally independent, given the latent individuals, and that fields are independent within individuals.", "We formulate the following Bayesian parametric model, where the joint posterior is in closed form and we sample from the full conditionals using a hybrid MCMC algorithm: $\\mathbf {x}_{ij\\ell }\\mid \\lambda _{ij},\\mathbf {y}_{\\lambda _{ij}\\ell },z_{ij\\ell },\\mathbf {\\theta }_\\ell &\\stackrel{\\text{ind}}{\\sim }{\\left\\lbrace \\begin{array}{ll}\\delta _{\\mathbf {y}_{\\lambda _{ij}\\ell }}&\\text{ if }z_{ij\\ell }=0\\\\\\text{MN}(1,\\mathbf {\\theta }_\\ell )&\\text{ if }z_{ij\\ell }=1\\end{array}\\right.", "}\\\\z_{ij\\ell }&\\stackrel{\\text{ind}}{\\sim }\\text{Bernoulli}(\\beta _\\ell )\\\\\\mathbf {y}_{j^{\\prime }\\ell }\\mid \\mathbf {\\theta }_{j\\ell }&\\stackrel{\\text{ind}}{\\sim }\\text{MN}(1,\\mathbf {\\theta }_\\ell )\\\\\\mathbf {\\theta }_\\ell &\\stackrel{\\text{ind}}{\\sim }\\text{Dirichlet}(\\mathbf {\\mu }_\\ell )\\\\\\beta _\\ell &\\stackrel{\\text{ind}}{\\sim }\\text{Beta}(a_\\ell ,b_\\ell ) \\\\\\pi (\\mathbf {\\Lambda }) &\\propto 1,$ where $a_\\ell ,b_\\ell ,$ and $\\mathbf {\\mu }_\\ell $ are all known, and MN denotes the Multinomial distribution.", "Remark 2.1 We assume that every legal configuration of the $\\lambda _{ij}$ is equally likely a priori.", "This implies a non-uniform prior on related quantities, such as the number of individuals in the data.", "The uniform prior on $\\mathbf {\\Lambda }$ is convenient, since constructing either a subjective or an alternative objective prior is unclear.", "A uniform distribution on one quantity, i.e.", "$\\Lambda $ , implies a non-uniform distribution on other, related quantities (such as $N$ ).", "Making every entry in the matrix $\\lambda _{ij}$ uniformly distributed on $1, 2, \\ldots N_{\\max }$ implies that the distribution of $N$ , a function of $\\Lambda $ , is not uniform on $1, 2, \\ldots N_{\\max }$ .", "This is a long-standing problem with “non-informative priors\" [10].", "Deriving the joint posterior and conditional distributions is now mostly straightforward.", "One subtlety, however, is that $\\mathbf {y}$ , $\\mathbf {z}$ and $\\mathbf {\\Lambda }$ are all related, since if $z_{ij\\ell }=0$ , then it must be the case that $y_{\\lambda _{ij}\\ell }=x_{ij\\ell }$ .", "Taking this into account, the joint posterior is $&\\pi (\\mathbf {\\Lambda },\\mathbf {y},\\mathbf {z},\\mathbf {\\theta },\\mathbf {\\beta }\\mid \\mathbf {x})\\\\&\\propto \\prod _{i,j,\\ell , m}\\left[(1-z_{ij\\ell })\\delta _{y_{\\lambda _{ij}\\ell }}(x_{ij\\ell })\\;\\;+\\;\\;z_{ij\\ell }\\theta _{\\ell m}^{I(x_{ij\\ell } = m)}\\right]\\\\&\\phantom{\\propto {}}\\times \\prod _{\\ell , m}\\theta _{\\ell m}^{\\mu _{\\ell m}+\\sum _{j^{\\prime }=1}^N I(y_{j^{\\prime }l}=m)}\\\\&\\phantom{\\propto {}}\\times \\prod _{\\ell }\\beta _\\ell ^{a_\\ell -1+\\sum _{i=1}^k\\sum _{j=1}^{n_i}z_{ij\\ell }} \\\\&\\times (1-\\beta _\\ell )^{b_\\ell -1+\\sum _{i=1}^k\\sum _{j=1}^{n_i}(1-z_{ij\\ell })}.", "$ We suppress derivation of the full conditionals, but note that the full conditionals of $\\mathbf {y}$ , $\\mathbf {z}$ and $\\mathbf {\\Lambda }$ always obey their logical dependence, and therefore never condition on impossible events.", "The full conditional of $\\mathbf {\\Lambda }$ must reflect whether or not there are duplicates within files.", "If we define $R_{ij^{\\prime }} = \\left\\lbrace j : \\lambda _{ij} = j^{\\prime }\\right\\rbrace ,$ then not having within-file duplicates means that $R_{ij^\\prime }$ must be either $\\emptyset $ or a single record, for each $i$ and $j^{\\prime }$ .", "Graphically, this means allowing or forbidding links from a latent individual to multiple records within one file." ], [ "Split and MErge REcord linkage and De-duplication (SMERED)\nAlgorithm", "Our main goal is estimating the posterior distribution of the linkage (i.e., the clustering of records into individuals).", "The simplest way of accomplishing this is via Gibbs sampling.", "We could iterate through the records, and for each record, sample a new assignment to an individual (from among the individuals represented in the remaining records, plus an individual comprising only that record).", "However, this requires the quadratic-time checking of proposed linkages for every record.", "Thus, instead of Gibbs sampling, we use a hybrid MCMC algorithm to explore the space of possible linkage structures, which allows our algorithm to run in linear time.", "Our hybrid MCMC takes advantage of split-merge moves, as done in [9], which avoids the problems associated with Gibbs sampling, even though the number of parameters grows with the number of records.", "This is accomplished via proposals that can traverse the state space quickly and frequently visit high-probability modes, since the algorithm splits or merges records in each update, and hence, frequent updates of the Gibbs sampler are not necessary.", "Furthermore, a common technique in record linkage is to require an exact match in certain fields (e.g., birth year) if records are to be linked.", "This technique of blocking can greatly reduce the number of possible links between records (see e.g., [17]).", "Since blocking gives up on finding truly co-referent records which disagree on those fields, it is best to block on fields that have little or no distortion.", "We block on the fairly reliable fields of sex and birth year in our application to the NLTCS below.", "A strength of our model is that it incorporates blocking organically.", "Setting $b_\\ell =\\infty $ for a particular field $\\ell $ forces the distortion probability for that field to zero.", "This requires matching records to agree on the $\\ell $ th field, just like blocking.", "We now discuss how the split-merge process links records to records, which it does by assigning records to latent individuals.", "Instead of sampling assignments at the record level, we do so at the individual level.", "Initially, each record is assigned to a unique individual.", "On each iteration, we choose two records at random.", "If the pair belong to distinct latent individuals, then we propose merging those individuals to form a single new latent individual (i.e., we propose that those records are co-referent).", "On the other hand, if the two records belong to the same latent individual, then we propose splitting it into two new latent individuals, each seeded with one of the two chosen records, and the other records randomly divided between the two.", "Proposed splits and merges are accepted based on the Metropolis-Hastings ratio and rejected otherwise.", "To choose the pair of records, one option is to sample uniformly from among all possible pairs.", "However, this is not ideal, for two reasons.", "First, most pairs of records are extremely unlikely to match since they agree on few, if any, fields.", "Frequent proposals to merge such records are wasteful.", "Therefore, we employ blocking, and only consider pairs of records within the same block.", "Second, sampling from all possible pairs of records will sometimes lead to proposals to merge records in the same list.", "If we permit duplication within lists, then this is not a problem.", "However, if we know (or assume) there are no duplicates within lists, we should avoid wasting time on such pairs.", "The no-duplication version of our algorithm does precisely this.", "(See Algorithm REF for pseudocode.)", "When there are no duplicates within files, we call the SMERE (Split and MErge REcord linkage) algorithm, which enforces the restriction that $R_{ij^{\\prime }}$ must be either $\\emptyset $ or a single record.", "This is done through limiting the proposal of record pairs to those in distinct files; the algorithm otherwise matches SMERED.", "[t!]", "$\\mathbf {X}$ and hyperparameters Initialize the unknown parameters $\\mathbf {\\theta }, \\mathbf {\\beta }, \\mathbf {y}, \\mathbf {z},$ and $\\mathbf {\\Lambda }.$ $i \\leftarrow 1$ $S_G$ $j \\leftarrow 1$ $S_M$ $t \\leftarrow 1$ $S_T$ Draw records $R_1$ and $R_2$ uniformly and independently at random.", "$R_1$ and $R_2$ refer to the same individualpropose splitting that individual, shifting $\\mathbf {\\Lambda }$ to $\\mathbf {\\Lambda ^{\\prime }}$ propose merging the individuals $R_1$ and $R_2$ refer to, shifting $\\mathbf {\\Lambda }$ to $\\mathbf {\\Lambda ^{\\prime }}$ $r \\leftarrow \\min {\\left\\lbrace 1, \\frac{\\pi (\\mathbf {\\Lambda }^{\\prime },\\mathbf {y},\\mathbf {z},\\mathbf {\\theta },\\mathbf {\\beta }|\\mathbf {x}) }{ \\pi (\\mathbf {\\Lambda },\\mathbf {y},\\mathbf {z},\\mathbf {\\theta },\\mathbf {\\beta }|\\mathbf {x})} \\right\\rbrace } $ Resample $\\mathbf {\\Lambda }$ by accepting proposal with Metropolis probability $r$ or rejecting with probability $1-r.$ Resample $\\mathbf {y}$ and $\\mathbf {z}.$ Resample $\\mathbf {\\theta }, \\mathbf {\\beta }.$ $\\mathbf {\\theta }|\\mathbf {X}, \\mathbf {\\beta }\\mathbf {X},\\mathbf {y}|\\mathbf {X}, \\mathbf {z}|\\mathbf {X},$ and $\\mathbf {\\Lambda }|\\mathbf {X}.$ Split and MErge REcord linkage and De-duplication (SMERED)" ], [ "Time Complexity", "Scalability is crucial to any record linkage algorithm Current approaches typically run in polynomial (but super-linear) time in $N_{\\max }$ .", "(The method of [14] is $O(N_{\\max }^k)$ , while that of [4] finds the maximum flow in an $N_{\\max }$ -node graph, which is $O(N_{\\max }^3)$ , but independent of $k$ .)", "In contrast, our algorithm is linear in both $N_{\\max }$ and MCMC iterations.", "Our running time is proportional to the number of Gibbs iterations $S_G,$ so we focus on the time taken by one Gibbs step.", "Recall the notation from §, and define $M =\\frac{1}{p} \\sum _{\\ell =1}^p M_{\\ell }$ as the average number of possible values per field ($M \\ge 1$ ).", "The time taken by a Gibbs step is dominated by sampling from the conditional distributions.", "Specifically, sampling $\\mathbf {\\beta }$ and $\\mathbf {y}$ are both $O(p N_{\\max })$ ; sampling $\\mathbf {\\theta }$ is $O(pMN) + O(pN_{\\max }) = O(pMN)$ , as is sampling $\\mathbf {z}$ .", "Sampling $\\mathbf {\\Lambda }$ is $O(pN_{\\max }M)$ if done carefully.", "Thus, all these samples can be drawn in time linear in $N_{\\max }$ .", "Since there are $S_M$ Metropolis steps within each Gibbs step and each Metropolis step updates $\\mathbf {y}$ , $\\mathbf {z}$ , and $\\mathbf {\\Lambda }$ , the time needed for the Metropolis part of one Gibbs step is $O(S_MpN_{\\max }) + O(S_MpMN) + O(S_MpN_{\\max }M).$ Since $N \\le N_{\\max },$ the run time becomes $O(pS_M N_{\\max }) + O(MpS_M N_{\\max }) = O(MpS_M N_{\\max }).", "$ On the other hand, the updates for $\\mathbf {\\theta }$ and $\\mathbf {\\beta }$ occur once each Gibbs step implying the run time is $O(pMN) + O(pN_{\\max }).$ Since $N \\le N_{\\max },$ the run time becomes $O(pM N_{\\max } + p N_{\\max }) = O(p M N_{\\max } ).$ The overall run time of a Gibbs step is $O(pM N_{\\max } S_M) + O(p M N_{\\max } ) = O(pM N_{\\max } S_M).$ Furthermore, for $S_G$ iterations of the Gibbs sampler, the algorithm is order $O(pM N_{\\max } S_G S_M).$ If $p$ and $M$ are all much less than $N_{\\max }$ , we find that the runtime is $O( N_{\\max } S_G S_M).$ Another important consideration is the number of MCMC steps needed to produce Gibbs samples that form an adequate approximation of the true posterior.", "This issue depends on the convergence properties (actual rate of convergence) of the hybrid Markov chain used by the algorithm, which are beyond the scope of the present work.", "Convergence diagnostics for our application to the NLTCS and hyperparameter sensitivity is discussed in Appendix ." ], [ "Posterior Matching Sets and Linkage Probabilities", "In a Bayesian framework, the output of record linkage is not a deterministic set of matches between records, but a probabilistic description of how likely records are to be co-referent, based on the observed data.", "Since we are linking multiple files at once, we propose a range of posterior matching probabilities: the posterior probability of linkage between two arbitrary records and more generally among $k$ records, the posterior probability given a set of records that they are linked, and the posterior probability that a given set of records is a maximal matching set (which will be defined later).", "Two records $(i_1,j_1)$ and $(i_2,j_2)$ match if they point to the same latent individual, so $\\lambda _{i_1j_1} = \\lambda _{i_2j_2}.$ The posterior probability of a match can be computed from the $S_G$ MCMC samples: $P(\\lambda _{i_1j_1} = \\lambda _{i_2j_2} | \\mathbf {X}) = \\frac{1}{S_G}\\sum _{h=1}^{S_G}I(\\lambda _{i_1j_1}^{(h)} = \\lambda _{i_1j_2}^{(h)}).$ A one-way match is when an individual appears in only one of the $k$ files, while a two-way match is when an individual appears in exactly two of the $k$ files, and so on (up to $k$ -way matches).", "We approximate the posterior probability of arbitrary one-way, two-way, ..., $k$ -way matches as the ratio of the number of times those matches happened in the posterior sample to $S_G$ .", "Although probabilistic results and interpretations provided by the Bayesian paradigm are useful both quantitatively and conceptually, we often report a point estimate of the linkage structure.", "Thus, we face the question of how to condense the overall posterior distribution of $\\mathbf {\\Lambda }$ into a single estimated linkage structure.", "Perhaps the most obvious approach is to set some threshold $v$ , where $0<v<1$ , and to declare (i.e., estimate) that two records match if and only if their posterior matching probability exceeds $v$ .", "This strategy is useful if only a few specific pairs of records are of interest, but its flaws are exposed when we consider the coherence of the overall estimated linkage structure implied by such a thresholding strategy.", "Note that the true linkage structure is transitive in the following sense: if records A and B are the same individual, and records B and C are the same individual, then records A and C must be the same individual as well.", "However, this requirement of transitivity is in no way enforced by the simple thresholding strategy described above.", "Thus, a more sophisticated approach is required if the goal is to produce an estimated linkage structure that preserves transitivity.", "To this end, it is useful to define a new concept.", "A set of records $\\mathcal {A}$ is a maximal matching set (MMS) if every record in the set has the same value of ${\\lambda }_{ij}$ and no record outside the set has that value of ${\\lambda }_{ij}.$ Define $\\mathbf {\\Omega }(\\mathcal {A}, \\mathbf {\\Lambda }):=\\mathbf {\\Omega }_{\\mathcal {A}, \\mathbf {\\Lambda }} $ to be 1 if $\\mathcal {A}$ is an MMS in $\\mathbf {\\Lambda }$ and 0 otherwise: $\\mathbf {\\Omega }_{\\mathcal {A}, \\mathbf {\\Lambda }} = \\sum _{j^{\\prime }}{\\left( \\prod _{(i,j) \\in \\mathcal {A}}{I(\\lambda _{i j}=j^{\\prime })}\\prod _{(i,j)\\notin \\mathcal {A}}{I(\\lambda _{ij}\\ne j^{\\prime }})\\right)}.$ Essentially, the MMS contains all the records which match some particular latent individual, though which individual is irrelevant.", "Given a set of records $\\mathcal {A}$ , the posterior probability that it is an MMS in $\\mathbf {\\Lambda }$ is simply $P(\\mathbf {\\Omega }_{\\mathcal {A}, \\mathbf {\\Lambda }} =1)&=\\frac{1}{S_G}\\sum _{h=1}^{S_G}{\\mathbf {\\Omega }(\\mathcal {A},\\mathbf {\\Lambda }^{(h)})}.$ The MMSs allow a sophisticated method of preserving transitivity when estimating a single overall linkage structure.", "For any record $(i,j)$ , its most probable MMS $\\mathcal {M}_{ij}$ is the set containing $(i,j)$ with the highest posterior probability of being an MMS, i.e., $\\mathcal {M}_{ij}:=\\operatornamewithlimits{arg\\,max}_{\\mathcal {A}:(i,j)\\in \\mathcal {A}}P(\\mathbf {\\Omega }_{\\mathcal {A}, \\mathbf {\\Lambda }} =1).$ Next, a shared most probable MMS is a set that is the most probable MMS of all records it contains, i.e., a set $\\mathcal {A}^\\star $ such that $\\mathcal {M}_{ij}=\\mathcal {A}^\\star $ for all $(i,j)\\in \\mathcal {A}^\\star $ .", "We then estimate the overall linkage structure by linking records if and only if they are in the same shared most probable MMS.", "The resulting estimated linkage structure is guaranteed to have the transitivity property since (by construction) each record is an element of at most one shared most probable MMS." ], [ "Functions of Linkage Structure ", "The output of the Gibbs sampler also allows us to estimate the value of any function of the variables, parameters, and linkage structure by computing the average value of the function over the posterior samples.", "For example, estimated summary statistics about the population of latent individuals are straightforward to calculate.", "Indeed, the ease with which such estimates can be obtained is yet another benefit of the Bayesian paradigm, and of MCMC in particular." ], [ "Assessing Accuracy of Matching and Application to NLTCS", "We test our model on data from the NLTCS, a longitudinal study of the health of elderly (65+) individuals (http://www.nltcs.aas.duke.edu/).", "The NLTCS was conducted approximately every six years, with each wave containing roughly 20,000 individuals.", "Two aspects of the NLTCS make it suitable for our purposes: individuals were tracked from wave to wave with unique identifiers, but at each wave, many patients had died (or otherwise left the study) and were replaced by newly-eligible patients.", "We can test the ability of our model to link records across files by seeing how well it is able to track individuals across waves, and compare its estimates to the ground truth provided by the unique identifiers.", "To show how little information our method needs to find links across files, we gave it access to only four variables, all known to be noisy: full date of birth, sex, state of residence, and the regional office at which the subject was interviewed.", "We treat all fields as categorical.", "We linked individuals across the 1982, 1989 and 1994 survey waves.The other three waves used different questionnaires and are not strictly comparable.", "Our model had little information on which to link, and not all of its assumptions strictly hold (e.g., individuals can move between states across waves).", "We demonstrate our method's validity using error rates, confusion matrices, posterior matching sets and linkage probabilities, and estimation of the unknown number of observed individuals from the population.", "Appendix provides a simulation study of the NLTCS with varying levels of distortion at the field level.", "We conclude from this that SMERE is able to handle low to moderate levels of distortion (Figure REF ).", "Furthermore, as distortion increases, so do the false negative rate (FNR) and false positive rate (FPR) (Figure REF )." ], [ "Error Rates and Confusion Matrix", "Since we have unique identifiers for the NLTCS, we can see how accurately our model matches records.", "A true link is a match between records which really do refer to the same latent individual; a false link is a match between records which refer to different latent individuals; and a missing link is a match which is not found by the model.", "Table REF gives posterior means for the number of true, false, and missing links.", "For the NLTCS, the FNR is $0.11,$ while the FPR is $0.046,$ when we block by date of birth year (DOB) and sex.", "More refined information about linkage errors comes from a confusion matrix, which compares records' estimated and actual linkage patterns (Figure REF and Appendix , Table REF ).", "Every row in the confusion matrix is diagonally dominated, indicating that correct classifications are overwhelmingly probable.", "The largest off-diagonal entry, indicating a mis-classification, is $0.07$ .", "For instance, if a record is estimated to be in both the 1982 and 1989 waves, it is 90% probable that this estimate is correct.", "If the estimate is wrong, the truth is most probably that the record is in all waves (4.4%), followed by the 1982 wave alone (1.4%) and waves 1982 and 1994 (0.15%), and then other patterns with still smaller probability.", "Figure: Heatmap of relative probabilities from the confusion matrix, runningfrom yellow (most probable) to dark red (probability 0).", "The largestprobabilities are on the diagonal, showing that the linkage patternsestimated for records are correct with high probability.", "Mis-classificationrates are low and show a tendency to under-link rather than over-link." ], [ "Example of Posterior Matching Probabilities", "We wish to search for sets of records that match record 10084 in 1982.", "In the posterior samples of $\\mathbf {\\Lambda }$ , this record is part of three maximal matching sets that occur with nonzero estimated posterior probability, one with high and two with low posterior matching probabilities (Table REF ).", "This record has a posterior probability of $0.995$ of simultaneously matching both record 6131 in 1989 and record 5583 in 1994.", "All three records denote a male, born 07/01/1910, visiting office 25 and residing in state 14.", "The unique identifiers show that these three records are in fact the same individual.", "If we threshold matching sets, ignoring ones of low posterior probability, we would simply return the set of records in last column of Table REF ." ], [ "Estimation of Attributes of Observed Individuals from the Population", "The number of observed unique individuals $N$ is easily inferred from the posterior of $\\mathbf {\\Lambda }|\\mathbf {X},$ since $N$ is simply the number of unique values in $\\mathbf {\\Lambda }.$ Defining $N|\\mathbf {X}$ to be the posterior distribution of $N,$ we can find this by applying a function to the posterior distribution on $\\mathbf {\\Lambda }$ , as discussed in §REF .", "(Specifically, $N=|\\#\\mathbf {\\Lambda }|$ , where $\\#\\mathbf {\\Lambda }$ maps $\\mathbf {\\Lambda }$ to its set of unique entries, and $|A|$ is the cardinality of the set $A$ .)", "Doing so, the posterior distribution of $N|\\mathbf {X}$ is given in Figure (REF ).", "Also, $\\hat{N}$ := $E(N|\\mathbf {X}) = 35,992$ with a posterior standard error of 19.08.", "Since the true number of observed unique individuals is 34,945, we are overmatching, which leads to an overestimate of $N$ .", "This phenomenon most likely occurs due to patients migrating between states across the three different waves.", "It is difficult to improve this estimate since we do not have additional information as just described above.", "Figure: Posterior density of the number of observed unique individuals N.N.We can also estimate attributes of sub-groups.", "For example, we can estimate the number of individuals within each wave or combination of waves—that is, the number of individuals with any given linkage pattern.", "(We summarize these estimates here with posterior expectations alone, but the full posterior distributions are easily computed.)", "For example, the posterior expectation for the number of individuals appearing in lists $i_i$ and $i_2$ but not $i_3$ is approximately $\\frac{1}{S_G} \\sum _{h=1}^{S_G} \\sum _{j^{\\prime }}I\\left( \\left| R_{i_1j^{\\prime }}^{(h)} \\right| = 1 \\right)I\\left( \\left| R_{i_2j^{\\prime }}^{(h)} \\right| = 1 \\right)I\\left( \\left| R_{i_3j^{\\prime }}^{(h)} \\right| = 0\\right).$ (Note that the inner sum is a function of $\\mathbf {\\Lambda }^{(h)}$ , but a very complicated one to express without the $R_{ij}$ .)", "Table REF reports the posterior means for the overlapping waves and each single wave of the NLTCS and compares this to the ground truth.", "In the first wave (1982), our estimates perform exceedingly well with relative error of 0.11%, however, as waves cross and we try to match people based on limited information, the relative errors range from 8% to 15%.", "This is not surprising, since as patients age, we expect their proxies to respond, making patient data more prone to errors.", "Also, older patients may move across states, creating further matching dilemmas.", "We are unaware of any alternative algorithm that does better on this data with only these fields available.", "Given these results, and considering how little field information we allowed it to use for matching, we find that our model performs overall very well." ], [ "De-duplication", "Our application of SMERE to the NLTCS assumes that each list had no duplicates, however, many other applications will contain duplicates within lists.", "We showed in §REF that we can theoretically handle de-duplication across and within lists.", "We apply SMERE with de-duplication (SMERED) to the NLTCS by (i) running SMERED on the three waves to show that the algorithm does not falsely detect duplicates when there really are none, and (ii) combining all the lists into one file, hence creating many duplicates, to show that SMERED can find them." ], [ "Application for NLTCS", "We combine the three files of the NLTCS mentioned in § which contain 22,132 duplicate records out of 57,077 total records.", "We run SMERED on settings (i) and (ii), evaluating accuracy with the unique IDs.", "In the the case of running SMERED on the three waves, we compare our results of SMERED and SMERE to that under ground truth (Table REF ).", "In the case of the NLTCS, compiling all three files together and running the three waves separately under SMERED yields similar results, since we match on similar covariate information.", "There is no covariate information to add to from thorough investigation to improve our results, except under simulation study.", "Specifically, when running SMERED for the three files, the FNR is 0.11 and is 0.38 for FPR, while its FNR and FPR is 0.11 AND 0.37 for the one compiled file.", "We contrast this with the FNR of 0.11 and FPR of 0.046 under SMERE for the three waves (Table REF ).", "The dramatic increase in the FPR and number of false links shown in Table REF is explained by how few field variables we match on.", "Their small number means that there are many records for different individuals that have identical or near-identical values.", "On examination, there are $2,558$ possible matches among “twins,” records which agree exactly on all attributes but have different unique IDs.", "Moreover, there are 353,536 “near-twins,” pairs of records that have different unique IDs but match on all but one attribute.", "This illustrates why the matching problem is so hard for the NLTCS and other data sources like it, where survey-responder information like name and address are lacking.", "However, if it is known that each file contains no duplicates, there is no need to consider most of these twins and near-twins as possible matches." ], [ "Discussion", "We have made two contributions in this paper.", "The first is to frame record linkage and de-duplication simultaneously, namely linking observed records to latent individuals and representing the linkage structure via $\\mathbf {\\Lambda }$ .", "The second contribution is our specific parametric Bayesian model, which, combined with the linkage structure, allows for efficient inference and exact error rate calculation.", "Moreover, this allows for easy integration with capture-recapture methods, where error propogation is exact.", "As with any parametric model, its assumptions only apply to certain problems, but it also serves as a starting point for more elaborate models, e.g., with missing fields, data fusion, complicated string fields, population heterogeneity, or dependence across fields, across time, or across individuals.", "Within the Bayesian paradigm, such model expansions will lead to larger parameter spaces, and therefore call for computational speed-ups, perhaps via online learning, variational inference, or approximate Bayesian computation.", "Our work serves as a first basis for solving record linkage problems using a noisy Bayesian model, a linkage structure that can handle large-scale databases, and a model that simultaneously combines record linkage and de-duplication for arbitrarily many files.", "We hope that our approach will encourage the emergence of new record linkage approaches, extensions of our method to non-categorical fields, and applications along with more state-of-the-art algorithms for this kind of high-dimensional data." ], [ "Acknowledgements", "This research was supported by NSF Census Research Network (NCRN), Research Training Grant (NSF), Singapore National Research Foundation (NRF) under its International Research Centre @ Singapore Funding Initiative and the Interactive Digital Media Programme Office (IDMPO) to the Living Analytics Research Centre (LARC).", "We thank the referees, the NCRN research node at CMU, Chris Genovese, Cosma Shalizi, Doug Sparks for providing helpful comments.", "Table: Example of posterior matching probabilities for record 10084 in 1982Table: Comparing NLTCS (ground truth) to the Bayes estimates of matches for SMERE and SMEREDTable: False, True, and Missing Links for NLTCS under blocking sex and DOB year where the Bayes estimates are calculated in the absence of duplicates per file and when duplicates are present (when combining all three waves).", "Also, reported FNR and FPR for NLTCS, Bayes estimates." ], [ "Simulation Study", "We provide a simulation study based on the model in §REF and we simulate data from the NLTCS based on our model, with varying levels of distortion.", "The varying levels of distortion (0, 0.25%, 0.5%, 1%, 2%, 5%) associated with the simulated data are then run using our MCMC algorithm to assess how well we can match under “noisy data.” Figure REF illustrates an approximate linear relationship with FNR and the distortion level, while we see an near-exponential relationship between FPR and the distortion level.", "Figure REF demonstrates that for moderate distortion levels (per field), we can estimate the true number of observed individuals extremely well via estimated posterior densities.", "However, once the distortion is too noisy, our model has trouble recovering this value.", "In summary, as records become more noisy or distorted, our matching algorithm typically matches less than 80% of the individuals.", "Furthermore, once the distortion is around 5%, we can only hope to recover approximately 65% of the individuals.", "Nevertheless, this degree of accuracy is in fact quite encouraging given the noise inherent in the data and given the relative lack of identifying variables on which to base the matching.", "Figure: Posterior density estimates for 6 levels of distortion (none, 0.25%, 0.5%, 1%, 2%, and 5%) compared to ground truth (in red).", "As distortion increases (and approaches 2% per field), we undermatch NN, however as distortion quickly increases to high levels (5% per field), the model overmatches.", "This behavior is expected to increase for higher levels of distortion.", "The simulated data illustrates that under our model, we are able to capture the idea of moderate distortion (per field) extremely well." ], [ "Convergence Diagnostics and Hyperparameter Sensitivity", "As for convergence diagnostics, for $S_G,$ our standard for NLTCS when running SMERE was to set $S_G = S_M = 10^5$ , after fixing on a burn-in of 1000 steps and a thinning the chain by 100 iterations from pilot runs.", "For SMERED, we used $S_G = 10^5$ and $S_M = 10000.$ Moreover, our simulation study (Appendix A) varies $a_{\\ell }$ and $b_{\\ell }$ but we do not varying $\\mu _l$ away from a uniform; if users have a priori knowledge regarding some idea about the expected distribution of categories, though, this could be incorporated fairly directly.", "For the NLTCS study itself, we set the parameters of $\\beta $ are $a_{\\ell } = 5 $ and $b_{\\ell } = 10$ and took $\\mu _{\\ell } = 1,$ corresponding to equivalent to a uniform distribution over the $M_{\\ell }-1$ simplex." ] ]
1403.0211
[ [ "Extended versus localized vibrations: the case of L-cysteine and\n L-cystine amino acids" ], [ "Abstract A detailed quantitative analysis of the specific heat in the $1.8-300$ K temperature range for L-cysteine and L-cystine amino acids was presented.", "We observed not extended but a sharp transition at $\\sim 76$ K for L-cysteine.", "This transition was associated to the thiol group ordering and the order-disorder transition was adequately modeled by a 2D Ising model.", "The energy difference among two thiol configurations was found to be $-J=\\varepsilon_{A}-\\varepsilon_{B}=-66.6$ cal/mole.", "Besides, we conducted a study of phonon and rotor contributions to the specific heat and we proposed a generalization of Debye model.", "It was possible to evaluate the exponent of the $g(\\omega)$, leading to the result that it corresponds to the Debye model for L-cysteine, which implies that the boson peak in this system is due to a maximum in the $C_{coup}(\\omega)$ and also that the plane wave of wave-vector $\\vec{q}$ is a good approximation to describe the phonons.", "On the other hand the origin of the boson peak for L-cystine correlates to a peak in $g(\\omega)$ and phonons in L-cystine could be well represented by strongly attenuated plane waves or localized vibrations.", "Lastly, the analysis at very low temperature ($T<3$ K) indicated that L-cysteine presented a nearly temperature independent behaviour which is opposite to which is widely observed in systems with glassy characteristics within the Two-Level System (TLS) framework." ], [ "Introduction", "It has been reported that biological macromolecules present two dynamical transitions at $T_{D}\\sim 200-230$ K and $T^{*}\\sim 80-100$ K [1], [2], [3], [4], [5], [6], [7].", "The first one occurs at hydration levels greater than $\\sim 18\\%$ , and it is related to a deviation from anharmonic to harmonic behavior of the mean squared atomic displacement with decrease of the temperature [4].", "According to some authors (see e.g.", "Ref.", "[8]) $T_{D}$ corresponds to the onset of a glass transition, $T_{g}$ , although some researchers [9] pointed out that $T_{D}$ and $T_{g}$ have different physical origin.", "Additionally, it has been suggested that $T_{D}$ is correlated to onset of biochemical activities of the macromolecule [1], [3], [2], [4], [10].", "Recent work [11] reported that some physical properties of hydrated L-cysteine resemble those of quantum glass materials.", "Furthermore, a universal feature of such systems is that the vibrational density of state ($g(\\omega $ )) departs from the squared-frequency Debye-law, displaying an excess of states, the boson peak [12].", "Transition at $T^{*}$ is hydration level independent [4].", "Some works [4], [13] interpreted this transition as related to the thermal activation of methyl groups rotation.", "However, neither the microscopic nature nor the biological relation of these transitions is completely understood.", "A detailed investigation of the crystal structures of amino acids and their dynamics is very important to understand complexes biological molecules [14], [15].", "Besides, it was shown [16], [17] that both transitions do not require the protein polypeptide chain as well as the protein secondary and tertiary structure.", "Intramolecular motions and intermolecular interactions could be probed by experimental techniques where temperature and pressure are tuning parameters [15].", "Several studies based on the calorimetric measurements of amino acids that revealed phase transition can be mentioned.", "Wang et al.", "[18] observed a $\\lambda -$ transition at 272 K by differential scanning calorimetry (DSC) for D-valine.", "It was proposed that the shape of the jump for D-valine is due to electron coupling.", "For taurine, Lima et al.", "[19] found the existence of a first-order transition at 251 K with temperature-dependent Raman spectroscopy that was confirmed by DSC data.", "Besides, it was observed by Drebushchak et al.", "[20] a second order phase transition near 252 K for $\\beta $ polymorph of glycine comparing the results with the data for $\\alpha $ -glycine.", "Such transition was considered as ferroelectric-paraelectric transition.", "The orthorhombic polymorph of the amino acid L-cysteine has been also focus of recent interest.", "This amino acid possesses a very simple chemical structure and high biological relevance.", "The thiol or sulfurous group in the residues of L-cysteine is the most chemically reactive site in proteins under physiological conditions [21].", "This compound presents a tiny specific heat anomaly near $\\sim 76$ K [22], [23].", "These authors presented a qualitative interpretation of this anomaly to an order-disorder phase transition bearing in mind the results of ref.", "[24] where it was shown that the thiol groups are ordered at 30 K. Paukov et al.", "[22], [23] measured the L-cysteine specific heat in pulse and continuous modes in the region of the anomaly, however no sharp order-disorder transition was observed.", "Kolesov et al.", "[15] utilizing variable-temperature polarized Raman spectroscopy verified the dynamic transition related to switching from $S-H\\cdots S$ hydrogen bonds to the $S-H\\cdots O$ contacts is not sharp, but is extended in a wide temperature range as observed by Paukov et al.", "[22], [23].", "Another qualitative propose concerning the nature of the transition near $\\sim 76$ K for L-cysteine relies to the rotation of $CH_{2}$ group [16].", "In the present work a detailed quantitative analysis of the specific heat in the $1.8-300$ K temperature range for L-cysteine is presented.", "A comparison with L-cystine amino acid, was also performed.", "L-cystine is formed by two cysteine molecules linked via a disulfide bond which prohibit thiol ordering [25].", "The samples used were the commercial powder of orthorhombic crystalline L-cysteine and hexagonal crystalline L-cystine from Sigma-Aldrich (purity of 97% and 98%, respectively).", "According to X-ray diffraction data obtained with STOE STADI-P diffractometer and performing the Rietveld method analysis (using GSAS+EXPGUI software [26], [27]), it was possible to verify that the crystal structure of L-cysteine is orthorhombic as previously determined in ref.", "[28].", "The space group is $P2_12_12_1$ with $Z=4$ and unit cell lattice parameters $a=8.11639(7)$ Å, $b=12.17169(11)$ Å, and $c=5.42266(4)$ Å.", "The crystal structure of L-cystine was refined as belonging to space group $P6_122$ (hexagonal) $Z=6$ with unit cell lattice parameters $a=b=5.42264(5)$ Å, and $c=56.2908(5)$ Å, in accordance with Ref.", "[29]." ], [ "Calorimetric measurements", "The calorimetric measurements were performed in the Physical Properties Measurements System (PPMS) with Evercool-II®option from Quantum Design Inc.", "This system employs a thermal-relaxation calorimeter that determine the specific heat of the sample by measuring the thermal response to a change in heating conditions [30]." ], [ "Theoretical models", "It has been pointed out that a superposition of complexes contributions need to be considered to explain the specific heat of L-cysteine.", "Due to the strong anharmonicity of the system and the glassy behavior, the usual low temperature Debye contribution to specific heat $c_{D}\\propto T^{3}$ need to be revised.", "Moreover, the methylene group rotors and the order-disorder contributions need also to be taken into account." ], [ "Generalized Debye model (GDM)", "Commonly, the approach used to describe the acoustic phonon contributions to the specific heat is the Debye model.", "However it has been shown that amino acids such as L-cysteine presents glass-like behavior, e.g.", "an excess contribution to the usual $g(\\omega )$ that can be observed in the scaled specific heat $c_{p}(T)/T^3$ at low temperatures [11].", "Within the phonon localization picture model for boson peak [31] one expects that the linear phonon dispersion law breaks down.", "We proposed a power-law for dispersion relation $\\omega \\left(q\\right)=vq^{\\alpha }$ where $v$ is the sound speed of a transverse or longitudinal phonon.", "Therefore, the $g(\\omega )$ will be written as $g(\\omega )=\\frac{3\\mathcal {V}}{2\\pi ^2 \\alpha v}\\left( \\frac{\\omega }{v}\\right)^{3/\\alpha - 1},$ where $\\mathcal {V}$ is the unit cell volume.", "The total internal energy is $U=\\frac{3\\mathcal {V}\\hbar }{2\\pi ^2 \\alpha v^{3/\\alpha }}\\int _{0}^{\\infty }\\frac{\\omega ^{3/\\alpha }}{e^{\\frac{\\hbar \\omega }{k_{B}T}}-1}d\\omega ,$ where $\\hbar $ is the reduced Planck constant, $k_{B}$ is the Boltzmann constant.", "A sharp cutoff at $\\omega _{c}$ is chosen that the total number of modes equals the number of vibrational degrees of freedom, $3\\mathcal {N}$ .", "Thus, $\\omega _{c}=v\\left(\\frac{6\\pi ^{2}\\mathcal {N}}{\\mathcal {V}}\\right)^{\\alpha /3}.$ The molar specific heat is calculated by $c_{\\mathcal {V}}^{phonons}=\\frac{N_{A}}{\\mathcal {N}\\mathcal {V}}\\left(\\frac{\\partial U}{\\partial T }\\right)_{\\mathcal {V}}= \\nonumber \\\\n\\frac{9R}{\\alpha }\\left(\\frac{T}{\\theta _{c}}\\right)^{3/\\alpha }\\int _{0}^{\\theta _{c}/{T}}\\frac{x^{3/\\alpha +1}e^{x}}{\\left(e^{x}-1\\right)^2}dx.$ where $R$ is the gas constant, $\\hbar \\omega _{c}=k_{B}\\Theta _{c}$ and $n$ is the number of atoms per unit formula.", "In the case of the L-cysteine and the L-cystine $n=14$ and 28, respectively.", "Note that the usual Debye model is the particular $\\alpha =1$" ], [ "Specific heat of anisotropic rigid rotors", "Other relevant contribution to specific heat of biomolecules to be considered arises from the methyl or methylene groups rotations.", "Once approximating this rotating side chain as an anisotropic rigid body one could use the results of Caride and Tsallis [32] and compute this contribution to the molar specific heat as $c_{\\mathcal {V}}^{rotors}=R\\frac{1}{T^2}\\left\\lbrace \\frac{V}{Z}-\\left(\\frac{W}{Z}\\right)^2\\right\\rbrace $ with $V\\equiv \\sum _{l=0}^{\\infty } (2l+1)e^{\\frac{-l(l+1)}{t}}\\times \\\\\\sum _{m=-l}^{l} \\left\\lbrace l(l+1)+\\left(\\frac{I_{xy}}{I_z}-1\\right)m^2\\right\\rbrace ^2 e^{\\frac{\\frac{I_{xy}}{I_z}-1}{t}},$ $W\\equiv \\sum _{l=0}^{\\infty } (2l+1)e^{\\frac{-l(l+1)}{t}}\\times \\\\\\sum _{m=-l}^{l} \\left\\lbrace l(l+1)+\\left(\\frac{I_{xy}}{I_z}-1\\right)m^2\\right\\rbrace e^{\\frac{\\frac{I_{xy}}{I_z}-1}{t}},$ $Z\\equiv \\sum _{l=0}^{\\infty } (2l+1)e^{\\frac{-l(l+1)}{t}}\\sum _{m=-l}^{l} e^{\\frac{\\frac{I_{xy}}{I_z}-1}{t}},$ where $l$ is the angular momentum, $I_x=I_y=I_{xy}$ is the moment of inertia about $x$ and $y$ axes with the same module, $I_z$ the moment of inertia about $z$ axis, $m$ is the magnetic quantum number ranging from $-l,-l+1,\\dots ,l$ ." ], [ "Order-disorder and Ising model", "Once making the analogy of the two possible states of the spins with the two possible states of the thiol groups in the plane the hypothesis of order-disorder transition at $\\sim 76$ K for L-cysteine could be modeled by using the 2D Ising model [33].", "Thereby, the molar specific heat at the order-disorder transition is given by $c_{\\mathcal {V}}^{Ising}(T)=\\frac{2R}{\\pi } K^{2}\\coth ^2(2K)\\left\\lbrace 2K(\\kappa )-2E(\\kappa )-2\\operatorname{sech}^2(2K)\\left(\\frac{\\pi }{2}+(2\\tanh (2K)-1)K(\\kappa )\\right)\\right\\rbrace ,$ with $\\kappa =2\\sinh (2K)/\\cosh ^2(2K)$ , $K=J/k_{B}T$ , $K(\\kappa )$ , and $E(\\kappa )$ are elliptic integrals of the first and second type, respectively.", "The parameter $J$ corresponds to the average energy between the two average ordered and disordered structures.", "The first question that will be addressed concerns the nature of the transition at $\\sim 76$ K. As pointed by Lashley et al.", "[30], it is important to stress that there are many difficulties in analyzing the individual contributions from various degrees of freedom to the specific heat at low temperatures.", "Consequently, there is a great deal of effort that goes into controlling the details that are part of calorimetric measurements.", "Usually the control of details as thermometry, temperature-scale issues, and the creation and control of heat leaks specially in sharper transitions of first order is needed.", "In a commercially available calorimeter a large number of these details might be hidden from the user [30].", "The specific heat can be determined in the vicinity of a phase transition by analyzing the thermal-relaxation data point-by-point rather than by obtaining a single $C_{\\mathcal {P}}$ value for the entire temperature region spanned by the decay [30].", "From the time-dependent relaxation data $T(t)$ the specific heat near a phase transition is obtained by $C_{\\mathcal {P}}[T(t)]=-K\\frac{(T-T_{0})}{dT(t)/dt},$ where $K$ is the thermal conductivity of the calorimeter wires and $T_{0}$ is thermal bath temperature.", "Figure: a) Relaxation data used to determine C p (T)C_{p}(T) around T c ∼76T_{c}\\sim 76 K. b) The specific heat data calculated in the vicinity of the first-order transition.Figure REF a) presents the relaxation data $T(t)$ around 76 K. The $C_{\\mathcal {P}}[T(t)]$ data obtained by using Eq.", "REF is shown on Fig.", "REF b).", "In the narrow temperature window of $\\sim 0.20$ K the sharpness of the transition becomes clear.", "The transition jump starts to develops at $T\\sim 75.96$ K ending at $T\\sim 76.96$ K with maximum of $\\sim 100$ J/mole K which is limited by experimental resolution.", "The jump in the specific heat compared to the overall $1.8-300$ K data is shown on Fig.", "REF a).", "Figure REF b) shows the molar specific heat data for L-cystine amino acid.", "The absence of phase transition in this case corroborates the hypothesis that thiol order-disorder transition is responsible for the sharp peak observed at 76 K for L-cysteine.", "The ordered configuration corresponds to the (A) scheme shown on Fig.", "REF .", "The Ising model was able to reproduce the peak corresponding to ordering of thiol groups with acceptable accordance.", "The jump observed experimentally appeared to be narrow than the Ising simulation peak.", "This fact could be explained remembering that Eq.", "REF does not take into the several possible thermally activated disordered configurations (B).", "Usually, Ising Monte Carlo simulations sampling several configurational possibilities furnish better accordance with experimental data.", "The energy cost of the thiol ordering is $-J=\\varepsilon _{A}-\\varepsilon _{B}=-66.6$ cal/mole.", "Besides the ordering transition, contributions from phonons and rotors also need be considered.", "The quantity experimentally accessed by our experiments is $C_{\\mathcal {P}}$ .", "Since for solids $C_{\\mathcal {V}}\\approx C_{\\mathcal {P}}$ and based on Eqs.", "REF , REF , REF the total molar specific heat of L-cysteine could be modeled according to $c_{\\mathcal {P}}=ac^{phonons}(\\Theta _{c},\\alpha ,T)+c^{Ising}(J,T)\\nonumber \\\\+bc^{rotors}(L, I_{xy}, I_{z}, T),$ For the L-cystine case the contribution due to order-disorder of the thiol group was not considered.", "Selected simulations and each contribution to specific heat are shown in Fig.", "REF for L-cysteine and L-cystine.", "For both amino acids, it was performed simulations taking $\\alpha =1$ (Debye model) and $\\alpha \\ne 1$ .", "The goodness of simulations was evaluated by computing the difference $\\Delta ^2 c=(c_{experimental}-c_{simulated})^2$ (Fig.", "REF ).", "Table REF summarizes all obtained parameters.", "Moments of inertia of $CH_{2}$ are in accordance to prolate symmetric top ($I_{xy}<I_z$ ), which is consistent with the result found by Lima et al.", "[16].", "$I_{xy}$ and $I_{z}$ for L-cystine were more sensible to $\\alpha $ presenting $\\sim 11$ % of variation.", "The obtained values are of the same magnitude order of those calculated from atom masses and distances.", "Table: Table of the parameters obtained by simulation for L-cysteine and L-cystine.Phonon contributions to the simulations need to be analyzed in more details due to implications on the $g(\\omega )$ .", "It is the main contribution to low temperature $c_{\\mathcal {P}}$ .", "Confining our analysis to $T\\le 50$ K, one could conclude from Fig.", "REF that L-cysteine data were best simulated with Debye model ($\\alpha =1$ ).", "On the other hand, the $\\alpha =1.5$ gave the best results for L-cystine.", "From values of $\\alpha $ obtained by simulations we could infer that $g(\\omega ) \\propto \\omega ^{2}$ ($\\alpha =1$ ) and $g(\\omega ) \\propto \\omega ^{2.75}$ ($\\alpha =1.5$ ) for L-cysteine and L-cystine, respectively.", "These findings have very important implications to the boson peak origin comprehension.", "Shuker et al.", "[34] have shown that the low frequency Raman scattering intensity of amorphous solid is $I(\\omega , T)=C_{coup}(\\omega )\\frac{g(\\omega )}{\\omega }\\left[1+n(\\omega ,T)\\right],$ where $C_{coup}(\\omega )$ is the coupling constant, $n(\\omega ,T)$ is the Bose-Einstein occupation factor.", "This expression have been widely used to explain the boson peak in glasses.", "Figure: Experimental data and best simulated curve using Eq.", "for L-cysteine (a) and L-cystine (b).", "The Ising contribution was not considered for L-cystine.Figure: Squared difference plot Δ 2 C\\Delta ^2C for L-cysteine (a) and L-cystine for some selected α\\alpha values.Since $g(\\omega )$ corresponds to the Debye model for L-cysteine, one could infer that the boson peak in this system does not have its origin due to a peak in the vibrational density of states but due a maximum in the $C_{coup}(\\omega )$ .", "Thereby, for L-cysteine the dispersion relation $\\omega =vq$ is expected to be valid up to higher frequencies.", "Thus the plane wave of wave-vector $\\vec{q}$ is a good approximation to describe the phonons [35].", "However, for L-cystine, the excess of vibrational density of states compared to Debye model is clear from the exponent dependence of $g(\\omega )$ and the origin of the boson peak for L-cystine correlates to a peak in $g(\\omega )$ .", "Therefore, phonons in L-cystine could be well represented by strongly attenuated plane waves or localized vibrations.", "This very distinct behavior has direct impact on $\\Theta _{c}$ estimation since the localized vibrations results an increase of 75 K for this parameter.", "Figure: c p /T 3 c_{p}/T^{3} vs TT specific heat data of L-cysteine (closed circles) and L-cystine (open circles).", "Dashed lines represent the TLS contribution T -0.005 T^{-0.005} and T -0.65 T^{-0.65} for L-cysteine and L-cystine, respectively.At very low temperatures ($\\sim 1$ K) the specific heat of glasses is usually described in the two-level systems (TLS) model framework [36].", "This model assumes the glass state as “frozen liquid\", with a large number of metastable states.", "A very low temperatures thermally activated processes between these states are highly improbable.", "Therefore one is left with the idea of tunneling process between two states that correspondS to the two local minima of configuration [36].", "Fig.", "REF shows the log-log plot of $c_{p}/T^{3}$ vs $T$ for L-cysteine and L-cystine samples.", "The Debye plus TLS contribution was fitted to $T^{-0.005}$ and $T^{-0.65}$ for L-cysteine and L-cystine, respectively." ], [ "Conclusion", "From our quantitative analysis of the specific heat results for L-cysteine and L-cystine we conclude that the transition at $\\sim 76$ K for L-cysteine is due to thiol group ordering.", "We show that this transition is not extended as presented in literature, but is a sharp first order phase transition.", "We elaborate that its sharpness prevented others researchers to clearly observe the emergence of the peak.", "The order-disorder transition was adequately modeled by Ising model.", "The energy cost of the thiol ordering was obtained as $-J=\\varepsilon _{A}-\\varepsilon _{B}= - 66.6$ cal/mole.", "Phonon and rotor contributions were also analyzed.", "From the conjugated analysis it was possible estimate the exponent of the $g(\\omega )$ .", "It was found that it corresponds to the Debye model for L-cysteine, which imply that the boson peak in this system is due to a maximum in the $C_{coup}(\\omega )$ and also that the plane wave of wave-vector $\\vec{q}$ is a good approximation to describe the phonons.", "On the other hand, for L-cystine, the origin of the boson peak correlates to a peak in $g(\\omega )$ and phonons in L-cystine could be well represented by strongly attenuated plane waves or localized vibrations.", "Analysis at very low temperature ($T<1$ K) indicates that L-cysteine presented a nearly temperature independent behavior of is a remarkable finding for a system with glass characteristics since does not follow the prevision of TLS model.", "The authors would like to thank the Brazilian agencies CNPq and FAPESP for their financial support and the Multiuser Central Facilities at UFABC (CEM-UFABC) for providing conditions to perform the experiments described in this work." ] ]
1403.0029
[ [ "Entropy of thin shells in a (2+1)-dimensional asymptotically AdS\n spacetime and the BTZ black hole limit" ], [ "Abstract The thermodynamic equilibrium states of a static thin ring shell in a (2+1)-dimensional spacetime with a negative cosmological constant are analyzed.", "Inside the ring, the spacetime is pure anti-de Sitter (AdS), whereas outside it is a Ba\\~nados-Teitelbom-Zanell$ (BTZ) spacetime and thus asymptotically AdS.", "The first law of thermodynamics applied to the thin shell, plus one equation of state for the shell's pressure and another for its temperature, leads to a shell's entropy, which is a function of its gravitational radius alone.", "A simple example for this gravitational entropy, namely, a power law in the gravitational radius, is given.", "The equations of thermodynamic stability are analyzed, resulting in certain allowed regions for the parameters entering the problem.", "When the Hawking temperature is set on the shell and the shell is pushed up to its own gravitational radius, there is a finite quantum backreaction that does not destroy the shell.", "One then finds that the entropy of the shell at the shell's gravitational radius is given by the Bekenstein-Hawking entropy." ], [ "Introduction", "Due to the long-range interaction of the gravitational field, gravitating systems have important and interesting thermodynamic properties, such as negative specific heat, making the systems unstable with consequent gravitational collapse or energy loss through evaporation.", "This happens both in Newtonian gravitation and in general relativity.", "A well-known instance of this fact is given by the black hole system, whose thermodynamic properties were understood by Bekenstein [1] and put on a firm basis by Hawking, by discovering that through quantum effects it radiates at a definite temperature [2].", "Refinement of the study of black hole thermodynamics appeared in many guises, in particular by the introduction of a formalism useful for studying general relativistic systems in a canonical ensemble [3], [4].", "Another gravitating system in general relativity prone to a thermodynamic study is a thin shell and the spacetime it generates.", "Spurred by the interest in black hole thermodynamics, some studies have analyzed the thermodynamics of thin shells in black hole spacetimes [5], or of pure thin shells in 3+1 spacetimes, notably in [6], where several thermodynamic quantities of thin shells are discussed and a stability analysis of them is performed.", "Other studies on the thermodynamics of thin shells are [7], [8].", "For related studies of thermodynamics of gravitating matter, especially quasiblack holes, i.e., stars on the verge of becoming a black hole, see [9], [10].", "All of these works are in the usual 3+1 dimensions.", "Now, in many senses, it is interesting to reduce the spatial dimension by 1 and study general relativity in 2+1 dimensions.", "This plays an important role in the understanding of systems in curved spacetime, as the decrease in dimensionality with respect to the usual 3+1 spacetime reduces the degrees of freedom to a few.", "This leaves possible complications aside and keeps the essential physical features.", "The interest in (2+1)-dimensional general relativity underwent a boost after a black hole solution was found in spacetimes with negative cosmological constant, i.e., spacetimes with an anti-de Sitter (AdS) background [11], [12].", "This (2+1)-dimensional black hole, the Bañados-Teitelbom-Zanelli (BTZ) black hole, belongs to a family of solutions, which, depending on the parameters of the solution, includes the BTZ black holes themselves, positive mass naked singularities, the AdS spacetime, and negative mass naked singularities.", "The BTZ black hole, the most important solution in the family, is a black hole solution in its simplest form.", "The singularity it hides is not a curvature singularity, but rather is a much milder topological singularity, akin to the conical singularities [11], [12].", "In the realm of thermodynamics and its connection to the quantum world, the BTZ black hole has a Bekenstein-Hawking entropy $S_{\\rm BH}= \\frac{1}{4}\\,\\frac{A_{\\rm h}}{l_{\\rm p}}$ , where $A_{\\rm h}$ is the horizon area, in 2+1 dimensions a circumference, $A_{\\rm h}=2\\,\\pi \\,r_+$ , $r_+$ is the horizon radius, and $l_{\\rm p}$ is the Planck length given by $l_{\\rm p}=G_3\\,\\hbar $ , $G_3$ being the three-dimensional gravitational constant and $\\hbar $ Planck's constant, and a Hawking temperature given by $T_{\\rm H} = \\frac{l_{\\rm p}}{2 \\pi G_3\\,l^2}\\,r_+$ [11] (we put $k_{\\rm B}=1$ and $c=1$ ).", "The thermodynamic and entropy properties of the BTZ black hole have been further explored in, e.g., [13], [14], [15], [16].", "In 2+1 dimensions, as in the 3+1 case, it is also interesting to study the thermodynamics of self-gravitating thin shells, since the BTZ black hole can form from the gravitational collapse of such thin shells [17], [18], [19] which, when static, can be stable or unstable according to their intrinsic parameters [20].", "In 2+1 dimensions, the thermodynamics of thin shells in spacetimes with zero cosmological constant has been studied [21].", "Motivated in part by this study [21] and also from the fact that in 2+1 dimensions in general relativity with a cosmological constant there are BTZ black holes with interesting thermodynamic properties, we want to study in this article the thermodynamics of static thin matter shells in 2+1 dimensions.", "In particular, we intend to find the shell's entropy and analyze their thermodynamic stability, as well as scrutinize the limit when the radius of the shell $R$ goes into its own gravitational radius.", "To find the spacetime solution we employ the junction conditions formalism for a thin shell [22], where in this 2+1 case the shell is simply a ring.", "One can then determine the pressure and the mass density of the shell in order for it to be static in a spacetime with a negative cosmological constant, in which the interior to the shell is pure AdS and the exterior is asymptotically AdS.", "Employing the first law of thermodynamics and using the formalism for the usual thermodynamic systems [23], which was developed by Martinez to apply to thin matter shell systems in 3+1 general relativity [6] (see, also, [3], [4]), one finds then the entropy for these gravitating systems, the desired thermodynamic properties, and the quasiblack hole limit.", "The paper is organized as follows.", "In Sec.", ", we compute the components of the extrinsic curvature of the ring shell that leads to the shell's linear density and pressure.", "We also discuss the no-trapped-surface condition and the dominant energy condition.", "In Sec.", "we review the thermodynamics of the shell.", "We use the entropy representation and assume that the state variables are the proper mass of the shell and its perimeter, or radius.", "We then use the first law of thermodynamics for a one-dimensional system to display the integrability and the stability conditions of the thermodynamic system.", "In Sec.", "we present the equation of state for the pressure in terms of the proper mass and radius of the shell, and we derive the equation of state for the temperature of the shell as a function of the state variables.", "In Sec.", "we use the previously obtained integrability conditions for the first law of thermodynamics in order to simplify the entropy differential of the shell.", "This allows us to obtain an expression for the entropy up to an arbitrary function of the gravitational radius.", "In Sec.", "we consider a phenomenological expression for the arbitrary function, consisting in a power law of the gravitational radius, which allows us to obtain an explicit expression for the entropy.", "We then analyze the thermodynamic stability of the system by calculating the permitted intervals of the free parameters in order for the shell to remain thermodynamically stable.", "In Sec.", ", the arbitrary function will be equated to the inverse Hawking temperature, and it will be found that the Bekenstein-Hawking entropy of the BTZ black hole naturally arises when the shell is pushed up to its gravitational radius.", "Finally, in Sec.", "we draw some conclusions." ], [ "The thin shell spacetime", "Einstein's equation in 2+1 dimensions is written as $G_{\\alpha \\beta }-\\Lambda g_{\\alpha \\beta }=8\\pi G_3T_{\\alpha \\beta }\\,,$ where $G_{\\alpha \\beta }$ is the Einstein tensor, $\\Lambda $ is the cosmological constant, $g_{\\alpha \\beta }$ is the spacetime metric, $8\\pi G_3$ is the coupling with $G_3$ being the gravitational constant in 2+1 dimensions, and $T_{\\alpha \\beta }$ is the energy-momentum tensor.", "We keep units where the velocity of light is $c=1$ , and thus $G_3$ has units of the inverse of mass.", "Greek indices are spacetime indices and run as $\\alpha ,\\beta =0,1,2$ , with 0 being the time index.", "Since we want to work in an AdS background where the cosmological constant is negative, we define the AdS length $l$ through the equation $-\\Lambda =\\frac{1}{l^2}\\,.$ We now consider a one-dimensional timelike shell, i.e., a ring, with radius $R$ in a (2+1)-dimensional spacetime.", "The ring divides spacetime into two parts, an inner region $\\mathcal {V}_-$ and an outer region $\\mathcal {V}_+$ .", "To find the corresponding spacetime solution, we follow [22].", "In the inner region $\\mathcal {V}_-$ ($r\\le R$ ), inside the ring, we consider a spherically symmetric AdS metric, with cosmological length $l$ , given by $ds_-^2 & = g_{\\alpha \\beta }^- dx^\\alpha dx^\\beta =\\nonumber \\\\&-\\frac{r^2}{l^2}\\,dt^2 +\\frac{dr^2}{\\frac{r^2}{l^2}}+ r^2\\, d\\phi ^2\\,,\\quad r\\le R\\,,$ where polar coordinates $x^{\\alpha -}=(t,r,\\phi )$ are used.", "In the outer region $\\mathcal {V}_+$ ($r\\ge R$ ), outside the shell, the spacetime is described by the BTZ line element $ds_+^2 & = g_{\\alpha \\beta }^+ dx^\\alpha dx^\\beta =-\\left(\\frac{r^2}{l^2} - 8G_3m\\right)\\,dt^2 +\\nonumber \\\\& \\hspace{76.82243pt}\\frac{dr^2}{\\left(\\frac{r^2}{l^2} - 8G_3 m\\right)} + r^2 \\,d\\phi ^2\\,,\\quad r\\ge R\\,,$ written also in polar $x^{\\alpha +}=(t,r,\\phi )$ coordinates.", "Here, $m$ is a constant which is interpreted as the Arnowitt-Deser-Misner (ADM) mass, or energy.", "At $r\\rightarrow \\infty $ the spacetime is asymptotically AdS.", "On the hypersurface itself, the induced metric $h_{ab}$ yields the line element $ds_{\\Sigma }^2 = h_{ab} dy^a dy^b =-d\\tau ^2 + R^2(\\tau ) d\\phi ^2,$ where we have chosen $y^a=(\\tau ,\\phi )$ as coordinates on the shell and where we have adopted the convention to use latin indexes for the components on the hypersurface.", "The shell ring is at radius $R=R(\\tau )$ , and the parametric equations of the ring hypersurface for both the $\\mathcal {V}_-$ and $\\mathcal {V}_+$ are $r=R(\\tau )$ and $t = T(\\tau )$ .", "The induced metric $h_{ab}$ is written in terms of the metrics $g^{\\pm }_{\\alpha \\beta }$ as $h^{\\pm }_{ab} = g^{\\pm }_{\\alpha \\beta } \\, e^{\\alpha }_{\\pm }{}_a \\,e^{\\beta }_{\\pm }{}_b$ where $e^{\\alpha }_{\\pm }{}_a$ are tangent vectors to the hypersurface viewed from each side of it.", "The formalism employed in [22] uses two conditions in order to assure the smoothness of the metric across the hypersurface.", "These are the junction conditions.", "The first junction condition states that $[h_{ab}]=0\\,,$ where the parentheses symbolize the jump in the quantity across the hypersurface, here the induced metric.", "This condition leads to the relation $ \\left(\\frac{r^2}{l^2} - 8G_3m\\right)\\dot{T}^2 - & \\frac{\\dot{R}^2}{\\left(\\frac{r^2}{l^2} - 8G_3m \\right)}\\nonumber \\\\&= \\frac{r^2}{l^2}\\dot{T}^2 -\\left(\\frac{r^2}{l^2}\\right)^{-1} \\dot{R}^2 = 1\\,,$ where a dot denotes differentiation with respect to $\\tau $ .", "The second junction condition makes use of the extrinsic curvature $K^{a}{}_{b}$ defined as $K^a_{\\pm }{}_{b} = \\nabla _{\\beta } n_{\\alpha } \\, e^{\\alpha }_{\\pm }{}_c\\, e^{\\beta }_{\\pm }{}_b \\, h^{ca}_{\\pm }\\,$ where $\\nabla _{\\beta }$ denotes the covariant derivative and $n_\\alpha $ is the normal to the shell.", "When the jump in this quantity is non-null, there exists a thin matter shell with stress-energy tensor $S^{a}{}_{b}$ given by $S^{a}{}_{b}=-\\frac{1}{8 \\pi G_3}\\left([K^{a}{}_{b}]-[K]h^{a}{}_{b}\\right)\\,,$ where $K = h^{b}{}_{a} K^{a}{}_{b}$ .", "For the line elements (REF ) and (REF ), and using Eq.", "(REF ), one can compute the nonzero components of $K^{a}{}_{b}$ .", "They are $K^{\\tau }_{+}{}_{\\tau } &= \\frac{\\frac{R}{l^2}+\\ddot{R}}{\\sqrt{-8G_3m+\\frac{R^2}{l^2}+\\dot{R}^2}}\\,, \\\\K^{\\tau }_{-}{}_{\\tau } &= \\frac{\\frac{R}{l^2}+\\ddot{R}}{\\sqrt{\\frac{R^2}{l^2}+\\dot{R}^2}}\\,, \\\\K^{\\phi }_{+}{}_{\\phi } &= \\frac{1}{R}\\sqrt{-8G_3 m+\\frac{R^2}{l^2}+\\dot{R}^2}\\,, \\\\K^{\\phi }_{-}{}_{\\phi } &= \\frac{1}{R}\\sqrt{\\frac{R^2}{l^2}+\\dot{R}^2}\\,.$ Imposing that the shell is static, i.e., $\\dot{R}=0$ and $\\ddot{R}=0$ , one finds the non-null components of the stress-energy tensor for a static shell, $S^{\\tau }{}_{\\tau } & = \\frac{\\sqrt{-8G_3m + \\frac{R^2}{l^2}}-\\frac{R}{l}}{8 \\pi G_3R} \\\\S^{\\phi }{}_{\\phi } & = \\frac{1}{8 \\pi G_3}\\frac{R}{l^2}\\bigg (\\frac{1}{\\sqrt{-8G_3m+ \\frac{R^2}{l^2}}} - \\frac{1}{\\frac{R}{l}}\\bigg ).$ If, in addition, we consider the shell to be made of a fluid with linear energy density $\\lambda $ and pressure $p$ , the stress-energy tensor will have the form $S^{a}{}_{b} = (\\lambda +p) u^a u_b + ph^{a}{}_{b}\\,,$ where $u^a$ is the 3-velocity of a shell element.", "Thus, for such a fluid, we find $S^{\\tau }{}_{\\tau } =-\\lambda \\,,$ $S^{\\phi }{}_{\\phi } = p\\,.$ Note that for a one-dimensional fluid with linear energy density and pressure in a (2+1)-dimensional spacetime, there are only two possible degrees of freedom to characterize the system.", "Therefore, the stress-energy tensor (REF ) is the most general one that one can consider in this setting, and it is thus seen to be the stress-energy tensor for a perfect fluid.", "Equations (REF )-() together with Eqs.", "(REF )-(REF ) yield $\\lambda & = \\frac{1}{8\\pi G_3\\, R}\\left({\\frac{R}{l}-\\sqrt{-8G_3m + \\frac{R^2}{l^2}}}\\right)\\,,\\, \\\\p & = \\frac{1}{8 \\pi G_3} \\frac{R}{l^2}\\left(\\frac{1}{\\sqrt{-8G_3m + \\frac{R^2}{l^2}}} -\\frac{1}{\\frac{R}{l}}\\right).$ Note that $m=0$ means no shell, i.e., $\\lambda =0$ and $p=0$ .", "Now, from Eq.", "(REF ), one finds that the gravitational radius $r_+$ of the shell is given by $r_+ = \\sqrt{8G_3 m}\\, l\\,.$ It is useful to define a variable $k$ as $k\\equiv \\sqrt{1-\\frac{r_+^2}{R^2}}\\,.$ Then Eqs.", "(REF )-() can be rewritten as $\\lambda = \\frac{1}{8\\pi G_3 l}\\left(1-k\\right)\\,,$ $p = \\frac{1}{8 \\pi G_3 l}\\left(\\frac{1}{k} -1\\right)\\,.$ Note that when $\\lambda =0$ and $p=0$ , Eqs.", "(REF ) and (REF ) [or, if one prefers, Eqs.", "(REF ) and (] give $m=0$ , which from Eq.", "(REF ) implies that the outside spacetime is pure AdS.", "Since the inside is also pure AdS there is no shell in this case, only AdS spacetime.", "Having treated the static problem and having found $\\lambda $ and $p$ , there are mechanical constraints on the shell that should be imposed.", "One constraint is that the shell must be outside any trapped surface, so that the spacetime defined by Eqs.", "(REF )-(REF ) makes sense.", "Imposing that there are no trapped surfaces gives $R\\ge r_+\\,,$ i.e., the shell is outside its own gravitational radius.", "One can also see what the energy conditions lead to.", "The weak energy condition is automatically satisfied as we impose $\\lambda $ and $p$ non-negative.", "On the other hand the dominant energy condition $p \\le \\lambda $ is equivalent to the relation $k^2 - \\frac{2+\\frac{l^2}{R^2}}{\\sqrt{1+\\frac{l^2}{R^2}}}\\,k +1 \\le 0$ .", "This is satisfied for $\\frac{1}{\\sqrt{1+\\frac{l^2}{R^2}}} \\le k \\le \\sqrt{1+\\frac{l^2}{R^2}}$ .", "The right inequality is trivially obeyed, the left inequality leads to $R\\ge \\frac{1}{\\sqrt{1-\\frac{r_+^2}{l^2}}}\\,r_+\\,,$ which is the equation the shell must obey in order that the dominant energy condition holds.", "It is new.", "It is more stringent than the no-trapped-surface condition Eq.", "(REF ).", "The condition (REF ) is plausible on physical grounds.", "Indeed, for large $l$ , the spacetime is weakly AdS, and so it is an almost flat spacetime, which in three dimensions means no, or negligible, gravity.", "The condition Eq.", "(REF ) gives then that $R>\\over \\sim r_+$ , i.e., any shell that makes sense, in the sense of Eq.", "(REF ), is possible.", "As $l$ decreases, the spacetime becomes strongly AdS, there is strong gravitational attraction and the shell can satisfy the dominant energy condition only for sufficient large $R$ .", "When $l=r_+$ , the shell has to have infinite radius in order to obey the dominant energy condition and for even smaller $l$ there is no shell that obeys the dominant energy condition.", "There are also stability conditions as Eiroa and Simeone have shown [20].", "Although not explicitly shown in [20], presumably the radius $R=R(r_+,l)$ at which the shell becomes unstable is slightly larger than the $R$ given in Eq.", "(REF )." ], [ "Thermodynamics\nand stability conditions for the thin shell: Generics", "Now, we assume that the shell is a hot shell, i.e., it possesses a temperature $T$ as measured locally and has an entropy $S$ .", "In the entropy representation, as stated in [23], the entropy $S$ of a system is given in terms of the state independent variables.", "Following [23], when $S$ is known, the thermodynamical system is known.", "We consider as the natural state independent variables the proper local mass $M$ of the shell, and its size, here denoted by the perimeter of the ring shell $A$ .", "Thus, for the shell, $S = S(M,A)\\,.$ Thus, when $S$ in Eq.", "(REF ) is known, the thermodynamical properties of the system follow.", "Using these variables, the first law of thermodynamics can be written as $TdS = dM + p \\,dA\\,,$ where $T$ and $p$ are the temperature and the pressure conjugate to $A$ .", "In order to find $S$ , one has to know the equations of state for these quantities, i.e., $p=p(M,A)\\,,$ and $\\beta =\\beta (M,A)\\,,$ where $\\beta =1/T$ is the inverse temperature.", "Given Eq.", "(REF ), one can find its integrability condition for the differential of the entropy.", "It is given by $\\left(\\frac{\\partial \\beta }{\\partial A}\\right)_M =\\left(\\frac{\\partial \\beta p}{\\partial M}\\right)_A\\,.$ There is then the possibility of studying the local intrinsic stability of the shell at a thermodynamical level, which is guaranteed as long as the inequalities $\\left(\\frac{\\partial ^2 S}{\\partial M^2}\\right)_A \\le 0\\,,$ $\\left(\\frac{\\partial ^2 S}{\\partial A^2}\\right)_M \\le 0\\,,$ $\\left(\\frac{\\partial ^2 S}{\\partial M^2}\\right)\\left(\\frac{\\partial ^2 S}{\\partial A^2}\\right) - \\left(\\frac{\\partial ^2 S}{\\partial M \\partial A}\\right)^2 \\ge 0\\,,$ are satisfied.", "For a derivation of this type of equations see [23]." ], [ "The two independent\nthermodynamic variables", "In order to find the entropy one needs to have the equations of state $p=p(M,A)$ and $\\beta =\\beta (M,A)$ , see Eqs.", "(REF )-(REF ), with $M$ and $A$ being the independent variables.", "Note, however, that $A=2 \\pi R\\,,$ so that the perimeter $A$ and the radius $R$ can be swapped at will as the independent variables.", "The definition of the shell's rest mass $M$ is $M = 2 \\pi R \\lambda \\,,$ where $\\lambda $ is given above in Eq.", "(REF ), and so $M = \\frac{R}{4 G_3 l}\\left(1-k\\right)\\,.$ We should now put some of the basic quantities, $m$ , $r_+$ , and $k$ , in terms of $M$ and $R$ .", "Equation (REF ) together with Eqs.", "(REF ) and (REF ) implies that the ADM mass $m$ is given in terms of the shell's proper mass $M$ and radius $R$ by $m(M,R) =\\frac{-2G^2_3 M^2 + G_3 M \\frac{R}{l}}{G_3}\\,,$ so that when there is no shell, i.e., $M=0$ , one has that the ADM mass of the spacetime is zero, $m=0$ .", "Also now Eq.", "(REF ) should be written as $r_+(M,R) = \\sqrt{8G_3 m(M,R)}\\, l\\,,$ where $m(M,R)$ is given in Eq.", "(REF ), and $k$ should be also seen as $k=k(M,R)$ , i.e., $k(M,R)\\equiv \\sqrt{1-\\frac{r_+^2(M,R)}{R^2}}\\,,$ where $r_+(M,R)$ is given in Eq.", "(REF )." ], [ "The pressure equation of state", "With this rationale in mind, and following the notation of Eq.", "(REF ), we write the equation for the pressure () in the form $p(M,R) = \\frac{1}{8 \\pi G_3 l}\\left(\\frac{1}{k(M,R)} - 1\\right)\\,.$ It is clear that Eq.", "(REF ) yields the equation we aimed for as an equation of state for the shell.", "This equation is an exclusive effect of the gravitational equations, and the junction conditions on the ring, and it does not depend on the essence of the fields of matter composing the shell.", "In brief, it is compulsory that the matter fields obey this equation of state so that mechanical equilibrium is maintained." ], [ "The temperature equation of state", "Now we turn to the other equation of state, Eq.", "(REF ), the equation for $p(M,R)$ .", "Inserting Eq.", "(REF ) and the differential of Eq.", "(REF ) into the first law (REF ), using Eq.", "(REF ), and changing variables from $(M,A)$ to $(r_+,R)$ to simplify the calculations, where $r_+$ is the gravitational radius of the ring given in Eq.", "(REF ), we obtain $dS = \\beta (r_+,R)\\frac{r_+}{4G_3 \\,l\\,R\\, k} dr_+\\,,$ where now $S$ can be seen as $S=S(r_+,R)$ , and the same for $\\beta $ , $\\beta (r_+,R)\\equiv 1/T(r_+,R)$ .", "Equation (REF ) is integrable as long as an appropriate form for $\\beta $ is given.", "To find this appropriate form for $\\beta $ we use the integrability condition Eq.", "(REF ), which upon changing to the $(r_+,R)$ variables reads $\\left(\\frac{\\partial \\beta }{\\partial R}\\right)_{r_+} = \\frac{\\beta }{R k^2}\\,,$ where $k$ is envisaged now as $k=k(r_+,R)$ , see Eq.", "(REF ).", "It can be shown that Eq.", "(REF ) has the following analytic solution, $\\beta (r_+,R) =\\frac{R}{l}\\, k(r_+,R)\\, b(r_+)\\,,$ where $b(r_+)$ is an arbitrary function of the gravitational radius $r_+$ .", "Note that $b(r_+)$ has units of inverse temperature and can be interpreted as the inverse of the temperature the shell would possess if located at $R =\\sqrt{l^2+r_+^2}$ , as can be seen from Eq.", "(REF ).", "Equation (REF ) follows from the integrability condition for the entropy and is directly related to the equivalence principle for systems at a temperature different from zero.", "It is the Tolman relation for the temperature in a gravitational system.", "Note that $b$ is forced to depend on the state variables $(M,R)$ through the specific function $r_+(M,R)$ .", "However, the integrability condition does not yield a precise form for $b$ .", "As stated in [6] (see also [5]), this is expected on physical grounds as the Euclideanized AdS geometry inside the ring can be identified with any period in the partition function stemming from a path integral approach, and thus the hot AdS space inside the shell can have any temperature, not fixing $b$ a priori.", "Any specific function $b(r_+(M,R))$ must resort to the specificity of the matter itself contained in the shell.", "This state of affairs is common in thermodynamics.", "In order to find the equation of state of a gas one can resort to generic and consistency considerations, and find for instance that the temperature $T$ is $T=T(\\rho ,p,V)$ quite generally, where $\\rho $ , $p$ , and $V$ are the density, pressure, and volume of the gas, respectively.", "To have then a concrete form for the function $T=T(\\rho ,p,V)$ , one has to know the specificities of the gas, whether it is an ideal gas or a Van der Waals gas with its two specifying constants, or any other gas, see, e.g., [23]." ], [ "Entropy of the thin shell", "We are now in a position to find the entropy $S$ of the thin shell spacetime.", "Inserting Eq.", "(REF ) into Eq.", "(REF ), one is led to the specific form for the differential of the entropy $dS(r_+) = b(r_+) \\frac{r_+ }{4 G_3\\,l^2} dr_+\\,.$ Integrating Eq.", "(REF ) yields the following entropy, $S(r_+) = \\frac{1}{4 G_3\\,l^2} \\, \\int b(r_+)\\, r_+\\,dr_+ + S_0$ , where $S_0$ is an integration constant.", "By noting that a zero ADM mass shell, i.e., $m=0$ or equivalently $r_+ = 0$ , should naturally have zero entropy, for any regular integrand in the entropy formula just given we must have, $ S(r_+\\rightarrow 0) \\rightarrow 0$ , i.e., $S_0= 0$ .", "So, $S(r_+) = \\frac{1}{4 G_3\\,l^2} \\, \\int b(r_+)\\, r_+\\,dr_+\\,.$ In the same way as $b(r_+)$ , $S$ is also forced to depend on the state variables $(M,R)$ through the specific function $b(r_+(M,R))$ .", "This dependence of $S$ on $r_+(M,R)$ seen in the formula (REF ) comes directly from the self-gravitating nature of the setup.", "It is the result of the matching conditions (REF ) and (REF ) which determine the pressure, and of the equivalence principle in the form of the redshift factor of the Tolman temperature given in (REF ).", "As explained above, a precise shape for the function $b(r_+(M,R))$ has to emanate from definite, thermodynamic or otherwise, configurations for the matter fields.", "Equation (REF ) opens the possibility of studying the local intrinsic stability of the shell at the thermodynamic level, which is guaranteed as long as the inequalities (REF )-(REF ) are satisfied.", "It also permits us to study the spacetime thermodynamics in the limit the shell approaches its own gravitational radius." ], [ "The temperature equation\nand the entropy", "In order to implement the calculation for the entropy, one must resort to a specific fluid or a specific gas and give exactly the function $b(r_+(M,R))$ .", "One could think of many, and not wanting to treat here the specificities of the matter, we resort to the most simple suggestion for $b(r_+)$ as given in [6], i.e., a power law equation of the form $b(r_+) = 4\\,\\alpha \\, G_3 \\,l^2\\frac{r_+^a}{l_{\\rm p}^{(2+a)}}\\,,$ where $a$ is a free parameter, essentially a number, the factors $4G_3$ , $l^2$ , and $l_{\\rm p}=G_3 \\hbar $ appear for dimensional and useful reasons, with $l_{\\rm p}$ being the Planck's length in a three-dimensional spacetime, and $\\hbar $ Planck's constant, and $\\alpha $ is another free parameter without units that can be some function of $l/l_{\\rm p}$ .", "For instance, one can choose $\\alpha ={\\bar{\\alpha }}\\,\\,\\frac{l_{\\rm p}^{(2+a)}}{l^{(2+a)}}$ , with $\\bar{\\alpha }$ a number, but many other choices are possible.", "Boltzmann's constant is taken as equal to 1.", "In order to further justify the choice of the form of $b(r_+)$ in Eq.", "(REF ), we recall that in many thermodynamic instances one recurs to power law functions, most notably near or at a phase transition point, where the temperature goes as the power of the density (or the mass) of the fluid and a power of its specific volume (or the volume itself), for instance.", "These thermodynamic treatments do not even need to know the details of the fine grain constituency of the fluid.", "Such power laws are assumed and indeed represent well the fluid behavior.", "Here, since the temperature, or what is the same, $b$ , cannot be any function of $M$ nor any function of $R$ , and so not a power of $M$ times a power of $R$ as one could be led to think from the usual thermodynamic treatments, but has to be a function such that $M$ and $R$ appear through $r_+(M,R)$ , a natural and simple choice for $b$ is that $b(r_+(M,R))$ is a power of $r_+(M,R)$ , as we have written in Eq.", "(REF ).", "Inserting Eq.", "(REF ) into (REF ) and integrating, we get $S(r_+)= \\frac{\\alpha }{a+2}\\left(\\frac{r_+}{l_{\\rm p}}\\right)^{(a+2)}\\,,$ valid for any $a$ as long as we consider the case $a=-2$ as yielding a logarithmic function, $S(r_+) = \\alpha \\ln r_+/l$ , as it should.", "Using this formula for the entropy, one is able to analyze the stability conditions imposed on the free parameters, and despite the fact that the values of the parameters $\\alpha $ and $a$ do not have specific values as long as some type of nature of the matter fields is not prescribed, it is possible to constrain $a$ nonetheless, such that thermodynamic equilibrium states of the shell are possible." ], [ "The stability conditions for the specific temperature\nansatz", "As for the stability equations, we leave the detailed analysis for the Appendix and present here the main results.", "It is then possible to show that Eqs.", "(REF )-(REF ) altogether applied to the temperature and entropy formulas, Eqs.", "(REF ) and (REF ), respectively, imply $-1 \\le a \\le a_c\\,,$ with $a_c$ being a root of a polynomial equation having the value $a_c=0.255$ .", "Now, Eq.", "(REF ) implies the inequality $\\frac{a}{a+1} R^2 \\le r_+^2 \\,,$ which, together with the condition (REF ), that the shell is above or at its own gravitational radius, restricts the values for $R$ as $\\frac{a}{a+1} R^2 \\le r_+^2 \\le {R^2}\\,.$ Clearly, for $a<-1$ , the left half of the inequality will exceed the right half and thus $a<-1$ is excluded, that is the reason for the the lower bound in Eq.", "(REF ).", "Turning to Eq.", "(REF ), and combining it with Eq.", "(REF ), one finds that in the interval $-1 \\le a \\le 0$ one has $0 \\le \\frac{R}{l} \\le \\left(\\frac{(2a+3) +\\sqrt{\\frac{(5a + 9)}{(a+1)}}}{2(-a)}\\right)^{\\hspace{-7.11317pt}1/2}\\hspace{-2.84544pt},\\,-1 \\le a \\le 0\\,.$ Still, from Eqs.", "(REF ) and (REF ), one finds that in the interval $0< a<a_c$ , for some critical $a_c$ , $R/l$ has to obey the equation $a\\, \\left(1+\\frac{R^2}{l^2}\\right)^{3/2} +\\frac{a-1}{\\sqrt{a+1}}\\frac{R}{l} \\le 0\\,,\\quad 0< a\\le a_c\\,.$ This means that in the interval $0< a\\le a_c$ , $R/l$ has to be in between two values $R_{\\rm min}(a)$ and $R_{\\rm max}(a)$ , i.e., $\\frac{R_{\\rm min}(a)}{l}\\le \\frac{R}{l} \\le \\frac{R_{\\rm max}(a)}{l}\\,,\\quad 0< a\\le a_c\\,.$ The critical value $a_c$ , where the equality ${R_{\\rm max}(a_c)}/{l}={R_{\\rm min}(a_c)}/{l}\\equiv R_c/l$ is obtained, is $a_c=0.255$ for which $R_c/l={\\sqrt{2}}/2$ .", "Finally, through Eq.", "(REF ), Eqs.", "(REF )-(REF ) automatically establish a limit on $r_+$ in order that the shell is thermodynamically stable.", "It is also interesting to note the case where we take the shell to its gravitational radius, $R=r_+$ .", "In that situation, and supposing that the backreaction of whatever kind is small or negligible, the shell is thermodynamically stable if $r_+/l$ satisfies the stability conditions given in Eqs.", "(REF )-(REF ), upon substitution of $R/l$ by $r_+/l$ in those.", "This result does not take into account the possible backreaction that might arise due to quantum effects appearing when the ring is at its own gravitational radius.", "In addition, in this $R=r_+$ limit, although the energy density is finite, the pressure blows up.", "In deriving the stability conditions Eqs.", "(REF )-(REF ) the mechanical condition we have imposed was the no-trapped-surface condition Eq.", "(REF ).", "One can also rework the stability conditions by imposing the tighter dominant energy condition Eq.", "(REF ) or even the mechanical stability condition that can be extracted from [20].", "We do not dwell on these conditions as our main interest here is on the thermodynamic properties of the shell." ], [ "The entropy of the thin shell in the BTZ black hole limit", "A case of particular interest is the marginal case $a=-1$ .", "In this case, the inverse temperature $b(r_+)$ of the shell, taken from Eq.", "(REF ), is $b(r_+) = \\frac{4\\,\\alpha \\, G_3 \\,l^2}{l_{\\rm p}} \\frac{1}{r_+ }\\,,$ and the entropy of the shell, taken from Eq.", "(REF ), has the explicit form $S(r_+) = \\alpha \\,\\frac{r_+}{l_{\\rm p}}\\,.$ This is valid for any radius $R$ of the shell ($R\\ge r_+$ ), since from the integrability condition the entropy does not depend on the radius of the shell $R$ .", "In particular, if we take the limit $R\\rightarrow r_+$ , then the shell hovers at its own gravitational radius.", "One then expects that quantum fields are present [2], and the backreaction will diverge unless one chooses the matter to be at the Hawking temperature [11] $T_{\\rm H} = \\frac{l_{\\rm p}}{2 \\pi G_3\\,l^2}\\,r_+ \\,.$ This fixes the function $b=1/T_{\\rm H}$ to be $b(r_+) = \\frac{2 \\pi G_3\\,l^2}{l_{\\rm p}} \\frac{1}{r_+ }\\,,$ which means that $\\alpha =\\pi /2$ in Eq.", "(REF ).", "Then the entropy (REF ) of the shell at its own gravitational radius is $S_{\\rm BH} = \\frac{\\pi }{2} \\frac{r_+}{l_{\\rm p}}\\,.$ The area $A_{\\rm h}$ of the horizon, which is actually a perimeter in 2+1 dimensions, is $A_{\\rm h}=2\\pi \\,r_+$ .", "Therefore $S_{\\rm BH} = \\frac{1}{4} \\frac{A_{\\rm h}}{l_{\\rm p}}\\,.$ This is precisely the Bekenstein-Hawking entropy of the (2+1)-dimensional BTZ black hole [11], now derived from the properties of the spacetime of the shell of matter and from the fact that the shell is at its own gravitational radius.", "At $R=r_+$ one has from Eq.", "(REF ) that $k=0$ , and so from Eqs.", "(REF ) and (REF ) one finds that $\\lambda = \\frac{1}{8\\pi \\,G_3\\,l}$ (i.e., $M = \\frac{1}{4\\,G_3}\\frac{r_+}{l}$ ) and $p= \\frac{1}{8\\pi \\,G_3\\,l}\\,\\frac{1}{k}\\rightarrow \\infty $ , characteristic of certain quasiblack holes, objects which also yield the Bekenstein-Hawking entropy [9], [10].", "Indeed, the shell at its own gravitational radius is a quasiblack hole." ], [ "Conclusions", "In this paper we have considered the thermodynamics and entropy of a (2+1)-dimensional shell, a ring, in an AdS spacetime.", "Inside the ring the spacetime is given by the AdS metric, characterized by a negative cosmological constant $-\\Lambda =\\frac{1}{l^2}$ , and outside it is given by the BTZ metric, characterized by a mass $m$ , by the same negative cosmological constant $-\\Lambda =\\frac{1}{l^2}$ , and by being asymptotically AdS.", "The ring shell at radius $R$ has a mass density $\\lambda $ (or equivalently a mass $M=2\\pi R \\lambda $ ) and a pressure $p$ associated with it required to achieve a static equilibrium.", "The first law of thermodynamics implies that we need to equations of state: one for the pressure $p$ of the shell and another for its temperature $T$ .", "The pressure $p$ is given in terms of the state variables $M$ and $R$ through the junction conditions.", "The temperature $T$ , or its inverse, is found to be a function of the gravitational radius $r_+$ of the system alone.", "This $r_+$ is itself a particular known function of the state variables $M$ and $R$ .", "The entropy of the shell can then be found as a function of $r_+$ alone.", "To give an example of a hot shell, inspired in several usual thermodynamics systems which have the temperature as given in power laws of the state variables, we have chosen the inverse temperature $b$ to be proportional to a power law in $r_+$ , $r_+^a$ , for some number $a$ .", "The computation of the specific form of the entropy led to an analysis of the parameter regions for which the ring is thermodynamically stable.", "We have found $a$ must be in the range $-1\\le a\\le 0.255$ .", "In the case $a=-1$ , the shell can be chosen to have a Hawking type temperature at the outset.", "One can then tune the temperature to be precisely the Hawking temperature, including all numerical factors, and push the shell up to its gravitational radius, since at this temperature there is a finite backreaction at the horizon that does not destroy the solution.", "The entropy found is then the Bekenstein-Hawking entropy as it is appropriate for a quasiblack hole.", "We thank Oleg Zaslavskii for conversations.", "We thank FCT-Portugal for financial support through Project No.", "PEst-OE/FIS/UI0099/2013." ], [ "Analysis of the stability equations", "We use here the stability conditions Eqs.", "(REF )-(REF ) in the temperature and entropy formulas, Eqs.", "(REF ) and (REF ), respectively, to show the results presented in Sec.", "REF .", "It is possible to show that Eq.", "(REF ) implies the inequality $\\frac{a}{a+1} R^2 \\le r_+^2 \\,,$ which, together with the condition that the shell is above or at its own gravitational radius, i.e., $r_+^2 \\le {R^2}$ , sets up the restricted values for $R$ relative to $r_+$ , namely, $\\frac{a}{a+1} R^2 \\le r_+^2 \\le {R^2}\\,.$ For $0\\le a<\\infty $ this inequality always holds.", "If $-1 \\le a < 0$ , the lower limit will assume negative values, but the inequality is satisfied nonetheless.", "For $a<-1$ , the left half of the inequality will exceed the right half and thus $a<-1$ is excluded.", "Thus, Eq.", "(REF ) restricts the interval of $a$ to $-1 \\le a < \\infty \\,.$ Turning now to Eq.", "(REF ), it leads to the relation $r_+^2 + & a\\,{R^2} \\left(1+ \\frac{R^2}{l^2}\\right)\\nonumber \\\\& \\le a\\, {R^2} \\sqrt{1+\\frac{R^2}{l^2}}\\sqrt{\\frac{R^2}{l^2}-\\frac{r_+^2}{l^2}}\\,,$ which, when used in conjunction with Eq.", "(REF ), leads to $a(a+1) \\frac{R^4}{l^4} + (a+1)(2a+3) \\frac{R^2}{l^2} + (a+2)^2 \\ge 0.$ Depending on $a$ , Eq.", "(REF ) gives further that it can only be verified for a certain set of values for $\\frac{R}{l}$ .", "Indeed, for $-1 \\le a < 0$ one has $0 \\le \\frac{R}{l} \\le \\left(\\frac{(2a+3) +\\sqrt{\\frac{(5a + 9)}{(a+1)}}}{2(-a)}\\right)^{\\hspace{-7.11317pt}1/2}\\hspace{-2.84544pt},\\,-1 \\le a \\le 0\\,.$ For $0\\le a<\\infty $ one has that any $\\frac{R}{l}$ satisfies the inequality (REF ).", "Through Eq.", "(REF ), this automatically establishes a limit on $r_+$ .", "Finally, from Eq.", "(REF ) one obtains the inequality $& (a+1)\\,\\frac{r_+^2}{l^2}\\left(1+\\frac{R^2}{l^2}\\right)^{3/2}+\\nonumber \\\\& + \\left( a\\, \\frac{R^2}{l^2}-(a+1)\\,\\frac{r_+^2}{l^2}\\right)\\sqrt{\\frac{R^2}{l^2}-\\frac{r_+^2}{l^2} } \\le 0.$ Imposing at the same time the condition Eq.", "(REF ) in Eq.", "(REF ), we are left with $a\\, \\left(1+\\frac{R^2}{l^2}\\right)^{3/2} +\\frac{a-1}{\\sqrt{a+1}}\\frac{R}{l} \\le 0\\,.$ It is clear that for $-1\\le a\\le 0$ the inequality (REF ) is always satisfied.", "It is also clear that there is a critical $a$ , $a_c$ , such that for $0< a\\le a_c$ , $R/l$ has to be in between two values, and for $a_c< a<\\infty $ , the inequality (REF ) can never be satisfied.", "The value of $a_c$ and the corresponding value $R_c/l$ are found as follows.", "Consider $f(R)\\equiv a\\, \\left(1+\\frac{R^2}{l^2}\\right)^{3/2} +\\frac{a-1}{\\sqrt{a+1}}\\frac{R}{l}$ .", "From the form of the function $f(R)$ , one can check that there is an $a_c$ above which the inequality (REF ) has no solution.", "To this $a_c$ there is a correspondent $R_c/l$ .", "Imposing $f(R)=0$ and $df(R)/dR=0$ , one is led to one equation for $a$ and one for $R/l$ , namely, $27\\,a^3+23\\,a^2+8\\,a-4=0$ and $R^2/l^2-1/2=0$ .", "The solutions are $a_c=0.255$ , up to the third decimal place, and $R_c/l={\\sqrt{2}}/2$ .", "Collecting the results given in this Appendix, we have for thermodynamic stability that the following equations must be satisfied: $-1 \\le a \\le a_c\\,,$ with $a_c = 0.255$ , $\\frac{a}{a+1} R^2 \\le r_+^2 \\le {R^2}\\,,$ $0 \\le \\frac{R}{l} \\le \\left(\\frac{(2a+3) +\\sqrt{\\frac{(5a + 9)}{(a+1)}}}{2(-a)}\\right)^{\\hspace{-7.11317pt}1/2}\\hspace{-2.84544pt},\\,-1 \\le a \\le 0\\,,$ and $a\\, \\left(1+\\frac{R^2}{l^2}\\right)^{3/2} +\\frac{a-1}{\\sqrt{a+1}}\\frac{R}{l} \\le 0\\,,\\quad 0< a\\le a_c\\,,$ i.e., $\\frac{R_{\\rm min}(a)}{l}\\le \\frac{R}{l} \\le \\frac{R_{\\rm max}(a)}{l}\\,,\\quad 0< a\\le a_c\\,.$ The critical value of $R/l$ , namely, ${R_{\\rm max}(a_c)}/{l}={R_{\\rm min}(a_c)}/{l}\\equiv R_c/l$ , is obtained for $a_c=0.255$ , yielding $R_c/l={\\sqrt{2}}/2$ .", "Equation (REF ), with the help of Eqs.", "(REF )-(REF ), automatically establishes a limit on $r_+$ in order that the shell is thermodynamically stable.", "These are the results presented in Sec.", "REF ." ] ]
1403.0579
[ [ "Ground States of a Bose-Hubbard Ladder in an Artificial Magnetic Field:\n Field-Theoretical Approach" ], [ "Abstract We consider a Bose-Hubbard ladder subject to an artificial magnetic flux and discuss its different ground states, their physical properties, and the quantum phase transitions between them.", "A low-energy effective field theory is derived, in the two distinct regimes of a small and large magnetic flux, using a bosonization technique starting from the weak-coupling limit.", "Based on this effective field theory, the ground-state phase diagram at a filling of one particle per site is investigated for a small flux and for a flux equal to $\\pi$ per plaquette.", "For $\\pi$-flux, this analysis reveals a tricritical point which has been overlooked in previous studies.", "In addition, the Mott insulating state at a small magnetic flux is found to display Meissner currents." ], [ "Introduction", "Recent developments in ultra-cold atom physics allow for studies of a wide range of many-body quantum systems of bosons, fermions and their mixtures, involving strong correlations and/or frustration.", "One of the remarkable recent advances is the so-called artificial gauge fields, [1] which allow one to generate spin-orbit couplings and magnetic fields, opening the way for example to quantum Hall and spin Hall effects.", "These effects are also related to studies on topological phases of matters.", "In addition, the control of interactions between atoms by the Feshbach resonance technique allows for the controlled study of quantum systems under the combined effects of an artificial gauge field and strong correlations.", "The key to artificial gauge fields is Berry's phases [2] tuned by atom-light interactions, [3] in which atoms acquire a geometric phase in their motion because of an adiabatic spatial change of the dressed states.", "Using Raman transitions based on these ideas, the synthesis of an effective magnetic field [4] and spin-orbit coupling [5], [6], [7] have been experimentally achieved in Bose condensates of $^{87}$ Rb atoms, and the spin-Hall effect in Bose condensates has been also successfully observed [8].", "The realization of artificial gauge fields in optical lattice potentials has also been intensively discussed.", "The pioneering theory making use of photo-assisted tunneling techniques has been established by Jaksch and Zoller [9].", "Subsequently other schemes for effective uniform magnetic fields [10], [11] and for staggered magnetic fields [12], [13] have also been proposed.", "In experiments, several types of artificial magnetic fields using photo-assisted tunneling have been subsequently realized in recent years: effective magnetic fluxes inhomogeneously set in stripes [14], [15], uniform magnetic fluxes [16], and spin-orbit couplings without spin flips [17], [18] in two-dimensional optical lattice systems.", "In addition to the above realizations, other schemes for artificial gauge fields have been invented using Zeeman lattice techniques [19] and shaking of optical lattice potentials [20], [21], [22].", "Optical lattices also allow for the control of dimensionality, so that one-dimensional quantum systems can be realized.", "A quasi-one-dimensional “ladder” geometry plays the role of a minimal model for studying the effect of gauge fields.", "In these low-dimensional systems, the whole range of interaction strengths from weak to strong coupling can be investigated using powerful numerical and analytic techniques, such as bosonization and the density-matrix renormalization group (DMRG).", "Because of the peculiar critical nature of Tomonaga-Luttinger (TL) liquids which describe their low-energy properties, such quasi-one-dimensional quantum systems subject to artificial gauge fields or high magnetic fields are expected to display interesting phenomena.", "In studies on ladder systems subjected to magnetic fields, fermion systems have been discussed in the context of strongly correlated electron systems [23], [24], [23], [25], [26].", "The study of bosonic ladders subject to magnetic fields has also been motivated by the Josephson junction ladders and ultra-cold Bose atoms in optical lattices [27], [28], [29], [30], [31], [32], [33], [34], [35], [36].", "In addition to common features of Bose-Hubbard models such as a one-dimensional superfluid (SF), Mott insulator (MI), and phase transition between them, the bosonic ladders exhibit interesting phenomena induced by the magnetic field: chiral superfluid phases (CSF) and chiral Mott insulating phases (CMI) displaying Meissner currents [31], [33], [34], [32].", "Attention to the topic of bosonic ladders subject to an artificial magnetic field has been reinforced recently by the first experimental realization of such a system.", "[37] In this article, we study the low-energy physics of Bose-Hubbard ladders subject to an artificial uniform magnetic flux, from the viewpoint of field theory.", "So far, field theoretical approaches to bosonic ladder systems have usually considered starting with the strong coupling limit, in which the rung hopping is treated perturbatively [38], [31], [32] In contrast, we derive an effective field theory from a weak coupling perspective, in which the effect of a rung hopping is fully taken into account, and the interaction is included perturbatively.", "In this approach, typical strong correlation phenomena such as the MI state and the MI-SF phase transition can nonetheless be investigated, by taking into account backscattering and umklapp scattering processes.", "The low-energy effective field theories in the two cases of a large and small magnetic flux are separately derived, for an arbitrary filling.", "In addition we also apply the constructed effective field theory and investigate the ground-state phase diagram in two limiting cases, namely that of a large magnetic flux equal to $\\pi $ per plaquette, and that of a small magnetic flux, with one particle per site.", "For the $\\pi $ magnetic flux, we compare our phase diagram to the one previously obtained numerically in Ref.", "[34], and all the phases found there are well described by our approach.", "Furthermore, more importantly, the presence of a tricritical point in the phase diagram is predicted by the analysis presented here, which has not been emphasized previously.", "In the limit of a small magnetic flux, we show that a SF state with Meissner current appears, and transits to the MI state for strong interactions.", "In addition we also find that the Meissner current can survive also in the MI state while the system is fully gapped.", "A similar fully gapped charge-ordered state with Meissner currents has also been found for the different filling of one particle per two sites by Petrescu and Le Hur [32].", "This paper is organized as follows.", "In Sec.", "we first define the considered Bose-Hubbard ladder with a magnetic flux.", "Next we analyze the single-particle band structure in the non-interacting case in Sec.", "REF , and derive the general form of the low-energy effective field theory for a large magnetic flux in Sec.", "REF , and for a small magnetic flux in Sec.", "REF .", "Furthermore, based on the derived field theory, we investigate the ground-state phase diagrams in Sec. .", "A summary and conclusion are provided in Sec. .", "In Appendix , the mean-field analysis which is used to construct the effective field theories is presented." ], [ "Model and effective theory", "Let us define the Bose-Hubbard ladder Hamiltonian with an applied uniform artificial magnetic field, $H&=H_{0}+H_{\\mathrm {loc}},\\\\H_{0}&= -J\\sum _{p=1,2}\\sum _{j}\\left[e^{iA^{\\parallel }_{j,p}}b^{\\dagger }_{j+1,p}b_{j,p}+\\mathrm {h.c.}\\right]\\nonumber \\\\& \\quad -J_{\\perp }\\sum _{j}\\left[e^{iA^{\\perp }_{j}}b^{\\dagger }_{j,1}b_{j,2} + \\mathrm {h.c.}\\right],\\\\H_{\\mathrm {loc}}&=\\sum _{j}\\sum _{p=1,2}\\left[-\\mu n_{j,p}+\\frac{U}{2}n_{j,p}\\left(n_{j,p} + 1\\right)\\right],$ where $n_{j,p}=b^{\\dagger }_{j,p}b_{j,p}$ is a number operator, and the index $p=1,2$ denotes the upper and lower chain, respectively.", "The model Hamiltonian is schematically illustrated in Fig.", "REF .", "Figure: The Bose-Hubbard ladder Hamiltonian considered in this paper.Due to the magnetic field, hopping involves a phase factor associatedwith the corresponding gauge field.The phases gained in the hopping processes are displayed, correspondingto the gauge choice defined by Eq.", "().This model has the two different hoppings, intrachain $J>0$ and interchain $J_{\\perp }>0$ , and only a repulsive on-site Hubbard interaction $U>0$ is considered.", "The artificial magnetic field is introduced via the Peierls substitution, and the corresponding gauge field along the chain direction on the chain $p$ , and along the rung direction are denoted by $A^{\\parallel }_{j,p}$ and $A^{\\perp }_{j}$ , respectively.", "This produces the applied artificial magnetic flux $\\phi $ piercing a plaquette as $\\oint _{\\Box }\\mathbf {A}\\cdot d\\mathbf {l}= A^{\\parallel }_{j,1}-A^{\\perp }_{j+1}-A^{\\parallel }_{j,2}+A^{\\perp }_{j}= \\phi .$ In this paper, we choose the following gauge: $\\left\\lbrace \\begin{aligned}& A^{\\parallel }_{j,1}=\\phi /2, \\\\& A^{\\parallel }_{j,2}=-\\phi /2, \\\\& A^{\\perp }_{j}=0,\\end{aligned}\\right.$ which obviously obeys Eq.", "(REF ).", "The Hamiltonian (REF ) is invariant under the transformation, $(\\phi ,b_{j,1},b_{j,2})\\rightarrow (-\\phi ,b_{j,2},b_{j,1})$ .", "Thus the magnetic flux $\\phi $ to be considered can be primitively reduced, and we restrict the magnetic flux to be $0<\\phi \\le \\pi $ throughout this paper." ], [ "Single-particle energy band structure", "Let us look at the single-particle spectrum.", "The single-particle Hamiltonian () is easily diagonalized by the following unitary transformation as $\\left\\lbrace \\begin{aligned}& b_{1}(k)=v_{k}\\alpha (k)+u_{k}\\beta (k),\\\\& b_{2}(k)=-u_{k}\\alpha (k)+v_{k}\\beta (k), \\\\\\end{aligned}\\right.$ where $b_{p}(k)$ with $p=1,2$ is a Fourier transformation of $b_{j,p}$ .", "The coefficients are given as $\\left\\lbrace \\begin{aligned}u_{k}&=\\sqrt{\\frac{1}{2}\\left(1-\\frac{\\sin (\\phi /2)\\sin {k}}{\\sqrt{(J_\\perp /2J)^2+\\sin ^{2}(\\phi /2)\\sin ^{2}{k}}}\\right)},\\\\v_{k}&=\\sqrt{\\frac{1}{2}\\left(1+\\frac{\\sin (\\phi /2)\\sin {k}}{\\sqrt{(J_\\perp /2J)^2+\\sin ^{2}(\\phi /2)\\sin ^{2}{k}}}\\right)}.\\end{aligned}\\right.$ Consequently the single-particle Hamiltonian () has a two-band structure: $H_0&= \\sum _{k}\\left[E_{+}(k)\\alpha ^{\\dagger }(k)\\alpha (k)+ E_{-}(k)\\beta ^{\\dagger }(k)\\beta (k)\\right],$ with the single-particle energy bands being $E_{\\pm }(k)&=-2J \\cos \\left(\\frac{\\phi }{2}\\right)\\cos {k}\\nonumber \\\\&\\quad \\pm \\sqrt{J_{\\perp }^2+(2J)^2\\sin ^2\\left(\\frac{\\phi }{2}\\right)\\sin ^2{k}}.$ The energy dispersions $E_{\\pm }(k)$ correspond to the upper and lower band, respectively.", "The band structures for certain values of $\\phi $ and $J_{\\perp }/J$ are shown in Fig.", "REF .", "Figure: (Color online) The single-particle energy bands of thetwo-leg ladder with a uniform magnetic flux: (a) φ=π\\phi =\\pi , (b)φ=π/2\\phi =\\pi /2, (c) φ=π/3\\phi =\\pi /3 and (d) φ=π/5\\phi =\\pi /5.The red, blue and green lines denote J ⊥ /2J=0.05J_{\\perp }/2J=0.05, 0.50.5 and0.750.75, respectively.Here only the commensurate magnetic fluxes are shown, but the structureof the spectrum is continuously deformed by varying the flux φ\\phi .Comprehensive results on the dependence of the band structure on the magnetic field $\\phi $ and hopping ratio $J_{\\perp }/J$ can be found in Refs.", "[23], [25].", "In addition, the band structure of the Hamiltonian $H_{0}$ is known to be analogous to that of spin-$1/2$ particles with a spin-orbit coupling in the presence of a magnetic field [39].", "The band structure around the lowest energy is the most important feature for low-energy physics, since bosons tend to populate states around energy minima.", "Thus we focus here only on the features at the bottom of the lower band.", "In the regime of a large magnetic flux per plaquette the lower band $E_{-}(k)$ has two separate minima, and the corresponding wave numbers $k_{\\mathrm {min}}$ at the band minima are given analytically as $k_{\\mathrm {min}}=\\pm Q$ with $Q=\\arccos \\left[\\cot \\left(\\frac{\\phi }{2}\\right)\\sqrt{\\left(\\frac{J_{\\perp }}{2J}\\right)^2+\\sin ^2\\left(\\frac{\\phi }{2}\\right)}\\right],$ which is shown in Fig.", "REF .", "Figure: (color online)The position of the momentum corresponding to the band minima,k min =±Qk_{\\min }=\\pm Q, as a function of the magnetic flux per plaquetteφ\\phi for several hopping ratios J ⊥ /2JJ_{\\perp }/2J.A finite QQ corresponds to the case when the lower band displays twominima, while Q=0Q=0 means that the band has a single minimumstructure.The critical value of the magnetic flux, at which QQ becomes zero,increases with J ⊥ /JJ_\\perp /J as in Eq.", "().The two band minima are maximally separated for $\\phi =\\pi $ , and located exactly at $k_{\\mathrm {min}}=\\pm \\pi /2$ .", "As the flux $\\phi $ decreases, the two minima approach one another, and eventually merge at a critical value of the magnetic flux $\\phi _{\\mathrm {c}}=2\\arccos \\left[\\sqrt{\\left(\\frac{J_\\perp }{4J}\\right)^2+1}-\\frac{J_\\perp }{4J}\\right].$ A single minimum structure is formed for small $\\phi $ .", "On the other hand, the hopping ratio $J_{\\perp }/J$ works so as to enlarge the distance between the two bands, and so as to narrow the band width.", "Thus the increase of the ratio $J_{\\perp }/J$ suppresses the height of the barrier between the two minima in the lower band, which also leads to the increase of the critical $\\phi _{c}$ with $J_{\\perp }/J$ .", "In what follows, we separately consider the two different cases.", "The first is the case of a sufficiently large magnetic flux $\\phi \\gg \\phi _\\mathrm {c}$ , in which the bottom of the lower band exhibits double minima, and they are clearly separated.", "The second is the case of a small magnetic flux $\\phi <\\phi _c$ , in which the band bottom forms a single minimum structure." ], [ "Effective Hamiltonian for large magnetic flux", "Let us derive the low-energy effective theory based on the single-particle spectrum obtained above.", "For a large enough magnetic flux $\\phi \\gg \\phi _\\mathrm {c}$ , we have a double-minimum structure in the lower energy band, and in the ground state the bosons dominantly populate the two energy minima even in the presence of the interaction.", "Thus, the spectrum relevant to the low-energy physics can be approximated by expanding the band-structure around the energy minima: $E_{-}(k=\\pm Q+q)\\approx -E_{0}+\\frac{q^2}{2m^{*}},$ where $E_{0}=-E(k=\\pm Q)$ and $m^{*}$ are, respectively, a minimum energy offset and effective mass, and they are given as $& E_{0}= \\frac{2J}{\\sin \\left(\\frac{\\phi }{2}\\right)}\\sqrt{\\left(\\frac{J_{\\perp }}{2J}\\right)^2+\\sin ^2\\left(\\frac{\\phi }{2}\\right)},\\\\& \\frac{1}{m^{*}}=\\frac{d^2E(k=\\pm Q)}{dk^2}.$ The wave number $q$ denotes the variation from the minima $k_{\\mathrm {min}}=\\pm Q$ , and is assumed to be small enough, $q\\ll 1$ .", "The minimum energy offset $-E_{0}$ shifts the chemical potential $\\mu $ in the Hamiltonian (REF ).", "Thus under this long-wave-length approximation we need to fix the chemical potential including this energy offset to reproduce the required density.", "Correspondingly to the long-wave-length expansion, the unitary transformation (REF ) is also approximated as follows.", "For the upper chain, $\\left\\lbrace \\begin{aligned}& b_{1}(k=Q+q)= V_{-}\\beta _{+}(q),\\nonumber \\\\& b_{1}(k=-Q+q)= V_{+}\\beta _{-}(q),\\end{aligned}\\right.,$ and for the lower chain, $\\left\\lbrace \\begin{aligned}& b_{2}(k=Q+q)= V_{+}\\beta _{+}(q),\\nonumber \\\\& b_{2}(k=-Q+q)= V_{-}\\beta _{-}(q).\\end{aligned}\\right.,$ where the weight factors, $V_{+}=u_{-Q}=v_{Q}$ and $V_{-}=u_{Q}=v_{-Q}$ , have been introduced, and they are explicitly given as $V_{\\pm }= \\sqrt{\\frac{1}{2}\\left[1\\pm \\sqrt{\\frac{\\sin ^2(\\phi /2)-(J_\\perp /2J)^2\\cot ^2(\\phi /2)}{(J_\\perp /2J)^2+\\sin ^2(\\phi /2)}}\\right]}.$ This approximation means that all the energy states except for the low-energy states near the band bottom are projected out.", "The approximated boson operators are represented in real space as $\\left\\lbrace \\begin{aligned}& b_{j,1}\\approx e^{-iQ j}V_{-}\\beta _{+,j}+e^{iQ j}V_{+}\\beta _{-,j},\\\\& b_{j,2}\\approx e^{-iQ j}V_{+}\\beta _{+,j}+e^{iQ j}V_{-}\\beta _{-,j},\\end{aligned}\\right.$ which lead to the following representation of the number operators as $\\left\\lbrace \\begin{aligned}n_{j,1}& \\approx V_{-}^2\\tilde{n}_{+,j}+V_{+}^2\\tilde{n}_{-,j}+V_{+}V_{-}\\left(e^{i2Qj}\\beta _{+,j}^{\\dagger }\\beta _{-,j}+\\mathrm {h.c.}\\right),\\\\n_{j,2}& \\approx V_{+}^2\\tilde{n}_{+,j}+V_{-}^2\\tilde{n}_{-,j}+V_{+}V_{-}\\left(e^{i2Qj}\\beta _{+,j}^{\\dagger }\\beta _{-,j}+\\mathrm {h.c.}\\right),\\end{aligned}\\right.$ where the density operators for the separate quadratic energy dispersions have been defined as $\\tilde{n}_{\\pm ,j}=\\beta _{\\pm ,j}^{\\dagger }\\beta _{\\pm ,j}$ .", "The above representation of the field operators in the long-wave-length approximation has a similar form to that of fermions.", "Namely the wave numbers $\\pm Q$ giving the minima of the energy band are analogous to Fermi points.", "From the above, the Hamiltonian in the long-wave-length-approximation are derived.", "The single particle Hamiltonian () is rewritten as $H_{0}\\approx -E_{0}\\sum _{j,\\sigma =\\pm }\\tilde{n}_{\\sigma ,j}+\\sum _{q,\\sigma =\\pm }\\frac{q^2}{2m^{*}}\\beta _{\\sigma ,q}^{\\dagger }\\beta _{\\sigma ,q},$ where $\\beta _{\\sigma }(q)$ is a Fourier transform of $\\beta _{\\sigma ,j}$ .", "Using the representation of Eq.", "(REF ), the local Hamiltonian () is rewritten as $H_{\\mathrm {loc}}& \\approx -\\left(\\mu +\\frac{U}{2}-UV_{+}^2V_{-}^2\\right)\\sum _{j,\\sigma =\\pm }\\tilde{n}_{\\sigma ,j}\\nonumber \\\\& \\quad +\\frac{U}{4}\\sum _{j}\\biggl [\\left(1+2V_{+}^2V_{-}^{2}\\right)\\left(\\tilde{n}_{+,j}+\\tilde{n}_{-,j}\\right)^2\\nonumber \\\\& \\quad +\\left(1-6V_{+}^2V_{-}^2\\right)\\left(\\tilde{n}_{+,j}-\\tilde{n}_{-,j}\\right)^2\\nonumber \\\\& \\quad +2V_{+}V_{-}\\left(\\tilde{n}_{+,j}+\\tilde{n}_{-,j}\\right)\\left(e^{i2Qj}\\beta _{+,j}^{\\dagger }\\beta _{-,j}+\\mathrm {H.c.}\\right)\\nonumber \\\\& \\quad +2V_{+}V_{-}\\left(e^{i2Qj}\\beta _{+,j}^{\\dagger }\\beta _{-,j}+\\mathrm {H.c.}\\right)\\left(\\tilde{n}_{+,j}+\\tilde{n}_{-,j}\\right)\\nonumber \\\\& \\quad +4V_{+}^2V_{-}^2\\left(e^{i4Qj}\\beta _{+,j}^{\\dagger }\\beta _{+,j}^{\\dagger }\\beta _{-,j}\\beta _{-,j}+\\mathrm {H.c.}\\right)\\biggl ].$ The $4Q$ -oscillating terms in Eq.", "(REF ) are regarded as the umklapp scattering between the particles in the two band minima.", "Thus the commensurability of the magnetic flux is determined by $Q$ .", "In order to fix the chemical potential for a given particle number per site, $\\bar{n}_{p}=\\langle {n_{j,p}}\\rangle $ ($p=1,2$ ), we use mean-field analysis.", "As discussed in Appendix , the mean-field theory leads to the balanced densities on the chains, $\\bar{n}_{1}=\\bar{n}_{2}=\\tilde{n}_{+}=\\tilde{n}_{-}=\\bar{n}$ where $\\tilde{n}_{\\pm }=\\langle {\\tilde{n}_{\\pm ,j}}\\rangle $ , and the density is controlled by the chemical potential as in Eq.", "(REF ).", "Based on the mean-field solution, we use the following bosonization formula as $\\left\\lbrace \\begin{aligned}& \\beta _{\\pm ,j}\\sim \\sqrt{\\bar{n}}e^{i\\theta _{\\pm }(x_{j})},\\\\& n_{\\pm ,j}\\sim \\bar{n}-\\frac{a}{\\pi }\\nabla \\varphi _{\\pm }(x_j)+2\\bar{n}\\cos \\left[\\frac{2\\pi \\bar{n}}{a}x_{j}-2\\varphi _{\\pm }(x_j)\\right],\\end{aligned}\\right.$ where we have introduced the continuum coordinate $x_j=a \\times j$ with the lattice length $a$ .", "Note that from the bosonization formula, the fields $\\varphi _{\\sigma }$ and $\\theta _{\\sigma }$ are compactified, respectively, as $\\varphi _{\\sigma }\\sim \\varphi _{\\sigma }+\\pi $ and $\\theta _{\\sigma }\\sim \\theta _{\\sigma }+2\\pi $ .", "In other words, the fields are uniquely defined in the regime, $\\left\\lbrace \\begin{aligned}& -\\frac{\\pi }{2} < \\varphi _{\\pm }(x) \\le \\frac{\\pi }{2},\\\\& -\\pi < \\theta _{\\pm }(x) \\le \\pi ,\\end{aligned}\\right.$ Applying Eq.", "(REF ) into Eqs.", "(REF ) and (REF ), the low-energy effective Hamiltonian is derived as $H_{\\mathrm {eff}} &=H_{\\mathrm {TL}}+\\sum _{i=0}^{4}V_{i},\\\\H_{\\mathrm {TL}} &=\\frac{v_\\mathrm {s}}{2\\pi }\\int \\!\\!dx\\,\\left[K_\\mathrm {s}\\left(\\nabla \\theta _\\mathrm {s}(x)\\right)^2+\\frac{1}{K_\\mathrm {s}}\\left(\\nabla \\varphi _\\mathrm {s}(x)\\right)^2\\right]\\nonumber \\\\&\\quad +\\frac{v_\\mathrm {a}}{2\\pi }\\int \\!\\!dx\\,\\left[K_\\mathrm {a}\\left(\\nabla \\theta _\\mathrm {a}(x)\\right)^2+\\frac{1}{K_\\mathrm {a}}\\left(\\nabla \\varphi _\\mathrm {a}(x)\\right)^2\\right],\\\\V_{0}&=g_{0}\\int \\!\\!", "\\frac{dx}{a}\\,\\cos \\left(\\frac{2Q}{a}x-\\sqrt{2}\\theta _\\mathrm {a}(x)\\right),\\\\V_{1}&=g_{1}\\int \\!\\!", "\\frac{dx}{a}\\,\\cos \\left(\\frac{4Q}{a}x-\\sqrt{8}\\theta _\\mathrm {a}(x)\\right),\\\\V_{2}&=g_{2}\\int \\!\\!", "\\frac{dx}{a}\\,\\cos \\left(\\frac{4\\pi \\bar{n}}{a}x-\\sqrt{8}\\varphi _\\mathrm {s}(x)\\right),\\\\V_{3}&=g_{3}\\int \\!\\!", "\\frac{dx}{a}\\,\\cos \\left(\\sqrt{8}\\varphi _\\mathrm {a}(x)\\right),\\\\V_{4}&=g_{4}\\int \\!\\!", "\\frac{dx}{a}\\,\\cos \\left(\\frac{2\\pi \\bar{n}}{a}x-\\sqrt{2}\\varphi _\\mathrm {s}(x)\\right)\\cos \\left(\\sqrt{2}\\varphi _\\mathrm {a}(x)\\right),$ where the symmetric and antisymmetric fields have been introduced as $\\left\\lbrace \\begin{aligned}\\varphi _{\\mathrm {s},\\mathrm {a}}(x)&=\\frac{\\varphi _{+}(x)\\pm \\varphi _{-}(x)}{\\sqrt{2}},\\\\\\theta _{\\mathrm {s},\\mathrm {a}}(x)&=\\frac{\\theta _{+}(x)\\pm \\theta _{-}(x)}{\\sqrt{2}}.\\end{aligned}\\right.$ The Hamiltonian $H_\\mathrm {TL}$ stands for that of TL liquids in the symmetric and antisymmetric sectors.", "Note that due to the redefinition of the fields, the compactification of the fields changes.", "[40], [41], [42] The redefined fields can not be independently compactified, and the identification of the fields are as follows: $\\varphi _{\\mathrm {s},\\mathrm {a}}\\sim \\varphi _{\\mathrm {s},\\mathrm {a}}+\\pi N_{\\mathrm {s},\\mathrm {a}}/\\sqrt{2}$ with $N_\\mathrm {s}\\equiv N_\\mathrm {a}$ (modulo 2), and $\\theta _{\\mathrm {s},\\mathrm {a}}\\sim \\theta _{\\mathrm {s},\\mathrm {a}}+\\sqrt{2}\\pi M_{\\mathrm {s},\\mathrm {a}}$ with $M_\\mathrm {s}\\equiv M_\\mathrm {a}$ (modulo 2).", "In other words, the symmetric and antisymmetric fields are uniquely defined in the following regime, $\\left\\lbrace \\begin{aligned}& -\\frac{\\pi }{\\sqrt{2}}< \\varphi _\\mathrm {s}(x)\\pm \\varphi _\\mathrm {a}(x)\\le \\frac{\\pi }{\\sqrt{2}},\\\\& -\\sqrt{2}\\pi < \\theta _\\mathrm {s}(x)\\pm \\theta _\\mathrm {a}(x)\\le \\sqrt{2}\\pi ,\\end{aligned}\\right.$ which is important in discussing the degeneracy of the ground states.", "The parameters introduced are roughly estimated as $\\left\\lbrace \\begin{aligned}& v_{\\mathrm {s}}\\sim a\\sqrt{\\frac{\\bar{n}U}{m^{*}}\\left(1+2V_{+}^{2}V_{-}^{2}\\right)},\\\\& v_{\\mathrm {a}}\\sim a\\sqrt{\\frac{\\bar{n}U}{m^{*}}\\left(1-6V_{+}^{2}V_{-}^{2}\\right)},\\\\& K_{\\mathrm {s}}\\sim \\pi \\sqrt{\\frac{\\bar{n}/m^{*}U}{1+2V_{+}^{2}V_{-}^{2}}},\\\\& K_{\\mathrm {a}}\\sim \\pi \\sqrt{\\frac{\\bar{n}/m^{*}U}{1-6V_{+}^{2}V_{-}^{2}}},\\\\& g_{0}\\sim 4\\bar{n}^2 UV_{+}V_{-},\\\\& g_{1}\\sim 2\\bar{n}^{2}U V_{+}^{2}V_{-}^{2},\\\\& g_{2}\\sim 8\\bar{n}^{2}UV_{+}^{2}V_{-}^{2},\\\\& g_{3}\\sim 8\\bar{n}^{2}UV_{+}^{2}V_{-}^{2},\\\\& g_{4}\\sim 2\\bar{n}^{2}U\\left(1+V_{+}^{2}V_{-}^{2}\\right),\\end{aligned}\\right.$ where $V_{+}^{2}V_{-}^{2}=\\frac{1}{4}\\left(\\frac{J_{\\perp }}{2J}\\right)^2/\\sin ^2\\left(\\frac{\\phi }{2}\\right)\\left[\\left(\\frac{J_{\\perp }}{2J}\\right)^2+\\sin ^2\\left(\\frac{\\phi }{2}\\right)\\right]$ .", "The above estimation is valid for a finite but sufficiently small interaction $U\\ll J_{\\perp }$ since a small interaction preserves the nature of the two minima in the single particle spectrum.", "At stronger coupling, the parameters will be strong influenced by the renormalization effect due to the irrelevant terms omitted in Eq.", "(REF ).", "However, the following qualitative tendency of the parameters controlled by the microscopic parameters is expected to be captured.", "In the limit of decoupled chains $J_\\perp \\rightarrow 0$ , the velocities $v_{\\mathrm {s},\\mathrm {a}}$ and Luttinger parameters $K_{\\mathrm {s},\\mathrm {a}}$ in the symmetric and antisymmetric sectors become identical: $v_\\mathrm {s}/v_\\mathrm {a}\\rightarrow 1$ , and $K_\\mathrm {s}/K_\\mathrm {a}\\rightarrow 1$ as $J_\\perp /J\\rightarrow 0$ .", "For the finite rung hopping $J_\\perp $ , $K_\\mathrm {s}/K_\\mathrm {a}<1$ and $v_\\mathrm {s}/v_\\mathrm {a}>1$ .", "In addition, the velocities and Luttinger parameters are controlled, respectively, to be enhanced and suppressed by the increase of the interaction $U$ .", "It is worthwhile showing the bosonized form of the physical quantity operators, which is useful when we discuss the physical meaning of the ordered phases caused by the lock of the fields $\\varphi _{\\mathrm {s},\\mathrm {a}}$ and $\\theta _{\\mathrm {s},\\mathrm {a}}$ .", "The density operators are represented in the bosonized form as $n_{j,1}&\\sim \\bar{n}-\\frac{a}{\\pi }\\left[V_{-}^{2}\\nabla \\varphi _{+}(x)+V_{+}^{2}\\nabla \\varphi _{-}(x)\\right]\\nonumber \\\\& \\quad +2\\bar{n}\\biggr [V_{-}^{2}\\cos \\left(\\frac{2\\pi \\bar{n}}{a}x -2\\varphi _{+}(x)\\right)\\nonumber \\\\& \\quad \\qquad \\qquad +V_{+}^{2}\\cos \\left(\\frac{2\\pi \\bar{n}}{a}x-2\\varphi _{-}(x)\\right)\\biggl ]\\nonumber \\\\& \\quad +2\\bar{n}V_{+}V_{-}\\cos \\left(\\frac{2Q}{a}x-\\sqrt{2}\\theta _{\\mathrm {a}}(x)\\right),\\\\n_{j,2}&\\sim \\bar{n}-\\frac{a}{\\pi }\\left[V_{+}^{2}\\nabla \\varphi _{+}(x)+V_{-}^{2}\\nabla \\varphi _{-}(x)\\right]\\nonumber \\\\& \\quad +2\\bar{n}\\biggl [V_{+}^{2}\\cos \\left(\\frac{2\\pi \\bar{n}}{a}x-2\\varphi _{+}(x)\\right)\\nonumber \\\\& \\quad \\qquad \\qquad +V_{-}^{2}\\cos \\left(\\frac{2\\pi \\bar{n}}{a}x-2\\varphi _{-}(x)\\right)\\biggr ]\\nonumber \\\\& \\quad +2\\bar{n}V_{+}V_{-}\\cos \\left(\\frac{2Q}{a}x-\\sqrt{2}\\theta _{\\mathrm {a}}(x)\\right).$ The current operators are defined as $j^{(\\parallel )}_{p,j}=-\\partial {H}/\\partial {A^{(\\parallel )}_{j,p}}$ at the $j$ th site on the $p$ th chain, and $j^{\\perp }_{j}=-\\partial {H}/\\partial {A^{(\\perp )}_{j}}$ on the $j$ th rung.", "Thus the bosonized form are given as $j^{(\\parallel )}_{j,1}&\\sim 2\\bar{n}J\\biggl [V_{+}^2\\sin \\left(Q-\\frac{\\phi }{2}\\right)-V_{-}^2\\sin \\left(Q+\\frac{\\phi }{2}\\right)\\nonumber \\\\&\\quad +aV_{-}^2\\cos \\left(Q+\\frac{\\phi }{2}\\right)\\nabla \\theta _{+}(x)\\nonumber \\\\&\\quad +aV_{+}^2\\cos \\left(Q-\\frac{\\phi }{2}\\right)\\nabla \\theta _{-}(x)\\nonumber \\\\&\\quad -2V_{+}V_{-}\\sin \\left(\\frac{\\phi }{2}\\right)\\cos \\left(\\frac{Q}{a}(2x+a)-\\sqrt{2}\\theta _{\\mathrm {a}}(x)\\right)\\biggr ],\\\\j^{(\\parallel )}_{j,2}&\\sim -2\\bar{n}J\\biggl [V_{+}^2\\sin \\left(Q-\\frac{\\phi }{2}\\right)-V_{-}^2\\sin \\left(Q+\\frac{\\phi }{2}\\right)\\nonumber \\\\&\\quad -aV_{+}^2\\cos \\left(Q-\\frac{\\phi }{2}\\right)\\nabla \\theta _{+}(x)\\nonumber \\\\&\\quad -aV_{-}^2\\cos \\left(Q+\\frac{\\phi }{2}\\right)\\nabla \\theta _{-}(x)\\nonumber \\\\&\\quad -2V_{+}V_{-}\\sin \\left(\\frac{\\phi }{2}\\right)\\cos \\left(\\frac{Q}{a}(2x+a)-\\sqrt{2}\\theta _{\\mathrm {a}}(x)\\right)\\biggr ],\\\\j^{(\\perp )}_{j}& \\sim 2\\bar{n}J_{\\perp }\\left(V_{+}^2-V_{-}^2\\right)\\sin \\left(\\frac{2Q}{a}x-\\sqrt{2}\\theta _{\\mathrm {a}}(x)\\right).$ Here in order to somewhat simplify the expression of the density and current operators, we have mixed the notation of $\\varphi _{\\pm }$ and $\\theta _{\\pm }$ with that of $\\varphi _{\\mathrm {s},\\mathrm {a}}$ and $\\theta _{\\mathrm {s},\\mathrm {a}}$ .", "The constant terms of the current operators imply the existence of Meissner currents, which are non-zero except for $Q\\pm \\phi /2=\\pi N$ ($N\\in \\mathbb {Z}$ )." ], [ "Effective Hamiltonian for small magnetic flux", "Let us consider the case for a small magnetic flux $\\phi <\\phi _\\mathrm {c}$ .", "As seen in Figs.", "REF and REF , the lower single-particle energy band then forms a single minimum at $k=0$ , and the low-energy physics would be governed by the band bottom since the bosons are expected to dominantly populate the energy minimum.", "Thus, similarly to the discussion in Sec.", "REF , we use the long-wave-length expansion around the energy minima at $k=0$ .", "Then the low-energy single-particle spectrum is approximated as $E_{-}(k)\\approx -E_{0} + \\frac{k^2}{2m^{*}},$ where the energy offset and the effective mass have been defined as $& E_{0}=J_{\\perp }+2J\\cos \\left(\\frac{\\phi }{2}\\right),\\\\& \\frac{1}{m^{*}}=2J\\left[\\cos \\left(\\frac{\\phi }{2}\\right)-\\frac{2J}{J_{\\perp }}\\sin ^2\\left(\\frac{\\phi }{2}\\right)\\right].$ The effective mass () diverges at the critical magnetic flux $\\phi _{\\mathrm {c}}$ given by Eq.", "(REF ), at which the two band minima merge as in Fig.", "REF .", "In such a flux regime near $\\phi _{\\mathrm {c}}$ , we would need higher orders of $k$ in the approximated dispersion (REF ), but we do not consider such a case in this paper.", "We look at the bosonic operators in the long-wave-length approximation.", "Projecting out the upper band states, and only considering the small wave length around the minimum of the lower energy band, i.e., $k=0$ , the bosonic operators (REF ) are approximated as $b_{j,1} \\approx b_{j,2} \\approx \\frac{1}{\\sqrt{2}}\\beta _{j},$ where $\\beta _{j}=\\frac{1}{\\sqrt{N}}\\sum _{k}e^{-ik j}\\beta (k)$ .", "It immediately leads to the approximate form of the density operators as $n_{j,1} \\approx n_{j,2} \\approx \\frac{1}{2}\\tilde{n}_{j},$ where $\\tilde{n}_{j}=\\beta ^{\\dagger }_{j}\\beta _{j}$ .", "This approximate form implies that the densities on the upper and lower chain are balanced as long as the bosons occupy only the vicinity of the energy minima.", "Using the formulas (REF ) and (REF ) in the long-wave-length approximation, the Hamiltonian (REF ) is rewritten as $H&\\approx \\sum _{k}\\frac{k^2}{2m^{*}}\\beta ^{\\dagger }_{k}\\beta _{k}-\\left(\\mu +E_{0}+\\frac{U}{2}\\right)\\sum _{j}\\tilde{n}_{j}+\\frac{U}{4}\\sum _{j}\\tilde{n}_{j}^{2}.$ As in Appendix , in this approximation, the chemical potential to reproduce the density of the original bosons $\\mathinner {\\langle {n_{j,1}}\\rangle }=\\mathinner {\\langle {n_{j,2}}\\rangle }=\\bar{n}$ should be controlled as Eq.", "(REF ), and the corresponding mean density of $\\tilde{n}_j$ is $\\langle {\\tilde{n}_j}\\rangle =2\\bar{n}$ .", "Based on this mean-field solution, we apply the bosonization, $\\left\\lbrace \\begin{aligned}& \\beta _{j}\\sim \\sqrt{2\\bar{n}}e^{i\\theta (x)},\\\\& \\tilde{n}_{j}\\sim 2\\bar{n}- \\frac{a}{\\pi }\\nabla \\varphi (x)+4\\bar{n}\\cos \\left[\\frac{4\\pi \\bar{n}}{a}x-2\\varphi (x)\\right].\\end{aligned}\\right.$ Then the effective theory of the Hamiltonian (REF ), in which only the fluctuation terms are retained, is straightforwardly found to be a simple sine-Gordon model: $H_{\\mathrm {eff}}&=\\frac{v}{2\\pi }\\int \\!\\!dx\\,\\left[K\\left(\\nabla \\theta (x)\\right)^2+\\frac{1}{K}\\left(\\nabla \\varphi (x)\\right)^2\\right]\\nonumber \\\\&\\quad +g\\int \\!\\!\\frac{dx}{a}\\,\\cos \\left(\\frac{4\\pi \\bar{n}}{a}x-2\\varphi (x)\\right),$ where the parameters are approximately estimated as $\\left\\lbrace \\begin{aligned}&v\\sim a\\sqrt{\\frac{\\bar{n}U}{m^{*}}},\\\\& K\\sim \\pi \\sqrt{\\frac{\\bar{n}}{m^{*}U}},\\\\& g\\sim 4\\bar{n}^2U.\\end{aligned}\\right.$ The estimation (REF ) applies only at finite but small $U\\ll \\max [J_{\\perp }, J]$ as mentioned in Sec.", "REF , but the qualitative tendency such as an increase and decrease of $v$ and $K$ with $U$ , respectively, is expected to be seen even if $U$ is not in the limit, as seen in other cases.", "The form of the effective theory (REF ) looks very similar to that of the one-dimensional Bose-Hubbard chain [43], but the underlying physics is different.", "To see this, it is useful to look at the bosonized form of the physical quantities.", "The density and current operators of the original bosons are found to be represented by the bosonization formula (REF ) as $\\begin{aligned}n_{j,1}&\\sim \\bar{n}-\\frac{a}{2\\pi }\\nabla \\varphi (x)+2\\bar{n}\\cos \\left[\\frac{4\\pi \\bar{n}}{a}x-2\\varphi (x)\\right],\\\\n_{j,2}&\\sim \\bar{n}-\\frac{a}{2\\pi }\\nabla \\varphi (x)+2\\bar{n}\\cos \\left[\\frac{4\\pi \\bar{n}}{a}x-2\\varphi (x)\\right],\\\\j^{(\\parallel )}_{j,1}&\\sim -\\bar{n}J\\sin \\left(\\frac{\\phi }{2}\\right)+a\\bar{n}J\\cos \\left(\\frac{\\phi }{2}\\right)\\nabla \\theta (x),\\\\j^{(\\parallel )}_{j,2}&\\sim \\bar{n}J\\sin \\left(\\frac{\\phi }{2}\\right)+a\\bar{n}J\\cos \\left(\\frac{\\phi }{2}\\right)\\nabla \\theta (x),\\\\j^{(\\perp )}_{j}&\\sim 0.\\end{aligned}$ Note that the currents on the two chains have finite constant terms, which are proportional to $\\sin (\\phi /2)$ and have opposite signs, while the rung current is always zero.", "This implies that for the small magnetic flux $\\phi <\\phi _\\mathrm {c}$ , finite counter-flowing currents are induced on the chains, which correspond to Meissner currents.", "Let us discuss the relation to the argument given in Ref.", "[31] in which a similar problem is considered, but a different approach is used.", "Orignac and Giamarchi have introduced the independent two phase fluctuations in the upper and lower chain, i.e., $b_{j,p}\\propto \\exp (i\\theta _{j,p})$ for $p=1$ , 2.", "In their scenario, the relative phase fluctuation, $\\theta _{j,1}-\\theta _{j,2}$ , turns out to be gapful because of the interchain hopping $J_{\\perp }$ .", "On the other hand, in our approach, the higher energy states irrelevant to the low-energy physics are projected out, which allows us to effectively identify the bosonic operators, i.e., $b_{j,1}\\approx b_{j,2}$ .", "Namely, it means that within our approximation only the in-phase fluctuation, $\\theta _{j,1}+\\theta _{j,2}$ , is considered as the phase field here, $\\theta (x)$ , and the relative phase fluctuation is omitted in projecting out the higher energy states.", "Therefore, the gapful excitation of the relative phase, pointed out by Orignac and Giamarchi, is associated with the upper band which is projected out in our treatment.", "We discuss here the ground-state properties based on the obtained effective theories (REF ) and (REF ).", "We consider separately two different limits: the case of a large magnetic flux $\\phi =\\pi $ and the case of a small magnetic flux $\\phi <\\phi _\\mathrm {c}$ .", "For the latter, the low-energy single-particle energy band has a single minimum.", "In this section, we only consider a filling of one particle per site." ], [ "General discussion", "The unity filling Bose-Hubbard ladder for a magnetic flux $\\phi =\\pi $ has been previously discussed by DMRG in Refs.", "[34], [33], and the ground-state phase diagram is known to show the following features.", "At weak coupling, the system is in a gapless SF state with staggered loop currents (chiral superfluid, CSF), while a Mott insulator (MI) is found in strong-coupling regime.", "In between, a MI phase with staggered loop currents (chiral Mott insulator CMI) is found.", "In addition, the criticalities between these phases have also been numerically studied: the CSF-CMI and CMI-MI transitions exhibit Berezinskii-Kosterlitz-Thouless (BKT) [44], [45], [46] and Ising criticality, respectively.", "Here we discuss this ground-state phase diagram from the viewpoint of the effective field theory.", "The momentum giving the energy minima becomes $\\pm Q=\\pm \\pi /2$ for $\\phi =\\pi $ (Fig.", "REF ).", "In the perturbation $V_{0}$ in the effective theory, an oscillation remains in the form of $\\cos \\left[\\frac{\\pi }{2a}x-\\sqrt{2}\\theta _{\\mathrm {a}}(x)\\right]$ , and $V_{0}$ turns out to be irrelevant, while the oscillation in $V_{1}$ is canceled.", "If one considers the second-order perturbation theory in $V_{0}$ , the oscillation cancels: $V_{0}^{2}\\sim {g^{\\prime }}_{0}\\int \\!\\!\\frac{dx}{a}\\cos \\left[\\sqrt{8}\\theta _{\\mathrm {a}}(x)\\right],$ where $g^{\\prime }_{0}$ is a coupling constant proportional to $g_{0}^{2}$ .", "The form of the higher-order contribution (REF ) is identical to that of $V_{1}$ , which means that the effect due to $V_{0}$ can be fully absorbed into $V_{1}$ .", "Thus let us ignore $V_{0}$ in this discussion.", "Setting $\\bar{n}=1$ , the effective Hamiltonian in the $\\pi $ magnetic flux case turns out to be slightly simplified as $H_{\\mathrm {eff}}&=H_{\\mathrm {TL}}+\\int \\!\\!\\frac{dx}{a}\\,\\biggl [g_{1}\\cos \\left(\\sqrt{8}\\theta _{\\mathrm {a}}(x)\\right)\\nonumber \\\\&\\quad +g_{2}\\cos \\left(\\sqrt{8}\\varphi _{\\mathrm {s}}(x)\\right)+g_{3}\\cos \\left(\\sqrt{8}\\varphi _{\\mathrm {a}}(x)\\right)\\nonumber \\\\&\\quad +g_{4}\\cos \\left(\\sqrt{2}\\varphi _{\\mathrm {a}}(x)\\right)\\cos \\left(\\sqrt{2}\\varphi _{\\mathrm {s}}(x)\\right)\\biggr ],$ where all the coupling constants are assumed to be positive from the estimation Eq.", "(REF ).", "The derived effective theory (REF ) is still complicated to analyze.", "We thus discuss the possible phases from the viewpoint of a scaling analysis.", "Let us consider a perturbative renormalization-group treatment of all the cosine terms in the effective Hamiltonian (REF ), and identify the scaling dimension of those cosine terms around the Gaussian fixed point.", "Denoting by $x_{\\mathcal {O}}$ the scaling dimension of a perturbation $\\mathcal {O}$ , we obtain for the effective theory (REF ) the following values: $\\left\\lbrace \\begin{aligned}& x_{\\cos (\\sqrt{8}\\theta _{\\mathrm {a}})}=\\frac{2}{K_{\\mathrm {a}}},\\\\& x_{\\cos (\\sqrt{8}\\varphi _{\\mathrm {s}})}=2K_{\\mathrm {s}},\\\\& x_{\\cos (\\sqrt{8}\\varphi _{\\mathrm {a}})}=2K_{\\mathrm {a}},\\\\& x_{\\cos (\\sqrt{2}\\varphi _{\\mathrm {a}})\\cos (\\sqrt{2}\\varphi _{\\mathrm {s}})}=\\frac{K_{\\mathrm {s}}+K_{\\mathrm {a}}}{2}.\\end{aligned}\\right.$ Up to first-order perturbative renormalization group, relevant perturbations $\\mathcal {O}$ are those for which $x_{\\mathcal {O}}<2$ .", "To derive the effective field theory depending on the possible values of the Luttinger parameters, we take the following steps: First the possible relevant terms, whose scaling dimensions are $<2$ , are written down depending on the parameter regime of $K_\\mathrm {s}$ and $K_\\mathrm {a}$ .", "Referring to the relevancy of the perturbations, we divide the parameter space into several subspaces.", "In each subspace the low-energy physics is described by an effective theory consisting of a different set of relevant perturbations.", "If there are several relevant perturbations in the subspace, each perturbation tends to lock the fields of $\\varphi _{\\mathrm {s},\\mathrm {a}}$ and $\\theta _{\\mathrm {s},\\mathrm {a}}$ to be different values.", "Then, if some of the relevant perturbations compete so as to fix the fields to the different values, e.g., the pairs of $\\cos (\\sqrt{8}\\theta _\\mathrm {a})$ and $\\cos (\\sqrt{8}\\varphi _\\mathrm {a})$ , and of $\\cos (\\sqrt{8}\\varphi _\\mathrm {s})$ and $\\cos (\\sqrt{2}\\varphi _\\mathrm {s})\\cos (\\sqrt{2}\\varphi _\\mathrm {a})$ , we retain only the most relevant perturbation, and omit the competing less relevant ones.", "If some of the relevant terms do not compete, e.g., $\\cos (\\sqrt{8}\\theta _{\\mathrm {a}})$ and $\\cos (\\sqrt{8}\\varphi _{\\mathrm {s}})$ , we retain all of them.", "Following this procedure, the parameter space spanned by the Luttinger parameters $K_\\mathrm {s}$ and $K_\\mathrm {a}$ is found to be separated into five different regimes, as displayed on Fig.", "REF , and each regime is governed by a particular form of the low-energy effective theory.", "Figure: The ground-state phase diagram in the parameter space spannedby the Luttinger parameters K s K_\\mathrm {s} and K a K_\\mathrm {a}.The different phases are identified as: chiral superfluid (CSF, Regime I),chiral Mott insulator (CMI, Regime II), conventional superfluid withouta current pattern (SF, Regime III), conventional Mott insulator withouta vortex current pattern (MI, Regime IV).The phase boundary between regimes I and IV is given byK a =-K s /2+(K s /2) 2 +4K_\\mathrm {a}=-K_\\mathrm {s}/2+\\sqrt{(K_\\mathrm {s}/2)^2+4}, and the one between IV and Vby K a =3K s K_\\mathrm {a}=3K_\\mathrm {s} and K a =K s /3K_\\mathrm {a}=K_\\mathrm {s}/3.In Regime I, $K_\\mathrm {s}>1$ , $K_\\mathrm {a}>1$ and $K_\\mathrm {a}>-K_\\mathrm {s}/2+\\sqrt{(K_\\mathrm {s}/2)^2+4}$ , the low-energy effective theory is given as $H^{\\mathrm {(I)}}_{\\mathrm {eff}}=H_{\\mathrm {TL}}+g_1\\int \\!\\!\\frac{dx}{a}\\cos \\left(\\sqrt{8}\\theta _\\mathrm {a}(x)\\right).$ Due to the cosine term, the relative phase $\\theta _\\mathrm {a}$ is locked in the ground states as $\\langle {\\theta _\\mathrm {a}}\\rangle =\\pm \\pi /\\sqrt{8}$ , which generates the finite energy gap in the antisymmetric field sector, while the unbounded symmetric phase sector remains gapless.", "According to the bosonized form of the current operators (REF )-(), the lock of the field $\\theta _\\mathrm {a}$ leads to the local currents: In the case of the fixed relative phase $\\langle {\\theta _\\mathrm {a}}\\rangle =\\pi /\\sqrt{8}$ , $\\left\\lbrace \\begin{aligned}& \\langle {j^{(\\parallel )}_{j,1}}\\rangle \\sim -4\\bar{n}J V_{+}V_{-}(-1)^{j},\\\\& \\langle {j^{(\\parallel )}_{j,2}}\\rangle \\sim 4\\bar{n}J V_{+}V_{-}(-1)^{j},\\\\& \\langle {j^{(\\perp )}_{j}}\\rangle \\sim -2\\bar{n}J_{\\perp }(V_{+}^2-V_{-}^2)(-1)^{j},\\end{aligned}\\right.$ and for $\\langle {\\theta _\\mathrm {a}}\\rangle =-\\pi /\\sqrt{8}$ the sign of all the currents becomes opposite.", "Because $V_{+}V_{-}=\\frac{1}{2}(J_\\perp /2J)^2/\\sqrt{(J_\\perp /2J)^2+1}$ and $J_\\perp (V_{+}^2-V_{-}^2)=J_\\perp /\\sqrt{1+(J_\\perp /2J)^2}$ for $\\phi =\\pi $ , the currents in Eq.", "(REF ) disappear at the limit of $J_\\perp /J\\rightarrow 0$ , and the strength grows as $J_\\perp /J$ goes up.", "The currents on the $j$ th bond in the upper and lower chain point oppositely, and the rung current are staggered along the chain direction.", "Based on the representation (REF ), the current pattern is illustrated on Fig.", "REF , in which staggered loop currents are found to appear.", "Therefore, Regime I should be interpreted as a CSF phase.", "The two-fold degeneracy is caused by the spontaneous breaking of translation symmetry Note that the choice of gauge (REF ) preserves translational symmetry of the Hamiltonian.", "If we take another choice of gauge, $A^{\\parallel }_{j,1}=A^{\\parallel }_{j,2}=0$ and $A^{\\perp }_{j}=-\\phi \\times j$ , the translation symmetry preserved in the choice of gauge (REF ) is explicitly broken..", "This physical description of the CSF phase agrees with that given in Refs.", "[34], [33].", "Figure: (Color online) A current pattern associated with the CSF phase(Regime I) and CMI phase (Regime II) for a π\\pi magnetic flux perplaquette.The black arrows denote the local currents given byEq.", "().The red circular arrows denote the local staggered vortices deducedfrom the local current pattern.The local currents vanish at small J ⊥ /JJ_\\perp /J, and their strengthincreases with the rung hopping J ⊥ /JJ_\\perp /J.In Regime II, $K_\\mathrm {s}<1$ , $K_\\mathrm {a}>1$ , $K_\\mathrm {a}>3K_\\mathrm {s}$ and $K_\\mathrm {a}>-K_\\mathrm {s}/2+\\sqrt{(K_\\mathrm {a}/2)^2+4}$ , the low-energy effective theory is given as $H^{\\mathrm {(II)}}_{\\mathrm {eff}}&=H_{\\mathrm {TL}}+\\int \\!\\!\\frac{dx}{a}\\biggl [g_1\\cos \\left(\\sqrt{8}\\theta _\\mathrm {a}(x)\\right)+g_2\\cos \\left(\\sqrt{8}\\varphi _\\mathrm {s}(x)\\right)\\biggr ].$ The two cosine terms in the effective theory separately lock both the relative phase $\\theta _\\mathrm {a}$ and the symmetric field $\\varphi _\\mathrm {s}$ in the ground state: $\\langle {\\theta _\\mathrm {a}}\\rangle =\\pm \\pi /\\sqrt{8}$ and $\\langle {\\varphi _\\mathrm {s}}\\rangle =\\pm \\pi /\\sqrt{8}$ .", "The two locked values $\\langle {\\varphi _\\mathrm {s}}\\rangle =\\pm \\pi /\\sqrt{8}$ cannot be distinguished by the compactification condition (REF ), but do not lead to any difference in the physical quantities (REF )-().", "Thus we can identify the two locks $\\langle {\\varphi _\\mathrm {s}}\\rangle =\\pm \\pi /\\sqrt{8}$ from the physical viewpoint, and in total the ground states are found to be two-fold degenerate.", "Due to the locking of the two fields, an energy gap opens both in the symmetric and antisymmetric sector, and thus the low-energy excitations in Regime II are fully gapped.", "As discussed in Regime I, the values of the locked relative phase, $\\langle {\\theta _\\mathrm {a}}\\rangle =\\pm \\pi /\\sqrt{8}$ , result in the current pattern (REF ) illustrated by Fig.", "REF .", "On the other hand, the locking of the field $\\varphi _\\mathrm {s}$ physically means that the density fluctuation is frozen, which means that the system behaves like a MI.", "Thus Regime II should be identified with the CMI phase which involves the current pattern shown in Fig.", "REF .", "From this current pattern, we can physically expect a two-fold degeneracy of the ground state, and this degeneracy comes from the two possible locks of $\\theta _\\mathrm {a}$ .", "In Regime III, $K_\\mathrm {s}>1$ , $K_\\mathrm {a}<1$ and $K_\\mathrm {a}<K_\\mathrm {s}/3$ , the low-energy effective theory is given by $H^{\\mathrm {(III)}}_{\\mathrm {eff}}=H_{\\mathrm {TL}}+g_3\\int \\!\\!\\frac{dx}{a}\\cos \\left(\\sqrt{8}\\varphi _\\mathrm {a}(x)\\right).$ The antisymmetric field $\\varphi _\\mathrm {a}$ is fixed in the ground state, i.e., $\\langle {\\varphi _\\mathrm {a}}\\rangle =\\pm \\pi /\\sqrt{8}$ , and the excitation in this antisymmetric sector becomes gapful, while the symmetric sector remains gapless.", "The physical meaning of this lock of the field $\\varphi _\\mathrm {a}$ is not clear because both the density and currents do not show a signature of the corresponding order.", "The two possible locks of $\\varphi _\\mathrm {a}$ result in the double degeneracy of the ground states, but they do not give any difference in the physical quantities (REF )-().", "From the above, we can conclude that the ground state in Regime III is unique and some kind of SF phase with one gapless excitation mode.", "In Regime IV, $K_\\mathrm {a}>K_\\mathrm {s}/3$ , $K_\\mathrm {a}<3K_\\mathrm {s}$ and $K_\\mathrm {a}<-K_\\mathrm {s}/2+\\sqrt{(K_\\mathrm {a}/2)^2+4}$ , the low-energy effective theory is given as $H^{(\\mathrm {IV})}_{\\mathrm {eff}}&=H_{\\mathrm {TL}}+g_4\\int \\!\\!\\frac{dx}{a}\\cos \\left(\\sqrt{2}\\varphi _\\mathrm {s}(x)\\right)\\cos \\left(\\sqrt{2}\\varphi _\\mathrm {a}(x)\\right).$ Thus in the ground state both the fields $\\varphi _\\mathrm {a}$ and $\\varphi _\\mathrm {s}$ are naively found to be locked in the following two ways: $\\langle {\\varphi _\\mathrm {a}}\\rangle =\\pi /\\sqrt{2}$ and $\\langle {\\varphi _\\mathrm {s}}\\rangle =0$ , or $\\langle {\\varphi _\\mathrm {a}}\\rangle =0$ and $\\langle {\\varphi _\\mathrm {s}}\\rangle =\\pi /\\sqrt{2}$ .", "However, from the compactification (REF ), these two locked points are identical, and thus the ground state is unique.", "Due to the locking of the two fields $\\varphi _\\mathrm {s}$ and $\\varphi _\\mathrm {a}$ , which means that all the density fluctuations are frozen, the ground state is fully gapped.", "Therefore Regime IV corresponds to the conventional MI phase.", "In Regime V, $K_\\mathrm {s}<1$ and $K_\\mathrm {a}<K_\\mathrm {s}/3$ or $K_\\mathrm {a}<1$ and $K_\\mathrm {a}>3K_\\mathrm {s}$ , the low-energy effective theory is given as $H^{(\\mathrm {V})}_{\\mathrm {eff}}&=H_{\\mathrm {TL}}+\\int \\!\\!\\frac{dx}{a}\\biggl [g_{2}\\cos \\left(\\sqrt{8}\\varphi _\\mathrm {s}(x)\\right)+g_{3}\\cos \\left(\\sqrt{8}\\varphi _\\mathrm {a}(x)\\right)\\biggr ].$ Thus in the ground state, $\\varphi _\\mathrm {s}$ and $\\varphi _\\mathrm {a}$ are fixed to be $\\langle {\\varphi _\\mathrm {s}}\\rangle =\\pm \\pi /\\sqrt{8}$ and $\\langle {\\varphi _\\mathrm {a}}\\rangle =\\pm \\pi /\\sqrt{8}$ .", "However, due to the compactification (REF ), $(\\langle {\\varphi _\\mathrm {s}}\\rangle ,\\langle {\\varphi _\\mathrm {a}}\\rangle )=(-\\pi /\\sqrt{8},\\pm \\pi /\\sqrt{8})$ are identified with $(\\pi /\\sqrt{8},\\mp \\pi /\\sqrt{8})$ , respectively.", "Thus the two distinguishable states minimize the cosine terms in the effective theory.", "In order to clarify the physical meaning of these ground states in this phase, we look at the bosonized form of the density difference between the chains from Eq.", "(REF ).", "Then the mean values of the density difference is found to give $\\langle {n_{j,1}-n_{j,2}}\\rangle \\sim 4\\left(V_{+}^{2}-V_{-}^{2}\\right)\\sin \\left(\\sqrt{2}\\langle {\\varphi _\\mathrm {s}}\\rangle \\right)\\sin \\left(\\sqrt{2}\\langle {\\varphi _\\mathrm {a}}\\rangle \\right).$ The density difference is found to be finite in the obtained two states: $\\langle {n_{j,1}-n_{j,2}}\\rangle >0$ for $\\langle {\\varphi _\\mathrm {s}}\\rangle =\\pi /\\sqrt{8}$ and $\\langle {\\varphi _\\mathrm {a}}\\rangle =\\pi /\\sqrt{8}$ , and $\\langle {n_{j,1}-n_{j,2}}\\rangle <0$ for $\\langle {\\varphi _\\mathrm {s}}\\rangle =\\pi /\\sqrt{8}$ and $\\langle {\\varphi _\\mathrm {a}}\\rangle =-\\pi /\\sqrt{8}$ .", "Such a density imbalance is inconsistent with the balanced density situation based on the mean-field analysis in Appendix .", "Thus the simultaneous lock of both fields $\\varphi _\\mathrm {s}$ and $\\varphi _\\mathrm {a}$ signals the instability of the state with balanced densities between the two chains, leading to a state with density imbalance (DI)." ], [ "Physical phase diagram as a function of\n$U/J$ and {{formula:8c10e73f-7a03-4a12-8558-0c34ee54c97b}}", "We have discussed the general structure of the phase diagram (Fig.", "REF ), but the SF and DI phase may not be realized in the original Bose-Hubbard model due to the two following reasons.", "The first is that the regime $K_\\mathrm {a}< K_\\mathrm {s}$ would be forbidden in terms of the microscopic parameters $(U/J,J_\\perp /J)$ .", "This is predicted by the naive parameter estimation (REF ).", "Thus the SF phase (Regime III) and a part of the DI phase (Regime V) would not be realistic.", "The other reason is that the Luttinger parameters in these regimes would be too small to reach.", "Naively a Luttinger parameter for bosons with short-range interaction such as the Lieb-Liniger model [48] and the Bose-Hubbard model at an incommensurate filling [43] can run only from infinity to unity as interaction increases, in which the infinite and unity limit of the Luttinger parameter correspond to the non-interacting and hard-core boson limit, respectively.", "Strictly speaking, these constraints do not necessarily apply the present ladder model, but $K_\\mathrm {s}<1/3$ or $K_\\mathrm {a}<1/3$ for the DI phase (Regime V) is still considered to be extremely small for a bosonic system.", "Indeed the numerically determined phase diagram given in Refs.", "[34], [33] does not show such SF and DI phases.", "The obtained phase diagram Fig.", "REF is parametrized by the phenomenological parameters $K_\\mathrm {s}$ and $K_\\mathrm {a}$ .", "Thus in order to estimate the phase diagram in terms of the microscopic parameters, $U/J$ and $J_\\perp /J$ , we need to clarify the behavior of the Luttinger parameters as a function of these microscopic parameters.", "The general field theory analysis, which applies only at low energy, is insufficient to fully answer to this microscopic question.", "Thus we make use of other general arguments and constraints to figure out qualitatively the phase diagram in terms of the microscopic parameters.", "The following qualitative features of the Luttinger parameters can be deduced from the estimation in Eq.", "(REF ).", "The Luttinger parameter of the symmetric sector is smaller than that of the antisymmetric sector for given $U/J$ and $J_\\perp /J$ , i.e., $K_\\mathrm {s}< K_\\mathrm {a}$ .", "In addition, $K_\\mathrm {s}/K_\\mathrm {a}\\rightarrow 1$ as the ladder is decoupled $J_\\perp /2J\\rightarrow 0$ .", "The Luttinger parameters must be large at small interaction $U$ and decrease as the interaction goes stronger.", "From these assumptions, we can expect the following evolution of the trajectory between $K_\\mathrm {s}$ and $K_\\mathrm {a}$ by controlling the interaction $U$ : At the limit $J_\\perp /2J\\rightarrow 0$ , $K_\\mathrm {a}=K_\\mathrm {s}$ , and this trajectory continuously deforms keeping $K_\\mathrm {a}>K_\\mathrm {s}$ as $J_\\perp /2J$ grows.", "The expected trajectories are shown in the left panel of Fig.", "REF .", "As clear from Fig.", "REF , the trajectory $K_\\mathrm {a}=K_\\mathrm {s}$ implies that the system is in the CSF phase in the weakly interacting regime, and becomes MI at a critical value of $U/J$ without an intervening CMI phase.", "Since the deformation of the trajectory by a change of $J_\\perp /J$ should be continuous, the above SF-MI transition must remain up to a certain value of $J_\\perp /J$ .", "At a specific value of $J_\\perp /J$ , the trajectory passes the tricritical point at which the phase boundaries among the CSF (Regime I), CMI (Regime II), and MI (Regime IV) phase meet.", "Beyond this value of $J_\\perp /J$ , a CMI phase (Regime II) opens up in between the CSF and MI phases for intermediate interaction strengths $U/J$ .", "We summarize this description and the deduced phase diagram in the space of microscopic parameters in Fig.", "REF .", "The important point of the deduced phase diagram Fig.", "REF is the presence of the tricritical point.", "In the DMRG study of Ref.", "[34], [33] this tricritical point was not found, presumably because of the limited number of values of the coupling constants that were investigated.", "The absence of the CMI phase for small $J_\\perp /J$ can also be established from another field-theoretical approach.", "As in Ref.", "[31], if we bosonize the Hamiltonian (REF ) in the limit of $J_\\perp /J=0$ , and take into account the rung hopping perturbatively, the first-order contribution of the rung hopping Hamiltonian involves a $\\pi $ -oscillating term, and thus we need to take into account at least the second-order perturbation in order to see the finite rung hopping contribution.", "It means that for sufficiently small rung hopping, the Bose-Hubbard ladder in the presence of a magnetic flux can be effectively identified to decoupled Bose-Hubbard chains.", "Thus, in such a small rung hopping regime, one can only observe the SF-MI transition by controlling the interaction $U/J$ as in the case of the single Bose-Hubbard chain.", "In addition, from this argument, the SF-MI transition line drawn by controlling $J_\\perp /J$ is inferred to be independent of $J_\\perp /J$ .", "Namely the boundary between the CSF and MI phase rises up from $J_\\perp /J=0$ perpendicularly to the $U/J$ axis, and eventually bifurcates into the two lines of the CSF-CMI and CMI-MI transitions.", "Figure: (Color online) A sketch of the trajectories of K s K_\\mathrm {s} andK a K_\\mathrm {a} for different values of J ⊥ /JJ_\\perp /J (left panel), andschematic phase diagram as a function of the microscopic parameters(right panel).Each arrow in the left panel indicates the evolution of K s K_\\mathrm {s} andK a K_\\mathrm {a} as the interaction strength U/JU/J is increased.The rightmost arrow whose trajectory is described byK s ≃K a K_\\mathrm {s}\\simeq K_\\mathrm {a} corresponds to the limit of decoupled chainsJ ⊥ /J→0J_\\perp /J\\rightarrow 0, and other arrows correspond to the gradualincrease of J ⊥ /JJ_\\perp /J.Following the different trajectories in the generic phase diagramallows one to establish qualitatively the physical phase diagramdisplayed in the right panel." ], [ "Critical behavior", "Closing the discussion on the ground-state phase diagram of the $\\pi $ magnetic flux case at unity filling, we discuss the nature of the quantum critical behavior between the different phases.", "Let us first consider the CSF-CMI transition.", "As in the effective theories, Eq.", "(REF ) for CSF, and Eq.", "(REF ) for CMI, the symmetric and antisymmetric sectors are decoupled in both regimes, and the transition is found to be characterized by the locking of the symmetric field $\\varphi _\\mathrm {s}$ .", "Hence, focusing only on the symmetric sector in these two regimes, the phase transition from CSF to CMI is analogous to that of the sine-Gordon model.", "Thus the CSF-CMI transition is concluded to be of BKT transition nature, which is in agreement with the statement made in the numerical study of Ref.", "[34], [33].", "The nature of the CMI-MI transition is a more complicated issue, because the symmetric and antisymmetric sectors are coupled in the effective theory (REF ) of the MI regime.", "Comparing the effective theories $H^{(\\mathrm {II})}_{\\mathrm {eff}}$ and $H^{(\\mathrm {IV})}_{\\mathrm {eff}}$ , two phenomena are found to occur at the transition from CMI to MI.", "One is the switch of the locked field in the antisymmetric sector, from $\\theta _\\mathrm {a}$ to $\\varphi _\\mathrm {a}$ , and the other is the change of the locking value of the symmetric field $\\varphi _\\mathrm {s}$ .", "In addition, the two-fold degeneracy caused by the fixed $\\theta _\\mathrm {a}$ in the CMI phase is found to change to a non-degenerate state in the MI phase.", "This change of the degeneracy means that the translation symmetry which is spontaneously broken in the CMI phase is restored in the MI phase.", "The nature is thus analogous to the $\\mathbb {Z}_{2}$ Ising transition, which was also predicted for the CMI-MI transition for the frustrated bosonic ladder system [49].", "The previous numerical studies [34], [33] has pointed out the $\\mathbb {Z}_{2}$ Ising criticality of the CMI-MI transition, and our field theoretical approach is thus consistent with this.", "Finally we consider the direct phase transition between CSF and MI, shown in Fig.", "REF .", "Because of the coupling of the symmetric and antisymmetric sector in the MI phase, the analysis of this transition is not simple.", "Two simultaneous phenomena occur: the switch of the bound field from $\\theta _\\mathrm {a}$ to $\\varphi _\\mathrm {a}$ in the asymmetric sector and the locking of the field $\\varphi _\\mathrm {s}$ in the symmetric sector.", "This phase boundary is intriguing because two different symmetries are simultaneously involved: the continuous $O(2)$ symmetry associated with the SF and the $\\mathbb {Z}_{2}$ symmetry associated with the breaking of translational invariance in the CSF phase.", "From usual considerations based on the Landau-Ginzburg-Wilson approach to critical phenomena, one may conclude that this phase transition is first-order.", "Indeed, according to Eq.", "(REF ), the local currents in the CSF phase do not depend on the interaction $U$ , and the loop current is thus expected to discontinuously vanish when crossing the phase boundary from the CSF to MI phase.", "However, because the discussion which leads to the loop currents (REF ) is a kind of mean-field approach, a more in-depth discussion would be needed to obtain a crucial conclusion on the criticality.", "A more intriguing possibility on this criticality is that it might nonetheless be second-order, despite breaking simultaenously two unrelated symmetries [50], [51]." ], [ "The ground state for small magnetic flux at unity filling", "Next we discuss the phase diagram of the Bose-Hubbard ladder for a sufficiently small magnetic flux in which the bottom of the single particle spectrum shows a single energy minimum structure.", "In addition, we fix the filling at one particle per site.", "Then, setting $\\bar{n}=1$ , we can write the effective Hamiltonian (REF ) as $H_{\\mathrm {eff}}&=\\frac{v}{2\\pi }\\int \\!\\!dx\\,\\left[K\\left(\\nabla \\theta (x)\\right)^2+\\frac{1}{K}\\left(\\nabla \\varphi (x)\\right)^2\\right]\\nonumber \\\\&\\quad +g\\int \\!\\!\\frac{dx}{a}\\,\\cos \\left(2\\varphi (x)\\right).$ It has the same form as that of the single Bose-Hubbard chain.", "Thus a BKT transition is found when the Luttinger parameter as defined here reaches $K=2$ .", "This transition is identical to the SF-Mott insulator transition [43], [52].", "In the weakly interacting regime, the Luttinger parameter is larger than the critical value $K=2$ , and the system is a gapless TL liquid, i.e.", "a one-dimensional SF.", "As the interaction is tuned to be larger, the Luttinger parameter becomes smaller, and the system becomes a MI for $K<2$ .", "This behavior can be captured by the approximately estimated parameters (REF ): $K\\propto U^{-1/2}$ .", "The Luttinger parameter should be determined in terms of the interaction $U/J$ and the rung hopping $J_\\perp /J$ , but the corresponding critical value of these microscopic parameters can not be determined just from the field theoretical argument.", "Thus in this paper we do not discuss further quantitatively the ground-state phase diagram of the effective theory (REF ) in the microscopic parameter space.", "Let us look at the ground-state physical properties of the gapless SF and MI phase predicted by the effective theory (REF ).", "As mentioned in Sec.", "REF , the bosonized form of the current operators in Eq.", "(REF ) implies a non-zero constant current, which is displayed in Fig.", "REF .", "As given in the form, $\\pm J\\sin (\\phi /2)$ , this persistent current is induced by the magnetic flux, and is thus interpreted to be a Meissner current in the case of the ladder geometry.", "What is interesting is that the presence of this Meissner current is independent of the physics of the density fluctuation $\\varphi $ .", "On the other hand, even when the system is in the MI phase, in which $\\varphi $ is locked by the cosine term, the Meissner current remains.", "Figure: A pattern of Meissner currents appearing for a small magneticflux.", "The strength of the Meissner current increases with the magneticflux φ\\phi .This current also remains even in the Mott insulator phase, but thechiral state is not gapless.The physical reason of the presence of the Meissner current in the MI phase can be understood as follows.", "As well known, in the MI state, the phase of each bosons is completely disordered since the canonically conjugate density fluctuations are frozen.", "However, this statement does not forbid the lock of the relative phase between the bosons of the different component.", "Thus, in this MI case, each phase of the bosons on the upper and on the lower chain is disordered, but the relative phase between them is kept to be locked like that of the SF phase.", "Indeed, as mentioned in Ref.", "[31], the Meissner current is a consequence of the lock of the relative phase between the bosons on the upper and on the lower chain.", "A similar nature of the Meissner current in the fully gapped ground state was also pointed out in Ref. [32].", "A question which naturally arises is what happens to the MI with Meissner currents in the limit of strong interactions.", "In our effective field theory approach (REF ), only two phases (SF and MI with Meissner currents) have been obtained.", "However, if we turn back to the original microscopic Hamiltonian, we do expect the current-carrying Mott state to be eventually unstable in favor of a conventional Mott state without currents.", "Indeed, in the weak-coupling effective field-theory, one first establishes the two-band structure of the non-interacting Hamiltonian and then turns on an interaction within the lowest band only (protected by a gap from the upper one).", "For a strong interaction, however, matrix elements of the interaction will couple the two bands, which may break the relative phase coherence between the two chains and lead to a conventional MI.", "This regime is away from the range of applicability of the effective field-theory approach.", "From the above, we can describe the ground state of the unity-filling Bose-Hubbard ladder at a small magnetic flux as follows.", "In the weakly interacting regime, the system is a SF with Meissner currents, and the low-energy excitations carry chiral current, i.e., the directions of the carried currents on the upper and on the lower chain are opposite each other.", "On the other hand, in the strong interaction regime, the system is a MI in which the density fluctuations are completely suppressed, and there are no gapless excitations.", "However, this MI state still includes the Meissner current background.", "The transition between these two SF and MI phase is a BKT transition, as seen in a simple one-dimensional Bose-Hubbard chain at an integer filling.", "Furthermore, at strong interaction $U\\gg J, J_\\perp $ , the MI with the Meissner current should turn into a conventional MI without currents.", "However, the transition and criticality between these two MI phases are not easily addressed within the present analysis." ], [ "Summary and perspectives", "In this paper, we have discussed the Bose-Hubbard model with a uniform magnetic flux in ladder geometry.", "Discussing the small and large magnetic flux limits separately, we have constructed in each case the appropriate low-energy effective field theory by using bosonization techniques based on the nature of the single-particle spectrum.", "The key difference between the two cases is the number of lowest energy band minima.", "For a small magnetic flux, the bottom of the lowest band displays a single minimum.", "Increasing the magnetic flux beyond a critical value $\\phi _\\mathrm {c}$ , this single minimum splits into two degenerate minima, which leads to a different structure of the low-energy field theory.", "As an application of the derived effective field theories, we have discussed in detail the phases and physical properties of the system with one particle per site in the two cases of $\\phi =\\pi $ and small $\\phi <\\phi _\\mathrm {c}$ .", "For the $\\pi $ magnetic flux, we have established the general ground-state phase diagram as a function of the Luttinger parameters characterizing the low-energy field theory.", "Several phases appear: a superfluid (SF), chiral superfluid (CSF), Mott insulator (MI), chiral Mott insulator (CMI), as well as a regime of density imbalance (DI).", "Furthermore, we have also discussed the mapping of this generic phase diagram in terms of the two microscopic parameters of the Bose-Hubbard model (the interaction strength $U/J$ and ratio of rung to in-chain hopping $J_\\perp /J$ ).", "We have established that the CMI phase only occurs beyond a critical value of $J_\\perp /J$ , and to reveal the existence of a tricritical point at which the CSF, CMI and MI phases meet together.", "We also discussed the zero-temperature transitions and critical behavior separating these phases, and pointed out that the precise nature of the critical behavior for the direct transition between the CSF and MI phase is an interesting open issue to be addressed in future studies.", "In the small magnetic flux case, we have clarified the possible ground states and their properties.", "The SF and MI states have been, respectively, found to appear at weak and strong interaction strength, with a BKT transition between them.", "We found that not only the SF state but also the MI state displays Meissner currents.", "In a remarkable recent experiment [37], Atala et al.", "realized a two-leg ladder optical lattice in which bosonic atoms are confined and subject to an artificial uniform magnetic field.", "The Meissner currents and vortex currents in the SF phase were successfully probed by using a site-resolved local current measurement [53], [54].", "These achievements should make it possible to investigate experimentally the various phases (SF, CSF, MI, CMI) discussed in the present work and to probe the Meissner currents and vortex structure.", "In addition, the tricritical point found in our study, and the nature of the CSF-MI transition could be put to the test in such experiments.", "We thank Thierry Giamarchi, Masaaki Nakamura, Masaki Oshikawa and Alexandru Petrescu for fruitful discussions.", "We acknowledge the support of the DARPA-OLE program, of the Swiss National Science Foundation under MaNEP and Division II, and of a grant from the European Research Council (ERC-319286 QMAC)." ], [ "Mean-field analysis to the long-wave-length effective\nHamiltonian", "Here we present the mean-field approach to determine the chemical potential in the effective Hamiltonian given by the long-wave-length approximation." ], [ "Large magnetic flux case", "First let us see the case of the large magnetic flux, in which the single-particle spectrum forms the double minima in the lower energy band.", "Based on the Hamiltonian (REF ) and (REF ) derived by the long-wave-length approximation the mean-field energy per site, in which the quantum fluctuations are ignored, is assumed to be $E_{\\mathrm {MF}}&=-\\frac{1}{2}\\left(\\mu +E_{0}+\\frac{U}{2}-UV_{+}^2V_{-}^2\\right)\\left(\\tilde{n}_{+}+\\tilde{n}_{-}\\right)\\nonumber \\\\& \\quad +\\frac{U\\left(1+2V_{+}^2V_{-}^{2}\\right)}{8}\\left(\\tilde{n}_{+}+\\tilde{n}_{-}\\right)^2\\nonumber \\\\& \\quad +\\frac{U\\left(1-6V_{+}^2V_{-}^2\\right)}{8}\\left(\\tilde{n}_{+}-\\tilde{n}_{-}\\right)^2,$ where $\\tilde{n}_{\\pm }=\\langle {\\tilde{n}_{\\pm ,j}}\\rangle $ .", "From the mean-field energy, the mean-field equations for the density of the bosons populating at each band minima are derived by $\\partial {E_{\\mathrm {MF}}}/{\\partial {\\tilde{n}_{\\pm }}}=0$ , and lead to the mean-field solution $\\bar{n}=\\tilde{n}_{+}=\\tilde{n}_{-}$ with $& \\mu = -E_{0}+U\\left(\\bar{n}-\\frac{1}{2}\\right)+V^{2}_{+}V^{2}_{-}U\\left(2\\bar{n}+1\\right),$ which determines the chemical potential for the given density $\\tilde{n}_{\\pm }$ .", "The obtained mean-field density $\\bar{n}$ can be associated with those of the chains, $\\bar{n}_{p}=\\langle {n_{j,p}}\\rangle $ ($p=1,2$ ), in the original representation.", "The approximated form of the densities (REF ) leads to $\\bar{n}_{1}=V^{2}_{-}\\tilde{n}_{+}+V^{2}_{+}\\tilde{n}_{-},\\nonumber \\\\\\bar{n}_{2}=V^{2}_{+}\\tilde{n}_{+}+V^{2}_{-}\\tilde{n}_{+},$ and $\\bar{n}_{1}=\\bar{n}_{2}=\\bar{n}$ is immediately concluded since $\\tilde{n}_{\\pm }=\\bar{n}$ and $V^{2}_{+}+V^{2}_{-}=1$ .", "In addition, the stability of the mean-field solution is confirmed by the positive definiteness of Hessian matrix $H_{\\alpha ,\\beta }=\\partial ^{2}{E_{\\mathrm {MF}}}/\\partial {\\tilde{n}_{\\alpha }}\\partial {\\tilde{n}_{\\beta }}>0$ ($\\alpha ,\\beta =\\pm $ ).", "From the straightforward calculation of the eigenvalues of the Hessian matrix, the condition of the stable mean-field solution is found to be reduced to $\\left(\\frac{J_{\\perp }}{2J}\\right)^2<\\frac{2\\sin ^4\\left(\\phi /2\\right)}{3-2\\sin ^2\\left(\\phi /2\\right)}.$ This condition needs the smaller rung hopping as $\\phi $ decreases.", "For example, for the largest magnetic flux case $\\phi =\\pi $ , it leads to $J_{\\perp }^{2}<8J^2$ , and for the less flux $\\phi =\\pi /2$ , $J_{\\perp }^2<J^2$ is needed." ], [ "Small magnetic flux case", "Next we consider the small magnetic flux case, in which the low-energy single-particle spectrum shows a single minimum in the bottom of the lower energy band.", "Neglecting the quantum fluctuations in the approximated long-wave-length Hamiltonian (REF ), the mean-field energy is given by $E_{\\mathrm {MF}}=-\\frac{1}{2}\\left(\\mu +E_{0}+\\frac{U}{2}\\right)\\tilde{n}+\\frac{U}{8}\\tilde{n}^2,$ where $\\tilde{n}=\\langle {\\tilde{n}_{j}}\\rangle $ in Eq.", "(REF ).", "Thus the mean-field solution is given by $\\partial {E_{\\mathrm {MF}}}/\\partial {\\tilde{n}}=0$ , which is $\\mu = -E_0+\\frac{U}{2}\\left(\\tilde{n}-1\\right).$ In addition, from the second-order derivative of the mean-field energy, the above mean-field solution is immediately found to be stable.", "The mean density on the chains is balanced as in Eq.", "(REF ), i.e., $\\langle {n_{j,1}}\\rangle =\\langle {n_{j,2}}\\rangle =\\tilde{n}/2$ .", "Thus, supposing the balanced mean density on the chains to be $\\bar{n}$ , this density is controlled by the chemical potential as $\\bar{n}=\\frac{1}{U}\\left(\\mu +E_{0}+\\frac{U}{2}\\right).$" ] ]
1403.0413
[ [ "An extension of Herglotz's theorem to the quaternions" ], [ "Abstract A classical theorem of Herglotz states that a function $n\\mapsto r(n)$ from $\\mathbb Z$ into $\\mathbb C^{s\\times s}$ is positive definite if and only there exists a $\\mathbb C^{s\\times s}$-valued positive measure $d\\mu$ on $[0,2\\pi]$ such that $r(n)=\\int_0^{2\\pi}e^{int}d\\mu(t)$for $n\\in \\mathbb Z$.", "We prove a quaternionic analogue of this result when the function is allowed to have a number of negative squares.", "A key tool in the argument is the theory of slice hyperholomorphic functions, and the representation of such functions which have a positive real part in the unit ball of the quaternions.", "We study in great detail the case of positive definite functions." ], [ "Introduction", "The main purpose of this paper is to prove a version of a theorem of Herglotz on positive functions in the quaternionic and indefinite setting.", "To set the framework we first recall some definitions and results pertaining to the complex numbers setting.", "A function $n\\mapsto r(n)$ from $\\mathbb {Z}$ into $\\mathbb {C}^{s\\times s}$ is called positive definite if the associated function (also called kernel) $K(n-m)$ is positive definite on $\\mathbb {Z}$ .", "This means that for every choice of $N\\in \\mathbb {N}$ and $n_1,\\ldots , n_N\\in \\mathbb {Z}$ , the $N\\times N$ block matrix with $(j,\\ell )$ block entry equal to the matrix $r(n_j-n_\\ell )$ is non-negative, that is, all the block Toeplitz matrices $\\mathbb {T}_N\\stackrel{\\rm def.", "}{=}\\begin{pmatrix}r(0)&r(1)&\\cdots &r(N)\\\\r(-1)&r(0)&\\cdots &r(N-1)\\\\& & &\\\\& & &\\\\r(-N)&r(1-N)&\\cdots & r(0)\\end{pmatrix}$ are non-negative.", "We will use the notation $\\mathbb {T}_N \\succeq 0$ .", "The positivity implies in particular that $r(-n)=r(n)^*$ , where $r(n)^*$ denotes the adjoint of $r(n)$ .", "A result of Herglotz, also known as Bochner's theorem, asserts that: Theorem 1.1 The function $n \\mapsto r(n)$ from $\\mathbb {Z}$ into $\\mathbb {C}^{s \\times s}$ is positive definite if and only if there exists a unique positive $\\mathbb {C}^{s \\times s}$ -valued measure $\\mu $ on $[0, 2\\pi ]$ such that $r(n) = \\int _0^{2\\pi } e^{i n t} d\\mu (t), \\quad n \\in \\mathbb {Z}.$ See for instance [17], [13] and the discussion in [18].", "We note that (REF ) can be rewritten as $r(n)=C^*U^nC,\\quad n\\in \\mathbb {Z},$ where $U$ denotes the unitary operator of multiplication by $e^{it}$ in $\\mathbf {L}_2([0,2\\pi ],d\\mu )$ , and $C$ denotes the operator defined by $C\\xi =\\xi $ from $\\mathbb {C}^s$ into $\\mathbf {L}_2([0,2\\pi ],d\\mu )$ .", "A result of Carathéodory [9], which is related to Theorem REF asserts that: Theorem 1.2 If the function $n \\mapsto r(n)$ from ${ -N, \\ldots , N}$ into $\\mathbb {C}^{s \\times s}$ is positive definite, i.e., $\\mathbb {T}_N \\succeq 0$ , then exists a function $n \\mapsto \\tilde{r}(n)$ from $\\mathbb {Z}$ to $\\mathbb {C}^{s \\times s}$ which is positive definite and satisfies $r(n) = \\tilde{r}(n), \\quad n \\in \\lbrace -N, \\ldots , N\\rbrace .$ In commutative harmonic analysis, Theorem REF is a special case of a general result of Weil [23] on the representation of positive definite functions on a group in terms of the characters of the group.", "See for instance [12] or [22].", "A key result in one of the proofs (see for instance [17]) of Theorem REF is Herglotz's representation theorem, which states that a $\\mathbb {C}^{s\\times s}$ -valued function $\\varphi $ is analytic and with a real positive part in the open unit disk $\\mathbb {D}$ if and only if it can be written as $\\varphi (z)=\\int _0^{2\\pi }\\frac{e^{it}+z}{e^{it}-z}d\\mu (t)+ia,$ where $d\\mu $ is as in Theorem REF and $a\\in \\mathbb {C}^{s\\times s}$ satisfies $a+a^*=0$ .", "There are a number of ways to prove (REF ).", "It can be obtained from Cauchy's formula and from the weak-$*$ compactness of the family of finite variation measures on $[0,2\\pi ]$ ; see for instance the discussion in [1].", "Krein extended the notion of positive definite functions to the notion of functions having a number of negative squares; see [15].", "We first recall the definition of this notion in the present setting: Definition 1.3 The function $n\\mapsto r(n)$ from $\\mathbb {Z}$ into $\\mathbb {C}^{s\\times s}$ satisfying $r(n)=r(-n)^*$ has a finite number of negative squares, say $\\kappa $ , if by definition the function $K(n,m)=r(n-m)$ has $\\kappa $ negative squares, that is, if all the block Toeplitz matrices $\\mathbb {T}_N$ defined in (REF ) (which are Hermitian since $r(n)=r(-n)^*$ ) have at most $\\kappa $ strictly negative eigenvalues and exactly $\\kappa $ strictly negative eigenvalues for some choice of $N$ and $n_1,\\ldots n_N$ .", "Theorem REF was extended, in the scalar case, by Iohvidov [16] to case where the function $K(n,m)$ has a finite number of negative squares.", "Formula (REF ) is then replaced by a more involved expression.", "More precisely, he obtained the following extension of (REF ) (there is a minus sign with respect to [16] and [14] because they work there with positive squares rather than negative squares): $r(n) =& \\; \\int _{0}^{2\\pi }\\frac{e^{int}-S_n(t)}{\\prod _{k=1}^u \\left(\\sin \\left(\\frac{t-\\varphi _k}{2} \\right)^{2\\rho _k} \\right)}d\\mu (t) \\nonumber \\\\& \\; \\; -\\left(\\sum _{j=1}^rQ_j(in)\\lambda _j^n+\\overline{Q_j(in)\\lambda _j}^{-n}+\\sum _{k=1}^uR_k(in)e^{in\\varphi _k}\\right).", "$ In this expression, the $\\lambda _j$ are of modulus strictly bigger than 1, the $Q_j$ and $R_j$ are polynomials and $S_n$ is a regularizing correction.", "These terms follow from the structure of a contraction in a Pontryagin space, and in particular from the fact that such an operator has always a strictly negative invariant subspace, on which it is one-to-one.", "See [14], where Iohvidov and Krein prove that such a representation is unique.", "In this paper we shall prove in particular a quaternionic analogue of Theorem REF , where $\\mathbb {H}$ denote the quaternions: Theorem 1.4 Let $(r(n))_{n\\in \\mathbb {Z}}$ be a sequence of $s\\times s$ matrices with quaternionic entries.", "Then: $(1)$ The function $K(n,m)=r(n-m)$ has a finite number of negative squares $\\kappa $ if and only if there exists a right quaternionic Pontryagin space $\\mathcal {P}$ , a unitary operator $U\\in \\mathbf {L}(\\mathcal {P})$ and a linear operator $C\\in \\mathbf {L}(\\mathbb {H}^s,\\mathcal {P})$ such that $r(n)=C^*U^nC,\\quad n\\in \\mathbb {Z}.$ $(2)$ Assume that $\\bigcup _{n\\in \\mathbb {Z}}{\\rm ran}\\, U^nC$ is dense in $\\mathcal {P}$ where ran $U^nC$ denotes the range of $U^nC$ .", "Then, the realization (REF ) is unique up to a unitary map.", "Some remarks: $(1)$ The sufficiency of condition (REF ) follows from the inner product representation $c^*r(n-m)d=c^*C^*U^{n-m}Cd=\\langle \\,U^{-m}Cd\\,,\\,U^{-n}Cc\\rangle _{\\mathcal {P}},\\quad n,m\\in \\mathbb {Z},\\quad c,d\\in \\mathbb {H}^s.$ The proof of the necessity is done using the theory of slice hyperholomorphic functions.", "We use in particular a representation theorem from [2] for functions $\\varphi $ slice-hyperholomorphic in some open subset of the unit ball and with a certain associated kernel $K_\\varphi (p,q)$ (defined by (REF ) below) having a finite number of negative squares there.", "We also note that the arguments in [2] rely on the theory of linear relations in Pontryagin spaces.", "$(2)$ The more precise integral representation of Iohvidov and Krein relies on the theory of unitary operators in Pontryagin spaces.", "Such results are still lacking in the setting of quaternionic Pontryagin spaces.", "$(3)$ We also consider the positive definite case.", "There, the lack of a properly established spectral theorem for unitary operators in quaternionic Hilbert spaces prevents to get a direct counterpart of the integral representation (REF ).", "The outline of the paper is as follows.", "The paper consists of seven sections, besides the introduction.", "In Section we review some results from the theory of slice hyperholomorphic functions.", "Some definitions and results on quaternionic Pontryagin spaces are recalled in Section as well as Herglotz-type theorem for matrix valued functions.", "The proof of the necessity and uniqueness in Theorem REF is done in Section .", "Section deals with the analogue of Herglotz's theorem in the quaternionic setting.", "In Section we prove a quaternionic analogue of Theorem REF .", "Section 7 contains the characterization of quaternionic, bounded, Hermitian sequences of matrices with $\\kappa $ negative squares.", "It uses results proved in Section .", "In Section we prove an Herglotz representation theorem for scalar valued functions slice hyperholomophic in the unit ball of the quaternions, and with a positive real part there." ], [ "Slice hyperholomorphic functions", "The kernels we will use in this paper are slice hyperholomorphic, so we recall their definition.", "For more details and the proofs of the results in this section see [10].", "The imaginary units in $\\mathbb {H}$ are denoted by $i$ , $j$ and $k$ , and an element in $\\mathbb {H}$ is of the form $p=x_0+ix_1+jx_2+kx_3$ , for $x_\\ell \\in \\mathbb {R}$ .", "The real part, the imaginary part and conjugate of $p$ are defined as ${\\rm Re}(p)=x_0$ , ${\\rm Im}(p)=i x_1 +j x_2 +k x_3$ and by $\\bar{p}=x_0-i x_1-j x_2-k x_3$ , respectively.", "The unit sphere of purely imaginary quaternions $\\mathbb {S}$ is defined by $\\mathbb {S}=\\lbrace q=ix_1+jx_2+kx_3\\ {\\rm such \\ that}\\ x_1^2+x_2^2+x_3^2=1\\rbrace .$ Note that if $I\\in \\mathbb {S}$ , then $I^2=-1$ ; for this reason the elements of $\\mathbb {S}$ are also called imaginary units.", "Note that $\\mathbb {S}$ is a 2-dimensional sphere in $\\mathbb {R}^4$ .", "Given a nonreal quaternion $p=x_0+{\\rm Im} (p)=x_0+I |{\\rm Im} (p)|$ , $I={\\rm Im} (p)/|{\\rm Im} (p)|\\in \\mathbb {S}$ , we can associate to it the 2-dimensional sphere defined by $[p]=\\lbrace x_0+I|{\\rm Im} (p)|\\ :\\ I\\in \\mathbb {S}\\rbrace .$ We will denote an element in the complex plane $\\mathbb {C}_I:=\\mathbb {R}+I\\mathbb {R}$ by $x+Iy$ .", "Definition 2.1 (Slice hyperholomorphic functions) Let $\\Omega $ be an open set in $\\mathbb {H}$ and let $f:\\Omega \\rightarrow \\mathbb {H}$ be a real differentiable function.", "Denote by $f_I$ the restriction of $f$ to the complex plane $\\mathbb {C}_I$ .", "We say that $f$ is (left) slice hyperholomorphic (or (left) slice regular) if, on $\\Omega \\cap \\mathbb {C}_I$ , $f_I$ satisfies $\\frac{1}{2}\\left(\\frac{\\partial }{\\partial x}+I\\frac{\\partial }{\\partial y}\\right)f_I(x+Iy)=0,$ for all $I \\in \\mathbb {S}$ .", "We say that $f$ is right slice hyperholomorphic (or right slice regular) if, on $\\Omega \\cap \\mathbb {C}_I$ , $f_I$ satisfies $\\frac{1}{2}\\left(\\frac{\\partial }{\\partial x}f_I(x+Iy)+\\frac{\\partial }{\\partial y}f_I(x+Iy)I\\right)=0,$ for all $I \\in \\mathbb {S}$ .", "An immediate consequence of the definition of slice regularity is that the monomial $p^na$ , with $a\\in \\mathbb {H}$ , is left slice regular, so power series with quaternionic coefficients written on the right are left slice regular where they converge.", "As one can easily verify only power series with center at real points are slice regular.", "We introduce a class of domains, which includes the balls with center at a real point, on which slice regular functions have good properties.", "Definition 2.2 (Axially symmetric domain) Let $U \\subseteq \\mathbb {H}$ .", "We say that $U$ is axially symmetric if, for all $x+Iy \\in U$ , the whole 2-sphere $[x+Iy]$ is contained in $U$ .", "Definition 2.3 (Slice domain) Let $U \\subseteq \\mathbb {H}$ be a domain in $\\mathbb {H}$ .", "We say that $U$ is a slice domain (s-domain for short) if $U \\cap \\mathbb {R}$ is non empty and if $ U\\cap \\mathbb {C}_I$ is a domain in $\\mathbb {C}_I$ for all $I \\in \\mathbb {S}$ .", "Lemma 2.4 (Splitting Lemma) Let $\\Omega $ be an s-domain in $\\mathbb {H}$ .", "If $f:\\Omega \\rightarrow \\mathbb {H}$ is left slice hyperholomorphic, then for every $I \\in \\mathbb {S}$ , and every $J\\in \\mathbb {S}$ , perpendicular to $I$ , there are two holomorphic functions $F,G:\\Omega \\cap \\mathbb {C}_I \\rightarrow \\mathbb {C}_I$ such that for any $z=x+Iy$ , it is $f_I(z)=F(z)+G(z)J.$ Note that the decomposition given in the Splitting Lemma is highly non-canonical.", "In fact, for any $I\\in \\mathbb {S}$ there is an infinite number of choices of $J\\in \\mathbb {S}$ orthogonal to it.", "Theorem 2.5 (Representation Formula) Let $\\Omega $ be an axially symmetric s-domain $\\Omega \\subseteq \\mathbb {H}$ and let $f:\\Omega \\rightarrow \\mathbb {H}$ be a slice hyperholomorphic function on $\\Omega $ .", "Then the following equality holds for all $x+yI, x\\pm Jy \\in \\Omega $ : $f(x+Iy) =\\frac{1}{2}\\Big [ f(x+Jy)+f(x-Jy)\\Big ] +I\\frac{1}{2}\\Big [ J[f(x-Jy)-f(x+Jy)]\\Big ].$" ], [ "Quaternionic Pontryagin spaces", "A Hermitian form on a right quaternionic vector space $\\mathcal {P}$ is an $\\mathbb {H}$ -valued map $[\\cdot , \\cdot ]$ defined on $\\mathcal {P}\\times \\mathcal {P}$ and such that $\\begin{split}[a,b]&=\\overline{[b,a]}\\\\[ap,bq]&=\\overline{q}[a,b]p,\\quad \\forall a,b\\in \\mathcal {P}\\,\\,{\\rm and}\\,\\,p,q\\in \\mathbb {H}.\\end{split}$ $\\mathcal {P}$ is called a (right quaternionic) Pontryagin space if it can written as $\\mathcal {P}=\\mathcal {P}_+\\stackrel{\\cdot }{[+]}\\mathcal {P}_-,$ where: $(a)$ The space $\\mathcal {P}_+$ endowed with the form $[\\cdot ,\\cdot ]$ is a Hilbert space.", "$(b)$ The space $\\mathcal {P}_-$ endowed with the form $-[\\cdot ,\\cdot ]$ is a finite dimensional Hilbert space.", "$(c)$ The sum is direct and orthogonal, meaning that $\\mathcal {P}_+\\cap \\mathcal {P}_-=\\left\\lbrace 0\\right\\rbrace $ and $[a,b]=0,\\quad \\forall (a,b)\\in \\mathcal {P}_+\\times \\mathcal {P}_-.$ The decomposition (REF ) is called a fundamental decomposition.", "It not unique unless one of the components reduces to $\\left\\lbrace 0\\right\\rbrace $ .", "The dimension of $\\mathcal {P}_-$ is the same for all fundamental decompositions, and is called the index of the Pontryagin space.", "The space $\\mathcal {P}$ endowed with the form $\\langle a,b\\rangle =[a_+,b_+]-[a_-,b_-]$ with $a_\\pm $ and $b_\\pm \\in \\mathcal {P}_\\pm $ is a Hilbert space.", "The inner product (REF ) depends on the given fundamental decomposition, but all the associated norms are equivalent,and hence define the same topology.", "We refer to [5] for a proof of these facts.", "We refer to [5], [3] for more details on quaternionic Pontryagin spaces and to[6], [8], [15] for the theory of Pontryagin spaces in the complex case.", "A reproducing kernel Pontryagin space will be a Pontryagin space of functions for which the point evaluations are bounded.", "The definition of negative squares makes sense in the quaternionic setting since an Hermitian quaternionic matrix $H$ is diagonalizable: It can be written as $T=UDU^*$ , where $U$ is unitary and $D$ is unique and with real entries.", "The number of strictly negative eigenvalues of $T$ is exactly the number of strictly negative elements of $D$ .", "See [24].", "The one-to-one correspondence between reproducing kernel Pontryagin spaces and functions with a finite number of negative squares, proved in the classical case by [20], [21], extends to the Pontryagin space setting, see [5].", "Definition 3.1 An $\\mathbb {H}^{s\\times s}$ -valued function $\\varphi $ slice hyperholomorphic in a neighborhood $\\mathcal {V}$ of the origin is called a generalized Carathéodory function if the kernel $k_{\\varphi }(p,q)=\\Sigma _{\\ell =0}^\\infty p^\\ell (\\varphi (p)+\\overline{\\varphi (q)})\\overline{q}^\\ell $ has a finite number of negative squares in $\\mathcal {V}$ .", "The following result is Theorem 10.2 in [2].", "Theorem 3.2 A $\\mathbb {H}^{s\\times s}$ -valued function $\\varphi $ is a generalized Carathéodory function if and only if it can be written as $\\varphi (p)=\\frac{1}{2} C\\star (I_{\\mathcal {P}}+pV)\\star (I_{\\mathcal {P}}-pV)^{-\\star } C^*J+\\frac{\\varphi (0)-\\varphi (0)^*}{2}$ where ${\\mathcal {P}}$ is a right quaternionic Pontryagin space of index $\\kappa $ , $V$ is a coisometry in ${\\mathcal {P}}$ , and $C$ is a bounded operator from ${\\mathcal {P}}$ to $\\mathbb {H}^N$ , and the pair $(C, A)$ is observable.", "Remark 3.3 When $\\kappa =0$ the representation (REF ) is the counterpart of Herglotz representation theorem for functions slice hyperholomorphic in the open unit ball and with a positive real part.", "In the last section we shall discuss a scalar version of this result." ], [ "Proof of the necessity and uniqueness of the realization", "In this section we assume that the function $K(n,m)=r(n-m)$ has a finite number of negative squares for $n,m\\in \\mathbb {Z}$ , and prove that the function $r(n)$ has a representation of the form (REF ).", "We also prove the uniqueness of this representation under hypothesis (REF ).", "We begin with a preliminary proposition.", "Proposition 4.1 There exists $C>0$ and $K>0$ such that $\\Vert r(n)\\Vert \\le K\\cdot C^{|n|},\\quad n\\in \\mathbb {Z}.$ The claim is true in the scalar complex-valued case and follows from (REF ); see [14].", "The idea is to reduce the problem to this case.", "We write $r(n)=a(n)+jb(n)$ where $a(n)$ and $b(n)$ are $\\mathbb {C}^{s\\times s}$ -valued.", "We obtain a bound of the required form for every entry of $a(n)$ and $b(n)$ .", "The coefficients $K$ and $C$ in (REF ) will depend on the given entry.", "Since there are $2s^2$ entries, we obtain a bound independent of the entry.", "STEP 1: For every choice of $(e,f)\\in \\mathbb {H}^s\\times \\mathbb {H}^s$ , the function $K_{e,f}(n,m)=e^*a(n-m)e+f^*\\overline{a(n-m)}f+e^*b(n-m)f-f^*\\overline{b(n-m)}e$ has at most $2\\kappa $ negative squares.", "Indeed, the $\\mathbb {C}^{2s\\times 2s}$ function $K_1(n,m)=\\begin{pmatrix}a(n-m)&b(n-m)\\\\-\\overline{b(n-m)}&\\overline{a(n-m)}\\end{pmatrix}$ has $2\\kappa $ negative squares (See [5]), and so, for every fixed choice of $(e,f)\\in \\mathbb {H}^s\\times \\mathbb {H}^s$ , the function $K_{e,f}(n-m)=\\begin{pmatrix}e^*&f^*\\end{pmatrix}\\begin{pmatrix}a(n-m)&b(n-m)\\\\-\\overline{b(n-m)}&\\overline{a(n-m)}\\end{pmatrix}\\begin{pmatrix}e\\\\f\\end{pmatrix}$ has at most $2\\kappa $ negative squares.", "STEP 2: The claim holds for every diagonal entry of $a(n)$.", "Take $e=e_j\\in \\mathbb {H}^s$ to be the vector with all entries equal to 0, except the $j$ -th one equal to 1 and $f=0$ .", "We have $K_{e,f}(n,m)=a_{jj}(n-m),$ and the result follows from [14].", "STEP 3: The claim holds for all the entries of $a(n)$ .", "Let $\\ell \\ne j\\in \\left\\lbrace 1,\\ldots , s\\right\\rbrace $ .", "We now take $e=e_{\\ell j}(\\varepsilon )\\in \\mathbb {H}^s$ to be the vector with all entries equal to 0, except the $\\ell $ -th one equal to 1, and the $j$ -th entry equal to $\\epsilon $ (where $\\varepsilon $ will be determined) and $f=0$ .", "We have $K_{e,f}(n,m)=a_{\\ell \\ell }(n-m)+a_{jj}(n-m)+\\overline{\\varepsilon }a_{j\\ell }(n-m)+\\varepsilon a_{\\ell j}(n-m).$ This function has at most $\\kappa $ negative squares and so the sequence $a_{\\ell \\ell }(n)+a_{jj}(n)+\\overline{\\varepsilon }a_{j\\ell }(n)+\\varepsilon a_{\\ell j}(n).$ has a bound of the form (REF ) (where $K$ and $C$ depend on $\\ell ,j$ and $\\epsilon $ ).", "The choices $\\epsilon =1$ and $\\varepsilon =i$ gives that the functions $a_{\\ell \\ell }(n-m)+a_{jj}(n-m)+a_{j\\ell }(n-m)+a_{\\ell j}(n-m)$ and $a_{\\ell \\ell }(n-m)+a_{jj}(n-m)+i(-a_{j\\ell }(n-m)+ a_{\\ell j}(n-m))$ have at most $\\kappa $ negative squares and so the functions $|a_{\\ell \\ell }(n)+a_{jj}(n)+a_{j\\ell }(n)+a_{\\ell j}(n)|\\le K_1C_1^{|n|}$ and $|a_{\\ell \\ell }(n)+a_{jj}(n)+i(-a_{j\\ell }(n)+ a_{\\ell j}(n))|\\le K_2C_2^{|n|}$ where the constants depend on $(\\ell ,j)$ .", "Since $a_{\\ell \\ell }(n)$ and $a_{jj}(n)$ admit similar bounds we get that both $a_{\\ell j}(n)$ and $a_{j\\ell }(n)$ admit bounds of the form (REF ).", "STEP 4: The claim holds for the diagonal entries of $b(n)$ .", "We now take $e=e_\\ell $ and $f=\\varepsilon e_j$ , where $\\varepsilon $ is of modulus 1.", "We have $K_{e,f}(n-m)=A(n-m)+B(n-m)$ where $\\begin{split}A(n-m)&=a_{\\ell \\ell }(n-m)+a_{jj}(n-m),\\\\B(n-m)&=\\overline{\\varepsilon }b_{\\ell \\ell }(n-m)+\\varepsilon b_{jj}(n-m).\\end{split}$ The choice $\\varepsilon =1$ and $\\varepsilon =i$ lead to the conclusion that $b_{\\ell \\ell }(n)$ admits a bound of the form (REF ) since, as follows from the previous step, $A(n-m)$ admits such a bound.", "STEP 5: The claim holds for all the entries of $b(n)$ .", "We now take $e=e_{\\ell j}(\\varepsilon _1)$ and $f=\\varepsilon e_{\\ell j}(e_2)$ , where $\\varepsilon _1$ and $\\varepsilon _2$ are of modulus 1.", "We have now $K_{e,f}(n-m)=A(n-m)+B(n-m)$ with $\\begin{split}A(n-m)&=e^*a(n-m)e+f^*\\overline{a(n-m)}f^*\\\\B(n-m)&=e^*b(n-m)f-f^*\\overline{b(n-m)}e\\\\&=b_{\\ell \\ell }(n-m)+\\overline{\\varepsilon _1}\\varepsilon _2 b_{jj}(n-m)+\\overline{\\varepsilon _1}b_{\\ell j}(n-m)+\\varepsilon _2b_{j\\ell }(n-m)-\\\\&\\hspace{14.22636pt}-\\overline{b_{\\ell \\ell }(n-m)}-\\overline{\\varepsilon _2}\\varepsilon _1\\overline{b_{jj}(n-m)} -\\overline{\\varepsilon _2}\\overline{b_{\\ell j}(n-m)}-\\varepsilon _1\\overline{b_{\\ell j}(n-m)}.\\end{split}$ In view of the previous steps the sequence $\\overline{\\varepsilon _1}b_{j\\ell }(n)+\\varepsilon _2b_{\\ell j}(n)-\\overline{\\varepsilon _2}\\overline{b_{j\\ell }(n)}-\\varepsilon _1\\overline{b_{\\ell j}(n)}$ admits a bound of the form (REF ).", "The choices $(\\varepsilon _1,\\varepsilon _2)\\in \\left\\lbrace (1,1),(1,-1),(i,i),(i,-i)\\right\\rbrace $ lead to the functions $\\begin{split}&(\\overline{b_{j\\ell }(n)}-b_{j\\ell }(n))+(b_{ \\ell j}(n)-\\overline{b_{ \\ell j}(n)})\\\\&(\\overline{b_{j\\ell }(n)}+b_{j\\ell }(n))+(b_{\\ell j}(n)+\\overline{b_{ \\ell j}(n)})\\\\&(\\overline{b_{j\\ell }(n)}-b_{j\\ell }(n))+(b_{\\ell j}(n)-\\overline{b_{\\ell j}(n)})\\\\&-(\\overline{b_{j\\ell }(n)}+b_{j\\ell }(n))+i(b_{\\ell j}(n)-\\overline{b_{\\ell j}(n)})\\\\&-i(\\overline{b_{j\\ell }(n)}+b_{j\\ell }(n))-i(b_{\\ell j}(n)+\\overline{b_{\\ell j}(n)})\\end{split}$ all admit a bound of the form (REF ).", "We proceed in a number of steps to prove the necessity part of the theorem.", "The first step is a direct computation which is omitted.", "STEP 1: Let $V$ be a coisometry (that is, $VV^*=I$ ) in the quaternionic Pontryagin space $\\mathcal {P}$ .", "Then, $U=\\begin{pmatrix}V^*&I-V^*V\\\\0&V\\end{pmatrix}$ is unitary from $\\mathcal {P}^2$ into itself, and is such that $V^n=\\begin{pmatrix}0&I\\end{pmatrix}U^n\\begin{pmatrix}0 \\\\I\\end{pmatrix},\\quad n=0,1,2,\\ldots $ STEP 2: The series $\\begin{split}\\varphi (p)&=r(0)+2\\sum _{n=1}^\\infty p^n r(n),\\\\K_\\varphi (p,q)&=\\sum _{n,m\\in \\mathbb {Z}}p^nr(n-m)\\overline{q}^m,\\end{split}$ converge for $p$ and $q$ in a neighborhood $\\Omega $ of the origin, and it holds that $K_\\varphi (p,q)-pK_\\varphi (p,q)\\overline{q}=\\frac{\\varphi (p)+\\varphi (q)^*}{2},\\quad p,q\\in \\Omega .$ The asserted convergences follow from (REF ), while (REF ) is a direct computation.", "STEP 4: It holds that $K_\\varphi (p,q)=\\sum _{n=0}^\\infty p^n\\left(\\frac{\\varphi (p)+\\varphi (q)^*}{2}\\right)\\overline{q}^n$ This is because equation (REF ) has a unique solution, and that the right side of (REF ) solves (REF ).", "STEP 5: $K_\\varphi (p,q)$ is has a finite number of negative squares in $\\Omega $ .", "Note that for every $N\\in \\mathbb {N}$ the function $K_{\\varphi ,N}(p,q)=\\begin{pmatrix}I_s&I_sp&\\cdots &I_sp^N\\end{pmatrix}\\mathbb {T}_N\\begin{pmatrix}I_s\\\\ I_s\\overline{q}\\\\\\vdots \\\\ I_s\\overline{q}^N\\end{pmatrix}$ has a finite number of negative squares, uniformly bounded by $\\kappa $ in $\\Omega $ .", "The claim then follows from $K_\\varphi (p,q)=\\lim _{N\\rightarrow \\infty } K_{\\varphi ,N}(p.q)$ STEP 6: There exist a right quaternionic Pontryagin space $\\mathcal {P}$ , a unitary operator $U\\in \\mathbf {L}(\\mathcal {P})$ and a linear operator $C\\in \\mathbf {L}(\\mathbb {H}^s,\\mathcal {P})$ such that $\\varphi (p)=\\frac{CC^*}{2}+\\sum _{n=1}^\\infty p^{n}C^*U^nC,\\quad p\\in \\Omega $ Indeed, since the expression in the right side of (REF ) defines a kernel with a finite number of negative squares, we can apply [2] to see that there exists a right quaternionic Pontryagin space $\\mathcal {P}_1$ , a coisometric operator $V\\in \\mathbf {L}(\\mathcal {P}_1)$ and a bounded operator $C_1\\in \\mathbf {L}(\\mathcal {P},\\mathcal {P}_1)$ such that $r(n)=C_1^*V^nC_1,\\quad n=0,1,\\ldots $ We now apply STEP 1 to write $r(n)=C_1^*\\begin{pmatrix}0&I\\end{pmatrix}U^n\\begin{pmatrix}0 \\\\I\\end{pmatrix}C_1,\\quad n=0,1,\\ldots ,$ which concludes the proof with $C=\\begin{pmatrix}0 \\\\ C_1\\end{pmatrix}$ and $n\\ge 0$ .", "That the formula still holds for negative $n$ follows from $r(-n)=r(n)^*$ and from the unitarity of $U$ .", "To conclude the proof, we turn to the uniqueness of the representation (REF ).", "Consider two representations (REF ), $r(n)=C_1^*U_1^nC_1=C_2^*U_2^nC_2,\\quad n\\in \\mathbb {Z},$ where $U_1$ and $U_2$ are unitary operators in quaternionic Pontryagin spaces $\\mathcal {H}_1$ and $\\mathcal {H}_2$ respectively.", "Consider the space of pairs $R=\\left\\lbrace (U_1^nC_1c\\,,\\, U_2^nC_2c)\\,,\\, n\\in \\mathbb {Z},\\,\\,c\\in \\mathbb {H}^s\\right\\rbrace .$ When condition (REF ) is in force for both representations $R$ , defines a linear isometric relation with dense domain and range, and hence, by the quaternionic version of a theorem of Shmulyan (see [2] and see [4] for the complex version of this theorem and for the definition of a linear relation in Pontryagin spaces), $R$ extends to the graph of a unitary operator, say $S$ , from $\\mathcal {H}_1$ into $\\mathcal {H}_2$ : $SU_1^nC_1c=U_2^nC_2c,\\quad n\\in \\mathbb {Z},\\,\\, c\\in \\mathbb {H}^s$ Setting $n=0$ we get $SC_1=C_2$ .", "Then, taking $n=1$ leads to $(SU_1)C_1=(U_2S)C_1$ , and more generally $(SU_1)U_1^nC_1=(U_2S)U_1^nC_1,\\quad n\\in \\mathbb {Z},$ and so $SU_1=U_2S$ ." ], [ "Herglotz's theorem in the quaternionic setting", "Herglotz's theorem has been already recalled in Section 1, see Theorem REF .", "Here we state a related result which will be useful in the sequel (see [18]).", "Theorem 5.1 Let $\\mu $ and $\\nu $ be $\\mathbb {C}^{s \\times s}$ -valued measures on $[0, 2\\pi ] $ .", "If $\\int _0^{2\\pi } e^{i n t} d\\mu (t) = \\int _0^{2\\pi } e^{i n t} d\\nu (t), \\quad n \\in \\mathbb {Z},$ then $\\mu = \\nu $ .", "Given $P \\in \\mathbb {H}^{s \\times s}$ , there exist unique $P_1, P_2 \\in \\mathbb {C}^{s \\times s}$ such that $P = P_1 + P_2 j$ .", "Thus there is a bijective homomorphism $\\chi : \\mathbb {H}^{s \\times s} \\rightarrow \\mathbb {C}^{2s \\times 2s}$ given by $\\chi \\hspace{1.42262pt} P = \\begin{pmatrix} P_1 & P_2 \\\\ - \\overline{P}_2 & \\overline{P}_1 \\end{pmatrix} \\quad {\\rm where} \\; P = P_1 + P_2 j ,$ Definition 5.2 Given an $\\mathbb {H}^{s \\times s}$ -valued measure $\\nu $ , write $\\nu = \\nu _1 + \\nu _2 j$ , where $\\nu _1$ and $\\nu _2$ are uniquely determined $\\mathbb {C}^{s \\times s}$ -valued measures.", "We call a measure $\\nu $ on $[0, 2\\pi ]$ q-positive if the $\\mathbb {C}^{2s \\times 2s}$ -valued measure $\\mu = \\begin{pmatrix} \\nu _1 & \\nu _2 \\\\ \\nu ^*_2 & \\nu _3 \\end{pmatrix}, \\quad {\\rm where}\\; d\\nu _3(t) = d\\bar{\\nu }_1(2\\pi - t),\\;\\; t \\in [0, 2\\pi )$ is positive and, in addition, $d\\nu _2(t) = -d\\nu _2(2\\pi -t)^T, \\quad t \\in [0, 2\\pi ),$ Remark 5.3 If $\\nu $ is $q$ -positive, then $\\nu = \\nu _1 + \\nu _2 j$ , where $\\nu _1$ is a uniquely determined positive $\\mathbb {C}^{s \\times s}$ -valued measure and $\\nu _2$ is a uniquely determined $\\mathbb {C}^{s \\times s}$ -valued measure.", "Remark 5.4 If $r = (r(n))_{n \\in \\mathbb {Z}}$ is a $\\mathbb {H}^{s \\times s}$ -valued sequence on $\\mathbb {Z}$ such that $r(n) = \\int _0^{2\\pi } e^{i n t} d\\nu (t),$ where $\\nu $ is a $q$ -positive measure, then $r$ is Hermitian, i.e., $r(-n)^* = r(n)$ .", "Indeed, write $\\nu = \\nu _1 + \\nu _2 j$ , where $\\nu _1$ and $\\nu _2$ are as in Definition REF .", "Then $r(-n)^* =& \\; \\int _0^{2\\pi } (d\\nu _1(t) - j d\\nu _2(t)^*) e^{i n t} \\\\=& \\; \\int _0^{2\\pi } e^{i n t} d\\nu _1(t) - \\int _0^{2\\pi } e^{- i n t} (-d\\nu _2(t)^T) j \\\\=& \\; \\int _0^{2\\pi } e^{i n t}d\\nu _1(t) + \\int _0^{2\\pi } e^{i n t} (-d\\nu _2(2\\pi - t)^T) j \\\\=& \\; \\int _0^{2\\pi } e^{i n t} d\\nu _1(t) + \\int _0^{2\\pi } e^{i n t} d\\nu _2(t) \\\\=& \\; r(n), \\quad \\quad n \\in \\mathbb {Z}$ Theorem 5.5 The function $n \\mapsto r(n)$ from $\\mathbb {Z}$ into $\\mathbb {H}^{s \\times s}$ is positive definite if and only if there exists a unique $q$ -positive measure $\\nu $ on $[0, 2\\pi ]$ such that $r(n) = \\int _0^{2\\pi } e^{i n t} d\\nu (t), \\quad n \\in \\mathbb {Z}.$ Let $(r(n))_{n \\in \\mathbb {Z}}$ be a positive definite sequence and write $r(n) = r_1(n) + r_2(n)j$ , where $r_1(n),r_2(n) \\in \\mathbb {C}^{s \\times s}$ , $n \\in \\mathbb {Z}$ .", "Put $R(n) = \\chi \\hspace{1.42262pt} r(n)$ , $n \\in \\mathbb {Z}$ .", "It is easily seen that $(R(n))_{n \\in \\mathbb {Z}}$ is a positive definite $\\mathbb {C}^{2s \\times 2s} $ -valued sequence if and only if $(r(n))_{n \\in \\mathbb {Z}}$ is a positive definite $\\mathbb {H}^{s \\times s}$ -valued sequence.", "Thus, by Theorem REF there exists a unique positive $\\mathbb {C}^{2s \\times 2s}$ -valued measure $\\mu $ on $[0, 2\\pi ] $ such that $R(n) = \\int _0^{2\\pi } e^{i n t} d\\mu (t),\\quad n \\in \\mathbb {Z}.$ Write $\\mu = \\begin{pmatrix} \\mu _{11} & \\mu _{12} \\\\ {\\mu }^*_{12} & \\mu _{22} \\end{pmatrix}: \\begin{array}{ccc} \\mathbb {C}^{s} & & \\mathbb {C}^s \\\\ \\oplus & \\rightarrow & \\oplus \\\\ \\mathbb {C}^s & & \\mathbb {C}^s \\end{array}.$ It follows from $R(n) = \\begin{pmatrix} r_1(n) & r_2(n) \\\\ -\\overline{r_2(n)} & \\overline{r_1(n)} \\end{pmatrix}, \\quad n \\in \\mathbb {Z}$ and (REF ) that $r_1(n) = \\int _0^{2\\pi } e^{i n t} d\\mu _{11}(t) = \\int _0^{2\\pi } e^{-i n t} d\\bar{\\mu }_{22}(t), \\quad n \\in \\mathbb {Z}$ and hence $\\int _0^{2\\pi } e^{i n t} d\\mu _{11}(t) = \\int _0^{2\\pi } e^{i n t} d\\bar{\\mu }_{22}(2\\pi - t),\\quad n \\in \\mathbb {Z}.$ Thus, Theorem REF yields that $d\\mu _{11}(t) = d\\bar{\\mu }_{22}(2\\pi -t)$ for $t \\in [0, 2\\pi )$ .", "Similarly, $r_2(n) = \\int _0^{2\\pi } e^{i n t } d\\mu _{12}(t) = - \\int _0^{2\\pi } e^{-i n t} d\\mu _{12}(t)^T,\\quad n \\in \\mathbb {Z}$ and hence $\\int _0^{2\\pi } e^{i n t} d\\mu _{12}(t) = \\int _0^{2\\pi } e^{i n t} (-d\\mu _{12}(2\\pi - t)^T),\\quad n \\in \\mathbb {Z} .$ Thus, Theorem REF yields that $d\\mu _{12}(t) = -d\\mu _{12}(2\\pi - t)^T$ for $t \\in [0, 2\\pi )$ .", "It is easy to show that $\\begin{pmatrix} I_s & -j I_s \\end{pmatrix} R(n) \\begin{pmatrix} I_s \\\\ j I_s \\end{pmatrix} = 2r(n)$ and hence (REF ) yields $2r(n) =& \\; \\int _0^{2\\pi } \\begin{pmatrix} e^{i n t} & -j e^{i n t} \\end{pmatrix} \\begin{pmatrix} d\\mu _{11}(t) + d\\mu _{12}(t) j \\\\ d\\mu _{12}(t)^* + d{\\mu }_{22}(t) j \\end{pmatrix} \\\\=& \\; \\int _0^{2\\pi } e^{i n t} d\\mu _{11}(t) + \\int _0^{2\\pi } e^{i n t} d\\mu _{12}(t)j - \\int _0^{2\\pi } e^{-i n t} d\\mu _{12}(t)^T j \\\\& \\;\\;\\;\\; + \\int _0^{2\\pi } e^{-i n t} d\\bar{\\mu }_{22}(t) \\\\=& \\; \\int _0^{2\\pi } e^{i n t} d\\mu _{11}(t) + \\int _0^{2\\pi } e^{i n t} d\\mu _{12}(t)j - \\int _0^{2\\pi } e^{i n t} d\\mu _{12}(2\\pi - t)^T j \\\\& \\;\\;\\;\\; + \\int _0^{2\\pi } e^{i n t} d\\bar{\\mu }_{22}(2\\pi - t) \\\\=& \\; 2 \\int _0^{2\\pi } e^{i n t} d\\mu _{11}(t) + 2 \\int _0^{2\\pi } e^{i n t} d\\mu _{12}(t)j , \\quad n \\in \\mathbb {Z},$ where the last line follows from $d\\mu _{11}(t) = d\\bar{\\mu }_{22}(2\\pi -t)$ and $d\\mu _{12}(t) = -d\\mu _{12}(2\\pi -t)^T$ .", "If we put $\\nu = \\mu _{11} + \\mu _{12} j$ , then $\\nu $ is a $q$ -positive measure which satisfies (REF ).", "Conversely, suppose $\\nu = \\nu _1 + \\nu _2 j$ is a $q$ -positive measure on $[0, 2\\pi ] $ and put $r(n) = \\int _0^{2\\pi } e^{i n t} d\\nu (t),\\quad n \\in \\mathbb {Z}.$ Since $\\nu $ is $q$ -positive, $\\mu = \\begin{pmatrix} \\nu _1 & \\nu _2 \\\\ \\nu ^*_2 & \\nu _3 \\end{pmatrix},\\quad {\\rm where}\\;\\; d\\nu _3(t) = d\\bar{\\nu }_1(2\\pi - t), \\;\\; t \\in [0, 2\\pi ),$ is a positive $\\mathbb {C}^{2s \\times 2s}$ -valued measure on $[0, 2\\pi ] $ and $d\\nu _2(t) = -d\\nu _2(2\\pi - t)^T,\\quad t \\in [0,2 \\pi ).$ Since $\\mu $ is a positive $\\mathbb {C}^{2s \\times 2s}$ -valued measure, $(R(n))_{n \\in \\mathbb {Z}}$ is a positive definite $\\mathbb {C}^{2s \\times 2s}$ -valued sequence, where $R(n) := \\int _0^{2\\pi } e^{i n t } d\\mu (t), \\quad n \\in \\mathbb {Z},$ Moreover, $R(n)$ can be written in form $R(n) = \\begin{pmatrix} r_1(n) & r_2(n) \\\\ -\\overline{r_2(n)} & \\overline{r_1(n)} \\end{pmatrix},\\quad n \\in \\mathbb {Z},$ where $r_1(n) =& \\; \\int _0^{2\\pi } e^{i n t} d\\nu _1(t) ,\\quad n \\in \\mathbb {Z}{\\rm ;} \\\\r_2(n) =& \\; \\int _0^{2\\pi } e^{i n t} d\\nu _2(t),\\quad n \\in \\mathbb {Z}.$ Thus, $R(n) = \\chi \\hspace{1.42262pt} r(n)$ , where $r(n) = r_1(n) + r_2(n) j = \\int _0^{2\\pi } e^{i n t} d\\nu (t).$ Since $(R(n))_{n \\in \\mathbb {Z}}$ is a positive definite $\\mathbb {C}^{2s \\times 2s}$ -valued sequence we get that $(r(n))_{n \\in \\mathbb {Z}}$ is a positive definite $\\mathbb {H}^{s \\times s}$ -valued sequence.", "Finally, suppose that the $q$ -positive measure $\\nu $ were not unique, i.e., there exists $\\tilde{\\nu }$ so that $\\tilde{\\nu } \\ne \\nu $ and $r(n) = \\int _0^{2\\pi } e^{i n t} d\\nu (t) = \\int _0^{2\\pi } e^{i n t} d\\tilde{\\nu }(t), \\quad n \\in \\mathbb {Z}.$ Write $\\nu = \\nu _1 + \\nu _2 j$ and $ \\tilde{\\nu } = \\tilde{\\nu }_1 + \\tilde{\\nu }_2 j$ as in Remark REF .", "If we consider $R(n) = \\chi \\hspace{1.42262pt} r(n), n \\in \\mathbb {Z}$ , then it follows from Theorem REF that $\\nu _1 = \\tilde{\\nu }_1$ and $\\nu _2 = \\tilde{\\nu }_2$ and hence that $\\nu = \\tilde{\\nu }$ , a contradiction.", "Remark 5.6 The statement and proof of Herglotz's theorem have been written using an exponential involving the imaginary unit $i$ of the quaternions.", "Analogous statements can be written using the imaginary units $j$ or $k$ in the basis or with respect to new basis elements chosen in $\\mathbb {S}$ ." ], [ "A theorem of Carathéodory in the quaternionic setting", "Definition 6.1 A function $r: \\lbrace -N, \\ldots , N \\rbrace \\rightarrow \\mathbb {H}^{s \\times s}$ is called positive definite if $\\mathbb {T}_N\\succeq 0$ , where $\\mathbb {T}_N$ is the matrix defined in (REF ).", "Definition 6.2 Let $r: \\lbrace -N, \\ldots , N \\rbrace \\rightarrow \\mathbb {H}^{s \\times s}$ be positive definite.", "We will say that $r$ has a positive definite extension if there exists a positive definite function $\\tilde{r}: \\mathbb {Z} \\rightarrow \\mathbb {H}^{s \\times s}$ such that $\\tilde{r}(n) = r(n), \\quad \\quad n = -N, \\ldots , N.$ Theorem 6.3 If $r: \\lbrace -N, \\ldots , N \\rbrace \\rightarrow \\mathbb {H}^{s \\times s}$ is positive definite, then $r$ has a positive definite extension.", "Remark 6.4 The strategy for proving Theorem REF is to establish the existence of $r(N+1), r(N+2), \\ldots $ so that the block matrices $\\mathbb {T}_{N+1} =& \\; \\begin{pmatrix} r(0) & \\cdots & r(N+1) \\\\\\vdots & \\ddots & \\vdots \\\\r(-N-1) & \\cdots & r(0) \\end{pmatrix} \\succeq 0 \\\\\\mathbb {T}_{N+2} =& \\; \\begin{pmatrix} r(0) & \\cdots & r(N+2) \\\\\\vdots & \\ddots & \\vdots \\\\r(-N-2) & \\cdots & r(0) \\end{pmatrix} \\succeq 0, \\quad \\ldots $ Here we let $r(-N-1) = r(N+1)^*$ , $r(-N-2) = r(N+2)^*, \\ldots $ .", "We must first establish some lemmas before proving Theorem REF .", "The proofs of Lemmas REF , REF and REF are adapted from Lemma 2.4.2, Corollary 2.4.3 and Theorem 2.4.5 in Bakonyi and Woerdeman [7], respectively.", "Lemma 6.5 If $A \\in \\mathbb {H}^{t \\times s}$ and $B \\in \\mathbb {H}^{u \\times s}$ , then $B^* B \\succeq A^* A$ if and only if there exists a contraction $G: {\\rm ran}\\,B \\rightarrow {\\rm ran}\\,A$ such that $A = GB$ .", "Moreover, $G$ is unique and an isometry if and only if $B^* B = A^* A$ .", "If there exists a contraction $G: {\\rm ran}\\,B \\rightarrow {\\rm ran}\\,A$ such that $A = GB$ , then it is easy to verify that $B^* B \\succeq A^* A$ .", "Conversely, if $B^* B \\succeq A^* A$ , then let $y \\in {\\rm ran}\\,B$ , i.e $y = Bx$ for some $x \\in \\mathbb {H}^{s}$ .", "Let $G: {\\rm ran}\\,B \\rightarrow {\\rm ran}\\,A$ be given by $Gy = Ax.$ To check that $G$ is well-defined, suppose that $y = Bx = B \\tilde{x},$ where $\\tilde{x} \\in \\mathbb {H}^s$ .", "Using $B^* B \\succeq A^* A$ we get that $0 \\le (x - \\tilde{x})^* A^* A (x- \\tilde{x}) \\le (x - \\tilde{x})^* B^* B (x - \\tilde{x}) = 0$ and hence $Ax = A \\tilde{x}$ .", "Therefore, $G$ is well-defined.", "We will now show that $G$ is a contraction.", "Let $\\lbrace y_n \\rbrace _{n=1}^{\\infty }$ be a convergent sequence in ${\\rm ran}\\,B$ .", "If $\\lbrace x_n \\rbrace _{n=1}^{\\infty }$ in $\\mathbb {H}^s$ so that $B x_n = y_n,$ then $(G y_n - G y_m )^* (G y_n - G y_m ) =& \\; [A(x_n - x_m)]^* [A(x_n - x_m)] \\nonumber \\\\\\le & \\; [B(x_n - x_m)]^* [B(x_n - x_m)] \\nonumber \\\\=& \\; (y_n - y_m)^* (y_n - y_m).", "$ Since $\\lbrace y_n \\rbrace _{n=1}^{\\infty }$ is a convergent sequence in ${\\rm ran}\\,B$ , $\\lbrace y_n \\rbrace _{n=1}^{\\infty }$ is also a Cauchy sequence in ${\\rm ran}\\,B$ and hence $\\lbrace G y_n \\rbrace _{n=1}^{\\infty }$ is a Cauchy sequence as well.", "Thus, $\\lim _{n\\uparrow \\infty } G y_n$ exists.", "The inequality given in (REF ) readily yields that $y^* G^* G y \\le y^* y$ , whence $G$ is a contraction.", "Note that $G$ is unique by construction, since the equation $A = GB$ requires that whenever $y = B x $ we get that $G y = Ax$ .", "Finally, if $G$ is an isometry then it follows from the equality $A = GB$ that $A^* A = B^* B$ .", "Conversely, if $A^* A = B^* B$ , then $ y = Bx$ and $G y = Ax$ yield that $ y^* G^* G y = x^* A^* A x = x^* B^* B x = y^* y.", "$ Thus, $u^* G^* G y = y^* y$ for $y \\in {\\rm ran}\\,B$ .", "Lemma 6.6 If $K = \\begin{pmatrix} A & B \\\\ B^* & C \\end{pmatrix} \\in \\mathbb {H}^{ (t+u) \\times (s+u) },$ then $K \\succeq 0$ if and only if the following conditions hold: $A \\succeq 0$ and $C \\succeq 0$ ; $B = A^{1/2} G C^{1/2}$ for some contraction $G: {\\rm ran}\\,C \\rightarrow {\\rm ran}\\,A$ .", "Suppose conditions (i) and (ii) are in force.", "It follows from (i) that there exist $P$ and $Q$ such that $A = P^* P$ and $C = Q^* Q$ .", "Thus, $K = \\begin{pmatrix} P^* & 0 \\\\ 0 & Q^* \\end{pmatrix} \\begin{pmatrix} I & G \\\\ G^* & I \\end{pmatrix} \\begin{pmatrix} P & 0 \\\\ 0 & Q \\end{pmatrix} \\succeq 0,$ since $G$ is a contraction.", "Conversely, suppose $K \\succeq 0$ and let $P$ and $Q$ be given by $ \\begin{pmatrix} P^* \\\\ Q^* \\end{pmatrix} \\begin{pmatrix} P & Q \\end{pmatrix} = \\begin{pmatrix} A & B \\\\ B^* & C \\end{pmatrix}.$ Thus, $P^* P = A^{1/2} A^{1/2}$ and $ Q^* Q = C^{1/2} C^{1/2}$ .", "Using Lemma REF we arrive at the isometries $G_1: {\\rm ran}\\,A \\rightarrow {\\rm ran}\\,P$ and $G_2: {\\rm ran}\\,C \\rightarrow {\\rm ran}\\,Q$ which satisfy $P = G_1 A^{1/2}$ and $Q = G_2 C^{1/2}$ , respectively.", "Therefore, $B = P^* Q = A^{1/2} G_1^* G_2 C^{1/2}$ and thus $B = A^{1/2} G C^{1/2}$ , where $G = G_1^* G_2$ is a contraction.", "Definition 6.7 We will call a block matrix, with quaternionic entries, $K = \\begin{pmatrix} A & B & ?", "\\\\ B^* & C & D \\\\ ?", "& D^* & E \\end{pmatrix}$ partially positive semidefinite if all principle specified minors are nonnegative.", "We will say that $K$ has a positive semidefinite completion if there exists a quaternionic matrix $X$ so that $\\begin{pmatrix} A & B & X \\\\ B^* & C & D \\\\ X^* & D^* & E \\end{pmatrix} \\succeq 0.$ Lemma 6.8 If $K = \\begin{pmatrix} A & B & ?", "\\\\ B^* & C & D \\\\ ?", "& D^* & E \\end{pmatrix}$ is partially positive semidefinite, then $K$ has a positive semidefinite completion given as follows.", "Let $G_1: {\\rm ran}\\,C \\rightarrow {\\rm ran}\\,A$ and $G_2: {\\rm ran}\\,E \\rightarrow {\\rm ran}\\,C$ be contractions so that $B = A^{1/2} G_1 C^{1/2}$ and $D = C^{1/2} G_2 E^{1/2}$ .", "Choosing the $(1,3)$ block entry of $K$ to be $A^{1/2} G_1 G_2 E^{1/2}$ results in a positive semidefinite completion.", "Since $K$ is partially positive semidefinite, $K_1 = \\begin{pmatrix} A & B \\\\ B^* & C \\end{pmatrix} \\succeq 0 \\quad \\quad {\\rm and} \\quad \\quad K_2 = \\begin{pmatrix} C & D \\\\ D^* & E \\end{pmatrix} \\succeq 0.$ Use Lemma REF on $K_1$ and $K_2$ to produce contractions $G_1$ and $G_2$ , resepectively, so that $B = A^{1/2} G_1 C^{1/2}$ and $D = C^{1/2} G_2 E^{1/2}$ .", "Since $G_1$ and $G_2$ are contractions, the factorization $& \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\; \\widetilde{K} = \\begin{pmatrix} A & B & A^{1/2} G_1 G_2 E^{1/2} \\\\B^* & C & D \\\\E^{1/2} (G_2)^* (G_1)^* A^{1/2} & D^* & E \\end{pmatrix} \\\\=& \\; \\begin{pmatrix} A & A^{1/2} G_1 C^{1/2} & A^{1/2} G_1 G_2 E^{1/2} \\\\C^{1/2} (G_1)^* A^{1/2} & C & C^{1/2} G_2 E^{1/2} \\\\E^{1/2} (G_2)^* (G_1)^* A^{1/2} & E^{1/2} (G_2)^* C^{1/2} & E \\end{pmatrix} \\\\=& \\; \\begin{pmatrix} A^{1/2} & 0 & 0 \\\\ 0 & C^{1/2} & 0 \\\\ 0 & 0 & E^{1/2} \\end{pmatrix}\\begin{pmatrix} I & G_1 & G_1 G_2 \\\\ (G_1)^* & I & G_2 \\\\ (G_2)^*(G_1)^* & (G_2)^* & I \\end{pmatrix} \\begin{pmatrix} A^{1/2} & 0 & 0 \\\\ 0 & C^{1/2} & 0 \\\\ 0 & 0 & E^{1/2} \\end{pmatrix} \\\\=& \\; \\begin{pmatrix} A^{1/2} & 0 & 0 \\\\ 0 & C^{1/2} & 0 \\\\ 0 & 0 & E^{1/2} \\end{pmatrix}\\begin{pmatrix} I & 0 & 0 \\\\ (G_1)^* & I & 0 \\\\ (G_1G_2)^* & (G_2)^* & I \\end{pmatrix}\\begin{pmatrix} I & 0 & 0 \\\\ 0 & I - (G_1)^*G_1 & 0 \\\\ 0 & 0 & I - (G_2)^* G_2 \\end{pmatrix} \\\\& \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\; \\times \\begin{pmatrix} I & G_1 & G_1 G_2 \\\\ 0 & I & G_2 \\\\ 0 & 0 & I \\end{pmatrix} \\begin{pmatrix} A^{1/2} & 0 & 0 \\\\ 0 & C^{1/2} & 0 \\\\ 0 & 0 & E^{1/2} \\end{pmatrix},$ shows that $\\widetilde{K}$ is a positive semidefinite completion of $K$ .", "We are now ready to prove Theorem REF .", "Let $r: \\lbrace -N, \\ldots , N \\rbrace \\rightarrow \\mathbb {H}^{s \\times s}$ be positive definite.", "It follows from Lemma REF with $A =& \\; r(0){\\rm ;} \\\\B =& \\; \\begin{pmatrix} r(1) & \\cdots & r(N) \\end{pmatrix}{\\rm ;} \\\\C =& \\; \\begin{pmatrix} r(0) & \\cdots & r(N) \\\\ \\vdots & \\ddots & \\vdots \\\\ r(-N) & \\cdots & r(0) \\end{pmatrix} {\\rm ;} \\\\D =& \\; \\begin{pmatrix} r(N)^T & \\cdots & r(1)^T \\end{pmatrix}^T{\\rm ;} \\\\E =& \\; A,$ that there exist contractions $G_1$ and $G_2$ so that if we put $r(N+1) = A^{1/2} G_1 G_2 E^{1/2}$ and $r(-N-1) = r(N+1)^*$ , then $\\begin{pmatrix}r(0) & \\cdots & r(N+1) \\\\\\vdots & \\ddots & \\vdots \\\\r(-N-1) & \\cdots & r(0)\\end{pmatrix} \\succeq 0.$ Continuing in this fashion, we can choose $r(N+2), r(N+3), \\ldots $ so that $\\begin{pmatrix} r(0) & \\cdots & r(N+2) \\\\ \\vdots & \\ddots & \\vdots \\\\ r(-N-2) & \\cdots & r(0)\\end{pmatrix} \\succeq 0, \\quad \\begin{pmatrix} r(0) & \\cdots & r(N+3) \\\\ \\vdots & \\ddots & \\vdots \\\\ r(-N-3) & \\cdots & r(0)\\end{pmatrix} \\succeq 0, \\ldots .$ Thus we have contructed $\\tilde{r}: \\mathbb {Z} \\rightarrow \\mathbb {H}^{s \\times s}$ which is positive definite and satisfies $\\tilde{r}(n) = r(n), \\quad \\quad n = -N, \\ldots , N.$" ], [ "A theorem of Krein and Iohvidov in the quaternionic setting", "Sasvári [19] attributes the following theorem to Krein and Iohvidov [14].", "Theorem 7.1 Let $a = (a(n))_{n \\in \\mathbb {Z}}$ be a bounded Hermitian complex-valued sequence on $\\mathbb {Z}$ .", "The sequence $a$ has $\\kappa $ negative squares if and only if there exist measures $\\mu _+$ and $\\mu _-$ on $[0, 2\\pi ]$ and mutually distinct points $t_1, \\ldots , t_\\kappa \\in [0, 2\\pi ]$ satisfying $\\mu _+(t_j) = 0$ for $j =1, \\ldots , \\kappa $ and ${\\rm supp}\\hspace{1.42262pt}(\\mu _-) = \\lbrace t_1, \\ldots , t_k \\rbrace $ and such that $a(n) = \\int _0^{2\\pi } e^{i n t} d\\mu _+(t) - \\int _0^{2\\pi } e^{i n t} d\\mu _-(t),\\quad \\quad n \\in \\mathbb {Z}.$ A proof for this result when $\\mathbb {Z}$ is replaced by an arbitrary locally compact Abelian group can be found in [19].", "It will be our goal in this section to obtain a direct analogue of Theorem REF when $(a(n))_{n \\in \\mathbb {Z}}$ is a bounded Hermitian $\\mathbb {H}^{s \\times s}$ -valued sequence.", "To achieve this goal, we will first generalize Theorem REF to the case when $(a(n))_{n \\in \\mathbb {Z}}$ is $\\mathbb {C}^{s \\times s}$ -valued sequence and then the desired result will follow.", "Definition 7.2 Let $ M = \\sum _{q=1}^k P_q \\delta _{t_q}$ be a $\\mathbb {C}^{s \\times s}$ -valued measure on $[0, 2\\pi ] $ , where $\\delta _t$ denotes the usual Dirac point measure at $t$ .", "We let ${\\rm card}\\hspace{1.42262pt}{\\rm supp}\\hspace{1.42262pt} M = \\sum _{q=1}^k {\\rm rank}\\hspace{1.42262pt} P_q.$ Theorem 7.3 Let $A = (A(n))_{n \\in \\mathbb {Z}}$ be a bounded Hermitian $\\mathbb {C}^{s \\times s}$ -valued sequence on $\\mathbb {Z}$ .", "$A$ has $\\kappa $ negative squares if and only if there exist positive $\\mathbb {C}^{s \\times s}$ -valued measures $ M_+$ and $ M_-$ on $[0, 2\\pi ]$ and mutually distinct points $t_1, \\ldots , t_k \\in [0, 2\\pi ] $ satisfying $ M_+(t_j) = 0$ for $j =1, \\ldots , k$ , ${\\rm supp}\\hspace{1.42262pt}( M_-) = \\lbrace t_1, \\ldots , t_k \\rbrace $ and ${\\rm card}\\hspace{1.42262pt}{\\rm supp}\\hspace{1.42262pt} M_- = \\kappa $ and such that $A(n) = \\int _0^{2\\pi } e^{i n t} d M_+(t) - \\int _0^{2\\pi } e^{i n t} d M_-(t),\\quad \\quad n \\in \\mathbb {Z}.$ If $A$ has $\\kappa $ negative squares, then $a_v = (v^* A(n) v )_{n \\in \\mathbb {Z}}$ will be a complex-valued sequence with at most $\\kappa $ negative squares for any $v \\in \\mathbb {C}^s$ .", "It follows then from Theorem REF that there exist measures $\\mu _+^{(v)}$ and $\\mu _-^{(v)}$ on $[0, 2\\pi ] $ and mutually distinct points $t_1^{(v)}, \\ldots , t_{k_v}^{(v)} \\in [0, 2\\pi ]$ satisfying $\\mu _+^{(v)}(t_j^{(v)}) = 0$ for $j =1, \\ldots , k_v$ and ${\\rm supp}\\hspace{1.42262pt}(\\mu _-^{(v)}) = \\lbrace t_1^{(v)}, \\ldots , t_{k_v}^{(v)} \\rbrace $ , where $k_v \\le \\kappa $ , and such that $a_v(n) = \\int _0^{2\\pi } e^{i n t} d\\mu _+^{(v)}(t) - \\int _0^{2\\pi } e^{i n t} d\\mu _-^{(v)}(t),\\quad \\quad n \\in \\mathbb {Z}.$ Let $4 \\mu _{\\pm }^{(v,w)} = \\mu _{\\pm }^{(v+w)} - \\mu _{\\pm }^{(v-w)} + i \\mu _{\\pm }^{(v + i w)} - i \\mu _{\\pm }^{(v-iw)},\\quad \\quad v,w \\in \\mathbb {C}^s.$ Then there exist positive $\\mathbb {C}^{s \\times s}$ -valued measures $ M_{\\pm }$ such that $\\langle M_{\\pm } v, w \\rangle = \\mu _{\\pm }^{(v,w)}$ and $A(n) = \\int _0^{2\\pi } e^{i n t} dM_+(t) - \\int _0^{2\\pi } e^{i n t} d M_-(t), \\quad \\quad n \\in \\mathbb {Z}.$ It follows from (REF ) together with the fact that $A$ has $\\kappa $ negative squares that ${\\rm card}\\hspace{1.42262pt} {\\rm supp}\\hspace{1.42262pt} M_- = \\kappa .$ By construction, $M_+(t_j) = 0$ for all $t_j \\in {\\rm supp}\\hspace{1.42262pt} M_+$ .", "Conversely, suppose (REF ) is in force.", "It is easy to check that $A$ is a bounded Hermitian sequence with at most $\\kappa $ negative squares.", "The fact that $A$ has exactly $\\kappa $ negative follows from the uniqueness of the measure $ M = M_+ - M_-$ in (REF ) (see Theorem REF ).", "Definition 7.4 Let $ \\nu = \\nu _1 + \\nu _2 j$ be a $q$ -positive measure on $[0, 2\\pi ] $ with finite support.", "We let ${\\rm card}\\hspace{1.42262pt}{\\rm supp}\\hspace{1.42262pt} \\nu = (1/2){\\rm card}\\hspace{1.42262pt}{\\rm supp} \\hspace{1.42262pt} \\mu ,$ where $\\mu $ is an in (REF ).", "Theorem 7.5 Let $a = (a(n))_{n \\in \\mathbb {Z}}$ be a bounded Hermitian $\\mathbb {H}^{s \\times s}$ -valued sequence on $\\mathbb {Z}$ .", "The sequence $a$ has $\\kappa $ negative squares if and only if there exist $q$ -positive measures $ \\nu _+$ and $ \\nu _-$ on $[0, 2\\pi ] $ and mutually distinct points $t_1, \\ldots , t_k \\in [0, 2\\pi ] $ satisfying satisfying $ \\nu _+(t_j) = 0$ for $j =1, \\ldots , k$ , ${\\rm supp}\\hspace{1.42262pt}(d \\nu _-) = \\lbrace t_1, \\ldots , t_k \\rbrace $ and ${\\rm card}\\hspace{1.42262pt}{\\rm supp}\\hspace{1.42262pt} \\nu _- = \\kappa $ and such that $a(n) = \\int _0^{2\\pi } e^{i n t} d \\nu _+(t) - \\int _0^{2\\pi } e^{i n t} d \\nu _-(t),\\quad \\quad n \\in \\mathbb {Z}.$ If $a$ is a bounded Hermitian $\\mathbb {H}^{s \\times s}$ -valued sequence with $\\kappa $ negative squares, then $A = (A(n))_{n \\in \\mathbb {Z}}$ , where $A(n) = \\chi \\hspace{1.42262pt} a(n)$ , has $2\\kappa $ negative squares (see Proposition 11.4 in [5]).", "Thus, Theorem REF guarantees the existence of positive $\\mathbb {C}^{2s \\times 2s}$ -valued measures $ M_+$ and $ M_-$ on $[0, 2\\pi ] $ and mutually distinct points $t_1, \\ldots , t_k \\in [0, 2\\pi ] $ satisfying satisfying $ M_+(t_j) = 0$ for $j =1, \\ldots , k$ , ${\\rm supp}\\hspace{1.42262pt}( M_-) = \\lbrace t_1, \\ldots , t_k \\rbrace $ and ${\\rm card}\\hspace{1.42262pt}{\\rm supp}\\hspace{1.42262pt} M_- = 2\\kappa $ and such that $A(n) = \\int _0^{2\\pi } e^{i n t} d M_+(t) - \\int _0^{2\\pi } e^{i n t} d M_-(t),\\quad \\quad n \\in \\mathbb {Z}.$ If we write $dM_{\\pm } = \\begin{pmatrix} dM_{\\pm }^{(11)} & dM_{\\pm }^{(12)} \\\\ (dM_{\\pm }^{(12)})^* & dM_{\\pm }^{(22)} \\end{pmatrix} \\begin{array}{ccc} \\mathbb {C}^s & & \\mathbb {C}^s \\\\ \\oplus & \\rightarrow & \\oplus \\\\ \\mathbb {C}^s & & \\mathbb {C}^s \\end{array}$ and proceed as in the proof of Theorem REF we get that $dM_+^{(11)}(t) - dM_-^{(11)}(t) = dM_+^{(22)}(2\\pi -t) - dM_-^{(22)}(2\\pi - t),\\quad \\quad t \\in [0, 2\\pi )$ and $dM_+^{(12)}(t) - dM_-^{(12)}(t) = -(dM_+^{(12)}(2\\pi -t)^T - dM_-^{(12)}(2\\pi - t)^T),\\quad \\quad t \\in [0, 2\\pi ).$ Consequently, it follows from $d M_+(t_j) = 0$ for $j =1, \\ldots , k$ and ${\\rm supp}\\hspace{1.42262pt}(d M_-) = \\lbrace t_1, \\ldots , t_k \\rbrace $ that $dM_-^{(11)}(t) = dM_-^{(22)}(2\\pi - t),\\quad \\quad t \\in [0, 2\\pi )$ and $dM_-^{(12)}(t) = -dM_-^{(12)}(2\\pi - t)^T,\\quad \\quad t \\in [0, 2\\pi ).$ Thus, $dM_+^{(11)}(t) = dM_+^{(22)}(2\\pi - t),\\quad \\quad t \\in [0, 2\\pi )$ and $dM_+^{(12)}(t) = -dM_+^{(12)}(2\\pi - t)^T,\\quad \\quad t \\in [0, 2\\pi ).$ Taking advantage of the above equalities we can obtain $a(n) = \\int _0^{2\\pi } e^{i n t} d\\nu _+(t) - \\int _0^{2\\pi } e^{i n t} d\\nu _-(t),\\quad \\quad n \\in \\mathbb {Z},$ where $\\nu _{\\pm }(t) = M_{\\pm }^{(11)}(t) + M_{\\pm }^{(12)}(t) j$ .", "It is readily checked that $\\nu _{\\pm }$ are $q$ -positive measures.", "Moreover, $a$ has $\\kappa $ negative squares since $A$ has $2\\kappa $ negative squares and $\\nu _+(t_j) = 0$ for all $t_j \\in {\\rm supp}\\hspace{1.42262pt}\\nu _-$ and ${\\rm card}\\hspace{1.42262pt}{\\rm supp}\\hspace{1.42262pt} \\nu _- = \\kappa $ .", "Conversely, suppose that $a$ is a $\\mathbb {H}^{s \\times s}$ -valued sequence which obeys (REF ).", "Consequently, $a$ is bounded.", "The fact that $a$ is Hermitian follows from Remark REF .", "To see that $a$ has $\\kappa $ negative squares, one can consider the $\\mathbb {C}^{2s \\times 2s}$ -valued sequence $(A(n))_{n \\in \\mathbb {Z}}$ , where $A(n) = \\chi \\hspace{1.42262pt} a(n)$ , $n \\in \\mathbb {Z}$ and use the converse statement in Theorem REF to see that $A$ has $2\\kappa $ negative squares.", "The fact that $a$ has $\\kappa $ negative squares then follows by definition." ], [ "Herglotz's integral representation theorem in the scalar case", "In this section we present an analogue of Herglotz's theorem in the quaternionic scalar case.", "Even though this is a byproduct of the preceding discussion, it may be useful to have the result stated for scalar valued slice hyperholomorphic functions.", "We begin by proving an integral representation formula which holds on $\\mathbb {B}_r=\\lbrace p\\in \\mathbb {H}\\ : \\ |p|<r\\rbrace $ , namely on the quaternionic open ball centered at 0 and with radius $r>0$ .", "A similar formula which is based on a different representation of a slice hyperholomorphic function, less useful to determine the real part of a function, is discussed in [11].", "Note also that, unlike what happens in the complex case, the real part of a slice hyperholomorphic function is not harmonic.", "Lemma 8.1 Let $f:\\mathbb {B}_{1+\\varepsilon }\\rightarrow \\mathbb {H}$ be a slice hyperholomorphic function, for some $\\varepsilon >0$ .", "Let $I,J\\in \\mathbb {S}$ with $J$ orthogonal to $I$ and let $F,G: \\mathbb {B}_{1+\\varepsilon }\\cap \\mathbb {C}_I \\rightarrow \\mathbb {C}_I$ be holomorphic functions such that for any $z=x+Iy$ the restriction $f_I$ can be written as $f_I(z)=F(z)+G(z)J.$ Then, on $\\mathbb {B}_1\\cap \\mathbb {C}_I$ the following formula holds: $f_I(z)=I [{\\rm Im} F(0) +{\\rm Im} G(0)J]+\\frac{1}{2\\pi }\\int _0^{2\\pi }\\frac{e^{It}+z}{e^{It}-z} \\, [{\\rm Re}(F(e^{It}))+{\\rm Re}(G(e^{It}))J] dt.$ Moreover ${\\rm Re }\\Big (\\frac{e^{It}+z}{e^{It}-z} \\, [{\\rm Re}(F(e^{It}))+{\\rm Re}(G(e^{It}))J]\\Big )=\\frac{1-|z|^2}{|e^{It}-z|^2}{\\rm Re}(F(e^{It})).$ The proof is an easy consequence of the Splitting Lemma REF : for every fixed $I$ , $J \\in \\mathbb {S}$ such that $J$ is orthogonal to $I$ , there are two holomorphic functions $F,G: \\mathbb {B}_{1+\\varepsilon }\\cap \\mathbb {C}_I \\rightarrow \\mathbb {C}_I$ such that for any $z=x+Iy$ , it is $f_I(z)=F(z)+G(z)J.$ It is immediate that these two holomorphic functions $F$ , $G$ satisfy (see p. 206 in [1]) $F(z)= I{\\rm Im}\\, F(0)+\\frac{1}{2\\pi }\\int _0^{2\\pi }\\frac{e^{It}+z}{e^{It}-z} \\, {\\rm Re}(F(e^{It})) dt \\ \\ \\ z\\in \\mathbb {B}_{1}\\cap \\mathbb {C}_I,$ $G(z)= I{\\rm Im}\\,G(0)+\\frac{1}{2\\pi }\\int _0^{2\\pi }\\frac{e^{It}+z}{e^{It}-z} \\, {\\rm Re}(G(e^{It})) dt \\ \\ \\ z\\in \\mathbb {B}_{1}\\cap \\mathbb {C}_I,$ and the first part of the statement follows.", "The second part is a consequence of the equality $\\frac{e^{It}+z}{e^{It}-z}=\\frac{1-|z|^2}{|e^{It}-z|^2}+2I\\frac{y\\cos t-x\\sin t}{|e^{It}-z|^2}$ leading to ${\\rm Re }\\Big (\\frac{e^{It}+z}{e^{It}-z} \\, [{\\rm Re}(F(e^{It}))+{\\rm Re}(G(e^{It}))J]\\Big )=\\frac{1-|z|^2}{|e^{It}-z|^2}{\\rm Re}(F(e^{It})).$ Remark 8.2 Let $f: \\Omega \\rightarrow \\mathbb {H}$ be a slice hyperholomorphic function and write $f(p)=f_0(x_0,\\ldots , x_3)+f_1(x_0,\\ldots , x_3)i+f_2(x_0,\\ldots , x_3)j+f_3(x_0,\\ldots , x_3)k,$ with $f_\\ell :\\Omega \\rightarrow \\mathbb {R}$ , $\\ell =0,\\ldots ,3$ , $p=x_0+x_1i+x_2j+x_3k$ .", "It is easily seen that the restriction $f_i=f_{|\\mathbb {C}_i}$ can be written as $\\begin{split}f_i(x+iy)&=(f_0(x+iy)+f_1(x+iy)i)+(f_2(x+iy)+f_3(x+iy)i)j\\\\&=F(x+iy)+G(x+iy)j\\end{split}$ and so ${\\rm Re}(f_{|\\mathbb {C}_i})(x+iy)={f_0}_{|\\mathbb {C}_i}(x+iy))={\\rm Re}(F)(x+iy).$ More in general, consider $I,J\\in \\mathbb {S}$ with $I$ orthogonal to $J$ , and rewrite $i,j,k$ in terms of the imaginary units $I,J,IJ=K$ .", "Then $f(p)=f_0(x_0,\\ldots , x^{\\prime }_3)+\\tilde{f}_1(x_0,\\ldots , x^{\\prime }_3)I+\\tilde{f}_2(x_0,\\ldots , x^{\\prime }_3)J+\\tilde{f}_3(x_0,\\ldots , x^{\\prime }_3)K,$ where $p=x_0+x^{\\prime }_1I+x^{\\prime }_2J+ x^{\\prime }_3K$ and the $x^{\\prime }_\\ell $ are linear combinations of the $x_\\ell $ , $\\ell =1,2,3$ .", "The restriction of $f$ to the complex plane $\\mathbb {C}_I$ is then $f_I(x+Iy)=\\tilde{F}(x+Iy)+\\tilde{G}(x+Iy)J$ and reasoning as above we have ${\\rm Re}(f_I(x+Iy))={f_0}_{|\\mathbb {C}_I}(x_0,\\ldots , x_3^{\\prime })={\\rm Re}(\\tilde{F}(x+Iy)).$ We conclude that the real part of the restriction $f_I$ of $f$ to a complex plane $\\mathbb {C}_I$ is the restriction of $f_0$ to the given complex plane.", "Thus if ${\\rm Re}(f)$ is positive also the real part of the restriction $f_I$ to any complex plane is positive.", "Theorem 8.3 (Herglotz's theorem on a slice) Let $f:\\mathbb {B}_{1}\\rightarrow \\mathbb {H}$ be a slice hyperholomorphic function with ${\\rm Re}(f(p))\\ge 0$ in $\\mathbb {B}_1$ .", "Fix $I,J\\in \\mathbb {S}$ with $J$ be orthogonal to $I$ .", "Let $f_I$ be the restriction of $f$ to the complex plane $\\mathbb {C}_I$ and let $F,G: \\mathbb {B}_{1}\\cap \\mathbb {C}_I \\rightarrow \\mathbb {C}_I$ be holomorphic functions such that for any $z=x+Iy$ , it is $f_I(z)=F(z)+G(z)J.$ Then $f_I$ can be written in $\\mathbb {B}_1\\cap \\mathbb {C}_I$ as $f_I(z)=I [{\\rm Im} F(0) +{\\rm Im} G(0)J]+\\int _0^{2\\pi }\\frac{e^{It}+z}{e^{It}-z} \\, d\\mu _J(t),$ where $\\mu _J(t)=\\mu _1+\\mu _2 J$ is a finite variation complex measure on $\\mathbb {C}_J$ with $\\mu _1$ positive and $\\mu _2$ real and of finite variation on $[0,2\\pi ]$ .", "The proof follows [1].", "First, we note that by Remark REF , ${\\rm Re}(f(p))\\ge 0$ in $\\mathbb {B}_1$ implies that ${\\rm Re}(f_I(z))\\ge 0$ for $z\\in \\mathbb {B}_1\\cap \\mathbb {C}_I$ .", "Let $\\rho $ be a real number such that $0<\\rho <1$ .", "Then $f_I(\\rho z)$ is slice hyperholomorphic in the disc $|z|<1/\\rho $ and so by Lemma REF the restriction $f_I(z)$ may be written in $|z|<1$ as $f_I(\\rho z)=I [{\\rm Im} F(0) +{\\rm Im} G(0)J]+\\frac{1}{2\\pi }\\int _0^{2\\pi }\\frac{e^{It}+z}{e^{It}-z} \\, [{\\rm Re}(F(\\rho e^{It}))+{\\rm Re}(G(\\rho e^{It}))J] dt.$ where $d\\mu _J(t,\\rho )= \\frac{1}{2\\pi }[{\\rm Re}(F(\\rho e^{It}))+{\\rm Re}(G(\\rho e^{It}))J] dt$ has real positive part, since it is immediate that ${\\rm Re}(f_I(\\rho e^{It}))={\\rm Re}(F(\\rho e^{It}))$ and $\\int _0^{2\\pi }d\\mu _{J}(t,\\rho )= [{\\rm Re} F(0) +{\\rm Re} G(0)J].$ Let us set $\\Lambda _I(z,t):= \\frac{e^{It}+z}{e^{It}-z}$ and consider $\\int _0^{2\\pi }\\Lambda _I(z,t)d\\mu _J(t;\\rho ).$ Let $\\lbrace \\rho _n\\rbrace _{n\\in \\mathbb {N}}$ be a sequence of real numbers with $0<\\rho _n<1$ such that $\\rho _n\\rightarrow 1$ when $n$ goes to infinity.", "To conclude the proof we need Helly's theorem in the complex case.", "This result assures that the family of finite variation real-valued $d\\nu (t;\\rho _n)$ contains a convergent subsequence which tends to $d\\nu (t)$ which is of finite variation, in the sense that $\\lim _{n\\rightarrow \\infty } \\int _0^{2\\pi }\\lambda (w,t) d\\nu (t,\\rho _n)= \\int _0^{2\\pi }\\lambda (w,t) d\\nu (t)$ for every continuous complex-valued function $\\lambda (w,t)$ .", "In the slice hyperholomorphic setting the integrand is the product of the continuous $\\mathbb {C}_I$ -valued function $\\Lambda _I(z,t)=\\Lambda _1(z,t)+I\\Lambda _2(z,t)$ where $\\Lambda _1$ and $\\Lambda _2$ are real valued, and of the $\\mathbb {C}_J$ -valued $d\\mu _{J}(t,\\rho _n)= d\\mu _1(t,\\rho _n)+d\\mu _2(t,\\rho _n) J$ (since both $d\\mu _1(t,\\rho _n)$ and $d\\mu _2(t,\\rho _n)$ are real-valued).", "Then $\\Lambda _I(z,t)d\\mu _{J}(t)$ can be split in components to which we apply Helly's theorem.", "The positivity of $d\\mu _1$ follows from the positivity of $d\\mu _1(t,\\rho _n)$ , and this completes the proof.", "Corollary 8.4 Let $f$ be slice hyperholomorphic function on $\\mathbb {B}_1$ such that $f(0)=1$ .", "Suppose that $f$ has real positive part on $\\mathbb {B}_1$ .", "Then its restriction $f_I$ can be represented as $f_I(z)=\\int _0^{2\\pi }\\frac{e^{It}+z}{e^{It}-z} \\, d\\mu _J(t),$ where $\\mu _J(t)=\\mu _1(t)+\\mu _2(t)J$ is a finite variation complex measure on $\\mathbb {C}_J$ with $\\mu _1(t)$ positive for $t\\in [0,2\\pi ] $ .", "Moreover, the power series expansion of $f$ $f(p)=1+\\sum _{n=1}^\\infty p^n a_n$ is such that $|a_n|\\le k$ , for some $k\\in \\mathbb {R}$ , for every $n\\in \\mathbb {N}$ .", "The first part of the corollary immediately follows from Theorem REF .", "Then observe that $\\frac{e^{It}+z}{e^{It}-z}=1+2\\sum _{n=1}^\\infty z^n e^{-Int}$ and so the coefficients $a_n$ in the power series expansion are given by $a_n=2\\int _0^{2\\pi } e^{-Int}d\\mu _J(t).$ Moreover $|a_n|\\le 2 \\int _0^{2\\pi }|d\\mu _{J}(t)|\\le k,$ for some $k\\in \\mathbb {R}$ since $d\\mu _{J}(t)$ is of finite variation and so it is bounded.", "Remark 8.5 Formula (REF ) expresses $a_n$ in integral form.", "However, there is an infinite number of ways of writing $a_n$ with a similar expression, depending on the choices of $I$ and $J$ made to write (REF ).", "An important difference with the result in Section is that the measure $d\\mu $ in formula (REF ) is quaternionic valued, while in this case it is complex valued (with values in $\\mathbb {C}_J$ ).", "One may wonder is there are choices of $I,J$ for which formula (REF ) would allow to define $a_{-n}$ via (REF ) and then obtain $a_{-n}=\\bar{a}_n$ .", "Since $a_{-n}=2\\int _0^{2\\pi } e^{Int}d\\mu _J(t),\\qquad \\qquad \\bar{a}_n=2\\int _0^{2\\pi } \\overline{d\\mu _J(t)} e^{Int},$ and $\\begin{split}\\bar{a}_n&=2\\int _0^{2\\pi } (d\\mu _1(t)-d\\mu _2(t)J) e^{Int}\\\\&=2\\int _0^{2\\pi } e^{Int} d\\mu _1(t)-e^{-Int}d\\mu _2(t)J \\\\&=2\\int _0^{2\\pi } e^{Int} (d\\mu _1(t)-d\\mu _2(2\\pi - t)J), \\\\\\end{split}$ the condition $a_{-n}=\\bar{a}_n$ translates into ${\\rm Re}(G)(e^ {It})=- {\\rm Re}(G)(e^ {I(2\\pi -t)})$ .", "If one writes the power series expansion of $f_I$ in the form $f_I(x+Iy)=\\sum _{n=0}^\\infty (x+Iy)^n a_n =\\sum _{n=0}^\\infty (x+Iy)^n(a_{0n}+Ia_{1n} +(a_{2n}+Ia_{3n})J)$ then it follows that $F(x+Iy)=\\sum _{n= 0}^\\infty (x+Iy)^n(a_{0n}+Ia_{1n})\\ \\ \\ {\\rm and}\\ \\ \\ G(x+Iy)=\\sum _{n = 0}^\\infty (x+Iy)^n (a_{2n}+Ia_{3n}).$ Then ${\\rm Re}(G)(x+Iy)=\\sum _{n = 0}^\\infty u_n(x,y) a_{2n} -v_n(x,y) a_{3n}$ where $(x+Iy)^n=u_n(x,y)+Iv_n(x,y)$ and $u_n(x,y)=\\sum _{k=0,\\, k\\, even}^n {n\\atopwithdelims ()k}(-1)^{k/2}x^{n-k}y^k,$ $v_n(x,y)=\\sum _{k=1,\\, k\\, odd}^n {n\\atopwithdelims ()k}(-1)^{(k-1)/2}x^{n-k}y^k.$ It is immediate that $u_n$ and $v_n$ are even and odd in the variable $y$ , respectively, thus ${\\rm Re}(G)$ is odd in the variable $y$ if and only if $a_{2n}=0$ for all $n\\in \\mathbb {N}$ .", "In general, given a slice hyperholomorphic function $f$ on $\\mathbb {B}_1$ there is no change of basis for which one can have all the coefficients $a_{2n}=0$ for all $n\\in \\mathbb {N}$ .", "Thus, formula (REF ) does not allow to define $a_{-n}$ in order to obtain the desired equality $a_{-n}=\\bar{a}_n$ .", "The formula is however one of the several possibilities to assign the coefficients of $f$ in integral form.", "We conclude this section with a global integral representation.", "Theorem 8.6 Let $f:\\mathbb {B}_{1}\\rightarrow \\mathbb {H}$ be a slice hyperholomorphic function.", "Let $f_I$ be the restriction of $f$ to the complex plane $\\mathbb {C}_I$ and let $F,G: \\mathbb {B}_{1}\\cap \\mathbb {C}_I \\rightarrow \\mathbb {C}_I$ be holomorphic functions $f_I(z)=F(z)+G(z)J$ , $z=x+Iy$ .", "Then $f(q)=I[{\\rm Im} F(0) +{\\rm Im} G(0)J] +\\frac{1}{2\\pi }\\int _0^{2\\pi }K(q,e^{It}) \\, d\\mu _J(t),$ where $\\mu _J(t)$ is a finite variation complex measure on $\\mathbb {C}_J$ for $t\\in [0,2\\pi ] $ and $\\begin{split}K(q,e^{It})&=\\frac{1}{2}\\left(\\frac{e^{It}+z}{e^{It}-z}+\\frac{e^{It}+\\overline{z}}{e^{It}-\\overline{z}}\\right)+\\frac{1}{2} I_qI\\left(\\frac{e^{It}+\\overline{z}}{e^{It}-\\overline{z}} -\\frac{e^{It}+z}{e^{It}-z}\\right)\\\\&=(1+q^2-2q{\\rm Re}(e^{It}))^{-1}(1+2q{\\rm Im}(e^{It})-q^2).\\end{split}$ From Theorem REF the restriction of $f$ to the complex plane $\\mathbb {C}_I$ is $f_I(z)=I [{\\rm Im} F(0) +{\\rm Im} G(0)J]+\\int _0^{2\\pi }\\frac{e^{It}+z}{e^{It}-z} \\, d\\mu _J(t),$ where $\\mu _J(t)$ is a finite variation complex measure on $\\mathbb {C}_J$ .", "Consider $f_I(z)+f_I(\\overline{z})=2[{\\rm Im} F(0) +{\\rm Im} G(0)J]\\left(\\frac{e^{It}+z}{e^{It}-z}+\\frac{e^{It}+\\overline{z}}{e^{It}-\\overline{z}}\\right) \\,d\\mu _J(t)$ and $f_I(\\overline{z})-f_I(z)=\\int _0^{2\\pi }\\left(\\frac{e^{It}+\\overline{z}}{e^{It}-\\overline{z}} -\\frac{e^{It}+z}{e^{It}-z}\\right) \\,d\\mu _J(t);$ by applying the Representation Formula we obtain the kernel $K(q,e^{It})=\\frac{1}{2}\\left(\\frac{e^{It}+z}{e^{It}-z}+\\frac{e^{It}+\\overline{z}}{e^{It}-\\overline{z}}\\right)+\\frac{1}{2} I_qI\\left(\\frac{e^{It}+\\overline{z}}{e^{It}-\\overline{z}} -\\frac{e^{It}+z}{e^{It}-z}\\right).$ written in the first form.", "Now we write it in an equivalent way observing that the slice hyperholomorphic extension of the function $K(z,e^{It})=\\frac{e^{It}+z}{e^{It}-z}, \\ \\ \\ z=x+Iy$ is (for the $\\star $ -inverse see Ch.", "4 in [10]) and the $K(q,e^{It})=(e^{It}-q)^{-*}*(e^{It}+q)$ so that $K(q,e^{It})=(1+q^2-2q{\\rm Re}(e^{It}))^{-1}(1+2q{\\rm Im}(e^{It})-q^2),$ and the statement follows." ] ]
1403.0079
[ [ "Tractable Epistemic Reasoning with Functional Fluents, Static Causal\n Laws and Postdiction" ], [ "Abstract We present an epistemic action theory for tractable epistemic reasoning as an extension to the h-approximation (HPX) theory.", "In contrast to existing tractable approaches, the theory supports functional fluents and postdictive reasoning with static causal laws.", "We argue that this combination is particularly synergistic because it allows one not only to perform direct postdiction about the conditions of actions, but also indirect postdiction about the conditions of static causal laws.", "We show that despite the richer expressiveness, the temporal projection problem remains tractable (polynomial), and therefore the planning problem remains in NP.", "We present the operational semantics of our theory as well as its formulation as Answer Set Programming." ], [ "INTRODUCTION", " Epistemic reasoning about action and change is a crucial requirement for systems that deal with incomplete knowledge in the presence of abnormalities, unobservable processes, human-computer interaction and other real-world considerations.", "A particular type of epistemic inference which is required for diagnosis tasks in an epistemic context is postiction [3].", "Postdiction determines the condition of an action by observing its effect, and is therefore a fundamental requirement to determine the context under which actions occur or to diagnose abnormalities.", "As an example, consider a smart home with a robotic wheelchair that can navigate autonomously within the environment.Such scenarios are currently investigated within the smart home BAALL and the autonomous wheelchair Rolland .", "If the smart home recognizes that the wheelchair does not arrive at its destination after executing a driving request, it can postdict that there must be an abnormality (such as a blocked corridor or a flat tire) that prevents the wheelchair from driving.", "The scenario becomes more complicated if a person is sitting on the wheelchair because here ramifications (i.e.", "indirect side-effects of actions) are involved: if the person is sitting on the wheelchair, it will move as the wheelchair moves.", "In this work, we are interested in the backward direction of such epistemic ramification chains.", "A smart home assistance system should be able to postdict that the person is sitting on the wheelchair if the system observes that the person's location changes.", "This inference is useful to trigger other directly linked inferences about the conditions of the sit-down action: if the person is not sitting on the wheelchair, this could mean that the patient is unconscious and help is required.", "In addition to incomplete knowledge and ramifications, the scenario suggests to model knowledge (e.g.", "knowledge about the location of the wheelchair and the person) in a functional manner, which allows for simpler and more elaboration tolerant modeling of the reasoning domain, as compared to a boolean knowledge model.", "In order to model and to solve such reasoning problems, we extend the h-approximation ($\\mathcal {HPX}$ ) theory [3] and present an extended theory called $\\mathcal {HPX}_{\\mathcal {F}}$ .", "The extended theory is provided in terms of an operational semantics and an implementation as Answer Set Programming (ASP) .", "It covers ramification, postdiction and functional fluents in an elaboration tolerant manner.", "The theory is particularly useful in practice because it does not require an exponential number of state variables to model the knowledge-state of an agent.", "The projection problem is therefore tractable, as opposed to existing action theories that are based on a possible-worlds semantics ($\\mathcal {PWS}$ ).", "Another improvement of $\\mathcal {HPX}_{\\mathcal {F}}$ compared to $\\mathcal {HPX}$ is that the new ASP implementation allows one to model domain-specific causal laws directly in terms of ASP.", "This was not possible in the previous formulation, where a set of translation rules was required to generate the domain-specific causal laws from a high-level input language." ], [ "PRELIMINARIES AND RELATED WORK", "The field of Reasoning about Action and Change (RAC) emanated from the seminal work on the Situation Calculus by [14].", "Action theories developed in this research area usually employ discrete state transition semantics, where a state transition emerges from the occurrence of an event.", "In this work we are primarily interested in epistemic action theories, where an action does not only change the world state, but also the knowledge state of an agent." ], [ "Epistemic Action Theory", "Most epistemic action theories are based on a possible-world semantics ($\\mathcal {PWS}$ ) of knowledge [16].", "Such theories support postdiction, but they require an exponential number of state variables to represent the epistemic state of an agent, This exponential blowup leads to a high computational complexity.", "For example, [2] have shown that under certain conditions the planning problem for the $\\mathcal {PWS}$ -based action language $\\mathcal {A}_k$ is $\\Sigma _2^P$ -complete.", "To this end, [17] provide approximations of epistemic action languages, and [2] has shown that the 0-approximation, reduces the planning problem to NP-completeness, but it does not support postdiction.", "Functional fluents are also not supported in most epistemic action theories.It is possible to “emulate” functional fluents with boolean fluents, but this approach is not elaboration tolerant.", "A recent exception is [12], providing a functional $\\mathcal {PWS}$ -based epistemic extension to the Event Calculus, but without SCL." ], [ "Ramifications Static Causal Laws", "Ramification has been thoroughly investigated in the field of reasoning about action and change (RAC) throughout the last decades, and static causal laws (SCLs) have been proposed as additional language elements to capture indirect side-effect of actions [13].", "The authors argue that a causal theory of ramifications should account for implications, but not necessarily for the contrapositive of implications.", "[13] gives the example of Fred the turkey, who can be made not to walk by making him dead, but making him walk does not make him alive.", "The authors formalize this thought by describing a result function that computes possible next states, thereby also considering inconsistent and disjunctive effects of actions.", "Thereupon, a least fixed point procedure is applied to capture the least possible change generated by SCL.", "There are various non-epistemic action languages that support SCLs [7], but there exists only little work on SCLs within epistemic action theory.", "The action language $\\mathcal {A}_k^c$[18] is a successful exception that realizes efficient epistemic reasoning with SCLs.", "But since this approach is based on the 0-approximation of knowledge, it does not consider postdictive reasoning in an elaboration tolerant manner,Postdiction be realized in $\\mathcal {A}_k^c$ with ad-hoc definitions of SCLs.", "This is often a convenient workaround, but in general not elaboration tolerant (see [5] for details).", "and it does not account for functional fluents." ], [ "The h-approximation", "The h-approximation [6], [3] ($\\mathcal {HPX}$ ) is an epistemic action theory based on discrete state transitions.", "It does not use the possible-worlds model of knowledge, and therefore it does not require an exponential number of state variables.", "Instead of $\\mathcal {PWS}$ , it uses a simple three-valued knowledge model, i.e.", "something is either known to be true, known to be false or unknown.", "Complexity results in [3] show that the temporal projection problem can be solved in polynomial time and the plan existence problem is solvable in NP, as opposed to $\\Sigma _2^P$ for the $\\mathcal {PWS}$ -based action language $\\mathcal {A}_k$ .", "This comes at the cost that $\\mathcal {HPX}$ is incomplete, i.e.", "not all knowledge generated with a $\\mathcal {PWS}$ -based approach is also generated with the $\\mathcal {HPX}$ theory.", "The $\\mathcal {HPX}$ formalism has also been formulated in terms of ASP, and it has been successfully integrated in a robotic assistance system for a Smart Home [4].", "The extended formalism involves functional fluents and SCLs.", "Functional fluents require a modification of knowledge histories and inference mechanisms, and SCLs are additional language elements that are compiled into conditional action effects." ], [ "Language Elements", "A reasoning domain $\\mathcal {D}$ is an 8-tuple $\\left\\langle \\mathcal {FR}, \\mathcal {VP}, \\mathcal {EP}, \\mathcal {SCL}, \\mathcal {KP}, \\mathcal {EXC}, \\mathcal {G}^{strong}, \\mathcal {G}^{weak} \\right\\rangle $ that consists of the following language elements: Fluent range specifications.", "$\\mathcal {FR}$ is a set of tuples $\\left\\langle f, v \\right\\rangle $ to denote that $v$ is in the functional range of $f$ .", "Value propositions.", "$\\mathcal {VP}$ is a set of tuples $\\left\\langle f,v \\right\\rangle $ , denoting that initially (at $t=0$ ) fluent $f$ has the value $v$ .", "Effect propositions represents conditional action effects.", "Formally, $\\mathcal {EP}$ is a set of triples $\\left\\langle a,\\lbrace \\left\\langle f^c_1,v^c_1 \\right\\rangle , \\hdots , \\left\\langle f^c_k,v^c_k \\right\\rangle \\rbrace , \\left\\langle f^e, v^e \\right\\rangle \\right\\rangle $ , denoting that $a$ causes $f^e$ to have the value $v^e$ under the condition that fluents $f^c_1, \\hdots , f^c_k$ have the values $v^c_1, \\hdots , v^c_k$ .", "For an effect proposition $ep$ , we write $c(ep)$ to denote denotes the set of condition fluents $\\lbrace \\left\\langle f^c_1,v^c_1 \\right\\rangle , \\hdots , \\left\\langle f^c_k,v^c_k \\right\\rangle \\rbrace $ and $e(ep)$ to denote the effect fluent $\\left\\langle f^e, v^e \\right\\rangle $ .", "Static causal laws are used to reason about indirect action effects.", "Formally, $\\mathcal {SCL}$ is a set of tuples $\\left\\langle \\lbrace \\left\\langle f^c_1,v^c_1 \\right\\rangle , \\hdots , \\left\\langle f^c_k,v^c_k \\right\\rangle \\rbrace , \\left\\langle f^e, v^e \\right\\rangle \\right\\rangle $ , denoting that $f^e$ is caused to have the value $v^e$ if fluents $f^c_1, \\hdots , f^c_k$ are caused to have the values $v^c_1, \\hdots , v^c_k$ .", "For a SCL $scl$ , we write $c(scl)$ to denote the set of condition fluent-value-pairs $\\lbrace \\left\\langle f^c_1,v^c_1 \\right\\rangle , \\hdots , \\left\\langle f^c_k,v^c_k \\right\\rangle \\rbrace $ and $e(scl)$ to denote the effect fluent-value-pair $\\left\\langle f^e, v^e \\right\\rangle $ .", "Knowledge propositions represent sensing actions.", "$\\mathcal {KP}$ is a set of tuples $\\left\\langle a,f \\right\\rangle $ , denoting that a sensing action $a$ will determine the value of $f$ .", "Executability conditions denote what an agent must know to execute an action.", "$\\mathcal {EXC}$ is a set of tuples $\\left\\langle a,\\lbrace \\left\\langle f^x_1,v^x_1 \\right\\rangle , \\hdots , \\left\\langle f^x_l,v^x_l \\right\\rangle \\rbrace \\right\\rangle $ , denoting that an agent must know that fluents $f^c_1, \\hdots , f^c_k$ have the values $v^c_1, \\hdots , v^c_k$ in order to execute $a$ .", "We say that $a$ is executable in $\\mathbf {\\mathfrak {h}}$ if all fluent-value pairs in the executability condition of $a$ are known to hold.", "Goal propositions ($\\mathcal {G}^{strong}$ , $\\mathcal {G}^{weak}$ ) are sets of fluent-value pairs that denote strong and weak goals.", "Weak goals denote that a plan has to be found which possibly achieves the goal.", "That is, there must be at least one leaf state in the transition tree where the goal is achieved.", "A strong goal must be achieved in all leaf states, i.e.", "a plan must necessarily achieve a goal." ], [ "Knowledge Histories with Functional Fluents", "The operational semantics of $\\mathcal {HPX}_{\\mathcal {F}}$ is based on so-called h-states (denoted as $\\mathbf {\\mathfrak {h}}$ ) that represent historical knowledge about past and present.", "Formally, an h-state is a pair $\\left\\langle \\alpha , \\mathbf {\\kappa } \\right\\rangle $ , where $\\alpha $ is the action history and $\\mathbf {\\kappa }$ is the knowledge history of an agent.", "The operational semantics explicitly considers knowledge that a fluent does not have a certain value, even if the actual value is unknown.", "Functional knowledge histories $\\mathbf {\\kappa }$ consist of triples $\\left\\langle f,v,t \\right\\rangle $ and $\\left\\langle f,\\lnot v,t \\right\\rangle $ , which denote that it is known that a fluent $f$ has the value $v$ at a step $t$ , respectively, that it is known that a fluent $f$ does not have the value $v$ at step $t$ .", "Definition 1 (Functional h-states) A functional h-state $\\mathbf {\\mathfrak {h}}$ is a pair $\\left\\langle \\alpha , \\mathbf {\\kappa } \\right\\rangle $ .", "An action history $\\alpha $ is a set of pairs of actions and time steps, and a knowledge history $\\mathbf {\\kappa }$ is a set of triples of fluents $f$ , values $v$ or $\\lnot v$ and time steps $t$ .", "A knowledge history $\\mathbf {\\kappa }$ is valid if it holds that for all triples $\\left\\langle f,v,t \\right\\rangle \\in \\mathbf {\\kappa }$ (i) there exists no triple $\\left\\langle f,v^{\\prime },t \\right\\rangle \\in \\mathbf {\\kappa }$ with $v \\ne v^{\\prime }$ and (ii) there exists no triple $\\left\\langle f, \\lnot v, t \\right\\rangle \\in \\mathbf {\\kappa }$ and (iii) $v$ is in the range of $f$ , according to the fluent range specifications ($\\mathcal {FR}$ ) of a given domain $\\mathcal {D}$ .", "To simplify our model of concurrent conditional effects we also define the effect history $\\mathbf {\\epsilon }$ of an h-state (see Definition REF ).", "As notational convention we write $\\alpha (\\mathbf {\\mathfrak {h}})$ , $\\mathbf {\\kappa }(\\mathbf {\\mathfrak {h}})$ and $\\mathbf {\\epsilon }(\\mathbf {\\mathfrak {h}})$ to denote the action history, knowledge history and effect history of an h-state $\\mathbf {\\mathfrak {h}}$ .", "To simplify notation, we sometimes transfer sub- and superscripts from $\\mathbf {\\mathfrak {h}}$ to $\\mathbf {\\epsilon }$ , $\\mathbf {\\kappa }$ and $\\alpha $ (if clear from the context).", "For instance we write $\\mathbf {\\epsilon }_n$ to denote $\\mathbf {\\epsilon }(\\mathbf {\\mathfrak {h}}_n)$ .", "Definition 2 (Effect history $\\mathbf {\\epsilon }$ ) Let $\\alpha =\\lbrace \\left\\langle a_1,t_1 \\right\\rangle ,\\hdots ,\\left\\langle a_n,t_n \\right\\rangle \\rbrace $ be an action history and let $\\mathcal {EP}^{a}$ denote the set of effect proposition of an action $a$ .", "Then the effect history $\\mathbf {\\epsilon }({\\mathbf {\\mathfrak {h}}})$ of the h-state $\\mathbf {\\mathfrak {h}}$ is given by (REF ).", "$\\mathbf {\\epsilon }({\\mathbf {\\mathfrak {h}}}) = \\lbrace \\left\\langle ep,t \\right\\rangle | \\exists \\left\\langle a,t \\right\\rangle \\in \\alpha (\\mathbf {\\mathfrak {h}}) : ep \\in \\mathcal {EP}^{a}\\rbrace $ Effect histories are used to simplify our model of concurrent action execution and used in the inference mechanisms described in Section REF .", "The behavior of an action is modeled via a transition function $\\Psi $ that takes a set of actions $\\mathcal {A}$ and an h-state $\\mathbf {\\mathfrak {h}}$ as input and returns a set of h-states as output.", "The transition function involves eight inference mechanisms IM.1 – IM.8, namely forward inertia, backward inertia, causation, positive postdiction, negative postdiction, positive exclusion, negative exclusion and static causal consequence.", "The inference mechanisms implement certain epistemic effects (in particular postdiction) that emerge from a possible-worlds model of knowledge.", "The advantage of implementing these effects manually with the eight IM is that they avoid the exponential blowup of the epistemic state space.", "Before presenting details of the IM in Section REF , we describe our approach to handle SCLs which allows us to apply the IM on SCLs as well." ], [ "Compiling Static Causal Laws to Effect Propositions", "Opposed to the approach by [13], our theory does not consider disjuctive and inconsistent effects, i.e.", "the effects of actions are always deterministic (see [3]).", "Therefore $\\mathcal {HPX}_{\\mathcal {F}}$ always generates one single possible successor state per possible sensing result, so that we do not need to employ a fixed point approach as proposed in [13].", "Instead of using a corresponding closure function as defined in [13], we compile SCLs into effect propositions (EPs) to reason about indirect effects of actions.", "By compiling SCLs away, we do not need to implement additional inference mechanisms for SCLs, because the transition function is based on EPs and therefore already performs all necessary reasoning tasks.", "The following recursive function generates additional effect propositions that are used in the inference mechanisms.", "$\\begin{aligned}&genEP(\\mathcal {EP}) ={\\left\\lbrace \\begin{array}{ll}\\mathcal {EP} & \\text{ if } addEP(\\mathcal {EP}) = \\emptyset \\\\genEP(\\mathcal {EP} \\cup addEP(\\mathcal {EP})) & \\text{ otherwise }\\end{array}\\right.}", "\\\\&\\text{ where }\\\\&addEP(\\mathcal {EP}) = \\lbrace \\left\\langle a,\\lbrace \\left\\langle f^{scl}_1,v^{scl}_1 \\right\\rangle , \\hdots , \\left\\langle f^{scl}_k,v^{scl}_k \\right\\rangle , \\left\\langle f^{ep}_1,v^{ep}_1 \\right\\rangle , \\hdots , \\left\\langle f^{ep}_l,v^{ep}_l \\right\\rangle \\rbrace \\setminus \\left\\langle f^{trig}, v^{trig} \\right\\rangle , \\left\\langle f^{e}, v^{e} \\right\\rangle \\right\\rangle | \\\\& ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\left\\langle \\lbrace \\left\\langle f^{scl}_1,v^{scl}_1 \\right\\rangle , \\hdots , \\left\\langle f^{scl}_k,v^{scl}_k \\right\\rangle \\rbrace , \\left\\langle f^e, v^e \\right\\rangle \\right\\rangle \\in \\mathcal {SCL} \\wedge \\\\& ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\left\\langle f^{trig}, v^{trig} \\right\\rangle \\subseteq \\lbrace \\left\\langle f^{scl}_1,v^{scl}_1 \\right\\rangle , \\hdots , \\left\\langle f^{scl}_k,v^{scl}_k \\right\\rangle \\rbrace \\wedge \\\\& ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \\left\\langle a,\\lbrace \\left\\langle f^{ep}_1,v^{ep}_1 \\right\\rangle , \\hdots , \\left\\langle f^{ep}_l,v^{ep}_l \\right\\rangle \\rbrace , \\left\\langle f^{trig}, v^{trig} \\right\\rangle \\right\\rangle \\in \\mathcal {EP}\\rbrace \\end{aligned}$ Intuitively, $addEP$ looks for fluent-value pairs $\\left\\langle f^{trig}, v^{trig} \\right\\rangle $ that trigger the condition of a SCL to become true, such that the effect of the SCL becomes true as well.", "The recursive nature of $genEP$ is required to also cope for “chained” SCL triggering.", "For example, consider an EP $\\left\\langle a, \\lbrace \\rbrace , f^{trig}, v^{trig} \\right\\rangle $ and two SCL $\\left\\langle \\lbrace \\left\\langle f^{trig},v^{trig} \\right\\rangle \\rbrace , f^{scl1}, v^{scl1} \\right\\rangle $ and $\\left\\langle \\lbrace \\left\\langle f^{scl1},v^{scl1} \\right\\rangle \\rbrace , f^{scl2}, v^{scl2} \\right\\rangle $ .", "The first call will produce an EP $\\left\\langle a, \\lbrace \\rbrace , f^{scl1}, v^{scl1} \\right\\rangle $ , and the second call will produce another EP $\\left\\langle a, \\lbrace \\rbrace , f^{scl2}, v^{scl2} \\right\\rangle $ .", "It is also possible to trigger a SCL with two trigger conditions.", "For example, consider two EPs $\\left\\langle a, \\lbrace \\rbrace , \\left\\langle f^{trig1}, v^{trig1} \\right\\rangle \\right\\rangle $ and $\\left\\langle a, \\lbrace \\rbrace , f^{trig2}, v^{trig2} \\right\\rangle $ and a SCL $\\left\\langle \\lbrace \\left\\langle f^{trig1},v^{trig1} \\right\\rangle , \\left\\langle f^{trig2},v^{trig2} \\right\\rangle \\rbrace , f^{scl}, v^{scl} \\right\\rangle $ .", "It is clearly the case, that this should result in another EP $\\left\\langle a, \\lbrace \\rbrace , \\left\\langle f^{scl}, v^{scl} \\right\\rangle \\right\\rangle $ .", "The first call of $addEP$ will generate the EPs $\\left\\langle a, \\lbrace \\left\\langle f^{trig1}, v^{trig1} \\right\\rangle \\rbrace , f^{scl}, v^{scl} \\right\\rangle $ and $\\left\\langle a, \\lbrace \\left\\langle f^{trig2}, v^{trig2} \\right\\rangle \\rbrace , f^{scl}, v^{scl} \\right\\rangle $ .", "In the second call, $addEP$ will generate the desired EP $\\left\\langle a, \\lbrace \\rbrace , \\left\\langle f^{scl}, v^{scl} \\right\\rangle \\right\\rangle $ ." ], [ "State transitions and Inference Mechanisms with Functional Fluents", "The transition function (REF ) adds a set of actions $A$ to the action history $\\alpha $ and then evaluates the knowledge-level effects of these actions.", "As an auxiliary notion we write $now(\\mathbf {\\mathfrak {h}})$ to refer to the current step number.", "$now(\\mathbf {\\mathfrak {h}}) ={\\left\\lbrace \\begin{array}{ll}0 & \\text{ if } \\alpha ({\\mathbf {\\mathfrak {h}}}) = \\emptyset \\\\t+1 & \\text{ if } \\exists \\left\\langle a,t \\right\\rangle \\in \\alpha ({\\mathbf {\\mathfrak {h}}}) : \\forall \\left\\langle a^{\\prime },t^{\\prime } \\right\\rangle \\in \\alpha ({\\mathbf {\\mathfrak {h}}}): t^{\\prime } \\le t\\end{array}\\right.", "}$ This allows as to define the transition function (REF ).", "$\\begin{aligned}&\\Psi (Ȃ,\\mathbf {\\mathfrak {h}}) = \\underset{k \\in sense(Ȃ^{ex},\\mathbf {\\mathfrak {h}})}{\\overset{}{\\bigcup }}eval(\\left\\langle \\alpha ^{\\prime }, \\mathbf {\\kappa }({\\mathbf {\\mathfrak {h}}}) \\cup k \\right\\rangle )\\\\&\\text{where }\\\\&\\bullet \\alpha ^{\\prime } = \\alpha ({\\mathbf {\\mathfrak {h}}}) \\cup \\lbrace \\left\\langle a,t \\right\\rangle |a \\in Ȃ^{ex} \\wedge t = now(\\mathbf {\\mathfrak {h}})\\rbrace \\\\&\\bullet \\text{$A^{ex}$ is the subset of actions of $A$ which are executable in $\\mathbf {\\mathfrak {h}}$}\\end{aligned}$ The transition function calls two other function, $sense$ and $eval$ .", "$eval$ (REF ) is a re-evaluation function that refines the knowledge-history of an h-state by determining the knowledge-level effects of non-sensing actions using the eight inference mechanisms described in Sections REF – REF .", "$sense$ adds sensing results to the knowledge history.", "It is formally defined as follows.", "Let $t^s = now(\\mathbf {\\mathfrak {h}})$ , let $\\mathcal {FR}$ be the fluent range specification, and let $\\mathcal {KP}$ be the knowledge propositions of a reasoning domain, then: $sense(Ȃ,\\mathbf {\\mathfrak {h}}) = \\underset{ \\lbrace \\left\\langle f,v \\right\\rangle \\in \\mathcal {FR} | \\exists a: \\left\\langle a,f \\right\\rangle \\in \\mathcal {KP} \\wedge \\left\\langle f,\\lnot v,t^s \\right\\rangle \\notin \\mathbf {\\kappa }(\\mathbf {\\mathfrak {h}}) \\rbrace }{\\overset{}{\\bigcup }} \\left\\langle f,v,t^s \\right\\rangle $ Note that we restrict $sense$ (REF ) (and thereby the $\\mathcal {HPX}_{\\mathcal {F}}$ theory) to the case where there is only one fluent to sense per state transition.", "Without this restriction, $sense$ would generate an exponential number of successor states and the tractability of $\\mathcal {HPX}_{\\mathcal {F}}$ would be destroyed.", "Intuitively, $sense$ describes that knowledge is added to the h-state if it is not known that the possible sensing results does not hold.", "The re-evaluation function $eval$ (REF ) consists of eight inference mechanisms, namely forward inertia (REF ), backward inertia (REF ), causation (REF ), positive postdiction (REF ) , negative postdiction (REF ), positive exclusion (REF ), negative exclusion (REF ) and static causal consequence (REF ) that constitute the re-evaluation process.", "To collectively apply the seven inference mechanisms in one function we define an $evalOnce$ function that successively applies each of the inference mechanisms.", "$\\begin{aligned}evalOnce(\\mathbf {\\mathfrak {h}}) = scl(ex^{neg}(ex^{pos}(pd^{neg}(pd^{pos}(cause(back(fwd(\\mathbf {\\mathfrak {h}}))))))))\\end{aligned}$ A problem is that inference mechanism may trigger each other in any order, so it is often not sufficient to apply IM.1 – IM.8 only once.", "To this end, re-evaluation is defined recursively (REF ) until convergence is reached.", "$eval(\\mathbf {\\mathfrak {h}}) ={\\left\\lbrace \\begin{array}{ll}\\mathbf {\\mathfrak {h}}& \\text{ if } evalOnce(\\mathbf {\\mathfrak {h}}) = \\mathbf {\\mathfrak {h}}\\\\eval(evalOnce(\\mathbf {\\mathfrak {h}})) & \\text{ otherwise}\\end{array}\\right.", "}$ $eval$ converges in linear time because there exists only a linear number of elements in the knowledge history and because no element is ever removed from the knowledge history (see also Lemma B.5 in [3] for the case of boolean fluents)." ], [ " – Forward and Backward Inertia", "To describe the laws of forward and backward knowledege propagation by inertia we first state when a fluent value pair $\\left\\langle f,v \\right\\rangle $ is inertial wrt.", "an h-state $\\mathbf {\\mathfrak {h}}$ and a step $t$ .", "$\\begin{aligned}inertial(f,v,t,\\mathbf {\\mathfrak {h}}) \\Leftrightarrow & \\left\\langle f,v \\right\\rangle \\in \\mathcal {FR} \\wedge && \\forall \\left\\langle ep,t \\right\\rangle \\in \\mathbf {\\epsilon }(\\mathbf {\\mathfrak {h}}) : \\\\&&&\\left(e(ep) = \\left\\langle f,v^{\\prime } \\right\\rangle \\wedge v \\ne v^{\\prime } \\right) \\Rightarrow \\\\&&&\\left(\\exists \\left\\langle f^c,v^c \\right\\rangle \\in c(ep) : \\left\\langle f^c,\\lnot v^c,t \\right\\rangle \\in \\mathbf {\\kappa }(\\mathbf {\\mathfrak {h}}) \\right)\\end{aligned}$ We also define inertia of a fluent and a negative value $\\lnot v$ .", "$\\begin{aligned}inertial(f,\\lnot v,t,\\mathbf {\\mathfrak {h}}) \\Leftrightarrow & \\left\\langle f,v \\right\\rangle \\in \\mathcal {FR} \\wedge && \\forall \\left\\langle ep,t \\right\\rangle \\in \\mathbf {\\epsilon }(\\mathbf {\\mathfrak {h}}) : \\\\&&&\\left(e(ep) = \\left\\langle f,v \\right\\rangle \\right) \\Rightarrow \\\\&&& \\left(\\exists \\left\\langle f^c,v^c \\right\\rangle \\in c(ep) : \\left\\langle f^c,\\lnot v^c,t \\right\\rangle \\in \\mathbf {\\kappa }(\\mathbf {\\mathfrak {h}}) \\right)\\end{aligned}$ Forward inertia describes that a fluent $f$ is known to have a value $v$ at a step $t$ if it is known that $f$ has the value $v$ already at step $t-1$ and that $\\left\\langle f,v \\right\\rangle $ is inertial at $t-1$ .", "$\\begin{aligned}&fwd(\\mathbf {\\mathfrak {h}}) = \\left\\langle \\alpha (\\mathbf {\\mathfrak {h}}), \\mathbf {\\kappa }({\\mathbf {\\mathfrak {h}}}) \\cup add_{fwd}^{pos}(\\mathbf {\\mathfrak {h}}) \\cup add_{fwd}^{neg}(\\mathbf {\\mathfrak {h}}) \\right\\rangle \\\\& \\text{ where } \\\\&add_{fwd}^{pos}(\\mathbf {\\mathfrak {h}}) = \\lbrace \\left\\langle f,v,t \\right\\rangle | \\left\\langle f,v,t-1 \\right\\rangle \\in \\mathbf {\\kappa }({\\mathbf {\\mathfrak {h}}}) \\wedge inertial(f,v,t-1,\\mathbf {\\mathfrak {h}}) \\wedge t \\le now(\\mathbf {\\mathfrak {h}}) \\rbrace \\\\&add_{fwd}^{neg}(\\mathbf {\\mathfrak {h}}) = \\lbrace \\left\\langle f,\\lnot v,t \\right\\rangle | \\left\\langle f,\\lnot v,t-1 \\right\\rangle \\in \\mathbf {\\kappa }({\\mathbf {\\mathfrak {h}}}) \\wedge inertial(f,\\lnot v,t-1,\\mathbf {\\mathfrak {h}}) \\wedge t \\le now(\\mathbf {\\mathfrak {h}}) \\rbrace \\end{aligned}$ Backward inertia describes that a fluent $f$ is known to have a value $v$ at a step $t$ if it is known that $f$ has the value $v$ at step $t+1$ and that $\\left\\langle f,v \\right\\rangle $ was not set at $t$ .", "$\\begin{aligned}&back(\\mathbf {\\mathfrak {h}}) = \\left\\langle \\alpha (\\mathbf {\\mathfrak {h}}), \\mathbf {\\kappa }({\\mathbf {\\mathfrak {h}}}) \\cup add_{back}^{pos}(\\mathbf {\\mathfrak {h}}) \\cup add_{back}^{neg}(\\mathbf {\\mathfrak {h}}) \\right\\rangle \\\\&\\text{ where } \\\\&add_{back}^{pos}(\\mathbf {\\mathfrak {h}}) = \\lbrace \\left\\langle f,v,t \\right\\rangle | \\left\\langle f,v,t+1 \\right\\rangle \\in \\mathbf {\\kappa }({\\mathbf {\\mathfrak {h}}}) \\wedge inertial(f,\\lnot v,t,\\mathbf {\\mathfrak {h}}) \\wedge t \\ge 0\\rbrace \\\\&add_{back}^{neg}(\\mathbf {\\mathfrak {h}}) = \\lbrace \\left\\langle f,\\lnot v,t \\right\\rangle | \\left\\langle f,\\lnot v,t+1 \\right\\rangle \\in \\mathbf {\\kappa }({\\mathbf {\\mathfrak {h}}}) \\wedge inertial(f,v,t,\\mathbf {\\mathfrak {h}}) \\wedge t \\ge 0\\rbrace \\end{aligned}$" ], [ " – Positive and Negative Postdiction", "Positive postdiction is the inference that knowledge about the conditions of an effect proposition is gained if (i) the effect is known to hold after the action and (ii) known not to hold before the action and (iii) no other effect proposition could have triggered the effect.", "$\\begin{aligned}&pd^{pos}(\\mathbf {\\mathfrak {h}}) = \\left\\langle \\alpha (\\mathbf {\\mathfrak {h}}),\\mathbf {\\kappa }({\\mathbf {\\mathfrak {h}}}) \\cup add_{pd^{pos}}(\\mathbf {\\mathfrak {h}}) \\right\\rangle \\\\&\\text{where } \\\\&add_{pd^{pos}}(\\mathbf {\\mathfrak {h}}) = \\lbrace \\left\\langle f^c,v^c,t \\right\\rangle | \\exists \\left\\langle ep,t \\right\\rangle \\in \\mathbf {\\epsilon }(\\mathbf {\\mathfrak {h}}) : \\\\&~~~~~~~~~~~~~~~~~~~~~~~~~~\\left\\langle f^c,v^c \\right\\rangle \\in c(ep) \\wedge \\left\\langle f^e,v^e,t+1 \\right\\rangle \\in \\mathbf {\\kappa }(\\mathbf {\\mathfrak {h}}) \\wedge \\left\\langle f^e,\\lnot v^e,t \\right\\rangle \\in \\mathbf {\\kappa }(\\mathbf {\\mathfrak {h}}) \\\\&~~~~~~~~~~~~~~~~~~~~~~~~~~\\wedge \\left( \\forall \\left\\langle ep^{\\prime },t \\right\\rangle \\in \\mathbf {\\epsilon }(\\mathbf {\\mathfrak {h}}) : \\left(ep^{\\prime } = ep \\vee e(ep^{\\prime }) \\ne l^e \\right) \\right) \\rbrace \\end{aligned}$ Negative postdiction generates knowledge that the condition of an EP does not hold if the effect does not hold after the EP is applied.", "Formally, negative postdiction is defined with (REF ).", "$\\begin{aligned}&pd^{neg}(\\mathbf {\\mathfrak {h}}) = \\left\\langle \\alpha (\\mathbf {\\mathfrak {h}}), \\mathbf {\\kappa }({\\mathbf {\\mathfrak {h}}}) \\cup add_{pd^{neg}}(\\mathbf {\\mathfrak {h}}) \\right\\rangle \\\\&\\text{where } \\\\&add_{pd^{neg}} = \\lbrace \\left\\langle f^c_u,\\lnot v^c_u,t \\right\\rangle | \\exists \\left\\langle ep,t \\right\\rangle \\in \\mathbf {\\epsilon }(\\mathbf {\\mathfrak {h}}) : \\\\&~~~~~~~~~~~~~~~~~~~~~~~~~~ f^c_u,v^c_u \\in c(ep) \\wedge \\left\\langle f^e,\\lnot v^e,t+1 \\right\\rangle \\in \\mathbf {\\kappa }(\\mathbf {\\mathfrak {h}}) \\\\&~~~~~~~~~~~~~~~~~~~~~~~~~~\\wedge \\left( \\forall \\left\\langle f^c,v^c \\right\\rangle \\in c(ep) \\setminus \\left\\langle f^c_u,v^c_u \\right\\rangle : \\left\\langle f^c,v^c,t \\right\\rangle \\in \\mathbf {\\kappa }(\\mathbf {\\mathfrak {h}}) \\right) \\rbrace \\end{aligned}$" ], [ " – Positive and Negative Exclusion", "Positive exclusion determines that a pair $\\left\\langle f,v \\right\\rangle $ holds if all values in the range of $f$ except $v$ are known not to hold.", "$\\begin{aligned}&ex^{pos}(\\mathbf {\\mathfrak {h}}) = \\left\\langle \\alpha (\\mathbf {\\mathfrak {h}}), \\mathbf {\\kappa }({\\mathbf {\\mathfrak {h}}}) \\cup add_{ex^{pos}}(\\mathbf {\\mathfrak {h}}) \\right\\rangle \\\\&\\text{where } \\\\&add_{ex^{pos}} = \\lbrace \\left\\langle f,v,t \\right\\rangle | \\forall \\left\\langle f,v^{\\prime } \\right\\rangle \\in \\mathcal {FR}: \\left(v \\ne v^{\\prime } \\Rightarrow \\left\\langle f,\\lnot v^{\\prime },t \\right\\rangle \\in \\mathbf {\\kappa }(\\mathbf {\\mathfrak {h}}) \\right) \\rbrace \\end{aligned}$ Negative exclusion generates knowledge that a pair $\\left\\langle f,v \\right\\rangle $ does not hold if it is known that $\\left\\langle f,v^{\\prime } \\right\\rangle $ holds where $v \\lnot v^{\\prime }$ .", "$\\begin{aligned}&ex^{neg}(\\mathbf {\\mathfrak {h}}) = \\left\\langle \\alpha (\\mathbf {\\mathfrak {h}}), \\mathbf {\\kappa }({\\mathbf {\\mathfrak {h}}}) \\cup add_{ex^{neg}}(\\mathbf {\\mathfrak {h}}) \\right\\rangle \\\\&\\text{where } \\\\&add_{ex^{neg}} = \\lbrace \\left\\langle f,\\lnot v,t \\right\\rangle | \\left\\langle f,v \\right\\rangle \\in \\mathcal {FR} \\wedge \\exists \\left\\langle f,v^{\\prime } \\right\\rangle \\in \\mathbf {\\kappa }(\\mathbf {\\mathfrak {h}}) : v^{\\prime } \\lnot v\\rbrace \\end{aligned}$" ], [ " – Static Causal Consequences", "The described approach to compile SCLs into EPs is not sufficient to cover all aspects of SCLs.", "It may also be possible to produce knowledge via applying SCLs on initial knowledge or knowledge generated by sensing.", "To this end, we require another inference mechanism that generates indirect knowledge from SCLs by considering immediate consequences.", "This is captured in (REF ).", "$\\begin{aligned}&scl(\\mathbf {\\mathfrak {h}}) = \\left\\langle \\alpha (\\mathbf {\\mathfrak {h}}), \\mathbf {\\kappa }({\\mathbf {\\mathfrak {h}}}) \\cup add_{scl}(\\mathbf {\\mathfrak {h}}) \\right\\rangle \\\\&\\text{where } \\\\&add_{scl} = \\lbrace \\left\\langle f^e,v^e,t \\right\\rangle | \\exists \\left\\langle \\lbrace \\left\\langle f^c_1,v^c_1 \\right\\rangle , \\hdots , \\left\\langle f^c_k,v^c_k \\right\\rangle \\rbrace , \\left\\langle f^e, v^e \\right\\rangle \\right\\rangle \\in \\mathcal {SCL} : \\left\\langle f^c_1,v^c_1,t \\right\\rangle , \\hdots , \\left\\langle f^c_k,v^c_k,t \\right\\rangle \\subseteq \\mathbf {\\kappa }(\\mathbf {\\mathfrak {h}}) \\rbrace \\end{aligned}$" ], [ "FORMALIZATION AS ANSWER SET PROGRAMMING", "The formalization is based on a foundational theory $\\Gamma _{hpx}$ , and a domain-specific theory $\\Gamma _{world}$ , based on the langauge elements specified in Section REF .", "Domain-dependent theory ($\\Gamma _{world}$ ): It consists of a set of rules $\\Gamma _{ini}$ representing initial knowledge; $\\Gamma _{act}$ representing actions; $\\Gamma _{scl-world}$ representing SCLs; and $\\Gamma _{goals}$ representing goals.", "Domain-independent theory ($\\Gamma _{hpx}$ ): This consists of a set of rules to handle inertia ($\\Gamma _{in}$ ); postdiction ($\\Gamma _{post}$ ); SCLs ($\\Gamma _{scl-hpx}$ ); sensing ($\\Gamma _{sen}$ ); concurrency ($\\Gamma _{conc}$ ), plan verification ($\\Gamma _{verify}$ ) as well as plan-generation & optimization ($\\Gamma _{plan}$ ).", "The resulting Logic Program for a reasoning domain $\\mathcal {D}$ is given as: $&LP(\\mathcal {D}) = \\\\&[~\\Gamma _{ini}\\cup \\Gamma _{act}\\cup \\Gamma _{scl-world}\\cup \\Gamma _{goal}] \\cup \\\\&[~\\Gamma _{in}\\cup \\Gamma _{post}\\cup \\Gamma _{scl-hpx}\\cup \\Gamma _{sen}\\cup \\Gamma _{conc}\\cup \\Gamma _{verify}\\cup \\Gamma _{plan}~]\\\\[-23pt]$" ], [ "$\\Gamma _{world}$ – Domain Specific Theory (", "The domain specific theory $\\Gamma _{world}$ is a set of facts that correspond to the reasoning domain specification $\\mathcal {D}$ , i.e.", "the language elements described in Section REF ." ], [ "Fluent Range Specification ($\\mathcal {FR}$ ).", "   For every pair $\\left\\langle f,v \\right\\rangle \\in \\mathcal {FR}$ , LP($\\mathcal {D}$ ) contains the fact: $\\mathit {possVal}(f,v).$    For every pair $\\left\\langle f,v \\right\\rangle \\in \\mathcal {VP}$ , LP($\\mathcal {D}$ ) contains the fact: $\\mathit {knows}(f,v,0,0,0).$    For every triple $\\left\\langle a,\\lbrace \\left\\langle f^c_1,v^c_1 \\right\\rangle , \\hdots , \\left\\langle f^c_k,v^c_k \\right\\rangle \\rbrace , \\left\\langle f^e, v^e \\right\\rangle \\right\\rangle $ , LP($\\mathcal {D}$ ) contains the facts: $\\begin{aligned}act(a).\\\\\\mathit {hasEP}(a,ep).\\\\\\mathit {hasEff}(ep,f^e,v^e).\\\\\\mathit {hasCond}(ep,f^c_1,v^c_1).", "\\cdots \\mathit {hasCond}(ep,f^c_k,v^c_k).\\\\\\text{where $ep$ is an arbitrary unique identifier for the particular EP.", "}\\end{aligned}$    For every tuple $\\left\\langle \\lbrace \\left\\langle f^c_1,v^c_1 \\right\\rangle , \\hdots , \\left\\langle f^c_k,v^c_k \\right\\rangle \\rbrace , \\left\\langle f^e, v^e \\right\\rangle \\right\\rangle \\in \\mathcal {SCL}$ , LP($\\mathcal {D}$ ) contains the facts: $\\begin{aligned}\\mathit {sclHasEff}(scl,f^e,v^e).\\\\\\mathit {sclHasCond}(scl,f^c_1,v^c_1).", "\\cdots \\mathit {hasCond}(scl,f^c_k,v^c_k).\\\\\\text{where $scl$ is an arbitrary unique identifier for the particular SCL.", "}\\end{aligned}$    For every tuple $\\left\\langle a,f \\right\\rangle \\in \\mathcal {KP}$ , LP($\\mathcal {D}$ ) contains the fact: $\\begin{aligned}\\mathit {hasKP}(a,f).\\end{aligned}$    For every tuple $\\left\\langle a,\\lbrace \\left\\langle f^x_1,v^x_1 \\right\\rangle , \\hdots , \\left\\langle f^x_l,v^x_l \\right\\rangle \\rbrace \\right\\rangle \\in \\mathcal {EXC}$ , LP($\\mathcal {D}$ ) contains the integrity constraints: $\\begin{aligned}\\leftarrow & \\mathit {occ}(a,N,B), \\mathit {not~knows}(f^x_1,v^x_1,N,N,B).", "\\\\& \\vdots \\\\\\leftarrow & \\mathit {occ}(a,N,B), \\mathit {not~knows}(f^x_l,v^x_l,N,N,B).\\end{aligned}$    For every tuple $\\left\\langle l^{wg}, v^{wg} \\right\\rangle \\in \\mathcal {G}^{weak}$ , resp.", "$\\left\\langle l^{sg}, v^{sg} \\right\\rangle \\in \\mathcal {G}^{strong}$ , LP($\\mathcal {D}$ ) contains the facts: $\\begin{aligned}\\mathit {wGoal}(f^{wg},v^{wg}).", "\\\\\\mathit {sGoal}(f^{sg},v^{sg}).\\end{aligned}$" ], [ "$\\Gamma _{hpx}$ – Foundational Theory (", "The foundational domain-independent $\\mathcal {HPX}$ -theory is constituted by rules (REF ) – (REF ).", "It covers concurrency, the eight inference mechanisms, sensing, goals, plan-generation and plan optimization." ], [ "   Auxiliaries ($\\Gamma _{aux}$ )", "   The following facts are auxiliary definitions to declare step numbers and branch labels.", "$s(0..\\text{\\ttfamily maxS}).", "~~~~ br(0..\\text{\\ttfamily maxB}).$ where $\\text{\\ttfamily maxS}$ and $\\text{\\ttfamily maxB}$ are constants that denote the maximum number of steps and branches respectively.", "Rules (REF ) handle concurrency.", "$\\mathit {apply}(EP,N,N,B) \\leftarrow & \\mathit {hasEP}(A,EP), \\mathit {occ}(A,N,B).", "\\\\\\leftarrow & \\mathit {apply}(EP_1,T,N,B), \\mathit {hasEff}(EP_1,F,V), \\mathit {apply}(EP_2,T,N,B), \\\\& \\mathit {hasEff}(EP_2,F,V), EP_1 \\ne EP_2, \\mathit {possVal}(F,V).", "\\\\\\mathit {apply}(EP,T,N+1,B) \\leftarrow & \\mathit {apply}(EP,T,N,B), N < \\text{\\ttfamily maxS}.", "$ (REF ) states that all EPs of an action are applied if the action occurs.", "() is a restriction that forbids the concurrent application of similar EPs.", "Two effect propositions are similar if they have the same effect.", "This restriction is necessary for rules capturing positive postdiction () and inertia (REF ).", "Inertia is applied in both forward and backward direction.", "We model inertia for knowledge that a fluent-value pair holds and knowledge that a fluent-value pair does not hold with rules (REF ).", "We first define a notion for knowing that a fluent-value pair $\\left\\langle f,t \\right\\rangle $ is not set, i.e.", "that $\\left\\langle f, \\lnot v \\right\\rangle $ is inertial at a step $t$ .", "This is possible for two reasons; (i) if no effect proposition with the efect $\\left\\langle f,v \\right\\rangle $ is applied (REF ), (), and (ii) if an effect proposition with the effect $\\left\\langle f,v \\right\\rangle $ is applied but it is known that a condition does not hold ().", "Note that the latter is only possible because of restriction ().", "Having defined when $\\left\\langle f,v \\right\\rangle $ is not set, i.e.", "that $\\left\\langle f, \\lnot v \\right\\rangle $ is inertial, we can define that $\\left\\langle f,v \\right\\rangle $ is inertial by counting the number of possible values $v^{\\prime }$ of $f$ and assuring that for all possible values $v^{\\prime } \\ne v$ the pair $\\left\\langle f, \\lnot v^{\\prime } \\right\\rangle $ is inertial ().", "Note that to this end we employ the auxiliary predicate $\\mathit {numPossVal}/2$ to count the size of the range of a fluent (see ()).", "$\\mathit {kMaySet}(F,V,T,N,B) \\leftarrow & \\mathit {apply}(EP,T,N,B), \\mathit {hasEff}(EP,F,V).", "\\\\\\mathit {kInertial}(F,\\lnot V,T,N,B) \\leftarrow & \\mathit {not~kMaySet}(F,V,T,N,B), \\mathit {uBr}(N,B), s(T), \\mathit {possVal}(F,V) \\\\\\mathit {kInertial}(F,\\lnot V,T,N,B) \\leftarrow & \\mathit {apply}(EP,T,N,B), \\mathit {hasEff}(EP,F,V), \\\\& \\mathit {hasCond}(EP,F^{\\prime },V_1), \\mathit {knows}(F^{\\prime },V_1,T,N,B), V_1 \\ne V_2, s(T).", "\\\\\\mathit {kInertial}(F,V,T,N,B) \\leftarrow & N_V := \\lbrace \\mathit {kInertial}(F,\\lnot V^{\\prime },T,N,B) : \\mathit {possVal}(F,V^{\\prime }) : V^{\\prime } \\ne V \\rbrace , \\\\& \\mathit {uBr}(N,B), s(T), \\mathit {numPossVal}(F,N_V+1), \\mathit {possVal}(F,V).", "$ Having defined inertia of fluent-value pairs, we can define forward and backward propagation of knowledge as follows.", "$\\mathit {knows}(F,V,T,N,B) \\leftarrow & \\mathit {knows}(F,V,T-1,N,B), \\mathit {kInertial}(F,V,T-1,N,B), \\\\& T \\le N, s(T), \\mathit {possVal}(F,V).", "\\\\\\mathit {knows}(F,V,T,N,B) \\leftarrow & \\mathit {knows}(F,V,T+1,N,B), \\mathit {kInertial}(F, \\lnot V,T,N,B),\\\\& T < N, \\mathit {possVal}(F,V).", "\\\\\\mathit {knowsNot}(F,V,T,N,B) \\leftarrow & \\mathit {knowsNot}(F,V,T-1,N,B), \\mathit {kInertial}(F, \\lnot V,T-1,N,B), \\\\& T \\le N, s(T), \\mathit {possVal}(F,V).", "\\\\\\mathit {knowsNot}(F,V,T,N,B) \\leftarrow & \\mathit {knowsNot}(F,V,T+1,N,B), \\mathit {kInertial}(F,V,T,N,B), \\\\& T < N , \\mathit {possVal}(F,V).", "\\\\\\mathit {knows}(F,V,T,N,B) \\leftarrow & \\mathit {knows}(F,V,T,N-1,B), s(N).", "\\\\\\mathit {knowsNot}(F,V,T,N,B) \\leftarrow & \\mathit {knowsNot}(F,V,T,N-1,B), s(N).", "$ (REF ) defines forward propagation of knowledge that a fluent $f$ has value $v$ .", "() defines backward propagation of knowledge that a fluent $f$ has value $v$ .", "() defines forward propagation of knowledge that a fluent $f$ does not have value $v$ .", "() defines backward propagation of knowledge that a fluent $f$ does not have value $v$ .", "Rules (), () capture forward propagation of knowledge itself.", "If an agent knows that fluent $f$ has value $v$ at a step $t$ while being in state $n-1$ , then it will still have this knowledge at a step $n$ .", "Causation and Postdiction are the primary knowledge-level effects of actions (REF ).", "For their implementation we first define two auxiliary predicates $\\mathit {numKnownCond}/5$ (REF ) and $\\mathit {hasNumCond}/2$ () to count the number of (known) conditions of EPs.", "$\\mathit {numKnownCond}(EP,C,T,N,B) \\leftarrow & C := {\\mathit {knows}(F,V,T,N,B) : \\mathit {hasCond}(EP,F,V)}, \\\\&\\mathit {uBr}(N,B), \\mathit {apply}(EP,T,N,B).", "\\\\\\mathit {hasNumCond}(EP,C) \\leftarrow & C := \\lbrace \\mathit {hasCond}(EP,F,V) \\rbrace , \\mathit {hasCond}(EP,\\_,\\_).", "$ Knowledge is produced by causation if all conditions of an EP are known to hold (REF ).", "Positive postdiction generates knowledge that the conditions of an EP hold, if the effect of an EP was known not to hold before the EP is applied and if the effect is known to hold after the application of the EP ().", "Note that this implementation of positive postdiction is only valid under restriction () that forbids the concurrent application of two EPs with the same effect.", "Negative postdiction produces knowledge that a fluent $f$ does not have a value $v$ if the effect of an EP is known not to hold after the EP is applied ().", "$\\mathit {kCause}(F,V,T+1,N,B) \\leftarrow & \\mathit {apply}(EP,T,N,B), \\mathit {numKnownCond}(EP,C,T,N,B), \\\\& \\mathit {hasNumCond}(EP,C), \\mathit {hasEff}(EP,F,V), \\mathit {uBr}(N,B), \\\\& N > T. \\\\\\mathit {kPosPost}(F,V,T,N,B) \\leftarrow & \\mathit {apply}(EP,T,N,B), \\mathit {uBr}(N,B), \\mathit {hasCond}(EP,F,V), \\\\&\\mathit {hasEff}(EP,F^{\\prime },V^{\\prime }), \\mathit {knows}(F^{\\prime },V^{\\prime },T+1,N,B), \\\\& \\mathit {knowsNot}(F^{\\prime },V^{\\prime },T,N,B), \\mathit {not~knowsNot}(F,V,T,N,B), N > T. \\\\\\mathit {kNotNegPost}(F,V,T,N,B) \\leftarrow & \\mathit {apply}(EP,T,N,B), \\mathit {hasEff}(EP,F^{\\prime },V^{\\prime }), \\\\& \\mathit {knowsNot}(F^{\\prime },V^{\\prime },T+1,N,B), \\mathit {uBr}(N,B), N > T, \\\\& \\mathit {hasCond}(EP,F,V), \\mathit {hasNumCond}(EP,C+1), \\\\& \\mathit {numKnownCond}(EP,C,T,N,B), \\mathit {not~knows}(F,V,T,N,B).", "$ Rules (REF ),(),() assign knowledge generated by causation and postdiction to the $\\mathit {knows}/5$ resp.", "$\\mathit {knowsNot}/5$ predicates.", "$\\mathit {knows}(F,V,T,N,B) \\leftarrow & \\mathit {kCause}(F,V,T,N,B).", "\\\\\\mathit {knows}(F,V,T,N,B) \\leftarrow & \\mathit {kPosPost}(F,V,T,N,B).", "\\\\\\mathit {knowsNot}(F,V,T,N,B) \\leftarrow & \\mathit {kNotNegPost}(F,V,T,N,B).", "$    To define rules that generate knowledge by exclusion we first define two auxiliary rules to count the number values of a fluent that are not known (REF ) and to count the total number of possible values of a fluent () .", "$\\mathit {numKNF}(F,KN,T,N,B) \\leftarrow & KN := \\lbrace \\mathit {knowsNot}(F,V,T,N,B) : \\mathit {possVal}(F,V), \\\\& \\mathit {uBr}(N,B), s(T), \\mathit {possVal}(F,\\_).", "\\\\\\mathit {numPossVal}(F,NV) \\leftarrow & NV := \\lbrace \\mathit {possVal}(F,V)\\rbrace , \\mathit {possVal}(F,\\_).", "$ We are now ready to define the rules that generate knowledge by positive exclusion (REF ) and negative exclusion ().", "Rules (),() assign knowledge generate by exclusion to the $\\mathit {knows}/5$ (resp.", "$\\mathit {knowsNot}/5$ ) predicate.", "$\\mathit {kPosEx}(F,V,T,N,B) \\leftarrow & \\mathit {numKNF}(F,KN,T,N,B), \\mathit {numPossVal}(F,KN+1), \\\\& \\mathit {not~knowsNot}(F,V,T,N,B), \\mathit {possVal}(F,V).", "\\\\\\mathit {kNotNegEx}(F,V,T,N,B) \\leftarrow & \\mathit {knows}(F,V^{\\prime },T,N,B), V \\ne V^{\\prime }, \\mathit {possVal}(F,V^{\\prime }).", "\\\\\\mathit {knows}(F,V,T,N,B) \\leftarrow & \\mathit {kPosEx}(F,V,T,N,B).", "\\\\\\mathit {knowsNot}(F,V,T,N,B) \\leftarrow & \\mathit {kNotNegEx}(F,V,T,N,B).", "$    Sensing is modeled for contingent planning (e.g.", "[9]) purposes, i.e.", "when during plan generation a sensing action is considered, then all possible outcomes of the sensing action are accounted for in separate branches.", "Branches are generated whenever a sensing action occurs in the plan.", "Potential sensing outcomes are modeled via the $\\mathit {sRes/5}$ predicate, i.e.", "$mi{sRes}(f,v,n,b,b^{\\prime })$ denotes that in node $\\left\\langle n,b \\right\\rangle $ a sensing action occurs that assigns the value $v$ to a fluent $f$ in the child-branch $b^{\\prime }$ .", "First, we state rules (REF ) – () to denote that branch 0 is valid in the initial step, and that if no sensing action occurs in a certain node of the transition tree, then the branch is marked as valid in the successor node without branching.", "$\\mathit {uBr}(0,0).", "& \\\\\\mathit {sNextBr}(N,B1) \\leftarrow & \\mathit {sRes}(\\_,\\_,N,B1,B2).", "\\\\\\mathit {uBr}(N,B)\\leftarrow & \\mathit {uBr}(N-1,B), \\mathit {not~sNextBr}(N-1,B), s(N).", "$ Next, we generate sensing results.", "Rule REF generates the actual $\\mathit {sRes}/5$ predicates by assigning one branch to each value in the range of the sensed fluent which is not known not to be actual value of the fluent.", "However, we have to be careful not to assign potential sensing outcomes to child branches that are already used.", "This is realized with different integrity constraints.", "() states that only one sensing result can be assigned to one branch.", "() assures that no used branch (except the current branch) is assigned.", "() prohibits that if multiple sensing actions happen in different nodes, a free branch can be double assigned with different sensing outcomes.", "() is an optional constraint that assures that there is a sensing result assigned to the original branch in any case.", "This reduces the number of different possible branch-assignments and therefore the search space.", "$1\\lbrace \\mathit {sRes}(F,V,N,B_1,B_2) : br(B_2)\\rbrace 1 \\leftarrow & \\mathit {occ}(A,N,B_1), \\mathit {hasKP}(A,F), s(N), \\mathit {possVal}(F,V), \\\\& \\mathit {not~ knowsNot}(F,V,N,N,B_1).", "\\\\\\leftarrow & 2\\lbrace \\mathit {sRes}(F,\\_,N,B_1,B_2)\\rbrace ,br(B_1),br(B_2), s(N).", "\\\\\\leftarrow & \\mathit {sRes}(F,V,N,B_1,B_2), uBr(N,B_2), B_1 \\ne B_2.", "\\\\\\leftarrow & \\mathit {sRes}(F,V,N,B^P_1,B^C), \\mathit {sRes}(F^{\\prime },V^{\\prime },N,B^P_2,B^C), \\\\& B^P_1 \\ne B^P_2.", "\\\\\\leftarrow & \\lbrace \\mathit {sRes}(F,\\_,N,B,B)\\rbrace 0, occ(A,N,B), hasKP(A,F), \\\\& s(N).", "$ Having defined how sensing results are generated, we mark new branches as used (REF ) and assign the sensing result to the knowledge () .", "Finally, we restrict that not more than one fluent can be sensed at a time ().", "$\\mathit {uBr}(N,B_2) \\leftarrow & \\mathit {sRes}(F,V,N-1,B_1,B_2), s(N).", "\\\\knows(F,V,N-1,N,B_2) \\leftarrow & sRes(F,V,N-1,B_1,B_2), s(N).", "\\\\\\leftarrow & 2\\lbrace occ(A,N,B) : hasKP(A,\\_)\\rbrace , br(B), s(N).", "$ When a new branch is generated, then the knowledge of the original branch has to be transferred to the new branch.", "Towards this we implement inheritance rules that assign knowledge (REF ) as well as application of effect propositions () from the original to the child branches.", "$\\mathit {knows}(F,V,T,N,B_2) \\leftarrow & \\mathit {sRes}(\\_,\\_,N-1,B_1,B_2), \\mathit {knows}(F,V,T,N-1,B_1), N \\ge T, s(N).", "\\\\\\mathit {apply}(EP,T,N,B_2) \\leftarrow & \\mathit {sRes}(\\_,\\_,N,B_1,B_2), \\mathit {apply}(EP,T,N,B_1), N \\ge T, s(N).", "$    As discussed wrt.", "the operational semantics of $\\mathcal {HPX}_{\\mathcal {F}}$ , we compile static causal laws into EPs.", "Towards this, we define rule (REF ) that generates a new effect proposition for actions that have an effect proposition with an effect that is identical to the condition of a SCL, and hence can trigger the SCL to cause an indirect effect.", "Rule assigns the effect to the new effect proposition.", "Rule () adds the conditions from the original EP to the new EP, and adds the conditions from the SCL to the new EP.", "$\\mathit {hasEP}(A,(EP,SCL)) \\leftarrow & \\mathit {hasEP}(A,EP), \\mathit {hasEff}(EP,F^{trig},V^{trig}), \\\\& \\mathit {sclHasCond}(SCL,F^{trig},V^{trig}).", "\\\\\\mathit {hasEff}((EP,SCL), F^e,V^e) \\leftarrow & \\mathit {hasEP}(A,EP), \\mathit {hasEff}(EP,F^{trig},V^{trig}), \\\\& \\mathit {sclHasCond}(SCL,F^{trig},V^{trig}), \\mathit {sclHasEff}(SCL,F^e,V^e).", "\\\\\\mathit {hasCond}((EP,SCL), F^c,V^c) \\leftarrow & \\mathit {hasEff}(EP,F^{trig},V^{trig}), \\mathit {hasCond}(EP,F^c,V^c), \\\\& \\mathit {sclHasCond}(SCL,F^{trig},V^{trig}).", "\\\\\\mathit {hasCond}((EP,SCL), F^c,V^c) \\leftarrow & hasEff(EP,F^{trig},V^{trig}), sclHasCond(SCL,F^c,V^c), \\\\& sclHasCond(SCL,F^{trig},V^{trig}), F^{trig} \\ne F^c, V^{trig} \\ne V^c.", "$ Compiling SCLs to EPs causes knowledge to be produced as indirect action effects.", "This however does not account for knowledge that is indirectly produced by considering SCLs in combination with sensing outcomes or initial knowledge.", "This is captured by three more rules.", "(REF ) and () are auxiliary rules that count how many conditions of a SCL are known to hold, and how many conditions a SCL has in total.", "Finally, rule () generates indirect knowledge that does not emerge via causation due to EPs.", "$\\mathit {sclNumKnownCond}(SCL,C,T,N,B) \\leftarrow & C := \\lbrace \\mathit {knows}(F,V,T,N,B) : \\mathit {sclHasCond}(SCL,F,V)\\rbrace , \\\\& \\mathit {uBr}(N,B), s(T), \\mathit {sclHasEff}(SCL,\\_,\\_), T \\le N. \\\\\\mathit {sclNumCond}(SCL,C) \\leftarrow & C := \\lbrace \\mathit {sclHasCond}(SCL,F,V)\\rbrace , \\\\& \\mathit {sclHasEff(SCL,\\_,\\_)}.", "\\\\\\mathit {knows}(F,V,T,N,B) \\leftarrow & \\mathit {sclHasEff}(SCL,F,V) , \\\\& \\mathit {sclNumKnownCond}(SCL,C,T,N,B), \\\\& \\mathit {sclNumCond}(SCL,C).", "$    The ASP formalization supports both weak and strong goals.", "For weak goals there must exist one leaf where all goal literals are achieved and for strong goals the goal literals must be achieved in all leafs.", "Weak or strong goals are declared with the $\\mathit {wGoal}$ and $\\mathit {sGoal}$ predicates and defined through declarations (REF ) in the domain-specific part of an $\\mathcal {HPX}_{\\mathcal {F}}$ program.", "(REF ) defines atoms $\\mathit {notWG}(n,b)$ which denote that a weak goal is not achieved at step $n$ in branch $b$ .", "An atom $\\mathit {allWGAchieved}(n)$ reflects whether all weak goals are achieved at a step $n$ ().", "If they are not achieved at step $\\text{\\ttfamily maxS}$ , then a corresponding model is not stable ( ).", "$\\mathit {notWG}(N,B) \\leftarrow & \\mathit {wGoal}(F,V), \\mathit {uBr}(N,B), \\mathit {not~ knows}(F,V,N,N,B), \\mathit {possVal}(F,V).", "\\\\\\mathit {allWGAchieved}(N) \\leftarrow & \\mathit {not~notWG}(N,B), \\mathit {uBr}(N,B).", "\\\\\\leftarrow & \\mathit {not~allWGAchieved}(\\text{\\ttfamily maxS}).", "$ Similarly, $\\mathit {notSG}(n,b)$ denotes that a strong goal is not achived at step $n$ in branch $b$ (REF ).", "In contrast to weak goals, strong goals must be achieved in all used branches at the final step $\\text{\\ttfamily maxS}$ ().", "$\\mathit {notSG}(N,B) \\leftarrow & \\mathit {sGoal}(F,V), \\mathit {uBr}(N,B), \\mathit {not~ knows}(F,V,N,N,B), \\mathit {possVal}(F,V).", "\\\\\\leftarrow & \\mathit {notSG}(\\text{\\ttfamily maxS},B), \\mathit {uBr}(\\text{\\ttfamily maxS},B).", "$ Information about nodes where goals are not yet achieved is also generated (REF ), ().", "This is used in the plan generation part for pruning (REF )–().", "$\\mathit {notGoal}(N,B) \\leftarrow & \\mathit {notWG}(N,B).", "\\\\\\mathit {notGoal}(N,B) \\leftarrow & \\mathit {notSG}(N,B).", "$ In the generation part of the Logic Program, (REF ) and () implement sequential and concurrent planning respectively: for concurrent planning the choice rule's upper bound “1” is simply removed.In an actual implementation the LP may of course only contain one of these two choice rules, depending on which kind of planning is desired.", "$1\\lbrace \\mathit {occ}(A,N,B): act(A)\\rbrace 1 \\leftarrow & \\mathit {uBr}(N,B), \\mathit {notGoal}(N,B), N < \\text{\\ttfamily maxS}.", "\\\\1\\lbrace \\mathit {occ}(A,N,B) : act(A)\\rbrace \\leftarrow & \\mathit {uBr}(N,B), \\mathit {notWG}(N,B), N < \\text{\\ttfamily maxS}.", "$" ], [ "COMPUTATIONAL PROPERTIES", "We are interested in the number of state variables that constitute the knowledge state of an agent, and the computational worst-time complexity for the temporal projection problem.", "$n_\\mathcal {F}$ is the number of fluents, $n_\\mathcal {V}$ an upper limit for rabge size of fluents and $n_{\\mathcal {S}}$ the max.", "number of steps.", "Theorem 1 (Number of state variables) $n_{\\mathcal {F}} \\cdot n_{\\mathcal {V}} \\cdot n_{\\mathcal {S}}$ is the maximal number of epistemic state variables per h-state $\\mathbf {\\mathfrak {h}}$ , i.e.", "$|\\mathbf {\\kappa }(\\mathbf {\\mathfrak {h}})| \\le n_{\\mathcal {F}} \\cdot n_{\\mathcal {V}} \\cdot n_{\\mathcal {S}}$ .", "Within the transition function, only the eight IM functions and $sense$ (REF ) can generate triples $\\left\\langle f,v,t \\right\\rangle $ .", "Herein, $t$ is limited to $0 \\le t \\le now(\\mathbf {\\mathfrak {h}})$ , where $now(\\mathbf {\\mathfrak {h}})$ equals the number of steps $n_{\\mathcal {s}}$ .", "Since a set can not have duplicate entries and the number of fluents and possible values is finite due to language element $\\mathcal {FR}$ , the theorem holds.", "Theorem 2 (Complexity of temporal projection) Let $\\mathcal {A}$ be a set of actions and $\\mathbf {\\mathfrak {h}}$ a valid h-state.", "It holds for all $\\mathbf {\\mathfrak {h}}^{\\prime } \\in \\Psi (\\mathcal {A}, \\mathbf {\\mathfrak {h}})$ that determining whether $\\left\\langle f,v,t \\right\\rangle \\in \\mathbf {\\mathfrak {h}}^{\\prime }$ is polynomial.", "The transition function (REF calls $sense$ (REF ) and $eval$ (REF ).", "$sense$ iterates over $\\mathcal {FR}$ , $\\mathcal {KP}$ and $\\mathbf {\\kappa }(\\mathbf {\\mathfrak {h}})$ , and since $|\\mathbf {\\kappa }(\\mathbf {\\mathfrak {h}})| \\le n_{\\mathcal {F}} \\cdot n_{\\mathcal {V}} \\cdot n_{\\mathcal {S}}$ (Theorem REF ) we have that $sense$ is polynomial.", "$eval$ converges with a linear number of applications of the inference mechanisms, because due to Theorem REF there exists only a linear number of elements in the knowledge history and because no element is ever removed from the knowledge history with any of the inference mechanisms (see also Lemma B.5 in [3] for the case of boolean fluents).", "All inference mechanisms execute in polynomial time, because they only quantify over language elements (sets) from the domain specification and the knowledge history $\\mathbf {\\kappa }(\\mathbf {\\mathfrak {h}})$ , which are all of linear size wrt.", "$\\mathcal {D}$ (see also Lemma B.4 in [3]).", "Note that the number of additional EP generated by $scl2ep$ (REF ) is also polynomial wrt.", "the size of $\\mathcal {SCL}$ and $\\mathcal {EP}$ .", "Applying the transition function polynomially often (e.g.", "for determining the outcome of a conditional plan of polynomial size) does not change the complexity class." ], [ "EXAMPLE SCENARIO", "We have integrated the $\\mathcal {HPX}$ -approach as an assistance planning system for the robotic wheelchair Rolland within the smart home BAALL .", "A typical use case within this environment involves a person calling the wheelchair to bring him to a destination.", "For example, a (sub-) problem of getting on the wheelchair and driving from the bath to the corridor can be modeled within $\\mathcal {HPX}_{\\mathcal {F}}$ as follows: $Ȓ = &\\lbrace {bath},{kit}\\rbrace , B̑ = \\lbrace {true},{false}\\rbrace \\\\\\mathcal {FR} = &\\lbrace \\left\\langle wcAt, r \\right\\rangle | r \\in Ȓ \\rbrace \\cup \\lbrace \\left\\langle pAt, r \\right\\rangle | r \\in Ȓ \\rbrace \\\\& \\cup \\lbrace \\left\\langle sitting, b \\right\\rangle | b \\in B̑ \\rbrace \\cup \\lbrace \\left\\langle ab\\_sit, b \\right\\rangle | b \\in B̑ \\rbrace \\\\\\mathcal {VP} = & \\lbrace \\left\\langle pAt,bath \\right\\rangle , \\left\\langle wcAt,bath \\right\\rangle , \\left\\langle sitting, false \\right\\rangle , \\\\\\mathcal {EP} = & \\lbrace \\langle drv(bath,kit), \\lbrace \\left\\langle wcAt(bath) \\right\\rangle \\rbrace , \\left\\langle wcAt, kit \\right\\rangle \\rangle \\rbrace \\\\& \\cup \\lbrace \\langle {sit}, \\lbrace \\left\\langle wcAt, bath \\right\\rangle , \\left\\langle pAt,bath \\right\\rangle , \\\\& \\left\\langle ab\\_sit,false \\right\\rangle \\rbrace , \\left\\langle sitting \\right\\rangle \\rangle \\rbrace \\\\\\mathcal {SCL} = & \\lbrace \\left\\langle \\lbrace \\left\\langle wcAt, r \\right\\rangle , \\left\\langle sitting, {true} \\right\\rangle \\rbrace , \\left\\langle pAt ,r \\right\\rangle \\right\\rangle | r \\in Ȓ \\rbrace \\\\\\mathcal {KP} = & \\lbrace \\left\\langle senseLoc, pAt \\right\\rangle \\rbrace \\\\[-20pt]$ (REF ) declares objects of the types room (Ȓ), and boolean (B̑).", "$\\mathcal {FR}$ specifies the fluents $wcAt \\mapsto Ȓ$ and $pAt \\mapsto Ȓ$ to denote that wheelchair and person are at a location, $sitting : P̑ \\mapsto B̑$ to denote that a person can sit on a wheelchair and $ab\\_sit \\mapsto B̑$ to denote that the sitting action may be abnormal.", "() specifies that initially the person and the wheelchair are in the bath and the person is not sitting on the wheelchair.", "The effect propositions describe the $drive$ () and the $sit$ () action, where the latter is only successful if there is no abnormality.", "The SCL () states that a person will move with the wheelchair when sitting on it.", "In this setting, we would like to infer that an abnormality occurred if we observe that a person is not in the corridor after he was supposed to sit down on it and driving the wheelchair from bathroom to corridor.", "The additional EP generated by $scl2ep(\\mathcal {EP})$ (REF ) is $\\left\\langle drv, \\lbrace \\left\\langle sitting,true \\right\\rangle , \\left\\langle pAt,bath \\right\\rangle , \\left\\langle wcAt,bath \\right\\rangle \\rbrace , \\left\\langle pAt, corr \\right\\rangle \\right\\rangle $ .", "This additional EP will make it possible to perform the desired inference through the $pd^{neg}$ function.", "For the transition tree consider Example .", "The initial h-state $\\mathbf {\\mathfrak {h}}_0$ corresponds to ().", "Then the person executes the $sit$ action to sit down on the wheelchair.", "In the successor state $\\mathbf {\\mathfrak {h}}_1$ it is unknown whether the person sits on the wheelchair because it is unknown whether there was an abnormality with the sitting.", "The next state $\\mathbf {\\mathfrak {h}}_2$ results from driving to $corr$ .", "Here it is unknown where the person is located because it was unknown whether the person is sitting on the wheelchair.", "The next action is the sensing of the person's location.", "In the figure we consider only the case where the person is still located in the corridor.", "This triggers the indirect postdiction through (REF ) that the person is not sitting on the wheelchair, which in turn again is used to postdict that there must be an abnormality.", "Information about the abnormality can be used to call the care personnel.", "Figure: NO_CAPTION" ], [ "CONCLUSION", "We present an epistemic action theory $\\mathcal {HPX}_{\\mathcal {F}}$ that accounts for epistemic ramification, postdiction, and functional fluents.", "We improve the original $\\mathcal {HPX}$ , in the sense that translation rules described earlier approaches [5] are not required anymore; the domain specification can now be given entirely in terms of ASP, and is therefore not restricted to a fixed input syntax anymore.", "We have demonstrated the action planning and reasoning capabilities of our approach in the backdrop of a smart home scenario, but we would like to emphasize that many other application domains require postdictive reasoning in combination with ramifications and functional knowledge.", "Examples for such applications include narrative interpretation, continuity checking in plot writing for novels and movies, as well as forensics and criminal reasoning (e.g. ).", "These applications usually involve many unknown world properties, and epistemic action theories that are based on $\\mathcal {PWS}$ will require a number of epistemic state variables that is exponential with the number of unknown world properties.", "$\\mathcal {HPX}_{\\mathcal {F}}$ only requires a linear number of state variables to model the knowledge state of an agent, and is therefore more appropriate than $\\mathcal {PWS}$ -based approaches in many practical application domains." ], [ "ACKNOWLEDGEMENTS", "I thank Mehul Bhatt for the invaluable help with the original version of the $\\mathcal {HPX}$ theory [5]." ] ]
1403.0034
[ [ "Determining pure discrete spectrum for some self-affine tilings" ], [ "Abstract By the algorithm implemented in the paper [2] by Akiyama-Lee and some of its predecessors, we have examined the pure discreteness of the spectrum for all irreducible Pisot substitutions of trace less than or equal to $2$, and some cases of planar tilings generated by boundary substitutions due to the paper [17] by Kenyon." ], [ "Introduction", "Self-affine tilings are often studied as examples of tiling dynamics.", "Many equivalent conditions are known for the spectrum of the tiling dynamics to be pure discrete [6], but none of them is known to hold in general.", "For particular instances of tilings, there are algorithms by which one can check whether the spectrum is pure discrete.", "The overlap algorithm [22] and the balanced pair algorithm [22] are practically usable mostly in one dimension, but the potential overlap algorithm of Akiyama and Lee [3] is of practical use in all dimensions, even if the tiles have complicated geometries.", "Here we use these algorithms to check the pure discreteness of the spectrum of the tiling dynamics for special cases of self-affine tilings.", "One of these cases is the 1-dimensional irreducible Pisot substitution tilings.", "There is a long-standing conjecture [6] that these tilings have pure discrete spectrum.", "For the cases with two tiles, it is known that the conjecture is true [15], but not much is known for more than two tiles.", "Already the case with three tiles is computationally involved.", "We concentrate here on the cases with substitution matrices of small trace.", "More precisely, all inequivalent substitution matrices $M$ with $Tr(M)\\le 2$ have been generated, and for all substitutions with these matrices (446683 substitutions in total), we have checked that the spectrum is pure discrete.", "The other type of tilings we consider are the self-affine tilings constructed from the endomorphisms of free groups by Kenyon [17].", "We have looked at the cubic polynomials whose coefficients are all less than or equal to 3, except for a single case, whose computation is beyond our computer capability.", "All other examples turn out to be pure discrete.", "We also give some non pure discrete examples of self-affine tilings in §.", "The construction of the first one is due to Bandt [7].", "The second arises from 4 interval exchanges studied by Arnoux-Ito-Furukado.", "Both satisfy the Pisot family condition, so that their translation actions are not weakly mixing.", "We provide the programs in [4], [13]." ], [ "Pisot substitutions with small trace", "In this section, we wish to computationally confirm the Pisot substitution conjecture to be true for a class of simple, irreducible Pisot substitutions with three tiles.", "Consider a monoid $\\mathcal {A}^*$ over finite alphabets $\\mathcal {A}$ equipped with concatenation and write the identity as $\\epsilon $ , the empty word.", "A symbolic substitution $\\sigma $ is a non-erasing homomorphism of $\\mathcal {A}^*$ , defined by $\\sigma (a)\\in \\mathcal {A}^+=\\mathcal {A}^* \\setminus $ {$\\epsilon $ } for $a\\in \\mathcal {A}$ .", "The set $\\mathcal {A}^{\\mathbb {Z}}$ of two sided sequences is compact by the product topology of the discrete topology on $\\mathcal {A}$ .", "The substitution $\\sigma $ acts naturally on $\\mathcal {A}^{\\mathbb {Z}}$ by $\\sigma (\\dots a_{-1}a_0a_1a_2\\dots )=\\dots \\sigma (a_{-1})\\sigma (a_0)\\sigma (a_1)\\sigma (a_2)\\dots $ .", "Let $M_{\\sigma }$ be the incidence matrix $(|\\sigma (j)|_i)_{ij}$ where $i,j\\in \\mathcal {A}$ .", "Here $|w|_j$ is the cardinality of $j$ appearing in a word $w\\in \\mathcal {A}^*$ .", "Denote by $\\chi _{\\sigma }$ the characteristic polynomial of $M_{\\sigma }$ .", "The substitution $\\sigma $ is primitive if $M_{\\sigma }$ is primitive and it is irreducible if $\\chi _{\\sigma }$ is irreducibleWe always assume the irreducibility of $M_{\\sigma }$ in the sense of Perron-Frobenius theory.", "So the irreducibility in this article is for $\\chi _{\\sigma }$ .. A Pisot number is an algebraic integer $\\lambda > 1$ whose all the other algebraic conjugates of $\\lambda $ lie strictly inside the unit circle.", "If the Perron-Frobenius root of $M_{\\sigma }$ is a Pisot number then we say that $\\sigma $ is a Pisot substitution.", "A word $w\\in \\mathcal {A}^*$ is admissible if there exist $k\\in \\mathbb {N}$ and $a\\in \\mathcal {A}$ such that $w$ is a subword of $\\sigma ^k(a)$ .", "Let $X_{\\sigma }=\\lbrace (a_n)_{n\\in \\mathbb {Z}}\\in \\mathcal {A}^{\\mathbb {Z}}\\ |\\ a_{k}a_{k+1}\\dots a_{\\ell }\\text{ is admissible for all } k, \\ell \\text{ with } k<\\ell \\rbrace .$ Then $(X_{\\sigma }, s)$ forms a topological dynamical system where $s$ is the shift map defined by $s( (a_n)_{n\\in \\mathbb {Z}})=(a_{n+1})_{n\\in \\mathbb {Z}}$ .", "By the primitivity, the system is minimal and uniquely ergodic with the unique invariant measure $\\mu $ .", "Therefore we can discuss the spectrum of the unitary operator $U_{\\sigma }$ acting on $L^2(X_{\\sigma }, \\mu )$ for which $(U_{\\sigma }(f))(x)=f(s(x))$ .", "The substitution $\\sigma $ has pure discrete dynamical spectrum if the spectral measure associated to $U_{\\sigma }$ consists only of point spectra, or equivalently, the linear span of eigenfunctions is dense in $ L^2(X_{\\sigma },\\mu )$ .", "It is conjectured [6] that this $\\mathbb {Z}$ -action by $U_{\\sigma }$ is pure discrete if $\\sigma $ is an irreducible Pisot substitution – so called Pisot substitution conjecture.", "For the primitive substitution $\\sigma $ , we can also discuss a natural suspension of $(X_{\\sigma },s)$ by associating to each letter the length determined by the associated entries of the left eigenvector of $M_{\\sigma }$ .", "The sequence then defines a tiling of $\\mathbb {R}$ with an inflation matrix $Q=(\\beta )$ , where $\\beta $ is the Perron-Frobenius root of $M_{\\sigma }$ .", "This gives a tiling dynamical system $(X_{\\mathcal {T}},\\mathbb {R})$ which is also minimal and uniquely ergodic.", "It is known [8] that if $\\sigma $ is an irreducible Pisot substitution, this $\\mathbb {R}$ -action on $(X_{\\mathcal {T}},\\mathbb {R})$ is pure discrete if and only if the $\\mathbb {Z}$ -action on $(X_{\\sigma },s)$ is pure discrete.", "The following assertion may be known but we did not find it in the literature.", "It gives a bound for the number of irreducible primitive substitutions over $m$ letters.", "Lemma 2.1 Let $B > 0$ .", "The cardinality of the set of primitive substitutions over $m$ letters, whose Perron Frobenius root is less than or equal to $B$ , is less than $m^{m^4 B^{2(m-1)^2+2}}$ .", "Let $\\sigma $ be a substitution over $m$ letters whose incidence matrix is $M_{\\sigma }$ .", "Then there is a positive integer $k$ that $M_{\\sigma }^k=M_{\\sigma ^k}$ is a positive matrix.", "In fact, one can take $k \\le (m-1)^2+1$ for all primitive matrix $M_{\\sigma }$ (see [14], [21]).", "Denote the characteristic polynomial by $\\Phi _{\\sigma ^k}(x)=x^m-c_{m-1}x^{m-1}-c_{m-2}x^{m-2}-\\dots -c_0$ .", "Then we have a bound $|c_{m-i}|< {m \\atopwithdelims ()i} \\beta ^{ki}$ , because other roots of $\\Phi _{\\sigma ^k}$ are less than $\\beta ^k$ in modulus.", "Our aim is to show that there are only finitely many matrices $M_{\\sigma }^k=(a_{ij})$ .", "Indeed this implies that all entries of $M_{\\sigma }$ are bounded by $\\max \\lbrace a_{ij} \\ | \\ i, j \\le m \\rbrace $ , since otherwise there is an entry of $M_{\\sigma ^k}$ larger than this bound.", "From $0\\le \\sum _i a_{ii}= c_{m-1}\\le m \\beta ^k$ , we have $a_{ii}< m \\beta ^k$ and it suffices to show that $a_{ij}$ is bounded for $i\\ne j$ .", "Using ${m \\atopwithdelims ()2}\\beta ^{2k}> c_{m-2}= \\sum _{i<j} a_{ij}a_{ji} - \\sum _{i<j} a_{ii}a_{jj}$ and $0<\\sum _{i<j} a_{ii}a_{jj} = \\frac{1}{2} \\left(\\left(\\sum _{i} a_{ii} \\right)^2- \\sum _{i} a_{ii}^2 \\right) \\le \\frac{m^2}{2} \\beta ^{2k}$ we see that $0\\le a_{ij}a_{ji}< \\sum _{i<j} a_{ij}a_{ji} \\le {m \\atopwithdelims ()2} \\beta ^{2k} + \\frac{m^2}{2} \\beta ^{2k}\\le m^2 \\beta ^{2k}$ Since $a_{ji} \\in \\mathbb {N}$ , we have the bound $a_{ij}\\le m^2 B^{2k}$ .", "Thus we have $b_{ij} \\le m^2 B^{2k}$ where $M_{\\sigma }=(b_{ij})$ and the number of possible $\\sigma (i)$ 's for each $i$ is bounded by the multinomial coefficient: ${m^3 B^{2k} \\atopwithdelims ()m^2 B^{2k}, m^2 B^{2k}, \\dots , m^2 B^{2k}}\\le m^{m^3 B^{2k}}\\le m^{m^3 B^{2(m-1)^2+2}}$ which gives the required bound.", "Although Lemma REF gives a bound for the number of substitutions whose Perron-Frobenius root is less than $B$ , it is too large to be useful in practice.", "In the sequel, we deduce a practical estimate by the property of the Pisot number to narrow the range of computation.", "If $\\sigma $ is a Pisot substitution of degree $d$ whose incidence matrix has a Pisot number $\\beta $ as the Perron-Frobenius root, we have $\\beta -(d-1)< {\\rm Tr} (M_{\\sigma }) < \\beta +(d-1)$ .", "Thus it is meaningful to check the Pisot conjecture for irreducible substitutions whose incidence matrix $M_{\\sigma }$ has small trace with a fixed degree.", "Our first result is the following proposition.", "Proposition 2.2 Let $\\sigma $ be a primitive, irreducible, cubic Pisot substitution with ${\\rm Tr}(M_{\\sigma }) \\le 2$ .", "Then the spectrum of $\\sigma $ is pure discrete.", "Note that ${\\rm Tr}(M_{\\sigma })\\ge 0$ since $M_{\\sigma }$ is a non-negative matrix.", "Let $ x^3-px^2-q x-r, \\ \\ \\ \\mbox{where} \\ p, q, r \\in \\mathbb {Z}$ be the characteristic polynomial of $\\sigma $ .", "It is the minimal polynomial of a cubic Pisot number if and only if ${\\rm max}\\lbrace 2-p-r,\\ r^2-{\\rm sign}(r)(1+pr)+1\\rbrace \\le q \\le p+r \\text{ and } r\\ne 0$ holds (see [2]).", "We see $p = {\\rm Tr}(M_{\\sigma }) \\ge 0$ .", "Let $M_{\\sigma }:=\\begin{pmatrix}k_1& a & b\\\\c& k_2 & d\\\\e& f & k_3\\\\\\end{pmatrix}$ be the incidence matrix of the primitive substitution $\\sigma $ .", "Then the coefficients $p, q, r$ of the characteristic polynomial (REF ) can be written as $p&=&k_1+k_2+k_3 \\ \\ \\mbox{and} \\ \\ p\\ge 0, \\\\q&=&ac+be+df-k_1k_2-k_2k_3-k_1k_3, \\\\r&=&k_1k_2k_3+dcf+ade-dfk_1-bek_2-ack_3.$ We claim that $a,b,c,d,e,f$ are not greater than $L$ , where $L := \\max \\lbrace q+k_1k_2+k_2k_3+k_3k_1,\\ r-k_1k_2k_3 + \\max \\lbrace k_1,k_2,k_3\\rbrace (q+k_1k_2+k_2k_3+k_1k_3) \\rbrace $ using the idea of Lemma REF .", "Note that by the primitivity, none of six vectors $(a,b)$ , $(c,d)$ , $(e,f)$ , $(c,e)$ , $(a,f)$ , $(b,d)$ is $(0,0)$ .", "By symmetry, we only prove this bound (REF ) for $a$ .", "Consider $q$ and $r$ as a linear polynomial on $a$ .", "Then the leading coefficients are $c$ and $de-ck_3$ and we see that either $c\\ne 0$ or $de-ck_3\\ne 0$ holds.", "If $c\\ne 0$ , then the formula for $q$ gives $a\\le ac \\le q+k_1k_2+k_2k_3+k_1k_3$ .", "If $c=0$ then, the formula for $q$ implies $be+df\\le q+k_1k_2+k_2k_3+k_1k_3$ and the formula for $r$ gives $a\\le ade \\le r-k_1k_2k_3 + df k_1+ be k_2 \\le r-k_1k_2k_3 + \\max \\lbrace k_1,k_2\\rbrace (q+k_1k_2+k_2k_3+k_1k_3).$ So the claim follows.", "Since $k_1,k_2,k_3$ are non-negative integers, from (REF ) one can also deduce $a,b,c,d,e,f \\le r+p(q+p^2)$ .", "In the case $p=0$ we easily have $q=r=1$ from (REF ).", "By using the bound of $a,b,c,d,e,f$ , there are 6 substitutions matrices with the characteristic polynomial $x^3-x-1$ .", "However, these matrices form a single orbit under the group $S_3$ which permutes the symbols, so that only one matrix with 2 substitutions needs to be checked.", "Moreover, there is a further symmetry which can be taken into account in order to avoid duplicate work.", "Let $\\xi :\\mathcal {A}^* \\rightarrow \\mathcal {A}^*$ be the `mirror' map, i.e., $\\xi (a_1a_2\\dots a_{n-1}a_n)=a_na_{n-1}\\dots a_2a_1$ .", "For a given substitution $\\sigma $ , we define $\\overline{\\sigma }$ by $\\overline{\\sigma }(a)=\\xi (\\sigma (a))$ for $a\\in \\mathcal {A}$ .", "Then the dynamical systems $X_{\\sigma }$ and $X_{\\overline{\\sigma }}$ are clearly isomorphic and we only have to compute one of them.", "By using this symmetry, the two substitutions of the $p=0$ case reduce to one, which turns out to be pure discrete.", "Next we study the case $p>0$ .", "Changing the order of letters, for $p=1$ we may assume that the incidence matrix $M_{\\sigma }$ is of the form: $\\begin{pmatrix}1& a & b\\\\c& 0 & d\\\\e& f & 0\\\\\\end{pmatrix} \\text{ with } a\\ge b$ and for $p=2$ we have $\\begin{pmatrix}2& a & b\\\\c& 0 & d\\\\e& f & 0\\\\\\end{pmatrix} \\text{ with } a\\ge b, \\text{ or }\\begin{pmatrix}1& a & b\\\\c& 1 & d\\\\e& f & 0\\\\\\end{pmatrix} \\text{ with } a\\ge c.$ For each case we can deduce bounds for $a,b,c,d,e,f$ from the inequality (REF ).", "The list of matrices satisfying these bounds is finite, but still contains some pairs which are equivalent under a permutation of the symbols.", "Only one matrix of each such pair is retained.", "For any substitution matrix $M$ in the list, there are only finitely many substitutions $\\sigma $ with $M_{\\sigma }=M$ (of which we keep only one per mirror pair), so that we obtain a finite list of irreducible Pisot substitutions to be checked.", "Then we use the `potential overlap algorithm' of [3] and a new implementation [13] of the classical overlap algorithm [22], [23] to check all the substitutions in the list.", "Moreover, since the substitutions in the list are irreducible (Prop.", "1.2.8 of [11]), we may check by the balanced pair algorithm [22] as well.", "Up to symbol relabelling and mirror symmetry as above, there are 7377 irreducible cubic Pisot unit substitutions, as given in Table 1, and all of them are pure discrete.", "The number of non-unit substitutions, especially with trace $p=2$ , is much larger, and on average each computation is much harder.", "These non-unit cases could mostly be checked only with a new implementation [13] of the classical overlap algorithm in the GAP language [12], in which a special effort has been made to keep the memory requirements small.", "Also the non-unit substitutions turned out to be pure discrete.", "The results are summarized in Table 1.", "Table: The columns labelled M σ M_{\\sigma } and σ\\sigma contain the numberof inequivalent substitution matrices and substitutions, respectively.In the trace 2 case, these numbers are split into the contributionsfrom matrices with diagonal (2,0,0)(2,0,0) and (1,1,1)(1,1,1), respectively.In principle, the above method of listing all cubic Pisot substitutions with a fixed trace can be used also for matrices $M_\\sigma $ with ${\\rm Tr}(M_{\\sigma })= 3$ .", "However, for this case there would be far too many substitutions to be checked, which currently seems beyond reach.", "Experiments with an implementation of the balanced pair algorithm in GAP [13], similar to that of the classical overlap algorithm [22], [23], suggests that the latter has an advantage for more complex substitutions.", "This is likely due to the balanced pairs not only growing in number, but also in length, which requires more memory to store them, but especially also more work to compare them.", "The overlap types needed in the overlap algorithm, on the other hand, grow only in number, and each of them requires only a fixed (small) amount of memory." ], [ "Substitution tilings in $\\mathbb {R}^2$", "In this section, we check the pure discreteness of self-affine tilings constructed by Kenyon [17].", "Before explaining his construction, we start with general notations.", "A tile in $\\mathbb {R}^2$ is defined as a pair $T=(A,i)$ where $A=\\mbox{\\rm supp}(T)$ (the support of $T$ ) is a compact set in $\\mathbb {R}^2$ which is the closure of its interior, and $i=l(T)\\in \\lbrace 1,\\ldots ,m\\rbrace $ is the color of $T$ .", "We say that a set $P$ of tiles is a patch if the number of tiles in $P$ is finite and the tiles of $P$ have mutually disjoint interiors.", "A tiling of $\\mathbb {R}^2$ is a set ${\\mathcal {T}}$ of tiles such that $\\mathbb {R}^2 = \\bigcup \\lbrace \\mbox{\\rm supp}(T) : T \\in {\\mathcal {T}}\\rbrace $ and distinct tiles have disjoint interiors.", "Let ${\\mathcal {A}}= \\lbrace T_1,\\ldots ,T_m\\rbrace $ be a finite set of tiles in $\\mathbb {R}^2$ such that $T_i=(A_i,i)$ ; we will call them prototiles.", "Denote by ${\\mathcal {P}}_{{\\mathcal {A}}}$ the set of non-empty patches.", "A substitution is a map $\\Omega : {\\mathcal {A}}\\rightarrow {\\mathcal {P}}_{{\\mathcal {A}}}$ with a $2 \\times 2$ expansive matrix $Q$ if there exist finite sets ${\\mathcal {D}}_{ij}\\subset \\mathbb {R}^2$ for $i,j \\le m$ such that $\\Omega (T_j)=\\lbrace u+T_i:\\ u\\in {\\mathcal {D}}_{ij},\\ i=1,\\ldots ,m\\rbrace $ with $ Q A_j = \\bigcup _{i=1}^m ({\\mathcal {D}}_{ij}+A_i) \\ \\ \\ \\mbox{for} \\ j\\le m.$ Here all sets in the right-hand side must have disjoint interiors; it is possible for some of the ${\\mathcal {D}}_{ij}$ to be empty.", "We say that ${\\mathcal {T}}$ is a substitution tiling if ${\\mathcal {T}}$ is a tiling and $\\Omega ({\\mathcal {T}}) = {\\mathcal {T}}$ with some substitution $\\Omega $ .", "We say that ${\\mathcal {T}}$ has finite local complexity (FLC) if $\\forall \\ R > 0$ , $\\exists $ finitely many translational classes of patches whose support lies in some ball of radius $R$ .", "A tiling ${\\mathcal {T}}$ is repetitive if for any compact set $K \\subset \\mathbb {R}^2$ , $\\lbrace t \\in \\mathbb {R}^2 : {\\mathcal {T}}\\cap K = (t + {\\mathcal {T}}) \\cap K\\rbrace $ is relatively dense.", "A repetitive fixed point of a primitive substitution with FLC is called a self-affine tiling.", "Let $\\lambda >1$ be the Perron-Frobenius eigenvalue of the substitution matrix $S$ .", "Let $ D =\\lbrace \\lambda _1,\\ldots ,\\lambda _{d}\\rbrace $ be the set of (real and complex) eigenvalues of $Q$ .", "We say that $Q$ (or the substitution $\\Omega $ ) fulfills the Pisot family property if, for every $\\lambda \\in D$ and every Galois conjugate $\\lambda ^{\\prime }$ of $\\lambda $ , $\\lambda ^{\\prime } \\notin D$ , then $|\\lambda ^{\\prime }| < 1$ ." ], [ "Endomorphisms of free group", "Generalizing the idea of Dekking [9], [10], Kenyon [17] introduced a class of self-similar tilings generated by the endomorphisms of a free group over three letters $a,b,c$ : $\\theta (a)&=&b\\\\\\theta (b)&=&c\\\\\\theta (c)&=&c^{p}a^{-r}b^{-q},$ where $p,q\\ge 0, r\\ge 1$ are integers for which $x^3-p x^2+qx+r$ has exactly two roots $\\lambda _1, \\lambda _2$ with modulus greater than one.Kenyon [17] studied the cases when $\\lambda _i$ are complex numbers.", "The letters $a,b,c$ are identified with vectors $(1,1), (\\lambda _1,\\lambda _2)$ , $(\\lambda _1^2,\\lambda _2^2)\\in \\mathbb {R}^2$ respectively if $\\lambda _1$ , $\\lambda _2$ are real numbers, and with $1, \\lambda _1, \\lambda _1^2 \\in \\mathbb {C}$ if they are complex conjugates.", "The endomorphism $\\theta $ acts naturally on the boundary word $ aba^{-1}b^{-1},aca^{-1}c^{-1}, bcb^{-1}c^{-1},$ which represent three fundamental parallelograms, and gives a substitution rule on the parallelograms.", "The associated tile equations are $Q T_1&=&T_2\\\\Q T_2&=&\\left(\\bigcup _{i=1}^q (T_2+ p c -i b -r a)\\right)\\cup \\left(\\bigcup _{i=1}^r (T_3+p c - i a) \\right) \\\\Q T_3&=&\\left(\\bigcup _{i=1}^r (T_1 +p c - i a) \\right) \\cup \\left(\\bigcup _{i=1}^{p-1} (T_2 + i c) \\right)$ where $Q$ is either $\\begin{pmatrix} \\lambda _1 & 0 \\cr 0 & \\lambda _2\\end{pmatrix}$ or $\\lambda _1$ depending on whether $\\lambda _1$ is real or complex, respectively.", "Figure: TilesIt is known [18], [19] that if the expansion map $Q$ of a self-affine tiling in $\\mathbb {R}^2$ is diagonalizable and the tiling has pure discrete dynamical spectrum, then $Q$ should fulfill the Pisot family property.", "So we are interested in considering self-affine tilings with the Pisot family property on the expansion map $Q$ .", "For this construction, we require that two roots of the polynomial $x^3-p x^2+qx+r$ are greater than one, and one root is smaller than one in modulus.", "In this case, we can note that there are no roots on the unit circle.", "We adapt the Schur-Cohn criterion (see [2] or [20]), which says that the number of roots within the unit circle coincides with the number of sign changes of the following sequence: $ 1, \\Delta _1, \\Delta _2, \\Delta _3, $ where $\\Delta _1 &=& - \\left| \\begin{array}{rr}1 & r \\\\r & 1\\end{array}\\right|= (r-1) (r+1), \\\\\\Delta _2 &=& \\left| \\begin{array}{rrrr}1 & -p & r & 0 \\\\0 & 1 & q & r \\\\r & q & 1 & 0 \\\\0 & r & -p & 1\\end{array}\\right|= -\\left(p r+q-r^2+1\\right) \\left(p r+q+r^2-1\\right), \\\\\\Delta _3 &=& -\\left| \\begin{array}{rrrrrr}1 & -p & q & r & 0 & 0 \\\\0 & 1 & -p & q & r & 0 \\\\0 & 0 & 1 & -p & q & r \\\\r & q & -p & 1 & 0 & 0 \\\\0 & r & q & -p & 1 & 0 \\\\0 & 0 & r & q & -p & 1\\end{array}\\right| \\\\&=& (p-q-r-1)(p+q-r+1) \\left(p r+q+r^2-1\\right)^2,$ if all entries are non-zero.", "Notice that it cannot be that $p = q = 0$ , otherwise all the roots of the polynomial have the same modulus.", "Thus from $p, q \\ge 0$ and $r \\ge 1$ , $p r+q+r^2-1>0$ .", "So the signs come from $1, (r-1) (r+1), -(p r+q-r^2+1), (p-q-r-1)(p+q-r+1)$ The last term is not zero, since $\\pm 1$ cannot be a root of $x^3-px^2+qx+r$ .", "When $r=1$ , the second term vanishes and the third term is negative (because $p$ or $q$ is positive).", "Since the roots of the polynomial are continuous with respect to the coefficients, the small perturbation of $r$ does not change the number of roots inside/outside of the unit circle.", "Therefore we may assume that the second coefficients are non-zero and use the Schur-Cohn criterion (c.f.", "[2]).", "As a result, the number of zeroes within the unit circle is 1 when $(p-q-r-1)(p+q-r+1)<0$ and it is 2 when $(p-q-r-1)(p+q-r+1)>0$ .", "If $r>1$ , then the second term is positive.", "In this case, applying the small perturbation argument when the third term vanishes, the number of zeroes within the unit circle is 1 when $(p-q-r-1)(p+q-r+1)<0$ and 0 or 2 when $(p-q-r-1)(p+q-r+1)>0$ .", "Overall we obtain a unified conclusion that the number of zeroes within the unit circle is 1 if and only if $(p-q-r-1)(p+q-r+1)<0$ .", "Our tiling exists when $|p-r|<q+1$ .", "For $\\max \\lbrace p,q,r\\rbrace \\le 3$ , there are 34 cases: $\\lbrace 0, 1, 1\\rbrace , \\lbrace 0, 2, 1\\rbrace , \\lbrace 0, 2, 2\\rbrace , \\lbrace 0, 3, 1\\rbrace , \\lbrace 0, 3, 2\\rbrace , \\lbrace 0, 3, 3\\rbrace ,\\lbrace 1, 0, 1\\rbrace , \\lbrace 1, 1, 1\\rbrace , \\lbrace 1, 1, 2\\rbrace ,$ $\\lbrace 1, 2, 1\\rbrace , \\lbrace 1, 2, 2\\rbrace , \\lbrace 1, 2, 3\\rbrace , \\lbrace 1, 3, 1\\rbrace , \\lbrace 1, 3, 2\\rbrace , \\lbrace 1, 3, 3\\rbrace ,\\lbrace 2, 0, 2\\rbrace , \\lbrace 2, 1, 1\\rbrace , \\lbrace 2, 1, 2\\rbrace ,$ $\\lbrace 2, 1, 3\\rbrace , \\lbrace 2, 2, 1\\rbrace , \\lbrace 2, 2, 2\\rbrace , \\lbrace 2, 2, 3\\rbrace , \\lbrace 2, 3, 1\\rbrace , \\lbrace 2, 3, 2\\rbrace ,\\lbrace 2, 3, 3\\rbrace , \\lbrace 3, 0, 3\\rbrace , \\lbrace 3, 1, 2\\rbrace ,$ $\\lbrace 3, 1, 3\\rbrace , \\lbrace 3, 2, 1\\rbrace , \\lbrace 3, 2, 2\\rbrace , \\lbrace 3, 2, 3\\rbrace , \\lbrace 3, 3, 1\\rbrace , \\lbrace 3, 3, 2\\rbrace ,\\lbrace 3, 3, 3\\rbrace $ We determined the spectral type of the tiling dynamical systems except for the case $\\lbrace 3,0,3\\rbrace $ .", "All computed systems admit an overlap coincidence, and thus have a pure discrete spectrum.", "In the remaning case $\\lbrace 3,0,3\\rbrace $ , the tiles are very thin, so that the number of initial overlaps seems to be beyond the capability of our program." ], [ "Examples with non pure discrete spectrum", "In this section, we wish to give two intriguing examples of self-affine tilings whose dynamical spectrum is not pure discrete.", "Both tiling dynamics are not weakly mixing and therefore there exist non-trivial eigenvalues of their translation actions.", "Bandt discovered a non-periodic tiling in [7], whose setting comes from crystallographic tiles, and called it the fractal chair tiling.", "The tile satisfies the following set equation $- I \\omega \\sqrt{3} A = A \\cup (A+1) \\cup (\\omega A +\\omega )$ where $\\omega =(1+\\sqrt{-3})/2$ is the 6-th root of unity.", "It is called 3-rep-tile, because it is a non-overlapping union of three similar contracted copies, i.e., the associated iterated function system satisfies the open set condition [7].", "Figure: Fractal chair tilingApplying the substitution rule, we obtain a tiling of the plane by six tiles $T_i= \\omega ^i A\\ (i=0,1,2,3,4,5)$ and their translates as given in Figure REF .", "This tiling is non-periodic, i.e., the only translation which sends the tiling exactly to itself is the zero vector.", "We can confirm that this tiling is repetitive and the corresponding expansion map satisfies the Pisot family property.", "So we can apply the potential overlap algorithm from [3].", "Our program shows that the fractal chair tiling is not purely discrete.", "The overlap graph with multiplicity contains a strongly connected component of spectral radius 3, being equal to the spectral radius of the substitution matrix, which does not lead to a coincidence.", "In fact, the existence of such a component is shown in [3] to be equivalent to having non-pure discrete spectrum.", "Geometrically this means that there is a finite set $C$ of overlaps not containing any coincidence, which is mapped to itself under the substitution.", "We call such a set $C$ a non-coincident component.", "We visualize in Figure REF how each overlap in the non-coincident component does not lead to a coincidence.", "Each figure represents an overlap of two fractal chair tiles, and the support of the overlap is depicted in thick color.", "The destinations of outgoing arrows show that we obtain three overlaps from each overlap by substitution.", "Figure: Non-coincident component of fractal chair tilingAs another example, the following substitution $\\nonumber 1&\\rightarrow & 1241224,\\\\2&\\rightarrow & 1224,\\\\\\nonumber 3&\\rightarrow & 1243334,\\\\\\nonumber 4&\\rightarrow & 124334$ with the characteristic polynomial $(x^2-3x+1)(x^2-6x+1)$ arose in the study of self-inducing 4 interval exchanges by P. Arnoux, S. Ito and M. Furukado [16].", "As it is a Pisot substitution, the natural suspension tiling satisfies the Pisot family condition and our potential overlap algorithm readily applies.", "It is of interest to check this non-weakly mixing system, as it is shown by [5] that a generic interval exchange transformation which is not a rotation is weakly mixing.", "Our algorithm shows that its suspension tiling dynamics is not pure discrete.", "Although there are several known reducible Pisot substitutions which are not pure discrete, this example is noteworthy in the sense that the non-coincident component is much more intricate than in other known examples.", "We describe it by another substitution on the 12 letters: $\\nonumber a&\\rightarrow & afdeEafdc, \\\\\\nonumber b&\\rightarrow & fBCFbcfdeE, \\\\\\nonumber c&\\rightarrow & afdc, \\\\\\nonumber A&\\rightarrow & AFDEeAFDC, \\\\\\nonumber B&\\rightarrow & FbcfBCFDEe, \\\\C&\\rightarrow & AFDC, \\\\\\nonumber d&\\rightarrow & FbcfdeE, \\\\\\nonumber D&\\rightarrow & fBCFDEe, \\\\\\nonumber e&\\rightarrow & afdeEe, \\\\\\nonumber E&\\rightarrow & AFDEeE, \\\\\\nonumber f&\\rightarrow & fBC, \\\\\\nonumber F&\\rightarrow & Fbc$ The associated tile equation of this substitution gives the non-coincident component of the suspension tiling corresponding to (REF ).", "In other words, (REF ) is the exact analogue of Figure REF , once we associate the intervals of canonical lengths given by the left eigenvector of the substitution matrix.", "Since the original tiling is a factor of this new suspension tiling, the system is again not pure discrete.", "The above two substitution examples have reducible characteristic polynomials.", "So the assumption of the Pisot substitution conjecture in higher dimensions of [1] does not hold." ], [ "Acknowledgment", "We are grateful to P. Arnoux, Sh.", "Ito and M. Furukado for permitting us to include the last example for the study of its spectrum.", "This work was supported by the National Research Foundation of Korea(NRF) Grant funded by the Korean Government(MSIP)(2014004168), the Japanese Society for the Promotion of Science (JSPS), Grant in aid 21540012, and the German Research Foundation (DFG) through the CRC 701 Spectral Structures and Topological Methods in Mathematics.", "The first author is partially supported by the TÁMOP-4.2.2.C-11 /1/KONV-2012-0001 project.", "(The project is implemented through the New Hungary Development Plan, co-financed by the European Social Fund and the European Regional Development Fund.)", "The third author is grateful for the support of Korea Institute for Advanced Study(KIAS) for this research.", "a: Institute of Mathematics, University of Tsukuba, 1-1-1 Tennodai,        Tsukuba, Ibaraki, Japan (zip:350-0006); b: Faculty of Mathematics, Bielefeld University,        Postfach 100131, 33501 Bielefeld, Germany; c: Dept.", "of Math.", "Edu., Catholic Kwandong University, 24, 579 Beon-gil, Beomil-ro, Gangneung,        Gangwon-do, 210-701 Republic of Korea;" ] ]
1403.0362
[ [ "Stacks and sheaves of categories as fibrant objects" ], [ "Abstract We show that the category of categories fibred over a site is a generalized Quillen model category in which the weak equivalences are the local equivalences and the fibrant objects are the stacks, as they were defined by J. Giraud.", "The generalized model category restricts to one on the full subcategory whose objects are the categories fibred in groupoids.", "We show that the category of sheaves of categories is a model category that is Quillen equivalent to the generalized model category for stacks and to the model category for strong stacks due to A. Joyal and M. Tierney." ], [ "Introduction", "The idea that stacks are the fibrant objects of a model category was developed by A. Joyal and M. Tierney in [19] and by S. Hollander in [15].", "The former paper uses internal groupoids and categories in a Grothendieck topos instead of fibred categories, and the latter only considers categories fibred in groupoids.", "The fibrant objects of the Joyal-Tierney model category are called strong stacks (of groupoids or categories), and the fibrant objects of Hollander's model category are the stacks of groupoids.", "Using some elaborate results from the homotopy theory of simplicial presheaves on a site, Hollander shows that her model category is Quillen equivalent to the model category for strong stacks of groupoids.", "The purpose of this paper is to extend Hollander's work to general stacks and to show that the category of internal categories in a Grothendieck topos admits another model category structure that is Quillen equivalent to the model category for strong stacks of categories.", "Our approach is different from both [15] and [19], and it was entirely inspired by J. Giraud's book [11].", "In fact, the influence of Giraud's work on ours cannot be overestimated.", "Concerning general stacks, we give a realization of the thought that parts of Giraud's presentation of the theory of stacks [11] hint at a connection with left Bousfield localizations of model categories as presented by P.S.", "Hirschhorn [14].", "In more detail, let $E$ be a site, that is, a category $E$ equipped with a Grothendieck topology and let Fib$(E)$ be the category of fibred categories over $E$ and cartesian functors between them.", "Let $\\mathcal {C}$ be the class of maps $R\\subset E_{/S}$ of Fib$(E)$ , where $S$ ranges through the objects of $E$ , $E_{/S}$ is the category of objects of $E$ over $S$ and $R$ is a covering sieve (or, refinement) of $S$ .", "Then Giraud's definition of stack resembles that of a $\\mathcal {C}$ -local object and his characterization of bicovering (bicouvrant in French) maps resembles the $\\mathcal {C}$ -local equivalences of [14].", "The bicovering maps are better known under the name `local equivalences'.", "The realization goes as follows.", "In order to deal with the absence of all finite limits and colimits in Fib$(E)$ we introduce, following a suggestion of A. Joyal, the notion of generalized model category (see Definition 8).", "Many concepts and results from the theory of model categories can be defined in the same way and have an exact analogue for generalized model categories.", "We disregard that $E$ has a topology and we show that Fib$(E)$ is naturally a generalized model category with the weak equivalences, cofibrations and fibrations defined on the underlying functors (see Theorem 13).", "Then we show that `the left Bousfield localization of Fib$(E)$ with respect to $\\mathcal {C}$ exists', by which we mean that there is a generalized model category structure on Fib$(E)$ having the bicovering maps as weak equivalences and the stacks over $E$ as fibrant objects (see Theorem 29).", "We call this generalized model category the generalized model category for stacks over $E$ and we denote it by Champ$(E)$ .", "To construct Champ$(E)$ we make essential use of the functorial construction of the stack associated to a fibred category (or, stack completion) and some of its consequences [11], and of a special property of bicovering maps (see Lemma 33).", "We adapt the method of proof of the existence of Champ$(E)$ to show that Fibg$(E)$ , the full subcategory of Fib$(E)$ whose objects are the categories fibred in groupoids, is a generalized model category in which the weak equivalences are the bicovering maps and the fibrant objects are the stacks of groupoids over $E$ (see Theorem 46).", "Concerning internal categories in a Grothendieck topos, let $\\widetilde{E}$ be the category of sheaves on $E$ .", "We show that the category Cat$(\\widetilde{E})$ of internal categories and internal functors in $\\widetilde{E}$ (or, sheaves of categories) is a model category that is Quillen equivalent to Champ$(E)$ (see Theorem 48).", "We denote this model category by Stack$(\\widetilde{E})_{proj}$ .", "The fibrant objects of Stack$(\\widetilde{E})_{proj}$ are the sheaves of categories that are taken to stacks by the Grothendieck construction functor.", "To construct Stack$(\\widetilde{E})_{proj}$ we make essential use of the explicit way in which Giraud constructs the stack associated to a fibred category—a way that highlights the role of sheaves of categories, and of a variation of Quillen's path object argument (see Lemma 49).", "The model category Stack$(\\widetilde{E})_{proj}$ is also Quillen equivalent via the identity functors to the model category for strong stacks [19] (see Proposition 52) and it behaves as expected with respect to morphisms of sites (see Proposition 53).", "The paper contains a couple of other results, essentially easy consequences of some of the results we have proved so far: the bicovering maps and the natural fibrations make Fib$(E)$ a category of fibrant objects [7] (see Proposition 40), and the 2-pullback (or, iso-comma object) of fibred categories is a model for the homotopy pullback in Champ$(E)$ (see Lemma 42).", "Appendix 1 is a review of Hollander's characterization of stacks of groupoids in terms of the homotopy sheaf condition [15].", "Appendix 2 studies the behaviour of left Bousfield localizations of model categories under change of cofibrations.", "The result contained in it is needed in Appendix 3, which is a review of the model category for strong stacks of categories [19] made with the hope that it sheds some light on the nature of strong stacks.", "I wish to express my gratitude to the referee whose comments and suggestions greatly improved the content of the paper.", "I wish to thank Jean Bénabou and Claudio Hermida for very useful correspondence related to fibred categories." ], [ "Fibred categories", "In this section we recall, for completeness and to fix notations, some results from the theory of fibred categories.", "We shall work in the setting of universes, as in [11], although we shall not mention the universe in which we shall be working.", "We shall also use the axiom of choice.", "We denote by $SET$ the category of sets and maps, by $CAT$ the category of categories and functors and by $GRPD$ its full subcategory whose objects are groupoids.", "Let $E$ be a category.", "We denote by $E^{op}$ the opposite category of $E$ .", "We let $CAT_{/E}$ be the category of categories over $E$ .", "Arrows of $CAT_{/E}$ will be called $E$ -functors.", "If $S$ is an object of $E$ , $E_{/S}$ stands for the category of objects of $E$ over $S$ .", "If $A$ and $B$ are two categories, we denote by $[A,B]$ the category of functors from $A$ to $B$ and natural transformations between them.", "We denote by $\\ast $ the terminal object of a category, when it exists.", "We denote by $J$ the groupoid with two objects and one isomorphism between them." ], [ "Isofibrations", "One says that a functor $A\\rightarrow B$ is an isofibration (called transportable in [13]) if it has the right lifting property with respect to one of the maps $\\ast \\rightarrow J$ .", "A functor is both an isofibration and an equivalence of categories if and only if it is an equivalence which is surjective on objects (surjective equivalence, for short).", "Given a commutative diagram in $CAT$ ${{A}[r] [d]_{f}&{C} [d]^{g}\\\\{B} [r]&{D}}$ in which the horizontal arrows are surjective equivalences, if $f$ is an isofibration then so is $g$ ." ], [ "Fibrations and isofibrations", "Let $E$ be a category.", "Let $f\\colon F\\rightarrow E$ be a functor.", "We denote by $F_{S}$ the fibre category over $S\\in Ob(E)$ .", "An $E$ -functor $u\\colon F\\rightarrow G$ induces a functor $u_{S}\\colon F_{S}\\rightarrow G_{S}$ for every $S\\in Ob(E)$ .", "Lemma 1 (1) Let $u\\colon F\\rightarrow G$ be an $E$ -functor with $F\\rightarrow E$ an isofibration.", "Then the underlying functor of $u$ is an isofibration if and only if for every $S\\in Ob(E)$ , the map $F_{S}\\rightarrow G_{S}$ is an isofibration.", "(2) Every fibration is an isofibration.", "Every surjective equivalence is a fibration.", "(3) Let $u\\colon F\\rightarrow G$ be an $E$ -functor such that the underlying functor of $u$ is an equivalence.", "If $F$ is a fibration then so is $G$ .", "(4) Let $u\\colon F\\rightarrow G$ be an $E$ -functor such that the underlying functor of $u$ is an equivalence.", "If $G$ is a fibration and $F\\rightarrow E$ is an isofibration then $F$ is a fibration.", "(5) Let $F$ and $G$ be two fibrations and $u\\colon F\\rightarrow G$ be an $E$ -functor.", "If the underlying functor of $u$ is full and faithful then $u$ reflects cartesian arrows.", "(6) Let ${{F}[r] [d]_{u}&{H} [d]^{v}\\\\{G} [r]&{K}}$ be a commutative diagram in $CAT_{/E}$ with $F,G,H$ and $K$ fibrations.", "If the underlying functors of the horizontal arrows are equivalences, then $u$ is a cartesian functor if and only if $v$ is cartesian.", "(1) We prove sufficiency.", "Let $\\beta \\colon u(x)\\rightarrow y$ be an isomorphism and let $S=g(y)$ .", "Then $g(\\beta )\\colon f(x)\\rightarrow S$ is an isomorphism therefore there is an isomorphism $\\alpha \\colon x\\rightarrow x_{0}$ such that $f(\\alpha )=g(\\beta )$ since $f$ is an isofibration.", "The composite $\\beta u(\\alpha ^{-1})\\colon u(x_{0})\\rightarrow y$ lives in $G_{S}$ hence there is an isomorphism $\\alpha ^{\\prime }\\colon x_{0}\\rightarrow x_{1}$ such that $u(\\alpha ^{\\prime })=\\beta u(\\alpha ^{-1})$ since $u_{S}$ is an isofibration.", "One has $u(\\alpha ^{\\prime }\\alpha )=\\beta $ .", "(2) is straightforward.", "(3) and (4) are consequences of [13].", "(5) Let $f\\colon F\\rightarrow E$ and $g\\colon G\\rightarrow E$ be the structure maps.", "Let $\\alpha \\colon x\\rightarrow y$ be a map of $F$ such that $u(\\alpha )$ is cartesian.", "We can factorize $\\alpha $ as $c\\gamma $ , where $c\\colon z\\rightarrow y$ is a cartesian map over $f(\\alpha )$ and $\\gamma \\colon x\\rightarrow z$ is a vertical map.", "Since $u(\\alpha )$ is cartesian and $gu(\\alpha )=gu(c)$ , there is a unique $\\epsilon \\colon u(z)\\rightarrow u(x)$ such that $u(\\alpha )\\epsilon =u(c)$ .", "Then $\\epsilon =u(\\beta )$ since $u$ is full, where $\\beta \\colon z\\rightarrow x$ .", "Hence $c=\\alpha \\beta $ since $u$ is faithful.", "Since $c$ is cartesian it follows that $\\gamma \\beta $ is the identity, and since $u(\\alpha )$ is cartesian it follows that $\\beta \\gamma $ is the identity.", "Thus, $\\gamma $ is a cartesian map.", "(6) is a consequence of (5) and [13].", "A sieve of $E$ is a collection $R$ of objects of $E$ such that for every arrow $X\\rightarrow Y$ of $E$ , $Y\\in R$ implies $X\\in R$ .", "Let $F\\rightarrow E$ be a fibration and $R$ a sieve of $F$ .", "The composite $R\\subset F\\rightarrow E$ is a fibration and $R\\subset F$ is a cartesian functor.", "A surjective equivalence takes sieves to sieves." ], [ "The 2-categories\n${F}ib(E)$ and {{formula:b2a24c29-08e3-41f5-bccf-ea232ef835d3}}", "Let $E$ be a category.", "We denote by Fib$(E)$ the category whose objects are the categories fibred over $E$ and whose arrows are the cartesian functors." ], [ "Let $F$ and $G$ be two objects of Fib$(E)$ .", "The cartesian functors from $F$ to $G$ and the cartesian (sometimes called vertical) natural transformations between them form a category which we denote by ${\\bf Cart}_{E}(F,G)$ .", "This defines a functor ${{{\\bf Cart}_{E}(-,-)\\colon {\\rm Fib}(E)^{op}\\times {\\rm Fib}(E)}[r] &{CAT}}$ so that the fibred categories over $E$ , the cartesian functors and cartesian natural transformations between them form a 2-category which we denote by ${F}ib(E)$ .", "The category Fib$(E)$ has finite products.", "The product of two objects $F$ and $G$ is the pullback $F\\times _{E}G$ .", "Let $A$ be a category and $F\\in {\\rm Fib}(E)$ .", "We denote by $A\\times F$ the pullback of categories ${{A\\times F}[r] [d]&{F} [d]\\\\{A\\times E} [r]&{E}}$ $A\\times F$ is the product in Fib$(E)$ of $F$ and $A\\times E$ .", "The construction defines a functor ${{-\\times -\\colon CAT\\times {\\rm Fib}(E)}[r] &{{\\rm Fib}(E)}}$ We denote by $F^{(A)}$ the pullback of categories ${{F^{(A)}}[r] [d]&{[A,F]} [d]\\\\{E} [r]&{[A,E]}}$ so that $(F^{(A)})_{S}=[A,F_{S}]$ .", "The functor $-\\times F$ is left adjoint to ${\\bf Cart}_{E}(F,-)$ and the functor $A\\times -$ is left adjoint to $(-)^{(A)}$ .", "These adjunctions are natural in $F$ and $A$ .", "There are isomorphisms that are natural in $F$ and $G$ ${\\bf Cart}_{E}(A\\times F,G)\\cong [A,{\\bf Cart}_{E}(F,G)]\\cong {\\bf Cart}_{E}(F,G^{(A)})$ so that ${F}ib(E)$ is tensored and cotensored over the monoidal category $CAT$ .", "Let $F$ and $G$ be two objects of Fib$(E)$ .", "We denote by $\\mathsf {CART}(F,G)$ the object of Fib$(E)$ associated by the Grothendieck construction to the functor $E^{op}\\rightarrow CAT$ which sends $S\\in Ob(E)$ to ${\\bf Cart}_{E}(E_{/S}\\times F,G)$ , so that $\\mathsf {CART}(F,G)_{S}={\\bf Cart}_{E}(E_{/S}\\times F,G)$ There is a natural equivalence of categories ${\\bf Cart}_{E}(F\\times G,H)\\simeq {\\bf Cart}_{E}(F,\\mathsf {CART}(G,H))$ The Grothendieck construction functor ${{[E^{op},CAT]}[r]^{\\Phi }&{{\\rm Fib}(E)}}$ has a right adjoint $\\mathcal {S}$ given by $\\mathcal {S}F(S)={\\bf Cart}_{E}(E_{/S},F)$ .", "$\\Phi $ and $\\mathcal {S}$ are 2-functors and the adjoint pair $(\\Phi ,\\mathcal {S})$ extends to a 2-adjunction betweeen the 2-categories $[E^{op},CAT]$ and ${F}ib(E)$ .", "$\\mathcal {S}F$ is a split fibration and $\\mathcal {S}$ sends maps in Fib$(E)$ to split functors.", "The composite $\\mathsf {S}=\\Phi \\mathcal {S}$ sends fibrations to split fibrations and maps in Fib$(E)$ to split functors.", "The counit of the 2-adjunction $(\\Phi ,\\mathcal {S})$ is a 2-natural transformation $\\mathsf {v}\\colon \\mathsf {S}\\rightarrow Id_{{F}ib(E)}$ .", "For every object $F$ of Fib$(E)$ and every $S\\in Ob(E)$ the map $(\\mathsf {v}F)_{S}\\colon {\\bf Cart}_{E}(E_{/S},F)\\rightarrow F_{S}$ is a surjective equivalence.", "$\\Phi $ has also a left adjoint $\\mathsf {L}$ , constructed as follows.", "For any category $A$ , the functor $-\\times A\\colon CAT\\rightarrow {\\rm Fib}(A)$ has a left adjoint $\\underset{\\longrightarrow }{\\rm Lim}(-/A)$ which takes $F$ to the category obtained by inverting the cartesian morphisms of $F$ .", "If $F\\in {\\rm Fib}(E)$ , $\\mathsf {L}F(S)=\\underset{\\longrightarrow }{\\rm Lim}(E^{/S}\\times _{E}F/E^{/S})$ where $E^{/S}$ is the category of objects of $E$ under $S$ .", "We denote by $\\mathsf {l}$ the unit of the adjoint pair $(\\mathsf {L},\\Phi )$ .", "For every $S\\in Ob(E)$ , the map $(\\mathsf {l}F)_{S}\\colon F_{S}\\rightarrow \\mathsf {L}F(S)$ is an equivalence of categories.", "The adjoint pair $(\\mathsf {L},\\Phi )$ extends to a 2-adjunction betweeen the 2-categories $[E^{op},CAT]$ and ${F}ib(E)$ .", "Let $F$ be an object of Fib$(E)$ .", "We denote by $F^{cart}$ the subcategory of $F$ which has the same objects and whose arrows are the cartesian arrows.", "The composite $F^{cart}\\subset F\\rightarrow E$ is a fibration and $F^{cart}\\subset F$ is a map in Fib$(E)$ .", "For each $S\\in Ob(E)$ , $(F^{cart})_{S}$ is the maximal groupoid associated to $F_{S}$ .", "A map $u\\colon F\\rightarrow G$ of Fib$(E)$ induces a map $u^{cart}\\colon F^{cart}\\rightarrow G^{cart}$ of Fib$(E)$ .", "In all, we obtain a functor $(-)^{cart}\\colon {\\rm Fib}(E)\\rightarrow {\\rm Fib}(E)$ .", "One says that $F$ is fibred in groupoids if the fibres of $F$ are groupoids.", "This is equivalent to saying that $F^{cart}=F$ and, if $f\\colon F\\rightarrow E$ is the structure map of $F$ , to saying that for every object $x$ of $F$ , the induced map $f_{/x}\\colon F_{/x}\\rightarrow E_{/f(x)}$ is a surjective equivalence.", "We denote by Fibg$(E)$ the full subcategory of Fib$(E)$ consisting of categories fibred in groupoids.", "The inclusion functor ${\\rm Fibg}(E)\\subset {\\rm Fib}(E)$ has $(-)^{cart}$ as right adjoint.", "Fibg$(E)$ is a full subcategory of $CAT_{/E}$ .", "We denote by ${F}ibg(E)$ the full sub-2-category of ${F}ib(E)$ whose objects are the categories fibred in groupoids.", "${F}ibg(E)$ is a $GRPD$ -category.", "If $F$ and $G$ are two objects of ${F}ibg(E)$ , we denote the $GRPD$ -hom between $F$ and $G$ by ${\\bf Cartg}_{E}(F,G)$ .", "${F}ibg(E)$ is tensored and cotensored over $GRPD$ with tensor and cotensor defined by the same formulas as for Fib$(E)$ .", "${F}ib(E)$ becomes a $GRPD$ -category by change of base along the maximal groupoid functor $max\\colon CAT\\rightarrow GRPD$ .", "Then the inclusion ${F}ibg(E)\\subset {F}ib(E)$ becomes a $GRPD$ -functor which has $(-)^{cart}$ as right $GRPD$ -adjoint.", "In particular, we have a natural isomorphism ${\\bf Cartg}_{E}(F,G^{cart})\\cong max{\\bf Cart}_{E}(F,G)$ Let $m\\colon A\\rightarrow B$ be a functor.", "There is a 2-functor $m_{\\bullet }^{fib}\\colon {F}ib(B)\\rightarrow {F}ib(A)$ given by $m_{\\bullet }^{fib}(F)=F\\times _{B}A$ .", "If $A$ is fibred in groupoids with structure map $m$ , $m_{\\bullet }^{fib}$ has a left 2-adjoint $m^{\\bullet }$ that is given by composing with $m$ .", "Let $P$ be a presheaf on $A$ and $D\\colon SET\\rightarrow CAT$ be the discrete category functor.", "The functor $D$ induces a functor $D\\colon [E^{op},SET]\\rightarrow [E^{op},CAT]$ .", "We denote the category $\\Phi DP$ by $A_{/P}$ , often called the category of elements of $P$ .", "As a consequence of the above 2-adjunction we have a natural isomorphism ${\\bf Cart}_{A}(A_{/P},m_{\\bullet }^{fib}(F))\\cong {\\bf Cart}_{B}(A_{/P},F)$ Let $E$ be a category and $P$ a presheaf on $E$ .", "Let $m$ be the canonical map $E_{/P}\\rightarrow E$ .", "We denote $m_{\\bullet }^{fib}(F)$ by $F_{/P}$ .", "As a consequence of the above 2-adjunction we have a natural isomorphism ${\\bf Cart}_{E_{/P}}(E_{/P},F_{/P})\\cong {\\bf Cart}_{E}(E_{/P},F)$" ], [ "We shall need to work with a more general notion of (Quillen) model category than in the current literature (like [14]).", "In this section we shall introduce the notion of generalized model category.", "Many concepts and results from the theory of model categories can be defined in the same way and have an exact analogue for generalized model categories.", "We shall review below some of them.", "Definition 8 A generalized model category is a category $\\mathcal {M}$ together with three classes of maps W, C and F (called weak equivalences, cofibrations and fibrations) satisfying the following axioms: A1: $\\mathcal {M}$ has initial and terminal objects.", "A2: The pushout of a cofibration along any map exists and the pullback of a fibration along any map exists.", "A3: W has the two out of three property.", "A4: The pairs $({\\rm C},{\\rm F}\\cap {\\rm W})$ and $({\\rm C}\\cap {\\rm W},{\\rm F})$ are weak factorization systems.", "If follows from the definition that the classes C and ${\\rm C}\\cap {\\rm W}$ are closed under pushout and that the classes F and ${\\rm F}\\cap {\\rm W}$ are closed under pullback.", "The opposite of the underlying category of a generalized model category is a generalized model category.", "Let $\\mathcal {M}$ be a generalized model category.", "A map of $\\mathcal {M}$ is a trivial fibration if it is both a fibration and a weak equivalence, and it is a trivial cofibration if it is both a cofibration and a weak equivalence.", "An object of $\\mathcal {M}$ is cofibrant if the map to it from the initial object is a cofibration, and it is fibrant if the map from it to the terminal object is a fibration.", "Let $X$ be an object of $\\mathcal {M}$ .", "For every cofibrant object $A$ of $\\mathcal {M}$ , the coproduct $A\\sqcup X$ exists and the map $A\\rightarrow A\\sqcup X$ is a cofibration.", "Dually, for every fibrant object $Z$ , the product $Z\\times X$ exists and the map $Z\\times X\\rightarrow X$ is a fibration.", "The class of weak equivalences of a generalized model category is closed under retracts [20]." ], [ "Let $\\mathcal {M}$ be a generalized model category with terminal object $\\ast $ .", "Let $f\\colon X\\rightarrow Y$ be a map of $\\mathcal {M}$ between fibrant objects.", "We review the construction of the mapping path factorization of $f$ [7].", "Let ${{Y}[r]^{s} &{PathY}[r]^{p_{0}\\times p_{1}} & {Y\\times Y}}$ be a factorization of the diagonal map $Y\\rightarrow Y\\times Y$ into a weak equivalence $s$ followed by a fibration $p_{0}\\times p_{1}$ .", "Consider the following diagram ${&{Pf}[r]^{q} [d]_{\\pi _{f}}&{X\\times Y} [d]_{f\\times Y}[r]^{p_{X}} &X[d]^{f}\\\\{Y} [r]_{s} &PathY [r]_{p_{0}\\times p_{1}}& Y\\times Y [r]_{p_{0}} [d]_{p_{1}}& Y[d]\\\\& &Y[r] &{\\ast }}$ in which all squares are pullbacks.", "The object $Pf$ is fibrant.", "There is a unique map $j_{f}\\colon X\\rightarrow Pf$ such that $\\pi _{f}j_{f}=sf$ and $p_{X}qj_{f}=1_{X}$ .", "The map $p_{X}q$ is a trivial fibration, hence the map $j_{f}$ is a weak equivalence.", "Put $q_{f}=p_{1}(f\\times Y)q$ .", "Then $q_{f}$ is a fibration and $f=q_{f}j_{f}$ .", "In a generalized model category, the pullback of a weak equivalence between fibrant objects along a fibration is a weak equivalence [7]." ], [ "Let $\\mathcal {M}$ be a generalized model category.", "A left Bousfield localization of $\\mathcal {M}$ is a generalized model category L$\\mathcal {M}$ on the underlying category of $\\mathcal {M}$ having the same class of cofibrations as $\\mathcal {M}$ and a bigger class of weak equivalences.", "Lemma 9 Let $\\mathcal {M}$ be a generalized model category with W, C and F as weak equivalences, cofibrations and fibrations.", "Let W$^{\\prime }$ be a class of maps of $\\mathcal {M}$ that contains W and has the two out of three property.", "We define F$^{\\prime }$ to be the class of maps having the right lifting property with respect to every map of ${\\rm C}\\cap {\\rm W^{\\prime }}$ .", "Then ${\\rm L}\\mathcal {M}=({\\rm W}^{\\prime },{\\rm C},{\\rm F}^{\\prime })$ is a left Bousfield localization of $\\mathcal {M}$ if and only if the pair $({\\rm C}\\cap {\\rm W}^{\\prime },F^{\\prime })$ is a weak factorization system.", "Moreover, $({\\rm C}\\cap {\\rm W}^{\\prime },F^{\\prime })$ is a weak factorization system if and only if the class ${\\rm C}\\cap {\\rm W}^{\\prime }$ is closed under codomain retracts and every arrow of $\\mathcal {M}$ factorizes as a map in ${\\rm C}\\cap {\\rm W^{\\prime }}$ followed by a map in F$^{\\prime }$ .", "We prove the first statement.", "The necessity is clear.", "Conversely, since ${\\rm C}\\cap {\\rm W}\\subset {\\rm C}\\cap {\\rm W}^{\\prime }$ it follows that ${\\rm F}^{\\prime }\\subset {\\rm F}$ .", "This implies that the second part of Axiom A2 is satisfied.", "To complete the proof it suffices to show that ${\\rm F}\\cap {\\rm W}={\\rm F}^{\\prime }\\cap {\\rm W}^{\\prime }$ .", "Since ${\\rm C}\\cap {\\rm W}^{\\prime }\\subset {\\rm C}$ , it follows that ${\\rm F}\\cap {\\rm W}\\subset {\\rm F}^{\\prime }$ and hence that ${\\rm F}\\cap {\\rm W}\\subset {\\rm F}^{\\prime }\\cap {\\rm W}^{\\prime }$ .", "We show that ${\\rm F}^{\\prime }\\cap {\\rm W}^{\\prime }\\subset {\\rm F}\\cap {\\rm W}$ .", "Let $X\\rightarrow Y$ be a map in ${\\rm F}^{\\prime }\\cap {\\rm W}^{\\prime }$ .", "We factorize it into a map $X\\rightarrow Z$ in C followed by a map $Z\\rightarrow Y$ in ${\\rm F}\\cap {\\rm W}$ .", "Since W$^{\\prime }$ has the two out of three property, the map $X\\rightarrow Z$ is in ${\\rm C}\\cap {\\rm W}^{\\prime }$ .", "It follows that the commutative diagram ${{X}@{=}[r][d]&{X} [d]\\\\{Z} [r]&{Y}}$ has a diagonal filler, hence $X\\rightarrow Y$ is a (domain) retract of $Z\\rightarrow Y$ .", "Thus, the map $X\\rightarrow Y$ is in ${\\rm F}\\cap {\\rm W}$ .", "The second statement follows from a standard characterization of weak factorization systems.", "Let L$\\mathcal {M}$ be a left Bousfield localization of $\\mathcal {M}$ .", "A map of $\\mathcal {M}$ between fibrant objects in L$\\mathcal {M}$ is a weak equivalence (fibration) in L$\\mathcal {M}$ if and only if it is a weak equivalence (fibration) in $\\mathcal {M}$ .", "Let $X\\rightarrow Y$ be a weak equivalence in $\\mathcal {M}$ between fibrant objects in $\\mathcal {M}$ .", "Then $X$ is fibrant in L$\\mathcal {M}$ if and only if $Y$ is fibrant in L$\\mathcal {M}$ ." ], [ "A generalized model category is left proper if every pushout of a weak equivalence along a cofibration is a weak equivalence.", "Dually, a generalized model category is right proper if every pullback of a weak equivalence along a fibration is a weak equivalence.", "A generalized model category is proper if it is left and right proper.", "A left Bousfield localization of a left proper generalized model category is left proper.", "Let $\\mathcal {M}$ be a right proper generalized model category.", "Let ${{X}[r]^{g} & {Z} & {Y}[l]_{f}}$ be a diagram in $\\mathcal {M}$ .", "We factorize $f$ as a trivial cofibration $Y\\overset{i_{f}}{\\rightarrow }E(f)$ followed by a fibration $E(f)\\overset{p_{f}}{\\rightarrow }Z$ .", "We factorize $g$ as a trivial cofibration $X\\overset{i_{g}}{\\rightarrow }E(g)$ followed by a fibration $E(f)\\overset{p_{g}}{\\rightarrow }Z$ .", "The homotopy pullback of diagram (10) is defined to be the pullback of the diagram ${{E(g)}[r]^{p_{g}} & {Z} & {E(f)}[l]_{p_{f}}}$ The analogue of [14] holds in this context.", "If $X,Y$ and $Z$ are fibrant, a model for the homotopy pullback is $X\\times _{Z}Pf$ , where $Pf$ is the mapping path factorization of $f$ described in Section 3.2.", "Let L$\\mathcal {M}$ be a left Bousfield localization of $\\mathcal {M}$ that is right proper.", "We denote by $X\\times _{Z}^{h}Y$ the homotopy pullback in $\\mathcal {M}$ of diagram (10) and by $X\\times _{Z}^{{\\rm L}h}Y$ the homotopy pullback of the same diagram, but in L$\\mathcal {M}$ .", "Proposition 11 (1) Suppose that $X,Y$ and $Z$ are fibrant in L$\\mathcal {M}$ .", "Then $X\\times _{Z}^{h}Y$ is weakly equivalent in $\\mathcal {M}$ to $X\\times _{Z}^{{\\rm L}h}Y$ .", "(2) Suppose that the pullback of a map between fibrant objects in $\\mathcal {M}$ that is both a fibration in $\\mathcal {M}$ and a weak equivalence in L$\\mathcal {M}$ is a weak equivalence in L$\\mathcal {M}$ .", "Suppose that $X,Y$ and $Z$ are fibrant in $\\mathcal {M}$ .", "Then $X\\times _{Z}^{h}Y$ is weakly equivalent in L$\\mathcal {M}$ to $X\\times _{Z}^{{\\rm L}h}Y$ .", "(1) is a consequence of [14].", "To prove (2) we first factorize $f$ in L$\\mathcal {M}$ as a trivial cofibration $Y\\rightarrow Y_{0}$ in followed by a fibration $Y_{0}\\rightarrow Z$ .", "Then we factorize $Y\\rightarrow Y_{0}$ in $\\mathcal {M}$ as a trivial cofibration $Y\\rightarrow Y^{\\prime }$ in followed by a fibration $Y^{\\prime }\\rightarrow Y_{0}$ .", "By assumption the map $X\\times _{Z}Y^{\\prime }\\rightarrow X\\times _{Z}Y_{0}$ is a weak equivalence in L$\\mathcal {M}$ ." ], [ "Let $\\mathcal {M}$ and $\\mathcal {N}$ be generalized model categories and $F\\colon \\mathcal {M}\\rightarrow \\mathcal {N}$ be a functor having a right adjoint $G$ .", "The adjoint pair $(F,G)$ is a Quillen pair if $F$ preserves cofibrations and trivial cofibrations.", "Equivalently, if $G$ preserves fibrations and trivial fibrations.", "If the classes of weak equivalences of $\\mathcal {M}$ and $\\mathcal {N}$ have the two out of six property, then $(F,G)$ is a Quillen pair if and only if $F$ preserves cofibrations between cofibrant objects and trivial cofibrations if and only if $G$ preserves fibrations between fibrant objects and trivial fibrations (a result due to Joyal).", "The adjoint pair $(F,G)$ is a Quillen equivalence if $(F,G)$ is a Quillen pair and if for every cofibrant object $A$ in $\\mathcal {M}$ and every fibrant object $X$ in $\\mathcal {N}$ , a map $FA\\rightarrow X$ is a weak equivalence in $\\mathcal {N}$ if and only if its adjunct $A\\rightarrow GX$ is a weak equivalence in $\\mathcal {M}$ ." ], [ "The natural generalized\nmodel category on Fib$(E)$", "We recall [19] that $CAT$ is a model category in which the weak equivalences are the equivalences of categories, the cofibrations are the functors that are injective on objects and the fibrations are the isofibrations.", "Therefore, for every category $E$ , $CAT_{/E}$ is a model category in which a map is a weak equivalence, cofibration or fibration if it is one in $CAT$ .", "Let $E$ be a category.", "Definition 12 Let $u\\colon F\\rightarrow G$ be a map of Fib$(E)$ .", "We say that $u$ is an $E$ -equivalence (isofibration) if the underlying functor of $u$ is an equivalence of categories (isofibration).", "We say that $u$ is a trivial fibration if it is both an $E$ -equivalence and an isofibration.", "Theorem 13 The category Fib$(E)$ is a proper generalized model category with the $E$ -equivalences as weak equivalences, the maps that are injective on objects as cofibrations and the isofibrations as fibrations.", "The proof of Theorem 13 will be given after some preparatory results.", "Proposition 14 Let $u\\colon F\\rightarrow G$ be a map of Fib$(E)$ .", "The following are equivalent: (1) $u$ is an $E$ -equivalence.", "(2) For every $S\\in Ob(E)$ , the map $u_{S}\\colon F_{S}\\rightarrow G_{S}$ is an equivalence of categories.", "(3) $u$ is an equivalence in the 2-category ${F}ib(E)$ .", "(4) ${\\bf Cart}_{E}(u,X)\\colon {\\bf Cart}_{E}(G,X)\\rightarrow {\\bf Cart}_{E}(F,X)$ is an equivalence for all $X\\in {\\rm Fib}(E)$ .", "(5) ${\\bf Cart}_{E}(X,u)\\colon {\\bf Cart}_{E}(X,F)\\rightarrow {\\bf Cart}_{E}(X,G)$ is an equivalence for all $X\\in {\\rm Fib}(E)$ .", "All is contained in [13].", "Corollary 15 A map $u\\colon F\\rightarrow G$ of Fib$(E)$ is a trivial fibration if and only if for every $S\\in Ob(E)$ , $u_{S}\\colon F_{S}\\rightarrow G_{S}$ is a surjective equivalence.", "This follows from Lemma 1((1) and (2)) and Proposition 14.", "For part (2) of the next result, let ${M}$ be a class of functors that is contained in the class of injective on objects functors.", "In our applications ${M}$ will be the class of injective on objects functors or the set consisting of one of the inclusions $\\ast \\rightarrow J$ .", "Let ${M}^{\\perp }$ be the class of functors that have the right lifting property with respect to every element of ${M}$ .", "Proposition 16 Let $u\\colon F\\rightarrow G$ be a map of Fib$(E)$ .", "(1) $u$ is an isofibration if and only if $u^{cart}$ is an isofibration.", "(2) If $u$ has the right lifting property with respect to the maps $f\\times E_{/S}$ , where $f\\in {M}$ and $S\\in Ob(E)$ , then $u_{S}\\in {M}^{\\perp }$ for every $S\\in Ob(E)$ .", "(1) This is a consequence of Lemma 1(1) and of the fact that for every $S\\in Ob(E)$ and every object $F$ of Fib$(E)$ , $(F^{cart})_{S}$ is the maximal groupoid associated to $F_{S}$ .", "(2) Let $S\\in Ob(E)$ and $A\\rightarrow B$ be an element of ${M}$ .", "Consider the commutative solid arrow diagram $@=4ex{&&{{\\bf Cart}_{E}(E_{/S},F)}[dr][dd]\\\\&&&{{\\bf Cart}_{E}(E_{/S},G)}[dd]\\\\{A}[rr][dr]@{.>}[uurr]&&{F_{S}}[dr]\\\\&{B}[rr]@{.>}[uurr]&&{G_{S}}\\\\}$ We recall that the category of arrows of $CAT$ is a model category in which the weak equivalences and fibrations are defined objectwise.", "A functor is cofibrant in this model category if and only if it is injective on objects.", "If we regard the previous diagram as a diagram in the category of arrows of $CAT$ , then it has by 2.3(5) and the assumption on ${M}$ a diagonal filler, the two dotted arrows.", "From Section 2.3 and hypothesis this diagonal filler has itself a diagonal filler, hence the bottom face diagram has one.", "Lemma 17 (1) Let ${{F\\times _{H}G}[r][d]&{G} [d]^{v}\\\\{F} [r]^{u}&{H}}$ be a pullback diagram in $CAT_{/E}$ .", "If $F,G$ and $H$ are fibrations, $u$ and $v$ are cartesian functors and $u$ is an isofibration, then $F\\times _{H}G$ is a fibration and the diagram is a pullback in Fib$(E)$ .", "(2) Let ${{F}[r]^{u}[d]_{v}&{G}[d]\\\\{H} [r]&{G\\sqcup _{F}H}}$ be a pushout diagram in $CAT_{/E}$ .", "If $F,G$ and $H$ are fibrations, $u$ and $v$ are cartesian functors and $u$ is injective on objects, then $G\\sqcup _{F}H$ is a fibration and the diagram is a pushout in Fib$(E)$ .", "(1) The objects of $F\\times _{H}G$ are pairs $(x,y)$ with $x\\in Ob( F),y\\in Ob(G)$ such that $u(x)=v(y)$ .", "We shall briefly indicate how the composite map $F\\times _{H}G\\rightarrow F\\overset{p}{\\rightarrow }E$ is a fibration.", "Let $S\\in Ob(E)$ , $(x,y)\\in F\\times _{H}G$ and $f\\colon S\\rightarrow p(x)$ .", "A cartesian lift of $f$ is obtained as follows.", "Let $y^{f}\\rightarrow y$ and $x^{f}\\rightarrow x$ be cartesian lifts of $f$ .", "Since $H$ is a fibration, $u$ and $v$ are cartesian functors and $u$ is an isofibration, there is $x_{0}^{f}\\in Ob(F_{S})$ such that $x^{f}\\cong x_{0}^{f}$ and $u(x_{0}^{f})=v(y^{f})$ .", "Then the obvious map $(x_{0}^{f},y^{f})\\rightarrow (x,y)$ is a cartesian lift of $f$ .", "The universal property of the pullback is easy to see.", "(2) The set of objects of $G\\sqcup _{F}H$ can be identified with $Ob(H)\\sqcup (Ob(G)\\setminus ImOb(u))$ .", "Since the structure functors $G\\rightarrow E$ and $H\\rightarrow E$ are isofibrations, one can easily check that the canonical map $G\\sqcup _{F}H\\rightarrow E$ is an isofibration.", "We shall use Lemma 1(4) to show that it is a fibration.", "Consider the following cube in $CAT_{/E}$ $@=2ex{{F}[rr]^{u}[dr] [dd]&& {G} [drr] ^{\\prime }[d][dd]\\\\& {H} [rrr] [dd]&&& {G\\sqcup _{F}H} [dd]\\\\{\\Phi \\mathsf {L}F} ^{\\prime }[r][rr] [dr]&& {\\Phi \\mathsf {L}G} [drr]\\\\& {\\Phi \\mathsf {L}H} [rrr]&&& {\\Phi \\mathsf {L}G\\sqcup _{\\Phi \\mathsf {L}F}\\Phi \\mathsf {L}H}}$ (see Section 2.3 for the functors $\\Phi $ and $\\mathsf {L}$ ).", "The top and bottom faces are pushouts and the vertical arrows having sources $F,G$ and $H$ are weak equivalences.", "The map $\\Phi \\mathsf {L}u$ is a cofibration since $u$ is one.", "By [14] the map $G\\sqcup _{F}H\\rightarrow \\Phi \\mathsf {L}G\\sqcup _{\\Phi \\mathsf {L}F}\\Phi \\mathsf {L}H$ is a weak equivalence.", "Since $\\Phi $ is a left adjoint, the target of this map is in the image of $\\Phi $ , hence it is a fibration.", "It follows from Lemma 1(4) that $G\\sqcup _{F}H$ is a fibration.", "The canonical maps $H\\rightarrow G\\sqcup _{F}H$ and $G\\rightarrow G\\sqcup _{F}H$ are cartesian functors by Lemma 1(6) applied to the front and right faces of the above cube diagram.", "Finally, it remains to prove that if ${{F}[r]^{u}[d]_{v}&{G}[d]\\\\{H} [r]&{K}}$ is a commutative diagram in Fib$(E)$ , then the resulting functor $G\\sqcup _{F}H\\rightarrow K$ is cartesian.", "This follows from Lemma 1(6) applied to the diagram ${{G\\sqcup _{F}H}[r][d]&{\\Phi \\mathsf {L}G\\sqcup _{\\Phi \\mathsf {L}F}\\Phi \\mathsf {L}H}[d]\\\\{K} [r]&{\\Phi \\mathsf {L}K}}$ Remark 18 A consequence of Lemma 17(1) is that the fibre category $(F\\times _{H}G)_{S}$ is the pullback $F_{S}\\times _{H_{S}}G_{S}$ .", "Thus, if $F,G$ and $H$ are fibred in groupoids then so is $F\\times _{H}G$ .", "A consequence of Lemma 17(2) is that if $F,G$ and $H$ are fibred in groupoids then so is $G\\sqcup _{F}H$ .", "Example 19 (1) Let $u\\colon F\\rightarrow G$ be a map of Fib$(E)$ and $H$ an object of Fib$(E)$ .", "Then the diagram ${{F\\times H}[r]^{F\\times u}[d]&{G\\times H} [d]\\\\{F} [r]^{u}&{G}}$ in a pullback in Fib$(E)$ .", "(2) Let $E_{/}\\colon E\\rightarrow CAT_{/E}$ be the functor which takes $S$ to $E_{/S}$ .", "The functor $E_{/}$ preserves all the limits that exist in $E$ .", "Therefore, if ${{U\\times _{S}T}[r][d]&{T} [d]\\\\{U} [r]&{S}}$ is a pullback diagram in $E$ , then ${{E_{/U\\times _{S}T}}[r][d]&{E_{/T}} [d]\\\\{E_{/U}} [r]&{E_{/S}}}$ is a pullback diagram in Fib$(E)$ .", "Axioms A1 and A3 from Definition 8 are clear.", "Axiom A2 was dealt with in Lemma 17.", "We prove Axiom 4.", "Any map $u\\colon F\\rightarrow G$ of Fib$(E)$ admits a factorization $u=vi\\colon F\\rightarrow H\\rightarrow G$ in $CAT_{/E}$ , where $i$ is injective on objects and the underlying functor of $v$ is a surjective equivalence.", "By Lemma 1(2) $H$ is an object of Fib$(E)$ .", "By Lemma 1(5) $i$ is a cartesian functor.", "By [13] $v$ is a cartesian functor.", "Any commutative diagram ${{F}[r] [d]_{u}&{H} [d]^{v}\\\\{G} [r]&{K}}$ Fib$(E)$ in which the underlying functor of $u$ is injective on objects and $v$ is a trivial fibration has a diagonal filler in $CAT_{/E}$ .", "By Lemma 1(5) (or [13], for example) this diagonal filler is a cartesian functor.", "Thus, the first part of Axiom 4 is proved.", "The rest of the Axiom 4 is proved similarly, using Lemma 1((3) and (5)) and [13].", "Properness is easy to see.", "Remark 20 Let $F$ be an object of Fib$(E)$ .", "Let $D2$ be the discrete category with two objects.", "By cotensoring the sequence $D2\\rightarrow J\\rightarrow \\ast $ with $F$ we obtain a natural factorization of the diagonal $F\\rightarrow F\\times F$ as ${{F}[r] &{F^{(J)}} [r] & {F\\times F}}$ in which the map $F\\rightarrow F^{(J)}$ is an $E$ -equivalence and the map $F^{(J)}\\rightarrow F\\times F$ is an isofibration.", "We obtain the following model for the mapping path factorization (Section 3.2) of a map $u\\colon F\\rightarrow G$ of Fib$(E)$ .", "The objects of a fibre category $(Pu)_{S}$ are triples $(x,y,\\theta )$ with $x\\in Ob(F_{S}),y\\in Ob(G_{S})$ and $\\theta \\colon y\\rightarrow u(x)$ an isomorphism in $G_{S}$ .", "The arrows are pairs of arrows making the obvious diagram commute.", "Proposition 21 (Compatibility with the 2-category structure) Let $u\\colon F\\rightarrow G$ be a cofibration and $v\\colon H\\rightarrow K$ an isofibration.", "Then the canonical map ${{{\\bf Cart}_{E}(G,H)} [r] &{{\\bf Cart}_{E}(G,K)\\times _{{\\bf Cart}_{E}(F,K)} {\\bf Cart}_{E}(F,H)}}$ is an isofibration that is a surjective equivalence if either $u$ or $v$ is an $E$ -equivalence.", "By Section 2.3 the diagram ${{\\ast } [r] [d] &{{\\bf Cart}_{E}(G,H)}[d]\\\\{J} [r] &{{\\bf Cart}_{E}(G,K)\\times _{{\\bf Cart}_{E}(F,K)}{\\bf Cart}_{E}(F,H)}}$ has a diagonal filler if and only if the diagram ${{F} [r] [d] &{H^{(J)}}[d]\\\\{G} [r] &{K^{(J)}\\times _{K}H}}$ has one (the pullback exists by Lemma 17(1)).", "The latter is true since the map $H^{(J)}\\rightarrow K^{(J)}\\times _{K}H$ is a trivial fibration using Corollary 15.", "Suppose that $u$ is an $E$ -equivalence.", "By Proposition 14, the functors ${\\bf Cart}_{E}(u,H)$ and ${\\bf Cart}_{E}(u,K)$ are surjective equivalences.", "Since surjective equivalences are stable under pullback, the functor ${\\bf Cart}_{E}(G,H)\\rightarrow {\\bf Cart}_{E}(G,K)\\times _{{\\bf Cart}_{E}(F,K)} {\\bf Cart}_{E}(F,H)$ is an equivalence by the two out of three property of equivalences.", "Suppose that $v$ is an $E$ -equivalence.", "Then ${\\bf Cart}_{E}(F,v)$ and ${\\bf Cart}_{E}(G,v)$ are equivalences the functor ${{{\\bf Cart}_{E}(G,K)\\times _{{\\bf Cart}_{E}(F,K)}{\\bf Cart}_{E}(F,H)} [r]&{{\\bf Cart}_{E}(G,K)}}$ is an equivalence being the pullback of an equivalence along an isofibration.", "Therefore the canonical map is an equivalence.", "Corollary 22 A map $u\\colon F\\rightarrow G$ is an isofibration if and only if for every object $X$ of Fib$(E)$ , the map ${\\bf Cart}_{E}(X,u)\\colon {\\bf Cart}_{E}(X,F)\\rightarrow {\\bf Cart}_{E}(X,G)$ is an isofibration.", "One half is a consequence of Proposition 21.", "The other half follows by putting $X=E_{/S}$ , where $S\\in Ob(E)$ , and using 2.3(5), Lemma 1(1) and Section 2.1.", "Corollary 23 (Compatibility with the `internal hom') Let $u:F\\rightarrow G$ be a cofibration and $v:H\\rightarrow K$ an isofibration.", "Then the canonical map ${{\\mathsf {CART}(G,H)}[r] &{\\mathsf {CART}(G,K)\\times _{\\mathsf {CART}(F,K)} \\mathsf {CART}(F,H)}}$ is an isofibration that is a trivial fibration if either $u$ or $v$ is an $E$ -equivalence.", "The map $\\mathsf {CART}(u,K)$ is an isofibration by 2.3(3) and Proposition 21, therefore the pullback in the displayed arrow exists by Lemma 17(1).", "The result follows from Remark 18, 2.3(3) and Proposition 21 applied to $v$ and the cofibration $E_{/S}\\times u$ , $S\\in Ob(E)$ .", "We recall [19] that the category $[E^{op},CAT]$ is a model category in which a map is a weak equivalence or cofibration if it is objectwise an equivalence of categories or objectwise injective on objects.", "We denote this model category by $[E^{op},CAT]_{inj}$ .", "We recall that the category $[E^{op},CAT]$ is a model category in which a map is a weak equivalence or fibration if it is objectwise an equivalence of categories or objectwise an isofibration.", "We denote this model category by $[E^{op},CAT]_{proj}$ .", "The identity functors form a Quillen equivalence between $[E^{op},CAT]_{proj}$ and $[E^{op},CAT]_{inj}$ .", "Recall from Section 2.3 the adjoint pairs $(\\Phi ,\\mathcal {S})$ and $(\\mathsf {L},\\Phi )$ .", "Proposition 24 The adjoint pair $(\\Phi ,\\mathcal {S})$ is a Quillen equivalence between Fib$(E)$ and $[E^{op},CAT]_{inj}$ .", "The adjoint pair $(\\mathsf {L},\\Phi )$ is a Quillen equivalence between Fib$(E)$ and $[E^{op},CAT]_{proj}$ .", "The functor $\\Phi \\colon [E^{op},CAT]_{inj}\\rightarrow {\\rm Fib}(E)$ preserves and reflects weak equivalences and preserves cofibrations.", "Since the map $\\mathsf {v}F$ is a weak equivalence (2.3(5)), the pair $(\\Phi ,\\mathcal {S})$ is a Quillen equivalence.", "The functor $\\Phi \\colon [E^{op},CAT]_{proj}\\rightarrow {\\rm Fib}(E)$ preserves fibrations.", "Since the map $\\mathsf {l}F$ is a weak equivalence, the pair $(\\mathsf {L},\\Phi )$ is a Quillen equivalence.", "Let $m\\colon A\\rightarrow B$ be a category fibred in groupoids.", "Recall from Section 2.3 that the functor $m_{\\bullet }^{fib}\\colon {\\rm Fib}(B)\\rightarrow {\\rm Fib}(A)$ has a left adjoint $m^{\\bullet }$ .", "The proof of the next result is straightforward.", "Proposition 25 Let $m\\colon A\\rightarrow B$ be a category fibred in groupoids.", "The adjoint pair $(m^{\\bullet },m_{\\bullet }^{fib})$ is a Quillen pair.", "Let $f\\colon T\\rightarrow S$ be a map of $E$ .", "The functor $f_{\\bullet }^{fib}\\colon {\\rm Fib}(E_{/S})\\rightarrow {\\rm Fib}(E_{/T})$ has a left adjoint $f^{\\bullet }$ .", "Corollary 26 Let $f\\colon T\\rightarrow S$ be a map of $E$ .", "The adjoint pair $(f^{\\bullet },f_{\\bullet }^{fib})$ is a Quillen pair." ], [ "The generalized model\ncategory for stacks over a site", "We briefly recall from [11] the notion of site.", "Let $E$ be a category.", "A topology on $E$ is an application which associates to each $S\\in Ob(E)$ a non-empty collection $J(S)$ of sieves of $E_{/S}$ .", "This data must satisfy two axioms.", "The elements of $J(S)$ are called refinements of $S$ .", "A site is a category endowed with a topology.", "Every category $E$ has the discrete topology (only $E_{/S}$ is a refinement of the object $S$ ) and the coarse topology (every sieve of $E_{/S}$ is a refinement of $S$ ).", "Any other topology on $E$ is `in between' the discrete one and the coarse one.", "Let $E$ be a site.", "Let $\\mathcal {C}$ be the collection of maps $R\\subset E_{/S}$ of Fib$(E)$ , where $S$ ranges through $Ob(E)$ and $R$ is a refinement of $S$ .", "Since $CAT$ is a model category, the theory of homotopy fiber squares [14] is available.", "Definition 27 A map $F\\rightarrow G$ of Fib$(E)$ has property $P$ if for every element $R\\subset E_{/S}$ of $\\mathcal {C}$ , the diagram ${{{\\bf Cart}_{E}(E_{/S},F)}[r] [d]&{{\\bf Cart}_{E}(R,F)}[d]\\\\{{\\bf Cart}_{E}(E_{/S},G)}[r]&{{\\bf Cart}_{E}(R,G)}}$ in which the horizontal arrows are the restriction functors, is a homotopy fiber square.", "The map $F\\rightarrow G$ is a $\\mathcal {C}$ -local fibration if it is an isofibration and it has property $P$ .", "An object $F$ of Fib$(E)$ is $\\mathcal {C}$ -local if the map $F\\rightarrow E$ is a $\\mathcal {C}$ -local fibration.", "The map $F\\rightarrow G$ is a $\\mathcal {C}$ -local equivalence if for all $\\mathcal {C}$ -local objects $X$ , the map ${\\bf Cart}_{E}(u,X)\\colon {\\bf Cart}_{E}(G,X)\\rightarrow {\\bf Cart}_{E}(F,X)$ is an equivalence of categories.", "It follows directly from Definition 27 and a standard property of homotopy fiber squares that a $\\mathcal {C}$ -local object is the same as a stack (=($E$ -)champ) in the sense of [11].", "Example 28 We shall recall that `sheaves are stacks'.", "Let $\\widehat{E}$ be the category of presheaves on $E$ and $\\eta $ be the Yoneda embedding.", "Let $D\\colon SET\\rightarrow CAT$ denote the discrete category functor; it induces a functor $D\\colon \\widehat{E}\\rightarrow [E^{op},CAT]$ .", "For every objects $X,Y$ of $\\widehat{E}$ there is a natural isomorphism ${\\bf Cart}_{E}(\\Phi DX,\\Phi DY)\\cong D{\\rm Fib}(E)(\\Phi DX,\\Phi DY)$ The composite functor $\\Phi D\\colon \\widehat{E}\\rightarrow {\\rm Fib}(E)$ is full and faithful, hence we obtain a natural isomorphism ${\\bf Cart}_{E}(\\Phi DX,\\Phi DY)\\cong D\\widehat{E}(\\Phi DX,\\Phi DY)$ Let now $S\\in Ob(E)$ and $R$ be a refinement of $S$ .", "Let $R^{\\prime }$ be the sub-presheaf of $\\eta (S)$ which corresponds to $R$ .", "Since $E_{/S}=\\Phi D\\eta (S)$ and $R=\\Phi DR^{\\prime }$ , the previous natural isomorphism shows that a presheaf $X$ on $E$ is a sheaf if and only if $\\Phi DX$ is a stack.", "In particular, $\\eta (S)$ is a sheaf if and only if $E_{/S}$ is a stack.", "Theorem 29 There is a proper generalized model category Champ$(E)$ on the category Fib$(E)$ in which the weak equivalences are the $\\mathcal {C}$ -local equivalences and the cofibrations are the maps that are injective on objects.", "The fibrant objects of Champ$(E)$ are the stacks.", "The proof of Theorem 29 will be given after some preparatory results.", "Proposition 30 (1) Every $E$ -equivalence is a $\\mathcal {C}$ -local equivalence.", "(2) The class of maps having property $P$ is invariant under $E$ -equivalences.", "(3) The class of maps having property $P$ contains $E$ -equivalences and all maps between stacks.", "(4) The class of maps having property $P$ is closed under compositions, pullbacks along isofibrations and retracts.", "(1) follows from Proposition 14.", "(2) says that for every commutative diagram ${{F}[r] [d]_{u}&{H} [d]^{v}\\\\{G} [r]&{K}}$ in which the horizontal maps are $E$ -equivalences, $u$ has property $P$ if and only if $v$ has it.", "This is so by Proposition 14 and [14].", "(3) follows from a standard property of homotopy fiber squares.", "(4) follows from standard properties of homotopy fiber squares and the fact that equivalences are closed under retracts.", "Lemma 31 A map between stacks has the right lifting property with respect to all maps that are both cofibrations and $\\mathcal {C}$ -local equivalences if and only if it is an isofibration.", "Let $H\\rightarrow K$ be an isofibration between stacks and $F\\rightarrow G$ a map that is both a cofibration and a $\\mathcal {C}$ -local equivalence.", "A commutative diagram ${{F}[r] [d]&{H} [d]\\\\{G} [r]&{K}}$ has a diagonal filler if and only if the functor ${\\bf Cart}_{E}(G,H)\\rightarrow {\\bf Cart}_{E}(G,K)\\times _{{\\bf Cart}_{E}(F,K)} {\\bf Cart}_{E}(F,H)$ is surjective on objects.", "We show that it is a surjective equivalence.", "The functor is an isofibration by Proposition 21.", "Hence it suffices to show that it is an equivalence.", "The maps ${\\bf Cart}_{E}(G,K) \\rightarrow {\\bf Cart}_{E}(F,K)$ and ${\\bf Cart}_{E}(G,H) \\rightarrow {\\bf Cart}_{E}(F,H)$ are surjective equivalences by assumption and Proposition 21.", "Since surjective equivalences are stable under pullback, the required functor is an equivalence by the two out of three property of equivalences.", "For the notion of bicovering (=bicouvrant) map in Fib$(E)$ we refer the reader to [11].", "As in [loc.", "cit., Chapitre II 1.4.1.1], we informally say that a map is bicovering if it is `locally bijective on arrows' and `locally essentially surjective on objects'.", "Example 32 For every $S\\in Ob(E)$ and every refinement $R$ of $S$ , $R\\subset E_{/S}$ is a bicovering map.", "By [11] there are a 2-functor $\\mathsf {A}\\colon {F}ib(E)\\rightarrow {F}ib(E)$ and a 2-natural transformation $a:Id_{{F}ib(E)}\\rightarrow \\mathsf {A}$ such that $\\mathsf {A}F$ is a stack and $aF$ is bicovering for every object $F$ of ${F}ib(E)$ .", "By [11] the class of bicovering maps coincides with the class of $\\mathcal {C}$ -local equivalences in the sense of Definition 27.", "Lemma 33 Bicovering maps are closed under pullbacks along isofibrations.", "Let ${{F\\times _{H}G}[r]^-{u^{\\prime }}[d]&{G} [d]^{v}\\\\{F} [r]^{u}&{H}}$ be a pullback diagram in Fib$(E)$ with $v$ an isofibration (see Lemma 17(1)).", "Step 1.", "Suppose that the above pullback diagram is a pullback diagram of split fibrations and split functors with $v$ an arbitrary split functor and $u$ `locally bijective on arrows'.", "We prove that $u^{\\prime }$ is `locally bijective on arrows'.", "Let $S\\in Ob(E)$ and $(x,y),(x^{\\prime },y^{\\prime })$ be two objects of $(F\\times _{H}G)_{S}$ .", "Then, in the notation of [11] and the terminology of [11] we have to show that the map ${{{\\rm Hom}_{S}((x,y),(x^{\\prime },y^{\\prime }))}[r] &{{\\rm Hom}_{S}(y,y^{\\prime })}}$ of presheaves on $E_{/S}$ is bicovering, where $E_{/S}$ has the induced topology [11].", "This map is the pullback of the map ${{{\\rm Hom}_{S}(x,x^{\\prime })}[r] &{{\\rm Hom}_{S}(u(x),u(x^{\\prime }))}}$ which is by assumption bicovering.", "But bicovering maps of presheaves are stable under pullbacks [11].", "Step 2.", "Suppose that in the above pullback diagram the map $u$ is `locally essentially surjective on objects'.", "We prove that $u^{\\prime }$ is `locally essentially surjective on objects'.", "Let $S\\in Ob(E)$ and $y\\in Ob(G_{S})$ .", "Let $R^{\\prime }$ be the set of maps $f\\colon T\\rightarrow S$ such that there are $x\\in Ob(F_{T})$ and $y^{\\prime }\\in Ob(G_{T})$ with $u_{T}(x)=v_{T}(y^{\\prime })$ and $y^{\\prime }\\cong f^{\\ast }(y)$ in $G_{T}$ .", "We have to show that $R^{\\prime }$ is a refinement of $S$ .", "Let $R$ be the set of maps $f\\colon T\\rightarrow S$ such that there is $x\\in Ob(F_{T})$ with $u_{T}x\\cong f^{\\ast }v_{S}(y)$ in $H_{T}$ .", "By assumption $R$ is a refinement of $S$ .", "Since $v_{T}f^{\\ast }(y)\\cong f^{\\ast }v_{S}(y)$ we have $R^{\\prime }\\subset R$ .", "Conversely, let $f\\colon T\\rightarrow S$ be in $R$ and $x$ as above.", "Let $\\xi $ be the isomorphism $u_{T}(x)\\cong v_{T}f^{\\ast }(y)$ .", "By assumption there are $y^{\\prime }\\in Ob(G_{T})$ and an isomorphism $y^{\\prime }\\cong f^{\\ast }(y)$ in $G_{T}$ which is sent by $v_{T}$ to $\\xi $ .", "In particular $u_{T}(x)=v_{T}(y^{\\prime })$ and so $R\\subset R^{\\prime }$ .", "Step 3.", "Suppose that in the above pullback diagram the map $u$ is bicovering.", "We can form the cube diagram $@=2ex{{\\mathsf {S}F\\times _{\\mathsf {S}H}\\mathsf {S}G}[rr]^-{(\\mathsf {S}u)^{\\prime }} [dr] [dd]&& {\\mathsf {S}G} [drr]^{\\mathsf {S}v} ^{\\prime }[d][dd]\\\\& {\\mathsf {S}F} [rrr]^{\\mathsf {S}u} [dd]&&& {\\mathsf {S}H} [dd]\\\\{F\\times _{H}G} ^{\\prime }[r][rr] [dr]&& {G} [drr]^{v}\\\\& {F} [rrr]^{u}&&& {H}}$ One clearly has $\\mathsf {S}(F\\times _{H}G)\\cong \\mathsf {S}F\\times _{\\mathsf {S}H}\\mathsf {S}G$ .", "By 2.3(5) the vertical arrows of the cube diagram are trivial fibrations and the map $\\mathsf {S}v$ is an isofibration.", "By Proposition 30(1) $\\mathsf {S}u$ is bicovering, hence by Steps 1 and 2 the map $(\\mathsf {S}u)^{\\prime }$ is bicovering, so $u^{\\prime }$ is bicovering.", "Corollary 34 Let $F\\in {\\rm Fib}(E)$ and $u$ be a bicovering map.", "Then $F\\times u$ is a bicovering map.", "This follows from Example 19(1) and Lemma 33.", "The next result is the first part of [11], with a different proof.", "Corollary 35 If $G$ is a stack then so is $\\mathsf {CART}(F,G)$ for every $F\\in {\\rm Fib}(E)$ .", "This follows from 2.3(4), Example 32 and Corollary 34.", "Lemma 36 An object of Fib$(E)$ that has the right lifting property with respect to all maps that are both cofibrations and $\\mathcal {C}$ -local equivalences is a stack.", "Let $F$ be as in the statement of the Lemma.", "Using Theorem 13 we factorize the map $aF\\colon F\\rightarrow \\mathsf {A}F$ as a cofibration $F\\rightarrow G$ followed by a trivial fibration $G\\rightarrow {\\rm A}F$ .", "By Proposition 30(2) $G$ is a stack.", "By hypothesis the diagram ${{F}@{=} [r] [d]&{F}\\\\{G}\\\\}$ has a diagonal filler, therefore $F$ is a retract of $G$ .", "By Proposition 30(4) $F$ is a stack.", "We shall apply Lemma 9 to the natural generalized model category Fib$(E)$ (Theorem 13).", "Since we have Proposition 30(1), it only remains to prove that every map $F\\rightarrow G$ of Fib$(E)$ can be factorized as a map that is both a cofibration and a $\\mathcal {C}$ -local equivalence followed by a map that has the right lifting property with respect to all maps that are both cofibrations and $\\mathcal {C}$ -local equivalences.", "Consider the diagram ${{F}[r]^{aF} [d]&{\\mathsf {A}F} [d]\\\\{G} [r]^{aG}&{\\mathsf {A}G}}$ We can factorize the map $\\mathsf {A}F\\rightarrow \\mathsf {A}G$ as a map $\\mathsf {A}F\\rightarrow H$ that is an $E$ -equivalence followed by an isofibration $H\\rightarrow \\mathsf {A}G$ .", "By Proposition 30(2) $H$ is a stack, so by Lemma 31 the map $H\\rightarrow \\mathsf {A}G$ has the right lifting property with respect to all maps that are both cofibrations and $\\mathcal {C}$ -local equivalences.", "Therefore the pullback map $G\\times _{\\mathsf {A}G}H\\rightarrow G$ has the right lifting property with respect to all maps that are both cofibrations and $\\mathcal {C}$ -local equivalences.", "By Lemma 33 the map $G\\times _{\\mathsf {A}G}H\\rightarrow H$ is bicovering, therefore the canonical map $F\\rightarrow G\\times _{\\mathsf {A}G}H$ is bicovering.", "We factorize it as a cofibration $F\\rightarrow K$ followed by a trivial fibration $K\\rightarrow G\\times _{\\mathsf {A}G}H$ .", "The desired factorization is $F\\rightarrow K$ followed by the composite $K\\rightarrow G\\times _{\\mathsf {A}G}H\\rightarrow G$ .", "The fact that the fibrant objects of Champ$(E)$ are the stacks follows from Lemmas 31 and 36.", "Left properness of Champ$(E)$ is a consequence of the left properness of Fib$(E)$ and right properness is a consequence of Lemma 33.", "Proposition 37 Every fibration of Champ$(E)$ is a $\\mathcal {C}$ -local fibration.", "Let $F\\rightarrow G$ be a fibration of Champ$(E)$ .", "The argument used in the proof of Theorem 29 shows that $F\\rightarrow G$ is a retract of the composite $K\\rightarrow G\\times _{\\mathsf {A}G}H\\rightarrow G$ .", "We conclude by Proposition 30((3) and (4)).", "Proposition 38 (Compatibility with the 2-category structure) Let $u\\colon F\\rightarrow G$ be a cofibration and $v\\colon H\\rightarrow K$ a fibration in Champ$(E)$ .", "Then the canonical map ${{{\\bf Cart}_{E}(G,H)} [r] &{{\\bf Cart}_{E}(G,K)\\times _{{\\bf Cart}_{E}(F,K)} {\\bf Cart}_{E}(F,H)}}$ is an isofibration that is a surjective equivalence if either $u$ or $v$ is a $\\mathcal {C}$ -local equivalence.", "The first part is contained in Proposition 21 since every fibration of Champ$(E)$ is an isofibration.", "If $v$ is a $\\mathcal {C}$ -local equivalence then $v$ is a trivial fibration and the Proposition is contained in Proposition 21.", "Suppose that $u$ is a $\\mathcal {C}$ -local equivalence.", "By adjunction it suffices to prove that for every injective on objects functor $A\\rightarrow B$ , the canonical map ${{A\\times G\\sqcup _{A\\times F}B\\times F}[r] & {B\\times G}}$ is a cofibration and a $\\mathcal {C}$ -local equivalence (the pushout in the displayed arrow exists by Lemma 17(2)).", "This follows, for example, from Corollary 34.", "Corollary 39 (Compatibility with the `internal hom') Let $u\\colon F\\rightarrow G$ be a cofibration and $v\\colon H\\rightarrow K$ a fibration in Champ$(E)$ .", "Then the canonical map ${{\\mathsf {CART}(G,H)}[r] &{\\mathsf {CART}(G,K)\\times _{\\mathsf {CART}(F,K)} \\mathsf {CART}(F,H)}}$ is a trivial fibration if either $u$ or $v$ is a $\\mathcal {C}$ -local equivalence.", "If $v$ is a $\\mathcal {C}$ -local equivalence then $v$ is a trivial fibration and the Corollary is Corollary 23.", "If $u$ is a $\\mathcal {C}$ -local equivalence the result follows from Proposition 38.", "Proposition 40 The classes of bicoverings and isofibrations make Fib$(E)$ a category of fibrant objects [7].", "A path object was constructed in Remark 20.", "Since we have Lemma 17(1), we conclude by the next result.", "Lemma 41 The maps that are both bicoverings and isofibrations are closed under pullbacks.", "A proof entirely similar to the proof of Lemma 33 can be given.", "We shall give a proof that uses Lemma 33.", "Let ${{F\\times _{H}G}[r]^-{u^{\\prime }}[d]&{G} [d]^{v}\\\\{F} [r]^{u}&{H}}$ be a pullback diagram in Fib$(E)$ with $u$ both an isofibration and a bicovering map.", "We factorize $v$ as $v=pj\\colon G\\rightarrow K\\rightarrow H$ , where $j$ is an $E$ -equivalence and $p$ is an isofibration and then we take successive pullbacks.", "The map $F\\times _{H}K\\rightarrow K$ is a bicovering map by Lemma 33 and an isofibration.", "The map $F\\times _{H}G\\rightarrow F\\times _{H}K$ is an $E$ -equivalence.", "By Proposition 30(1) the map $u^{\\prime }$ is bicovering.", "We give now, as Lemma 42, the analogues, in our context, of [16].", "Let $E$ be a category.", "Let ${{F}[r]^{u} & {H} & {G}[l]_{v}}$ be a digram in Fib$(E)$ .", "The discussion from Section 3.4 and Remark 20 suggest the following model for the homotopy pullback of the previous diagram.", "The objects of the fibre category over $S\\in Ob(E)$ are triples $(x,y,\\theta )$ with $x\\in Ob(F_{S}),y\\in Ob(G_{S})$ and $\\theta \\colon u(x)\\rightarrow v(y)$ an isomorphism in $H_{S}$ .", "The arrows are pairs of arrows making the obvious diagram commute.", "This model is commonly known as the 2-pullback or the iso-comma object of $u$ and $v$ and from now on we shall designate it by $F\\times _{H}^{h}G$ .", "Lemma 42 (Homotopy pullbacks in Champ$(E)$ ) Suppose that $E$ is a site.", "(1) If $F,G$ and $H$ are stacks, then $F\\times _{H}^{h}G$ is weakly equivalent in Fib$(E)$ to the homotopy pullback in Champ$(E)$ of the previous diagram.", "(2) $F\\times _{H}^{h}G$ is weakly equivalent in Champ$(E)$ to the homotopy pullback in Champ$(E)$ of the previous diagram.", "(1) follows from Proposition 11(1).", "(2) follows from Proposition 11(2) and Lemma 41.", "Let $E$ and $E^{\\prime }$ be two sites and $f\\colon E\\rightarrow E^{\\prime }$ be a category fibred in groupoids.", "Then for every $S\\in Ob(E)$ , the induced map $f_{/S}\\colon E_{/S}\\rightarrow E^{\\prime }_{/f(S)}$ sends sieves to sieves.", "Recall from Section 2.3 the adjoint pair $(f^{\\bullet },f_{\\bullet }^{fib})$ .", "Proposition 43 (A change of base) Let $E$ and $E^{\\prime }$ be two sites and $f\\colon E\\rightarrow E^{\\prime }$ be a category fibred in groupoids.", "Suppose that for every $S\\in Ob(E)$ , the map $f_{/S}$ sends a refinement of $S$ to a refinement of $f(S)$ .", "Then the adjoint pair $(f^{\\bullet },f_{\\bullet }^{fib})$ is a Quillen pair between Champ$(E)$ and Champ$(E^{\\prime })$ .", "Since we have Proposition 25, it suffices to show that $f_{\\bullet }^{fib}$ preserves stacks (Sections 3.5 and 3.3).", "Let $F$ be a stack in Fib$(E^{\\prime })$ , $S\\in Ob(E)$ and $R$ be a refinement of $S$ .", "The map $f_{/S}$ is an $E^{\\prime }$ -equivalence and its restriction to $R$ is an $E^{\\prime }$ -equivalence $f_{/S}\\colon R\\rightarrow f_{/S}(R)$ .", "It follows from Proposition 14 that the maps ${\\bf Cart}_{E^{\\prime }}(f_{/S},F)$ are equivalences.", "We have the following commutative diagram (see 2.3(7)) ${{{\\bf Cart}_{E^{\\prime }}(E^{\\prime }_{/f(S)},F)}[r][d]&{{\\bf Cart}_{E^{\\prime }}(E_{/S},F)}[d][r]^{\\cong }&{{\\bf Cart}_{E}(E_{/S},f_{\\bullet }^{fib}F)}[d]\\\\{{\\bf Cart}_{E^{\\prime }}(f_{/S}(R),F)}[r]&{{\\bf Cart}_{E^{\\prime }}(R,F)}[r]^{\\cong }&{{\\bf Cart}_{E}(R,f_{\\bullet }^{fib}F)}}$ The left vertical arrow is an equivalence by assumption, hence $f_{\\bullet }^{fib}F$ is a stack." ], [ "Categories fibred in groupoids", "Let $E$ be a category.", "In this section we give the analogues of Theorems 13 and 29 for the category Fibg$(E)$ defined in Section 2.3.", "Theorem 44 The category Fibg$(E)$ is a proper generalized model category with the $E$ -equivalences as weak equivalences, the maps that are injective on objects as cofibrations and the isofibrations as fibrations.", "It only remains to check axiom A2 from Definition 8.", "This is satisfied by Remark 18.", "Suppose now that $E$ is a site.", "Notice that for every $S\\in Ob(E)$ and every refinement $R$ of $S$ , $E_{/S}$ and $R$ are objects of Fibg$(E)$ .", "We recall from [11] that an object $F$ of Fib$(E)$ is a prestack if for every $S\\in Ob(E)$ and every refinement $R\\subset E_{/S}$ of $S$ , the restriction functor ${\\bf Cart}_{E}(E_{/S},F)\\rightarrow {\\bf Cart}_{E}(R,F)$ is full and faithful.", "Lemma 45 ([11] and [21]) If an object $F$ of Fib$(E)$ is a stack, so is $F^{cart}$ .", "The converse holds provided that $F$ is a prestack.", "We recall that $max\\colon CAT\\rightarrow GRPD$ denotes the maximal groupoid functor.", "We recall that an arbitrary functor $f$ is essentially surjective if and only if the functor $max(f)$ is so and that if $f$ is full and faithful then so is $max(f)$ .", "The Lemma follows then from the following commutative diagram (see 2.3(6)) ${{{\\bf Cartg}_{E}(E_{/S},F^{cart})}[r]^{\\cong } [d]&{max{\\bf Cart}_{E}(E_{/S},F)} [d]\\\\{{\\bf Cartg}_{E}(R,F^{cart})} [r]^{\\cong }&{max{\\bf Cart}_{E}(R,F)}}$ Let $F$ be an object of Fibg$(E)$ , $G$ an object of Fib$(E)$ and $u\\colon F\\rightarrow G$ a bicovering map.", "We claim that $u^{cart}$ is a bicovering map as well.", "For, consider the diagram ${{F^{cart}}@{=}[r][d]_{u^{cart}}&{F} [d]^{u}\\\\{G^{cart}} [r]&{G}}$ One can readily check that the inclusion map $G^{cart}\\rightarrow G$ is an isofibration and that by Lemma 17(1) the above diagram is a pullback.", "We conclude by Lemma 33.", "If, in addition, $G$ is a stack, then the map $G^{cart}\\rightarrow G$ is a bicovering map between stacks (see Lemma 45), hence by [11] it is an $E$ -equivalence.", "It follows that $G$ is an object of Fibg$(E)$ .", "Theorem 46 There is a proper generalized model category Champg$(E)$ on the category Fibg$(E)$ in which the weak equivalences are the bicovering maps, the cofibrations are the maps that are injective on objects and the fibrantions are the fibrations of Champ$(E)$ .", "Using Theorem 44 and Lemma 9 it only remains to prove the factorization of an arbitrary map of Fibg$(E)$ into a map that is both a cofibration and bicovering followed by a map that has the right lifting property with respect to all maps that are both cofibrations and bicoverings.", "The argument is the same as the one given in the proof of Theorem 29.", "For it to work one needs the functor $\\mathsf {A}$ to send objects of Fibg$(E)$ to objects of Fibg$(E)$ .", "This is so by the considerations preceding the statement of the Theorem, applied to the map $F\\rightarrow \\mathsf {A}F$ ." ], [ "Sheaves of categories", "We begin by recalling the notion of sheaf of categories.", "Let $E$ be a small site.", "We recall that $\\widehat{E}$ is the category of presheaves on $E$ and $\\eta \\colon E\\rightarrow \\widehat{E}$ is the Yoneda embedding.", "We denote by $\\widetilde{E}$ the category of sheaves on $E$ and by $a$ the associated sheaf functor, left adjoint to the inclusion functor $i\\colon \\widetilde{E}\\rightarrow \\widehat{E}$ .", "We denote by $\\underline{Hom}$ the internal $CAT$ -hom of the 2-category $[E^{op},CAT]$ and by $X^{(A)}$ the cotensor of $X\\in [E^{op},CAT]$ with a category $A$ .", "Let $Ob\\colon CAT\\rightarrow SET$ denote the set of objects functor; it induces a functor $Ob\\colon [E^{op},CAT]\\rightarrow \\widehat{E}$ .", "Lemma 47 Let $X$ be an object of $[E^{op},CAT]$ .", "The following are equivalent.", "$(a)$ For every category $A$ , $ObX^{(A)}$ is a sheaf [1].", "$(b)$ For every $S\\in Ob(E)$ and every refinement $R$ of $S$ , the natural map $\\underline{Hom}(D\\eta (S),X)\\rightarrow \\underline{Hom}(DR^{\\prime },X)$ is an isomorphism, where $R^{\\prime }$ is the sub-presheaf of $\\eta (S)$ which corresponds to $R$ .", "$(c)$ For every $S\\in Ob(E)$ and every refinement $R$ of $S$ , the natural map $X(S)\\rightarrow \\underset{R^{op}}{\\rm lim}(X|R)$ is an isomorphism, where $(X|R)$ is the composite $R^{op}\\rightarrow (E_{/S})^{op}\\rightarrow E^{op}\\overset{X}{\\rightarrow }CAT$ .", "An object $X$ of $[E^{op},CAT]$ is a sheaf on $E$ with values in $CAT$ (simply, sheaf of categories) if it satisfies one of the conditions of Lemma 47.", "We denote by Faisc$(E;CAT)$ the full subcategory of $[E^{op},CAT]$ whose objects are the sheaves of categories.", "The category $[E^{op},CAT]$ is equivalent to the category Cat$(\\widehat{E})$ of internal categories and internal functors in $\\widehat{E}$ and Faisc$(E;CAT)$ is equivalent to the category Cat$(\\widetilde{E})$ of internal categories and internal functors in $\\widetilde{E}$ [1].", "Consider now the adjunctions ${{{\\rm Fib}(E)}[r]^{\\mathsf {L}}&{{\\rm Cat}(\\widehat{E})}@<1ex>[l]^{\\Phi }[r]^{a}&{{\\rm Cat}(\\widetilde{E})}@<1ex>[l]^{i}}$ (see Section 2.3 for the adjoint pair $(\\mathsf {L},\\Phi )$ ).", "We denote the unit of the adjoint pair $(a,i)$ by $\\mathsf {k}$ .", "Theorem 48 There is a right proper model category Stack$(\\widetilde{E})_{proj}$ on the category Cat$(\\widetilde{E})$ in which the weak equivalences and the fibrations are the maps that $\\Phi $ takes into weak equivalences and fibrations of Champ$(E)$ .", "The adjoint pair $(a\\mathsf {L},\\Phi i)$ is a Quillen equivalence between Champ$(E)$ and Stack$(\\widetilde{E})_{proj}$ .", "The prove the existence of the model category Stack$(\\widetilde{E})_{proj}$ we shall use Lemma 49 below and the following facts: (1) if $X$ is a sheaf of categories then $\\Phi X$ is a prestack [11]; (2) for every $X\\in [E^{op},CAT]$ , the natural map $\\Phi \\mathsf {k}(X)\\colon \\Phi X\\rightarrow \\Phi iaX$ is bicovering [11]; (3) if $X$ is a sheaf of categories then $\\Phi ia\\mathcal {S}\\Phi X$ is a stack (which is a consequence of) [11].", "See the end of this section for another proof of (1).", "We recall that the weak equivalences of Stack$(\\widetilde{E})_{proj}$ have a simplified description.", "Let $f$ be a map of Cat$(\\widetilde{E})$ .", "By [11] the map $\\Phi f$ is bicovering if and only if $\\Phi f$ is full and faithful and $\\Phi f$ is `locally essentially surjective on objects'.", "Given any map $u$ of Fib$(E)$ , the underlying functor of $u$ is full and faithful if and only if for every $S\\in Ob(E)$ , $u_{S}$ is full and faithful [13].", "Hence $f$ is a weak equivalence if and only if $f$ is full and faithful and $\\Phi f$ is `locally essentially surjective on objects'.", "Lemma 49 Let $\\mathcal {M}$ be a generalized model category.", "Suppose that there is a set $I$ of maps of $\\mathcal {M}$ such that a map of $\\mathcal {M}$ is a trivial fibration if and only if it has the right lifting property with respect to every element of $I$ .", "Let $\\mathcal {N}$ be a complete and cocomplete category and let $F\\colon \\mathcal {M}\\rightleftarrows \\mathcal {N}\\colon G$ be a pair of adjoint functors.", "Assume that (1) the set $F(I)=\\lbrace F(u)\\ |\\ u\\in I\\rbrace $ permits the small object argument [14]; (2) $\\mathcal {M}$ is right proper; (3) $\\mathcal {N}$ has a fibrant replacement functor, which means that there are $(i)$ a functor $\\widehat{{\\rm F}}\\colon \\mathcal {N}\\rightarrow \\mathcal {N}$ such that for every object $X$ of $\\mathcal {N}$ the object $G\\widehat{{\\rm F}}X$ is fibrant and $(ii)$ a natural transformation from the identity functor of $\\mathcal {N}$ to $\\widehat{{\\rm F}}$ such that for every object $X$ of $\\mathcal {N}$ the map $GX\\rightarrow G\\widehat{{\\rm F}}X$ is a weak equivalence; (4) every fibrant object of $\\mathcal {N}$ has a path object, which means that for every object $X$ of $\\mathcal {N}$ such that $GX$ is fibrant there is a factorization ${{X}[r]^{s} &{PathX}[r]^{p_{0}\\times p_{1}} & {X\\times X}}$ of the diagonal map $X\\rightarrow X\\times X$ such that $G(s)$ is a weak equivalence and $G(p_{0}\\times p_{1})$ is a fibration.", "Then $\\mathcal {N}$ becomes a right proper model category in which the weak equivalences and the fibrations are the maps that $G$ takes into weak equivalences and fibrations.", "The adjoint pair $(F,G)$ is a Quillen equivalence if and only if for every cofibrant object $A$ of $\\mathcal {M}$ , the unit map $A\\rightarrow GFA$ of the adjunction is a weak equivalence.", "Let $f$ be a map of $\\mathcal {N}$ .", "We say that $f$ is a trivial fibration if $G(f)$ is a trivial fibration and we say that $f$ is a cofibration if it is an $F(I)$ -cofibration in the sense of [14].", "By (1) and [14] every map of $\\mathcal {N}$ can be factorized into a cofibration followed by a trivial fibration and every cofibration has the left lifting property with respect to every trivial fibration.", "Let $f\\colon X\\rightarrow Y$ be a map of $\\mathcal {N}$ such that $GX$ and $GY$ are fibrant.", "Then (4) implies that we can construct the mapping path factorization of $f$ (see Section 3.0.6, for instance), that is, $f$ can be factorized into a map $X\\rightarrow Pf$ that is a weak equivalence followed by a map $Pf\\rightarrow Y$ that is a fibration.", "Moreover, $GPf$ is fibrant.", "We show that every map $f\\colon X\\rightarrow Y$ of $\\mathcal {N}$ can be factorized into a map that is both a cofibration and a weak equivalence followed by a map that is a fibration.", "By (3) we have a commutative diagram ${{X} [r] [d]_{f} &{\\widehat{{\\rm F}}X}[d]^{\\widehat{{\\rm F}}f}\\\\{Y} [r] &{\\widehat{{\\rm F}}Y}}$ The map $\\widehat{{\\rm F}}f$ can be factorized into a map $\\widehat{{\\rm F}}X\\rightarrow P\\widehat{{\\rm F}}f$ that is a weak equivalence followed by a map $P\\widehat{{\\rm F}}f\\rightarrow \\widehat{{\\rm F}}Y$ that is a fibration.", "Let $Z$ be the pullback of $P\\widehat{{\\rm F}}f\\rightarrow \\widehat{{\\rm F}}Y$ along $Y\\rightarrow \\widehat{{\\rm F}}Y$ .", "By (2) the map $Z\\rightarrow P\\widehat{{\\rm F}}f$ is a weak equivalence, therefore the canonical map $X\\rightarrow Z$ is a weak equivalence.", "We factorize $X\\rightarrow Z$ into a map $X\\rightarrow X^{\\prime }$ that is a cofibration followed by a map $X^{\\prime }\\rightarrow Z$ that is a trivial fibration.", "The desired factorization of $f$ is $X\\rightarrow X^{\\prime }$ followed by the composite $X^{\\prime }\\rightarrow Z\\rightarrow Y$ .", "We show that every commutative diagram in $\\mathcal {N}$ ${{A} [r] [d]_{j} & {X} [d]^{p}\\\\{B} [r] & {Y}}$ where $j$ is both a cofibration and a weak equivalence and $p$ is a fibration has a diagonal filler.", "We shall construct a commutative diagram ${{A} [r] [d]_{j} & {X^{\\prime }}[r] [d]^{q} & {X} [d]^{p}\\\\{B} [r] & {Y^{\\prime }} [r] & {Y}}$ with $q$ a trivial fibration.", "We factorize the map $B\\rightarrow Y$ into a map $B\\rightarrow Y^{\\prime }$ that is both a cofibration and a weak equivalence followed by a map $Y^{\\prime }\\rightarrow Y$ that is a fibration.", "Similarly, we factorize the canonical map $A\\rightarrow Y^{\\prime }\\times _{Y}X$ into a map $A\\rightarrow X^{\\prime }$ that is both a cofibration and a weak equivalence followed by a map $X^{\\prime }\\rightarrow Y^{\\prime }\\times _{Y}X$ that is a fibration.", "Let $q$ be the composite map $X^{\\prime }\\rightarrow Y^{\\prime }$ .", "Then $q$ is a trivial fibration.", "The model category $\\mathcal {N}$ is right proper since $\\mathcal {M}$ is right proper.", "Suppose that $(F,G)$ is a Quillen equivalence.", "Let $A$ be a cofibrant object of $\\mathcal {M}$ .", "We can find a weak equivalence $f\\colon FA\\rightarrow X$ with $X$ fibrant.", "The composite map $A\\rightarrow GFA\\rightarrow GX$ is the adjunct of $f$ , hence it is a weak equivalence.", "Thus, $A\\rightarrow GFA$ is a weak equivalence.", "Conversely, let $A$ be a cofibrant object of $\\mathcal {M}$ and $X$ a fibrant object of $\\mathcal {N}$ .", "If $FA\\rightarrow X$ is a weak equivalence then its adjunct is the composite $A\\rightarrow GFA\\rightarrow GX$ , which is a weak equivalence.", "If $f\\colon A\\rightarrow GX$ is a weak equivalence, then it factorizes as $A\\rightarrow GFA\\overset{Gf^{\\prime }}{\\rightarrow }GX$ , where $f^{\\prime }$ is the adjunct of $f$ .", "Hence $Gf^{\\prime }$ is a weak equivalence, which means that $f^{\\prime }$ is a weak equivalence.", "In Lemma 49 we take $\\mathcal {M}={\\rm Champ}(E)$ , $\\mathcal {N}={\\rm Cat}(\\widetilde{E})$ , $F=a\\mathsf {L}$ , $G=\\Phi i$ and $I$ to be the set of maps $\\lbrace f\\times E_{/S}\\rbrace $ with $S\\in Ob(E)$ and $f\\in {M}$ , where ${M}$ is the set of functors such that a functor is a surjective equivalence if and only if it has the right lifting property with respect to every element of ${M}$ .", "By Theorem 13 and Proposition 16(2) a map of Fib$(E)$ is a trivial fibration if and only if it has the right lifting property with respect to every element of $I$ .", "We shall now check the assumptions (1)-(4) of Lemma 49.", "(1) and (2) are clear.", "We check (3).", "Let $X$ be a sheaf of categories.", "We put $\\widehat{F}X=ia\\mathcal {S}\\Phi X$ and the natural transformation from the identity functor of Cat$(\\widetilde{E})$ to $\\widehat{F}$ to be the composite map ${{X}[r]&{\\mathcal {S}\\Phi X}[r]^{\\mathsf {k}(\\mathcal {S}\\Phi X)}&{ia\\mathcal {S}\\Phi X}}$ Assumption (3) of Lemma 49 is fulfilled by the facts (2) and (3) mentioned right after the statement of Theorem 48.", "We check (4).", "Let $X$ be a sheaf of categories such that $\\Phi X$ is a stack.", "Let $J$ be the groupoid with two objects and one isomorphism between them.", "The diagonal $X\\rightarrow X\\times X$ factorizes as ${{X}[r]^{s} &{X^{(J)}}[r]^{p_{0}\\times p_{1}} & {X\\times X}}$ Since $\\Phi $ preserves cotensors and the cotensor of a stack and a category is a stack (2.3(2)), (4) follows from Remark 20.", "We now prove that $(a\\mathsf {L},\\Phi i)$ is a Quillen equivalence.", "For this we use Lemma 49.", "For every object $F$ of Fib$(E)$ , the unit $F\\rightarrow \\Phi ia\\mathsf {L}F$ of this adjoint pair is the composite ${{F}[r]^{\\mathsf {l}_{F}}&{\\Phi \\mathsf {L}F}[r]^{\\Phi \\mathsf {k}(\\mathsf {L}F)}&{\\Phi ia\\mathsf {L}F}}$ which is a bicovering map.", "Theorem 50 The model category Cat$(\\widehat{E})_{proj}$ admits a proper left Bousfield localization Stack$(\\widehat{E})_{proj}$ in which the weak equivalences and the fibrations are the maps that $\\Phi $ takes into weak equivalences and fibrations of Champ$(E)$ .", "The adjoint pair $(\\mathsf {L},\\Phi )$ is a Quillen equivalence between Champ$(E)$ and Stack$(\\widehat{E})_{proj}$ .", "The proof is similar to the proof of Theorem 48, using the adjoint pair $(\\mathsf {L},\\Phi )$ and the fibrant replacement functor ${{X}[r]&{\\mathcal {S}\\Phi X}[r]^{\\mathcal {S}(a\\Phi X)}&{\\mathcal {S}\\mathsf {A}\\Phi X}}$ Proposition 51 (Compatibility with the 2-category structure) Let $A\\rightarrow B$ be an injective on objects functor and $X\\rightarrow Y$ a fibration of Stack$(\\widetilde{E})_{proj}$ .", "Then the canonical map ${{X^{(B)}} [r] &{X^{(A)}\\times _{Y^{(A)}}Y^{(B)}}}$ is a fibration that is a trivial fibration if $A\\rightarrow B$ is an equivalence of categories or $X\\rightarrow Y$ is a weak equivalence.", "Since $\\Phi $ preserves cotensors, the Proposition follows from Proposition 38.", "We recall [19] that Cat$(\\widetilde{E})$ is a model category in which the weak equivalences are the maps that $\\Phi $ takes into bicovering maps and the cofibrations are the internal functors that are monomorphisms on objects.", "See Appendix 3 for another approach to this result.", "We denote this model category by Stack$(\\widetilde{E})_{inj}$ .", "Proposition 52 The identity functors on ${\\rm Cat}(\\widetilde{E})$ form a Quillen equivalence between Stack$(\\widetilde{E})_{proj}$ and Stack$(\\widetilde{E})_{inj}$ .", "We show that the identity functor ${\\rm Stack}(\\widetilde{E})_{proj}\\rightarrow {\\rm Stack}(\\widetilde{E})_{inj}$ preserves cofibrations.", "For that, it suffices to show that for every object $F$ of Fib$(E)$ and every injective on objects functor $f$ , the map $a\\mathsf {L}(f\\times F)$ is a cofibration of Stack$(\\widetilde{E})_{inj}$ .", "The map $\\mathsf {L}(f\\times F)$ is objectwise injective on objects (see the proof of Proposition 24), which translates in Cat$(\\widehat{E})$ as: $\\mathsf {L}(f\\times F)$ is an internal functor having the property that is a monomorphism on objects.", "But the associated sheaf functor $a$ is known to preserve this property.", "Since the classes of weak equivalences of the two model categories are the same, the result follows.", "Let $E^{\\prime }$ be another small site and $f^{-1}\\colon E\\rightarrow E^{\\prime }$ be the functor underlying a morphisms of sites $f\\colon E^{\\prime }\\rightarrow E$ [11].", "The adjoint pair $f^{\\ast }\\colon \\widetilde{E}\\rightleftarrows \\widetilde{E^{\\prime }}\\colon f_{\\ast }$ induces an adjoint pair $f^{\\ast }\\colon {\\rm Cat}(\\widetilde{E})\\rightleftarrows {\\rm Cat}(\\widetilde{E^{\\prime }})\\colon f_{\\ast }$ .", "Proposition 53 (Change of site) The adjoint pair $(f^{\\ast },f_{\\ast })$ is a Quillen pair between Stack$(\\widetilde{E})_{proj}$ and Stack$(\\widetilde{E^{\\prime }})_{proj}$ .", "Consider the diagram ${{{\\rm Fib}(E)}&{{\\rm Fib}(E^{\\prime })}[l]_{f_{\\bullet }^{fib}}\\\\{[E^{op},CAT]}[u]^{\\Phi }&{[E^{\\prime op},CAT]}[u]_{\\Phi ^{\\prime }}[l]^{f_{\\ast }}\\\\{{\\rm Cat}(\\widetilde{E})}[u]^{i}&{{\\rm Cat}(\\widetilde{E^{\\prime }})}[l]^{f_{\\ast }}[u]_{i^{\\prime }}}$ where $f_{\\bullet }^{fib}$ was defined in Section 2.3 and $f_{\\ast }\\colon [E^{\\prime op},CAT]\\rightarrow [E^{op},CAT]$ is the functor obtained by composing with $f$ .", "It is easy to check that the functor $f_{\\bullet }^{fib}$ preserves isofibrations and trivial fibrations.", "By [11] it also preserves stacks.", "Since $f_{\\bullet }^{fib}\\Phi ^{\\prime }=\\Phi f_{\\ast }$ , it follows that $f_{\\ast }$ preserves trivial fibrations and the fibrations between fibrant objects.", "Let $p\\colon C\\rightarrow I$ be a fibred site [2] and $\\tilde{p}\\colon \\tilde{C}^{/I}\\rightarrow I$ be the (bi)fibred topos associated to $p$ [2].", "Using the above considerations we obtain a bifibration ${\\rm Cat}(\\tilde{C}^{/I})\\rightarrow I$ whose fibres are isomorphic to ${\\rm Cat}(\\widetilde{C_{i}})$ , hence by they are model categories.", "Moreover, by Proposition 53 the inverse and direct image functors are Quillen pairs.", "An elementary example of a fibred site is the Grothendieck construction associated to the functor that sends a topological space $X$ to the category $\\mathcal {O}(X)$ whose objects are the open subsets of $X$ and whose arrows are the inclusions of subsets.", "Proposition 54 Let $E$ and $E^{\\prime }$ be two small sites and $f\\colon E\\rightarrow E^{\\prime }$ be a category fibred in groupoids.", "Suppose that for every $S\\in Ob(E)$ , the map $E_{/S}\\rightarrow E^{\\prime }_{/f(S)}$ sends a refinement of $S$ to a refinement of $f(S)$ .", "Then $f$ induces a Quillen pair between Stack$(\\widetilde{E})_{proj}$ and Stack$(\\widetilde{E^{\\prime }})_{proj}$ .", "The proof is similar to the proof of Proposition 53.", "Consider the solid arrow diagram ${{{\\rm Fib}(E)}&{{\\rm Fib}(E^{\\prime })}[l]_{f_{\\bullet }^{fib}}\\\\{[E^{op},CAT]}@<.5ex>[r]^{f_{!", "}}[u]^{\\Phi }@<.5ex>[d]^{a}&{[E^{\\prime op},CAT]}@<.5ex>[d]^{a^{\\prime }}[u]_{\\Phi ^{\\prime }}@<.5ex>[l]^{f^{\\ast }}\\\\{{\\rm Cat}(\\widetilde{E})}@<.5ex>[u]^{i}&{{\\rm Cat}(\\widetilde{E^{\\prime }})}@{.>}[l]^{f^{\\ast }}@<.5ex>[u]^{i^{\\prime }}}$ where $f_{!", "}$ is the left adjoint to the functor $f^{\\ast }$ obtained by composing with $f$ .", "We claim that the composition with $f$ functor $f^{\\ast }\\colon \\widehat{E^{\\prime }}\\rightarrow \\widehat{E}$ preserves sheaves.", "Let $X$ be a sheaf on $E^{\\prime }$ .", "By Example 28 it suffices to show that $\\Phi Df^{\\ast }X$ is a stack.", "But $\\Phi Df^{\\ast }X=f_{\\bullet }^{fib}\\Phi ^{\\prime }DX$ , so $f^{\\ast }X$ is a sheaf by Proposition 43.", "Therefore, $f^{\\ast }$ induces a functor $f^{\\ast }\\colon {\\rm Cat}(\\widetilde{E^{\\prime }})\\rightarrow {\\rm Cat}(\\widetilde{E})$ .", "Since $f^{\\ast }i^{\\prime }=if^{\\ast }$ , a formal argument implies that $a^{\\prime }f_{!", "}i$ is left adjoint to $f^{\\ast }$ .", "The fact that $(a^{\\prime }f_{!", "}i,f^{\\ast })$ is a Quillen pair follows from Proposition 43.", "Here is an application of Proposition 54.", "For every $S\\in Ob(E)$ , the category $E_{/S}$ has the induced topology [11].", "A map $T\\rightarrow S$ of $E$ induces a category fibred in groupoids $E_{/T}\\rightarrow E_{/S}$ .", "The assumption of Proposition 54 is satisfied.", "By [11] we obtain a stack over $E$ whose fibres are model categories and such that the inverse and direct image functors are Quillen pairs." ], [ "Sheaves of categories\nare prestacks", "Let $E$ be a site.", "We recall that an object $F$ of Fib$(E)$ is a prestack if for every $S\\in Ob(E)$ and every refinement $R\\subset E_{/S}$ of $S$ , the restriction functor ${\\bf Cart}_{E}(E_{/S},F)\\rightarrow {\\bf Cart}_{E}(R,F)$ is full and faithful.", "We give here an essentially-from-the-definition proof of [11], namely that if $X\\in [E^{op},CAT]$ is a sheaf of categories then $\\Phi X$ is a prestack.", "Let first $X\\in [E^{op},CAT]$ .", "Let $D\\colon SET\\rightarrow CAT$ be the discrete category functor; it induces a functor $D\\colon \\widehat{E}\\rightarrow [E^{op},CAT]$ .", "Let $R^{\\prime }$ be the sub-presheaf of $\\eta (S)$ which corresponds to $R$ .", "We have the following commutative diagram ${{{\\bf Cart}_{E}(E_{/S},\\Phi X)}[r][dd]&{(\\Phi X)_{S}=\\underline{Hom}(D\\eta (S),X)}[d]^{(I)}\\\\&{\\underline{Hom}(DR^{\\prime },X)}[d]^{(II)}\\\\{{\\bf Cart}_{E}(R,\\Phi X)}[r]&{\\underline{Hom}(DR^{\\prime },\\mathcal {S}\\Phi X)}}$ The top horizontal arrow is a surjective equivalence (2.3(5)).", "Since $(\\Phi ,\\mathcal {S})$ is a 2-adjunction, the bottom horizontal arrow is an isomorphism.", "We will show below that the map $(II)$ is full and faithful.", "If $X$ is now a sheaf of categories, then the map $(I)$ is an isomorphism by Lemma 47, therefore in this case $\\Phi X$ is a prestack.", "Let $P\\in \\widehat{E}$ .", "We denote by $E_{/P}$ the category $\\Phi DP$ .", "Let $m\\colon E_{/P}\\rightarrow E$ be the canonical map.", "The natural functor $m^{\\ast }\\colon [E^{op},CAT]\\rightarrow [(E_{/P})^{op},CAT]$ has a left adjoint $m_{!", "}$ that is the left Kan extension along $m^{op}$ .", "Since $m^{op}$ is an opfibration, $m_{!", "}$ has a simple description.", "For example, let $A$ be a category and let $cA\\in [(E_{/P})^{op},CAT]$ be the constant object at $A$ ; then $m_{!", "}cA$ is the tensor between $A$ and $DP$ in the 2-category $[E^{op},CAT]$ .", "It follows that for every $X\\in [E^{op},CAT]$ we have an isomorphism $\\underset{(E_{/P})^{op}}{\\rm lim}m^{\\ast }X\\cong \\underline{Hom}(DP,X)$ The map $X\\rightarrow \\mathcal {S}\\Phi X$ is objectwise both an equivalence of categories and injective on objects, hence so is the map $m^{\\ast }X\\rightarrow m^{\\ast }\\mathcal {S}\\Phi X$ .", "Therefore the map $\\underset{(E_{/P})^{op}}{\\rm lim}m^{\\ast }X\\rightarrow \\underset{(E_{/P})^{op}}{\\rm lim}m^{\\ast }\\mathcal {S}\\Phi X$ is both full and faithful and injective on objects." ], [ "Appendix 1: Stacks vs.\nthe homotopy sheaf condition", "Throughout this section $E$ is a site whose topology is generated by a pretopology." ], [ "We recall that the model category $CAT$ is a simplicial model category.", "The cotensor $A^{(K)}$ between a category $A$ and a simplicial set $K$ is constructed as follows.", "Let S be the category of simplicial sets.", "Let $cat\\colon {\\bf S}\\rightarrow CAT$ be the fundamental category functor, left adjoint to the nerve functor.", "Let $(-)_{1}^{-1}\\colon CAT\\rightarrow GRPD$ be the free groupoid functor, left adjoint to the inclusion functor.", "Then $A^{(K)}=[(cat K)_{1}^{-1},A]$ One has $A^{(\\Delta [n])}=[J^{n},A]$ , where $J^{n}$ is the free groupoid on $[n]$ .", "Let X be a cosimplicial object in $CAT$ .", "The total object of X [14] is calculated as ${\\rm Tot}{\\bf X}=\\underline{Hom}(J,{\\bf X})$ where $\\underline{Hom}$ is the $CAT$ -hom of the 2-category $[\\Delta ,CAT]$ and $J$ is the cosimplicial object in $CAT$ that $J^{n}$ defines.", "The category $\\underline{Hom}(J,{\\bf X})$ has a simple description.", "For $n\\ge 2$ , $J^{n}$ is constructed from $J^{1}$ by iterated pushouts, so by adjunction an object of $\\underline{Hom}(J,{\\bf X})$ is a pair $(x,f)$ , where $x\\in Ob({\\bf X}^{0})$ and $f\\colon d^{1}(x)\\rightarrow d^{0}(x)$ is an isomorphism of ${\\bf X}^{1}$ such that $s^{0}(f)$ is the identity on $x$ and $d^{1}(f)=d^{0}(f)d^{2}(f)$ .", "An arrow $(x,f)\\rightarrow (y,g)$ is an arrow $u\\colon x\\rightarrow y$ of ${\\bf X}^{0}$ such that $d^{0}(u)f=gd^{1}(u)$ .", "If X moreover a coaugmented cosimplicial object in $CAT$ with coaugmentation ${\\bf X}^{-1}$ , there is a natural map ${\\bf X}^{-1}\\rightarrow {\\rm Tot}{\\bf X}$ We recall [14] that if X is Reey fibrant in $[\\Delta ,CAT]$ , then the natural map ${\\rm Tot}{\\bf X}\\rightarrow {\\rm holim}{\\bf X}$ is an equivalence of categories." ], [ "For each $S\\in Ob(E)$ and each covering family ${S}=(S_{i}\\rightarrow S)_{i\\in I}$ there is a simplicial object $E_{/{S}}$ in Fib$(E)$ given by $(E_{/{S}})_{n}=\\underset{i_{0},...,i_{n}\\in I^{n+1}}{\\coprod }E_{/S_{i_{0},...,i_{n}}}$ where $S_{i_{0},...,i_{n}}=S_{i_{0}}\\times _{S}...\\times _{S}S_{i_{n}}$ .", "$E_{/{S}}$ is augmented with augmentation $E_{/S}$ .", "Proposition 55 An object $F$ of Fib$(E)$ is a stack if and only if for every $S\\in Ob(E)$ and every covering family ${S}=(S_{i}\\rightarrow S)$ , the natural map ${\\bf Cart}_{E}(E_{/S},F)\\rightarrow {\\rm Tot}{\\bf Cart}_{E}(E_{/{S}},F)$ is an equivalence of categories.", "The proof consists of unraveling the definitions.", "Following [15], we say that an object $F$ of Fib$(E)$ satisfies the homotopy sheaf condition if for every $S\\in Ob(E)$ and every covering family ${S}=(S_{i}\\rightarrow S)$ , the natural map ${\\bf Cart}_{E}(E_{/S},F)\\rightarrow {\\rm holim}{\\bf Cart}_{E}(E_{/{S}},F)$ is an equivalence of categories.", "Proposition 56 [15] An object of Fib$(E)$ satisfies the homotopy sheaf condition if and only if it is a stack.", "This follows from Proposition 55 and Lemma 57.", "Lemma 57 For every $S\\in Ob(E)$ , every covering family ${S}=(S_{i}\\rightarrow S)_{i\\in I}$ and every object $F$ of Fib$(E)$ , the cosimplicial object in $CAT$ ${\\bf Cart}_{E}(E_{/{S}},F)$ is Reedy fibrant.", "By Proposition 21 the adjoint pair $F^{(-)}\\colon CAT\\rightleftarrows {\\rm Fib}(E)^{op}\\colon {\\bf Cart}_{E}(-,F)$ is a Quillen pair.", "Therefore, to prove the Lemma it suffices to show that $E_{/{S}}$ is Reedy cofibrant, by which we mean that for every $[n]\\in Ob(\\Delta )$ the latching object of $E_{/{S}}$ at $[n]$ , denoted by $L_{n}E_{/{S}}$ , exists and the natural map $L_{n}E_{/{S}}\\rightarrow (E_{/{S}})_{n}$ is injective on objects.", "A way to prove this is by using Lemma 58." ], [ "Latching objects\nof simplicial objects", "In general, the following considerations may help deciding whether a simplicial object in a generalized model category is Reedy cofibrant.", "Let $\\mathcal {M}$ be a category and X a simplicial object in $\\mathcal {M}$ .", "We recall that the latching object of X at $[n]\\in Ob(\\Delta )$ is $L_{n}{\\bf X}=\\underset{\\partial ([n]\\downarrow \\overleftarrow{\\Delta })^{op}}{\\rm colim}{\\bf X}$ provided that the colimit exists.", "Here $\\overleftarrow{\\Delta }$ is the subcategory of $\\Delta $ consisting of the surjective maps and $\\partial ([n]\\downarrow \\overleftarrow{\\Delta })$ is the full subcategory of $([n]\\downarrow \\overleftarrow{\\Delta })$ containing all the objects except the identity map of $[n]$ .", "Below we shall review the construction of $L_{n}{\\bf X}$ .", "The category $([n]\\downarrow \\overleftarrow{\\Delta })$ has the following description [12].", "The identity map of $[n]$ is its initial object.", "Any other object is of the form $s^{i_{1}}...s^{i_{k}}\\colon [n]\\rightarrow [n-k]$ , where $s^{j}$ denotes a codegeneracy operator, $1\\le k\\le n$ and $0\\le i_{1}\\le ...\\le i_{k}\\le n-1$ .", "For $n\\ge 0$ we let $\\underline{n}$ be the set $\\lbrace 1,2,...,n\\rbrace $ , with the convention that $\\underline{0}$ is the empty set.", "We denote by $\\mathcal {P}(\\underline{n})$ the power set of $\\underline{n}$ .", "$\\mathcal {P}(\\underline{n})$ is a partially ordered set.", "We set $\\mathcal {P}_{0}(\\underline{n})=\\mathcal {P}(\\underline{n})\\setminus \\lbrace \\emptyset \\rbrace $ and $\\mathcal {P}_{1}(\\underline{n})=\\mathcal {P}(\\underline{n})\\setminus \\lbrace \\underline{n}\\rbrace $ .", "There is an isomorphism $([n]\\downarrow \\overleftarrow{\\Delta })\\cong \\mathcal {P}(\\underline{n})$ which sends the identity map of $[n]$ to $\\emptyset $ and the object $s^{i_{1}}...s^{i_{k}}\\colon [n]\\rightarrow [n-k]$ as above to $\\lbrace i_{1}+1,...,i_{k}+1\\rbrace $ .", "Under this isomorphism the category $\\partial ([n]\\downarrow \\overleftarrow{\\Delta })$ corresponds to $\\mathcal {P}_{0}(\\underline{n})$ , therefore $\\partial ([n]\\downarrow \\overleftarrow{\\Delta })^{op}$ is isomorphic to $\\mathcal {P}_{1}(\\underline{n})$ .", "The displayed isomorphism is natural in the following sense.", "Let $Dec^{1}\\colon \\Delta \\rightarrow \\Delta $ be $Dec^{1}([n])=[n]\\sqcup [0]\\cong [n+1]$ .", "Then we have a commutative diagram ${{([n]\\downarrow \\overleftarrow{\\Delta })}[r]^{\\cong }[d]_{Dec^{1}}&{\\mathcal {P}(\\underline{n})}[d]\\\\{([n+1]\\downarrow \\overleftarrow{\\Delta })}[r]^{\\cong }&{\\mathcal {P}(\\underline{n+1})}}$ in which the unlabelled vertical arrow is the inclusion.", "Restricting the arrow $Dec^{1}$ to $\\partial $ and then taking the opposite category we obtain a commutative diagram in which the unlabelled vertical arrow becomes $-\\cup \\lbrace n+1\\rbrace \\colon \\mathcal {P}_{1}(\\underline{n})\\rightarrow \\mathcal {P}_{1}(\\underline{n+1})$ .", "For $n\\ge 1$ the category $\\mathcal {P}_{1}(\\underline{n})$ is constructed inductively as the Grothendieck construction applied to the functor $(2\\leftarrow 1\\rightarrow 0)\\rightarrow CAT$ given by $\\ast \\leftarrow \\mathcal {P}_{1}(\\underline{n-1})=\\mathcal {P}_{1}(\\underline{n-1})$ .", "Therefore colimits indexed by $\\mathcal {P}_{1}(\\underline{n})$ have the following description.", "Let ${\\bf Y}\\colon \\mathcal {P}_{1}(\\underline{n})\\rightarrow \\mathcal {M}$ .", "We denote by Y the precomposition of Y with the inclusion $\\mathcal {P}_{1}(\\underline{n-1})\\subset \\mathcal {P}_{1}(\\underline{n})$ ; then $\\underset{\\mathcal {P}_{1}(\\underline{n})}{\\rm colim}{\\bf Y}$ is the pushout of the diagram ${{\\underset{\\mathcal {P}_{1}(\\underline{n-1})}{\\rm colim}{\\bf Y}}[r][d] &{\\underset{\\mathcal {P}_{1}(\\underline{n-1})}{\\rm colim}{\\bf Y}(-\\cup \\lbrace n\\rbrace )}\\\\{{\\bf Y}_{\\underline{n-1}}}}$ provided that the pushout and all the involved colimits exist.", "Let $\\overleftarrow{{\\bf X}}$ be the restriction of X to $(\\overleftarrow{\\Delta })^{op}$ .", "Notice that the definition of the latching object of X uses only $\\overleftarrow{{\\bf X}}$ .", "Summing up, $L_{n}{\\bf X}$ is the pushout of the diagram ${{L_{n-1}\\overleftarrow{{\\bf X}}}[r][d] &{L_{n-1}Dec^{1}(\\overleftarrow{{\\bf X}})}\\\\{{\\bf X}_{n-1}}}$ provided that the pushout and all the involved colimits exist, where $\\overleftarrow{{\\bf X}}\\rightarrow Dec^{1}(\\overleftarrow{{\\bf X}})$ is induced by $s^{n}\\colon [n]\\sqcup [0]\\rightarrow [n]$ .", "Thus, we have Lemma 58 Let $\\mathcal {M}$ be a generalized model category and X a simplicial object in $\\mathcal {M}$ .", "Let $n\\ge 1$ .", "If $L_{n-1}\\overleftarrow{{\\bf X}}$ and $L_{n-1}Dec^{1}(\\overleftarrow{{\\bf X}})$ exist and the map $L_{n-1}\\overleftarrow{{\\bf X}}\\rightarrow {\\bf X}_{n-1}$ is a cofibration, then $L_{n}{\\bf X}$ exists." ], [ "Appendix 2: Left Bousfield localizations\nand change of cofibrations", "In this section we essentially propose an approach to the existence of left Bousfield localizations of `injective'-like model categories.", "The approach is based on the existence of both the un-localized `injective'-like model category and the left Bousfield localization of the `projective'-like model category.", "We give a full description of the fibrations of these localized `injective'-like model categories; depending on one's taste, the description may or may not be satisfactory.", "The approach uses only simple factorization and lifting arguments.", "Let $\\mathcal {M}_{1}=({\\rm W},{\\rm C}_{1},{\\rm F}_{1})$ and $\\mathcal {M}_{2}=({\\rm W},{\\rm C}_{2},{\\rm F}_{2})$ be two model categories on a category $\\mathcal {M}$ , where, as usual, W stands for the class of weak equivalences, C stands for the class of cofibrations, and F for the class of fibrations.", "We assume that ${\\rm C}_{1}\\subset {\\rm C}_{2}$ .", "Let ${\\rm W}^{\\prime }$ be a class of maps of $\\mathcal {M}$ that contains W and has the two out of three property.", "We define ${\\rm F}_{1}^{\\prime }$ to be the class of maps having the right lifting property with respect to every map of ${\\rm C}_{1}\\cap {\\rm W}^{\\prime }$ , and we define ${\\rm F}_{2}^{\\prime }$ to be the class of maps having the right lifting property with respect to every map of ${\\rm C}_{2}\\cap {\\rm W}^{\\prime }$ .", "One can think of $\\mathcal {M}_{1}$ as the `projective' model category, of $\\mathcal {M}_{2}$ as the `injective' model category, and of ${\\rm W}^{\\prime }$ as the class of `local', or `stable, equivalences'.", "Of course, other adjectives can be used.", "Recall from Section 3.3 the notion of left Bousfield localization of a (generalized) model category.", "Theorem 59 (1) (Restriction) If ${\\rm L}\\mathcal {M}_{2}=({\\rm W}^{\\prime },{\\rm C}_{2},{\\rm F}_{2}^{\\prime })$ is a left Bousfield localization of $\\mathcal {M}_{2}$ , then the class of fibrations of ${\\rm L}\\mathcal {M}_{2}$ is the class ${\\rm F}_{2}\\cap {\\rm F}_{1}^{\\prime }$ and ${\\rm L}\\mathcal {M}_{1}=({\\rm W}^{\\prime },{\\rm C}_{1},{\\rm F}_{1}^{\\prime })$ is a left Bousfield localization of $\\mathcal {M}_{1}$ .", "(2) (Extension) If ${\\rm L}\\mathcal {M}_{1}=({\\rm W}^{\\prime },{\\rm C}_{1},{\\rm F}_{1}^{\\prime })$ is a left Bousfield localization of $\\mathcal {M}_{1}$ that is right proper, then ${\\rm L}\\mathcal {M}_{2}=({\\rm W}^{\\prime },{\\rm C}_{2},{\\rm F}_{2}^{\\prime })$ is a left Bousfield localization of $\\mathcal {M}_{2}$ .", "For future purposes we display the conclusion of Theorem 59(2) in the diagram $@=2ex{&{\\mathcal {M}_{2}}@{-}[dr]\\\\{{\\rm L}\\mathcal {M}_{2}}@{.}[ur]@{.", "}[dr]&&{\\mathcal {M}_{1}}\\\\&{{\\rm L}\\mathcal {M}_{1}}@{-}[ur]}$ The proofs of the existence of the left Bousfield localizations in parts (1) and (2) are different from one another.", "As it will be explained below, the existence of the left Bousfield localization in part (1) is actually well-known, but perhaps it has not been formulated in this form.", "Also, the right properness assumption in part (2) is dictated by the method of proof.", "We prove part (1).", "We first show that ${\\rm F}_{2}^{\\prime }={\\rm F}_{2}\\cap {\\rm F}_{1}^{\\prime }$ .", "Clearly, we have ${\\rm F}_{2}^{\\prime }\\subset {\\rm F}_{2}\\cap {\\rm F}_{1}^{\\prime }$ .", "Conversely, we must prove that every commutative diagram in $\\mathcal {M}$ ${{A}[r][d]_{j}&{X}[d]^{p}\\\\{B} [r]&{Y}}$ where $j$ is in ${\\rm C}_{2}\\cap {\\rm W}^{\\prime }$ and $p$ is in ${\\rm F}_{2}\\cap {\\rm F}_{1}^{\\prime }$ , has a diagonal filler.", "The idea, which we shall use again, is very roughly that a commutative diagram ${{\\bullet }[r][d]&{\\bullet }[d]\\\\{\\bullet } [r]&{\\bullet }}$ in an arbitrary category has a diagonal filler when, for example, viewed as an arrow going from left to right in the category of arrows, it factors through an isomorphism.", "We first construct a commutative diagram ${{A} [r] [d]_{j} & {X^{\\prime }}[r] [d]^{q} & {X} [d]^{p}\\\\{B} [r] & {Y^{\\prime }} [r] & {Y}}$ with $q$ in ${\\rm F}_{2}\\cap {\\rm W}$ .", "Then, since $j$ is in ${\\rm C}_{2}$ , the left commutative square diagram has a diagonal filler.", "We factorize the map $B\\rightarrow Y$ into a map $B\\rightarrow Y^{\\prime }$ in ${\\rm C}_{2}\\cap {\\rm W}^{\\prime }$ followed by a map $Y^{\\prime }\\rightarrow Y$ in ${\\rm F}_{2}^{\\prime }$ .", "We factorize the canonical map $A\\rightarrow Y^{\\prime }\\times _{Y}X$ into a map $A\\rightarrow X^{\\prime }$ in ${\\rm C}_{2}\\cap {\\rm W}^{\\prime }$ followed by a map $X^{\\prime }\\rightarrow Y^{\\prime }\\times _{Y}X$ in ${\\rm F}_{2}^{\\prime }$ .", "Let $q$ be the composite map $X^{\\prime }\\rightarrow Y^{\\prime }$ ; then $q$ is in ${\\rm F}_{2}$ being the composite of two maps in ${\\rm F}_{2}$ .", "On the other hand, $q$ is in ${\\rm F}_{1}^{\\prime }$ since ${\\rm F}_{2}^{\\prime }\\subset {\\rm F}_{1}^{\\prime }$ and since ${\\rm F}_{1}^{\\prime }$ is stable under pullbacks and compositions.", "By the two out of three property $q$ is in ${\\rm W}^{\\prime }$ , therefore $q$ belongs to ${\\rm F}_{1}^{\\prime }\\cap {\\rm W}^{\\prime }={\\rm F}_{1}\\cap {\\rm W}$ .", "In all, $q$ is in ${\\rm F}_{2}\\cap {\\rm W}$ .", "We now prove the existence of ${\\rm L}\\mathcal {M}_{1}$ .", "This can be seen as a consequence of a result of M. Cole [8] (or of B.A.", "Blander [4]).", "In our context however, since we have Lemma 9 we only need to check the factorization of an arbitrary map of $\\mathcal {M}$ into a map in ${\\rm C}_{1}\\cap {\\rm W}^{\\prime }$ followed by a map in ${\\rm F}_{1}^{\\prime }$ .", "This proceeds as in [8], [4]; for completeness we reproduce the argument.", "Let $f\\colon X\\rightarrow Y$ be a map of $\\mathcal {M}$ .", "We factorize it as a map $X\\rightarrow Z$ in ${\\rm C}_{2}\\cap {\\rm W}^{\\prime }$ followed by a map $Z\\rightarrow Y$ in ${\\rm F}_{2}^{\\prime }$ .", "We further factorize $X\\rightarrow Z$ into a map $X\\rightarrow Z^{\\prime }$ in ${\\rm C}_{1}$ followed by a map $Z^{\\prime }\\rightarrow Z$ in ${\\rm F}_{1}\\cap {\\rm W}$ .", "The desired factorization of $f$ is $X\\rightarrow Z^{\\prime }$ followed by the composite $Z^{\\prime }\\rightarrow Y$ .", "We prove part (2).", "By Lemma 9, it only remains to check the factorization of an arbitrary map of $\\mathcal {M}$ into a map in ${\\rm C}_{2}\\cap {\\rm W}^{\\prime }$ followed by a map in ${\\rm F}_{2}^{\\prime }$ .", "Mimicking the argument given in part (1) for the existence of ${\\rm L}\\mathcal {M}_{1}$ does not seem to give a solution.", "We shall instead expand on an argument due to A.K.", "Bousfield [6], that's why we assumed right properness of ${\\rm L}\\mathcal {M}_{1}$ .", "Step 1.", "We give an example of a map in ${\\rm F}_{2}^{\\prime }$ .", "We claim that every commutative diagram in $\\mathcal {M}$ ${{A}[r][d]_{j}&{X}[d]^{p}\\\\{B} [r]&{Y}}$ where $j$ is in ${\\rm C}_{2}\\cap {\\rm W}^{\\prime }$ , $p$ is in ${\\rm F}_{2}$ , and $X$ and $Y$ are fibrant in ${\\rm L}\\mathcal {M}_{1}$ , has a diagonal filler.", "For this we shall construct a commutative diagram ${{A} [r] [d]_{j} & {X^{\\prime }}[r] [d]^{q} & {X} [d]^{p}\\\\{B} [r] & {Y^{\\prime }} [r] & {Y}}$ with $q$ in W. Factorizing then $q$ as a map in ${\\rm C}_{2}$ followed by a map in ${\\rm F}_{2}\\cap {\\rm W}$ and using two diagonal fillers, we obtain the desired diagonal filler.", "We factorize the map $B\\rightarrow Y$ into a map $B\\rightarrow Y^{\\prime }$ in ${\\rm C}_{1}\\cap {\\rm W}^{\\prime }$ followed by a map $Y^{\\prime }\\rightarrow Y$ in ${\\rm F}_{1}^{\\prime }$ .", "We factorize the canonical map $A\\rightarrow Y^{\\prime }\\times _{Y}X$ into a map $A\\rightarrow X^{\\prime }$ in ${\\rm C}_{1}\\cap {\\rm W}^{\\prime }$ followed by a map $X^{\\prime }\\rightarrow Y^{\\prime }\\times _{Y}X$ in ${\\rm F}_{1}^{\\prime }$ .", "Let $q$ be the composite map $X^{\\prime }\\rightarrow Y^{\\prime }$ .", "By the two out of three property $q$ is in ${\\rm W}^{\\prime }$ .", "Since $Y$ is fibrant in ${\\rm L}\\mathcal {M}_{1}$ , so is $Y^{\\prime }$ .", "The map $Y^{\\prime }\\times _{Y}X\\rightarrow X$ is in ${\\rm F}_{1}^{\\prime }$ and $X$ is fibrant in ${\\rm L}\\mathcal {M}_{1}$ , therefore $Y^{\\prime }\\times _{Y}X$ , and hence $X^{\\prime }$ , are fibrant in ${\\rm L}\\mathcal {M}_{1}$ .", "It follows that the map $q$ is in W. The claim is proved.", "Step 2.", "Let $f\\colon X\\rightarrow Y$ be a map of $\\mathcal {M}$ .", "We can find a commutative diagram ${{X}[r][d]_{f}&{X^{\\prime }}[d]^{f^{\\prime }}\\\\{Y} [r]&{Y^{\\prime }}}$ in which the two horizontal arrows are in ${\\rm W}^{\\prime }$ and both $X^{\\prime }$ and $Y^{\\prime }$ are fibrant in ${\\rm L}\\mathcal {M}_{1}$ .", "We can find a commutative diagram ${{X^{\\prime }}[r][d]_{f^{\\prime }}&{X^{\\prime \\prime }}[d]^{g}\\\\{Y^{\\prime }} [r]&{Y^{\\prime \\prime }}}$ in which the two horizontal arrows are in W, $g$ is in ${\\rm F}_{2}$ , and both $X^{\\prime \\prime }$ and $Y^{\\prime \\prime }$ are fibrant in $\\mathcal {M}_{1}$ .", "It follows that both $X^{\\prime \\prime }$ and $Y^{\\prime \\prime }$ are fibrant in ${\\rm L}\\mathcal {M}_{1}$ .", "The map $g$ is a fibration in ${\\rm L}\\mathcal {M}_{1}$ , since ${\\rm F}_{2}\\subset {\\rm F}_{1}$ .", "Putting the two previous commutative diagrams side by side we obtain a commutative diagram ${{X}[r][d]_{f}&{X^{\\prime \\prime }}[d]^{g}\\\\{Y} [r]&{Y^{\\prime \\prime }}}$ in which the two horizontal arrows are in ${\\rm W}^{\\prime }$ .", "Since ${\\rm L}\\mathcal {M}_{1}$ is right proper, the map $Y\\times _{Y^{\\prime \\prime }}X^{\\prime \\prime }\\rightarrow X^{\\prime \\prime }$ is in ${\\rm W}^{\\prime }$ , therefore the canonical map $X\\rightarrow Y\\times _{Y^{\\prime \\prime }}X^{\\prime \\prime }$ is in ${\\rm W}^{\\prime }$ .", "By the claim, the map $Y\\times _{Y^{\\prime \\prime }}X^{\\prime \\prime }\\rightarrow Y$ is in ${\\rm F}_{2}^{\\prime }$ .", "We factorize the map $X\\rightarrow Y\\times _{Y^{\\prime \\prime }}X^{\\prime \\prime }$ into a map $X\\rightarrow Z$ that is in ${\\rm C}_{2}$ followed by a map $Z\\rightarrow Y\\times _{Y^{\\prime \\prime }}X^{\\prime \\prime }$ that is in ${\\rm F}_{2}\\cap {\\rm W}$ .", "Since ${\\rm F}_{2}\\cap {\\rm W}\\subset {\\rm F}_{2}^{\\prime }$ , we obtain the desired factorization of $f$ into a map in ${\\rm C}_{2}\\cap {\\rm W}^{\\prime }$ followed by a map in ${\\rm F}_{2}^{\\prime }$ .", "The proof of the existence of ${\\rm L}\\mathcal {M}_{2}$ is complete.", "Some results in the subject of `homotopical sheaf theory' can be seen as consequences of Theorem 59.", "Here are a couple of examples.", "Let $\\mathcal {C}$ be a small category.", "The category of presheaves on $\\mathcal {C}$ with values in simplicial sets is a model category in two standard ways: it has the so-called projective and injective model structures.", "The class of cofibrations of the projective model category is contained in the class of cofibrations of the injective model category.", "If $\\mathcal {C}$ is moreover a site, a result of Dugger-Hollander-Isaksen [10] says that the projective model category admits a left Bousfield localization $U\\mathcal {C}_{\\mathcal {L}}$ at the class $\\mathcal {L}$ of local weak equivalences.", "The fibrations of $U\\mathcal {C}_{\\mathcal {L}}$ are the objectwise fibrations that satisfy descent for hypercovers [10].", "The model category $U\\mathcal {C}_{\\mathcal {L}}$ is right proper (for an interesting proof, see [9]).", "Therefore, by Theorem 59(2), Jardine's model category, denoted by $sPre(\\mathcal {C})_{\\mathcal {L}}$ in [10], exists.", "Moreover, by Theorem 59(1) its fibrations are the injective fibrations that satisfy descent for hypercovers: this is exactly the content of the first part of [10].", "As suggested in [9], this approach to $sPre(\\mathcal {C})_{\\mathcal {L}}$ reduces the occurence of stalks and Boolean localization technique.", "The category of presheaves on $\\mathcal {C}$ with values in simplicial sets also admits the so-called flasque model category [18].", "The class of cofibrations of the projective model category is contained in the class of cofibrations of the flasque model category [18].", "Using $U\\mathcal {C}_{\\mathcal {L}}$ and Theorem 59 it follows that the local flasque model category [18] exists.", "Other examples can be found on page 199 of [17]: the existence of both therein called the $S$ model and the injective stable model structures, together with the description of their fibrations, can be seen as consequences of Theorem 59." ], [ "Appendix 3: Strong stacks of\ncategories revisited", "Let $E$ be a small site.", "Recall from Theorem 48 the model category $Stack(\\widetilde{E})_{proj}$ .", "Lemma 60 The class of weak equivalences of $Stack(\\widetilde{E})_{proj}$ is accessible.", "Let $f\\colon X\\rightarrow Y$ be a map of ${\\rm Cat}(\\widetilde{E})$ .", "Consider the commutative square diagram ${{X}[r] [d]_{f}&{\\widehat{F}X}[d]^{\\widehat{F}f}\\\\{Y} [r]&{\\widehat{F}Y}}$ with $\\widehat{F}X$ defined in the proof of Theorem 48.", "Then $f$ is a weak equivalence if and only if $i\\widehat{F}f$ is a weak equivalence in $[E^{op},CAT]_{proj}$ .", "The functors $i$ and $\\widehat{F}$ preserve $\\kappa $ -filtered colimits for some regular cardinal $\\kappa $ .", "Since the class of weak equivalences of $[E^{op},CAT]_{proj}$ is accessible, the result follows.", "Recall from Theorem 50 the (right proper) model category $Stack(\\widehat{E})_{proj}$ .", "By Theorem 59(2) we have the model category $Stack(\\widehat{E})_{inj}$ , which we display in the diagram $@=2ex{&{[E^{op},CAT]_{inj}}@{-}[dr]\\\\{Stack(\\widehat{E})_{inj}}@{.}[ur]@{.", "}[dr]&&{[E^{op},CAT]_{proj}}\\\\&{Stack(\\widehat{E})_{proj}}@{-}[ur]}$ By Theorem 59(1), an object $X$ of $[E^{op},CAT]$ is fibrant in $Stack(\\widehat{E})_{inj}$ if and only if $\\Phi X$ is a stack and $X$ is fibrant in $[E^{op},CAT]_{inj}$ .", "Theorem 61 [19] There is a model category Stack$(\\widetilde{E})_{inj}$ on the category Cat$(\\widetilde{E})$ in which the weak equivalences are the maps that $\\Phi $ takes into bicovering maps and the cofibrations are the internal functors that are monomorphisms on objects.", "A sheaf of categories $X$ is fibrant in Stack$(\\widetilde{E})_{inj}$ (aka $X$ is a strong stack) if and only if $\\Phi X$ is a stack and $X$ is fibrant in $[E^{op},CAT]_{inj}$ .", "We shall use J. Smith's recognition principle for model categories [3].", "We take in op.", "cit.", "the class W to be the class of weak equivalences of $Stack(\\widetilde{E})_{proj}$ .", "By Lema 60, W is accessible.", "Let $I_{0}$ be a generating set for the class C of cofibrations of $[E^{op},CAT]_{inj}$ , so that ${\\rm C}={\\rm cof}(I_{0})$ .", "We put $I=aI_{0}$ .", "The functors $a$ and $i$ preserve the property of internal functors of being a monomorphism on objects.", "Using that $i$ is full and faithful it follows that $a{\\rm C}$ is the class of internal functors that are monomorphisms on objects, and that moreover $a{\\rm C}={\\rm cof}(I)$ .", "By adjunction, every map in ${\\rm inj}(I)$ is objectwise an equivalence of categories, so in particular every such map is in W. Recall that for every object $X$ of $[E^{op},CAT]$ , the natural map $X\\rightarrow iaX$ is a weak equivalence (see fact (2) stated below Theorem 48).", "Thus, by Lemma 62 all the assumptions of Smith's Theorem are satisfied, so Cat$(\\widetilde{E})$ is a model category, which we denote by Stack$(\\widetilde{E})_{inj}$ .", "Let $f$ be a fibration in this model category.", "Then clearly $if$ is a fibration in $Stack(\\widehat{E})_{inj}$ .", "Conversely, if $if$ is a fibration in $Stack(\\widehat{E})_{inj}$ , then, since $i$ is full and faithful, $f$ is a fibration in Stack$(\\widetilde{E})_{inj}$ .", "Lemma 62 Let $\\mathcal {M}$ be a model category.", "We denote by C the class of cofibrations of $\\mathcal {M}$ .", "Let $\\mathcal {N}$ be a category and let $R\\colon \\mathcal {M}\\rightleftarrows \\mathcal {N}\\colon K$ be a pair of adjoint functors with $K$ full and faithful.", "We denote by W the class of maps of $\\mathcal {N}$ that $K$ takes into weak equivalences.", "Assume that (1) $KR{\\rm C}\\subset {\\rm C}$ and (2) for every object $X$ of $\\mathcal {M}$ , the unit map $X\\rightarrow KRX$ is a weak equivalence.", "Then the class ${\\rm W}\\cap R{\\rm C}$ is stable under pushouts and transfinite compositions.", "We first remark that by (2), the functor $R$ takes a weak equivalence to an element of W. Let ${{X}[r]^{f} [d]&{Y}[d]\\\\{Z} [r]_{g}&{P}}$ be a pushout diagram in $\\mathcal {N}$ with $f\\in {\\rm W}\\cap R{\\rm C}$ .", "Then $g$ is obtained by applying $R$ to the pushout diagram ${{KX}[r]^{Kf} [d]&{KY}[d]\\\\{KZ} [r]&{P^{\\prime }}}$ By the assumptions it follows that $g\\in {\\rm W}\\cap R{\\rm C}$ .", "The case of transfinite compositions is dealt with similarly." ] ]
1403.0536
[ [ "A packing problem approach to energy-aware load distribution in Clouds" ], [ "Abstract The Cloud Computing paradigm consists in providing customers with virtual services of the quality which meets customers' requirements.", "A cloud service operator is interested in using his infrastructure in the most efficient way while serving customers.", "The efficiency of infrastructure exploitation may be expressed, amongst others, by the electrical energy consumption of computing centers.", "We propose to model the energy consumption of private Clouds, which provides virtual computation services, by a variant of the Bin Packing problem.", "This novel generalization is obtained by introducing such constraints as: variable bin size, cost of packing and the possibility of splitting items.", "We analyze the packing problem generalization from a theoretical point of view.", "We advance on-line and off-line approximation algorithms to solve our problem to balance the load either on-the-fly or on the planning stage.", "In addition to the computation of the approximation factors of these two algorithms, we evaluate experimentally their performance.", "The quality of the results is encouraging.", "This conclusion makes a packing approach a serious candidate to model energy-aware load balancing in Cloud Computing." ], [ "Introduction ", "The Cloud Computing paradigm consists in providing customers with virtual services of the quality which meets customers' requirements.", "A cloud service operator is interested in using his infrastructure in the most efficient way while serving customers.", "Namely, he wishes to diminish the environmental impact of his activities by reducing the amount of energy consumed in his computing servers.", "Such an attitude allows him to lower his operational cost (electricity bill, carbon footprint tax, etc.)", "as well.", "Three elements are crucial in the energy consumption on a cloud platform: computation (processing), storage, and network infrastructure [1], [2], [3], [4].", "We intend to study different techniques to reduce the energy consumption regarding these three elements.", "We tempt to consolidate applications on servers to keep their utilization at hundred per cent.", "The consolidation problem was discussed in [5] through an experimental approach based on the intuition as its authors did not propose any formal problem definition.", "In this paper we address the challenge of the minimization of energy required for processing by means of proper mathematical modeling and we propose algorithmic solutions to minimize the energy consumption on Cloud Computing platforms.", "We address here a private Cloud infrastructure which operates with knowledge of resource availability.", "We study a theoretical problem adjacent to the minimization of energy required to execute computational tasks.", "Our working hypotheses are as follows: any computional task is parallelizable, i.e.", "it may be executed on several servers; there is, however, a restriction on the number of servers on which a task can be launched, available servers have different computing capacities, the computation cost of a server in terms of its energy consumption is monotone, i.e.", "a unity of computation power is cheaper on a voluminous server than on a less capacious one.", "The assumption that all tasks are divisible may sound unrealistic as in practice some tasks cannot be split.", "We make it in order to formulate theoretical problems and analyze them.", "In the real world scenario one will rather cope with jobs which either cannot be cut at all or which can be cut one, twice, up to $D$ times.", "Such a situation corresponds to a problem which is “somewhere between” two extremal cases: no jobs can be split and all jobs can be split $D$ times.", "As the reader will notice going through this paper, the “real life” problem performance bounds can be deducted from those of the extremal problems.", "The three assumptions above lead us to formulate a generalization of the Bin Packing problem [6], which we refer to as the Variable-Sized Bin Packing with Cost and Item Fragmentation Problem (VS-CIF-P).", "In the general case considered a cost of packing is monotone.", "This problem models a distribution of computational tasks on Cloud servers which ensures the lowest energy consumption.", "Its definition is given in Subsection REF .", "We point out that the approach through packing problems to the energy-aware load distribution has not yet been proposed.", "Confronted with numerous constraints of the VS-CIF-P we decided to start, however, by studying in Subsection REF a less constrained problem, without an explicit cost function, the Variable-Sized Bin Packing and Item Fragmentation Problem (VS-IF-P).", "This has not yet been studied either.", "This gradual approach allows us to deduce several theoretical properties of the VS-IF-P which can be then extended to the principal problem.", "In Section  we propose customized algorithms to solve the VS-CIF-P.", "Willing to treat users' demands in bulk, what corresponds to regular dispatching of collected jobs (for instance, hourly) we propose an off-line method (Subsection REF ).", "An on-line algorithm, dealing with demands on-the-fly is given in Subsection REF .", "This treatment allows one to launch priority jobs which have to be processed upon their arrivals.", "Expecting an important practical potential of the VS-CIF-P we also furnish results concerning the theoretical performance bounds of the algorithms we elaborated.", "Despite the fact that the problem is approximate with a constant factor, we go further with the performance evaluation of the algorithms we come up with.", "The empirical performance evaluation is discussed in Section .", "The list of our contributions given above also partially constitutes the description of the paper's organization.", "We complete this description by saying that in Section  we present a survey of related works concerning definitions of the family of bin-packing problems together with their known approximation factors.", "We also give there an outline of algorithmic approaches used to solve packing problems.", "Our special attention is put on those which inspired us in our study.", "We point out that the notation used in the article is also introduced in that section while carrying out our survey.", "After giving our contributions in the order announced above we draw conclusions and give directions of our further work." ], [ "Related Works ", "Let $L$ be a list of $n$ items numbered from 1 to $n$ , $L=(s_1, s_2, \\ldots , s_n)$ , where $s_i$ indicates an item size.", "Let us also assume for a moment that for all $i=1,2,\\ldots , n$ $s_i\\in [0,1]$ .", "The classical Bin Packing Problem (BPP) consists in grouping the items of $L$ into $k$ disjoint subsets, called bins, $B_1, B_2, \\ldots , B_k$ , $\\bigcup _{l=1}^m B_l = L$ , such that for any $j$ , $j=1,2,\\ldots ,k$ , $\\sum _{l\\in B_j}s_l \\le 1$ .", "The question 'Can I pack all items of $L$ into $K$ , $K\\le k$ , bins?'", "defines the BPP in the decision form.", "Put differently, we ask whether a packing $B_1, B_2, \\ldots , B_k$ for which $k$ is less or equal to a given value $K$ exists.", "The corresponding optimization problem aims to find the minimal $k$ .", "Due to its numerous practical applications the BPP, which is NP-hard, was studied exhaustively.", "The current basic on-line approaches, Next Fit (NF) and First Fit (FF) give satisfactory results.", "The asymptotic approximation factor for any on-line algorithm cannot be less than $1.54$  [6].", "A widely used off-line approach consists in sorting items in decreasing order of their size before packing them.", "The tight bound for First Fit Decreasing (FFD) is given in [7]." ], [ "Variable-Sized Bin Packing with Cost", "In the initial problem the capacity of all bins is unitary.", "The problem may thus be modified by admitting different bin capacities.", "A bin can have any of $m$ possible capacities $b_j$ , $B= (b_1, b_2, \\ldots , b_m)$ .", "In other words, we have $m$ bin classes.", "If any bin is as good as the others, putting items inside a solution to this problem is trivial, as one will always be interested in using the largest bin.", "We thus suppose that bin utilization induces a certain cost associated to this bin.", "This assumption leads to the Variable Sized Bin Packing with Cost Problem (VSBPCP).", "The reader might already observe that in the BPP the cost of packing is always the same regardless of the bins chosen.", "This fact explains that the new problem is NP-hard [8].", "Intuitively, one can consider that the cost of packing varies in function of a bin capacity.", "From this point of view we are no longer interested in minimizing the number of bins used but in minimizing the global packing cost as a voluminous bin which remains 'almost empty' may be more expensive in use than several little bins 'almost totally' full.", "Solving the VSBPCP we have at our disposal $m$ classes of bins and the infinite number of bins of any class available.", "In the simplest case, the cost is a linear function of a capacity.", "Packing into bin $i$ costs $c_i$ , $C= (c_1, c_2, \\ldots , c_m)$ and we have as many costs as bin classes available.", "Similarly to the notation introduced above, we denote a cost of a bin $B_j$ taking part in a packing as $\\mbox{cost}(B_j) = c_l$ (a bin in position $j$ in a packing costs $c_l$ ).", "The goal is thus to find a packing $B_1, B_2, \\ldots , B_k$ such that $\\sum _{l\\in B_j}s_l \\le \\mbox{capacity}(B_j)$ for which the overall cost, $\\sum _{l=1}^k \\mbox{cost}(B_l)$ , is minimal.", "Without loss of generality one may assume that the cost of using bin $i$ is equal to its capacity, $c_i=b_i$ , $i=1,2,\\ldots , m$ .", "For a packing we thus have: $\\mbox{cost}(B_j) = \\mbox{capacity}(B_j)$ .", "A monotone cost function signifies that a unity in a bigger bin $i$ is not more expensive than a unity in a smaller one, bin $j$ : $\\frac{c_i}{b_i}\\le \\frac{c_j}{b_j}$ , $i,j=1,2,\\ldots ,m$ , $b_i>b_j$ and the cost of a smaller bin is not greater than the cost of a bigger bin, $c_j\\le c_i$ .", "A linear cost function is a special case of a monotone one.", "Staying in the context of a monotone cost, our attention was attracted by the off-line algorithms from [8].", "Their main idea consists in applying an iterative approach to a well-known algorithm, for example, FFD which leads to IFFD.", "In a nutshell, at the beginning IFFD performs the classical FFD with identical bins of the greatest capacity, $\\max _{b_i}(B)$ .", "The packing obtained is next modified by trying to move (again with FFD) all the items from the last bin into the next biggest bin.", "The repacking procedure continues by transferring items entirely from the bin of capacity $b_j$ , which was the last one filled up, into a bin of size $b_i$ , $b_i<b_j$ and there is not any $l$ such that $b_i<b_l<b_j$ .", "It stops when any further repacking becomes impossible.", "Their authors showed that the solutions are approximated with $1.5$ ." ], [ "Linear cost, on-line approach ", "We pay special attention to the on-line algorithm to solve the VSBPCP described in [9].", "This algorithm deals with a linear cost.", "Its authors proposed an approach 'in between' the First Fit, using Largest possible bin (FFL)A largest possible bin is here a unit-capacity bin.", "and the First Fit, using Smallest possible bin (FFS) trying to take advantage of both methods with regard to the size of the item to be packed.", "Their idea is to determine whether an item to be packed occupies a lot of space or not.", "The decision is taken upon a fill factor $f$ , $f\\in [0.5,1]$ .", "Their algorithm, called FFf, operates in the following way.", "If item $i$ is small (i.e.", "$s_i\\le 0.5$ ), it will be inserted into the first bin in which it enters or into a new bin of unitary capacity when it does not fit into any opened bin (FFL).", "Otherwise, it will be inserted into the first opened bin into which it enters.", "If the use of an opened bin is impossible, it will be packed into the smallest bin among those whose capacity is between $s_i$ and $\\frac{s_i}{f}$ , if it fits inside, or into a new unit-capacity bin if it does not (FFS).", "The authors of FFf proved that the result it furnishes is approximated by $1.5 + \\frac{f}{2}$ .", "In another variant of the classical BPP one is allowed to fragment items while the identical bin size and bin cost remain preserved, the Bin Packing with Item Fragmentation Problem (BPIFP).", "Item cutting may reduce the number of bins required.", "On the other hand, if the item fragmentation is not for free, it may increase the overall cost of packing.", "In [10], [11], [12], for instance, its authors investigated two possible expenses: the item size growth which results from segmentation and the global limit on the number of items cut.", "We point out that one may also consider a limit on the number of items permitted to be packed into a bin.", "A variant of the BPP fixing such a limit was introduced and studied in [13], [14].", "It models task scheduling in multiprogramming systems and is known as the Bin Packing with Cardinality Constraints Problem (BPCCP).", "From our particular perspective, founded upon the virtualization of computing services in Clouds, we opt to restrain the number of fragments into which an item can be cut.", "The maximal number of cuts for any item, which models a computation task, is limited to $D$ .", "As certain items from list $L$ should be fragmented before packing, we do not cope with items $i$ but with their fragments whose sizes are noted as $s_{i_d}$ , where $i$ indicates an original item $i$ from $L$ and $d$ enumerates fragments of item $i$ .", "Let $D_i$ be a number of cuts of item $i$ made.", "Obviously, $D_i=0$ signifies that item $i$ has not been fragmented at all.", "Moreover, for any $i$ we have $D_i\\le D$ and $\\sum _{l=1}^{D_i +1} s_{i_l} = s_i $ .", "Thus the solution to the BPIFP with limit $D$ consists in finding an appropriate fragmentation first, which results in a new list of sizes of items to be packed $\\begin{array}{lll}L_D & = & \\left((s_{1_1}, s_{1_2}, \\ldots , s_{1_{D_1 +1}}), (s_{2_1}, s_{2_2}, \\ldots , s_{2_{D_2 +1}}), \\ldots , \\right.", "\\\\& & \\left.", "\\ldots , (s_{n_1}, s_{n_2}, \\ldots , s_{n_{D_n +1}})\\right)\\end{array}$ (actually, this a list of lists).", "The number of items to be packed is now $\\sum _{i=1}^{n}(D_i + 1)$ .", "Next, the BPP is to be solved with $L_D$ as input data." ], [ "Survey conclusions ", "To the best of our knowledge, neither the problem being in the center of our interest, the Variable-Sized Bin Packing with Cost and Item Fragmentation Problem (VS-CIF-P), which we propose to model an energy-aware load distribution nor the less constrained one, the Variable-Sized Bin Packing and Item Fragmentation Problem (VS-IF-P), have been studied yet.", "As announced above, we start by treating the auxiliary problem.", "It will be generalized after its analysis." ], [ "Auxiliary Problem ", "We suppress the explicit cost function in the general problem.", "By doing this we expect to be able to find an optimal solution to the auxiliary VS-IF-P with polynomial complexity for certain particular cases.", "We recall to the reader here that we limit the number of cuts of any individual item.", "Definition 1 Variable-Sized Bin Packing with Item Fragmentation Problem (VS-IF-P) Input: $n$ items to be packed, sizes of items to be packed $L=(s_1, s_2, \\ldots , s_n)$ , $s_i \\in \\mathbb {N^+}$ , $i=1,2,\\ldots n$ , capacities of bins available $B= (b_1, b_2, \\ldots , b_m)$ , $ b_j \\in \\mathbb {N^+}$ , $j=1,2,\\ldots m$ , a constant $D$ which limits the number of splits authorized for each item, $D \\in \\mathbb {N^+}$ , a constant $k$ , $k \\in \\mathbb {N^+}$ which signifies the number of bins used.", "Question: Is it possible to find a packing $B_1, B_2, \\ldots , B_K$ of items $L$ whose fragment sizes are in $L_D$ defined as in Eq.", "(REF ) such that $K\\le k$ ?", "In the analysis of the VS-IF-P we appeal to a variant of the BPP coming from the memory allocation modeling [15], [16].", "Its particularity consists in having a limit on the number of items in any bin.", "This limit holds without regard to whether an item inserted is 'an original one' or results from the item fragmentation itself.", "Thus this problem may be considered as a variant of the BPCCP (see Subsection REF ) with item fragmentation.", "For our purposes we call it the Memory Allocation with Cuts Problem (MACP) and we give below its formal definition using our notation.", "We believe that this formal presentation allows the reader to discover a palpable duality existing between the VS-IF-P and the MACP: in the first one there is a limit on the cut number of any item, in the latter we have a limit on the number of 'cuts' in any bin.", "As the MACP admits the item fragmentation, the objects which it packs are picked from the following list: $\\begin{array}{lll}L^{\\prime }_{D^{\\prime }} & = & \\left((s^{\\prime }_{1_1}, s^{\\prime }_{1_2}, \\ldots , s^{\\prime }_{1_{d_1 +1}}), (s^{\\prime }_{2_1}, s^{\\prime }_{2_2}, \\ldots , s^{\\prime }_{2_{d_2 +1}}), \\ldots , \\right.\\\\& &\\left.", "\\ldots , (s^{\\prime }_{n_1}, s^{\\prime }_{n_2}, \\ldots , s^{\\prime }_{n_{d_n +1}})\\right),\\end{array}$ where $\\sum _{l=1}^{d_i + 1}s^{\\prime }_{i_l} = s^{\\prime }_i$ , $i=1,2,\\ldots , n$ .", "The number of splits of item $i$ , $d_i$ , is not a priori limited but one cannot cut items at will.", "The values of $d_i$ will be determined later on by the constraint restricting the number of pieces in a bin, $D^{\\prime }$ .", "Definition 2 Memory Allocation with Cuts Problem (MACP) Input: $n^{\\prime }$ items to be packed, sizes of items to be packed $L^{\\prime }=(s^{\\prime }_1, s^{\\prime }_2, \\ldots , s^{\\prime }_n)$ , $s^{\\prime }_i \\in \\mathbb {N^+}$ , $i=1,2,\\ldots n^{\\prime }$ , a capacity $b^{\\prime }$ of each bin, a constant $D^{\\prime }$ which limits the number of items authorized inside each bin (no more than $D^{\\prime }+1$ pieces inside a bin), $D^{\\prime } \\in \\mathbb {N^+}$ , a constant $k^{\\prime }$ , $k^{\\prime } \\in \\mathbb {N^+}$ which signifies the number of bins used.", "Question: Is it possible to find a packing $B^{\\prime }_1, B^{\\prime }_2, \\ldots , B^{\\prime }_{K^{\\prime }}$ of elements whose sizes are in $L^{\\prime }_{D^{\\prime }}$ defined as in Eq.", "(REF ) such that for each $l$ , $l=1,2,\\ldots K^{\\prime }$ , $|B^{\\prime }_l|\\le D^{\\prime }$ and $K^{\\prime }\\le k^{\\prime }$ ?", "Theorem 1 The VS-IF-P is NP-complete in the strong sense.", "It is easy to see that the VS-IF-P is in NP.", "In this proof, as in Def.", "REF , we consequently use a prime symbol when referring to an instance of the MACP.", "Figure: MACP/VS-IF-P instance transformationFirst, we demonstrate that the VS-IF-P is NP-complete in the strong sense by reducing the MACP to it as the NP-completeness in the strong sense of the MACP was shown in [16].", "For both instances, $I$ and $I^{\\prime }$ , we put $D=D^{\\prime }$ .", "Bins of the MACP become $k^{\\prime }$ items to be packed in the VS-IF-P, $L=(b^{\\prime },b^{\\prime },\\ldots , b^{\\prime })$ , $n=k^{\\prime }$ .", "Items of the MACP are transformed into variable-sized bins, $L^{\\prime }=B$ .", "Finally, we require that all available bins of the VS-IF-P are used: $|L^{\\prime }|= |B| =k$ .", "Obviously, this transformation, also illustrated in Figure REF , can be performed in polynomial time.", "We focus our attention on a particular case $k^{\\prime }b^{\\prime }=\\sum _{b_i\\in L^{\\prime }}b_i$ in which there is no empty space left in bins forming a solution.", "It is evident that in this situation the packing of items into bins corresponds to 'inserting' bins into items.", "If the verification gives a positive answer for one instance, it will give a positive answer for another, too.", "An illustrative example, $D=d^{\\prime }=1$ , is given in Figure REF .", "The overall items' mass is 40.", "Thus we need four bins of capacity 10 to pack $L^{\\prime }=(16,15,9)$ for the instance $I^{\\prime }$ of the MACP depicted on the left of Figure REF .", "The instance $I$ of the VS-IF-P (on the right of Figure REF ) is composed of four items of size 10 and $B=(16,15,9)$ .", "Figure: Example of solutions to MACP and VS-IF-P instances (I ' I^{\\prime } and II, respectively).", "Instance I ' I^{\\prime } is composed of three items whose sizes are 16, 15, 9 and bins of capacity 10.", "Instance II has four items of size 10 and three bin classes of capacity: 16, 15, and 9.Second, we estimate the computational effort required to obtain a positive response to the question whether 'a candidate to be a solution' is a solution.", "In order to do this we determine a size $N_i$ of the VS-IF-P instances and a size $N_s$ of solutions to it.", "The greatest elements of both lists determine $N_i$ , which is thus in $\\mathcal {O} (\\log k + (m+n)\\log \\max (\\max _{s_i}(L), \\max _{b_i}(B)))$ .", "Solutions are made up of bins and 'quantities' of items, possibly split, selected in $L$ .", "Similarly, we take into account the most voluminous elements of the lists, which leads us to $N_s$ not greater than $nm(\\log m + \\log \\max (\\max _{s_i}(L), \\max _{b_i}(B)))$ , being in $\\mathcal {O}( N_i^2)$ (a polynomial verification time).", "In order to analyze the feasibility of solutions to the VS-IF-P we assume for a moment that fragments resulting from the item splitting are equal in size.", "We have to be assured that any fragment of the greatest item can be inserted entirely into the highest capacity bin: $ \\frac{\\max _{s_l} (L)}{D+1}\\le \\max _{b_i} (B).", "$ If we can pack items of an instance of the VS-IF-P with such a fragmentation, we can do the same for the VSBPP instance, whose items to be packed are simply those of the VS-IF-P split.", "This reasoning allows us to adapt numerous algorithms existing for the VSBPP to solve the VS-IF-P by incorporating cutting.", "Eq.", "(REF ) guarantees the existence of a solution even in cases when items are split into 'almost equal' pieces because a single 'over-sized' fragment will be not greater than $\\frac{\\max _{s_l} (L)}{D+1}$ .", "We propose to admit cutting in FF (Cut and FF, CFF).", "Despite sorting the bin classes in decreasing order of their capacity, the on-line principle is preserved as items are not reordered before their cut and insertion.", "An illustration of a CFF execution with 'imperfect cuts' is presented in Figure REF .", "Figure: Example CFF solving an VS-IF-P instance; observe that an imperfect cut do not compromise the solution feasibilityThe discussion above, which exhibits a relationship between instances of the VS-IF-P and the VSBPP allows us to apply FF (or NF).", "In solutions obtained with these algorithms one bin at most is less than half full.", "On the other hand, if bins are of capacity $2s$ , where $s$ is a natural number, items are of size $2(s+1)$ and any item can be cut no more than once, $D=1$ , we cannot expect a better approximation factor than 2.", "We visualize this example by imagining in Figure REF the item size equal to 12 and keeping bin capacity equal to 10.", "This observation leads us to formulate: Theorem 2 A solution to the VS-IF-P for any $D$ with CFF is tightly bounded by 2." ], [ "Main Problem ", "As stated above, the problem which models the distribution of computation tasks within a private Cloud infrastructure is a bin packing in which bins are of different sizes and tasks can be split over several servers.", "We assume that all numerical data (item sizes, bin capacities, costs) are natural numbers.", "We also assume that a task is parallelizable (the discussion of our hypothesis can be found in Section ).", "We allow an item to be cut into no more than $D+1$ pieces, i.e.", "any item may be split at most $D$ times.", "Indeed, if any number of cuts was admitted, we might cut all tasks into unitary pieces and end up with a trivial packing of $\\sum _{i=1}^n s_i$ unitary objects being able to fill up any used bin entirely.", "As the reader might have already notice while passing through Sections  and , our problem, the Variable-Sized Bin Packing with Cost and Item Fragmentation Problem (VS-CIF-P), puts together the three problems announced above.", "To be more precise, we 'mix up' the VSBPCP and BPIFP, the latter with the constraints which have just been discussed.", "Unless stated differently, the cost function is monotone.", "Definition 3 Variable-Sized Bin Packing with Cost and Item Fragmentation Problem (VS-CIF-P) Input: $n$ items to be packed, sizes of items to be packed $L=(s_1, s_2, \\ldots , s_n)$ ,$s_i \\in \\mathbb {N^+}$ , $i=1,2,\\ldots n$ , capacities of bins available $B= (b_1, b_2, \\ldots , b_m)$ , $ b_j \\in \\mathbb {N^+}$ , $j=1,2,\\ldots m$ , costs of using of bins available $C= (c_1, c_2, \\ldots , c_m)$ , $ c_j \\in \\mathbb {N^+}$ , $j=1,2,\\ldots m$ , a constant $D$ which limits the number of splits authorized for each item, $D \\in \\mathbb {N^+}$ , a constant $e$ , $e \\in \\mathbb {N^+}$ which signifies the cost limit of a packing.", "Question: Is it possible to find a packing $B_1, B_2, \\ldots , B_k$ of items $L$ whose fragment sizes are in $L_D$ defined as in Eq.", "(REF ) such that $\\sum _{l=1}^k \\mbox{cost}(B_l)\\le e$ ?" ], [ "Algorithms ", "Before introducing our methods to solve the VS-CIF-P (Def.", "REF ) we will discuss the solution to the auxiliary VS-IF-P (Def.", "REF ), in order to select the algorithmic approaches the best adapted to treat our principal problem.", "Indeed, as we presumed (Subsection REF ), the VS-IF-P can be solved exactly and in polynomial time under a certain hypothesis." ], [ "Next Fit with Cuts (NFC) for the auxiliary problem ", "Let us hypothesize that we are dealing with the instances for which we can always find a bin to pack any entire item (without cutting it).", "This hypothesis may be expressed by: $\\max _{s_i} (L) \\le \\max _{b_i} (B).$ If this hypothesis is satisfied, we propose a variant of NF, Next Fit with Cuts (NFC) as an algorithmic solution.", "A similar approach was used for another purpose under the name of NF$_{\\mbox{\\scriptsize f}}$ in [10], [11].", "First, the capacities of bin classes are sorted in decreasing order.", "Next, for each item, if there is some room in a current bin, we pack the item inside, cutting it if necessary, and inserting the second fragment of the item into the next bin.", "We observe that with Hypothesis (REF ) valid, NFC fragments any item at most once.", "Moreover, when this hypothesis is satisfied, the packing problem with variable-sized bins and item fragmentation is in $P$ .", "Indeed, the verification whether Hypothesis (REF ) holds or not can be performed in $\\mathcal {O} (m)$ or $\\mathcal {O}(n)$ depending upon the relationship existing between $m$ and $n$ .", "An execution of NFC requires $\\mathcal {O}(m \\log m + \\max (n,m))$ operations.", "Lastly, we observe that the bins used are all totally filled up and they are of the greatest available capacities, which proves the algorithm's optimality.", "This discussion leads us to the conclusion: Theorem 3 NFC is optimal and polynomial to solve the VS-IF-P when Hypothesis (REF ) holds." ], [ "Off-line Approach to the Main Problem with Monotone Cost ", "The approach presented here is based upon IFFD (Paragraph REF ) combined with item cutting.", "For this reason we refer to it as CIFFD.", "As before, we assume that bin classes are sorted in decreasing capacity order, $b_1>b_2> \\cdots b_m$ .", "The initial idea of our algorithm to solve the VS-CIF-P (Algorithm REF ) consists in dividing items of $L$ into two categories: those items $i$ which may be possibly packed without fragmentation and those which undoubtedly may not.", "Such an approach makes our algorithm off-line.", "We propose to reason here upon item sizes $s_i$ , not upon their indices $i$ .", "This mental operation enables us to avoid tedious renumbering of items to be packed, which might deteriorate the text limpidity.", "At the same time, it does not introduce any ambiguity.", "We formally note the two categories as $T_1$ and $T^+_1$ , respectively: $T_1 \\cup T^+_1 = \\lbrace s_1,s_2, \\ldots , s_n\\rbrace $ , for all $s_i\\in T_1$ we have $s_i\\le \\max _{b_i}(B)=b_1$ , and for all $s_i\\in T^+_1$ we have $ s_i > \\max _{b_i}(B)=b_1$ .", "Items from $T^+_1$ are cut naturally up to $D$ times in order to completely fill up a bin of capacity $b_1$ , the remaining fragment whose size is inferior to $b_1$ is stored in $T^-_1$ (lines – of Algorithm REF ).", "This loop also allows us to detect the instance infeasibility, i.e.", "the number of cuts allowed $D$ is too small to insert an item fragmented into the largest bins.", "The items whose sizes are in $T_1 \\cup T^-_1$ are then packed according an appropriate algorithm to solve the BPP as we use momentarily bins of identical capacity $b_1$ .", "We have thus a solution which we try to improve iteratively, taking bins in decreasing order of their capacity (for a bin of capacity $b_j$ items are divided into $T_j$ and $T_j^+$ ), by consecutive repacking of the contents of the less efficiently used bin into a smaller empty one, if possible (as IFFD described in Paragraph REF does).", "As our algorithm can fragment an item to fill up a bin, its iterative descent may stop when the number of cuts allowed has been reached.", "A solution which offers the lowest cost among the obtained ones is returned.", "Finally, an attempt is made to squeeze this solution more (lines – of Algorithm REF ).", "[!h] VS-CIF-P data as in Def.", "REF with $B$ sorted in decreasing order $B_{\\min } = (B_1, B_2, \\ldots , B_k)$ whose cost is as minimal as possible; cost $e_{\\min }$ of this packing $e\\leftarrow 0$ ; divide $L$ into $T_1$ and $T^+_1$ $t$ in $T^+_1$ feasibilityAlgoOFFLoopBegin $t - b_1 \\le b_1$ split $t$ into a fragment $b_1$ and the remainder $t - b_1$ pack the fragment $b_1$ into a bin of capacity $b_1$ $e\\leftarrow e +$ cost of filling up a bin $b_1$ put the remainder $t - b_1$ into $T^-_1$ feasibilityAlgoOFFLoopEnd pack items from $ T_1 \\cup T^-_1$ to bins $b_1$ with any BPP algorithm $e\\leftarrow e +$ cost of this packing ; $e_{\\min } \\leftarrow e$ ; $B_{\\min } \\leftarrow $ a current packing $j\\leftarrow 2$ $m-1$ take out items from the less filled bin of size $b_{j-1}$ of packing $B_{\\min }$ divide them into two categories $T_j$ and $T^+_j$ $t$ in $T^+_j$ which has not yet reached the limit of cuts $D$ split $t$ into $b_j$ and $t - b_j$ ; pack fragment $b_j$ into a bin $b_j$ $e\\leftarrow e +$ cost of filling up a bin $b_j$ ; put $t - b_j$ into $T^-_j$ pack items from $ T_j \\cup T^-_j$ to bins $b_j$ with any BPP algorithm $e\\leftarrow e +$ cost of this packing $e<e_{\\min }$ $e_{\\min } \\leftarrow e$ ; $B_{\\min } \\leftarrow $ a current packing take $B_{\\min }$ for repacking $B_j$ of $B_{\\min }$ taken in decreasing order repackingAlgoOFFLoopBegin $B_j$ is not full $\\wedge $ its content may enter into bins of certain capacities $b_l$ find $b_{l_{\\min }}$ , the smallest of these $b_l$ repack the contents of $B_j$ into an empty bin of capacity $b_{l_{\\min }}$ repackingAlgoOFFLoopEnd CIFFD solving the VS-CIF-P For any item CIFFD looks for an appropriate opened bin.", "If it does not find one, it will open up the smallest bin into which the item enters.", "Its complexity is thus $\\mathcal {O}(mn\\log n)$ .", "Theorem 4 The VS-CIF-P is 2-approximable with CIFFD.", "CIFFD is based upon the consecutive executions of FFD and possibly improving their result due to successful repacking.", "Taking advantage of Theorem REF and the fact that the cost is monotone (i.e.", "a cost of packing is not less than the sum of items to be packed) we obtain also 2-approximation for CIFFD." ], [ "On-line Approach to the Main Problems with Linear Cost ", "The algorithmic on-line method we propose now is founded upon FFf (see Paragraph REF ) with item cuts incorporated (CFFf).", "As in Subsection REF and for the same reason, we operate on item sizes, not on item indices.", "To keep the notation brief, we put $b_{\\max } = \\max _{b_i}(B)$ .", "In a nutshell, the CFFf idea is as follows.", "For items whose sizes are smaller than the largest bin capacity Hypothesis REF holds.", "These items, which form set $A$ , can be therefore packed optimally with NFC (Subsection REF ).", "Other items, which constitute set $A^+=L-A$ , require a split before packing.", "For any element $t$ of $A$ we perform a cut into $b_{\\max }$ and $t-b_{\\max }$ fragments.", "The remainders $t-b_{\\max }$ form set $A^-$ and they are packed according to FFf with $f$ indicating a fill factor (Paragraph REF ).", "We believe that this explanation is sufficient to implement the algorithm.", "We propose, however, in Algorithm REF , a more detailed description which shows explicitly a classification of items from set $A^-$ (i.e.", "items which are the remainders of cuts) into categories which are induced by different manners of item packing.", "These three categories of items from set $A^-$ , which we enumerate and comment on below, play an important role in the approximability proof: $X$ — items packed individually into bins of capacity $b_{\\max }$ , $Y$ — items packed into bins of capacity $b_{\\max }$ sharing them with other items, $Z$ — items packed into bins of any capacity $b$ , $b<b_{\\max }$ .", "[!h] VS-CIF-P data as in Def.", "REF , a fill factor $f$ a packing whose cost is as minimal as possible $t$ in $L$ (*[f]$t$ from $A$ )$t\\le b_{\\max }$ pack $t$ into bins of capacity $b_{\\max }$ with NFCNFCinCFFf continue (*[f]$t$ from $A^+$ ) $t\\le b_{\\max }$ split $t$ into a fragment $b_{\\max }$ and the remainder $t - b_{\\max }$ beginRepeatCFFf pack the fragment $b_{\\max }$ into a bin $b_{\\max }$ $t \\leftarrow t - b_{\\max }$ endRepeatCFFf *[h]$t$ is from $A^-$ there is room for $t$ in an opened binbeginIntelligentPackCFFf $b \\leftarrow $ the first opened bin into which $t$ can be packed pack $t$ into b *[f]$t\\in Y$ when the bin $b_{\\max }$ is used and $t\\in Z$ otherwise *[h]an empty bin has to be opened for $t$ (*[f]$t$ is small)$t\\le 0.5\\cdot b_{\\max }$ pack $t$ into an empty bin of capacity $b_{\\max }$ *[f]$t\\in Y$ (*[f]$t$ is big) there are bins of capacity between $t$ and $\\frac{t}{f}$ $b \\leftarrow $ the smallest empty bin of capacity between $t$ and $\\frac{t}{f}$ $b \\leftarrow $ a bin of capacity $b_{\\max }$ pack $t$ into bin $b$ *[f]$t\\in Z$ if $b<b_{\\max }$ and $t\\in X$ or $t\\in Y$ if $b=b_{\\max }$ endIntelligentPackCFFf CFFf solving the VS-CIF-P The computational effort of CFFf is concentrated upon searching an appropriate bin among those which have been already opened and selecting an empty bin with respect to a given fill factor $f$ .", "For $f=0.5$ the complexity of CFFf is $\\mathcal {O}(n(\\log n + \\log m) +m\\log m)$ .", "We estimate the quality of solution obtained with CFFf for an instance $I$ with list $L$ of items to be packed, $\\mbox{CFFf}(L)$ .", "We assume here that the cost is linear and, moreover, a bin cost is equal to its capacity, $b_i=c_i$ , $i=1,2,\\ldots , m$ as stated in Subsection REF .", "Any solution cost is always less than or equal to the overall mass of items from $L$ : $S_L=\\sum _{t\\in L}t$ .", "The notation $S_C$ indicates later on a sum of item sizes from any set $C$ .", "Theorem 5 $\\mbox{CFFf}(L)\\le \\frac{4}{3}S_L + 2b_{\\max } $ .", "As NFC, which packs items from $A$ is exact and polynomial (Theorem REF ), $\\mbox{CFFf}(A)\\le S_A + b_{\\max }$ .", "We have to estimate the packing quality for items from $A^-$ which are divided into three categories: $X$ , $Y$ , and $Z$ (see Algorithm REF ).", "Obviously, as items of category $X$ result from splitting and they occupy bins singly, $\\mbox{CFFf}(X) = 2|X|b_{\\max }$ and $S_X\\ge 1.5 |X|b_{\\max }$ with exception to at most a single bin, which gives: $\\mbox{CFFf}(X) \\le \\frac{4}{3}S_X + b_{\\max }.$ Let $Y_B$ stand for these items of Y which are packed into bin B. Analogously, $\\mbox{CFFf}(Y_B) = (|Y_B|+1)b_{\\max }$ and $S_Y \\ge (|Y_B|+\\frac{2}{3})b_{\\max }$ with exception to at most a single bin.", "This leads to: $\\mbox{CFFf}(Y) \\le \\frac{9}{8}S_Y + b_{\\max }.$ For a bin of capacity $b$ in which items $Z_b$ of $Z$ , $Z_b\\subset Z$ , are packed we have $\\mbox{CFFf}(Z_b) = |Z_b| b_{\\max } + b$ and $S_{Z_b} > b_{\\max } +fb$ .", "Consequently, $\\mbox{CFFf}(Z) \\le \\frac{3}{2+f}S_Z.$ Combining the inequalities (REF )–(REF ) with $f=0.5$ we get $\\mbox{CFFf}(A^-) \\le \\frac{4}{3}S_{A^-} + 2b_{\\max }$ which proves the theorem as $ \\mbox{CFFf}(L) = \\mbox{CFFf}(A) + \\mbox{CFFf}(A^-)$ as the number of completely filled bins $b_{\\max }$ has already been counted." ], [ "Performance Evaluation ", "The goal of the performance analysis is to estimate the difference of results obtained with our approximation algorithms relative to the exact solutions.", "Moreover, we oppose our methods to two simple reference algorithms, less “intelligent” and less costly in terms of computational effort.", "We also analyze the impact of the number of bin classes available and the number of cuts allowed $D$ on the approximation ratio of the obtained results." ], [ "Experimental Setup ", "In order to estimate algorithm's approximation ratios we created VS-CIF-P instances from exact solutions artificially made.", "We also proceeded with the comparaison of algorithms' results for instances whose exact solutions are unknown.", "All instances treated are feasible, i.e.", "the largest item fragmented at most $D$ times can be inserted into bins of the greatest capacity.", "The numerical experiments were conducted for instances with few ($m=3$ ) and many bin classes ($m=10$ ).", "We arbitrarily fixed the largest capacity $b_{\\max }$ to 100.", "The capacities of other $m-1$ classes are chosen uniformly in the natural interval $[1,b_{\\max }-1]$ .", "In the case of a linear cost we took a bin cost equal to its capacity, $c_i = b_i$ , $i=1\\ldots , m$ .", "When dealing with a monotone cost we assume that the largest bin cost is also equal to its capacity, $c_{\\max } = b_{\\max }$ .", "Assuming that bin classes are sorted in decreasing order of their size, the cost $c_{i+1}$ is chosen uniformly in $[b_{i+1},c_i -1]$ .", "Initial items are generated uniformly in $[1,99]$ and their average size is 50.", "The number of initial items is arbitrarily fixed to 200.", "We have thus the expected total volume of $10^{\\prime }000$ to be packed.", "The construction of exact solutions consists in putting initial items into bins with FF and filling the opened bins entirely with extra items.", "Items to feed the algorithms which admit fragmentation are made up of initial and extra items put together: up to $D+1$ items (initial or extra) can be glued to form an individual item.", "Consequently, in the experiments we made, for $D=1$ the average item size is equal to 100, for $D=2$ the average item size is equal to 200, etc.", "The exact solutions which we took as a base of the instance creation are composed typically of numerous large bins and a single little one.", "Not willing to restrict ourselves to such a solution form we also realized the direct confrontation of results obtained from initial items, possibly glued up to $D$ times, as explained above.", "We did not add, however, any extra items as we did not fill up the opened bins.", "By giving up fixing an exact solution as a starting point of the instance construction we award instances with more flexibility.", "Despite the fact that the approximation factor we give for CFFs in Theorem REF holds for the linear cost, we decided to run CFFf with the monotone cost, too.", "We argue that at this stage we can evaluate empirically its performance with the monotone cost.", "The results are averaged for series of 1000 instances with which the algorithms are fed.", "The confidence intervals depicted in all figures illustrating the following subsection are computed with the confidence level $\\alpha =0.05$ ." ], [ "Results ", "Before presenting the results we explain the simple greedy algorithms which will be brought face to face with our algorithms.", "The first of them, called CNFL (this abbreviation is straightforward and will be explained below) is on-line.", "Its operating mode is two-fold.", "First, it cuts items up to $D$ times to fit their fragments into largest bins.", "This is a “modulo $b_{\\max }$ cut”: $D$ bins are filled up, the last one may be partially filled.", "Next, it packs them according to the Next Fit principle (Cut and Next Fit, CNF).", "The reader may refer to Subsection REF and Figure REF to recall the discussion of the similar CFF based upon “almost equal” cuts.", "Cost minimizing is obtained by always using the Largest bin (the dual principle to the one seen in Paragraph REF ).", "Assuming that bin classes are preliminarily sorted, the CNFL complexity is $\\mathcal {O}(n)$ .", "Its performance will be compared with that of CFFf.", "Figure: Estimation of the approximation ratio for the off-line algorithms, CIFFD and CDNFL, with D=1D=1 and linear cost for different numbers of bin classesFigure: Estimation of the approximation ratio for the off-line algorithms, CIFFD and CDNFL, with D=1D=1 and monotone cost for different numbers of bin classesThe second one is an off-line mutation of CNFL in which the items, after the preliminary “modulo $b_{\\max }$ cut” made as explained above, are sorted in decreasing order.", "This off-line algorithm, to be confronted with CIFFD, is obviously called CDNFL.", "Intuitivelly, the great number of cuts allowed may facilitate packing procedures.", "We opted thus to confront the algorithms for $D=1$ .", "Figures REF and REF present the results of the comparison of the approximation ratios obtained with two off-line algorithms in function of the number of bin classes for linear and monotone costs, respectively.", "Figures REF and REF do the same for both on-line methods.", "Figures REF and REF show at a glance that the algorithms perform much better the theoretical performance bounds given in Theorems REF and REF for CIFFD with monotone cost and CFFf with linear cost, respectively.", "As one may expect, the approximation ratio obtained with CIFFD is significantly better comparing with the one produced by the naive approach for both the analyzed costs (Figures REF and REF ).", "As our algorithm is based upon consecutive repacking of a single, the least filled bin, the impact of the number of bin classes available is considerable.", "CIDDF packs better when having many bin classes at its disposal, in contrast to CDNFL which is insensitive to this parameter.", "The on-line approach, CFFf, is not significantly influenced by the number of bin classes.", "Figures REF –REF put in evidence its strikingly good performance.", "CFFf, despite being on-line, outperforms even the off-line CIFFD method in certain situations.", "This interesting phenomenon will be explained below while studying the influence of the number of cuts allowed.", "Figure: Estimation of the approximation ratio for the on-line algorithms, CFFf and CNFL, with D=1D=1 and linear cost for different numbers of bin classesFigure: Estimation of the approximation ratio for the on-line algorithms, CFFf and CNFL, with D=1D=1 and monotone cost for different numbers of bin classesFigures REF and REF show the performance of the algorithms for the same series of instances whose exact solutions are a priori unknown, with linear and monotone cost, respectively.", "This experience allowed us to compare the algorithm quality for instances which do not suffer from the imposed form of an exact solution.", "The smaller value of the average packing cost signifies a higher packing efficiency.", "As the average total volume of items to be packed is preserved and equal to $10^{\\prime }000$ , the reader may observe the similar tendency as in the case of the comparison with optimal solutions.", "It is not astonishing that all algorithms behave better for the linear cost.", "CIFFD becomes more efficient when the number of bin classes goes up.", "Again, CFFf performs much better that a greedy off-line method CNFL.", "The impact of the limit set on the number of splits permitted is illustrated in Figures REF and REF for the off-line and on-line approaches, respectively.", "This analysis reveals a secret of the excellent performance of CFFf.", "As the results for the linear and monotone costs exhibit the same tendency, we restrict the graphical presentation to the latter only.", "The fragmentation ban ($D=0$ ) signifies that the problem solved is simply the VSBPCP.", "The graph in Figure REF confirms the intuition that more splits allowed make packing easier.", "For instance, CIFFD with monotone cost, 10 bin classes and up to 8 cuts often reaches “an almost exact solution”.", "The behavior of CFFf depicted in Figure REF does not, however, exhibit the same trend.", "Before explaining this phenomenon we recall to the reader that the items which satisfy Hypothesis (REF ) are packed optimally with NFC according to Theorem REF .", "We also call up that in our experiments the average total volume of items to be packed is preserved regardless the value of $D$ (see Subsection REF ).", "It means that for a great $D$ value an instance has less items but they are bigger.", "As we see in Figure REF , admitting one cut ($D=1$ ) drastically lowers the solution cost comparing with the situation when the fragmentation is forbidden (the VSBPCP for $D=0$ ).", "In our experiments, the average item size for $D=1$ is 100.", "The largest bin has the same capacity $b_{\\max }=100$ .", "A relatively large part of items is therefore inserted into largest bins by NFC (line  in Algorithm REF ).", "When more cuts are allowed, for example $D=2$ , the average item size is greater, 200, while the largest bin capacity stays unchanged, $b_{\\max }=100$ , the “modulo $b_{\\max }$ splitting” made in the repeat loop (lines – of Algorithm REF ) takes over.", "This loop may potentially open up too many largest bins than necessary.", "Finally, when $D$ is increasing (starting from $D=7$ in our experiments) the negative impact of the repeat loop is compensated by the intelligent packing realized in lines – of Algorithm REF and the CFFf performance stabilizes close to an exact solution (between five and ten per cent).", "We thus draw a conclusion that the more items CFFf inserts with NFC, the better its performance is.", "Figure: Average result of the off-line (CIFFD and CDNFL) and on-line algorithms (CFFf and CNFL) with D=1D=1 and linear cost for different numbers of bin classesFigure: Average result of the off-line (CIFFD and CDNFL) and on-line algorithms (CFFf and CNFL) with D=1D=1 and monotone cost for different numbers of bin classesFigure: Estimation of the approximation ratio for the off-line CIFFD with monotone cost for different numbers of bin classes available in function of the number of splits allowedFigure: Estimation of the approximation ratio for the on-line CFFf with monotone cost for different numbers of bin classes available in function of the number of splits allowed" ], [ "Conclusions and Perspectives ", "We proposed the modeling of the energy-aware load balancing of computing servers in networks providing virtual services by a generalization of the Bin Packing problem.", "As a member of the Bin Packing family, our problem is also approximable with a constant factor.", "In addition to its theoretical analysis we proposed two algorithms, one off-line and another one on-line, giving their theoretical performance bounds.", "The empirical performance evaluation we realized showed that the results they provide are significantly below the approximation factor.", "On the one hand, this practical observation encourages us to continue looking for a better approximation factor, especially for CIFFD whose performance is much better than predicted theoretically.", "We will also try to extend the analysis of CFFf to the monotone cost.", "On the other hand, we gave the performance evaluation which allows us to consider CIFFD and CFFf as the algorithms with a very good potential for practical applications like energy-aware load balancing, which motivated our work.", "We emphasize the remarkable efficiency of our on-line approach (CFFf) which outperforms considerably simple off-line algorithms.", "Notwithstanding, the theoretical result proven only for the linear cost CFFf behaves very well when bin costs are monotone.", "The approach presented in this paper is centralized and adapted to private Cloud infrastructures.", "Another challenge calling into question is the application of our packing approach into a distributed environment when information about resource availability is incomplete.", "We believe that the approach through packing is a powerful tool allowing one to perform a load-balancing in Clouds which ensures the realization of tasks with respect to their requirements while consuming the smallest quantity of electrical energy.", "For this reason we think to use this approach in our further multi-criteria optimization of a Cloud infrastructure.", "Among other criteria which we find essential to study in this context are the efficient utilization of resources of a telecommunication network (the principle “network-aware Clouds”) and the guarantee of meeting the QoS requirements expressed in customers' contracts, concerning, for instance, the task termination before a given dead-line or the task execution time limited by a given make-span." ], [ "Acknowledgment", "Stéphane Henriot participation in this work was partially financed by the grant 2013-22 of PRES UniverSud Paris." ] ]
1403.0493
[ [ "Henon-Heiles potential as a bridge between nontopological solitons of\n different types" ], [ "Abstract We apply the Hubbard-Stratanovich transformation to the Lagrangian for nontopological solitons of the Coleman type in a two-dimensional theory.", "The resulted theory with an extra real scalar field can be supplemented with a cubic term to obtain a model with exact analytical solution." ], [ "Introduction", "Internal symmetry may provide a mechanism of stabilization of extended objects in a scalar field theory with an appropriate self-interaction [1], [2].", "Constructions with extra scalar fields are also possible [3] depending on the existence of a second symmetry which is spontaneously brokenSee also references on earlier works in [3]..", "Following [2] we will say that the latter theories and theories with only one scalar field are of different genus.", "In four space-time dimensions such spherically symmetric solutions are usually referred to as Q-balls of the Coleman and Friedberg-Lee-Sirlin types respectively.", "In this paper we will study nontopological solitons in two-dimensional theories providing analytical solutions.", "We will start from the theory of one complex scalar field with self-interaction.", "Consideration of this self-interaction as an effective interaction due to the massive field leads to the theory with an extra scalar field similar to the Friedberg-Lee-Sirlin case.", "We show that the theory of a single complex scalar field can be completed in a simple way to obtain an explicit solution for the nontopological soliton." ], [ "Nontopological soliton of the Coleman type", "Let us start with the simplest model of a scalar field in two dimensions.", "The $U(1)$ -invariant Lagrangian density is ${\\cal L}\\equiv T-V= \\partial _\\mu \\phi ^*\\partial ^\\mu \\phi -m^2(\\phi ^*\\phi )+\\frac{\\lambda }{2}(\\phi ^*\\phi )^2$ with positiveThe potential $V$ is unbounded from below.", "However one can add positive term $\\epsilon (\\phi ^*\\phi )^3$ with $\\epsilon \\ll \\lambda $ to cure the model for large values of the field.", "$\\lambda $ .", "Decay of the vacuum $\\phi =0$ is exponentially suppressed in the case of $\\lambda /m^2\\ll 1$ , thus we can use density (REF ) for the effective description.", "Q-balls in this model were examined in [4] and the earlier works on soliton interaction.", "For the usual anzatz $\\phi (t,z)={\\rm e}^{{\\rm i}\\omega t}f(z)$ one can obtain a solution of the form $f=\\sqrt{\\frac{2(m^2-\\omega ^2)}{\\lambda }}\\frac{1}{\\cosh (\\sqrt{m^2-\\omega ^2}z)}.$ For $\\omega =0$ it corresponds to the bounce for the tunnelling problemThis feature can be used as a starting point for the demonstration of the existence of continuius family of solutions [5].. Energy $E$ and charge $Q$ are finite for all $\\omega \\in [0,m]$ : $E=\\frac{8}{\\lambda }\\sqrt{m^2-\\omega ^2}\\left(m^2-\\frac{2}{3}(m^2-\\omega ^2)\\right),$ $Q=\\frac{8\\omega }{\\lambda }\\sqrt{m^2-\\omega ^2}.$ On Fig.REF we plot the $E(Q)$ dependence for $\\lambda =m^2$ .", "To obtain this dependence for arbitrary $\\lambda $ one can use the following scaling to dimensionless parameters: $\\omega \\rightarrow \\frac{\\omega }{m},\\qquad E\\rightarrow \\frac{E}{m}\\frac{\\lambda }{m^2}\\qquad Q\\rightarrow Q\\frac{\\lambda }{m^2}.$ It is useful to introduce an analogue of the grand potential $\\Omega (\\omega )\\equiv E(Q(\\omega ))-\\omega Q(\\omega )=\\frac{8}{3\\lambda }\\left(\\sqrt{m^2-\\omega ^2}\\right)^3.$ From obvious relation $Q=-d\\Omega /d\\omega $ one can obtain the equality $dE/dQ=\\omega $ , which may be used to verify our result.", "Figure: The E(Q)E(Q) dependence for λ=m 2 \\lambda =m^2 (solid line).", "Dashed linecorresponds to free particles with E=mQE=mQ.This relation is also useful for applying the classical stability criterion $d^2E/dQ^2<0$ (or, using (REF ), $\\partial ^2\\Omega /\\partial \\omega ^2>0$ ) of [6]: $\\frac{d^2E}{dQ^2}=\\frac{d\\omega }{dQ}=\\frac{8}{\\lambda }\\sqrt{m^2-\\omega ^2}\\frac{1}{m^2-2\\omega ^2}.$ It means that the solution is stable classically for $\\omega /m\\in (1/\\sqrt{2},1)$ .", "This range of parameters corresponds to the lower branch of Fig.REF with $Q_{min}=0$ and $Q_{max}=4m^2/\\lambda $ .", "Thus, for the small coupling regime $m^2/\\lambda \\gg 1$ one can consider nontopological solitons with large $Q$ as quasiclassical objects." ], [ "Analytical solution with extra field", "Interaction in (REF ) can be considered as an effective interaction due to an extra very massive real scalar field $\\chi $ [7].", "Let us consider the Lagrangian $\\partial _\\mu \\phi ^*\\partial ^\\mu \\phi -m^2(\\phi ^*\\phi )+\\frac{1}{2}\\partial _\\mu \\chi \\partial ^\\mu \\chi -\\frac{M^2}{2}\\chi ^2+g\\chi (\\phi ^*\\phi ).$ In the limit $M^2\\gg m^2$ the heavy field $\\chi $ can be integrated out to obtain an effective low-energy theory of a single field $\\phi $ .", "The tree-level result is equivalent to (REF ) with self-interaction coupling constant $\\lambda _{eff}=\\frac{g^2}{M^2}$ .", "One can find a solution in the system of coupled fields using the perturbation theory.", "However, note that an exact analytical solution exists in the following model ${\\cal L} = \\partial _\\mu \\phi ^*\\partial ^\\mu \\phi -m^2(\\phi ^*\\phi )+\\frac{1}{2}\\partial _\\mu \\chi \\partial ^\\mu \\chi -\\frac{M^2}{2}\\chi ^2+g\\chi (\\phi ^*\\phi )+g\\chi ^3.$ Here the coupling constant $g$ is arbitrary, but the cubic interaction is tuned.", "The solution for a nontopological soliton looks like $\\phi ={\\rm e}^{{\\rm i}\\omega t}\\frac{A}{{\\rm cosh} (\\sqrt{m^2-\\omega ^2}z)}, \\qquad \\chi =\\frac{B}{({\\rm cosh} (\\sqrt{m^2-\\omega ^2}z))^2},$ where $\\begin{array}{c}A=\\frac{\\sqrt{m^2-\\omega ^2}}{g}\\sqrt{2(M^2-4(m^2-\\omega ^2))},\\\\B=\\frac{2(m^2-\\omega ^2)}{g}.\\end{array}$ The charge of this configuration $Q=\\frac{8\\omega \\sqrt{m^2-\\omega ^2}}{g^2}(M^2-4(m^2-\\omega ^2))$ and the energy $E=\\frac{8\\sqrt{m^2-\\omega ^2}}{g^2}\\left(\\frac{2}{3}\\omega ^2M^2+\\frac{1}{3}m^2M^2-2(m^4-\\omega ^4)+\\frac{6}{5}(m^2-\\omega ^2)^2\\right).$ also satisfy the constraintThis relation also holds for solitons with arbitrary $V(\\chi )$ in the Lagrangian (REF ).", "$dE/dQ=\\omega $ .", "The Compton length of our solution is small compared to its characteristic size for the intermediate values of $\\omega /m$ for $g/M^2\\ll 1$ .", "The reason of integrability of equations of motion lies in the formal correspondence to the problem of classical mechanics with the Hénon-Heiles potential [8] (see also [9] for an explicit integral of motion for our choice of the cubic interaction).", "One can see from (REF ) that the range of allowed values is $\\omega /m\\in [0,1]$ for $M/m\\ge 2$ and $\\omega /m\\in [\\sqrt{1-(M/(2m))^2},1]$ for $M/m\\le 2$ .", "From Fig.REF one can see that the $E(Q)$ dependence reproduces the shape of Fig.REF even for the value $m/M=0.33$ which is not very small.", "One can also obtain the values of the charge and the energy for the coupling constant $g\\ne m^2$ using an analogue of reparametrization (REF ).", "Figure: The E(Q)E(Q) plots for M/m≥2M/m\\ge 2, g=m 2 g=m^2 (left plot, M/m=2M/m=2 – thick line,M/m=2.5M/m=2.5 – dashed line, M/m=3M/m=3 – dotted line) and m/M≤2m/M\\le 2, g=m 2 g=m^2 (right plot, M/m=1.5M/m=1.5 – thick line,M/m=1.7M/m=1.7 – dashed line, M/m=2M/m=2 – dotted line)." ], [ "Conclusion", "New analytical solution for nontopological soliton with an extra scalar field is obtained.", "The integrability of equations of motion is determined by the formal correspondence to the problem of classical mechanics with the Hénon-Heiles symmetry.", "There is no additional discrete symmetry in (REF ) which can be spontaneously broken.", "In this sense our soliton is not of the Friedberg-Lee-Sirlin type.", "However, the absence of self-interaction for the field $\\phi $ in (REF ) is crucial, because the existence of solution is determined by the extra field $\\chi $ .", "We examined the limit $m/M\\rightarrow 0$ in which the theory with the simplest self-interaction is reproduced in the tree-level approximation.", "A remarkable feature of our model that is its renormalizability in theories with extra dimensions, i.e.", "cubic interactions in (REF ) are irrelevant operators in theories with more than three spatial dimension.", "The author is indebted to M. N. Smolyakov and S. V. Troitsky for reading the text and providing comments that improved the manuscript, and M. V. Libanov and S. Yu.", "Vernov for helpful discussions and correspondence.", "This work was supported by grant NS-2835.2014.2 of the President of Russian Federation and by RFBR grant 14-02-31384." ] ]
1403.0434
[ [ "Moduli spaces of bundles over non-projective K3 surfaces" ], [ "Abstract We study moduli spaces of sheaves over non-projective K3 surfaces.", "More precisely, if $v=(r,\\xi,a)$ is a Mukai vector on a K3 surface $S$ with $r$ prime to $\\xi$ and $\\omega$ is a \"generic\" K\\\"ahler class on $S$, we show that the moduli space $M$ of $\\mu_{\\omega}-$stable sheaves on $S$ with associated Mukai vector $v$ is an irreducible holomorphic symplectic manifold which is deformation equivalent to a Hilbert scheme of points on a K3 surface.", "If $M$ parametrizes only locally free sheaves, it is moreover hyperk\\\"ahler.", "Finally, we show that there is an isometry between $v^{\\perp}$ and $H^{2}(M,\\mathbb{Z})$ and that $M$ is projective if and only if $S$ is projective." ], [ "Introduction", "Moduli spaces of sheaves on projective K3 surfaces have been studied since the '80s.", "In [9] Fujiki considered the Hilbert scheme $Hilb^{2}(S)$ of 2 points on a K3 surface $S$ ; his result was widely generalized by Beauville in [4], who studied $Hilb^{n}(S)$ for any $n\\in \\mathbb {N}$ , showing that it is an irreducible hyperkähler manifold, i.e.", "a compact Kähler manifold which is simply connected, holomorphically symplectic and has $h^{2,0}=1$ .", "Moduli spaces of $\\mu -$ stable sheaves are a generalization of Hilbert schemes of points, and they have been extensively studied when the base surface $S$ is a projective K3 surface.", "In [28] Mukai showed that on the moduli space $M$ of simple sheaves of Mukai vector $v=(r,c_{1}(L),a)$ (i.e.", "of rank $r$ , determinant $L$ and second Chern character $a-r$ ), there is a natural holomorphic symplectic form associated to the one on $S$ .", "This moduli space $M$ is a non-separated scheme containing as a smooth open subset the moduli space $M^{\\mu }_{v}(S,H)$ of $\\mu _{H}-$ stable sheaves (with respect to some ample line bundle $H$ on $S$ ) of Mukai vector $v$ ; Mukai's construction thus produces a holomorphic symplectic form on $M^{\\mu }_{v}(S,\\omega )$ .", "If $H$ is generic and $r$ and $L$ are prime to each other, then $M_{v}^{\\mu }(S,H)$ is a projective holomorphically symplectic manifold.", "Moreover, it is an irreducible hyperkähler manifold deformation equivalent to a Hilbert scheme of points on $S$ (see [30] and [42]).", "If $S$ is a non-projective K3 surface and $\\omega $ is a Kähler class on it, one still defines the notion of $\\mu _{\\omega }-$ stable sheaf and constructs the moduli space $M^{\\mu }_{v}(S,\\omega )$ of $\\mu _{\\omega }-$ stable sheaves of Mukai vector $v$ .", "In [36] it is shown that $M^{\\mu }_{v}(S,\\omega )$ is a smooth complex manifold carrying a holomorphic symplectic form.", "If $\\omega $ is generic and $r$ is prime with $c_{1}(L)$ , then $M^{\\mu }_{v}(S,\\omega )$ is even compact (see subsection REF for the precise notion of genericity we use for Kähler classes, called $v-$ genericity in analogy to the projective case).", "It is natural to ask if $M^{\\mu }_{v}(S,\\omega )$ is irreducible symplectic, and in this case what is its deformation class.", "We first show the following: Theorem 1.1 Let $S$ be a K3 surface, $v=(r,\\xi ,a)\\in H^{*}(S,\\mathbb {Z})$ where $\\xi \\in NS(S)$ , $r>1$ prime with $\\xi $ and $v^{2}\\ge 0$ .", "Suppose $\\omega $ to be $v-$ generic.", "The moduli space $M^{\\mu }_{v}(S,\\omega )$ is a compact, connected complex manifold of dimension $v^{2}+2$ which is holomorphically symplectic and deformation equivalent to a Hilbert scheme of points on a projective K3 surface.", "On $H^{2}(M^{\\mu }_{v}(S,\\omega ),\\mathbb {Z})$ there is a non-degenerate quadratic form, and there is an isometry between $H^{2}(M^{\\mu }_{v},\\mathbb {Z})$ and $v^{\\perp }$ if $v^{2}>0$ (resp.", "$v^{\\perp }/\\mathbb {Z}v$ if $v^{2}=0$ ).", "The condition $v^{2}\\ge 0$ implies that $M^{\\mu }_{v}(S,\\omega )\\ne \\emptyset $ (see [2], [33], [23]).", "As recalled above, if $S$ is projective and $\\omega =c_{1}(H)$ for a generic ample line bundle $H$ we even know that $M^{\\mu }_{v}(S,\\omega )$ is an irreducible symplectic manifold.", "To prove Theorem REF , we study the two remaining cases: $S$ is projective and $\\omega \\notin NS(S)$ ; and $S$ is non-projective.", "When $S$ is projective and $\\omega $ is not the first Chern class of an ample line bundle, we show that there is a $v-$ generic ample line bundle $H$ such that $M^{\\mu }_{v}(S,\\omega )=M_{v}^{\\mu }(S,H)$ .", "This is done by showing that the $v-$ chamber in which $\\omega $ lies intersects the ample cone, and that moving the polarization inside a $v-$ chamber does not affect the moduli space.", "When $S$ is non-projective, the strategy to prove Theorem REF is to deform $M^{\\mu }_{v}(S,\\omega )$ along the twistor family $\\mathcal {X}\\longrightarrow \\mathbb {P}^{1}$ of $(S,\\omega )$ : even if the sheaves in $M^{\\mu }_{v}(S,\\omega )$ do not necessarily deform along such a twistor family, we can still deform them as twisted sheaves.", "We then provide a construction of a relative moduli space of stable twisted sheaves extending Yoshioka's construction in [43] to non-projective base manifolds and we show that we can connect the K3 surface $S$ to a projective K3 surface $S^{\\prime }$ only by means of twistor lines, in such a way that $M^{\\mu }_{v}(S,\\omega )$ deforms to $M^{\\mu }_{v}(S^{\\prime },\\omega ^{\\prime })$ for some $v-$ generic polarization $\\omega ^{\\prime }$ on $S^{\\prime }$ .", "Theorem REF holds true even if we replace $M^{\\mu }_{v}(S,\\omega )$ with a moduli space of stable twisted sheaves.", "The non-degenerate quadratic form on $H^{2}(M^{\\mu }_{v}(S,\\omega ),\\mathbb {Z})$ is defined as a quadratic form on the second complex cohomology using the same definition of the Beauville form, the only difference being that we have to fix one holomorphic symplectic form to define it as a priori we have $h^{2,0}\\ge 1$ .", "We then show that it is non-degenerate.", "The construction of the isometry with $v^{\\perp }$ is standard, and uses the same strategy as in the projective case.", "As one might see from the statement on Theorem REF , there is only one missing property for $M^{\\mu }_{v}(S,\\omega )$ to be an irreducible symplectic manifold; namely, we don't know if $M^{\\mu }_{v}(S,\\omega )$ is Kähler.", "This is a longstanding problem: on the open subset $M^{\\mu -lf}_{v}(S,\\omega )$ of $M^{\\mu }_{v}(S,\\omega )$ parametrizing locally free sheaves we have a natural Kähler metric, the Weil-Petersson metric, cf.", "[19], [20], but at present nothing is known as to how this metric could extend to a Kähler metric on the whole $M^{\\mu }_{v}(S,\\omega )$ .", "The strategy to prove Theorem REF together with [13] may be employed to obtain another proof of the existence of a Kähler metric on $M^{\\mu -lf}_{v}(S,\\omega )$ and of a description of a twistor family for such a hyperkähler metric.", "But, as pointed to us by Daniel Huybrechts, this strategy does not allow to show that $M_{v}^{\\mu }(S,\\omega )$ carries a Kähler metric too.", "Let us remark however that there are choices of Mukai vectors for which $M_{v}^{\\mu }(S,\\omega )$ coincides with $M^{\\mu -lf}_{v}(S,\\omega )$ and is therefore a compact irreducible hyperkähler manifold.", "Moreover such compact moduli spaces of stable locally free sheaves may acquire any positive even complex dimension; see Proposition REF .", "As an application of the previous result, we will show the following projectivity criterion for the moduli spaces of slope-stable sheaves on a K3 surface: Theorem 1.2 Let $S$ be a K3 surface, $v=(r,\\xi ,a)\\in H^{2*}(S,\\mathbb {Z})$ where $\\xi \\in NS(S)$ , $r\\ge 2$ , $(r,\\xi )=1$ and $v^{2}\\ge 0$ .", "If $\\omega $ is a $v-$ generic polarization, the moduli space $M^{\\mu }_{v}(S,\\omega )$ is projective if and only if $S$ is projective." ], [ "Acknowledgements", "We are grateful to Daniel Huybrechts for pointing out to us a mistake in a previous version of this paper.", "We also thank the referee for his remarks and suggestions which significantly contributed to improve the exposition." ], [ "Moduli spaces of stable sheaves", "In the following $S$ will be a K3 surface, possibly non-projective.", "If ${F}$ is a coherent sheaf on $S$ , we let the Mukai vector of ${F}$ be $v({F}):=ch({F})\\cdot \\sqrt{td(S)}\\in H^{2*}(S,\\mathbb {Z}).$ If $v_{i}$ is the component of $v({F})$ in $H^{2i}(S,\\mathbb {Z})$ , we have $v_{0}=rk({F})$ , $v_{1}=c_{1}({F})$ and $v_{2}=ch_{2}({F})+rk({F})=\\frac{1}{2}c_1^2({F})-c_2({F})+rk({F})$ , which will be viewed as an integer (i.e.", "we fix an isomorphism $H^{4}(S,\\mathbb {Z})\\simeq \\mathbb {Z}$ ).", "We recall that on $H^{2*}(S,\\mathbb {Z})$ we have a pure weight-two Hodge structure and a lattice structure with respect to the Mukai pairing (see Definitions 6.1.5 and 6.1.11 of [15]): the obtained lattice will be referred to as Mukai lattice, and we will write $v^{2}$ for the square of $v\\in H^{2*}(S,\\mathbb {Z})$ with respect to the Mukai pairing.", "Explicitly, $v^2=v_1^2-2v_0v_2$ .", "When $v_0\\ne 0$ we define the discriminant of $v$ , or respectively of ${F}$ in case $v=v({F})$ , as $\\Delta (v):=\\frac{1}{2v_0^2}v^2+1,$ This coincides with the definition of [2] for instance, where $\\Delta ({F})=\\Delta (v({F}))=\\frac{1}{rk({F})}\\bigg (c_{2}({F})-\\frac{rk({F})-1}{2rk({F})}c_1^2({F})\\bigg ).$" ], [ "The stability condition", "Let $g$ be a Kähler metric on $S$ and $\\omega $ the associated Kähler class, that will be called a polarization on $S$ .", "If ${F}\\in Coh(S)$ has positive rank, the slope of ${F}$ with respect to $\\omega $ is $\\mu _{\\omega }({F}):=\\frac{c_{1}({F})\\cdot \\omega }{rk({F})}.$ Definition 2.1 A torsion-free coherent sheaf ${F}$ is $\\mu _{\\omega }-$ stable if for every coherent subsheaf ${E}\\subseteq {F}$ such that $0< rk({E})< rk({F})$ we have $\\mu _{\\omega }({E})<\\mu _{\\omega }({F})$ .", "If $\\mu _{\\omega }({E})\\le \\mu _{\\omega }({F})$ for all such subsheaves ${E}$ , then we say that ${F}$ is $\\mu _{\\omega }-$ semistable.", "The family of $\\mu _{\\omega }-$ stable sheaves of Mukai vector $v$ admits a moduli space $M^{\\mu }_{v}(S,\\omega )$ .", "If $S$ is projective and $\\omega $ is the first Chern class of an ample line bundle $H$ , then $M^{\\mu }_{v}(S,\\omega )$ is the moduli space $M^{\\mu }_{v}(S,H)$ of $\\mu _{H}-$ stable sheaves on $S$ with Mukai vector $v$ .", "We have the following proposition dealing also with the non-projective case (see [36]).", "Proposition 2.2 Let $S$ be a K3 surface, $v\\in H^{2*}(S,\\mathbb {Z})$ a Mukai vector and $\\omega $ a polarization on $S$ .", "The moduli space $M^{\\mu }_{v}(S,\\omega )$ is a smooth, holomorphically symplectic manifold (possibly non-compact) and, if it is not empty, its dimension is $v^{2}+2$ .", "In the following we will restrict to the case of those $M^{\\mu }_{v}(S,\\omega )$ which are non-empty and compact.", "We introduce in the next section some hypothesis on $v$ and $\\omega $ under which $M^{\\mu }_{v}(S,\\omega )$ is compact.", "We now present a condition which guarantees its non-emptyness, and even the existence of a stable vector bundle with respect to any polarization.", "Recall that over any non-algebraic surface there exist non-filtrable holomorphic rank two vector bundles (see [2], [34] p.18).", "By definition they do not admit coherent subsheaves of rank one, hence they are stable with respect to any polarization.", "We now extend this type of result to arbitrary rank in the case of Kähler surfaces.", "Following [2] we say that a coherent sheaf on the surface $S$ is irreducible if its only coherent subsheaf of lower rank is the zero sheaf.", "In particular, an irreducible sheaf is stable with respect to any polarization.", "We have the following result, about the existence of locally free irreducible vector bundles.", "Proposition 2.3 Let $S$ be a Kähler non-algebraic compact complex surface, $r$ a positive integer and $\\xi \\in NS(S)$ .", "Then there exists a bound $b:=b(r,\\xi )\\in \\mathbb {Z}$ depending on $r$ and on $\\xi $ such that for any integer $c\\ge b$ there is on $S$ an irreducible locally free sheaf ${F}$ of rank $r$ , $c_{1}({F})=\\xi $ and $c_{2}({F})=c$ .", "If $r=2$ , a statement of this type is proved in [2] and in [34] without the Kähler assumption.", "The idea there was to look at the versal deformation space of a rather arbitrary coherent sheaf ${F}$ and show that if $c_{2}\\gg 0$ then ${F}$ must contain irreducible objects.", "For $r>2$ we shall this time consider deformations of suitably chosen coherent sheaves and make essential use of the fact that $S$ is Kähler.", "In this way we shall reduce ourselves to the argument used by Bănică and Le Potier in the case when the algebraic dimension of $S$ is zero, [2].", "We proceed by induction on $r$ .", "The statement is trivial for $r=1$ and already proven for $r=2$ .", "Let then $r\\ge 3$ and suppose that the statement is true for rank $r-1$ .", "Take an irreducible locally free sheaf ${E}$ on $S$ of rank $r-1$ , $c_{1}({E})=\\xi $ and $c_{2}({E})=c$ .", "Consider an irreducible component $B$ of the versal deformation space of ${F}_0:={O}_{S}\\oplus {E}$ and the corresponding family ${F}$ of coherent sheaves over $S\\times B$ .", "We shall check that if $c\\gg 0$ , the relative Douady space $D_{(X\\times B)/B}({F},k)$ of flat quotients of rank $k$ of ${F}$ over $B$ does not cover $B$ for $1\\le k\\le r-1$ .", "Let $b:D_{(X\\times B)/B}({F},k)\\rightarrow B$ be the natural morphism and $Q\\subset B$ a relatively compact subdomain of $B$ containing the origin $0\\in B$ .", "Fujiki proved in [10] that any irreducible component of $b^{-1}(Q)$ is proper over $Q$ .", "By another result of Fujiki in [8], there are countably many such components.", "The idea is to show by a dimension count that very general neighbours of ${F}_0$ are not in the image of $D_{(X\\times B)/B}({F},k)$ for $2\\le k\\le r-2$ .", "Remark that if ${F}_b$ is such a neighbour sitting in a short exact sequence $0\\rightarrow F^{\\prime }\\rightarrow {F}_b\\rightarrow F^{\\prime \\prime }\\rightarrow 0$ with $F^{\\prime \\prime }$ torsion-free, then $F^{\\prime }$ and $F^{\\prime \\prime }$ are irreducible of different ranks, hence $\\mbox{Hom}(F^{\\prime },F^{\\prime \\prime })=0=\\mbox{Hom}(F^{\\prime \\prime }, F^{\\prime })$ .", "This remark makes the arguments in the proof of [2] work by replacing the corresponding inequality in loc.", "cit.", "Lemme 5.12.", "Hence our statement." ], [ "The $v-$ genericity for Kähler forms", "Let $S$ be a K3 surface and ${K}_{S}$ its Kähler cone, which is an open and convex cone in $H^{1,1}(S)$ .", "For $v=(r,\\xi ,a)$ with $r\\ge 2$ and $\\xi \\in NS(S)$ , we define a system of hyperplanes in $H^{1,1}(S)$ , which is locally finite in ${K}_{S}$ and has the property that for any $\\omega \\in {K}_{S}$ not lying on such hyperplanes, a torsion free sheaf ${F}$ on $S$ with $v({F})=v$ is $\\mu _{\\omega }$ -stable if and only if it is $\\mu _{\\omega }$ -semistable.", "Polarizations verifying this will be called $v-$ generic." ], [ "The notion of $v-$ genericity", "To start, let $S$ be any compact Kähler surface and fix $r,c_{2}\\in \\mathbb {Z}$ , $c_{1}\\in NS(S)$ and suppose $r>0$ .", "We let $\\tau :=(r,c_{1},c_{2})$ , and if ${F}\\in Coh(S)$ of rank $r$ and Chern classes $c_{1}$ and $c_{2}$ , we call $\\tau $ the topological type of ${F}$ .", "If $S$ is a K3 surface and ${F}\\in Coh(S)$ has Mukai vector $v=(r,\\xi ,a)$ , its topological type is $\\tau _{v}=(r,\\xi ,\\xi ^{2}/2+r-a)$ .", "Notice that the discriminant $\\Delta ({F})$ only depends on the topological type of ${F}$ , hence we can talk about the discriminant $\\Delta (\\tau )$ of $\\tau $ : more precisely, if $\\tau =(r,c_{1},c_{2})$ then $\\Delta (\\tau )=\\frac{1}{r}\\bigg (c_{2}-\\frac{r-1}{2r}c_1^2\\bigg ).$ We set $W_{\\tau }:=\\lbrace D\\in NS(S)\\,|\\,-\\frac{r^{4}}{2}\\Delta (\\tau )\\le D^{2}<0\\rbrace $ and for every $\\alpha \\in H^{1,1}(S)$ we write $\\alpha ^{\\perp }:=\\lbrace \\beta \\in H^{1,1}(S)\\,|\\,\\alpha \\cdot \\beta =0\\rbrace .$ When $\\alpha \\ne 0$ , the set $\\alpha ^{\\perp }$ is a hyperplane in $H^{1,1}(S)$ .", "Using the same argument of Lemma 4.C.2 of [15], one shows that if $\\beta \\in H^{1,1}(S)$ , then there is a open neighbourhood $U$ of $\\beta $ in $H^{1,1}(S)$ such that $U\\cap D^{\\perp }\\ne \\emptyset $ for at most a finite number of $D\\in W_{\\tau }$ .", "If the surface $S$ is K3, we will use the notation $W_{v}$ for $W_{\\tau _{v}}$ .", "Definition 2.4 For every $D\\in W_{\\tau }$ , the hyperplane $D^{\\perp }\\cap {K}_{S}$ will be called $\\tau -$wall in the Kähler cone of $S$ .", "A connected component of ${K}_{S}\\setminus \\bigcup _{D\\in W_{\\tau }}D^{\\perp }$ is an open convex cone called $\\tau -$chamber in the Kähler cone of $S$ .", "A Kähler class in a $\\tau -$ chamber of ${K}_{S}$ is called $\\tau -$generic polarization.", "If $S$ is a K3 surface and $v$ is a Mukai vector, we will call $v-$wall in the Kähler cone (resp.", "$v-$chamber in the Kähler cone, $v-$generic polarization) a $\\tau _{v}-$ wall in the Kähler cone (resp.", "a $\\tau _{v}-$ chamber in the Kähler cone, a $\\tau _{v}-$ generic polarization).", "Recall that the ample cone of $S$ is $Amp(S)={K}_{S}\\cap NS_{\\mathbb {R}}(S)$ (where $NS_{\\mathbb {R}}(S)=NS(S)\\otimes \\mathbb {R}$ ): if $S$ is a projective K3 surface and $\\mathcal {C}\\subseteq {K}_{S}$ is a $v-$ chamber in the Kähler cone of $S$ , then $\\mathcal {C}\\cap NS_{\\mathbb {R}}(S)$ is a $v-$ chamber in the ample cone of $S$ in the usual terminology: if $H$ is an ample line bundle on $S$ , then $c_{1}(H)$ is a $v-$ generic polarization if and only if $H$ is $v-$ generic as in [15]." ], [ "Compactness of $M^{\\mu }_{v}(S,\\omega )$ when {{formula:fdfd68d6-68a0-4cf8-bed3-50d472644e85}} is {{formula:c84a630f-9124-48bc-b25b-a8394a31d59e}} generic", "Using the same proof as in the projective case (see Theorem 4.C.3 of [15]), we show that $v-$ generic polarizations enjoy the above stated property concerning the existence of properly semistable sheaves.", "Lemma 2.5 Let $\\omega $ be a Kähler class on a compact Kähler surface $S$ , and ${F}$ a $\\mu _{\\omega }-$ semistable sheaf of topological type $\\tau =(r,\\xi ,c_{2})$ .", "Suppose that there is ${E}\\subseteq {F}$ of rank $0<s<r$ , first Chern class $\\zeta $ and such that $\\mu _{\\omega }({E})=\\mu _{\\omega }({F})$ .", "Then $D:=r\\zeta -s\\xi $ is such that $-\\frac{r^4}{2}\\Delta (\\tau )\\le D^{2}\\le 0,$ and $D^{2}=0$ if and only if $D=0$ .", "We can suppose that ${E}$ is saturated, so that ${G}:={F}/{E}$ is torsion free, $\\mu _{\\omega }-$ semistable and of rank $r-s$ .", "Notice that as $\\mu _{\\omega }({E})=\\mu _{\\omega }({F})$ , we have $D\\cdot \\omega =0$ .", "As $\\omega $ is a Kähler class, from the Hodge Index Theorem we then have $D^{2}\\le 0$ , and $D^{2}=0$ if and only if $D=0$ .", "We then just need to show that $D^{2}\\ge -\\frac{r^4}{2}\\Delta (\\tau )$ .", "By definition of the discriminant, it follows that $\\Delta ({F})-\\frac{s}{r}\\Delta ({E})-\\frac{r-s}{r}\\Delta ({G})=-\\frac{D^{2}}{2s(r-s)r^2}.$ Now, recall that the Bogomolov inequality is surely satisfied by ${E}$ and ${G}$ , so that $\\Delta ({E}),\\Delta ({G})\\ge 0$ .", "But this implies that $-D^{2}\\le 2s(r-s)r^2\\Delta ({F})=2s(r-s)r^2\\Delta (\\tau )\\le \\frac{r^4}{2}\\Delta (\\tau ),$ and we are done.", "Using the main result of [36] we then get the following: Proposition 2.6 Let $S$ be a K3 surface, $r\\ge 2$ an integer and $\\xi \\in NS(S)$ such that $(r,\\xi )=1$ .", "Let $a\\in \\mathbb {Z}$ , $v:=(r,\\xi ,a)$ and $\\omega $ a $v-$ generic polarization.", "If $M^{\\mu }_{v}(S,\\omega )\\ne \\emptyset $ , then it is a smooth, compact, holomorphically symplectic manifold.", "The statement follows from the main result of [36] if $S$ is non-algebraic.", "When $S$ is projective we shall show in section 3.1 that there exists some integer ample class $H$ in the same $v$ -chamber as $\\omega $ .", "The (semi)stability with respect to $\\omega $ or with respect to $H$ will then come down to the same thing and $M^{\\mu }_{v}(S,\\omega )$ will coincide with the Gieseker moduli space $M_{v}(S,H)$ of $H-$ semistable sheaves, which is known to be smooth, projective and holomorphically symplectic (see [15])." ], [ "Projective K3 surfaces with non-ample polarizations", "In this section we prove that if $S$ is a projective K3 surface, $v=(r,\\xi ,a)$ is a Mukai vector with $(r,\\xi )=1$ and $\\omega $ is a $v-$ generic polarization, then $M^{\\mu }_{v}(S,\\omega )$ is an irreducible holomorphically symplectic manifold, deformation equivalent to a Hilbert scheme of points on $S$ ." ], [ "Changing polarization in a chamber", "We first show that changing polarization inside a chamber does not affect the moduli space.", "The following adaptation of Lemma 4.C.5 from [15] to the case of Kähler polarizations works also on Kähler manifolds; see [11] Lemma 6.2.", "Lemma 3.1 Let $\\omega $ , $\\omega ^{\\prime }$ be two Kähler classes on a compact Kähler manifold $X$ and ${F}$ be a torsion free sheaf on $X$ which is $\\mu _{\\omega }-$ stable but not $\\mu _{\\omega ^{\\prime }}-$ stable.", "Denote by $[\\omega ,\\omega ^{\\prime }]:=\\lbrace \\omega _{t}:=t\\omega ^{\\prime }+(1-t)\\omega \\,|\\,t\\in [0,1]\\rbrace $ the segment from $\\omega $ to $\\omega ^{\\prime }$ .Then there is a Kähler class $\\omega _t\\in [\\omega , \\omega ^{\\prime }]$ such that ${F}$ is properly $\\mu _{\\omega _t}-$ semistable.", "As a consequence of this, changing the polarization inside a chamber does not affect the moduli space.", "This is well-known for $v-$ generic ample line bundles, and requires the same proof.", "We let $M^{\\mu }_{\\tau }(S,\\omega )$ be the moduli space of $\\mu _{\\omega }-$ stable sheaves whose topological type is $\\tau $ .", "If $S$ is a K3 surface, then $M^{\\mu }_{\\tau _{v}}(S,\\omega )=M^{\\mu }_{v}(S,\\omega )$ Proposition 3.2 Let $S$ be a smooth projective surface and $\\tau =(r,\\xi ,c_{2})$ such that $r\\ge 2$ and $\\xi \\in NS(S)$ .", "Let $\\mathcal {C}$ be a $\\tau -$ chamber in the Kähler cone of $S$ , and $\\omega ,\\omega ^{\\prime }\\in \\mathcal {C}$ .", "Then $M^{\\mu }_{\\tau }(S,\\omega )=M^{\\mu }_{\\tau }(S,\\omega ^{\\prime })$ .", "We show that if ${F}$ is a $\\mu _{\\omega }-$ stable sheaf of topological type $\\tau $ , then it is $\\mu _{\\omega ^{\\prime }}-$ stable as well.", "Indeed, suppose that ${F}$ is not $\\mu _{\\omega ^{\\prime }}-$ stable.", "By Lemma REF this implies that there is $\\omega _{t}\\in [\\omega ,\\omega ^{\\prime }]$ such that ${F}$ is properly $\\mu _{\\omega _{t}}-$ semistable.", "Hence there is ${E}\\subseteq {F}$ of rank $0<s<r$ and first Chern class $\\zeta $ , such that $\\mu _{\\omega _t}({E})=\\mu _{\\omega _t}({F})$ .", "Let $D:=r\\zeta -s\\xi $ : hence $D\\cdot \\omega _{t}=0$ , and by Lemma REF we have $D\\in W_{\\tau }\\cup \\lbrace 0\\rbrace $ .", "Notice that as ${F}$ is $\\mu _{\\omega }-$ stable, we have $D\\cdot \\omega <0$ , so that $D\\in W_{\\tau }$ .", "It follows that $\\omega _{t}\\notin \\mathcal {C}$ which is not possible as $\\mathcal {C}$ is convex.", "In conclusion, ${F}$ is $\\mu _{\\omega ^{\\prime }}-$ stable." ], [ "Conclusion for projective K3 surfaces", "We first introduce some notations: if $S$ is a projective surface, we let $NS_{\\mathbb {R}}(S)$ be the real Néron-Severi space of $S$ , which is a linear subspace of $H^{1,1}(S)$ .", "Recall that on $H^{1,1}(S)$ we have a non-degenerate intersection product whose restriction to $NS_{\\mathbb {R}}(S)$ remains non-degenerate.", "Let $T_{\\mathbb {R}}(S)$ be the orthogonal of $NS_{\\mathbb {R}}(S)$ in $H^{1,1}(S)$ , so that we have $H^{1,1}(S)=NS_{\\mathbb {R}}(S)\\oplus T_{\\mathbb {R}}(S)$ .", "Finally, for every $\\alpha \\in H^{1,1}(S)$ we let $p_{NS}:H^{1,1}(S)\\longrightarrow NS_{\\mathbb {R}}(S)$ and $p_{T}:H^{1,1}(S)\\longrightarrow T_{\\mathbb {R}}(S)$ be the two projections.", "Moreover, for every $\\alpha \\in H^{1,1}(S)$ we let $\\alpha _{NS}:=p_{NS}(\\alpha )$ and $\\alpha _{T}:=p_{T}(\\alpha )$ .", "The first result we show is the following: Lemma 3.3 Let $S$ be a projective surface and $\\omega $ a Kähler class on $S$ .", "The class $\\omega _{NS}$ is an ample class on $S$ .", "For every $\\xi \\in NS_{\\mathbb {R}}(S)$ we have $\\xi \\cdot \\omega =\\xi \\cdot \\omega _{NS}$ .", "Recall that $\\omega =\\omega _{NS}+\\omega _{T}$ : it follows that for every non-zero effective curve class $C$ we have $\\omega _{NS}\\cdot C=\\omega \\cdot C-\\omega _{T}\\cdot C=\\omega \\cdot C>0,$ since $\\omega _{T}$ is orthogonal to $NS_{\\mathbb {R}}(S)$ (where $C$ lies), and $\\omega $ is a Kähler class.", "This implies that $\\omega _{NS}$ is a nef class on $S$ .", "In particular, this means that $\\omega _{NS}$ is a class in the closure of the ample cone of $S$ .", "Now, recall that the projection $p_{NS}$ is an open map; moreover, the previous part of the proof shows that the image of the Kähler cone of $S$ under $p_{NS}$ is contained in the nef cone of $S$ .", "As the Kähler cone is open in $H^{1,1}(S)$ and the interior of the nef cone is the ample cone, it follows that the image of the Kähler cone by projection is contained in the ample cone.", "The last point of the statement is simply the fact that $\\omega _{T}$ is orthogonal to $NS_{\\mathbb {R}}(S)$ .", "Using the previous Lemma, we can finally prove the following, which shows part (1) of Theorem REF .", "Theorem 3.4 Let $S$ be a projective K3 surface and $v=(r,\\xi ,a)\\in H^{2*}(S,\\mathbb {Z})$ such that $r\\ge 2$ , $\\xi \\in NS(S)$ and $(r,\\xi )=1$ .", "If $\\omega $ is $v-$ generic and $M^{\\mu }_{v}(S,\\omega )\\ne \\emptyset $ , then $M^{\\mu }_{v}(S,\\omega )$ is a projective irreducible hyperkähler manifold deformation equivalent to a Hilbert scheme of points on $S$ .", "The class $\\omega _{NS}$ is ample by Lemma REF , and $\\omega _{NS}\\cdot \\xi =\\omega \\cdot \\xi $ for every $\\xi \\in NS_{\\mathbb {R}}(S)$ .", "It follows that for every ${F}\\in Coh(S)$ we have $\\mu _{\\omega }({F})=\\mu _{\\omega _{NS}}({F})$ .", "In particular, a coherent sheaf is $\\mu _{\\omega }-$ stable if and only if it is $\\mu _{\\omega _{NS}}-$ stable, so that $M^{\\mu }_{v}(S,\\omega )=M^{\\mu }_{v}(S,\\omega _{NS})$ .", "Moreover, if $D\\in W_{v}$ , then $\\omega _{NS}\\cdot D=\\omega \\cdot D$ : as $\\omega $ is $v-$ generic, it follows that $\\omega _{NS}$ is $v-$ generic.", "Let $\\mathcal {C}$ be the $v-$ chamber of the ample cone where $\\omega _{NS}$ lies.", "As $\\mathcal {C}$ is open in $Amp(S)$ , there is $\\epsilon >0$ such that the ball $B_{\\epsilon }(\\omega _{NS})\\subseteq Amp(S)$ of ray $\\epsilon $ and centred at $\\omega _{NS}$ is contained in $\\mathcal {C}$ .", "Let $\\omega ^{\\prime }\\in B_{\\epsilon }(\\omega _{NS})\\cap H^{2}(S,\\mathbb {Q})$ : by Proposition REF we have $M^{\\mu }_{v}(S,\\omega _{NS})=M^{\\mu }_{v}(S,\\omega ^{\\prime })$ .", "As $\\omega ^{\\prime }\\in H^{2}(S,\\mathbb {Q})\\cap H^{1,1}(S)$ , there are $p\\in \\mathbb {N}$ and $H\\in Pic(S)$ such that $p\\omega ^{\\prime }=c_{1}(H)$ .", "As $\\omega ^{\\prime }\\in \\mathcal {C}$ and $\\mathcal {C}$ is a cone, we have $c_{1}(H)\\in \\mathcal {C}$ : hence $H$ is a $v-$ generic ample line bundle, and $M^{\\mu }_{v}(S,\\omega ^{\\prime })=M^{\\mu }_{v}(S,H)$ .", "By [30] and [42] $M^{\\mu }_{v}(S,H)$ is an irreducible hyperkähler manifold deformation equivalent to a Hilbert scheme of points, and we are done.", "Remark 3.5 A useful corollary of Lemma REF is that if $\\mathcal {C}$ is a $v-$ chamber in the Kähler cone of $S$ , then $\\mathcal {C}$ intersects the ample cone (and the intersection is clearly a $v-$ chamber in the ample cone of $S$ ).", "Indeed, consider the segment $[\\omega ,\\omega _{NS}]$ : as the projection $p_{NS}$ is a linear map, we have that $[\\omega ,\\omega _{NS}]\\cap NS_{\\mathbb {R}}(S)=\\lbrace \\omega _{NS}\\rbrace $ , and that $p_{NS}([\\omega ,\\omega _{NS}])=\\lbrace \\omega _{NS}\\rbrace $ .", "We show that $\\omega _{NS}\\in \\mathcal {C}$ (showing that $\\mathcal {C}$ intersects the ample cone by Lemma REF ).", "Indeed, suppose that $\\omega _{NS}$ does not lie in $\\mathcal {C}$ : it follows that there is $\\omega ^{\\prime }\\in [\\omega ,\\omega _{NS}]$ lying on a $v-$ wall.", "This means that there is $D\\in W_{v}$ such that $\\omega ^{\\prime }\\cdot D=0$ .", "But as $p_{NS}(\\omega ^{\\prime })=p_{NS}(\\omega )=\\omega _{NS}$ , it follows that $\\omega \\cdot D=0$ , which is not possible." ], [ "Moduli spaces of stable twisted sheaves", "In this section we recall the notion of twisted sheaf on a complex manifold, and we introduce the notion of stability for coherent twisted sheaves.", "We will then construct (relative) moduli spaces of stable twisted sheaves on a K3 surface (not necessarily projective): they will be used to show that the moduli spaces $M^{\\mu }_{v}(S,\\omega )$ of $\\mu _{\\omega }-$ stable sheaves with Mukai vector $v=(r,\\xi ,a)$ such that $r$ and $\\xi $ are prime to each other are compact, connected, simply connected and deformation equivalent to a Hilbert scheme of points on a projective K3 surface (whenever the polarization $\\omega $ is $v-$ generic)." ], [ "Twisted sheaves and stability", "We recall some basic facts about twisted sheaves on a complex manifold $X$ (we refer the interested reader to [6] or [25] for more details).", "There are several definitions of twisted sheaves, giving equivalent categories.", "We use three of them: the first one is due to Căldăraru [6], and presents twisted sheaves as a twisted gluing of local coherent sheaves on $X$ ; the second one (to be found again in [6]) presents twisted sheaves as modules over an Azumaya algebra on $X$ ; the last one, due to Yoshioka [43], presents twisted sheaves as a full subcategory of the category of coherent sheaves on some projective bundle over $X$ .", "We begin by recalling these definitions.", "As our aim are moduli spaces of stable twisted sheaves, we need to introduce several notions: first, we recall the Chern character and the slope of a twisted sheaf (for projective manifolds, this was done in [18] and [43]); we then introduce $\\mu _{\\omega }-$ stability for twisted sheaves (with respect to a Kähler form $\\omega $ )." ], [ "Twisted sheaves following Căldăraru", "Let ${U}=\\lbrace U_{i}\\rbrace _{i\\in I}$ be an open covering of $X$ , and let $U_{ij}:=U_{i}\\cap U_{j}$ and $U_{ijk}:=U_{i}\\cap U_{j}\\cap U_{k}$ .", "Choose a $2-$ cocyle $\\lbrace \\alpha _{ijk}\\rbrace $ , where $\\alpha _{ijk}\\in {O}_{X}^{*}(U_{ijk})$ , defining a class $\\alpha \\in H^{2}(X,{O}_{X}^{*})$ .", "A $\\lbrace \\alpha _{ijk}\\rbrace -$twisted coherent sheaf is a collection ${F}=\\lbrace {F}_{i},\\phi _{ij}\\rbrace $ , where ${F}_{i}\\in Coh(U_{i})$ for every $i\\in I$ , and for every $i,j\\in I$ $\\phi _{ij}:{F}_{j|U_{ij}}\\longrightarrow {F}_{i|U_{ij}}$ is an isomorphism in $Coh(U_{ij})$ such that $\\phi _{ii}=id_{{F}_{i}}$ for every $i\\in I$ ; $\\phi _{ij}=\\phi _{ji}^{-1}$ for every $i,j\\in I$ ; $\\phi _{ij}\\circ \\phi _{jk}\\circ \\phi _{ki}=\\alpha _{ijk}\\cdot id$ for every $i,j,k\\in I$ .", "By a morphism of $\\lbrace \\alpha _{ijk}\\rbrace -$ twisted sheaves $f:{F}=\\lbrace {F}_{i},\\phi _{ij}\\rbrace \\longrightarrow {G}=\\lbrace {G}_{i},\\psi _{ij}\\rbrace $ we mean a collection $f=\\lbrace f_{i}\\rbrace $ of morphisms $f_{i}:{F}_{i}\\longrightarrow {G}_{i}$ of ${O}_{U_{i}}-$ modules such that $\\psi _{ij}\\circ f_{j}=f_{i}\\circ \\phi _{ij}$ for every $i,j\\in I$ .", "The $\\lbrace \\alpha _{ijk}\\rbrace -$ twisted coherent sheaves form an abelian category $Coh(X,\\lbrace \\alpha _{ijk}\\rbrace )$ .", "If $\\lbrace \\alpha _{ijk}\\rbrace $ and $\\lbrace \\alpha ^{\\prime }_{ijk}\\rbrace $ are two representatives of the same class $\\alpha \\in H^{2}(X,{O}_{X}^{*})$ , then there is an equivalence between $Coh(X,\\lbrace \\alpha _{ijk}\\rbrace )$ and $Coh(X,\\lbrace \\alpha ^{\\prime }_{ijk}\\rbrace )$ , so that we can speak of the category $Coh(X,\\alpha )$ of coherent $\\alpha -$ twisted sheaves.", "If ${F}\\in Coh(X,\\alpha )$ and ${G}\\in Coh(X,\\beta )$ , we can define in a natural way ${F}\\otimes {G}$ and ${H}om({F},{G})$ : the first one is a coherent $\\alpha \\beta -$ twisted sheaf, while the second is a coherent $\\alpha ^{-1}\\beta -$ twisted sheaf.", "We now recall an important definition: a sheaf $\\mathcal {A}$ of ${O}_{X}-$ modules is said to be an Azumaya algebra if it is a sheaf of ${O}_{X}-$ algebras whose generic fibre is a central simple algebra.", "Equivalence classes of Azumaya algebras form a group $Br(X)$ , the Brauer group of $X$ , which has an injection into $H^{2}(X,{O}_{X}^{*})$ .", "One of the main properties we will use in the following is (see Theorem 1.3.5 of [6]): Proposition 4.1 Let $X$ be a complex manifold and $\\alpha \\in Br(X)$ .", "Then there exist a locally free $\\alpha -$ twisted sheaf on $X$ .", "For the rest of this section, we suppose $\\alpha \\in Br(X)$ and define the twisted Chern character and twisted Mukai vector for $\\alpha -$ twisted sheaves.", "More precisely, let ${F}$ be an $\\alpha -$ twisted coherent sheaf on $X$ and $E$ a locally free $\\alpha -$ twisted coherent sheaf.", "The Chern character of ${F}$ with respect to $E$ is $ch_{E}({F}):=\\frac{ch({F}\\otimes E^{\\vee })}{\\sqrt{ch(E\\otimes E^{\\vee })}}.$ The Mukai vector of ${F}$ with respect to $E$ is $v_{E}({F}):=ch_{E}({F})\\cdot \\sqrt{td(X)}.$ The slope of a torsion-free $\\alpha -$ twisted sheaf ${F}$ with respect to $E$ and to a Kähler class $\\omega $ is $\\mu _{E,\\omega }({F}):=\\frac{c_{E,1}({F})\\cdot \\omega }{rk({F})},$ where $c_{E,1}({F})$ is the component of $ch_{E}({F})$ lying in $H^{2}(S,\\mathbb {Q})$ .", "We collect some explicit formulas when $X=S$ is a K3 surface.", "Let $r:=rk({F})$ , $s:=rk(E)$ , $\\xi :=c_{1}({F}\\otimes E^{\\vee })$ , $a:=ch_{2}({F}\\otimes E^{\\vee })$ and $b:=ch_{2}(E\\otimes E^{\\vee })$ .", "Then $ch_{E}({F})=(r,\\xi /s,(2as-rb)/2s^{2}),$ $v_{E}({F})=(r,\\xi /s,r+(2as-rb)/2s^{2})$ so that $\\mu _{E,\\omega }({F})=\\frac{\\xi \\cdot \\omega }{rs}=\\frac{c_{1}({F}\\otimes E^{\\vee })\\cdot \\omega }{rk({F})rk(E)}=\\mu _{\\omega }({F}\\otimes E^{\\vee })$ and $v_{E}^{2}({F})=\\frac{\\xi ^{2}}{s^{2}}-\\frac{2ra}{s}+\\frac{r^{2}b}{s^{2}}-2r^{2}.$ If $\\alpha =0$ , then one easily sees that $\\mu _{E,\\omega }({F})=\\mu _{\\omega }({F})-\\mu _{\\omega }(E)$ and that $v^{2}_{E}({F})=v^{2}({F}).$ If ${F}$ is a torsion free $\\alpha -$ twisted sheaf on $S$ , we let $ch_{\\alpha }({F}):=ch_{{F}^{\\vee \\vee }}({F}),\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,v_{\\alpha }({F}):=v_{{F}^{\\vee \\vee }}({F}),$ called twisted Chern character and twisted Mukai vector of ${F}$ .", "The twisted slope of ${F}$ with respect to $\\omega $ is $\\mu _{\\alpha ,\\omega }({F}):=\\frac{c_{\\alpha ,1}({F})\\cdot \\omega }{rk({F})},$ where $c_{\\alpha ,1}({F})$ is the component of $ch_{\\alpha }({F})$ in $H^{2}(S,\\mathbb {Q})$ .", "Using twisted slopes, we introduce the notion of stability for twisted sheaves.", "Fix $\\alpha \\in Br(X)$ and $E$ an $\\alpha -$ twisted locally free sheaf.", "Definition 4.2 We say that a torsion-free ${F}\\in Coh(X,\\alpha )$ is $\\mu _{E,\\omega }-$stable if for every $\\alpha -$ twisted coherent subsheaf ${E}\\subseteq {F}$ such that $0<rk({E})<rk({F})$ we have $\\mu _{E,\\omega }({E})<\\mu _{E,\\omega }({F})$ .", "If $\\mu _{E,\\omega }({E})\\le \\mu _{E,\\omega }({F})$ for every such subsheaf, we say that ${F}$ is $\\mu _{E,\\omega }-$semistable.", "A $\\mu _{{F}^{\\vee \\vee },\\omega }-$ (semi)stable sheaf will be called $\\mu _{\\alpha ,\\omega }-$(semi)stable.", "To conclude this section, we show that the $\\mu _{E,\\omega }-$ stability does not depend on $E$ .", "Lemma 4.3 Let $\\alpha \\in Br(S)$ , ${F}\\in Coh(S,\\alpha )$ and $\\omega \\in {K}_{S}$ .", "If $E^{\\prime },E\\in Coh(S,\\alpha )$ are locally free, then ${F}$ is $\\mu _{E,\\omega }-$ stable if and only if it is $\\mu _{E^{\\prime },\\omega }-$ stable.", "In particular, ${F}$ is $\\mu _{E,\\omega }-$ stable if and only if it is $\\mu _{\\alpha ,\\omega }-$ stable.", "If $\\alpha =0$ , the sheaf ${F}$ is $\\mu _{0,\\omega }-$ stable if and only if it is $\\mu _{\\omega }-$ stable.", "Let ${F}\\in Coh(S,\\alpha )$ , ${G}$ an $\\alpha -$ twisted coherent subsheaf of ${F}$ , and $H$ a locally free $\\alpha -$ twisted coherent sheaf.", "Then $rk(H)rk({F})c_{1}({G}\\otimes H^{\\vee })-rk(H)rk({G})c_{1}({F}\\otimes H^{\\vee })=$ $=c_{1}({G}\\otimes {F}^{\\vee }\\otimes H\\otimes H^{\\vee })=rk^{2}(H)c_{1}({G}\\otimes {F}^{\\vee }).$ Suppose now that ${F}$ is $\\mu _{E,\\omega }-$ stable but not $\\mu _{E^{\\prime },\\omega }-$ stable.", "Hence there is an $\\alpha -$ twisted coherent subsheaf ${G}$ of ${F}$ of rank $0<s<rk({F})$ such that $\\mu _{E^{\\prime },\\omega }({G})\\ge \\mu _{E^{\\prime },\\omega }({F})$ .", "By $\\mu _{E,\\omega }-$ stability of ${F}$ we even have $\\mu _{E,\\omega }({G})<\\mu _{E,\\omega }({F})$ .", "Writing these two inequalities explicitly we have $\\omega \\cdot (rk(E^{\\prime })rc_{1}({G}\\otimes (E^{\\prime })^{\\vee })-rk(E^{\\prime })sc_{1}({F}\\otimes (E^{\\prime })^{\\vee }))\\ge 0,$ $\\omega \\cdot (rk(E)rc_{1}({G}\\otimes E^{\\vee })-rk(E)sc_{1}({F}\\otimes E^{\\vee }))< 0.$ Using equation (REF ) for $H=E^{\\prime }$ , equation (REF ) becomes $\\omega \\cdot c_{1}({G}\\otimes {F}^{\\vee })\\ge 0$ .", "Using equation (REF ) for $H=E$ , equation (REF ) becomes $\\omega \\cdot c_{1}({G}\\otimes {F}^{\\vee })<0$ , getting a contradiction." ], [ "Twisted sheaves as $\\mathcal {A}-$ modules", "Let again $X$ be a complex manifold and $\\mathcal {A}$ an Azumaya algebra on $X$ .", "We let $Coh(X,\\mathcal {A})$ be the abelian category of coherent sheaves on $X$ having the structure of $\\mathcal {A}-$ module.", "The following is Proposition 1.3.6 of [6]: Proposition 4.4 Let $X$ be a complex manifold, $\\mathcal {A}$ an Azumaya algebra on $X$ and $\\alpha $ its class in $Br(X)$ .", "If $E$ is a locally free $\\alpha -$ twisted coherent sheaf such that ${E}nd(E)\\simeq \\mathcal {A}$ , we have an equivalence $Coh(X,\\alpha )\\longrightarrow Coh(X,\\mathcal {A}),\\,\\,\\,\\,\\,\\,\\,\\,\\,{F}\\mapsto {F}\\otimes E^{\\vee }.$ We now define Chern characters, Mukai vectors and slopes for the objects of $Coh(X,\\mathcal {A})$ , which allow us to define a notion of stability.", "For ${F}\\in Coh(X,\\mathcal {A})$ we define $ch_{\\mathcal {A}}({F}):=\\frac{ch({F})}{\\sqrt{ch(\\mathcal {A})}},\\,\\,\\,\\,\\,\\,\\,\\,v_{\\mathcal {A}}({F}):=ch_{\\mathcal {A}}({F})\\cdot \\sqrt{td(X)},$ and if $\\omega $ is a Kähler class and ${F}$ is torsion-free, we let $\\mu _{\\mathcal {A},\\omega }({F}):=\\frac{c_{\\mathcal {A},1}({F})\\cdot \\omega }{rk({F})},$ where $c_{\\mathcal {A},1}({F})$ is the component of $ch_{\\mathcal {A}}({F})$ in $H^{2}(S,\\mathbb {Q})$ .", "We now introduce the notion of stability for $\\mathcal {A}-$ modules: Definition 4.5 A torsion-free ${F}\\in Coh(X,\\mathcal {A})$ is $\\mu _{\\mathcal {A},\\omega }-$stable if for every ${E}\\subseteq {F}$ in $Coh(X,\\mathcal {A})$ such that $0<rk({E})<rk({F})$ , we have $\\mu _{\\mathcal {A},\\omega }({E})<\\mu _{\\mathcal {A},\\omega }({F})$ .", "If $\\mu _{\\mathcal {A},\\omega }({E})\\le \\mu _{\\mathcal {A},\\omega }({F})$ for every such subobject, we say that ${F}$ is $\\mu _{\\mathcal {A},\\omega }-$semistable.", "We notice that if ${G}\\in Coh(X,\\alpha )$ and $E$ is a locally free $\\alpha -$ twisted sheaf such that ${E}nd(E)\\simeq \\mathcal {A}$ , then $ch_{E}({G})=ch_{\\mathcal {A}}({G}\\otimes E^{\\vee })$ .", "It follows that $v_{E}({G})=v_{\\mathcal {A}}({G}\\otimes E^{\\vee }),\\,\\,\\,\\,\\,\\,\\,\\mu _{E,\\omega }({G})=\\mu _{\\mathcal {A},\\omega }({G}\\otimes E^{\\vee }),$ so that ${F}\\in Coh(X,\\alpha )$ is $\\mu _{E,\\omega }-$ stable if and only if ${F}\\otimes E^{\\vee }$ is $\\mu _{\\mathcal {A},\\omega }-$ stable.", "Remark 4.6 We notice that $\\Lambda :=({O}_{X},\\mathcal {A})$ is a sheaf of rings of differential operators following the definition of [32], and $Coh(X,\\mathcal {A})$ is the category of $\\Lambda -$ modules (always in the sense of [32]).", "Moreover, $\\mu _{\\mathcal {A},\\omega }-$ stable $\\mathcal {A}-$ modules are exactly $\\mu -$ stable $\\Lambda -$ modules (always in the sense of [32]).", "Even if the definitions of [32] are given only for projective manifolds, they can immediately be extended to compact complex manifolds." ], [ "Twisted sheaves following Yoshioka", "In [43] Yoshioka introduces twisted sheaves as a full subcategory of the category of coherent sheaves on a projective bundle.", "More precisely, let $X$ be a complex manifold, $\\alpha \\in Br(X)$ and $E$ a locally free $\\alpha -$ twisted sheaf.", "On an open cover ${U}=\\lbrace U_{i}\\rbrace _{i\\in I}$ we represent $E$ by $\\lbrace E_{i},\\phi _{ij}\\rbrace _{i,j\\in I}$ .", "Let $\\mathbb {P}_{i}:=\\mathbb {P}(E_{i})$ , together with the map $\\pi _{i}:\\mathbb {P}_{i}\\longrightarrow U_{i}$ .", "The twisted gluing data $\\phi _{ij}$ turn to a gluing data $\\varphi _{ij}$ of the $\\mathbb {P}_{i}$ and of the $\\pi _{i}$ , getting a projective bundle $\\pi :\\mathbb {P}\\longrightarrow X$ (whose class in $Br(X)$ is $\\alpha $ ).", "As shown in Lemma 1.1 of [43], we have $Ext^{1}(T_{\\mathbb {P}/X},{O}_{\\mathbb {P}})=\\mathbb {C}$ , hence, up to scalars, there is a unique non-trivial extension $0\\longrightarrow {O}_{\\mathbb {P}}\\longrightarrow G\\longrightarrow T_{\\mathbb {P}/X}\\longrightarrow 0.$ The vector bundle $G$ can be described in another way.", "Fix a tautological line bundle ${O}(\\lambda _{i})$ over $\\mathbb {P}_{i}$ , so that the twisted gluing data $\\phi _{ij}$ give isomorphisms $\\psi _{ij}:{O}(\\lambda _{i})\\longrightarrow {O}(\\lambda _{j})$ , and $L:=\\lbrace {O}(\\lambda _{i}),\\psi _{ij}\\rbrace $ is an $\\pi ^*(\\alpha ^{-1})-$ twisted line bundle on $\\mathbb {P}$ .", "Then the vector bundles $\\pi _{i}^{*}E_{i}(\\lambda _{i})$ glue together to give a locally free sheaf which is isomorphic to $G$ .", "Definition 4.7 A coherent sheaf ${F}$ on $\\mathbb {P}$ is called $\\mathbb {P}-$sheaf if the canonical morphism $\\pi ^{*}\\pi _{*}(G^{\\vee }\\otimes {F})\\longrightarrow G^{\\vee }\\otimes {F}$ is an isomorphism.", "We let $Coh(\\mathbb {P},X)$ be the full subcategory of $Coh(\\mathbb {P})$ given by $\\mathbb {P}-$ sheaves.", "Lemma 1.5 of [43] shows that ${F}\\in Coh(\\mathbb {P},X)$ if and only if ${F}_{|\\mathbb {P}_{i}}\\simeq \\pi ^{*}{E}_{|U_i}\\otimes {O}(\\lambda _{i})$ for some ${E}\\in Coh(U_{i})$ .", "Using this, one shows: Proposition 4.8 Let $X$ be a complex manifold and $\\pi :\\mathbb {P}\\longrightarrow X$ a projective bundle whose class in $Br(X)$ is $\\alpha $ .", "Then there is an equivalence of categories $P:Coh(\\mathbb {P},X)\\longrightarrow Coh(X,\\alpha ),\\,\\,\\,\\,\\,\\,\\,\\,\\,P({F}):=\\pi _{*}({F}\\otimes L^{\\vee }).$ Following Yoshioka, we have a definition of Chern character, Mukai vector and slope of a $\\mathbb {P}-$ sheaf ${F}$ .", "More precisely, we have $ch_{\\mathbb {P}}({F}):=\\frac{ch(\\pi _{*}(G^{\\vee }\\otimes {F}))}{\\sqrt{ch(\\pi _{*}(G^{\\vee }\\otimes G))}},$ so that $v_{\\mathbb {P}}({F})=ch_{\\mathbb {P}}({F})\\cdot \\sqrt{td(S)},\\,\\,\\,\\,\\,\\,\\,\\mu _{\\mathbb {P},\\omega }({F}):=\\frac{c_{\\mathbb {P},1}({F})\\cdot \\omega }{rk({F})},$ where $c_{\\mathbb {P},1}({F})$ is the component of $ch_{\\mathbb {P}}({F})$ in $H^{2}(S,\\mathbb {Q})$ .", "We now introduce the notion of stability for $\\mathbb {P}-$ sheaves.", "Definition 4.9 We say that a torsion-free ${F}\\in Coh(\\mathbb {P},X)$ is $\\mu _{\\mathbb {P},\\omega }-$stable if for every subobject ${E}$ of ${F}$ in $Coh(\\mathbb {P},X)$ such that $0<rk({E})<rk({F})$ , we have $\\mu _{\\mathbb {P},\\omega }({E})<\\mu _{\\mathbb {P},\\omega }({F})$ .", "If $\\mu _{\\mathbb {P},\\omega }({E})\\le \\mu _{\\mathbb {P},\\omega }({F})$ for every such subobject, we say that ${F}$ is $\\mu _{\\mathbb {P},\\omega }-$semistable.", "If $\\mathbb {P}=\\mathbb {P}(E)$ for some locally free $\\alpha -$ twisted sheaf $E$ , the equivalence $P$ gives $ch_{\\mathbb {P}}({F})=ch_{E}(P({F})),\\,\\,\\,\\,\\,\\,\\,v_{\\mathbb {P}}({F})=v_{E}(P({F})),$ $\\mu _{\\mathbb {P},\\omega }({F})=\\mu _{E,\\omega }(P({F})).$ It follows that ${F}\\in Coh(\\mathbb {P},X)$ is $\\mu _{\\mathbb {P},\\omega }-$ stable if and only if $P({F})$ is $\\mu _{E,\\omega }-$ stable.", "If ${F}$ is a $\\mu _{\\mathbb {P},\\omega }-$ stable $\\mathbb {P}-$ sheaf, as $Coh(\\mathbb {P},S)$ is a full subcategory of $Coh(\\mathbb {P})$ and as the functor $P$ is an equivalence, we have that $Ext^{1}_{Coh(\\mathbb {P},S)}({F},{F})\\simeq Ext^{1}_{Coh(S,\\alpha )}(P({F}),P({F})),$ and $Ext^{2}_{Coh(\\mathbb {P},S)}({F},{F})\\simeq Ext^{2}_{Coh(S,\\alpha )}(P({F}),P({F})).$" ], [ "Chern classes following Huybrechts and Stellari", "If we consider twisted sheaves following Căldăraru, there is another possible definition of their Chern classes and character, introduced by Huybrechts and Stellari in [17], that we recall here.", "Consider a complex manifold $X$ and $\\alpha \\in H^{2}(X,\\mathcal {O}^{*}_{X})$ , and fix a Čech $2-$ cocycle $\\lbrace \\alpha _{ijk}\\rbrace $ representing $\\alpha $ on an open covering $\\lbrace U_{i}\\rbrace _{i\\in I}$ of $X$ .", "Moreover, choose a Čech $2-$ cocyle $\\lbrace B_{ijk}\\rbrace $ , where $B_{ijk}\\in \\Gamma (U_{ijk},\\mathbb {Q})$ , such that $\\alpha _{ijk}=\\exp (B_{ijk})$ (viewed as local sections of $\\mathbb {R}/\\mathbb {Z}=U(1)\\subseteq \\mathcal {O}_{X}^{*}$ ).", "As the sheaf $\\mathcal {C}^{\\infty }_{X}$ of $C^{\\infty }-$ functions on $X$ is acyclic, up to supposing the covering $\\lbrace U_{i}\\rbrace _{i\\in I}$ is sufficiently fine, there are $a_{ij}\\in \\Gamma (U_{ij},C^{\\infty })$ such that $B_{ijk}=-a_{ij}+a_{ik}-a_{jk}.$ Now, let us consider an $\\alpha -$ twisted sheaf given by ${F}=\\lbrace {F}_{i},\\phi _{ij}\\rbrace $ and let $\\psi _{ij}:=\\phi _{ij}\\cdot \\exp (a_{ij}),$ which is clearly an isomorphism between the restrictions of ${F}_{i}$ and ${F}_{j}$ to $U_{ij}$ .", "It is moreover easy to show that $\\psi _{ij}\\circ \\psi _{jk}\\circ \\psi _{ki}=id,$ hence the sheaf ${F}_{B}=\\lbrace {F}_{i},\\psi _{ij}\\rbrace _{i,j\\in I}$ is an untwisted sheaf.", "We then let $ch^{B}({F}):=\\mbox{ch}({F}_{B}).$ The definition given in this way depends only on $B$ .", "The relation between $ch^{B}$ and the previous Chern characters is explained in [18], and goes as follows, supposing that $\\alpha \\in Br(X)$ .", "Let $E$ be a locally free $\\alpha -$ twisted sheaf and $B_{E}:=\\frac{c_{1}^{B}(E)}{rk(E)},$ where $c_{1}^{B}(E)$ is the degree two part of $ch^{B}(E)$ .", "Then we have $ch^{B}({F})=ch_{E}({F})\\cdot \\exp (B_{E}).$" ], [ "Genericity for polarizations", "We now extend the notion of genericity for polarization to the twisted case.", "As we did in section 2.2, we first introduce a notion of discriminant for twisted sheaves, which depends on the choice of a locally free $E\\in Coh(S,\\alpha )$ ." ], [ "Discriminant of a twisted sheaf", "If ${F}$ is an $\\alpha -$ twisted coherent sheaf, we call discriminant of ${F}$ with respect to $E$ the number $\\Delta _{E}({F}):=\\frac{1}{2rk^{2}({F})}v_{E}^{2}({F})+1.$ If $\\alpha =0$ , this is just $\\Delta ({F})$ by equation (REF ).", "More generally, the discriminant does not depend on $E$ , as shown in the following: Lemma 4.10 Let $\\alpha \\in Br(S)$ and ${F}\\in Coh(S,\\alpha )$ .", "If $E_{1},E_{2}\\in Coh(S,\\alpha )$ are locally free, then $\\Delta _{E_{1}}({F})=\\Delta _{E_{2}}({F})$ .", "Let $E\\in Coh(S,\\alpha )$ be locally free of rank $s$ , and pose $r:=rk({F})$ , $\\xi :=c_{1}({F}\\otimes E^{\\vee })$ , $a:=ch_{2}({F}\\otimes E^{\\vee })$ and $b:=ch_{2}(E\\otimes E^{\\vee })$ .", "By equation (REF ) we have $\\Delta _{E}({F})=\\frac{1}{2r^{2}}\\bigg (\\frac{\\xi ^{2}}{s^{2}}-\\frac{2ra}{s}+\\frac{r^{2}b}{s^{2}}-2r^{2}\\bigg )+1.$ An easy computation shows that $\\frac{\\xi ^{2}}{s}-\\frac{2ra}{s}+\\frac{r^{2}b}{s^{2}}=-\\frac{ch_{2}({F}\\otimes {F}^{\\vee }\\otimes E\\otimes E^{\\vee })}{s^{2}}+\\frac{r^{2}ch_{2}(E\\otimes E^{\\vee })}{2s^{2}}=$ $=-ch_{2}({F}\\otimes {F}^{\\vee }),$ so that $\\Delta _{E}({F})=\\frac{1}{2r^{2}}(-ch_{2}({F}\\otimes {F}^{\\vee })-2r^{2})+1,$ which does not depend on $E$ , implying the statement.", "For $v\\in H^{2*}(S,\\mathbb {Q})$ and $\\alpha \\in Br(S)$ , we let $\\Delta _{\\alpha }(v):=\\Delta _{{F}^{\\vee \\vee }}({F}),$ where ${F}\\in Coh(S,\\alpha )$ is torsion free and $v_{\\alpha }({F})=v$ .", "By Lemma REF this is well defined and if $\\alpha =0$ , then $\\Delta _{0}(v)=\\Delta (v)$ .", "We now prove a generalization to twisted sheaves of the Bogomolov inequality: Proposition 4.11 Let $\\alpha \\in Br(S)$ , ${F}\\in Coh(S,\\alpha )$ and $\\omega $ a Kähler class on $S$ .", "If ${F}$ is $\\mu _{\\alpha ,\\omega }-$ semistable, then $\\Delta _{\\alpha }({F})\\ge 0$ .", "It is easy to see that ${F}$ is $\\mu _{\\alpha ,\\omega }-$ semistable if and only if ${F}^{\\vee }$ is $\\mu _{\\alpha ^{-1},\\omega }-$ semistable.", "In particular, this implies that ${F}$ is $\\mu _{\\alpha ,\\omega }-$ semistable if and only if ${F}^{\\vee \\vee }$ is $\\mu _{\\alpha ,\\omega }-$ semistable.", "Now, notice that ${F}^{\\vee \\vee }\\otimes {F}^{\\vee }=({F}\\otimes {F}^{\\vee })^{\\vee \\vee }$ , hence if $l$ is the length of the singular locus of ${F}\\otimes {F}^{\\vee }$ , it follows that $ch_{2}({F}^{\\vee \\vee }\\otimes {F}^{\\vee })=ch_{2}({F}\\otimes {F}^{\\vee })+l.$ By equation (REF ), it then follows that $v_{\\alpha }^{2}({F}^{\\vee \\vee })=v_{\\alpha }^{2}({F})-2l\\le v_{\\alpha }^{2}({F}).$ As $rk({F})=rk({F}^{\\vee \\vee })$ it follows that $\\Delta _{\\alpha }({F})\\ge \\Delta _{\\alpha }({F}^{\\vee \\vee })$ , hence we just need to show the statement for ${F}^{\\vee \\vee }$ .", "Let now $F:={F}^{\\vee \\vee }$ , which is locally free and $\\mu _{\\alpha ,\\omega }-$ semistable.", "By the Kobayashi-Hitchin correspondence for twisted sheaves as proved by Wang in [41], the sheaf ${E}nd(F)=F\\otimes F^{\\vee }$ is $\\mu _{\\omega }-$ polystable, so that $\\Delta (F\\otimes F^{\\vee })\\ge 0$ by the Bogomolov inequality.", "Choose now a locally free $E\\in Coh(S,\\alpha )$ of rank $s$ , and let $b:=ch_{2}(E\\otimes E^{\\vee })$ .", "By Lemma REF we have $\\Delta (F\\otimes F^{\\vee })=\\Delta _{E\\otimes E^{\\vee }}(F\\otimes F^{\\vee })$ , so that $\\Delta _{E\\otimes E^{\\vee }}(F\\otimes F^{\\vee })\\ge 0$ .", "If $\\xi =c_{1}(F\\otimes E^{\\vee })$ and $a=ch_{2}(F\\otimes E^{\\vee })$ , it follows from equation (REF ) that $v_{E\\otimes E^{\\vee }}^{2}(F\\otimes F^{\\vee })=\\frac{2r^{2}\\xi ^{2}}{s^{2}}-\\frac{4r^{3}a}{s}+\\frac{2r^{2}b}{s^{2}}-2r^{4}=2r^{2}v_{E}^{2}(F)+2r^{4}.$ Hence $0\\le \\Delta _{E\\otimes E^{\\vee }}(F\\otimes F^{\\vee })=\\frac{1}{2r^{4}}v^{2}_{E\\otimes E^{\\vee }}(F\\otimes F^{\\vee })+1=2\\Delta _{E}(F).$ Hence $\\Delta _{\\alpha }(F)=\\Delta _{E}(F)\\ge 0$ , and we are done." ], [ "Walls and chambers", "Now, let $W_{\\alpha ,v}:=\\lbrace D\\in NS(S)\\,|\\,-\\frac{r^{4}}{2}\\Delta _{\\alpha }(v)\\le D^{2}<0\\rbrace .$ If $\\alpha =0$ , we have $W_{0,v}=W_{v}$ .", "Definition 4.12 If $D\\in W_{\\alpha ,v}$ , we call the hyperplane $D^{\\perp }$ an $(\\alpha ,v)-$wall.", "A connected component of ${K}_{S}\\setminus \\bigcup _{D\\in W_{\\alpha ,v}}D^{\\perp }$ is called $(\\alpha ,v)-$chamber.", "A polarization $\\omega \\in {K}_{S}$ is $(\\alpha ,v)-$generic if it lies in a $(\\alpha ,v)-$ chamber.", "A polarization $\\omega $ is $(0,v)-$ generic if and only if it is $v-$ generic.", "We are now ready to prove one of the main results of this section about changing polarization inside a chamber.", "The argument is the same one for untwisted sheaves, here adapted to the twisted case.", "Proposition 4.13 Let $\\alpha \\in Br(S)$ , $v\\in H^{2*}(S,\\mathbb {Q})$ and $\\omega ,\\omega ^{\\prime }$ two $(\\alpha ,v)-$ generic polarizations lying in the same $(\\alpha ,v)-$ chamber.", "If ${F}\\in Coh(S,\\alpha )$ is a torsion free sheaf such that $v_{\\alpha }({F})=v$ , then ${F}$ is $\\mu _{\\alpha ,\\omega }-$ stable if and only if it is $\\mu _{\\alpha ,\\omega ^{\\prime }}-$ stable.", "The proof is divided in two steps.", "Step 1.", "Choose an $\\alpha -$ twisted locally free sheaf $E$ , and let $r:=rk({F})$ , $\\xi :=c_{1}({F}\\otimes E^{\\vee })$ , $a:=ch_{2}({F}\\otimes E^{\\vee })$ , $s:=rk(E)$ and $b:=ch_{2}(E\\otimes E^{\\vee })$ .", "Let $\\omega $ be any polarization, and suppose that ${F}$ is properly $\\mu _{\\alpha ,\\omega }-$ semistable: hence there is an $\\alpha -$ twisted subsheaf ${E}\\subseteq {F}$ such that $\\mu _{E,\\omega }({E})=\\mu _{E,\\omega }({F})$ .", "We let $r^{\\prime }:=rk({E})$ , $\\xi ^{\\prime }:=c_{1}({E}\\otimes E^{\\vee })$ and $a^{\\prime }:=ch_{2}({E}\\otimes E^{\\vee })$ , where $0<r^{\\prime }<r$ and $\\xi ^{\\prime }\\in NS(S)\\otimes \\mathbb {Q}$ .", "Moreover, let $D:=r\\frac{\\xi ^{\\prime }}{s}-r^{\\prime }\\frac{\\xi }{s},$ so that $D\\cdot \\omega =0$ .", "Hence $D^{2}\\le 0$ , as $\\omega $ is a Kähler form.", "Now, let ${G}:={F}/{E}$ , and we suppose without loss of generality that ${E}$ is saturated and that ${E}$ and ${G}$ are $\\mu _{\\alpha ,\\omega }$ -semistable.", "Moreover, let $r^{\\prime \\prime }:=rk({G})$ , $\\xi ^{\\prime \\prime }:=c_{1}({G}\\otimes E^{\\vee })$ and $a^{\\prime \\prime }:=ch_{2}({G}\\otimes E^{\\vee })$ , so that $r^{\\prime \\prime }=r-r^{\\prime }$ , $\\xi ^{\\prime \\prime }=\\xi -\\xi ^{\\prime }$ and $a^{\\prime \\prime }=a-a^{\\prime }$ .", "Finally, let $K:=\\frac{v_{\\alpha }^{2}({F})}{r}-\\frac{v_{\\alpha }^{2}({E})}{r^{\\prime }}-\\frac{v_{\\alpha }^{2}({G})}{r^{\\prime \\prime }}.$ We notice that as ${E}$ and ${G}$ are $\\mu _{\\alpha ,\\omega }-$ semistable, by Proposition REF we have $\\Delta _{\\alpha }({E}),\\Delta _{\\alpha }({G})\\ge 0$ , meaning $v_{\\alpha }^{2}({E})\\ge -2(r^{\\prime })^{2}$ and $v_{\\alpha }^{2}({G})\\ge -2(r^{\\prime \\prime })^{2}$ .", "Hence we get $K\\le \\frac{v_{\\alpha }^{2}({F})}{r}+2r.$ On the other hand, by equation (REF ) we have $K=\\frac{\\xi ^{2}}{rs^{2}}-\\frac{(\\xi ^{\\prime })^{2}}{r^{\\prime }s^{2}}-\\frac{(\\xi ^{\\prime \\prime })^{2}}{r^{\\prime \\prime }s^{2}}=-\\frac{r^{2}(\\xi ^{\\prime })^{2}+(r^{\\prime })^{2}\\xi ^{2}-2rr^{\\prime }\\xi \\xi ^{\\prime }}{s^{2}rr^{\\prime }r^{\\prime \\prime }}.$ By definition of $D$ we have $D^{2}=\\frac{r^{2}(\\xi ^{\\prime })^{2}+(r^{\\prime })^{2}\\xi ^{2}-2rr^{\\prime }\\xi \\xi ^{\\prime }}{s^{2}},$ so that the inequality (REF ) implies $D^{2}=-rr^{\\prime }r^{\\prime \\prime }K\\ge -r^{\\prime }(r-r^{\\prime })v_{\\alpha }^{2}({F})-2r^{2}r^{\\prime }(r-r^{\\prime }).$ But as $r^{\\prime }(r-r^{\\prime })\\le r^{2}/4$ , we finally get $D^{2}\\ge -\\frac{r^{2}}{4}v_{\\alpha }^{2}({F})-\\frac{r^{4}}{2}=-\\frac{r^{4}}{2}\\Delta _{\\alpha }({F})=-\\frac{r^{4}}{2}\\Delta _{\\alpha }(v).$ In conclusion, $D\\in W_{\\alpha ,v}\\cup \\lbrace 0\\rbrace $ .", "Step 2.", "Suppose that ${F}$ is $\\mu _{\\alpha ,\\omega }-$ stable but not $\\mu _{\\alpha ,\\omega ^{\\prime }}-$ stable.", "Let $[\\omega ,\\omega ^{\\prime }]:=\\lbrace \\omega _{t}:=t\\omega ^{\\prime }+(1-t)\\omega \\,|\\,t\\in [0,1]\\rbrace $ be the segment from $\\omega $ to $\\omega ^{\\prime }$ , and let $B_{\\alpha }$ be the family of subsheaves of ${F}$ whose slope with respect to $E$ and $\\omega ^{\\prime }$ is bounded from below.", "If ${E}\\in B_{\\alpha }$ , then ${E}\\otimes E^{\\vee }$ is a subsheaf of ${F}\\otimes E^{\\vee }$ , and $\\mu _{E,\\omega }({E})=\\mu _{\\omega }({E}\\otimes E^{\\vee })$ .", "This implies that ${E}\\otimes E^{\\vee }$ is in the family $B$ of subsheaves of ${F}\\otimes E^{\\vee }$ whose slope with respect to $\\omega $ is bounded from below.", "As the family $B$ is bounded, it follows that the family $B_{\\alpha }$ is bounded.", "Using the same argument as in the proof of Lemma REF , one can then conclude that there is $t\\in ]0,1]$ such that ${F}$ is properly $\\mu _{\\alpha ,\\omega _{t}}-$ semistable.", "Hence there is a subsheaf ${E}$ of ${F}$ of rank $0<s<r$ such that $\\mu _{E,\\omega _{t}}({E})=\\mu _{E,\\omega _{t}}({F})$ .", "If $D=rc_{E,1}({E})-sc_{E,1}({F})$ , it follows that $D\\cdot \\omega _{t}=0$ .", "As $D\\cdot \\omega \\ne 0$ , we have $D\\ne 0$ , hence $D^{2}<0$ .", "But as ${F}$ is $\\mu _{E,\\omega _{t}}-$ semistable, Step 1 implies that $D\\in W_{\\alpha ,v}$ , which is not possible as $\\omega _{t}$ is in the same $(\\alpha ,v)-$ chamber as $\\omega $ and $\\omega ^{\\prime }$ .", "In conclusion, the sheaf ${F}$ has to be $\\mu _{E,\\omega ^{\\prime }}-$ stable, and we are done." ], [ "Moduli space of stable twisted sheaves", "We now introduce (relative) moduli spaces of stable twisted sheaves.", "On projective manifolds these were constructed by Yoshioka in [43] (viewing twisted sheaves as $\\mathbb {P}-$ sheaves, and using a GIT construction), and by Lieblich in [25] (viewing twisted sheaves as sheaves on some ${O}^{*}-$ gerbe).", "Here we first provide a relative moduli space of simple twisted sheaves by viewing them as simple $\\mathbb {P}-$ sheaves.", "The relative moduli space of stable sheaves will then be an open subset of it." ], [ "The relative moduli space of simple twisted sheaves", "Consider a smooth and proper morphism $\\pi :{X}\\longrightarrow T$ such that for every $t\\in T$ the fibre $X_{t}$ over $t$ is a K3 surface.", "We assume for simplicity that $T$ is a complex manifold, although the constructions work over complex spaces as well.", "Suppose moreover that we are given a complex manifold ${P}$ together with a morphism $f:{P}\\longrightarrow {X}$ of $T-$ complex spaces such that for every $t\\in T$ , the morphism $f_{t}:P_{t}\\longrightarrow X_{t}$ is a projective bundle, where $P_{t}=f^{-1}(X_{t})$ .", "For every $t\\in T$ the projective bundle $P_{t}\\longrightarrow X_{t}$ defines a class $\\alpha _{t}$ in the Brauer group $Br(X_{t})$ .", "Now, let $f^{\\prime }:=\\pi \\circ f$ , so that we get a map $f^{\\prime }:{P}\\longrightarrow T$ .", "By Theorem (6.4) of [22], there is a complex space ${M}({P}/T)$ together with a holomorphic surjective map $\\phi :{M}({P}/T)\\longrightarrow T$ which is a relative moduli space of simple coherent sheaves on ${P}$ : for every $t\\in T$ the fibre ${M}_{t}$ of $\\phi $ over $t$ is the moduli space of simple coherent sheaves on $P_{t}$ .", "Now, ${F}\\in Coh(P_{t})$ is simple if and only if $End({F})\\simeq \\mathbb {C}$ .", "As $Coh(P_{t},X_{t})$ is a full subcategory of $Coh(P_{t})$ , a $P_{t}-$ sheaf ${F}$ is simple in $Coh(P_{t},X_{t})$ if and only if it is simple in $Coh(P_{t})$ .", "Hence simple $P_{t}-$ sheaves form a subset ${M}^{s}({P}/T)$ of ${M}({P}/T)$ .", "As the condition defining $\\mathbb {P}-$ sheaves is open (see Lemma 1.5 of [43]), it follows that ${M}^{s}({P}/T)$ is open in ${M}({P}/T)$ , hence it is a complex space together with a holomorphic map $\\psi :{M}^{s}({P}/T)\\longrightarrow T$ .", "This is the relative moduli space of simple ${P}-$ sheaves on ${X}$ .", "The relative projective bundle $f:{P}\\longrightarrow {X}$ corresponds to the existence of a relative Azumaya algebra $\\mathcal {A}$ on ${X}$ : for every $t\\in T$ , we have $P_{t}=\\mathbb {P}(E_{t})$ for some locally free $\\alpha _{t}-$ twisted sheaf on $X_{t}$ , and we let $\\mathcal {A}_{t}:=E_{t}\\otimes E_{t}^{\\vee }$ .", "The previous equivalence of categories of twisted sheaves then shows that ${M}^{s}({P}/T)$ is the relative moduli space of simple $\\mathcal {A}-$ modules on ${X}$ or, equivalently, the relative moduli space of simple twisted sheaves on ${X}$ ." ], [ "The relative moduli space of stable twisted sheaves", "We now produce out of $\\psi :{M}^{s}({P}/T)\\longrightarrow T$ the relative moduli space of stable twisted sheaves.", "Choose $v=(v_{0},v_{1},v_{2})\\in H^{2*}(S,\\mathbb {Q})$ such that $v_{1}\\in NS(S_{t})$ for every $t\\in T$ , and $v_{0}\\ge 2$ .", "We let ${M}^{s}_{v}({P}/T)$ be the component of ${M}^{s}({P}/T)$ parametrizing simple $\\mathbb {P}-$ sheaves of Mukai vector $v$ , and we write $\\psi _{v}:{M}^{s}_{v}({P}/T)\\longrightarrow T$ for $\\psi _{|{M}^{s}_{v}({P}/T)}$ .", "In order to define the moduli space of stable twisted sheaves of Mukai vector $v$ , we need a section $\\widetilde{\\omega }\\in R^{2}\\pi _{*}\\mathbb {C}$ such that $\\omega _{t}:=\\widetilde{\\omega }_{|X_{t}}$ is a Kähler class for every $t\\in T$ , which is used to define stability on every fibre.", "As stable twisted sheaves are simple, we let ${M}^{\\mu }_{v}({P}/T,\\widetilde{\\omega })$ be the subset of ${M}^{s}_{v}({P}/T)$ whose fibre over $t\\in T$ is given by the simple $P_{t}-$ sheaves which are $\\mu _{P_{t},\\omega _{t}}-$ stable and whose Mukai vector is $v$ .", "We then have a natural map (of sets) $p:{M}^{\\mu }_{v}({P}/T,\\widetilde{\\omega })\\longrightarrow T.$ The main result of this section is the following Proposition 4.14 Let $\\pi :{X}\\longrightarrow T$ , $f:{P}\\longrightarrow {X}$ , $v$ and $\\widetilde{\\omega }$ be as before.", "Then ${M}^{\\mu }_{v}({P}/T,\\widetilde{\\omega })$ is an open subset of ${M}^{s}_{v}({P}/T)$ .", "Hence it is a complex manifold, and the map $p$ is holomorphic.", "By Remark REF , the openness can be proved as in Lemma 3.7 of [32].", "Indeed, if ${F}\\in Coh(\\mathbb {P},S)$ and $F:=P({F})^{\\vee \\vee }$ , then ${F}$ is $\\mu _{\\mathbb {P},\\omega }-$ stable if and only if $P({F})\\otimes P(F)^{\\vee }$ is $\\mu _{\\mathcal {A},\\omega }-$ stable in $Coh(S,\\mathcal {A})$ , where $\\mathcal {A}=P(F)\\otimes P(F)^{\\vee }$ .", "Moreover, the openness of stability in the analytic case may be proved in the usual way, by using boundedness results which are contained in [37] and [38].", "The separatedness follows from Proposition (6.6) of [22] since the parameterized sheaves are stable.", "Standard deformation arguments following [5] allow us to show that if $p:{M}:={M}^{\\mu }_{v}({P}/{X},\\widetilde{\\omega })\\longrightarrow T$ is the relative moduli space of twisted stable sheaves, then for every $t\\in T$ and for every ${F}\\in p^{-1}(t)={M}_{t}$ we have $T_{[{F}]}{M}_{t}\\simeq Ext^{1}_{Coh(X_{t},\\alpha _{t})}(P({F}),P({F})),$ that the obstruction for the existence of deformations of ${F}$ live in $Ext^{2}_{Coh(X_{t},\\alpha _{t})}(P({F}),P({F})),$ and that we have an exact sequence $Ext^{1}_{Coh(X_{t},\\alpha _{t})}(P({F}),P({F}))\\longrightarrow T_{[{F}]}{M}\\longrightarrow $ $\\longrightarrow T_{t}T\\longrightarrow Ext^{2}_{Coh(X_{t},\\alpha _{t})}(P({F}),P({F}))_{0}.$ It follows from this exact sequence and by the previous discussion, that the morphism $p:{M}\\longrightarrow T$ is smooth.", "If $T$ is reduced to a point, then $X$ is just a K3 surface $S$ and $P\\longrightarrow S$ is a projective bundle whose class in $Br(S)$ is $\\alpha $ .", "The moduli space of $\\mu _{\\alpha ,\\omega }-$ stable $\\alpha -$ twisted sheaves of twisted Mukai vector $v$ on $S$ will then be denoted $M^{\\mu }_{\\alpha ,v}(S,\\omega )$ .", "Remark 4.15 Suppose that $\\alpha =0$ and let $\\gamma :=\\frac{ch({F}^{\\vee })}{\\sqrt{ch({F}\\otimes {F}^{\\vee })}}$ for ${F}\\in M^{\\mu }_{0,v}(S,\\omega )$ .", "Then $v_{0}({F})=v$ if and only if $v({F})=v/\\gamma $ , so that $M^{\\mu }_{0,v}(S,\\omega )\\simeq M^{\\mu }_{v/\\gamma }(S,\\omega )$ .", "We even notice that $\\omega $ is $(0,v)-$ generic if and only if it is $v/\\gamma -$ generic." ], [ "Moduli spaces of stable twisted sheaves over projective K3 surfaces", "If the base K3 surface $S$ is projective, from [43] we have some informations more about the moduli spaces of stable twisted sheaves.", "We make use of the following notation: let $\\alpha \\in Br(S)$ and ${F}$ a torsion free $\\alpha -$ twisted sheaf whose twisted Mukai vector is $w=(r,0,a)$ .", "We let $F$ be a locally free $\\alpha -$ twisted sheaf and $\\xi $ be a representative of the class of $\\mathbb {P}(E)$ in $H^{2}(S,\\mathbb {Z})$ .", "We let $e^{\\xi /r}:=(1,\\xi /r,\\xi ^{2}/2r^{2})$ and $w_{\\xi }:=e^{\\xi /r}\\cdot w$ , so that $w_{\\xi }=(r,\\xi ,a+\\xi ^{2}/2r).$ It is worthwhile to notice that there is a topological vector bundle $E_{\\xi }$ on $S$ such that $v(E_{\\xi })=w_{\\xi }$ .", "As shown in [43], we have $w_{\\xi }\\in H^{2}(S,\\mathbb {Z})$ (while in general we have $w\\in H^{2}(S,\\mathbb {Q})$ ).", "Remark 4.16 If $\\alpha =0$ and ${F}$ is a $\\mu _{\\omega }-$ stable sheaf whose Mukai vector is $v=(r,\\xi ,a)$ , write $a=c+r$ where $c=ch_{2}({F})$ .", "The $0-$ twisted Mukai vector of ${F}$ is then $w=(r,0,r+a^{\\prime }/2r)$ , where $a^{\\prime }=ch_{2}({F}\\otimes {F}^{\\vee \\vee })$ .", "We notice that $a^{\\prime }=2rc-\\xi ^{2}$ , hence $w=(r,0,r+c-\\xi ^{2}/2r)$ .", "A representative of the class of $\\mathbb {P}(E)$ in this case can be chosen to be $\\xi $ itself.", "Hence we have $w_{\\xi }=e^{\\xi /r}w=(r,\\xi ,r+c)=v.$ The following is Theorem 3.16 of [43]: Theorem 4.17 Let $S$ be a projective K3 surface, $w=(r,\\zeta ,b)\\in H^{2}(S,\\mathbb {Q})$ and $\\alpha \\in Br(S)$ .", "Choose a representative $\\xi $ of $\\alpha $ in $H^{2}(S,\\mathbb {Z})$ , and suppose that $w_{\\xi }$ is primitive.", "Moreover, let $H$ be a $(\\alpha ,w)-$ generic ample line bundle on $S$ .", "Then the moduli space $M^{s}_{\\alpha ,w}(S,H)$ is an irreducible symplectic manifold which is deformation equivalent to a Hilbert scheme of points on $S$ .", "We have the following result, which is the twisted version of Theorem REF : Proposition 4.18 Let $S$ be a projective K3 surface, $w=(r,\\zeta ,b)$ a Mukai vector and $\\alpha \\in Br(S)$ .", "Choose $\\xi $ to be a representative of $\\alpha $ in $H^{2}(S,\\mathbb {Z})$ , and suppose that $r$ and $\\xi $ are prime to each other.", "If $\\omega $ is a $(\\alpha ,w)-$ generic polarization, then $M^{\\mu }_{\\alpha ,w}(S,\\omega )$ is an irreducible symplectic manifold which is deformation equivalent to a Hilbert scheme of points on $S$ .", "By Lemma REF , Proposition REF and following the same proof of Theorem REF , we see that there is an ample line bundle $H$ such that $M^{\\mu }_{\\alpha ,w}(S,\\omega )=M^{\\mu }_{\\alpha ,w}(S,H)$ .", "This last moduli space is an irreducible symplectic manifold which is deformation equivalent to a Hilbert scheme of points on $S$ by Theorem REF ." ], [ "Quasi-universal families", "We conclude this section with the following result about the existence of a quasi-universal family; cf.", "[1] for the absolute untwisted case.", "Proposition 4.19 Let $\\pi :{X}\\longrightarrow T$ , $f:{P}\\longrightarrow {X}$ , $v=(v_{0},v_{1},v_{2})$ and $\\widetilde{\\omega }$ be as before.", "Let $\\mathcal {A}$ be a relative Azumaya algebra corresponding to ${P}$ , and for every $t\\in T$ let $\\alpha _{t}\\in Br(X_{t})$ be the class of $\\mathcal {A}_{t}$ .", "Suppose that there is a locally free $\\mathcal {A}$ -module $\\mathcal {V}$ verifying the two following properties for every $t\\in T$ : the restriction $\\mathcal {V}_{t}$ of $\\mathcal {V}$ to $X_{t}$ is $\\mu _{\\alpha _{t},\\omega _{t}}-$ stable; the twisted Mukai vector of $\\mathcal {V}_{t}$ is $(v_{0},v_{1},w_{2})$ , where $w_{2}<v_{2}$ .", "Then there is a quasi-universal family on ${M}^{\\mu }_{v}({P}/T,\\widetilde{\\omega })\\times _{T}{X}$ .", "Let ${M}:={M}^{\\mu }_{\\widetilde{v}}({P}/T,\\widetilde{\\omega })$ .", "As for stable coherent sheaves, there is an open covering ${U}=\\lbrace U_{i}\\rbrace _{i\\in I}$ of ${M}$ given by analytic subsets endowed with universal $\\mathcal {A}$ -modules ${F}_{i}$ .", "Let $p_i:U_i\\times _T{X}\\rightarrow U_i$ and $q_i:U_i\\times _T{X}\\rightarrow {X}$ denote the two projections.", "We put ${E}_{i}:={F}_{i}\\otimes _{q_i^*\\mathcal {A}} q_{i}^{*}\\mathcal {V}^{\\vee }$ .", "By the choice of $\\mathcal {V}$ we have $R^0p_{i,*}{E}_i=0=R^2p_{i,*}{E}_i$ and $W_{i}:=R^1p_{i,*}{E}_i$ is a non-trivial locally free ${O}_{U_i}$ -module whose rank is independent of $i$ .", "It is easy to check now that the $\\mathcal {A}$ -modules ${F}_i\\otimes _{{O}}p_i^*W_i^{\\vee }$ glue together to give the desired quasi-universal family." ], [ "Deformation of stable twisted sheaves along twistor lines", "In this subsection we describe and generalize a construction used by several authors in the case of stable locally free sheaves of slope zero, cf.", "[35], [39], [40], [27].", "Let $(S, I, \\omega )$ be a polarized K3 surface and $\\pi :Z(S)\\longrightarrow \\mathbb {P}^{1}$ its twistor family.", "We suppose that the fibre over 0 is $S_{0}=(S,I)$ , and we write $S_{t}=(S,I_{t})$ for the fibre over $t$ .", "Here $I=I_0$ and $I_t$ denote the complex structures on $S$ .", "With this convention we have $S_\\infty =(S, I_\\infty )=(S, -I)$ .", "Recall that the choice of $\\omega $ on $(S,I)$ is equivalent to the choice of a Riemannian metric $g$ which is compatible with $I$ and whose associated Kähler class is $\\omega $ .", "Along the twistor line the metric $g$ remains compatible with $I_{t}$ , the associated class $\\omega _{t}$ is Kähler, and we get a section $\\widetilde{\\omega }$ of $R^{2}\\pi _{*}\\mathbb {C}$ which is $\\omega _{t}$ on $S_{t}$ .", "Slope stability on $S_t$ will be considered with respect to $\\omega _t$ .", "Before we introduce deformations of sheaves along twistor lines we make an observation on $(1,1)$ -forms on the twistor space of $S$ .", "Recall that, as a differentiable manifold, $Z(S)$ is the product $S\\times \\mathbb {P}^1$ , which is endowed with a complex structure in the following way (see [13]): cover $\\mathbb {P}^1$ by two charts (each isomorphic to $) and take $$ the complex coordinate function on one of them and $ -1$ on the other.", "Further, let $ I,J,K$ be the complex structures on $ S$ which make it into a hyperkähler manifold.", "If $ IP1$ is the complex structure on $ P1$ then put the following complex structure to act on the tangent space $ TSTP1$ of $ SP1$:$ $\\mathfrak {I}:=\\bigg (\\frac{1-\\zeta \\bar{\\zeta }}{1+\\zeta \\bar{\\zeta }}I+ \\frac{\\zeta +\\bar{\\zeta }}{1+\\zeta \\bar{\\zeta }}J+ i\\frac{\\zeta -\\bar{\\zeta }}{1+\\zeta \\bar{\\zeta }}K, I_{\\mathbb {P}^1}\\bigg ).", "$ With respect to this complex structure the projection $q: S\\times \\mathbb {P}^1\\rightarrow S$ is not holomorphic but only $C^\\infty $ .", "Lemma 4.20 Let $\\psi $ be a $(1,1)$ -form on $(S, I,\\omega )$ .", "Its pull-back $q^*\\psi $ is a $(1,1)$ -form on $Z(S)$ if and only if $\\psi $ is anti-self-dual on $(S, I,\\omega )$ .", "Let $\\Psi :=q^*\\psi $ .", "It is a 2-form on $Z(S)$ , so it is of type $(1,1)$ if and only if $\\Psi (\\mathfrak {I}v,\\mathfrak {I}w)=\\Psi (v,w)$ for any pair $(v,w)$ of real tangent vectors at a point of $Z(S)$ .", "As $\\mathfrak {I}$ preserves the horizontal and the vertical directions on $Z(S)=S\\times \\mathbb {P}^1$ , and as $\\Psi (v,w)=0$ if one of the tangent vectors $v$ or $w$ is horizontal, it suffices to check $\\Psi (\\mathfrak {I}v,\\mathfrak {I}w)=\\Psi (v,w)$ only on vertical vectors, meaning that the restrictions of $\\Psi $ to the fibres of $\\pi :Z(S)\\longrightarrow \\mathbb {P}^{1}$ are of type $(1,1)$ .", "Suppose that $\\psi $ is anti-self-dual.", "This property only depends on $g$ and on the orientation of $S$ : as $g$ is compatible with each complex structure $I_t$ , it follows that the restriction of $\\Psi $ to each fibre of $\\pi $ is then anti-self-dual.", "In particular, it is of type $(1,1)$ , hence also $\\Psi $ is of type $(1,1)$ on $Z(S)$ .", "Conversely, if $\\psi $ is not anti-self-dual, then it decomposes as $\\psi =\\psi ^{SD}+\\psi ^{ASD}$ , where the self-dual part is of the form $\\psi ^{SD}=f\\omega _I$ for some non-zero function $f$ .", "But $\\omega _I $ is not of type $(1,1)$ with respect to $J$ so neither will be $\\Psi $ .", "We now turn to deformations of sheaves along the twistor line." ], [ "Deformation of a locally free polystable sheaf with trivial slope", "Let $E_0$ be a polystable vector bundle on $S_0$ whose slope is zero, and denote by $E^\\infty $ the $C^\\infty -$ vector bundle underlying $E_{0}$ .", "The Kobayashi-Hitchin correspondence provides $E^\\infty $ with an ASD-connection.", "By Lemma REF the curvature of the connection is of $(1,1)$ -type on each $S_t$ .", "We therefore obtain holomorphic structures $E_t$ on $E^\\infty $ over each $S_t$ , induced by the structure $E_0$ in a canonical way.", "In fact we even get a holomorphic structure on $q^{*}E^\\infty $ ; denote by $\\tilde{E}$ the corresponding sheaf of holomorphic sections over $Z(S)$ .", "As $E_{t}$ is holomorphic and carries an ASD-connection, it is polystable for every $t\\in \\mathbb {P}^{1}$ .", "It is easy to see that if $E_{0}$ is stable, then $E_{t}$ is stable for every $t\\in \\mathbb {P}^{1}$ ." ], [ "Deformation of an Azumaya algebra", "Let now $\\mathcal {A}_{0}$ be an Azumaya algebra on $S_{0}$ , and let $\\alpha _{0}$ be its class in $Br(S_{0})$ .", "Choose a locally free $\\alpha _{0}-$ twisted sheaf $E_{0}$ such that $\\mathcal {A}_{0}\\simeq {E}nd(E_{0})$ .", "We will suppose that $E_{0}$ is $\\mu _{\\alpha _{0},\\omega _{0}}-$ stable.", "The Kobayashi-Hitchin correspondence for twisted sheaves established by Wang in [41] shows that $\\mathcal {A}_{0}$ is $\\mu _{\\omega _{0}}-$ polystable.", "Notice that $\\mu _{\\omega _{0}}(\\mathcal {A}_{0})=0$ , hence by section 4.4.1 the vector bundle $\\mathcal {A}:=q^{*}\\mathcal {A}_{0}$ carries a holomorphic structure, and for every $t\\in \\mathbb {P}^{1}$ its restriction $\\mathcal {A}_{t}$ to the fibre $S_{t}$ is a $\\mu _{\\omega _{t}}-$ polystable vector bundle with trivial slope.", "We need to show that $\\mathcal {A}_{t}$ is an Azumaya algebra.", "To do so, we argue as in the proof of Lemma 6.5 in [27], point (3).", "The Azumaya algebra structure on $\\mathcal {A}_{0}$ is given by a holomorphic map $m_{0}:\\mathcal {A}_{0}\\otimes \\mathcal {A}_{0}\\longrightarrow \\mathcal {A}_{0}$ verifying some identities among holomorphic sections.", "This means that $m_{0}$ is a holomorphic section of the vector bundle ${H}om(\\mathcal {A}_{0}\\otimes \\mathcal {A}_{0},\\mathcal {A}_{0})$ .", "But this is $\\mu _{\\omega _{0}}-$ polystable as $\\mathcal {A}_{0}$ is, hence it carries an ASD-connection, and $m_{0}$ is parallel with respect to it.", "As a consequence, $m_{0}$ defines a parallel section of ${H}om(\\mathcal {A}_{t}\\otimes \\mathcal {A}_{t},\\mathcal {A}_{t})$ , hence a holomorphic map $m_{t}:\\mathcal {A}_{t}\\otimes \\mathcal {A}_{t}\\longrightarrow \\mathcal {A}_{t}$ .", "Hence $\\mathcal {A}_{t}$ is an ${O}_{S_{t}}-$ algebra: as the same identities among sections which are verified on $\\mathcal {A}_{0}$ are verified even on $\\mathcal {A}_{t}$ , it follows that $\\mathcal {A}_{t}$ is an Azumaya algebra.If $E_{0}$ is an untwisted sheaf, we can give a more direct proof.", "The multiplication of two holomorphic sections $\\phi _1$ , $\\phi _2$ of $\\mathcal {A}_t$ remains holomorphic (hence $\\mathcal {A}_{t}$ is a sheaf of algebras on $S_{t}$ ): this is a consequence of the formula $\\hat{D}(\\phi _1\\circ \\phi _2)=\\hat{D}\\phi _1\\circ \\phi _2+\\phi _1\\circ \\hat{D}\\phi _2$ , where $\\hat{D}$ is the connection induced by $D$ on $\\mathcal {A}_{0}$ .", "By [6] we just need to show that $\\mathcal {A}_t$ is locally of the form ${E}nd(F)$ for some locally free sheaf $F$ of ${O}_{S_t}$ -modules.", "To do so, consider the self-dual part $R_{SD}$ of the curvature $R$ of $D$ .", "We have $R_{SD}=c \\cdot Id\\cdot \\omega _0$ for a suitable constant $c$ .", "By solving the equation $dd^c\\phi = -\\frac{c}{r}\\omega _0$ on a open subset $U$ , we find a holomorphic hermitian line bundle $(L,h)$ on $U$ whose curvature is $-\\frac{c}{r}\\omega _0$ .", "Hence $F^{\\infty }:=E_0\\otimes L$ is a rank $r$ vector bundle on $U$ with a Hermite-Einstein connection, and $\\mathcal {A}^\\infty \\cong {E}nd^\\infty (F^\\infty )$ as ASD-vector bundles.", "Hence on $F^\\infty $ we have a holomorphic structure $F_t$ compatible with the corresponding $I_t$ , and $\\mathcal {A}_t\\cong {E}nd(F_t)$ ." ], [ "Deformation of a stable twisted vector bundle", "Let $\\alpha _{0}\\in Br(S_{0})$ and $F_{0}$ an $\\alpha _{0}-$ twisted locally free sheaf which is $\\mu _{\\alpha _{0},\\omega _{0}}-$ stable.", "Choose an $\\alpha _{0}-$ twisted locally free sheaf $E_{0}$ which is $\\mu _{\\alpha _{0},\\omega _{0}}-$ stable in such a way that $c_{E_{0},1}({F}_{0})=0$ .", "We let $G_{0}:=F_{0}\\otimes E_{0}^{\\vee }$ and $\\mathcal {A}_{0}:=E_{0}\\otimes E_{0}^{\\vee }$ : then $\\mathcal {A}_{0}$ is an Azumaya algebra, and as we saw in section 4.4.2 it is a polystable sheaf.", "Moreover, $G_{0}$ is a locally free sheaf of trivial slope and it has the structure of $\\mathcal {A}_{0}-$ module.", "The Kobayashi-Hitchin correspondence for twisted sheaves in [41] shows that $G_{0}$ is a polystable sheaf.", "Following section 4.4.2, $q^{*}\\mathcal {A}_{0}$ is a holomorphic vector bundle, and for every $t\\in \\mathbb {P}^{1}$ its restriction $\\mathcal {A}_{t}$ to $S_{t}$ is a polystable sheaf having the structure of Azumaya algebra.", "We let $\\alpha _{t}$ be its class in $Br(S_{t})$ .", "By section 4.4.2 the polystable sheaf $G_{0}$ gives rise, for every $t\\in \\mathbb {P}^{1}$ , to a polystable sheaf $G_{t}$ with trivial slope.", "The same argument used in section 4.4.2 to show that $\\mathcal {A}_{t}$ is an Azumaya algebra, applied this time to $m_{t}:\\mathcal {A}_{t}\\otimes G_{t}\\longrightarrow G_{t}$ , shows that the sheaf $G_{t}$ has the structure of an $\\mathcal {A}_{t}-$ module.", "As $G_{t}$ is an $\\mathcal {A}_{t}-$ module, it corresponds to an $\\alpha _{t}-$ twisted locally free sheaf $F_{t}$ on $S_{t}$ .", "In particular $E_{0}$ gives rise to an $\\alpha _{t}-$ twisted locally free sheaf $E_{t}$ on $S_{t}$ such that ${E}nd(E_{t})\\simeq \\mathcal {A}_{t}$ and $F_{t}\\otimes E_{t}^{\\vee }\\simeq G_{t}$ .", "Lemma 4.21 The sheaves $F_{t}$ and $E_{t}$ are $\\mu _{\\alpha _{t},\\omega _{t}}-$ stable.", "We show that $E_{t}$ is $\\mu _{\\alpha _{t},\\omega _{t}}-$ stable.", "The proof for $F_{t}$ is similar.", "Suppose that $E_{t}$ is not $\\mu _{\\alpha _{t},\\omega _{t}}-$ stable, and let ${E}_{t}\\subseteq E_{t}$ in $Coh(S_{t},\\alpha _{t})$ with $\\mu _{E_{t},\\omega _{t}}({E}_{t})\\ge \\mu _{E_{t},\\omega _{t}}(E_{t})$ .", "We suppose that ${E}_{t}$ is $\\mu _{\\alpha _{t},\\omega _{t}}-$ stable.", "We let $\\mathcal {H}_{t}:={E}_{t}\\otimes E_{t}^{\\vee }$ , which is an $\\mathcal {A}_{t}-$ module and we have $\\mathcal {H}_{t}\\subseteq \\mathcal {A}_{t}$ .", "The inequality $\\mu _{E_{t},\\omega _{t}}({E}_{t})\\ge \\mu _{E_{t},\\omega _{t}}(E_{t})$ gives $\\mu _{\\mathcal {A}_{t},\\omega _{t}}(\\mathcal {H}_{t})\\ge \\mu _{\\mathcal {A}_{t},\\omega _{t}}(\\mathcal {A}_{t})$ , so that $\\mu _{\\omega _{t}}(\\mathcal {H}_{t})\\ge \\mu _{\\omega _{t}}(\\mathcal {A}_{t})=0$ .", "As $\\mathcal {A}_{t}$ is $\\mu _{\\omega _{t}}-$ polystable, this implies that $\\mu _{\\omega _{t}}(\\mathcal {H}_{t})=0$ , and that it is a direct summand of $\\mathcal {A}_{t}$ .", "In particular, it is $\\mu _{\\omega _{t}}-$ polystable.", "Using the same argument given before, the sheaf $\\mathcal {H}_{t}$ gives rise to a $\\mu _{\\omega _{0}}-$ polystable sheaf $\\mathcal {H}_{0}$ on $S_{0}$ , which is contained in $\\mathcal {A}_{0}$ , has the structure of $\\mathcal {A}_{0}-$ module, and $\\mu _{\\omega _{0}}(\\mathcal {H}_{0})=\\mu _{\\omega _{0}}(\\mathcal {A}_{0})=0$ .", "The equivalence between $Coh(S_{0},\\alpha _{0})$ and $Coh(S_{0},\\mathcal {A}_{0})$ given by tensoring with $E_{0}^{\\vee }$ produces then a subsheaf ${E}_{0}$ of $E_{0}$ such that $\\mu _{E_{0},\\omega _{0}}({E}_{0})=\\mu _{E_{0},\\omega _{0}}(E_{0})$ .", "But this is not possible as $E_{0}$ is $\\mu _{\\alpha _{0},\\omega _{0}}-$ stable.", "In conclusion, the sheaf $E_{t}$ is $\\mu _{\\alpha _{t},\\omega _{t}}-$ stable." ], [ "Relative moduli space of twisted sheaves on twistor lines", "In this section we show that the relative moduli space of stable twisted sheaves gives us a way to deform the moduli spaces $M^{\\mu }_{\\alpha ,w}(S,\\omega )$ to an irreducible symplectic manifold (which is moreover deformation equivalent to a Hilbert scheme of points on projective K3 surface).", "We let $S$ be a K3 surface, $w=(r,0,a)\\in H^{2*}(S,\\mathbb {Z})$ with $r\\ge 2$ , $\\alpha \\in Br(S)$ and $\\omega $ an $(\\alpha ,w)-$ generic polarization.", "The Kähler class $\\omega $ corresponds to the choice of a Riemannian metric $g$ which is compatible with the complex structure $I$ of $S$ , and whose associated Kähler class is $\\omega $ .", "Let $\\pi :Z(S)\\longrightarrow \\mathbb {P}^{1}$ be the twistor family of $g$ : we denote $S_{t}$ the fibre of $\\pi $ over $t$ , which corresponds to a complex structure $I_{t}$ on $S$ associated with $t$ .", "The metric $g$ is compatible with $I_{t}$ , the associated class $\\omega _{t}$ is Kähler, and $w$ is a Mukai vector on $S_{t}$ for every $t\\in \\mathbb {P}^{1}$ .", "Choose now a $\\mu _{\\alpha ,\\omega }-$ stable $\\alpha -$ twisted sheaf ${E}$ on $S$ of rank $r$ , and let $E:={E}^{\\vee \\vee }$ : this is a $\\mu _{\\alpha ,\\omega }-$ stable $\\alpha -$ twisted vector bundle of rank $r$ , and we let $\\mathcal {A}_{0}:={E}nd(E)$ the corresponding Azumaya algebra.", "We suppose that $v_{E}({E})=w$ .", "By section 4.4.2, there is holomorphic vector bundle $\\mathcal {A}$ on $Z(S)$ whose restriction $\\mathcal {A}_{t}$ to $S_{t}$ is an Azumaya algebra on $S_{t}$ for every $t\\in \\mathbb {P}^{1}$ .", "We let $\\alpha _{t}\\in Br(S_{t})$ be its class and $\\mathcal {A}_{t}\\simeq {E}nd(E_{t})$ , where $E_t$ is the deformation of $E$ along the twistor line (see section 4.4.3).", "By section 4.3.2 there is then a relative moduli space of stable twisted sheaves $p:{M}\\longrightarrow \\mathbb {P}^{1}$ such that for every $t\\in \\mathbb {P}^{1}$ the fibre over $t$ is the moduli space $M^{\\mu }_{\\alpha _{t},w}(S_{t},\\omega _{t})$ of $\\mu _{\\alpha _{t},\\omega _{t}}-$ stable $\\alpha _{t}-$ twisted sheaves whose twisted Mukai vector with respect to $E_{t}$ is $w$ .", "Remark 4.22 On ${M}\\times _{\\mathbb {P}^{1}}Z(S)$ we have a quasi-universal family: if ${F}\\in M^{\\mu }_{\\alpha ,w}(S,\\omega )$ , let $F:={F}^{\\vee \\vee }$ and $\\mathcal {V}_{0}:=F\\otimes E^{\\vee }$ .", "We let $\\mathcal {V}$ in Proposition REF be $\\mathcal {V}:=q^{*}\\mathcal {V}_{0}$ .", "We first prove some geometrical properties of the relative moduli space $p:{M}\\longrightarrow \\mathbb {P}^{1}$ .", "Proposition 4.23 Let $S$ be a K3 surface, $w=(r,0,a)\\in H^{2*}(S,\\mathbb {Z})$ with $r\\ge 2$ , $\\alpha \\in Br(S)$ and $\\omega $ a $(\\alpha ,w)-$ generic polarization.", "The relative moduli space $p:{M}\\longrightarrow \\mathbb {P}^{1}$ of stable twisted sheaves verifies the following properties: the morphism $p$ is submersive; if $T_{p}^{*}$ denotes the relative cotangent bundle of $p$ , there is a holomorphic global section of $\\wedge ^{2}T^{*}_{p}\\otimes {O}_{\\mathbb {P}^{1}}(2)$ whose restriction to any fibre is a holomorphic symplectic form; We divide the proof in several parts.", "Step 1: submersivity.", "As every ${E}\\in {M}_{t}$ is simple and the canonical bundle of a K3 surface is trivial, we have $Ext^{2}({E},{E})_{0}=0$ .", "The exact sequence (REF ) implies then that the map $p$ is submersive, so that condition (1) of the statement is proved.", "Step 2: section through locally free sheaves.", "Let $t_{0}\\in \\mathbb {P}^{1}$ , and choose $F\\in M^{\\mu }_{\\alpha _{t_{0}},w}(S_{t_{0}},\\omega _{t_{0}})$ a locally free sheaf.", "As we saw in section 4.4.3, the sheaf $F$ gives rise to a sheaf $F_{t}\\in M^{\\mu }_{\\alpha _{t},w}(S_{t},\\omega _{t})$ for every $t\\in \\mathbb {P}^{1}$ .", "This produces a section $s_{F}:\\mathbb {P}^{1}\\longrightarrow {M},\\,\\,\\,\\,\\,\\,\\,\\,s_{F}(t):=F_{t}$ of $p$ , which is holomorphic.", "If we let $E_{t}$ be the $\\alpha _{t}-$ twisted $\\mu _{\\alpha _{t},\\omega _{t}}-$ stable sheaf such that $\\mathcal {A}_{t}={E}nd(E_{t})$ (an Azumaya algebra on $S_{t}$ whose class in $Br(S_{t})$ is $\\alpha _{t}$ ), and $G_{t}:=F_{t}\\otimes E_{t}^{\\vee }$ , we let $\\mathcal {G}:=q^{*}G_{t}$ , which is a holomorphic vector bundle on $Z(S)$ .", "The restriction of the relative tangent bundle $T_{p}$ of $p$ to the section $s$ is $s^{*}T_{p}\\simeq R^{1}\\pi _{*}{E}nd(\\mathcal {G}).$ Step 3: relative symplectic form.", "We prove that the condition (2) is fulfilled.", "We first notice that for every $t\\in \\mathbb {P}^{1}$ the restriction $T_{p|t}$ of $T_{p}$ to ${M}_{t}$ is the tangent bundle of $M^{\\mu }_{\\alpha _{t},w}(S_{t},\\omega _{t})$ , and similarly the restriction $(T_{p}^{*})_{|t}$ of $T_{p}^{*}$ to ${M}_{t}$ is the cotangent bundle of $M^{\\mu }_{\\alpha _{t},w}(S_{t},\\omega _{t})$ .", "As on $M^{\\mu }_{\\alpha _{t},w}(S_{t},\\omega _{t})$ we have a holomorphic symplectic form (if $S_{t}$ is projective, this is done in [43]; the proof in the general case is similar), we get an isomorphism $T_{p|t}\\simeq (T_{p}^{*})_{t}$ .", "This implies the existence of a line bundle ${O}_{\\mathbb {P}^{1}}(d)$ for some $d\\in \\mathbb {Z}$ together with an isomorphism $T_{p}\\longrightarrow T_{p}^{*}\\otimes p^{*}{O}_{\\mathbb {P}^{1}}(d)$ .", "We then just need to show that $d=2$ .", "To do so, consider a locally free sheaf $F\\in {M}_{0}$ : as seen in Step 2 we have a holomorphic section $s:\\mathbb {P}^{1}\\longrightarrow {M}$ of $p$ , and $s^{*}T_{p}\\simeq R^{1}p_{*}{E}nd(\\mathcal {G})$ where $\\mathcal {G}=q^{*}(F\\otimes E_{0}^{\\vee })$ .", "By the relative Serre duality we get $R^{1}p_{*}{E}nd(\\mathcal {G})\\simeq (R^{1}p_{*}{E}nd(\\mathcal {G})^{*}\\otimes K_{\\pi })^{*},$ where $K_{\\pi }$ is the relative canonical bundle of $\\pi :Z(S)\\longrightarrow \\mathbb {P}^{1}$ .", "Now, as $\\mathcal {G}$ is locally free, we have ${E}nd(\\mathcal {G})\\simeq {E}nd(\\mathcal {G})^{*}$ .", "Moreover, $K_{\\pi }\\simeq {O}_{\\mathbb {P}^{1}}(-2)$ (see [13]), hence $R^{1}p_{*}{E}nd(\\mathcal {G})\\simeq R^{1}p_{*}{E}nd(\\mathcal {G})^{*}\\otimes {O}_{\\mathbb {P}^{1}}(2).$ In conclusion, $s^{*}T_{p}\\simeq s^{*}T_{p}^{*}\\otimes {O}_{\\mathbb {P}^{1}}(2).$ As $s^{*}T_{p}\\simeq s^{*}T^{*}_{p}\\otimes {O}_{\\mathbb {P}^{1}}(d)$ , it follows $d=2$ .", "This shows that condition (2) is fulfilled.", "We now prove some geometrical properties of the moduli spaces of stable twisted sheaves we are considering: in particular, we show that they are all compact and connected.", "Proposition 4.24 Let $S$ be a K3 surface, $w=(r,0,a)\\in H^{2*}(S,\\mathbb {Z})$ with $r\\ge 2$ , $\\alpha \\in Br(S)$ and $\\omega $ a $(\\alpha ,w)-$ generic polarization.", "Moreover, let $\\xi $ be a representative of $\\alpha $ in $H^{2}(S,\\mathbb {Z})$ which is prime with $r$ .", "The moduli space $M^{\\mu }_{\\alpha ,w}(S,\\omega )$ is a compact, connected manifold.", "The compactness of $M^{\\mu }_{\\alpha ,w}(S,\\omega )$ is well known when $S$ is projective and a proof in the non-projective and non-twisted case has been given in [36].", "This proof uses in an essential way the comparison map from the moduli space of stable sheaves to the corresponding Donaldson-Uhlenbeck compactification of the moduli space of anti-self-dual connections in a hermitian vector bundle on $S$ .", "These arguments may be extended to the twisted case.", "We refer the reader to [36] and [41] for the ingredients.", "To show that $M^{\\mu }_{\\alpha ,w}(S,\\omega )$ is connected, we will follow the strategy used by Mukai and by Kaledin, Lehn and Sorger to prove the analogous result when $S$ is projective, $\\omega $ is the first Chern class of an ample line bundle, and the sheaves are untwisted (see the proof of Theorem 4.1 in [21]).", "We first suppose that the moduli space $M^{\\mu }_{\\alpha ,w}(S,\\omega )$ is not connected, and we choose a connected component $Y$ .", "Moreover, we fix a sheaf $F\\in Y$ and a sheaf $G\\in M^{\\mu }_{\\alpha ,w}(S,\\omega )\\setminus Y$ .", "Let $p:Y\\times S\\longrightarrow Y$ and $q:Y\\times S\\longrightarrow S$ be the two projections, and consider a $p^{*}\\beta \\cdot q^{*}\\alpha -$ twisted universal family ${F}$ on $Y\\times S$ .", "We then define two complexes $K^{\\bullet }_{F}:={E}xt_{p}^{\\bullet }(q^{*}F,{F}),\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,K^{\\bullet }_{G}:={E}xt_{p}^{\\bullet }(q^{*}G,{F})$ of $\\beta -$ twisted sheaves on $Y$ .", "As the sheaves $F$ and $G$ have the same topological invariants (since their Mukai vectors are equal), letting $d:=dim(Y)$ , by the Grothendieck-Riemann-Roch Theorem we have $c^{B}_{d}(K^{\\bullet }_{F})=c^{B}_{d}(K^{\\bullet }_{G})$ , where $c^{B}_{d}$ is the component of degree $2d$ of $c^{B}$ (for some $B-$ field giving the twist $\\beta $ ).", "We now compute more explicitely these twisted Chern classes, and we start from $K^{\\bullet }_{G}$ .", "We notice that if $E\\in Y$ , then $E$ is a stable twisted sheaf having the same slope of $G$ , but which is not isomorphic to $G$ .", "It follows that $Ext^{0}(G,F)=Ext^{2}(G,F)=0.$ As ${E}xt_{p}^{j}(q^{*}G,{F})_{E}\\simeq Ext^{j}(G,E),$ it follows that ${E}xt_{p}^{j}(q^{*}G,{F})=0$ if $j=0,2$ , and that ${E}xt_{p}^{1}(q^{*}G,{F})$ is a locally free $\\beta -$ twisted sheaf whose rank is $d-2$ .", "As a consequence we have $c^{B}_{d}(K^{\\bullet }_{G})=-c^{B}_{d}({E}xt_{p}^{1}(q^{*}G,{F}))=0,$ as ${E}xt_{p}^{1}(q^{*}G,{F})$ is a locally free $\\beta -$ twisted vector bundle of rank $d-2<d$ : recall that $c^{B}$ of ${E}xt_{p}^{1}(q^{*}G,{F})$ is defined as the Chern class of some untwisted vector bundle of the same rank, hence, as this rank is smaller then the dimension of $Y$ , the $d-$ th $B-$ twisted Chern class is trivial.", "We now need to compute $c^{B}_{d}(K^{\\bullet }_{F})$ .", "To do so, we first recall that by [3] there is locally on $Y$ a complex $A^{\\bullet }=\\cdots \\stackrel{a_{-1}}{\\longrightarrow }A^{0}\\stackrel{a_{0}}{\\longrightarrow }A^{1}\\stackrel{a_{1}}{\\longrightarrow }A^{2}\\longrightarrow 0$ of free sheaves such that for every $\\sigma :Y^{\\prime }\\longrightarrow Y$ and for every $j\\in \\mathbb {Z}$ we have ${E}xt^{j}_{p^{\\prime }}(\\sigma ^{*}(q^{\\prime })^{*}F,\\sigma ^{*}{F})\\simeq \\mathcal {H}^{j}(\\sigma ^{*}A^{\\bullet }),$ where $p^{\\prime }:Y^{\\prime }\\times S\\longrightarrow Y^{\\prime }$ and $q^{\\prime }:Y^{\\prime }\\times S\\longrightarrow S$ are the two projections, and where $\\mathcal {H}^{j}$ denotes the $j-$ th cohomology of the complex.", "Let us now cover $Y$ with open subsets $U_{i}$ so that $F$ is contained in only one of them, and let us moreover suppose that the previous complex $A^{\\bullet }$ exists over $U_{i}$ .", "If $E\\in U_{i}$ and $E$ is not $F$ , then $\\mathcal {H}^{j}(A^{\\bullet })_{E}=0$ , hence the rank of all the maps $a_{i}$ of the complex $A^{\\bullet }$ is constant on $Y\\setminus \\lbrace F\\rbrace $ .", "But we have $\\mathcal {H}^{0}(A^{\\bullet })_{F}\\simeq \\mathcal {H}^{0}(A^{\\bullet })_{F}\\simeq \\mathbb {C},$ hence the rank of $a_{0}$ and $a_{1}$ at $F$ drops by 1, while the rank of $a_{i}$ is constant on $Y$ for $i\\le -1$ .", "The same proof of Lemma 4.3 of [21] shows that the degeneracy locus of the $a_{0}$ and $a_{1}$ is the reduced point $F$ , while $a_{i}$ does not degenerate if $i\\le -1$ .", "Let us now consider the blow-up $\\sigma :Z\\longrightarrow Y$ of $Y$ at $F$ with reduced structure, and let $D$ be the exceptional divisor on $Z$ .", "Consider the complex $\\sigma ^{*}A^{\\bullet }=\\cdots \\stackrel{\\sigma ^{*}a_{-1}}{\\longrightarrow }\\sigma ^{*}A^{0}\\stackrel{\\sigma ^{*}a_{0}}{\\longrightarrow }\\sigma ^{*}A^{1}\\stackrel{\\sigma ^{*}a_{1}}{\\longrightarrow }\\sigma ^{*}A^{2}\\longrightarrow 0.$ The degeneracy locus of $\\sigma ^{*}a_{0}$ and $\\sigma ^{*}a_{1}$ is then the exceptional divisor $D$ with reduced structure, while the $\\sigma ^{*}a_{i}$ 's do not degenerate on $Z$ for $i\\le -1$ .", "The maps $\\sigma ^{*}a_{0}$ and $\\sigma ^{*}a_{1}$ hence factor through $(A^{\\prime })^{0}\\stackrel{a^{\\prime }_{0}}{\\longrightarrow }\\sigma ^{*}A^{1}\\stackrel{a^{\\prime }_{1}}{\\longrightarrow }(A^{\\prime })^{2}$ where $\\sigma ^{*}A^{0}\\subseteq (A^{\\prime })^{0}$ , $(A^{\\prime })^{2}\\subseteq \\sigma ^{*}A^{2}$ , and the sheaves $M:=(A^{\\prime })^{0}/\\sigma ^{*}A^{0},\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,L:=\\sigma ^{*}A^{2}/(A^{\\prime })^{2}$ are supported on $D$ .", "As in the proof of Theorem 4.1 of [21], Step 4, the sheaves $L$ and $M$ are characterized by canonical isomorphisms $L\\otimes \\mathcal {O}_{D}\\simeq Ext^{2}(F,F)\\otimes \\mathcal {O}_{D},\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,Tor_{1}^{\\mathcal {O}_{D}}(M,\\mathcal {O}_{D})\\simeq Hom(F,F)\\otimes \\mathcal {O}_{D}.$ Here the computation is done in a neighborhood of the divisor $D$ .", "As in [21], it follows from this that ${E}xt^{0}_{p^{\\prime }}(\\sigma ^{*}q^{*}F,\\sigma ^{*}{F})\\simeq \\mathcal {O}_{D}(D),\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{E}xt^{2}_{p^{\\prime }}(\\sigma ^{*}q^{*}F,\\sigma ^{*}{F})\\simeq \\mathcal {O}_{D},$ viewed as $\\sigma ^{*}\\beta -$ twisted sheaves, and that ${E}xt^{1}_{p^{\\prime }}(\\sigma ^{*}q^{*}F,\\sigma ^{*}{F})$ is a locally free $\\sigma ^{*}\\beta -$ twisted sheaf of rank $d-2$ .", "It follows that $c^{B}_{d}(\\sigma ^{*}K^{\\bullet }_{F})=D^{d}=-1.$ But remark that $c^{B}_{d}(\\sigma ^{*}K^{\\bullet }_{F})=\\sigma ^{*}c^{B}_{d}(K^{\\bullet }_{F})=\\sigma ^{*}c^{B}(K^{\\bullet }_{G})=0,$ getting a contradiction.", "In conclusion the moduli space $M^{\\mu }_{\\alpha ,w}(S,\\omega )$ has to be connected.", "We can now prove the following result, which is the main result of this section, and which concludes the proof of part (1) of Theorem REF : Proposition 4.25 Let $S$ be a K3 surface, $w=(r,0,a)\\in H^{2*}(S,\\mathbb {Z})$ with $r\\ge 2$ , $\\alpha \\in Br(S)$ and $\\omega $ a $(\\alpha ,w)-$ generic polarization.", "Moreover, let $\\xi $ be a representative of $\\alpha $ in $H^{2}(S,\\mathbb {Z})$ which is prime with $r$ .", "Consider the relative moduli space of stable twisted sheaves $p:{M}\\longrightarrow \\mathbb {P}^{1}$ along the twistor family of $(S,\\omega )$ .", "There is a $\\overline{t}\\in \\mathbb {P}^{1}$ such that ${M}_{\\overline{t}}$ is an irreducible symplectic manifold which is deformation equivalent to a Hilbert scheme of points on a projective K3 surface $S$ .", "The moduli space $M^{\\mu }_{\\alpha ,w}(S,\\omega )$ is a compact, connected complex manifold which is simply connected and carries a holomorphic symplectic form.", "We let $\\pi :Z(S)\\longrightarrow \\mathbb {P}^{1}$ be the twistor family of $(S,\\omega )$ .", "By [16] there is a $\\overline{t}$ such that $S_{\\overline{t}}$ is a projective K3 surface.", "The polarization $\\omega _{\\overline{t}}$ is $(\\alpha _{\\overline{t}},w)-$ generic, and $w_{\\xi }=v(E_{\\xi })$ for some topological vector bundle $E_{\\xi }$ : such a topological vector bundle remains constant along $\\mathbb {P}^{1}$ , hence $w_{\\xi }=(r,\\xi ,b)$ where $r$ and $\\xi $ are prime to each other.", "It follows from Proposition REF that $M^{\\mu }_{\\alpha _{\\overline{t}},w}(S_{\\overline{t}},\\omega _{\\overline{t}})$ is an irreducible symplectic manifold which is deformation equivalent to a Hilbert scheme of points on $S_{\\overline{t}}$ .", "By Proposition REF , all the fibers are compact, connected manifolds, and by point (a) of Proposition REF the morphism $p$ is submersive.", "By the Proposition in section 1 of [7], it follows that $p$ is a smooth and proper morphism, hence it is a deformation of $M^{\\mu }_{\\alpha ,w}(S,\\omega )$ , and we are done." ], [ "Moduli spaces of locally free sheaves", "The previous results can be largely improved if we suppose something more on $M^{\\mu }_{v}(S,\\omega )$ , namely that it parametrizes only locally free sheaves.", "However this case has already been considered by differential geometers.", "We therefore only state the following result and refer the reader to [19] and [20] for the proof.", "Proposition 4.26 Let $S$ be a K3 surface, $v=(r,\\xi ,a)$ a Mukai vector such that $r$ and $\\xi $ are prime to each other, and $\\omega $ a $v-$ generic polarization.", "Then the open part ${M}^{lf}$ of the relative moduli space $p:{M}\\longrightarrow \\mathbb {P}^{1}$ along the twistor family of $(S,\\omega )$ , parameterizing locally free sheaves, is the twistor family of the moduli space $M^{\\mu -lf}_{v}(S,\\omega )$ of $\\omega $ -stable locally free sheaves with Mukai vector $v$ on $S$ .", "If morever $v^{2}=0$ , a standard argument shows that every sheaf in $M^{\\mu }_{v}(S,\\omega )$ is locally free (see Remark 6.1.9 of [15]), and thus the previous proposition applies to $M^{\\mu }_{v}(S,\\omega )$ which is moreover compact.", "The next proposition shows that compact moduli spaces of stable locally free sheaves as above may attain any even complex dimension.", "Proposition 4.27 Let $r$ be a positive integer, $d\\in [0, 2r-2]$ be an even integer and $g$ be an integer such that $g\\le -(r^2-1)(r-1)$ and $g$ congruent to $\\frac{d}{2}$ modulo $r$ .", "Then there exists a K3 surface $X$ with $NS(X)$ generated by one element $\\xi $ such that $\\xi ^2=2g-2$ and there exist torsion-free coherent sheaves $E$ on $X$ of rank $r$ , $c_1(E)=\\xi $ and such that $2r^2\\Delta (E)-2(r^2-1)=d$ .", "Moreover all such sheaves are locally free and irreducible.", "In particular they are stable with respect to any polarization on $X$ and their moduli space is a compact irreducible holomorphic symplectic manifold of dimension $d$ .", "The existence of K3 surfaces $X$ with cyclic Néron-Severi groups was proved in [24] whereas the existence of torsion-free sheaves $E$ with the above invariants follows from [23].", "We shall check that such sheaves are irreducible and locally free.", "Suppose $0\\rightarrow E_1\\rightarrow E\\rightarrow E_2\\rightarrow 0$ is an exact sequence with $E_i$ coherent sheaves without torsion on $X$ of ranks $r_i$ and with $c_1(E_i)=\\xi _i$ , $(i=1,2)$ .", "Then $\\xi _1+\\xi _2=\\xi $ , $r_1+r_2=r$ and we directly compute $\\Delta (E)=\\frac{1}{2r}(\\frac{\\xi ^2}{r}-\\frac{\\xi _1^2}{r_1}-\\frac{\\xi _2^2}{r_2})+\\frac{r_1}{r}\\Delta (E_1)+\\frac{r_2}{r}\\Delta (E_2).$ Since $g\\le 0$ , $X$ is non-algebraic hence $\\Delta (E_i)\\ge 0$ and thus $\\Delta (E)\\ge \\frac{1}{2r}\\left(\\frac{\\xi ^2}{r}-\\frac{\\xi _1^2}{r_1}-\\frac{\\xi _2^2}{r_2}\\right)=$ $-\\frac{1}{2r_1r_2}\\left(\\frac{r_2\\xi }{r}-\\xi _2\\right)^2\\ge -\\frac{\\xi ^2}{2r^2(r-1)}=\\frac{1-g}{(r-1)r^2}>\\frac{r^2-1}{r^2}=1-\\frac{1}{r^2}.$ But this implies $d>2r^2$ which contradicts our choice of $d$ .", "Hence $E$ is irreducible.", "If $E$ were not locally free an easy computation would imply that the discriminant of its double dual would be negative: a contradiction to the non-algebraicity of $X$ ." ], [ "The second integral cohomology", "We now study the second integral cohomology of $M^{\\mu }_{v}(S,\\omega )$ .", "We will show that it carries a non-degenerate quadratic form of signature $(3,20)$ , and that we have an isometry between $H^{2}(M^{\\mu }_{v},\\mathbb {Z})$ and $v^{\\perp }$ .", "If $M^{\\mu }_{v}(S,\\omega )$ is Kähler, it is even a Hodge isometry: as a consequence, we will show that the moduli space is projective if and only if $S$ is projective." ], [ "The quadratic form", "All along this section we will let $X:=M^{\\mu }_{v}(S,\\omega )$ for a choice of a K3 surface $S$ , a Mukai vector $v=(r,\\xi ,a)$ with $r$ and $\\xi $ prime to each other, and a $v-$ generic polarization $\\omega $ .", "We let $2n$ be its complex dimension.", "We start by defining a quadratic form on $H^{2}(X,\\mathbb {C})$ for every holomorphic symplectic form $\\sigma $ on $X$ , by using the same formula as for the Beauville form of an irreducible symplectic manifold: for every $\\alpha \\in H^{2}(X,\\mathbb {C})$ , we let $q_{\\sigma }(\\alpha ):=\\frac{n}{2} \\int _{X}\\alpha ^{2}\\wedge \\sigma ^{n-1}\\wedge \\overline{\\sigma }^{n-1}\\int _X\\sigma ^n\\wedge \\overline{\\sigma }^n+$ $(1-n)\\int _{X}\\alpha \\wedge \\sigma ^{n}\\wedge \\overline{\\sigma }^{n-1}\\int _{X}\\alpha \\wedge \\sigma ^{n-1}\\wedge \\overline{\\sigma }^{n}.$ Note that the symplectic form is always supposed to be closed so the above definition does not depend on representatives.", "Note also that $q_\\sigma (\\sigma +\\overline{\\sigma })= (\\int _X\\sigma ^n\\wedge \\overline{\\sigma }^n)^2\\ne 0$ so $q_\\sigma $ is non-trivial.", "Recall next the definition of the \"topological\" quadratic form $\\tilde{q}_X(\\alpha ):=c_n\\int _X\\alpha ^2\\sqrt{\\mbox{td}(X)}$ where $c_n$ is a constant depending only on $n$ chosen so that the form becomes integral on $H^2(X,\\mathbb {Z})$ (see [12], Definition 26.19 in Part III.", "Compact Hyperkähler Manifolds).", "It is known that $q_\\sigma $ and $\\tilde{q}_X$ are proportional when $X$ is moreover supposed to be Kähler.", "We finally define $\\tilde{H}^{2,0}:=\\mbox{Im}((\\lbrace \\tau \\in H^0(\\Omega ^2) \\ | \\ \\mbox{d}\\tau =0\\rbrace \\rightarrow H^2(X, )$ and $\\tilde{h}^{2,0}(X):=\\dim \\tilde{H}^{2,0}(X)$ .", "We first prove the following: Proposition 5.1 Let $p:X\\rightarrow C$ be a proper submersion of relative dimension $2n$ over a connected curve $C$ such that there exists a point $0\\in C$ with $X_0:=p^{-1}(0)$ irreducible holomorphic symplectic.", "Suppose moreover that there exists a relative non-degenerate symplectic form $\\sigma \\in H^0(X,\\Omega ^2_{X/C}\\otimes p^* L)$ with values in a line bundle $L$ over $C$ .", "Let $q_t:=q_{\\sigma _t}$ be the quadratic form defined by $\\sigma $ on $H^2(X_t,$ for each $t\\in C$ .", "Then for all $t\\in C$ the quadratic form $q_{t}$ is a positive multiple of $\\tilde{q}_{X_0}$ .", "In particular $q_t$ is non-degenerate of signature $(3,b_{2}(X)-3)$ and $\\tilde{h}^{2,0}(X_t)=1$ .", "We may suppose that $L$ is the trivial line bundle on $C$ .", "Indeed, for the general case it will suffice to take trivializations of $L$ over Zariski open subsets of $C$ containing 0.", "Fix some $\\alpha \\in H^2(X_t,$ and define for $t_1,t_2\\in C$ : $q_{t_1,t_2}(\\alpha ):=\\frac{n}{2}\\int _{X}\\alpha ^{2}\\wedge \\sigma ^{n-1}_{t_1}\\wedge \\overline{\\sigma }^{n-1}_{t_2}\\int _X\\sigma ^n_{t_1}\\wedge \\overline{\\sigma }^n_{t_2}+$ $+(1-n)\\int _{X}\\alpha \\wedge \\sigma ^{n}_{t_1}\\wedge \\overline{\\sigma }^{n-1}_{t_2}\\int _{X}\\alpha \\wedge \\sigma ^{n-1}_{t_1}\\wedge \\overline{\\sigma }^{n}_{t_2}.$ (Note again that the above formula does not depend on representatives since the syplectic forms $\\sigma _{t_1}$ , $\\sigma _{t_2}$ are closed.)", "This defines a complex function on $C\\times C$ which is holomorphic in $t_1$ and antiholomorphic in $t_2$ .", "It becomes holomorphic on $C\\times C^-$ , where $C^-$ denotes the curve $C$ with the opposite complex structure.", "Over an analytical open neighbourhood $U$ of 0 in $C$ all fibers $X_t$ are Kähler.", "Hence for $t\\in U$ the quadratic form $q_t$ is proportional to $\\tilde{q}$ .", "Take now $\\alpha , \\alpha ^{\\prime }\\in H^2(X_0,$ such that $ q_0(\\alpha )\\ne 0$ .", "Then the meromorphic function $(t_1,t_2)\\mapsto \\frac{q_{t_1,t_2}(\\alpha ^{\\prime })}{q_{t_1,t_2}(\\alpha )}$ on $C\\times C^-$ is constant on the diagonal $\\Delta _U\\subset U\\times U^-\\subset C\\times C^-$ .", "But $\\Delta _U$ is Zariski dense in $C\\times C^-$ .", "To see this consider the system of local holomorphic curves $C_t$ on $C\\times C^-$ given as images of the maps $z\\mapsto (t+z,t+\\bar{z})$ .", "Each curve $C_t$ passes through the reference point $(t,t)\\in \\Delta _U$ but its intersection with $\\Delta _U$ is a piece of a \"real line\".", "Hence by the principle of isolated zeroes any holomorphic function vanishing locally on $\\Delta _U$ will also vanish on the curves $C_t$ and thus also on the three dimensional real submanifold of $C\\times C^-$ they cover.", "Therefore the function $(t_1,t_2)\\mapsto \\frac{q_{t_1,t_2}(\\alpha ^{\\prime })}{q_{t_1,t_2}(\\alpha )}$ is constant on $C\\times C^-$ .", "From this it follows that $q_t$ is proportional to $\\tilde{q}$ for any $t\\in C$ .", "It remains to check that $\\tilde{h}^{2,0}(X_t)=1$ for all $t\\in C$ .", "For this we will show that the kernel $K$ of the linear map $\\lbrace \\tau \\in H^0(X_t,\\Omega ^2) \\ | \\ \\mbox{d}\\tau =0\\rbrace \\rightarrow H^0(X_t, K_{X_t}), \\ \\tau \\mapsto \\tau \\wedge \\sigma ^{n-1},$ consists of $\\mbox{d}$ -exact forms only.", "Let $b_t$ be the associated bilinear form to $q_t$ .", "Then for any $\\tau \\in K$ and $\\alpha \\in H^2(X_t,$ we have $b_t(\\tau ,\\alpha )=\\frac{n}{2}\\int _{X}\\tau \\wedge \\alpha \\wedge \\sigma _t^{n-1}\\wedge \\overline{\\sigma }_t^{n-1}\\int _X\\sigma _t^n\\wedge \\overline{\\sigma }_t^n+$ $+\\frac{1-n}{2}\\int _{X}\\tau \\wedge \\sigma ^{n}_t\\wedge \\overline{\\sigma }^{n-1}_t\\int _{X}\\alpha \\wedge \\sigma _t^{n-1}\\wedge \\overline{\\sigma }^{n}_t+$ $+\\frac{1-n}{2}\\int _{X}\\alpha \\wedge \\sigma ^{n}_t\\wedge \\overline{\\sigma }^{n-1}_t\\int _{X}\\tau \\wedge \\sigma _t^{n-1}\\wedge \\overline{\\sigma }^{n}_t=0$ and our assertion follows since $q_t$ is non-degenerate." ], [ "Isometry with $v^{\\perp }$", "We now show that there is an isometry between $H^{2}(M^{\\mu }_{v}(S,\\omega ),\\mathbb {Z})$ and $v^{\\perp }$ if $v^{2}>0$ , and with $v^{\\perp }/\\mathbb {Z}\\cdot v$ if $v^{2}=0$ .", "We introduce some notations.", "If $v\\in H^{2*}(S,\\mathbb {Z})$ , we let $v^{\\perp }$ be the orthogonal of $v$ with respect to the Mukai pairing.", "If $v=(r,\\xi ,a)$ and $\\xi \\in NS(S)$ , then the pure weight-two Hodge structure on $H^{2*}(S,\\mathbb {Z})$ induces a pure weight-two Hodge structure on $v^{\\perp }$ : namely, a class $\\alpha =(\\alpha _{0},\\alpha _{1},\\alpha _{2})\\in v^{\\perp }$ is of $(1,1)-$ type if and only if $\\alpha _{1}\\in NS(S)$ .", "If $\\alpha =(\\alpha _{0},\\alpha _{1},\\alpha _{2})\\in H^{2*}(S,\\mathbb {Q})$ , we write $\\alpha ^{\\vee }:=(\\alpha _{0},-\\alpha _{1},\\alpha _{2})$ .", "If $\\alpha =ch(F)$ for some locally free sheaf $F$ , then $\\alpha ^{\\vee }=ch(F^{\\vee })$ .", "It is immediate to see that if $\\alpha ,\\beta \\in H^{2*}(S,\\mathbb {Q})$ , then $(\\alpha \\cdot \\beta )^{\\vee }=\\alpha ^{\\vee }\\cdot \\beta ^{\\vee }$ .", "In particular, this implies that $(\\beta /\\alpha )^{\\vee }=\\beta ^{\\vee }/\\alpha ^{\\vee }$ and $(\\sqrt{\\alpha })^{\\vee }=\\sqrt{\\alpha ^{\\vee }}$ , whenever these expressions make sense.", "We now introduce a morphism associating to any class in $v^{\\perp }$ a rational cohomology class on the moduli space of stable (twisted) sheaves.", "The construction is inspired from the similar morphism which is used in the projective case (see [30], [42], [26], [31]).", "Let $\\alpha \\in Br(S)$ , $w\\in H^{2*}(S,\\mathbb {Q})$ a Mukai vector and $\\omega $ a $w-$ generic polarization.", "Suppose moreover that $M^{\\mu }_{\\alpha ,w}(S,\\omega )$ is compact, and let $p:M^{\\mu }_{\\alpha ,w}(S,\\omega )\\times S\\longrightarrow M^{\\mu }_{\\alpha ,w}(S,\\omega )$ and $q:M^{\\mu }_{\\alpha ,w}(S,\\omega )\\times S\\longrightarrow S$ be the projections.", "Choosing a quasi-universal family ${E}$ on $M^{\\mu }_{\\alpha ,w}(S,\\omega )\\times S$ of similitude $\\rho $ (which exists by Remark REF ), we define a morphism $\\lambda _{S,\\alpha ,w}:w^{\\perp }\\longrightarrow H^{2}(M^{\\mu }_{\\alpha ,w}(S,\\omega ),\\mathbb {Q})$ by letting $\\lambda _{S,\\alpha ,w}(\\beta ):=\\frac{1}{\\rho }[p_{*}(q^{*}(\\beta ^{\\vee }\\cdot \\sqrt{td(S)})\\cdot ch({E}))]_{1},$ where $[\\cdot ]_{1}$ is the part lying in $H^{2}(M^{\\mu }_{\\alpha ,w}(S,\\omega ),\\mathbb {Q})$ .", "As $\\beta \\in w^{\\perp }$ , the class $\\lambda _{S,\\alpha ,w}(\\beta )$ does not depend on the chosen quasi-universal family.", "If $\\alpha =0$ we simply write $\\lambda _{S,w}$ for $\\lambda _{S,0,w}$ .", "We now show the following, which is a generalization of known results in the projective case (see [29], [30], [42]): Proposition 5.2 Let $S$ be a K3 surface, $v=(r,\\xi ,a)\\in H^{2*}(S,\\mathbb {Z})$ where $r\\ge 2$ , $\\xi \\in NS(S)$ , $(r,\\xi )=1$ and $v^{2}\\ge 0$ .", "Moreover, let $\\omega $ be a $v-$ generic polarization.", "Then the image of $\\lambda _{S,v}$ is contained in $H^{2}(M^{\\mu }_{v}(S,\\omega ),\\mathbb {Z})$ , and if $v^{2}=0$ , then $\\lambda _{S,v}$ defines an isometry $\\overline{\\lambda }_{S,v}:v^{\\perp }/\\mathbb {Z}\\cdot v\\longrightarrow H^{2}(M^{\\mu }_{v}(S,\\omega ),\\mathbb {Z});$ if $v^{2}>0$ , then $\\lambda _{S,v}$ is an isometry.", "If $v^{2}>0$ , we just need to show the following properties: the image of $\\lambda _{S,v}$ is contained in $H^{2}(M^{\\mu }_{v}(S,\\omega ),\\mathbb {Z})$ ; the morphism $\\lambda _{S,v}$ is bijective; the morphism $\\lambda _{S,v}$ is an isometry.", "Let ${E}$ be a quasi-universal family of similitude $\\rho $ on $M^{\\mu }_{v}(S,\\omega )\\times S$ , and fix a locally free $\\mu _{\\omega }-$ stable vector bundle $F$ of Mukai vector $v$ .", "Let $w:=v_{F}(F)=(r,0,a-\\xi ^{2}/2r)$ and $f:M^{\\mu }_{v}(S,\\omega )\\longrightarrow M^{\\mu }_{0,w}(S,\\omega ),\\,\\,\\,\\,\\,\\,\\,\\,f({F}):={F}\\otimes F^{\\vee }$ which is an isomorphism (see Remark REF ).", "We let $q:M_{0,w}^{\\mu }(S,\\omega )\\times S\\longrightarrow S$ be the projection, and ${E}^{\\prime }:=(f\\times id_{S})_{*}{E}\\otimes q^{*}F^{\\vee },$ which is a quasi-universal family of similitude $\\rho $ on $M^{\\mu }_{0,w}(S,\\omega )\\times S$ .", "Moreover, as $f$ is an isomorphism, the morphism $f_{*}:H^{2}(M^{\\mu }_{v}(S,\\omega ),\\mathbb {Z})\\longrightarrow H^{2}(M^{\\mu }_{0,w}(S,\\omega ),\\mathbb {Z})$ is easily checked to be an isometry.", "Now, we let $h:H^{2*}(S,\\mathbb {Z})\\longrightarrow H^{2*}(S,\\mathbb {Q}),\\,\\,\\,\\,\\,\\,\\,h(\\beta ):=\\frac{\\beta \\cdot ch(F^{\\vee })}{\\sqrt{ch(F\\otimes F^{\\vee })}}.$ We let $(\\cdot ,\\cdot )_{S}$ be the Mukai pairing on $S$ and $[\\cdot ]_{2}$ the part lying in $H^{4}(S,\\mathbb {Q})$ .", "If $\\beta \\in v^{\\perp }$ we have $(h(\\beta ),w)_{S}=-\\bigg [\\frac{\\beta ^{\\vee }\\cdot ch(F)}{\\sqrt{ch(F\\otimes F^{\\vee })}}\\cdot v_{F}(F)\\bigg ]_{2}=$ $=-[\\beta ^{\\vee }\\cdot ch(F)\\cdot \\sqrt{td(S)}]_{2}=(\\beta ,v)_{S}=0,$ so that $h:v^{\\perp }\\longrightarrow w^{\\perp }.$ The same argument shows that it is an isometry.", "We even have $f_{*}(\\lambda _{S,v}(\\beta ))=\\lambda _{S,w}(h(\\beta ))$ .", "Indeed $f_{*}(\\lambda _{S,v}(\\beta ))=\\frac{1}{\\rho }[f_{*}p_{*}(q^{*}(\\beta ^{\\vee }\\sqrt{td(S)})\\mbox{ch}({E}))]_{1}=$ $=\\frac{1}{\\rho }[p_{*}((f\\times id_{S})_{*}q^{*}(\\beta ^{\\vee }\\sqrt{td(S)})ch({E}^{\\prime }))]_{1}=$ $=\\frac{1}{\\rho }[p_{*}(q^{*}(h(\\beta )^{\\vee }\\sqrt{td(S)})ch({E}^{\\prime }))]_{1}=\\lambda _{S,w}(h(\\beta )).$ In conclusion, we see that $\\lambda _{S,v}$ verifies the properties a), b) and c) above if and only if $\\lambda _{S,w}$ verifies them.", "Now, consider the twistor line of $(S,\\omega )$ and let $p:{M}\\longrightarrow \\mathbb {P}^{1}$ be the associated relative moduli space.", "As we can define $\\lambda _{S,v}$ in a relative way using relative quasi-universal families (which exist by Remark REF ), properties a), b) and c) above are verified on a fibre if and only if they are verified all along the twistor line.", "It follows that $\\lambda _{S,w}$ verifies a), b) and c) if and only $\\lambda _{S_{t},w_{t}}$ verifies them for some $t\\in \\mathbb {P}^{1}$ .", "As we saw before, there is $t$ such that $S_{t}$ is projective, and in this case $\\lambda _{S_{t},w_{t}}$ is an isometry by [43], hence we are done.", "If $v^{2}=0$ , the proof is similar: the only difference is about the fact that $\\mathbb {Z}\\cdot v$ is the kernel of $\\lambda _{S,v}$ , which holds in the general case as it holds over a projective K3 surface (see [29]).", "An immediate corollary of the previous Proposition is the following: Corollary 5.3 Let $S$ be a K3 surface, $v=(r,\\xi ,a)\\in H^{2*}(S,\\mathbb {Z})$ where $\\xi \\in NS(S)$ , $r\\ge 2$ , $(r,\\xi )=1$ and $v^{2}\\ge 0$ .", "If $\\omega $ is a $v-$ generic polarization and $M^{\\mu }_{v}(S,\\omega )$ is Kähler, then the morphism $\\lambda _{v}$ is a Hodge isometry.", "Theorem REF can now be seen as a corollary of the previous results: Corollary 5.4 Let $S$ be a K3 surface, $v=(r,\\xi ,a)\\in H^{2*}(S,\\mathbb {Z})$ where $\\xi \\in NS(S)$ , $r\\ge 2$ , $(r,\\xi )=1$ and $v^{2}\\ge 0$ .", "If $\\omega $ is a $v-$ generic polarization, then $M^{\\mu }_{v}(S,\\omega )$ is projective if and only if $S$ is projective.", "First, notice that if $S$ is projective, then $M^{\\mu }_{v}(S,\\omega )$ is projective by Theorem REF .", "Suppose now that $S$ is not projective, we want to prove that $M^{\\mu }_{v}(S,\\omega )$ is not projective as well.", "Suppose that $M^{\\mu }_{v}(S,\\omega )$ is projective: in particular this implies that it is Kähler, hence by part (1) of Theorem REF it follows that it is an irreducible symplectic manifold.", "Recall that an irreducible symplectic manifold $X$ is projective if and only if there is a line bundle $L$ on $X$ such that $q(L)>0$ , where $q$ is the Beauville form of $X$ (see [14]).", "Hence there is a line bundle $L$ on $M^{\\mu }_{v}(S,\\omega )$ such that $q(L)>0$ , where $q$ is the Beauville form on $M^{\\mu }_{v}(S,\\omega )$ , which coincides with the non-degenerate quadratic form we defined in the previous section.", "Moreover, by Corollary REF , as $M^{\\mu }_{v}(S,\\omega )$ is Kähler we have that $\\lambda _{v}$ is a Hodge isometry.", "There is then $\\alpha \\in v^{\\perp }$ of type $(1,1)$ (with respect to the Hodge structure on $v^{\\perp }$ ) such that $\\lambda _{v}(\\alpha )=c_{1}(L)$ , and $(\\alpha ,\\alpha )_{S}>0$ .", "Let us now describe $v^{\\perp }\\otimes \\mathbb {Q}$ .", "First, an element $(0,\\zeta ,b)\\in \\widetilde{H}(S,\\mathbb {Q})$ is in $v^{\\perp }\\otimes \\mathbb {Q}$ if and only if $b=\\zeta \\cdot \\xi $ .", "As $(0,\\zeta ,\\zeta \\cdot \\xi )=e^{\\xi /r}\\cdot (0,\\zeta ,0)$ , we have $e^{\\xi /r}\\cdot H^{2}(S,\\mathbb {Q})\\subseteq v^{\\perp }.$ It is easy to see that $e^{\\xi /r}\\cdot (2r^{2},0,v^{2})\\in v^{\\perp }\\otimes \\mathbb {Q}$ , hence $e^{\\xi /r}\\cdot \\mathbb {Q}(2r^{2},0,v^{2})\\subseteq v^{\\perp }\\otimes \\mathbb {Q}.$ This implies that $v^{\\perp }\\otimes \\mathbb {Q}=e^{\\xi /r}\\cdot (H^{2}(S,\\mathbb {Q})\\oplus \\mathbb {Q}(2r^{2},0,v^{2})),$ so that the $(1,1)-$ part $(v^{\\perp })^{1,1}$ of $v^{\\perp }\\otimes \\mathbb {Q}$ is $(v^{\\perp })^{1,1}=e^{\\xi /r}\\cdot (NS_{\\mathbb {Q}}(S)\\oplus \\mathbb {Q}(2r^{2},0,v^{2})),$ where $NS_{\\mathbb {Q}}(S):=NS(S)\\otimes \\mathbb {Q}$ .", "The direct sum is orthogonal with respect to the Mukai pairing, and it is easy to see that $(e^{\\xi /r}(2r^{2},0,v^{2}))^{2}=-4r^{2}v^{2}\\le 0,$ as $v^{2}\\ge 0$ .", "Moreover, as $S$ is non-projective the lattice $e^{\\xi /r}NS_{\\mathbb {Q}}(S)$ is negative semi-definite.", "It follows that $(v^{\\perp })^{1,1}$ is negative semi-definite, hence for every $\\alpha \\in (v^{\\perp })^{1,1}$ we have $(\\alpha ,\\alpha )_{S}\\le 0$ , which is not possible.", "In conclusion, if $S$ is not projective, the moduli space cannot be projective, and we are done.", "Institut Élie Cartan, UMR 7502, Université de Lorraine, CNRS, INRIA, Boulevard des Aiguillettes, B.P.", "70239, 54506 Vandoeuvre-lès-Nancy Cedex, France" ] ]
1403.0104
[ [ "Uniform Error Estimation for Convection-Diffusion Problems" ], [ "Abstract Let us consider the singularly perturbed model problem $Lu:=-\\varepsilon\\Delta u-bu_x+c u =f$ with homogeneous Dirichlet boundary conditions on $\\Gamma=\\partial\\Omega$ $u|_\\Gamma =0$ on the unit-square $\\Omega=(0,1)^2$.", "Assuming that $b>0$ is of order one, the small perturbation parameter $0<\\varepsilon\\ll 1$ causes boundary layers in the solution.", "In order to solve above problem numerically, it is beneficial to resolve these layers.", "On properly layer-adapted meshes we can apply finite element methods and observe convergence.", "We will consider standard Galerkin and stabilised FEM applied to above problem.", "Therein the polynomial order $p$ will be usually greater then two, i.e.", "we will consider higher-order methods.", "Most of the analysis presented here is done in the standard energy norm.", "Nevertheless, the question arises: Is this the right norm for this kind of problem, especially if characteristic layers occur?", "We will address this question by looking into a balanced norm.", "Finally, a-posteriori error analysis is an important tool to construct adapted meshes iteratively by solving discrete problems, estimating the error and adjusting the mesh accordingly.", "We will present estimates on the Green's function associated with $L$, that can be used to derive pointwise error estimators." ], [ "=7 =1 0.8cm 0.27cm 3 english Uniform Error Estimation for Convection-Diffusion Problems H A B I L I T A T I O N S S C H R I F T vorgelegt der Fakultät Mathematik und Naturwissenschaften der Technischen Universität Dresden von Dr. Sebastian Franz geboren am 29.12.1978 in Dresden Tag der Disputation  Eingereicht am : 31.05.2013 Tag der Disputation : 20.01.2014 Die Habilitationsschrift wurde in der Zeit von Juni 2012 bis Mai 2013 im Institut für Numerische Mathematik angefertigt.", "[n] Acknowledgement I would like to thank all my colleagues whom I had the pleasure to work with over the recent years.", "This includes especially the group of Prof. Hans-Görg Roos and Prof. Torsten Linß (now in Hagen) in Dresden, the Irish guys Dr. Natalia Kopteva and Prof. Martin Stynes, and Prof. Gunar Matthies in Kassel.", "Life is not only mathematics — although a good part of it is.", "I'm very grateful that Anja chose to follow me to Ireland and back.", "Thanks for staying at my side, keeping me down-to-earth and becoming my wife!", "Abstract Let us consider the singularly perturbed model problem $Lu:=-\\varepsilon \\Delta u-bu_x+c u & =f\\\\\\multicolumn{2}{l}{\\text{with homogeneous Dirichlet boundary conditions on $\\Gamma =\\partial \\Omega $}}\\\\u|_\\Gamma & =0$ on the unit-square $\\Omega =(0,1)^2$ .", "Assuming that $b>0$ is of order one, the small perturbation parameter $0<\\varepsilon \\ll 1$ causes boundary layers in the solution.", "In order to solve above problem numerically, it is beneficial to resolve these layers.", "On properly layer-adapted meshes we can apply finite element methods and observe convergence.", "We will consider standard Galerkin and stabilised FEM applied to above problem.", "Therein the polynomial order $p$ will be usually greater then two, i.e.", "we will consider higher-order methods.", "Most of the analysis presented here is done in the standard energy norm.", "Nevertheless, the question arises: Is this the right norm for this kind of problem, especially if characteristic layers occur?", "We will address this question by looking into a balanced norm.", "Finally, a-posteriori error analysis is an important tool to construct adapted meshes iteratively by solving discrete problems, estimating the error and adjusting the mesh accordingly.", "We will present estimates on the Green's function associated with $L$ , that can be used to derive pointwise error estimators.", "Simple model problems are often helpful in understanding the behaviour of numerical methods in presence of layers for more complicated problems.", "We will consider the singularly perturbed convection-diffusion problem with an exponential layer at the outflow boundary and two characteristic layers, given by $-\\varepsilon \\Delta u-b u_x+c u & =f &\\qquad & \\text{in} \\ \\Omega =(0,1)^2,\\\\u & =0 &\\qquad & \\text{on}\\ \\partial \\Omega .$ We assume $f\\in C(\\bar{\\Omega })$ , $b\\in W^1_\\infty (\\bar{\\Omega })$ and $c\\in L_\\infty (\\bar{\\Omega })$ .", "Furthermore, let $b(x,y)\\ge \\beta $ for $(x,y)\\in \\bar{\\Omega }$ with some positive constant $\\beta $ of order one, while $0<\\varepsilon \\ll 1$ is a small perturbation parameter.", "For further assumptions on $f$ see Remark REF .", "This combination gives rise to an exponential layer of width $\\mathcal {O}\\left(\\varepsilon |\\ln \\varepsilon |\\right)$ at $x=0$ and to two characteristic layers of width $\\mathcal {O}\\left(\\sqrt{\\varepsilon }|\\ln \\varepsilon |\\right)$ at $y=0$ and $y=1$ .", "Figure  REF Figure: Typical solution of () with two parabolic layersand an exponential layer.shows a typical example of a solution $u$ to (REF ).", "Under the condition $c+\\textstyle \\frac{1}{2} b_x\\ge \\gamma >0$ problem (REF ) possesses a unique solution in $H^1_0(\\Omega )\\cap H^2(\\Omega )$ .", "Note that (REF ) can always be satisfied by a transformation $\\tilde{u}(x,y) = u(x,y) e^{\\varrho x}$ with a suitably chosen constant $\\varrho $ .", "In our case $\\varrho $ with $\\varrho (b-\\varepsilon \\varrho )\\ge c+\\frac{1}{2}b_x+\\gamma $ suffices.", "When quasi uniform meshes are used, numerical methods do not give accurate approximations of (REF ) unless the mesh size is of the order of the perturbation parameter $\\varepsilon $ .", "On the one hand this constitutes a prohibitive restriction for a practical treatment of singularly perturbed problems.", "But on the other hand, the mesh sizes do only have to be small in the layer region.", "Therefore, layer-adapted meshes are often used to obtain efficient discretisations.", "Based on a priori knowledge of the layer behaviour, we apply a-priori adapted meshes.", "Early ideas on layer-adapted meshes can be found in [4], [48], [55], [61].", "We will use generalisations of Shishkin meshes, so called S-type meshes [52], [40], [41], that resolve the layers and yield robust (or uniform) convergence.", "In Figure REF the layer-resolving effect of Shishkin's idea can be seen clearly.", "We have condensed meshes near the characteristic boundaries ($y=1$ and $y=0$ , resp.)", "and the outflow boundary ($x=0$ ).", "Even on layer-adapted meshes the standard Galerkin method shows instabilities, see [42], [58].", "Therefore, stabilised discretisations have to be considered.", "The recent book by Roos, Stynes and Tobiska [54] gives an overview of many stabilisation ideas.", "We will apply and analyse two stabilisation techniques.", "The first one will be the streamline-diffusion finite element method (SDFEM), introduced by Hughes and Brooks [29].", "For problems with characteristic layers, the SDFEM with bilinear elements was analysed in [22].", "Here we will look into higher-order finite element methods.", "A disadvantage of the SDFEM accounts in particular for discretisations with higher-order elements.", "Several additional terms like second order derivatives have to be assembled in order to ensure Galerkin orthogonality of the resulting method.", "The second stabilisation technique does not fulfil the Galerkin orthogonality.", "It is the Local Projection Stabilisation method, proposed originally for the Stokes problem in [5].", "Although, the Galerkin orthogonality is not valid, the remainder can be bounded such that the optimal order of convergence is maintained.", "Again, we will look into higher-order methods.", "The main focus of our analysis will be the uniform convergence and supercloseness of the numerical methods with respect to $\\varepsilon $ .", "Most of it is done in the so-called energy norm $\\left|\\!\\!\\;\\left|\\!\\!\\;\\left| {v}\\right|\\!\\!\\;\\right|\\!\\!\\;\\right|_\\varepsilon := \\left(\\varepsilon \\Vert {\\nabla v}\\Vert _{0}^2 + \\gamma \\Vert {v}\\Vert _{0}^2\\right)^{1/2}.$ We denote by $\\Vert {\\cdot }\\Vert _{L_p(D)}$ the standard $L_p$ -norm over $D\\subset \\mathbb {R}^2$ .", "Whenever $p=2$ we write $\\Vert {\\cdot }\\Vert _{0,D}$ and if $D=\\Omega $ we skip the reference to the domain.", "Not all norms are equally adequate in measuring errors for problems with layers.", "Although the energy-norm is the associated norm to the weak formulation of (REF ), not all features of the solution are “seen”.", "Especially for small $\\varepsilon $ the characteristic layer term is less represented then the exponential one.", "Therefore,s we will also consider a balanced norm, where both types of layer are equally well represented.", "Another norm that is suitable in recognising the layer behaviour is the $L_\\infty $ -norm.", "We will not present a-priori results in the maximum-norm but an approach to uniform pointwise a-posteriori error estimation using the Green's function.", "This habilitation treatise is structured as follows.", "In Chapter REF the basics are given, i.e.", "a solution decomposition of $u$ is assumed, meshes, polynomial spaces and interpolation operators defined, and finally the numerical methods are given.", "In Chapter REF we present several analytical and numerical results on the convergence and supercloseness of the numerical methods in the energy and related norms.", "In Chapter REF we consider the question, whether a different norm then the energy norm could and should be used in the analysis.", "Finally, in Chapter REF we present $L_1$ -norm estimates of the Green's function associated with problems like (REF ).", "Moreover, they are applied in a first a-posteriori error-estimator for a simple finite difference method.", "Most of the results of the Chapters REF -REF are from already published work.", "Eight of the papers, whose content is contained in these chapters, are given in the appendix.", "Notation.", "Throughout this treatise, $C$ denotes a generic constant that is independent of both the perturbation parameter $\\varepsilon $ and the mesh parameter $N$ .", "The dependence of any constant on the polynomial order $p$ will not be elaborated.", "This chapter contains results from [23], [24], [16] that are also given in Appendix REF , REF and REF .", "Our uniform numerical analysis is based on a decomposition of the solution $u$ of (REF ).", "To be more precise: We suppose the existence of a decomposition of $u$ into a regular solution component and various layer parts.", "Assumption 2.1 The solution $u$ of problem (REF ) can be decomposed as $u =v+w_1+w_2+w_{12},$ where we have for all $x,y\\in [0,1]$ and $0\\le i+j\\le p+1$ the pointwise estimates $\\left.\\begin{aligned}\\left|\\frac{\\partial ^{i+j} v}{\\partial x^i \\partial y^j}(x,y)\\right|&\\le C,\\quad \\left|\\frac{\\partial ^{i+j} w_1}{\\partial x^i \\partial y^j}(x,y)\\right|\\le C\\varepsilon _{}^{-i}e_{}^{-\\beta x/\\varepsilon },\\\\[0.2cm]\\left|\\frac{\\partial ^{i+j} w_2}{\\partial x^i \\partial y^j}(x,y)\\right|&\\le C\\varepsilon _{}^{-j/2}\\left(e_{}^{-y/\\sqrt{\\varepsilon }}+e_{}^{-(1-y)/\\sqrt{\\varepsilon }} \\right), \\\\[0.2cm]\\left|\\frac{\\partial ^{i+j} w_{12}}{\\partial x^i \\partial y^j}(x,y)\\right|&\\le C\\varepsilon ^{-(i+j/2)} e^{-\\beta x/\\varepsilon }\\left(e^{-y/\\sqrt{\\varepsilon }}+e^{-(1-y)/\\sqrt{\\varepsilon }}\\right).\\end{aligned}\\right\\rbrace $ Here $w_1$ is the exponential boundary layer, $w_2$ covers the characteristic boundary layers, $w_{12}$ the corner layers, and $v$ is the regular part of the solution.", "Remark 2.2 The validity of Assumption REF is proved in [31], [32] for constant functions $b,\\, c$ under certain compatibility and smoothness conditions on the right-hand side $f$ .", "In [19] the Green's function associated with problem (REF ) was analysed.", "It was shown, that the Green's function $G$ in the variable-coefficient case and the Green's function $\\bar{G}$ in the constant coefficient case show a similar behaviour and the same estimates.", "As a Green's function can be used to represent the solution $u$ of its associated problem by $u(x,y)=\\iint _\\Omega G(x,y;\\xi ,\\eta )f(\\xi ,\\eta )d\\xi d\\eta ,$ it is reasonable to assume the validity of Assumption REF in the variable-coefficient case too.", "A discretisation of a singularly perturbed problem on an equidistant mesh results in oscillatory solutions unless the mesh-size is of order $\\varepsilon $ .", "A loophole lies in layer-adapted meshes that are fine in layer regions and coarse in regions, where the solution and its derivatives are uniformly bounded.", "Back in 1969 Bakhvalov [4] proposed one of the first layer-adapted meshes.", "Analysis on these kinds of graded meshes is somewhat difficult.", "The piecewise uniform Shishkin meshes [48] proposed in 1996 are easier to handle.", "The first analysis of finite element methods on Shishkin meshes was published in [55].", "For a detailed discussion of properties of Shishkin meshes and their uses see [54] and also [41] for a survey on layer-adapted meshes.", "Here we use a tensor-product mesh that is constructed by taking in both $x$ - and $y$ -direction so called S-type meshes [52] with $N$ mesh intervals in each direction.", "These meshes condense in the layer regions and are equidistant outside the layer region.", "The points, where the mesh-character changes, are called transition points.", "We define them by $\\lambda _x :=\\frac{\\sigma \\varepsilon }{\\beta }\\ln N \\le \\frac{1}{2}\\quad \\text{and}\\quad \\lambda _y :=\\sigma \\sqrt{\\varepsilon }\\ln N \\le \\frac{1}{4},$ with some user-chosen positive parameter $\\sigma >0$ .", "In (REF ) we assumed $\\varepsilon \\le \\min \\left\\lbrace \\frac{\\beta }{2\\sigma }(\\ln N)^{-1},\\frac{1}{16\\sigma ^2}(\\ln N)^{-2}\\right\\rbrace \\le C(\\ln N)^{-2}$ which is typically for (REF ) as otherwise $N$ would be exponentially large in $\\varepsilon $ .", "Using these transition points, the domain $\\Omega $ is divided into the subdomains $\\Omega _{11}$ , $\\Omega _{12}$ , $\\Omega _{21}$ and $\\Omega _{22}$ as shown in Fig.", "REF .", "Here $\\Omega _{12}$ covers the exponential layer, $\\Omega _{21}$ the characteristic layers, $\\Omega _{22}$ the corner layers and $\\Omega _{11}$ the remaining non-layer region.", "Figure: Decomposition of Ω\\Omega into subregions.By choosing the transition points $\\lambda _x$ and $\\lambda _y$ according to (REF ), the layer terms $w_1$ , $w_2$ , and $w_{12}$ of $u$ are of size $\\mathcal {O}\\left(N^{-\\sigma }\\right)$ on $\\Omega _{11}$ , i.e., $\\big |w_1(x,y)\\big | + \\big |w_2(x,y)\\big | + \\big |w_{12}(x,y)\\big |\\le C N^{-\\sigma } \\quad \\text{for} \\ (x,y)\\in \\Omega _{11}.$ The parameter $\\sigma $ is typically equal to the formal order of the numerical method or is chosen slightly larger to accommodate the error analysis.", "The precise definition of $\\sigma $ will be given later.", "The domain $\\Omega _{11}$ will be dissected uniformly while the dissection in the other subdomains depends on the mesh generating function $\\phi $ .", "This function is monotonically increasing and satisfies $\\phi (0)=0$ and $\\phi (1/2)=\\ln N$ .", "The precise definition of the tensor product mesh $T^N$ is given by the mesh points $x_i&:={\\left\\lbrace \\begin{array}{ll}\\frac{\\sigma \\varepsilon }{\\beta }\\phi \\left(\\frac{i}{N}\\right),&i=0,\\dots ,N/2,\\\\1-2(1-\\lambda _x)(1-\\frac{i}{N}),&i=N/2,\\dots ,N,\\end{array}\\right.", "}\\\\y_j&:={\\left\\lbrace \\begin{array}{ll}\\sigma \\sqrt{\\varepsilon }\\phi \\left(\\frac{2j}{N}\\right), &j=0,\\dots ,N/4,\\\\\\frac{1}{2}+(1-2\\lambda _y)(\\frac{2j}{N}-1), &j=N/4,\\dots ,3N/4,\\\\1-\\sigma \\sqrt{\\varepsilon }\\phi \\left(2-\\frac{2j}{N}\\right),&j=3N/4,\\dots ,N.\\end{array}\\right.", "}$ Now with an arbitrary function $\\phi $ fulfilling above conditions, an S-type mesh is defined.", "Figure: Layer-adapted mesh T 8 T^8 of Ω\\Omega .Fig.", "REF shows an example of such a mesh.", "Related to the mesh-generating function $\\phi $ , we define by $\\psi =\\mathrm {e}^{-\\phi }$ the mesh-characterising function $\\psi $ which is monotonically decreasing with $\\psi (0)=1$ and $\\psi (1/2)=N^{-1}$ .", "In Table REF some representatives of S-type meshes from [52] are given.", "The polynomial S-mesh has an additional parameter $m>0$ to adjust the grading inside the layer.", "Table: Some examples of mesh-generating and mesh-characterising functions ofS-type meshes.In order to provide sufficient properties for our convergence analysis, the meshes need to fulfil some additional assumptions.", "Assumption 2.3 Let the mesh-generating function $\\phi $ be piecewise differentiable such that $\\max _{t\\in [0,\\frac{1}{2}]} \\phi ^{\\prime }(t)\\le C N\\,\\text{ or equivalently }\\max _{t\\in [0,\\frac{1}{2}]} \\frac{|\\psi ^{\\prime }(t)|}{\\psi (t)}\\le C N$ is fulfilled.", "Moreover, let $\\phi $ fulfil $\\min _{i=0,\\dots ,N/2-1}\\left(\\phi \\left(\\frac{i+1}{N}\\right)-\\phi \\left(\\frac{i}{N}\\right)\\right)\\ge C N^{-1}.$ Finally we assume $\\max |\\psi ^{\\prime }|:=\\max _{t\\in [0,\\frac{1}{2}]}|\\psi ^{\\prime }(t)|\\le C \\left(\\frac{N}{\\ln N}\\right)^{1/2}.$ Remark 2.4 Note that (REF ) is satisfied for all meshes given in Table REF .", "Assumption (REF ) allows to bound the mesh width in the layer regions from below while applying an inverse inequality.", "This additional assumption restricts the use of S-type meshes from Table REF .", "For the original Shishkin mesh, we have $\\min _{i=0,\\dots ,N/2-1}\\left(\\phi \\left(\\frac{i+1}{N}\\right)-\\phi \\left(\\frac{i}{N}\\right)\\right)= C N^{-1}\\ln N\\ge C N^{-1}.$ The Bakhvalov S-mesh and its modification both fulfil $\\min _{i=0,\\dots ,N/2-1}\\left(\\phi \\left(\\frac{i+1}{N}\\right)-\\phi \\left(\\frac{i}{N}\\right)\\right)\\ge C N^{-1}.$ But the polynomial S-type mesh yields $\\min _{i=0,\\dots ,N/2-1}\\left(\\phi \\left(\\frac{i+1}{N}\\right)-\\phi \\left(\\frac{i}{N}\\right)\\right)\\ge C N^{-m}$ such that Assumption (REF ) fails for $m>1$ .", "The restriction (REF ) is fulfilled for all meshes of Table REF .", "Nevertheless, S-meshes fulfilling the other two assumptions such that (REF ) is violated are possible, see [23].", "The quantity $1+(N^{-1}\\ln N)^{1/2}\\max |\\psi ^{\\prime }|$ arises in the convergence analysis of the Galerkin FEM, see [23], and can be bounded by a constant $C$ with the help of (REF ).", "Using (REF ) we bound the mesh width inside the layers from above.", "Let $h_i:=x_i-x_{i-1}$ and $t_i=i/N$ .", "Then, it holds for $i=1,\\dots ,N/2$ and $t\\in [t_{i-1},t_i]$ (with $\\max \\phi ^{\\prime }$ taken over $t\\in [t_{i-1},t_i]$ ) $\\begin{aligned}\\psi (t_i) = \\mathrm {e}^{-\\phi (t_i)}= \\mathrm {e}^{-(\\phi (t_i)-\\phi (t))}\\mathrm {e}^{-\\phi (t)}& \\ge \\mathrm {e}^{-(\\phi (t_i)-\\phi (t_{i-1}))}\\psi (t)\\\\& \\ge \\mathrm {e}^{-N^{-1}\\max \\phi ^{\\prime }}\\psi (t)\\ge C\\psi (t)\\end{aligned}$ where we used (REF ) for the last estimate.", "Furthermore, we have $x=\\frac{\\sigma \\varepsilon }{\\beta }\\phi (t)=-\\frac{\\sigma \\varepsilon }{\\beta }\\ln \\psi (t)\\quad \\mbox{which gives}\\quad \\psi (t)=\\mathrm {e}^{-\\beta x/(\\sigma \\varepsilon )}.$ Using this, the monotonicity of $\\psi $ , and (REF ), we obtain for $i=1,\\ldots ,N/2$ and $x\\in [x_{i-1},x_i]$ $h_i & = \\frac{\\sigma \\varepsilon }{\\beta }(\\phi (t_i)-\\phi (t_{i-1}))\\le \\frac{\\sigma }{\\beta } \\varepsilon N^{-1}\\max _{t\\in [t_{i-1},t_i]}\\phi ^{\\prime }(t)\\le \\frac{\\sigma }{\\beta } \\varepsilon N^{-1}\\left(\\max _{t\\in [t_{i-1},t_i]}|\\psi ^{\\prime }(t)|\\right)/\\psi (t_i)\\\\& \\le C \\varepsilon N^{-1}\\left(\\max _{t\\in [t_{i-1},t_i]}|\\psi ^{\\prime }(t)|\\right)/\\psi (t)\\le C \\varepsilon N^{-1}\\max |\\psi ^{\\prime }|\\mathrm {e}^{\\beta x/(\\sigma \\varepsilon )}$ where again $\\max |\\psi ^{\\prime }|:=\\max \\limits _{t\\in [0,1/2]}|\\psi ^{\\prime }(t)|$ .", "Similarly, we get for $j=1,\\dots ,N/4$ and $j=3N/4+1,\\dots ,N$ $k_j:=y_j-y_{j-1}&\\le C \\varepsilon ^{1/2}N^{-1}\\max |\\psi ^{\\prime }|{\\left\\lbrace \\begin{array}{ll}\\mathrm {e}^{y/(\\sigma \\varepsilon ^{1/2})}, & j\\le N/4,\\\\\\mathrm {e}^{(1-y)/(\\sigma \\varepsilon ^{1/2})},& j>3N/4,\\end{array}\\right.", "}$ with $y\\in [y_{j-1},y_j]$ .", "Of course, the simpler bounds $h_i & \\le C \\varepsilon N^{-1}\\max \\phi ^{\\prime } && \\le C \\varepsilon ,&\\qquad & i=1,\\ldots ,N/2,\\\\k_j & \\le C \\varepsilon ^{1/2}N^{-1}\\max \\phi ^{\\prime } && \\le C \\varepsilon ^{1/2},&& j=1,\\ldots ,N/4, \\; 3N/4+1,\\ldots ,N,$ follow also from (REF ).", "For the maximal mesh sizes inside the layer regions $h:=\\max _{i=1,\\dots ,N/2} h_i\\quad \\mbox{and}\\quad k:=\\max _{j=1,\\dots ,N/4} k_j$ we assume for simplicity of the presentation $h\\le k\\le N^{-1}\\max |\\psi ^{\\prime }|$ which represents for some meshes a restriction on $\\varepsilon $ .", "With this assumption convergence results like $\\mathcal {O}\\left(h+k+N^{-1}\\max |\\psi ^{\\prime }|\\right)$ become $\\mathcal {O}\\left(N^{-1}\\max |\\psi ^{\\prime }|\\right)$ .", "We denote by $\\tau _{ij}=[x_{i-1},x_i]\\times [y_{j-1},y_j]$ a specific element and by $\\tau $ a generic mesh rectangle.", "Note that the mesh cells are assumed to be closed.", "Having a discretisation of the domain $\\Omega $ , let us come to discretising the infinite-dimensional function space $H_0^1(\\Omega )$ by higher-order, finite-dimensional polynomial spaces.", "Let our discrete space be given by $V^N:=\\Big \\lbrace v\\in H_0^1(\\Omega ):v|_\\tau \\in \\mathcal {E}(\\tau )\\;\\forall \\tau \\in T^N\\Big \\rbrace $ with an yet unspecified local finite element space $\\mathcal {E}(\\tau )$ .", "Let $\\hat{\\tau }=[-1,1]^2$ denote the reference element.", "We will look at two different polynomial spaces, the full $\\mathcal {Q}_p$ -space given locally by $\\mathcal {Q}_p(\\hat{\\tau })=\\text{span}\\Big \\lbrace \\lbrace 1,\\xi ,\\dots ,\\xi ^p\\rbrace \\times \\lbrace 1,\\eta ,\\dots ,\\eta ^p\\rbrace \\Big \\rbrace ,$ and the Serendipity-space $\\mathcal {Q}_p^\\oplus $ defined locally by enriching the polynomial space $\\mathcal {P}_p$ with two edge-bubble functions: $\\mathcal {Q}_p^{\\oplus }(\\hat{\\tau })=\\mathcal {P}_p(\\hat{\\tau })&\\oplus \\text{span}\\Big \\lbrace &&(1+\\xi )(1-\\eta ^2)\\eta ^{p-2},(1+\\eta )(1-\\xi ^2)\\xi ^{p-2}\\Big \\rbrace .$ This polynomial space is also known as “trunk element” [59], [47], [3], [2].", "It is the continuous quadrilateral element with the fewest degrees of freedom containing $\\mathcal {P}_p$ .", "Both spaces can be written in the general form $\\mathcal {Q}_p^\\clubsuit (\\hat{\\tau })=\\text{span}\\bigg \\lbrace \\lbrace 1,\\xi \\rbrace \\times \\lbrace 1,\\eta ,\\dots ,\\eta ^p\\rbrace \\cup \\lbrace 1,\\xi ,\\dots ,\\xi ^p\\rbrace \\times \\lbrace 1,\\eta \\rbrace \\cup \\xi ^2\\eta ^2 \\widetilde{\\mathcal {Q}}(\\hat{\\tau })\\bigg \\rbrace $ with $\\widetilde{\\mathcal {Q}}(\\hat{\\tau })=\\mathcal {Q}_{p-2}(\\hat{\\tau })$ for the full space and $\\widetilde{\\mathcal {Q}}(\\hat{\\tau })={\\left\\lbrace \\begin{array}{ll}\\emptyset ,&\\text{ for }{p=2,\\,3},\\\\\\mathcal {P}_{p-4}(\\hat{\\tau }),&\\text{ for }{p\\ge 4}\\end{array}\\right.", "}$ for the Serendipity space.", "Note that in both cases we find $s_0\\ge s_1\\ge \\dots \\ge s_{p-2}$ , such that $\\widetilde{\\mathcal {Q}}(\\hat{\\tau }) = \\text{span} \\Big \\lbrace \\xi ^i\\eta ^j\\::\\:i=0,\\dots ,p-2,\\:j=0,\\dots ,s_i\\Big \\rbrace .$ Therein the $s_i$ can also be negative.", "Figure  REF Figure: Full space 𝒬 p (τ ^)\\mathcal {Q}_p(\\hat{\\tau }) (left) andSerendipity space 𝒬 p ⊕ (τ ^)\\mathcal {Q}_p^{\\oplus }(\\hat{\\tau }) (right) for p=9p=9.gives a graphical representation of the two spaces.", "Therein a square at position $(i,j)$ stands for a basis function $\\xi ^i\\eta ^j$ of $\\mathcal {Q}_p^\\clubsuit (\\hat{\\tau })$ .", "The darker squares correspond to those functions present in both spaces, while the lighter ones represent $\\xi ^2 \\eta ^2\\widetilde{\\mathcal {Q}}(\\hat{\\tau })$ .", "Note that it holds $\\mathcal {P}_p\\subset \\mathcal {Q}_p^\\oplus \\subset \\mathcal {Q}_p$ and that $\\mathcal {Q}_p^\\oplus $ uses only about half the number of degrees of freedom that $\\mathcal {Q}_p$ uses.", "Now let us come to defining interpolation operators for these two spaces.", "We will consider two types of interpolation: vertex-edge-cell interpolation and Lagrange interpolation.", "The first interpolation operator is based on point evaluation at the vertices, line integrals along the edges and integrals over the cell interior, see [27], [38].", "Let $\\hat{a}_i$ and $\\hat{e}_i$ , $i=1,\\ldots ,4$ , denote the vertices and edges of $\\hat{\\tau }$ , respectively.", "We define the vertex-edge-cell interpolation operator $\\hat{\\pi }:C(\\hat{\\tau })\\rightarrow \\mathcal {Q}_p^\\clubsuit (\\hat{\\tau })$ by $\\hat{\\pi } \\hat{v}(\\hat{a}_i)&=\\hat{v}(\\hat{a}_i),\\,\\quad i=1,\\dots ,4,&&\\\\\\int _{\\hat{e}_i}(\\hat{\\pi }\\hat{v})\\hat{q} &= \\int _{\\hat{e}_i} \\hat{v} \\hat{q},\\quad i=1,\\dots ,4,\\quad &&\\hat{q}\\in \\mathcal {P}_{p-2}(\\hat{e}_i),\\\\\\iint _{\\hat{\\tau }} (\\hat{\\pi }\\hat{v})\\hat{q} &= \\iint _{\\hat{\\tau }} \\hat{v} \\hat{q},&&\\hat{q}\\in \\widetilde{\\mathcal {Q}}(\\hat{\\tau }).$ This operator is uniquely defined and can be extended to the globally defined interpolation operator $\\pi ^N:C(\\overline{\\Omega })\\rightarrow V^N$ by $(\\pi ^N v)|_\\tau := \\big (\\hat{\\pi }(v\\circ F_\\tau )\\big )\\circ F_\\tau ^{-1}\\quad \\forall \\tau \\in T^N,\\,v\\in C(\\overline{\\Omega }),$ with the bijective reference mapping $F_\\tau :\\hat{\\tau }\\rightarrow \\tau $ .", "Lemma 2.5 For the interpolation operator $\\pi ^N:C(\\overline{\\Omega })\\rightarrow V^N$ the stability property $\\left\\Vert {\\pi ^N w} \\right\\Vert _{L_\\infty (\\tau )}\\le C\\Vert {w}\\Vert _{L_\\infty (\\tau )}\\quad \\forall w\\in C(\\tau ),\\,\\forall \\tau \\subset \\overline{\\Omega },$ holds and we have the anisotropic error estimates $\\left\\Vert {w-\\pi ^N w} \\right\\Vert _{L_q(\\tau _{ij})}&\\le C \\sum _{r=0}^s\\left\\Vert {h_i^{s-r}k_j^r\\frac{\\partial ^{s}w}{\\partial x^{s-r}\\partial y^{r}}} \\right\\Vert _{L_q(\\tau _{ij})},\\\\\\left\\Vert {(w-\\pi ^N w)_x} \\right\\Vert _{L_q(\\tau _{ij})}&\\le C \\sum _{r=0}^t\\left\\Vert {h_i^{t-r}k_j^r\\frac{\\partial ^{t+1}w}{\\partial x^{t-r+1}\\partial y^{r}}} \\right\\Vert _{L_q(\\tau _{ij})},\\\\\\left\\Vert {(w-\\pi ^N w)_y} \\right\\Vert _{L_q(\\tau _{ij})}&\\le C \\sum _{r=0}^t\\left\\Vert {h_i^{t-r}k_j^r\\frac{\\partial ^{t+1}w}{\\partial x^{t-r}\\partial y^{r+1}}} \\right\\Vert _{L_q(\\tau _{ij})}$ for $\\tau _{ij}\\subset \\overline{\\Omega }$ and $q\\in [1,\\infty ]$ , $2\\le s\\le p+1$ , $1\\le t\\le p$ .", "The proof for arbitrary $\\mathcal {Q}^\\clubsuit _p$ can be found in [24] and for the full space $\\mathcal {Q}_p$ also in e.g.", "[57].", "The second interpolation type we consider is the Lagrange type, i.e.", "it uses only point-value information.", "Let $-1=\\xi _0<\\xi _1<\\dots <\\xi _{p-1}<\\xi _p=+1$ and $-1=\\eta _0<\\eta _1<\\dots <\\eta _{p-1}<\\eta _p=+1$ be two increasing sequences of $p+1$ points of $[-1,+1]$ which include both end points.", "We define the Lagrange-type interpolation operator $\\hat{J}:C(\\hat{\\tau })\\rightarrow \\mathcal {Q}_p^\\clubsuit (\\hat{\\tau })$ by values at the vertices $\\begin{alignedat}{2}(\\hat{J}\\hat{v})(\\pm 1,-1) & := \\hat{v}(\\pm 1,-1),&\\qquad (\\hat{J}\\hat{v})(\\pm 1,+1) & := \\hat{v}(\\pm 1,+1)\\end{alignedat}\\\\\\multicolumn{2}{l}{\\text{values on the edges}}\\\\\\left.\\begin{alignedat}{2}(\\hat{J}\\hat{v})(\\xi _i,\\pm 1) & := \\hat{v}(\\xi _i,\\pm 1),&\\qquad & i=1,\\dots ,p-1,\\\\(\\hat{J}\\hat{v})(\\pm 1,\\eta _j) & := \\hat{v}(\\pm 1,\\eta _j),&\\qquad & j=1,\\dots ,p-1,\\\\\\end{alignedat}\\qquad \\right\\rbrace &\\\\\\multicolumn{2}{l}{\\text{and values in the interior}}\\\\\\begin{alignedat}{2}(\\hat{J}\\hat{v})(\\xi _{i+1},\\eta _{j+1})&:= \\hat{v}(\\xi _{i+1},\\eta _{j+1}),&\\qquad &i=0,\\dots ,p-2, j=0,\\dots ,s_i,\\end{alignedat}$ where the $s_i$ are those given in (REF ).", "In [24] it is shown that this operator is uniquely defined.", "What is left to specify are the sequences $\\lbrace \\xi _i\\rbrace $ and $\\lbrace \\eta _j\\rbrace $ .", "Here we consider two choices: 1) equidistant distribution: We define the operator $J^N:C(\\overline{\\Omega })\\rightarrow V^N$ by $(J^N v)|_\\tau := \\big (\\hat{J}(v\\circ F_\\tau )\\big )\\circ F_\\tau ^{-1}\\quad \\forall \\tau \\in T^N,\\,v\\in C(\\overline{\\Omega }),$ with the bijective reference mapping $F_\\tau :\\hat{\\tau }\\rightarrow \\tau $ and the local sequences $\\xi _i=\\eta _i=-1+2i/p,\\,i=0,\\dots ,p.$ 2) distribution according to the Gauß-Lobatto quadrature rule: Let $-1=t_0<t_1<\\dots <t_p=1$ , be the zeros of $(1-t^2)L_{p}^{\\prime }(t)=0,\\quad t\\in [-1,1],$ where $L_p$ is the Legendre polynomial of degree $p$ .", "These points are also used in the Gauß-Lobatto quadrature rule of approximation order $2p-1$ .", "Therefore, we refer to them as Gauß-Lobatto points.", "In literature they are also named Jacobi points [37] as they are also the zeros of the orthogonal Jacobi-polynomials $P_p^{(1,1)}$ of order $p$ .", "Now we define the operator $I^N:C(\\overline{\\Omega })\\rightarrow V^N$ by $(I^N v)|_\\tau := \\big (\\hat{J}(v\\circ F_\\tau )\\big )\\circ F_\\tau ^{-1}\\quad \\forall \\tau \\in T^N,\\,v\\in C(\\overline{\\Omega }),$ with the bijective reference mapping $F_\\tau :\\hat{\\tau }\\rightarrow \\tau $ and the local sequences $\\xi _i=\\eta _i=t_i,\\,i=0,\\dots ,p.$ Lemma 2.6 The interpolation operators $J^N:C(\\overline{\\Omega })\\rightarrow V^N$ and $I^N:C(\\overline{\\Omega })\\rightarrow V^N$ yield the stability property $\\left\\Vert {J^N w} \\right\\Vert _{L_\\infty (\\tau )}+\\left\\Vert {I^N w} \\right\\Vert _{L_\\infty (\\tau )}\\le C\\Vert {w}\\Vert _{L_\\infty (\\tau )}\\quad \\forall w\\in C(\\tau ),\\,\\forall \\tau \\subset \\overline{\\Omega },\\\\$ and we have the anisotropic error estimates $\\left\\Vert {w-J^N w} \\right\\Vert _{L_q(\\tau _{ij})}+\\left\\Vert {w-I^N w} \\right\\Vert _{L_q(\\tau _{ij})}&\\le C \\sum _{r=0}^s\\left\\Vert {h_i^{s-r}k_j^r\\frac{\\partial ^{s}w}{\\partial x^{s-r}\\partial y^{r}}} \\right\\Vert _{L_q(\\tau _{ij})},\\\\\\left\\Vert {(w-J^N w)_x} \\right\\Vert _{L_q(\\tau _{ij})}+\\left\\Vert {(w-I^N w)_x} \\right\\Vert _{L_q(\\tau _{ij})}&\\le C \\sum _{r=0}^t\\left\\Vert {h_i^{t-r}k_j^r\\frac{\\partial ^{t+1}w}{\\partial x^{t-r+1}\\partial y^{r}}} \\right\\Vert _{L_q(\\tau _{ij})},\\\\\\left\\Vert {(w-J^N w)_y} \\right\\Vert _{L_q(\\tau _{ij})}+\\left\\Vert {(w-I^N w)_y} \\right\\Vert _{L_q(\\tau _{ij})}&\\le C \\sum _{r=0}^t\\left\\Vert {h_i^{t-r}k_j^r\\frac{\\partial ^{t+1}w}{\\partial x^{t-r}\\partial y^{r+1}}} \\right\\Vert _{L_q(\\tau _{ij})}$ for $\\tau _{ij}\\subset \\overline{\\Omega }$ and $q\\in [1,\\infty ]$ , $2\\le s\\le p+1$ , $1\\le t\\le p$ .", "The proof for arbitrary $\\mathcal {Q}^\\clubsuit _p$ can be found in [24] and for the full space $\\mathcal {Q}_p$ also in e.g.", "[1].", "There is a strong connection between $\\pi ^N$ and $I^N$ in the case of $\\mathcal {Q}_p$ -elements.", "Let us spend a subscript for the polynomial order $p$ , i.e.", "we write $\\pi _p^N$ and $I_p^N$ for the interpolation operators mapping into $V^N$ with local polynomial spaces $\\mathcal {Q}_p$ .", "Then it holds the identity $\\pi _p^N=I_p^N\\pi _{p+1}^N,$ see [16].", "A direct consequence is the additional identity $\\pi _p^Nv=I_p^Nv+(\\pi ^N_{p+1}v-v)+\\left(I^N_p(\\pi ^N_{p+1}v-v)-(\\pi ^N_{p+1}v-v)\\right)$ for arbitrary $v\\in C(\\bar{\\Omega })$ .", "It shows the distance between both interpolation operators to be proportional to terms of order $p+1$ .", "The identity (REF ) (with the properly redefinition of the interpolation operators therein) does also hold for the Serendipity spaces $\\mathcal {Q}_2^\\oplus $ and $\\mathcal {Q}_3^\\oplus $ , but not for $\\mathcal {Q}_p^\\oplus $ with $p\\ge 4$ .", "This can be shown analogously to the proof of [16].", "The reason for the failed identity lies in the definition of the interior degrees of freedom () and (REF ).", "For $\\mathcal {Q}_2^\\oplus $ and $\\mathcal {Q}_3^\\oplus $ these conditions are not existent and therefore always fulfilled, while for higher order $p$ they do not match any more.", "Let us come to the numerical methods that we will consider in the next chapter.", "The first method will be the unstabilised Galerkin FEM given by: Find $u_{Gal}^N\\in V^N$ such that $a_{Gal}(u_{Gal}^N, v^N) = (f,v^N)\\qquad \\forall v^N\\in V^N.$ This problem possesses a unique solution due to (REF ).", "Furthermore, the Galerkin orthogonality $ a_{Gal}(u - u_{Gal}^N, v^N) = 0\\qquad \\forall v^N\\in V^N$ holds true and we have coercivity $a_{Gal}(v,v)\\ge \\left|\\!\\!\\;\\left|\\!\\!\\;\\left| {v}\\right|\\!\\!\\;\\right|\\!\\!\\;\\right|_\\varepsilon ^2,\\qquad v\\in H^1_0(\\Omega )$ where the energy norm is defined by (REF ) $\\left|\\!\\!\\;\\left|\\!\\!\\;\\left| {v}\\right|\\!\\!\\;\\right|\\!\\!\\;\\right|_\\varepsilon := \\left(\\varepsilon \\Vert {\\nabla v}\\Vert _{0}^2 + \\gamma \\Vert {v}\\Vert _{0}^2\\right)^{1/2}.$ Since the standard Galerkin discretisation lacks stability even on S-type meshes, see the numerical results given in [43], [58], we will also consider stabilised methods.", "A survey of several different stabilised method for singularly perturbed problems can be found in the book [54].", "In 1979 Hughes and Brooks [29] introduced the streamline-diffusion finite element method (SDFEM), sometimes also called streamline upwind Petrov Galerkin finite element method (SUPG-FEM).", "This method provides highly accurate solutions outside the layers and good stability properties.", "Its basic idea is to add weighted local residuals to the variational formulation, i.e.", "to add $\\delta _\\tau (Lu-f,-bw_x)_\\tau =0$ where the constant parameters $\\delta _\\tau =\\delta _{ij}\\ge 0$ for $\\tau \\subset \\Omega _{ij}$ are user chosen and influence both stability and convergence.", "A slightly different approach will be used in Chapter REF .", "Defining $a_{stabSD}(v,w):=\\sum _{\\tau \\in T^N}\\delta _\\tau (\\varepsilon \\Delta v+bv_x-c v,bw_x)_\\tau ,\\qquad \\mbox{for all }v,\\,w\\in H^1_0(\\Omega )$ and $f_{SD}(w):=(f,w)-\\sum _{\\tau \\in T^N}\\delta _\\tau (f,bw_x)_\\tau ,\\quad \\mbox{for all }w\\in H_0^1(\\Omega )$ we obtain the streamline diffusion formulation of (REF ) by: Find $u_{SD}^N\\in V^N$ such that $a_{SD}(u_{SD}^N,v^N):=a_{Gal}(u_{SD}^N, v^N)+a_{stabSD}(u_{SD}^N, v^N) = f_{SD}(v^N),\\quad \\mbox{for all }v^N\\in V^N.$ Associated with this method is the streamline diffusion norm, defined by $\\left|\\!\\!\\;\\left|\\!\\!\\;\\left| {v}\\right|\\!\\!\\;\\right|\\!\\!\\;\\right|_{SD}:=\\left(\\varepsilon \\Vert {\\nabla v}\\Vert _{0}^2+\\gamma \\Vert {v}\\Vert _{0}^2+\\sum _{\\tau \\in T^N}\\delta _\\tau \\Vert {bv_x}\\Vert _{0,\\tau }^2\\right)^{1/2}.$ We have Galerkin orthogonality, and for $0\\le \\delta _\\tau \\le \\frac{1}{2}\\min \\left\\lbrace \\frac{\\gamma }{\\Vert {c}\\Vert _{L_\\infty (\\tau )}^2},\\frac{h_\\tau ^2}{\\mu ^2\\varepsilon }\\right\\rbrace ,$ where $\\mu \\ge 0$ is a fixed constant such that the inverse inequality $\\Vert {\\Delta v^N}\\Vert _{0,\\tau }\\le \\mu h_\\tau ^{-1}\\Vert {\\nabla v^N}\\Vert _{0,\\tau },\\qquad \\forall v^N\\in V^N,\\,\\tau \\in T^N$ holds with $h_{\\tau _{ij}}:=\\min \\lbrace h_i,k_j\\rbrace $ , we have coercivity $a_{SD}(v,v)\\ge \\frac{1}{2}\\left|\\!\\!\\;\\left|\\!\\!\\;\\left| {v}\\right|\\!\\!\\;\\right|\\!\\!\\;\\right|_{SD}^2,\\qquad v\\in H^1_0(\\Omega ).$ A disadvantage of the SDFEM are several additional terms including second order derivatives that have to be assembled in order to ensure the Galerkin orthogonality of the resulting method.", "Moreover, for systems of differential equations additional coupling between different species occurs.", "An alternative stabilisation technique overcoming some drawbacks of the SDFEM is the Local Projection Stabilisation method LPSFEM.", "Instead of adding weighted residuals, only weighted fluctuations $(id-\\pi )$ of the streamline derivatives are added.", "Therein $\\pi $ denotes a projection into a discontinuous finite element space.", "Originally the method was introduced for Stokes and transport problems [5], [6], but also applied to the Oseen problem in [7], [46].", "In its original definition, the local projection method was proposed as a two-level method, where the projection space is defined on a coarser mesh consisting of patches of elements [5], [6], [7].", "In this case, standard finite element spaces can be used for both the approximation space and the projection space.", "Based on the existence of a special interpolation operator [46], the one level approach using enriched spaces was constructed.", "It was shown in [46] that it suffices to enrich the standard $\\mathcal {Q}_p$ -element, $p\\ge 2$ , in 2d by just two additional bubble functions of higher order.", "For its application on layer-adapted meshes for problems with exponential boundary layers see [45], [44].", "Here we will use the one level approach without enriching the polynomial spaces.", "Let $\\pi _{\\tau }$ denote the $L_2$ -projection into the finite dimensional function space $D(\\tau )=\\mathcal {P}_{p-2}(\\tau )$ .", "The fluctuation operator $\\kappa _{\\tau }:L_2(\\tau )\\rightarrow L_2(\\tau )$ is defined by $\\kappa _{\\tau } v:= v - \\pi _{\\tau }v$ .", "In order to get additional control on the derivative in streamline direction, we define the stabilisation term $s(u,v) := \\sum _{\\tau \\in T^N}\\delta _{\\tau }\\big (\\kappa _{\\tau }(b u_x),\\kappa _{\\tau }(b v_x)\\big )_{\\tau }$ with the parameters $\\delta _\\tau =\\delta _{ij}\\ge 0$ , $\\tau \\subset \\Omega _{ij}$ , which will be specified later.", "It was stated in [12], [13] for different stabilisation methods that stabilisation is best if only applied in $\\Omega _{11}\\cup \\Omega _{21}$ .", "Therefore, we set $\\delta _{12}=\\delta _{22}=0$ in the following.", "The stabilised bilinear form $a_{LPS}$ is defined by $a_{LPS}(u,v) := a_{Gal}(u,v) + s(u,v),\\qquad u,v\\in H^1_0(\\Omega ),$ and the stabilised discrete problem reads: Find $u_{LPS}^N\\in V^N$ such that $ a_{LPS}(u_{LPS}^N,v^N) = (f,v^N)\\qquad \\forall v^N\\in V^N.$ Associated with this bilinear form is the LPS norm $\\left|\\!\\!\\;\\left|\\!\\!\\;\\left| {v}\\right|\\!\\!\\;\\right|\\!\\!\\;\\right|_{LPS} := \\left(\\varepsilon \\Vert {\\nabla v}\\Vert _{0}^2 + \\gamma \\Vert {v}\\Vert _{0}^2 + s(v,v)\\right)^{1/2}.$ The bilinear form is coercive w.r.t.", "this norm $ a_{LPS}(v,v)\\ge \\left|\\!\\!\\;\\left|\\!\\!\\;\\left| {v}\\right|\\!\\!\\;\\right|\\!\\!\\;\\right|_{LPS}^2, \\qquad v\\in H^1_0(\\Omega ).$ Moreover, the solutions $u$ of (REF ) and $u_{LPS}^N$ of (REF ) do not fulfil the Galerkin orthogonality, but $ a_{LPS}(u - u_{LPS}^N, v^N) = s(u, v^N) \\qquad \\forall v^N\\in V^N.$ The LPSFEM gives control over the fluctuations of the streamline derivative.", "In [33] a slight variation of the formulation is considered and an inf-sup condition w.r.t.", "the SDFEM norm is shown on a quasi-regular mesh.", "Thus, this LPSFEM gives control over the full streamline derivative.", "Whether such a result holds on S-type meshes is not known.", "This chapter contains results from [23], [24], [14], [15] that are also given in Appendix REF , REF , REF and REF .", "All theoretical results will be accompanied by a numerical study using the singularly perturbed convection-diffusion problem $-\\varepsilon \\Delta u - (2-x) u_x + \\frac{3}{2} u & = f&\\quad &\\text{in }\\Omega =(0,1)^2,\\\\u & = 0 && \\text{on }\\partial \\Omega ,$ where the right-hand side $f$ is chosen such that $u(x,y) = \\left(\\cos \\frac{\\pi x}{2} - \\frac{ \\mathrm {e}^{-x/\\varepsilon } - \\mathrm {e}^{-1/\\varepsilon }}{1-\\mathrm {e}^{-1/\\varepsilon }}\\right)\\frac{\\left(1-\\mathrm {e}^{-y/\\sqrt{\\varepsilon }}\\right)\\left( 1-\\mathrm {e}^{-(1-y)/\\sqrt{\\varepsilon }} \\right)}{1-\\mathrm {e}^{-1/\\sqrt{\\varepsilon }}}$ is the solution.", "We will used a fixed perturbation parameter $\\varepsilon =10^{-6}$ .", "Computations verifying the uniformity w.r.t.", "$\\varepsilon $ were also done.", "Figure  REF Figure: Typical solution of () with two parabolic layersand an exponential layer.shows the resulting solution.", "For comparison, the energy norm of $u$ is in this case $\\left|\\!\\!\\;\\left|\\!\\!\\;\\left| {u}\\right|\\!\\!\\;\\right|\\!\\!\\;\\right|_\\varepsilon \\approx 0.9975$ .", "Let us start with results for the standard Galerkin FEM.", "In [21], [13] results for bilinear elements are presented.", "If the mesh parameter $\\sigma $ fulfils $\\sigma \\ge 2$ , then the convergence result of [51] holds $\\left|\\!\\!\\;\\left|\\!\\!\\;\\left| {u-u_{Gal}^N}\\right|\\!\\!\\;\\right|\\!\\!\\;\\right|_\\varepsilon \\le C(N^{-1}\\max |\\psi ^{\\prime }|)$ with $\\max |\\psi ^{\\prime }|$ from e.g.", "Table REF and for $\\sigma \\ge 5/2$ the supercloseness result [21] $\\left|\\!\\!\\;\\left|\\!\\!\\;\\left| {u^I-u_{Gal}^N}\\right|\\!\\!\\;\\right|\\!\\!\\;\\right|_\\varepsilon \\le C(N^{-1}\\max |\\psi ^{\\prime }|)^2$ where $u^I$ denotes the nodal bilinear interpolant.", "In the higher-order case with either the full space $\\mathcal {Q}_p$ or the serendipity space $\\mathcal {Q}_p^\\oplus $ results can be found in [24].", "Theorem 3.1 (Theorem 6 of [24]) Let the solution $u$ of (REF ) satisfy Assumption REF and let $u_{Gal}^N$ denote the Galerkin solution of (REF ).", "Then, we have for $\\sigma \\ge p+1$ $\\left|\\!\\!\\;\\left|\\!\\!\\;\\left| {u-u_{Gal}^N}\\right|\\!\\!\\;\\right|\\!\\!\\;\\right|_\\varepsilon \\le C\\big (N^{-1}\\max |\\psi ^{\\prime }|\\big )^p.$ Thus, similar to the bilinear case, we achieve convergence of order $p$ in the energy norm.", "To our knowledge, no supercloseness result is available in literature in the higher-order case.", "Nevertheless, it can be observed numerically for the full space $\\mathcal {Q}_p$ .", "Let us come to the numerical example  (REF ).", "We will use a Bakhvalov-S-mesh, as here $|\\max \\psi ^{\\prime }|$ is bounded by a constant, see Table REF , and the convergence rates can be observed easiest.", "According to Theorem REF we expect $\\left|\\!\\!\\;\\left|\\!\\!\\;\\left| {u-u_{Gal}^N}\\right|\\!\\!\\;\\right|\\!\\!\\;\\right|_\\varepsilon \\le CN^{-p}.$ Table  REF Table: Convergence errors of Galerkin FEM for 𝒬 p \\mathcal {Q}_p- and 𝒬 p ⊕ \\mathcal {Q}_p^\\oplus -elements, and p=4,5p=4,\\,5confirms our expectation.", "In this table the errors and their estimated orders of convergence are given for $\\sigma =p+3/2$ .", "We see for the spaces $\\mathcal {Q}_4$ and $\\mathcal {Q}_4^\\oplus $ a convergence of order four, while for the spaces $\\mathcal {Q}_5$ and $\\mathcal {Q}_5^\\oplus $ we obtain order five.", "Moreover, the switch from the full space to Serendipity-space does increase the error only by a factor of two for $p=4$ and four for $p=5$ .", "Thus the error is increased, but at the same time only about half the number of degrees of freedom are used.", "Let us also look at supercloseness.", "Although no analytical result is given, Table  REF Table: Supercloseness property of Galerkin FEM for p=5p=5shows for $p=5$ a supercloseness property of order $p+1$ for the Galerkin FEM with $\\mathcal {Q}_p$ -elements and the two interpolation operators $\\pi ^N$ (vertex-edge-cell interpolation) and $I^N$ (Gauß-Lobatto interpolation).", "No such property is evident for $J^N$ (equidistant Lagrange interpolation) or the Serendipity-elements.", "For other polynomial degrees similar tables and conclusions can be given and are therefore omitted.", "We come back to the behaviour of $\\pi ^N$ and $I^N$ in the next section.", "One of the most popular stabilisation methods is the SDFEM.", "This method can also be used in connection with the general higher-order elements.", "Under certain restrictions on the stabilisation parameters convergence of order $p$ can be proved.", "Theorem 3.2 (Theorem 8 of [14]) Let $\\delta _{11}\\le C,\\quad \\delta _{21}\\le C\\max \\lbrace 1,\\varepsilon ^{-1/2}(N^{-1}\\max |\\psi ^{\\prime }|)^{2/3}\\rbrace (N^{-1}\\max |\\psi ^{\\prime }|)^{4/3},\\quad \\delta _{12}=\\delta _{22}=0,$ and (REF ) be satisfied.", "Let $u$ be the solution of (REF ) fulfilling Assumption REF and $u^N_{SD}$ be the streamline diffusion solution of (REF ).", "Then it holds for $\\sigma \\ge p+1$ that $\\left|\\!\\!\\;\\left|\\!\\!\\;\\left| {u-u^N_{SD}}\\right|\\!\\!\\;\\right|\\!\\!\\;\\right|_{\\varepsilon }\\le C(N^{-1}\\max |\\psi ^{\\prime }|)^p.$ For the standard Shishkin mesh the proof is given in [14] based mainly on Lemma 6 therein.", "For the Bakhvalov S-mesh the result is stated in [15].", "The proof for a general S-type mesh can be done in a very similar way to [14] and one obtains $a_{stabSD}(u-\\pi ^N u,\\chi )\\le C \\big [\\delta _{11}^{1/2}&N^{-p}+ \\delta _{12}\\varepsilon ^{-1}(N^{-1}\\max |\\psi ^{\\prime }|)^{p-1}\\\\+&\\min \\lbrace \\delta _{21}^{1/2}\\varepsilon ^{1/4},\\delta _{21}^{3/4}\\rbrace (N^{-1}\\max |\\psi ^{\\prime }|)^{p-1}\\\\+&\\min \\lbrace \\delta _{22}\\varepsilon ^{-3/4},\\delta _{22}^{1/2}\\varepsilon ^{-1/4}\\rbrace (\\ln N)^{1/2}(N^{-1}\\max |\\psi ^{\\prime }|)^{p-1}\\big ]\\left|\\!\\!\\;\\left|\\!\\!\\;\\left| {\\chi }\\right|\\!\\!\\;\\right|\\!\\!\\;\\right|_{SD}$ which together with the result for the Galerkin bilinear form [23], coercivity (REF ) and the interpolation error [23] gives above theorem.", "It can be seen quite nicely, that $|a_{stabSD}(u-\\pi ^Nu,\\chi )|$ becomes smaller, if the stabilisation parameters are reduced.", "But there is also an interaction between the Galerkin bilinear form $a_{Gal}(u-\\pi ^Nu,\\chi )$ and the SDFEM norm, that can be exploited to prove supercloseness.", "In order to do so, we will need an extension of Assumption REF on the solution decomposition.", "Assumption 3.3 Let the solution $u$ of (REF ) be decomposable according to Assumption REF into $u =v+w_1+w_2+w_{12}.$ In addition to the pointwise bounds for $i+j\\le p+1$ stated in Assumption REF we assume the $L_2$ -norm bounds $\\left\\Vert {\\frac{\\partial ^{p+2}v}{\\partial x^i\\partial y^j}} \\right\\Vert _{0}&\\le C,&\\left\\Vert {\\frac{\\partial ^{p+2}w_1}{\\partial x^i\\partial y^j}} \\right\\Vert _{0}&\\le C\\varepsilon ^{-i+1/2},\\\\\\left\\Vert {\\frac{\\partial ^{p+2}w_2}{\\partial x^i\\partial y^j}} \\right\\Vert _{0}&\\le C\\varepsilon ^{-j/2+1/4},&\\left\\Vert {\\frac{\\partial ^{p+2}w_{12}}{\\partial x^i\\partial y^j}} \\right\\Vert _{0}&\\le C\\varepsilon ^{-i-j/2+3/4}$ for $i+j=p+2$ with either $i=1$ or $j=1$ .", "Having this additional smoothness, the integral identities by Lin, see [57], [38], [62] can be used.", "Here we cite [57].", "Lemma 3.4 Let $w\\in H^{p+2}(\\tau _{ij})$ .", "Then for each $\\chi \\in \\mathcal {Q}_p(\\tau _{ij})$ we have $\\left|\\left((\\pi ^Nw-w)_x,\\chi _x\\right)_{\\tau _{ij}}\\right|&\\le C \\left\\Vert {k_j^{p+1}\\frac{\\partial ^{p+2}w}{\\partial x\\partial y^{p+1}}} \\right\\Vert _{0,\\tau _{ij}}\\Vert {\\chi _x}\\Vert _{0,\\tau _{ij}}\\\\\\mbox{and}\\hspace*{28.45274pt}\\left|\\left((\\pi ^Nw-w)_y,\\chi _y\\right)_{\\tau _{ij}}\\right|&\\le C \\left\\Vert {h_i^{p+1}\\frac{\\partial ^{p+2}w}{\\partial x^{p+1}\\partial y}} \\right\\Vert _{0,\\tau _{ij}}\\Vert {\\chi _y}\\Vert _{0,\\tau _{ij}}.\\hspace*{42.67912pt}~$ A different approach was used in [9], [10].", "Therein a method attributed to Zlámal [63] is applied by adding and subtracting a certain higher-order polynomial and using its approximation properties.", "Although only done for bilinear finite elements, it seems plausible that a similar technique might work in the higher-order case.", "Note that identities like those given in Lemma REF do not hold for proper subspaces $\\mathcal {Q}_p^\\clubsuit \\subset \\mathcal {Q}_p$ .", "Therefore, they cannot be used to prove a supercloseness property for spaces like the Serendipity space.", "This is not a real drawback, as for proper subspaces no supercloseness property is observed numerically.", "Under above assumptions, [14] gives a supercloseness result for the SDFEM method.", "Theorem 3.5 (Theorem 13 of [14]) For $\\mathcal {Q}_p^{\\clubsuit }=\\mathcal {Q}_p$ , $\\sigma \\ge p+1$ $\\delta _{11}=C N^{-1},\\quad \\delta _{21}\\le C\\max \\lbrace 1,\\varepsilon ^{-1/2}(N^{-1}\\max |\\psi ^{\\prime }|)\\rbrace (N^{-1}\\max |\\psi ^{\\prime }|)^{2},\\quad \\delta _{12}=\\delta _{22}=0$ and (REF ) we have $\\left|\\!\\!\\;\\left|\\!\\!\\;\\left| {\\pi ^N u-u^N_{SD}}\\right|\\!\\!\\;\\right|\\!\\!\\;\\right|_{SD}\\le C (N^{-1}\\max |\\psi ^{\\prime }|)^{p+1/2}(\\max |\\psi ^{\\prime }|\\ln N)^{1/2}.$ In [14] the proof for the standard Shishkin mesh can be found.", "The adaptation to general S-type meshes is straight-forward.", "The proof itself is based on the idea to estimate parts of the convective term of $a_{Gal}(\\cdot ,\\cdot )$ by the SDFEM norm instead of the energy norm, see [57].", "To be more precise, it's main step is $|(\\pi ^Nu-u,b\\chi _x)_{\\Omega _{11}}|&\\le C \\Vert {\\pi ^Nu-u}\\Vert _{0,\\Omega _{11}}\\Vert {b\\chi _x}\\Vert _{0,\\Omega _{11}}\\\\&\\le C N^{-(p+1)}\\Vert {b\\chi _x}\\Vert _{0,\\Omega _{11}}\\\\&\\le C \\min \\lbrace \\varepsilon ^{-1/2},\\delta _{11}^{-1/2}\\rbrace N^{-(p+1)}\\left|\\!\\!\\;\\left|\\!\\!\\;\\left| {\\chi }\\right|\\!\\!\\;\\right|\\!\\!\\;\\right|_{SD}$ that leads to $|((\\pi ^Nu-u),b\\chi _x)|\\le C \\bigg (\\min \\lbrace \\varepsilon ^{-1/2},\\delta _{11}^{-1/2}\\rbrace N^{-(p+1)}+\\\\(1+\\min \\lbrace \\delta _{21}^{-1/4}, N^{1/2}\\rbrace )(N^{-1}\\max |\\psi ^{\\prime }|)^{p+1}(\\ln N)^{1/2}\\bigg )\\left|\\!\\!\\;\\left|\\!\\!\\;\\left| {\\chi }\\right|\\!\\!\\;\\right|\\!\\!\\;\\right|_{SD}.$ The new bounds on the stabilisation parameters are consequences of (REF ).", "Remark 3.6 In order to achieve the supercloseness property we have to stabilise in $\\Omega _{11}$ .", "In the characteristic layer region we may stabilise, but this is not necessary for supercloseness.", "By choosing $\\delta _{21}=C(N^{-1}\\max |\\psi ^{\\prime }|)^2$ above result can be slightly improved to $\\left|\\!\\!\\;\\left|\\!\\!\\;\\left| {\\pi ^N u-u^N_{SD}}\\right|\\!\\!\\;\\right|\\!\\!\\;\\right|_{SD}\\le C (N^{-1}\\max |\\psi ^{\\prime }|)^{p+1/2}(\\ln N)^{1/2}.$ The bound (REF ) does also show, that for $\\varepsilon \\ge N^{-1}$ even the Galerkin FEM ($\\delta _{11}=\\delta _{21}=0$ ) fulfils a supercloseness property of order $p+1/2$ .", "Unfortunately, this case is of little interest in general.", "We have already seen in Section REF that the two interpolation operators $\\pi ^N$ (vertex-edge-cell interpolation) and $I^N$ (Gauß-Lobatto interpolation) show a similar numerical behaviour.", "Recalling (REF ) $\\pi _p^Nv=I_p^Nv+(\\pi ^N_{p+1}v-v)+\\left(I^N_p(\\pi ^N_{p+1}v-v)-(\\pi ^N_{p+1}v-v)\\right)$ we obtain $\\left|\\!\\!\\;\\left|\\!\\!\\;\\left| {I_p^Nu-u_{SD}^N}\\right|\\!\\!\\;\\right|\\!\\!\\;\\right|_\\varepsilon \\le \\left|\\!\\!\\;\\left|\\!\\!\\;\\left| {\\pi _p^Nu-u^N_{SD}}\\right|\\!\\!\\;\\right|\\!\\!\\;\\right|_\\varepsilon +\\left|\\!\\!\\;\\left|\\!\\!\\;\\left| {I_p^N(\\pi _{p+1}^Nu-u)-(\\pi _{p+1}^Nu-u)}\\right|\\!\\!\\;\\right|\\!\\!\\;\\right|_\\varepsilon +\\left|\\!\\!\\;\\left|\\!\\!\\;\\left| {\\pi _{p+1}^Nu-u}\\right|\\!\\!\\;\\right|\\!\\!\\;\\right|_\\varepsilon .$ Now the first term is estimated in Theorem REF , while the other two terms are interpolation errors of higher-order.", "Combining the results gives [16].", "Theorem 3.7 (Theorem 4.8 of [16]) Let $\\sigma \\ge p+2$ .", "Then it holds for the streamline-diffusion solution $u^N_{SD}$ under the restrictions on the stabilisation parameters given in Theorem  REF $\\left|\\!\\!\\;\\left|\\!\\!\\;\\left| {I^N u-u^N_{SD}}\\right|\\!\\!\\;\\right|\\!\\!\\;\\right|_\\varepsilon \\le C(N^{-1}\\max |\\psi ^{\\prime }|)^{p+1/2}(\\max |\\psi ^{\\prime }|\\ln N)^{1/2}.$ Thus, the Gauß-Lobatto interpolation inherits the supercloseness property from the vertex-edge-cell interpolation.", "A supercloseness property can be used to enhance the quality of the solution by a simple postprocessing routine.", "Figure: Macroelements MM of T ˜ N/2 \\tilde{T}^{N/2}constructed from T N T^NSuppose $N$ is divisible by 8.", "We construct a coarser macro mesh $\\tilde{T}^{N/2}$ composed of macro rectangles $M$ , each consisting of four rectangles of $T^N$ .", "The construction of these macro elements $M$ is done such that the union of them covers $\\Omega $ and none of them crosses the transition lines at $x=\\lambda _x$ and at $y=\\lambda _y$ or $y=1-\\lambda _y$ , see Figure REF .", "Remark that in general $\\tilde{T}^{N/2}\\ne T^{N/2}$ due to different transition points $\\lambda _x$ and $\\lambda _y$ , and the mesh generating function $\\phi $ .", "We now define local postprocessing operators for one macro element $M\\in \\tilde{T}^{N/2}$ .", "The precise definition can be found in [16], we will give only the basic ideas here.", "The first one was presented in 1d in [60] and is a modification of an operator given in [38].", "Let the local operator $\\widehat{P}_{vec}:C[-1,1]\\rightarrow \\mathcal {P}_{p+1}[-1,1]$ be given on the reference interval $[-1,1]$ by $\\widehat{P}_{vec} \\hat{v}(-1)&=v(x_{i-1}),\\qquad \\widehat{P}_{vec} \\hat{v}( a) =v(x_{i}),\\qquad \\widehat{P}_{vec} \\hat{v}( 1) =v(x_{i+1}),\\\\\\mbox{and for $p=2$:}\\hspace*{28.45274pt}\\int _{-1}^{1} (\\widehat{P}_{vec}\\hat{v}-\\hat{v})&=0,\\\\\\mbox{while for $p\\ge 3$:}\\hspace*{28.45274pt}\\int _{-1}^{a} (\\widehat{P}_{vec}\\hat{v}-\\hat{v})&=0,\\quad \\int _{a}^1 (\\widehat{P}_{vec}\\hat{v}-\\hat{v}) =0,\\\\\\int _{-1}^1 (\\widehat{P}_{vec}\\hat{v}-\\hat{v})q&=0,\\quad q\\in \\mathcal {P}_{p-2}[-1,1]\\setminus \\mathbb {R},$ where $\\hat{v}$ is a function $v|_{[x_{i-1},x_{i+1}]}$ linearly mapped onto the reference interval and $a\\in (-1,1)$ is the point that $x_i$ is mapped onto.", "By using the reference mapping and the tensor product structure, we obtain the full postprocessing operator $P_{vec,M}:C(M)\\rightarrow \\mathcal {Q}_{p+1}(M)$ on each macro element.", "Then, this piecewise projection is extended to a global, continuous operator $P_{vec}$ .", "The second postprocessing operator is defined by using the ordered sample of Gauß-Lobatto points $\\lbrace (\\tilde{x}_i,\\tilde{y}_j)\\rbrace $ , $i,j=0,\\dots ,2p$ of the four rectangles that $M$ consists of.", "Let $P_{GL,M}:C(M)\\rightarrow \\mathcal {Q}_{p+1}(M)$ denote the projection/interpolation operator fulfilling $P_{GL,M}v(\\tilde{x}_i,\\tilde{y}_j)=v(\\tilde{x}_i,\\tilde{y}_j),\\quad i,j=0,\\,1,\\,3,\\,5,\\dots ,2p-3,2p-1,\\,2p.$ Then, this piecewise projection is extended to a global, continuous operator $P_{GL}$ .", "We have for the postprocessed numerical solutions the following superconvergence result [16].", "Theorem 3.8 (Theorem 5.2 of [16]) Let $\\sigma \\ge p+2$ .", "Then it holds for the streamline-diffusion solution $u^N_{SD}$ under the restrictions on the stabilisation parameters given in Theorem  REF $\\left|\\!\\!\\;\\left|\\!\\!\\;\\left| {u-P_{GL}u^N_{SD}}\\right|\\!\\!\\;\\right|\\!\\!\\;\\right|_\\varepsilon +\\left|\\!\\!\\;\\left|\\!\\!\\;\\left| {u-P_{vec}u^N_{SD}}\\right|\\!\\!\\;\\right|\\!\\!\\;\\right|_\\varepsilon \\le C(N^{-1}\\max |\\psi ^{\\prime }|)^{p+1/2}(\\max |\\psi ^{\\prime }|\\ln N)^{1/2}.$ Let us come to the numerical example  (REF ).", "Although Theorems REF and REF assume $\\sigma \\ge p+2$ we will use a Bakhvalov-S-mesh with $\\sigma =p+3/2$ and $\\varepsilon =10^{-6}$ as numerical results suggest this to be enough.", "Note also, that in the bilinear case $\\sigma =1+3/2$ is a standard choice for superconvergence simulations, [62], [22], [21].", "For the stabilisation parameters we have two choices, according to Theorems REF and REF : $\\delta _{11}&=C_{SD},\\quad \\delta _{21}=C_{SD}\\varepsilon ^{-1/2}N^{-2},\\quad \\delta _{12}=\\delta _{22}=0,\\\\\\mbox{or}\\qquad \\delta _{11}&=C_{SD}N^{-1},\\quad \\delta _{21}=C_{SD}\\varepsilon ^{-1/2}N^{-3},\\quad \\delta _{12}=\\delta _{22}=0.$ For both our investigations into convergence and superconvergence we will use the smaller parameters, i.e.", "().", "The influence of $C_{SD}$ to various norms can be seen in Figure REF using $N=64$ and $\\varepsilon =10^{-6}$ .", "Figure: Influence of the stabilisation constant C SD C_{SD} onto the error behaviourTherein, the norms are not strongly influenced by the choice of moderate values of $C_{SD}$ .", "Thus, in the following we will use $C_{SD}=1$ .", "Table  REF Table: Convergence errors of SDFEM for 𝒬 p \\mathcal {Q}_p- and 𝒬 p ⊕ \\mathcal {Q}_p^\\oplus -elements,and p=4,5p=4,\\,5 with δ ij \\delta _{ij} according to ()shows the results for the polynomial spaces $\\mathcal {Q}_p$ and $\\mathcal {Q}_p^\\oplus $ in the cases $p=4$ and $p=5$ .", "As we can see, the convergence orders of $p$ are achieved and again we only have a constant factor of about 2 ($p=4$ ) and about 3 ($p=5$ ) in the errors when switching from the full to the Serendipity space.", "Table: Supercloseness property of SDFEM for p=4p=4 and δ ij \\delta _{ij} according to ()As predicted by Theorems REF and REF we observe in Table REF for the case $p=4$ a supercloseness property.", "But, the order is $p+1$ for both the vertex-edge-cell interpolation operator $\\pi ^N$ and the Gauß-Lobatto interpolation operator $I^N$ instead of the predicted $p+1/2$ .", "Thus the analytical results may not be sharp.", "Note that for the equidistant-interpolation operator $J^N$ and for the Serendipity space this property is not evident.", "Let us now come to exploiting the supercloseness property.", "Table  REF Table: Postprocessing of SDFEM for 𝒬 4 \\mathcal {Q}_4 and δ ij \\delta _{ij} according to ()gives the results of applying the postprocessing operators $P_{vec}$ and $P_{GL}$ to the SDFEM-solution.", "In correspondence with Theorem REF we observe an improved convergence behaviour.", "We see a superconvergence of order $p+1$ , half an order better than predicted.", "Note that simulations with other polynomial degrees show similar results.", "Finally, we analyse the LPSFEM.", "For its application to (REF ) with general higher-order elements we find a convergence result of order $p$ in [24].", "Theorem 3.9 (Theorem 6 of [24]) Let the solution $u$ of (REF ) satisfy Assumption REF .", "If the stabilisation parameters are chosen according to $\\delta _{11}\\le C_{LPS} N^{-2}\\big (\\max |\\psi ^{\\prime }|\\big )^{2p},\\,\\delta _{21}\\le C_{LPS} \\varepsilon ^{-1/2}\\ln ^{-1}N \\big (N^{-1}\\max |\\psi ^{\\prime }|\\big )^2,\\,\\delta _{12} =\\delta _{22} =0$ where $C_{LPS}>0$ is a constant and $\\sigma \\ge p+1$ , we have for the LPSFEM solution $u_{LPS}^N$ of (REF ) $\\left|\\!\\!\\;\\left|\\!\\!\\;\\left| {u-u_{LPS}^N}\\right|\\!\\!\\;\\right|\\!\\!\\;\\right|_\\varepsilon \\le C\\big (N^{-1}\\max |\\psi ^{\\prime }|\\big )^p.$ Thus, if the stabilisation parameters are not too large then the convergence order $p$ of the Galerkin FEM is not disturbed.", "Similarly to the Galerkin FEM, no supercloseness result is known in the higher-order case.", "A supercloseness property of order two was shown for bilinear elements in [23].", "When analysing the SDFEM we proved superconvergence by bounding the convective term of the Galerkin bilinear form against terms in the SDFEM norm.", "Unfortunately this trick does not help here with the LPSFEM.", "Basically, there are two problems.", "First, the convective term cannot easily be bounded by the stabilisation term, as the stabilisation terms only include fluctuations of the convection.", "Here the idea of [33] may help and we may use a stronger LPS-SDFEM norm, where the full weighted streamline derivative is included.", "But then the second problem comes into play.", "In order to estimate with the streamline derivative part of the norm we have to borrow half an order of the stabilisation parameter $\\delta _{11}$ , cf.", "(REF ).", "This costs us $\\delta _{11}^{-1/2}\\ge N/(\\max |\\psi ^{\\prime }|)^p$ by (REF ).", "Thus there would be no benefit in estimating with the stronger LPS norm.", "Let us now look at the numerical example  (REF ).", "Again we will use a Bakhvalov-S-mesh with $\\sigma =p+3/2$ and $\\varepsilon =10^{-6}$ .", "The stabilisation parameters are chosen according to Theorem REF .", "The influence of $C_{LPS}$ to various norms can be seen in Figure REF using $N=64$ and $\\varepsilon =10^{-6}$ .", "Figure: Influence of the stabilisation constant C LPS C_{LPS} onto the error behaviourClearly, for larger values of $C_{LPS}$ more stabilisation is introduced.", "On the downside, if $C_{LPS}$ is too large the stabilisation term dominates the weak formulation of (REF ) unless $N$ is large enough.", "Therefore we have chosen for the following simulations $C_{LPS}=0.001$ , i.e.", "$\\delta _{11}\\le 0.001 N^{-2}\\big (\\max |\\psi ^{\\prime }|\\big )^{2p},\\,\\delta _{21}\\le 0.001 \\varepsilon ^{-1/2}\\ln ^{-1}N \\big (N^{-1}\\max |\\psi ^{\\prime }|\\big )^2,\\,\\delta _{12} =\\delta _{22} =0.$ Table  REF Table: Convergence errors of LPSFEM for 𝒬 p \\mathcal {Q}_p- and 𝒬 p ⊕ \\mathcal {Q}_p^\\oplus -elements,and p=4,5p=4,\\,5 with δ ij \\delta _{ij} according to ()shows the convergence behaviour of the LPSFEM for the same polynomial spaces as Table REF .", "Again we see convergence of order $p$ for the full and the Serendipity spaces.", "Although the Serendipity spaces need only half the number of degrees of freedom, and are therefore much cheaper in computation, only a factor of 2-4 lies between the errors of the full space and those of the Serendipity space.", "Numerically, the LPSFEM does possess a supercloseness property too.", "Table  REF Table: Supercloseness property of LPSFEM for p=4p=4 and δ ij \\delta _{ij} according to ()shows it for the standard choice of the stabilisation parameters (REF ).", "Here for $p=4$ the vertex-edge-cell interpolation operator $\\pi ^N$ and the Gauß-Lobatto interpolation operator $I^N$ show for the full space $\\mathcal {Q}_p$ a supercloseness property of order $p+1$ .", "So far, there is no theoretical explanation known for this fact.", "Similarly to the SDFEM and the Galerkin method, the equidistant interpolation operator $J^N$ and the Serendipity space do not possess such a property.", "This chapter contains results from [25] that are also given in Appendix REF .", "Here we consider only bilinear elements, i.e.", "$V^N:=\\Big \\lbrace v\\in H_0^1(\\Omega ):v|_\\tau \\in \\mathcal {Q}_1(\\tau )\\;\\forall \\tau \\in T^N\\Big \\rbrace $ and restrict ourselves to the standard Shishkin mesh.", "See Remark REF for ideas about the general case.", "As assumed in Assumption REF , the solution $u$ of (REF ) has an exponential outflow layer of the type $e^{-x/\\varepsilon }$ and a characteristic layer of the type $e^{-y/\\sqrt{\\varepsilon }}$ .", "The energy norms of these two components are $\\left|\\!\\!\\;\\left|\\!\\!\\;\\left| {e^{-x/\\varepsilon }}\\right|\\!\\!\\;\\right|\\!\\!\\;\\right|_\\varepsilon =\\mathcal {O}\\left(1\\right)\\quad \\mbox{and}\\quad \\left|\\!\\!\\;\\left|\\!\\!\\;\\left| {e^{-y/\\sqrt{\\varepsilon }}}\\right|\\!\\!\\;\\right|\\!\\!\\;\\right|_\\varepsilon =\\mathcal {O}\\left(\\varepsilon ^{1/4}\\right).$ Thus, the last one, characterising the characteristic layer, is not well represented in the energy norm and is dominated by the exponential layer for small $\\varepsilon $ .", "In the following we will present results in the balanced norm $\\left|\\!\\!\\;\\left|\\!\\!\\;\\left| {v}\\right|\\!\\!\\;\\right|\\!\\!\\;\\right|_b:=\\sqrt{\\varepsilon \\Vert {v_x}\\Vert _{0}^2+\\varepsilon ^{1/2}\\Vert {v_y}\\Vert _{0}^2+\\gamma \\Vert {v}\\Vert _{0}^2}.$ Now it holds $\\left|\\!\\!\\;\\left|\\!\\!\\;\\left| {e^{-x/\\varepsilon }}\\right|\\!\\!\\;\\right|\\!\\!\\;\\right|_b=\\mathcal {O}\\left(1\\right)\\quad \\mbox{and}\\quad \\left|\\!\\!\\;\\left|\\!\\!\\;\\left| {e^{-y/\\sqrt{\\varepsilon }}}\\right|\\!\\!\\;\\right|\\!\\!\\;\\right|_b=\\mathcal {O}\\left(1\\right)$ and therefore both layer components are equally well represented in this norm.", "One possible application of balanced norms are uniform $L_\\infty $ -bounds of the error using a supercloseness result in a balanced norm, see [54].", "Therein the concept is shown for a convection-diffusion problem with exponential layers only where the standard energy norm suffices.", "Considering reaction-diffusion problems, the standard energy norm is not well balanced either.", "Here, first results in a balanced norm were obtained in [39] for a mixed finite element formulation and in [53] for a standard Galerkin approach.", "We will prove estimates in the balanced norm for a modified streamline diffusion method.", "Let us define the stabilisation bilinear form $a_{stab}(v,w)&:= \\sum _{\\tau \\in T^N}(\\varepsilon \\Delta v+b v_x-c v,\\delta _\\tau b w_x)_\\tau ,&& \\!\\!\\!v\\in H^1_0(\\Omega )\\cap H^2(\\Omega ),w\\in H^1_0(\\Omega ),$ and the linear form $f_{modSD}(v):=(f,v)-\\sum _{\\tau \\in T^N}(f,\\delta _\\tau bv_x)_\\tau ,\\qquad v\\in H^1_0(\\Omega ).$ Following the suggestion of [8] we choose $\\delta _\\tau $ as a stabilisation function on $\\tau $ given by $\\delta |_{\\tau _{ij}}:=\\delta _{\\tau _{ij}}:=\\min \\left\\lbrace \\frac{h_i}{2\\varepsilon },\\frac{1}{\\Vert {b}\\Vert _{\\infty ,\\tau _{ij}}}\\right\\rbrace h_i\\frac{(x_i-x)(x-x_{i-1})}{h_i^2}.$ Thus $\\delta _{\\tau }$ is a quadratic bubble function in $x$ -direction.", "This enables us to apply integration by parts in $x$ to some terms in our analysis without additional inner-boundary terms.", "Numerically, we see no difference to the standard SDFEM-formulation of Section REF with constant $\\delta _\\tau $ .", "Note, that by definition it holds $\\Vert {\\delta }\\Vert _{L_\\infty (\\Omega _{12}\\cup \\Omega _{22})}&\\le C\\varepsilon (N^{-1}\\ln N)^2,\\\\\\Vert {\\delta }\\Vert _{L_\\infty (\\Omega _{11}\\cup \\Omega _{21})}&\\le C N^{-1}\\quad \\mbox{and}\\quad \\Vert {\\varepsilon \\delta }\\Vert _{L_\\infty (\\Omega _{11}\\cup \\Omega _{21})}\\le C N^{-2}.$ We obtain the modified SDFEM formulation of (REF ): Find $u_{modSD}^N\\in V^N$ such that $a_{modSD}(u_{modSD}^N, v^N):=a_{Gal}(u_{modSD}^N,v^N)+a_{stab}(u_{modSD}^N,v^N)= f_{modSD}(v^N),\\quad \\forall v^N\\in V^N.$ Associated with this method is the modified streamline diffusion norm, defined by $\\left|\\!\\!\\;\\left|\\!\\!\\;\\left| {v}\\right|\\!\\!\\;\\right|\\!\\!\\;\\right|_{modSD}:=\\left(\\varepsilon \\Vert {\\nabla v}\\Vert _{0}^2+\\gamma \\Vert {v}\\Vert _{0}^2+\\sum _{\\tau \\in T^N}\\Vert {\\delta _\\tau ^{1/2}bv_x}\\Vert _{0,\\tau }^2\\right)^{1/2}.$ Under similar conditions on $\\delta _\\tau $ as in (REF ) we have coercivity in this norm: $a_{modSD}(v^N,v^N)\\ge \\frac{1}{2}\\left|\\!\\!\\;\\left|\\!\\!\\;\\left| {v^N}\\right|\\!\\!\\;\\right|\\!\\!\\;\\right|_{modSD},\\qquad \\forall v^N\\in V^N.$ Let us now come to the error analysis in the balanced norm.", "Although the modified SDFEM is coercive w.r.t.", "the modified SDFEM-norm, it is not uniformly coercive w.r.t.", "the balanced norm.", "Therefore, we use an additional projection to prove the error estimates.", "Let a projection operator $\\pi :H^1(\\Omega )\\cap C(\\Omega )\\rightarrow V^N$ be given by $a_{proj}(\\pi u-u,\\chi )=0\\quad \\mbox{for all }\\chi \\in V^N$ where $a_{proj}(v,w)=\\varepsilon (v_x,w_x) + (c v-b v_x, w)+\\sum _{\\tau \\in T^N}(\\varepsilon v_{xx}+b v_x-c v,\\delta _\\tau b w_x)_\\tau .$ The operator is defined in such a way, that for all $\\chi \\in V^N$ it holds $a_{modSD}(\\pi u-u,\\chi )= \\varepsilon ((\\pi u-u)_y,\\chi _y)+\\sum _{\\tau \\in T^N}(\\varepsilon (\\pi u-u)_{yy},\\delta _\\tau b \\chi _x)_\\tau .$ Combining coercivity (REF ), Galerkin orthogonality and (REF ) gives $\\frac{1}{2}\\left|\\!\\!\\;\\left|\\!\\!\\;\\left| {\\pi u-u_{modSD}^N}\\right|\\!\\!\\;\\right|\\!\\!\\;\\right|^2_{modSD}\\le a_{modSD}(\\pi u-u,\\pi u-u_{modSD}^N)\\\\= \\varepsilon ((\\pi u-u)_y,(\\pi u-u_{modSD}^N)_y)+\\sum _{\\tau \\in T^N}(\\varepsilon (\\pi u-u)_{yy},\\delta _\\tau b (\\pi u-u_{modSD}^N)_x)_\\tau .$ By omitting terms on the left-hand side, bounding the scalar product on the right-hand side by its $L_2$ -norms, multiplying by $\\varepsilon ^{-1/2}$ and setting $\\chi :=\\pi u-u_{modSD}^N\\in V^N$ we obtain $\\frac{1}{2}\\Vert {(\\pi u-u_{modSD}^N)_y}\\Vert _{0}\\left|\\!\\!\\;\\left|\\!\\!\\;\\left| {\\chi }\\right|\\!\\!\\;\\right|\\!\\!\\;\\right|_{modSD}\\le \\\\\\Vert {(u-\\pi u)_y}\\Vert _{0}\\left|\\!\\!\\;\\left|\\!\\!\\;\\left| {\\chi }\\right|\\!\\!\\;\\right|\\!\\!\\;\\right|_{modSD}+\\varepsilon ^{-1/2}\\left|\\sum _{\\tau \\in T^N}(\\varepsilon (\\pi u-u)_{yy},\\delta _\\tau b \\chi _x)_\\tau \\right|.$ The goal is to bound the right-hand side of (REF ) by $\\varepsilon ^{-1/4}$ times $\\left|\\!\\!\\;\\left|\\!\\!\\;\\left| {\\chi }\\right|\\!\\!\\;\\right|\\!\\!\\;\\right|_{modSD}$ and a term of order $N^{-1}$ .", "This can be done, as shown in [25] with one main ingredient being the $L_\\infty $ -stability of $\\pi $ .", "Theorem 4.1 (Theorem 1 of [25]) Let $\\sigma \\ge 2$ , $\\varepsilon \\le C(\\ln N)^{-2}$ , $u_{modSD}^N$ be the discrete solution of (REF ) and $u$ the weak solution of (REF ).", "Then it holds $\\left|\\!\\!\\;\\left|\\!\\!\\;\\left| {u-u_{modSD}^N}\\right|\\!\\!\\;\\right|\\!\\!\\;\\right|_{b}\\le C N^{-1}(\\ln N)^{3/2}.$ Remark 4.2 The result of Theorem REF can in theory be generalised in the following way for S-type meshes and higher-order polynomials.", "Let a consistent numerical method be given by: Find $\\tilde{u}^N\\in V^N=\\lbrace v\\in H_0^1(\\Omega ):v|_\\tau \\in \\mathcal {Q}_p(\\tau ),\\,\\tau \\in T^N\\rbrace $ with $a_{Gal}(\\tilde{u}^N,v^N)+a_{stab}(\\tilde{u}^N,v^N)=f(v^N)+f_{stab}(v^N)\\quad \\mbox{ for }v^N\\in V^N$ where $a_{stab}(\\cdot ,\\cdot )$ is a bilinear form and $f_{stab}(\\cdot )$ is a linear form.", "Suppose $a_{Gal}(\\cdot ,\\cdot )+a_{stab}(\\cdot ,\\cdot )$ is coercive w.r.t.", "a norm $\\left|\\!\\!\\;\\left|\\!\\!\\;\\left| {\\cdot }\\right|\\!\\!\\;\\right|\\!\\!\\;\\right|$ that contains the energy norm.", "Define the projection $\\pi u\\in V^N$ by $a_{proj}(\\pi u,\\chi )=a_{proj}(u,\\chi )\\quad \\mbox{for all }\\chi \\in V^N$ where $a_{proj}(u,v)=a_{Gal}(u,v)+a_{stab}(u,v)-\\varepsilon (u_y,v_y)-a_{rest}(u,v)$ for some suitable bilinear form $a_{rest}(\\cdot ,\\cdot )$ .", "Note that for our modified SDFEM we have $a_{rest}(u,v)=\\sum _{\\tau \\in T^N}(\\varepsilon u_{yy},\\delta _\\tau b v_x)_\\tau .$ In the general setting we obtain instead of (REF ) the estimate $\\Vert {(\\pi u-\\tilde{u}^N)_y}\\Vert _{0}\\left|\\!\\!\\;\\left|\\!\\!\\;\\left| {\\chi }\\right|\\!\\!\\;\\right|\\!\\!\\;\\right|\\le C\\left(\\Vert {(u-\\pi u)_y}\\Vert _{0}\\left|\\!\\!\\;\\left|\\!\\!\\;\\left| {\\chi }\\right|\\!\\!\\;\\right|\\!\\!\\;\\right|+\\varepsilon ^{-1/2}|a_{rest}(\\pi u-u,\\chi )|\\right).$ If we had the convergence result $\\left|\\!\\!\\;\\left|\\!\\!\\;\\left| {u-\\tilde{u}^N}\\right|\\!\\!\\;\\right|\\!\\!\\;\\right|_\\varepsilon \\le C (N^{-1}\\max |\\psi ^{\\prime }|)^p,$ the stability result $\\Vert {\\pi u}\\Vert _{L_\\infty }\\le C\\Vert {u}\\Vert _{L_\\infty }$ and the estimate $|a_{rest}(u-\\pi u,\\chi )|\\le C \\varepsilon ^{1/4}(N^{-1}\\max |\\psi ^{\\prime }|)^p(\\ln N)^{1/2}\\left|\\!\\!\\;\\left|\\!\\!\\;\\left| {\\chi }\\right|\\!\\!\\;\\right|\\!\\!\\;\\right|,$ then it would follow $\\left|\\!\\!\\;\\left|\\!\\!\\;\\left| {u-\\tilde{u}^N}\\right|\\!\\!\\;\\right|\\!\\!\\;\\right|_{b}\\le C (N^{-1}\\max |\\psi ^{\\prime }|)^p(\\ln N)^{1/2},$ thus convergence of order $p$ in the balanced norm.", "While the adaptation of the proof for our modified SDFEM to S-type meshes is straight-forward, higher-order polynomials are more problematic.", "To our knowledge, no result generalising the stability given in [8] for linear elements to higher-order elements is available in literature.", "Setting $\\delta _\\tau \\equiv 0$ everywhere gives the unstabilised Galerkin method.", "Unfortunately, the corresponding projection $\\pi $ is not known to be $L_\\infty $ -stable.", "Thus, our method of proof does not help with the pure Galerkin method.", "We use the test problem (REF ) from Chapter REF , i.e.", "$-\\varepsilon \\Delta u - (2-x) u_x + \\frac{3}{2} u = f$ with homogeneous Dirichlet boundary conditions and the right-hand side $f$ chosen such that $u = \\left(\\cos (\\pi x/2)-\\frac{e^{-x/\\varepsilon }-e^{-1/\\varepsilon }}{1-e^{-1/\\varepsilon }}\\right)\\frac{(1-e^{-y/\\sqrt{\\varepsilon }})(1-e^{-(1-y)/\\sqrt{\\varepsilon }})}{1-e^{-1/\\sqrt{\\varepsilon }}}$ is the exact solution.", "In the following, 'order' will always denote the exponent $\\alpha $ in a convergence order of form $\\mathcal {O}(N^{-\\alpha })$ while 'ln-order' corresponds to the exponent $\\alpha $ in a convergence order given by $\\mathcal {O}\\big ((N^{-1}\\ln N)^\\alpha \\big )$ .", "It is computed as usual using two consecutive numerical solutions.", "The experiments are carried out with $\\sigma =5/2$ and all integrations are approximated by a Gauss-Legendre quadrature of $6\\times 6$ -points.", "In our first experiment we look into the $\\varepsilon $ -uniformity of our calculations.", "Table  REF Table: ε\\varepsilon -uniformity of modSDFEM-errors for N=64N=64shows the results of the modified SDFEM for fixed $N=64$ and varying values of $\\varepsilon =10^{-1},\\dots ,10^{-8}$ .", "In both norms we can clearly see $\\varepsilon $ -uniformity, confirming Theorem REF .", "Note that the errors measured in the balanced norm are larger than those measured in the energy norm, but still bounded for decreasing $\\varepsilon $ .", "In the following we will always use the fixed value $\\varepsilon =10^{-6}$ that is small enough to bring out the layer behaviour of the solution $u$ of (REF ).", "Table  REF Table: Errors of the modSDFEM in the balanced and energy normshows the errors of the modified SDFEM in the given numerical example when $N$ is varied.", "Clearly we have convergence of almost order one in the balanced and the standard energy norm.", "Whether the exponent of the logarithmic factor is 1 or $3/2$ cannot be decided from this experiment, as the numerical behaviour of the two functions $N^{-1}\\ln N$ and $N^{-1}(\\ln N)^{3/2}$ is almost the same.", "Nevertheless, this table corresponds well with Theorem REF .", "Table  REF Table: Errors of the Galerkin FEM in the balanced and energy normshows the results of standard Galerkin FEM applied to our numerical example.", "Although we could not prove convergence for the Galerkin FEM in the balanced norm, we see convergence of almost order one in both norms.", "Let $u^I$ denote the standard bilinear interpolant of $u$ .", "Tables  REF Table: Supercloseness errors of the modSDFEM in the balanced and energy normand REF Table: Supercloseness errors of the Galerkin FEM in the balanced and energy normshow convergence of $u^N_{modSD}-u^I$ and $u^N-u^I$ in both norms to be of almost second order.", "Thus, we have supercloseness and via a simple postprocessing, e.g.", "biquadratic interpolation on a macro mesh, a numerical solution that is almost second order superconvergent can be constructed, see e.g.", "[56].", "For this purpose assume $N$ to be divisible by 8.", "We construct a macro mesh of the original mesh by fusing 2-by-2 elements such that the macro elements are pairwise disjoint and do not cross the boundaries of the subdomains $\\Omega _{ij}$ , $i,j=1,2$ , see also Figure REF .", "Tables REF Table: Superconvergence errors of the modSDFEM in the balanced and energy normand REF Table: Superconvergence errors of the Galerkin FEM in the balanced and energy normshow the resulting errors after applying a biquadratic interpolation $P$ to the discrete solutions on a macro mesh.", "It can be seen quite clearly, that $u-Pu^N_{modSD}$ and $u-Pu^N$ achieve (almost) second order convergence for both methods and in both norms.", "Another norm that “sees” all features of the solution is the $L_\\infty $ -norm.", "In this chapter we want to look into pointwise a-posteriori error estimation.", "A-priori error estimation in the $L_\\infty $ -norm for convection-diffusion problems is still an open field of research.", "Some results for stabilised methods can be found in e.g.", "[54] or [41] for an upwind finite difference method.", "This chapter contains results from [20], [19] that are also given in Appendix REF and REF .", "Let us rewrite problem (REF ) in a slightly different form: $L_{xy}u(x,y):=-\\varepsilon (u_{xx}+u_{yy})-(b(x,y)\\,u)_x+c(x,y)\\,u&=f(x,y)\\quad \\mbox{for }(x,y)\\in \\Omega ,\\\\u(x,y)&=0\\qquad \\quad \\;\\,\\mbox{for }(x,y)\\in \\partial \\Omega $ where the coefficients $b$ and $c$ are sufficiently smooth (e.g., $b,\\,c\\in C^\\infty (\\bar{\\Omega })$ ).", "Let us also assume, for some positive constant $\\beta $ , that $b(x,y)\\ge \\beta >0,\\qquad c(x,y)-b_x(x,y)\\ge 0\\qquad \\mbox{for~all~}(x,y)\\in \\bar{\\Omega }.$ Note that $b$ has a different meaning here compared with the previous chapters.", "We are interested in estimates of the Green's function $G(x,y;\\xi ,\\eta )$ associated with problem (REF ).", "For each fixed $(x,y)\\in \\Omega $ , it satisfies the adjoint problem $L^*_{\\xi \\eta }G(x,y;\\xi ,\\eta )=-\\varepsilon (G_{\\xi \\xi }+G_{\\eta \\eta })\\!+\\!b(\\xi ,\\eta )G_\\xi \\!+\\!c(\\xi ,\\eta )G&=\\delta (x-\\xi )\\,\\delta (y-\\eta ),\\,(\\xi ,\\eta )\\in \\Omega ,\\\\G(x,y;\\xi ,\\eta )&=0,\\hspace{75.0pt}(\\xi ,\\eta )\\in \\partial \\Omega .$ Here $L^*_{\\xi \\eta }$ is the adjoint differential operator to $L_{xy}$ , and $\\delta (\\cdot )$ is the one-dimensional Dirac $\\delta $ -distribution.", "Figure  REF Figure: Typical behaviour of the Green's function G(1 3,1 2;·,·)G(\\frac{1}{3},\\frac{1}{2};\\cdot ,\\cdot )for problem () with b=1b=1, c=0c=0 and ε=10 -3 \\varepsilon =10^{-3}.shows a representation of a Green's function $G(1/3,1/2;\\cdot ,\\cdot )$ for a small value of $\\varepsilon =10^{-3}$ and coefficients $b=1$ and $c=0$ .", "The singularity at $(\\xi ,\\eta )=(x,y)$ and strong anisotropic behaviour of $G$ can be seen quite nicely.", "Near the boundary $\\xi =1$ the Green's function has a strong boundary layer – an outflow boundary layer.", "The unique solution $u$ of (REF ) has the representation $u(x,y)=\\iint _{\\Omega }G(x,y;\\xi ,\\eta )\\,f(\\xi ,\\eta )\\,d\\xi \\, d\\eta .$ Our goal is to use (REF ) and $L_1$ -norm estimates of $G$ to obtain pointwise error bounds of $u-u^N$ , where $u^N$ is the numerical solution of a certain method.", "By using this idea, we get a-posteriori error bounds with computable terms.", "In a more general numerical-analysis context, we note that sharp estimates for continuous Green's functions (or their generalised versions) frequently play a crucial role in a priori and a posteriori error analyses [11], [28], [49].", "The main result for $L_1$ -norm bounds on $G$ is the following from [20].", "Theorem 5.1 (Theorem 2.2 of [20]) Let $\\varepsilon \\in (0,1]$ .", "The Green's function $G(x,y;\\xi ,\\eta )$ associated with (REF ) on the unit square $\\Omega =(0,1)^2$ satisfies, for any $(x,y)\\in \\Omega $ , the following bounds $\\Vert {\\partial _\\xi G(x,y;\\cdot )}\\Vert _{L_1(\\Omega )}&\\le C(1+|\\ln \\varepsilon |),&\\Vert {\\partial _\\eta G(x,y;\\cdot )}\\Vert _{L_1(\\Omega )}&\\le C\\varepsilon ^{-1/2}.$ Furthermore, for any ball $B(x^{\\prime },y^{\\prime };\\rho )$ of radius $\\rho $ centred at any $(x^{\\prime },y^{\\prime })\\in \\bar{\\Omega }$ , we have $\\Vert {G(x,y;\\cdot )}\\Vert _{W_1^1(B(x^{\\prime },y^{\\prime };\\rho ))}&\\le C\\varepsilon ^{-1}\\rho ,$ while for the ball $B(x,y;\\rho )$ of radius $\\rho $ centred at $(x,y)$ we have $\\Vert {\\partial ^2_{\\xi } G(x,y;\\cdot )}\\Vert _{L_1(\\Omega \\setminus B(x,y;\\rho ))}&\\le C\\varepsilon ^{-1}\\ln (2+\\varepsilon /\\rho ),\\\\\\Vert {\\partial ^2_{\\eta }G(x,y;\\cdot )}\\Vert _{L_1(\\Omega \\setminus B(x,y;\\rho ))}&\\le C\\varepsilon ^{-1}(\\ln (2+\\varepsilon /\\rho )+|\\ln \\varepsilon |).$ Let us compare the first order results to those obtained in one dimension, see e.g.", "[41].", "Here we have for the Green's function $g^{cd}$ of a convection-diffusion problem $\\Vert {\\partial _\\xi g^{cd}(x;\\cdot )}\\Vert _{L_1(\\Omega )}\\le C$ and for $g^{rd}$ of a reaction-diffusion problem $\\Vert {\\partial _\\xi g^{rd}(x;\\cdot )}\\Vert _{L_1(\\Omega )}\\le C\\varepsilon ^{-1/2}.$ Comparing these results with the results of Theorem REF , we see an additional dependence on $|\\ln \\varepsilon |$ in the streamline derivative.", "Thus the question for sharpness of these estimates is legitimate.", "In [19] it is shown that above bounds are sharp w.r.t.", "$\\varepsilon $ .", "Theorem 5.2 (Theorem 3 of [19]) Let $\\varepsilon \\in (0,c_0]$ for some sufficiently small positive $c_0$ .", "The Green's function $G$ associated with the constant-coefficient problem (REF ) in the unit square $\\Omega =(0,1)^2$ satisfies, for all $(x,y)\\in [\\frac{1}{4},\\frac{3}{4}]^2$ , the following lower bounds: There exists a constant $\\underline{c}>0$ independent of $\\varepsilon $ such that $\\Vert {\\partial _{\\xi } G(x,y;\\cdot )}\\Vert _{L_1(\\Omega )}&\\ge \\underline{c}|\\ln \\varepsilon |,&\\Vert {\\partial _{\\eta } G(x,y;\\cdot )}\\Vert _{L_1(\\Omega )}&\\ge \\underline{c}\\varepsilon ^{-1/2}.$ Furthermore, for any ball $B(x,y;\\rho )$ of radius $\\rho \\le \\frac{1}{8}$ , we have $\\Vert {G(x,y;\\cdot )}\\Vert _{W_1^1(\\Omega \\cap B(x,y;\\rho ))}&\\ge {\\left\\lbrace \\begin{array}{ll}\\underline{c}\\rho /\\varepsilon , & \\mbox{if~}\\rho \\le 2\\varepsilon ,\\\\\\underline{c}(\\rho /\\varepsilon )^{1/2},&\\mbox{otherwise},\\\\\\end{array}\\right.", "}\\\\\\Vert {\\partial ^2_{\\xi } G(x,y;\\cdot )}\\Vert _{L_1(\\Omega \\setminus B(x,y;\\rho ))}&\\ge \\underline{c}\\varepsilon ^{-1}\\ln (2+\\varepsilon /\\rho ),\\quad &&\\mbox{if~}\\rho \\le c_1\\varepsilon ,\\\\\\Vert {\\partial ^2_{\\eta } G(x,y;\\cdot )}\\Vert _{L_1(\\Omega \\setminus B(x,y;\\rho ))}&\\ge \\underline{c}\\varepsilon ^{-1}(\\ln (2+\\varepsilon /\\rho )+|\\ln \\varepsilon |),&&\\mbox{if~}\\rho \\le {\\textstyle \\frac{1}{8}},$ where $c_1$ is a sufficiently small positive constant.", "Not that the restriction $(x,y)\\in [\\frac{1}{4},\\frac{3}{4}]^2$ can be replaced by $(x,y)\\in [\\theta ,1-\\theta ]^2$ with $\\theta \\in (0,\\frac{1}{2})$ .", "Doing so, we have to replace $\\rho \\le \\frac{1}{8}$ by $\\rho \\le \\frac{1}{2}\\theta $ .", "Above results have been proved in 2d and 3d in [20], [19], [18].", "The basic idea is to look at a frozen coefficient version of (REF ) and to analyse the behaviour of its fundamental solution and of the difference to the fundamental solution of the original problem.", "This approach is sometimes called parametrix method.", "The results can be generalised to arbitrary dimensions, say $n\\in \\mathbb {N}$ .", "In order to do so, let us denote by $\\underline{\\mathbf {x}}=(x_1,x_2,\\dots ,x_n)$ a vector in $\\mathbb {R}^n$ and by $K_s$ the modified Bessel function of second kind and order $s$ with $s\\in \\mathbb {R}$ .", "The basic idea is to look at the fundamental solution of $\\bar{L}^*_{\\underline{\\mathbf {\\xi }}}\\bar{g}(\\underline{\\mathbf {x}};\\underline{\\mathbf {\\xi }})=-\\varepsilon \\Delta _{\\underline{\\mathbf {\\xi }}}\\bar{g}+b(\\underline{\\mathbf {x}})\\bar{g}_{\\xi _1}&=\\delta (\\underline{\\mathbf {x}}-\\underline{\\mathbf {\\xi }}),\\quad \\underline{\\mathbf {\\xi }}\\in \\mathbb {R}^n$ where $\\delta (\\cdot )$ is the $n$ -dimensional Dirac-distribution.", "For fixed $\\underline{\\mathbf {x}}$ the coefficient $b(\\underline{\\mathbf {x}})$ in (REF ) is constant and we can solve the problem explicitly.", "To simplify our presentation, let $q=\\frac{1}{2}b(\\underline{\\mathbf {x}})$ for fixed $\\underline{\\mathbf {x}}\\in (0,1)^n$ .", "Now a transformation, see [30], can be used to change the type of the problem from convection-diffusion to reaction-diffusion.", "For reaction-diffusion problems with constant coefficients the fundamental solution is known and we obtain the fundamental solution of (REF ) as $\\bar{g}(\\underline{\\mathbf {x}};\\underline{\\mathbf {\\xi }})=\\frac{1}{(2\\pi )^{n/2}\\varepsilon ^{n-1}}\\,\\left(\\frac{q}{\\hat{r}}\\right)^{n/2-1}e^{q\\hat{\\xi }_1}K_{n/2-1}(q\\hat{r}),\\qquad q=q(\\underline{\\mathbf {x}})={\\textstyle \\frac{1}{2}}b(\\underline{\\mathbf {x}})$ where $\\hat{r}=\\Vert {\\underline{\\mathbf {\\xi }}-\\underline{\\mathbf {x}}}\\Vert _{2}/\\varepsilon $ and $\\hat{\\xi }_k = (\\xi _k-x_k)/\\varepsilon $ .", "Note that for $n=2$ we obtain $\\bar{g}_2(x,y;\\xi ,\\eta )=\\frac{1}{2\\pi \\varepsilon }\\,e^{q\\hat{\\xi }_1}K_{0}(q\\hat{r}),\\qquad q=q(x,y)={\\textstyle \\frac{1}{2}}b(x,y),$ the fundamental solution used in [20] and for $n=3$ $\\bar{g}_3(\\underline{\\mathbf {x}};\\underline{\\mathbf {\\xi }})=\\frac{1}{(2\\pi )^{3/2}\\varepsilon ^2}\\,\\left(\\frac{q}{\\hat{r}}\\right)^{1/2}e^{q\\hat{\\xi }_1}K_{1/2}(q\\hat{r})=\\frac{1}{4\\pi \\varepsilon ^2}\\,\\frac{e^{q(\\xi _1-x_1-r)/\\varepsilon }}{\\hat{r}},\\qquad q=q(\\underline{\\mathbf {x}})={\\textstyle \\frac{1}{2}}b(\\underline{\\mathbf {x}}),$ the fundamental solution used in [18].", "The modified Bessel functions $K_s$ of order $s$ and those of order zero behave asymptotically very similar, see [50].", "Therefore, to modify the analysis presented in [20], [19], [18] to the $n$ -dimensional case is straightforward, though tedious and we obtain the analogue to Theorem REF and REF also in the $n$ -dimensional case.", "Here we want to apply the $L_1$ -norms of the Green's function and derive a-posteriori error estimates in the $L_\\infty $ -norm.", "The analysis following is from the forthcoming paper [17].", "Note, that in this section derivatives are to be understood in the sense of distributions.", "Let the domain $\\Omega $ be discretised by a rectangular tensor-product mesh $T$ with the nodes $(x_i,y_j)$ , where $0=x_0<x_1<\\ldots <x_N=1$ and $0=y_0<y_1<\\ldots <y_M=1$ for $N,M\\in \\mathbb {N}$ .", "On this mesh we derive the main ingredient of an a-posteriori error estimator, an $(L_\\infty , W_{-1,\\infty })$ -stability result, following [35].", "Theorem 5.3 Let $u$ be the unique solution of (REF ) for a given right-hand side $f$ satisfying $f(x,y)=\\bar{f}(x,y)-\\frac{\\partial }{\\partial x}[F_1(x,y)+\\bar{F}_1(x,y)]-\\frac{\\partial }{\\partial y}[F_2(x,y)+\\bar{F}_2(x,y)]$ where $F_1(x,y)|_{(x_{i-1},x_i)}&=A_i(y)(x-x_{i-1/2}),&i=1,\\dots ,N\\\\F_2(x,y)|_{(y_{j-1},y_j)}&=B_j(x)(y-y_{j-1/2}),&j=1,\\dots ,M$ and $\\bar{f},\\,\\bar{F}_1,\\,\\bar{F}_2,\\,A_i,$ and $B_j$ are arbitrary functions in $L_\\infty (\\Omega )$ .", "Then it holds that $\\Vert {u}\\Vert _{L_\\infty (\\Omega )}\\le C\\bigg [&\\Vert {\\bar{f}}\\Vert _{L_\\infty (\\Omega )}+(1+|\\ln \\varepsilon |)\\Vert {\\bar{F}_1}\\Vert _{L_\\infty (\\Omega )}+\\varepsilon ^{-1/2}\\Vert {\\bar{F}_2}\\Vert _{L_\\infty (\\Omega )}+\\\\&\\max _{i=1,\\dots ,N}\\left\\lbrace \\min \\left\\lbrace h_{i}^2\\frac{\\ln (2+\\varepsilon /\\kappa _h)}{\\varepsilon },h_{i}(1+|\\ln \\varepsilon |)\\right\\rbrace \\max _{y\\in [0,1]}|A_i(y)|\\right\\rbrace +\\\\&\\max _{j=1,\\dots ,M}\\left\\lbrace \\min \\left\\lbrace k_{j}^2\\frac{|\\ln \\varepsilon |+\\ln (2+\\varepsilon /\\kappa _k)}{\\varepsilon },\\frac{k_{j}}{\\varepsilon ^{1/2}}\\right\\rbrace \\max _{x\\in [0,1]}|B_j(x)|\\right\\rbrace \\bigg ]$ with $\\kappa _h=\\min h_i,\\,\\kappa _k=\\min k_j$ .", "Remark 5.4 The existence of $u\\in L_\\infty (\\Omega )$ for a given right-hand side $f$ of the form (REF ) follows from the classical results [36].", "[Proof of Theorem REF ] Using the linearity of the operator $L$ , we split $f$ into different parts and analyse them separately.", "For simplicity of the representation, denote by $g(\\xi ,\\eta )=G(x,y;\\xi ,\\eta )$ .", "1) Let $\\mathbf {\\bar{F}_1=\\bar{F}_2=F_1=F_2=0}$ , i.e.", "$f=\\bar{f}$ .", "The maximum principle (or (REF ) and $\\Vert {g}\\Vert _{L_1(\\Omega )}\\le C$ ) implies $\\Vert {u}\\Vert _{L_\\infty (\\Omega )}\\le C \\Vert {\\bar{f}}\\Vert _{L_\\infty (\\Omega )}.$ 2) Let $\\mathbf {\\bar{f}=F_1=F_2=0}$ , i.e.", "$f=-\\frac{\\partial }{\\partial x}\\bar{F}_1-\\frac{\\partial }{\\partial y}\\bar{F}_2$ .", "We represent $u$ using (REF ).", "Integration by parts and a Cauchy-Schwarz inequality give $u(x,y)&=\\iint _\\Omega g(\\xi ,\\eta )f(\\xi ,\\eta ) d\\xi d\\eta \\\\&=\\iint _\\Omega g_\\xi (\\xi ,\\eta )\\bar{F}_1(\\xi ,\\eta ) d\\xi d\\eta +\\iint _\\Omega g_\\eta (\\xi ,\\eta )\\bar{F}_2(\\xi ,\\eta ) d\\xi d\\eta \\\\&\\le \\Vert {G_\\xi }\\Vert _{L_1(\\Omega )}\\Vert {\\bar{F}_1}\\Vert _{L_\\infty (\\Omega )}+\\Vert {G_\\eta }\\Vert _{L_1(\\Omega )}\\Vert {\\bar{F}_2}\\Vert _{L_\\infty (\\Omega )}.$ With (REF ) we obtain $\\Vert {u}\\Vert _{L_\\infty (\\Omega )}\\le C \\left[(1+|\\ln \\varepsilon |)\\Vert {\\bar{F}_1}\\Vert _{L_\\infty (\\Omega )}+\\varepsilon ^{-1/2}\\Vert {\\bar{F}_2}\\Vert _{L_\\infty (\\Omega )}\\right].$ 3) Let $\\mathbf {\\bar{f}=\\bar{F}_1=\\bar{F}_2=F_2=0}$ , $f=-\\frac{\\partial }{\\partial x}F_1$ .", "Using (REF ) and integration by parts again, we have $u(x,y)&=\\iint _\\Omega F_1(\\xi ,\\eta ) g_\\xi (\\xi ,\\eta ) d\\xi d\\eta =\\sum _{i=1}^N\\iint _{\\Omega _i} A_i(\\eta )(\\xi -\\xi _{i-1/2})g_\\xi (\\xi ,\\eta )d\\xi d\\eta $ where $\\Omega _i=(x_{i-1},x_i)\\times [0,1]$ .", "The Green's function $g$ has a singularity at $(x,y)$ .", "Define $0<n<N$ where $x\\in [x_{n-1/2},x_{n+1/2}]$ and $\\Omega ^{\\prime }=(x_{n-1},x_{n+1})\\times (y-\\tilde{h}_n,y+\\tilde{h}_n)$ where $\\tilde{h}_n=\\min \\lbrace h_n,h_{n+1}\\rbrace /2$ .", "Note that the singularity now lies in $\\Omega ^{\\prime }$ .", "Defining the singularity-free function $\\tilde{g}$ by $\\tilde{g}=g$ in $\\Omega \\setminus \\Omega ^{\\prime }$ and $\\tilde{g}=0$ in $\\Omega ^{\\prime }$ we obtain $u(x,y)&=\\sum _{i=1}^N\\iint _{\\Omega _i}\\!\\!\\!\\!", "A_i(\\eta )(\\xi -\\xi _{i-1/2})\\tilde{g}_\\xi (\\xi ,\\eta )d\\xi d\\eta +\\sum _{i=n}^{n+1}\\iint _{\\Omega _i\\cap \\Omega ^{\\prime }}\\!\\!\\!\\!\\!", "A_i(\\eta )(\\xi -\\xi _{i-1/2})g_\\xi (\\xi ,\\eta )d\\xi d\\eta \\\\&=:S_1+S_2.$ The term $S_1$ can be estimated in two different ways.", "Either by $\\left|\\int _{x_{i-1}}^{x_i}\\!\\!", "(\\xi -x_{i-1/2})\\tilde{g}_\\xi (\\xi ,\\eta )d\\xi \\right|&\\le \\frac{h_i}{2}\\int _{x_{i-1}}^{x_i}|\\tilde{g}_\\xi (\\xi ,\\eta )|d\\xi \\multicolumn{2}{l}{\\text{or by}}\\\\\\left|\\int _{x_{i-1}}^{x_i}\\!\\!", "(\\xi -x_{i-1/2})\\tilde{g}_\\xi (\\xi ,\\eta )d\\xi \\right|&= \\left|\\int _{x_{i-1}}^{x_i}\\!\\!", "(\\xi -x_{i-1/2})\\int _{x_{i-1}}^\\xi \\!\\!\\tilde{g}_{\\xi \\xi }(s,\\eta )ds d\\xi \\right|\\le \\frac{h_i^2}{4}\\int _{x_{i-1}}^{x_i}\\!\\!|\\tilde{g}_{\\xi \\xi }(\\xi ,\\eta )|d\\xi .$ Note that $\\tilde{g}_{\\xi \\xi }$ is well defined.", "In order to use these two possibilities, decompose $A_i=A_i^1+A_i^2$ where $A_i^1={\\left\\lbrace \\begin{array}{ll}A_i, & 2h_i\\varepsilon (1+|\\ln \\varepsilon |)\\le h_i^2\\ln (2+\\varepsilon /\\kappa _h)\\\\0, &\\mbox{otherwise}\\end{array}\\right.", "}\\quad \\mbox{and }A_i^2=A_i-A_i^1.$ This yields by using Theorem  REF $|S_1|&\\le \\sum _{i=1}^N\\int _0^1 |A_i(\\eta )|\\left|\\int _{x_{i-1}}^{x_i}(\\xi -\\xi _{i-1/2})\\tilde{g}_\\xi (\\xi ,\\eta )d\\xi \\right| d\\eta \\\\&\\le \\max _{i=1,\\dots ,N}\\left\\lbrace \\frac{h_i}{2}\\max _{\\eta \\in [0,1]}|A_i^1(\\eta )|\\right\\rbrace \\iint _{\\Omega \\setminus \\Omega ^{\\prime }} |G_\\xi (\\xi ,\\eta )|d\\xi d\\eta +\\\\&\\hspace{14.22636pt}\\max _{i=1,\\dots ,N}\\left\\lbrace \\frac{h_i^2}{4}\\max _{\\eta \\in [0,1]}|A_i^2(\\eta )|\\right\\rbrace \\iint _{\\Omega \\setminus \\Omega ^{\\prime }} |G_{\\xi \\xi }(\\xi ,\\eta )|d\\xi d\\eta \\\\&\\le C\\max _{i=1,\\dots ,N}\\left\\lbrace \\min \\left\\lbrace h_i(1+|\\ln \\varepsilon |),\\frac{h_i^2}{\\varepsilon }\\ln (2+\\varepsilon /\\kappa _h)\\right\\rbrace \\max _{\\eta \\in [0,1]}|A_i(\\eta )|\\right\\rbrace .$ For $S_2$ we use either (REF ) $\\Vert {G_\\xi }\\Vert _{L_1(\\Omega )}\\le C(1+|\\ln \\varepsilon |)$ or (REF ) $\\Vert {G_\\xi }\\Vert _{L_1(B(a,b,\\rho ))}\\le C\\varepsilon ^{-1}\\rho .$ Let $A_i=\\tilde{A}_i^1+\\tilde{A}_i^2$ with $\\tilde{A}_i^1={\\left\\lbrace \\begin{array}{ll}A_i, & h_i\\varepsilon (1+|\\ln \\varepsilon |)\\le h_i^2\\\\0, &\\mbox{otherwise}\\end{array}\\right.", "}\\quad \\mbox{and }\\tilde{A}_i^2=A_i-\\tilde{A}_i^1.$ Then holds $|S_2|&\\le \\sum _{i=n}^{n+1}\\iint _{\\Omega _i\\cap \\Omega ^{\\prime }} |A_i(\\eta )||(\\xi -\\xi _{i-1/2})||g_\\xi (\\xi ,\\eta )|d\\xi d\\eta \\\\&\\le \\sum _{i=n}^{n+1}\\frac{h_i}{2}\\max _{\\eta \\in [0,1]}|A_i(\\eta )|\\iint _{B(x_{i-1/2},y,h_i)} |g_\\xi (\\xi ,\\eta )|d\\xi d\\eta \\\\&\\le C\\left(\\max _{i=n,n+1}\\left\\lbrace \\frac{h_i}{2}\\max _{\\eta \\in [0,1]}|\\tilde{A}_i^1(\\eta )|\\right\\rbrace |\\ln \\varepsilon |+\\max _{i=n,n+1}\\left\\lbrace \\frac{h_i^2}{2\\varepsilon }\\max _{\\eta \\in [0,1]}|\\tilde{A}_i^2(\\eta )|\\right\\rbrace \\right)\\\\&\\le C\\max _{i=n,n+1}\\left\\lbrace \\min \\left\\lbrace h_i(1+|\\ln \\varepsilon |),\\frac{h_i^2}{\\varepsilon }\\right\\rbrace \\max _{\\eta \\in [0,1]}|A_i(\\eta )|\\right\\rbrace .$ Thus we obtain $\\Vert {u}\\Vert _{L_\\infty (\\Omega )}\\le C\\max _{i=1,\\dots ,N}\\left\\lbrace \\min \\left\\lbrace h_i(1+|\\ln \\varepsilon |),\\frac{h_i^2}{\\varepsilon }\\ln (2+\\varepsilon /\\kappa _h)\\right\\rbrace \\max _{\\eta \\in [0,1]}|A_i(\\eta )|\\right\\rbrace .$ 4) Let $\\mathbf {\\bar{f}=\\bar{F}_1=F_1=\\bar{F}_2=0}$ , i.e.", "$f=-\\frac{\\partial }{\\partial y}F_2$ .", "This case can be treated similarly to the one above.", "Using a similar splitting of $u=\\widetilde{S}_1+\\widetilde{S}_2$ gives $|\\widetilde{S}_1|&\\le C\\max _{j=1,\\dots ,M}\\left\\lbrace \\min \\left\\lbrace \\frac{k_j}{\\varepsilon ^{1/2}},\\frac{k_j^2}{\\varepsilon }(|\\ln \\varepsilon |+\\ln (2+\\varepsilon /\\kappa _k))\\right\\rbrace \\max _{\\xi \\in [0,1]}|B_j(\\xi )|\\right\\rbrace \\multicolumn{2}{l}{\\text{and}}\\\\|\\widetilde{S}_2|&\\le C\\max _{j=m,m+1}\\left\\lbrace \\min \\left\\lbrace \\frac{k_j}{\\varepsilon ^{1/2}},\\frac{k_j^2}{\\varepsilon }\\right\\rbrace \\max _{\\xi \\in [0,1]}|B_j(\\xi )|\\right\\rbrace $ and therefore $\\Vert {u}\\Vert _{L_\\infty (\\Omega )}\\le C\\max _{j=1,\\dots ,M}\\left\\lbrace \\min \\left\\lbrace \\frac{k_j}{\\varepsilon ^{1/2}},\\frac{k_j^2}{\\varepsilon }(|\\ln \\varepsilon |+\\ln (2+\\varepsilon /\\kappa _k))\\right\\rbrace \\max _{\\xi \\in [0,1]}|B_j(\\xi )|\\right\\rbrace .$ By combining these estimates the stability result is proved.", "Note that in above results the global minima $\\kappa _h$ and $\\kappa _k$ can be replaced by local minima over two adjacent cells each.", "So far our Green's function estimates have not been applied to finite element methods.", "The Green's function $G$ is in general not in $H_0^1(\\Omega )$ which complicates the derivation of uniform a-posteriori error estimators via above approach.", "Further research is needed to apply this approach to finite element methods.", "Instead, we will apply the stability result to an upwind finite difference method.", "Let us start by rewriting (REF ) as $Lu&=-(A_1u)_x-(A_2u)_y-(Bu)_x+Cu=f\\\\\\multicolumn{2}{l}{\\text{where}}\\\\A_1u &=\\varepsilon u_x,\\quad A_2u =\\varepsilon u_y,\\quad Bu =bu\\quad \\mbox{and}\\quad Cu =cu.$ Using the index sets $I=\\lbrace 1,\\dots ,N-1\\rbrace $ , $\\bar{I}=\\lbrace 0,\\dots ,N\\rbrace $ , $J=\\lbrace 1,\\dots ,M-1\\rbrace $ and $\\bar{J}=\\lbrace 0,\\dots ,M\\rbrace $ , we define our discrete counterpart to (REF ): $L^N\\mathbf {u}_{ij}=&-\\widetilde{D}_x A_1^N\\mathbf {u}_{ij}-\\widetilde{D}_y A_2^N\\mathbf {u}_{ij}-\\widetilde{D}_x B^N\\mathbf {u}_{ij}+C^N\\mathbf {u}_{ij}\\\\=&-\\varepsilon (D_x^2\\mathbf {u}_{ij}+D_y^2\\mathbf {u}_{ij})-\\widetilde{D}_x (\\mathbf {b}_{ij}\\mathbf {u}_{ij})+\\mathbf {c}_{ij}\\mathbf {u}_{ij}\\quad =\\mathbf {f}_{ij},&&i\\in I,j\\in J\\\\\\mathbf {u}_{i,0}=&\\,\\mathbf {u}_{i,M}=0,&&i\\in \\bar{I}\\\\\\mathbf {u}_{0,j}=&\\,\\mathbf {u}_{N,j}=0,&&j\\in \\bar{J}.$ where $\\mathbf {f}_{ij}&=f(x_i,y_j)\\\\A_1^N\\mathbf {u}_{ij}&= \\varepsilon D^-_x\\mathbf {u}_{ij},\\quad A_2^N\\mathbf {u}_{ij} = \\varepsilon D^-_y\\mathbf {u}_{ij},\\quad B^N\\mathbf {u}_{ij} = \\mathbf {b}_{ij}\\mathbf {u}_{ij}\\quad \\mbox{and}\\quad C^N\\mathbf {u}_{ij} = \\mathbf {c}_{ij}\\mathbf {u}_{ij}$ with the standard backward difference operators $D^-$ .", "With $\\hbar _i=(h_i+h_{i+1})/2$ the other discrete operators are defined as $\\widetilde{D}_x\\mathbf {u}_{ij}&=\\frac{\\mathbf {u}_{i+1,j}-\\mathbf {u}_{ij}}{\\hbar _i},&D_x^2\\mathbf {u}_{ij}&=\\frac{1}{\\hbar _i}\\left[ \\frac{\\mathbf {u}_{i+1,j}-\\mathbf {u}_{ij}}{h_{i+1}}-\\frac{\\mathbf {u}_{i,j}-\\mathbf {u}_{i-1,j}}{h_{i}}\\right],&i\\in I,\\,j\\in J,$ and similarly in $y$ -direction.", "Note that (REF ) is a non-standard upwind finite difference method.", "The difference to the standard upwind FDM is the treatment of the convective term by $\\widetilde{D}_x$ instead of $D^+_x$ .", "The reason for this different treatment lies in the following analysis.", "Let us use the continuous residual, i.e.", "$L(\\mathbf {u}^\\mathcal {B}-u)$.", "Here $\\mathbf {u}^\\mathcal {B}$ denotes the piecewise bilinear interpolant of the discrete variable $\\mathbf {u}$ .", "With $\\mathbf {u}^I$ and $\\mathbf {u}^J$ as the one-dimensional piecewise linear interpolations in $x$ - and $y$ -direction, respectively we have $\\mathbf {u}^\\mathcal {B}=(\\mathbf {u}^I)^J=(\\mathbf {u}^J)^I.$ With a proper extension of our discrete operators to the boundary of $\\Omega $ , it holds for the residual $L(\\mathbf {u}^\\mathcal {B}-u)&= -\\left[(A_1\\mathbf {u}^I)_x+\\mathbf {F_1}^I\\right]^J-\\left[(A_2\\mathbf {u}^J)_y+\\mathbf {F_2}^J\\right]^I\\\\&\\quad -\\left[(B\\mathbf {u}^I)_x+\\mathbf {F_3}^I\\right]^J+\\left[C\\mathbf {u}^\\mathcal {B}-\\mathbf {F_4}^\\mathcal {B}\\right]-f+\\mathbf {f}^\\mathcal {B}$ where $\\mathbf {F_{1}}_{ij}&:=-\\widetilde{D}_x A_1^N\\mathbf {u}_{ij},&\\mathbf {F_{2}}_{ij}&:=-\\widetilde{D}_y A_2^N\\mathbf {u}_{ij},\\\\\\mathbf {F_{3}}_{ij}&:=-\\widetilde{D}_x B^N\\mathbf {u}_{ij},&\\mathbf {F_{4}}_{ij}&:=C^N\\mathbf {u}_{ij}.$ Let us start with the first term on the right-hand side of (REF ) for $x\\in [x_{i-1},x_i]$ and fixed $y=y_j$ .", "With the auxiliary terms $Q^1 &:= \\int _x^1 \\mathbf {F_{1}}^I,&Q^1_i&:= \\sum _{k=i}^{N-1}\\mathbf {F_{1}}_{kj}\\hbar _k$ we obtain $Q^1_i=A_1^N\\mathbf {u}_{ij}-A_1^N\\mathbf {u}_{Nj}=A_1^N\\mathbf {u}_{ij}$ and therefore $(A_1\\mathbf {u}^I)_x+\\mathbf {F_1}^I= \\partial _x(A_1\\mathbf {u}^I-Q^1)= \\partial _x(A_1\\mathbf {u}^I-A_1^N\\mathbf {u}_{ij})+\\partial _x(Q^1_i-Q^1)= \\partial _x(Q^1_i-Q^1).$ Now using the summation in $Q^1_i$ we can estimate further $\\partial _x(Q^1_i-Q^1)&= \\partial _x\\left( \\int _{x_i}^1 \\mathbf {F_1}^I-\\int _{x}^1\\mathbf {F_1}^I+\\mathbf {F_1}_{ij}\\frac{h_i}{2}-\\mathbf {F_1}_{Nj}\\frac{h_N}{2}\\right)\\\\&= \\partial _x\\left(-\\int _x^{x_i} \\mathbf {F_1}^I+\\mathbf {F_1}_{ij}\\frac{h_i}{2}\\right)\\\\&= \\partial _x\\left( \\mathbf {F_1}_{ij}\\frac{(x-x_{i-1})^2}{2h_i}-\\mathbf {F_1}_{i-1,j}\\frac{(x_i-x)^2}{2h_i}\\right).$ Thus we obtain $-\\left[(A_1\\mathbf {u}^I)_x+\\mathbf {F_1}^I\\right]^J\\bigg |_{x \\in [x_{i-1},x_i]}= -\\partial _x\\left( \\mathbf {F_1}_{ij}\\frac{(x-x_{i-1})^2}{2h_i}-\\mathbf {F_1}_{i-1,j}\\frac{(x_i-x)^2}{2h_i}\\right)^J.$ With similar techniques the other terms of (REF ) can be rewritten.", "Note that (REF ) can be further transformed to yield second-order terms in $h_i$ but then we obtain discrete third-order derivatives.", "We apply this for the $y$ -derivatives.", "Now using the stability result of Theorem REF to the right-hand side of the continuous residual (REF ) yields an a-posteriori error estimator.", "Theorem 5.5 Let $\\mathbf {u}$ be the solution of (REF ) and $u$ be the solution of (REF ).", "Then holds $\\Vert {\\mathbf {u}^\\mathcal {B}-u}\\Vert _{L_\\infty (\\Omega )}&\\le C\\bigg (\\max _{\\stackrel{i=0,\\dots ,N}{j=1,\\dots ,M}}M^1_{ij}+\\max _{\\stackrel{i=0,\\dots ,N}{j=1,\\dots ,M}}M^2_{ij}+\\max _{\\stackrel{i=0,\\dots ,N}{j=1,\\dots ,M}}M^3_{ij}+\\\\&\\hspace{34.14322pt}\\max _{\\stackrel{i=1,\\dots ,N}{j=0,\\dots ,M}}M^4_{ij}+\\max _{\\stackrel{i=1,\\dots ,N}{j=0,\\dots ,M}}M^5_{ij}+\\max _{\\stackrel{i=1,\\dots ,N}{j=0,\\dots ,M}}M^6_{ij}+\\max _{\\stackrel{i=1,\\dots ,N}{j=0,\\dots ,M}}M^7_{ij}\\bigg )\\multicolumn{2}{l}{\\text{with the terms depending on discrete $y$-derivatives}}\\\\M_{ij}^1 &:= \\min \\lbrace \\varepsilon ^{1/2} k_j,k_j^2(|\\ln \\varepsilon |+\\ln (2+\\varepsilon /\\kappa _k))\\rbrace \\min \\lbrace |D_y^2\\mathbf {u}_{i,j-1}|,|D_y^2\\mathbf {u}_{ij}|\\rbrace ,\\\\M_{ij}^2 &:= \\varepsilon ^{1/2} k_j^2|D_y^-D_y^2\\mathbf {u}_{i,j}|,\\\\M_{ij}^3 &:= k_j^2(1+|D_y^-\\mathbf {u}_{ij}|^2),\\multicolumn{2}{l}{\\text{and terms depending on discrete $x$-derivatives}}\\\\M_{ij}^4 &:= \\varepsilon h_i(1+|\\ln \\varepsilon |)\\max \\lbrace |[D_{x}^2\\mathbf {u}]_{i-1,j}|,|[D_{x}^2\\mathbf {u}]_{ij}|\\rbrace ,\\\\M_{ij}^5 &:= h_{i}^2(1+|[D_{x}^-\\mathbf {u}]_{ij}|^2),\\\\M_{ij}^6 &:= h_{i}(1+|\\ln \\varepsilon |)\\max \\lbrace |[\\widetilde{D}_{x}\\mathbf {u}]_{i-1,j}|,|[\\widetilde{D}_{x}\\mathbf {u}]_{ij}|\\rbrace ,\\\\M_{ij}^7 &:= h_{i}(1+|\\ln \\varepsilon |)(1+|[D_{x}^-\\mathbf {u}]_{ij}|).$ Note that formally, $M^1$ to $M^3$ are of order $k_j^2$ while $M^4$ , $M^6$ and $M^7$ are of order $h_i$ .", "Only $M^5$ is of order $h_i^2$ and therefore probably negligible.", "Thus, the estimator is formally of first order (if $k_j^2\\le h_i$ ) which is consistent with the formal order of an upwind method.", "The constant $C$ in the error bound of Theorem REF is unknown.", "By setting it to $C=1$ we obtain an error indicator that gives us information about the convergence behaviour, though not about the exact value of the error.", "For singularly perturbed problems the uniformity of the indicator is usually more important than the precise value of $C$ .", "Remark 5.6 The part $M^2$ contains discrete third-order derivatives.", "They are costly to evaluate and therefore an estimator with only second-order derivatives would be beneficial.", "In [34] an idea is used, that bounds the third-order derivative by a second-order derivative term.", "This approach could be used here too.", "We will tackle it in the forthcoming paper [17].", "Let us consider the numerical example $-\\varepsilon \\Delta u - u_x + \\frac{1}{2} u & = f&\\quad &\\text{in }\\Omega =(0,1)^2,\\\\u & = 0 && \\text{on }\\partial \\Omega ,$ where the right-hand side $f$ is chosen such that $u(x,y) = \\left(\\cos \\frac{\\pi x}{2} - \\frac{ \\mathrm {e}^{-x/\\varepsilon } - \\mathrm {e}^{-1/\\varepsilon }}{1-\\mathrm {e}^{-1/\\varepsilon }}\\right)\\frac{\\left(1-\\mathrm {e}^{-y/\\sqrt{\\varepsilon }}\\right)\\left( 1-\\mathrm {e}^{-(1-y)/\\sqrt{\\varepsilon }} \\right)}{1-\\mathrm {e}^{-1/\\sqrt{\\varepsilon }}}$ is the solution.", "In our first experiment we look into the uniformity w.r.t.", "$\\varepsilon $ of the indicator given in Theorem REF with $C=1$ .", "For given values of $\\varepsilon $ we compute the numerical solution on an a-priori chosen Shishkin mesh for $N=64$ and compare the results in Figure REF .", "Figure: Error and estimated error of () for N=64N=64 on a Shishkin meshTherein for each component of the indicator a line is shown.", "Additionally, a solid black line represents the real error, and a black dash-dot line represents a modified indicator.", "The modified indicator takes only the maxima of $M^1$ , $M^3$ , $M^4$ and $M^7$ .", "In numerical simulations this modification represents the behaviour of the error much better than the real indicator.", "For another motivation, see also Remark REF .", "In Figure REF both indicators behave like $|\\ln \\varepsilon |$ , which is also given for comparison as a line in magenta.", "But the real error stays almost constant for $\\varepsilon $ becoming smaller.", "Thus, there is a $|\\ln \\varepsilon |$ -dependence in our estimators coming from the Green's function estimates, although they are sharp.", "This behaviour was seen for several different examples.", "As a consequence we will use from now on the heuristic indicator $\\eta &:=\\bigg (\\max _{\\stackrel{i=0,\\dots ,N}{j=1,\\dots ,M}}M^1_{ij}+\\max _{\\stackrel{i=0,\\dots ,N}{j=1,\\dots ,M}}M^2_{ij}+\\max _{\\stackrel{i=0,\\dots ,N}{j=1,\\dots ,M}}M^3_{ij}+\\\\&\\hspace{34.14322pt}\\max _{\\stackrel{i=1,\\dots ,N}{j=0,\\dots ,M}}M^4_{ij}+\\max _{\\stackrel{i=1,\\dots ,N}{j=0,\\dots ,M}}M^5_{ij}+\\max _{\\stackrel{i=1,\\dots ,N}{j=0,\\dots ,M}}M^6_{ij}+\\max _{\\stackrel{i=1,\\dots ,N}{j=0,\\dots ,M}}M^7_{ij}\\bigg )\\multicolumn{2}{l}{\\text{and the modified indicator}}\\\\\\widetilde{\\eta }&:=\\bigg (\\max _{\\stackrel{i=0,\\dots ,N}{j=1,\\dots ,M}}M^1_{ij}+\\max _{\\stackrel{i=0,\\dots ,N}{j=1,\\dots ,M}}M^3_{ij}+\\max _{\\stackrel{i=1,\\dots ,N}{j=0,\\dots ,M}}M^4_{ij}+\\max _{\\stackrel{i=1,\\dots ,N}{j=0,\\dots ,M}}M^7_{ij}\\bigg )\\multicolumn{2}{l}{\\text{where}}\\\\M_{ij}^1 &:= \\min \\lbrace \\varepsilon ^{1/2} k_j,k_j^2\\ln (2+\\varepsilon /\\kappa _k)\\rbrace \\min \\lbrace |D_y^2\\mathbf {u}_{i,j-1}|,|D_y^2\\mathbf {u}_{ij}|\\rbrace ,\\\\M_{ij}^2 &:= \\varepsilon ^{1/2} k_j^2|D_y^-D_y^2\\mathbf {u}_{i,j}|,\\qquad \\qquad M_{ij}^3 := k_j^2(1+|D_y^-\\mathbf {u}_{ij}|^2),\\\\M_{ij}^4 &:= \\varepsilon h_i\\max \\lbrace |[D_{x}^2\\mathbf {u}]_{i-1,j}|,|[D_{x}^2\\mathbf {u}]_{ij}|\\rbrace ,\\\\M_{ij}^5 &:= h_{i}^2(1+|[D_{x}^-\\mathbf {u}]_{ij}|^2),\\qquad \\qquad M_{ij}^6 := h_{i}\\max \\lbrace |[\\widetilde{D}_{x}\\mathbf {u}]_{i-1,j}|,|[\\widetilde{D}_{x}\\mathbf {u}]_{ij}|\\rbrace ,\\\\M_{ij}^7 &:= h_{i}(1+|[D_{x}^-\\mathbf {u}]_{ij}|).$ Figure  REF Figure: Error and modified estimated error of () for N=64N=64 on a Shishkin meshshows the behaviour of these modified indicators.", "Obviously, there is no dependence on $\\varepsilon $ any longer and the errors are caught quite well.", "For our second experiment we let $\\varepsilon =10^{-6}$ be constant, chose a-priori adapted meshes, apply the modified upwind method and estimate the error with $\\eta $ and $\\widetilde{\\eta }$ .", "Figures  REF Figure: Error and estimated error of () for ε=10 -6 \\varepsilon =10^{-6} on Shishkin meshesand REF Figure: Error and estimated error of () for ε=10 -6 \\varepsilon =10^{-6} on a Bakhvalov S-meshesshow the results for the two indicators and variable $N$ .", "The principal behaviour of the errors is caught by both of them although the magnitude is wrong.", "We also observe the blue lines to fall much faster than the red lines.", "The reason behind is the formal second order convergence in $y$ -direction of $M^1$ to $M^3$ .", "This gives hope for a-posteriori mesh adaptation to behave better than a-priori adapted meshes.", "Let us consider the following simple, anisotropic mesh adaptation approach.", "We start with a coarse initial, tensor-product mesh.", "In each step we compute the numerical solution on the given mesh and use the error indicators to decide, whether and where the $x$ -part or the $y$ -part of the tensor product mesh should be refined.", "This will be done as follows: Compute $M_y:=\\max \\limits _{k=1,3}\\lbrace \\max \\lbrace M_{ij}^k\\rbrace \\rbrace $ and $M_x:=\\max \\limits _{k=4,5,6,7}\\lbrace \\max \\lbrace M_{ij}^k\\rbrace \\rbrace $ .", "If $M_x>M_y$ we refine in $x$ -direction, otherwise in $y$ -direction.", "Assuming $M_x>M_y$ we collect all $i$ with $M_{ij}^k\\ge \\alpha \\max \\lbrace M_{ij}^k\\rbrace $ for any $k=4,5,6,7$ and given $\\alpha \\in [0,1]$ , and divide $[x_{i-1},x_i]$ into two intervals of equal length.", "Similarly, we proceed in the other case and divide $[y_{j-1},y_j]$ into two intervals for all $j$ with $M_{ij}^k\\ge \\alpha \\max \\lbrace M_{ij}^k\\rbrace $ for any $k=1,2,3$ .", "Compute $M_y:=\\max \\limits _{k=1,3}\\lbrace \\max \\lbrace M_{ij}^k\\rbrace \\rbrace $ and $M_x:=\\max \\limits _{k=4,5,6,7}\\lbrace \\max \\lbrace M_{ij}^k\\rbrace \\rbrace $ .", "If $M_x>M_y$ we refine in $x$ -direction, otherwise in $y$ -direction.", "Assuming $M_x>M_y$ we collect all $i$ with $M_{ij}^k\\ge \\alpha \\max \\lbrace M_{ij}^k\\rbrace $ for any $k=4,5,6,7$ and given $\\alpha \\in [0,1]$ , and divide $[x_{i-1},x_i]$ into two intervals of equal length.", "Similarly, we proceed in the other case and divide $[y_{j-1},y_j]$ into two intervals for all $j$ with $M_{ij}^k\\ge \\alpha \\max \\lbrace M_{ij}^k\\rbrace $ for any $k=1,2,3$ .", "With these refined partitions we construct a new tensor product mesh and the cycle begins again.", "This refinement process has a parameter $\\alpha $ influencing the marking of elements to refine.", "We chose $\\alpha =0.9$ to refine only elements with large contributions.", "In Figure REF and REF the convergence results for $\\varepsilon =10^{-6}$ are shown until the number of degrees of freedom reaches approximately $512^2$ .", "In Figure REF initially a Shishkin mesh of 4-by-4 cells was taken and in the end we have $1014\\times 250$ cells.", "In Figure REF the initial mesh was equidistant with 4-by-4 cells, and the final mesh has $992\\times 252$ cells.", "We observe in both cases that our adaptation process reduces the error nicely.", "The observed overall order of convergence (after some initial phase) is $(\\#dof)^{-1/2}$ where $\\#dof$ is the number of degrees of freedom.", "Comparing the errors for the number of degrees of freedom taken to be about $512^2$ , Table  REF Table: Comparison of errors of a-priori and a-posteriori meshesshows the results on the a-posteriori adapted meshes to be comparable to the a-priori adapted meshes.", "With the different number of cells in each direction the a-posteriori adapted meshes can reduce the error much better than a Shishkin mesh.", "Still, the grading of the Bakhvalov S-mesh gives a mesh with the smallest error.", "Moreover, the costs for an adaptive algorithm are high due to the repeated solving of the numerical problems on the different meshes.", "We have presented convergence and supercloseness results for higher-order finite-element methods, including stabilised methods like LPSFEM and SDFEM.", "Having general polynomial spaces $\\mathcal {Q}^\\clubsuit _p$ , convergence of order $p$ can be proved.", "If we use proper subspaces of $\\mathcal {Q}_p$ , like the Serendipity space, we cannot apply the supercloseness techniques that are valid for the full space $\\mathcal {Q}_p$ .", "But numerical results do also indicate, that for proper subspaces no supercloseness property holds.", "While numerical simulations indicate supercloseness properties of order $p+1$ for many methods, numerical analysis provides proof only for order $p+1/2$ in the case of SDFEM.", "Further research is needed to improve this situation.", "Some preliminary results for the pure Galerkin method are topic of ongoing research.", "Here a supercloseness property in the case of exponential boundary layers and odd polynomial degree $p$ of order $p+1/4$ could be proved, [26].", "The proof therein can easily be adapted to the case of characteristic boundary layers too.", "Nevertheless, there is still a gap between theory and simulation of $3/4$ orders.", "Although convergence and supercloseness can be proved in the energy and related norms, these norms do not “see” the characteristic layers correctly.", "The layers are under-represented in the resulting terms.", "An alternative is shown in the balanced norm that has the right weighting of the norm components.", "But now the Galerkin FEM is no longer coercive w.r.t.", "this norm.", "Using certain stability arguments, for a modified bilinear SDFEM convergence in this norm is proved.", "How the proof can be modified for the standard Galerkin FEM and other stabilised methods, and for higher order methods in general is an open question.", "Numerically, all these methods show the same convergence and supercloseness behaviour in the energy and the balanced norm.", "The use of a-priori adapted meshes requires knowledge about the layer-structure of solutions to the considered problem.", "Alternatively, the mesh can be adapted after computation of an (approximative) numerical solution.", "For this a-posteriori mesh adaptation, uniform error estimators or indicators are needed.", "We presented estimates on the $L_1$ -norm of the Green's function as an ingredient for $L_\\infty $ -error estimators.", "A simple, first estimator for a finite difference method is also given and analysed.", "The optimisation of this estimator and an extension of this approach to finite element methods are open problems.", "Bibliography Appendix Appendix Appendix pages=1-1, pagecommand= tocsectionA.1   S. Franz, G. Matthies: Local projection stabilisation on S-type meshes for convection-diffusion problems with characteristic layers.", "Computing, 87(3-4), 135–167, 2010 S. Franz, G. Matthies: Local projection stabilisation on S-type meshes for convection-diffusion problems with characteristic layers.", "Computing, 87(3-4), 135–167, 2010 [The article is removed from this electronic version due to copyright reasons.]", "Abstract: Singularly perturbed convection-diffusion problems with exponential and characteristic layers are considered on the unit square.", "The discretisation is based on layer-adapted meshes.", "The standard Galerkin method and the local projection scheme are analysed for bilinear and higher order finite element where enriched spaces were used.", "For bilinears, first order convergence in the $\\varepsilon $ -weighted energy norm is shown for both the Galerkin and the stabilised scheme.", "However, supercloseness results of second orders hold for the Galerkin method in the $\\varepsilon $ -weighted energy norm and for the local projection scheme in the corresponding norm.", "For the enriched $\\mathcal {Q}_p$ -elements, $p\\ge 2$ , which already contain the space $\\mathcal {P}_{p+1}$ , a convergence order $p+1$ in the $\\varepsilon $ -weighted energy norm is proved for both the Galerkin method and the local projection scheme.", "Furthermore, the local projection methods provides a supercloseness result of order $p+1$ in local projection norm.", "Keywords: Singular perturbation, Characteristic layers, Shishkin meshes, Local projection Mathematics Subject Classification (2000): 65N12, 65N30, 65N50 DOI: 10.1007/s00607-010-0079-y tocsectionA.2   S. Franz, G. Matthies: Convergence on Layer-adapted Meshes and Anisotropic Interpolation Error Estimates of Non-Standard Higher Order Finite Elements.", "Appl.", "Numer.", "Math., 61, 723–737, 2011 S. Franz, G. Matthies: Convergence on Layer-adapted Meshes and Anisotropic Interpolation Error Estimates of Non-Standard Higher Order Finite Elements.", "Appl.", "Numer.", "Math., 61, 723–737, 2011 [The article is removed from this electronic version due to copyright reasons.]", "Abstract: For a general class of finite element spaces based on local polynomial spaces $\\mathcal {E}$ with $\\mathcal {P}_p\\subset \\mathcal {E}\\subset \\mathcal {Q}_p$ we construct a vertex-edge-cell and point-value oriented interpolation operators that fulfil anisotropic interpolation error estimates.", "Using these estimates we prove $\\varepsilon $ -uniform convergence of order $p$ for the Galerkin FEM and the LPSFEM for a singularly perturbed convection-diffusion problem with characteristic boundary layers.", "Keywords: singular perturbation, characteristic layers, exponential layers, Shishkin meshes, local-projection, higher-order FEM Mathematics Subject Classification (2000): 65N12, 65N30, 65N50 DOI: 10.1016/j.apnum.2011.02.001 tocsectionA.3   S. Franz: Superconvergence Using Pointwise Interpolation in Convection-Diffusion Problems.", "Appl.", "Numer.", "Math., 76, 132–144, 2014 S. Franz: Superconvergence Using Pointwise Interpolation in Convection-Diffusion Problems.", "Appl.", "Numer.", "Math., 76, 132–144, 2014 [The article is removed from this electronic version due to copyright reasons.]", "Abstract: Considering a singularly perturbed convection-diffusion problem, we present an analysis for a superconvergence result using pointwise interpolation of Gauß-Lobatto type for higher-order streamline diffusion FEM.", "We show a useful connection between two different types of interpolation, namely a vertex-edge-cell interpolant and a pointwise interpolant.", "Moreover, different postprocessing operators are analysed and applied to model problems.", "Keywords: singular perturbation, layer-adapted meshes, superconvergence, postprocessing Mathematics Subject Classification (2000): 65N12, 65N30, 65N50 DOI: 10.1016/j.apnum.2013.07.007 tocsectionA.4   S. Franz: SDFEM with non-standard higher-order finite elements for a convection-diffusion problem with characteristic boundary layers.", "BIT Numerical Mathematics, 51(3), 631–651, 2011 S. Franz: SDFEM with non-standard higher-order finite elements for a convection-diffusion problem with characteristic boundary layers.", "BIT Numerical Mathematics, 51(3), 631–651, 2011 [The article is removed from this electronic version due to copyright reasons.]", "Abstract: Considering a singularly perturbed problem with exponential and characteristic layers, we show convergence for non-standard higher-order finite elements using the streamline diffusion finite element method (SDFEM).", "Moreover, for the standard higher-order space $\\mathcal {Q}_p$ supercloseness of the numerical solution w.r.t.", "an interpolation of the exact solution in the streamline diffusion norm of order $p+1/2$ is proved.", "Keywords: singular perturbation, characteristic layers, exponential layers, Shishkin mesh, SDFEM, higher order Mathematics Subject Classification (2000): 65N12, 65N30, 65N50 DOI: 10.1007/s10543-010-0307-z tocsectionA.5   S. Franz: Convergence Phenomena of $Q_p$ -Elements for Convection-Diffusion Problems.", "Numer.", "Methods Partial Differential Equations, 29(1), 280–296, 2013 S. Franz: Convergence Phenomena of $Q_p$ -Elements for Convection-Diffusion Problems.", "Numer.", "Methods Partial Differential Equations, 29(1), 280–296, 2013 [The article is removed from this electronic version due to copyright reasons.]", "Abstract: We present a numerical study for singularly perturbed convection-diffusion problems using higher-order Galerkin and Streamline Diffusion FEM.", "We are especially interested in convergence and superconvergence properties with respect to different interpolation operators.", "For this we investigate pointwise interpolation and vertex-edge-cell interpolation.", "Keywords: singular perturbation, boundary layers, layer-adapted meshes, superconvergence Mathematics Subject Classification (2000): 65N12, 65N30, 65N50 DOI: 10.1002/num.21709 tocsectionA.6   S. Franz, H.-G. Roos: Error estimation in a balanced norm for a convection-diffusion problem with characteristic boundary layers.", "Calcolo, DOI:10.1007/s10092-013-0093-5, 2013 S. Franz, H.-G. Roos: Error estimation in a balanced norm for a convection-diffusion problem with characteristic boundary layers.", "Calcolo, DOI:10.1007/s10092-013-0093-5, 2013 [The article is removed from this electronic version due to copyright reasons.]", "Abstract: The $\\varepsilon $ -weighted energy norm is the natural norm for singularly perturbed convection-diffusion problems with exponential layers.", "But, this norm is too weak to recognise features of characteristic layers.", "We present an error analysis in a differently weighted energy norm—a balanced norm—that overcomes this drawback.", "Keywords: singular perturbation, characteristic and exponential layers, Shishkin mesh, SDFEM, balanced norm Mathematics Subject Classification (2000): 65N12, 65N30, 65N50 DOI: 10.1007/s10092-013-0093-5 tocsectionA.7   S. Franz, N. Kopteva: Green's function estimates for a singularly perturbed convection-diffusion problem.", "Journal of Differential Equations, 252, 1521–1545, 2012 S. Franz, N. Kopteva: Green's function estimates for a singularly perturbed convection-diffusion problem.", "Journal of Differential Equations, 252, 1521–1545, 2012 [The article is removed from this electronic version due to copyright reasons.]", "Abstract: We consider a singularly perturbed convection-diffusion problem posed in the unit square with a horizontal convective direction.", "Its solutions exhibit parabolic and exponential boundary layers.", "Sharp estimates of the Green's function and its first- and second-order derivatives are derived in the $L_1$ norm.", "The dependence of these estimates on the small diffusion parameter is shown explicitly.", "The obtained estimates will be used in a forthcoming numerical analysis of the considered problem.", "Keywords: Green's function, singular perturbations, convection-diffusion Mathematics Subject Classification (2000): 35J08, 35J25, 65N15 DOI: 10.1016/j.jde.2011.07.033 tocsectionA.8   S. Franz, N. Kopteva: On the Sharpness of Green's function estimates for a convection-diffusion problem.", "In Proceedings of the Fifth Conference on Finite Difference Methods: Theory and Applications (FDM: T&A 2010), 44–57, Rousse University Press, 2011 S. Franz, N. Kopteva: On the Sharpness of Green's function estimates for a convection-diffusion problem.", "In Proceedings of the Fifth Conference on Finite Difference Methods: Theory and Applications (FDM: T&A 2010), 44–57, Rousse University Press, 2011 [The article is removed from this electronic version due to copyright reasons.]", "Abstract: Linear singularly perturbed convection-diffusion problems with characteristic layers are considered in three dimensions.", "We demonstrate the sharpness of our recently obtained upper bounds for the associated Green's function and its derivatives in the $L_1$ norm.", "For this, in this paper we establish the corresponding lower bounds.", "Both upper and lower bounds explicitly show any dependence on the singular perturbation parameter.", "Keywords: Green's function, singular perturbations, convection-diffusion, a posteriori error estimates Mathematics Subject Classification (2000): 35J08, 35J25, 65N15 ISBN: 978-954-8467-44-5 Erklärung Hiermit versichere ich, dass ich die vorliegene Arbeit ohne unzulässige Hilfe Dritter und ohne Benutzung anderer als der angegebenen Hilsmittel angefertigt habe.", "Die aus fremden Quellen direkt oder indirekt übernommenen Gedanken sind als solche kenntlich gemacht.", "Die Arbeit wurde bisher weder im In- noch im Ausland in gleicher oder ähnlicher Form einer anderen Prüfungsbehörde vorgelegt.", "Dresden, den 31.05.2013." ] ]
1403.0407
[ [ "Spin filtering in a magnetic barrier structure: in-plane spin\n orientation" ], [ "Abstract We investigate ballistic spin transport in a two dimensional electron gas system through magnetic barriers of various geometries using the transfer matrix method.", "While most of the previous studies have focused on the effect of magnetic barriers perpendicular to the two dimensional electron gas plane, we concentrate on the case of magnetic barriers parallel to the plane.", "We show that resonant oscillation occurs in the transmission probability without electrostatic potential modulation which is an essential ingredient in the case of ordinary out-of-plane magnetic barriers.", "Transmission probability of the in-plane magnetic barrier structure changes drastically according to the number of barriers and also according to the electrostatic potential modulation applied in the magnetic barrier region.", "Using a hybrid model consisting of a superconductor, ferromagnets, and a two dimensional electron gas plane, we show that it can serve as a good in-plane oriented spin selector which can be operated thoroughly by electrical modulation without any magnetic control." ], [ "Spin filtering in a magnetic barrier structure: in-plane spin orientation Nammee KimElectronic mail: [email protected] and Heesang Kim Department of Physics, Soongsil University, Seoul 156-743, Korea We investigate ballistic spin transport in a two dimensional electron gas system through magnetic barriers of various geometries using the transfer matrix method.", "While most of the previous studies have focused on the effect of magnetic barriers perpendicular to the two dimensional electron gas plane, we concentrate on the case of magnetic barriers parallel to the plane.", "We show that resonant oscillation occurs in the transmission probability without electrostatic potential modulation which is an essential ingredient in the case of ordinary out-of-plane magnetic barriers.", "Transmission probability of the in-plane magnetic barrier structure changes drastically according to the number of barriers and also according to the electrostatic potential modulation applied in the magnetic barrier region.", "Using a hybrid model consisting of a superconductor, ferromagnets, and a two dimensional electron gas plane, we show that it can serve as a good in-plane oriented spin selector which can be operated thoroughly by electrical modulation without any magnetic control.", "Semiconductor device including magnetic barriers has recently attracted much attention as a spin device, because it circumvents the resistance mismatch problem in the spin injection process, which is one of the main obstacles in realization of the Datta-Das-type spin transistor[1].", "Very recently the magnetic barrier study is expanded to a graphene[2] and a topological insulator[3] with great interests.", "Magnetic barrier structure has been introduced by using vortices in superconductors, superconducting(SC) masks or ferromagnetic material stripes on a two dimensional electron gas(2DEG).", "Since the 2DEG having a perpendicular magnetic field has been studied intensively in experiments, for example, Quantum Hall effect, observation of Commensurability effects and Novel giant magnetoresistance, most of the previous theoretical studies[4], [5], [6] on a magnetic barrier structure have carried on the out-of-plane magnetic barrier system with a purpose to use the system as an out-of plane oriented spin filter or a spin injection device.", "However, in some experiments like those on Spin valve and Spin Hall Effect[7], [8], [9], [10], spin orientation is along the 2DEG plane and an in-plane oriented spin filter or a spin injection device is in order, i.e., an in-plane magnetic barrier system start to draw attention.", "[11], [12], [13].", "The aim of our work is to investigate the ballistic transport properties through an in-plane magnetic barrier system in 2DEG.", "In this system, it is easier to have tunneling process compared to an out-of-plane magnetic barrier system because the in-plane barrier system has no unwanted magnetic barrier due to vector potential[5].", "We calculate spin-dependent transmission coefficients for double and triple in-plane magnetic barrier systems with/without external electrostatic modulation across a barrier using transfer matrix method.", "Schematic illustration for the possible realization of the in-plane magnetic barrier system is shown in Fig.", "1(a).", "The device consists of ferromagnets(FM) and a superconducting(SC) mask having openings on top of a 2DEG system.", "The SC mask is used to provide the magnetic field profile demanded in the system as shown in Fig.", "1(b).", "Figure: (a) Schematic illustration for a possible realization of the in-plane magnetic field barriers.The device consists of ferromagnets and a SC mask on top ofa 2DEG system.", "LL is the distance between openings in the SC mask and dd is the opening width.", "(b) Dotted line indicates the in-plane magnetic field profile in the 2DEG under the SC mask.The Hamiltonian with effective mass $m^*$ , and effective $g$ -factor $g^*$ with electrostatic potential $U(x)$ is; $H= \\frac{(\\vec{p}+ e/c \\vec{A})^2}{2m^*}+U(x) + \\frac{e\\hbar g^* \\sigma }{4m_{0}c}B_{y}(x),$ where $\\sigma =+1/-1$ denotes spin up/spin down of the electron.", "The in-plane magnetic barrier, which is square-function like, assumes the magnetic field $B_{y}$ along the y direction at two locations $x=-L/2$ and $L/2$ ; $B_{y}(x)=[B\\Theta (|x|-L/2) \\Theta (L/2+d-|x|)].$ Here, $\\Theta (x)$ is the Heaviside step function, B is the magnetic field strength in barriers, $L$ is the distance between the openings in the SC mask and $d$ is the width of each opening.", "In Fig.", "1(b), the dotted line indicates the in-plane magnetic profile in the 2DEG under the SC mask as written in Eq.", "(REF ).", "Vector potential $\\vec{A}$ , in Landau gauge, is given by $\\vec{A}=\\int _{0}^{z} B_{y}(x) dz^\\prime \\,\\hat{i}=[B\\Theta (|x|-L/2) \\Theta (L/2+d-|x|)]z \\,\\hat{i}$ , which vanishes at the 2DEG plane ($z=0$ ).", "As a result, the transverse motion is decoupled from the longitudinal one.", "This formalism also applies to $\\vec{B}=B(x)\\hat{i}$ with $\\vec{A}=\\int _{0}^{z} B_{x} dz^{\\prime }\\,\\hat{j}$ , which provides longitudinal spin orientation in 2DEG.", "The system is translation-invariant along the y direction, and the Schrödinger equation $H\\Psi (x,y)=E\\Psi (x,y)$ in two dimensional space is simplified by using $\\Psi (x,y)=e^{i k_{y} y}\\psi (x)$ ; $[\\frac{d^2}{dx^2}-k_{y}^2 +2(E-U(x))-\\frac{m^*}{m_{0}}\\frac{\\sigma g^*}{2}B_{y}(x)]\\psi (x)=0,$ where we use units of length $l_{B}=\\sqrt{\\hbar c/eB_{0}}=1$ , energy $\\hbar \\omega _{c}=\\hbar e B_{0}/m^{*} c=1$ , and $B_{0}$ is the magnetic field scaling unit.", "In the out-of-plane magnetic barrier system, it is essential to include electrostatic potential $U(x)$ in order to compensate for unwanted step-like potential barriers coming from the vector potential $\\vec{A}$[5].", "However, the vector potential does not appear in Eq.", "(REF ) since it becomes zero at the 2DEG plane and, therefore, it is not essential to include electrostatic potential $U(x)$ in the in-plane magnetic barrier system.", "Notice that the Zeeman term in Eq.", "(REF ) plays a role of an effective potential barrier for spin-up($\\sigma =+1$ ) electrons, while it acts like an effective potential well for spin-down($\\sigma =-1$ ) electrons.", "Hereinafter, a magnetic barrier should be interpreted as an effective potential well for a spin-down($\\sigma =-1$ ) electron.", "Based on the above Shrödinger equation, transmission probability is calculated by the standard transfer matrix method.", "Transfer matrices for the magnetic barriers $T_b$ and for the well confined by the barriers $T_{w}$ are ; $T_{b} &=&\\left( \\begin{array}{cc}\\cosh \\kappa d & (1/\\kappa )\\sinh \\kappa d \\\\\\kappa \\sinh \\kappa d & \\cosh \\kappa d\\end{array} \\right),\\\\T_{w} &=&\\left( \\begin{array}{cc}\\cos k d & (1/k)\\sin k d \\\\-k \\sin k d & \\cos k d\\end{array} \\right),$ where $\\kappa =\\sqrt{2(U(x)-E)+k_{y}^2+\\frac{m^*}{m_{0}}\\frac{\\sigma g^*}{2}B}$ and $k=\\sqrt{2(E-U(x))+k_{y}^2}$ .", "The transmission probability $T^{\\sigma }(E, k_{y})$ is obtained from the transmission coefficient $\\tau $ of the wavefunction after tunneling through the magnetic barriers by using the transfer matrices.", "$\\tau &=&\\frac{2 i k(C_{12}C_{21}-C_{11}C_{22})}{(C_{21}-k^{2}C_{12})-i k (C_{11}+C_{22})},$ where $C_{ij}$ are elements of the transfer matrix $C=T_{b}\\cdot T_{w}\\cdot T_{b}$ for double barriers.", "In our numerical calculation, since the better spin filtering effect is expected in a material with large $g^{*}$ and $m^{*}$[5], [6], material parameters of HgCdTe are used as follows: the effective mass $m^{*}=0.01m_{0}$ , $g$ -factor $g^{*}=100$ , energy unit $E_{0}=2.32$ meV, magnetic length $l_{B}=57.5$ nm, and the magnetic scaling unit $B_{0}=0.2$ T. Figure: Transmission Probability T σ T^{\\sigma } for the in-plane double magnetic barrier structureas a function of incident electron's energy.", "(a) B=5B=5, L=0.7L=0.7 and d=0.7d=0.7, and(b) B=5B=5, L=0.7L=0.7 and d=1.4d=1.4 in dimensionless unit.Figure 2 shows the transmission probability $T^{\\sigma }(E,0)$ of the in-plane double magnetic barrier system with $U(x)=0$ for two different magnetic barrier widths.", "For minimum energy requirement for electron tunneling, $k_{y}=0$ is chosen for qualitative calculation.", "The transmission probability of both spins shows clear oscillating behavior as a function of incident electron's energy, even when the electrostatic potential $U(x)$ is absent.", "In Fig.", "2(a), sharp resonance peaks are clearly seen for both spin-up and spin-down electrons.", "As magnetic barrier width $d$ increases, however, the spin-down resonance peak is broadened due to the lack of barrier formation.", "In Fig.", "2(b), since the total structure length increases as the barrier width increases, energy difference between two resonant peaks for a spin-up electron decreases and the better negative spin polarization is achieved around $E/E_{0} \\sim 1.5$ compared to the case of Fig.", "2(a).", "Figure 3 shows the transmission probability $T^{\\sigma }(E,0)$ and corresponding tunneling spin polarization $P(E, 0)$ for an in-plane triple magnetic barrier system as a function of incident electron's energy with electrostatic potential $U(x)=U_{0}\\Theta (|x|-L/2) \\Theta (L/2+d-|x|)$ applied.", "Figure: (a) Transmission probability T σ T^{\\sigma },(b) corresponding tunneling spin polarization for a in-plane triple magnetic barrier structure with U(x)as a function of incident electron's energy E/E 0 E/E_{0}.Parameters, B=2.5B=2.5, L=0.7L=0.7, d=0.7d=0.7, k y =0k_{y}=0 and U 0 =±0.9BU_0=\\pm 0.9B are used in the dimensionless unit.The tunneling spin polarization is defined by $P(E, k_{y})&=&\\frac{T^{+} (E,k_{y})-T^{-} (E,k_{y})}{T^{+} (E,k_{y})+T^{-} (E,k_{y})}.$ As the number of barriers increases, a resonance peak splitting appears clearly.", "In the triple case, $C_{ij}$ in Eq.", "(REF ) are obtained from $C=T_{b}\\cdot T_{w}\\cdot T_{b}\\cdot T_{w}\\cdot T_{b}$ .", "Since $U(x)$ affects on the height of effective potential barrier, when positive electrostatic potential $U(x)$ is applied at the magnetic barrier region, resonant peaks move to higher energy than negative $U(x)$ case.", "When $U_{0}> |\\frac{m^*}{m_{0}}\\frac{\\sigma g^*}{2}B|$ , the resultant effective potential for spin-down electrons becomes positive, and the spin-down electrons experience a barrier instead of a well.", "As a result, the resonance peak splitting appears also in the spin-down case, which is shown as the thin dashed line in Fig.", "3(a).", "Figure 3(b) shows tunneling spin polarization corresponding to Fig.", "3(a).", "Notice that in the low energy regime of $E/E_{0} \\le 1.5$ , the spin polarization can be switched between spin-up and spin-down by reversing the sign of the electric potential $U(x)$ .", "This implies that in-plane spin orientation of injected currents through the triple magnetic barrier structure can be manipulated electrically.", "In conclusion, the transmission properties of a two-dimensional electron gas system with in-plane magnetic barriers are investigated.", "Spin dependent resonance oscillation occurs in the transmission probability even without electrostatic potential applied, although it can be used to control the spin current.", "The transmission property and the current spin polarization can be manipulated efficiently by the number of barriers as well as by electrostatic potential modulation.", "As a result of this work, the in-plane triple magnetic barrier structure can serve as a good in-plane oriented spin selector which can be operated thoroughly by electrical modulation without any magnetic control.", "We are grateful to J. W. Kim for helpful discussions.", "This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology (grant 2012R1A1A2006303 and 2010-0021328)." ] ]
1403.0067
[ [ "High-fidelity contact pseudopotentials and p-wave superconductivity" ], [ "Abstract We develop ultratransferable pseudopotentials for the contact interaction that are 100 times more accurate than contemporary approximations.", "The pseudopotential offers scattering properties very similar to the contact potential, has a smooth profile to accelerate numerics by a factor of up to 4,000, and, for positive scattering lengths, does not support an unwanted bound state.", "We demonstrate these advantages in a Diffusion Monte Carlo study of fermions with repulsive interactions, delivering the first numerical evidence for the formation of a p-wave superconducting state." ], [ "Derivation of the pseudopotential", "To construct the pseudopotential we study the two-body problem: two fermions in their center-of-mass frame with wavevector $k\\ge 0$ and angular momentum quantum number $\\ell $ .", "The Hamiltonian in atomic units ($\\hbar =m=1$ ) is $-\\frac{\\nabla ^2}{2}\\psi +V(r)\\psi =\\frac{k^2}{2}\\psi $ , with the contact potential $V(r)=2\\pi a\\delta (r)\\frac{\\partial }{\\partial r}r$  [35].", "The scattering states are $\\psi ^{\\mathrm {cont}}_{k,\\ell }=\\sin [kr-\\delta ^\\mathrm {cont}_\\ell ]/kr$ , where $\\delta _\\ell $ is the scattering phase shift in angular momentum channel $\\ell $ .", "We seek a pseudopotential that (i) reproduces the correct phase shifts over the range of wavevectors $0<k\\lesssim 2k_\\mathrm {F}$ , where $k_\\mathrm {F}$ is the Fermi wavevector, (ii) supports no superfluous bound states to be compatible with ground state methods and (iii) is smooth and broad to accelerate numerical calculations.", "We first focus on positive scattering lengths $a > 0$ , with no bound state.", "We describe four families of pseudopotentials: hard sphere, soft sphere (top hat), the Troullier-Martins form of norm-conserving pseudopotentials [36], [37] and the new ultratransferable pseudopotential.", "The usual approach [1], [8] starts from the low energy expansion for the $s$ -wave scattering phase shift $\\cot \\delta _0=-\\frac{1}{ka}+\\frac{1}{2}kr_\\mathrm {eff}+\\mathcal {O}(k^3)$ where $r_\\mathrm {eff}$ is the “effective range” of the potential.", "For a contact potential, $r_\\mathrm {eff}$ and all higher order terms are zero.", "Perhaps the simplest pseudopotential is a hard sphere potential with radius $a$ .", "This reproduces the correct scattering length $a$ , thus delivering the correct phase shift for $k=0$ .", "However, the hard sphere has an effective range $r_\\mathrm {eff}=2a/3$ .", "Fig.", "REF (c) shows that this causes significant deviations in scattering power for $k>0$ .", "To improve the scattering phase shift, Ref.", "[8] adopted a soft sphere potential: $V(r)=V_0\\Theta (r-R)$ , with $V_0$ and $R$ chosen to reproduce the correct scattering length $a=R(1-\\tanh \\gamma /\\gamma )$ and effective range $r_\\mathrm {eff}=R[1+\\frac{3\\tanh \\gamma -\\gamma (3+\\gamma ^2)}{3\\gamma (\\gamma -\\tanh \\gamma )^2}]=0$ , where $\\gamma =R\\sqrt{2V_0}$ .", "The first two terms in the low energy expansion of the phase shift are now correct, leading to a small reduction in phase shift error in Fig.", "REF (c).", "The two potentials considered so far display incorrect behavior for larger wavevectors due to the focus on reproducing the correct $k=0$ scattering behavior.", "To improve the accuracy we turn to the Troullier-Martins [36] formalism developed for constructing attractive electron-ion pseudopotentials.", "These pseudopotentials reproduce both the correct phase shift and its derivative with respect to energy at a prescribed calibration energy (when constructing an electron-ion pseudopotential, this is the bound state energy in an isolated atom [37], [38], [39], [40], [41], [42]).", "By calibrating at the energy corresponding to the median incident scattering wavevector $k=k_\\mathrm {F}$ , we reduce the errors in the scattering phase shift over a broad range of wavevectors.", "This delivers the pseudopotential shown in Fig.", "REF (a) that is smooth, leading to improved numerical stability and efficiency.", "Fig.", "REF (c) demonstrates that this potential is exact at the calibration wavevector $k=k_\\mathrm {F}$ and delivers a marked decrease in phase shift error across all wavevectors.", "The three potentials deliver a significant progression in accuracy.", "The hard sphere potential reproduces the correct scattering behavior at $k=0$ .", "Both the soft sphere and Troullier-Martins potential are transferable: the former producing correct scattering around $k=0$ and the latter around $k\\sim k_\\mathrm {F}$ The significant improvement delivered by the Troullier-Martins potential encourages us to develop the formalism to propose an ultratransferable pseudopotential that produces accurate phase shifts over all of the wavevectors occupied in a Fermi gas.", "To develop ultransferable pseudopotentials we continue to focus on the contact potential, though the methodology can be readily generalized to other interparticle interactions.", "We construct a pseudopotential that is identical to the contact potential outside of a cutoff radius $r_c$ , but inside has a continuous first derivative at both $r=0$ and $r=r_c$ , V(r)EF= {ll (1-rrc)2 [v1(12+rrc)+ i=2Nvvi(rrc)i]rrc 0r>rc ,.", "with $N_{\\text{v}}=9$ .", "We choose the cutoff radius to correspond to the first anti-node of the true wavefunction By choosing a cutoff that is beyond the first node in the wavefunction, we guarantee that the pseudopotential will not harbor a bound state, as demonstrated in Fig.", "REF (a).", "We calculate the scattering solution $\\psi ^\\mathrm {PP}_{k,\\ell }(r)$ of the pseudopotential numerically to determine the phase shift $\\delta _\\ell ^\\mathrm {PP}(k(k))$ .", "The difference in the scattering phase shift $\\delta _\\ell $ of the potentials is characterized by the mean squared error in the phase shifts at the cutoff radius, (PP-cont)2= 02kF[ PP(k)- cont(k) ]2dk , that is integrated over all wavevectors $0\\le k\\le 2k_\\mathrm {F}$ of interest.", "The integrand can be convolved with a density of states to emphasize $k$ values of interest.", "We seek the variational parameters $\\lbrace v_{i}\\rbrace $ that minimize the deviation $\\langle (\\delta _\\ell ^\\mathrm {PP}-\\delta _\\ell ^\\mathrm {cont})^2\\rangle $ to determine the pseudopotential that delivers the best approximation for the contact potential.", "As demonstrated in Fig.", "REF (c), this potential delivers an error in $\\delta _0$ of less than $10^{-3}$ for all wavevectors $0\\le k\\le 2k_\\mathrm {F}$ found in a Fermi gas, corresponding to an improvement of two orders of magnitude over previously used pseudopotentials.", "The pseudopotentials constructed will have finite scattering amplitude in the p-wave and higher angular momentum channels.", "The contact potential, by contrast, scatters only in the s-wave channel $| s \\rangle $ .", "This can be solved by using a non-local pseudopotential [43], [44] $\\hat{V}=| s \\rangle V(r)\\langle s |$ , where $\\langle \\mathbf {r} | s \\rangle =\\mathrm {Y}_0^0(\\mathbf {r})$ , with the spherical harmonic $\\mathrm {Y}_0^0$ centered on either of the interacting particles.", "This potential only acts on the s-wave component of the relative wavefunction.", "Additional accuracy could also be gained by using different projectors for different energy ranges [45], [46].", "Attractive branch: We can use a similar procedure to derive pseudopotentials for the attractive branch $a<0$ .", "For the attractive case, the cutoff can be arbitrarily reduced to generate a potential that tends to the contact limit, at the cost of computational efficiency.", "For example, in Monte Carlo simulations, the sampling efficiency is approximately proportional to $r_c^3$ .", "In Fig.", "REF we adopt a cutoff $r_c=1/2k_\\mathrm {F}$ , and compare to the square well potential with cutoff $r_c=0.01\\@root 3 \\of {3\\pi ^2}/k_\\mathrm {F}$ in Ref. [3].", "Both the Troullier-Martins pseudopotential and the ultratransferable pseudopotential have an average error approximately 10 times smaller than the square well potential, but their larger cutoff allows them to be sampled 4,000 times more efficiently.", "Bound state: To construct a pseudopotential for the bound state (corresponding to $a>0$ ), we follow the Troullier-Martins prescription [36].", "We calibrate the pseudopotential at the binding energy $E=-1/2a^2$ .", "The cutoff is constructed in the same manner as for the attractive branch, delivering a similar improvement in efficiency." ], [ "Atoms in a trap", "We have developed a pseudopotential that delivers the correct scattering phase shift for an isolated system.", "To test the pseudopotential we turn to an experimentally realizable configuration [47], [48]: two atoms in a spherical harmonic trap with frequency $\\omega $ and characteristic length $d=1/\\sqrt{\\omega }$ .", "For all three types of interaction shown in Fig.", "REF (d) this system has an analytical solution [35] that we can benchmark against, forming an ideal test in an inhomogeneous environment.", "Moreover, the exact solution extends to excited states, allowing us to test the performance of the pseudopotential across a wide range of energy levels to provide a firm foundation from which to study the many-body system.", "Ground state: We first compare the pseudopotential estimates of the ground state energy to the exact analytical solution [35].", "For the repulsive and attractive branches the hard/soft sphere potentials deliver $\\sim 1\\%$ error in the energy, whilst both the Troullier-Martins and ultratransferable pseudopotentials (shown in Fig.", "REF (a,b)) are significantly more accurate with a $\\sim 0.01\\%$ error.", "Finally we examine the bound state energy in Fig.", "REF (c).", "Both the square well and Troullier-Martins formalism give the exact ground state energy for two atoms in isolation.", "However, the trapping potential introduces inhomogeneity, so the square well potential gives a $\\sim 10\\%$ error in the ground state energy, whereas the Troullier-Martins pseudopotential gives a $\\sim 0.01\\%$ error.", "This affirms the benefits of using a pseudopotential that is robust against changes in the local environment.", "The success of the Troullier-Martins and ultratransferable formalism at describing the ground state is all the more significant considering these pseudopotentials aim to describe the correct scattering properties over a range of energies.", "We would therefore expect them to perform even better when modeling the excited states of the trap.", "Excited states: We now turn to examine the predictions for the excited states in the repulsive and attractive branches.", "Due to the shell structure, the excited states of a few-body system are related to the ground state of a many-body system [7], allowing us to probe the performance expected from the pseudopotential in a many-body setting.", "We consider states up to a maximum energy $E_\\mathrm {max}=7.5\\hbar \\omega $ , corresponding to 112 non-interacting atoms in the trap.", "In Fig.", "REF (a,b) the Troullier-Martins pseudopotential has a mean squared error averaged over all bands below $E_\\mathrm {max}$ that is between 10 and 100 times lower than existing pseudopotentials.", "The ultratransferable pseudopotential is a further factor of 2 more accurate.", "Additionally, when modeling the attractive branch, the Troullier-Martins and the ultratransferable formalism are 4,000 times more efficient, due to their larger cutoffs." ], [ "Repulsive fermions", "Having verified that the pseudopotentials reproduce the correct scattering phase shift and bound state energy for two harmonically trapped atoms, we now exploit their accuracy to study two unsolved questions in many-body ferromagnetic metals tuned near quantum criticality: the nature of the ferromagnetic phase transition and presence of p-wave superconducting correlations.", "Quantum Monte Carlo: We use fixed-node Diffusion Monte Carlo (DMC) [49] implemented in the casino code [50], with a trial wavefunction $\\Psi =\\mathrm {e}^JD_\\uparrow D_\\downarrow $ , where $D_\\alpha $ denotes a Slater determinant of $N_\\alpha $ plane waves.", "The Jastrow factor is taken to be J=ji ,{,} (1-|ri-rj|Lu)2 u(|ri-rj|)(Lu-|ri-rj|), where $u_{\\alpha \\beta }$ is a polynomial whose parameters we optimize in a Variational Monte Carlo (VMC) calculation and $L^u_{\\alpha \\beta }$ is a cutoff length [51].", "We model spin polarized systems by performing calculations for $N_\\uparrow =81$ and $N_\\downarrow \\in \\lbrace 81,57,33,27,19,7,1\\rbrace $ that correspond to filled shells.", "This guarantees that the trial wavefunction is an eigenstate of the total spin operator $\\hat{S}^2$ and the spatial symmetry operators of the cubic lattice.", "We use a backflow transformation [2], [52] in the construction of the orbitals that enter the Slater determinant, with the replacement $\\mathbf {r}_{i\\sigma } \\rightarrow \\mathbf {r}_{i\\sigma }+\\sum {\\begin{subarray}{l}j\\ne i\\\\\\alpha ,\\beta \\in \\lbrace \\uparrow ,\\downarrow \\rbrace \\end{subarray}}(\\mathbf {r}_i-\\mathbf {r}_j)\\eta ^{\\alpha \\beta }_{ij}(|\\mathbf {r}_i-\\mathbf {r}_j|)$ where $\\eta ^{\\alpha \\beta }_{ij}(r)=(1-r/L^\\eta _{\\alpha \\beta })^2\\Theta (L^\\eta _{\\alpha \\beta }-r) p_{\\alpha \\beta }(r)$ , $p_{\\alpha \\beta }$ is a polynomial whose parameters are optimized in VMC, and $L^\\eta _{\\alpha \\beta }$ is a cutoff length.", "We reduce finite size effects by twist averaging [54], [55], [53] and correct the non-interacting kinetic energy of the finite sized system with that of the corresponding infinite system [1].", "Ferromagnetic phase transition: In Fig.", "REF we observe a first order phase transition to a partially polarized state at $k_\\mathrm {F}a=0.71$ , markedly lower than previous DMC predictions of $k_\\mathrm {F}a\\sim 0.85$  [1], [8].", "The system becomes fully polarized at $k_\\mathrm {F}a=1.89$ , close to the theoretical prediction of 1.87 [56], [57].", "This is significantly larger than the values calculated previously using DMC [1], [8], demonstrating the quantitative benefits of using a high fidelity pseudopotential.", "The presence of the first order transition is consistent with theory [17], [8], [14], [16] and with the ferromagnetic transition seen in experiments on heavy fermion materials [58].", "P-wave superconductivity can be understood by considering two up-spin electrons in a fermionic gas with repulsive interactions, each surrounded by a fluctuating magnetic polarization cloud.", "As the electrons coalesce the magnetic fluctuations (that drove the first order ferromagnetic transition) reinforce to create an effective attractive interaction, inducing p-wave superconducting order [34], [59].", "The p-wave superconducting state has been observed in experiments on ferromagnetic superconductors [29], [30], [28], [31], [32], [33], [34], and has been modeled by a contact interaction in perturbation theory [22], [23], [24], [25], [26], [27], but has never been observed in numerics.", "Equipped with a pseudopotential that reproduces the contact interaction with high fidelity and whose broad profile leads to improved efficiency, we search for p-wave superconducting order.", "The p-wave superconducting order is defined by the order parameter $\\Delta _\\mathbf {k}=\\sum _{\\mathbf {k}^{\\prime }} V_{\\mathbf {k}\\mathbf {k}^{\\prime }}\\langle c_{\\mathbf {k}\\uparrow }c_{-\\mathbf {k}\\uparrow }\\rangle $ .", "This must be recast into an operator in the position representation and projected onto the p-wave channel.", "Effecting this transformation results in the projection of the off-diagonal long-range order in the two-body reduced density matrix onto the p-wave channel [61], [60], [10] |p|2=-(4kFa)2812 R rr'cr2 c-r2 cR+r'2 cR-r'2drdr', where $\\Omega $ is the simulation cell volume.", "The expectation value is zero for the Slater determinant trial wavefunction $D_{\\uparrow }D_{\\downarrow }$ with no electron-electron correlations.", "However, if we insert the full trial wavefunction $\\psi =\\text{e}^{J}D_{\\uparrow }D_{\\downarrow }$ into the expectation value and expand in the limit of small electron separation, we find that $\\langle |\\Delta _\\mathrm {p}|^2\\rangle \\approx 2^{10}3^{-15}5^{-2}7^{-1}(k_{\\text{F}}c)^{8}u_{\\uparrow \\uparrow }(0)$ , connecting the superconducting correlations to the up-spin correlation term in the Jastrow factor.", "This verifies that the trial wave function has the variational freedom to exhibit a superconducting instability.", "In Fig.", "REF we show the emergence of the p-wave superconducting order parameter with increasing interaction strength.", "The p-wave superconductor may be enhanced in the partially polarized phase [27], but is destroyed in the fully polarized state as there can be no magnetic fluctuations.", "The delicacy of the superconducting order requires a high-fidelity pseudopotential.", "The emergence of the p-wave superconducting order provides the first verification of the magnetic fluctuations theory valid at high interaction strengths, confirming the NMR measurements on UCoGe [34]." ], [ "Discussion", "We have developed a high fidelity pseudopotential for the contact interaction.", "The pseudopotential is ultratranserable, delivering accurate scattering properties over all wavevectors $0\\le k\\le 2k_\\mathrm {F}$ in a Fermi gas and its smoothness accelerates computation.", "This pseudopotential allowed us to characterize the first order itinerant ferromagnetic transition and present the first computational evidence for a p-wave superconducting state.", "The performance and portability of the pseudopotential makes it widely applicable across first principles methods including VMC, DMC, coupled cluster theory, and configuration interaction.", "The formalism developed can also be applied more widely in scattering problems in condensed matter to develop pseudopotentials, including the repulsive Coulomb interaction and dipolar interactions.", "The authors thank Stefan Baur, Andrew Green, Jesper Levinsen, Gunnar Möller, Michael Rutter, and Lukas Wagner for useful discussions, and acknowledge the financial support of the EPSRC and Gonville & Caius College." ] ]
1403.0047
[ [ "Observations and three-dimensional photoionization modelling of the\n Wolf-Rayet planetary nebula Abell 48" ], [ "Abstract Recent observations reveal that the central star of the planetary nebula Abell 48 exhibits spectral features similar to massive nitrogen-sequence Wolf-Rayet stars.", "This raises a pertinent question, whether it is still a planetary nebula or rather a ring nebula of a massive star.", "In this study, we have constructed a three-dimensional photoionization model of Abell 48, constrained by our new optical integral field spectroscopy.", "An analysis of the spatially resolved velocity distributions allowed us to constrain the geometry of Abell 48.", "We used the collisionally excited lines to obtain the nebular physical conditions and ionic abundances of nitrogen, oxygen, neon, sulphur and argon, relative to hydrogen.", "We also determined helium temperatures and ionic abundances of helium and carbon from the optical recombination lines.", "We obtained a good fit to the observations for most of the emission-line fluxes in our photoionization model.", "The ionic abundances deduced from our model are in decent agreement with those derived by the empirical analysis.", "However, we notice obvious discrepancies between helium temperatures derived from the model and the empirical analysis, as overestimated by our model.", "This could be due to the presence of a small fraction of cold metal-rich structures, which were not included in our model.", "It is found that the observed nebular line fluxes were best reproduced by using a hydrogen-deficient expanding model atmosphere as the ionizing source with an effective temperature of 70 kK and a stellar luminosity of 5500 L_sun, which corresponds to a relatively low-mass progenitor star (~ 3 M_sun) rather than a massive Pop I star." ], [ "Introduction", "The highly reddened planetary nebula Abell 48 (PN G029.0$+$ 00.4) and its central star (CS) have been the subject of recent spectroscopic studies [77], [10], [73], [21].", "The CS of Abell 48 has been classified as Wolf–Rayet [WN5] [73], where the square brackets distinguish it from the massive WN stars.", "Abell 48 was first identified as a planetary nebula (PN) by [1].", "However, its nature remains a source of controversy whether it is a massive ring nebula or a PN as previously identified.", "Recently, [77] described it as a spectral type of WN6 with a surrounding ring nebula.", "But, [73] concluded from spectral analysis of the CS and the surrounding nebula that Abell 48 is rather a PN with a low-mass CS than a massive (Pop I) WN star.", "Previously, [74] also associated the CS of PB 8 with [WN/C] class.", "Furthermore, IC 4663 is another PN found to possess a [WN] star [53].", "A narrow-band H$\\alpha $ +[N ii] image of Abell 48 obtained by [33] first showed its faint double-ring morphology.", "[81] identified it as a member of the elliptical morphological class.", "The H$\\alpha $ image obtained from the SuperCOSMOS Sky H$\\alpha $ Survey [54] shows that the angular dimensions of the shell are about 46$\\hbox{$^{\\prime \\prime }$}\\times $ 38$\\hbox{$^{\\prime \\prime }$}$ , and are used throughout this paper.", "The first integral field spectroscopy of Abell 48 shows the same structure in the H$\\alpha $ emission-line profile.", "But, a pair of bright point-symmetric regions is seen in [N ii] (see Fig.", "REF ), which could be because of the N$^{+}$ stratification layer produced by the photoionization process.", "A detailed study of the kinematic and ionization structure has not yet been carried out to date.", "This could be due to the absence of spatially resolved observations.", "Table: Journal of the IFU observations with the ANU 2.3-m Telescope.The main aim of this study is to investigate whether the [WN] model atmosphere from [73] of a low-mass star can reproduce the ionization structure of a PN with the features like Abell 48.", "We present integral field unit (IFU) observations and a three-dimensional photoionization model of the ionized gas in Abell 48.", "The paper is organized as follows.", "Section presents our new observational data.", "In Section  we describe the morpho-kinematic structure, followed by an empirical analysis in Section .", "We describe our photoionization model and the derived results in Sections  and , respectively.", "Our final conclusion is stated in Section ." ], [ "Observations and data reduction", "Integral field spectra listed in Table REF were obtained in 2010 and 2012 with the 2.3-m ANU telescope using the Wide Field Spectrograph [11], [12].", "The observations were done with a spectral resolution of $R\\sim 7000$ in the 441.5–707.0 nm range in 2010 and $R\\sim 3000$ in the 329.5–932.6 nm range in 2012.", "The WiFeS has a field-of-view of $25\\hbox{$^{\\prime \\prime }$}\\times 38\\hbox{$^{\\prime \\prime }$}$ and each spatial resolution element of $10\\times 05$ (or $1\\hbox{$^{\\prime \\prime }$}\\times 1\\hbox{$^{\\prime \\prime }$}$ ).", "The spectral resolution of $R\\,(=\\lambda /\\Delta \\lambda )\\sim 3000$ and $R\\sim 7000$ corresponds to a full width at half-maximum (FWHM) of $\\sim 100$ and 45 km s${}^{-1}$ , respectively.", "We used the classical data accumulation mode, so a suitable sky window has been selected from the science data for the sky subtraction purpose.", "Table: Observed and dereddened relative line fluxes of the PN Abell 48, on a scale where Hβ=100\\beta =100.Uncertain and very uncertain values are followed by `:' and `::', respectively.The symbol `*' denotes blended emission lines.The positions observed on the PN are shown in Fig.", "REF (a).", "The centre of the IFU was placed in two different positions in 2010 and 2012.", "The exposure time of 20 min yields a signal-to-noise ratio of $S/N \\gtrsim 10$ for the [O iii] emission line.", "Multiple spectroscopic standard stars were observed for the flux calibration purposes, notably Feige 110 and EG 274.", "As usual, series of bias, flat-field frames, arc lamp exposures, and wire frames were acquired for data reduction, flat-fielding, wavelength calibration and spatial calibration.", "Data reductions were carried out using the iraf pipeline wifes (version 2.0; 2011 Nov 21).IRAF is distributed by NOAO, which is operated by AURA, Inc., under contract to the National Science Foundation.", "The reduction involves three main tasks: WFTABLE, WFCAL and WFREDUCE.", "The iraf task WFTABLE converts the raw data files with the single-extension Flexible Image Transport System (FITS) file format to the Multi-Extension FITS file format, edits FITS file key headers, and makes file lists for reduction purposes.", "The iraf task WFCAL extracts calibration solutions, namely the master bias, the master flat-field frame (from QI lamp exposures), the wavelength calibration (from Ne–Ar or Cu–Ar arc exposures and reference arc) and the spatial calibration (from wire frames).", "The iraf task WFREDUCE applies the calibration solutions to science data, subtracts sky spectra, corrects for differential atmospheric refraction, and applies the flux calibration using observations of spectrophotometric standard stars.", "Figure: Maps of the PN Abell 48 in Hα\\alpha λ\\lambda 6563 Å (top) and [[N ii]] λ\\lambda 6584 Å (bottom) from the IFU ( PA =0 ∘ {\\rm PA}=0^{\\circ }) taken in 2010 April.", "Fromleft to right: spatial distribution maps of flux intensity, continuum, LSR velocity and velocitydispersion.", "Flux unit is in 10 -15 10^{-15} erg s -1 {}^{-1} cm -2 {}^{-2} spaxel -1 {}^{-1}, continuum in 10 -15 10^{-15} erg s -1 {}^{-1} cm -2 {}^{-2} Å -1 {}^{-1} spaxel -1 {}^{-1}, and velocities in km s -1 {}^{-1}.", "North is up and east is towards the left-hand side.The white contour lines show the distribution of the narrow-band emission of Hα\\alpha in arbitrary unit obtained from the SHS.A complete list of observed emission lines and their flux intensities are given in Table REF on a scale where H$\\beta $  = 100.", "All fluxes were corrected for reddening using $I(\\lambda )_{\\rm corr}=F(\\lambda )_{\\rm obs}10^{c({\\rm H}\\beta )[1+f(\\lambda )]}.$ The logarithmic $c({\\rm H}\\beta )$ value of the interstellar extinction for the case B recombination [70] has been obtained from the H$\\alpha $ and H$\\beta $ Balmer fluxes.", "We used the Galactic extinction law $f(\\lambda )$ of [29] for $R_V = A(V)/E(B-V)=3.1$ , and normalized such that $f({\\rm H}\\beta )=0$ .", "We obtained an extinction of $c({\\rm H}\\beta )=3.1$ for the total fluxes (see Table REF ).", "Our derived nebular extinction is in excellent agreement with the value derived by [73] from the stellar spectral energy (SED).", "The same method was applied to create $c({\\rm H}\\beta )$ maps using the flux ratio H$\\alpha $ /H$\\beta $ , as shown in Fig.", "REF (b).", "Assuming that the foreground interstellar extinction is uniformly distributed over the nebula, an inhomogeneous extinction map may be related to some internal dust contributions.", "As seen, the extinction map of Abell 48 depicts that the shell is brighter than other regions, and it may contain the asymptotic giant branch (AGB) dust remnants." ], [ "Kinematics", "Fig.", "REF shows the spatial distribution maps of the flux intensity, continuum, radial velocity and velocity dispersion of H$\\alpha $ $\\lambda $ 6563 and $[$ N ii$]$ $\\lambda $ 6584 for Abell 48.", "The white contour lines in the figures depict the distribution of the emission of H$\\alpha $ obtained from the SHS [54], which can aid us in distinguishing the nebular borders from the outside or the inside.", "The observed velocity $v_{\\rm obs}$ was transferred to the local standard of rest (LSR) radial velocity $v_{\\rm LSR}$ by correcting for the radial velocities induced by the motions of the Earth and Sun at the time of our observation.", "The transformation from the measured velocity dispersion $\\sigma _{\\rm obs}$ to the true line-of-sight velocity dispersion $\\sigma _{\\rm true}$ was done by $\\sigma _{\\rm true}=\\sqrt{\\sigma ^2_{\\rm obs}-\\sigma ^2_{\\rm ins}-\\sigma ^2_{\\rm th}}$ , i.e.", "correcting for the instrumental width (typically $\\sigma _{\\rm ins}\\approx 42$  km/s for $R\\sim 3000$ and $\\sigma _{\\rm ins}\\approx 18$  km/s for $R\\sim 7000$ ) and the thermal broadening ($\\sigma _{\\rm th}^2=8.3\\,T_{\\rm e}[{\\rm kK}]/Z$ , where $Z$ is the atomic weight of the atom or ion).", "Figure: (a) The shape mesh model before rendering at the best-fitting inclination and corresponding rendered model.", "(b) The normalized synthetic intensity map and the radial velocity map at the inclinationof -35 ∘ -35{}^{\\circ } and the position angle of 135 ∘ 135{}^{\\circ }, derived from the model (v sys =0v_{\\rm sys}=0), which can be compared directly with Fig.", ".We have used the three-dimensional morpho-kinematic modelling program shape (version 4.5) to study the kinematic structure.", "The program described in detail by [69] and [68], uses interactively moulded geometrical polygon meshes to generate the 3D structure of objects.", "The modelling procedure consists of defining the geometry, emissivity distribution and velocity law as a function of position.", "The program produces several outputs that can be directly compared with long slit or IFU observations, namely the position–velocity (P–V) diagram, the 2-D line-of-sight velocity map on the sky and the projected 3-D emissivity on the plane of the sky.", "The 2-D line-of-sight velocity map on the sky can be used to interpret the IFU velocity maps.", "For best comparison with the IFU maps, the inclination ($i$ ), the position angle `PA' in the plane of the sky, and the model parameters are modified in an iterative process until the qualitatively fitting 3D emission and velocity information are produced.", "We adopted a model, and then modified the geometry and inclination to conform to the observed H$\\alpha $ and [N ii] intensity and radial velocity maps.", "For this paper, the three-dimensional structure has then been transferred to a regular cell grid, together with the physical emission properties, including the velocity that, in our case, has been defined as radially outwards from the nebular centre with a linear function of magnitude, commonly known as a Hubble-type flow [67].", "The morpho-kinematic model of Abell 48 is shown in Fig.", "REF (a), which consists of a modified torus, the nebular shell, surrounded by a modified hollow cylinder and the faint outer halo.", "The shell has an inner radius of $10\\hbox{$^{\\prime \\prime }$}$ and an outer radius of $23\\hbox{$^{\\prime \\prime }$}$ and a height of $23\\hbox{$^{\\prime \\prime }$}$ .", "We found an expansion velocity of $v_{\\rm exp}=35\\pm 5$  km s${}^{-1}$ and a LSR systemic velocity of $v_{\\rm sys}=65 \\pm 5$  km s${}^{-1}$ .", "Our value of the LSR systemic velocity is in good agreement with the heliocentric systemic velocity of $v_{\\rm hel}=50.4\\pm 4.2$  km s${}^{-1}$ found by [73].", "Following [13], we estimated the nebula's age around 1.5 of the dynamical age, so the star left the top of the AGB around 8880 years ago.", "Fig.", "REF shows the orientation of Abell 48 on to the plane of the sky.", "The nebula has an inclination of $i=-35^{\\circ }$ between the line of sight and the nebular symmetry axis.", "The symmetry axis has a position angle of ${\\rm PA}=135^{\\circ }$ projected on to the plane of the sky, measured from the north towards the east in the equatorial coordinate system (ECS).", "The PA in the ECS can be transferred into the Galactic position angle (GPA) in the Galactic coordinate system (GCS), measured from the north Galactic pole (NGP; ${\\rm GPA}=0^{\\circ }$ ) towards the Galactic east (${\\rm GPA}=90^{\\circ }$ ).", "Note that ${\\rm GPA}=90^{\\circ }$ describes an alignment with the Galactic plane, while ${\\rm GPA}=0^{\\circ }$ is perpendicular to the Galactic plane.", "As seen in Table REF , Abell 48 has a GPA of 1978, meaning that the symmetry axis is approximately perpendicular to the Galactic plane.", "Based on the systemic velocity, Abell 48 must be located at less than 2 kpc, since higher distances result in very high peculiar velocities [46].", "However, it cannot be less than 1.5 kpc due to the large interstellar extinction.", "Using the infrared dust mapsWebsite: http://www.astro.princeton.edu/~schlegel/dust of [61], we found a mean reddening value of $E(B-V)=11.39 \\pm 0.64$ for an aperture of $10 \\hbox{$^\\prime $}$ in diameter in the Galactic latitudes and longitude of $(l,b)=(29.0,0.4)$ , which is within a line-of-sight depth of $\\lesssim 20$  kpc of the Galaxy.", "Therefore, Abell 48 with $E(B-V)\\simeq 2.14$ must have a distance of less than $3.3$ kpc.", "Considering the fact that the Galactic bulge absorbs photons overall 1.9 times more than the Galactic disc [14], the distance of Abell 48 should be around 2 kpc, as it is located at the dusty Galactic disc.", "Table: Kinematic results obtained for Abell 48 based on the morpho-kinematic model matched to the observed 2-D radial velocity map." ], [ "Plasma diagnostics", "The derived electron temperatures ($T_{\\rm e}$ ) and densities ($N_{\\rm e}$ ) are listed in Table REF , together with the ionization potential required to create the emitting ions.", "We obtained $T_{\\rm e}$ and $N_{\\rm e}$ from temperature-sensitive and density-sensitive emission lines by solving the equilibrium equations of level populations for a multilevel atomic model using equib code [30].", "The atomic data sets used for our plasma diagnostics from collisionally excited lines (CELs), as well as for abundances derived from CELs, are given in Table REF .", "The diagnostics procedure to determine temperatures and densities from CELs is as follows: we assume a representative initial electron temperature of 10 000 K in order to derive $N_{\\rm e}$ from $[$ S ii$]$ line ratio; then $T_{\\rm e}$ is derived from $[$ N ii$]$ line ratio in conjunction with the mean density derived from the previous step.", "The calculations are iterated to give self-consistent results for $N_{\\rm e}$ and $T_{\\rm e}$ .", "The correct choice of electron density and temperature is important for the abundance determination.", "We see that the PN Abell 48 has a mean temperature of $T_{\\rm e}([$ N ii$])=6980 \\pm 930 $  K, and a mean electron density of $N_{\\rm e}([$ S ii$])=750 \\pm 200$  cm${}^{-3}$ , which are in reasonable agreement with $T_{\\rm e}([$ N ii$])=7\\,200 \\pm 750$  K and $N_{\\rm e}([$ S ii$])=1000 \\pm 130$  cm${}^{-3}$ found by [73].", "The uncertainty on $T_{\\rm e}([$ N ii$])$ is order of 40 percent or more, due to the weak flux intensity of [N ii] $\\lambda $ 5755, the recombination contribution, and high interstellar extinction.", "Therefore, we adopted the mean electron temperature from our photoionization model for our CEL abundance analysis.", "Table: References for atomic data.Table REF also lists the derived He i temperatures, which are lower than the CEL temperatures, known as the ORL-CEL temperature discrepancy problem in PNe [43], [45].", "To determine the electron temperature from the He i $\\lambda \\lambda $ 5876, 6678 and 7281 lines, we used the emissivities of He I lines by [62], which also include the temperature range of $T_{\\rm e} < 5000$  K. We derived electron temperatures of $T_{\\rm e}({\\rm He~I})=5110$  K and $T_{\\rm e}({\\rm He~I})=4360$  K from the flux ratio He i $\\lambda \\lambda $ 7281/5876 and $\\lambda \\lambda $ 7281/6678, respectively.", "Similarly, we got $T_{\\rm e}({\\rm He~I})=6960$  K for He i $\\lambda \\lambda $ 7281/5876 and $T_{\\rm e}({\\rm He~I})=7510$  K for $\\lambda \\lambda $ 7281/6678 from the measured nebular spectrum by [73].", "Table: Diagnostics for the electron temperature, T e T_{\\rm e} and the electron density, N e N_{\\rm e}.", "References: D13 – this work; T13 – ." ], [ "Ionic and total abundances from ORLs", "Using the effective recombination coefficients (given in Table REF ), we determine ionic abundances, X${}^{i+}$ /H${}^{+}$ , from the measured intensities of optical recombination lines (ORLs) as follows: $\\frac{N({\\rm X}^{i+})}{N({\\rm H}^{+})}=\\frac{I({\\lambda })}{I({{\\rm H}\\beta })}\\frac{\\lambda ({\\rm {Å}})}{4861} \\frac{\\alpha _{\\rm eff}({\\rm H}\\beta )}{\\alpha _{\\rm eff}(\\lambda )},$ where $I({\\lambda })$ is the intrinsic line flux of the emission line $\\lambda $ emitted by ion ${\\rm X}^{i+}$ , $I({{\\rm H}\\beta })$ is the intrinsic line flux of H$\\beta $ , $\\alpha _{\\rm eff}({\\rm H}\\beta )$ the effective recombination coefficient of H$\\beta $ , and $\\alpha _{\\rm eff}(\\lambda )$ the effective recombination coefficient for the emission line $\\lambda $ .", "Abundances of helium and carbon from ORLs are given in Table REF .", "We derived the ionic and total helium abundances from He i $\\lambda $ 4471, $\\lambda $ 5876 and $\\lambda $ 6678 lines.", "We assumed the Case B recombination for the He i lines [57], [58].", "We adopted an electron temperature of $T_{\\rm e}=5\\,000$  K from He i lines, and an electron density of $N_{\\rm e}=1000$  cm${}^{-3}$ .", "We averaged the He${}^{+}$ /H${}^{+}$ ionic abundances from the He i $\\lambda $ 4471, $\\lambda $ 5876 and $\\lambda $ 6678 lines with weights of 1:3:1, roughly the intrinsic intensity ratios of these three lines.", "The total He/H abundance ratio is obtained by simply taking the sum of He${}^{+}$ /H${}^{+}$ and He${}^{2+}$ /H${}^{+}$ .", "However, He${}^{2+}$ /H${}^{+}$ is equal to zero, since He ii $\\lambda $ 4686 is not present.", "The C$^{2+}$ ionic abundance is obtained from C ii $\\lambda $ 6462 and $\\lambda $ 7236 lines.", "Table: Empirical ionic abundances derived from ORLs." ], [ "Ionic and total abundances from CELs", "We determined abundances for ionic species of N, O, Ne, S and Ar from CELs.", "To deduce ionic abundances, we solve the statistical equilibrium equations for each ion using equib code, giving level population and line sensitivities for specified $N_{\\rm e}=1000$  cm${}^{-3}$ and $T_{\\rm e}=10\\,000$  K adopted according to our photoionization modelling.", "Once the equations for the population numbers are solved, the ionic abundances, X${}^{i+}$ /H${}^{+}$ , can be derived from the observed line intensities of CELs as follows: $\\frac{N({\\rm X}^{i+})}{N({\\rm H}^{+})}=\\frac{I(\\lambda _{ij})}{I({{\\rm H}\\beta })}\\frac{\\lambda _{ij}({\\rm {Å}})}{4861} \\frac{\\alpha _{\\rm eff}({{\\rm H}\\beta })}{A_{ij}}\\frac{N_{\\rm e}}{n_i},$ where $I(\\lambda _{ij})$ is the dereddened flux of the emission line $\\lambda _{ij}$ emitted by ion ${\\rm X}^{i+}$ following the transition from the upper level $i$ to the lower level $j$ , $I({{\\rm H}\\beta })$ the dereddened flux of H$\\beta $ , $\\alpha _{\\rm eff}({{\\rm H}\\beta })$ the effective recombination coefficient of H$\\beta $ , $A_{ij}$ the Einstein spontaneous transition probability of the transition, $n_i$ the fractional population of the upper level $i$ , and $N_{\\rm e}$ is the electron density.", "Total elemental and ionic abundances of nitrogen, oxygen, neon, sulphur and argon from CELs are presented in Table REF .", "Total elemental abundances are derived from ionic abundances using the ionization correction factors ($icf$ ) formulas given by [37].", "The total O/H abundance ratio is obtained by simply taking the sum of the O$^{+}$ /H$^{+}$ derived from [O ii] $\\lambda \\lambda $ 3726,3729 doublet, and the O$^{2+}$ /H$^{+}$ derived from [O iii] $\\lambda \\lambda $ 4959,5007 doublet, since He ii $\\lambda $ 4686 is not present, so O${}^{3+}$ /H${}^{+}$ is negligible.", "The total N/H abundance ratio was calculated from the N$^{+}$ /H$^{+}$ ratio derived from the [N ii] $\\lambda \\lambda $ 6548,6584 doublet, correcting for the unseen N$^{2+}$ /H$^{+}$ using, $\\footnotesize \\frac{{\\rm N}}{{\\rm H}}=\\left(\\frac{{\\rm N}^{+}}{{\\rm H}^{+}}\\right) \\left(\\frac{{\\rm O}}{{\\rm O}^{+}}\\right).$ The Ne$^{2+}$ /H$^{+}$ is derived from [Ne iii] $\\lambda $ 3869 line.", "Similarly, the unseen Ne$^{+}$ /H$^{+}$ is corrected for, using $\\frac{{\\rm Ne}}{{\\rm H}}=\\left(\\frac{{\\rm Ne}^{2+}}{{\\rm H}^{+}} \\right)\\left(\\frac{{\\rm O}}{{\\rm O}^{2+}}\\right) .$ For sulphur, we have S$^{+}$ /H$^{+}$ from the [S ii] $\\lambda \\lambda $ 6716,6731 doublet and S$^{2+}$ /H$^{+}$ from the [S iii] $\\lambda $ 9069 line.", "The total sulphur abundance is corrected for the unseen stages of ionization using $\\footnotesize \\frac{{\\rm S}}{{\\rm H}}=\\left(\\frac{{\\rm S}^{+}}{{\\rm H}^{+}} + \\frac{{\\rm S}^{2+}}{{\\rm H}^{+}} \\right)\\left[1-\\left(1-\\frac{{\\rm O}^{+}}{{\\rm O}}\\right)^{3}\\right]^{-1/3}.$ The [Ar iii] 7136 line is only detected, so we have only Ar$^{2+}$ /H$^{+}$ .", "The total argon abundance is obtained by assuming Ar$^{+}$ /Ar = N$^{+}$ /N: $\\footnotesize \\frac{{\\rm Ar}}{{\\rm H}}=\\left(\\frac{{\\rm Ar}^{2+}}{{\\rm H}^{+}} \\right)\\left(1-\\frac{{\\rm N}^{+}}{{\\rm N}}\\right)^{-1}.$ As it does not include the unseen Ar$^{3+}$ , so the derived elemental argon may be underestimated.", "Figure: Ionic abundance maps of Abell 48.", "From left to right: spatial distribution maps of singly ionized Helium abundance ratio He + {}^{+}/H + {}^{+} from He i ORLs (4472, 5877, 6678); ionic nitrogen abundance ratio N + {}^{+}/H + {}^{+} (×10 -5 \\times 10^{-5}) from [[N ii]] CELs (5755, 6548, 6584); ionic oxygen abundance ratio O 2+ {}^{2+}/H + {}^{+} (×10 -4 \\times 10^{-4}) from [[O iii]] CELs (4959, 5007); and ionic sulphur abundance ratio S + {}^{+}/H + {}^{+} (×10 -7 \\times 10^{-7}) from [[S ii]] CELs (6716, 6731).", "North is up and east is towards the left-hand side.", "The white contour lines show the distribution of the narrow-band emission of Hα\\alpha in arbitrary unit obtained from the SHS.Fig.", "REF shows the spatial distribution of ionic abundance ratio He${}^{+}$ /H${}^{+}$ , N${}^{+}$ /H${}^{+}$ , O${}^{2+}$ /H${}^{+}$ and S${}^{+}$ /H${}^{+}$ derived for given $T_{\\rm e}=10000$  K and $N_{\\rm e}=1000$  cm$^{-3}$ .", "We notice that both O${}^{2+}$ /H${}^{+}$ and He${}^{+}$ /H${}^{+}$ are very high over the shell, whereas N${}^{+}$ /H${}^{+}$ and S${}^{+}$ /H${}^{+}$ are seen at the edges of the shell.", "It shows obvious results of the ionization sequence from the highly inner ionized zones to the outer low ionized regions.", "Table: Empirical ionic abundances derived from CELs." ], [ "Photoionization modelling", "The 3-D photoionization code mocassin [17], [15], [20] was used to study the best-fitting model for Abell 48.", "The code has been used to model a number of PNe, for example NGC 3918 [16], NGC 7009 [24], NGC 6302 [78], and SuWt 2 [7].", "The modelling procedure consists of defining the density distribution and elemental abundances of the nebula, as well as assigning the ionizing spectrum of the CS.", "This code uses a Monte Carlo method to solve self-consistently the 3-D radiative transfer of the stellar radiation field in a gaseous nebula with the defined density distribution and chemical abundances.", "It produces the emission-line spectrum, the thermal structure and the ionization structure of the nebula.", "It allows us to determine the stellar characteristics and the nebula parameters.", "The atomic data sets used for the calculation are energy levels, collision strengths and transition probabilities from the CHIANTI data base [40], hydrogen and helium free–bound coefficients of [18], and opacities from [76] and [75].", "The best-fitting model was obtained through an iterative process, involving the comparison of the predicted H$\\beta $ luminosity $L_{{\\rm H}\\beta }$ (erg s${}^{-1}$ ), the flux intensities of some important lines, relative to H$\\beta $ (such as $[$ O iii$]$ $\\lambda $ 5007 and $[$ N ii$]$ $\\lambda $ 6584), with those measured from the observations.", "The free parameters included distance and nebular parameters.", "We initially used the stellar luminosity ($L_{\\star }=6000$  L$_{\\bigodot }$ ) and effective temperature ($T_{\\rm eff}=70$ kK) found by [73].", "However, we slightly adjusted the stellar luminosity to match the observed line flux of $[$ O iii$]$ emission line.", "Moreover, we adopted the nebular density and abundances derived from empirical analysis in Section , but they have been gradually adjusted until the observed nebular emission-line spectrum was reproduced by the model.", "The best-fitting $L_{{\\rm H}\\beta }$ depends upon the distance and nebula density.", "The plasma diagnostics yields $N_{\\rm e} = 750$ –1000 cm$^{-3}$ , which can be an indicator of the density range.", "Based on the kinematic analysis, the distance must be less than 2 kpc, but more than 1.5 kpc due to the large interstellar extinction.", "We matched the predicted H$\\beta $ luminosity $L({\\rm H}\\beta )$ with the value derived from the observation by adjusting the distance and nebular density.", "Then, we adjusted abundances to get the best emission-line spectrum." ], [ "The ionizing spectrum", "The hydrogen-deficient synthetic spectra of Abell 48 was modelled using stellar model atmospheres produced by the Potsdam Wolf–Rayet (PoWR) models for expanding atmospheres [25], [26].", "It solves the non-local thermodynamic equilibrium (non-LTE) radiative transfer equation in the comoving frame, iteratively with the equations of statistical equilibrium and radiative equilibrium, for an expanding atmosphere under the assumptions of spherical symmetry, stationarity and homogeneity.", "The result of our model atmosphere is shown in Fig.", "REF .", "The model atmosphere calculated with the PoWR code is for the stellar surface abundances H:He:C:N:O = 10:85:0.3:5:0.6 by mass, the stellar temperature $T_{\\rm eff}$  = 70 kK, the transformed radius $R_{\\rm t}=0.54$  R${}_{\\bigodot }$ and the wind terminal velocity $v_{\\infty }=1000$  km s$^{-1}$ .", "The best photoionization model was obtained with an effective temperature of 70 kK [73] and a stellar luminosity of $L_{\\rm \\star }/$ L$_{\\bigodot }$ = 5500, which is close to $L_{\\star }/$ L$_{\\bigodot }$ = 6000 adopted by [73].", "This stellar luminosity was found to be consistent with the observed H$\\beta $ luminosity and the flux ratio of $[$ O iii$]$ /H$\\beta $ .", "A stellar luminosity higher than 5500 L$_{\\bigodot }$ produces inconsistent results for the nebular photoionization modelling.", "The emission-line spectrum produced by our adopted stellar parameters was found to be consistent with the observations.", "Table: Input parameters for the mocassin photoionization model." ], [ "The density distribution", "We initially used a three-dimensional uniform density distribution, which was developed from our kinematic analysis.", "However, the interacting stellar winds (ISW) model developed by [38] demonstrated that a slow dense superwind from the AGB phase is swept up by a fast tenuous wind during the PN phase, creating a compressed dense shell, which is similar to what we see in Fig.", "REF .", "Additionally, [34] extended the ISW model to describe a highly elliptical mass distribution.", "This extension later became known as the generalized interacting stellar winds theory.", "There are a number of hydrodynamic simulations, which showed the applications of the ISW theory for bipolar PNe [48], [49].", "As shown in Fig.", "REF , we adopted a density structure with a toroidal wind mass-loss geometry, similar to the ISW model.", "In our model, we defined a density distribution in the cylindrical coordinate system, which has the form $N_{\\rm H}(r) = N_{0}[ 1 + (r/r_{\\rm in})^{-\\alpha } ],$ where $r$ is the radial distance from the centre, $\\alpha $ the radial density dependence, $N_{0}$ the characteristic density, $r_{\\rm in} = r_{\\rm out}-\\delta r$ the inner radius, $r_{\\rm out}$ the outer radius and $\\delta r$ the thickness.", "The density distribution is usually a complicated input parameter to constrain.", "However, the values found from our plasma diagnostics ($N_{\\rm e}=750$ –1000 cm$^{-3}$ ) allowed us to constrain our density model.", "The outer radius and the height of the cylinder are equal to $r_{\\rm out}=23\\hbox{$^{\\prime \\prime }$}$ and the thickness is $\\delta r=13\\hbox{$^{\\prime \\prime }$}$ .", "The density model and distance (size) were adjusted in order to reproduce $I$ (H$\\beta )=1.355 \\times 10^{-10}$  erg s$^{-1}$  cm$^{-2}$ , dereddened using c(H$\\beta $ ) = 3.1 (see Section ).", "We tested distances, with values ranging from 1.5 to 2.0 kpc.", "We finally adopted the characteristic density of $N_{0}=600$  cm$^{-3}$ and the radial density dependence of $\\alpha =1$ .", "The value of 1.90 kpc found here, was chosen, because of the best predicted H$\\beta $ luminosity, and it is in excellent agreement with the distance constrained by the synthetic spectral energy distribution (SED) from the PoWR models.", "Once the density distribution and distance were identified, the variation of the nebular ionic abundances were explored.", "Table: Dereddened observed and predicted emission-line fluxes for Abell 48.", "References: D13 – this work; T13 – .", "Uncertain and very uncertain values are followed by `:' and `::', respectively.The symbol `*' denotes blended emission lines." ], [ "The nebular elemental abundances", "Table REF lists the nebular elemental abundances (with respect to H) used for the photoionization model.", "We used a homogeneous abundance distribution, since we do not have any direct observational evidence for the presence of chemical inhomogeneities.", "Initially, we used the abundances from empirical analysis as initial values for our modelling (see Section ).", "They were successively modified to fit the optical emission-line spectrum through an iterative process.", "We obtain a C/O ratio of 21 for Abell 48, indicating that it is predominantly C-rich.", "Furthermore, we find a helium abundance of 0.12.", "This can be an indicator of a large amount of mixing processing in the He-rich layers during the He-shell flash leading to an increase carbon abundance.", "The nebulae around H-deficient CSs typically have larger carbon abundances than those with H-rich CSs [9].", "The ${\\rm O}/{\\rm H}$ we derive for Abell 48 is lower than the solar value [2].", "This may be due to that the progenitor has a sub-solar metallicity.", "The enrichment of carbon can be produced in a very intense mixing process in the He-shell flash [28].", "Other elements seem to be also decreased compared to the solar values, such as sulphur and argon.", "Sulphur could be depleted on to dust grains [63], but argon cannot have any strong depletion by dust formation [64].", "We notice that the N/H ratio is about the solar value given by [2], but it can be produced by secondary conversion of initial carbon if we assume a sub-solar metallicity progenitor.", "The combined (C+N+O)/H ratio is by a factor of 3.9 larger than the solar value, which can be produced by multiple dredge-up episodes occurring in the AGB phase.", "Figure: Top: electron density and temperature as a function of radius along the equatorial direction.Bottom: ionic stratification of the nebula.", "Ionization fractions are shown for helium, carbon, oxygen, argon (left-hand panel), nitrogen, neon and sulphur (right-hand panel).Table: Fractional ionic abundances for Abell 48 obtained from the photoionization model." ], [ "Comparison of the emission-line fluxes", "Table REF compares the flux intensities predicted by the best-fitting model with those from the observations.", "Columns 2 and 3 present the dereddened fluxes of our observations and those from [73].", "The predicted emission-line fluxes are given in Column 4, relative to the intrinsic dereddened H$\\beta $ flux, on a scale where $I($ H$\\beta )$  = 100.", "The most emission-line fluxes presented are in reasonable agreement with the observations.", "However, we notice that the [O ii] $\\lambda $ 7319 and $\\lambda $ 7330 doublets are overestimated by a factor of 3, which can be due to the recombination contribution.", "Our photoionization code incorporates the recombination term to the statistical equilibrium equations.", "However, the recombination contribution are less than 30 per cent for the values of $T_{\\rm e}$ and $N_{\\rm e}$ found from the plasma diagnostics.", "Therefore, the discrepancy between our model and observed intensities of these lines can be due to inhomogeneous condensations such as clumps and/or colder small-scale structures embedded in the global structure.", "It can also be due to the measurement errors of these weak lines.", "The [O ii] $\\lambda \\lambda $ 3726,3729 doublet predicted by the model is around 25 per cent lower, which can be explained by either the recombination contribution or the flux calibration error.", "There is a notable discrepancy in the predicted [N ii] $\\lambda $ 5755 auroral line, being higher by a factor of $\\sim 3$ .", "It can be due to the errors in the flux measurement of the [N ii] $\\lambda $ 5755 line.", "The predicted [Ar iii] $\\lambda $ 7751 line is also 30 per cent lower, while [Ar iii] $\\lambda $ 7136 is about 20 per cent higher.", "The [Ar iii] $\\lambda $ 7751 line usually is blended with the telluric line, so the observed intensity of these line can be overestimated.", "It is the same for [S iii] $\\lambda $ 9069, which is typically affected by the atmospheric absorption band.", "Table: Integrated ionic abundance ratios for He, C, N, O, Ne, S and Ar, derived from model ionic fractions and compared to those from the empirical analysis." ], [ "Ionization and thermal structure", "The volume-averaged fractional ionic abundances are listed in Table REF .", "We note that hydrogen and helium are singly-ionized.", "We see that the O$^{+}$ /O ratio is higher than the N$^{+}$ /N ratio by a factor of 1.34, which is dissimilar to what is generally assumed in the $icf$ method.", "However, the O$^{2+}$ /O ratio is nearly a factor of 1.16 larger than the Ne$^{2+}$ /Ne ratio, in agreement with the general assumption for $icf$ (Ne).", "We see that only 19 per cent of the total nitrogen in the nebula is in the form of N$^{+}$ .", "However, the total oxygen largely exists as O$^{2+}$ with 70 per cent and then O$^{+}$ with 26 per cent.", "The elemental abundances we used for the photoionization model returns ionic abundances listed in Table REF , are comparable to those from the empirical analysis derived in Section .", "The ionic abundances derived from the observations do not show major discrepancies in He$^{+}$ /H$^{+}$ , C$^{2+}$ /H$^{+}$ , N$^{+}$ /H$^{+}$ , O$^{2+}$ /H$^{+}$ , Ne$^{2+}$ /H$^{+}$ and Ar$^{2+}$ /H$^+$ ; differences remain below 18 per cent.", "However, the predicted and empirical values of O$^{+}$ /H$^{+}$ , S$^{+}$ /H$^+$ and S$^{2+}$ /H$^+$ have discrepancies of about 45, 31 and 33 per cent, respectively.", "Fig.", "REF (bottom) shows plots of the ionization structure of helium, carbon, oxygen, argon (left-hand panel), nitrogen, neon and sulphur (right-hand panel) as a function of radius along the equatorial direction.", "As seen, ionization layers have a clear ionization sequence from the highly ionized inner parts to the outer regions.", "Helium is 97 percent singly-ionized over the shell, while oxygen is 26 percent singly ionized and 70 percent doubly ionized.", "Carbon and nitrogen are about $\\sim 20$ percent singly ionized $\\sim 80$ percent doubly ionized.", "The distribution of N$^{+}$ is in full agreement with the IFU map, given in Fig REF .", "Comparison between the He$^{+}$ , O$^{2+}$ and S$^{+}$ ionic abundance maps obtained from our IFU observations and the ionic fractions predicted by our photoionization model also show excellent agreement.", "Table: Mean electron temperatures (K) weighted by ionic species for the whole nebula obtained from the photoionization model.Table REF lists mean temperatures weighted by the ionic abundances.", "Both [N ii] and [O iii] doublets, as well as He i lines arise from the same ionization zones, so they should have roughly similar values.", "The ionic temperatures increasing towards higher ionization stages could also have some implications for the mean temperatures averaged over the entire nebula.", "However, there is a large discrepancy by a factor of 2 between our model and ORL empirical value of $T_{\\rm e}$ (He i$)$ .", "This could be due to some temperature fluctuations in the nebula [55], [56].", "The temperature fluctuations lead to overestimating the electron temperature deduced from CELs.", "This can lead to the discrepancies in abundances determined from CELs and ORLs [43].", "Nevertheless, the temperature discrepancy can also be produced by bi-abundance models [42], [44], containing some cold hydrogen-deficient material, highly enriched in helium and heavy elements, embedded in the diffuse warm nebular gas of normal abundances.", "The existence and origin of such inclusions are still unknown.", "It is unclear whether there is any link between the assumed H-poor inclusions in PNe and the H-deficient CSs." ], [ "Conclusion", "We have constructed a photoionization model for the nebula of Abell 48.", "This consists of a dense hollow cylinder, assuming homogeneous abundances.", "The three-dimensional density distribution was interpreted using the morpho-kinematic model determined from spatially resolved kinematic maps and the ISW model.", "Our aim was to construct a model that can reproduce the nebular emission-line spectra, temperatures and ionization structure determined from the observations.", "We have used the non-LTE model atmosphere from [73] as the ionizing source.", "Using the empirical analysis methods, we have determined the temperatures and the elemental abundances from CELs and ORLs.", "We notice a discrepancy between temperatures estimated from $[$ O iii$]$ CELs and those from the observed He i ORLs.", "In particular, the abundance ratios derived from empirical analysis could also be susceptible to inaccurate values of electron temperature and density.", "However, we see that the predicted ionic abundances are in decent agreement with those deduced from the empirical analysis.", "The emission-line fluxes obtained from the model were in fair agreement with the observations.", "We notice large discrepancies between He i electron temperatures derived from the model and the empirical analysis.", "The existence of clumps and low-ionization structures could solve the problems [43].", "Temperature fluctuations have been also proposed to be responsible for the discrepancies in temperatures determined from CELs and ORLs [55], [56].", "Previously, we also saw large ORL–CEL abundance discrepancies in other PNe with hydrogen-deficient CSs, for example Abell 30 [16] and NGC 1501 [19].", "A fraction of H-deficient inclusions might produce those discrepancies, which could be ejected from the stellar surface during a very late thermal pulse (VLTP) phase or born-again event [32].", "However, the VLTP event is expected to produce a carbon-rich stellar surface abundance [27], whereas in the case of Abell 48 negligible carbon was found at the stellar surface [73].", "The stellar evolution of Abell 48 still remains unclear and needs to be investigated further.", "But, its extreme helium-rich atmosphere (85 per cent by mass) is more likely connected to a merging process of two white dwarfs as evidenced for R Cor Bor stars of similar chemical surface composition by observations [6], [23] and hydrodynamic simulations [65], [80], [51].", "We derived a nebula ionized mass of $\\sim 0.8$  M$_{\\bigodot }$ .", "The high C/O ratio indicates that it is a predominantly C-rich nebula.", "The C/H ratio is largely over-abundant compared to the solar value of [2], while oxygen, sulphur and argon are under-abundant.", "Moreover, nitrogen and neon are roughly similar to the solar values.", "Assuming a sub-solar metallicity progenitor, a 3rd dredge-up must have enriched carbon and nitrogen in AGB-phase.", "However, extremely high carbon must be produced through mixing processing in the He-rich layers during the He-shell flash.", "The low N/O ratio implies that the progenitor star never went through the hot bottom burning phase, which occurs in AGB stars with initial masses more than 5M$_{\\bigodot }$ [35], [36].", "Comparing the stellar parameters found by the model, $T_{\\rm eff}$  = 70 kK and $L_{\\rm \\star }/$ L$_{\\bigodot }$ = 5500, with VLTP evolutionary tracks from [5], we get a current mass of $\\sim 0.62 {\\rm M}_{\\bigodot }$ , which originated from a progenitor star with an initial mass of $\\sim 3 {\\rm M}_{\\bigodot }$ .", "However, the VLTP evolutionary tracks by [52] yield a current mass of $\\sim 0.52 {\\rm M}_{\\bigodot }$ and a progenitor mass of $\\sim 1{\\rm M}_{\\bigodot }$ , which is not consistent with the derived nebula ionized mass.", "Furthermore, time-scales for VLTP evolutionary track [5] imply that the CS has a post-AGB age of about $\\sim $  9 000 yr, in agreement with the nebula's age determined from the kinematic analysis.", "We therefore conclude that Abell 48 originated from an $\\sim 3$  M$_{\\bigodot }$ progenitor, which is consistent with the nebula's features." ], [ "Acknowledgments", "AD warmly acknowledges the award of an international Macquarie University Research Excellence Scholarship (iMQRES).", "BE is supported by the German Research Foundation (DFG) Cluster of Excellence “Origin and Structure of the Universe”.", "AYK acknowledges the support from the National Research Foundation (NRF) of South Africa.", "We would like to thank Prof. Wolf-Rainer Hamann, Prof. Simon Jeffery and Dr. Amanda Karakas for illuminating discussions and helpful comments.", "We would also like to thank Dr. Kyle DePew for carrying out the 2010 ANU 2.3 m observing run.", "AD thanks Dr. Milorad Stupar for assisting with the 2012 ANU 2.3 m observing run and his guidance on the iraf pipeline wifes, Prof. Quentin A. Parker and Dr. David J. Frew for helping in the observing proposal writing stage, and the staff at the ANU Siding Spring Observatory for their support.", "We would also like to thank the anonymous referee for helpful suggestions." ] ]
1403.0567
[ [ "I Know Why You Went to the Clinic: Risks and Realization of HTTPS\n Traffic Analysis" ], [ "Abstract Revelations of large scale electronic surveillance and data mining by governments and corporations have fueled increased adoption of HTTPS.", "We present a traffic analysis attack against over 6000 webpages spanning the HTTPS deployments of 10 widely used, industry-leading websites in areas such as healthcare, finance, legal services and streaming video.", "Our attack identifies individual pages in the same website with 89% accuracy, exposing personal details including medical conditions, financial and legal affairs and sexual orientation.", "We examine evaluation methodology and reveal accuracy variations as large as 18% caused by assumptions affecting caching and cookies.", "We present a novel defense reducing attack accuracy to 27% with a 9% traffic increase, and demonstrate significantly increased effectiveness of prior defenses in our evaluation context, inclusive of enabled caching, user-specific cookies and pages within the same website." ], [ "Introduction", "HTTPS is far more vulnerable to traffic analysis than has been previously discussed by researchers.", "In a series of important papers, a variety of researchers have shown a number of traffic analysis attacks on SSL proxies [1], [2], SSH tunnels [3], [4], [5], [6], [7], Tor [3], [4], [8], [9], and in unpublished work, HTTPS [10], [11].", "Together, these results suggest that HTTPS may be vulnerable to traffic analysis.", "This paper confirms the vulnerability of HTTPS, but more importantly, gives new and much sharper attacks on HTTPS, presenting algorithms that decrease errors 3.6x from the best previous techniques.", "We show the following novel results: Novel attack technique capable of achieving 89% accuracy over 500 pages hosted at the same website, as compared to 60% with previous techniques Impact of caching and cookies on traffic characteristics and attack performance, affecting accuracy as much as 18% Novel defense reducing accuracy to 27% with 9% traffic increase; significantly increased effectiveness of packet level defenses in the HTTPS context We evaluate attack, defense and measurement techniques on websites for healthcare (Mayo Clinic, Planned Parenthood, Kaiser Permanente), finance (Wells Fargo, Bank of America, Vanguard), legal services (ACLU, Legal Zoom) and streaming video (Netflix, YouTube).", "We design our attack to distinguish minor variations in HTTPS traffic from significant variations which indicate distinct traffic contents.", "Minor traffic variations may be caused by caching, dynamically generated content, or user-specific content including cookies.", "Our attack applies clustering techniques to identify patterns in traffic.", "We then use a Gaussian distribution to determine similarity to each cluster and map traffic samples into a fixed width representation compatible with a wide range of machine learning techniques.", "Due to similarity with the Bag-of-Words approach to document classification, we refer to our technique as Bag-of-Gaussians (BoG).", "This approach allows us to identify specific pages within a website, even when the pages have similar structures and shared resources.", "After initial classification, we apply a hidden Markov model (HMM) to leverage the link structure of the website and further increase accuracy.", "We show our approach achieves substantially greater accuracy than attacks developed by Panchenko et al.", "(Pan) [8], Liberatore and Levine (LL) [6], and Wang et al. [9].", "We also present a novel defense technique and evaluate several previously proposed defenses.", "We consider deployability both in the design of our technique and the selection of previous techniques.", "Whereas the previous, and less effective, techniques could be implemented as stateless packet filters, our technique operates statelessly at the granularity of individual HTTP requests and responses.", "Our evaluation demonstrates that some techniques which are ineffective in other traffic analysis contexts have significantly increased impact in the HTTPS context.", "For example, although Dyer et al.", "report exponential padding as only decreasing accuracy of the Panchenko classifier from 97.2% to 96.6%, we observe a decrease from 60% to 22% [5].", "Our novel defense reduces the accuracy of the BoG attack from 89% to 27% while generating only 9% traffic overhead.", "We conduct our evaluations using a dataset of 463,125 page loads collected from 10 websites during December 2013 and January 2014.", "Our collection infrastructure includes virtual machines (VMs) which operate in four separate collection modes, varying properties such as caching and cookie retention across the collection modes.", "By training a model using data from a specific collection mode and evaluating the model using a different collection mode, we are able to isolate the impact of factors such as caching and user-specific cookies on analysis results.", "We present these results along with insights into the fundamental properties of the traffic itself.", "Section  presents the risks posed by HTTPS traffic analysis and adversaries who may be motivated and capable to conduct attacks.", "Section  reviews prior work, and in section  we present the core components of our attack.", "Section  presents the impact of evaluation conditions on reported attack accuracy, section  evaluates our attack, and section  presents and evaluates defense techniques.", "In Section  we discuss results and conclude." ], [ "Risks of HTTPS Traffic Analysis", "This section presents an overview of the potential risks and attackers we consider in analyzing HTTPS traffic analysis attacks.", "Section REF describes four categories of content, each of which we explore in this work, and potential consequences of a privacy violation in each category.", "Section REF discusses adversaries who may be motivated and capable to conduct the attacks discussed in section REF ." ], [ "Privacy Applications of HTTPS", "We present several categories of website in which the specific pages accessed by the user are more interesting than the mere fact that the user is visiting the website at all.", "This notion is present in traditional privacy concepts such as patient confidentiality or attorney-client privilege, where the content of a communication is substantially more sensitive than the simple presence of communication.", "Healthcare Many medical conditions or procedures are associated with significant social stigma.", "We examine the websites of Planned Parenthood, Mayo Clinic and Kaiser Permanente, a healthcare provider serving 9 million members in the US.", "The page views of these websites have the potential to reveal whether a pending procedure is an appendectomy or an abortion, or whether a chronic medication is for diabetes or HIV/AIDS.", "These types of distinctions and others can form the basis for discrimination or persecution and represent an easy opportunity to target advertising for products which consumers are highly motivated to purchase.", "Beyond personal risks, the health care details of corporate and political leaders can also have significant financial implications, as evidenced by Apple stock fluctuations in response to reports, both true and false, of Steve Jobs's health [12].", "Legal There are many common reasons for interaction with a lawyer, such as completing a will, filing taxes, or reviewing a contract.", "However, contacting a lawyer to investigate divorce, bankruptcy, or legal options as an undocumented immigrant may attract greater interest.", "Since some legal advice is relatively unremarkable while other advice may require strict privacy, the specific details of legal services are more interesting than mere interaction with a lawyer.", "Our work examines LegalZoom, a website offering legal services spanning the above themes and others.", "We additionally examine the American Civil Liberties Union (ACLU), which offers legal information and actively litigates on a wide range of sensitive topics including LGBT rights, human reproduction and immigration.", "Financial While most consumers utilize some form of financial products to manage their personal finances, the exact products a person uses reveal a great deal more about their personal circumstances.", "For example, a user with educational savings accounts likely has children, a joint account is an indicator of a long term relationship, and high volume mutual funds offering reduced fees likely indicate high levels of minimum net worth.", "Our work examines Bank of America and Wells Fargo, both large banks in the US, as well as Vanguard, a firm offering a range of investment vehicles and brokerage services.", "Streaming Video As demonstrated during the Netflix Prize contest and ensuing $9 million settlement, the video rental history of an individual can potentially reveal information as personal as sexual orientation [13], [14].", "Beyond any guarantees given in privacy policies, video rentals in the US are additionally protected by law [15].", "We examine YouTube and Netflix, both of which offer streaming videos covering a wide range of topics." ], [ "Attack Settings", "Having reviewed the possible consequences of traffic analysis attacks against HTTPS, we now examine situations in which an adversary may be motivated and capable to learn the types of private details previously discussed.", "Note that all capable adversaries must have at least two abilities.", "The adversary must be able to visit the same webpages as the victim, allowing the adversary to identify patterns in encrypted traffic indicative of different webpages.", "The adversary must also be able to observe victim traffic, allowing the adversary to match observed traffic with previously learned patterns.", "ISP Snooping ISPs are uniquely well positioned to target and sell advertising since they have the most comprehensive view of the consumer.", "Both ISPs [16], [17] and commercial chains of wi-fi access points [18], have shown efforts to mine customer data and/or sell advertising.", "Traffic analysis vulnerabilities would allow ISPs to conduct data mining despite the presence of encryption.", "Separate from electronic ad delivery, access points associated with businesses such as cafes and hotels could also deliver ads along with transaction receipts, physical mailings, or other special offers.", "Employee Monitoring Employers have the ability to monitor the online activities of employees connected to an employer provided network, regardless of whether the device in use is a personal or corporate device.", "This power has been abused by extensively monitoring the activities of employees [19], even extending to whistleblowers whose communications are protected by law [20].", "Traffic analysis would allow employers to remove many of the protections expected by employees using HTTPS to protect their sensitive communications from untrusted parties.", "Surveillance While revelations of NSA surveillance spanning from social media to World of Warcraft are an unwelcome surprise to many [21], [22], [23], [24], other governments around the world have long employed these practices [25], [26].", "When asked about the efficacy of encryption, Snowden maintained “Encryption works.", "Properly implemented strong crypto systems are one of the few things that you can rely on.", "Unfortunately, endpoint security is so terrifically weak that NSA can frequently find ways around it” [27].", "Despite this assertion, we still see NSA surveillance efforts specifically targeting HTTPS [28], indicating the value of removing side-channel attacks to ensure that HTTPS is “properly implemented.” Censorship Although the consequences of forbidden internet activity can include imprisonment and beyond in some settings, in other contexts broad filtering efforts have resulted in lower grade punishments designed to deter further transgression and encourage self-censorship.", "For example, Chinese social media firm Sina has recently punished more than 100,000 users through account suspensions and occasional public admonishment for violating the country's “Seven Bottom Lines” guidelines for internet use [29].", "Similarly, traffic analysis attacks could be used to degrade or block service for users suspected of viewing prohibited content over encrypted connections.", "Table: Prior works have focused almost exclusively on website homepages accessed via proxy.", "Cheng and Danezis work is preliminary and unpublished.", "Evaluations for both works parse object sizes from unencrypted traffic or server logs, which is not possible for actual encrypted traffic.", "Note that “?” indicates the author did not specify the property; several properties did not apply to Danezis as his evaluation used HTTP server logs.", "All evaluations used Linux with Firefox (FF) 2.0-3.6, except for Hintz and Sun (IE5), Cheng (Netscape), Wang (FF10) and Miller (FF22)." ], [ "Prior Work", "In this section we review attacks and defenses proposed in prior work, as well as the contexts in which work is evaluated.", "Comparisons with prior work are limited since much work has targeted specialized technologies such as Tor.", "Table REF presents an overview of prior attacks.", "The columns are as follows: Privacy Technology The encryption or protection mechanism analyzed for traffic analysis vulnerability.", "Note that some authors considered multiple mechanisms, and hence appear twice.", "Page Set Scope Closed indicates the evaluation used a fixed set of pages known to the attacker in advance.", "Open indicates the evaluation used traffic from pages both of interest and unknown to the attacker.", "Whereas open conditions are appropriate for Tor, closed conditions are appropriate for HTTPS.", "Page Set Size For closed scope, the number of pages used in the evaluation.", "For open scope, the number of pages of interest to the attacker and the number of background traffic pages, respectively.", "Accuracy For closed scope, the percent of pages correctly identified.", "For open scope, the true positive (TP) rate of correctly identifying a page as being within the censored set and false positive (FP) rate of identifying an uncensored page as censored.", "Cache Off indicates caching disabled.", "On indicates default caching behavior.", "Cookies Universal indicates that training and evaluation data were collected on the same machine or machines, and consequently with the same cookie values.", "Individual indicates training and evaluation data were collected on separate machines with distinct cookie values.", "Traffic Composition Single Site indicates the work identified pages within a website or websites.", "Homepages indicates all pages used in the evaluation were the homepages of different websites.", "Analysis Primitive The basic unit on which traffic analysis was conducted.", "Request indicates the analysis operated on the size of each object (e.g.", "image, style sheet, etc.)", "loaded for each page.", "Packet indicates meta-data observed from TCP packets.", "NetFlow indicates network traces anonymized using NetFlow.", "Active Content Indicates whether Flash, JavaScript, Java or any other plugins were enabled in the browser.", "Several works require discussion in addition to Table REF .", "Danezis focused on the HTTPS context, but evaluated his technique using HTTP server logs at request granularity, removing any effects of fragmentation, concurrent connections or pipelined requests [11].", "Cheng et al.", "also focused on HTTPS and conducted an evaluation using traffic from an HTTP website intentionally selected for its static content [10].", "Both works were unpublished, and operated on individual object sizes parsed from the unencrypted traffic rather than packet metadata.", "Likewise, the approaches of Sun et al.", "and Hintz et al.", "also assume the ability to parse entire object sizes from traffic [1], [2].", "For these reasons, we compare our work to Liberatore and Levine, Panchenko et al.", "and Wang et al.", "as these are more advanced and recently proposed techniques.", "Herrmann [3] and Cai [4] both conduct small scale preliminary evaluations which involve enabling the browser cache.", "In contrast to our evaluation, these evaluations only consider website homepages and all pages are loaded in a fixed, round-robin order.", "Herrmann additionally increases the cache size from the default to 2GB, reducing the likelihood of any cache evictions and stabilizing traffic.", "With caching enabled, Herrmann and Cai both observe approximately a 5% decrease in accuracy for their techniques, and Cai reports slightly improved performance for the Panchenko classifier.", "We evaluate the impact of caching on pages within the same website, where caching will have a greater effect than on the homepages of different websites due to increased page similarity, and load pages in a randomized order for greater cache state variation.", "Separate from attacks, we also review prior work relating to traffic analysis defense.", "Dyer et al.", "conduct a review of low level defenses operating on individual packets [5].", "Dyer evaluates defenses using data released by Liberatore and Levine and Herrmann et al.", "which collect traffic from website home pages on a single machine with caching disabled.", "In this context, Dyer finds that low level defenses are ineffective against attacks which examine features aggregated over multiple packets.", "For example, the linear and exponential padding defenses, which pad packet sizes to multiples of 128 and powers of 2 respectively, reduce the accuracy of the Panchenko classifier at most from 97.2% to 96.6%.", "In our evaluation, which considers pages within the same website, enabled caching and identification of traces collected on machines separate from the attacker, we find that low level, stateless defenses can be considerably more effective than initially indicated by Dyer.", "In addition to the packet level defenses evaluated by Dyer, many defenses have been proposed which operate at higher levels with additional cost and implementation requirements.", "These include HTTPOS [31], traffic morphing [32] and BuFLO [4], [5].", "HTTPOS, unlike most defenses, works from the client side to perturb the traffic generated by manipulating various features of TCP and HTTP to affect packet size, object size, pipelining behavior, packet timing and other properties.", "These manipulations require some degree of coordination and support from the server.", "BuFLO aims to provide provable defense against traffic analysis attacks by sending a constant stream of traffic at a fixed packet size for a pre-set minimum amount of time.", "Given the effectiveness and advantages of lower level defenses in our evaluation context, we do not further explore these higher level approaches in our work." ], [ "Attack Presentation", "In this section we present our attack.", "Figure REF presents an overview of the attack, depicting the anticipated workflow of the attacker as well as the subsections in which we discuss his efforts.", "In section REF , we present a formalism for identifying and labeling pages within a website and generating a site graph representing the website link structure.", "Section REF presents the core of our classification approach: Gaussian clustering techniques that capture standard variations in traffic and allow logistic regression to robustly identify key objects which reliably differentiate pages.", "Having generated isolated predictions, we then leverage the site graph and sequential nature of the data in section REF with a hidden Markov model (HMM) to further improve accuracy.", "Throughout this section we depend on several terms which we define as follows: Uniform Resource Locator (URL) A character string referencing a specific web resource, such as an image or HTML file.", "Webpage The set of resources loaded by a browser in response to the user clicking a link or otherwise entering a URL into the browser address bar.", "Two webpages are the same if a user could be reasonably expected to view their contents as substantially similar, regardless of the specific URLs fetched while loading the webpages or dynamic content such as advertising.", "Sample An instance of the traffic generated when a browser displays a webpage.", "Label A unique identifier assigned to each set of webpages which are the same.", "For example, two webpages which differ only in advertising will receive the same label, while webpages differing in core content are assigned different labels.", "Labels are assigned to samples in accordance with the webpage contained in the sample's traffic.", "Website A set of webpages such that the URLs which cause each webpage to load in the browser are hosted at the same domain.", "Site Graph A graph representing the link structure of a website.", "Nodes correspond to labels assigned to webpages within the website.", "Edges correspond to links between webpages, represented as the set of labels reachable from a given label." ], [ "Label and Site Graph Generation", "This section presents our approach to labeling and site graph generation.", "Merely treating the URL which causes a webpage to load as the label for the webpage is not sufficient for analyzing webpages within the same website.", "URLs may contain arguments, such as session IDs, which do not impact the content of the webpage and result in different labels aliasing webpages which are the same.", "This prevents accumulation of sufficient training samples for each label and hinders evaluation.", "URL redirection further complicates labeling; the same URL may refer to multiple webpages (e.g.", "error pages or A/B testing) or multiple URLs may refer to the same webpage.", "We present a labeling solution based on URLs and designed to accommodate these challenges.", "While URL redirection may be implemented within the web server or via JavaScript that alters webpage contents, allowing a single URL to represent many webpages, this behavior is limited in practice because website designers are motivated to allow search engines to link to webpages in search results.", "When labeling errors are inevitable, we prefer to have a single webpage aliased to multiple labels rather than have multiple distinct pages aliased to a single label.", "The former may result in lower accuracy ratings, but it allows our attacker to learn correct information.", "Our approach contains two phases.", "In the first phase, we conduct a preliminary crawl of the website, yielding many URLs from links encountered during the crawl.", "We then analyze these URLs to produce a canonicalization function which, given a URL, returns a canonical label for the webpage loaded as result of entering the URL into a browser address bar.Note that we label pages based on the final URL after any URL redirection occurs.", "We use the canonicalization function to produce a preliminary site graph, which guides further crawling activity.", "Our approach proceeds in two phases because the non-deterministic nature of URL redirections requires the attacker to conduct extensive crawling to observe the full breadth of both URLs and redirections, and crawling can not be conducted without a basic heuristic identifying URLs which likely alias the same webpage.", "As we describe below our approach allows, but does not require, the second crawl to be combined with training data collection.", "After the second phase is complete, both the labels and site graph are refined using the additional URLs and redirections observed during the crawl.", "We present our approach below.", "Execute Preliminary Crawl The first step in developing labels and a site graph is to crawl the website.", "The crawl can be implemented as either a depth- or breadth-first search, beginning at the homepage and exploring every link on a page up to a fixed maximum depth.", "We perform a breadth first search to depth 5.", "This crawl will produce a graph $G = (U, E)$ , where $U$ represents the set of URLs seen as links during the crawl, and $E = \\lbrace (u, u^{\\prime }) \\in U \\times U \\; | \\; u$ links to $u^{\\prime }\\rbrace $ represents links between URLs in $U$ .", "Produce Canonicalization Function Since multiple URLs may cause webpages which are effectively the same to load when entered into a browser address bar, the role of a canonicalization function is to produce a canonical label given a URL.", "The canonicalization function will be of the form $C: U \\rightarrow L$ , where $C$ denotes the canonicalization function, $U$ denotes the initial set of URLs, and $L$ denotes the set of labels.", "To maintain the criterion that we error on the side of multiple labels aliasing the same webpage, our approach forces any URLs with different paths to be assigned different labels and selectively identifies URL arguments that appreciably impact webpage content for inclusion in the label.", "We were able to execute this phase on all websites we surveyed.", "See Appendix B for our full approach, including several heuristics independent of URL arguments which further guide canonicalization.", "Canonicalize Initial Graph We use our canonicalization function to produce an initial site graph $G^{\\prime } = (L, E^{\\prime })$ where $L$ represents the set of labels on the website and $E^{\\prime }$ represents links.", "We construct $E^{\\prime }$ as follows: $E^{\\prime } = \\lbrace (C(u), C(u^{\\prime })) \\: | \\: (u, u^{\\prime }) \\in E\\rbrace $ We define a reverse canonicalization function $R: L \\rightarrow \\mathcal {P}(U)$ such that $R(l) = \\lbrace u \\in U \\: | \\: C(u) = l\\rbrace $ Note that $\\mathcal {P}(X)$ denotes the power set of $X$ , which is the set of all subsets of $X$ .", "Identify Browsing Sessions The non-deterministic nature of URL redirection requires the attacker to observe many redirection examples to finalize the site graph and canonicalization function.", "This process can also be used to collect training data.", "To collect training data and observe URL redirections the attacker builds a list of browsing sessions, each consisting of a fixed length sequence of labels.", "We fix the length of our browsing sessions to 75 labels.", "For cache accuracy, the attacker builds browsing sessions using a random walk through $G^{\\prime }$ .", "Since the graph structure prevents visiting all nodes evenly, the attacker prioritizes labels not yet visited.", "When the portion of duplicate labels reaches a fixed threshold (we used 0.6), the attacker visits the remaining labels regardless of the graph link structure until all labels have received at least single visit.", "This process is repeated until the attacker has produced enough browsing sessions to collect the desired amount of training data; we collected at least 64 samples of each label in total.", "Execute Browsing Sessions To generate traffic samples the attacker selects a URL $u$ for each label $l$ in a browsing session such that $u \\in R(l)$ and loads $u$ and (all supporting resources) by effectively entering $u$ into a browser address bar.", "The attacker records the value of document.location (once the entire webpage is done loading) to identify any URL redirections.", "$U^{\\prime }$ denotes the set of final URLs which are observed in document.location.", "We define a new function $T: U \\rightarrow \\mathcal {P}(U^{\\prime })$ such that $T(u) = \\lbrace u^{\\prime } \\in U^{\\prime } \\; | \\; u$ resolved at least once to $u^{\\prime }\\rbrace $ .", "We use this to define a new translation $T^{\\prime }: L \\rightarrow \\mathcal {P}(U^{\\prime })$ such that $T^{\\prime }(l) = \\bigcup _{u \\in R(l)} T(u)$ Refine Canonicalization Function Since the set of final URLs $U^{\\prime }$ may include arguments which were not present in the original set $U$ , we refine our canonicalization function $C$ to produce a new function $C^{\\prime }: U^{\\prime } \\rightarrow L^{\\prime }$ , where $L^{\\prime }$ denotes a new set of labels.", "The refinement is conducted using the same techniques as we used to produced $C$ .", "Samples are labeled as $C^{\\prime }(u^{\\prime }) \\in L^{\\prime }$ where $u^{\\prime }$ denotes the value of document.location when the sample finished loading.", "Refine Site Graph Since the final set of labels $L^{\\prime }$ may contain labels which are not in $L$ , the attacker must update $G^{\\prime }$ .", "The update must maintain the property that any sequence of labels $l^{\\prime }_0, l^{\\prime }_1, ... \\in L^{\\prime }$ observed during data collection must be a valid path in the final graph.", "Therefore, the attacker defines a new graph $G^{\\prime \\prime } = (U^{\\prime }, E^{\\prime \\prime })$ such that $E^{\\prime \\prime }$ is defined as $E^{\\prime \\prime } = \\lbrace (u, u^{\\prime }) \\: | \\: u \\in T^{\\prime }(l) \\: \\wedge \\: u^{\\prime } \\in T^{\\prime }(l^{\\prime }) \\: \\forall \\: (l, l^{\\prime }) \\in E^{\\prime } \\rbrace $ We apply our canonicalization function $C^{\\prime }$ to produce a final graph $G^{\\prime \\prime \\prime } = (L^{\\prime }, E^{\\prime \\prime \\prime })$ where $E^{\\prime \\prime \\prime } = \\lbrace (C^{\\prime }(u), C^{\\prime }(u^{\\prime })) \\: \\forall \\: (u, u^{\\prime }) \\in E^{\\prime \\prime }\\rbrace $ maintaining the property that any sequence of labels observed during training is a valid path in the final graph.", "Note that our evaluation generates a separate site graph for each model, using only redirections which occurred in training data.", "This leaves the possibility of a path in evaluation data which is not valid on the attacker site graph, but we did not find this to be an issue in practice.", "Table: Site graph and canonicalization summary.", "“Selected Subset” denotes the subset of the preliminary site graph which we randomly select for inclusion in our evaluation, “Avg.", "Links” denotes the average number of links per label, and “URLs” indicates the number of URLs seen as links in the preliminary site graph corresponding to an included label.For the purposes of this work, we augment the above approach to select a subset of the preliminary site graph for further analysis.", "By surveying a subset of each website, we are able to explore additional websites and browser configurations and remain within our resource constraints.", "We initialize the selected subset to include the label corresponding to the homepage, and iteratively expand the subset by adding a label reachable from the selected subset via the link structure of the preliminary site graph until 500 labels are selected.", "The set of links for the graph subset is defined as any links occurring between the 500 selected labels.", "Table REF presents properties of the preliminary site graph $G^{\\prime }$ , selected subset, and the final site graph $G^{\\prime \\prime \\prime }$ for each of the 10 websites we survey.", "We implement the preliminary crawl using Python and the second crawl (i.e.", "training data collection) using the browsing infrastructure described in Appendix A." ], [ "Feature Extraction and Machine Learning", "This section presents our individual sample classification technique.", "First, we describe the information which we extract from a sample, then we describe processing to produce features for machine learning, and finally describe the application of the learning technique itself.", "Figure: Table  displays the burst pairs extracted from three hypothetical samples.", "Figures  and  show the result of burst pair clustering.", "Figure  depicts the Bag-of-Gaussians features for each sample, where each feature value is defined as the total likelihood of all points from a given sample at the relevant domain under the Gaussian distribution corresponding to the feature.", "Our Gaussian similarity metric enables our attack to distinguish minor traffic variations from significant differences.We initially extract traffic burst pairs from each sample.", "Burst pairs are defined as the collective size of a contiguous outgoing packet sequence followed by the collective size of a contiguous incoming packet sequence.", "Intuitively, contiguous outgoing sequences correspond to requests, and contiguous incoming sequences correspond to responses.", "All packets must occur on the same TCP connection to minimize the effects of interleaving traffic.", "For example denoting outgoing packets as positive and incoming packets as negative, the sequence [+1420, +310, -1420, -810, +530, -1080] would result in the burst pairs [1730, 2230] and [530, 1080].", "Analyzing traffic bursts removes any fragmentation effects.", "Additionally, treating bursts as pairs allows the data to contain minimal ordering information and go beyond techniques which focus purely on packet size distributions.", "Once burst pairs are extracted from each TCP connection, the pairs are grouped using the second level domain of the host associated with the destination IP of the connection.", "All IPs for which the reverse DNS lookup fails are treated as a single “unknown” domain.", "Pairs from each domain undergo k-means clustering to identify commonly occurring and closely related tuples.", "Since tuples correspond to individual requests and pipelined series of requests, some tuple values will occur on multiple webpages while other tuples will occur only on individual webpages.", "Once clusters are formed we fit a Gaussian distribution to each cluster and treat each cluster as a feature dimension, producing our fixed-width feature vector.", "Features are extracted from samples by computing the extent to which each Gaussian is represented in the sample.", "Figure REF depicts the feature extraction process using a fabricated example involving three samples and two domains.", "Clustering results in five clusters, indexed 1–5, for Domain A and three clusters, indexed 6–8, for Domain B.", "The feature vector thus has eight dimensions, with one corresponding to each cluster.", "Sample x has traffic tuples in clusters 1, 2, 5, 6 and 7, but no traffic tuples in clusters 3, 4, 8, so its feature vector has non-zero values in dimensions 1, 2, 5, 6, 7, and zero values in dimensions 3, 4, 8.", "We create feature vectors for samples + and o in a similar fashion.", "We specify our approach formally as follows: Let $X$ denote the entire set of tuples from a trace, with $X^d \\subseteq X$ denoting set all tuples observed at domain $d$ .", "Let $\\Sigma ^d_i, \\mu ^d_i$ respectively denote the covariance and mean of Gaussian $i$ at domain $d$ .", "Let $F$ denote all features, with $F^d_i$ denoting feature $i$ from domain $d$ .", "$F^d_i = \\sum _{x \\in X^d} \\mathcal {N}(x | \\Sigma ^d_i, \\mu ^d_i)$ To determine the best number of Gaussian features for each domain, we train models using a range of values of $K$ and then select the best performing model for each domain.", "Analogously to the Bag-of-Words document processing technique, our approach projects a variable length vector of tuples into a finite dimensional space where each dimension “occurs” to some extent in the original sample.", "Whereas occurrence is determined by word count in Bag-of-Words, occurrence in our method is determined by Gaussian likelihood.", "For this reason, we refer to our approach as Bag-of-Gaussians (BoG).", "Once Gaussian features have been extracted from each sample the feature set is augmented to include counts of packet sizes observed over the entire trace.", "For example, if the lengths of all outgoing and incoming packets are between 1 and 1500 bytes, we add 3000 additional features where each feature corresponds to the total number of packets sent in a specific direction with a specific size.", "We linearly normalize all features to be in the range $[0, 1]$ and train a model using L2 regularized multi-class logistic regression with $C = 128$ using the liblinear package [33]." ], [ "Hidden Markov Model", "The basic attack presented in section REF classifies each sample independently.", "In practice, samples in a browsing session are not independent since the link structure of the website guides the browsing sequence.", "We leverage this ordering information, contained in the site graph produced in section REF , to improve results using a hidden Markov model (HMM).", "Recall that a HMM for a sequence of length $N$ is defined by a set of latent variables $Z = \\lbrace z_n \\; | \\; 1 \\le n \\le N\\rbrace $ , a set of observed variables $X = \\lbrace x_n \\; | \\; 1 \\le n \\le N\\rbrace $ , transition matrix $A$ such that $A_{i,j} = P(Z_{n+1} = j | Z_n = i)$ , an initial distribution $\\pi $ such that $\\pi _j = P(Z_1 = j)$ and an emission function $E(x_n, z_n) = P(x_n | z_n)$ .", "Applied to our context, the HMM is configured as follows: Latent variables $z_n$ correspond to labels $l^{\\prime } \\in L^{\\prime }$ visited by the victim during browsing sessions Observed variables $x_n$ correspond to observed feature vectors $X$ Initial distribution $\\pi $ assigns an equal likelihood to all pages Transition matrix $A$ encodes $E^{\\prime \\prime \\prime }$ , the set of links between pages in $L^{\\prime }$ , such that all links have equal likelihood Emission function $E(x_n, z_n) = P(z_n | x_n)$ determined by logistic regression After obtaining predictions with logistic regression, the attacker refines the predictions using the Viterbi algorithm to solve for the most likely values of the latent variables, each of which corresponds to a pageview by the user." ], [ "Impact of Evaluation Conditions", "In this section we demonstrate the impact of evaluation conditions on accuracy results and traffic characteristics.", "First, we present the scope and motivation of our investigation.", "Then, we describe the experimental methodology we use to determine the impact of evaluation conditions.", "Finally, we present the results of our experiments on four attack implementations.", "All attacks are impacted by the evaluation condition variations we consider, with the most affected attack decreasing accuracy from 68% to 50%.", "We discuss attack accuracy in this section only insofar as is necessary to understand the impact of evaluation conditions; we defer a full attack evaluation to section .", "Cache Configuration The browser cache improves page loading speed by storing web resources which were previously loaded, potentially posing two challenges to traffic analysis.", "Providing content locally decreases the total amount of traffic, reducing the information available for use in an attack.", "Additionally, differences in browsing history can result in differences in cache contents and further vary network traffic.", "Since privacy tools such as Tor frequently disable caching, many prior evaluations have disabled caching as well [34].", "But in practice, general users of HTTPS typically do not modify default cache settings, so we evaluate the impact of enabling caching to default settings.", "User-Specific Cookies If an evaluation collects all data with either the same browser instance or repeatedly uses a fresh browser image (such as the Tor browser bundle), there are respective assumptions that the attacker and victim will either share the same cookies or use none at all.", "While a traffic analysis attacker will not have access to victim cookies, privacy technologies which begin all browsing sessions from a clean browsing image effectively share the null cookie.", "We compare the performance of evaluations which use the same (non-null) cookie value in all data, different (non-null) cookie values in training and evaluation, a null cookie in all data, and evaluations which mix null and non-null cookies.", "Pageview Diversity Many evaluations collect data by repeatedly visiting a fixed set of URLs from a single machine and dividing the collected data for training and evaluation.", "This approach introduces an unrealistic assumption that, during training, an attacker will be able to visit the same set of webpages as the victim.", "Note that this would require collecting separate training data for each victim given that each victim visits a unique set of pages.", "We examine the impact of allowing the victim to intersperse browsing of multiple websites, including websites outside our attacker's monitoring focus.Note that this is different from an open-world vs. closed-world distinction as described in section , as we assume that the attacker will train a model for each website in its entirety and be able to identify the correct model based on traffic destination.", "Here, we are concerned with any effects on browser cache or personalized content which may impact traffic analysis.", "Webpage Similarity Since HTTPS will usually allow an eavesdropper to learn the domain a user is visiting, our evaluation focuses on efforts to differentiate individual webpages within a website protected by HTTPS.", "Differentiating webpages within the same website may pose a greater challenge than differentiating website homepages.", "Webpages within the same website share many resources, increasing the effect of caching and making webpages within a website harder to distinguish.", "We examine the relative traffic volumes of browsing both website homepages and webpages within a website.", "To quantify the impact of evaluation conditions on accuracy results, we design four modes for data collection designed to isolate specific effects.", "Our approach assumes that data will be gathered in a series of browsing sessions, where each session consists of loading a fixed number of URLs in a browser.", "The four modes are as follows: Cache disabled, new virtual machine (VM) for each browsing session Cache enabled, new VM for each browsing session Cache enabled, persistent VM for all browsing sessions, single site per VM Cache enabled, persistent VM for all browsing sessions, all sites on same VM Figure: Disabling the cache significantly increases the number of unique packet sizes for samples of a given label.", "For each label ll we determine the mean number l m l_m of unique packet sizes for samples of ll with caching enabled, and normalize the unique packet size counts of all samples of label ll using l m l_m.", "We present the normalized values for all labels separated by cache configuration.In our experiments we fixed the session length to 75 URLs and collect at least 16 samples of each label under each collection mode.", "We begin each browsing session in the first two configurations with a fresh VM image to eliminate the possibility of cookie persistence in browser or machine state.", "The first and second modes differ only with respect to cache configuration, allowing us to quantify the impact of caching.", "In effect the second, third and fourth modes each represent a distinct cookie value, with the second mode representing a null cookie and the third and fourth modes having actual, distinct, cookie values set by the site.", "The third and fourth modes differ in pageview diversity.", "In the context of HTTPS evaluations, the fourth mode most closely reflects the behavior of actual users and hence serves as evaluation data, while the second and third modes generate training data.", "Figure: “Train: X, Eval: Y” indicates training data from mode X and evaluation data from mode Y as shown in Table .", "For evaluations which use a privacy tool such as the Tor browser bundle and assume a closed world, training and evaluating using mode 1 is most realistic.", "However, in the HTTPS context training using mode 2 or 3 and evaluating using mode 4 is most realistic.", "Figure  presents differences as large as 18% between these conditions, demonstrating the importance of evaluation conditions when measuring attack accuracy.Our analysis reveals that caching significantly decreases the number of unique packet sizes observed for samples of a given label.", "We focus on the number of unique packet sizes since packet size counts are a commonly used feature in traffic analysis attacks.", "A reduction in the number of unique packet sizes reduces the number of non-zero features and creates difficulty in distinguishing samples.", "Figure REF contrasts samples from the first and second collection modes, presenting the effect of caching on the number of unique packet sizes observed per label for each of the 10 websites we evaluate.", "Note that the figure only reflects TCP data packets.", "We use a normalization process to present average impact of caching on a per-label basis across an entire website, allowing us to depict for each website the expected change in number of unique packet sizes for any given label as a result of disabling the cache.", "Figure: Decrease in traffic volume caused by browsing webpages internal to a website as compared to website homepages.", "Similar to the effect of caching, the decreased traffic volume is likely to increase classification errors.", "Note that packet count ranges are selected to divide internal pages into 5 even ranges.Beyond examining traffic characteristics, our analysis shows that factors such as caching, user-specific cookies and pageview diversity can cause variations as large as 18% in measured attack accuracy.", "We examine each of these factors by training a model using data from a specific collection mode, and comparing model accuracy when evaluated on a range of collection modes.", "Since some models must be trained and evaluated using data from the same collection mode, we use 8 training samples per label and leave the remaining 8 samples for evaluation.", "Figure REF presents the results of our analysis: Cache Effect Figure REF compares the performance of models trained and evaluated using mode 1 to models trained and evaluated using mode 2.", "As these modes differ only by enabled caching, we see that caching has moderate impact and can influence reported accuracy by as much as 10%.", "Cookie Effect Figure REF measures the impact of user-specific cookies by comparing the performance of models trained and evaluated using browsing modes 2 and 3.", "We observe that both the null cookie in mode 2 and the user-specific cookie in mode 3 perform 5-10 percentage points better when the evaluation data is drawn from the same mode as the training data.", "This suggests that any difference in cookies between training and evaluation conditions will impact accuracy results.", "Total Effect Figure REF presents the combined effects of enabled caching, user-specific cookies and increased pageview diversity.", "Recalling Figure REF , notice that models trained using mode 2 perform similarly on modes 3 and 4, and models trained using mode 3 perform similarly on modes 2 and 4, confirming the importance of user-specific cookies.", "In total, the combined effect of enabled caching, user-specific cookies and pageview diversity can influence reported accuracy by as much as 19%.", "Figure REF suggests that the effect is primarily due to caching and cookies since mode 2 generally performs better on mode 4, which includes visits to other websites, than on mode 3, which contains traffic from only a single website.", "Since prior works have focused largely on website homepages but HTTPS requires identification of webpages within the same website, we present data demonstrating a decrease in traffic when browsing webpages within a website.", "Figure REF presents the results of browsing through the Alexa top 1,000 websites, loading the homepage of each site, and then loading nine additional links on the site at random with caching enabled.", "By partitioning the total count of data packets transferred in the loading of webpages internal to a website into five equal size buckets, we see that there is a clear skew towards homepages generating more traffic, reflecting increased content and material for traffic analysis.", "This increase, similar to the increase caused by disabled caching, is likely to increase classification errors." ], [ "Attack Evaluation", "In this section we evaluate the performance of our attack.", "We begin by presenting the selection of previous techniques for comparison and the implementation of each attack.", "Then, we present the aggregate performance of each attack across all 10 websites we consider, the impact of training data on attack accuracy, and the performance each attack at each individual website.", "We select the Liberatore and Levine (LL), Panchenko et al.", "(Pan), and Wang et al.", "attacks for evaluation in addition to the BoG attack.", "The LL attack offers a view of baseline performance achievable from low level packet inspection, applying naive Bayes to a feature set consisting of packet size counts [6].", "We implemented the LL attack using the naive Bayes implementation in scikit-learn [35].", "The Pan attack extends size count features to include additional features related to burst length as measured in both packets and bytes as well as total traffic volume [8].", "For features aggregated over multiple packets, the Pan attack rounds feature values to predetermined intervals.", "We implement the Pan attack using the libsvm [36] implementation of the RBF kernel support vector machine with the $C$ and $\\gamma $ parameters specified by Panchenko.", "We select the Pan attack for comparison to demonstrate the significant benefit of Gaussian similarity rather than predetermined rounding thresholds.", "The BoG attack functions as described in section .", "We implement the BoG attack using the k-means package from sofia-ml [37] and logistic regression with class probability output from liblinear [33], with Numpy [38] performing intermediate computation.", "The Wang attack assumes a fundamentally different approach from LL, Pan and BoG based on string edit distance [9].", "There are several variants of the Wang attack which trade computational cost for accuracy by varying the string edit distance function.", "Wang reports that the best distance function for raw packet traces is the Optimal String Alignment Distance (OSAD) originally proposed by Cai et al. [4].", "Unfortunately, the edit distance must be computed for each pair of samples, and OSAD is extremely expensive.", "Therefore, we implement the Fast Levenshtein-Like (FLL) distance,Note that the original attack rounded packet sizes to multiples of 600; we operate on raw packet sizes as we found this improves attack accuracy in our evaluation.", "an alternate edit distance function proposed by Wang which runs approximately 3000x faster.OSAD has $O(mn)$ runtime where $m$ and $n$ represent the length of the strings, whereas FLL runs in $O(m+n)$ .", "Wang et al.", "report completing an evaluation with 40 samples of 100 labels each in approximately 7 days of CPU time.", "Since our evaluation involves 10 websites with approx.", "500 distinct labels each and 16 samples of each label for training and evaluation, we would require approximately 19 months of CPU time (excluding any computation for sections  or ).", "Since Wang demonstrates that FLL achieves 46% accuracy operating on raw packet traces, as compared to 74% accuracy with OSAD, we view FLL as a rough indicator of the potential of the OSAD attack.", "We implement the Wang - FLL attack using scikit-learn [35].", "We now examine the performance of each attack implementation.", "We evaluate attacks using data collected in mode 4 since this mode is most similar to the behavior of actual users.", "We consider both modes 2 and 3 for training data to avoid any bias introduced by using the same cookies as seen in evaluation data or browsing the exact same websites.", "As shown in Figure REF , mode 2 performs slightly better so we train all models using data from mode 2.", "Consistent with prior work, our evaluation finds that accuracy of each attack improves with increased training data, as indicated by Figure REF .", "Notice that the Pan attack is most sensitive to variations in the amount of training data, and the BoG attack continues to perform well even at low levels of training data.", "In some cases an attacker may have ample opportunity to collect training data, although in other cases the victim website may attempt to actively resist traffic analysis attacks by detecting crawling behavior and engaging in cloaking, rate limiting or other forms of blocking.", "These defenses would be particularly feasible for special interest, low volume websites where organized, frequent crawling would be hard to conceal.", "The BoG attack derives significant performance gains from the application of the HMM.", "Figure REF presents the BoG attack accuracy as a function of the browsing session length.", "Although we collect browsing sessions which each contain 75 samples, we simulate shorter browsing sessions by applying the HMM to randomly selected subsets of browsing sessions and observing impact on accuracy.", "At session length 1 the HMM has no effect and the BoG attack achieves 72% accuracy, representing the improvement over the Pan attack resulting from the Gaussian feature extraction.", "The HMM accounts for the remaining performance improvement from 72% accuracy to 89% accuracy.", "We achieve most of the benefit of the HMM after observing two samples in succession, and the full benefit after observing approximately 15 samples.", "Although any technique which assigns a likelihood to each label for each sample could be extended with a HMM, applying a HMM requires solving the labeling and site graph challenges which we present novel solutions for in section .", "Figure: Performance of BoG attack and prior techniques.", "Figure  presents the performance of all four attacks as a function of training data.", "Figure  presents the accuracy of the BoG attack trained with 16 samples as a function of browsing session length.", "Note that the BoG attack achieves 89% accuracy as compared to 60% accuracy with the best prior work.Figure: Accuracy of each attack for each website.", "Note that the BoG attack performs the worst at Kaiser Permanente, Mayo Clinic and Netflix, which each have approx.", "1000 labels in their final site graphs according to Table .", "The increase in graph size during finalization suggests potential for improved performance through better canonicalization to eliminate labels aliasing the same webpages.Although the BoG attack averages 89% accuracy overall, only 4 of the 10 websites included in evaluation have accuracy below 92%.", "Figure REF presents the accuracy of each attack at each website.", "The BoG attack performs the worst at Mayo Clinic, Netflix and Kaiser Permanente.", "Notably, the number of labels in the site graphs corresponding to each of these websites approximately doubles during the finalization process summarized in Table REF of section .", "URL redirection causes the increase in labels, as new URLs appear whose corresponding labels were not included in the preliminary site graph.", "Some new URLs may have been poorly handled by the canonicalization function, resulting in labels which alias the same content.", "Although we collected supplemental data to gather sufficient training samples for each label, the increase in number of labels and label aliasing behavior degrade measured accuracy for all attacks.", "Despite the success of string edit distance based attacks against Tor, the Wang - FLL attack struggles in the HTTPS setting.", "Wang's evaluation is confined to Tor, which pads all packets into fixed size cells, and does not effectively explore edit distance approaches applied to unpadded traffic.", "Consistent with the unpadded nature of HTTPS, we observe that Wang's attack performs best on unpadded traffic in the HTTPS setting.", "Despite this improvement, the Wang - FLL technique may struggle because edit distance treats all unique packet sizes as equally dissimilar; for example, 310 byte packets are equally similar to 320 byte packets and 970 byte packets.", "Additionally, the application of edit distance at the packet level causes large objects sent in multiple packets to have proportionally large impact on edit distance.", "This bias may be more apparent in the HTTPS context than with website homepages since webpages within the same website are more similar than homepages of different websites.", "Replacing the FLL distance metric with OSAD or Damerau-Levenshtein would improve attack accuracy, although the poor performance of FLL suggests the improvement would not justify the cost given the alternative techniques available." ], [ "Defense", "This section presents and evaluates several defense techniques, including our novel Burst defense which operates between the application and TCP layers to obscure high level features of traffic while minimizing overhead.", "Figure REF presents the effectiveness and cost of the defenses we consider.", "We find that evaluation context significantly impacts defense performance, as we observe increased effectiveness of low level defenses in our evaluation as compared to prior work [5].", "Additionally, we find that the Burst defense offers significant performance improvements while maintaining many advantages of low level defense techniques.", "We select defenses for evaluation on the combined basis of cost, deployability and effectiveness.", "We select the linear and exponential padding defenses from Dyer et al.", "as they are reasonably effective, have the lowest overhead, and are implemented statelessly below the TCP layer.", "The linear defense pads all packet sizes up to multiples of 128, and the exponential defense pads all packet sizes up to powers of 2.", "Stateless implementation at the IP layer allows for easy adoption across a wide range of client and server software stacks.", "Additionally, network overhead is limited to minor increases in packet size with no new packets generated, keeping costs low in the network and on end hosts.", "We also introduce the fragmentation defense which randomly splits any packet which is smaller than the MTU, similar to portions of the strategy adopted by HTTPOS [31].", "Fragmentation offers the deployment advantages of not introducing any additional data overhead, as well as being entirely supported by current network protocols and hardware.", "We do not consider defenses such as BuFLO or HTTPOS given their complexity, cost and the effectiveness of the alternatives we do consider [5], [31].", "The exponential defense slightly outperforms the linear defense, decreasing the accuracy of the Pan attack from 60% to 22% and the BoG attack from 89% to 59%.", "Notice that the exponential defense is much more effective in our evaluation context than Dyer's context, which focuses on comparing website homepages loaded over an SSH tunnel with caching disabled and evaluation traffic collected on the same machine as training traffic.", "The fragmentation defense is extremely effective against the LL and Wang - FLL attacks, reducing accuracy to below 1% for each attack, but less effective against the Pan and BoG attacks.", "The Pan and BoG attacks each perform TCP stream re-assembly, aggregating fragmented packets, while LL and Wang - FLL do not.", "Since TCP stream re-assembly is expensive and requires complete access to victim traffic, the fragmentation defense may be a superior choice against many adversaries in practice.", "Although the fragmentation, linear and exponential defenses offer the deployment advantages of functioning statelessly below the TCP layer, their effectiveness is limited.", "The Burst defense offers greater protection, operating between the TCP layer and application layer to pad contiguous bursts of traffic up to pre-defined thresholds uniquely determined for each website.", "The Burst defense allows for a natural tradeoff between performance and cost, as fewer thresholds will result in greater privacy but at the expense of increased padding.", "Unlike the BoG attack which considers bursts as a tuple, padding by the Burst defense is independent in each direction.", "We determine Burst thresholds as shown in Algorithm , repeating the algorithm for each direction.", "We pad traffic bursts as shown in Algorithm .", "Precondition: Threshold Calculation [1] $bursts$ is a set containing the length of each burst in a given direction in defense training traffic $threshold$ specifies the maximum allowable cost of the Burst defense $thresholds \\leftarrow set()$ $bucket \\leftarrow set()$ $b$ in sorted$(bursts)$ $inflation \\leftarrow \\texttt {len}(bucket + b) * \\texttt {max}(bucket + b) / \\texttt {sum}(bucket + b)$ $inflation \\ge threshold$ $thresholds \\leftarrow thresholds + max(bucket)$ $bucket \\leftarrow set() + b$ $bucket \\leftarrow bucket + b$ $thresholds + \\texttt {max}(bucket)$ Burst Padding [1] $burst$ specifies the size of a directed traffic burst $thresholds$ specifies the thresholds obtained in Algorithm  $t$ in sorted$(thresholds)$ $t \\ge burst$ $t$ $burst$ We evaluate the Burst defense for $threshold$ values 1.10 and 1.40, with the resulting cost and performance shown in Figure REF .", "The Burst defense outperforms defenses which operate solely at the packet level by obscuring features aggregated over entire TCP streams.", "Simultaneously, the Burst defense offers deployability advantages over techniques such as HTTPOS since the Burst defense is implemented between the TCP and application layers.", "The cost of the Burst defense compares favorably to defenses such as HTTPOS, BuFLO and traffic morphing, reported to cost at least 37%, 94% and 50% respectively [4], [32].", "Having demonstrated the performance and favorable cost of the Burst defense, we plan to address further comparative evaluation in future work." ], [ "Discussion and Conclusion", "This work examines the vulnerability of HTTPS to traffic analysis attacks, focusing on evaluation methodology, attack and defense.", "Although we present novel contributions in each of these areas, many open problems remain.", "Our examination of evaluation methodology focuses on caching and user-specific cookies, but does not explore factors such as browser differences, operating system differences, mobile/tablet devices or network location.", "Each of these factors may contribute to traffic diversity in practice, likely degrading attack accuracy.", "In the event that differences in browser, operating system or device type negatively influence attack results, we suggest that these differences may be handled by collecting separate training data for each client configuration.", "We then suggest constructing a HMM that contains isolated site graphs for each client configuration.", "During attack execution, the classifier will likely assign higher likelihoods to samples from the client configuration matching the actual client, and the HMM will likely focus prediction within a single isolated site graph.", "By identifying the correct set of training data for use in prediction, the HMM may effectively minimize the challenge posed by multiple client configurations.", "We leave refinement and evaluation of this approach as future work.", "Additional future work remains in the area of attack development.", "To date, all approaches have assumed that the victim browses the web in a single tab and that successive page loads can be easily delineated.", "Future work should investigate actual user practice in these areas and impact on analysis results.", "For example, while many users have multiple tabs open at the same time, it is unclear how much traffic a tab generates once a page is done loading.", "Additionally, we do not know how easily traffic from separate page loadings may be delineated given a contiguous stream of user traffic.", "Lastly, our work assumes that the victim actually adheres to the link structure of the website.", "In practice, it may be possible to accommodate users who do not adhere to the link structure by introducing strong and weak transitions rather than a binary transition matrix, where strong transitions are assigned high likelihood and represent actual links on a website and weak transitions join all unlinked pages and are assigned low likelihood.", "In this way the HMM will allow transitions outside of the site graph provided that the classifier issues a very confident prediction.", "Defense development and evaluation also require further exploration.", "Attack evaluation conditions and defense development are somewhat related since conditions which favor attack performance will simultaneously decrease defense effectiveness.", "Defense evaluation under conditions which favor attack creates the appearance that defenses must be complex and expensive, effectively discouraging defense deployment.", "To increase likelihood of deployment, future work must investigate necessary defense measures under increasingly realistic conditions since realistic conditions may substantially contribute to effective defense.", "This work has involved substantial implementation, data collection and computation.", "To facilitate future comparative work, our data collection infrastructure, traffic samples, attack and defense implementations and results are available upon request.", "Figure: Data collection infrastructure." ], [ "Appendix A", "In this appendix we describe the software system used to collect traffic samples.", "To allow parallel collection of data, we collected traffic inside a VirtualBox 4.2.16 VM running Linux 12.04 and Firefox 22.", "We ran 4 VMs at a time on the same workstation with a quad core Xeon processor and 12GB of RAM.", "We ran only 4 VMs to allow each VM sufficient processor, memory, disk and network resources to prevent any dropped packets or delays in website loading.", "Since some collection modes require a fresh VM image for each browsing session, we disabled automatic updates in Ubuntu and Firefox as these would have consistently downloaded updates and contaminated traffic samples.", "Figure REF depicts the software components of the collection infrastructure.", "The HostDriver managed the collection process, including booting VMs and assigning workloads.", "The HostDriver shared a folder with each VM containing a user.js file used to configure Firefox caching behavior and list of URLs to visit.", "To disable the Firefox cache, we set 15 caching related configuration options listed in Firefox source file all.js.", "Each VM launched the GuestDriver script after boot, which then launched Firefox using the supplied user.js configuration file.", "A Greasemonkey script (version 0.9.20) installed in Firefox then successively visited each of the listed URLs.", "After each webpage had fully loaded the Greasemonkey script waited 3 seconds to allow for any JavaScript URL redirection.", "Once any redirection had finished, the script waited an additional 4 seconds to ensure all traffic generated by the page was collected.", "Greasemonkey then issued a blocking request to a server running locally on the VM which caused the server to stop and restart TCPDUMP, thereby creating separate PCAP files for each sample.", "Note that we disabled Content Security Policy as several sites had policies which prevented our Greasemonkey script.", "In this appendix we further describe the techniques we use to produce the canonicalization function.", "Recall that the canonicalization function transforms the URL displayed in the browser address bar after a webpage loads into a label identifying the contents of the webpage.", "The most basic heuristics we apply in the canonicalization function handle differences in the URL which rarely impact content.", "We remove the www subdomain at the beginning of the full domain name, convert all URLs to be lower case and remove any trailing “/” at the end of the URL.", "Beyond these, we assume that any two URLs which differ prior to the query string will correspond to webpages which are not the same.", "This assumption is not inherent to the concept of a canonicalization function and could be removed by modifying the canonicalization function to also operate on the domain, port and path of a URL.", "Having canonicalized the domain, port and path of the URL, we now identify any arguments appearing in the query string which appreciably impact page content.", "We enumerate all arguments which appear in any URL at the website, and then for each argument enumerate all values associated with that argument and a list of all URLs in which each (argument, value) pair appears.", "We then iterate through arguments to identify arguments that significantly impact page content.", "For each argument, we randomly select 6 URL paths in which the argument appears and up to six distinct values of the argument for each URL path.", "Note that the impact of an argument can normally be determined by viewing simply a pair of argument values for a single URL path; we consider additional samples as described below to provide a margin of safety.", "The decision process for deciding whether a particular argument significantly influences content is as follows: If all pages with the same base URL appear the same, then the argument does not influence content.", "If pages with the same base URL appear different, and the argument being examined is the only difference in the URL, then the argument does influence content.", "In the case that pages with the same base URL do appear different and multiple arguments are different, additional investigation is necessary.", "If removal of the argument causes page content to change, then the argument influences page content.", "Alternately, if substitution of alternate argument values causes page content to change, then the argument influences page content.", "Once we have identified all arguments which do not impact page content, we canonicalize URLs by removing these arguments and their associated values.", "This approach to canonicalization makes several assumptions.", "The approach assumes that the impact of the argument is independent of the URL path.", "Additionally, the approach assumes that the effect of each argument can be observed by manipulating that argument independent of any other arguments.", "To provide limited validation of our assumptions, we perform a “safety check” for each website which randomly selects labels and compares URLs corresponding to the label to verify that page contents are comparable." ] ]
1403.0297
[ [ "Integral representations and complete monotonicity related to the\n remainder of Burnside's formula for the gamma function" ], [ "Abstract In the paper, the authors establish integral representations of some functions related to the remainder of Burnside's formula for the gamma function and find the (logarithmically) complete monotonicity of these and related functions.", "These results extend and generalize some known conclusions." ], [ "Motivation and main results", "A function $f$ is said to be completely monotonic on an interval $I$ if $f$ has derivatives of all orders on $I$ and $(-1)^{n}f^{(n)}(x)\\ge 0$ for all $x \\in I$ and $n \\ge 0$ .", "A function $f$ is said to be absolutely monotonic on an interval $I$ if $f$ has derivatives of all orders on $I$ and $f^{(n)}(x)\\ge 0$ for all $x \\in I$ and $n \\ge 0$ .", "A positive function $f(x)$ is said to be logarithmically completely monotonic on an interval $I\\subseteq \\mathbb {R}$ if it has derivatives of all orders on $I$ and its logarithm $\\ln f(x)$ satisfies $(-1)^k[\\ln f(x)]^{(k)}\\ge 0$ for all $k\\in \\mathbb {N}$ on $I$ .", "For more information on these kinds of functions, please refer to the papers and monographs [7], [18], [27], [32], [36], [40], [44] and plenty of references cited therein.", "It is well known that the classical Euler's gamma function may be defined by $\\Gamma (x)=\\int ^\\infty _0t^{x-1} e^{-t}\\textup {\\,d}t$ for $x>0$ .", "The logarithmic derivative of $\\Gamma (x)$ , denoted by $\\psi (x)=\\frac{\\Gamma ^{\\prime }(x)}{\\Gamma (x)}$ , is called the psi or digamma function, and $\\psi ^{(k)}(x)$ for $k\\in \\mathbb {N}$ are called the polygamma functions.", "The noted Binet's formula [17] states that $\\ln \\Gamma (x)= \\biggl (x-\\frac{1}{2}\\biggr )\\ln x-x+\\ln \\sqrt{2\\pi }\\,+\\theta (x)$ for $x>0$ , where $\\Gamma (x)=\\int ^\\infty _0t^{x-1} e^{-t}\\operatorname{d}t$ stands for Euler's gamma function and $\\theta (x)=\\int _{0}^{\\infty }\\left(\\frac{1}{e^{t}-1}-\\frac{1}{t}+\\frac{1}{2}\\right)\\frac{e^{-xt}}{t}\\operatorname{d}t$ is called the remainder of Binet's formula (REF ).", "By the way, some functions related to the function $\\frac{1}{e^{t}-1}-\\frac{1}{t}+\\frac{1}{2}$ in the formula (REF ) have been investigated, applied, and surveyed in [8], [9], [10], [12], [23], [25], [26], [37] and many references listed therein.", "For real numbers $p>0$ , $q\\in \\mathbb {R}$ , and $r\\ne 0$ , define $f_{p,q,r}(x)=r[\\theta (px)-q\\theta (x)]$ on $(0,\\infty )$ .", "In [4], [12] and [5], the complete monotonicity of $f_{p,q,r}(x)$ and the star-shaped and subadditive properties of $\\theta (x)$ were established.", "In [6] and [17], it is given that $\\psi (x)=\\ln x-\\frac{1}{2x}-2\\int _{0}^{\\infty }\\frac{t\\operatorname{d}t}{(t^2 +x^2)(e^{2\\pi t}-1)}$ and $\\psi \\biggl (x+\\frac{1}{2}\\biggr )=\\ln x+2\\int _{0}^{\\infty }\\frac{t\\operatorname{d}t}{(t^2 +4x^2)(e^{\\pi t}+1)} $ for $x>0$ .", "For $p>0$ and $q\\in \\mathbb {R}$ , let $\\Lambda _{p,q}(x)=\\lambda (px)-q\\lambda (x)\\quad \\text{and}\\quad \\Phi _{p,q}(x)=\\phi (px)-q\\phi (x)$ on $(0,\\infty )$ , where $\\lambda (x)=\\int _{0}^{\\infty }\\frac{t\\operatorname{d}t}{(t^2 +x^2)(e^{2\\pi t}-1)}\\quad \\text{and}\\quad \\phi (x)=\\int _{0}^{\\infty }\\frac{t\\operatorname{d}t}{(t^2 +4x^2)(e^{\\pi t}+1)}.$ In [38], [39], it was obtained that the function $\\Lambda _{p,q}(x)$ is positive and decreasing in $x\\in (0,\\infty )$ if either $q\\le 0$ , or $0<p<1$ and $pq\\le 1$ , or $0<q=\\frac{1}{p^2}\\le 1$ ; the function $\\Lambda _{p,q}(x)$ is negative and increasing in $x\\in (0,\\infty )$ if either $p\\ge 1$ and $pq\\ge 1$ , or $\\frac{1}{p^2}=q\\ge 1$ ; the function $\\Phi _{p,q}(x)$ is positive and decreasing in $x\\in (0,\\infty )$ if either $p\\ge 1$ and $q\\le 0$ , or $0<p<1$ and $q\\le 1$ , or $p^2q<1$ and $q(p^2-1)[(1+3q)p^2-4]\\le 0$ , or $p^2q=1$ and $0<q\\le 1$ ; the function $\\Phi _{p,q}(x)$ is negative and increasing in $x\\in (0,\\infty )$ if either $4\\le p^2(1+3q)\\le 1+3q$ , or $p>1$ and $q\\ge 1$ .", "In [31] and its preprint [30], some more properties on the remainder of Binet's formula (REF ) were further obtained.", "Binet's formula (REF ) may be reformulated as $\\Gamma (x+1)=\\sqrt{2\\pi x}\\,\\Bigl (\\frac{x}{e}\\Bigr )^{x}e^{\\theta (x)}.$ We also call $\\theta (x)$ the remainder of Stirling's formula $n!\\sim \\sqrt{2\\pi n}\\,\\Bigl (\\frac{n}{e}\\Bigr )^{n}, \\quad n\\rightarrow \\infty $ established by James Stirling in 1764.", "When replacing $\\theta (x)$ by $\\frac{1}{12}\\psi ^{\\prime }(x+\\alpha )$ , it was proved in [21], [22], [41] that the function $F_{\\alpha }(x) =\\frac{e^x\\Gamma (x+1)}{x^x\\sqrt{2\\pi x}\\,e^{\\psi ^{\\prime }(x+\\alpha )/12}}$ is logarithmically completely monotonic on $(0,\\infty )$ if and only if $\\alpha \\ge \\frac{1}{2}$ and that the function $\\frac{1}{F_{\\alpha }(x)}$ is logarithmically completely monotonic on $(0,\\infty )$ if and only if $\\alpha =0$ .", "Consequently, the double inequality $\\exp \\biggl (\\frac{1}{12}\\psi ^{\\prime }\\biggl (x+\\frac{1}{2}\\biggr ) \\biggr ) <\\frac{e^x\\Gamma (x+1)}{x^x\\sqrt{2\\pi x}\\,} <\\exp \\biggl (\\frac{1}{12}\\psi ^{\\prime }(x) \\biggr )$ was derived in [41].", "In [20], the following conclusions were obtained.", "As $n\\rightarrow \\infty $ , the asymptotic formula $ n!", "\\sim \\frac{n^n}{e^n}\\sqrt{2\\pi n}\\,\\exp \\biggl (\\frac{1}{12}\\psi ^{\\prime }\\biggl (n+\\frac{1}{2}\\biggr ) \\biggr )$ is the most accurate one among all approximations of the form $n!", "\\sim \\frac{n^n}{e^n}\\sqrt{2\\pi n}\\,\\exp \\biggl (\\frac{1}{12}\\psi ^{\\prime }(n+a)\\biggr ),$ where $a\\in \\mathbb {R}$ ; As $x\\rightarrow \\infty $ , we have $\\Gamma (x+1)\\sim \\sqrt{2\\pi }\\,x^{x+1/2}\\exp \\biggl (\\frac{1}{12}\\psi ^{\\prime }\\biggl (x+\\frac{1}{2}\\biggr )-x+\\frac{1}{240}\\frac{1}{x^{3}}\\\\*-\\frac{11}{6720}\\frac{1}{x^{5}}+\\frac{107}{80640}\\frac{1}{x^{7}} -\\frac{2911}{1520640}\\frac{1}{x^{9}}+\\cdots \\biggr );$ For every integer $x\\ge 1$ , we have $\\exp \\biggl (\\frac{1}{240x^{3}}-\\frac{11}{6720x^{5}}\\biggr )<\\frac{e^{x}\\Gamma (x+1)}{x^{x}\\sqrt{2\\pi x}\\exp \\bigl ( \\frac{1}{12}\\psi ^{\\prime }\\bigl (x+\\frac{1}{2}\\bigr ) \\bigr )}<\\exp \\biggl (\\frac{1}{240x^{3}}\\biggr ).$ For $\\alpha \\in \\mathbb {R}$ , let $g_{\\alpha }(x)=\\frac{e^x\\Gamma (x+1)}{(x+\\alpha )^{x+\\alpha }}$ on the interval $(\\max \\lbrace 0,-\\alpha \\rbrace ,\\infty )$ .", "In [11], [13], it was showed that the function $g_{\\alpha }(x)$ is logarithmically completely monotonic if and only if $\\alpha \\ge 1$ and that the function $\\frac{1}{g_{\\alpha }(x)}$ is logarithmically completely monotonic if and only if $\\alpha \\le \\frac{1}{2}$ .", "In [2], the double inequality $\\biggl (x-\\frac{1}{2}\\biggr )\\biggl [\\ln \\biggl (x-\\frac{1}{2}\\biggr )-1\\biggr ]+\\ln \\sqrt{2\\pi }\\, -\\frac{1}{24(x-1)}\\le \\ln \\Gamma (x)\\\\*\\le \\biggl (x-\\frac{1}{2}\\biggr )\\biggl [\\ln \\biggl (x-\\frac{1}{2}\\biggr )-1\\biggr ] +\\ln \\sqrt{2\\pi }\\,-\\frac{1}{24\\bigl (\\sqrt{x^2+x+1/2}\\,-1/2\\bigr )}, \\quad x>1$ was obtained, which may be rewritten as $e^{-1/24x}< \\frac{e^{x+1/2}\\Gamma (x+1)}{\\sqrt{2\\pi }\\,(x+1/2)^{x+1/2}} \\le e^{-1/24\\bigl (\\sqrt{x^2+3x+5/2}\\,-1/2\\bigr )},\\quad x>0.$ In [41], the function $H(x)=\\frac{e^{x+1/24(x+1/2)}\\Gamma (x+1)}{(x+1/2)^{x+1/2}}$ was proved to be logarithmically completely monotonic on $(0,\\infty )$ .", "Consequently, it was deduced in [41] that the double inequality $\\alpha _1<\\frac{e^{x+1/24(x+1/2)}\\Gamma (x+1)}{(x+1/2)^{x+1/2}} \\le \\alpha _2$ holds for $x>0$ if and only if $\\alpha _1\\le \\sqrt{\\frac{2\\pi }{e}}\\,$ and $\\alpha _2\\ge \\sqrt{2}\\,e^{1/12}$ .", "It is clear that the left hand side inequality in (REF ) is stronger than the corresponding one in (REF ), but, when $x\\ge 1$ , the right hand side inequality in (REF ) is weaker than the corresponding one in (REF ).", "The double inequality in [15] is weaker than (REF ).", "In [35], among other things, some necessary and sufficient conditions on $\\lambda \\ge 0$ for the function $H_\\lambda (x)=\\frac{e^{x+1/24(x+\\lambda )}\\Gamma (x+1)}{(x+1/2)^{x+1/2}}$ to be logarithmically completely monotonic on $(0,\\infty )$ were discovered.", "For more information on inequalities for bounding the gamma function $\\Gamma $ and on the (logarithmically) complete monotonicity of functions involving $\\Gamma $ , please refer to the survey articles [24], [33], [34] and plenty of references collected therein.", "When replacing $\\theta (x)$ by $\\frac{\\vartheta (x)}{12x}$ , Binet's formulas (REF ) and (REF ) become $\\Gamma (x+1) =\\sqrt{2\\pi x}\\Bigl (\\frac{x}{e}\\Bigr )^{x}e^{\\vartheta (x)/12x}, \\quad x>0.$ We call $\\vartheta (x)$ the variant remainder of Stirling's formula.", "In [42], it was proved that the function $\\vartheta (x)$ is strictly increasing on $[1,\\infty )$ .", "In [19], it was further proved that $\\vartheta (x)$ is strictly decreasing on $(0,\\beta )$ and strictly increasing on $(\\beta ,\\infty )$ , where $\\beta =0.34142\\cdots $ is the unique positive real number satisfying $\\ln \\Gamma (\\beta +1) +\\beta \\psi (\\beta +1) -\\ln \\sqrt{2\\pi }\\,-2\\beta \\ln \\beta +\\beta =0.$ For $x>-\\frac{1}{2}$ , let $\\Gamma (x+1) =\\sqrt{2\\pi }\\,\\biggl (\\frac{x+1/2}{e}\\biggr )^{x+1/2}e^{b(x)}$ and $\\Gamma (x+1) =\\sqrt{2\\pi }\\,\\biggl (\\frac{x+1/2}{e}\\biggr )^{x+1/2}e^{w(x)/12x}.$ We call $b(x)$ and $w(x)$ the remainder of Burnside's formula and the variant remainder of Burnside's formula respectively.", "$n!\\sim \\sqrt{2\\pi }\\,\\biggl (\\frac{n+1/2}{e}\\biggr )^{n+1/2}, \\quad n\\rightarrow \\infty $ established in [3].", "In [43], it was proved that the functions $-b(x)$ , $xb(x) +\\frac{1}{24}$ , and $w(x)+\\frac{1}{2}$ are completely monotonic on $\\bigl (\\frac{1}{2},\\infty \\bigr )$ .", "It is clear that the complete monotonicity of $-b(x)$ on $\\bigl (\\frac{1}{2},\\infty \\bigr )$ may be derived from the logarithmically complete monotonicity of the function $\\frac{1}{g_{\\alpha }(x)}$ on $(0,\\infty )$ .", "The aim of this paper is to extend and generalize some results mentioned above.", "Our main results may be formulated as the following theorems.", "Theorem 1.1 The remainder $b(x)$ and the variant remainder $w(x)$ of Burnside's formula have the following properties.", "For $x>-\\frac{1}{2}$ , the remainders $b(x)$ and $w(x)$ of Burnside's formula have the integral representation $b(x)=\\frac{w(x)}{12x}=\\int _0^\\infty \\biggl (1-\\frac{e^t}{2t}+\\frac{1}{e^{2t}-1}\\biggr ) \\frac{1}{t}e^{-2(x+1)t}\\operatorname{d}t$ and the function $-b(x)=-\\frac{w(x)}{12x}$ is completely monotonic on $\\bigl (-\\frac{1}{2},\\infty \\bigr )$ ; For $x>0$ , the function $xb(x)+\\frac{1}{24}=\\frac{1}{12}\\biggl [w(x)+\\frac{1}{2}\\biggr ]=\\frac{1}{2}\\int _0^\\infty \\frac{f_1(t)}{t^3(e^{2t}-1)^2}e^{-(2x+1)t}\\operatorname{d}t$ and is completely monotonic on $(0,\\infty )$ , where $f_1(t)=(t+2)e^{4 t}-2 t(2 t+1) e^{3 t}-2 (t+2)e^{2 t}+2t e^t +t+2$ is absolutely monotonic on $(0,\\infty )$ ; For $x>0$ , the function $\\frac{1}{6}-\\frac{x}{3}-8x^2b(x)=\\frac{2}{3}\\biggl [\\frac{1}{4}-\\frac{x}{2}-xw(x)\\biggr ]=\\int _0^\\infty \\frac{f_2(t)}{t^4(e^{2t}-1)^3} e^{-(2x+1)t}\\operatorname{d}t$ and is completely monotonic on $(0,\\infty )$ , where $\\begin{split}f_2(t)&=(t^2+4t+6)e^{6 t}-4 t(2t^2+2 t+1) e^{5 t}-3(t^2+4 t+6)e^{4 t}\\\\&\\quad -8t(t^2-t-1) e^{3 t}+3(t^2+4 t+6) e^{2 t}-4t e^t -t^2-4 t-6\\end{split}$ is absolutely monotonic on $(0,\\infty )$ ; For $x>0$ , the function $\\begin{split}16x^3b(x)+\\frac{2}{3}x^2-\\frac{1}{3}x+\\frac{23}{180}&=\\frac{4}{3}\\biggl [x^2w(x)+\\frac{1}{2}x^2-\\frac{1}{4}x+\\frac{23}{240}\\biggr ]\\\\&=\\int _0^\\infty \\frac{f_3(t)}{t^5(e^{2t}-1)^4} e^{-(2x+1)t}\\operatorname{d}t\\end{split}$ and is completely monotonic on $(0,\\infty )$ , where $\\begin{aligned}f_3(t)&=(t^3+6 t^2+18 t+24)e^{8 t}-4t(4 t^3+6t^2+6 t+3) e^{7 t}\\\\&\\quad -4 (t^3+6 t^2+18 t+24)e^{6 t}-4 t (16 t^3-12 t-9)e^{5 t}\\\\&\\quad +6 (t^3+6 t^2+18 t+24) e^{4 t}-4t (4 t^3-6 t^2+6 t+9) e^{3 t}\\\\&\\quad -4 (t^3+6 t^2+18t+24)e^{2 t} +12 te^t +t^3+6 t^2+18 t+24\\end{aligned}$ is absolutely monotonic on $(0,\\infty )$ ; For $x>-\\frac{1}{2}$ , the function $(2x+1)b(x)+\\frac{1}{12}$ has the integral representation $(2x+1)b(x)+\\frac{1}{12}=\\int _0^\\infty \\frac{h_1(t)}{\\left(e^{2 t}-1\\right)^2 t^3} e^{-(2x+1)t}\\operatorname{d}t$ and is completely monotonic on $\\bigl (-\\frac{1}{2},\\infty \\bigr )$ , where $h_1(t)=e^{4 t}- t(t+1)e^{3 t}-2 e^{2 t}- t(t-1)e^t +1$ is absolutely monotonic on $(0,\\infty )$ ; For $x>-\\frac{1}{2}$ , the function $-(x+1)b(x)-\\frac{1}{24}$ has the integral representation $-(x+1)b(x)-\\frac{1}{24}=\\frac{1}{4}\\int _0^\\infty \\frac{h_2(t)}{t^3(e^{2t}-1)^2}e^{-(2x+1)t}\\operatorname{d}t$ and is completely monotonic on $\\bigl (-\\frac{1}{2},\\infty \\bigr )$ , where $h_2(t)=e^{4 t} (t-2)+2 e^{3 t} t-2 e^{2 t} (t-2)+2 e^t t (2 t-1)+t-2$ is absolutely monotonic on $(0,\\infty )$ ; For $x>-\\frac{1}{2}$ , the function $-(x+1)^2b(x)-\\frac{1}{24}x-\\frac{1}{16}$ has the integral representation $-(x+1)^2b(x)-\\frac{1}{24}x-\\frac{1}{16} =\\frac{1}{8}\\int _0^\\infty \\frac{h_3(t)}{(e^{2 t}-1)^3 t^4} e^{-(2x+1)t}\\operatorname{d}t$ and is completely monotonic on $\\bigl (-\\frac{1}{2},\\infty \\bigr )$ , where $\\begin{split}h_3(t)&=e^{6 t} (t^2-4 t+6)-4 e^{5 t} t-3 e^{4 t} (t^2-4 t+6)-8 e^{3 t} (t^2+t-1) t\\\\&\\quad +3 e^{2 t} (t^2-4 t+6)-4 e^t (2 t^2-2 t+1) t-t^2+4 t-6\\end{split}$ is absolutely monotonic on $(0,\\infty )$ ; For $x>-\\frac{1}{2}$ , the function $-(x+1)^3b(x)-\\frac{x^2}{24}-\\frac{5 x}{48}-\\frac{203}{2880}$ has the integral representation $-(x+1)^3b(x)-\\frac{x^2}{24}-\\frac{5 x}{48}-\\frac{203}{2880}=\\int _0^\\infty \\frac{h_4(t)}{(e^{2 t}-1)^4 t^5} e^{-(2x+1)t}\\operatorname{d}t$ and is completely monotonic on $\\bigl (-\\frac{1}{2},\\infty \\bigr )$ , where $\\begin{aligned}h_4(t)&=e^{8 t} (t^3-6 t^2+18 t-24)+12 e^{7 t} t-4 e^{6 t} (t^3-6 t^2+18 t-24)\\\\&\\quad +4 e^{5 t} (4 t^3+6 t^2+6 t-9) t+6 e^{4 t} (t^3-6 t^2+18 t-24)\\\\&\\quad +4 e^{3 t} (16 t^3-12 t+9) t-4 e^{2 t} (t^3-6 t^2+18 t-24)\\\\&\\quad +4 e^t (4 t^3-6 t^2+6 t-3) t+t^3-6 t^2+18 t-24\\end{aligned}$ is absolutely monotonic on $(0,\\infty )$ .", "Theorem 1.2 The functions $\\biggl [\\frac{\\Gamma (x+1)}{\\sqrt{2\\pi }\\,} \\biggl (\\frac{e}{x+1/2}\\biggr )^{x+1/2}\\biggr ]^x,\\quad \\biggl [\\frac{\\sqrt{2\\pi }\\,}{\\Gamma (x+1)} \\biggl (\\frac{x+1/2}{e}\\biggr )^{x+1/2}\\biggr ]^{8x^2} e^{-x/3},$ and $\\biggl [\\frac{\\Gamma (x+1)}{\\sqrt{2\\pi }\\,} \\biggl (\\frac{e}{x+1/2}\\biggr )^{x+1/2}\\biggr ]^{16x^3} e^{x(2x-1)/3}$ are logarithmically completely monotonic on $(0,\\infty )$ .", "The functions $\\frac{\\sqrt{2\\pi }\\,}{\\Gamma (x+1)} \\biggl (\\frac{x+1/2}{e}\\biggr )^{x+1/2},\\quad \\biggl [\\frac{\\Gamma (x+1)}{\\sqrt{2\\pi }\\,} \\biggl (\\frac{e}{x+1/2}\\biggr )^{x+1/2}\\biggr ]^{2x+1},\\\\\\biggl [\\frac{\\sqrt{2\\pi }\\,}{\\Gamma (x+1)} \\biggl (\\frac{x+1/2}{e}\\biggr )^{x+1/2}\\biggr ]^{x+1},\\quad \\biggl [\\frac{\\sqrt{2\\pi }\\,}{\\Gamma (x+1)} \\biggl (\\frac{x+1/2}{e}\\biggr )^{x+1/2}\\biggr ]^{(x+1)^2} e^{-x/24},$ and $\\biggl [\\frac{\\sqrt{2\\pi }\\,}{\\Gamma (x+1)} \\biggl (\\frac{x+1/2}{e}\\biggr )^{x+1/2}\\biggr ]^{(x+1)^3} e^{-(2x+5)/48}$ are logarithmically completely monotonic on $\\bigl (-\\frac{1}{2},\\infty \\bigr )$ ." ], [ "Proofs of main results", "Now we start out to prove Theorems REF and REF .", "From (REF ) and (REF ), it follows that $w(x) =12xb(x)=12x\\biggl [\\ln \\Gamma (x+1)-\\frac{1}{2}\\ln (2\\pi ) -\\biggl (x+\\frac{1}{2}\\biggr )\\ln \\biggl (x+\\frac{1}{2}\\biggr )+x +\\frac{1}{2}\\biggr ].$ By Binet's formula (REF ), we have $\\ln \\Gamma (x+1)= \\biggl (x+\\frac{1}{2}\\biggr )\\ln (x+1)-x-1+\\ln \\sqrt{2\\pi }\\,+\\theta (x+1).$ Substituting this into (REF ) results in $b(x)=\\frac{w(x)}{12x} =\\frac{1}{2}\\biggl [(2x+1)\\ln \\biggl (1+\\frac{1}{2x+1}\\biggr ) -1\\biggr ]+\\theta (x+1).$ It was listed in [40] that $x^2 \\ln \\biggl (1+\\frac{1}{x}\\biggr ) -x+\\frac{1}{2}=\\int _0^\\infty \\frac{2-(t^2+2 t+2) e^{-t}}{t^3}e^{-xt}\\operatorname{d}t, \\quad x>0.$ Making use of (REF ) in (REF ) produces $b(x) &=\\frac{1}{2(2x+1)}\\biggl [\\int _0^\\infty \\frac{2-(t^2+2t+2) e^{-t}}{t^3}e^{-(2x+1)t}\\operatorname{d}t -\\frac{1}{2}\\biggr ]+\\theta (x+1)\\\\&=\\frac{1}{2(2x+1)}\\biggl \\lbrace \\int _0^\\infty \\frac{\\operatorname{d}}{\\operatorname{d}t}\\biggl [\\frac{e^{-t} (1+t-e^t)}{t^2}\\biggr ] e^{-(2x+1)t}\\operatorname{d}t -\\frac{1}{2}\\biggr \\rbrace +\\theta (x+1)\\\\&=\\frac{1}{2(2x+1)}\\biggl [\\frac{e^{-t} (1+t-e^t)}{t^2}e^{-(2x+1)t}\\bigg |_{t=0}^{t=\\infty } \\\\&\\quad +(2x+1)\\int _0^\\infty \\frac{e^{-t} (1+t-e^t)}{t^2} e^{-(2x+1)t}\\operatorname{d}t -\\frac{1}{2}\\biggr ] +\\theta (x+1)\\\\&=\\frac{1}{2}\\int _0^\\infty \\frac{1+t-e^t}{t^2} e^{-2(x+1)t}\\operatorname{d}t+\\theta (x+1)\\\\&=\\frac{1}{2}\\int _0^\\infty \\frac{1+t-e^t}{t^2} e^{-2(x+1)t}\\operatorname{d}t +\\int _{0}^{\\infty }\\biggl (\\frac{1}{e^{t}-1}-\\frac{1}{t}+\\frac{1}{2}\\biggr )\\frac{e^{-(x+1)t}}{t}\\operatorname{d}t\\\\&=\\frac{1}{2}\\int _0^\\infty \\frac{1+t-e^t}{t^2} e^{-2(x+1)t}\\operatorname{d}t +\\int _{0}^{\\infty }\\biggl (\\frac{1}{e^{2t}-1}-\\frac{1}{2t}+\\frac{1}{2}\\biggr )\\frac{e^{-2(x+1)t}}{t}\\operatorname{d}t\\\\&=\\int _0^\\infty \\biggl [\\frac{1+t-e^t}{2t}+\\biggl (\\frac{1}{e^{2t}-1}-\\frac{1}{2t}+\\frac{1}{2}\\biggr )\\biggr ]\\frac{e^{-2(x+1)t}}{t}\\operatorname{d}t\\\\&=\\int _0^\\infty \\biggl (1-\\frac{e^t}{2t}+\\frac{1}{e^{2t}-1}\\biggr )\\frac{e^{-t}}{t}e^{-(2x+1)t}\\operatorname{d}t.$ Since the function $1-\\frac{e^t}{2t}+\\frac{1}{e^{2t}-1}=-\\frac{e^t (e^{2 t}-2t e^t-1)}{2t (e^{2t}-1)}<0$ on $(0,\\infty )$ , from the integral representation (REF ) of $b(x)$ , it may be easily deduced that the function $-b(x)$ is completely monotonic on $\\bigl (-\\frac{1}{2},\\infty \\bigr )$ .", "Utilizing the integral representation (REF ) of $b(x)$ and integrating by parts reveal $-xb(x)&=\\frac{1}{2}\\int _0^\\infty \\biggl (1-\\frac{e^t}{2t}+\\frac{1}{e^{2t}-1}\\biggr ) \\frac{e^{-2t}}{t}\\frac{\\operatorname{d}e^{-2xt}}{\\operatorname{d}t}\\operatorname{d}t\\\\&=\\frac{1}{2}\\biggl [\\frac{1}{12}-\\int _0^\\infty \\frac{e^{-t}f_1(t)}{2 t^3(e^{2t}-1)^2}e^{-2xt}\\operatorname{d}t\\biggr ]\\\\&=\\frac{1}{2}\\biggl [\\frac{1}{12}+\\frac{1}{4x}\\int _0^\\infty \\frac{e^{-t}f_1(t)}{t^3(e^{2t}-1)^2}\\frac{\\operatorname{d}e^{-2xt}}{\\operatorname{d}t}\\operatorname{d}t\\biggr ]\\\\&=\\frac{1}{2}\\biggl \\lbrace \\frac{1}{12}+\\frac{1}{4x}\\biggl [-\\frac{1}{6}+\\int _0^\\infty \\frac{e^{-t} f_2(t)}{t^4(e^{2t}-1)^3} e^{-2xt}\\operatorname{d}t\\biggr ]\\biggr \\rbrace \\\\&=\\frac{1}{2}\\biggl \\lbrace \\frac{1}{12}+\\frac{1}{4x}\\biggl [-\\frac{1}{6}-\\frac{1}{2x} \\int _0^\\infty \\frac{e^{-t}f_2(t)}{t^4(e^{2t}-1)^3} \\frac{\\operatorname{d}e^{-2xt}}{\\operatorname{d}t}\\operatorname{d}t\\biggr ]\\biggr \\rbrace \\\\&=\\frac{1}{2}\\biggl \\lbrace \\frac{1}{12}+\\frac{1}{4x}\\biggl [-\\frac{1}{6}-\\frac{1}{2x} \\biggl (-\\frac{23}{180}+\\int _0^\\infty \\frac{e^{-t}f_3(t)}{t^5(e^{2t}-1)^4} e^{-2xt}\\operatorname{d}t\\biggr )\\biggr ]\\biggr \\rbrace .$ By straightforward computation, we have $f_1^{\\prime }(t)&=e^{4 t} (4 t+9)-2 e^{3 t} (6 t^2+7 t+1)-2 e^{2 t} (2 t+5)+2 e^t (t+1)+1\\\\&\\rightarrow 0\\quad \\text{as $t\\rightarrow 0$},\\\\f_1^{\\prime \\prime }(t)&=2 e^t [4 e^{3 t} (2 t+5)-e^{2 t} (18 t^2+33 t+10)-4 e^t (t+3)+t+2]\\\\&\\triangleq 2 e^tf_{11}(t)\\\\&\\rightarrow 0\\quad \\text{as $t\\rightarrow 0$},\\\\f_{11}^{\\prime }(t)&=4 e^{3 t} (6 t+17)-e^{2 t} (36 t^2+102 t+53)-4 e^t (t+4)+1\\\\&\\rightarrow 0\\quad \\text{as $t\\rightarrow 0$},\\\\f_{11}^{\\prime \\prime }(t)&=4 e^t [3 e^{2 t} (6 t+19)-e^t (18 t^2+69 t+52)-t-5]\\\\&\\triangleq 4 e^t f_{12}(t)\\\\&\\rightarrow 0\\quad \\text{as $t\\rightarrow 0$},\\\\f_{12}^{\\prime }(t)&=12 e^{2 t} (3 t+11)-e^t (18 t^2+105 t+121)-1\\\\&\\rightarrow 10\\quad \\text{as $t\\rightarrow 0$},\\\\f_{12}^{\\prime \\prime }(t)&=e^t [12 e^t (6 t+25)-18 t^2-141 t-226]\\\\&\\triangleq e^tf_{13}(t)\\\\&\\rightarrow 74\\quad \\text{as $t\\rightarrow 0$},\\\\f_{13}^{\\prime }(t)&=3 [4 e^t (6 t+31)-12 t-47]\\\\&\\rightarrow 231\\quad \\text{as $t\\rightarrow 0$},\\\\f_{13}^{\\prime \\prime }(t)&=12[e^t (6 t+37)-3]\\\\&\\rightarrow 408\\quad \\text{as $t\\rightarrow 0$},\\\\f_{13}^{\\prime \\prime \\prime }(t)&=12 e^t (6 t+43).$ As a result, since the product of finitely many absolutely monotonic functions is still absolutely monotonic, the function $f_1(t)$ is absolutely monotonic on $(0,\\infty )$ .", "This means that the function $xb(x)+\\frac{1}{24}$ is completely monotonic on $(0,\\infty )$ .", "By direct calculation, we have $f_2^{\\prime }(t)&=e^{6 t} (6 t^2+26 t+40)-4 e^{5 t} (10 t^3+16 t^2+9 t+1)-6 e^{4 t} (2 t^2+9 t+14)\\\\&\\quad -e^{3 t} (24 t^3-40 t-8)+6 e^{2 t} (t^2+5 t+8)-4 e^t (t+1)-2 (t+2)\\\\&\\rightarrow 0\\quad \\text{as $t\\rightarrow 0$},\\\\f_2^{\\prime \\prime }(t)&=2 \\bigl [e^{6 t} (18 t^2+84 t+133)-2 e^{5 t} (50 t^3+110 t^2+77 t+14)\\\\&\\quad -3 e^{4 t} (8 t^2+40 t+65)-4 e^{3 t} (9 t^3+9 t^2-15 t-8)\\\\&\\quad +e^{2 t} (6 t^2+36 t+63)-2 e^t (t+2)-1\\bigr ]\\\\&\\rightarrow 0\\quad \\text{as $t\\rightarrow 0$},\\\\f_2^{\\prime \\prime \\prime }(t)&=4 e^t \\bigl [9 e^{5 t} (6 t^2+30 t+49)-e^{4 t} (250 t^3+700 t^2+605 t+147)\\\\&\\quad -6 e^{3 t} (8 t^2+44 t+75)-6 e^{2 t} (9 t^3+18 t^2-9 t-13)\\\\&\\quad +e^t (6 t^2+42 t+81)-t-3\\bigr ]\\\\&\\triangleq 4 e^tf_{21}(t)\\\\&\\rightarrow 0\\quad \\text{as $t\\rightarrow 0$},\\\\f_{21}^{\\prime }(t)&=9 e^{5 t} (30 t^2+162 t+275)-e^{4 t} (1000 t^3+3550 t^2+3820 t+1193)\\\\&\\quad -6 e^{3 t} (24 t^2+148 t+269)-6 e^{2 t} (18 t^3+63 t^2+18 t-35)\\\\&\\quad +3 e^t (2 t^2+18 t+41)-1\\\\&\\rightarrow 0\\quad \\text{as $t\\rightarrow 0$},\\\\f_{21}^{\\prime \\prime }(t)&=e^t \\bigl [9 e^{4 t} (150 t^2+870 t+1537)-4 e^{3 t} (1000 t^3+4300 t^2+5595 t+2148)\\\\&\\quad -6 e^{2 t} (72 t^2+492 t+955)-12 e^t (18 t^3+90 t^2+81 t-26)\\\\&\\quad +6 t^2+66 t+177\\bigr ]\\\\&\\triangleq e^tf_{22}(t)\\\\&\\rightarrow 0\\quad \\text{as $t\\rightarrow 0$},\\\\f_{22}^{\\prime }(t)&=2 \\bigl [9 e^{4 t} (300 t^2+1890 t+3509)-2 e^{3 t}(3000 t^3+15900 t^2+25385 t\\\\&\\quad +12039)-6 e^{2 t} (72 t^2+564 t+1201)-6 e^t (18 t^3+144 t^2+261 t+55)\\\\&\\quad +6 t+33\\bigr ]\\\\&\\rightarrow 0\\quad \\text{as $t\\rightarrow 0$},\\\\f_{22}^{\\prime \\prime }(t)&=4 \\bigl [9 e^{4 t} (600 t^2+4080 t+7963)-e^{3 t}(9000 t^3+56700 t^2\\\\&\\quad +107955 t+61502)-6 e^{2 t} (72 t^2+636 t+1483)\\\\&\\quad -3 e^t (18 t^3+198 t^2+549 t+316)+3\\bigr ]\\\\&\\rightarrow 1288\\quad \\text{as $t\\rightarrow 0$},\\\\f_{22}^{\\prime \\prime \\prime }(t)&=12 e^t \\bigl [12 e^{3 t}(600 t^2+4380 t+8983)-e^{2 t}(9000 t^3+65700 t^2+145755 t\\\\&\\quad +97487)-4 e^t (72 t^2+708 t+1801)-18 t^3-252 t^2-945 t-865\\bigr ]\\\\&\\triangleq 12 e^tf_{23}(t)\\\\&\\rightarrow 26880\\quad \\text{as $t\\rightarrow 0$},\\\\f_{23}^{\\prime }(t)&=36 e^{3 t} (600 t^2+4780 t+10443)-e^{2 t}(18000 t^3+158400 t^2+422910 t\\\\&\\quad +340729)-4 e^t (72 t^2+852 t+2509)-9 (6 t^2+56 t+105)\\\\&\\rightarrow 24238\\quad \\text{as $t\\rightarrow 0$},\\\\f_{23}^{\\prime \\prime }(t)&=4 \\bigl [9 e^{3 t} (1800 t^2+15540t+36109)-e^{2t}(9000 t^3+92700 t^2+290655 t\\\\&\\quad +276092)-e^t (72 t^2+996 t+3361)-9 (3 t+14)\\bigr ]\\\\&\\rightarrow 181608\\quad \\text{as $t\\rightarrow 0$},\\\\f_{23}^{\\prime \\prime \\prime }(t)&=4 \\bigl [81 e^{3 t} (600 t^2+5580 t+13763)-e^{2 t} (18000 t^3+212400 t^2+766710 t\\\\&\\quad +842839)-e^t (72 t^2+1140 t+4357)-27\\bigr ]\\\\&\\rightarrow 1070320\\quad \\text{as $t\\rightarrow 0$},\\\\f_{23}^{(4)}(t)&=4 e^t \\bigl [243 e^{2 t} (600 t^2+5980 t+15623)-4 e^t (9000 t^3+119700 t^2+489555 t\\\\&\\quad +613097)-72 t^2-1284 t-5497\\bigr ]\\\\&\\triangleq 4 e^tf_{24}(t)\\\\&\\rightarrow 5354016\\quad \\text{as $t\\rightarrow 0$},\\\\f_{24}^{\\prime }(t)&=2 \\bigl [243 e^{2 t} (600 t^2+6580 t+18613)-2e^t(9000 t^3+146700 t^2+728955 t\\\\&\\quad +1102652)-72 t-642\\bigr ]\\\\&\\rightarrow 4634026\\quad \\text{as $t\\rightarrow 0$},\\\\f_{24}^{\\prime \\prime }(t)&=4 \\bigl [243 e^{2 t} (600 t^2+7180 t+21903)-e^t (9000 t^3+173700 t^2\\\\&\\quad +1022355 t+1831607)-36\\bigr ]\\\\&\\rightarrow 13963144\\quad \\text{as $t\\rightarrow 0$},\\\\f_{24}^{\\prime \\prime \\prime }(t)&=4 e^t \\bigl [486 e^t (600 t^2+7780 t+25493)-9000 t^3-200700 t^2\\\\&\\quad -1369755 t-2853962\\bigr ]\\\\&\\triangleq 4 e^tf_{25}(t)\\\\&\\rightarrow 38142544\\quad \\text{as $t\\rightarrow 0$},\\\\f_{25}^{\\prime }(t)&=9 [54 e^t (600 t^2+8980 t+33273)-5 (600 t^2+8920 t+30439)]\\\\&\\rightarrow 14800923\\quad \\text{as $t\\rightarrow 0$},\\\\f_{25}^{\\prime \\prime }(t)&=18[27 e^t(600 t^2+10180 t+42253)-100 (30 t+223)]\\\\&\\rightarrow 20133558\\quad \\text{as $t\\rightarrow 0$},\\\\f_{25}^{\\prime \\prime \\prime }(t)&=54[9 e^t(600 t^2+11380 t+52433)-1000]\\\\&\\rightarrow 25428438\\quad \\text{as $t\\rightarrow 0$},\\\\f_{25}^{(4)}(t)&=486 e^t(600 t^2+12580 t+63813).$ Therefore, by the fact that the product of finitely many absolutely monotonic functions is still absolutely monotonic, we see that the function $f_2(t)$ is absolutely monotonic on $(0,\\infty )$ .", "This implies that the function $\\frac{1}{6}-\\frac{x}{3}-8x^2b(x)$ is completely monotonic on $(0,\\infty )$ .", "By similar arguments to proofs of the absolute monotonicity of $f_1$ and $f_2$ , we may verify that the function $f_3$ is also absolutely monotonic on $(0,\\infty )$ .", "Consequently, the function $16x^3b(x)+\\frac{2}{3}x^2-\\frac{1}{3}x+\\frac{23}{180}$ is completely monotonic on $(0,\\infty )$ .", "Employing the integral representation (REF ) and integrating by parts gives $-(2x+1)b(x)&=\\int _0^\\infty \\biggl (1-\\frac{e^t}{2t}+\\frac{1}{e^{2t}-1}\\biggr ) \\frac{e^{-t}}{t}\\frac{\\operatorname{d}e^{-(2x+1)t}}{\\operatorname{d}t}\\operatorname{d}t\\\\&=\\frac{1}{12}-\\int _0^\\infty \\frac{h_1(t)}{(e^{2 t}-1)^2 t^3} e^{-(2x+1)t}\\operatorname{d}t.$ The verification of the absolute monotonicity of $h_1(t)$ is same as that of $f_1$ and $f_2$ .", "Accordingly, the function $(2x+1)b(x)+\\frac{1}{12}$ is completely monotonic on $\\bigl (-\\frac{1}{2},\\infty \\bigr )$ .", "Using the integral representation (REF ) and integrating by parts acquires $-2(x+1)b(x)=\\int _0^\\infty \\biggl (1-\\frac{e^t}{2t}+\\frac{1}{e^{2t}-1}\\biggr ) \\frac{1}{t}\\frac{\\operatorname{d}e^{-2(x+1)t}}{\\operatorname{d}t}\\operatorname{d}t\\\\=\\frac{1}{12}+\\frac{1}{2}\\int _0^\\infty \\frac{e^th_2(t)}{t^3(e^{2t}-1)^2}e^{-2(x+1)t}\\operatorname{d}t\\\\=\\frac{1}{12}-\\frac{1}{4(x+1)} \\int _0^\\infty \\frac{e^th_2(t)}{t^3(e^{2t}-1)^2} \\frac{\\operatorname{d}e^{-2(x+1)t}}{\\operatorname{d}t}\\operatorname{d}t\\\\=\\frac{1}{12}-\\frac{1}{4(x+1)} \\biggl [-\\frac{1}{6}-\\int _0^\\infty \\frac{e^th_3(t)}{(e^{2 t}-1)^3 t^4} e^{-2(x+1)t}\\operatorname{d}t\\biggr ]\\\\=\\frac{1}{12}-\\frac{1}{4(x+1)} \\biggl [-\\frac{1}{6}+\\frac{1}{2(x+1)}\\int _0^\\infty \\frac{e^th_3(t)}{(e^{2 t}-1)^3 t^4} \\frac{\\operatorname{d}e^{-2(x+1)t}}{\\operatorname{d}t}\\operatorname{d}t\\biggr ]\\\\=\\frac{1}{12}-\\frac{1}{4(x+1)} \\biggl \\lbrace -\\frac{1}{6}+\\frac{1}{2(x+1)}\\biggl [-\\frac{23}{180}-\\int _0^\\infty \\frac{e^th_4(t)}{(e^{2 t}-1)^4 t^5} e^{-2(x+1)t}\\operatorname{d}t\\biggr ]\\biggr \\rbrace .$ By similar arguments to proofs of the absolute monotonicity of $f_1$ and $f_2$ , we may verify that the functions $h_2$ , $h_3$ , and $h_4$ are absolutely monotonic on $(0,\\infty )$ .", "Consequently, the functions $-(x+1)b(x)-\\frac{1}{24}$ , $-(x+1)^2b(x)-\\frac{1}{24}x-\\frac{1}{16}$ , and $-(x+1)^3b(x)-\\frac{x^2}{24}-\\frac{5 x}{48}-\\frac{203}{2880}$ are completely monotonic on $\\bigl (-\\frac{1}{2},\\infty \\bigr )$ .", "The proof of Theorem REF is complete.", "This follows from reformulation of the functions involving the remainder $b(x)$ in Theorem REF ." ], [ "Remarks", "For better understanding our main results, we list several remarks as follows.", "Remark 3.1 In [18], [40], [44], we may find the classical Bernstein-Widder theorem which reads that a function $f$ is completely monotonic on $(0,\\infty )$ if and only if it is a Laplace transform of some nonnegative measure $\\mu $ , that is, $ f(x)=\\int _0^\\infty e^{-xt}\\operatorname{d}\\mu (t),$ where $\\mu (t)$ is non-decreasing and the integral converges for $0<x<\\infty $ .", "In [1], [7], [28], [29], [40], we may search out that any logarithmically completely monotonic function must be a completely monotonic function, but not conversely.", "To some extent, these reveal the significance and meanings of our main results.", "Remark 3.2 By the monotonicity in Theorem REF , we may derive some double inequalities for bounding the function $\\frac{\\Gamma (x+1)}{\\sqrt{2\\pi }\\,} \\biggl (\\frac{e}{x+1/2}\\biggr )^{x+1/2}$ or its reciprocal on the intervals $(0,\\infty )$ and $\\bigl (-\\frac{1}{2},\\infty \\bigr )$ .", "These inequalities would be better than (REF ) and (REF ).", "Remark 3.3 From (REF ) and (REF ), it follows that $\\sqrt{x}\\,\\Bigl (\\frac{x}{e}\\Bigr )^{x}e^{\\vartheta (x)/12x}=\\biggl (\\frac{x+1/2}{e}\\biggr )^{x+1/2}e^{w(x) /12x}$ which may be rewritten as $\\vartheta (x) =w(x)-6x+12x\\biggl (x+\\frac{1}{2}\\biggr )\\ln \\biggl (1+\\frac{1}{2x}\\biggr )$ and $\\theta (x) =b(x)+\\biggl (x+\\frac{1}{2}\\biggr )\\ln \\biggl (1+\\frac{1}{2x}\\biggr )-\\frac{1}{2}.$ Combining these two identities with Theorem REF , we may deduce the complete monotonicity of some functions related to the remainder $\\theta (x)$ and the variant remainder $\\vartheta (x)$ of Binet's formula.", "Furthermore, we may deduce some double inequalities for bounding the function $\\frac{e^x\\Gamma (x+1)}{x^x\\sqrt{2\\pi x}\\,}$ or its reciprocal on the interval $(0,\\infty )$ .", "Remark 3.4 In [14], it was given that the inequality $\\Gamma (x+1)<\\sqrt{2\\pi }\\,\\biggl (\\frac{x+1/2}{e}\\biggr )^{x+1/2} \\biggr [1-\\frac{k}{24x}+\\biggl (\\frac{k^2}{1152}+\\frac{k}{48}\\biggr )\\frac{1}{x^2}\\biggr ]^{1/k}$ is valid for $x\\ge m_1$ and for any positive integer $k$ , where $m_1\\ge 0$ is a constant depending on $k$ .", "To compare the right hand side inequality in (REF ) with  (REF ), it suffices to discuss the inequality $\\frac{e^{7/12}}{\\sqrt{\\pi }\\,}\\le e^{1/24(x+1/2)} \\biggr [1-\\frac{k}{24x} +\\biggl (\\frac{k^2}{1152}+\\frac{k}{48}\\biggr )\\frac{1}{x^2}\\biggr ]^{1/k}.$ If letting $x\\rightarrow \\infty $ , the above inequality becomes $e^{7/12}=1.79\\cdots \\le \\sqrt{\\pi }\\,=1.77\\cdots $ .", "This contradiction implies that, when $x$ is sufficiently large, the inequality (REF ) is better than the right hand side inequality in (REF ).", "In [16], it was also deduced that the inequality $\\Gamma (x+1)<\\sqrt{2\\pi x}\\,\\Bigl (\\frac{x}{e}\\Bigr )^x \\biggl (1+\\frac{k}{12x}+\\frac{k^2}{288x^2}\\biggr )^{1/k}$ holds for $x\\ge m_1$ and $1\\le k\\le 5$ and reverses for $x\\ge m_2$ and $k\\ge 6$ , where $m_1$ and $m_2$ are constants depending on $k$ .", "Because the inequalities (REF ) and (REF ) are derived from an asymptotic approximation of the gamma function $\\Gamma (x+1)$ , they may be more accurate when $x$ is sufficiently large, but not, even invalid, when $x$ is close to zero.", "Remark 3.5 Now we consider the functions $F(x) =1+4x-8x\\biggl (x+\\frac{1}{2}\\biggr ) \\ln \\biggl (1+\\frac{1}{2x}\\biggr )$ and $G(x)=\\biggl (x+\\frac{1}{2}\\biggr )\\ln \\biggl (1+\\frac{1}{2x}\\biggr )-\\frac{1}{2}$ appeared in (REF ) and (REF ).", "It is obvious that $F(x)=1-8xG(x)$ .", "We claim that the functions $F(x)$ and $G(x)$ are completely monotonic on $(0,\\infty )$ .", "This may be verified as follows.", "A direct computation gives $F^{\\prime }(x)&=4\\biggl [2-(4 x+1) \\ln \\biggl (1+\\frac{1}{2 x}\\biggr )\\biggr ],\\\\F^{\\prime \\prime }(x)&=\\frac{4(4 x+1)}{x (2 x+1)}-16\\ln \\biggl (1+\\frac{1}{2 x}\\biggr ),\\\\F^{\\prime \\prime \\prime }(x)&=-\\frac{4}{x^2 (2 x+1)^2}.$ It is clear that $\\lim _{x\\rightarrow \\infty }F^{\\prime \\prime }(x)=0$ .", "It is not difficult to see that $F^{\\prime }(x)=4\\biggl [2-2\\ln \\biggl (1+\\frac{1}{2 x}\\biggr )^{2x}-\\ln \\biggl (1+\\frac{1}{2 x}\\biggr )\\biggr ]\\rightarrow 4(2-2\\ln e-0)=0$ and $F(x) &=1-2\\ln \\biggl (1+\\frac{1}{2x}\\biggr )^{2x}+8x^2\\biggl [\\frac{1}{2x}-\\ln \\biggl (1+\\frac{1}{2x}\\biggr )\\biggr ]\\\\&\\rightarrow 1-2\\ln e+2\\lim _{u\\rightarrow 0^+}\\frac{u-\\ln (1+u)}{u^2}\\\\&=0$ as $x\\rightarrow \\infty $ , where $u=\\frac{1}{2x}$ .", "Hence, by virtue of $F^{\\prime \\prime \\prime }(x)>0$ on $(0,\\infty )$ , we may conclude $F(x)>0$ , $F^{\\prime }(x)<0$ , and $F^{\\prime \\prime }(x)>0$ .", "Furthermore, by the facts that the functions $\\frac{1}{x}$ and $\\frac{1}{1+2x}$ are completely monotonic on $(0,\\infty )$ and that the product of finitely many completely monotonic functions is also a completely monotonic function, we may see that the function $-F^{\\prime \\prime \\prime }(x)$ is completely monotonic on $(0,\\infty )$ .", "In a word, the function $F(x)$ is completely monotonic on $(0,\\infty )$ .", "The complete monotonicity of $G(x)$ may also be verified straightforwardly." ], [ "Acknowledgements", "The author thanks the anonymous referee for his/her careful corrections to and valuable comments on the original version of this paper.", "This work was partially supported by the NNSF under Grant No.", "11361038 of China and the Foundation of the Research Program of Science and Technology at Universities of Inner Mongolia Autonomous Region under Grant number No.", "NJZY14191 and No.", "NJZY14192, China." ] ]
1403.0278
[ [ "Continuous phase transition and critical behaviors of 3D black hole with\n torsion" ], [ "Abstract We study the phase transition and the critical behavior of the BTZ black hole with torsion obtained in $(1+2)$-dimensional Poincar\\'{e} gauge theory.", "According to Ehrenfest's classification, when the parameters in the theory are arranged properly the BTZ black hole with torsion may posses the second order phase transition which is also a smaller mass/larger mass black hole phase transition.", "Nevertheless, the critical behavior is different from the one in the van der Waals liquid/gas system.", "We also calculated the critical exponents of the relevant thermodynamic quantities, which are the same as the ones obtained in the Ho\\v{r}ava-Lifshitz black hole and the Born-Infeld black hole." ], [ "introduction", "The laws of black hole thermodynamics and the usual laws of thermodynamics have very similar forms.", "The good agreement indicates that black hole is also a thermodynamic system.", "Phase transitions and critical phenomena are important characteristics of usual thermodynamics.", "Thus, the natural question to ask is whether there also exists phase transition in the black hole thermodynamics.", "In fact, since the pioneering work of Davies[1] and the well-known Hawking-Page phase transition[2] were proposed, the question has been answered partly.", "The phase transitions and critical phenomena in four and higher dimensional AdS black holes have been studied extensively[3], [4], [5], [6].", "Recently, some interesting works on asymptotically anti-de Sitter black holes have been done, which show that there exists phase transition similar to the van der Waals liquid/gas phase transition[7], [8], [9], [12], [10], [11], [14], [13], [15].", "The properties of black hole are relevant to the dimension of spacetime.", "Thus we want to know whether the similar phase transition exists for lower dimensional black holes.", "The key advantage of lower dimensional black holes lies in the simplicity of the construction.", "Although just mathematical abstraction, lower dimensional black holes can be applied to physical reality in some special cases.", "Hence it can be interesting to investigate the possibility that a lower dimensional black hole does exhibit a phase transition.", "BTZ black hole is an important solution of general relativity with negative cosmological constant in three-dimensional spacetime[16], [17].", "The BTZ black hole is free of singularity and closely related with the recent developments in gravity, gauge theory and string theory[18].", "In general relativity and many other theories of gravity, curvature plays an essential role , while torsion has received less attention.", "However, torsion also has its geometrical meaning and plays some roles in gravitation theory.", "Since the 1970s, many theories of gravity with torsion have been proposed, such as Poincaré gauge gravity, de Sitter gauge gravity, teleparallel gravity, $f(T)$ gravity, etc.", "In particular, Mielke and Baekler proposed a model of three-dimensional gravity with torsion ( the MB model), which also has BTZ black hole as solution[19], [20], [21], [22], [23].", "This model aroused the following research on the thermodynamics of the BTZ black hole with torsion (BTZT black hole for short) and AdS/CFT with torsion[24], [25], [26].", "In [27] we have verified that for the BTZT black hole phase transition may exist.", "In this paper we will investigate the type of the phase transition and calculate the critical exponents.", "Although the BTZ solution is the same as the one obtained in GR, the different actions will make their thermodynamics very different.", "The modified action in the MB model will modify the conserved charges such as mass and angular momentum.", "Correspondingly the entropy of BTZ black hole and the first law of black hole thermodynamics will also been changed.", "It is shown that the heat capacity of the BTZ black hole in the MB model is not always positive any more, but changes signs at some points and may diverge at the critical point.", "Thus for the BTZT black hole in the MB model phase transition exists.", "According to Ehrenfest's classification we also consider the Gibbs free energy, the isothermal compressibility and the expansion coefficient as functions of temperature.", "It is shown that the kind of phase transition for the BTZT black hole belong to the second-order one or continuous one.", "The paper is arranged as follows: in the next section we simply introduce the MB model and its BTZ-like solution and the corresponding thermodynamic quantities.", "In section 3 according to Ehrenfest's classification we will analyze the type of the phase transition of the BTZT black hole in the extended phase space.", "In section 4 the case with non-extended phase space will be discussed and the critical behaviors are investigated.", "We make some concluding remarks in section 5." ], [ "Mielke-Baekler model and the 3D black hole with torsion", "First we should review the topological three-dimensional gravity model with torsion proposed by Mielke and Baekler[19], [20], which is a natural generalization of Riemannian GR with a cosmological constant.", "Defining curvature and torsion 2-forms out of $\\omega ^a_{~b}$ and coframe $e^a$ by $T^a&=&de^a+\\omega ^a_{~b}\\wedge e^b, \\\\R^a_{~b}&=&d\\omega ^a_{~b}+\\omega ^a_{~c}\\wedge \\omega ^c_{~b},$ the gravitational action is written as $I=\\int 2\\chi e^{a}\\wedge R_{a}-\\frac{\\Lambda }{3}\\epsilon _{abc}e^{a}\\wedge e^{b}\\wedge e^{c}+\\alpha _{3}\\left( \\omega ^{a}\\wedge d\\omega _{a}+\\frac{1}{3}\\varepsilon _{abc}\\omega ^{a}\\wedge \\omega ^{b}\\wedge \\omega ^{c}\\right) +\\alpha _{4}e^{a}\\wedge T_{a}~,$ where the dual expression, $R_a$ and $\\omega _a$ are defined by $R^{ab}=\\epsilon ^{abc}R_c$ and $\\omega ^{ab}=\\epsilon ^{abc}\\omega _c$ .", "In Eq.", "(REF ) the first term corresponds to the Einstein-Cartan action, with $\\chi =\\frac{1}{16\\pi G}$ .", "The second one is the cosmological term.", "The last two terms are the Chern-Simons term and the Nieh-Yan term, which should be given particular attention.", "The Nieh-Yan(N-Y) form is a special 2-form only for the Riemann-Cartan geometry[28], [29].", "On the four-dimensional manifold $M$ it can be written as $&&N=T^a\\wedge T_a+R_{ab}\\wedge e^a\\wedge e^b=dQ_{NY}, \\nonumber \\\\&&Q_{NY}=e^a\\wedge T_a$ The N-Y form is a kind of Chern-Simons form and will have its application to manifolds with boundaries and reflect the role of torsion in geometry.", "After variation to $\\omega ^a_{~b}$ and $e^a$ , two vacuum equations can be obtained from the MB action (REF ) $T^a&=&\\frac{p}{2}\\varepsilon ^a_{~bc}e^b\\wedge e^c, \\\\R^a&=&\\frac{q}{2}\\varepsilon ^a_{~bc}e^b\\wedge e^c,$ with the two constant coefficients $p, ~q$ defined by $p=\\frac{\\alpha _{3}\\Lambda +\\alpha _{4}\\chi }{\\alpha _{3}\\alpha _{4}-\\chi ^{2}}$ and $q=-\\frac{\\alpha _{4}^{2}+\\chi \\Lambda }{\\alpha _{3}\\alpha _{4}-\\chi ^{2}}$ .", "The curvatures in Einstein-Cartan geometry can be connected to their counterparts in Riemannian geometry.", "In particular, in three-dimensional spacetime, the equations above can be simplified to equations without torsion $\\tilde{R}^a=\\frac{\\Lambda _{eff}}{2}\\varepsilon ^a_{~bc}e^b\\wedge e^c,$ where $\\tilde{R}^a$ is the curvature without torsion and $\\Lambda _{eff}=q-\\frac{1}{4}p^{2}$ is the effective cosmological constant.", "One can let $\\Lambda _{eff}=-\\frac{1}{l^{2}}<0$ to construct an asymptotically anti-de Sitter space.", "As in the three-dimensional Einstein equation, Eq.", "(REF ) has the well-known BTZ solution.", "But in this case, torsion is contained in the gravitational action.", "The metric is $ds^{2}=-N\\left( r\\right) ^{2}dt^{2}+\\frac{1}{N\\left( r\\right) ^{2}}dr^{2}+r^{2}\\left( d\\phi +N_{\\phi }dt\\right) ^{2}$ where $N\\left( r\\right) ^{2}=\\frac{r^{2}}{l^{2}}-M_0+\\frac{J_0^{2}}{4r^{2}},\\text{\\ }N_{\\phi }\\left( r\\right) =\\frac{J_0}{2r^{2}}~.$ Here we have considered $8G=1$ .", "This metric is the same as the one in GR, except that $l=1/\\sqrt{-\\Lambda _{eff}}$ here and a constant torsion[21].", "For this metric there are two horizons: the outer one $r_+$ and the inner one $r_{-}$ .", "From $N^2(r)=0$ , one can obtain the expressions of both horizons: $r^2_{\\pm }=\\frac{M_0l^2}{2}(1\\pm \\Delta ), \\quad \\Delta =[1-(J_0/M_0l)^2]^{1/2} $ Conversely, $M_0$ and $J_0$ can be expressed as follows: $M_0=\\frac{r_{+}^2+r_{-}^2}{l^2}, \\quad J_0=\\frac{2r_+r_{-}}{l}.", "$ Hawking radiation is just a kinematic effect, which only depends on the event horizon and is irrelevant to the dynamical equations and the gravitational theories.", "Therefore the temperature of BTZ black hole in the MB model has the similar form as in GR, which is $T=\\frac{r_{+}^2-r_{-}^2}{2\\pi l^2r_{+}}$ Certainly because of the existence of $l$ , the temperature is relevant to the coefficients $\\alpha _3,\\alpha _4, \\Lambda $ of MB Lagrangian.", "Define $\\Omega _H=-\\frac{g_{t\\phi }}{g_{\\phi \\phi }}|_{r_{+}}=\\frac{J_0}{2r_+^2}$ which can be regarded as the angular velocity of BTZ black hole.", "Because of the existence of the topological terms, the asymptotically behavior is different from the one for Einstein-Cartan theory.", "Blagojevic et.al have proved that the gravitational conserved charges in the MB model should be[24], [25], [30] $M=M_0+2\\pi \\alpha _3(\\frac{pM_0}{2}-\\frac{J_0}{l^2})=aM_0-\\frac{\\displaystyle b}{\\displaystyle l^2}J_0, \\quad J=J_0+2\\pi \\alpha _3(\\frac{p J_0}{2}-M_0)=aJ_0-b M_0$ where we have defined $a=1+ \\pi \\alpha _3p,~b=2\\pi \\alpha _3$ .", "Obviously when $\\alpha _3=0$ they will return to their conventional interpretation as energy and angular momentum, as with the BTZ metric in general relativity.", "Correspondingly the entropy can be derived $S=4\\pi r_{+}+4\\pi ^2\\alpha _3(pr_{+}-\\frac{2r_{-}}{l})=4\\pi (ar_{+}-\\frac{\\displaystyle b}{\\displaystyle l}r_{-})$ It differs from the Bekenstein-Hawking result by an additional term and will coincides with Solodukhin's result if $p=0$[31].", "Black hole entropy is not always equated with one quarter of the event horizon area.", "In fact it is related to the gravitational theory under consideration.", "It can be easily verified that in the MB model the entropy, temperature, and the conserved charges not only satisfy the first law of thermodynamics $dM=TdS+\\Omega _{H} dJ $ but also fulfill the Smarr-like formula $M=\\frac{1}{2}TS+\\Omega _{H}J $ This further implicates that with torsion the BTZ black hole can still be treated as a thermodynamic system and the thermodynamic laws still hold.", "It should be noted that in the expression of the entropy of the BTZ black hole no torsion exists explicitly, only $\\alpha _4$ in $p$ implicitly.", "In particular, when $\\alpha _3=0$ the entropy in Eq.", "(REF ) returns to the usual BTZ black hole entropy.", "It means that the N-Y term $\\alpha _{4}e^{a}\\wedge T_{a}$ influences the conserved charges and the exact form of entropy only when the CS term exists." ], [ "phase transition in extended phase space", "Ehrenfest had ever attempted to classify the phase transitions .", "Phase transitions connected with an entropy discontinuity are called discontinuous or first order phase transitions, and phase transitions where the entropy is continuous are called continuous or second/higher order phase transitions.", "More precisely, for the first-order phase transition the Gibbs free energy $G(T,P,...)$ should be continuous and its first derivative with respect to the external fields: $S=-\\left.", "{\\frac{\\partial G}{\\partial T}}\\right|_{(P,...)}, \\quad V=\\left.", "{\\frac{\\partial G}{\\partial P}}\\right|_{(T,...)}$ are discontinuous at the phase transition points.", "For the second-order phase transition the Gibbs free energy $G(T,P,...)$ and its first derivative are both continuous, but the second derivative of $G$ will diverge at the phase transition points like the specific heat $C_P$ , the compressibility $\\kappa $ , the expansion coefficient $\\alpha $ : $C_P=T\\left.", "{\\frac{\\partial S}{\\partial T}}\\right|_P=-T\\left.", "{\\frac{\\partial ^2 G}{\\partial T^2}}\\right|_P,\\kappa =-\\frac{1}{V}\\left.", "{\\frac{\\partial V}{\\partial P}}\\right|_T=-\\frac{1}{V}\\left.", "{\\frac{\\partial ^2 G}{\\partial P^2}}\\right|_T ,\\alpha =-\\frac{1}{V}\\left.", "{\\frac{\\partial V}{\\partial T}} \\right|_P=-\\frac{1}{V}\\frac{\\partial ^2 G}{\\partial P\\partial T}$ In this sense, because the heat capacity is always positive, there is no second order phase transition for BTZ black hole obtained in GR.", "This property of BTZ black hole can also be verified by the method of thermodynamic curvature[32], [33].", "To utilize Ehrenfest's classification, we consider variable cosmological constant and relate it to the pressure[10], [11], [14], [15], [34], [35].", "The first law of thermodynamics for the BTZT black hole should be $dM=TdS+\\Omega dJ+V dP$ where $P=\\frac{\\displaystyle 1}{\\displaystyle 8\\pi l^2}=-\\frac{\\displaystyle \\Lambda _{eff}}{\\displaystyle 8\\pi }$ , and $V=\\left.", "{\\frac{\\displaystyle \\partial M}{\\displaystyle \\partial P}}\\right|_{S,J}$ is the corresponding thermodynamic volume.", "Therefore the mass of black hole is no more internal energy, but should be interpreted as the thermodynamic enthalpy, namely $H=M(S,P,J)$[10], [11], [34], [35].", "The first law of black hole thermodynamics represented by the internal energy $U(S,V,J)$ reads $dU=TdS+\\Omega dJ-PdV$ where $U=H-PV$ .", "For BTZT black hole one can express mass $M$ as functions of $S, J, P$ , which are $H_{\\pm }=M_{\\pm }=\\frac{1}{8\\pi ^2b^2}\\left[{a S^2+8\\pi ^2ab J }\\pm {S\\sqrt{(a^2-8\\pi b^2P)(S^2+16\\pi ^2b J)}}\\right]$ One can substitute the expressions of $S,M,J$ into Eq.", "(REF ) to test and verify it directly.", "This result can also be verified easily by differentiating the mass $M_{\\pm }$ with the entropy $S$ to get the Hawking temperature, Eq.", "(REF ).", "It should be noted that when $M$ is expressed with $r_{+}, r_{-},l$ , its form is unique.", "When we express it with the thermodynamic quantities, two different forms $M_{\\pm }$ appear.", "In fact $M_{\\pm }$ depend on the relation between $al$ and $b$ .", "They are established under different conditions: $b^2 \\le a^2l^2$ or $b^2 \\le \\frac{\\displaystyle a^2}{\\displaystyle 8\\pi P}$ , the expression $M_{-}$ is right.", "At this time, to keep the expression in square root have physical meaning, $ S^2+16\\pi ^2b J \\ge 0$ should also be satisfied.", "$b^2 \\ge a^2l^2$ or $b^2 \\ge \\frac{\\displaystyle a^2}{\\displaystyle 8\\pi P}$ , the expression $M_{+}$ should be used.", "Similarly, to keep the expression in square root have physical meaning, $ S^2+16\\pi ^2b J \\le 0$ should be satisfied.", "According to Eqs.", "(REF ),(REF ),when $|b|\\ge |a|l$ , $M,J, S$ may be negative.", "In fact for gravities with higher derivative terms there is the possibility for negative entropy and energy which depend on the parameters of higher derivative terms[36].", "Although black holes behave as thermodynamic systems, they also show some exotic behaviors, the most known one is the entropy of black holes is proportional to area and not the volume.", "Therefore it is understandable if black holes exhibit some strange thermodynamic properties.", "Below we will show that the condition $|b|\\ge |a|l$ is the key for the BTZT black hole to have phase transition.", "The temperature of BTZT black hole can be evaluated according to $S,P,J$ : $ T_{-}(S,J,P)=\\left.\\frac{\\displaystyle \\partial M_{-}}{\\displaystyle \\partial S}\\right|_{P,J} \\text{or} \\quad T_{+}(S,J,P)=\\left.\\frac{\\displaystyle \\partial M_{+}}{\\displaystyle \\partial S}\\right|_{P,J}$ They look different when expressed with the thermodynamic quantities $S,P,J$ because they correspond to different conditions for the parameters $a,b,l$ .", "When replacing the thermodynamic variables $(S,J,P)$ with the geometric ones $(r_+, r_{-},l)$ , the two expressions can be unified and Eq.", "(REF ) can turn up.", "According to Eqs.", "(REF ),(REF ), the specific heats at constant pressure and constant angular momentum can be calculated easily: $C_{-}=\\left.\\frac{\\displaystyle \\partial M_{-}}{\\displaystyle \\partial T{-}}\\right|_{P,J}=T_{-}\\frac{1}{\\left.", "{\\frac{\\partial T_{-}}{\\partial S}}\\right|_{P,J}},\\quad C_{+}=\\left.\\frac{\\displaystyle \\partial M_{+}}{\\displaystyle \\partial T{+}}\\right|_{P,J}=T_{+}\\frac{1}{\\left.", "{\\frac{\\partial T_{+}}{\\partial S}}\\right|_{P,J}}$ With the geometric quantities, the heat capacity can be written as $C_P=C_{\\pm }=\\frac{4 \\pi r_{+}^2 \\left(r_{+}^2-r_{-}^2\\right) \\left(b^2-a^2 l^2\\right)}{l \\left(b~ r_{-} \\left(3 r_{+}^2+r_{-}^2\\right)-a l r_{+} \\left(r_{+}^2+3 r_{-}^2\\right)\\right)}$ It is more appropriate to study the phase transition of the BTZT black holes according to geometric quantities $r_{+},~r_{-}$ .", "Because the conditions $r_{+}\\ge r_{-}, ~M_0>0, ~J_0>0$ must be fulfilled.", "If employing the thermodynamic quantities completely, one may omits these conditions.", "Below we will set $l=1$ and analyze the phase transition of the BTZT black hole numerically.", "According to Ehrenfest's classification, we should first derive the Gibbs free energy $G$ : $G=H-TS=M-TS$" ], [ "$a^2l^2\\ge b^2$", "One can easily plot the $G_{-}-T_{},~S-T_{},~ C_{-}-T_{}$ curves as shown in Fig.REF .", "Obviously $G_{-},~ S,~ C_{-}$ are all continuous function of temperature $T$ .", "No turning point and divergence turn up, which means no second-order phase transition happens in this case.", "This conclusion can also be supported by the method of thermodynamic curvature[33], [37].", "Figure: The Gibbs free energy, entropy and heat capacity at constant pressure as functions of temperature for BTZTblack hole for the choices of l=1,r - =1,a=±1,b=±0.5 ~l=1, ~r_{-} =1,~a=\\pm 1,~b=\\pm 0.5 and r + ≥r - r_{+}\\ge r_{-}.", "For C - C_{-} no divergent point exists." ], [ "$a^2l^2\\le b^2$", "In this case we first plot the $C_{+}-r_{+}$ curves for different values of $a,~b$ .", "In Fig.REF (a) and Fig.REF (b) we show that there is no divergent point when $a,~b$ take opposite signs.", "The phase transition may happen only when $a,~b$ are both positive or negative.", "Under the given conditions one can easily derive the position of the divergent point, $r_{c}\\approx 5.522$ .", "Obviously the phase transition is a smaller mass/larger mass black hole phase transition.", "When $a>0,~b>0$ , the smaller black hole is stable because the heat capacity is positive, when $a<0,~b<0$ , the other way around, the larger black hole is stable.", "Figure:  a=-1,b=-2 a=-1, ~b=-2 One should further analyze the value of $M, ~J,~S$ .", "We only consider the two cases which have the phase transitions.", "In Fig.REF it is shown that under the condition $a^2l^2\\le b^2$ when $a,~b$ are both positive, the corrected mass $M$ and the entropy $S$ are positive at the divergent point $r_{c}$ , while $J$ is negative; for the case with negative $a,~b$ , the situation is just the opposite.", "Thus we conclude that for the BTZT black hole if the phase transition can happen, the $M,~J,~S$ cannot be all positive.", "Figure:  a=-1,b=-2 a=-1, ~b=-2 Now we investigate the type of the phase transition for the BTZT black hole according to Ehrenfest's classification.", "To calculate the isothermal compressibility $\\kappa $ and the expansion coefficient $\\alpha $ we should first obtain the thermodynamic volume.", "$V_{+}=\\left.", "{\\frac{\\partial M_{+}}{\\partial P}}\\right|_{S,J},\\quad V_{-}=\\left.", "{\\frac{\\partial M_{-}}{\\partial P}}\\right|_{S,J}$ Although $M_{+}$ and $M_{-}$ are different, when considering the conditions for $M_{\\pm }$ one can find that $V=V_{+}=V_{-}&=&\\frac{S }{2 \\pi }\\sqrt{\\frac{16\\pi ^2 b J+S^2}{a^2-8\\pi b^2 P }}$ Inversely, one can derive the pressure $P=\\frac{-16 \\pi ^2 b J S^2-S^4+4 \\pi ^2 a^2 V^2}{32 \\pi ^3 b^2V^2}$ which must be greater than zero.", "Thus $ 4 \\pi ^2 a^2 V^2-16\\pi ^2 b J S^2-S^4 \\ge 0 $ should be satisfied.", "Because $V=V(S,P,J)$ , $\\left.", "{\\frac{\\partial V}{\\partial P}}\\right|_T=\\left.", "{\\frac{\\partial V}{\\partial P}}\\right|_S+\\frac{\\partial V}{\\partial S}\\left.", "{\\frac{\\partial S}{\\partial P}}\\right|_T$ and $\\left.", "{\\frac{\\partial V}{\\partial T}}\\right|_P=\\frac{\\partial V}{\\partial S}\\left.", "{\\frac{\\partial S}{\\partial T}}\\right|_P$ According to the above equations one can derive the isothermal compressibility $\\kappa $ and the expansion coefficient $\\alpha $ .", "We can plot the curves of $G-T,~S-T,~C-T,~\\kappa -T,~\\alpha -T$ in Fig.REF .", "Similarly, only when the parameters $a,~b$ have the same sign, the phase transition for the BTZT black hole can turn up.", "The critical temperature lies at $T_{c}=0.85$ .", "As is shown in the figure, the Gibbs free energy and the entropy are continuous functions of temperature.", "While the heat capacity, the isothermal compressibility and the expansion coefficient all diverge at the critical point.", "Therefore the phase transition at this critical point is the second-order phase transition or continuous one.", "Figure: The Gibbs free energy, entropy, the heat capacity at constant pressure and the isothermal compressibility and the expansion coefficientas functions of temperature for BTZTblack hole for the choices of l=1,r - =1,a=±1,b=±2 ~l=1, ~r_{-} =1,~a=\\pm 1,~b=\\pm 2 and r + ≥r - r_{+}\\ge r_{-}.", "For the two cases a=1,b=2a=1,~b=2 and a=-1,b=-2a=-1,~b=-2,the κ-T\\kappa -T and the α-T\\alpha -T curves are both concurrent respectively.According to Eqs.", "(REF ), (REF ) and $U=H-PV$ , we can obtain the internal energy $U=U(S,V,J)=-\\frac{\\left(S^2-2 \\pi a V\\right) \\left(-2 \\pi a V+16 \\pi ^2 b J+S^2\\right)}{32 \\pi ^3 b^2 V}$ From which one can easily derive the temperature as functions of $S,~V,~J$ : $T(S,V,J)=\\frac{\\displaystyle S (2 a \\pi V-8\\pi ^2 b J - S^2)}{\\displaystyle 8 b^2 \\pi ^3 V}$ From Eq.", "(REF ) and Eq.", "(REF ), one can derive the equation of state between the pressure $P$ , the temperature $T$ and the volume $V$ by eliminating $S$ .", "Thus one can obtain the pressure $P$ as function of $V,T,J$ in principle.", "But the expression is too lengthy and obviously it will depend on the value of $J$ .", "Below we will analyze the $P-V$ relation by means of the static scaling law[38].", "Dimensional analysis implies that the $P$ and $T$ are both homogeneous functions of the variables $S, V, J$ , since $P\\rightarrow P$ , $T\\rightarrow \\lambda T$ when $V \\rightarrow \\lambda ^2 V, S \\rightarrow \\lambda S, J \\rightarrow \\lambda ^2 J$ .", "Thus $P$ and $T$ are in fact the functions of two independent variables.", "The same logic also applies to the internal energy $U$ .", "So we can take advantage of the scaling character to redefine the functions and the variables.", "One can take $t=\\frac{T}{S},\\quad p=P, \\quad v=\\frac{V}{S^2}, \\quad j=\\frac{J}{S^2}$ In this way the entropy $S$ can be eliminated in Eqs.", "(REF ), (REF ) and they are simplified to be $t&=&\\frac{2\\pi v-8 \\pi ^2\\beta j -1}{8 \\pi ^3\\beta ^2 v}\\nonumber \\\\p&=&-\\frac{16 \\pi ^2\\beta j +1-4 \\pi ^2 v^2}{32 \\pi ^3\\beta ^2 v^2}$ Further removing the $j$ and combing the two equations together, one can get $p=\\frac{t}{2 v}+\\frac{1}{8 b^2 \\pi ^2}-\\frac{1}{8 b^2 \\pi ^2 v}+\\frac{1}{32 b^2 \\pi ^3 v^2}$ The great advantage of the above relation lies at it is irrelevant to $J$ and is more universal.", "Although the $p-v$ structure is not the same as the $P-V$ structure, it can reflect some properties of the system.", "The critical point occurs at the point where $\\frac{\\partial p}{\\partial v}=0, \\quad \\frac{\\partial ^2 p}{\\partial v^2}=0$ But unfortunately, the above equations do not have solutions.", "One can also easily plot the $p-v$ curves of Eq.", "(REF ) at different temperatures.", "We can draw conclusions that for the BTZT black hole there is no the similar phase structure and critical behavior to the van der Waals liquid-gas system." ], [ "critical behaviors in non-extended phase space", "In the non-extended phase space, the $l$ or $\\Lambda $ should be considered as constant.", "Thus Eq.", "(REF ) is modified to $M_{\\pm }=\\frac{1}{8\\pi ^2\\beta ^2}\\left[{a S^2+8\\pi ^2ab J }\\pm {\\frac{\\displaystyle S}{\\displaystyle l}\\sqrt{(a^2l^2-b^2)(S^2+16\\pi ^2b J)}}\\right]$ When considering the $J-\\Omega $ relations some rotating black holes such as Kerr-AdS black hole will exhibit similar critical behavior to the van der Waals liquid-gas system[39], [40].", "Thus we analyze for the BTZT black hole whether there are similar conclusion.", "The $J-\\Omega $ relation can be easily obtained: $J=-\\frac{4 \\pi ^2 T^2 \\left(-2 a l^2 \\Omega +b l^2 \\Omega ^2+b\\right)}{\\left(l^2 \\Omega ^2-1\\right)^2} $ One may try to derive the critical point according to $\\left.", "{\\frac{\\partial J}{\\partial \\Omega }}\\right|_T=0, \\quad \\left.", "{\\frac{\\partial ^2 J}{\\partial \\Omega ^2}}\\right|_T=0$ But it can be easily verified that also no solution exists.", "In this case the heat capacity $C$ is the same as Eq.", "(REF ).", "However, the isothermal compressibility should be defined as[40] $\\kappa _{T}=\\left.", "{\\frac{\\partial \\Omega }{\\partial J}}\\right|_T=-\\frac{\\left(l^2 \\Omega ^2-1\\right)^3}{8 \\pi ^2 l^2 T^2 \\left(3 a l^2 \\Omega ^2+a-b l^2 \\Omega ^3-3 b \\Omega \\right)}$ We plot the $\\kappa _{T}-T$ curves as in Fig.5.", "In this case the critical temperature still lies at $T_{c}=0.85$ .", "Figure:  a=-1,b=-2 a=-1, ~b=-2 In order to see the thermodynamic behavior near the critical point, the critical exponents can be introduced as $&&J-J_c\\sim |\\Omega -\\Omega _c|^{\\delta }, \\quad \\Omega -\\Omega _c\\sim |T-T_c|^{\\beta }\\nonumber \\\\&&C_J\\sim |T-T_c|^{-\\alpha }, \\quad \\kappa _T\\sim |T-T_c|^{-\\gamma }$ From Eq.", "(REF ), at the critical point $T=T_c$ , the first derivative of $J$ over $\\Omega $ satisfy $\\left.", "{\\frac{\\partial J}{\\partial \\Omega }}\\right|_{T_{c}} =0$ But the second derivative can be calculated as $\\left.", "{\\frac{\\partial ^2 J}{\\partial \\Omega ^2}}\\right|_{T_{c}}=\\pm 322.187\\ne 0$ for the two cases $a=1,~b=2,~l=1$ and $a=-1,~b=-2,~l=1$ .", "Thus $J-J_c=\\left.", "{\\frac{\\partial ^2 J}{\\partial \\Omega ^2}}\\right|_{T_{c}}(\\Omega -\\Omega _c)^2+O((\\Omega -\\Omega _c)^3)$ which means $\\delta =2$ .", "According to Eq.", "(REF ), $&&C_J=\\nonumber \\\\&&-\\frac{4 \\pi r_{+}^2 \\left(b^2-a^2 l^2\\right) \\left(-2 a l r_{+} \\left(\\sqrt{a^2 l^2 r_{+}^2-b^2 r_{+}^2-b J l^2}+a l r_{+}\\right)+2 b^2 r_{+}^2+b J l^2\\right)}{l \\left(2 a^3 l^3 r_{+}^3+2 a^2 l^2 r_{+}^2 \\sqrt{a^2 l^2 r_{+}^2-b^2 r_{+}^2-b J l^2}+b \\left(J l^2-2 b r_{+}^2\\right) \\sqrt{a^2 l^2 r_{+}^2-b^2 r_{+}^2-b J l^2}-2 a b^2 l r_{+}^3\\right)}$ One can set $T=T_c(1+\\epsilon ), \\quad r_+=r_c(1+\\Delta )$ where $|\\epsilon |,~|\\Delta |\\ll 1$ .", "Because the entropy $S$ and the temperature $T$ can both be expressed as $S=S(r_{+},J),~T=T(r_{+},~J)$ , and $C_J=T\\left.", "{\\frac{\\partial S}{\\partial T}}\\right|_{J}=T\\frac{\\left.", "{\\frac{\\partial S}{\\partial r_{+}}}\\right|_{J}}{\\left.", "{\\frac{\\partial T}{\\partial r_{+}}}\\right|_{J}}$ According to Fig.REF , at the critical point $r=r_c$ $\\left.\\left({\\frac{\\partial T}{\\partial r_{+}}}\\right)_{J}\\right|_{r=r_c}=0$ Moreover one can easily verify $\\left.\\left({\\frac{\\partial ^2 T}{\\partial r_{+}^2}}\\right)_{J}\\right|_{r=r_c}\\ne 0$ .", "Therefore, in a sufficiently small neighborhood of $r_c$ , one can expand $T$ in terms of $r_{+}$ as $T(r_{+})=T(r_c)+\\frac{\\displaystyle 1}{\\displaystyle 2}\\left.\\left({\\frac{\\partial ^2 T}{\\partial r_{+}^2}}\\right)_{J}\\right|_{r=r_c}r_c^2\\Delta ^2+O(\\Delta ^3)$ from which we obtain $\\Delta =\\frac{\\displaystyle \\epsilon ^{1/2}}{\\displaystyle D^{1/2}}$ where $D=\\frac{\\displaystyle r_c^2}{\\displaystyle 2T_c}\\left.\\left({\\frac{\\partial ^2 T}{\\partial r_{+}^2}}\\right)_{J}\\right|_{r=r_c}$ Substitute Eq.", "(REF ) into Eq.", "(REF ), we can derive that the critical behavior of $C_J$ is described by $C_J\\approx \\frac{\\displaystyle A}{\\displaystyle \\epsilon ^{1/2}}$ where $A$ is a function of $a,~b,~l,~J_c,~D$ and very complicated.", "Here we do not give the detailed expression.", "Comparing Eq.", "(REF ) with Eq.", "(REF ), one can find that $\\alpha =1/2$ .", "To calculate $\\beta $ we first derive the $\\Omega $ as function of $r_{+},~J$ .", "$\\Omega (r_{+},J)=\\frac{\\sqrt{a^2 l^2 r_{+}^2-b^2 r_{+}^2-b J l^2}+a l r_{+}}{b l r_{+}}$ For fixed $a,~b,~l$ and the critical $J_c$ , $\\Omega (r_{+},J)=\\Omega (r_{c},J_c)+\\left.\\left({\\frac{\\partial \\Omega }{\\partial r_{+}}}\\right)_{J}\\right|_{r=r_c}(r_{+}-r_{c})+\\text{higher order terms}$ Ignoring the higher order terms, we finally obtain $\\Omega (r_{+},J)-\\Omega (r_{c},J_c)=\\frac{J_c l}{r_{c}^2 \\sqrt{a^2 l^2 r_{c}^2-b^2 r_{c}^2-b J_c l^2}T_c^{1/2}D^{1/2}}|T-T_c|^{1/2}$ Therefore $\\beta =1/2$ .", "Following the previous approach one can express the $\\kappa _T$ as function of $r_{+},~J$ .", "Utilizing Eq.", "(REF ) we can obtain $\\kappa _T\\approx \\frac{\\displaystyle B}{\\displaystyle \\Delta }=\\frac{\\displaystyle BD^{1/2}}{\\displaystyle \\epsilon ^{1/2}}=\\frac{\\displaystyle BD^{1/2T_c^{1/2}}}{\\displaystyle |T-T_c|^{1/2}}$ which means $\\gamma =1/2$ .", "Therefore the critical exponents $\\alpha ,~\\beta ,~\\gamma ,~\\delta $ have the same values as the ones obtained in the Hořava-Lifshitz black hole and the Born-Infeld black hole[41], [42].", "Obviously they obey the scaling symmetry like the ordinary thermodynamic systems $&&\\alpha +2\\beta +\\gamma =2, \\quad \\alpha +\\beta (\\delta +1)=2\\nonumber \\\\&&\\gamma (\\delta +1)=(2-\\alpha )(\\delta -1), \\quad \\gamma =\\beta (\\delta -1)$" ], [ "Discussion and conclusion", "In this paper, we adopted Ehrenfest's classification to study the phase transition of the BTZ black hole with torsion obtained in the MB topological gravitational model.", "Although the gravitational action contains torsion, the metric part of the BTZ black hole with torsion looks like the usual BTZ solution.", "Because of the existence of the Chern-Simons term and the Nieh-Yan term, the conserved charges for the BTZ black hole should be modified.", "Inclusion of these topological terms makes the thermodynamic properties and critical behaviors of BTZ black hole with torsion very different from the ones of the usual BTZ black hole obtained in GR.", "By treating the effective cosmological constant as a thermodynamic pressure, in the extended phase space we completely followed the standard of Ehrenfest to explore the type of the phase transition of the BTZ black hole with torsion.", "It is shown that when $|a|l\\le |b|$ the Gibbs free energy and entropy are continuous functions of temperature, however the heat capacity $C_P$ , the isothermal compressibility $\\kappa $ and the expansion coefficient $\\alpha $ are all divergent at the critical point.", "This means this kind of phase transition for the BTZ black hole with torsion is continuous or second order.", "Nevertheless, the phase transition and critical behavior are different from the ones in the van der Waals liquid/gas system.", "Because $a,~b$ here are related to the parameters $\\alpha _3,~\\alpha _4$ in the action of MB model.", "Thus whether phase transition can happen depends not only upon the black hole solutions, but also upon the gravitational actions.", "Moreover we also considered the non-extended phase space.", "In this case, no direct thermodynamic analogy for the the isothermal compressibility exists.", "Thus we employed another form and named it $\\kappa _T$ .", "It is shown that the $\\kappa _T$ also diverge at the same critical point.", "Therefore in the non-extended phase space, the phase transition is also the second order.", "The critical exponents for the BTZT black hole are also calculated, which are the same as the ones obtained in the Hořava-Lifshitz black hole and the Born-Infeld black hole.", "Is this just a coincidence, or is there some inherent reason, still need consideration further.", "Although we discussed the three-dimensional topological model with torsion, the results have included the torsion-free case which corresponds to the topologically massive gravity (TMG)[43].", "For the TMG, the field equations of which are also solved by the BTZ metric (CS-BTZ solution).", "The conserved charges and the entropy are modified to be [31], [44], [45] $M=M_0-\\frac{\\beta }{L^2}J_0, \\quad J=J_0-\\beta M_0,\\quad S=4\\pi (r_+-\\frac{\\beta }{L}r_{-})$ Here $~\\beta ~$ is the Chern-Simons coupling constant and $L$ is the usual cosmological radius.", "Obviously the phase transition and the critical behaviors for the BTZ black hole in the TMG correspond to the $a=1$ case of the BTZT black hole in the MB model.", "Therefore when $|\\beta |>L$ phase transition also exists in the CS-BTZ black hole.", "Similarly, in this time the mass, angular momentum and the entropy cannot all be positive." ], [ "Acknowledgements", "MSM thanks H. H. Zhao and H. F. Li for useful discussion.", "This work is supported in part by NSFC under Grant Nos.", "(11247261;11175109;11075098;11205097)." ] ]
1403.0449
[ [ "Identifying non-Abelian topological ordered state and transition by\n momentum polarization" ], [ "Abstract Using a method called momentum polarization, we study the quasiparticle topological spin and edge-state chiral central charge of non-Abelian topological ordered states described by Gutzwiller-projected wave functions.", "Our results verify that the fractional Chern insulator state obtained by Gutzwiller projection of two partons in bands of Chern number 2 is described by $SU(2)_2$ Chern-Simons theory coupled to fermions, rather than the pure $SU(2)_2$ Chern-Simons theory.", "In addition, by introducing an adiabatic deformation between one Chern number 2 band and two Chern number 1 bands, we show that the topological order in the Gutzwiller-projected state does not always agree with the expectation of topological field theory.", "Even if the parton mean-field state is adiabatically deformed, the Gutzwiller projection can introduce a topological phase transition between Abelian and non-Abelian topologically ordered states.", "Our approach applies to more general topologically ordered states described by Gutzwiller-projected wave functions." ], [ "Introduction", "Topologically ordered states (TOSs) are unconventional states of matter with ground-state degeneracy, elementary quasiparticle excitations with fractional statistics, and long-range quantum entanglement[1].", "The non-Abelian TOSs are a subcategory of TOSs in which quasiparticles carry nonlocal topological degeneracy and have received much recent attention due to their potential applications in topological quantum computations[2], [3], [4].", "The braiding processes of quasiparticles within a non-Abelian TO induce noncommuting unitary transformations in the ground-state space instead of merely incurring a $U(1)$ phase factor as in the Abelian case.", "Candidates for non-Abelian TOSs include the $\\nu =5/2$ and $\\nu =12/5$ fractional quantum Hall states[5], which are proposed to be the Moore-Read state[6] and the Read-Rezayi states[7].", "Unlike conventional states of matter characterized by the symmetries preserved or those broken spontaneously, TOSs are characterized by topological properties such as ground-state degeneracy and fusion and braiding of topological quasiparticles.", "Except for some exactly solvable models, most candidate systems for TOSs can be studied only by numerical methods such as the density-matrix renormalization group (DMRG)[8] and the variational Monte Carlo method[9].", "To determine the topological order in a numerically studied system, it is essential to develop numerical probes of topological properties.", "The search for more efficient and general numerical methods has attracted much recent attention.", "Various methods have been developed to characterize quasiparticle statistics based on direct calculation of the Berry phase [1], explicit braiding of excitations[10] and modular transformation of ground states with minimum entanglement entropy[11].", "Recently, an additional approach has been proposed for numerically extracting two topological properties of a given TOS, the topological spins of quasiparticles $h_a$ and the edge-state chiral central charge $c$[12].", "Physically, the topological spin determines the phase factor $\\theta _a=e^{i2\\pi h_a}$ obtained by the system when a quasiparticle spins through $2\\pi $ .", "The chiral central charge of the edge state determines the thermal current $I_E=\\frac{c}{6}T^2$ at temperature $T$[13].", "These two quantities are essential in determining the TOS.", "The proposal is based on the concept of momentum polarization defined for cylindrical systems.", "For a cylindrical lattice system with periodic boundary condition along the $y$ direction, one can define a unitary “partial translation operator\" $T_y^L$ which translates the lattice sites along the $y$ direction by one lattice constant for all sites that are in the left half of the system.", "For a topological ground state $\\left|\\Phi _a\\right\\rangle $ with quasiparticle type $a$ in the cylinder, the expectation value of $T_y^L$ is proposed to have the following asymptotic form[12] $\\lambda _a\\equiv \\left\\langle \\Phi _a\\right|T_y^L\\left|\\Phi _a\\right\\rangle \\simeq \\exp \\left[\\frac{2\\pi i}{L_y}p_a-\\alpha L_y\\right]$ where $L_y$ is the number of lattice sites in the $\\hat{y}$ direction, $\\alpha $ is a nonuniversal complex constant for the leading contribution and independent of the specific topological sector $a$ , and remarkably, the fractional part of the momentum polarization $p_a$ has a universal value $p_a=h_a-\\frac{c}{24}$ , which measures the combination of topological spin $h_a$ (modulo 1) and central charge $c$ (modulo 24).", "Since $T_y^L$ only acts only on the left half of the system, the momentum polarization is a quantum entanglement property determined by the reduced density matrix of the left half of the system.", "The average value $\\lambda _a$ has the merit of being relatively simple to evaluate in comparison with the previous methods based on entanglement entropy[11].", "The calculation of the Renyi entanglement entropy involves a swap operator and requires a minimum of two replicas of the system, while for momentum polarization the evaluation of $T_y^L$ does not need a replica so the Hilbert space for Monte Carlo sampling is much smaller for the same system size.", "In Ref.", "mompol, the momentum polarization was studied for two simple TOSs, the Laughlin 1/2 state in fractional Chern insulators (the definition of which will be given in the next paragraph) and the honeycomb lattice Kitaev model[14].", "The former is an Abelian state, while the latter has a special non-Abelian state that can be solved by mapping to free Majorana fermions.", "In this paper, we apply the momentum polarization approach to more generic non-Abelian TOSs.", "More specifically, we study non-Abelian states described by Gutzwiller-projected wave functions[15] of fractional Chern insulators (FCIs).", "An (integer) Chern insulator is a band insulator with nonzero quantized Hall conductance.", "The Hall conductance $\\sigma _H=n\\frac{e^2}{h}$ carried by an occupied band is determined by a topological invariant of the energy band, known as the Chern number $C=n$ .", "FCIs are generalizations of Chern insulators to interacting systems, which have fractional Hall conductance and topological order.", "One way to understand FCIs is through the parton construction, in which the electron is considered as a composite particle of several “partons\" carrying fractional quantum numbers.", "For example, an electron can be split into three fermionic partons, with each parton in an integer Chern insulator with $C=1$ .", "The corresponding electron state has Hall conductance $\\frac{1}{3}\\frac{e^2}{h}$ and is the $\\frac{1}{3}$ Laughlin state.", "Gauge fields are coupled to partons to enforce the constraint that all physical states are electron states and no individual parton will be observed.", "The parton construction can be expressed in ansatz ground state wave functions constructed by the procedure of Gutzwiller projection [15], which is a projection of the parton ground state into the physical electron Hilbert space.", "Gutzwiller-projected wave functions have been constructed for FCI[16].", "When two partons are glued together to form a bosonic “electron\", and each parton is in a state with Chern number $C=1$ , from topological effective field theory (which we will review later in the paper) one expects to find a $1/2$ bosonic Laughlin state.", "In contrast, if each parton is in a state with Chern number $C=2$ , the resulting electron TOS is expected to be non-Abelian, related to $SU(2)$ level-2 Chern-Simons (CS) theory [6].", "The non-Abelian nature of this state has been verified by calculation of the modular $\\mathcal {S}$ matrix for the projected wave functions[17].", "In this paper, we study the momentum polarization of the Gutzwiller-projected wave function for the state of two partons with Chern number $C=2$ .", "In addition to confirming the non-Abelian topological order of this state, our result contains the following two points.", "First, the spin and central charge obtained from momentum polarization clearly distinguish two related but distinct topological states, the $SU(2)_{2}$ CS theory and the $SU(2)_{2}$ CS theory coupled to fermions[18].", "The particle fusion, braiding, and modular $\\mathcal {S}$ matrix of these two theories are identical, but they are distinct TOSs with different edge-state chiral central charge $c=\\frac{3}{2}$ and $c=\\frac{5}{2}$ , respectively.", "The momentum polarization calculation clearly demonstrates that the Gutzwiller-projected parton wave function has the topological order of the latter theory.", "Second, there is an apparent paradox in the statement that Gutzwiller projection of parton $C=2$ states leads to $SU(2)_2$ CS theory coupled to fermions.", "Since Chern number is the only topological invariant of a fermion energy band, a Chern number $C=2$ band can be adiabatically deformed to two decoupled $C=1$ bands, as long as translation symmetry breaking is allowed.", "Since the Gutzwiller projection of two $C=1$ partons is known to give the Laughlin $1/2$ state, it appears that one can adiabatically deform the non-Abelian TOS obtained from partons occupying the $C=2$ band to the Abelian TOS of two decoupled Laughlin $1/2$ states.", "This is clearly in contradiction with the topological stability of TOSs.", "By introducing an explicit adiabatic deformation between a $C=2$ band structure and two decoupled $C=1$ bands, we study the quasiparticle topological spin during the adiabatic interpolation.", "Our result shows that there is a topological phase transition between the Abelian phase of the bilayer Laughlin state and the non-Abelian phase of the $SU(2)_2$ CS coupled to fermions.", "The topological phase transition occurs at a finite coupling between the two $C=1$ bands.", "In other words, the TOS obtained from Gutzwiller projection of $C=2$ parton bands is not completely determined by the Chern number of the parton band structure, but may depend on details of the Chern bands and the projection.", "The argument based on parton “mean-field theory\", i.e., integrating over partons to obtain CS gauge theory, may not predict the correct phase.", "This example further emphasizes the importance of numerical approaches such as momentum polarization in identifying TOSs.", "Based on this numerical observation, we will also discuss theoretically the effective theory interpretation of this topological phase transition.", "The remaining of the paper is organized as follows: In Sec.", ", we present our momentum polarization calculation in the Gutzwiller-projected wave function of non-Abelian FCIs, after reviewing the relevant background knowledge.", "Sec.", "REF presents our projective construction and the $C=2$ Chern insulator model; Sec.", "REF gives a brief field theory discussion of the corresponding TOS; Sec.", "REF shows our numerical results from momentum polarization.", "We obtain the topological spin of the non-Abelian quasiparticle $h_\\sigma =0.321\\pm 0.013$ and the fermion quasiparticle $h_\\psi =0.520\\pm 0.026$ and edge central charge $c=2.870\\pm 0.176$ , in agreement with the $SU(2)$ CS theory coupled to fermions ($h_\\sigma =5/16$ and $c=5/2$ ).", "In Sec.", ", we introduce the adiabatic deformation between two $C=1$ bands and one $C=2$ band, and study the topological phase transition between the two TOSs.", "In Sec.", "REF , we present an adiabatic interpolation of the parton tight-binding Hamiltonian.", "Sec.", "REF presents the results for the quasiparticle topological spin and ground-state degeneracy which indicate the transition between the non-Abelian and Abelian TOS; In Sec.", "REF , we discuss the physical interpretation of this topological transition.", "Finally, Sec is devoted to a conclusion from our main results and discussion of open questions.", "The projective construction is a powerful formalism for ansatz wave functions of many TOS[19].", "For our projective construction, we first introduce several species of partons $\\psi _a$ as free fermions in a Chern insulator, and then constrain the partons to recombine into physical “electrons\" (which may be bosons or fermions).", "In the simple Gutzwiller-projected states we will discuss in this work, the projected wave function is defined in first quantized language by $\\Phi \\left(\\left\\lbrace z_i\\right\\rbrace \\right)=\\underset{a}{\\prod }\\psi _a\\left(\\left\\lbrace z_i\\right\\rbrace \\right)$ .", "Here $\\left\\lbrace z_i\\right\\rbrace $ with $i=1,2,....,N$ are the coordinates of all particles, and $\\psi _a\\left(\\left\\lbrace z_i\\right\\rbrace \\right)$ is the wave function of the $a$ -th parton.", "$N$ is the number of each parton type, which is the same as the total electron number of the system.", "The properties of the resulting states can be numerically computed through variational Monte Carlo calculations.", "Figure: An illustration of the hopping Hamiltonian in Eq..", "The two orbitals on each lattice site are shown asdifferent layers and colored in black and blue, respectively.", "Thehopping is +1+1 (-1-1) along the solid (dashed) lines, and i/2i/\\sqrt{2} (-i/2-i/\\sqrt{2}) along (against) the red arrows.For our focused non-Abelian TOS, we start with the following parton mean-field Hamiltonian on a two-dimensional square lattice $H_{C=2}&=&\\underset{<ij>,I,s}{\\sum }(-1)^{I-1}c_{jIs}^{\\dagger }c_{iIs}+\\underset{<ij>,s}{\\sum }e^{i2\\theta _{ij}}\\left(c_{j2s}^{\\dagger }c_{i1s}+c_{j1s}^{\\dagger }c_{i2s}\\right)\\nonumber \\\\&+&\\frac{1}{\\sqrt{2}}\\underset{<<ik>>,s}{\\sum }e^{i2\\theta _{ik}}\\left(c_{k2s}^{\\dagger }c_{i1s}-c_{k1s}^{\\dagger }c_{i2s}\\right)+\\mbox{H.C.}\\nonumber \\\\&=&\\underset{<ij>_{y},s}{\\sum }\\left[\\left(c_{j1s}^{\\dagger }c_{i1s}-c_{j2s}^{\\dagger }c_{i2s}\\right)-\\left(c_{j2s}^{\\dagger }c_{i1s}+c_{j1s}^{\\dagger }c_{i2s}\\right)\\right]\\nonumber \\\\&+&\\underset{<ij>_{x},s}{\\sum }\\left[\\left(c_{j1s}^{\\dagger }c_{i1s}-c_{j2s}^{\\dagger }c_{i2s}\\right)+\\left(c_{j2s}^{\\dagger }c_{i1s}+c_{j1s}^{\\dagger }c_{i2s}\\right)\\right]\\nonumber \\\\&+&\\frac{1}{\\sqrt{2}}\\underset{<<ik>>,s}{\\sum }e^{i2\\theta _{ik}}\\left(c_{k2s}^{\\dagger }c_{i1s}-c_{k1s}^{\\dagger }c_{i2s}\\right)+\\mbox{H.C.}$ where $I=1,2$ are the two orbitals on each lattice site and $s=\\uparrow , \\downarrow $ labels the two flavors of partons.", "$\\theta _{ij}$ is the azimuthal angle for the vector connecting $i$ and $j$ .", "$<ij>$ and $<<ik>>$ label nearest neighbor and next nearest neighbor links, while $\\left\\langle ij\\right\\rangle _{x}$ and $\\left\\langle ij\\right\\rangle _{y}$ denote nearest neighbors along the $\\hat{x}$ and $\\hat{y}$ directions, respectively, as is illustrated in Fig.", "REF .", "A previous study[17] has shown that at half filling the system is a Chern insulator with $C=2$ .", "The correlation length $\\xi $ is on the order of a lattice constant, and therefore the finite-size effects are generally suppressed for the system sizes we study.", "In real space, the parton wave function $\\psi _a\\left(\\left\\lbrace z_{i}\\right\\rbrace \\right)$ is a Slater determinant for a completely filled valence band, where $z=(i, I)$ labels both the position and orbital indices of a parton.", "Next we apply the Gutzwiller projection imposing the constraint $n_{iI\\uparrow }=n_{iI\\downarrow }$ , with $n_{iIs}=c_{iIs}^\\dagger c_{iIs}$ the number of partons at each site and orbital.", "The states satisfying this constraint have two partons bound at each site and orbital, and are physical electron states with electron number $n_{iI}^e=n_{iI\\uparrow }$ .", "This leads to the following many-body wave function $\\Phi \\left(\\left\\lbrace z_{i}\\right\\rbrace \\right)=\\psi _{\\uparrow }\\left(\\left\\lbrace z_{i}\\right\\rbrace \\right)\\psi _{\\downarrow }\\left(\\left\\lbrace z_{i}\\right\\rbrace \\right)=\\psi ^2_{\\uparrow }\\left(\\left\\lbrace z_{i}\\right\\rbrace \\right)$ This state is the major focus of the paper.", "Previously, the three topological sectors on a torus for this projective construction were obtained by tuning the boundary condition of the parton mean-field Hamiltonian in Eq.", "REF and their connection to the corresponding threaded quasiparticle has been established[17].", "For our momentum polarization calculations, we need to generalize the projective construction to a cylinder.", "To resolve the complication from the gapless chiral edge modes on the open edges, we start from a torus and adiabatically lower all hopping amplitudes across the open boundary until they are much smaller than the edge modes' finite size gap.", "The residue hoppings effectively couple only the zero energy states at $k_y=\\pm \\pi /2$ on the two edges of the cylinder, therefore the original boundary conditions of topological sectors on the torus lead to linear combinations of the zero energy states[12].", "Since such a process involves no level crossing, we can obtain the topological sectors on a cylinder by allowing occupation of different parton zero-energy states on the two edges." ], [ "Topological Field Theory Description", "To understand the TOS described by the above projective construction, we briefly review the topological field theory description of this state.", "The electron operator can be expressed in partons as $f_{iI}=c_{iI\\uparrow }c_{iI\\downarrow }$ .", "This decomposition has an $SU(2)$ gauge symmetry: for any $SU(2)$ matrix with $\\alpha ,\\beta \\in \\mathbb {C}$ and $\\left|\\alpha \\right|^2+\\left|\\beta \\right|^2=1$ $\\left(\\begin{array}{c}c_{iI\\uparrow }\\\\c_{iI\\downarrow }\\end{array}\\right)\\rightarrow \\left(\\begin{array}{cc}\\alpha & \\beta \\\\-\\beta ^{*} & \\alpha ^{*}\\end{array}\\right)\\left(\\begin{array}{c}c_{iI\\uparrow }\\\\c_{iI\\downarrow }\\end{array}\\right)$ this transformation preserves the electron operator $f_{iI}\\rightarrow \\left(\\alpha c_{iI\\uparrow }+\\beta c_{iI\\downarrow }\\right)\\left(-\\beta ^{*}c_{iI\\uparrow }+\\alpha ^{*}c_{iI\\downarrow }\\right)=c_{iI\\uparrow }c_{iI\\downarrow }=f_{iI}$ , and therefore the effective theory of partons should also be gauge invariant.", "The simplest possible effective theory satisfying the gauge invariant condition is obtained by a minimal coupling of the mean-field Hamiltonian (REF ) to an $SU(2)$ gauge field[20].", "A lattice $SU(2)$ gauge field is described by gauge connection $e^{ia_{ij}}\\in SU(2)$ defined along each link $ij$ .", "The Hamiltonian is written as $H_{eff}&=&\\underset{<ij>,I}{\\sum }(-1)^{I-1}e^{ia^{ji}_{sr}}c_{jIs}^{\\dagger }c_{iIr}+\\underset{<ij>}{\\sum }e^{i2\\theta _{ij}}e^{ia^{ji}_{sr}}\\left(c_{j2s}^{\\dagger }c_{i1r}+c_{j1s}^{\\dagger }c_{i2r}\\right)\\nonumber \\\\&+&\\frac{1}{\\sqrt{2}}\\underset{<<ik>>}{\\sum }e^{i2\\theta _{ik}}e^{ia^{ki}_{sr}}\\left(c_{k2s}^{\\dagger }c_{i1r}-c_{k1s}^{\\dagger }c_{i2r}\\right)+\\mbox{H.C.}$ where $s,r=\\uparrow ,\\downarrow $ denote the two parton species, and repeated indices are summed over.", "Since the partons are gapped, it is straightforward to integrate them out.", "Due to the Chern number $C=2$ of each parton band, integrating over the parton results in an $SU(2)_2$ non-Abelian CS theory $\\mathcal {L}= \\frac{2}{4\\pi } \\epsilon _{\\mu \\nu \\rho } \\mbox{tr}\\left[a_\\mu \\partial _\\nu a_\\rho +\\frac{2}{3} a_\\mu a_\\nu a_\\rho \\right]$ However, it is not accurate to say that the topological field theory describing the TOS of this parton construction is $SU(2)_2$ CS gauge theory, because the partons have nontrivial contribution to topological properties such as edge theory.", "The edge theory of $SU(2)_2$ CS theory is a chiral $SU(2)_2$ Weiss-Zumino-Witten (WZW) model[21], [22], while the edge theory of the FCI described above consists of four chiral fermions (two from each flavor of parton) coupled to the $SU(2)_2$ WZW model.", "Technically, the edge state of fermions coupled to the WZW model is described by a quotient of two conformal field theories $\\frac{U(4)_1}{SU(2)_2}$ , in which $U(4)_1$ describes four free chiral fermions and $SU(2)_2$ describes the gauge degrees of freedom which are removed from physical excitations.", "[18] Although they both have three quasiparticles with the same fusion rule and braiding statistics, these two theories are not topologically equivalent.", "In particular, the topological spin differs by a fermionic sign for quasiparticles which correspond to an odd number of holes in the parton Chern insulator state.", "For comparison purpose, we list the theoretical values for the quasiparticle topological spins and edge central charges for the two theories in Table REF .", "Table: Theoretical values of topological properties including theedge central charge cc, the topological spins for quasiparticlesh 1 h_1, h σ h_\\sigma , h ψ h_\\psi and the ground-state degeneracy DD forthe pure SU(2) 2 SU(2)_2 CS theory and the ν=2\\nu =2 fermions coupled to anSU(2) 2 SU(2)_2 gauge field (or equivalently, U(4) 1 SU(2) 2 \\frac{U(4)_1}{SU(2)_2}theory).In summary, we have seen that the effective topological field theory analysis suggests that the topological order in the Gutzwiller-projected state is $\\frac{U(4)_1}{SU(2)_2}$ instead of $SU(2)_2$ .", "However, it is essential to verify that directly for the Gutzwiller-projected wave function, as there is no guarantee that the effect of Gutzwiller projection is completely equivalent to the coupling to a gauge field in the effective field theory.", "This is achieved in the next section by studying the momentum polarization." ], [ "Topological spin and edge central charge from momentum polarization calculations", "Quasiparticle braiding from previous studies has determined that the TOS for $\\Phi \\left(\\left\\lbrace z_i\\right\\rbrace \\right)$ is necessarily non-Abelian.", "However, both theories in Table 1 are consistent with the braiding, and therefore additional information is necessary to make a complete identification.", "We numerically extract the quasiparticle topological spin and edge central charge from momentum polarization calculations for the model in Eq.", "REF defined on a cylinder.", "Care should be taken about the non-Abelian topological sector, which consists of parton states with an overall difference of momentum $\\pi $ on the left edge.", "For the expectation value of the partial translation operator $T^L_y$ that translates the left half of the cylinder by one lattice constant along the $\\hat{y}$ direction, see Fig.", "REF for illustration, this $\\pi $ momentum difference will result in contributions with opposite signs.", "To overcome this difficulty, we generalize $T^L_y$ to twist the left half of the cylinder by $l$ lattice constants, so that the overall phase difference vanishes for a partial translation of $l=2$ lattice constants.", "For this purpose, we take $\\frac{L_y}{l}$ to be integer, consider $l$ sites along the $\\hat{y}$ direction as one unit cell, and replace $L_y$ by $\\frac{L_y}{l}$ in the formula proposed in Ref.", "mompol.", "Consequently, the average value of $T_y^L$ defined by $\\lambda _{a}=\\left\\langle \\Phi _{a}\\right|T_{y}^{L}\\left|\\Phi _{a}\\right\\rangle $ has the following leading contributions $\\lambda _{a}=\\exp \\left[i\\frac{2\\pi l}{L_{y}}p_a-\\alpha \\frac{L_{y}}{l}\\right] $ in which $\\alpha $ is a nonuniversal complex constant independent of the specific topological sector $a$ , while $p_a$ has a universal topological value $p_a=h_{a}-c/24$ determined by the topological spin $h_{a}$ and the edge central charge $c$ .", "The quantity in Eq.", "REF can be efficiently evaluated for the projected wave functions with the variational Monte Carlo method.", "For a cylinder with $L_{x}=8$ , $L_{y}=16$ and $T_{y}^{L}$ translating the left half by $l=2$ lattice constants for the aforementioned reason, numerical calculations yield $\\arg \\left(\\lambda _{1}\\right)=-3.4449\\pm 0.0063$ for the identity sector, $\\arg \\left(\\lambda _{\\sigma }\\right)=-3.1929\\pm 0.0082$ for the sector associated with the non-Abelian quasiparticle, and $\\arg \\left(\\lambda _{\\psi }\\right)=-3.0366 \\pm 0.0257$ for the fermion sector.", "With $h_1=0$ by definition of the identity particle, we obtain $h_{\\sigma }=\\frac{L_{y}}{2\\pi l}\\left[\\arg \\left(\\lambda _{\\sigma }\\right)-\\arg \\left(\\lambda _{1}\\right)\\right]=0.321\\pm 0.013\\\\h_{\\psi }=\\frac{L_{y}}{2\\pi l}\\left[\\arg \\left(\\lambda _{\\psi }\\right)-\\arg \\left(\\lambda _{1}\\right)\\right]=0.520\\pm 0.026$ This is fully consistent with the theoretical value of $h_{\\sigma }^{th}=5/16=0.3125$ for the non-Abelian quasiparticle and $h_{\\psi }^{th}=1/2=0.5$ for the fermion quasiparticle of a theory of $\\nu =2$ fermions couple to an $SU(2)$ gauge field .", "In addition, we calculate $\\lambda _{1}$ for $L_{x}=8$ , $l=1$ , and various values of $L_{y}$ .", "The numerical results are shown in Fig.", "REF .", "To compare with Eq.", "REF , note that $-L_{y}\\arg \\left(\\lambda _{1}\\right)=\\mbox{Im}\\alpha L_{y}^{2}-2\\pi p_1$ , so the intercept of this linear fitting gives the value of $-2\\pi p_1=2\\pi c/24=0.7513\\pm 0.046$ .", "The resulting value of $c=2.870\\pm 0.176$ is also fairly consistent with the prediction of $c^{th}=5/2$ according to the theory of $\\nu =2$ fermions coupled to an $SU(2)$ gauge field.", "Although there is a deviation between the numerical value and the theoretical value $5/2$ which is probably due to the finite-size effect, the accuracy of the result is sufficient to completely distinguish this system from the bare $SU(2)_{2}$ CS theory with $h_\\sigma =3/16$ and $c=3/2$ .", "This result also provides further evidence that the momentum polarization method for computing topological quantities is applicable to non-Abelian TOSs.", "Figure: The value of -L y argλ 1 -L_y \\arg \\left(\\lambda _1\\right) versusL y 2 L^2_y for the identity sector a=1a=1.", "The intercept at L y 2 =0L^2_y=0 ofthe linear fitting gives -2πp 1 =0.7513±0.046-2\\pi p_1=0.7513\\pm 0.046.", "We set L x =8L_x=8and l=1l=1 for all calculations.From the results discussed in the last section, it seems that the TOS of the Gutzwiller-projected wave function agrees well with the expectation from the topological field theory approach.", "However, there is a hidden paradox in this result.", "Since the Chern number is the only topological invariant for a generic energy band in two dimensions, a band with Chern number $C=2$ is topologically equivalent to two $C=1$ bands.", "More explicitly, an exact mapping has been constructed between a $C=2$ band and two decoupled Landau level systems which are related by a lattice translation operation[23], [24], [25].", "Therefore one would naively expect that a state with each parton in a $C=2$ Chern insulator is adiabatically equivalent to one in which each parton occupies two $C=1$ bands.", "However, this statement seems to contradict the fact that the Gutzwiller-projected wave function of the latter state is Abelian.", "It is known that the Gutzwiller-projected wave function of two partons each in a $C=1$ band gives a Laughlin $\\nu =\\frac{1}{2}$ Abelian TOS[26], [27], [28], [16], [11], [12], which is also denoted $SU(2)_1$ Chern-Simons theory.", "Therefore one would expect that when each parton occupies two decoupled $C=1$ bands, which can be viewed as two decoupled layers, the Gutzwiller-projected wave function of the whole system is simply two copies of the Laughlin $\\nu =\\frac{1}{2}$ state, i.e.", "$SU(2)_1\\times SU(2)_1$ , which is an Abelian state clearly distinct from the $\\frac{U(4)_1}{SU(2)_2}$ theory we obtained earlier from both effective theory and numerical results.", "To resolve this apparent paradox, in this section we introduce an explicit interpolation between the $C=2$ model used in last section and a model with two decoupled $C=1$ bands.", "By studying the momentum polarization of the corresponding Gutzwiller-projected wave functions during this interpolation, we find a topological phase transition between the Abelian and non-Abelian phases." ], [ "An adiabatic interpolation of the parent Hamiltonian", "As an explicit example of the interpolation between a $C=2$ band and two $C=1$ bands, we consider the following parton mean-field Hamiltonian on a two-dimensional square lattice[29] $H_\\Theta &=&\\sqrt{2}\\underset{<ij>_{y},s}{\\sum }\\left[\\cos \\Theta \\left(c_{j1s}^{\\dagger }c_{i1s}-c_{j2s}^{\\dagger }c_{i2s}\\right)-\\sin \\Theta \\left(c_{j2s}^{\\dagger }c_{i1s}+c_{j1s}^{\\dagger }c_{i2s}\\right)\\right]\\nonumber \\\\&+&\\sqrt{2}\\underset{<ij>_{x},s}{\\sum }\\left[\\sin \\Theta \\left(c_{j1s}^{\\dagger }c_{i1s}-c_{j2s}^{\\dagger }c_{i2s}\\right)+\\cos \\Theta \\left(c_{j2s}^{\\dagger }c_{i1s}+c_{j1s}^{\\dagger }c_{i2s}\\right)\\right]\\nonumber \\\\&+&\\frac{1}{\\sqrt{2}}\\underset{<<ik>>,s}{\\sum }e^{i2\\theta _{ik}}\\left(c_{k2s}^{\\dagger }c_{i1s}-c_{k1s}^{\\dagger }c_{i2s}\\right)+\\mbox{H.C.}$ where the label definition is the same as in Eq.", "REF , and $\\Theta $ is a continuous parameter.", "For $\\Theta =\\pi /4$ , Eq.", "REF returns to the Hamiltonian in Eq.", "REF with a $C=2$ band.", "For $\\Theta =0$ , the Hamiltonian becomes $H_{\\Theta =0} &=&\\sqrt{2}\\underset{<ij>_{y},s}{\\sum }\\left(c_{j1s}^{\\dagger }c_{i1s}-c_{j2s}^{\\dagger }c_{i2s}\\right)+\\sqrt{2}\\underset{<ij>_{x},s}{\\sum }\\left(c_{j2s}^{\\dagger }c_{i1s}+c_{j1s}^{\\dagger }c_{i2s}\\right)\\nonumber \\\\&+&\\frac{1}{\\sqrt{2}}\\underset{<<ik>>,s}{\\sum }e^{i2\\theta _{ik}}\\left(c_{k2s}^{\\dagger }c_{i1s}-c_{k1s}^{\\dagger }c_{i2s}\\right)+\\mbox{H.C.}$ The hopping matrix elements are drawn in Fig.", "REF .", "Since hoppings exist only between $I=1$ ($I=2$ ) orbitals on the $x_i$ odd sites and $I=2$ ($I=1$ ) orbitals on the $x_i$ even sites, the system can be directly decomposed into two uncoupled subsystems with even and odd values of $x_i+I$ .", "The two subsystems are related by a translation by one lattice constant along the $\\hat{x}$ direction.", "Suppressing the orbital index, each of the two subsystems has the following Hamiltonian, which is a Chern insulator with $C=1$ for each parton flavor $s$ $H_{C=1}=\\underset{\\left\\langle ij\\right\\rangle ,s}{\\sum }t_{i,j}c_{is}^{\\dagger }c_{js}+ \\underset{\\left\\langle \\left\\langle ik\\right\\rangle \\right\\rangle ,s}{\\sum }\\Delta _{i,k}c_{is}^{\\dagger }c_{ks} + \\mbox{H.C.}$ where the nearest neighbor hopping amplitude $t_{i,j}$ is $\\sqrt{2}$ along the $\\hat{x}$ direction and alternates between $\\sqrt{2}$ and $-\\sqrt{2}$ along the $\\hat{y}$ direction, and the next nearest neighbor is $\\Delta _{i,k} = i/\\sqrt{2}$ along the arrow and $\\Delta _{i,k} = -i/\\sqrt{2}$ against the arrow, see Fig.", "REF for an illustration.", "The unit cell contains two lattice sites.", "Therefore, Eq.", "REF defines an interpolation between one Chern insulator with $C=2$ and two decoupled Chern insulators each with $C=1$ .", "It it also verified that the interpolation is adiabatic and the band gap remains finite for all $\\Theta $ .", "Actually, the Hamiltonians with different $\\Theta $ can be related by a global unitary transformation on the orbital space $H_\\Theta &=&U^{-1}H_0U\\nonumber \\\\U&=&\\exp \\underset{s}{\\sum }\\left[\\frac{\\Theta }{2}\\left(c_{i1s}^\\dagger c_{i2s}-c_{i2s}^\\dagger c_{i1s}\\right)\\right]$ The effect of the rotation on annihilation operators is $U^{-1}\\left(\\begin{array}{c}c_{i1s}\\\\c_{i2s}\\end{array}\\right)U&=&\\left(\\begin{array}{cc}\\cos \\frac{\\Theta }{2}&-\\sin \\frac{\\Theta }{2}\\\\\\sin \\frac{\\Theta }{2}&\\cos \\frac{\\Theta }{2}\\end{array}\\right)\\left(\\begin{array}{c}c_{i1s}\\\\c_{i2s}\\end{array}\\right)$ Consequently, the dispersion and band gap are intact with respect to the variation of $\\Theta $ .", "Figure: An illustration of the hopping Hamiltonian in Eq..", "The two orbitals on each lattice site are shown indifferent layers and colored in black and blue, respectively.", "Thehoppings along the solid (dashed) lines are +2+\\sqrt{2} (-2-\\sqrt{2}),and along (against) the red arrows are i/2i/\\sqrt{2} (-i/2-i/\\sqrt{2}).It is straightforward to separate the system into two uncoupledzigzag subsystems with odd and even values of x i +Ix_i+I.Figure: Illustration of a C=1C=1 Chern insulator model on atwo-dimensional square lattice.", "The nearest neighbor hoppingamplitudes are 2\\sqrt{2} along the square edges and -2-\\sqrt{2} alongthe dashed lines.", "The next nearest neighbor hoppings are along thesquare diagonal with amplitude +i/2+i/\\sqrt{2} along (-i/2-i/\\sqrt{2}against) the arrow.", "The two lattice sites in the unit cell aremarked as AA and BB.Table: Theoretical values of topological properties including theedge central charge cc, ground-state degeneracy DD, andquasiparticle topological spins for the SU(2) 1 ×SU(2) 1 SU(2)_1\\times SU(2)_1 CStheory and the ν=2\\nu =2 fermions coupled to an SU(2) 2 SU(2)_2 gaugefield.Now we study the Gutzwiller-projected state corresponding to the parton mean-field Hamiltonian $H_\\Theta $ .", "We have shown that $H_{\\Theta =\\frac{\\pi }{4}}$ leads to the $\\frac{U(4)_1}{SU(2)_2}$ state.", "On the other hand, $H_{\\Theta =0}$ describes two decoupled “layers\", each with two partons in $C=1$ bands.", "The Gutzwiller projection also applies separately to the two layers, so that the resulting state is a decoupled bilayer of the projected $C=1$ states.", "The projected wave functions from a Chern insulator with $C=1$ have been confirmed to be consistent with the $SU(2)_{1}$ CS theory[26], [27], [28], [16], [11], [12].", "Correspondingly, the projected wave function of two uncoupled Chern insulators each with $C=1$ should be describable by an Abelian $SU(2)_{1}\\times SU(2)_{1}$ CS theory, which has four Abelian particles and is clearly distinct from the non-Abelian TOS established for $\\Theta =\\frac{\\pi }{4}$ .", "There are major differences in their topological properties including the torus ground-state degeneracy, edge central charge and quasiparticle topological spins, as listed in Table REF .", "Due to this topological difference between $\\Theta =0$ and $\\Theta =\\frac{\\pi }{4}$ , a topological phase transition must occur for some intermediate $\\Theta $ .", "Since the parton ground states before Gutzwiller projection with different $\\Theta $ are related by a local unitary transformation, one has to conclude that the topological phase transition is introduced by the Gutzwiller projection procedure.", "We study this topological phase transition numerically in the next section." ], [ "The quasiparticle topological spin as a signature for topological phase transition", "First of all, we would like to determine whether there is a first-order phase transition at some $\\Theta $ .", "Even though the interpolation of the parton ground state before projection is clearly adiabatic, the same is not necessarily true for the projected wave function.", "Numerically, for $H_\\Theta $ defined on a system of size $L_x=L_y=12$ with periodic boundary conditions, we study the evolution of the projected wave functions with steps of $\\Theta $ as small as $\\delta \\Theta = \\frac{\\pi }{400}$ .", "Variational Monte Carlo calculations[16] indicate that for all values of $\\Theta \\in [0, \\frac{\\pi }{4}]$ , the overlap between neighboring steps' wave functions $\\left|\\left\\langle \\Phi (\\Theta +\\delta \\Theta )|\\Phi (\\Theta )\\right\\rangle \\right|=1-O\\left(10^{-3}\\right)$ , which clearly suggests that $\\left\\langle \\delta \\Phi (\\Theta )|\\Phi (\\Theta )\\right\\rangle \\rightarrow 0$ for small $\\delta \\Theta \\rightarrow 0$ and excludes the presence of singularities.", "Therefore the quantum phase transition must be continuous..", "Figure: The topological spin hh for the semion (non-Abelianquasiparticle) sector versus various values of Θ∈[0,π/4]\\Theta \\in [0,\\pi /4]for the projected Chern insulator in Eq.", "frommomentum polarization calculations.", "The red dashed line and the bluedotted line are the theoretical values of hh for theSU(2) 1 ×SU(2) 1 SU(2)_{1}\\times SU(2)_{1} CS theory (h s =1/4h_s=1/4) and ν=2\\nu =2fermions coupled to an SU(2)SU(2) gauge field (h σ =5/16h_\\sigma =5/16),respectively.In particular, the open boundary conditions are equivalent for the semion sector in the Abelian TOS and the non-Abelian quasiparticle sector in the non-Abelian TOSs, as well as for the identity sectors in both TOS, making an adiabatic interpolation possible within each sector.", "To determine the topological phase transition point, we compute the momentum polarization with $l=2$ for the identity and semion (non-Abelian quasiparticle) sectors of the projected wave functions on a cylinder of $L_{x}=8$ and $L_{y}=12, 16$ for each interpolation of Eq.", "REF .", "The results of topological spin $h$ for the semion (non-Abelian quasiparticle) sector versus $\\Theta \\in [0,\\pi /4]$ are shown in Fig.", "REF .", "For small value of $\\Theta =0.05\\pi $ , the topological spin starts to deviate from the semionic statistics of $h_s=1/4$ for the Abelian TOS and evolve towards $h_\\sigma =5/16$ for the non-Abelian TOS, see Table REF .", "Still, there is a finite region of $\\Theta $ where the value of $h$ represents an Abelian TOS.", "For further verification, for a smaller value of $\\Theta =0.025\\pi $ , we numerically calculated the overlaps between projected wave functions of various boundary conditions on an $L_x=L_y=12$ torus[16] and find that there are four linearly independent candidate ground-state wave functions by projective construction, consistent with the Abelian $SU(2)_{1}\\times SU(2)_{1}$ CS theory.", "In contrast, for values such as $\\Theta = \\pi /4$ and $\\Theta = 3\\pi /8$ fully in the parameter region of the non-Abelian topological order, such linear independence is only three fold.", "Our numerical results show that a topological phase transition occurs at finite $\\Theta $ , which is consistent with the fact that the $\\Theta =0$ Abelian state is topologically stable and should persist for a finite region of $\\Theta $ : the fractional Chern insulator is an intrinsic topological ordered state protected by an excitation gap that is stable against small local perturbations of arbitrary form such as weak couplings between the subsystems.", "Since the two subsystems are coupled for all nonzero $\\Theta $ , the mean-field Hamiltonian at nonzero $\\Theta $ can be viewed only as a Chern insulator with a $C=2$ band.", "Therefore the topological field theory approach will predict that the TOS of the system is described by $SU(2)_2$ Chern-Simons theory coupled to $C=2$ partons, as we discussed in Sec.", ".", "In contrast, our numerical result for small $\\Theta $ finds an Abelian TOS, which provides a concrete example of a case when the TOS of the Gutzwiller-projected wave function is different from the prediction of topological field theory." ], [ "Theoretical interpretation of the topological phase transition", "To understand physically the topological phase transition, we first ask why the derivation of the effective field theory in Sec REF does not apply to $\\Theta =0$ .", "For general $\\Theta $ , the constraints on the partons induces an $SU(2)$ gauge field along all lattice edges in Fig.", "REF that dominates the low-energy theory after the partons are integrated out.", "In the $\\Theta =0$ limit, however, the Hamiltonian becomes Eq.", "REF , and all hoppings between the two subsystems vanish.", "Therefore there are two well-defined $SU(2)$ gauge fields in the long wavelength limit, one for each subsystem.", "As is clear in Fig.", "REF , these two $SU(2)$ gauge fields exist on independent pieces and remain independent after the partons are integrated out.", "Integrating out the $C=1$ band of the parton gives the $SU(2)$ level 1 Chern-Simons theory, so that the topological field theory of the $\\Theta =0$ system consists of fermions coupling to $SU(2)_1\\times SU(2)_1$ .", "At finite $\\Theta $ , coupling is turned on between the two effective “layers\" and breaks the separate $SU(2)\\times SU(2)$ gauge symmetry into one single $SU(2)$ .", "As an alternative view of the symmetry breaking, one can carry out the unitary rotation in Eq.", "REF in reverse to transform the Hamiltonian $H_\\Theta $ back to $H_0$ .", "In the new basis, the partons occupy the two decoupled $C=1$ bands before projection.", "The only way the two independent layers are coupled is through the constraint.", "In the original basis the constraint is written as $n_{iI\\uparrow }=n_{iI\\downarrow }$ ($c_{iI\\uparrow }^{\\dagger }c_{iI\\uparrow }=c_{iI\\downarrow }^{\\dagger }c_{iI\\downarrow }$ ) in real space.", "After the inverse unitary transformation for a finite $\\Theta $ , the resulting constraints are $c_{i1\\uparrow }^{\\dagger }c_{i1\\uparrow }+c_{i1\\uparrow }^{\\dagger }c_{i1\\uparrow }=c_{i1\\downarrow }^{\\dagger }c_{i1\\downarrow }+c_{i1\\downarrow }^{\\dagger }c_{i1\\downarrow }$ and $\\cos \\Theta \\left( c_{i1\\uparrow }^{\\dagger }c_{i1\\uparrow }-c_{i1\\uparrow }^{\\dagger }c_{i1\\uparrow }\\right)+\\sin \\Theta \\left(c_{i2\\uparrow }^{\\dagger }c_{i1\\uparrow }+c_{i1\\uparrow }^{\\dagger }c_{i2\\uparrow }\\right)=\\cos \\Theta \\left(c_{i1\\downarrow }^{\\dagger }c_{i1\\downarrow }-c_{i1\\downarrow }^{\\dagger }c_{ i1\\downarrow }\\right)+\\sin \\Theta \\left(c_{i2\\downarrow }^{\\dagger }c_{i1\\downarrow }+c_{i1\\downarrow }^{\\dagger }c_{i2\\downarrow }\\right)$ .", "The latter explicitly breaks the intra-layer charge conservation symmetry of the parent Hamiltonian in Eq.REF , defined by $c_{iIs}^\\dagger \\rightarrow e^{-i\\phi }c_{iIs}^\\dagger ,c_{iIs} \\rightarrow e^{i\\phi }c_{iIs}$ , $x_i+I\\in \\mbox{odd}$ .", "As a consequence of this inter-layer coupling, the two $SU(2)$ gauge fields in the effective theory are coupled and only a diagonal $SU(2)$ gauge symmetry is preserved.", "Physically, the holes in the two $C=1$ bands are no longer distinguishable so that the two semionic quasiparticles originating from the holes in the two bands now merge to one particle.", "Consequently, the ground-state degeneracy on a torus, effectively labeled by the quasiparticle content, also decreases from four fold to three fold.", "Figure: A histogram of the number of sampled configurations versusthe parton number around its average N 1 -N ¯ 1 N_1- \\bar{N}_1 in one of thedecomposed C=1C=1 bands.", "While the N 1 =N ¯ 1 N_1=\\bar{N}_1 central peakcontains more than 98%98\\% of the configurations for Θ=0.05π\\Theta =0.05\\pi (red), the spread for Θ=0.125π\\Theta =0.125\\pi (black) is much wider andthe percentage of the N 1 =N ¯ 1 N_1=\\bar{N}_1 configurations is only 15%15\\%suggesting that N 1 N_1 is no longer a good quantum number.", "Theresults are obtained on system size L x =L y =28L_x=L_y=28 with periodicboundary conditions.The discussion above suggests that the Abelian and non-Abelian phases are distinguished by whether the two layers (in the rotated parton basis) have separately conserved particle numbers.", "In the Abelian (non-Abelian) phase, the separate particle number conservation of the two layers is effective preserved (broken).", "To verify this scenario, we numerically calculate the fluctuations of parton number in one of the $C=1$ layers (in the rotated parton basis): $N_1=\\underset{I+x_i \\in \\mbox{odd}}{\\sum } c^\\dagger _{iI\\uparrow } c_{iI\\uparrow }$ .", "In the $\\Theta =0$ limit, the two bands are independent, therefore $N_1=\\bar{N}_1$ and the fluctuation is exactly zero.", "As $\\Theta $ increases, the intra-band charge conservation is broken, and therefore one may expect an increase in the $N_1$ fluctuation.", "Fig.", "REF is a histogram of the number of sampled configurations in the projected wave function versus the parton number $N_1$ fluctuation around its average value $\\bar{N}_1$ in one of the $C=1$ bands at $\\Theta =\\pi /20$ (red) and $\\Theta =\\pi /8$ (black).", "While such fluctuation is still largely suppressed and the $N_1$ conservation approximately holds at $\\Theta =\\pi /20$ on the Abelian TOS side of the transition, it proliferates at $\\Theta =\\pi /8$ and the intralayer charge conservation no longer exists for a non-Abelian TOS.", "To see further the connection between the parton number fluctuation and the non-Abelian TOS, we show in Fig.", "REF the mean squared deviation $\\sqrt{\\left\\langle \\left(N_1-\\bar{N}_1\\right)^2\\right\\rangle /\\bar{N}_1}$ versus $\\Theta $ for various system sizes.", "Figure: The mean squared deviation N 1 -N ¯ 1 2 /N ¯ 1 \\sqrt{\\left\\langle \\left(N_1-\\bar{N}_1\\right)^2\\right\\rangle /\\bar{N}_1} versus Θ\\Theta for system sizes L x =L y =8,12,16,20,24,28L_x=L_y=8,12,16,20,24,28.In reality, for a multiband TOS such as the topological nematic states[24], band mixing, be it hopping or interaction, is hard to eliminate.", "The existence of a finite $\\Theta _c$ suggests that the Abelian TOS is stable against weak band-mixing perturbations.", "Intuitively, this is because the TOS are protected by excitation gaps.", "For small band-mixing perturbations, the charge conservation within the bands can appear as an emergent symmetry.", "Nevertheless, in comparison with integer Chern insulators protected by the band gap, the TOS are relatively vulnerable.", "A topological phase transition can occur even if the band structure remains adiabatically equivalent." ], [ "Conclusions", "In conclusion, we study topological properties of non-Abelian TOS using Gutzwiller-projected wave functions and the momentum polarization approach.", "Our numerical results on the topological spin and edge central charge confirm that projected wave functions of two partons in Chern bands with Chern number $C=2$ are described by the field theory of $\\nu =2$ fermions coupled to an $SU(2)$ gauge field, and clearly distinguish it from the pure $SU(2)_2$ CS theory.", "In addition, we adiabatically interpolate the parent Chern insulator with $C=2$ with two Chern insulators each with $C=1$ , and track the variation of topological quantities such as the topological spin and ground-state degeneracy for their corresponding TOS projected wave functions.", "We show that the topological phase transition between the non-Abelian and Abelian TOS is marked by the breaking down of charge conservation within each of the $C=1$ Chern bands.", "The transition point is close to but apart from the completely decoupled limit, in consistency with the intuition that the corresponding Abelian TOS is protected by a gap and stable against small band-mixing perturbations.", "Our result demonstrates explicitly that the topological order in a Gutzwiller-projected state does not always agree with the prediction of topological field theory, and generically has to be determined by numerical calculations of topological properties.", "Our numerical methods based on momentum polarization and the variational Monte Carlo method are generalizable to more complicated non-Abelian TOSs described by Gutzwiller-projected wave functions.", "Compared to previous approaches, momentum polarization provides an efficient way to extract characteristic quantities given the many-body wave functions of a chiral topological ordered state.", "One open question left for future work is whether the critical behavior of momentum polarization across a topological phase transition can be studied numerically and compared with any field theory description.", "Another open question is whether there is a more generic proof of the momentum polarization formula in Eq.", "REF , which has been verified numerically in several TOS, but has not been proved analytically except for arguments based on edge-state conformal field theory[12].", "We would like to thank Maissam Barkeshli, Chao-Ming Jian, Ching Hua Lee and Peng Ye for insightful discussions.", "This work is supported by the Stanford Institute for Theoretical Physics (YZ) and the National Science Foundation through the grant No.", "DMR-1151786 (XLQ)." ] ]
1403.0164
[ [ "Noise prevents infinite stretching of the passive field in a stochastic\n vector advection equation" ], [ "Abstract A linear stochastic vector advection equation is considered; the equation may model a passive magnetic field in a random fluid.", "When the driving velocity field is rough but deterministic, in particular just H\\\"{o}lder continuous and bounded, one can construct examples of infinite stretching of the passive field, arising from smooth initial conditions.", "The purpose of the paper is to prove that infinite stretching is prevented if the driving velocity field contains in addition a white noise component." ], [ "Introduction", "Consider the linear stochastic vector advection equation in $\\mathbb {R}^{3}$ : $d\\mathbf {B}+\\mathrm {curl}({\\mathbf {v}}\\times \\mathbf {B})dt+\\sigma \\sum _{k=1}^{3}\\mathrm {curl}(\\mathbf {e}_{k}\\times \\mathbf {B})\\circ dW^{k}=0,$ where $\\mathbf {v}$$:\\left[ 0,T\\right] \\times \\mathbb {R}^{3}\\rightarrow \\mathbb {R}^{3}$ is a given divergence-free vector field, the solution $\\mathbf {B}$ is a divergence-free vector field, $\\mathbf {e}_{1},\\mathbf {e}_{2},\\mathbf {e}_{3}$ is the canonical basis of $\\mathbb {R}^{3}$ , $\\mathbf {W}=\\left( W^{1},W^{2},W^{3}\\right) $ is a Brownian motion in $\\mathbb {R}^{3}$ , $\\sigma $ is a real number.", "The initial condition, at time $t=0$ , will be denoted by $\\mathbf {B}_{0}$ .", "The driving vector field (the velocity field of the fluid, in the usual interpretation) is modeled by the Gaussian field ${\\mathbf {v}}+\\sigma \\sum _{k=1}^{3}\\mathbf {e}_{k}\\frac{dW^{k}}{dt}={\\mathbf {v}}+\\sigma \\frac{d\\mathbf {W}}{dt}$ where $\\mathbf {v}$ is deterministic, a sort of average or slow-varying component, and $\\sigma d\\mathbf {W}$ is the fast-varying random component, white noise in time.", "This equation may model a passive vector field $\\mathbf {B}$ , like a magnetic field, in a turbulent fluid with a non-trivial average component $\\mathbf {v}$ .", "The intensity $\\sigma $ of the noise can be arbitrarily small, in the sequel, to model real situations when the noise (which always exists) is usually neglected in first approximation.", "However, the trajectories of $\\mathbf {W}$ are only Hölder continuous with exponent smaller than $\\frac{1}{2}$ and not differentiable at any point, so that the impulses given by the term $\\sigma \\frac{d\\mathbf {W}}{dt}$ are small when cumulated in time ($\\sigma \\mathbf {W}$ ) but istantaneously very strong.", "We aim at studying existence, uniqueness, representation formula and regularity under low regularity assumption on $\\mathbf {v}$ .", "The key point of this work is the fact that the noise prevents blow-up, under assumptions on $\\mathbf {v}$ such that blow-up may occur in the deterministic case.", "When $\\sigma =0$ , we give an example of Hölder continuous vector field $\\mathbf {v}$ such that infinite values of $\\mathbf {B}$ arise in finite time from a bounded continuous initial field $\\mathbf {B}_{0}$ ; then we prove that Hölder continuity and boundedness of $\\mathbf {v}$ is sufficient, in the stochastic case ($\\sigma \\ne 0$ ), to prove that continuous initial field $\\mathbf {B}_{0}$ produces continuous fields $\\mathbf {B}_{t}$ for all $t\\ge 0$ .", "The singularity in the deterministic case is associated to infinite stretching of $\\mathbf {B}$ ; randomness prevents stretching to blow-up to infinity.", "Precisely, we prove (see the notations below): Theorem 1 i) For $\\sigma =0$ , there exists $\\mathbf {v}$$\\in C_{b}^{\\alpha }(\\mathbb {R}^{3};\\mathbb {R}^{3})$ and $\\mathbf {B}_{0}\\in C^{\\infty }(\\mathbb {R}^{3};\\mathbb {R}^{3})$ such that $\\sup _{\\left|x\\right|\\le 1}\\left|\\mathbf {B}\\left( t,x\\right) \\right|=+\\infty $ for all $t>0$ .", "ii) For $\\sigma \\ne 0$ , for all $\\mathbf {v}$$\\in C([0,T];C_{b}^{\\alpha }(\\mathbb {R}^{3};\\mathbb {R}^{3}))$ and $\\mathbf {B}_{0}\\in C(\\mathbb {R}^{3};\\mathbb {R}^{3})$ one has $\\mathbf {B}\\in C([0,T]\\times \\mathbb {R}^{3};\\mathbb {R}^{3})$ , with probability one.", "Clearly, linear vector advection equation is a very idealized model of fluid dynamics but this result opens the question whether noise may prevent blow-up of the vorticity field of 3D Euler equations.", "The emergence of singularity seems to require a certain degree of organization of the fluid structures and perhaps this organization is lost, broken, under the influence of randomness.", "With further degree of speculation, one could even think that a turbulent regime may contain the necessary degree of randomness to prevent blow-up; if so, singularities of the vorticity could more likely be associated to strong transient phases, instead of established turbulent ones.", "From the mathematical side, this is not the first result of this nature, see [13], [3], [18], [9], and also [17], [24], [2], , [20], [26] for uniqueness of weak solutions due to noise (the other face of the celebrated open problem presented by [14]).", "However, these papers deal with scalar problems, like linear transport equations, linear continuity equations, vorticity in 2D Euler equations, 1D Vlasov-Poisson equations.", "The result of the present paper is the first one dealing with vector valued PDEs like 3D Euler equations; the kind of singularity in the vectorial case is different, related to rotations and stretching instead of shocks or mass concentration.", "Several new technical difficulties arise due to the vectorial nature of the equation (for instance, the proof of uniqueness of non-regular solutions, Lemma REF , usually involving commutator estimates, here is more difficult and is obtained by special cancellations, also inspired to [26]).", "Let us mention also the improvement of well-posedness due to noise proved for dispersive equations, [10], [5].", "In all the works mentioned so far the noise is multiplicative, and often of transport type like in the present paper.", "The role of additive noise in preventing singularities is more obscure.", "For uniqueness under poor drift, additive noise is very powerful see [28], [29], [21], [7], [8]; however, its relevance in fluid dynamics is still under investigation.", "See [6], [19], [15], [27] for partial results.", "For additional details on vector advection equations see for instance [4] and references therein.", "For advanced results on the differentiability of stochastic flow generated by rough drift (key ingredient of the representation formula (REF )), see [1], [3], [13], [25].", "For a general reference on passive advection driven by random velocity fields see [12], where also the case of a passive magnetic field is discussed; the structure of the noise term in the present work is very simplified with respect to [12] but the point here is to prove that noise has a depleting effect on $\\mathbf {B}$ and this fact is true also under this simple noise; generalization to space-homogeneous noise with more complex space structure is possible, if $Q\\left( 0\\right) $ , the covariance matrix at $x=0$ , is non-degenerate.", "The model described here is clearly too idealized for a direct interest in fluid dynamics but once the phenomenon of depletion of stretching is rigorously proved in this particular framework, there is more motivation to investigate generalizations which could become closer to reality.", "One of them would be the case when $\\mathbf {v}$ contains (also just small) high frequency fluctuations, although not being white noise.", "This extension looks very difficult but potentially not impossible." ], [ "Notations", "We denote by $C(\\mathbb {R}^{3};\\mathbb {R}^{3})$ (resp.", "$C^{\\infty }(\\mathbb {R}^{3};\\mathbb {R}^{3})$ ) the space of all continuous (resp.", "infinitely differentiable) vector fields $\\mathbf {v}:\\mathbb {R}^{3}\\rightarrow \\mathbb {R}^{3}$ .", "We denote by $C_{b}(\\mathbb {R}^{3};\\mathbb {R}^{3})$ the space of all $\\mathbf {v}\\in C(\\mathbb {R}^{3};\\mathbb {R}^{3})$ such that $\\left\\Vert \\mathbf {v}\\right\\Vert _{0}:=\\sup _{x\\in \\mathbb {R}^{3}}\\left|\\mathbf {v}\\left( x\\right) \\right|<\\infty $ .", "For any $\\alpha \\in \\left( 0,1\\right)$ we denote by $C_{b}^{\\alpha }(\\mathbb {R}^{3};\\mathbb {R}^{3})$ the space of all $\\mathbf {v}\\in C_{b}(\\mathbb {R}^{3};\\mathbb {R}^{3})$ such that $\\left[\\mathbf {v}\\right] _{\\alpha }:=\\sup _{x,y\\in \\mathbb {R}^{3},x\\ne y}\\frac{\\left|\\mathbf {v}\\left( x\\right) -\\mathbf {v}\\left( y\\right) \\right|}{\\left|x-y\\right|^{\\alpha }}<\\infty $ ; the space $C_{b}^{\\alpha }(\\mathbb {R}^{3};\\mathbb {R}^{3})$ is endowed with the norm $\\left\\Vert \\mathbf {v}\\right\\Vert _{\\alpha }=\\left\\Vert \\mathbf {v}\\right\\Vert _{0}+\\left[\\mathbf {v}\\right] _{\\alpha }$ .", "We denote by $C_{c}^{\\infty }(\\mathbb {R}^{3};\\mathbb {R}^{3})$ the space of all $\\mathbf {v}\\in C^{\\infty }(\\mathbb {R}^{3};\\mathbb {R}^{3})$ which have compact support.", "For $p\\ge 1$ , we denote by $L_{loc}^{p}(\\mathbb {R}^{3};\\mathbb {R}^{3})$ the space of measurable vector fields $\\mathbf {v}:\\mathbb {R}^{3}\\rightarrow \\mathbb {R}^{3}$ such that $\\int _{\\left|x\\right|\\le R}\\left|\\mathbf {v}\\left( x\\right) \\right|^{p}dx<\\infty $ for all $R>0$ ; we write $\\mathbf {v}\\in L^{p}(\\mathbb {R}^{3};\\mathbb {R}^{3})$ when $\\int _{\\mathbb {R}^{3}}\\left|\\mathbf {v}\\left( x\\right) \\right|^{p}dx<\\infty $ .", "The notation $\\left\\langle {\\mathbf {v}},{\\mathbf {w}}\\right\\rangle $ stands for $\\int _{\\mathbb {R}^{3}}{\\mathbf {v}}\\left( x\\right)\\cdot {\\mathbf {w}}\\left( x\\right) dx$ , when ${\\mathbf {v}},{\\mathbf {w\\in }}L^{2}(\\mathbb {R}^{3};\\mathbb {R}^{3})$ .", "If ${\\mathbf {v}}:\\left[ 0,T\\right] \\times \\mathbb {R}^{3}\\rightarrow \\mathbb {R}^{3}$ , we usually write ${\\mathbf {v}}\\left( t,x\\right) $ , but also $\\mathbf {v}_{t}$ to denote the function $x\\mapsto {\\mathbf {v}}\\left( t,x\\right) $ at given $t\\in \\left[ 0,T\\right] $ .", "If ${\\mathbf {v}}\\in \\mathbb {R}^{3}$ we write ${\\mathbf {v}}\\cdot \\nabla $ for the differential operator $\\sum _{i=1}^{3}v^{i}\\partial _{x_{i}}$ .", "If ${\\mathbf {v}},\\mathbf {B}:\\mathbb {R}^{3}\\rightarrow \\mathbb {R}^{3}$ the notation $\\left( {\\mathbf {v}}\\cdot \\nabla \\right) \\mathbf {B}$ stands for the vector field with components $\\left( {\\mathbf {v}}\\cdot \\nabla \\right) B^{i}$ .", "Similarly, we interpret componentwise operations like $\\partial _{k}\\mathbf {B}$ , $\\Delta \\mathbf {B}$ ." ], [ "Example of blow-up in the deterministic case", "In this section we consider equation (REF ) in the deterministic case $\\sigma =0$ .", "We give an example of Hölder continuous bounded vector field $\\mathbf {v}$ such that $\\sup _{\\left|x\\right|\\le 1}\\left|\\mathbf {B}\\left( t,x\\right) \\right|=+\\infty $ for all $t>0$ , although $\\sup _{x\\in \\mathbb {R}^{3}}\\left|\\mathbf {B}_{0}\\left(x\\right) \\right|<\\infty $ and $\\mathbf {B}_{0}$ is smooth.", "Let us also remark that, on the contrary, when $\\mathbf {v}$ is of class $C\\left( \\left[ 0,T\\right] ;C_{b}^{1}(\\mathbb {R}^{3};\\mathbb {R}^{3})\\right) $ , for every $\\mathbf {B}_{0}\\in C(\\mathbb {R}^{3};\\mathbb {R}^{3})$ there exists a unique continuous weak solution $\\mathbf {B}$ (the definition is analogous to Definition REF below and the proof is similar to the one of Theorem REF ); it satisfies identity (REF ) below where $\\Phi _{t}(x)$ is the deterministic flow given by the equation of characteristics $\\frac{d}{dt}\\Phi _{t}(x)={\\mathbf {v}}(t,\\Phi _{t}(x)),\\qquad \\Phi _{0}(x)=x.$ When $\\mathbf {v}$ is of class $C\\left( \\left[ 0,T\\right] ;C_{b}^{2}(\\mathbb {R}^{3};\\mathbb {R}^{3})\\right) $ and $\\mathbf {B}_{0}\\in C^{1}(\\mathbb {R}^{3};\\mathbb {R}^{3})$ the solution $\\mathbf {B}$ is of class $C\\left( \\left[0,T\\right] ;C^{1}(\\mathbb {R}^{3};\\mathbb {R}^{3})\\right) $ , and so on, from identity (REF ).", "The idea of the example of blow-up comes from identity (REF ): one has to construct a flow $\\Phi _{t}(x)$ , corresponding to a vector field $\\mathbf {v}$ less regular than $C\\left( \\left[ 0,T\\right] ;C_{b}^{1}(\\mathbb {R}^{3};\\mathbb {R}^{3})\\right) $ , such that $D\\Phi _{t}(x)$ blows-up at some point." ], [ "Preliminaries on cylindrical coordinates", "Limited to this and next subsection, we denote points of $\\mathbb {R}^{3}$ by $\\left( x,y,z\\right) $ instead of $x$ (and analogous notations for Euclidea coordinates).", "Let us recall that the material derivative, in cylindrical coordinates, for vectors $\\mathbf {A}=\\mathbf {A}(r,\\theta ,z)$ , $\\mathbf {B}=\\mathbf {B}(r,\\theta ,z)$ , $\\mathbf {A}=A_{r}e_{r}+A_{\\theta }e_{\\theta }+A_{z}e_{z}$ , $\\mathbf {B}=B_{r}e_{r}+B_{\\theta }e_{\\theta }+B_{z}e_{z}$ (where $e_{r}=\\frac{x}{r}e_{x}+\\frac{y}{r}e_{y}$ , $e_{\\theta }=-\\frac{y}{r}e_{x}+\\frac{x}{r}e_{y}$ ) are given by the formula $\\left( {\\mathbf {A}}\\cdot \\nabla \\right) \\mathbf {B}& =(A_{r}\\frac{\\partial B_{r}}{\\partial r}+\\frac{A_{\\theta }}{r}\\frac{\\partial B_{r}}{\\partial \\theta }+A_{z}\\frac{\\partial B_{r}}{\\partial z}-\\frac{A_{\\theta }B_{\\theta }}{r})e_{r} \\\\& +(A_{r}\\frac{\\partial B_{\\theta }}{\\partial r}+\\frac{A_{\\theta }}{r}\\frac{\\partial B_{\\theta }}{\\partial \\theta }+A_{z}\\frac{\\partial B_{\\theta }}{\\partial z}+\\frac{A_{\\theta }B_{r}}{r})e_{\\theta } \\\\& +(A_{r}\\frac{\\partial B_{z}}{\\partial r}+\\frac{A_{\\theta }}{r}\\frac{\\partial B_{z}}{\\partial \\theta }+A_{z}\\frac{\\partial B_{z}}{\\partial z})e_{z} $ Consequently, $\\left( {\\mathbf {A}}\\cdot \\nabla \\right) \\mathbf {B}& -\\left( {\\mathbf {B}}\\cdot \\nabla \\right) \\mathbf {A}=(A_{r}\\frac{\\partial B_{r}}{\\partial r}-B_{r}\\frac{\\partial A_{r}}{\\partial r}+\\frac{A_{\\theta }}{r}\\frac{\\partial B_{r}}{\\partial \\theta }-\\frac{B_{\\theta }}{r}\\frac{\\partial A_{r}}{\\partial \\theta }+A_{z}\\frac{\\partial B_{r}}{\\partial z}-B_{z}\\frac{\\partial A_{r}}{\\partial z})e_{r} \\\\& +(A_{r}\\frac{\\partial B_{\\theta }}{\\partial r}-B_{r}\\frac{\\partial A_{\\theta }}{\\partial r}+\\frac{A_{\\theta }}{r}\\frac{\\partial B_{\\theta }}{\\partial \\theta }-\\frac{B_{\\theta }}{r}\\frac{\\partial A_{\\theta }}{\\partial \\theta }+A_{z}\\frac{\\partial B_{\\theta }}{\\partial z}-B_{z}\\frac{\\partial A_{\\theta }}{\\partial z}+\\frac{A_{\\theta }B_{r}-B_{\\theta }A_{r}}{r})e_{\\theta } \\\\& +(A_{r}\\frac{\\partial B_{z}}{\\partial r}-B_{r}\\frac{\\partial A_{z}}{\\partial r}+\\frac{A_{\\theta }}{r}\\frac{\\partial B_{z}}{\\partial \\theta }-\\frac{B_{\\theta }}{r}\\frac{\\partial A_{z}}{\\partial \\theta }+A_{z}\\frac{\\partial B_{z}}{\\partial z}-B_{z}\\frac{\\partial A_{z}}{\\partial z})e_{z}$ With these preliminaries, let us consider a vector field $\\mathbf {v}$ of the form $\\mathbf {v}=v_{\\theta }e_{\\theta },\\quad v_{\\theta }=v_{\\theta }(r)$ and assume that $\\mathbf {B}(t)=B_{r}(t)e_{r}+B_{\\theta }(t)e_{\\theta }+B_{z}(t)e_{z},t\\ge 0$ is a vector field of class $C^{1}$ on $\\mathbb {R}^{3}\\backslash \\left\\lbrace 0\\right\\rbrace $ which satisfies (on $\\mathbb {R}^{3}\\backslash \\left\\lbrace 0\\right\\rbrace $ ) the equation $\\frac{\\partial \\mathbf {B}}{\\partial t}+{curl}(\\mathbf {v}\\times \\mathbf {B})=0$ with divergence-free initial condition $\\mathbf {B}_{0}$ .", "Notice that ${div}$$\\mathbf {v}$$={div}$$\\mathbf {B}$$=0$ .", "Indeed, $\\mathbf {v}$ is divergence free vector field by definition and $\\frac{\\partial {div}{\\mathbf {B}}}{\\partial t}=-{div}{curl}(\\mathbf {v}\\times \\mathbf {B})=0$ .", "Hence we can rewrite equation for $\\mathbf {B}$ as follows $\\frac{\\partial \\mathbf {B}}{\\partial t}+\\left( {\\mathbf {v}}\\cdot \\nabla \\right) \\mathbf {B}-\\left( \\mathbf {B}\\cdot \\nabla \\right) \\mathbf {v}=0.$ Consequently, in cylindrical coordinates we have $\\frac{\\partial B_{r}}{\\partial t}+\\frac{v_{\\theta }}{r}\\frac{\\partial B_{r}}{\\partial \\theta }=0,$ $\\frac{\\partial B_{\\theta }}{\\partial t}=B_{r}\\frac{\\partial v_{\\theta }}{\\partial r}-\\frac{v_{\\theta }}{r}\\frac{\\partial B_{\\theta }}{\\partial \\theta }-\\frac{v_{\\theta }}{r}B_{r}=-\\frac{v_{\\theta }}{r}\\frac{\\partial B_{\\theta }}{\\partial \\theta }+B_{r}(\\frac{\\partial v_{\\theta }}{\\partial r}-\\frac{v_{\\theta }}{r}),$ $\\frac{\\partial B_{z}}{\\partial t}+\\frac{v_{\\theta }}{r}\\frac{\\partial B_{z}}{\\partial \\theta }=0.$" ], [ "The example", "Choose, for some $\\alpha \\in (0,1)$ , $v_{\\theta }(r)=r^{\\alpha },\\qquad \\text{\\ for }r\\in \\left[ 0,1\\right]$ and define $v_{\\theta }$ for $r>1$ in a such way that $v_{\\theta }\\in C^{\\infty }$ , $v_{\\theta }>0$ and $v_{\\theta }(r)\\le e^{-\\gamma r}$ , $\\gamma >0$ , $r\\ge A>1$ for some $\\gamma ,A$ .", "Then we have, for $r\\in (0,1)$ , $\\frac{\\partial B_{r}}{\\partial t}+r^{\\alpha -1}\\frac{\\partial B_{r}}{\\partial \\theta }& =0, \\\\\\frac{\\partial B_{\\theta }}{\\partial t}+r^{\\alpha -1}\\frac{\\partial B_{\\theta }}{\\partial \\theta }+(1-\\alpha )B_{r}r^{\\alpha -1}& =0 \\\\\\frac{\\partial B_{z}}{\\partial t}+r^{\\alpha -1}\\frac{\\partial B_{z}}{\\partial \\theta }& =0.", "$ Hence we can deduce that (we write $B_{r}^{0},B_{z}^{0},B_{\\theta }^{0}$ for the coordinates of $\\mathbf {B}_{0}$ ), for $r\\in (0,1)$ , $B_{r}(t,r,\\theta ,z)=B_{r}^{0}(r,\\theta -r^{\\alpha -1}t,z)$ $B_{z}(t,r,\\theta ,z)=B_{z}^{0}(r,\\theta -r^{\\alpha -1}t,z)$ $B_{\\theta }(t,r,\\theta ,z)=B_{\\theta }^{0}(r,\\theta -r^{\\alpha -1}t,z)-(1-\\alpha )\\fbox{r$^{\\alpha -1}$}tB_{r}^{0}(r,\\theta -r^{\\alpha -1}t,z).$ A non-zero radial component $B_{r}^{0}$ of the initial condition near the vertical axis for the origin ($r=0$ ) yields a blow-up of the angular component $B_{\\theta }$ .", "Thus we see that any smooth bounded initial condition $\\mathbf {B}_{0}$ , such that $B_{r}^{0}(r,\\theta ,z)>0$ for all values of the arguments, gives rise to a solution $\\mathbf {B}$ such that $\\lim _{r\\rightarrow 0}\\left|B_{\\theta }(t,\\theta ,r,z)\\right|=\\infty $ for any $t>0$ , at any point $\\left( \\theta ,z\\right) $ .", "From this one deduces $\\lim _{\\left( x,y,z\\right) \\rightarrow 0}\\left|\\mathbf {B}\\left(t,x,y,z\\right) \\right|=+\\infty $ for all $t>0$ (since $B_{\\theta }$ is the projection on $e_{\\theta }$ at $\\left( x,y,z\\right) $ ; similarly for $B_{r},B_{z}$ ; thus the divergence of $B_{\\theta }$ and boundedness of $B_{r},B_{z}$ imply the divergence of $\\mathbf {B}$ ).", "Remark 2 With more work, taking a time-dependent vector field $\\mathbf {v}$ which is smooth until time $t_{0}>0$ when it develops an Hölder singularity of the form above, one can construct an example of solution $\\mathbf {B}$ which is smooth on $[0,t_{0})$ but infinite at some point at time $t_{0}$ .", "Such example would mimic more closely what maybe could happen in a non-passive version of the vector advection equation." ], [ "The Lagrangian picture", "We summarize here the features of this example, with the following items and some pictures (just to give a graphical intuition of what happens).", "i) The fluid rotates around the vertical $z$ -axis $\\zeta $ at the origin; the Lagrangian particles describe circles around $\\zeta $ , the Cauchy problem $\\frac{d}{dt}X_{t}={\\mathbf {v}}(t,X_{t}),\\qquad X_{0}=x $ is uniquely solvable and generate a continuous flow $\\Phi _{t}(x)$ .", "Figure 1 shows a number of Lagrangian trajectories (solutions of the Cauchy problem (REF )) starting on the $x$ -axis, in the regular case $\\alpha =1$ , where the velocity produces a rigid motion (no singularity).", "Figure 2 shows the case $\\alpha =0.2$ , where the velocity of rotation near the origin is so large (still infinitesimal, so that the velocity field is Hölder continuous) that very close initial particles are displaced a lot; and the ratio between the displacement at time $t$ and that at time zero diverges when the particles approach zero.", "2 Figure: NO_CAPTION Fig.1.", "Lagrangian trajectories for $\\protect \\alpha =1$ .", "Figure: NO_CAPTION Fig.2.", "Lagrangian trajectories for $\\protect \\alpha =0.2$ .", "ii) The flow $\\Phi _{t}(x)$ is however not differentiable at the vertical axis $\\zeta $ (it is smooth outside $\\zeta $ ), as it may be guesses from Figure 2; ideal lines of Lagrangian points in a plane orthogonal to $\\zeta $ are stretched near $\\zeta $ and the stretching becomes infinite at $\\zeta $ , see Figure 3 below.", "iii) The passive field $\\mathbf {B}$ is also stretched by the fluid and the stretching blows-up at $\\zeta $ .", "With this picture in mind, we may anticipate the behavior when we add noise.", "As we shall see below, the transport type noise, in Stratonovich form, introduced in equation (REF ), corresponds at the Lagrangian level to the addition of a random shift to all Lagrangian particles (see equation (REF )).", "Figures 3 and 4 below show the time evolution of the ideal line initially equal to the $x$ axis.", "In the deterministic case (Figure 3) this line is infinitely stretched near the origin.", "In the stochastic case (Figure 4), even with very small noise intensity ($\\sigma =0.1)$ , the line is shifted by noise a little bit in all possible directions and thus it passes through the origin only for a negligible amount of time.", "Stretching still occurs but not with infinite strength and the visible result is that the line at the forth time instant looks still smooth although strongly curved.", "Stretching still exists but it is smeared-out, distributed among different portions of fluid; the deterministic concentration of stretching at $\\zeta $ is broken.", "2 Figure: NO_CAPTION Fig.3.", "Ideal lines evolution, no noise.", "Figure: NO_CAPTION Fig.4.", "Ideal lines evolution, noise with $\\protect \\sigma =0.1$ ." ], [ "The regular case ", "In this section we study the regular case.", "Let $\\mathbf {W}=\\left(W^{1},W^{2},W^{3}\\right) $ be a 3-dimensional Brownian motion on a probability space $(\\Omega ,\\mathcal {A},P)$ and let $(\\mathcal {F}_{t})_{t}$ be its natural completed filtration.", "Let $\\mathbf {v}$ be a divergence-free vector field in $C^{1}([0,T];C_{c}^{\\infty }(\\mathbb {R}^{3};\\mathbb {R}^{3}))$ and $\\mathbf {B}_{0}$ be a divergence-free vector field in $C_{c}^{\\infty }(\\mathbb {R}^{3};\\mathbb {R}^{3})$ .", "For a divergence-free solution $\\mathbf {B}$ , equation (REF ) reads formally $d\\mathbf {B}+\\left( \\left( {\\mathbf {v}}\\cdot \\nabla \\right) \\mathbf {B}-\\left(\\mathbf {B}\\cdot \\nabla \\right) \\mathbf {v}\\right) dt+\\sigma \\sum _{k}\\partial _{k}\\mathbf {B}\\circ dW^{k}=0.", "$ We will always use (REF ) in this form.", "Remark 3 The Stratonovich operation $\\partial _{k}\\mathbf {B}\\circ dW^{k}$ is the natural one from the physical viewpoint, because of Wong-Zakai principle, see the Appendix of [17] for an example, and because of the formal validity of conservation laws.", "More rigorously, it is responsible for the validity of relation (REF ) between $\\mathbf {B}$ and the Lagrangian motion, relation which extends to the stochastic case a well know deterministic relation.", "For mathematical convenience, we translate Stratonovich in Itô form.", "Formally, the martingale part of $\\partial _{k}\\mathbf {B}$ is (from equation (REF ) itself) equal to $\\sigma \\sum _{j}\\partial _{k}\\partial _{j}\\mathbf {B}dW^{j}$ and thus the quadratic variation $d\\left[ \\partial _{k}\\mathbf {B,}W^{k}\\right] $ is equal to $\\sigma \\partial _{k}\\partial _{k}\\mathbf {B}$ ; therefore $\\sigma \\sum _{k}\\partial _{k}\\mathbf {B}\\circ dW^{k}=\\sigma \\sum _{k}\\partial _{k}\\mathbf {B}dW^{k}+\\frac{\\sigma ^{2}}{2}\\Delta \\mathbf {B.", "}$ This is the heuristic justification of the following rigorous definition.", "Definition 4 A regular solution to (REF ) is a vector field $\\mathbf {B}:[0,T]\\times \\mathbb {R}^{3}\\times \\Omega \\rightarrow \\mathbb {R}^{3}$ such that i) $\\mathbf {B}(t,x)$ and its derivatives in $x$ up to fourth order are continuous in $(t,x)$ ii) for every $i,j=1,...,d$ and $x\\in \\mathbb {R}^{3}$ , $\\mathbf {B}(t,x)$ , $\\partial _{x_{i}}\\mathbf {B}(t,x)$ , $\\partial _{x_{j}}\\partial _{x_{i}}\\mathbf {B}(t,x)$ are adapted processes ii) for every $(t,x)$ , $\\mathrm {div}\\mathbf {B}(t,x)=0$ and $\\mathbf {B}(t,x)& =\\mathbf {B}_{0}(x)+\\int _{0}^{t}\\left[ \\left( \\mathbf {B}(r,x)\\cdot \\nabla \\right) \\mathbf {v}(r,x)-\\left( {\\mathbf {v}}(r,x)\\cdot \\nabla \\right) \\mathbf {B}(r,x)\\right] dr \\\\& -\\sigma \\sum _{k=1}^{3}\\int _{0}^{t}\\partial _{k}\\mathbf {B}(r,x)dW_{r}^{k}+\\frac{\\sigma ^{2}}{2}\\int _{0}^{t}\\Delta \\mathbf {B}(r,x)dr.$ Remark 5 In order to give a meaning to the equation it is not necessary to ask $C^{4}$ regularity in $x$ in point (i); the requirement is imposed to apply Itô-Kunita-Wentzell formula (Theorem 3.3.1 in [23]) below.", "Remark 6 For the purpose of this paper, one can simplify and ask that $\\mathbf {B}$ is $C^{\\infty }$ in $x$ , with all derivatives continuous in $(t,x)$ ; the results below remain true.", "Consider now the SDE on $\\mathbb {R}^{3}$ $dX_{t}={\\mathbf {v}}(t,X_{t})dt+\\sigma d\\mathbf {W}_{t},\\qquad X_{0}=x.$ It is a classical result (see [23]) that there exists a stochastic flow $\\Phi $ of $C^{\\infty }$ diffeomorphisms (see Definition REF in Section REF ) solving the above SDE.", "Since $\\mathbf {v}$ is divergence-free, $\\Phi _{t}$ and $\\Phi _{t}^{-1}$ are also measure-preserving for every $t$ , i.e.", "$det(D\\Phi _{t})=1$ .", "We can now prove the representation formula for the regular solution to equation (REF ), which will be the key ingredient of our work.", "Proposition 7 Suppose $\\mathbf {B}_{0}\\in C_{c}^{\\infty }(\\mathbb {R}^{3};\\mathbb {R}^{3})$ and $\\mathbf {v}$$\\in C^{1}([0,T];C_{c}^{\\infty }(\\mathbb {R}^{3};\\mathbb {R}^{3}))$ , both divergence free.", "Then equation (REF ) admits a unique regular solution, satisfying the identity $\\mathbf {B}(t,\\Phi _{t}(x))=D\\Phi _{t}(x)\\mathbf {B}_{0}(x).", "$ Remark 8 Notice that $D\\Phi _{t}(\\Phi _{t}^{-1}(x))=(D\\Phi _{t}^{-1}(x))^{-1}$ .", "This inverse matrix is the transpose of the cofactor matrix of $D\\Phi _{t}^{-1}$ , multiplied by the inverse of the determinant of $D\\Phi _{t}^{-1}(x)$ , which is 1 since $\\Phi _{t}$ is measure-preserving; the cofactor matrix of a given $3\\times 3$ matrix $A$ is a polynomial function $H(A)^{T}$ , of degree 2, of $A$ .", "So we have $D\\Phi _{t}(\\Phi _{t}^{-1}(x))=H(D\\Phi _{t}^{-1}(x))$ and formula (REF ) also reads $\\mathbf {B}(t,x)=H(D\\Phi _{t}^{-1}(x))\\mathbf {B}_{0}(\\Phi _{t}^{-1}(x)).$ Proof.", "Step 1 (chain rule).", "Let us recall the so called Itô-Kunita-Wentzell formula (Theorem 8.1 in [22]; see also Theorem 3.3.1 in [23] for a variant).", "We state it with the notations of interest for us.", "Assume that $F\\left( t,x\\right) $ , $t\\in \\left[ 0,T\\right] $ , $x\\in \\mathbb {R}^{d}$ , is a continuous random field, twice differentiable in $x$ with second derivatives continuous in $\\left( t,x\\right) $ , of the form $F\\left( t,x\\right) =F_{0}\\left( x\\right) +\\int _{0}^{t}f_{0}\\left( s,x\\right)ds+\\sum _{k=1}^{n}\\int _{0}^{t}f_{k}\\left( s,x\\right) dW_{s}^{k}$ where $W^{k}$ , $k=1,...,n$ are independent Brownian motions and $f_{k}$ , $k=0,1,...,n$ are twice differentiable in $x$ , continuous in $\\left(t,x\\right) $ with their second space derivatives, and for each $x$ the processes $t\\mapsto f_{k}\\left( t,x\\right) $ are adapted.", "Let $X_{t}$ be a continuous semimartingale in $\\mathbb {R}^{d}$ .", "Then $F\\left( t,X_{t}\\right) &=&F_{0}\\left( X_{0}\\right) +\\int _{0}^{t}f_{0}\\left(s,X_{s}\\right) ds+\\sum _{k=1}^{n}\\int _{0}^{t}f_{k}\\left( s,X_{s}\\right)dW_{s}^{k} \\\\&&+\\int _{0}^{t}\\nabla F\\left( t,X_{s}\\right) \\cdot dX_{s}+\\frac{1}{2}\\sum _{k,h=1}^{n}\\int _{0}^{t}\\partial _{x_{k}}\\partial _{x_{h}}F\\left(t,X_{s}\\right) d\\left[ X^{h},X^{k}\\right] _{s} \\\\&&+\\sum _{k=1}^{n}\\sum _{i=1}^{d}\\int _{0}^{t}\\partial _{x_{i}}f_{k}\\left(s,X_{s}\\right) d\\left[ X^{i},W^{k}\\right] _{s}$ where $\\left[ X^{h},X^{k}\\right] _{t}$ denotes the quadratic mutual variation between the components of $X$ and similarly for $\\left[ X^{h},W^{k}\\right] _{t}$ .", "Step 2 (uniqueness).", "Fix $x$ in $\\mathbb {R}^{3}$ ; observe that $D\\Phi _{t}(x)\\mathbf {B}_{0}(x)$ is the unique solution to $\\frac{d\\mathbf {Z}_{t}}{dt}=\\left( \\mathbf {Z}_{t}\\cdot \\nabla \\right) \\mathbf {v}(t,\\Phi _{t}(x)).", "$ with $\\mathbf {Z}_{0}=\\mathbf {B}_{0}(x)$ (uniqueness follows from the fact the the stochastic drift for this ODE, namely $(t,y)\\rightarrow D$$\\mathbf {v}$$(t,\\Phi _{t}(x))y$ is in $C_{b}^{1}$ ).", "Thus, in order to get uniqueness for equation (REF ) and prove formula (REF ), it is enough to prove that, for any regular solution $\\mathbf {B}$ to (REF ), $\\mathbf {B}(t,\\Phi _{t}(x))$ satisfies equation (REF ).", "For this purpose, we use the chain rule of Step 1 (the assumptions in the definition of regular solution above are imposed precisely in order to apply this result).", "For each component $j=1,2,3$ we apply the formula with $F=B^{j},f_{0}=\\left( \\left( \\mathbf {B}\\cdot \\nabla \\right) \\mathbf {v}-\\left({\\mathbf {v}}\\cdot \\nabla \\right) \\mathbf {B}\\right) ^{j}+\\frac{\\sigma ^{2}}{2}\\Delta B^{j},f_{k}=-\\sigma \\partial _{k}B^{j},X_{t}=\\Phi _{t}(x).$ The result, rewritten in vector form, is $d[\\mathbf {B}(t,\\Phi _{t}(x))] &=&\\left( d\\mathbf {B}\\right) (t,\\Phi _{t}(x))\\\\&&+\\sum _{i=1}^{3}\\partial _{x_{i}}\\mathbf {B}(t,\\Phi _{t}(x))d\\Phi _{t}^{i}(x)\\\\&&+\\frac{\\sigma ^{2}}{2}\\Delta \\mathbf {B}(t,\\Phi _{t}(x))dt-\\sigma ^{2}\\Delta \\mathbf {B}(t,\\Phi _{t}(x))dt$ because $\\sum _{k=1}^{3}\\sum _{i=1}^{3}\\int _{0}^{t}\\partial _{x_{i}}f_{k}d\\left[X^{i},W^{k}\\right] _{s}=-\\sigma \\sum _{k=1}^{3}\\sum _{i=1}^{3}\\int _{0}^{t}\\partial _{x_{i}}\\partial _{k}B^{j}\\sigma \\delta _{ik}ds=-\\sigma ^{2}\\int _{0}^{t}\\Delta B^{j}ds$ (since $d\\left[ X^{i},W^{k}\\right] _{s}=\\sigma \\delta _{ik}ds$ ).", "Therefore $d[\\mathbf {B}(t,\\Phi _{t}(x))] &=&\\left( \\left( \\mathbf {B}\\cdot \\nabla \\right) \\mathbf {v}-\\left( {\\mathbf {v}}\\cdot \\nabla \\right) \\mathbf {B}\\right)dt+\\frac{\\sigma ^{2}}{2}\\Delta \\mathbf {B}dt-\\sigma \\sum _{k=1}^{3}\\partial _{k}\\mathbf {B}dW_{t}^{k} \\\\&&+\\left( {\\mathbf {v}}\\cdot \\nabla \\right) \\mathbf {B}dt+\\sigma \\sum _{k=1}^{3}\\partial _{k}\\mathbf {B}dW_{t}^{k} \\\\&&-\\frac{\\sigma ^{2}}{2}\\Delta \\mathbf {B}(t,\\Phi _{t}(x))dt$ $=\\left( \\left( \\mathbf {B}\\cdot \\nabla \\right) \\mathbf {v}\\right) (t,\\Phi _{t}(x))dt.$ Therefore $\\mathbf {B}(t,\\Phi _{t}(x))$ satisfies (REF ).", "Uniqueness and formula (REF ) are proved.", "Step 3 (existence).", "Conversely, given $\\mathbf {B}$ defined by (REF ), let us prove that it is a regular solution to equation (REF ).", "Properties (i)-(ii) of the definition of regular solution are obvious from (REF ) (indeed $\\mathbf {B}$ is $C^{\\infty }$ in $x$ ).", "It also follows, from Itô-Kunita-Wentzell formula, that $\\mathbf {B}\\left( t,x\\right) $ has the form $d\\mathbf {B}\\left( t,x\\right) =\\mathbf {A}\\left( t,x\\right) dt+\\sum _{k=1}^{3}\\mathbf {S}_{k}\\left( t,x\\right) dW_{t}^{k} $ where $\\mathbf {B}\\left( t,x\\right) ,\\mathbf {A}\\left( t,x\\right) ,\\mathbf {S}_{k}\\left( t,x\\right) $ are continuous in $\\left( t,x\\right) $ with their second space derivatives, and are adapted in $t$ for every $x$ .", "Notice that we do not need to compute explicitly $\\mathbf {A}\\left( t,x\\right) $ and $\\mathbf {S}_{k}\\left( t,x\\right) $ (by Itô-Kunita-Wentzell formula) from the identity (REF ) (this would involve too complex expressions with derivatives of the flow).", "We just need to realize that Itô-Kunita-Wentzell formula can be applied and gives a decomposition of the form (REF ) with $\\mathbf {B}\\left( t,x\\right) ,\\mathbf {A}\\left(t,x\\right) ,\\mathbf {S}_{k}\\left( t,x\\right) $ having the regularity stated above.", "Thanks to this regularity, we may apply Itô-Kunita-Wentzell formula to $\\mathbf {B}(t,\\Phi _{t}(x))$ , where now we only know that identities (REF ) and (REF ) are satisfied by $\\mathbf {B}$ .", "On one side, from (REF ) and the fact that $D\\Phi _{t}(x)\\mathbf {B}_{0}(x)$ is the unique solution to (REF ) we get $d[\\mathbf {B}(t,\\Phi _{t}(x))]=\\left( \\mathbf {B}(t,\\Phi _{t}(x))\\cdot \\nabla \\right) \\mathbf {v}(t,\\Phi _{t}(x))dt.$ On the other side, similarly to the computation of Step 2, from Itô-Kunita-Wentzell formula applied to the function $F=B^{j},f_{0}=A^{j},f_{k}=S_{k}^{j},X_{t}=\\Phi _{t}(x)$ we get $d[\\mathbf {B}(t,\\Phi _{t}(x))] &=&\\left( d\\mathbf {B}\\right) (t,\\Phi _{t}(x))\\\\&&+\\sum _{i=1}^{3}\\partial _{x_{i}}\\mathbf {B}(t,\\Phi _{t}(x))d\\Phi _{t}^{i}(x)\\\\&&+\\frac{\\sigma ^{2}}{2}\\Delta \\mathbf {B}(t,\\Phi _{t}(x))dt+\\sigma \\sum _{k=1}^{3}\\partial _{x_{k}}\\mathbf {S}_{k}(t,\\Phi _{t}(x))dt$ because now $\\sum _{k=1}^{3}\\sum _{i=1}^{3}\\int _{0}^{t}\\partial _{x_{i}}f_{k}d\\left[X^{i},W^{k}\\right] _{s}=\\sum _{k=1}^{3}\\sum _{i=1}^{3}\\int _{0}^{t}\\partial _{x_{i}}S_{k}^{j}\\sigma \\delta _{ik}ds=\\sigma \\sum _{k=1}^{3}\\int _{0}^{t}\\partial _{x_{k}}S_{k}^{j}ds.$ Therefore $d[\\mathbf {B}(t,\\Phi _{t}(x))] &=&{\\mathbf {A}}(t,\\Phi _{t}(x))dt+\\sum _{k=1}^{3}\\mathbf {S}_{k}(t,\\Phi _{t}(x))dW_{t}^{k} \\\\&&+\\left( \\left( {\\mathbf {v}}\\cdot \\nabla \\right) \\mathbf {B}\\right) (t,\\Phi _{t}(x))dt+\\sigma \\sum _{k=1}^{3}\\partial _{k}\\mathbf {B}(t,\\Phi _{t}(x))dW_{t}^{k} \\\\&&+\\frac{\\sigma ^{2}}{2}\\Delta \\mathbf {B}(t,\\Phi _{t}(x))dt+\\sigma \\sum _{k=1}^{3}\\partial _{x_{k}}\\mathbf {S}_{k}(t,\\Phi _{t}(x))dt.$ Equating the two identities satisfied by $d[\\mathbf {B}(t,\\Phi _{t}(x))]$ , and using the invertibility of $\\Phi _{t}$ , we get $\\mathbf {S}_{k} &=&-\\sigma \\partial _{k}\\mathbf {B} \\\\\\left( {\\mathbf {B}}\\cdot \\nabla \\right) \\mathbf {v} &=&{\\mathbf {A}}+\\left( {\\mathbf {v}}\\cdot \\nabla \\right) \\mathbf {B}+\\frac{\\sigma ^{2}}{2}\\Delta \\mathbf {B}+\\sigma \\sum _{k=1}^{3}\\partial _{x_{k}}\\mathbf {S}_{k}.$ Thus ${\\mathbf {A}}=\\left( {\\mathbf {B}}\\cdot \\nabla \\right) \\mathbf {v-}\\left( {\\mathbf {v}}\\cdot \\nabla \\right) \\mathbf {B}+\\frac{\\sigma ^{2}}{2}\\Delta \\mathbf {B}$ which completes the proof that $\\mathbf {B}$ satisfies the SPDE.", "It remains to prove the divergence-free property.", "For this, since $\\mathbf {B}$ is regular, it is enough to show that, for every fixed $t$ , for a.e.", "$\\omega $ , $\\mathrm {div}\\mathbf {B}(t,\\cdot ,\\omega )$ is 0 in the sense of distributions.", "For this, take $\\varphi $ in $C_{c}^{\\infty }(\\mathbb {R}^{3})$ ; then, using integration by parts (notice that also $\\mathbf {B}(\\omega )$ has compact support and remember that $\\Phi _{t}$ is measure-preserving) $\\int _{\\mathbb {R}^{3}}\\mathbf {B}_{t}\\cdot \\nabla \\varphi dx=\\int _{\\mathbb {R}^{3}}D\\Phi _{t}\\mathbf {B}_{0}\\cdot \\nabla \\varphi (\\Phi _{t})dx$ $=\\int _{\\mathbb {R}^{3}}\\mathbf {B}_{0}\\cdot (D\\Phi _{t})^{T}\\nabla \\varphi (\\Phi _{t})dx=\\int _{\\mathbb {R}^{3}}\\mathbf {B}_{0}\\cdot \\nabla [\\varphi (\\Phi _{t})]dx=0$ since $\\mathbf {B}_{0}$ is divergence-free.", "The proof is complete." ], [ "The case when $\\mathbf {v}$ is only Hölder continuous and\nbounded", "In this section we shall always assume $\\sigma \\ne 0$ and the following condition.", "Condition 9 The vector field $\\mathbf {v}$ is in $C([0,T];C_{b}^{\\alpha }(\\mathbb {R}^{3};\\mathbb {R}^{3}))$ for some $\\alpha \\in \\left(0,1\\right) $ and it is divergence-free.", "Definition 10 Let $\\mathbf {B}_{0}$ be divergence-free and in $C(\\mathbb {R}^{3};\\mathbb {R}^{3})$ .", "A continuous weak solution to equation (REF ) is a vector field $\\mathbf {B}:[0,T]\\times \\mathbb {R}^{3}\\times \\Omega \\rightarrow \\mathbb {R}^{3}$ , with a.e.", "path in $C([0,T]\\times \\mathbb {R}^{3};\\mathbb {R}^{3})$ , weakly adapted to $(\\mathcal {F}_{t})_{t}$ (namely such that $\\langle \\mathbf {B},\\varphi \\rangle $ is adapted for all $\\varphi $ in $C_{c}^{\\infty }(\\mathbb {R}^{3};\\mathbb {R}^{3})$ ) such that: i) for every $\\varphi $ in $C_{c}^{\\infty }(\\mathbb {R}^{3};\\mathbb {R}^{3})$ , the continuous adapted process $\\langle \\mathbf {B},\\varphi \\rangle $ satisfies $\\langle \\mathbf {B}_{t},\\varphi \\rangle =\\langle \\mathbf {B}_{0},\\varphi \\rangle +\\int _{0}^{t}\\langle (D\\varphi )_{r}^{A}\\mathbf {v},\\mathbf {B}_{r}\\rangle dr+\\sigma \\sum _{k=1}^{d}\\int _{0}^{t}\\langle (D\\varphi )\\mathbf {e}_{k},\\mathbf {B}_{r}\\rangle dW_{r}^{k}+\\frac{\\sigma ^{2}}{2}\\int _{0}^{t}\\langle \\Delta \\varphi ,\\mathbf {B}_{r}\\rangle dr,$ where $((D\\varphi )(x))^{A}=D\\varphi (x)-(D\\varphi (x))^{T}$ is the antisymmetric part of the matrix $D\\varphi (x)$ ; ii) $\\mathbf {B}_{t}$ is divergence-free, in the sense that $P\\lbrace \\mathrm {div}\\mathbf {B}_{t}=0,\\ \\forall t\\in \\left[ 0,T\\right] \\rbrace =1$ .", "Notice that the Itô integrals are well defined since the processes $\\langle (D\\varphi )\\mathbf {e}_{k},\\mathbf {B}_{r}\\rangle $ are continuous and adapted.", "Remark 11 One can define a similar notion of $L^{p}$ weak solution and, at least for $p>1$ , existence and uniqueness should remain true with a more elaborated proof.", "We restrict ourselves to continuous solution to emphasize the no blow-up result.", "The aim of this section is to prove the following main result.", "Theorem 12 Assume that $\\sigma \\ne 0$ .", "Let $\\mathbf {B}_{0}$ be divergence-free in $C(\\mathbb {R}^{3};\\mathbb {R}^{3})$ and suppose Condition REF .", "Then there exists a unique continuous weak solution $\\mathbf {B}$ to equation (REF ), starting from $\\mathbf {B}_{0}$ .", "In particular no blow-up occurs.", "Let us recall the notion of stochastic flow of $C^{1,\\beta }$ diffeomorphisms, limited to the properties of interest to us.", "Definition 13 A stochastic flow $\\Phi $ of $C^{1,\\beta }$ diffeomorphisms (on $\\mathbb {R}^{3}$ ), $\\beta \\in \\left( 0,1\\right) $ , is a map $[0,T]\\times \\mathbb {R}^{3}\\times \\Omega \\rightarrow \\mathbb {R}^{3}$ such that for every $x$ in $\\mathbb {R}^{3}$ , $\\Phi (x)$ is adapted to $(\\mathcal {F}_{t})_{t}$ ; for a.e.", "$\\omega $ in $\\Omega $ , $\\Phi (\\omega )$ is a flow of $C^{1,\\beta }$ diffeomorphisms, i.e.", "$\\Phi _{0}(\\omega )=id$ , for every $t$ , $\\Phi _{t}(\\omega )$ is a diffeomorphism, $\\Phi _{t}$ , $\\Phi ^{-1}_{t}$ , $D\\Phi _{t}$ and $D\\Phi ^{-1}_{t}$ are jointly continuous on $[0,T]\\times \\mathbb {R}^{3}$ , $\\beta $ -Hölder continuous in space uniformly in time.", "In the definition of flows we did not mention the cocycle property, since it is not useful for our purposes.", "We need the following result (valid more in general in $\\mathbb {R}^{d}$ ), see [17], Theorem 5.", "Theorem 14 Assume that $\\sigma \\ne 0$ .", "Let $\\mathbf {v}$ satisfy Condition REF and consider the SDE (REF ) on $\\mathbb {R}^{3}$ .", "For every $x$ in $\\mathbb {R}^{3}$ , there exists a unique strong solution to the SDE (REF ) starting from $x$ .", "There exists a stochastic flow of $C^{1,\\alpha ^{\\prime }}$ diffeomorphisms, for every $\\alpha ^{\\prime }<\\alpha $ , solving the SDE and belonging to $L_{loc}^{\\infty }([0,T]\\times \\mathbb {R}^{3};L^{m}(\\Omega ))$ for every finite $m$ .", "Let $(\\mathbf {v}^{\\epsilon })_{\\epsilon >0}$ be a family of divergence-free vector fields in $C([0,T];C_{b}^{\\alpha }(\\mathbb {R}^{3};\\mathbb {R}^{3}))$ converging to $\\mathbf {v}$ in this space, as $\\epsilon \\rightarrow 0$ .", "For every $\\epsilon >0$ , let $\\Phi ^{\\epsilon }$ be the stochastic flow of diffeomorphisms solving (REF ) with drift $\\mathbf {v}^{\\epsilon }$ .", "Then, for every $R>0$ and every $m\\ge 1$ , the following results hold: $\\lim _{\\epsilon \\rightarrow 0}\\sup _{t\\in [0,T]}\\sup _{\\left|x\\right|\\le R}E[|\\Phi _{t}^{\\epsilon }(x)-\\Phi _{t}(x)|^{m}]& =0, \\\\\\lim _{\\epsilon \\rightarrow 0}\\sup _{t\\in [0,T]}\\sup _{\\left|x\\right|\\le R}E[|D\\Phi _{t}^{\\epsilon }(x)-D\\Phi _{t}(x)|^{m}]& =0$ and the same for the inverse flow $\\Phi _{t}^{-1}$ and its derivative in space.", "for every $t$ , $\\Phi _{t}$ is measure-preserving, i.e.", "$det(D\\Phi _{t}(x))=1$ for every $x$ in $\\mathbb {R}^{3}$ .", "We split the proof of Theorem REF in two lemmata, one of existence and the other of uniqueness.", "Lemma 15 Let $\\mathbf {B}_{0}$ be divergence-free and in $C(\\mathbb {R}^{3};\\mathbb {R}^{3})$ and suppose Condition REF hold; let $\\Phi $ be the flow of diffeomorphisms solving the SDE (REF ) (as given in Theorem REF ).", "Define the random vector field $\\mathbf {B}$ as $\\mathbf {B}(t,x)=D\\Phi _{t}(\\Phi _{t}^{-1}(x))\\mathbf {B}_{0}(\\Phi _{t}^{-1}(x)).", "$ Then $\\mathbf {B}$ is a continuous weak solution to equation (REF ).", "Proof.", "Step 1 (regularity).", "By definition (REF ), the assumption on $\\mathbf {B}_{0}$ and the continuity properties in $\\left(t,x\\right) $ of $\\Phi _{t}^{-1}(x)$ and $D\\Phi _{t}(x)$ it follows that $\\mathbf {B}\\in C([0,T]\\times \\mathbb {R}^{3};\\mathbb {R}^{3})$ with probability one; since $\\Phi _{t}^{-1}(x)$ and $D\\Phi _{t}(x)$ are $\\mathcal {F}_{t}$ measurable, for every $x$ the process $\\mathbf {B}(t,x)$ is adapted to $(\\mathcal {F}_{t})_{t}$ , hence also weakly adapted.", "It remains to prove properties (i) and (ii) of Definition REF .", "Step 2 (property (i)).", "Let $(\\mathbf {v}^{\\epsilon })_{\\epsilon >0}$ be a family of $C^{1}([0,T];C_{c}^{\\infty }(\\mathbb {R}^{3};\\mathbb {R}^{3}))$ divergence-free vector fields, approximating $\\mathbf {v}$ in $C([0,T];C_{b}^{\\alpha }(\\mathbb {R}^{3};\\mathbb {R}^{3}))$ ; let $(\\mathbf {B}_{0}^{\\epsilon })_{\\epsilon }$ be a family of $C_{c}^{\\infty }(\\mathbb {R}^{3};\\mathbb {R}^{3})$ divergence-free vector fields, approximating $\\mathbf {B}_{0}$ in $C_{b}(\\mathbb {R}^{3};\\mathbb {R}^{3})$ .", "We know from Lemma REF that, for every $\\epsilon >0$ , $\\mathbf {B}^{\\epsilon }(t,x)=D\\Phi _{t}^{\\epsilon }((\\Phi _{t}^{\\epsilon })^{-1}(x))\\mathbf {B}_{0}^{\\epsilon }((\\Phi _{t}^{\\epsilon })^{-1}(x))$ solves equation (REF ), where $\\Phi ^{\\epsilon }$ is the regular stochastic flow solving the SDE (REF ) with drift $\\mathbf {v}^{\\epsilon }$ .", "Let us first show that for every $(t,x)$ , $(\\mathbf {B}^{\\epsilon }(t,x))_{\\epsilon }$ converges to $\\mathbf {B}(t,x)$ , defined by (REF ), in $L^{m}(\\Omega ;\\mathbb {R}^{3})$ , for every finite $m$ .", "Fix $(t,x)$ and $m\\ge 1$ .", "Using Remark REF (which also applies to $\\Phi $ , since $\\det (D\\Phi _{t})=1$ ), We have $&&|\\mathbf {B}^{\\epsilon }(t,x)-\\mathbf {B}(t,x)| \\\\&\\le &|\\mathbf {B}_{0}^{\\epsilon }((\\Phi _{t}^{\\epsilon })^{-1}(x))||H((D\\Phi _{t}^{\\epsilon })^{-1}(x))-H(D\\Phi _{t}^{-1}(x))| \\\\&&+|H((D\\Phi _{t}^{\\epsilon })^{-1}(x))||\\mathbf {B}_{0}^{\\epsilon }((\\Phi _{t}^{\\epsilon })^{-1}(x))-\\mathbf {B}_{0}((\\Phi _{t}^{\\epsilon })^{-1}(x))|\\\\&&+|H((D\\Phi _{t}^{\\epsilon })^{-1}(x))||\\mathbf {B}_{0}((\\Phi _{t}^{\\epsilon })^{-1}(x))-\\mathbf {B}_{0}(\\Phi _{t}^{-1}(x))|$ so, by Hölder inequality, we get ${E[\\left|\\mathbf {B}^{\\epsilon }(t,x)-\\mathbf {B}(t,x)\\right|^{m}]} $ $&\\le &C\\Vert \\mathbf {B}_{0}^{\\epsilon }\\Vert _{0}E[|H((D\\Phi _{t}^{\\epsilon })^{-1}(x))-H(D\\Phi _{t}^{-1}(x))|^{m}] \\\\&&+CE[|H(D\\Phi _{t}^{-1}(x))|^{m}]\\Vert \\mathbf {B}_{0}^{\\epsilon }-\\mathbf {B}_{0}\\Vert _{0} \\\\&&+CE[|H(D\\Phi _{t}^{-1}(x))|^{2m}]^{1/2}E[|\\mathbf {B}_{0}((\\Phi _{t}^{\\epsilon })^{-1}(x))-\\mathbf {B}_{0}(\\Phi _{t}^{-1}(x))|^{2m}]^{1/2}.$ We will prove that every term on the right-hand-side of (REF ) tends to 0.", "First notice that $\\Vert \\mathbf {B}_{0}^{\\epsilon }\\Vert _{0}$ and $E[|H(D\\Phi _{t}^{-1}(x)|^{m}]$ are bounded uniformly in $\\epsilon $ , for every $m$ , since $H$ is a polynomial function.", "The convergence of $\\Vert \\mathbf {B}_{0}^{\\epsilon }-\\mathbf {B}_{0}\\Vert _{0}$ is ensured by our assumptions, that of $|\\mathbf {B}_{0}((\\Phi _{t}^{\\epsilon })^{-1}(x))-\\mathbf {B}_{0}(\\Phi _{t}^{-1}(x))|$ by Theorem REF and dominated convergence theorem ($\\mathbf {B}_{0}$ is bounded).", "Also the convergence of $|H(D(\\Phi _{t}^{\\epsilon })^{-1}(x))-H(D\\Phi _{t}^{-1}(x))|$ in $L^{m}(\\Omega )$ is a consequence of Theorem REF and the fact that $H$ is a polynomial.", "To see this in detail, we can write this term as ${H(D(\\Phi _{t}^{\\epsilon })^{-1}(x))-H(D\\Phi _{t}^{-1}(x))} \\\\& =\\sum _{i,j=1}^{3}\\left( \\int _{0}^{1}\\frac{\\partial H}{\\partial x_{ij}}(\\xi D(\\Phi _{t}^{\\epsilon })^{-1}(x)+(1-\\xi )D\\Phi _{t}^{-1}(x))d\\xi \\right)\\left( D(\\Phi _{t}^{\\epsilon })^{-1}(x)-D\\Phi _{t}^{-1}(x)\\right) _{ij}.$ The function $H^{\\prime }$ is linear (because $H$ is quadratic), so we can use Hölder inequality and get ${E[|H((D\\Phi _{t}^{\\epsilon })^{-1}(x))-H(D\\Phi _{t}^{-1}(x))|^{m}]}\\\\& \\le CE[|(D\\Phi _{t}^{\\epsilon })^{-1}(x)|^{2m}+|D\\Phi _{t}^{-1}(x)|^{2m}]^{1/2}E[|(D\\Phi _{t}^{\\epsilon })^{-1}(x)-D\\Phi _{t}^{-1}(x)|^{2m}]^{1/2}.$ The term $E[|(D\\Phi _{t}^{\\epsilon })^{-1}(x)|^{2m}+|D\\Phi _{t}^{-1}(x)|^{2m}]$ is uniformly bounded in $\\epsilon $ , thanks to Theorem REF , and the term $E[|(D\\Phi _{t}^{\\epsilon })^{-1}(x)-D\\Phi _{t}^{-1}(x)|^{2m}]$ tends to 0 by Theorem REF .", "Putting all together, we get convergence of $\\mathbf {B}^{\\epsilon }(t,x)$ to $\\mathbf {B}(t,x)$ in $L^{m}(\\Omega )$ .", "Now, with the help of this convergence, we may prove that $\\mathbf {B}$ solves equation (REF ).", "We know that (REF ) is satisfied by $\\mathbf {B}^{\\epsilon }$ , pointwise and thus in the distributional form (formula (REF )), by integration by parts.", "Let us prove that, for every $\\varphi $ in $C_{c}^{\\infty }(\\mathbb {R}^{3};\\mathbb {R}^{3})$ , for every $t$ , every term of (REF ) for $\\mathbf {B}^{\\epsilon }$ converges in $L^{m}(\\Omega )$ ,for any fixed finite $m$ , to the corresponding term for $\\mathbf {B}$ .", "We will use the previous convergence result and the uniform estimate $\\sup _{\\epsilon }\\sup _{t\\in [0,T]}\\sup _{\\left|x\\right|\\le R}E[|\\mathbf {B}^{\\epsilon }(t,x)|^{m}]<+\\infty ,\\ \\ \\sup _{t\\in [0,T]}\\sup _{\\left|x\\right|\\le R}E[|\\mathbf {B}(t,x)|^{m}]<+\\infty $ which again follows from Theorem REF .", "Take the term $\\langle \\mathbf {B}_{t},\\varphi \\rangle $ .", "Since $E[|\\langle \\mathbf {B}_{t}^{\\epsilon }-\\mathbf {B}_{t},\\varphi \\rangle |^{m}]\\le C\\langle E[|\\mathbf {B}_{t}^{\\epsilon }-\\mathbf {B}_{t}|^{m}],|\\varphi |^{m}\\rangle ,$ (in the last term $\\left\\langle .,.\\right\\rangle $ denotes the scalar product in $L^{2}\\left( \\mathbb {R}^{3}\\right) $ between real-valued functions, not vector fields as usual), $\\mathbf {B}^{\\epsilon }(t,x)$ tends to $\\mathbf {B}(t,x)$ in $L^{m}(\\Omega )$ for every $x$ and $E[|\\mathbf {B}_{t}^{\\epsilon }|^{m}+|\\mathbf {B}_{t}|^{m}]$ is bounded uniformly in $\\epsilon $ and $x$ , the convergence of this term follows from dominated convergence theorem.", "Similarly one can prove the convergence of the terms $\\int _{0}^{t}\\langle \\mathbf {B}_{r},\\Delta \\varphi \\rangle dr$ and $\\int _{0}^{t}\\langle (D\\varphi )e_{k},\\mathbf {B}_{r}\\rangle dW_{r}^{k}$ , $k=1,2,3$ , the last ones using Burkholder inequality $E\\left[ \\left|\\int _{0}^{t}\\langle (D\\varphi )e_{k},\\mathbf {B}_{r}^{\\epsilon }-\\mathbf {B}_{r}\\rangle dW_{r}^{k}\\right|^{m}\\right]\\le C\\int _{0}^{t}\\langle |(D\\varphi )e_{k}|^{m},E[|\\mathbf {B}_{r}^{\\epsilon }-\\mathbf {B}_{r}|^{m}]\\rangle dr.$ For the last term, $\\int _{0}^{t}\\langle (D\\varphi )^{A}\\mathbf {v}_{r},\\mathbf {B}_{r}\\rangle dr$ , we have ${E\\left[ \\left|\\int _{0}^{t}\\left\\langle (D\\varphi )^{A}\\mathbf {v}_{r}^{\\epsilon },\\mathbf {B}_{r}^{\\epsilon }\\right\\rangle dr-\\int _{0}^{t}\\left\\langle (D\\varphi )^{A}\\mathbf {v}_{r},\\mathbf {B}_{r}\\right\\rangle dr\\right|^{m}\\right] } \\\\& \\le C\\int _{0}^{t}\\langle |D\\varphi |^{m}|\\mathbf {v}_{r}^{\\epsilon }-\\mathbf {v}_{r}|^{m},E[|\\mathbf {B}_{r}^{\\epsilon }|^{m}]\\rangle dr+C\\int _{0}^{t}\\langle |D\\varphi |^{m}|\\mathbf {v}_{r}|^{m},E[|\\mathbf {B}_{r}^{\\epsilon }-\\mathbf {B}_{r}|^{m}]\\rangle dr.$ Both the two addends in the right-hand-side of this equation tend to 0 by dominated convergence theorem, because $\\mathbf {v}^{\\epsilon }\\rightarrow \\mathbf {v}$ and $E[|\\mathbf {B}_{r}^{\\epsilon }-\\mathbf {B}_{r}|^{m}]\\rightarrow 0$ for every $(t,x)$ and $|\\mathbf {v}^{\\epsilon }|+|\\mathbf {v}|$ , $E[|\\mathbf {B}_{t}^{\\epsilon }|^{m}+|\\mathbf {B}_{t}|^{m}]$ are uniformly bounded.", "Since all the terms of (REF ) converge, (REF ) holds for $\\mathbf {B}$ .", "Thus $\\mathbf {B}$ solves (REF ) in the sense of distributions.", "Step 3 (property (ii)).", "Concerning property (ii) of Definition REF , we will prove that, for every $t$ , for every $\\varphi $ in $C_{c}^{\\infty }(\\mathbb {R}^{3};\\mathbb {R}^{3})$ , $E\\left[ \\int _{\\mathbb {R}^{3}}\\mathbf {B}_{t}\\cdot \\nabla \\varphi dx\\right] =0.$ Since, for a.e.", "$\\omega $ , $\\mathbf {B}$ is continuous in $(t,x)$ and since $C_{c}^{\\infty }(\\mathbb {R}^{3};\\mathbb {R}^{3})$ is separable, (REF ) implies that, outside a negligible set in $\\Omega $ , $B_{t}$ is divergence-free for every $t$ .", "We know that (REF ) is satisfied for $\\mathbf {B}^{\\epsilon }$ and that $\\mathbf {B}^{\\epsilon }$ tends to $\\mathbf {B}$ in $L^{m}(\\Omega )$ , for every $m$ , with $L^{m}$ -norm bounded uniformly in $x$ (in a ball).", "Then, applying dominated convergence theorem as before, we get (REF ).", "The proof is complete.", "Finally, let us prove that the solution given by the previous theorem is unique.", "Lemma 16 Let $\\mathbf {B}_{0}$ be divergence-free and in $C(\\mathbb {R}^{3};\\mathbb {R}^{3})$ and suppose Condition REF hold.", "The there is at most one continuous weak solution to equation (REF ), given by formula (REF ).", "Proof.", "Step 1 (origin of the proof).", "Since the equation is linear, it is sufficient to consider the case $\\mathbf {B}_{0}=0$ and prove that, if $\\mathbf {B}$ is a continuous weak solution to equation (REF ) with $\\mathbf {B}_{0}=0$ , then $\\mathbf {B}=0$ .", "In Proposition REF we proved, by Itô-Kunita-Wentzell formula, that a regular solution $\\mathbf {B}$ satisfies the identity $\\frac{d}{dt}[\\mathbf {B}(t,\\Phi _{t}(x))]=\\left( \\mathbf {B}(t,\\Phi _{t}(x))\\cdot \\nabla \\right) {\\mathbf {v}}(t,\\Phi _{t}(x))$ and thus, by uniqueness for equation (REF ), we got $\\mathbf {B}(t,\\Phi _{t}(x))=D\\Phi _{t}(x)\\mathbf {B}_{0}(x)$ , namely $\\mathbf {B}(t,\\Phi _{t}(x))=0$ in the present case (hence $\\mathbf {B}=0$ ).", "We may also go further and drop the step involving equation (REF ): it is sufficient to differentiate $\\left( D\\Phi _{t}(x)\\right) ^{-1}\\mathbf {B}(t,\\Phi _{t}(x))$ : $\\frac{d}{dt}\\left[ \\left( D\\Phi _{t}(x)\\right) ^{-1}\\mathbf {B}(t,\\Phi _{t}(x))\\right] =0$ which readily implies $\\left( D\\Phi _{t}(x)\\right) ^{-1}\\mathbf {B}(t,\\Phi _{t}(x))=\\mathbf {B}_{0}(x)=0$ , hence $\\mathbf {B}(t,\\Phi _{t}(x))=0$ and thus $\\mathbf {B}=0$ .", "We have used the fact that $\\frac{d}{dt}\\left( D\\Phi _{t}(x)\\right) ^{-1}=-\\left( D\\Phi _{t}(x)\\right)^{-1}D{\\mathbf {v}}(t,\\Phi _{t}(x))$ which comes from the computation (in the regular case) $\\frac{d}{dt}\\left( D\\Phi _{t}(x)\\right) ^{-1} &=&\\lim _{h\\rightarrow 0}\\frac{\\left( D\\Phi _{t+h}(x)\\right) ^{-1}-\\left( D\\Phi _{t}(x)\\right) ^{-1}}{h} \\\\&=&\\lim _{h\\rightarrow 0}\\left( D\\Phi _{t+h}(x)\\right) ^{-1}\\frac{\\left(D\\Phi _{t}(x)-D\\Phi _{t+h}(x)\\right) }{h}\\left( D\\Phi _{t}(x)\\right) ^{-1} \\\\&=&-\\left( D\\Phi _{t}(x)\\right) ^{-1}\\frac{d}{dt}D\\Phi _{t}(x)\\left( D\\Phi _{t}(x)\\right) ^{-1} \\\\&=&-\\left( D\\Phi _{t}(x)\\right) ^{-1}D\\mathbf {v}(t,\\Phi _{t}(x)).$ These are proofs of uniqueness for regular solutions.", "If $\\mathbf {B}$ is only a continuous weak solution, Itô-Kunita-Wentzell formula cannot be applied.", "Moreover, $D$$\\mathbf {v}$ is a distribution, hence everywhere it enters the computations it may cause troubles (for instance, the meaning of equation (REF ) is less clear; although in mild form it is meaningful because $D\\Phi _{t}(x)$ , which exists also in the non-regular case, is formally its fundamental solution).", "Thus we regularize both $\\mathbf {B}$ and the flow $\\Phi _{t}(x)$ .", "Usually, with this procedure, the regularized field $\\mathbf {B}^{\\epsilon }$ satisfies an equation similar to (REF ) but with a remainder, a commutator; this has been a successful procedure for linear transport equations with non-smooth coefficients, see [11]; in the stochastic case one has a commutator composed with the flow and the approach works again well due to variants of the commutator lemma, see [17].", "The commutator estimates are the central tool in this approach, both deterministic and stochastic.", "When special cancellations apply, in particular due to divergence free conditions, it is possible to follow an interesting variant of this approach, not based on commutator estimates, developed by [26].", "We follow this approach and exploit special cancellations; in absence of them, the vectorial case proper of this paper could not be treated (see below the argument about second space derivatives of the flow).", "Step 2 (approximation).", "Let $\\rho $ be a $C^{\\infty }$ compactly supported even function on $\\mathbb {R}^{3}$ and define the approximations of identity as $\\rho _{\\epsilon }(x):=\\epsilon ^{-3}\\rho (\\epsilon ^{-1}x)$ , for $\\epsilon >0$ .", "Call $\\mathbf {B}^{\\epsilon }=\\mathbf {B}\\ast \\rho _{\\epsilon }$ , $\\mathbf {v}^{\\epsilon }=\\mathbf {v}\\ast \\rho _{\\epsilon }$ (and similarly for other fields).", "Then, using $\\rho ^{\\epsilon }$ as test function, we get the following equation for $\\mathbf {B}^{\\epsilon }$ , satisfied pointwise (actually, for a.e.", "$\\omega $ , for every $(t,x)$ , up to a suitable modification): $\\mathbf {B}^{\\epsilon }(t,x) &=&\\int _{0}^{t}[(\\left( {\\mathbf {B}}\\cdot \\nabla \\right) \\mathbf {v})^{\\epsilon }(r,x)-(\\left( {\\mathbf {v}}\\cdot \\nabla \\right) \\mathbf {B})^{\\epsilon }(r,x)]dr \\\\&&-\\sigma \\sum _{k=1}^{3}\\int _{0}^{t}\\partial _{k}\\mathbf {B}^{\\epsilon }(r,x)dW_{r}^{k}+\\frac{\\sigma ^{2}}{2}\\int _{0}^{t}\\Delta \\mathbf {B}^{\\epsilon }(r,x)dr$ where we have used the fact that $\\mathbf {B}_{0}^{\\epsilon }=0$ .", "Let $\\Phi _{t}^{\\epsilon }(x)$ be the regular flow associated to $\\mathbf {v}^{\\epsilon }$ .", "Since $\\mathbf {B}^{\\epsilon }$ is regular, we can now apply Itô-Kunita-Wentzell formula to $\\mathbf {B}^{\\epsilon }(t,\\Phi _{t}^{\\epsilon }(x))$ and get (as in the proof of Proposition REF ): $d[\\mathbf {B}^{\\epsilon }(t,\\Phi _{t}^{\\epsilon }(x))] &=&\\left( d\\mathbf {B}^{\\epsilon }\\right) (t,\\Phi _{t}^{\\epsilon }(x)) \\\\&&+\\sum _{i=1}^{3}\\partial _{x_{i}}\\mathbf {B}^{\\epsilon }(t,\\Phi _{t}^{\\epsilon }(x))d\\left( \\Phi _{t}^{\\epsilon }\\right) ^{i}(x) \\\\&&+\\frac{\\sigma ^{2}}{2}\\Delta \\mathbf {B}^{\\epsilon }(t,\\Phi _{t}^{\\epsilon }(x))dt-\\sigma ^{2}\\Delta \\mathbf {B}^{\\epsilon }(t,\\Phi _{t}^{\\epsilon }(x))dt$ $&=&\\left[ (\\left( {\\mathbf {B}}\\cdot \\nabla \\right) \\mathbf {v})^{\\epsilon }-(\\left( {\\mathbf {v}}\\cdot \\nabla \\right) \\mathbf {B})^{\\epsilon }(t,\\Phi _{t}^{\\epsilon }(x))\\right] dt+\\frac{\\sigma ^{2}}{2}\\Delta \\mathbf {B}^{\\epsilon }(t,\\Phi _{t}^{\\epsilon }(x))dt-\\sigma \\sum _{k=1}^{3}\\partial _{k}\\mathbf {B}^{\\epsilon }(t,\\Phi _{t}^{\\epsilon }(x))dW_{t}^{k} \\\\&&+\\left( \\mathbf {v}^{\\epsilon }\\cdot \\nabla \\right) \\mathbf {B}^{\\epsilon }(t,\\Phi _{t}^{\\epsilon }(x))dt+\\sigma \\sum _{k=1}^{3}\\partial _{k}\\mathbf {B}^{\\epsilon }(t,\\Phi _{t}^{\\epsilon }(x))dW_{t}^{k} \\\\&&-\\frac{\\sigma ^{2}}{2}\\Delta \\mathbf {B}^{\\epsilon }(t,\\Phi _{t}^{\\epsilon }(x))dt$ $=\\left( \\left( {\\mathbf {B}}\\cdot \\nabla \\right) \\mathbf {v})^{\\epsilon }-(\\left( {\\mathbf {v}}\\cdot \\nabla \\right) \\mathbf {B})^{\\epsilon }\\right)(t,\\Phi _{t}^{\\epsilon }(x))dt+\\left( \\mathbf {v}^{\\epsilon }\\cdot \\nabla \\right) \\mathbf {B}^{\\epsilon }(t,\\Phi _{t}^{\\epsilon }(x))dt.$ Since $\\frac{d}{dt}\\left( D\\Phi _{t}^{\\epsilon }(x)\\right) ^{-1}=-\\left(D\\Phi _{t}^{\\epsilon }(x)\\right) ^{-1}D\\mathbf {v}^{\\epsilon }(t,\\Phi _{t}^{\\epsilon }(x))$ , we get $&&\\frac{d}{dt}\\left[ \\left( D\\Phi _{t}^{\\epsilon }(x)\\right) ^{-1}\\mathbf {B}^{\\epsilon }(t,\\Phi _{t}^{\\epsilon }(x))\\right] \\\\&=&\\left( D\\Phi _{t}^{\\epsilon }(x)\\right) ^{-1}\\left[ (\\left( {\\mathbf {B}}\\cdot \\nabla \\right) \\mathbf {v})^{\\epsilon }-(\\left( {\\mathbf {v}}\\cdot \\nabla \\right) \\mathbf {B})^{\\epsilon }+\\left( \\mathbf {v}^{\\epsilon }\\cdot \\nabla \\right) \\mathbf {B}^{\\epsilon }-\\left( \\mathbf {B}^{\\epsilon }\\cdot \\nabla \\right) \\mathbf {v}^{\\epsilon }\\right] (t,\\Phi _{t}^{\\epsilon }(x)).$ Fix $\\varphi $ in $C_{c}^{\\infty }(\\mathbb {R}^{3};\\mathbb {R}^{3})$ .", "We multiply the previous formula by $\\varphi $ , integrate in space and change variable $x=\\Phi _{t}^{\\epsilon }(x^{\\prime })$ recalling that $\\Phi _{t}^{\\epsilon }$ is measure preserving: $\\int _{\\mathbb {R}^{3}}\\left( D\\Phi _{t}^{\\epsilon }(x)\\right) ^{-1}\\mathbf {B}^{\\epsilon }(t,\\Phi _{t}^{\\epsilon }(x))\\varphi \\left( x\\right) dx$ $=\\int _{0}^{t}\\int _{\\mathbb {R}^{3}}\\left[ (\\left( {\\mathbf {B}}\\cdot \\nabla \\right) \\mathbf {v})^{\\epsilon }-(\\left( {\\mathbf {v}}\\cdot \\nabla \\right)\\mathbf {B})^{\\epsilon }+\\left( \\mathbf {v}^{\\epsilon }\\cdot \\nabla \\right)\\mathbf {B}^{\\epsilon }-\\left( \\mathbf {B}^{\\epsilon }\\cdot \\nabla \\right)\\mathbf {v}^{\\epsilon }\\right] (s,x)\\psi ^{\\epsilon }\\left( s,x\\right) dxds$ where we have introduced the random field $\\psi ^{\\epsilon }\\left( s,x\\right) :=\\left( D\\Phi _{s}^{\\epsilon }((\\Phi _{s}^{\\epsilon })^{-1}\\left( x\\right) )\\right) ^{-1}\\varphi \\left( (\\Phi _{s}^{\\epsilon })^{-1}\\left( x\\right) \\right) .$ By integration by parts we get: $\\int _{\\mathbb {R}^{3}}\\left( D\\Phi _{t}^{\\epsilon }(x)\\right) ^{-1}\\mathbf {B}^{\\epsilon }(t,\\Phi _{t}^{\\epsilon }(x))\\varphi \\left( x\\right)dx=-\\sum _{i,j=1}^{3}\\int _{0}^{t}\\int _{\\mathbb {R}^{3}}\\left[(v^{j}B^{i})^{\\epsilon }-v^{\\epsilon ,j}B^{\\epsilon ,i}\\right] (s,x)(D\\psi ^{\\epsilon })_{ij}^{A}(s,x)dxds.", "$ Step 3 (support and convergence of $\\psi ^{\\epsilon }$ ).", "In the next step we need a technical fact about the support of $x\\mapsto \\psi ^{\\epsilon }\\left( t,x,\\omega \\right) $ .", "Let $R^{\\prime }>0$ be such that the support of $\\varphi $ is contained in $B\\left( 0,R^{\\prime }\\right) $ .", "Define $R^{\\epsilon }\\left( \\omega \\right) $ as $R_{t}^{\\epsilon }\\left( \\omega \\right) =\\max _{x\\in B\\left( 0,R^{\\prime }\\right) }\\left|\\Phi _{t}^{\\epsilon }(x,\\omega )\\right|.$ Then the support of $x\\mapsto \\psi ^{\\epsilon }\\left( t,x,\\omega \\right) $ is contained in $\\overline{B\\left( 0,R_{t}^{\\epsilon }\\left( \\omega \\right)\\right) }$ .", "We have $\\Phi _{t}^{\\epsilon }(x,\\omega )=x+\\int _{0}^{t}\\mathbf {v}^{\\epsilon }\\left(s,\\Phi _{s}^{\\epsilon }(x,\\omega )\\right) ds+\\sigma \\mathbf {W}_{t}\\left(\\omega \\right)$ and there is a constant $C>0$ such that $\\left|\\mathbf {v}^{\\epsilon }\\left( s,\\Phi _{s}^{\\epsilon }((\\Phi _{t}^{\\epsilon })^{-1}\\left( x,\\omega \\right) ,\\omega )\\right) \\right|\\le C$ ; thus $\\left|\\Phi _{t}^{\\epsilon }(x,\\omega )\\right|\\le \\left|x\\right|+Ct+\\sigma \\max _{t\\in \\left[ 0,T\\right] }\\left|\\mathbf {W}_{t}\\left( \\omega \\right) \\right|.$ It implies that $R_{t}^{\\epsilon }\\left( \\omega \\right) \\le \\overline{R}\\left( \\omega \\right) :=R^{\\prime }+CT+\\sigma \\max _{t\\in \\left[ 0,T\\right] }\\left|\\mathbf {W}_{t}\\left( \\omega \\right) \\right|$ for all $\\epsilon >0$ , $t\\in \\left[ 0,T\\right] $ .", "The r.v.", "$\\overline{R}\\left( \\omega \\right) $ is finite a.s. and thus we have proved that the function $x\\mapsto \\psi ^{\\epsilon }\\left( t,x,\\omega \\right) $ has a random support which is contained in $B\\left( 0,\\overline{R}\\left( \\omega \\right)\\right) $ for all $\\epsilon >0$ , $t\\in \\left[ 0,T\\right] $ , with probability one.", "The same result is true replacing $\\Phi _{t}^{\\epsilon }(x,\\omega )$ with $\\Phi _{t}(x,\\omega )$ .", "About the convergence of $\\psi ^{\\epsilon }$ , we shall use the following fact: for a.e.", "$\\omega $ , possibly passing to a subsequence, $\\psi ^{\\epsilon }\\left( t,\\cdot ,\\omega \\right) $ tends to $\\psi \\left( t,\\cdot ,\\omega \\right) $ in $L_{loc}^{m}(\\mathbb {R}^{3})$ and $\\psi ^{\\epsilon }\\left( \\cdot ,\\cdot ,\\omega \\right) $ tends to $\\psi \\left( \\cdot ,\\cdot ,\\omega \\right) $ in $L_{loc}^{m}([0,T]\\times \\mathbb {R}^{3})$ , for every finite $m$ .", "Indeed, first notice that $\\left( D\\Phi _{s}^{\\epsilon }((\\Phi _{s}^{\\epsilon })^{-1}\\left( x\\right) )\\right) ^{-1}=D(\\Phi _{s}^{\\epsilon })^{-1}\\left( x\\right) $ , so that $\\psi ^{\\epsilon }\\left( s,x\\right) =D(\\Phi _{s}^{\\epsilon })^{-1}\\left(x\\right) \\varphi \\left( (\\Phi _{s}^{\\epsilon })^{-1}\\left( x\\right) \\right) .$ By Theorem REF and standard arguments like in the proof of Lemma REF , Step 2, $\\psi ^{\\epsilon }\\left( t,\\cdot ,\\cdot \\right) $ converges in $L^{m}(B_{R}\\times \\Omega )$ for every finite $m$ and every $R>0$ ; this implies that, for a.e.", "$\\omega $ , possibly passing to a subsequence, for a.e.", "$\\omega $ it converges in $L^{m}(B_{R})$ for every finite $m$ and every $R>0$ ; by a diagonal procedure we can choose this subsequence independently of $m$ and $R$ .", "The proof of the convergence of $\\psi ^{\\epsilon }\\left( \\cdot ,\\cdot ,\\omega \\right) $ in $L_{loc}^{m}([0,T]\\times \\mathbb {R}^{3})$ is similar.", "Step 4 (passage to the limit).", "Now we fix $t>0$ and let $\\epsilon $ go to 0 in formula (REF ).", "We will prove we obtain in the limit $\\int _{\\mathbb {R}^{3}}\\left( D\\Phi _{t}(x)\\right) ^{-1}\\mathbf {B}(t,\\Phi _{t}(x))\\varphi \\left( x\\right) dx=0 $ which implies $\\mathbf {B}=0$ as already explained above.", "The term on the left-hand-side of (REF ) converges, possibly up to subsequences, to the one on the left-hand-side of (REF ).", "Indeed, by the change variable $x=\\Phi _{t}^{\\epsilon }(x^{\\prime })$ and the support result of the previous step we have (recall that $\\overline{R}$ is random but independent of $\\epsilon >0$ ) $\\int _{\\mathbb {R}^{3}}\\left( D\\Phi _{t}^{\\epsilon }(x^{\\prime })\\right) ^{-1}\\mathbf {B}^{\\epsilon }(t,\\Phi _{t}^{\\epsilon }(x^{\\prime }))\\varphi \\left(x^{\\prime }\\right) dx^{\\prime } &=&\\int _{\\mathbb {R}^{3}}\\mathbf {B}^{\\epsilon }(t,x)\\psi ^{\\epsilon }\\left( t,x\\right) dx \\\\&=&\\int _{B\\left( 0,\\overline{R}\\right) }\\mathbf {B}^{\\epsilon }(t,x)\\psi ^{\\epsilon }\\left( t,x\\right) dx$ With probability one, for every $R>0$ the function $\\mathbf {B}^{\\epsilon }(t,x)$ converges to $\\mathbf {B}(t,x)$ uniformly on $\\left[ 0,T\\right]\\times B\\left( 0,R\\right) $ , by classical mollifiers arguments.", "We have seen in Step 3 that, for a.e.", "$\\omega $ , possibly passing to a subsequence, $\\psi ^{\\epsilon }\\left( t,\\cdot ,\\omega \\right) $ tends to $\\psi \\left(t,\\cdot ,\\omega \\right) $ in $L_{loc}^{1}(\\mathbb {R}^{3})$ .", "Hence we may pass to the limit in $\\int _{B\\left( 0,\\overline{R}\\right) }\\mathbf {B}^{\\epsilon }(t,x)\\psi ^{\\epsilon }\\left( t,x\\right) dx$ , for a.e.", "$\\omega $ ; the limit is $\\int _{B\\left( 0,\\overline{R}\\right) }\\mathbf {B}(t,x)\\psi \\left( t,x\\right) dx$ which gives the left-hand-side of (REF ) by going backwards with the same computations.", "Let us consider now the term on the right-hand-side of (REF ); we want to prove that it converges to zero.", "It is not difficult to show that, for a.e.", "$\\omega $ , both $(v^{j}B^{i})^{\\epsilon }$ and $v^{\\epsilon ,j}B^{\\epsilon ,i}$ converge to $v^{j}B^{i}$ in $C([0,T]\\times \\mathbb {R}^{3})$ (namely, uniformly on compact sets) so $(v^{j}B^{i})^{\\epsilon }-v^{\\epsilon ,j}B^{\\epsilon ,i}$ tends to 0 in that space.", "The term $(D\\psi ^{\\epsilon })_{ij}^{A}(s,x)$ could look problematic at a first view, since it seems to involve the second derivatives of the flow $\\Phi ^{\\epsilon }$ , which are not under control.", "But this is not the case, because we only need the antisymmetric part of the derivative.", "Indeed, differentiating $\\psi ^{\\epsilon }=(D((\\Phi ^{\\epsilon })^{-1}))^{T}\\varphi ((\\Phi ^{\\epsilon })^{-1})$ , we get $(D\\psi ^{\\epsilon })_{ij}=\\sum _{k=1}^{3}\\partial _{j}\\partial _{i}((\\Phi ^{\\epsilon })^{-1})^{k}\\varphi _{k}((\\Phi ^{\\epsilon })^{-1})+\\sum _{k=1}^{3}\\partial _{i}((\\Phi ^{\\epsilon })^{-1})^{k}\\partial _{i}[\\varphi _{k}((\\Phi ^{\\epsilon })^{-1})].$ The possible problem is only with the first addend.", "Its antisymmetric part however is $\\sum _{k=1}^{3}\\partial _{j}\\partial _{i}((\\Phi ^{\\epsilon })^{-1})^{k}\\varphi _{k}((\\Phi ^{\\epsilon })^{-1})-\\sum _{k=1}^{3}\\partial _{i}\\partial _{j}((\\Phi ^{\\epsilon })^{-1})^{k}\\varphi _{k}((\\Phi ^{\\epsilon })^{-1})=0.$ So $(D\\psi ^{\\epsilon })^{A}$ involves only powers of first derivatives of $\\Phi ^{\\epsilon }$ .", "Hence, using again arguments like in proof of Lemma REF , up to subsequences, $(D\\psi ^{\\epsilon })^{A}$ converges to $(D\\psi )^{A}$ in $L_{loc}^{1}([0,T]\\times B_{R})$ , with probability one.", "Using again the uniform random support of $\\psi ^{\\epsilon } $ we see that the term on the right-hand-side of (REF ), equal to $-\\sum _{i,j=1}^{3}\\int _{0}^{t}\\int _{B\\left( 0,\\overline{R}\\right) }\\left[(v^{j}B^{i})^{\\epsilon }-v^{\\epsilon ,j}B^{\\epsilon ,i}\\right] (s,x)(D\\psi ^{\\epsilon })_{ij}^{A}(s,x)dxds$ converges to zero, with probability one.", "Then (REF ) is proved and the proof is complete." ] ]
1403.0022
[ [ "A Stellar Census of the Tucana-Horologium Moving Group" ], [ "Abstract We report the selection and spectroscopic confirmation of 129 new late-type (K3-M6) members of the Tuc-Hor moving group, a nearby (~40 pc), young (~40 Myr) population of comoving stars.", "We also report observations for 13/17 known Tuc-Hor members in this spectral type range, and that 62 additional candidates are likely to be unassociated field stars; the confirmation frequency for new candidates is therefore 129/191 = 67%.", "We have used RVs, Halpha emission, and Li6708 absorption to distinguish contaminants and bona fide members.", "Our expanded census of Tuc-Hor increases the known population by a factor of ~3 in total and by a factor of ~8 for members with SpT>K3, but even so, the K-M dwarf population of Tuc-Hor is still markedly incomplete.", "The spatial distribution of members appears to trace a 2D sheet, with a broad distribution in X and Y, but a very narrow distribution (+/-5 pc) in Z.", "The corresponding velocity distribution is very small, with a scatter of +/-1.1 km/s about the mean UVW velocity.", "We also show that the isochronal age (20--30 Myr) and the lithium depletion age (40 Myr) disagree, following a trend seen in other PMS populations.", "The Halpha emission follows a trend of increasing EW with later SpT, as seen for young clusters.", "We find that members have been depleted of lithium for spectral types of K7.0-M4.5.", "Finally, our purely kinematic and color-magnitude selection procedure allows us to test the efficiency and completeness for activity-based selection of young stars.", "We find that 60% of K-M dwarfs in Tuc-Hor do not have ROSAT counterparts and would be omitted in Xray selected samples.", "GALEX UV-selected samples using a previously suggested criterion for youth achieve completeness of 77% and purity of 78%.", "We suggest new selection criteria that yield >95% completeness for ~40 Myr populations.", "(Abridged)" ], [ "Introduction", "Over the past 20 years, co-moving associations of young stars ($\\tau 100$ Myr) have been identified among the nearby field population [23], [80], [39], [76], [89], [88].", "These moving groups represent the dispersed remnants of coeval stellar populations [81] that apparently formed in the same star-forming region, and might be older analogs to unbound associations like Taurus-Auriga and Upper Scorpius [27].", "Most of these populations are associated with well-known isolated classical T Tauri stars (such as the TW Hya Association, or TWA) or debris disk hosts (such as the $\\beta $ Pic moving group, or BPMG), an association which provided the first indication that they were post-T Tauri associations.", "Surveys to identify active young stars within the solar neighborhood ($d 50$ pc) have subsequently identified several additional populations, including the AB Dor, Carina-Near, Hercules-Lyra, and Tucana-Horologium associations [88], [86], [78], [15].", "Young moving groups ($\\tau \\sim 8$ –300 Myr) provide a critical link between star-forming regions (which can be recognized by the presence of molecular cloud material and the preponderance of protoplanetary disk hosts) and the old field population.", "The close proximity of these young moving groups makes them especially advantageous for the study of circumstellar processes that depend on angular resolution (such as multiple star formation; [9], [8], [17]) and searches for extrasolar planets [40], [29].", "The low distances also result in additional sensitivity for flux-limited studies of disks [33], [7], [50] and the (sub)stellar mass function [19], [36], [45], [68].", "Finally, these stellar populations record a key epoch of planet formation, representing the end of giant planet formation and the onset of terrestrial planet assembly.", "The Tucana-Horologium moving group (hereafter Tuc-Hor) is a particularly intriguing stellar population.", "Its members were first identified separately as the Tucana association and the Horologium association [76], [89], but were subsequently recognized to represent a single comoving population with an age of $\\tau \\sim 30$ Myr.", "Tuc-Hor is host to at least 12 BAF-type stars [88], [78], similar in size to the BPMG (also with 12 BAF-type stars) and much larger than the TWA (with a single BAF-type member).", "Tuc-Hor is likely one of the largest young stellar populations within $d < 100$ pc, making it a robust site for measuring population statistics (the IMF, multiplicity properties, disk frequencies, and activity rates).", "As an “intermediate age” moving group, Tuc-Hor represents a key calibration for age indicators like H$\\alpha $ emission, UV and X-ray excesses, rotational velocities, and age-dependent spectral features like $Li_{6708}$ , and $Na_{8189}$ .", "If these indicators can be robustly calibrated for the age of Tuc-Hor, then their measurement for stars unaffiliated with any moving group can distinguish analogs to stars in star-forming regions ($\\tau = 1$ –20 Myr) from the young ($\\tau = 50$ –300 Myr) field population [67] and old field stars [56].", "The current census of Tuc-Hor is largely restricted to the higher-mass (AFGK-type) stars, which can be selected via all-sky activity indicators like ROSAT and confirmed with high-quality proper motions from Hipparcos.", "There are $<$ 10 spectroscopically confirmed M dwarfs in Tuc-Hor, even though these stars represent the peak of the IMF and thus should comprise the majority of the population by both number and mass.", "The reason for this paucity is straightforward.", "M dwarfs are fainter both optically and in the Xray/UV, so they have been more difficult to mine out of all-sky surveys.", "[37] and [59] have begun to identify significant samples of low-mass candidate members, based on proper motions and ROSAT/GALEX excesses, but they spectroscopically confirmed only a handful of late-type members.", "In this paper, we present the discovery and spectroscopic confirmation of 129 new K3–M6 members of the Tuc-Hor moving group, along with the recovery of most known $\\ge $ K3 members, and compute isochronal sequences for several spectroscopic signatures of youth.", "We also use this sample to characterize the age, mass function, spatial and velocity distribution, disk population, and activity of Tuc-Hor members.", "In Section 2, we describe our candidate selection procedures.", "In Section 3, we describe our high-resolution optical spectroscopic observations, and in Section 4, we summarize the analysis methods used to measure each candidate's spectroscopic properties.", "In Section 5, we combine all of the signatures of youth and membership to identify a sample of bona fide moving group members with spectral types mid-K to mid-M, and compare our results to those of previous surveys.", "Finally, in Section 6, we discuss the population statistics of the Tuc-Hor moving group." ], [ "Candidate Selection", "Pre-main sequence stars in a stellar population can be distinguished by three features: common movement through space, overluminosity compared to the zero-age main sequence (ZAMS), and the presence of various spectroscopic, UV/X-ray, or mid-IR signatures of youth.", "For data mining of candidates across large swaths of the sky, the most cost-effective features to select against are the proper motion and the overluminosity, since both can be computed from existing all-sky survey data.", "As we describe below, we have combined multiple surveys in this work, using methods first described in [26].", "Broadband optical-NIR photometry and proper motions also are largely unbiased against activity and disk existence, allowing for robust population statistics in the resulting member census.", "In particular, we did not use any activity criteria (such as Xray or UV emission) to select candidates because one of our primary goals was to test the efficiency and completeness of those selection methods (Sections 5.2 and 6.7).", "In the following subsections, we list the all-sky surveys that contribute to our work, describe the calculation of proper motions, describe the calculation of bolometric fluxes and spectral types, and explain the selection of candidate members of Tuc-Hor.", "We selected our candidates from the entire southern sky ($\\delta < 0^o$ ) between right ascensions of $20^h < \\alpha < 6^h$ , encompassing the entire spatial distribution of previously known members." ], [ "USNO-B1.0", "The USNO-B1.0 survey (USNOB; [43]) is a catalog based on the digitization of photographic survey plates from five epochs.", "For fields north of $\\delta = -30^o$ , these plates are drawn from the two Palomar Observatory Sky Surveys, which observed the entire available sky in the 1950s with photographic B and R plates and the 1990s with photographic B, R, and I plates.", "For fields south of $\\delta = -30^o$ , including most of Tuc-Hor, the corresponding observations were taken by the UK Schmidt telescope in the 1970s-1980s and 1980s-1990s, respectively.", "The approximate detection limits of the USNOB catalog are $B \\sim 20$ , $R \\sim 20$ , and $I \\sim 19$ , and the observations saturate for stars brighter than $R \\sim 11$ .", "The typical astrometric accuracy at each epoch is $\\sim $ 200 mas, dominated by systematic uncertainty due to its uncertain calibration into the International Celestial Reference System (ICRS) via the unpublished USNO YS4.0 catalog (as tested for specific pointings by [26], and verified across the entire sky by [60])." ], [ "2MASS", "The Two-Micron All-Sky Survey [71] observed the entire sky in the J, H, and Ks bands over the interval of 1998–2002.", "Each point on the sky was imaged six times and the coadded total integration time was 7.8 s, yielding 10$\\sigma $ detection limits of $K_s = 14.3$ , $H = 15.1$ , and $J = 15.8$ .", "The typical astrometric accuracy is $\\sim $ 70 mas for any source detected at $S/N > 20$ , as for all the sources considered in this work.", "The absolute astrometry calibration was calculated with respect to stars from Tycho-2; subsequent tests have shown that systematic errors are typically $<$ 30 mas [85]." ], [ "DENIS", "The Deep Near Infrared Survey of the Southern Sky [16] observed $\\sim $ 84% of the southern sky (with some gaps) in the optical (with a Gunn $i$ filter) and the near-infrared (with $J$ , and $K_s$ filters) during 1995-2001.", "It observed 3662 strips that each spanned 30o in declination and 12  in right ascension, reaching limiting magnitudes of $i=18.5$ , $J=16.5$ , and $K_s=14$ and saturating at $i=9.8$ , $J=7.5$ , and $K_s=6.0$ .", "The photometry is nominally redundant with 2MASS (for $J$ and $K_s$ ), SDSS (for SDSS $i^{\\prime }$ ) and USNOB (for $I$ , though the DENIS $i$ magnitude is more precise than the USNOB $I2$ magnitude), but is still useful for reducing stochastic errors and accounting for potential variability.", "Our experiments indicate that some strips from DENIS have significant systematic discrepancies in the tie-in to the ICRS, so we do not use DENIS astrometry in our proper motion calculations." ], [ "SDSS", "The Sloan Digital Sky Survey [83] is an ongoing deep optical imaging and spectroscopic survey of the northern galactic cap and selected regions of the southern cap.", "The most recent data release [1] reported imaging results in five filters (ugriz) for 14,555 deg$^2$ .", "The 10$\\sigma $ detection limits in each filter are $u = 22.0$ , $g = 22.2$ , $r = 22.2$ , $i = 21.3$ , and $z = 20.5$ ; the saturation limit in all filters is $m \\sim 14$ .", "The typical absolute astrometric accuracy is $\\sim $ 45 mas rms for sources brighter than $r \\sim 20$ , declining to 100 mas at $r \\sim 22$ [49]; absolute astrometry was calibrated with respect to stars from UCAC2, which is calibrated to the ICRS.", "The default astrometry reported by the SDSS catalog is the r-band measurement, not the average of all five filters.", "However, the residuals for each filter (with respect to the default value) are available, so we used these residuals to construct a weighted mean value for our analysis.", "We adopted a conservative saturation limit of $m=15$ in all filters, even though the nominal saturation limit is $m=14$ , because we found that many photometric measurements were mildly saturated for $14 < m < 14.5$ .", "We also neglect measurements which are flagged by the SDSS database as having one or more saturated pixels.", "Finally, we removed all sources which did not have at least one measurement above the nominal 10$\\sigma $ detection limits.", "Any moving group members fainter than this limit will not have counterparts in other catalogs, and the presence of excess sources can complicate attempts to match counterparts between data sets." ], [ "UCAC3", "The astrometric quality of the above surveys could be compromised for bright, saturated stars, so proper motions calculated from those observations could be unreliable.", "Many of the brightest stars are saturated in all epochs, so we have no astrometry with which to compute proper motions.", "We have addressed this problem by adopting proper motions for bright stars as measured by the Third USNO CCD Astrograph Catalog [84].", "UCAC3 was compiled from a large number of photographic sky surveys and a complete reimaging of the sky by the U.S.", "Naval Observatory Twin Astrograph.", "UCAC3 extends to $R = 16$ , though the proper motion errors become quite large at $R > 13$ –14.", "The typical errors in the reported proper motions are $\\sim $ 1–3 mas/yr down to $R=12$ and $\\sim $ 6 mas/yr down to $R =16$ ." ], [ "Proper Motions", "Many recent efforts have employed various combinations of all-sky surveys in order to systematically measure proper motions of both clusters and field stars; USNOB is itself a product of such analysis, [21] combined SDSS and USNOB, and the PPMXL survey [60] combined 2MASS and USNOB.", "However, there has been no systematic attempt to combine USNOB, 2MASS, and SDSS using a single algorithm to produce a single unified set of proper motions.", "All catalogs are calibrated into the ICRS, so in principle their coordinate lists can be adopted without any need for further calibration.", "In practice, this choice incurs a systematic error of $\\sim $ 200 mas on each USNOB epoch (as described above), though 2MASS and SDSS appear highly consistent.", "We obtained the astrometry for all sources from the Vizier archive using the IDL routine queryvizier.pro, and then combined the coordinate lists for each source using a weighted least-squares fit.", "Our algorithm tested the goodness of each fit and rejected all outliers at $> 3 \\sigma $ ; most of these outliers were found in the photographic survey data, not in 2MASS or SDSS, due to the heavy weight assigned to the modern CCD-based epochs.", "We find that the addition of at least one high-quality modern data point from 2MASS or SDSS reduces the uncertainties on a given proper motion by a factor of $\\sim $ 2 compared to standard USNOB proper motions; the use of a sigma clip also substantially reduces the number of extremely erroneous measurements, since USNOB used no sigma clip.", "As we showed in [26], this procedure led to a recovery rate of $>$ 90% for the known members of Praesepe, and hence any population mined out of this dataset should be nearly complete.", "Finally, we supplemented our measurements for bright stars with the proper motions from the UCAC3 catalog, which typically yields more precise proper motions for $R 12$ mag.", "In any case where the reported uncertainty from UCAC3 was less than the inferred uncertainty in our measurement, we simply adopted its measurement instead.", "A comparison of these measurements shows that they are consistent within the uncertainties.", "Combined with the $>$ 90% yield we found for recovering open cluster members [26], our proper motions appear broadly robust." ], [ "Photometry and SED Fits", "Most population membership surveys select candidates based on agreement with the expected isochrone sequence in one or more color-magnitude diagrams.", "However, when many photometry points are available (i.e., 16 points for 2MASS+SDSS+DENIS+USNOB), then this procedure is unwieldy and neglects important covariances in the data.", "A superior method is to fit all available data with a model drawn from a grid of template SEDs, using all available photometry to find the best-fit stellar parameters ($T_{eff}$ and $f_{bol}$ , or equivalently SpT and $m_{bol}$ ).", "If a star is assumed to fall on the main sequence, then a comparison of the inferred $m_{bol}$ with the expected $M_{bol}$ also directly yields a spectrophotometric distance modulus $DM_{phot}$ .", "We describe the motivation and details in [26].", "Our definition of the main sequence was tied to the Praesepe open cluster sequence, so the uncertainty in the corresponding value of $DM_{phot}$ is most likely dominated by the uncertainty in the distance modulus for Praesepe ($\\sim $ 0.1 mag) and variations in the photometric calibration of USNOB for the northern and southern skies ($\\sim $ 0.2–0.3 mag), since our USNOB-2MASS SEDs were bootstrapped from SDSS-2MASS SEDS for Praesepe.", "The corresponding uncertainty is therefore $\\pm $ 0.2–0.3 mag in $M_{bol}$ or $\\pm $ 0.2 subclasses in spectral type.", "As for the astrometry, we obtained the photometry for all sources using queryvizier.pro and then computed the $\\chi ^2$ goodness of fit against a set of 541 main-sequence spectral type templates spanning B8.0-L5.0 in steps of 0.1 subclass.", "We describe the SED library and its construction in more detail in [26] and in Bowsher et al.", "(in prep).", "We rejected potentially erroneous observations by identifying any measurement that disagreed with the best-fit SED by more than 3$\\sigma $ , where $\\sigma $ is the photometric error reported by the CCD-based surveys or is adopted to be $\\pm $ 0.25 mag for USNOB, and then calculating a new fit.", "The uncertainties in the spectral type and distance modulus were estimated from the $\\Delta \\chi ^2 = 1$ interval of the $\\chi ^2$ fit for each object.", "The distribution of reduced $\\chi ^2$ values for our fits had too many sources with $\\chi _{\\nu }^2<1$ and $\\chi _{\\nu }^2>1$ , and too few with $\\chi _{\\nu }^2 \\sim 1$ .", "Further investigation revealed the source of this discrepancy to be apparent non-Gaussianity of the errors of USNOB photometry.", "While the “typical” uncertainty is indeed $\\pm $ 0.25 mag, this uncertainty actually consists of a stochastic component with $\\sigma < 0.25$ mag and a systematic uncertainty (most likely due to the photometric calibration for individual photographic plates) which can exceed 1 magnitude.", "A global recalibration of USNOB would likely reduce this systematic uncertainty, but is beyond the scope of the current work.", "As we discuss in Section 4.1, a comparison of spectral types from SED fits and from spectroscopy yields excellent agreement, with dispersion and systematic offsets of $$ 1 subclass for bona fide members across the K-M spectral type range.", "Given a typical uncertainty of at least 0.5 subclass for most spectroscopic spectral types (including those of our standard stars), then the observed dispersion in the relation indicates that photometric SpTs can be measured with similar accuracy.", "However, the HR diagram that we show in Figure 1 suggests that systematic errors remain for some spectral type ranges.", "In particular, it appears that M0-M1 stars might be systematically pulled to a classification of $\\sim $ K7.5, perhaps due to an error in the SED grid.", "This error was not seen in northern populations (e.g., [26]), but those fields also had SDSS data that dominated the fits.", "Without SDSS data, then USNOB and 2MASS colors become more significant in the fits.", "We are producing updated SED templates for use in future surveys, but since we selected our input sample for this study with the old templates, then we have retained them for the present analysis.", "As we discuss further in Section 6.1, this systematic error could account for the absence of M0-M2 members within our sample." ], [ "Selection Criteria", "After computing proper motions and SED fits, our input set consisted of $1.2 \\times 10^8$ sources spanning 8300 deg$^2$ to consider as potential cluster members.", "However, almost all have proper motions inconsistent with membership or are too faint and blue to sit on the moving group sequence, so this number can be efficiently winnowed down.", "Unlike for compact clusters, moving group members span a large range of distances and a large area of the sky, and hence do not share a single proper motion.", "The kinematic selection must instead be made against the projection of the moving group's $UVW$ space velocity onto the plane of the sky at each source's position.", "Furthermore, the unknown distance means that the magnitude of the proper motion is a free parameter.", "Our selection of candidates from this input set can be divided into two main stages.", "First, for each source we computed the angle of the expected proper motion, given the $UVW$ space motion of Tuc-Hor ($U = -9.9 \\pm 1.5$ km/s, $V = -20.9 \\pm 0.8$ km/s; $W = -1.4 \\pm 0.9$ km/s; [78]).", "We then found the magnitude of the proper motion (i.e., the assumed distance) which minimizes the difference between the observed proper motion and the expected proper motion.", "We were then left with two quantities for each star: the discrepancy $\\Delta $ (in mas/yr) between the observed proper motion and the best-fit proper motion it needed to have if it were a member, and the corresponding best-fit kinematic distance modulus $DM_{kin}$ corresponding to the magnitude of that best-fit proper motion vector.", "This method is similar to that used by [32] and [63].", "We identified 1813 sources as kinematic candidates where the observed and expected proper motions agreed within 3$\\sigma $ ($\\Delta / \\sigma _{\\mu } < 3$ , where $\\sigma _{\\mu }$ is the observational uncertainty in the proper motion) or the total magnitude of the discrepancy was $\\Delta < 10$ mas/yr, as well as requiring the kinematic distance to be $d < 80$ pc ($DM_{kin} \\le 4.5$ ).", "Second, for these 1813 kinematic candidates we computed the difference between the kinematic distance modulus $DM_{kin}$ and the spectrophotometric distance modulus $DM_{phot}$ to test whether, if an object were a member with the appropriate $DM_{kin}$ , then it would sit at an appropriate height above the ZAMS.", "For our initial reconnaissance of this sample, we used the known members of Tuc-Hor [78] to set the criterion for rejection at $DM_{kin}-DM_{phot} \\le 0.0$ .", "We also rejected any star with $DM_{kin}-DM_{phot} \\ge 4.0$ since it would be too high above the ZAMS to be a member, and hence most likely is a field interloper with spurious $DM_{kin}$ and/or a giant with spurious $DM_{phot}$ .", "We found that most candidates which were ultimately confirmed sat $\\sim $ 0.5–1.0 mag above the main sequence, suggesting that our cuts should only miss a small number of candidates that sit low in the HR diagram.", "These cuts yielded 768 photometric/kinematic candidates.", "Finally, to produce a manageable sample for our observing time, we omitted any sources earlier than K3 (which should have been identified via HIPPARCOS in previous searches, and are heavily contaminated by subgiants) or later than M6 (which are too faint for efficient optical spectroscopy), leaving a total sample of 497 potential low-mass Tuc-Hor members.", "In Figure 1, we illustrate the astrometric and photometric selection procedures for an area of 16 deg$^2$ near the central locus of the known and new Tuc-Hor nenvers.", "In Figure 2, we show a map of our candidates on the sky, including those which we observed and ultimately found to be either bona fide Tuc-Hor members or field star interlopers.", "Within the full 8300 deg$^2$ area, we prioritized those stars which were closest to the known members and which were bright enough to be observed with $\\le $ 15 minute integration times.", "Our survey did not clearly reach a boundary for the distribution of Tuc-Hor members on the sky, and indeed, it is unclear whether even our initial area of 8300 deg$^2$ is sufficient to encompass all Tuc-Hor members, so future reconnaissance of a wider area might be necessary.", "Given the flux limits of the input all-sky surveys ($R 18$ mag, $K_s 14$ mag), then our input sample should have included $\\tau = 30$ Myr Tuc-Hor members down to the substellar boundary ($M = 0.07 M_{\\odot }$ ) at a distance of $d = 80$ pc.", "The flux limit for our optical spectroscopy ($R \\sim 15$ mag) raises this limit to $M = 0.15 M_{\\odot }$ (SpT$\\sim $ M5) at $d = 80$ pc.", "The SACY search of Tuc-Hor [77], [78] found 17 members with spectral types of K3-M0, of which we recovered 15 as candidates.", "Our search missed HD 222259 B because it is a close companion to the G6V star DS Tuc, which affected its measurements in the input all-sky catalogs.", "CD-35 1167 would have been selected by our photometric selection procedures, but it has no UCAC3 counterpart and was too bright for our procedure to calculate a new proper motion, so our astrometric selection procedure missed it.", "We therefore estimate that our candidate list is $88^{+4}_{-12}\\%$ complete in this spectral type range.", "There are few known members of Tuc-Hor later than M0, so we can not estimate the completeness of our sample, though we discuss the recovery of these members in Section 5.2.", "However, our results for Praesepe and Coma Ber remained $\\ge $ 90% complete to nearly the flux limit of USNO-B1 ($R\\sim 19$ mag), so we do not expect the completeness to differ substantially to the limit of our current Tuc-Hor sample ($R \\sim 15$ mag).", "We identified CD-46 1064, BD-19 1062, and CD-30 2310 as candidates, but did not obtain spectra for them, so for uniformity we will not include them in the analysis summarized in Section 6.", "We obtained a spectrum of CD-35 1167 (which we missed as a candidate), and so we report its measurements and its confirmation as a member, but also will not include it in our analysis.", "We also obtained spectra for the other 12 previously known members in this list and will include them in our analysis.", "Finally, we note that the Columba association is nearly comoving with Tuc-Hor in $UVW$ space and is coincident with the eastern end of Tuc-Hor on the sky, and hence we might expect some confusion between the two populations.", "However, the radial velocities differ by $\\sim $ 4 km/s near the center of the point of overlap ($\\alpha = 60^o$ , $\\delta = -45^o$ ), which can be distinguished at 3$\\sigma $ for most of our candidates.", "Furthermore, the populations are distinct in XYZ space, with most known Columba members falling at $d > 60$ pc ([88], [78].", "There is no clear evidence of a parallel population meeting the RV or XYZ values of Columba (Sections 5.1 and 6.4), though this possibility should be considered after future surveys have more robustly determined the spatial and kinematic distributions off Columba.", "The handful of young stars that we find with $Z < -50$ , which fall closer to the Columba locus, could be preliminary evidence of this overlap." ], [ "Observations and Data Reduction", "Our candidates were selected by their proper motions and photometry, so we require independent measurements in order to confirm their membership in Tuc-Hor.", "The traditional methods for confirmation of young stars in stellar populations are to measure their radial velocity (testing for comovement in the dimension perpendicular to proper motion) and the identification of spectral signatures that can be associated with youth (such as lithium absorption, low-gravity diagnostics like shallow alkali lines, H$\\alpha $ emission, and rapid rotation).", "In both cases, we can obtain the necessary measurements from high-resolution optical spectroscopy.", "Many of the diagnostics of youth have only been firmly calibrated for higher-mass stars [42], so our measurements also yield the first robust calibration of isochronal parameter sequences for mid-K to mid-M dwarfs in Tuc-Hor.", "We obtained high-resolution spectra for our targets in three observing runs in 2012 July, 2012 September, and 2013 February.", "We observed our targets using the Magellan Inamori Kyocera Echelle (MIKE) optical echelle spectrograph on the Clay telescope at Magellan Observatory.", "For all observations we used the 0.7  slit, which yields spectral resolution of $R=35,000$ across a range of $\\lambda = 3350$ -9500Å.", "Since our targets are relatively red, most only had useable signal on the red chip ($\\lambda > 5000$ Å).", "The pixel scale oversamples the resolution with the 0.7  slit, so we observed with 2x binning in the spatial and spectral directions to reduce readout overheads.", "We also observed standard stars nightly from the list of [46], which serve as both RV and spectral type templates, as well as most of the known Tuc-Hor members with SpT$\\ge $ K3, numerous members of the Sco-Cen OB association, and a selection of known members from other nearby young moving groups.", "We reduced the raw spectra using the CarPy pipeline (Kelson 2003)http://code.obs.carnegiescience.edu/mike but used observations of spectrophotometric standard stars to measure and flatten the blaze function due to the uncertain temperature of the flat lamp.", "In order to correct for residual wavelength errors (due to flexure and uneven slit illumination), we then cross-correlated the 7600 Å  telluric A band for each spectrum against a well-exposed spectrum of a telluric standard, solving for the shift (typically $\\sim $ 1 km/s) that places each spectrum into a common wavelength system defined by the atmosphere.", "Finally, we calculated and applied heliocentric radial velocity corrections.", "Multiple observations of dwarf standards suggest a repeatability of $<$ 0.5 km/s for our observations, as do multi-night observations of the young star 1SWASP J140747.93-394542.6 that will be reported in a future publication (M. Kenworthy et al., in preparation).", "In Table 1, we list the epochs and exposure times for all of our MIKE observations of known or candidate Tuc-Hor members, as well as the $S/N$ for each spectrum at 6600 Å.", "We also list all other relevant measurements used in target selection: proper motion, SED-fit SpT and bolometric flux, photometric and kinematic distance modulus, and the proper motion residual $\\Delta $ ." ], [ "Spectral Types", "Our spectra provide a valuable check of the SpTs estimated from SED fitting (Section 2.3).", "Our SED SpTs broadly match known spectroscopic SpTs with a dispersion of 0.7 subclasses, but erroneous input photometry could significantly bias the results for some individual stars.", "To measure spectral types for our targets, we computed the $\\chi ^2$ goodness of fit with respect to standard stars of known spectral type.", "We applied this analysis using both primary standards (well-studied dwarfs with known RV and SpT, drawn from the sample of [46]) and secondary standards (67 candidate Tuc-Hor members for which spectroscopic SpTs are available in the literature).", "The absolute value of the $\\chi ^2$ statistic can not be easily interpreted because each pixel of spectrum conveys a different amount of information about a star (with much distinct information from temperature-sensitive lines or molecular bands, and little distinct information from temperature-independent lines or from continuum).", "However, the relative $\\chi ^2$ values can be used to determine which standard spectrum best matches a given science target.", "Each science target spectrum was compared to all standard star spectra.", "In this procedure, the standard spectra were shifted to the same radial velocity (using RV measurements from Section 4.2), the spectrum with narrower lines was convolved with a Gaussian kernel to match the more broadened spectrum (based on $v_{rot}$ measurements from Section 4.2), and then the reduced $\\chi ^2$ values were computed for each of seven orders ($\\lambda = 6100$ Å, 6700Å, 7000Å, 7100Å, 7400Å, 7900Å, and 8400Å).", "We adjusted the assumed uncertainties for each order so that the best-fitting standard yielded reduced $\\chi ^2 = 1$ , recomputed all fits, and then averaged the reduced $\\chi ^2$ values across all seven orders to find the standard star that best fit the entire spectrum.", "Finally, since the standard stars are quantized by 0.5 or 1.0 subclasses, we fit the set of reduced $\\chi ^2$ values (as a function of standard-star SpT) with a low-order polynomial in order to find the true minimum in the relation (indicating the ideal best-fit SpT).", "In Figure 3, we demonstrate the results of this fit for two stars in our sample.", "The known Tuc-Hor member CT Tuc was assessed to be an M0V star by Zuckerman & Song (2004), and our SED fit yielded a SpT of K7.6.", "Our spectroscopic analysis finds that the best-fit standard is an M0 star (HIP 1910, another Tuc-Hor member), while a polynomial fit of the reduced $\\chi ^2$ surface yields a best-fit SpT of K5.7.", "Similarly, 2MASS J02505959-3409050 (a previously unidentified candidate member) is found to be an M3.8 star in our SED fit, whereas our spectral analysis finds it is best fit by an M3.5 standard star and the polynomial fit of the reduced $\\chi ^2$ surface yields a best-fit SpT of M3.7.", "In Figure 4, we compare our spectroscopic SpTs with the SED SpTs computed in Section 2.3, and show that the two results broadly agree for $\\ge $ M0 dwarfs.", "We find that stars which appear spectroscopically to be K3-K7 dwarfs are fit with SED SpTs which are systematically $\\sim $ 1.5 subclasses later.", "Mid-late K dwarfs are defined in various ambiguous ways (with variable usage of the K6/K8/K9 types) and span a wide range of $T_{eff}$ , so this systematic uncertainty is not unexpected; until more independent spectroscopic studies of these stars are available, we can not determine whether this systematic offset results from the color-SpT relations that we used to compute SED fits or from our choice of spectroscopic standards.", "We also found that one Tuc-Hor member (2MASS J20474501-3635409) has a best-fit spectral type of G7.4, substantially earlier than the SED-fit spectral type of K4.3.", "This star is a very rapid rotator ($\\sigma _{vsin(i)} = 106$ km/s), so a spectral classification from spectral line strengths is highly uncertain; the SED-fit spectral type is likely to be more valid, so we retain it in our K3–M6 sample.", "In Table 2, we report the best-fitting polynomial-fit SpTs for each candidate Tuc-Hor member." ], [ "RVs, Rotational Broadening, and SB2s", "We measured radial velocities for our targets using broadening function deconvolution [61]; as Rucinski described, broadening functions have a flatter base than cross-correlations and are less susceptible to effects like “peak pulling” for spectroscopic binaries.", "S. Rucinski distributes an IDL pipeline that is designed to conduct BF for any input spectrum, and we adopted this pipeline as written.http://www.astro.utoronto.ca/~rucinski/ For each order of each spectrum, we used the BFD pipeline to calculate the broadening function with respect to a bright RV standard star that best matches the target star's spectral type (Sections 2.3 and 5.1).", "We then fit the peak of the broadening function with a Gaussian function in order to measure the RV in that order, and measured the average RV for that spectrum by calculating a mean of all orders with $S/N > 7$ at the order midpoint.", "We observed some dwarf templates and young stars multiple times, and we found that the typical standard deviation across multiple epochs was $\\sigma \\sim 0.4$ km/s, indicating that any systematic noise floor falls at or below this limit.", "Finally, we computed the rotational or instrumental broadening (the standard deviation $\\sigma $ of the Gaussian fits) by calculating the mean value of $\\sigma $ for all Gaussian fits across the same orders.", "The spectral resolution of $R=35000$ corresponds to a minimum detectable broadening of $\\sigma \\sim $ 4.5 km/s.", "Finally, we found six SB2s and three SB3s among the 206 candidates that we observed.", "We fit these targets with a two- and three-component versions of our BFD pipeline, and determined separate radial and rotational velocities for each component.", "We also fit the ratio of the cross-correlation areas (a proxy for flux ratio) as a function of wavelength and used a linear fit to estimate the flux ratio at 7600Å  (the SDSS $i^{\\prime }$ filter).", "This process is described in more detail in [28] and [31].", "For the SB3s, we assumed that blueshifted and redshifted components were in a close pair, while the intermediate-velocity component was likely the singleton tertiary; the velocity of this tertiary was then adopted as the best available estimate of the system velocity.", "For the SB2s, we used the flux ratio to assess an approximate mass ratio (and hence ratio of RV amplitudes) using the 30 Myr models of [3], and then combined the ratio of RV amplitudes with the total RV difference in order to estimate the system velocity.", "These system velocities are more uncertain due to the significant systematic uncertainties in pre-main sequence evolutionary models (e.g., Hillenbrand & White 2004), but based on the members which can be independently confirmed from lithium or H$\\alpha $ (and hence should be comoving with Tuc-Hor), we estimate that they should be reliable to within $\\pm $ 2 km/s.", "We list the stellar or system velocities, the discrepancy ($\\Delta v_{rad}$ ) compared to the expected values for Tuc-Hor members, and the rotational broadening values in Table 2, and report the properties of all SB2s in Table 3 and SB3s in Table 4.", "In Figure 5, we plot the difference $\\Delta v_{rad}$ between the observed RV and the expected RV (for a Tuc-Hor member at that position on the sky) as a function of spectral type; there is a clear excess at $\\Delta v = 0$ km/s, denoting the Tuc-Hor population.", "Some outliers can be identified as young stars via age diagnostics and hence are likely SB1s.", "Three apparently comoving stars appear old according to those same age diagnostics, and hence are likely field interlopers that happen to be comoving with Tuc-Hor." ], [ "H$\\alpha $ and Lithium", "Most young stars with SpT $\\ge $ M0 show the Balmer series in emission, and even earlier-type stars often show filled-in absorption lines (due to activity) or occasionally very broad emission (if they are still accreting from a disk).", "Due to the lack of other significant lines in this region of the spectrum, measurement of $EW[H\\alpha ]$ is quite straightforward for the vast majority of cases; we simply fit the absorption or emission line with a single Gaussian.", "In spectra where the morphology was complicated or $EW[H\\alpha ] \\sim 0$ , we instead measured the EW by setting the continuum level using sidebands spanning 2–5 Å  on either side, then summing the flux across the H$\\alpha $ line to directly determine the excess or deficit.", "We also used this procedure to measure the total H$\\alpha $ emission for all SB2s and SB3s, as the lines were generally too blended to confidently disentangle.", "Another key indicator of youth is the lithium line at 6708 Å, as lithium is rapidly burned at the base of the convective envelopes of late-K and M dwarfs as they age.", "At an age of $\\tau \\sim 30$ Myr, lithium should be completely depleted for stars of SpT K7–M4, but not yet burned for earlier and later types [54], [14].", "The measurement of $EW[Li_{6708}]$ is more complicated than for H$\\alpha $ , because it is blended with a weak Fe I line for G-K stars and has several other features in close proximity; for M dwarfs, the spectrum is quite complex.", "We therefore measured $EW[Li]$ by fitting that order of each spectrum with the dwarf template most closely resembling it (Section 4.1), then measuring the relative flux deficit for the science target within $\\pm $ 1.0Å  of the expected wavelength.", "Only one SB (2MASS J05332558-5117131) showed lithium absorption in its spectrum, and it was only detectable at the expected wavelength for the primary star, so there was no need to conduct multiple-line fits to determine individual component line strengths.", "For 4 rapid rotators, we measured the equivalent widths manually using the IRAF task splot.", "We found that when K stars otherwise appeared old (with discrepant RVs and no H$\\alpha $ emission), then the mean equivalent width was $EW[Li] = 3$ mÅ, with a standard deviation of 16 mÅ.", "Any measurement with $EW[Li] > 50$ mA is significant at a confidence level of 3$\\sigma $ , and hence can be regarded as a confident detection.", "For M1–M3 stars (which we do not expect to host lithium at this age), the mean equivalent width was $EW[Li] = -1$ mÅ, with a standard deviation of 23 mÅ.", "The corresponding 3$\\sigma $ limit is therefore $EW[Li] > 70$ mÅ.", "We report our measurements for H$\\alpha $ and $Li_{6708}$ in Table 2, and plot the equivalent widths as a function of (SED-fit) spectral type in Figures 6 and 7." ], [ "New Members", "Synthesizing our observations into a unified membership census is a complicated task, because any one measurement could yield a false positive or negative.", "Some field stars will be comoving in radial velocity by chance, and most short-period binaries among the bona fide Tuc-Hor members will not appear comoving in any single-epoch spectrum.", "Also, activity signatures show wide variations in any single-aged population [75], though some show a lower bound that allows for dispositive rejection of nonmembers.", "The most conclusively affirmative or dispositive measurement is the presence or absence of lithium, but this is only valid for a restricted range of spectral types.", "We use three criteria to select members of Tuc-Hor and reject likely field interlopers.", "In order of precedence, these criteria are: Lithium: For the assumed age of Tuc-Hor ($\\tau \\sim 20$ –50 Myr), lithium should be depleted in the atmospheres of stars with spectral types $$ K7 and $$ M5 [42].", "However, lithium is depleted across the entire spectral type range of our sample (K3–M6) by the age of $\\sim $ 125 Myr (as seen in the Pleiades and AB Dor; [73], [42]).", "We therefore use the presence of lithium (with $EW[Li] > 100$ mÅ) as a youth indicator across our entire spectral type range (confirming 34 members).", "Allowing for spectral type uncertainties of $\\sim $ 1–2 subclass, then we also use the absence of lithium (with $EW[Li] < 100$ mÅ) as a nonyouth indicator for spectral types $\\le $ K4 or $\\ge $ M6 (rejecting 19 $\\le $ K4 field interlopers).", "$H\\alpha $ emission: Old main sequence stars exhibit a wide range of activity levels [22], [82], and hence the presence of strong H$\\alpha $ emission can not be used as a positive criterion for determining membership.", "However, young stars exhibit a lower bound on their H$\\alpha $ emission as a function of spectral type, and this boundary has been well-determined for the similar-aged clusters IC 2602 and IC 2391 [72].", "We use that lower bound (which we show in Figure 6) as a nonyouth indicator (rejecting another 23 field interlopers).", "Radial velocities: By definition, young moving groups are comoving with a very small velocity dispersion ($\\sigma \\sim $ 1 km/s for TWA; [38]), and hence the radial velocities of Tuc-Hor members should correlate very well with the projection of the group $UVW$ velocity into the line of sight.", "However, short-period binaries could have large velocity discrepancies due to orbital motions.", "SB1s in particular are impossible to distinguish from non-members in single-epoch spectroscopy.", "We identify candidates as members if they agree with the expected velocity of a Tuc-Hor member to within $\\pm 3 \\sigma $ or $\\pm 3$ km/s (confirming 108 members).", "We label all remaining stars as likely non-members or SB1s (rejecting 20 likely field interlopers).", "Distinguishing the SB1s from the bona fide field stars will require additional RV measurements in the future to test for the variations denoting orbital motion.", "As we summarize in Table 2, these three criteria identify 129 new Tuc-Hor members and recover 13 previously known members, while rejecting 42 confirmed field stars (based on spectroscopic youth indicators) and 20 likely field stars or SB1s (based on RVs).", "The new member yield for our kinematic selection process is therefore $129/191 = 67\\%$ .", "The overlap between our selection criteria provides a check on their validity.", "Of the 19 interlopers rejected by Li, all are also rejected by H$\\alpha $ and only one has an RV consistent with membership.", "Conversely, of the 34 members confirmed by Li, none would be rejected by H$\\alpha $ and 5 would be rejected by RVs.", "Of the 23 interlopers rejected by H$\\alpha $ , only 4 have RVs consistent with membership.", "The nine objects with conflicting indicators should be observed in more detail to confirm their nature, as should the 19 objects which were rejected by their RVs.", "Given a short-period binary frequency of $F \\sim 10 \\%$ for $a < 2$ AU [53], then our 142 members should be matched by $\\sim $ 14 SBs that could have been rejected by their discrepant RVs, much as the 34 objects confirmed by Li include 5 RV-discrepant objects that likely are SB1s." ], [ "Comparison to Previous Surveys of Tuc-Hor", "As we discuss in Section 2.4, there are 17 well-studied members of Tuc-Hor with spectral types $\\ge $ K3 from [77], [78].", "We recovered 15 of them as candidates and re-confirmed 13 with our own spectroscopic observations.", "However, many other candidate members have been proposed that are not yet fully confirmed.", "Most programs have only identified a small number of candidates [25] or have concentrated on higher-mass membership [88], [77], [78], [87].", "The only large studies aimed at the low-mass population of Tuc-Hor were conducted by [37], who suggested 37 late-type stars (drawn from the active M dwarf sample of [57]) to be candidate members, and by [59], who suggested 58 late-type stars (identified based on GALEX excesses) to be candidate members.", "However, each group was only able to confirm one member based on the presence of lithium, and neither obtained RVs.", "Of the 37 late-type stars identified as candidates by [37], we identified 23 to be candidates in our own search as well and obtained spectra for 19 of them, confirming 15 new members and rejecting 4 interlopers.", "Of the 14 candidates that we did not recover, two were rejected for data quality issues: one star had a spurious proper motion in UCAC3, while the other fell among a compact clustering of candidates that we attributed to a bad photographic plate, and hence rejected as a group.", "One candidate fell outside the RA range we considered, and two candidates had a best-fit kinematic distances of $d >$ 80 pc.", "Two candidates failed our photometric cut, falling below the main sequence for the best-fit kinematic distance.", "Finally, eight candidates failed our astrometric selection criterion with large values of $\\Delta $ (6 with $\\Delta = 10$ –20 mas/yr, and 2 with $\\Delta > 20$ mas/yr).", "[64] have already shown that two of these stars are field interlopers, indicating that our more precise proper motions might be better at rejecting field stars that are only moderately discrepant from the proper motion of Tuc-Hor.Simultaneously with our results, Malo et al.", "(2014) reported spectroscopic observations of 30 previously unobserved candidate Tuc-Hor members with $P_{mem} > 25\\%$ , mostly drawn from the Xray-selected sample of Malo et al.", "(2012).", "They confirmed 23 new members (15 of which we also confirm) and reject 7 field stars (3 of which we also reject).", "Our results disagree regarding the membership of 2MASS J02224418-6022476, which they confirm and we reject.", "Of the 58 late-type stars identified as candidates by [59], we identified 35 as candidates and obtained spectra for 29 of them, confirming 26 new members and rejecting 3 interlopers.", "Of the 23 candidates that we did not recover as candidates, 2 were rejected for data quality issues: one had a poor reduced $\\chi ^2$ fit for its proper motion, and another for its SED fit.", "Another 11 candidates failed our astrometric selection criterion with large values of $\\Delta $ (9 with $\\Delta = 10$ –20 mas/yr, and 2 with $\\Delta > 20$ mas/yr).", "Four candidates failed our photometric selection criterion, falling below the main sequence at their nominal kinematic distance by 0.1–0.5 mag.", "Finally, one candidate had a spectral type which was too early ($<$ K3) and five candidates had spectrophotometric distances that were too large ($d \\ge 80$ pc), but otherwise would have been included in our sample.", "The question remains as to why [37] and [59] did not identify our remaining 90 new Tuc-Hor members.", "We ultimately trace this paucity of identified candidates to their choice of input samples, which were based on ROSAT or GALEX.", "Most of our new members are not found in the active M dwarf census of [57], and indeed, the majority do not have ROSAT counterparts at all; only 53 of our 129 previously unidentified members have an X-ray counterpart in the ROSAT All-Sky Survey within $<$ 30.", "Most searches for nearby young stars begin with a pre-selection of candidates that are active in ROSAT [66], [67], [37] or GALEX [18], [68], [58], [59], but given the wide range of activity levels for even very young stars [51], [52] and the limited sensitivity of these surveys for stars at $>$ 25 pc, then this pre-selection must be pursued with great caution or ultimately might need to be abandoned in favor of purely kinematic criteria (as in our program).", "We address the role of activity selection with GALEX data in more detail in Section 6.7." ], [ "Mass Function", "As we show in Figure 2 and discuss further in Section 6.4, it is unlikely that our survey is spatially complete, and it is not clear whether our survey even encompasses the same spatial volume as the surveys that identified the known higher-mass members.", "As a result, any mass function for the region must be considered extremely preliminary.", "However, plotting the mass function of the members discovered to date still can be very illustrative.", "Given the strong evidence that the IMF is universal for most young populations in the solar neighborhood (e.g., [5] and references therein), then a comparison of Tuc-Hor to the standard IMF can demonstrate which mass ranges of members are still incomplete.", "In Figure 8, we show the mass function ($dN/d\\log M$ ) for the known members of Tuc-Hor and our newly-discovered members.", "We inferred masses from the observed spectral types using the mass-$T_{eff}$ relations of [3] and [69] (for $\\le $ 1.4 $M_{\\odot }$ and $>$ 1.4 $M_{\\odot }$ members, respectively), combined with the dwarf $T_{eff}-SpT$ temperature scale of [65].", "We also show the Salpeter IMF [62] for $>$ 1 $M_{\\odot }$ stars and the Chabrier IMF [11] for $\\le $ 1 $M_{\\odot }$ stars, normalized to the observed number of Tuc-Hor members in the two bins straddling 1 $M_{\\odot }$ .", "The paucity of stars at $M = 0.2$ –0.7 $M_{\\odot }$ (SpT$=$ M0–M3) indicates that our survey is indeed incomplete.", "Given the apparently higher rate of completeness for VLM stars in the $M = 0.07$ –0.2 $M_{\\odot }$ range, then mere spatial incompleteness appears unlikely, given that we observed a similar fraction of candidates in both mass ranges.", "We instead speculate that this paucity of early-M stars might result from errors in our SED templates.", "As we discussed in Section 2.3, it appears that M0–M1 stars are being pulled to a spectral type of $\\sim $ K7.5.", "The inferred $m_{bol}$ (which is effectively set by the sum of the observed flux in all filters) would not change, and hence any candidates in the M0–M1 range would appear to be underluminous K7.5 stars and would tend to fall under our photometric selection criterion.", "A future reanalysis of the entire sky with updated SED templates should demonstrate if this is the case, yielding the missing candidates." ], [ "HR Diagram and Isochronal Age", "In Figure 9, we show an HR diagram for the confirmed members, plotting $M_{bol}$ as a function of spectroscopic spectral type.", "The absolute $M_{bol}$ for each star is calculated from the apparent $m_{bol}$ derived from the SED fit and the kinematic distance modulus derived from the proper motion.", "We also show the 10 Myr, 30 Myr, 100 Myr, and 1 Gyr models of [3], as derived with a convective scale length of 1.9 times the pressure scale height.", "We converted the model $T_{eff}$ values to spectral types using the dwarf temperature sequence of [65] for $\\le $ M0 stars and the dwarf sequence of [20] for $\\ge $ M1 stars.", "If we use the young-star temperature sequence of [35] for $\\ge $ M1, the sequence is shifted $\\sim $ 0.5 subclass later for a given mass or temperature.", "The dwarf and young-star temperature scales of [47] both fall between these limits.", "Using the dwarf sequence, we find that the median isochronal age for Tuc-Hor is $\\tau \\sim 20$ Myr.", "For the young-star temperature sequence, the age is shifted to $\\tau \\sim 30$ Myr.", "The results of [47] suggest that the young-star temperature sequence might be more appropriate even for intermediate-age populations like Tuc-Hor, and hence it remains unclear which sequence should be preferred.", "As we discuss below, both of these ages are younger than the lithium depletion boundary age of $\\tau \\sim 40$ Myr.", "This trend is consistent with the results of [48], who use several other age diagnostics to determine that the Upper Scorpius subgroup of the Sco-Cen OB association might be a factor of $\\sim $ 2 older ($\\tau \\sim 11$ Myr) than its traditionally accepted isochronal age for low-mass members ($\\tau \\sim 5$ Myr; [51]).", "[6] also have demonstrated a similar discrepancy for the BPMG." ], [ "Lithium Depletion Age", "As we discussed in Sections 4.3 and 5.1, lithium depletion is a key indicator of age for low-mass stars, being depleted on timescales of $$ 10 Myr for early-M stars and $\\sim $ 100 Myr for stars across the full range of spectral types we consider.", "Lithium also can be used to age-date stellar populations as a whole, placing them in a relative age sequence based on the location of the lithium depletion boundaries (for both late-K stars and early-M stars) as a function of spectral type or absolute magnitude.", "The depletion of lithium in K stars has long been used to age-date populations [24], but age-dating with mid-M stars is less widespread because these low-mass members have been more difficult to identify.", "The precise location of the boundary can be difficult to quantify, as the observed properties of moving group members can be blurred by observational uncertainties (most notably in distance or spectral type) or astrophysical effects (unresolved binarity, rotation, or genuine age spreads).", "We therefore have quantified the location of the lithium depletion boundaries by identifying the limit where equal numbers of lithium-depleted and lithium-bearing stars encroach onto the opposite side of the boundary.", "To avoid a bias from our use of lithium as a membership indicator, we only consider those lithium-bearing stars that were also selected based on RVs.", "Using this definition, we find that the late-K lithium boundary is at (spectroscopically determined) SpTs of K5.5 $\\pm $ 0.3 (where two earlier members are lithium depleted and another two later members are lithium-bearing).", "The mid-M boundary is at M4.5 $\\pm $ 0.3 (with 6 members violating each side of the boundary).", "The corresponding boundaries for SED-fot SpTs are K7.6 $\\pm $ 0.6 (4 members) and M4.7 $\\pm $ 0.7 (8 members).", "For absolute bolometric luminosities ($M_{bol} = m_{bol} + DM_{kin}$ ), the boundaries are at $M_{bol} = 6.64 \\pm 0.20$ (3 members) and $M_{bol} = 9.89 \\pm 0.10$ (5 members).", "Finally, for absolute $K_s$ magnitudes, the boundaries are at $M_{Ks} = 4.33 \\pm 0.15$ (3 members) and $M_{Ks} = 7.12 \\pm 0.16$ (5 members).", "In each case, we estimate the uncertainty from the range encompassing a number of non-encroaching objects equal to the number of encroaching objects.", "For absolute bolometric luminosities ($M_{bol} = m_{bol} + DM_{kin}$ ), the boundaries are at $M_{bol} = 6.64 \\pm 0.20$ (3 members) and $M_{bol} = 9.89 \\pm 0.10$ (5 members).", "Finally, for absolute $K_s$ magnitudes, the boundaries are at $M_{Ks} = 4.33 \\pm 0.15$ (3 members) and $M_{Ks} = 7.12 \\pm 0.16$ (5 members).", "In each case, we estimate the uncertainty from the range encompassing a number of non-encroaching objects equal to the number of encroaching objects.", "The late-K depletion boundary only changes subtly at ages of $$ 10 Myr, and hence it is challenging to construct an unambiguous sequence.", "[54] studied the $\\sim $ 50 Myr clusters IC 2602 and IC 2391, and for the same boundary definition as described above, they found it to fall at $T_{eff} = 4025$ K (or SpT = K7.1 from the temperature scale of [65]).", "[2] found that in the older ($\\sim $ 75 Myr) $\\alpha $ Per cluster, the boundary falls at $T_{eff} = 4735$ K (SpT = K3.2) with 3 interlopers.", "In the canonically $\\sim $ 125 Myr Pleiades cluster, [24] found the boundary to lie at $T_{eff} = 4420$ K (SpT = K4.7).", "For these and many other clusters, a more diagnostic estimate can be derived from examining the full sequence of $EW[Li]$ versus SpT for FGK stars, as lithium depletion occurs gradually across this full range.", "However, our census only adds a modest number of stars with SpT earlier than K5, so we refer the reader to a comprehensive analysis of the known higher-mass members by [42] and [12].", "The evolution of the mid-M lithium depletion boundary is more unambiguous due to the large dynamic range of $M_{bol}$ over which it varies in relevant age scales.", "[4] reported a boundary at $M_{bol} = 10.24 \\pm 0.15$ in IC 2391, as well as updating the results of [73], [74] for $\\alpha $ Per ($M_{bol} = 11.31 \\pm 0.15$ ) and the Pleiades ($M_{bol} = 12.14 \\pm 0.15$ ).", "[14] reported the boundary to fall at $M_K = 7.37 \\pm 0.20$ for IC 2602, which is equivalent to $M_{bol} = 10.22 \\pm 0.22$ given that $BC_K = 2.85 \\pm 0.10$ for M5.0-M5.5 stars [26].", "[10] reported a boundary for Blanco 1 at $M_{bol} = 11.99 \\pm 0.30$ .", "Finally, a very recent measurement for new members of the BPMG by [6] found the lithium depletion bound to fall at $M_{bol} = 8.3 \\pm 0.5$ , corresponding to an age of $\\tau = 21 \\pm 4$ Myr.", "As we summarize in Table 5, our measurement of the mid-M lithium depletion boundary for Tuc-Hor ($M_{bol} = 9.89 \\pm 0.10$ or $M_{Ks} = 7.12 \\pm 0.16$ ) indicates an age consistent with that of IC 2391 and IC 2602 ($\\sim $ 45 Myr), clearly older than BPMG ($\\tau = 21$ Myr) and clearly younger than the other reference populations.", "The evolutionary models of [3] and [13] imply lithium depletion ages of $41 \\pm 2$ Myr and $38 \\pm 2$ Myr respectively, where the uncertainties reflect only the uncertainty in the boundary location.", "The real error budget is most likely dominated by the uncalibrated nature of the models themselves." ], [ "The Spatial Structure of Tuc-Hor", "In Figure 10, we show the $XYZ$ spatial distributions for our observed members and for the SpT$<$ K3 members that were previously known [78].", "The two distributions broadly match and demonstrate that the main body of Tuc-Hor is compact in $Z$ , with a median value of $Z = -36$ pc and a total extent of $\\pm $ 5 pc for all but a few extreme outlying members.", "In contrast, the distribution is very broad in the ($X$ ,$Y$ ) plane.", "A similar spatial distribution can be seen in the activity-selected candidates reported by [59].", "Inspection of Figure 2 shows that we did not observe the candidates which would fall near the edges of the ($X$ ,$Y$ ) panel of Figure 10, and hence we can not comment on the total extent in this plane.", "Finally, a visually recognizable overdensity is located at ($XYZ$ ) $\\sim $ (+10,-25,-35) pc, corresponding to the traditionally identified “core” of Tuc-Hor which has on-sky coordinates of ($\\alpha $ ,$\\delta $ ) $\\sim $ (2$^h$ ,-60o) and which is equally recognizable in Figure 2.", "As for other moving groups, like TWA (e.g., [81]), our results demonstrate that the Tuc-Hor population is not distributed in an ellipsoid.", "TWA shows broadly filametary structure, while Tuc-Hor more closely resembles a sheet.", "These populations are too young to have been distorted by the Milky Way's tidal field, having only existed for $$ 1/8 of a galactic orbit, and hence this geometry must trace a combination of the primordial molecular cloud structure and specific forces (such as interactions with molecular clouds) that induced non-spherical velocity dispersions.", "If the former effect dominates, then it would indicate that geometric analyses are of limited use for determining trace-back ages, and more generally that these moving groups formed in a distributed manner [70], [27] rather than as compact clusters that have since become unbound [45].", "Finally, the small extent in $Z$ places a strong constraint on the internal velocity dispersion.", "Assuming that Tuc-Hor formed in a sheet with zero thickness in the $Z$ axis, then a typical member has moved by $<$ 5 pc in the moving group's lifetime of $\\sim $ 30 Myr.", "The corresponding one-dimensional velocity dispersion for individual members (or for substructures with typical scales of $$ 5 pc) is only $\\sigma _v \\sim 160$ m/s.", "This velocity dispersion is comparable to the small-scale velocity dispersion seen in Taurus-Auriga [27], further supporting Tuc-Hor's origin as a dynamically quiet T association." ], [ "The UVW Velocity and Dispersion of Tuc-Hor", "For any individual member of a stellar population, the radial velocity $v_{rad}$ indicates the one-dimensional projection of the population's space velocity $v_{UVW}$ onto the line of sight $d$ toward that member ($\\vec{v}_{rad} = \\vec{v}_{UVW} \\cdot \\hat{d}$ ).", "When a stellar population subtends a large solid angle of the sky, then the radial velocities of its members collectively trace a wide range of such projections, and hence can be used to tomographically reconstruct the full three-dimensional value of $v_{UVW}$ .", "This geometric reconstruction can place extremely tight constraints on the space motion, since the measurement of $v_{rad}$ is limited only by the intrinsic velocity dispersion of the cluster and the instrumental precision, whereas full $v_{UVW}$ measurements for individual stars are generally limited by the precision of the proper motion and by the distance (which is needed to convert the proper motion from an angular velocity to a spatial velocity).", "For Tuc-Hor, we computed this tomographic reconstruction by conducting a grid search of $UVW$ velocities, finding the mean $UVW$ velocity that minimizes the $\\chi ^2$ of the fit and determining confidence intervals in the $\\chi ^2$ surface around that minimum.", "For this calculation, we used 65 stars that have observational uncertainties $\\sigma _{vrad} < 1$ km/s (to reject fast rotators and other stars with noisy measurements) and for which their velocities agree with the expected value to within $<$ 3 km/s (to reject spectroscopic binaries).", "The resulting space motion at the minimum in the $\\chi ^2$ surface is $v_{UVW} = $ (-10.6, -21.0, -2.1) km/s, with 1$\\sigma $ uncertainties on each dimension of $\\pm $ 0.2 km/s.", "The reduced $\\chi ^2$ value for our best-fit value of $v_{UVW}$ is $\\chi ^2_{\\nu } = 7.4$ (with 62 degrees of freedom), indicating that the velocity dispersion is significantly resolved compared to our estimated uncertainties on the RVs.", "We therefore increased the uncertainties by a factor of $\\sqrt{7.4}$ before calculating the 1$\\sigma $ uncertainty on the mean $v_{UVW}$ .", "Our value for the mean $v_{UVW}$ is very close to the canonical velocity of $v_{UVW} = $ (-9.9, -20.9, -1.4) km/s [78], but is considerably more precise, even after increasing our RV uncertainties so that $\\chi _{\\nu }=1$ .", "If we compare the expected radial velocity for each star ($\\vec{v}_{rad} = \\vec{v}_{UVW} \\cdot \\hat{d}$ ) to the measured values, we find that the scatter of our measured RVs about the best-fit values is $\\pm $ 1.1 km/s.", "This scatter could result from either the noise floor of our RV measurements (which we have ruled out via multiple observations of a subset of standard stars; Section 3) or the intrinsic velocity dispersion across all of Tuc-Hor.", "The physical arguments from the previous section motivate a low velocity dispersion on small angular scales ($\\sigma _v \\sim 160$ m/s on scales of 5 pc).", "However, the total extent of Tuc-Hor is large ($$ 50 pc), and the velocity dispersion in molecular clouds (and the resulting stellar populations) should increase on large angular scales by $v \\propto d^{0.5}$ [30].", "We therefore expect the velocity dispersion on a spatial scale of 50 pc to be $\\sim $ 3 times larger than the velocity dispersion on a spatial scale of 5 pc, which would account for part of the larger dispersion in our RV measurements." ], [ "Disk Frequency", "Several members of the BPMG and TWA moving groups are known to host disks, either optically thick protoplanetary disks (as for TW Hya) or optically thin debris disks (as for $\\beta $ Pic).", "Since the Tuc-Hor moving group is only moderately older, it is plausible that some Tuc-Hor members might also host disks.", "To search for these disks, we cross-referenced our list of 143 K3-M6 members of Tuc-Hor with the all-sky catalog of the Wide-Field Infrared Survey Explorer (WISE), which observed the full sky in 4 bands spanning 3.4–22 $\\mu $ m. In all cases, our targets are sufficiently bright to be detected in the $W1$ (3.4 $\\mu $ m), $W2$ (4.5 $\\mu $ m), and $W3$ (12 $\\mu $ m) bands; only 24 were detected in the $W4$ (22 $\\mu $ m) band.", "In Figure 11, we show a plot of $W1-W3$ as a function of spectral type.", "Based on the criteria suggested by [34], all targets are Class III (diskless) sources with $W1-W3 < 1$ .", "The 24 stars that were detected in W4 also are consistent with photospheric colors ($W1 - W4 < 0.5$ ).", "We therefore conclude that the number of stars hosting significant quantities of warm circumstellar dust in this spectral type range is $F < 0.7\\%$ , with $F < 0.8\\%$ for M0.0–M6.0 stars and $F < 5\\%$ for K3.0–K7.9 stars." ], [ "Selection of Young Stars with GALEX", "Stellar activity is known to be an indicator of youth [51], [52], so X-ray and UV all-sky surveys are well-suited to finding active young stars.", "However, the ROSAT X-ray catalogs (e.g.", "[79]) are generally limited to the nearest, earliest-type M dwarfs, since their luminosities are $\\sim $ 10–300$\\times $ lower than solar-type stars.", "As we discussed in Section 5.2, this insensitivity to low-activity M dwarfs places a fundamental limit on ROSAT selection of young stars.", "Several teams have shown the higher completeness of using UV wavelengths to search for young M dwarfs [18], [68], [58], [59], making the NASA Galaxy Evolution Explorer (GALEX; [41]) a useful resource with which to expand the young low-mass census.", "The GALEX satellite has imaged most of the sky simultaneously in two bands: near-UV (NUV; 1750–2750 Å) and far-UV (FUV; 1350–1750 Å), with angular resolutions of 5  and 6.5.", "The full description of the instrumental performance is presented by [44].", "The GALEX mission produced a relatively shallow All-sky Imaging Survey as well as several deeper surveys which collectively cover $\\approx $ 3/4 of the sky.", "The NUV and FUV fluxes and magnitudes were produced by the standard GALEX Data Analysis Pipeline (ver.", "4.0) operated at the Caltech Science Operations Center [44]The data presented in this paper made use of the seventh data release (GR7).", "See details at http://www.galex.caltech.edu/researcher/techdoc-ch2.html.", "and archived at the Mikulski Archive at the Space Telescope Science Institute (MAST).", "Previous searches for low-mass YMG members using GALEX used color and proper motion cuts coupled with NUV and/or FUV selection criteria ([68] and [58] for TWA and [59] for Tuc-Hor) before acquiring optical spectra for candidate confirmation.", "In this work, we did not use GALEX to pre-select UV-active candidates, in order to avoid any bias against low-activity members.", "Our purely kinematic and color-magnitude selection procedure now allows us to test the efficiency and completeness of GALEX-selected surveys for young stars.", "In [68] we used the NStars 25-pc census ($\\approx $ 1500 M dwarfs; [55]) to calibrate our GALEX selection criteria.", "Namely, we identified young M dwarfs ($<300$ Myr) as having fractional flux densities $F_{NUV}/F_J$ $>$ 10$^{-4}$ and, if detected, $F_{FUV}/F_J$ $>$ 10$^{-5}$ , while the quiescent emission of old stars (those with $F_{FUV}/F_J$ $<$ 10$^{-5}$ and no ROSAT detection) traces out a clear sequence which lies below the young, ROSAT detected M dwarfs.", "For stars earlier than K2, the Tuc-Hor and field sequences converge, and hence the GALEX NUV cut is not a distinguishing criterion.", "However, in these cases, the FUV cut of $F_{FUV}/F_J$ $>$ 10$^{-5}$ can instead distinguish young stars.", "In Figure 12, we plot the GALEX NUV flux density (normalized to $J$ band flux density, $F_{NUV}/F_J$ ) as a function of spectral type for our Tuc-Hor candidates.", "Of the 204 candidates, 166 (80%) had a GALEX counterpart in the NUV bandpass within $<$ 10, while 26 were not observed by GALEX; 138 of the 204 candidates (69%) were observed and lie above the $F_{NUV}/F_J > 10^{-4}$ threshold used by [68].", "We found that 107 of the high-NUV emitters (78%) are confirmed members, while the remaining 31 (22%) presumably are either Tuc-Hor SBs that had discrepant RVs in our observation epoch, other young stars ($\\tau 300$ Myr), or old field SBs which are tidally-locked into fast rotation (and hence high activity).", "Only 80 of our candidates were detected in the FUV bandpass, with all but one having $F_{FUV}/F_J$ $>$ 10$^{-5}$ .", "We found that 68/80 are confirmed Tuc-Hor members.", "We therefore find that the FUV criterion works well for the more massive stars in Tuc-Hor, but fails at $d 40$ pc for a significant fraction of young stars with SpT$>$ M2, where they are too faint to be detected in the FUV.", "Of the 142 Tuc-Hor members observed in this paper, 13 were not observed by GALEX  5 were observed and not detected, 3 were not identified due to confusion with brighter neighbors, and 14 would have been rejected using the $F_{NUV}/F_J$ $>$ 10$^{-4}$ criterion.", "Therefore, had we pre-selected candidates using the NUV GALEX criterion from [68], we would have identified 107 of 142 (77%) of the confirmed members as Tuc-Hor candidates.", "Had we first applied the NUV criterion prior to collecting additional data, we would have needed spectra of 138 stars to confirm 107 new members, yielding a confirmation efficiency of 78%.", "Without the NUV criterion, we needed 204 spectra to confirm 142 members, yielding a confirmation efficiency of 70%.", "Therefore, adding the GALEX criterion to the candidate-selection process discussed in Section 2 is somewhat more efficient, but it limits the search to $\\approx $ 75% of the existing members due to incomplete sky coverage and the intrinsic spread in the intrinsic $NUV$ excesses of young stars.", "Finally, we can use the results of our kinematic+CMD selection procedure (which is unbiased toward stellar activity) to set a new SpT-dependent lower envelope for NUV fluxes of $\\tau = 40$ Myr young stars.", "We find that for K3–M2 stars, the lower envelope is defined by a linear relation connecting (SpT=K3, $F_{NUV}/F_J=2 \\times 10^{-4}$ ) and (SpT=M2, $F_{NUV}/F_J=5 \\times 10^{-5}$ ).", "For M2–M4 stars, the lower envelope is defined by $F_{NUV}/F_J > 5 \\times 10^{-5}$ .", "For stars later than M4, strong stellar activity can persist for a significant fraction of a Hubble time [82], limiting the usefulness of GALEX data.", "However, for stars with SpT $<$ M4, these criteria would only reject 3 of our newly-identified Tuc-Hor members (subject to the spatial completeness of GALEX), which could themselves be field interlopers that are comoving by chance.", "We suggest that the optimal strategy for completing the Tuc-Hor census would be to use GALEX selection to identify the spatial distribution of Tuc-Hor members and remove most contaminants, and then to use kinematic+CMD selection to achieve the highest possible completeness within the spatial locus of Tuc-Hor members.", "The authors thank Jason Curtis for obtaining many excellent observations as part of a time trade, Jason Wright for useful suggestions regarding the optimal map projection for plotting stars on the celestial sphere, and the anonymous referee for a helpful and thorough critique of the paper.", "ALK was supported in part by a Clay fellowship.", "ll rr l crr rr lrr 0pt Selection Criteria and Observations of Candidate Tuc-Hor Members 2MASS J Other Name $R_{USNOB}$ $K_s$ $\\mu $ SpT$_{SED}$ $m_{bol}$ $DM_{phot}$ $\\Delta $ $DM_{kin}$ UT date $t_{int}$ SNR (mag) (mag) (mas/yr) (mag) (mag) (mas/yr) (mag) (yyyymmdd) (sec) @6600Å Known Members 02414730-5259306 AF Hor 11.2 7.64 (92.2,-4.2)$\\pm $ 1.3 K7.2$\\pm $ 0.7 9.89$\\pm $ 0.09 2.86$\\pm $ 0.25 8.5 3.3 20120901 180 115 03190864-3507002 CD-35 1167 10.4 7.72 ... K7.4$\\pm $ 0.2 10.04$\\pm $ 0.03 2.86$\\pm $ 0.12 .. .. 20130202 360 242 03315564-4359135 CD-44 1173 10.4 7.47 (84.7,-8.4)$\\pm $ 1.8 M0.0$\\pm $ 0.6 9.87$\\pm $ 0.05 2.27$\\pm $ 0.24 1.4 3.2 20130202 240 190 02414683-5259523 CD-53 544 9.6 6.76 (98.5,-14.0)$\\pm $ 1.5 K7.7$\\pm $ 0.3 9.10$\\pm $ 0.04 1.72$\\pm $ 0.14 0.6 3.1 20120901 120 173 02423301-5739367 CD-58 553 10.2 7.78 (83.8,-8.8)$\\pm $ 1.1 K7.1$\\pm $ 0.2 10.01$\\pm $ 0.02 3.05$\\pm $ 0.11 2.5 3.4 20120902 120 136 02073220-5940210 CD-60 416 9.8 7.54 (92.3,-23.3)$\\pm $ 1.5 K7.1$\\pm $ 0.7 9.76$\\pm $ 0.03 2.80$\\pm $ 0.18 5.9 3.3 20120901 120 184 00422033-7747397 CD-78 24 9.7 7.53 (78.8,-30.7)$\\pm $ 1.1 K5.0$\\pm $ 0.5 9.69$\\pm $ 0.02 3.01$\\pm $ 0.07 1.1 3.6 20120901 180 215 00251465-6130483 CT Tuc 10.5 7.75 (85.9,-55.9)$\\pm $ 1.4 K7.6$\\pm $ 0.2 10.10$\\pm $ 0.04 2.79$\\pm $ 0.08 4.7 3.3 20120901 180 158 00345120-6154583 HD 3221 8.8 6.53 (86.0,-50.8)$\\pm $ 0.9 K5.8$\\pm $ 0.4 8.75$\\pm $ 0.02 1.90$\\pm $ 0.11 4.1 3.3 20120901 120 224 21443012-6058389 HIP 107345 10.7 7.87 (40.3,-94.9)$\\pm $ 1.6 M0.0$\\pm $ 0.2 10.31$\\pm $ 0.03 2.71$\\pm $ 0.08 6.6 3.4 20120901 180 137 00240899-6211042 HIP 1910 10.6 7.49 (83.2,-51.9)$\\pm $ 1.2 M0.2$\\pm $ 0.9 9.95$\\pm $ 0.06 2.28$\\pm $ 0.31 2.7 3.4 20120901 180 154 04480066-5041255 TYC 8083-0455 10.7 7.92 (54.3,14.1)$\\pm $ 1.8 K7.8$\\pm $ 0.2 10.29$\\pm $ 0.04 2.83$\\pm $ 0.11 2.2 3.7 20130202 240 111 23261069-7323498 TYC 9344-0293 10.9 7.94 (72.1,-66.8)$\\pm $ 1.0 M1.5$\\pm $ 1.2 10.50$\\pm $ 0.11 2.29$\\pm $ 0.48 5.5 3.4 20120902 180 117 Candidate Members 00123485-5927464 8.8 6.70 (58.6,-29.3)$\\pm $ 1.0 K5.8$\\pm $ 0.4 8.89$\\pm $ 0.02 2.04$\\pm $ 0.10 7.9 4.3 20120716 20 94 00125703-7952073 13.0 8.75 (80.9,-46.1)$\\pm $ 2.0 M3.4$\\pm $ 0.3 11.44$\\pm $ 0.03 2.02$\\pm $ 0.19 4.8 3.4 20120902 180 62 00144767-6003477 13.0 8.83 (91.3,-63.1)$\\pm $ 1.5 M3.5$\\pm $ 0.3 11.50$\\pm $ 0.02 2.00$\\pm $ 0.21 4.0 3.1 20120716 100 34 00152752-6414545 11.8 8.44 (80.2,-49.9)$\\pm $ 1.2 M1.5$\\pm $ 0.4 11.00$\\pm $ 0.02 2.80$\\pm $ 0.15 1.7 3.5 20120716 40 34 00155556-6137519 13.7 9.77 (70.1,-42.2)$\\pm $ 1.4 M2.8$\\pm $ 0.4 12.39$\\pm $ 0.02 3.43$\\pm $ 0.25 1.8 3.8 20120716 160 34 00173041-5957044 10.4 7.64 (112.8,-68.6)$\\pm $ 2.1 K7.9$\\pm $ 0.4 10.03$\\pm $ 0.03 2.50$\\pm $ 0.15 3.5 2.8 20120901 180 132 00220446-1016191 10.3 8.67 (63.0,-44.4)$\\pm $ 1.4 K7.5$\\pm $ 0.1 10.98$\\pm $ 0.02 3.73$\\pm $ 0.08 2.7 4.0 20120718 60 46 00235732-5531435 14.8 10.24 (91.9,-66.9)$\\pm $ 3.1 M4.0$\\pm $ 0.5 12.95$\\pm $ 0.05 3.03$\\pm $ 0.42 7.8 3.1 20120901 300 25 00273330-6157169 13.7 9.47 (87.5,-56.8)$\\pm $ 1.5 M3.5$\\pm $ 0.1 12.16$\\pm $ 0.02 2.65$\\pm $ 0.08 6.2 3.2 20120716 160 29 00275023-3233060 12.1 8.01 (97.8,-60.9)$\\pm $ 2.5 M3.0$\\pm $ 0.4 10.65$\\pm $ 0.03 1.56$\\pm $ 0.28 5.2 3.1 20120718 60 39 00284683-6751446 15.5 10.50 (92.0,-43.8)$\\pm $ 7.2 M4.7$\\pm $ 0.2 13.29$\\pm $ 0.02 2.61$\\pm $ 0.21 4.5 3.3 20120718 600 22 00302572-6236015 10.6 7.55 (95.5,-48.4)$\\pm $ 4.4 K7.9$\\pm $ 0.2 9.98$\\pm $ 0.04 2.45$\\pm $ 0.11 4.2 3.2 20120716 30 44 00305785-6550058 13.5 8.95 (83.6,-52.2)$\\pm $ 1.4 M4.0$\\pm $ 0.5 11.65$\\pm $ 0.04 1.73$\\pm $ 0.39 7.2 3.3 20120716 150 37 00332438-5116433 13.5 9.01 (94.7,-59.9)$\\pm $ 1.2 M2.4$\\pm $ 0.2 11.63$\\pm $ 0.02 2.93$\\pm $ 0.09 2.9 3.1 20120901 300 72 00382147-4611043 15.1 10.96 (60.9,-46.1)$\\pm $ 4.7 M4.1$\\pm $ 0.4 13.65$\\pm $ 0.03 3.62$\\pm $ 0.32 7.6 4.0 20120901 600 34 00393579-3816584 11.1 7.86 (100.2,-65.5)$\\pm $ 3.5 M1.8$\\pm $ 0.4 10.44$\\pm $ 0.03 2.10$\\pm $ 0.18 2.7 3.0 20130204 300 44 00394063-6224125 15.0 10.38 (104.1,-38.8)$\\pm $ 12.8 M4.2$\\pm $ 0.2 13.10$\\pm $ 0.04 2.96$\\pm $ 0.13 13.2 3.1 20120718 600 24 00421010-5444431 12.9 8.93 (89.4,-47.9)$\\pm $ 1.8 M3.0$\\pm $ 0.4 11.59$\\pm $ 0.02 2.50$\\pm $ 0.25 1.9 3.3 20120716 60 18 00421092-4252545 12.7 8.76 (83.3,-43.7)$\\pm $ 1.1 M2.1$\\pm $ 0.6 11.34$\\pm $ 0.04 2.83$\\pm $ 0.28 6.4 3.5 20130204 300 31 00425349-6117384 15.2 10.45 (89.0,-55.2)$\\pm $ 2.8 M4.5$\\pm $ 0.5 13.19$\\pm $ 0.05 2.73$\\pm $ 0.42 9.0 3.2 20120718 600 29 00485254-6526330 13.8 9.55 (82.3,-40.7)$\\pm $ 1.9 M3.2$\\pm $ 0.3 12.22$\\pm $ 0.02 2.96$\\pm $ 0.21 3.4 3.5 20120716 170 34 00493566-6347416 12.1 8.43 (86.9,-45.2)$\\pm $ 1.2 M1.8$\\pm $ 0.3 10.98$\\pm $ 0.02 2.63$\\pm $ 0.14 5.0 3.3 20120716 45 37 00502644-4628539 8.3 6.34 (62.0,-36.9)$\\pm $ 1.6 K4.7$\\pm $ 0.5 8.50$\\pm $ 0.02 1.86$\\pm $ 0.07 1.7 4.1 20130202 641 454 00514081-5913320 14.7 10.40 (98.0,-50.3)$\\pm $ 3.5 M4.1$\\pm $ 0.6 13.11$\\pm $ 0.04 3.08$\\pm $ 0.52 2.6 3.1 20120718 525 28 00555140-4938216 15.2 10.49 (101.0,-56.1)$\\pm $ 4.0 M4.8$\\pm $ 0.3 13.29$\\pm $ 0.03 2.50$\\pm $ 0.30 3.6 3.1 20120901 600 42 00590177-6054124 11.6 8.47 (96.2,-32.3)$\\pm $ 1.1 M0.1$\\pm $ 0.4 10.92$\\pm $ 0.04 3.28$\\pm $ 0.13 9.4 3.3 20120716 40 42 01024375-6235344 13.1 8.80 (89.0,-39.6)$\\pm $ 1.2 M3.8$\\pm $ 0.3 11.46$\\pm $ 0.02 1.71$\\pm $ 0.22 3.1 3.3 20120716 90 26 01033563-5515561 14.2 9.24 (100.3,-46.9)$\\pm $ 2.2 M5.1$\\pm $ 0.6 12.10$\\pm $ 0.05 0.99$\\pm $ 0.60 0.4 3.1 20120716 330 25 01101448-4715453 15.1 11.16 (66.3,-38.3)$\\pm $ 4.5 M4.7$\\pm $ 0.3 13.94$\\pm $ 0.04 3.26$\\pm $ 0.29 5.5 3.9 20120901 600 42 01125587-7130283 11.1 8.37 (76.6,-30.7)$\\pm $ 1.0 K7.7$\\pm $ 1.8 10.72$\\pm $ 0.15 3.33$\\pm $ 0.78 7.8 3.6 20120902 180 94 01134031-5939346 13.5 9.06 (96.0,-35.4)$\\pm $ 1.9 M4.0$\\pm $ 0.4 11.77$\\pm $ 0.03 1.85$\\pm $ 0.31 2.6 3.2 20120716 160 25 01160045-6747311 15.5 10.89 (66.7,-12.1)$\\pm $ 4.8 M4.2$\\pm $ 0.3 13.61$\\pm $ 0.02 3.47$\\pm $ 0.26 8.6 4.1 20120718 650 27 01180670-6258591 15.5 10.64 (109.9,-23.4)$\\pm $ 17.1 M4.8$\\pm $ 0.2 13.45$\\pm $ 0.02 2.66$\\pm $ 0.21 14.1 3.0 20120718 625 20 01211297-6117281 14.8 10.48 (80.7,-28.3)$\\pm $ 3.0 M4.1$\\pm $ 0.2 13.16$\\pm $ 0.02 3.13$\\pm $ 0.20 0.1 3.6 20120718 500 23 01211813-5434245 10.7 7.81 (81.7,-42.8)$\\pm $ 1.0 K7.4$\\pm $ 0.2 10.10$\\pm $ 0.03 2.93$\\pm $ 0.12 9.4 3.5 20120716 30 53 01224511-6318446 13.3 8.98 (94.5,-29.0)$\\pm $ 2.2 M3.3$\\pm $ 0.3 11.63$\\pm $ 0.02 2.29$\\pm $ 0.25 2.3 3.3 20120718 120 33 01233280-4113110 15.3 9.92 (109.3,-54.8)$\\pm $ 1.9 M5.6$\\pm $ 0.5 12.76$\\pm $ 0.02 1.12$\\pm $ 0.50 2.6 2.9 20120901 600 53 01245895-7953375 9.2 7.14 (74.8,-20.6)$\\pm $ 3.2 K3.8$\\pm $ 0.2 9.25$\\pm $ 0.04 2.75$\\pm $ 0.02 8.0 3.7 20130203 120 127 01253196-6646023 14.7 10.11 (89.1,-28.4)$\\pm $ 2.5 M4.5$\\pm $ 0.1 12.86$\\pm $ 0.03 2.39$\\pm $ 0.09 3.1 3.4 20120901 300 30 01275875-6032243 14.6 10.22 (88.4,-30.8)$\\pm $ 3.0 M4.0$\\pm $ 0.2 12.91$\\pm $ 0.02 2.99$\\pm $ 0.19 1.7 3.4 20120901 300 30 01283025-4921094 14.9 9.71 (101.4,-41.7)$\\pm $ 1.8 M4.0$\\pm $ 0.3 12.40$\\pm $ 0.02 2.48$\\pm $ 0.22 0.4 3.1 20120901 300 43 01301454-2949175 15.2 10.25 (161.7,-87.4)$\\pm $ 3.6 M5.3$\\pm $ 0.5 13.07$\\pm $ 0.02 1.75$\\pm $ 0.47 4.0 2.1 20130203 600 36 01321522-5034307 15.3 10.71 (71.5,-32.1)$\\pm $ 2.2 M4.0$\\pm $ 0.8 13.42$\\pm $ 0.05 3.50$\\pm $ 0.60 4.0 3.8 20120901 600 49 01344601-5707564 16.3 11.16 (68.2,-18.8)$\\pm $ 8.9 M4.6$\\pm $ 0.2 13.97$\\pm $ 0.03 3.40$\\pm $ 0.15 3.6 4.0 20130203 600 11 01351393-0712517 12.5 8.08 (97.7,-51.5)$\\pm $ 4.2 M4.1$\\pm $ 0.1 10.79$\\pm $ 0.02 0.76$\\pm $ 0.10 3.8 3.2 20120718 90 40 01372781-4558261 15.1 10.19 (116.0,-37.1)$\\pm $ 6.1 M5.4$\\pm $ 0.3 13.06$\\pm $ 0.02 1.63$\\pm $ 0.32 9.3 2.9 20120901 600 28 01375879-5645447 14.1 9.53 (92.5,-32.8)$\\pm $ 2.3 M3.8$\\pm $ 0.3 12.21$\\pm $ 0.02 2.45$\\pm $ 0.26 2.9 3.3 20120718 315 37 01380029-4603398 13.7 10.14 (69.4,-20.4)$\\pm $ 3.2 M2.6$\\pm $ 0.5 12.75$\\pm $ 0.03 3.92$\\pm $ 0.28 7.1 4.0 20120901 300 46 01380311-5904042 13.1 9.00 (100.9,-27.4)$\\pm $ 2.3 M3.2$\\pm $ 0.2 11.67$\\pm $ 0.03 2.41$\\pm $ 0.14 3.6 3.1 20120718 120 35 01504543-5716488 16.3 11.28 (93.2,-37.0)$\\pm $ 13.9 M4.9$\\pm $ 0.4 14.09$\\pm $ 0.03 3.19$\\pm $ 0.35 11.4 3.2 20130203 600 10 01505688-5844032 12.6 8.64 (91.3,-24.9)$\\pm $ 2.1 M2.9$\\pm $ 0.4 11.29$\\pm $ 0.02 2.26$\\pm $ 0.24 1.8 3.3 20120718 90 36 01521830-5950168 11.7 8.14 (107.8,-27.0)$\\pm $ 1.8 M1.6$\\pm $ 0.4 10.66$\\pm $ 0.02 2.41$\\pm $ 0.15 1.7 3.0 20120901 180 90 01532494-6833226 15.1 10.18 (98.0,-16.3)$\\pm $ 4.9 M5.0$\\pm $ 0.2 13.02$\\pm $ 0.03 2.01$\\pm $ 0.19 1.4 3.2 20120902 300 26 01570140-7721221 15.1 10.73 (78.5,-13.9)$\\pm $ 2.2 M4.4$\\pm $ 0.9 13.49$\\pm $ 0.07 3.14$\\pm $ 0.76 9.7 3.6 20130203 600 27 02000918-8025009 10.0 8.02 (77.7,6.7)$\\pm $ 1.0 K4.0$\\pm $ 0.5 10.16$\\pm $ 0.03 3.61$\\pm $ 0.07 7.6 3.7 20130203 120 95 02001277-0840516 11.7 7.87 (110.0,-65.7)$\\pm $ 3.2 M2.0$\\pm $ 0.2 10.46$\\pm $ 0.02 2.02$\\pm $ 0.12 4.5 2.9 20120718 80 59 02001992-6614017 14.6 9.88 (82.7,-14.1)$\\pm $ 4.5 M3.0$\\pm $ 0.8 12.56$\\pm $ 0.04 3.47$\\pm $ 0.61 2.2 3.5 20120902 300 35 02045317-5346162 14.2 9.56 (95.6,-30.9)$\\pm $ 3.0 M3.4$\\pm $ 0.2 12.23$\\pm $ 0.02 2.81$\\pm $ 0.12 6.5 3.2 20120718 300 32 02070176-4406380 11.8 8.40 (95.7,-32.8)$\\pm $ 8.5 M1.2$\\pm $ 0.3 10.94$\\pm $ 0.04 2.87$\\pm $ 0.07 1.2 3.2 20120718 75 38 02105538-4603588 10.3 8.61 (54.8,-21.2)$\\pm $ 1.2 K3.2$\\pm $ 0.7 10.64$\\pm $ 0.08 4.28$\\pm $ 0.09 4.4 4.4 20120718 60 57 02125819-5851182 12.0 8.44 (88.4,-16.1)$\\pm $ 1.3 M3.5$\\pm $ 0.5 11.11$\\pm $ 0.03 1.61$\\pm $ 0.36 0.8 3.4 20120718 60 38 02153328-5627175 16.3 10.95 (98.6,-29.3)$\\pm $ 14.9 M4.9$\\pm $ 0.6 13.81$\\pm $ 0.03 2.91$\\pm $ 0.61 10.3 3.1 20130203 600 6 02155892-0929121 11.2 7.55 (97.8,-44.4)$\\pm $ 3.2 M2.3$\\pm $ 0.2 10.17$\\pm $ 0.02 1.53$\\pm $ 0.10 7.9 3.2 20120718 70 56 02180960-6657524 14.9 9.97 (99.0,-14.7)$\\pm $ 2.4 M4.5$\\pm $ 0.2 12.72$\\pm $ 0.02 2.26$\\pm $ 0.20 7.4 3.1 20120902 300 25 02192210-3925225 15.5 10.40 (111.8,-44.0)$\\pm $ 8.3 M5.9$\\pm $ 0.3 13.32$\\pm $ 0.02 1.36$\\pm $ 0.25 6.1 2.8 20130203 600 15 02201988-6855014 15.0 10.66 (63.8,6.8)$\\pm $ 3.4 M3.5$\\pm $ 0.3 13.36$\\pm $ 0.02 3.85$\\pm $ 0.21 9.7 4.1 20120902 300 27 02205139-5823411 13.1 8.83 (110.5,-8.9)$\\pm $ 15.0 M3.2$\\pm $ 0.6 11.47$\\pm $ 0.03 2.22$\\pm $ 0.48 7.9 2.9 20120718 120 40 02224418-6022476 12.6 8.10 (136.9,-14.4)$\\pm $ 1.7 M4.2$\\pm $ 0.2 10.83$\\pm $ 0.04 0.70$\\pm $ 0.13 3.2 2.4 20120718 90 20 02234926-4238512 9.7 7.71 (56.3,-24.2)$\\pm $ 0.8 K4.3$\\pm $ 0.7 9.84$\\pm $ 0.04 3.25$\\pm $ 0.07 7.2 4.3 20130202 240 207 02242453-7033211 14.2 9.49 (93.8,-6.4)$\\pm $ 3.2 M4.0$\\pm $ 0.3 12.18$\\pm $ 0.02 2.26$\\pm $ 0.23 5.6 3.2 20120902 300 43 02294569-5541496 14.8 10.26 (89.5,-17.7)$\\pm $ 6.3 M4.3$\\pm $ 0.3 12.97$\\pm $ 0.03 2.73$\\pm $ 0.25 4.4 3.3 20120718 500 27 02294869-6906044 16.0 11.06 (91.5,-5.4)$\\pm $ 7.6 M4.6$\\pm $ 0.2 13.84$\\pm $ 0.02 3.27$\\pm $ 0.21 5.0 3.3 20130203 600 7 02303239-4342232 9.8 7.23 (81.4,-13.5)$\\pm $ 0.9 K7.0$\\pm $ 0.3 9.44$\\pm $ 0.02 2.55$\\pm $ 0.11 8.1 3.6 20120901 120 182 02304370-5811560 14.4 10.44 (59.6,-0.1)$\\pm $ 2.6 M3.4$\\pm $ 0.2 13.08$\\pm $ 0.03 3.66$\\pm $ 0.11 7.0 4.2 20120718 350 30 02321934-5746117 14.6 10.23 (84.9,-17.7)$\\pm $ 3.5 M3.8$\\pm $ 0.6 12.90$\\pm $ 0.05 3.15$\\pm $ 0.49 7.7 3.4 20120718 400 29 02341866-5128462 14.4 9.76 (100.6,-17.5)$\\pm $ 2.5 M4.2$\\pm $ 0.3 12.48$\\pm $ 0.04 2.34$\\pm $ 0.29 1.5 3.1 20120718 350 32 02351646-5049133 10.0 8.07 (75.8,-8.5)$\\pm $ 1.5 K4.8$\\pm $ 1.0 10.21$\\pm $ 0.05 3.56$\\pm $ 0.14 5.2 3.7 20130202 240 197 02372562-4912033 11.7 8.62 (63.8,-19.2)$\\pm $ 1.2 M0.0$\\pm $ 0.3 11.06$\\pm $ 0.03 3.46$\\pm $ 0.10 6.8 4.0 20120718 60 48 02383255-7528065 15.3 10.80 (77.1,5.5)$\\pm $ 10.5 M4.7$\\pm $ 0.2 13.61$\\pm $ 0.02 2.92$\\pm $ 0.20 2.8 3.6 20130203 600 14 02412721-3049149 14.8 10.26 (97.5,-32.2)$\\pm $ 2.2 M4.3$\\pm $ 0.2 12.99$\\pm $ 0.03 2.74$\\pm $ 0.20 4.0 3.1 20130203 600 30 02420204-5359147 14.7 9.98 (97.0,-20.8)$\\pm $ 2.2 M4.3$\\pm $ 0.2 12.70$\\pm $ 0.02 2.45$\\pm $ 0.21 8.6 3.1 20120718 450 29 02420404-5359000 14.0 9.29 (98.6,-10.3)$\\pm $ 3.7 M3.9$\\pm $ 0.1 11.97$\\pm $ 0.02 2.13$\\pm $ 0.09 2.0 3.1 20120718 250 28 02441466-5221318 9.0 7.24 (77.0,-7.6)$\\pm $ 0.9 K3.0$\\pm $ 0.6 9.25$\\pm $ 0.07 2.94$\\pm $ 0.08 3.1 3.6 20130202 240 270 02474639-5804272 11.8 8.45 (94.9,-4.0)$\\pm $ 1.5 M1.6$\\pm $ 0.3 11.00$\\pm $ 0.02 2.75$\\pm $ 0.13 2.3 3.2 20120718 70 38 02485260-3404246 12.7 8.40 (89.0,-23.8)$\\pm $ 1.4 M3.9$\\pm $ 0.4 11.09$\\pm $ 0.03 1.25$\\pm $ 0.30 4.8 3.3 20120718 80 32 02502222-6545552 13.3 9.44 (76.7,2.5)$\\pm $ 1.8 M2.8$\\pm $ 0.4 12.08$\\pm $ 0.02 3.12$\\pm $ 0.26 0.9 3.6 20120902 180 36 02505959-3409050 13.8 9.62 (85.8,-21.0)$\\pm $ 1.8 M3.8$\\pm $ 0.8 12.31$\\pm $ 0.04 2.56$\\pm $ 0.57 5.9 3.4 20120718 180 28 02523550-7831183 15.8 10.78 (67.6,15.9)$\\pm $ 5.1 M4.8$\\pm $ 0.2 13.59$\\pm $ 0.03 2.80$\\pm $ 0.14 2.6 3.9 20130203 600 14 02543316-5108313 11.1 7.78 (92.7,-13)$\\pm $ 1.2 M1.4$\\pm $ 0.7 10.34$\\pm $ 0.05 2.18$\\pm $ 0.25 2.3 3.2 20120718 70 59 02553178-5702522 15.0 10.22 (89.5,-5.8)$\\pm $ 3.0 M4.4$\\pm $ 0.1 12.98$\\pm $ 0.02 2.62$\\pm $ 0.10 2.5 3.3 20120901 300 21 02564708-6343027 12.7 9.01 (67.4,8.8)$\\pm $ 2.8 M3.2$\\pm $ 0.3 11.63$\\pm $ 0.02 2.38$\\pm $ 0.20 6.0 3.9 20120718 90 24 02572682-6341293 13.2 9.33 (64.3,12.2)$\\pm $ 2.4 M3.2$\\pm $ 0.4 11.97$\\pm $ 0.03 2.72$\\pm $ 0.30 9.4 4.0 20120902 180 32 02590284-6120000 14.6 10.74 (52.6,2.0)$\\pm $ 7.2 M3.5$\\pm $ 0.3 13.40$\\pm $ 0.02 3.90$\\pm $ 0.21 0.8 4.4 20130202 300 23 02591904-5122341 15.9 10.82 (81.7,-14.7)$\\pm $ 7.2 M5.3$\\pm $ 0.5 13.67$\\pm $ 0.02 2.34$\\pm $ 0.47 6.7 3.4 20130202 600 20 02592564-2947275 10.2 8.27 (67.1,-26.2)$\\pm $ 1.2 K5.4$\\pm $ 0.1 10.25$\\pm $ 0.04 3.49$\\pm $ 0.06 3.2 3.8 20130203 120 116 03050556-5317182 15.4 10.26 (89.4,-11.3)$\\pm $ 3.5 M5.1$\\pm $ 0.2 13.10$\\pm $ 0.02 1.98$\\pm $ 0.16 6.4 3.2 20130202 600 30 03050976-3725058 12.5 8.65 (51.6,-11.5)$\\pm $ 1.3 M2.5$\\pm $ 0.5 11.27$\\pm $ 0.03 2.50$\\pm $ 0.27 1.7 4.4 20120718 80 39 03083950-3844363 15.3 10.42 (68.3,-11.0)$\\pm $ 3.8 M4.7$\\pm $ 0.3 13.18$\\pm $ 0.03 2.50$\\pm $ 0.25 4.5 3.8 20130203 600 16 03093877-3014352 15.5 10.70 (88.7,-35.9)$\\pm $ 4.9 M4.7$\\pm $ 0.4 13.49$\\pm $ 0.04 2.81$\\pm $ 0.40 6.7 3.2 20130203 600 13 03104941-3616471 14.6 9.79 (90.5,-28.7)$\\pm $ 2.0 M4.5$\\pm $ 0.3 12.54$\\pm $ 0.03 2.07$\\pm $ 0.25 6.1 3.2 20130203 300 19 03114544-4719501 13.6 9.57 (88.4,-4.0)$\\pm $ 2.3 M3.7$\\pm $ 0.3 12.27$\\pm $ 0.02 2.60$\\pm $ 0.21 6.0 3.2 20120718 180 23 03204757-5041330 11.8 8.56 (81.8,6.1)$\\pm $ 2.8 M1.1$\\pm $ 0.4 11.06$\\pm $ 0.03 3.04$\\pm $ 0.13 9.2 3.4 20120718 70 42 03210395-6816475 13.4 9.30 (70.3,20.4)$\\pm $ 3.6 M3.4$\\pm $ 0.3 12.04$\\pm $ 0.02 2.62$\\pm $ 0.21 6.2 3.7 20130204 600 29 03244056-3904227 13.2 9.02 (86.5,-17.1)$\\pm $ 2.4 M4.1$\\pm $ 0.1 11.71$\\pm $ 0.03 1.68$\\pm $ 0.10 1.4 3.2 20120718 120 31 03271701-6128407 9.4 7.39 (82.6,6.5)$\\pm $ 1.5 K3.7$\\pm $ 0.2 9.47$\\pm $ 0.04 2.99$\\pm $ 0.02 4.9 3.3 20120902 60 140 03285469-3339192 6.4 4.19 (93.5,-19.9)$\\pm $ 8.4 K5.6$\\pm $ 0.5 6.44$\\pm $ 0.03 -0.37$\\pm $ 0.13 3.9 3.1 20130203 120 362 03291649-3702502 14.1 9.78 (89.8,-20.8)$\\pm $ 3.1 M3.7$\\pm $ 0.5 12.46$\\pm $ 0.03 2.79$\\pm $ 0.37 2.7 3.1 20130203 300 20 03312105-5955006 9.4 7.61 (50.6,6.1)$\\pm $ 1.2 K3.4$\\pm $ 0.3 9.67$\\pm $ 0.05 3.26$\\pm $ 0.03 1.1 4.4 20120902 60 121 03512287-5154582 14.1 9.77 (71.9,2.4)$\\pm $ 2.4 M4.4$\\pm $ 0.1 12.48$\\pm $ 0.02 2.13$\\pm $ 0.10 3.9 3.5 20130202 300 33 03561624-3915219 14.6 9.60 (68.6,-3.7)$\\pm $ 2.8 M4.5$\\pm $ 0.2 12.37$\\pm $ 0.02 1.91$\\pm $ 0.20 3.7 3.6 20130203 300 18 04000382-2902165 9.7 7.20 (78.7,-12.5)$\\pm $ 1.4 K7.2$\\pm $ 0.2 9.44$\\pm $ 0.03 2.41$\\pm $ 0.09 8.3 3.3 20120718 60 96 04000395-2902280 9.7 7.52 (77.7,-23.0)$\\pm $ 1.7 K5.3$\\pm $ 0.4 9.72$\\pm $ 0.02 2.98$\\pm $ 0.09 2.7 3.2 20120718 60 73 04013874-3127472 16.1 11.14 (59.3,-12.3)$\\pm $ 3.4 M4.7$\\pm $ 0.2 13.92$\\pm $ 0.03 3.24$\\pm $ 0.20 1.6 3.8 20130203 600 11 04021648-1521297 9.5 7.57 (68.7,-27.9)$\\pm $ 1.4 K4.1$\\pm $ 0.6 9.70$\\pm $ 0.04 3.14$\\pm $ 0.07 4.9 3.6 20130202 240 253 04074372-6825111 13.7 9.52 (57.8,22.0)$\\pm $ 2.8 M2.6$\\pm $ 0.1 12.17$\\pm $ 0.02 3.34$\\pm $ 0.06 2.8 3.9 20130204 300 36 04082685-7844471 11.6 8.40 (55.7,42.8)$\\pm $ 1.8 M0.3$\\pm $ 0.3 10.85$\\pm $ 0.03 3.14$\\pm $ 0.09 8.2 3.8 20120718 90 57 04133314-5231586 12.7 9.12 (65.7,14.8)$\\pm $ 1.5 M1.7$\\pm $ 0.3 11.67$\\pm $ 0.02 3.37$\\pm $ 0.11 2.7 3.5 20130204 300 47 04133609-4413325 14.0 9.91 (56.7,0.4)$\\pm $ 2.0 M3.3$\\pm $ 0.3 12.57$\\pm $ 0.02 3.24$\\pm $ 0.25 1.8 3.9 20130204 300 29 04213904-7233562 12.8 8.99 (62.2,25.4)$\\pm $ 1.3 M2.4$\\pm $ 0.4 11.59$\\pm $ 0.02 2.89$\\pm $ 0.21 9.5 3.8 20120718 120 32 04240094-5512223 12.7 8.95 (41.6,17.2)$\\pm $ 2.1 M2.3$\\pm $ 0.5 11.53$\\pm $ 0.03 2.90$\\pm $ 0.25 5.4 4.4 20120718 90 22 04274963-3327010 15.2 10.38 (61.8,-0.7)$\\pm $ 2.5 M4.5$\\pm $ 0.2 13.12$\\pm $ 0.03 2.65$\\pm $ 0.15 8.1 3.6 20130204 600 31 04334610-4511249 12.8 8.90 (58.6,8.1)$\\pm $ 1.4 M2.4$\\pm $ 0.9 11.54$\\pm $ 0.05 2.84$\\pm $ 0.56 1.3 3.6 20130204 300 64 04365738-1613065 12.4 8.26 (78.1,-32.8)$\\pm $ 3.6 M3.0$\\pm $ 0.3 10.89$\\pm $ 0.02 1.80$\\pm $ 0.19 5.0 3.0 20130202 300 88 04435860-3643188 14.8 9.87 (54.1,-2.1)$\\pm $ 2.4 M3.5$\\pm $ 0.2 12.52$\\pm $ 0.02 3.01$\\pm $ 0.17 0.2 3.7 20130204 300 31 04440099-6624036 11.6 8.58 (53.0,30.2)$\\pm $ 4.0 K7.6$\\pm $ 0.2 10.93$\\pm $ 0.02 3.61$\\pm $ 0.09 2.2 3.8 20120718 80 38 04440824-4406473 11.0 8.42 (39.6,5.9)$\\pm $ 2.2 M2.7$\\pm $ 2.4 11.04$\\pm $ 0.18 2.15$\\pm $ 1.38 1.0 4.4 20130204 180 44 04444511-3714380 10.5 8.37 (42.8,0.8)$\\pm $ 1.3 K4.0$\\pm $ 0.6 10.50$\\pm $ 0.04 3.95$\\pm $ 0.07 1.9 4.2 20130204 120 69 04470041-5134405 12.8 9.21 (54.9,14.4)$\\pm $ 3.1 M2.4$\\pm $ 0.4 11.81$\\pm $ 0.03 3.11$\\pm $ 0.24 2.8 3.7 20130204 180 41 04475779-5035200 14.3 10.02 (47.9,17.8)$\\pm $ 3.6 M4.1$\\pm $ 0.3 12.73$\\pm $ 0.03 2.70$\\pm $ 0.28 3.6 3.9 20130204 300 14 04515303-4647309 11.0 8.89 (30.2,11.4)$\\pm $ 1.7 M0.0$\\pm $ 0.3 11.33$\\pm $ 0.03 3.73$\\pm $ 0.10 6.6 4.5 20130202 360 91 05111098-4903597 14.2 9.77 (33,20.4)$\\pm $ 2.3 M3.7$\\pm $ 0.3 12.44$\\pm $ 0.03 2.77$\\pm $ 0.20 7.5 4.3 20130202 300 34 05233951-3227031 8.9 6.94 (38.7,5.1)$\\pm $ 0.9 K4.0$\\pm $ 0.5 9.10$\\pm $ 0.06 2.55$\\pm $ 0.04 7.6 4.0 20130204 138 90 05241818-3622024 12.6 8.90 (35.4,6.0)$\\pm $ 1.2 M2.7$\\pm $ 0.8 11.56$\\pm $ 0.04 2.66$\\pm $ 0.43 4.0 4.1 20130204 60 81 05332558-5117131 10.9 8.16 (44.0,24.2)$\\pm $ 1.8 K7.5$\\pm $ 0.2 10.48$\\pm $ 0.02 3.23$\\pm $ 0.10 1.5 3.6 20130202 720 192 05392505-4245211 12.3 8.60 (40.6,15.9)$\\pm $ 2.1 M1.8$\\pm $ 0.4 11.14$\\pm $ 0.02 2.79$\\pm $ 0.16 2.3 3.6 20130202 300 86 05421278-3738180 11.6 8.50 (27.6,9.3)$\\pm $ 0.9 M1.0$\\pm $ 0.3 11.01$\\pm $ 0.03 3.04$\\pm $ 0.10 4.8 4.3 20130204 60 68 20095193-5526509 14.6 10.32 (11.3,-66.2)$\\pm $ 2.2 M3.5$\\pm $ 0.3 12.96$\\pm $ 0.02 3.45$\\pm $ 0.21 3.9 4.3 20120902 300 38 20143542-5430588 14.1 9.50 (17.4,-171.5)$\\pm $ 13.0 M4.3$\\pm $ 0.2 12.24$\\pm $ 0.04 1.99$\\pm $ 0.18 6.0 2.3 20120902 300 46 20144598-2306214 13.6 8.94 (14.3,-127.0)$\\pm $ 2.5 M3.6$\\pm $ 0.3 11.61$\\pm $ 0.02 2.02$\\pm $ 0.21 6.6 2.4 20120718 120 39 20162190-4137359 9.3 7.27 (6.5,-60.9)$\\pm $ 1.0 K5.6$\\pm $ 0.4 9.48$\\pm $ 0.02 2.67$\\pm $ 0.09 2.2 4.4 20120718 5 36 20175858-5712583 10.4 8.57 (12.7,-61.9)$\\pm $ 1.9 K3.1$\\pm $ 0.5 10.56$\\pm $ 0.04 4.22$\\pm $ 0.07 4.1 4.4 20120902 180 155 20273570-4202324 15.1 10.90 (16.1,-60.7)$\\pm $ 3.2 M3.2$\\pm $ 0.5 13.56$\\pm $ 0.02 4.31$\\pm $ 0.43 4.7 4.4 20120718 600 31 20291446-5456116 15.2 10.36 (14.3,-111.1)$\\pm $ 3.3 M4.6$\\pm $ 0.3 13.13$\\pm $ 0.03 2.56$\\pm $ 0.25 6.3 3.2 20120902 300 27 20421624-5552074 13.0 9.17 (25.4,-85.3)$\\pm $ 1.5 M3.1$\\pm $ 0.3 11.83$\\pm $ 0.02 2.66$\\pm $ 0.22 5.0 3.7 20120902 300 68 20423672-5425263 14.2 9.86 (28.3,-96.5)$\\pm $ 2.8 M3.9$\\pm $ 0.3 12.57$\\pm $ 0.02 2.73$\\pm $ 0.27 5.2 3.4 20120718 220 24 20474501-3635409 8.9 6.79 (20.4,-80.8)$\\pm $ 0.9 K4.3$\\pm $ 0.6 8.95$\\pm $ 0.04 2.37$\\pm $ 0.07 2.3 3.7 20120718 30 102 20583990-4743489 13.3 9.55 (32.2,-83.4)$\\pm $ 0.9 M2.4$\\pm $ 0.4 12.15$\\pm $ 0.03 3.45$\\pm $ 0.24 6.4 3.7 20120718 100 32 21083826-4244540 14.0 9.24 (33.2,-100.7)$\\pm $ 1.6 M4.8$\\pm $ 0.2 12.06$\\pm $ 0.04 1.26$\\pm $ 0.13 3.4 3.2 20120718 360 39 21100614-5811483 14.4 10.07 (24.5,-89.5)$\\pm $ 2.6 M3.8$\\pm $ 0.5 12.75$\\pm $ 0.04 2.99$\\pm $ 0.45 6.5 3.6 20120718 400 33 21143354-4213528 14.9 10.53 (40.7,-82.7)$\\pm $ 3.2 M4.1$\\pm $ 0.2 13.22$\\pm $ 0.02 3.19$\\pm $ 0.19 8.6 3.6 20120718 475 28 21163528-6005124 13.5 9.31 (32.0,-98.1)$\\pm $ 1.3 M3.9$\\pm $ 0.4 12.00$\\pm $ 0.03 2.16$\\pm $ 0.34 4.8 3.4 20120902 300 52 21200112-5328347 12.3 8.25 (52.8,-99.4)$\\pm $ 15.9 M0.0$\\pm $ 0.5 10.70$\\pm $ 0.05 3.10$\\pm $ 0.16 12.8 3.2 20120718 60 57 21273697-4213021 13.8 10.36 (30.7,-55.9)$\\pm $ 1.4 M2.6$\\pm $ 0.2 12.96$\\pm $ 0.03 4.13$\\pm $ 0.08 5.7 4.4 20120718 180 30 21275054-6841033 14.1 9.58 (29.5,-85.5)$\\pm $ 2.4 M4.1$\\pm $ 0.3 12.27$\\pm $ 0.02 2.24$\\pm $ 0.24 7.1 3.6 20120902 300 36 21354554-4218343 15.7 10.81 (46.8,-67.9)$\\pm $ 7.6 M4.9$\\pm $ 0.4 13.64$\\pm $ 0.03 2.74$\\pm $ 0.35 13.1 3.8 20120902 600 23 21370885-6036054 13.1 8.76 (41.3,-91.3)$\\pm $ 1.8 M3.6$\\pm $ 0.4 11.44$\\pm $ 0.03 1.85$\\pm $ 0.32 2.1 3.4 20120718 160 44 21380269-5744583 14.7 9.99 (40.2,-97.3)$\\pm $ 1.5 M4.5$\\pm $ 0.2 12.76$\\pm $ 0.04 2.29$\\pm $ 0.14 5.3 3.3 20120718 475 37 21401098-5317466 13.8 10.19 (25.6,-68.6)$\\pm $ 3.9 M2.6$\\pm $ 0.4 12.82$\\pm $ 0.03 3.99$\\pm $ 0.24 6.7 4.1 20120718 180 27 21490499-6413039 14.2 9.47 (47.8,-96.5)$\\pm $ 2.0 M4.6$\\pm $ 0.3 12.26$\\pm $ 0.03 1.68$\\pm $ 0.25 3.0 3.3 20120718 320 31 21504048-5113380 14.0 9.51 (42.8,-93.7)$\\pm $ 1.6 M4.2$\\pm $ 0.3 12.21$\\pm $ 0.04 2.07$\\pm $ 0.29 6.1 3.4 20120718 300 39 22015342-4623115 8.5 6.30 (30.2,-60.8)$\\pm $ 1.1 K5.8$\\pm $ 0.4 8.50$\\pm $ 0.02 1.65$\\pm $ 0.10 4.8 4.3 20120718 30 130 22021626-4210329 11.3 7.99 (51.8,-93.3)$\\pm $ 0.9 M0.8$\\pm $ 0.6 10.51$\\pm $ 0.05 2.61$\\pm $ 0.17 3.8 3.3 20120718 60 61 22025453-6440441 11.7 8.16 (51.6,-95.3)$\\pm $ 1.7 M2.1$\\pm $ 0.6 10.76$\\pm $ 0.04 2.26$\\pm $ 0.29 5.0 3.3 20120718 60 47 22102820-4431480 13.9 9.95 (39.4,-68.9)$\\pm $ 2.1 M4.0$\\pm $ 0.8 12.63$\\pm $ 0.04 2.71$\\pm $ 0.60 3.6 3.9 20120901 300 40 22223966-6303258 13.6 9.35 (59.0,-87.6)$\\pm $ 2.0 M3.2$\\pm $ 0.3 11.99$\\pm $ 0.02 2.74$\\pm $ 0.21 1.5 3.3 20120718 180 34 22244102-7724036 14.9 10.53 (54.1,-67.2)$\\pm $ 3.3 M4.0$\\pm $ 0.3 13.23$\\pm $ 0.02 3.31$\\pm $ 0.23 2.7 3.6 20120902 300 23 22432875-5515068 14.0 9.73 (81.3,-103.0)$\\pm $ 7.9 M4.2$\\pm $ 0.6 12.46$\\pm $ 0.04 2.32$\\pm $ 0.52 3.0 2.9 20120718 300 35 22444835-6650032 15.2 10.14 (66.8,-80.4)$\\pm $ 3.0 M4.8$\\pm $ 0.3 12.94$\\pm $ 0.04 2.15$\\pm $ 0.30 0.3 3.3 20120718 600 27 22463471-7353504 12.8 8.81 (59.3,-68.2)$\\pm $ 1.1 M3.2$\\pm $ 0.3 11.46$\\pm $ 0.03 2.20$\\pm $ 0.19 2.0 3.6 20120902 180 64 22501826-4651310 8.7 6.65 (40.5,-40.2)$\\pm $ 0.9 K5.2$\\pm $ 0.5 8.85$\\pm $ 0.02 2.13$\\pm $ 0.09 7.1 4.5 20120901 251 254 22545651-7646072 10.6 8.12 (40.5,-43.4)$\\pm $ 1.0 K5.1$\\pm $ 0.5 10.30$\\pm $ 0.03 3.60$\\pm $ 0.08 1.5 4.4 20120902 180 158 23124644-5049240 12.8 8.30 (77.6,-75.7)$\\pm $ 1.2 M4.4$\\pm $ 0.3 11.02$\\pm $ 0.04 0.66$\\pm $ 0.30 3.5 3.3 20120901 180 64 23130558-6127077 14.6 10.05 (68.8,-72.1)$\\pm $ 2.6 M4.6$\\pm $ 0.5 12.84$\\pm $ 0.04 2.27$\\pm $ 0.44 2.4 3.4 20120716 400 30 23131671-4933154 13.4 8.92 (77.4,-90.4)$\\pm $ 1.6 M4.1$\\pm $ 0.3 11.60$\\pm $ 0.03 1.58$\\pm $ 0.28 7.5 3.1 20120716 180 48 23143165-5357285 10.2 7.97 (48.8,-57.1)$\\pm $ 2.0 K5.1$\\pm $ 0.2 10.17$\\pm $ 0.02 3.47$\\pm $ 0.04 5.4 4.1 20120901 120 138 23170011-7432095 13.9 9.56 (81.7,-78.3)$\\pm $ 1.7 M4.1$\\pm $ 0.2 12.25$\\pm $ 0.04 2.22$\\pm $ 0.14 5.0 3.0 20120716 180 35 23273447-8512364 14.5 10.01 (60.6,-50.8)$\\pm $ 3.2 M4.0$\\pm $ 0.3 12.69$\\pm $ 0.02 2.77$\\pm $ 0.23 6.3 3.7 20120902 300 28 23283419-5136527 11.7 9.18 (48.7,-41.6)$\\pm $ 1.0 K7.2$\\pm $ 0.4 11.43$\\pm $ 0.06 4.40$\\pm $ 0.17 2.8 4.4 20120901 180 86 23285763-6802338 12.4 8.38 (67.9,-65.6)$\\pm $ 2.5 M2.9$\\pm $ 0.5 11.04$\\pm $ 0.03 2.02$\\pm $ 0.34 5.8 3.5 20120716 60 45 23291752-6749598 14.1 9.89 (70.5,-67.4)$\\pm $ 2.2 M3.9$\\pm $ 0.4 12.60$\\pm $ 0.03 2.76$\\pm $ 0.30 5.5 3.4 20120718 300 30 23294775-7439325 10.5 9.70 (81.2,-53.8)$\\pm $ 2.1 M4.2$\\pm $ 0.6 12.42$\\pm $ 0.07 2.28$\\pm $ 0.53 8.8 3.4 20120718 300 40 23314492-0244395 13.3 8.67 (93.6,-66.6)$\\pm $ 3.4 M4.5$\\pm $ 0.3 11.42$\\pm $ 0.03 0.95$\\pm $ 0.25 0.9 2.9 20120716 140 36 23334224-4913495 14.7 10.52 (53.3,-36.5)$\\pm $ 3.7 M2.9$\\pm $ 0.4 13.16$\\pm $ 0.03 4.14$\\pm $ 0.27 8.7 4.4 20120901 300 37 23382851-6749025 14.6 10.05 (68.5,-56.4)$\\pm $ 3.2 M3.9$\\pm $ 0.3 12.73$\\pm $ 0.02 2.89$\\pm $ 0.22 1.5 3.6 20120718 500 41 23424333-6224564 14.9 10.41 (80.9,-61.6)$\\pm $ 5.2 M4.4$\\pm $ 0.2 13.18$\\pm $ 0.02 2.82$\\pm $ 0.21 3.5 3.3 20120718 500 30 23452225-7126505 13.9 9.32 (76.5,-64.0)$\\pm $ 1.8 M3.8$\\pm $ 0.3 12.01$\\pm $ 0.02 2.26$\\pm $ 0.26 6.8 3.3 20120718 225 39 23474694-6517249 11.6 8.17 (80.7,-66.4)$\\pm $ 1.2 M1.5$\\pm $ 0.3 10.74$\\pm $ 0.02 2.54$\\pm $ 0.13 4.5 3.3 20120716 40 45 23524562-5229593 15.3 10.71 (76.4,-82.4)$\\pm $ 10.4 M5.3$\\pm $ 0.3 13.54$\\pm $ 0.02 2.21$\\pm $ 0.27 17.1 3.2 20120901 300 16 23541799-8254492 14.0 10.00 (73.9,-35.5)$\\pm $ 2.0 M3.3$\\pm $ 0.6 12.64$\\pm $ 0.04 3.30$\\pm $ 0.36 6.3 3.6 20120902 300 40 23570417-0337559 14.4 10.03 (71.6,-53.5)$\\pm $ 3.3 M3.3$\\pm $ 0.3 12.69$\\pm $ 0.02 3.35$\\pm $ 0.21 5.4 3.6 20120716 300 34 23585674-8339423 11.0 8.25 (67.2,-37.9)$\\pm $ 1.4 K7.5$\\pm $ 0.2 10.58$\\pm $ 0.03 3.33$\\pm $ 0.12 1.8 3.8 20120902 180 109 lrrrcrrrccccccc 0pt Spectroscopic Results and Membership Assessments Name $v_{rad}$ $\\Delta v_{rad}$ $vsin(i)$ SpT $EW[H\\alpha ]$ $EW[Li_{6708}]$ 2cAssessments (km/s) (km/s) (km/s) (Å) (mÅ) RV,H$\\alpha $ ,Li Final AF Hor12.6$\\pm $ 0.7 0.30 6.6$\\pm $ 0.8 M2.1 -3.96 14.8 Y,?,?", "Y - RV CD-35 116713.2$\\pm $ 0.3 -0.33 4.6$\\pm $ 1.2 K4.7 -0.32 48.4 Y,?,?", "Y - RV CD-44 117315.1$\\pm $ 0.5 0.36 7.5$\\pm $ 0.9 K5.2 -1.17 213.0 Y,?,Y Y - Li CD-53 54411.9$\\pm $ 4.6 -0.46 41.7$\\pm $ 5.3 K5.3 -2.01 185.2 Y,?,Y Y - Li CD-58 55312.2$\\pm $ 0.3 -0.37 4.8$\\pm $ 0.2 K5.0 -0.68 130.2 Y,?,Y Y - Li CD-60 41610.4$\\pm $ 0.3 -0.67 6.7$\\pm $ 0.2 K4.1 -0.25 210.9 Y,?,Y Y - Li CD-78 2411.6$\\pm $ 0.6 1.51 10.2$\\pm $ 0.3 K3.8 -0.16 259.5 Y,?,Y Y - Li CT Tuc8.4$\\pm $ 0.4 1.45 6.8$\\pm $ 0.6 K5.7 -0.68 4.8 Y,?,?", "Y - RV HD 322110.7$\\pm $ 4.9 3.26 68.8$\\pm $ 4.0 K4.6 -1.07 375.0 Y,?,Y Y - Li HIP 1073453.3$\\pm $ 0.3 1.67 5.9$\\pm $ 0.5 K7.2 -1.48 21.6 Y,?,?", "Y - RV HIP 19106.1$\\pm $ 0.6 -0.91 11.5$\\pm $ 0.7 K7.2 -1.76 179.7 Y,?,Y Y - Li TYC 8083-045519.2$\\pm $ 0.3 1.03 5.6$\\pm $ 0.6 K5.7 -1.15 10.6 Y,?,?", "Y - RV TYC 9344-02939.2$\\pm $ 2.5 1.36 33.1$\\pm $ 2.4 K7.7 -3.38 132.9 Y,?,Y Y - Li J00123485-592746498.0$\\pm $ 0.2 91.99 4.0$\\pm $ 0.2 K3.1 1.19 -24.7 N,N,N N - Li J00125703-79520739.4$\\pm $ 0.7 -0.57 9.3$\\pm $ 0.7 M2.9 -4.95 15.5 Y,?,?", "Y - RV J00144767-60034775.1$\\pm $ 3.3 -1.13 29.8$\\pm $ 3.0 M3.6 -3.53 18.9 Y,?,?", "Y - RV J00152752-64145456.7$\\pm $ 0.3 -0.45 6.5$\\pm $ 0.4 M1.8 -3.22 48.5 Y,?,?", "Y - RV J00155556-613751919.9$\\pm $ 0.7 13.24 8.9$\\pm $ 0.5 M3.0 -4.03 -0.6 N,?,?", "N?", "- RV J00173041-5957044-1.9$\\pm $ 0.3 -8.19 4.6$\\pm $ 0.4 K7.1 0.59 -24.0 N,N,?", "N - H$\\alpha $ J00220446-10161911.0$\\pm $ 1.5 5.59 18.8$\\pm $ 1.9 K7.0 -1.76 17.4 N,?,?", "N?", "- RV J00235732-55314355.3$\\pm $ 0.7 -0.44 6.9$\\pm $ 0.5 M4.1 -5.05 24.4 Y,?,?", "Y - RV J00273330-61571696.5$\\pm $ 2.1 -0.66 20.8$\\pm $ 1.8 M4.0 -9.18 31.6 Y,?,?", "Y - RV J00275023-32330609.5$\\pm $ 0.3 8.59 5.5$\\pm $ 0.4 M2.6 -5.50 -0.8 N,?,?", "N?", "- RV J00284683-67514467.6$\\pm $ 1.5 -0.63 13.0$\\pm $ 2.1 M4.5 -8.37 264.7 Y,?,Y Y - Li J00302572-62360159.4$\\pm $ 0.7 2.07 11.2$\\pm $ 0.8 M2.2 -4.51 -21.7 Y,?,?", "Y - RV J00305785-655005813.6$\\pm $ 0.9 5.61 12.4$\\pm $ 1.4 M3.6 -5.96 -18.7 N,?,?", "N?", "- RV J00332438-51164337.3$\\pm $ 2.1 2.03 39.1$\\pm $ 2.6 M3.4 -4.72 -2.3 Y,?,?", "Y - RV J00382147-461104315.0$\\pm $ 0.8 10.53 9.5$\\pm $ 0.9 M3.8 -10.97 33.2 N,?,?", "N?", "- RV J00393579-38165843.3$\\pm $ 0.4 0.32 5.3$\\pm $ 0.4 M1.4 -3.52 20.6 Y,?,?", "Y - RV J00394063-62241256.7$\\pm $ 2.3 -0.99 19.1$\\pm $ 3.3 M4.9 -8.70 495.0 Y,?,Y Y - Li J00421010-54444315.9$\\pm $ 0.9 -0.55 6.7$\\pm $ 0.8 M2.9 -6.26 -31.0 Y,?,?", "Y - RV J00421092-42525458.7$\\pm $ 0.1 4.59 6.2$\\pm $ 0.5 M2.2 -5.24 -6.5 N,?,?", "N?", "- RV J00425349-61173846.9$\\pm $ 1.0 -0.70 7.1$\\pm $ 0.5 M4.2 -6.75 79.3 Y,?,?", "Y - RV J00485254-65263309.9$\\pm $ 0.4 1.32 12.1$\\pm $ 0.7 M3.2 -5.94 -59.3 Y,?,?", "Y - RV J00493566-63474168.1$\\pm $ 0.3 -0.18 5.5$\\pm $ 0.4 M1.7 -3.38 -24.4 Y,?,?", "Y - RV J00502644-462853913.6$\\pm $ 1.3 8.26 5.9$\\pm $ 2.7 K3.1 0.84 18.3 N,N,N N - Li J00514081-59133206.3$\\pm $ 1.3 -1.34 20.6$\\pm $ 2.4 M4.4 -6.01 257.2 Y,?,Y Y - Li J00555140-493821618.2$\\pm $ 1.5 11.96 14.9$\\pm $ 1.1 M3.9 -5.88 -23.6 N,?,?", "N?", "- RV J00590177-605412410.2$\\pm $ 0.2 1.94 4.4$\\pm $ 0.3 M0.5 0.29 -12.1 Y,N,?", "N - H$\\alpha $ J01024375-62353447.0$\\pm $ 2.0 -1.63 5.7$\\pm $ 0.1 M2.9 -3.61 -3.9 Y,?,?", "Y - RV J01033563-55155617.3$\\pm $ 2.6 -0.25 19.2$\\pm $ 2.4 M5.0 -18.71 -57.8 Y,?,?", "Y - RV J01071194-19353599.3$\\pm $ 0.5 8.25 6.8$\\pm $ 0.6 M1.0 -2.49 323.3 N,?,Y Y - Li J01101448-47154538.9$\\pm $ 0.7 2.33 4.5$\\pm $ 0.8 M2.0 0.38 -35.7 Y,N,?", "N - H$\\alpha $ J01125587-713028312.9$\\pm $ 0.3 2.78 4.5$\\pm $ 0.4 K7.3 0.53 -6.8 Y,N,?", "N - H$\\alpha $ J01134031-593934611.9$\\pm $ 6.7 3.19 46.8$\\pm $ 5.4 M3.7 -16.51 -14.3 Y,?,?", "Y - RV J01160045-674731111.5$\\pm $ 1.4 1.72 16.5$\\pm $ 1.7 M4.1 -6.82 -49.3 Y,?,?", "Y - RV J01180670-62585919.3$\\pm $ 1.3 -0.01 14.4$\\pm $ 2.6 M5.1 -8.68 360.5 Y,?,Y Y - Li J01211297-61172818.7$\\pm $ 5.5 -0.58 21.9$\\pm $ 5.1 M4.1 -6.77 1.2 Y,?,?", "Y - RV J01211813-543424524.0$\\pm $ 0.3 15.60 4.0$\\pm $ 0.2 K5.0 0.63 0.5 N,N,?", "N - H$\\alpha $ J01224511-63184467.8$\\pm $ 1.4 -1.76 16.9$\\pm $ 1.4 M3.5 -6.63 52.1 Y,?,?", "Y - RV J01233280-41131105.7$\\pm $ 1.4 -0.71 12.7$\\pm $ 0.7 M4.2 -5.55 -41.2 Y,?,?", "Y - RV J01245895-7953375170.7$\\pm $ 0.3 159.64 4.8$\\pm $ 0.4 F9.5 1.23 25.3 N,N,N N - Li J01253196-66460237.1$\\pm $ 5.1 -2.90 41.1$\\pm $ 5.1 M4.2 -4.92 -38.0 Y,?,?", "Y - RV J01275875-60322439.1$\\pm $ 2.5 -0.34 26.1$\\pm $ 2.9 M4.2 -7.58 24.8 Y,?,?", "Y - RV J01283025-49210946.5$\\pm $ 5.7 -1.49 59.9$\\pm $ 3.9 M4.1 -7.71 19.1 Y,?,?", "Y - RV J01301454-294917524.9$\\pm $ 0.7 20.02 4.9$\\pm $ 0.5 M2.3 0.24 4.8 N,N,?", "N - H$\\alpha $ J01321522-503430723.4$\\pm $ 1.4 14.97 11.5$\\pm $ 1.5 M2.2 -0.39 37.3 N,N,?", "N - H$\\alpha $ J01344601-570756411.1$\\pm $ 6.3 1.79 26.0$\\pm $ 8.3 M4.9 -20.43 526.1 Y,?,Y Y - Li J01351393-07125178.2$\\pm $ 2.5 7.27 27.3$\\pm $ 1.7 M3.8 -10.30 -13.8 Y,?,?", "Y - RV J01372781-455826113.5$\\pm $ 1.4 5.51 8.5$\\pm $ 0.8 M5.0 -6.61 249.4 N,?,Y Y - Li J01375879-56454478.5$\\pm $ 0.6 -0.92 9.8$\\pm $ 0.7 M3.9 -8.83 -36.4 Y,?,?", "Y - RV J01380029-460339811.6$\\pm $ 0.2 3.55 4.1$\\pm $ 0.2 M2.4 -0.45 -16.4 N,N,?", "N - H$\\alpha $ J01380311-590404210.1$\\pm $ 0.6 0.39 8.6$\\pm $ 0.5 M3.1 -13.30 -35.1 Y,?,?", "Y - RV J01504543-57164889.3$\\pm $ 1.7 -0.80 9.7$\\pm $ 1.3 M5.5 -9.46 673.2 Y,?,Y Y - Li J01505688-584403211.1$\\pm $ 0.5 0.85 10.1$\\pm $ 0.7 M3.0 -6.38 -17.5 Y,?,?", "Y - RV J01521830-595016810.3$\\pm $ 0.3 -0.12 5.0$\\pm $ 0.3 M1.6 -3.02 -12.3 Y,?,?", "Y - RV J01532494-68332269.8$\\pm $ 1.4 -1.26 11.6$\\pm $ 1.6 M4.5 -14.45 -54.6 Y,?,?", "Y - RV J01570140-772122154.4$\\pm $ 0.7 42.85 4.7$\\pm $ 0.6 M2.1 0.33 -18.7 N,N,?", "N - H$\\alpha $ J02000918-802500915.5$\\pm $ 0.2 3.92 4.8$\\pm $ 0.1 K3.7 0.53 126.8 N,?,Y Y - Li J02001277-08405164.5$\\pm $ 0.4 1.05 8.9$\\pm $ 0.4 M2.1 -3.97 -5.3 Y,?,?", "Y - RV J02001992-661401711.8$\\pm $ 1.1 0.63 16.2$\\pm $ 1.2 M4.0 -6.19 -41.8 Y,?,?", "Y - RV J02045317-53461628.4$\\pm $ 0.3 -2.08 6.0$\\pm $ 0.5 M3.8 -9.26 -29.8 Y,?,?", "Y - RV J02070176-440638011.1$\\pm $ 2.0 1.47 4.7$\\pm $ 0.3 M1.9 -2.62 -7.1 Y,?,?", "Y - RV J02105538-460358824.7$\\pm $ 0.9 14.60 19.8$\\pm $ 1.1 K4.2 -0.76 215.3 N,?,Y Y - Li J02125819-58511829.1$\\pm $ 0.8 -2.13 12.1$\\pm $ 0.9 M1.9 -5.43 28.9 Y,?,?", "Y - RV J02153328-562717511.3$\\pm $ 5.7 0.10 47.3$\\pm $ 19.5 M6.2 -13.69 177.7 Y,?,Y Y - Li J02155892-092912110.1$\\pm $ 0.6 5.18 9.9$\\pm $ 0.9 M1.9 -4.69 15.1 N,?,?", "N?", "- RV J02180960-665752411.0$\\pm $ 1.2 -0.85 14.0$\\pm $ 2.7 M4.5 -6.76 375.1 Y,?,Y Y - Li J02192210-392522510.6$\\pm $ 0.7 0.71 6.5$\\pm $ 0.4 M4.9 -7.02 639.9 Y,?,Y Y - Li J02201988-6855014-15.5$\\pm $ 0.4 -27.41 4.1$\\pm $ 0.3 M3.0 0.19 8.7 N,N,?", "N - H$\\alpha $ J02205139-582341112.1$\\pm $ 0.6 0.52 11.3$\\pm $ 0.9 M3.2 -8.94 -12.1 Y,?,?", "Y - RV J02224418-602247616.2$\\pm $ 1.5 4.42 16.1$\\pm $ 2.7 M3.4 -4.40 48.9 N,?,?", "N?", "- RV J02234926-4238512-17.9$\\pm $ 0.3 -28.42 4.0$\\pm $ 0.2 K1.0 1.09 -9.2 N,N,N N - Li J02242453-703321111.8$\\pm $ 0.3 -0.30 5.5$\\pm $ 0.7 M3.3 -8.61 -5.8 Y,?,?", "Y - RV J02294569-554149611.5$\\pm $ 1.0 -0.34 12.5$\\pm $ 1.3 M4.8 -6.41 499.6 Y,?,Y Y - Li J02294869-690604413.0$\\pm $ 1.2 0.77 14.8$\\pm $ 3.1 M4.6 -11.64 566.0 Y,?,Y Y - Li J02303239-434223212.2$\\pm $ 0.5 1.14 5.7$\\pm $ 0.2 K4.4 0.02 34.2 Y,?,?", "Y - RV J02304370-5811560-16.0$\\pm $ 2.0 -28.04 4.6$\\pm $ 0.4 M1.7 0.25 40.2 N,N,?", "N - H$\\alpha $ J02321934-574611711.2$\\pm $ 0.7 -0.94 11.7$\\pm $ 0.7 M4.1 -6.55 -20.6 Y,?,?", "Y - RV J02341866-512846210.9$\\pm $ 0.9 -1.02 11.2$\\pm $ 1.2 M4.3 -8.33 -40.1 Y,?,?", "Y - RV J02351646-5049133-14.9$\\pm $ 0.3 -26.80 4.2$\\pm $ 0.1 K3.8 0.82 -5.5 N,N,N N - Li J02372562-4912033-0.6$\\pm $ 0.4 -12.48 4.8$\\pm $ 0.3 M0.3 0.54 -7.9 N,N,?", "N - H$\\alpha $ J02383255-752806512.3$\\pm $ 0.6 0.00 8.7$\\pm $ 1.2 M4.1 -8.79 -88.9 Y,?,?", "Y - RV J02412721-304914918.2$\\pm $ 1.1 7.80 12.8$\\pm $ 1.3 M4.3 -8.83 0.4 N,?,?", "N?", "- RV J02420204-535914711.5$\\pm $ 2.3 -0.94 22.9$\\pm $ 2.4 M4.3 -9.41 -27.3 Y,?,?", "Y - RV J02420404-535900011.9$\\pm $ 1.6 -0.51 19.4$\\pm $ 1.9 M4.3 -8.94 -23.0 Y,?,?", "Y - RV J02441466-522131814.7$\\pm $ 0.3 2.24 4.6$\\pm $ 0.3 G2.0 1.22 29.1 Y,N,N N - Li J02474639-580427213.1$\\pm $ 0.5 0.26 4.8$\\pm $ 0.2 M1.8 -2.68 -8.6 Y,?,?", "Y - RV J02485260-340424623.3$\\pm $ 0.5 12.04 7.8$\\pm $ 1.4 M3.1 -9.97 5.1 N,?,?", "N?", "- RV J02502222-654555215.0$\\pm $ 0.5 2.12 8.7$\\pm $ 0.5 M3.2 -7.09 -18.8 Y,?,?", "Y - RV J02505959-34090509.9$\\pm $ 4.0 -1.52 26.1$\\pm $ 2.0 M3.7 -6.19 6.9 Y,?,?", "Y - RV J02523550-783118312.8$\\pm $ 1.3 0.32 11.6$\\pm $ 1.8 M4.4 -7.47 45.0 Y,?,?", "Y - RV J02543316-510831313.8$\\pm $ 0.4 0.89 5.2$\\pm $ 0.3 M1.1 -2.46 -1.2 Y,?,?", "Y - RV J02553178-570252213.3$\\pm $ 0.9 0.19 9.7$\\pm $ 1.0 M4.3 -6.81 -82.9 Y,?,?", "Y - RV J02564708-634302716.7$\\pm $ 4.7 3.49 27.0$\\pm $ 3.5 M3.6 -9.32 9.2 Y,?,?", "Y - RV J02572682-634129317.8$\\pm $ 2.1 4.56 25.9$\\pm $ 1.8 M3.6 -4.27 -22.6 Y,?,?", "Y - RV J02590284-612000012.1$\\pm $ 1.1 -1.21 12.7$\\pm $ 1.5 M3.9 -5.46 -43.3 Y,?,?", "Y - RV J02591904-512234111.0$\\pm $ 2.3 -2.21 18.2$\\pm $ 3.4 M5.4 -32.11 689.5 Y,?,Y Y - Li J02592564-294727525.2$\\pm $ 0.1 13.61 4.5$\\pm $ 0.2 K3.2 0.93 -11.3 N,N,N N - Li J03050556-531718212.1$\\pm $ 2.2 -1.40 20.7$\\pm $ 2.5 M4.7 -8.95 -44.8 Y,?,?", "Y - RV J03050976-372505814.2$\\pm $ 0.5 1.43 8.9$\\pm $ 1.0 M1.4 -3.33 11.3 Y,?,?", "Y - RV J03083950-384436317.5$\\pm $ 2.3 4.44 15.9$\\pm $ 0.8 M4.3 -6.35 39.5 Y,?,?", "Y - RV J03093877-301435212.5$\\pm $ 2.3 0.13 23.8$\\pm $ 3.0 M4.7 -5.97 540.6 Y,?,Y Y - Li J03104941-361647113.8$\\pm $ 1.6 0.78 16.4$\\pm $ 1.7 M3.9 -9.73 34.1 Y,?,?", "Y - RV J03114544-471950111.3$\\pm $ 0.5 -2.36 5.2$\\pm $ 0.2 M3.2 -4.27 7.4 Y,?,?", "Y - RV J03204757-504133017.4$\\pm $ 0.3 3.18 4.4$\\pm $ 0.2 M1.5 -0.79 -2.0 N,N,?", "N - H$\\alpha $ J03210395-681647513.4$\\pm $ 0.8 -0.43 9.5$\\pm $ 0.8 M4.0 -6.83 5.6 Y,?,?", "Y - RV J03244056-390422716.2$\\pm $ 4.1 2.17 36.1$\\pm $ 2.7 M3.7 -9.93 -22.8 Y,?,?", "Y - RV J03271701-612840727.0$\\pm $ 0.3 12.62 4.3$\\pm $ 0.2 G6.4 1.15 18.5 N,N,N N - Li J03285469-333919281.1$\\pm $ 0.1 67.09 4.2$\\pm $ 0.2 K3.1 1.15 -20.6 N,N,N N - Li J03291649-370250213.0$\\pm $ 2.0 -1.24 20.4$\\pm $ 2.7 M4.1 -10.01 -36.3 Y,?,?", "Y - RV J03312105-5955006-4.4$\\pm $ 0.3 -18.95 4.2$\\pm $ 0.2 K2.6 1.03 -2.7 N,N,N N - Li J03512287-515458217.2$\\pm $ 1.3 1.51 15.0$\\pm $ 1.4 M4.0 -8.50 42.7 Y,?,?", "Y - RV J03561624-391521916.7$\\pm $ 0.7 0.81 6.3$\\pm $ 0.9 M4.1 -10.97 46.3 Y,?,?", "Y - RV J04000382-290216514.2$\\pm $ 5.3 -1.42 69.8$\\pm $ 6.6 K4.6 -2.26 347.9 Y,?,Y Y - Li J04000395-290228015.8$\\pm $ 0.3 0.17 6.9$\\pm $ 0.3 K4.1 -0.66 207.3 Y,?,Y Y - Li J04013874-312747215.1$\\pm $ 1.6 -0.78 17.9$\\pm $ 1.8 M4.9 -9.28 528.4 Y,?,Y Y - Li J04021648-152129714.3$\\pm $ 0.4 0.01 5.4$\\pm $ 0.2 K3.4 0.06 218.0 Y,?,Y Y - Li J04074372-682511117.4$\\pm $ 1.1 2.27 15.9$\\pm $ 1.4 M3.2 -5.40 -29.7 Y,?,?", "Y - RV J04082685-784447116.8$\\pm $ 0.5 3.24 8.2$\\pm $ 0.6 K7.2 -1.58 7.5 N,?,?", "N?", "- RV J04133314-523158618.4$\\pm $ 0.2 1.69 4.4$\\pm $ 0.3 M2.4 -2.45 -18.1 Y,?,?", "Y - RV J04133609-441332516.4$\\pm $ 1.4 -0.48 15.7$\\pm $ 0.7 M3.6 -7.38 -9.5 Y,?,?", "Y - RVa J04213904-723356215.6$\\pm $ 0.4 0.75 5.6$\\pm $ 0.4 M2.1 -4.05 11.2 Y,?,?", "Y - RV J04240094-551222319.0$\\pm $ 0.7 2.12 6.9$\\pm $ 0.9 M2.0 -3.54 21.2 Y,?,?", "Y - RV J04274963-332701018.8$\\pm $ 1.4 1.25 15.8$\\pm $ 1.3 M4.0 -7.33 -23.3 Y,?,?", "Y - RV J04334610-451124921.0$\\pm $ 0.3 3.10 4.6$\\pm $ 0.2 M1.8 -1.36 -9.2 N,N,?", "N - H$\\alpha $ J04365738-161306516.6$\\pm $ 1.9 -0.05 27.5$\\pm $ 1.3 M3.3 -7.28 3.4 Y,?,?", "Y - RV J04435860-364318819.4$\\pm $ 0.5 0.98 8.5$\\pm $ 0.6 M3.6 -7.99 -3.3 Y,?,?", "Y - RV J04440099-662403616.0$\\pm $ 0.5 -0.26 5.6$\\pm $ 0.4 M0.0 -1.37 21.6 Y,?,?", "Y - RV J04440824-440647324.6$\\pm $ 0.5 6.23 5.2$\\pm $ 0.5 M0.9 -1.71 14.6 N,?,?", "N?", "- RV J04444511-371438064.6$\\pm $ 0.3 46.14 4.0$\\pm $ 0.2 G6.3 1.17 10.8 N,N,N N - Li J04470041-513440519.9$\\pm $ 0.3 1.86 5.1$\\pm $ 0.3 M1.9 -2.64 7.5 Y,?,?", "Y - RV J04475779-503520018.6$\\pm $ 0.9 0.39 12.2$\\pm $ 1.0 M4.0 -8.13 -16.6 Y,?,?", "Y - RV J04515303-464730923.2$\\pm $ 2.0 4.64 6.5$\\pm $ 0.9 K7.8 -1.33 -8.3 Y,?,?", "Y - RV J05111098-490359721.5$\\pm $ 0.4 2.37 8.2$\\pm $ 0.6 M3.2 -7.72 -15.0 Y,?,?", "Y - RV J05233951-322703174.0$\\pm $ 0.2 53.81 7.2$\\pm $ 0.3 G6.0 1.01 -9.3 N,N,N N - Li J05241818-362202439.5$\\pm $ 0.2 19.26 4.4$\\pm $ 0.3 G5.3 1.23 4.2 N,N,N N - Li J05332558-5117131-1.6$\\pm $ 2.0 -21.24 5.1$\\pm $ 0.2 K4.9 -0.90 179.1 N,?,Y Y - Li J05392505-424521121.7$\\pm $ 0.2 1.20 5.1$\\pm $ 0.3 M1.7 -2.88 -11.2 Y,?,?", "Y - RV J05421278-373818075.2$\\pm $ 0.6 54.35 5.3$\\pm $ 1.1 K3.0 1.05 -44.8 N,N,N N - Li J20095193-552650912.9$\\pm $ 2.1 14.80 21.2$\\pm $ 1.5 M3.6 -6.85 10.9 N,?,?", "N?", "- RV J20143542-5430588-4.5$\\pm $ 1.0 -2.28 11.5$\\pm $ 0.8 M3.8 -4.93 -7.6 Y,?,?", "Y - RV J20144598-2306214-18.7$\\pm $ 0.3 -4.91 5.0$\\pm $ 0.4 M3.0 -4.08 -1.7 N,?,?", "N?", "- RV J20162190-4137359-206.1$\\pm $ 1.0 -198.82 4.2$\\pm $ 1.3 K3.2 1.23 6.8 N,N,N N - Li J20175858-5712583-7.1$\\pm $ 0.3 -6.02 4.1$\\pm $ 0.1 K2.7 0.95 -4.5 N,N,N N - Li J20273570-42023245.7$\\pm $ 0.6 12.66 7.8$\\pm $ 1.3 M3.5 -9.02 -29.2 N,?,?", "N?", "- RV J20291446-5456116-1.4$\\pm $ 1.2 0.51 10.8$\\pm $ 0.9 M4.3 -7.69 -33.7 Y,?,?", "Y - RV J20421624-5552074-9.7$\\pm $ 0.2 -8.28 4.4$\\pm $ 0.2 M2.3 -3.41 -12.7 N,?,?", "N?", "- RV J20423672-5425263-1.4$\\pm $ 1.7 0.62 13.5$\\pm $ 2.3 M4.0 -5.69 67.8 Y,?,?", "Y - RV J20474501-3635409-8.3$\\pm $ 7.9 0.33 75.3$\\pm $ 8.2 G7.4 0.16 339.1 Y,?,Y Y - Li J20583990-474348928.2$\\pm $ 0.4 32.46 4.8$\\pm $ 0.2 M1.4 0.26 17.2 N,N,?", "N - H$\\alpha $ J21083826-4244540-4.9$\\pm $ 1.9 1.01 17.7$\\pm $ 1.9 M4.4 -10.72 -16.9 Y,?,?", "Y - RV J21100614-58114830.8$\\pm $ 1.1 0.93 15.1$\\pm $ 0.9 M4.0 -7.24 19.3 Y,?,?", "Y - RV J21143354-42135284.1$\\pm $ 3.5 10.06 20.1$\\pm $ 2.0 M3.9 -6.51 3.8 Y,?,?", "Y - RV J21163528-60051240.3$\\pm $ 0.9 -0.45 14.2$\\pm $ 0.9 M3.5 -5.20 9.9 Y,?,?", "Y - RV J21200112-5328347-50.2$\\pm $ 0.5 -48.5 5.1$\\pm $ 0.6 K4.8 1.14 -59.1 N,N,?", "N - H$\\alpha $ J21273697-42130212.5$\\pm $ 0.3 8.12 4.8$\\pm $ 0.2 M1.4 0.27 -46.0 N,N,?", "N - H$\\alpha $ J21275054-68410337.0$\\pm $ 3.4 2.83 31.5$\\pm $ 2.7 M4.2 -8.09 38.7 Y,?,?", "Y - RV J21354554-42183430.9$\\pm $ 1.4 6.25 9.1$\\pm $ 0.7 M5.2 -12.27 634.0 N,?,Y Y - Li J21370885-60360540.2$\\pm $ 0.4 -1.14 6.0$\\pm $ 0.5 M3.0 -7.02 15.5 Y,?,?", "Y - RV J21380269-5744583-0.5$\\pm $ 1.3 -0.80 15.6$\\pm $ 1.3 M3.7 -4.70 -33.1 Y,?,?", "Y - RV J21401098-5317466-19.1$\\pm $ 0.4 -17.83 4.7$\\pm $ 0.3 M1.9 0.31 6.2 N,N,?", "N - H$\\alpha $ J21490499-64130390.4$\\pm $ 5.1 -2.46 47.7$\\pm $ 7.5 M4.4 -7.22 -35.2 Y,?,?", "Y - RV J21504048-5113380-1.1$\\pm $ 0.8 0.59 12.5$\\pm $ 0.8 M3.7 -6.59 -7.0 Y,?,?", "Y - RV J22015342-462311523.4$\\pm $ 0.3 26.46 4.2$\\pm $ 0.3 K3.2 1.16 -12.5 N,N,N N - Li J22021626-4210329-2.8$\\pm $ 0.3 1.67 6.4$\\pm $ 0.3 M0.7 -1.95 -5.8 Y,?,?", "Y - RV J22025453-64404412.2$\\pm $ 5.3 -1.15 44.9$\\pm $ 4.3 M1.8 -3.10 -15.2 Y,?,?", "Y - RV J22102820-44314807.9$\\pm $ 1.6 11.28 18.2$\\pm $ 2.0 M3.4 -6.59 26.4 N,?,?", "N?", "- RV J22223966-63032584.5$\\pm $ 1.0 1.26 11.9$\\pm $ 0.9 M3.5 -9.40 -7.7 Y,?,?", "Y - RV J22244102-77240368.5$\\pm $ 1.4 0.61 13.6$\\pm $ 1.4 M4.2 -6.75 -18.7 Y,?,?", "Y - RV J22432875-5515068-18.4$\\pm $ 0.4 -19.82 4.5$\\pm $ 0.3 M2.9 0.20 17.1 N,N,?", "N - H$\\alpha $ J22444835-66500320.7$\\pm $ 1.7 -4.34 15.4$\\pm $ 1.7 M4.8 -7.21 510.7 Y,?,Y Y - Li J22463471-73535049.1$\\pm $ 0.6 1.90 10.5$\\pm $ 0.6 M2.3 -4.26 15.8 Y,?,?", "Y - RV J22501826-4651310-33.4$\\pm $ 0.3 -32.46 3.7$\\pm $ 0.2 K3.0 0.98 6.6 N,N,N N - Li J22545651-764607223.9$\\pm $ 0.5 15.83 3.9$\\pm $ 0.2 K2.7 0.95 -7.2 N,N,N N - Li J23124644-50492404.1$\\pm $ 11.9 2.88 99.3$\\pm $ 36.4 M3.9 -8.81 27.2 Y,?,?", "Y - RV J23130558-61270772.9$\\pm $ 2.3 -1.40 25.8$\\pm $ 2.8 M4.5 -7.59 185.3 Y,?,Y Y - Li J23131671-49331540.3$\\pm $ 0.7 -0.54 8.1$\\pm $ 0.9 M3.5 -12.64 31.4 Y,?,?", "Y - RV J23143165-535728514.6$\\pm $ 0.3 12.42 4.0$\\pm $ 0.1 K4.9 0.65 9.3 N,N,?", "N - Ha J23170011-74320958.3$\\pm $ 2.0 0.43 14.9$\\pm $ 1.2 M3.6 -6.57 22.8 Y,?,?", "Y - RV J23273447-851236411.7$\\pm $ 0.7 1.17 7.3$\\pm $ 0.5 M3.8 -8.77 16.5 Y,?,?", "Y - RV J23283419-51365279.5$\\pm $ 0.3 7.34 4.3$\\pm $ 0.2 K4.9 0.26 16.8 N,?,?", "N?", "- RV J23285763-68023388.0$\\pm $ 1.5 1.43 25.1$\\pm $ 1.8 M2.3 -5.33 20.2 Y,?,?", "Y - RV J23291752-67495986.1$\\pm $ 0.5 -0.43 6.2$\\pm $ 0.5 M3.5 -9.07 29.3 Y,?,?", "Y - RV J23294775-7439325-8.0$\\pm $ 0.3 -16.14 4.5$\\pm $ 0.2 M2.7 0.22 21.9 N,N,?", "N - H$\\alpha $ J23314492-0244395-5.9$\\pm $ 0.8 4.26 5.5$\\pm $ 0.7 M3.7 -17.67 -39.1 N,?,?", "N?", "- RV J23334224-491349514.9$\\pm $ 0.1 13.11 4.4$\\pm $ 0.1 M1.9 -0.89 -22.7 N,N,?", "N - H$\\alpha $ J23382851-67490256.8$\\pm $ 1.8 0.09 24.1$\\pm $ 1.9 M4.0 -5.75 -14.3 Y,?,?", "Y - RV J23424333-62245645.1$\\pm $ 4.6 -0.44 17.8$\\pm $ 1.9 M4.3 -36.76 34.6 Y,?,?", "Y - RV J23452225-71265058.0$\\pm $ 0.6 0.28 8.0$\\pm $ 0.7 M3.4 -5.98 10.1 Y,?,?", "Y - RV J23474694-65172496.1$\\pm $ 0.3 -0.37 5.3$\\pm $ 0.4 M1.0 -2.36 0.6 Y,?,?", "Y - RV J23524562-52295933.1$\\pm $ 0.7 -0.44 6.9$\\pm $ 0.6 M4.6 -8.34 528.6 Y,?,Y Y - Li J23541799-82544927.8$\\pm $ 0.6 -2.46 4.2$\\pm $ 0.2 M2.5 0.31 -14.1 Y,N,?", "N - H$\\alpha $ J23570417-0337559-5.5$\\pm $ 0.3 2.60 4.8$\\pm $ 0.4 M3.0 -4.60 24.8 Y,?,?", "Y - RV J23585674-833942311.1$\\pm $ 0.5 0.65 4.2$\\pm $ 0.2 K5.8 -0.28 10.6 Y,?,?", "Y - RV The spectroscopic spectral types are likely uncertain by $\\pm $ 1 subclass, based on the shape of the $\\chi _{\\nu }^2$ surfaces shown in Figure 3.", "Based on the scatter for field objects in Figure 6 and Figure 7, the uncertainties in equivalent widths are $\\pm $ 0.05 Å for Li$_{6708}$ and $\\pm $ 0.1 Å for H$\\alpha $ .", "The final column lists our assessment of an object's membership in Tuc-Hor, and the criterion used for making that judgement (Section 5.1).", "aPossible field interloper; see caption of Figure 12. lrrrrlcr 0pt Properties of SB2s Name $RV_A$ $v\\sin (i)_A$ $RV_B$ $v\\sin (i)_B$ $F_B/F_A$ $q$ $v_{sys}$ (km/s) (km/s) (km/s) (km/s) (@ 7600Å) ($M_B/M_A$ ) (km/s) J22170881-7159400 -16.29$\\pm $ 0.09 5.60$\\pm $ 0.04 12.84$\\pm $ 0.10 5.41$\\pm $ 0.11 0.40$\\pm $ 0.02 0.73 -4.0 J05332558-5117131 19.37$\\pm $ 0.07 7.47$\\pm $ 0.08 -29.36$\\pm $ 0.23 14.65$\\pm $ 0.20 0.47$\\pm $ 0.04 0.76 -1.6 J04515303-4647309 37.95$\\pm $ 0.05 6.81$\\pm $ 0.09 5.10$\\pm $ 0.17 10.20$\\pm $ 0.31 0.57$\\pm $ 0.04 0.81 23.2 J02070176-4406380 -14.11$\\pm $ 0.05 6.66$\\pm $ 0.07 66.01$\\pm $ 0.18 7.71$\\pm $ 0.39 0.27$\\pm $ 0.03 0.46 11.1 J02304370-5811560 -5.38$\\pm $ 0.15 6.71$\\pm $ 0.22 -28.33$\\pm $ 0.13 6.53$\\pm $ 0.11 0.80$\\pm $ 0.05 0.86 -16.0 J00582620-7544511 56.58$\\pm $ 0.10 7.67$\\pm $ 0.14 -36.81$\\pm $ 0.09 7.59$\\pm $ 0.16 0.94$\\pm $ 0.05 0.97 10.5 All measured radial velocities are systematically uncertain by $\\pm $ 0.3 km/s.", "We assess the uncertainties on the mass ratios to be $\\sigma _q 0.10$ , based on uncertainties in the flux ratios and the stellar evolutionary models themselves.", "The system velocities are uncertain by $\\pm 2$ km/s.", "lrrrrrrllr 0pt Properties of SB3s Name $RV_A$ $v\\sin (i)_A$ $RV_B$ $v\\sin (i)_B$ $RV_C$ $v\\sin (i)_C$ $F_B/F_A$ $F_C/F_A$ (km/s) (km/s) (km/s) (km/s) (km/s) (km/s) (@ 7600Å) (@ 7600Å) J23334224-4913495 14.87$\\pm $ 0.09 6.29$\\pm $ 0.08 34.15$\\pm $ 0.08 6.43$\\pm $ 0.15 -33.59$\\pm $ 0.19 6.56$\\pm $ 0.23 0.92$\\pm $ 0.06 0.41$\\pm $ 0.06 J01024375-6235344 7.01$\\pm $ 0.14 8.07$\\pm $ 0.12 27.86$\\pm $ 0.14 6.72$\\pm $ 0.16 -44.52$\\pm $ 0.38 7.05$\\pm $ 0.37 0.75$\\pm $ 0.07 0.20$\\pm $ 0.04 J20503576-4015473 -20.43$\\pm $ 0.07 5.90$\\pm $ 0.09 -50.57$\\pm $ 0.23 10.94$\\pm $ 0.33 46.21$\\pm $ 0.63 8.76$\\pm $ 0.52 0.92$\\pm $ 0.09 0.31$\\pm $ 0.07 All measured radial velocities are systematically uncertain by $\\pm $ 0.3 km/s.", "We adopt the velocity of the outer (tertiary) component as the best estimate for the system velocity, but it is likely to be uncertain by $$ 2 km/s, based on typical orbital velocities of unresolved pairs with $\\rho < 1 $ ($\\rho < 50$ AU).", "lc|cccc|cccc|l 0pt Lithium Depletion Boundaries Region Age 4cLate-K Depletion Boundary 4cMid-M Depletion Boundary References Spec-SpT SED-SpT $M_{Ks}$ $M_{bol}$ Spec-SpT SED-SpT $M_{Ks}$ $M_{bol}$ Tuc-Hor 40 K5.5$\\pm $ 0.3 K7.6$\\pm $ 0.6 6.64$\\pm $ 0.20 4.33$\\pm $ 0.15 M4.5$\\pm $ 0.3 M4.7$\\pm $ 0.7 9.89$\\pm $ 0.10 7.12$\\pm $ 0.16 1 BPMG 12–20 ... ... ... ... ... ... 8.3$\\pm $ 0.5 ... 2 IC 2391 45 K7.1a ... ... ... ... ... 10.24$\\pm $ 0.15 ... 3, 4 IC 2602 45 K7.1a ... ... ... ... ... ... 7.37$\\pm $ 0.20 3, 5 $\\alpha $ Per 75 K3.2 ... ... ... ... ... 11.31$\\pm $ 0.15 ... 4, 6, 7 Blanco 1 120 ... ... ... ... ... ... 11.99$\\pm $ 0.30 ... 8 Pleiades 125 K4.7 ... ... ... ... ... 12.14$\\pm $ 0.15 ... 4, 6, 9 1) This work, 2) [6], 3) [54], 4) [4], 5) [14], 6) [24], 7) [73], [74], 8) [10], 9) [2].", "aThe measurement by [54] was for a combined sample of both IC 2391 and IC 2602 members." ] ]
1403.0050
[ [ "Intensional RDB Manifesto: a Unifying NewSQL Model for Flexible Big Data" ], [ "Abstract In this paper we present a new family of Intensional RDBs (IRDBs) which extends the traditional RDBs with the Big Data and flexible and 'Open schema' features, able to preserve the user-defined relational database schemas and all preexisting user's applications containing the SQL statements for a deployment of such a relational data.", "The standard RDB data is parsed into an internal vector key/value relation, so that we obtain a column representation of data used in Big Data applications, covering the key/value and column-based Big Data applications as well, into a unifying RDB framework.", "We define a query rewriting algorithm, based on the GAV Data Integration methods, so that each user-defined SQL query is rewritten into a SQL query over this vector relation, and hence the user-defined standard RDB schema is maintained as an empty global schema for the RDB schema modeling of data and as the SQL interface to stored vector relation.", "Such an IRDB architecture is adequate for the massive migrations from the existing slow RDBMSs into this new family of fast IRDBMSs by offering a Big Data and new flexible schema features as well." ], [ "Introduction", "The term NoSQL was picked out in 2009 and used for conferences of advocates of non-relational databases.", "In an article of the Computerworld magazine [1], June 2009, dedicated to the NoSQL meet-up in San Francisco is reported the following: \"NoSQLers came to share how they had overthrown the tyranny of slow, expensive relational databases in favor of more efficient and cheaper ways of managing data\".", "In this article, the Computerworld summarizes the following reasons: High Throughput.", "The NoSQL databases provide a significantly higher data throuhput than traditional RDBMSs.", "Horizontal Scalability.", "In contrast to RDBMSs most NoSQL databases are designed to scale well in the horizontal direction and not rely on highly available hardware.", "Cost Setting and Complexity of Database Clusters.", "NoSQL does not need the complexity and cost of sharding which involves cutting up databases into multiple tables to run on large clusters or grids.", "\"One size fits all\" [2] Database Thinking Is Wrong.", "The thinks that the realization and the search for alternatives towards traditional RDBMs can be explained by the following two major trends: The continuous growth of data volumes and the growing need to process larger amounts of data in shorter time.", "Requirement of Cloud Computing.", "Are mentioned two major requirements: High until almost ultimate scalability (especially in the horizontal direction) and low administration overhead.", "Developed cloud Amazon's SimpleDB can store large collections of items which themselves are hashtables containing attributes that consist of key-value pairs.", "Avoidance of Unneeded Complexity.", "The reach feature set and the ACID properties implemented by RDBMSs might be more than necessary for particular applications.", "There are different scenarios where applications would be willing to compromise reliability for better performances.", "Moreover, the NoSQL movements advocate that relational fit well for data that is rigidly structured with relations and are designated for central deployments with single, large high-end machines, and not for distribution.", "Often they emphasize that SQL queries are expressed in a sophisticated language.", "But usually they do not tell that also the NoSQL databases often need the sophisticated languages (as object-oriented databases, or Graph-based databases) as well.", "Moreover, they do not say that SQL and RDB are based on sound logical framework (a subset of the First-Order Logic (FOL) language) and hence it is not a procedural language, but a higher level declarative language able to specify \"what\" we need instead of \"how\" to obtain what we need.", "Thus, from the point of view of the development of computer science, instead to go in the direction of the logically higher levels of knowledge representation and query languages, more appropriated to the human understanding, they propose the old technics as key-value representations or the simpler forms of object-oriented representations and their relative procedural query languages.", "They jumped into the past instead to jump in the future of the social and scientific human development.", "It happened because the current RDBMSs were obsolete and not ready to accept the new social-network Web applications in the last 10 years, so that the isolated groups of developers of these ad-hoc systems (e.g., Google, Amazon, LinkedIn, Facebook, etc..) could use only the ready old-known technics and development instruments in order to satisfy the highly urgent business market requirements.", "From the academic research side, instead, most of the work has been done in Sematic Web \"industrial-funded\" programs (e.g., the European IST projects) by considering the new knowledge and reasoning logic systems, whose impact to the existing RDB applications framework would be very hard, with difficult migration and implementation in these new semantics systems (it would need one or more decade of time).", "Instead, we needed a more fundamental theoretical research for the significative technological advances and evolutions of the existing RDB engine.", "Thus, the core RDB technology was in some way \"abandoned\" from both major development initiatives in the last 20 years.", "Nobody probably wanted to consider the most natural evolution of the RDBMSs and its FOL and SQL query framework, and the database industry tried only to cover \"with pieces\" and \"adding\" the new emergent customer's necessities, without a strong investment and the necessary efforts for the complete revision of their old System R based RDB engines of the 1970s.", "The world's economical crisis form 2007 did not help for such an effort.", "However, from the technical point of view, it is clear that if we would come back to make the application programs in Assembler, we probably will obtain better computational performances for some algorithms than with more powerful programming languages, but it is justifiable when we write the system infrastructures and parsers, and not when we have to develop the legacy software for user's requirements.", "Analogously, we may provide the BD infrastructure and physical level in a form of simpler structures, adequate to support the distributive and massive BigData query computations, by preserving the logically higher level interface to customer's applications.", "That is, it is possible to preserve the RDB interface to data, with SQL query languages for the programmers of the software applications, with the \"physical\" parsing of data in more simple structures, able to deal with Big Data scalability in a high distributive computation framework.", "The first step to maintain the logical declarative (non-procedural) SQL query language level, is done by the group (M.I.T.", "and Microsoft) and in widely adopted paper \"The End of an Architectural Era\" (cf.", "[3] Michael Stonebraker et all.)", "where the authors come to the conclusion \"that the current RDBMS code lines, while attempting to be a \"one size fits all\" solution, in fact excel at nothing\".", "At first, Stonebraker et all.", "argue that RDBMSs have been architected more than 25 years ago when the hardware characteristics, user requirements and database markets where very different from those today.", "The resulting revision of traditional RDBMSs is provided by developing H-store (M.I.T., Brown and Yale University), a next generation OLTP systems that operates on distributed clusters of shared-nothing machines where the data resides entirely in main memory, so that it was shown to significantly outperform (83 times) a traditional, disc-based DBMS.", "A more full-featured version of the system [4] that is able to execute across multiple machines within a local area cluster has been presented in August 2008.", "The data storage in H-store is managed by a single-thread execution engine that resides underneath the transaction manager.", "Each individual site executes an autonomous instance of the storage engine with a fixed amount of memory allocated from its host machine.", "Multi-side nodes do not share any data structures with collocated sites, and hence there is no need to use concurrent data structures (every read-only table is replicated on all nodes nd other tables are divided horizontally into disjoint partitions with a k-safety factor two).", "Thus, H-store (at http://hstore.cs.brown.edu/documentation/architecture-overview/) was designed as a parallel, row-storage relational DBMS that runs on a cluster of shared-nothing, main memory executor nodes.", "The commercial version of H-store's design is VoltDB.", "More recently, during 2010 and 2011, Stonebraker has been a critic of the NoSQL movement [5], [6]: \"Here, we argue that using MR systems to perform tasks that are best suited for DBMSs yields less than satisfactory results [7], concluding that MR is more like an extract-transform-load (ETL) system than a DBMS, as it quickly loads and processes large amounts of data in an ad hoc manner.", "As such, it complements DBMS technology rather than competes with it.\"", "After a number of arguments about MR (MapReduction) w.r.t.", "SQL (with GROUP BY operation), the authors conclude that parallel DBMSs provide the same computing model as MR (popularized by Google and Hadoop to process key/value data pairs), with the added benefit of using a declarative SQL language.", "Thus, parallel DBMSs offer great scalability over the range of nodes that customers desire, where all parallel DBMSs operate (pipelining) by creating a query plan that is distributed to the appropriate nodes at execution time.", "When one operator in this plan send data to next (running on the same or a different node), the data are pushed by the first to the second operator (this concept is analog to the process described in my book [8], February 2014, in Section 5.2.1 dedicated to normalization of SQL terms (completeness of the Action-relational-algebra category RA), so that (differently from MR), the intermediate data is never written to disk.", "The formal theoretical framework (the database category DB) of the parallel DBMSs and the semantics of database mappings between them is provided in Big Data integration theory as well [8].", "It is interesting that in [6], the authors conclude that parallel DBMSs excel at efficient querying of large data sets while MR key/value style systems excel at complex analytics and ETL tasks, and propose: \"The result is a much more efficient overall system than if one tries to do the entire application in either system.", "That is, “smart software” is always a good idea.\"", "The aim of this paper is to go one step in advance in developing this NewSQL approach, and to extend the \"classic\" RDB systems with both features: to offer, on user's side, the standard RDB database schema for SQL querying and, on computational side, the \"vectorial\" relational database able to efficiently support the low-level key/value data structures together, in the same logical SQL framework.", "Moreover, this parsing of the standard RDBs into a \"vectorial\" database efficiently resolves also the problems of NoSQL applications with sparse-matrix and \"Open schema\" data models.", "The plan of this paper is the following: In Section 2 we present the method of parsing of any RDB into a vector relation of the key/value structure, compatible with most Big Data structures, and Open schema solutions, but with preserving the RDB user-defined schema for the software applications.", "We show that such a parsing changes the standard semantics of the RDBs based on the FOL by introducing the intensional concepts for user-defined relational tables.", "Consequently, in Section 3 we introduce a conservative intensional extension of the FOL adequate to express the semantics for the IRDBs and the SQL.", "In Section 4 we define a new semantics for the IRDBSs and their canonical models based on the Data Integration systems, where the user-defined RDB is a global schema and the source schema is composed by the unique vector relations which contains the parsed data of the whole used-defined RDB.", "As in GAV (Global-As-View) Data Integration systems, we dematerialize the global schema (i.e., user-defined RDB) and, in Section 5, we define a query-rewriting algorithm to translate the original query written for the user-defined RDB schema into the source database composed by the vector relation containing the parsed data." ], [ "Vector databases with intensional FOL Semantics ", "In what follows, we denote by $B^A$ the set of all functions from $A$ to $B$ , and by $A^n$ a n-folded cartesian product $A \\times ...\\times A$ for $n \\ge 1$ , we denote by $\\lnot , \\wedge , \\vee ,\\Rightarrow $ and $\\Leftrightarrow $ the logical operators negation, conjunction, disjunction, implication and equivalence, respectively.", "For any two logical formulae $\\phi $ and $\\psi $ we define the XOR logical operator $\\underline{\\vee }$ by $\\phi \\underline{\\vee } \\psi $ logically equivalent to $(\\phi \\vee \\psi ) \\wedge \\lnot (\\phi \\wedge \\psi )$ .", "Then we will use the following RDB definitions, based on the standard First-Order Logic (FOL) semantics: A database schema is a pair $\\mathcal {A}= (S_A , \\Sigma _A)$ where $S_A$ is a countable set of relational symbols (predicates in FOL) $r\\in \\mathbb {R}$ with finite arity $n = ar(r) \\ge 1$ ($~ar:\\mathbb {R} \\rightarrow \\mathcal {N}$ ), disjoint from a countable infinite set $\\textbf {att}$ of attributes (a domain of $a\\in \\textbf {att}$ is a nonempty finite subset $dom(a)$ of a countable set of individual symbols $\\textbf {dom}$ ).", "For any $r\\in \\mathbb {R}$ , the sort of $r$ , denoted by tuple $\\textbf {a} = atr(r)= <atr_r(1),...,atr_r(n)>$ where all $a_i = atr_r(m) \\in \\textbf {att}, 1\\le m \\le n$ , must be distinct: if we use two equal domains for different attributes then we denote them by $a_i(1),...,a_i(k)$ ($a_i$ equals to $a_i(0)$ ).", "Each index (\"column\") $i$ , $1\\le i \\le ar(r)$ , has a distinct column name $nr_r(i) \\in SN$ where $SN$ is the set of names with $nr(r) =<nr_r(1),...,nr_r(n)>$ .", "A relation symbol $r \\in \\mathbb {R}$ represents the relational name and can be used as an atom $r(\\textbf {x})$ of FOL with variables in $\\textbf {x}$ assigned to its columns, so that $\\Sigma _A$ denotes a set of sentences (FOL formulae without free variables) called integrity constraints of the sorted FOL with sorts in $\\textbf {att}$ .", "An instance-database of a nonempty schema $\\mathcal {A}$ is given by $A = (\\mathcal {A},I_T) = \\lbrace R =\\Vert r\\Vert = I_T(r) ~|~r \\in S_A \\rbrace $ where $I_T$ is a Tarski's FOL interpretation which satisfies all integrity constraints in $\\Sigma _A$ and maps a relational symbol $r \\in S_A$ into an n-ary relation $R=\\Vert r\\Vert \\in A$ .", "Thus, an instance-database $A$ is a set of n-ary relations, managed by relational database systems.", "Let $A$ and $A^{\\prime } = (\\mathcal {A},I_T^{\\prime })$ be two instances of $\\mathcal {A}$ , then a function $h:A \\rightarrow A^{\\prime }$ is a homomorphism from $A$ into $A^{\\prime }$ if for every k-ary relational symbol $r \\in S_A$ and every tuple $<v_1,...,v_k>$ of this k-ary relation in $A$ , $<h(v_1),...,h(v_k)>$ is a tuple of the same symbol $r$ in $A^{\\prime }$ .", "If $A$ is an instance-database and $\\phi $ is a sentence then we write $A\\models \\phi ~$ to mean that $A$ satisfies $\\phi $ .", "If $\\Sigma $ is a set of sentences then we write $A \\models \\Sigma $ to mean that $A\\models \\phi $ for every sentence $\\phi \\in \\Sigma $ .", "Thus the set of all instances of $\\mathcal {A}$ is defined by $Inst(\\mathcal {A}) = \\lbrace A~|~ A \\models \\Sigma _A \\rbrace $ .", "We consider a rule-based conjunctive query over a database schema $\\mathcal {A}$ as an expression $ q(\\textbf {x})\\longleftarrow r_1(\\textbf {u}_1), ...,r_n(\\textbf {u}_n)$ , with finite $n\\ge 0$ , $r_i$ are the relational symbols (at least one) in $\\mathcal {A}$ or the built-in predicates (e.g.", "$\\le , =,$ etc.", "), $q$ is a relational symbol not in $\\mathcal {A}$ and $\\textbf {u}_i$ are free tuples (i.e., one may use either variables or constants).", "Recall that if $\\textbf {v} = (v_1,..,v_m)$ then $r(\\textbf {v})$ is a shorthand for $r(v_1,..,v_m)$ .", "Finally, each variable occurring in $\\textbf {x}$ is a distinguished variable that must also occur at least once in $\\textbf {u}_1,...,\\textbf {u}_n$ .", "Rule-based conjunctive queries (called rules) are composed of a subexpression $r_1(\\textbf {u}_1),...., r_n(\\textbf {u}_n)$ that is the body, and the head of this rule $q(\\textbf {x})$ .", "The $Yes/No$ conjunctive queries are the rules with an empty head.", "If we can find values for the variables of the rule, such that the body is logically satisfied, then we can deduce the head-fact.", "This concept is captured by a notion of \"valuation\".", "The deduced head-facts of a conjunctive query $q(\\textbf {x})$ defined over an instance $A$ (for a given Tarski's interpretation $I_T$ of schema $\\mathcal {A}$ ) are equal to $\\Vert q(x_1,...,x_k)\\Vert _A = \\lbrace <v_1,...,v_k> \\in \\textbf {dom}^k ~|~ A \\models \\exists \\textbf {y}(r_1(\\textbf {u}_1)\\wedge ...\\wedge r_n(\\textbf {u}_n))[x_i/v_i]_{1\\le i \\le k} \\rbrace = I_T^*(\\exists \\textbf {y}(r_1(\\textbf {u}_1)\\wedge ...\\wedge r_n(\\textbf {u}_n)))$ , where the $\\textbf {y}$ is a set of variables which are not in the head of query, and $I_T^*$ is the unique extension of $I_T$ to all formulae.", "We recall that the conjunctive queries are monotonic and satisfiable, and that a (Boolean) query is a class of instances that is closed under isomorphism [9].", "Each conjunctive query corresponds to a \"select-project-join\" term $t(\\textbf {x})$ of SPRJU algebra obtained from the formula $\\exists \\textbf {y}(r_1(\\textbf {u}_1)\\wedge ...\\wedge r_n(\\textbf {u}_n))$ , as explained in Section .", "We consider a finitary view as a union of a finite set $S$ of conjunctive queries with the same head $q(\\textbf {x})$ over a schema $\\mathcal {A}$ , and from the equivalent algebraic point of view, it is a \"select-project-join-rename + union\" (SPJRU) finite-length term $t(\\textbf {x})$ which corresponds to union of the terms of conjunctive queries in $S$ .", "In what follows we will use the same notation for a FOL formula $q(\\textbf {x})$ and its equivalent algebraic SPJRU expression $t(\\textbf {x})$ .", "A materialized view of an instance-database $A$ is an n-ary relation $R = \\bigcup _{q(\\textbf {x}) \\in S}\\Vert q(\\textbf {x})\\Vert _A$ .", "Notice that a finitary view can also have an infinite number of tuples.", "We denote the set of all finitary materialized views that can be obtained from an instance $A$ by $TA$ .", "The principal idea is to use an analogy with a GAV Data Integration [10], [8] by using the database schema $\\mathcal {A}= (S_A,\\Sigma _A)$ as a global relational schema, used as a user/application-program interface for the query definitions in SQL, and to represent the source database of this Data Integration system by parsing of the RDB instance $A$ of the schema $\\mathcal {A}$ into a single vector relation $\\overrightarrow{A}$ .", "Thus, the original SQL query $q(\\textbf {x})$ has to be equivalently rewritten over (materialized) source vector database $\\overrightarrow{A}$ .", "The idea of a vector relation $\\overrightarrow{A}$ for a given relational database instance $A$ comes from the investigation of the topological properties of the RDB systems, presented in Chapter 8 of the book [8].", "In order to analyze the algebraic lattice of all RDB database instances each instance database $A$ , composed by a set of finitary relations $R_i \\in A$ , $i = 1,...,n$ , in Lemma 21 is defined the transformation of the instance database $A$ into a vector relation $\\overrightarrow{A}$ , with $\\overrightarrow{A} =\\bigcup _{R \\in A} \\overrightarrow{R}$ where for each relation $R$ $ar(R) \\ge 1$ it the arity (the number of its columns) of this relational table and $\\pi _i$ is its $i$ -th column projection, and hence $\\overrightarrow{R} = \\bigcup _{1\\le i \\le ar(R)} \\pi _i(R)$ .", "Such vectorial representation of a given database $A$ in [8] is enough to define the lattice of RDB lattices, because we do not needed the converse process (to define a database $A$ from its vectorial representation).", "However, by considering that a database $A$ is seen by the users and their software applications (with the embedded SQL statements), while $\\overrightarrow{A}$ is its single-table internal representation, over which is executed a rewritten user's query, the extracted information has to be converted in the RDB form w.r.t.", "the relational schema of the original user's model.", "Consequently, we need a reacher version of the vector database, such that we can obtain an equivalent inverse transformation of it into the standard user defined RDB schema.", "In fact, each $i$ -th column value $d_i$ in a tuple $\\textbf {d} = \\langle d_1,...,d_i,...,d_{ar(r)} \\rangle $ of a relation $R_k = \\Vert r_k\\Vert , r_k \\in S_A$ , of the instance database $A$ is determined by the free dimensional coordinates: relational name $nr(r)$ , the attribute name $nr_r(i)$ of the i-th column, and the tuple index $Hash(\\textbf {d})$ obtained by hashing the string of the tuple $\\textbf {d}$ .", "Thus, the relational schema of the vector relation is composed by the four attributes, relational name, tuple-index, attribute name, and value, i.e., r-name, t-index, a-name and value, respectively, so that if we assume $r_V$ (the name of the database $\\mathcal {A}$ ) for the name of this vector relation $\\overrightarrow{A}$ then this relation can be expressed by the quadruple $r_V$ (r-name, t-index, a-name, value), and the parsing of any RDB instance $A$ of a schema $\\mathcal {A}$ can be defined as: Parsing RDB instances: Given a database instance $A =\\lbrace R_1,...,R_n\\rbrace $ , $n\\ge 1$ , of a RDB schema $\\mathcal {A}= (S_A,\\Sigma _A)$ with $S_A = \\lbrace r_1,...,r_n\\rbrace $ such that $R_k = \\Vert r_k\\Vert , k = 1,...,n$ , then the extension $\\overrightarrow{A} = \\Vert r_V\\Vert $ of the vector relational symbol (name) $r_V$ with the schema $r_V$ (r-name, t-index, a-name, value), and NOT NULL constraints for all its four attributes, and with the primary key composed by the first three attributes, is defined by: we define the operation PARSE for a tuple $\\textbf {d} =\\langle d_1,...,d_{ar(r_k)}\\rangle $ of the relation $r_k \\in S_A$ by the mapping $(r_k,\\textbf {d}) ~~\\mapsto ~~\\lbrace \\langle r_k,Hash(\\textbf {d}),nr_{r_k}(i),d_i \\rangle |~ d_i \\emph {NOT NULL},1\\le i \\le ar(r_k)\\rbrace $ , so that (1) $~~~\\overrightarrow{A} = \\bigcup _{r_k \\in S_A,\\textbf {d} \\in \\Vert r_k\\Vert } \\emph {\\underline{PARSE}}(r_k,\\textbf {d})$ .", "Based on the vector database representation $\\Vert r_V\\Vert $ we define a GAV Data Integration system $I= \\langle \\mathcal {A}, \\mathcal {S},\\mathcal {M}\\rangle $ with the global schema $\\mathcal {A}= (S_A, \\Sigma _A)$ , the source schema $\\mathcal {S}= (\\lbrace r_V\\rbrace ,\\emptyset )$ , and the set of mappings $\\mathcal {M}$ expressed by the tgds (tuple generating dependencies) (2) $~~~\\forall y,x_1,...,x_{ar(r_k)}(((r_V(r_k,y,nr_{r_k}(1),x_1)~\\underline{\\vee }~ x_1 \\emph {NULL})\\wedge ...\\\\...\\wedge (r_V(r_k,y,nr_{r_k}(ar(r_k)),x_{ar(r_k)})~\\underline{\\vee }~x_{ar(r_k)} \\emph {NULL})) \\Rightarrow r_k(x_1,...,x_{ar(r_k)}))$ , for each $r_k \\in S_A$ .", "The operation PARSE corresponds to the parsing of the tuple $\\textbf {v}$ of the relation $r_k \\in S_A$ of the user-defined database schema $\\mathcal {A}$ into a number of tuples of the vector relation $r_V$ .", "In fact, we can use this operation for virtual inserting/deleting of the tuples in the user defined schema $\\mathcal {A}$ , and store them only in the vector relation $r_V$ .", "This operation avoids to materialize the user-defined (global) schema, but only the source database $\\mathcal {S}$ , so that each user-defined SQL query has to be equivalently rewritten over the source database (i.e., the big table $\\overrightarrow{A} = \\Vert r_V\\Vert $ ) as in standard FOL Data Integration systems.", "Notice that this parsing defines a kind of GAV Data Integration systems, where the source database $\\mathcal {S}$ is composed by the unique vector relation $\\Vert r_V\\Vert = \\overrightarrow{A}$ (Big Data) which does not contain NULL values, so that we do not unnecessarily save the NULL values of the user-defined relational tables $r_k \\in S_A$ in the main memories of the parallel RDBMS used to horizontal partitioning of the unique big-table $\\overrightarrow{A}$ .", "Moreover, any adding of the new columns to the user-defined schema $\\mathcal {A}$ does not change the table $\\overrightarrow{A}$ , while the deleting of a $i$ -th column of a relation $r$ will delete all tuples $r_V(x,y,z,v)$ where $x =nr(r)$ and $z = nr_r(i)$ in the main memory of the parallel RDBMS.", "Thus, we obtain very schema-flexible RDB model for Big Data.", "Other obtained NoSQL systems properties are: Compatible with key/value systems.Note that the vector big-table $\\overrightarrow{A}$ is in the 6th normal form, that is with the primary key corresponding to the first three attributes (the free dimensional coordinates) and the unique value attribute.", "Thus we obtained the key/value style used for NoSQL Big Data systems.", "That is, the RDB parsing with resulting Data Integration system subsumes all Big Data key/value systems.", "Compatible with \"Open schema\" systems.", "Entity-attribute-value model (EAV) is a data model to describe entities where the number of attributes (properties, parameters) that can be used to describe them is potentially vast, but the number that will actually apply to a given entity is relatively modest.", "In mathematics, this model is known as a sparse matrix.", "EAV is also known as object-attribute-value model, vertical database model and open schema.", "We can use the special relational symbol with name \"OpenSchema\" in the user database schema $\\mathcal {A}$ so that its tuples in $\\overrightarrow{A}$ will corresponds to atoms $r_V({\\tt OpenSchema}, object,attribute, value)$ .", "In this case the software developed for the applications which use the Open schema data will directly access to the vector relation $\\overrightarrow{A}$ and DBMS will restrict all operations only to tuples where the first attribute has the value equal to OpenSchema (during an inserting of a new tuple $\\langle object, attribute, value \\rangle $ the DBMS inserts also the value OpenSchema in the first column of $\\overrightarrow{A}$ ).", "But this simple and unifying framework needs more investigation for the SQL and underlying logical framework.", "In fact, we can easy see that the mapping tgds used from the Big Data vector table $\\overrightarrow{A}$ (the source schema in Data Integration) into user-defined RDB schema $\\mathcal {A}$ (the global schema of this Data Integration system with integrity constraints) is not simple FOL formula.", "Because the same element $r_k$ is used as a predicate symbol (on the right-side of the tgd's implication) and as a value (on the left side of the implication as the first value in the predicate $r_V$ ).", "It means that the elements of the domain of this logic are the elements of other classes and are the classes for themselves as well.", "Such semantics is not possible in the standard FOL, but only in the intensional FOL, and hence the Data Integration $I$ is not a classic FOL Data Integration as in [10] but an Intensional Data Integration system.", "In the next sections we will investigate what is the proper logic framework for this class of RDBs, denominated as IRDBs (Intensional RDBs), and to show that the standard SQL is complete in this new logical framework." ], [ "Conservative intensional extension of the FOL for IRDBs", "The first conception of intensional entities (or concepts) is built into the possible-worlds treatment of Properties, Relations and Propositions (PRP)s. This conception is commonly attributed to Leibniz, and underlies Alonzo Church's alternative formulation of Frege's theory of senses (\"A formulation of the Logic of Sense and Denotation\" in Henle, Kallen, and Langer, 3-24, and \"Outline of a Revised Formulation of the Logic of Sense and Denotation\" in two parts, Nous,VII (1973), 24-33, and VIII,(1974),135-156).", "This conception of PRPs is ideally suited for treating the modalities (necessity, possibility, etc..) and to Montague's definition of intension of a given virtual predicate $\\phi (x_1,...,x_k)$ (a FOL open-sentence with the tuple of free variables $(x_1,...x_k)$ ), as a mapping from possible worlds into extensions of this virtual predicate.", "Among the possible worlds we distinguish the actual possible world.", "For example, if we consider a set of predicates, of a given Database, and their extensions in different time-instances, then the actual possible world is identified by the current instance of the time.", "The second conception of intensional entities is to be found in Russell's doctrine of logical atomism.", "In this doctrine it is required that all complete definitions of intensional entities be finite as well as unique and non-circular: it offers an algebraic way for definition of complex intensional entities from simple (atomic) entities (i.e., algebra of concepts), conception also evident in Leibniz's remarks.", "In a predicate logics, predicates and open-sentences (with free variables) expresses classes (properties and relations), and sentences express propositions.", "Note that classes (intensional entities) are reified, i.e., they belong to the same domain as individual objects (particulars).", "This endows the intensional logics with a great deal of uniformity, making it possible to manipulate classes and individual objects in the same language.", "In particular, when viewed as an individual object, a class can be a member of another class.", "The distinction between intensions and extensions is important (as in lexicography [11]), considering that extensions can be notoriously difficult to handle in an efficient manner.", "The extensional equality theory of predicates and functions under higher-order semantics (for example, for two predicates with the same set of attributes $p = q$ is true iff these symbols are interpreted by the same relation), that is, the strong equational theory of intensions, is not decidable, in general.", "For example, the second-order predicate calculus and Church's simple theory of types, both under the standard semantics, are not even semi-decidable.", "Thus, separating intensions from extensions makes it possible to have an equational theory over predicate and function names (intensions) that is separate from the extensional equality of relations and functions.", "Relevant recent work about the intension, and its relationship with FOL, has been presented in [12] in the consideration of rigid and non-rigid objects, w.r.t.", "the possible worlds, where the rigid objects, like \"George Washington\", and are the same things from possible world to possible world.", "Non-rigid objects, like \"the Secretary-General of United Nations\", are varying from circumstance to circumstance and can be modeled semantically by functions from possible worlds to domain of rigid objects, like intensional entities.", "However, Fitting substantially and ad-hock changes the syntax and semantics of FOL, and introduces the Higher-order Modal logics, differently from our approach.", "More about other relevant recent works are presented in [13], [14] where a new conservative intensional extension of the Tarski's semantics of the FOL is defined.", "Intensional entities are such concepts as propositions and properties.", "The term 'intensional' means that they violate the principle of extensionality; the principle that extensional equivalence implies identity.", "All (or most) of these intensional entities have been classified at one time or another as kinds of Universals [15].", "We consider a non empty domain $~ D_{-1} \\bigcup D_I$ , where a subdomain $D_{-1}$ is made of particulars (extensional entities), and the rest $D_I = D_0 \\bigcup D_1 ...\\bigcup D_n ...$ is made of universals ($D_0$ for propositions (the 0-ary concepts), and $D_n, n \\ge 1,$ for n-ary concepts).", "The fundamental entities are intensional abstracts or so called, 'that'-clauses.", "We assume that they are singular terms; Intensional expressions like 'believe', mean', 'assert', 'know', are standard two-place predicates that take 'that'-clauses as arguments.", "Expressions like 'is necessary', 'is true', and 'is possible' are one-place predicates that take 'that'-clauses as arguments.", "For example, in the intensional sentence \"it is necessary that $\\phi $ \", where $\\phi $ is a proposition, the 'that $\\phi $ ' is denoted by the $\\lessdot \\phi \\gtrdot $ , where $\\lessdot \\gtrdot $ is the intensional abstraction operator which transforms a logic formula into a term.", "Or, for example, \"x believes that $\\phi $ \" is given by formula $p_i(x,\\lessdot \\phi \\gtrdot )$ ( $p_i$ is binary 'believe' predicate).", "We introduce an intensional FOL [14], with slightly different intensional abstraction than that originally presented in [16], as follows: The syntax of the First-order Logic (FOL) language $\\mathcal {L}$ with intensional abstraction $\\lessdot \\gtrdot $ is as follows: Logical operators $(\\wedge , \\lnot , \\exists )$ ; Predicate letters $r_i,p_i \\in \\mathbb {R}$ with a given arity $k_i = ar(r_i) \\ge 1$ , $i = 1,2,...$ (the functional letters are considered as particular case of the predicate letters); a set PR of propositional letters (nullary predicates) with a truth $r_\\emptyset \\in PR \\bigcap \\mathbb {R}$ ; Language constants $\\overline{0}, \\overline{1},...,\\overline{c},\\overline{d}...$ ; Variables $x,y,z,..$ in $; Abstraction $ _ $, and punctuationsymbols (comma, parenthesis).With the following simultaneous inductive definition of \\emph {term} and\\emph {formula}:\\\\1.", "All variables and constants are terms.", "All propositional letters are formulae.\\\\2.", "If $  t1,...,tk$ are terms then $ ri(t1,...,tk)$ is a formulafor a k-ary predicate letter $ ri R$ .\\\\3.", "If $$ and $$ are formulae, then $ ()$, $$, and$ (x)$ are formulae.", "\\\\4.", "If $ (x)$ is a formula (virtual predicate) with a list of free variables in $x =(x1,...,xn)$ (with orderingfrom-left-to-right of their appearance in $$), and $$ isits sublist of \\emph {distinct} variables,then $$ is a term, where $$ is the remaining list of free variables preserving ordering in $x$ as well.", "The externally quantifiable variables are the \\emph {free} variables not in $$.When $ n =0,  $ is a term which denotes aproposition, for $ n 1$ it denotesa n-ary concept.\\\\An occurrence of a variable $ xi$ in a formula (or a term) is\\emph {bound} (\\emph {free}) iff it lies (does not lie) within aformula of the form $ (xi)$ (or a term of the form$$ with $ xi $).", "Avariable is free (bound) in a formula (or term) iff it has (does nothave) a free occurrence in that formula (or term).", "A \\emph {sentence}is a formula having no free variables.$ An interpretation (Tarski) $I_T$ consists of a nonempty domain $ D_{-1} \\bigcup D_I$ and a mapping that assigns to any predicate letter $r_i \\in \\mathbb {R}$ with $k = ar(r_i)\\ge 1$ , a relation $\\Vert r_i\\Vert = I_T(r_i) \\subseteq k$ ; to each individual constant $\\overline{c}$ one given element $I_T(\\overline{c}) \\in , with $ IT(0) = 0, IT(1) = 1$ fornatural numbers $ N={0,1,2,...}$, and to anypropositional letter $ p PR$ one truth value $ IT(p) {f,t}$, where $ f$ and $ t$ are the empty set $ {}$and the singleton set $ {<>}$ (with the empty tuple $ <> D-1$), as those used in the Codd^{\\prime }s relational-database algebra \\cite {Codd72} respectively,so that for any $ IT$, $ IT(r) = {<>}$(i.e., $ r$ is a tautology), while $ Truth D0$ denotes the concept (intension)of this tautology.", "\\\\Note that in the intensional semantics a k-ary functional symbol,for $ k 1$, in standard (extensional) FOL is considered as a$ (k+1)$-ary predicate symbols: let $ fm$ be such a $ (k+1)$-arypredicate symbol which represents a k-ary function denoted by$fm$ with standard Tarski^{\\prime }s interpretation$ IT(fm):k .", "Then $I_T(f_m)$ is a relation obtained from its graph, i.e., $I_T(f_m) = R =\\lbrace (d_1,...,d_k,I_T(\\underline{f}_m)(d_1,...,d_k)) ~| ~d_i \\in 1\\le i \\le k \\rbrace $ .", "The universal quantifier is defined by $\\forall = \\lnot \\exists \\lnot $ .", "Disjunction $\\phi \\vee \\psi $ and implication $\\phi \\Rightarrow \\psi $ are expressed by $\\lnot (\\lnot \\phi \\wedge \\lnot \\psi )$ and $\\lnot \\phi \\vee \\psi $ , respectively.", "In FOL with the identity $\\doteq $ , the formula $(\\exists _1 x)\\phi (x)$ denotes the formula $(\\exists x)\\phi (x) \\wedge (\\forall x)(\\forall y)(\\phi (x)\\wedge \\phi (y) \\Rightarrow (x \\doteq y))$ .", "We denote by $R_{=}$ the Tarski's interpretation of $\\doteq $ .", "In what follows any open-sentence, a formula $\\phi $ with non empty tuple of free variables $(x_1,...,x_m)$ , will be called a m-ary virtual predicate, denoted also by $\\phi (x_1,...,x_m)$ .", "This definition contains the precise method of establishing the ordering of variables in this tuple: such an method that will be adopted here is the ordering of appearance, from left to right, of free variables in $\\phi $ .", "This method of composing the tuple of free variables is the unique and canonical way of definition of the virtual predicate from a given formula.", "An intensional interpretation of this intensional FOL is a mapping between the set $\\mathcal {L}$ of formulae of the logic language and intensional entities in $, $ I:L, is a kind of \"conceptualization\", such that an open-sentence (virtual predicate) $\\phi (x_1,...,x_k)$ with a tuple $\\textbf {x}$ of all free variables $(x_1,...,x_k)$ is mapped into a k-ary concept, that is, an intensional entity $u =I(\\phi (x_1,...,x_k)) \\in D_k$ , and (closed) sentence $\\psi $ into a proposition (i.e., logic concept) $v =I(\\psi ) \\in D_0$ with $I(\\top ) = Truth \\in D_0$ for a FOL tautology $\\top $ .", "This interpretation $I$ is extended also to the terms (called as denotation as well).", "A language constant $\\overline{c}$ is mapped into a particular (an extensional entity) $a = I(\\overline{c}) \\in D_{-1}$ if it is a proper name, otherwise in a correspondent concept in $.For each $ k$-ary atom $ ri(x)$, $ I(ri(x) x)$ is the relation-name (symbol)$ ri R$ (only if $ ri$ is not defined as a languageconstant as well).The extension of $ I$ to the complex abstracted terms is given in \\cite {Majk12a} (in Definition 4).\\\\An assignment $ g: for variables in $ isapplied only to free variables in terms and formulae.", "Such anassignment $ g $ can be recursively uniquely extended intothe assignment $ g*:X̰ , where $X̰$ denotes the set of all terms with variables in $X \\subseteq (here $ I$ is anintensional interpretation of this FOL, as explainedin what follows), by :\\\\1.", "$ g*(tk) = g(x) if the term $t_k$ is a variable $x \\in .\\\\2.", "$ g*(tk) = I(c) if the term $t_k$ is a constant $\\overline{c}$ .", "3. if $t_k$ is an abstracted term $\\lessdot \\phi \\gtrdot _{\\alpha }^{\\beta }$ , then $g^*(\\lessdot \\phi \\gtrdot _{\\alpha }^{\\beta }) = I(\\phi [\\beta /g(\\beta )] ) \\in D_k, k =|\\alpha |$ (i.e., the number of variables in $\\alpha $ ), where $g(\\beta ) = g(y_1,..,y_m) = (g(y_1),...,g(y_m))$ and $[\\beta /g(\\beta )]$ is a uniform replacement of each i-th variable in the list $\\beta $ with the i-th constant in the list $g(\\beta )$ .", "Notice that $\\alpha $ is the list of all free variables in the formula $\\phi [\\beta /g(\\beta )]$ .", "We denote by $~t_k/g~$ (or $\\phi /g$ ) the ground term (or formula) without free variables, obtained by assignment $g$ from a term $t_k$ (or a formula $\\phi $ ), and by $\\phi [x/t_k]$ the formula obtained by uniformly replacing $x$ by a term $t_k$ in $\\phi $ .", "The distinction between intensions and extensions is important especially because we are now able to have an equational theory over intensional entities (as $\\lessdot \\phi \\gtrdot $ ), that is predicate and function \"names\", that is separate from the extensional equality of relations and functions.", "An extensionalization function $h$ assigns to the intensional elements of $ an appropriateextension as follows: for each proposition $ u D0$, $ h(u) {f,t} P(D-1)$ is itsextension (true or false value); for each n-aryconcept $ u Dn$, $ h(u)$ is a subset of $ n$(n-th Cartesian product of $ ); in the case of particulars $u \\in D_{-1}$ , $h(u) = u$ .", "We define $0 = \\lbrace <>\\rbrace $ , so that $\\lbrace f,t\\rbrace = \\mathcal {P}(0)$ , where $\\mathcal {P}$ is the powerset operator.", "Thus we have (we denote the disjoint union by '+'): $h = (h_{-1} + \\sum _{i\\ge 0}h_i):\\sum _{i\\ge -1}D_i \\longrightarrow D_{-1} + \\sum _{i\\ge 0}\\mathcal {P}(D^i)$ , where $h_{-1} = id:D_{-1} \\rightarrow D_{-1}$ is identity mapping, the mapping $h_0:D_0 \\rightarrow \\lbrace f,t\\rbrace $ assigns the truth values in $ \\lbrace f,t\\rbrace $ to all propositions, and the mappings $h_i:D_i \\rightarrow \\mathcal {P}(D^i)$ , $i\\ge 1$ , assign an extension to all concepts.", "Thus, the intensions can be seen as names of abstract or concrete entities, while the extensions correspond to various rules that these entities play in different worlds.", "Remark: (Tarski's constraints) This intensional semantics has to preserve standard Tarski's semantics of the FOL.", "That is, for any formula $\\phi \\in \\mathcal {L}$ with a tuple of free variables $(x_1,...,x_k)$ , and $h \\in \\mathcal {E}$ , the following conservative conditions for all assignments $g,g^{\\prime } \\in { has to be satisfied: \\\\(T)~~~~~~~h(I(\\phi /g)) = t~~ iff~~(g(x_1),...,g(x_k)) \\in h(I(\\phi ));\\\\and, if \\phi is a predicate letter p, k = ar(p) \\ge 2 whichrepresents a (k-1)-aryfunctional symbol f^{k-1} in standard FOL,\\\\(TF)~~~~~~~h(I(\\phi /g)) = h(I(\\phi /g^{\\prime })) = t and\\forall _{1\\le i \\le k-1}(g^{\\prime }(x_i) = g(x_i))~~ implies ~~g^{\\prime }(x_{k+1})=g(x_{k+1}).\\\\\\square \\\\Thus, intensional FOL has a simple Tarski^{\\prime }sfirst-order semantics, with a decidableunification problem, but we need also the actual world mappingwhich maps any intensional entity to its \\emph {actual worldextension}.", "In what follows we will identify a \\emph {possible world} by aparticular mapping which assigns, in such a possible world, the extensions to intensional entities.This is direct bridge betweenan intensional FOL and a possible worlds representation\\cite {Lewi86,Stal84,Mont70,Mont73,Mont74,Majk09FOL}, where the intension (meaning) of a proposition is a\\emph {function}, from a set of possible worlds \\mathcal {W} into the set oftruth-values.Consequently, \\mathcal {E} denotes the set of possible\\emph {extensionalization functions} h satisfying the constraint(T).", "Each h \\in \\mathcal {E} may be seen as a \\emph {possible world}(analogously to Montague^{\\prime }s intensional semantics for naturallanguage \\cite {Mont70,Mont74}), as it has been demonstrated in\\cite {Majk08in,Majk08ird}, and given by the bijection~~~is:\\mathcal {W}\\simeq \\mathcal {E}.\\\\Now we are able to formally define this intensional semantics\\cite {Majk09FOL}:\\begin{definition} \\textsc {Two-step \\textsc {I}ntensional\\textsc {S}emantics:}\\\\Let ~\\mathfrak {R} = \\bigcup _{k \\in \\mathbb {N}} \\mathcal {P}(k) =\\sum _{k\\in \\mathbb {N}}\\mathcal {P}(D^k) be the set of all k-ary relations,where k \\in \\mathbb {N} = \\lbrace 0,1,2,...\\rbrace .", "Notice that \\lbrace f,t\\rbrace =\\mathcal {P}(0) \\in \\mathfrak {R}, that is, the truth values are extensionsin \\mathfrak {R}.", "The intensional semantics of the logic languagewith the set of formulae \\mathcal {L} can be represented by the mapping\\begin{center}~~~ \\mathcal {L}~^{I} \\Longrightarrow _{w \\in \\mathcal {W}}~ \\mathfrak {R},\\end{center}where ^{I} is a \\emph {fixed intensional} interpretation I:\\mathcal {L}\\rightarrow and ~\\Longrightarrow _{w \\in \\mathcal {W}}~ is \\emph {the set}of all extensionalization functions h = is(w):\\mathfrak {R} in \\mathcal {E}, where is:\\mathcal {W}\\rightarrow \\mathcal {E} is the mappingfrom the set of possible worlds to the set ofextensionalization functions.\\\\We define the mapping I_n:\\mathcal {L}_{op} \\rightarrow \\mathfrak {R}^{\\mathcal {W}}, where \\mathcal {L}_{op} is the subset of formulae withfree variables (virtual predicates), such that for any virtualpredicate \\phi (x_1,...,x_k) \\in \\mathcal {L}_{op} the mappingI_n(\\phi (x_1,...,x_k)):\\mathcal {W}\\rightarrow \\mathfrak {R} is theMontague^{\\prime }s meaning (i.e., \\emph {intension}) of this virtualpredicate \\cite {Lewi86,Stal84,Mont70,Mont73,Mont74}, that is, themapping which returns with the extension of this (virtual) predicatein each possible world w\\in \\mathcal {W}.\\end{definition}Another relevant question w.r.t.", "this two-stepinterpretations of an intensional semantics is how in it is managedthe extensional identity relation \\doteq (binary predicate of theidentity) of the FOL.", "Here this extensional identity relation ismapped into the binary concept Id = I(\\doteq (x,y)) \\in D_2, suchthat (\\forall w \\in \\mathcal {W})(is(w)(Id) = R_{=}), where \\doteq (x,y)(i.e., p_1^2(x,y)) denotes an atom of the FOL of the binarypredicate for identity in FOL, usually written by FOL formula x\\doteq y.\\\\Note that here we prefer to distinguish this \\emph {formalsymbol} ~ \\doteq ~ \\in \\mathbb {R} of the built-in identity binarypredicate letter in the FOL, from the standard mathematicalsymbol ^{\\prime }=^{\\prime } used in all mathematical definitions in this paper.\\\\In what follows we will use the function f_{<>}:\\mathfrak {R}\\rightarrow \\mathfrak {R}, such that for any relation R \\in \\mathfrak {R}, f_{<>}(R) = \\lbrace <>\\rbrace if R \\ne \\emptyset ;\\emptyset otherwise.", "Let us define the following set of algebraicoperators forrelations in \\mathfrak {R}:\\begin{enumerate}\\item binary operator ~\\bowtie _{S}:\\mathfrak {R} \\times \\mathfrak {R} \\rightarrow \\mathfrak {R},such that for any two relations R_1, R_2 \\in \\mathfrak {R}~, the~R_1 \\bowtie _{S} R_2 is equalto the relation obtained by natural joinof these two relations ~ {\\tt if}S is a non emptyset of pairs of joined columns of respective relations (where thefirst argument is the column index of the relation R_1 while thesecond argument is the column index of the joined column of therelation R_2); {\\tt otherwise} it is equal to the cartesianproduct R_1\\times R_2.\\\\ For example, the logic formula\\phi (x_i,x_j,x_k,x_l,x_m) \\wedge \\psi (x_l,y_i,x_j,y_j) will betraduced by the algebraic expression ~R_1 \\bowtie _{S}R_2 whereR_1 \\in \\mathcal {P}(5), R_2\\in \\mathcal {P}(4) are the extensions for a givenTarski^{\\prime }s interpretation of the virtual predicate \\phi , \\psi relatively, so that S = \\lbrace (4,1),(2,3)\\rbrace and the resulting relationwill have the following ordering of attributes:(x_i,x_j,x_k,x_l,x_m,y_i,y_j).\\item unary operator ~ \\sim :\\mathfrak {R} \\rightarrow \\mathfrak {R}, such that for any k-ary (with k \\ge 0)relation R \\in \\mathcal {P}({k}) \\subset \\mathfrak {R}we have that ~ \\sim (R) = k \\backslash R \\in {k}, where ^{\\prime }\\backslash ^{\\prime } is the substraction of relations.", "For example, thelogic formula \\lnot \\phi (x_i,x_j,x_k,x_l,x_m) will be traduced bythe algebraic expression ~5 \\backslash R where R is theextensions for a given Tarski^{\\prime }s interpretation of the virtualpredicate \\phi .\\item unary operator ~ \\pi _{-m}:\\mathfrak {R} \\rightarrow \\mathfrak {R}, such that for any k-ary (with k \\ge 0) relation R \\in \\mathcal {P}({k}) \\subset \\mathfrak {R}we have that ~ \\pi _{-m} (R) is equal to the relation obtained byelimination of the m-th column of the relation R~ {\\tt if} 1\\le m \\le k and k \\ge 2; equal to ~f_{<>}(R)~ {\\tt if} m = k=1; {\\tt otherwise} it is equal to R. \\\\For example, the logicformula (\\exists x_k) \\phi (x_i,x_j,x_k,x_l,x_m) will be traducedby the algebraic expression ~\\pi _{-3}(R) where R is theextensions for a given Tarski^{\\prime }s interpretation of the virtualpredicate \\phi and the resulting relation will have the followingordering of attributes: (x_i,x_j,x_l,x_m).\\end{enumerate}Notice that the ordering of attributes of resulting relationscorresponds to the method used for generating the ordering ofvariables in the tuples of free variables adopted for virtualpredicates.\\begin{definition} Intensional algebra for the intensional FOL in Definition \\ref {def:bealer} is a structure ~\\mathcal {A}_{int}= ~( f, t, Id, Truth, \\lbrace conj_{S}\\rbrace _{ S \\in \\mathcal {P}(\\mathbb {N}^2)},neg, \\lbrace exists_{n}\\rbrace _{n \\in \\mathbb {N}}), ~~ withbinary operations ~~conj_{S}:D_I\\times D_I \\rightarrow D_I,unary operation ~~neg:D_I\\rightarrow D_I, unaryoperations ~~exists_{n}:D_{I}\\rightarrow D_I, such that for anyextensionalization function h \\in \\mathcal {E},and u \\in D_k, v \\in D_j, k,j \\ge 0,\\\\1.", "~h(Id) = R_=~ and ~h(Truth) = \\lbrace <>\\rbrace .\\\\2.", "~h(conj_{S}(u, v)) = h(u) \\bowtie _{S}h(v), where \\bowtie _{S}is the natural join operation defined above and conj_{S}(u, v) \\in D_m where m = k + j - |S|if for every pair (i_1,i_2) \\in S it holds that 1\\le i_1 \\le k, 1 \\le i_2 \\le j (otherwise conj_{S}(u, v) \\in D_{k+j}).\\\\3.", "~h(neg(u)) = ~\\sim (h(u)) = k \\backslash (h(u)),where ~\\sim ~ is the operationdefined above and neg(u) \\in D_k.\\\\4.", "~h(exists_{n}(u)) =\\pi _{-n}(h(u)), where \\pi _{-n} is the operation defined above and\\\\ exists_n(u) \\in D_{k-1} if 1 \\le n \\le k (otherwiseexists_n is the identity function).\\end{definition}Notice that for u \\in D_0, ~h(neg(u)) = ~\\sim (h(u)) = 0\\backslash (h(u)) = \\lbrace <>\\rbrace \\backslash (h(u)) \\in \\lbrace f,t\\rbrace .\\\\Intensional interpretation I:\\mathcal {L}\\rightarrow satisfies thefollowing homomorphic extension:\\begin{enumerate}\\item The logic formula \\phi (x_i,x_j,x_k,x_l,x_m) \\wedge \\psi (x_l,y_i,x_j,y_j) will be intensionally interpreted by the conceptu_1 \\in D_7, obtained by the algebraic expression ~conj_{S}(u,v) where u = I(\\phi (x_i,x_j,x_k,x_l,x_m)) \\in D_5, v =I(\\psi (x_l,y_i,x_j,y_j))\\in D_4 are the concepts of the virtualpredicates \\phi , \\psi , relatively, and S = \\lbrace (4,1),(2,3)\\rbrace .Consequently, we have that for any two formulae \\phi ,\\psi \\in \\mathcal {L}and a particular operator conj_S uniquely determined by tuples offree variables in these two formulae, I(\\phi \\wedge \\psi ) =conj_{S}(I(\\phi ),I(\\psi )).\\item The logic formula \\lnot \\phi (x_i,x_j,x_k,x_l,x_m) will beintensionally interpreted by the concept u_1 \\in D_5, obtained bythe algebraic expression ~neg(u) where u =I(\\phi (x_i,\\\\x_j,x_k,x_l,x_m)) \\in D_5 is the concept of thevirtual predicate \\phi .", "Consequently, we have that for any formula\\phi \\in \\mathcal {L}, ~I(\\lnot \\phi ) = neg(I(\\phi )).\\item The logic formula (\\exists x_k) \\phi (x_i,x_j,x_k,x_l,x_m) willbe intensionally interpreted by the concept u_1 \\in D_4, obtainedby the algebraic expression ~exists_{3}(u) where u =I(\\phi (x_i,x_j,x_k,x_l,x_m)) \\in D_5 is the concept of the virtualpredicate \\phi .", "Consequently, we have that for any formula \\phi \\in \\mathcal {L} and a particular operator exists_{n} uniquely determinedby the position of the existentially quantified variable in thetuple of free variables in \\phi (otherwise n =0 if thisquantified variable is not a free variable in \\phi ), ~I((\\exists x)\\phi ) = exists_{n}(I(\\phi )).\\end{enumerate}Once one has found a method for specifying the interpretations ofsingular terms of \\mathcal {L} (take in consideration the particularity ofabstracted terms), the Tarski-style definitions of truth andvalidity for \\mathcal {L} may be given in the customary way.What is proposed specifically in \\cite {Majk12a} is a method forcharacterizing the intensional interpretations of singular terms of\\mathcal {L} in such a way that a given singular abstracted term \\lessdot \\phi \\gtrdot _{\\alpha }^{\\beta } will denote an appropriate property,relation, or proposition, depending on the value of m =|\\alpha |.\\\\Notice than if \\beta = \\emptyset is the emptylist, then I(\\lessdot \\phi \\gtrdot _{\\alpha }^{\\beta } ) = I(\\phi ).Consequently, the denotation of \\lessdot \\phi \\gtrdot is equal to the meaning of a proposition \\phi , that is, ~I(\\lessdot \\phi \\gtrdot ) =I(\\phi )\\in D_0.", "In the case when \\phi is an atomp_i(x_1,..,x_m) then I (\\lessdot p_i(x_1,..,x_m)\\gtrdot _{x_1,..,x_m}) = I(p_i(x_1,..,x_m)) \\in D_m,while \\\\I (\\lessdot p_i(x_1,..,x_m)\\gtrdot ^{x_1,..,x_m}) = union(\\lbrace I(p_i(g(x_1),..,g(x_m)))~|~ g \\in {\\lbrace x_1,..,x_m\\rbrace } \\rbrace ) \\in D_0, with h(I (\\lessdot p_i(x_1,..,x_m)\\gtrdot ^{x_1,..,x_m})) =h(I((\\exists x_1)...(\\exists x_m)p_i(x_1,..,x_m))) \\in \\lbrace f,t\\rbrace .\\\\For example,\\\\ h(I(\\lessdot p_i(x_1) \\wedge \\lnot p_i(x_1)\\gtrdot ^{x_1})) = h(I((\\exists x_1)(\\lessdot p_i(x_1) \\wedge \\lnot p_i(x_1)\\gtrdot ^{x_1}))) = f.\\\\The interpretation of a more complex abstract \\lessdot \\phi \\gtrdot _\\alpha ^{\\beta } is defined in terms of the interpretationsof the relevant syntactically simpler expressions, because theinterpretation of more complex formulae is defined in terms of theinterpretation of the relevant syntactically simpler formulae, basedon the intensional algebra above.", "For example, I(p_i(x) \\wedge p_k(x)) = conj_{\\lbrace (1,1)\\rbrace }(I(p_i(x)), I(p_k(x))), I(\\lnot \\phi ) = neg(I(\\phi )), I(\\exists x_i)\\phi (x_i,x_j,x_i,x_k) = exists_3(I(\\phi )).\\\\Consequently, based on the intensional algebra in Definition\\ref {def:int-algebra} and on intensional interpretations ofabstracted terms, it holds that the interpretation of any formula in\\mathcal {L} (and any abstracted term) will be reduced to an algebraicexpression over interpretations of primitive atoms in \\mathcal {L}.", "Thisobtained expression is finite for any finite formula (or abstractedterm), and represents the \\emph {meaning} of such finite formula (or abstracted term).\\\\Let \\mathcal {A}_{FOL} = (\\mathcal {L}, \\doteq , \\top , \\wedge , \\lnot , \\exists ) be a freesyntax algebra for \"First-order logic with identity \\doteq \", withthe set \\mathcal {L} of first-order logic formulae, with \\top denotingthe tautology formula (the contradiction formula is denoted by \\lnot \\top ), with the set of variables in and the domain ofvalues in .", "\\\\Let us define the extensional relational algebra for the FOL by,\\\\\\mathcal {A}_{\\mathfrak {R}} = (\\mathfrak {R}, R_=, \\lbrace <>\\rbrace , \\lbrace \\bowtie _{S}\\rbrace _{ S\\in \\mathcal {P}(\\mathbb {N}^2)}, \\sim , \\lbrace \\pi _{-n}\\rbrace _{n \\in \\mathbb {N}}),\\\\where \\lbrace <>\\rbrace \\in \\mathfrak {R} is the algebraic valuecorrespondent to the logic truth, and R_= is the binary relationfor extensionally equal elements.We use ^{\\prime }=^{\\prime } for the extensional identity for relations in \\mathfrak {R}.\\\\Then, for any Tarski^{\\prime }s interpretation I_T its unique extension toall formulae I_T^*:\\mathcal {L}\\rightarrow \\mathfrak {R} is also thehomomorphism I_T^*:\\mathcal {A}_{FOL} \\rightarrow \\mathcal {A}_{\\mathfrak {R}} from thefree syntax FOL algebra into this extensional relational algebra.\\\\Consequently, we obtain the following Intensional/extensional FOL semantics\\cite {Majk09FOL}:\\\\For any Tarski^{\\prime }s interpretation I_T of the FOL, the followingdiagram of homomorphisms commutes,\\begin{diagram}&& & \\mathcal {A}_{int}~ (concepts/meaning) && &\\\\&^{intensional~interpret.~I} && \\frac{Frege/Russell}{semantics} &&^{h ~(extensionalization)} &\\\\\\mathcal {A}_{FOL}~(syntax)~~~~~~~~ && &_{I_T^*~(Tarski^{\\prime }s ~interpretation)}& && ~~~~~~~~\\mathcal {A}_{\\mathfrak {R}} ~(denotation) \\\\\\end{diagram}where h = is(w) where w = I_T \\in \\mathcal {W} is the explicit possibleworld (extensional Tarski^{\\prime }s interpretation).\\\\This homomorphic diagram formally express the fusion of Frege^{\\prime }s andRussell^{\\prime }s semantics \\cite {Freg92,Russe05,WhRus10} of meaning anddenotation of the FOL language, and renders mathematically correctthe definition of what we call an \"intuitive notion ofintensionality\", in terms of which a language is intensional ifdenotation is distinguished from sense: that is, if both adenotation and sense is ascribed to its expressions.", "This notion issimply adopted from Frege^{\\prime }s contribution (without its infinitesense-hierarchy, avoided by Russell^{\\prime }s approach where there is onlyone meaning relation, one fundamental relation between words andthings, here represented by one fixed intensional interpretationI), where the sense contains mode of presentation (here describedalgebraically as an algebra of concepts (intensions) \\mathcal {A}_{int}, andwhere sense determines denotation for any given extensionalizationfunction h (correspondent to a given Traski^{\\prime }s interpretaionI_T).", "More about the relationships between Frege^{\\prime }s and Russell^{\\prime }stheories of meaning may be found in the Chapter 7,\"Extensionality and Meaning\", in \\cite {Beal82}.\\\\As noted by Gottlob Frege and Rudolf Carnap (he uses termsIntension/extension in the place of Frege^{\\prime }s terms sense/denotation\\cite {Carn47}), the two logic formulae with the same denotation(i.e., the same extension for a given Tarski^{\\prime }s interpretation I_T)need not have the same sense (intension), thus such co-denotationalexpressions are not\\emph {substitutable} in general.\\\\In fact there is exactly \\emph {one} sense (meaning) of a given logicformula in \\mathcal {L}, defined by the uniquely fixed intensionalinterpretation I, and \\emph {a set} of possible denotations(extensions) each determined by a given Tarski^{\\prime }s interpretation ofthe FOL as follows from Definition \\ref {def:intensemant},\\begin{center}~~~ \\mathcal {L}~^I \\Longrightarrow _{h = is(I_T), I_T \\in ~\\mathcal {W}}~\\mathfrak {R}.\\end{center}Often ^{\\prime }intension^{\\prime } has been used exclusively in connection withpossible worlds semantics, however, here we use (as many others; asBealer for example) ^{\\prime }intension^{\\prime } in a more wide sense, that is as an\\emph {algebraic expression} in the intensional algebra of meanings(concepts) \\mathcal {A}_{int} which represents the structural composition ofmore complex concepts (meanings) from the given set of atomicmeanings.", "Consequently, not only the denotation (extension) iscompositional, but also the meaning (intension) is compositional.\\section {Canonical models for IRDBs}The application of the intensional FOL semantics to the DataIntegration system I= (\\mathcal {A},\\mathcal {S},\\mathcal {M}) in Definition \\ref {def:parsing}with the user defined RDB schema \\mathcal {A}= (S_A, \\Sigma _A) and thevector big table r_V can be summarized in what follows:\\begin{itemize}\\item Each relational name (symbol) r_k \\in S_A = \\lbrace r_1,...,r_n\\rbrace with the arity m =ar(r_k), is an intensional m-ary concept, so that r_k = I(\\lessdot r_k(\\textbf {x})\\gtrdot _{\\textbf {x}}) \\in D_m, for a tuple ofvariables \\textbf {x} = \\langle x_1,...,x_m \\rangle and any intensional interpretation I.\\\\For a given Tarski^{\\prime }s interpretation I_T, the extensionalizationfunction h is determined by h(r_k) = \\Vert r_k\\Vert = \\lbrace \\langle d_1,...,d_m\\rangle \\in m~|~I_T(r_k(d_1,...,d_m)) = t \\rbrace = I_T(r_k) \\in A.The instance database A of the user-defined RDB schema \\mathcal {A} is amodel of \\mathcal {A} if it satisfies all integrity constraints in\\Sigma _A.\\\\\\item The relational symbol (name) r_V of the vector big tableis a particular (extensional entity), r_V \\in D_{-1}, so that h(r_V) = r_V (the name of the database \\mathcal {A}).", "For a given modelA = \\lbrace \\Vert r_1\\Vert ,...,\\Vert r_n\\Vert \\rbrace of the user-defined RDB schema \\mathcal {A}, correspondent to a given Tarski^{\\prime }s interpretation I_T,its extension is determined by I_T(r_V) = \\Vert r_V\\Vert =\\overrightarrow{A}.\\\\\\item Intensional nature of the IRDB is evident in the fact thateach tuple \\langle r_k,Hash(d_1,...,\\\\d_m),nr_{r_k}(i),d_i\\rangle \\in \\overrightarrow{A}, corresponding to the atomr_V(y_1,y_2,y_3,y_4)/gfor an assignment g such that g(y_1) =r_k \\in D_m, g(y_3) =nr_{r_k}(i) \\in D_{-1}, g(y_2) = Hash(d_1,...,d_m)) \\in D_{-1} and g(y_4) = d_i \\in , is equal tothe intensional tuple \\langle I(\\lessdot r_k(\\textbf {x}) \\gtrdot _{\\textbf {x}},Hash(d_1,...,d_m),nr_{r_k}(i),d_i\\rangle .\\\\ Notice that the intensional tuples are different fromordinary tuples composed by only particulars (extensionalelements) in D_{-1}, what is the characteristics of the standardFOL (where the domain of values is equal to D_{-1}), while herethe \"value\" r_k = I(\\lessdot r_k(\\textbf {x}) \\gtrdot _{\\textbf {x}}) \\in D_mis an m-ary intensional concept, for which h(r_k) \\ne r_k is an m-aryrelation (while for all ordinary values d \\in D_{-1}, h(d) =d).\\end{itemize}The intensional Data Integration system I= (\\mathcal {A},\\mathcal {S},\\mathcal {M}) inDefinition \\ref {def:parsing} is used in the way that the globalschema is only virtual (empty) database with a user-defined schema\\mathcal {A}= (S_A, \\Sigma _A) used to define the SQL user-defined querywhich then has to be equivalently rewritten over the vector relationr_V in order to obtain the answer to this query.", "Thus, theinformation of the database is stored only in the big table\\Vert r_V\\Vert .", "Thus, the materialization of the original user-definedschema \\mathcal {A} can be obtained by the following operation:\\begin{definition} \\textsc {Materialization of the RDB} \\\\Given a user-defined RDB schema \\mathcal {A}= (S_A,\\Sigma _A) with S_A = \\lbrace r_1,...,r_n\\rbrace and a big vector table\\Vert r_V\\Vert , the non SQL operation \\emph {\\underline{MATTER}} whichmaterializes the schema \\mathcal {A} into its instance database A =\\lbrace R_1,...,R_n\\rbrace where R_k = \\Vert r_k\\Vert , for k = 1,...,n, is givenby the following mapping, forany R \\subseteq \\Vert r_V\\Vert :\\\\(r_k, R) ~~~\\mapsto ~~~ \\lbrace \\langle v_1,...,v_{ar(r_k)}\\rangle ~|~\\exists y \\in \\pi _2(R) ((r_V(r_k,y,nr_{r_k}(1),v_1)~\\underline{\\vee }~ v_1 \\emph {NULL})\\wedge ...\\\\...\\wedge (r_V(r_k,y,nr_{r_k}(ar(r_k)),v_{ar(r_k)})~\\underline{\\vee }~v_{ar(r_k)} \\emph {NULL})) \\rbrace ,\\\\so that the materialization of the schema \\mathcal {A} is defined by\\\\R_k =\\Vert r_k\\Vert \\triangleq \\emph {\\underline{MATTER}}(r_k, \\Vert r_V\\Vert ) for each r_k \\in S_A.\\end{definition}The canonical models of the intensional Data Integration system I= (\\mathcal {A},\\mathcal {S},\\mathcal {M}) in Definition \\ref {def:parsing} are the instances Aof the schema \\mathcal {A} such that\\\\\\Vert r_k\\Vert = \\underline{MATTER}(r_k,\\bigcup _{\\textbf {v}\\in \\Vert r_k\\Vert }\\underline{PARSE}(r_k,\\textbf {v})), that is,when\\\\A = \\lbrace \\underline{MATTER}(r_k, \\overrightarrow{A}) ~|~r_k \\in S_A\\rbrace .\\\\The canonical models of such intensional Data Integration system I= \\langle \\mathcal {A},\\mathcal {S},\\mathcal {M}\\rangle can be provided in a usual logicalframework as well:\\begin{propo} Let the IRDB be given by a Data Integration system I= \\langle \\mathcal {A},\\mathcal {S},\\mathcal {M}\\rangle for a used-defined global schema \\mathcal {A}=(S_A,\\Sigma _A) with S_A =\\lbrace r_1,...,r_n\\rbrace , the source schema \\mathcal {S}=(\\lbrace r_V\\rbrace ,\\emptyset ) with the vector big data relation r_V and theset of mapping tgds \\mathcal {M} from the source schema into he relationsof the global schema.", "Then a canonical model of I is any modelof the schema \\mathcal {A}^+ = (S_A \\bigcup \\lbrace r_V\\rbrace , \\Sigma _A \\bigcup \\mathcal {M}\\bigcup \\mathcal {M}^{OP}), where \\mathcal {M}^{OP} is an opposite mapping tgds from\\mathcal {A} into r_V given by the following set of tgds:\\\\\\mathcal {M}^{OP} = \\lbrace \\forall x_1,...,x_{ar(r_k)}((r_k(x_1,...,x_{ar(r_k)})\\wedge x_i \\emph {NOT NULL}) \\Rightarrow \\\\r_V(r_k, Hash(x_1,...,x_{ar(r_k)}),nr_{r_k}(i),x_i))~|~1\\le i \\le ar(r_k), r_k \\in S_A \\rbrace .\\end{propo}\\textbf {Proof}: It is enough to show that for each r_k \\in S_A, and\\textbf {x} = (x_1,...,x_{ar(r_k)}),\\\\r_k(\\textbf {x}) \\Leftrightarrow ((r_V(r_k,Hash(\\textbf {x}),nr_{r_k}(1),x_1) ~\\underline{\\vee }~ x_1NULL) \\wedge ...\\\\ \\wedge (r_V(r_k,Hash(\\textbf {x}),nr_{r_k}(ar(r_k)),x_{ar(r_k)})~\\underline{\\vee }~ x_{ar(r_k)} NULL) ).\\\\ From \\mathcal {M}^{OP} we have\\\\\\lnot r_k(\\textbf {x}) \\vee \\lnot x_i NOT NULL ~\\vee ~r_V(r_k,Hash(\\textbf {x}),nr_{r_k}(i),x_i), that is,\\\\(a) ~~~\\lnot r_k(\\textbf {x}) \\vee (x_i NULL ~\\vee ~r_V(r_k,Hash(\\textbf {x}),nr_{r_k}(i),x_i)).\\\\From the other side,from the fact that we have the constraint NOT NULL for theattribute {\\tt value} (in Definition \\ref {def:parsing}), then\\\\~~~\\lnot r_k(\\textbf {x}) \\vee \\lnot x_i NULL ~\\vee ~ \\lnot r_V(r_k,Hash(\\textbf {x}),nr_{r_k}(i),x_i)), that is\\\\(b) ~~~\\lnot r_k(\\textbf {x}) \\vee \\lnot (x_i NULL ~\\wedge ~r_V(r_k,Hash(\\textbf {x}),nr_{r_k}(i),x_i)), \\\\is true and also the conjunction of (a) and (b) has to be true,i.e.,\\\\ (\\lnot r_k(\\textbf {x}) \\vee (x_i NULL ~\\vee ~r_V(r_k,Hash(\\textbf {x}),nr_{r_k}(i),x_i))) \\wedge (\\lnot r_k(\\textbf {x}) \\vee \\lnot (x_i NULL ~\\wedge ~\\\\ ~r_V(r_k,Hash(\\textbf {x}),nr_{r_k}(i),x_i))), thus, by distributivity,\\\\(b) ~~~\\lnot r_k(\\textbf {x}) \\vee (x_i NULL ~\\underline{\\vee }~r_V(r_k,Hash(\\textbf {x}),nr_{r_k}(i),x_i)).\\\\If we repeat this for all 1\\le i \\le ar(r_k) and conjugate allthese true formula, again by distributive property of conjunction\\wedge , we obtain\\\\~~~\\lnot r_k(\\textbf {x}) \\vee ((x_1 NULL ~\\underline{\\vee }~r_V(r_k,Hash(\\textbf {x}),nr_{r_k}(1),x_1) \\wedge ...\\\\\\wedge (x_{ar(r_k)} NULL ~\\underline{\\vee }~r_V(r_k,Hash(\\textbf {x}),nr_{r_k}(ar(r_k)),x_{ar(r_k)})), thatis,\\\\(c) ~~~r_k(\\textbf {x}) \\Rightarrow ((r_V(r_k,Hash(\\textbf {x}),nr_{r_k}(1),x_1) ~\\underline{\\vee }~ x_1NULL) \\wedge ...\\\\ \\wedge (r_V(r_k,Hash(\\textbf {x}),nr_{r_k}(ar(r_k)),x_{ar(r_k)})~\\underline{\\vee }~ x_{ar(r_k)} NULL) ).\\\\Moreover, from Definition we also have\\\\(d) ~~~r_k(\\textbf {x}) \\Leftarrow ((r_V(r_k,Hash(\\textbf {x}),nr_{r_k}(1),x_1) ~\\underline{\\vee }~ x_1NULL) \\wedge ...\\\\ \\wedge (r_V(r_k,Hash(\\textbf {x}),nr_{r_k}(ar(r_k)),x_{ar(r_k)})~\\underline{\\vee }~ x_{ar(r_k)} NULL) ),\\\\That is, the logical equivalence of the formula on the left and onthe right side of the logical implication, and hence if the atomr_k(\\textbf {x}) is true for some assignment to the variables gso that the tuple \\langle g(x_1),...,g(x_{ar(r_k)})\\rangle is inrelation r_k \\in \\mathcal {A}, then for the same assignment g the everyformula\\\\ r_V(r_k,Hash(\\textbf {x}),nr_{r_k}(1),x_1)~\\underline{\\vee }~ x_1 NULL,\\\\...,\\\\ r_V(r_k,Hash(\\textbf {x}),nr_{r_k}(ar(r_k)),x_{ar(r_k)})~\\underline{\\vee }~ x_{ar(r_k)} NULL \\\\has to be true and hencegenerates the tuples in r_V for NOT NULL values of g(x_i),1\\le i \\le ar(r_k).\\\\Notice that the implication (c) corresponds to the nonSQL operation\\underline{PARSE}, while the implication (d) is the logicalsemantics of the non SQL operation \\underline{MATTER}.\\\\Consequently, we obtain \\Vert r_k\\Vert = \\underline{MATTER}(r_k,\\bigcup _{\\textbf {v}\\in \\Vert r_k\\Vert }\\underline{PARSE}(r_k,\\textbf {v})), that is,A = \\lbrace \\underline{MATTER}(r_k, \\overrightarrow{A}) ~|~r_k \\in S_A\\rbrace and the database instanceA which satisfies all integrity constraints \\Sigma _A \\bigcup \\mathcal {M}\\bigcup \\mathcal {M}^{OP} is the canonical model of the intensional DataIntegration system I= (\\mathcal {A},\\mathcal {S},\\mathcal {M}).\\\\\\square \\\\By joking with the words, we can say that \"by PARSEing the MATTER weobtain the pure energy\" of the big vector relation, and conversely,\"by condensing the PARSEd energy we obtain the common MATTER\" in a standard RDB.\\\\The fact is that we do not need both of them because they areequivalent, so instead of the more (schema) rigid RDB matter in \\mathcal {A}we prefer to use the non rigid pure energy of the big vector tabler_V.", "But we are also able to render more flexible this approachand to decide only a subset of relations to be the intensionalconcepts whose extension has to be parsed in to the vector big tabler_V.", "For standard legacy systems we can chose to avoid at all tohave the intensional concepts, thus to have the standard RDBs withstandard FOL Tarski^{\\prime }s semantics.", "By declaring any of the relationalnames r_k \\in S_A as an intensional concept, we conservativelyextend the Tarski^{\\prime }s semantics for the FOL in order to obtain a moreexpressive intensional FOL semantics for the IRDBs.\\\\The fact that we assumed r_V to be only a particular (extensionalentity) is based on the fact that it always will be materialized(thus non empty relational table) as standard tables in the RDBs.The other reason is that the extension h(r_V) has not to be equalto the vector relation (the set of tuples) \\Vert r_V\\Vert but to the\\emph {set of relations} in the instance database A. Consequently,we do not use the r_V (equal to the name of the database \\mathcal {A}) asa value in the tuples of other relations and we do not use theparsing used for all relations in the user-defined RDBschema \\mathcal {A} assumed to be the intensional concepts as well.\\\\If we would decide to use also r_V as an intensional concept inD_4 we would be able to parse it as all other intensional conceptsin \\mathcal {A} into itself, and such recursive definition will render (onlytheoretically) an infinite extension of the r_V, thus nonapplicable, as follows.", "Let r_k = Person \\in \\mathcal {A} be an user-definedrelational table, and Pname an attribute of this table, and letID be the t-index value obtained by Hash function from one tupleof r_k where the value of the attribute Pname is \"MarcoAurelio\", then we willhave this tuples in the vector table with (database) \\emph {name} r_V \\in D_4:\\\\1.", "\\langle Person, ID, Pname, Marco Aurelio\\rangle \\in \\Vert r_V\\Vert ;\\\\then by parsing this tuple 1, we will obtain for ID_1 =Hash(Person, ID,Pname, \\\\Marco Aurelio) the following new tuples\\\\2.", "\\langle r_V, ID_1, {\\tt r-name}, Person \\rangle \\in \\Vert r_V\\Vert ;\\\\3.", "\\langle r_V, ID_1,{\\tt t-index}, ID \\rangle \\in \\Vert r_V\\Vert ;\\\\4.", "\\langle r_V, ID_1,{\\tt a-name}, Pname \\rangle \\in \\Vert r_V\\Vert ;\\\\5.", "\\langle r_V, ID_1, {\\tt value}, Marco Aurelio \\rangle \\in \\Vert r_V\\Vert ;\\\\then by parsing this tuple 2, we will obtain for ID_2 =Hash(r_V, ID_1, {\\tt r-name},\\\\ Person) the following new tuples\\\\6.", "\\langle r_V, ID_2, {\\tt r-name}, {\\tt Vector} \\rangle \\in \\Vert r_V\\Vert ;\\\\7.", "\\langle r_V, ID_2, {\\tt t-index}, ID_1 \\rangle \\in \\Vert r_V\\Vert ;\\\\8.", "\\langle r_V, ID_2, {\\tt a-name}, {\\tt r-name} \\rangle \\in \\Vert r_V\\Vert ;\\\\9.", "\\langle r_V, ID_2, {\\tt value}, Person \\rangle \\in \\Vert r_V\\Vert ;\\\\then by parsing this tuple 2, we will obtain for ID_3 =Hash(r_V, ID_2, {\\tt r-name}, \\\\Person) the following new tuples\\\\10.", "\\langle r_V, ID_3, {\\tt r-name}, r_V \\rangle \\in \\Vert r_V\\Vert ;\\\\...\\\\Thus, as we see, the tuple 10 is equal to the tuple 6, but only withnew t-index (tuple index) value, so by continuing this process,theoretically (if we do not pose the limits for the values oft-indexes) we obtain an infinite process and an infinite extensionof r_V.", "Obviously, it can not happen in real RDBs, because thelength of the attribute {\\tt t-index} is finite so that at somepoint we will obtained the previously generated value for thisattribute (we reach a fixed point), and from the fact that thisattribute is a part of the primary key this tuple would not beinserted in r_V because r_V contains the same tuple already.\\\\Notice that this process is analogous to the selfreferencingprocess, where we try to use an intensional concept \\emph {as anelement of itself} that has to be avoided and hence there is nosense to render r_V an intensional concept.", "Consequently, the IRDBhas at least one relational table which is not an intensionalconcept and which will not be parsed: the vector big table, whichhas this singular built-in property in every IRDB.\\section {NewSQL property of the IRDBs }This last section we will dedicate to demonstrate that the IRDBs arecomplete w.r.t.", "the standard SQL.", "This demonstration is based on thefact that each SQL query, defined over the user-defined schema \\mathcal {A},which (in full intensional immersion) is composed by the intensionalconcepts, will be executed over standard relational tables that\\emph {are not} the intensional concepts.", "If a query is defined overthe non-intensional concepts (relations) in \\mathcal {A} in this case itwill be directly executed over these relational tables as in everyRDB.", "If a query is defined over the intensional concepts in \\mathcal {A}(which will remain \\emph {empty tables}, i.e., non materialized) thenwe need to demonstrate the existence of an effective query-rewritinginto an equivalent SQL query over the (non-intensional concept)vector big table r_V.", "In order to define this query-rewriting, wewill shortly introduce the abstract syntax and semantics ofCodd^{\\prime }s relational algebra, as follows.\\\\Five primitive operators of Codd^{\\prime }s algebra are: the selection, theprojection, the Cartesian product (also called the cross-product orcross-join), the set union, and the set difference.", "Anotheroperator, rename, was not noted by Codd, but the need for it isshown by the inventors of Information Systems Base Language (ISBL)for one of the earliest database management systems whichimplemented Codd^{\\prime }s relational model of data.", "These six operators arefundamental in the sense that if we omit any one of them, we willlose expressive power.", "Many other operators have been defined interms of these six.", "Among the most important are set intersection,division, and the natural join.", "In fact, ISBL made a compelling casefor replacing the Cartesian product with the natural join, of whichthe Cartesianproduct is a degenerate case.", "\\\\Recall that two relations r_1 and r_2 are union-compatible iff\\lbrace atr(r_1)\\rbrace = \\lbrace atr(r_2)\\rbrace , where for a given list (or tuple) ofthe attributes ~~\\textbf {a} = atr(r) = <a_1,...,a_k> =<atr_r(1),...,atr_r(k)>, we denote the \\emph {set} \\lbrace a_1,...,a_k\\rbrace by \\lbrace atr(r)\\rbrace , k = ar(r), with the injective functionnr_r:\\lbrace 1,...,k\\rbrace \\rightarrow SN which assigns distinct names toeach column of this relation.", "If a relation r_2 is obtained from agiven relation r_1 by permutating its columns, then we tell thatthey are not equal (in set theoretic sense) but that they areequivalent.", "Notice that in the RDB theory the two equivalentrelations are considered equal as well.", "In what follows, given anytwo lists (tuples), \\textbf {d} = <d_1,...,d_k> and \\textbf {b} =<b_1,...,b_m> their concatenation <d_1,...,d_k,b_1,...,b_m> isdenoted by \\textbf {d} \\& \\textbf {b}, where ^{\\prime }\\&^{\\prime } is the symbolfor concatenation of the lists.", "By \\Vert r\\Vert we denote the extensionof a given relation (relational symbol) r; it is extended to anyterm t_R of Codd^{\\prime }s algebra, so that \\Vert t_R\\Vert is the relationobtained by computation of this term.Let us briefly define thesebasic operators, and their correspondence with the formulae of FOL:\\index {First-Order Logic (FOL)}\\begin{enumerate}\\item Rename is a unary operation written as \\_~ RENAME ~name_1~AS~name_2 where the result is identical to input argument (relation) r except that thecolumn i with name nr_r(i) = name_1 in all tuples is renamed to nr_r(i) =name_2.\\\\This operation is neutral w.r.t.", "the logic, where we are using thevariables for the columns of relational tables and not theirnames.\\item Cartesian product is a binary operation \\_~ TIMES \\_~, written also as \\_~ \\bigotimes \\_~,such that for the relations r_1 and r_2, first we do the rename normalizationof r_2 (w.r.t.", "r_1), denoted by r^\\rho _2, such that:\\\\For each k-th copy of the attribute a_i (or, equivalently, a_i(0))of the m-th column of r_2 (with 1\\le m \\le ar(r_2)), denoted by a_i(k) = atr_{r_2}(m) \\in atr(r_2), such that the maximum index of the same attributea_i in r_1 is a_i(n), we change r_2 by:\\\\1.", "a_i(k) \\mapsto a_i(k+n);\\\\2.", "if name_1 = nr_{r_2}(m) is a name that exists in the set of the columnnames in r_1, then we change the naming functionnr_{r_2}:\\lbrace 1,...,ar(r_2)\\rbrace \\rightarrow SN, by nr_{r_2}(m) = name_2, wherename_2 \\in SN is a new name distinct from all other usednames, and we define the renaming normalization \\rho by mapping name_1 \\mapsto name_2.\\\\The relation obtained from r_2, after this renamingnormalization,will be denoted by r^\\rho _2.Then we define the new relation r (when both \\Vert r_1\\Vert \\ne \\lbrace <>\\rbrace and\\Vert r_2\\Vert \\ne \\lbrace <>\\rbrace , i.e., when are not empty relations)by~ r_1 \\bigotimes r^\\rho _2,\\\\with \\Vert r \\Vert \\triangleq \\lbrace \\textbf {d}_1\\& \\textbf {d}_2~|~\\textbf {d}_1 \\in \\Vert r_1\\Vert , \\textbf {d}_2 \\in \\Vert r_2\\Vert \\rbrace , with thenaming function nr_r:\\lbrace 1,...,ar(r_1)+ar(r_2)\\rbrace \\rightarrow SN,such that nr_{r}(i) = nr_{r_1}(i) for 1\\le i \\le ar(r_1) andnr_{r}(i) = nr_{r_2}(i) for 1+ar(r_1)\\le i \\le ar(r_1)+ar(r_2), and atr_r:\\lbrace 1,....,ar(r_1)+ar(r_2)\\rbrace \\rightarrow \\textbf {att} function defined by atr_{r}(i) = atr_{r_1}(i) for1\\le i \\le ar(r_1) and atr_{r}(i) = atr_{r_2}(i) for1+ar(r_1)\\le i \\le ar(r_1)+ar(r_2).\\\\This Cartesian product is given by the followinglogical equivalence, by considering the relational symbols as predicates,\\\\r(x_1,...,x_{ar(r_1)},y_1,...,y_{ar(r_2)})\\Leftrightarrow (r_1(x_1,...,x_{ar(r_1)}) \\wedge r_2(y_1,...,y_{ar(r_2)})), so that\\Vert r \\Vert = \\Vert r_1(x_1,...,x_{ar(r_1)}) \\wedge r_2(y_1,...,y_{ar(r_2)})\\Vert .\\\\(if \\Vert r_1\\Vert is empty then ~ r_1 \\bigotimes r^\\rho _2 = r_2;if \\Vert r_2\\Vert is empty then ~ r_1 \\bigotimes r^\\rho _2 = r_1).\\item Projection is a unary operation written as \\_~[S], whereS is a tuple of column names such that for a relation r_1and S = <nr_{r_1}(i_1),...,nr_{r_1}(i_k)>, with k \\ge 1 and1 \\le i_m \\le ar(r_1) for 1 \\le m \\le k, and i_m \\ne i_j if m \\ne j, we define the relation r by:~r_1[S],\\\\with \\Vert r\\Vert = \\Vert r_1\\Vert if \\exists name \\in S.name \\notin nr(r_1);otherwise \\Vert r\\Vert =\\pi _{<i_1,...,i_k>}(\\Vert r_1\\Vert ), where nr_r(m) =nr_{r_1}(i_m), atr_r(m) = atr_{r_1}(i_m), for1 \\le m \\le k.\\\\This projection is given by the followinglogical equivalence\\\\r(x_{i_1},...,x_{i_k}) \\Leftrightarrow \\exists x_{j_1}...x_{j_n}r_1(x_1,...,x_{ar(r_1)}),\\\\ where n = ar(r_1)-kand for all 1 \\le m \\le n, j_m \\notin \\lbrace i_1,...,i_k\\rbrace , so that\\\\\\Vert r \\Vert = \\Vert \\exists x_{j_1}...x_{j_n}r_1(x_1,...,x_{ar(r_1)}) \\Vert .\\item Selection is a unary operation written as \\_~ WHERE ~C, where a condition C is a finite-length logical formula that consists of atoms^{\\prime }(name_i ~\\theta ~name_j)^{\\prime }~ or ~^{\\prime }(name_i ~\\theta ~\\overline{d})~^{\\prime },with built-in predicates \\theta \\in \\Sigma _\\theta \\supseteq \\lbrace \\doteq ,>,< \\rbrace , a constant \\overline{d}^{\\prime },and the logical operators \\wedge (AND), \\vee (OR) and \\lnot (NOT), such that for a relation r_1 and name_i, name_j the names of its columns, we define the relation r by\\\\ r_1 ~WHERE~ C,\\\\as the relation with atr(r) = atr(r_1) and the function nr_r equal to nr_{r_1}, where \\Vert r\\Vert is composedby the tuples in \\Vert r_1\\Vert for which C is satisfied.\\\\This selection is given by the followinglogical equivalence:\\\\r(x_{i_1},...,x_{i_k}) \\Leftrightarrow (r_1(x_1,..., x_{ar(r_1)}) \\wedge C(\\textbf {x})),\\\\where C(\\textbf {x}) is obtained bysubstitution of each name_i = nr_{r_1}(j) (of the j-th column of r_1)in the formula C by the variable x_j, so that\\\\\\Vert r \\Vert = \\Vert r_1(x_1,..., x_{ar(r_1)}) \\wedge C(\\textbf {x}) \\Vert .\\\\4.1 We assume as an \\emph {identity unary operation}, the operation \\_~ WHERE~C when C is the atomic condition \\overline{1} \\doteq \\overline{1}(i.e., a tautology).\\item Union is a binary operation written as \\_~ UNION \\_~, such that for twounion-compatible relations r_1 and r_2, we define the relation r by:r_1~ UNION ~r_2,\\\\where \\Vert r\\Vert \\triangleq \\Vert r_1\\Vert \\bigcup \\Vert r_2\\Vert , with atr(r) =atr(r_1), and the functions atr_r = atr_{r_1}, and nr_r =nr_{r_1}.This union is given by the followinglogical equivalence:\\\\r(x_{1},...,x_{n}) \\Leftrightarrow (r_1(x_1,...,x_{n}) \\vee r_2(x_1,...,x_{n})),\\\\where n = ar(r) = ar(r_1) = ar(r_2), so that\\\\\\Vert r\\Vert = \\Vert r_1(x_1,...,x_{n}) \\vee r_2(x_1,...,x_{n})\\Vert .\\item Set difference is a binary operation written as \\_~ MINUS \\_~ such that for twounion-compatible relations r_1 and r_2, we define the relation r by:r_1 ~MINUS~ r_2,\\\\where \\Vert r\\Vert \\triangleq \\lbrace \\textbf {t}~|~\\textbf {t}\\in \\Vert r_1\\Vert suchthat \\textbf {t}\\notin \\Vert r_2\\Vert \\rbrace , with atr(r) = atr(r_1), and thefunctions atr_r = atr_{r_1}, andnr_r = nr_{r_1}.\\\\Let r_1 and r_2 be the predicates (relational symbols) for thesetwo relations.", "Then their difference is given by the followinglogical equivalence:\\\\r(x_{1},...,x_{n}) \\Leftrightarrow (r_1(x_1,...,x_{n}) \\wedge \\lnot r_2(x_1,...,x_{n})),\\\\where n = ar(r) = ar(r_1) = ar(r_2) and hence\\\\\\Vert r\\Vert = \\Vert r_1(x_1,...,x_{n}) \\wedge \\lnot r_2(x_1,...,x_{n})\\Vert .\\end{enumerate}Natural join \\bowtie _S is a binary operator, written as (r_1\\bowtie _S r_2), where r_1 and r_2 are the relations.", "The resultof the natural join is the set of all combinations of tuples inr_1 and r_2 that are equal on their common attribute names.", "Infact, (r_1 \\bowtie _S r_2) can be obtained by creating theCartesian product r_1\\bigotimes r_2 and then by execution of theSelection with the condition C defined as a conjunction of atomicformulae (nr_{r_1}(i) = nr_{r_2}(j)) with(nr_{r_1}(i),nr_{r_2}(j)) \\in S (where i and j are the columnsof the same attribute in r_1 and r_2, respectively, i.e.,satisfying atr_{r_1}(i) = atr_{r_2}(j)) that represents theequality of the common attribute names of r_1 and r_2.The natural join is arguably one of the mostimportant operators since it is the relational counterpart oflogical AND.", "Note carefully that if the same variable appears ineach of two predicates that are linked by AND, then that variablestands for the same thing and both appearances must always besubstituted by the same value.", "In particular, natural join allowsthe combination of relations that are associated by a foreign key.It can also be used to define composition of binary relations.", "\\\\Altogether, the operators of relational algebra have identicalexpressive power to that of domain relational calculus or tuplerelational calculus.", "However, relational algebra is less expressivethan first-order predicate calculus without function symbols.Relational algebra corresponds to \\emph {a subset} of FOL\\index {First-Order Logic (FOL)} (denominated \\emph {relationalcalculus}), namely Horn clauses without recursion and negation (orunion of conjunctive queries).", "Consequently, relational algebra isessentially equivalent in expressive power to \\emph {relationalcalculus} (and thus FOL and queries defined in Section\\ref {sec:vector}); this result is known as Codd^{\\prime }s theorem.", "However,the negation, applied to a formula of the calculus, constructs aformula that may be true on an infinite set of possible tuples.To overcome this difficulty, Codd restricted the operands ofrelational algebra to finite relations only and also proposedrestricted support for negation \\lnot (NOT) and disjunction \\vee (OR).", "Codd defined the term \"relational completeness\" to refer to alanguage that is complete with respect to first-order predicatecalculus apart from the restrictions he proposed.", "In practice, therestrictions have no adverse effect on the applicability of hisrelational algebra for database purposes.", "\\\\Several papers have proposed new operators of an algebraic nature ascandidates for addition to the original set.", "We choose theadditional unary operator ^{\\prime }EXTEND\\_~ADD a,name AS e^{\\prime } denotedshortly as \\_~\\langle a,name,e\\rangle , where a is a new addedattribute as the new column (at the end of relation) with a newfresh name name and e is an expression (in the most simple casesit can be the value NULL or a constant \\overline{d}, or the i-thcolumn name nr(i) of the argument (i.e., relation) of thisoperation), for \\emph {update} relational algebra operators, in orderto cover all of the basic features of data manipulation (DML)aspects of a relation models of data, so that\\begin{itemize}\\item We define a unary operator \\_~\\langle a,name,e\\rangle , for an attributea \\in \\textbf {att}, its name, and expression e, as a function with aset of column names, such thatfor a relation r_1 and expression e composed of thenames of the columns of r_1 with n = ar(r_1), we obtain the(ar(r_1)+1)-aryrelation r by~ \\langle a,name, e\\rangle (r_1),\\\\with naming function nr_r:\\lbrace ar(r_1)+1\\rbrace \\rightarrow SN suchthat nr_r(i) = nr_{r_1}(i) if i \\le ar(r_1); nr_r(ar(r_1)+1) = name otherwise, being afresh new name for this column; with the attribute function atr_r:\\lbrace ar(r_1)+1\\rbrace \\rightarrow \\textbf {att} suchthat atr_r(i) = atr_{r_1}(i) if i \\le ar(r_1); atr_r(ar(r_1)+1) = a otherwise, and\\\\ \\Vert r\\Vert = \\lbrace <>\\rbrace \\bigcup \\lbrace \\textbf {d}\\& e(\\textbf {d})~|~\\textbf {d} \\in \\Vert r_1\\Vert \\rbrace ,\\\\where e(\\textbf {d}) \\in dom(a) is a constant or the value obtained from the functione where each name nr_r(i) is substituted by the value d_iof the tuple \\textbf {d} = <d_1,...,d_n> \\in \\Vert r_1\\Vert ; in the specialcases, we can use nullary functions (constants) for the expressione (for example, for the NULL value).\\\\(note that r is empty if e is an expression and r_1 emptyas well).\\\\Then, for a nonempty relation r_1, the EXTEND~ r_1 ADD a,name AS e (i.e., r_1 \\langle a,name,e\\rangle )can be represented by the followinglogical equivalence:\\\\~r(x_1,...,x_{n+1}) \\Leftrightarrow (r_1(x_1,...,x_{n}) \\wedge (x_{n+1} =e(\\textbf {x})))\\\\where e(\\textbf {x}) is obtained bysubstituting each name_i = nr_{r_1}(j) (of the j-th columnof r_1) in the expression e by the variable x_j.\\end{itemize}We are able to define a new relation with a single tuple \\langle \\overline{d}_1,..,\\overline{d}_k \\rangle , k \\ge 1 with the givenlist of attributes \\langle a_1,..,a_k \\rangle , by the followingfinite length expression,\\\\EXTEND (...(EXTEND r_\\emptyset ADD a_1,name_1 AS\\overline{d}_1)...) ADD a_k,name_k AS \\overline{d}_k, orequivalently byr_\\emptyset \\langle a_1,name_1, \\overline{d}_1\\rangle \\bigotimes ...\\bigotimes r_\\emptyset \\langle a_k,name_k, \\overline{d}_k\\rangle ,\\\\where r_\\emptyset is the emptytype relation with \\Vert r_\\emptyset \\Vert = \\lbrace <>\\rbrace , ar(r_\\emptyset ) =0 introduced in Definition \\ref {def:bealer}, and empty functionsatr_{r_\\emptyset } and nr_{r_\\emptyset }.", "Such single tuplerelations can be used for an insertion in a given relation (with thesame list of attributes) inwhat follows.\\\\\\textbf {Update operators.}", "The three update operators,^{\\prime }UPDATE^{\\prime }, ^{\\prime }DELETE^{\\prime } and ^{\\prime }INSERT^{\\prime } of the Relational algebra, arederived operators from these previously defined operators in thefollowing way:\\begin{enumerate}\\item Each algebraic formulae ^{\\prime }DELETE FROM ~ r WHERE C^{\\prime } isequivalent to the formula ^{\\prime }r MINUS (r WHERE C)^{\\prime }.\\item Each algebraic expression (a term) ^{\\prime }INSERT INTO ~r[S] VALUES (list of values)^{\\prime }, ^{\\prime }INSERT INTO ~r[S] AS SELECT...^{\\prime },is equivalent to ^{\\prime }r~ UNION ~r_1^{\\prime } where the union compatiblerelation r_1 is a one-tuple relation(defined by list) in the first, or a relation defined by ^{\\prime }SELECT...^{\\prime } in the second case.\\\\In the case of a single tuple insertion (version with ^{\\prime }VALUES^{\\prime }) intoa given relation r, we can define a single tuple relation r_1 byusing ^{\\prime }EXTEND..^{\\prime } operations.\\item Each algebraic expression ^{\\prime }UPDATE ~r SET [nr_r(i_1)= e_{i_1} ,..., nr_r(i_k) =e_{i_k}] WHERE C^{\\prime }, for n =ar(r), where e_{i_m}, 1 \\le i_m\\le n for 1 \\le m\\le k are the expressions and C is acondition, is equal to the formula ^{\\prime }(rWHERE \\lnot C) UNION r_1^{\\prime } , where r_1 is a relation expressed by\\\\(EXTEND(...(EXTEND (r WHERE C) ADD att_r(1),name_1 AS e_1)...) ADD att_r(n),name_n ASe_{n})[S],\\\\such that for each 1 \\le m \\le n, if m \\notin \\lbrace i_1,...,i_k \\rbrace then e_m =nr_r(m), and S = <name_1,...,name_n>.\\end{enumerate}Consequently, all update operators of the relational algebra can beobtained by addition of these ^{\\prime }EXTEND \\_~ ADD a,name AS e^{\\prime }operations.", "\\\\Let us define the \\Sigma _R-algebras sa follows (\\cite {Majk14},Definition 31 in Section 5.1):\\begin{definition} We denote the algebra of the set of operations, introduced previously inthis section (points from 1 to 6 and \\emph {EXTEND} \\_~ \\emph {ADD}a,name \\emph {AS} e) with additional nullary operator(empty-relation constant) \\perp , by \\Sigma _{RE}.", "Its subalgebrawithout \\_~ \\emph {MINUS} \\_~ operator is denoted by\\Sigma _R^+, and without \\perp and unary operators \\emph {EXTEND}\\_~ \\emph {ADD} a,name \\emph {AS} e is denoted by \\Sigma _R(it is the \"select-project-join-rename+union\" (\\emph {SPJRU})subalgebra).", "We define the set of terms PX with variables inXof this \\Sigma _R-algebra (and analogously for the terms P^+Xof \\Sigma _R^+-algebra),inductively as follows:\\\\1.", "Each relational symbol (a variable) r \\in X \\subseteq \\mathbb {R} and a constant (i.e., a nullary operation) is a term inP X;\\\\2.", "Given any term t_R \\in P X and an unary operation o_i \\in \\Sigma _R, o_i(t_R)\\in P X;\\\\3.", "Given any two terms t_R, t^{\\prime }_R \\in P X and a binary operationo_i \\in \\Sigma _R, o_i(t_R,t^{\\prime }_R) \\in P X.\\\\We define the evaluation of terms in P X, for X = \\mathbb {R},by extending the assignment \\Vert \\_~\\Vert :\\mathbb {R} \\rightarrow \\underline{\\Upsilon }, which assigns a relation to each relationalsymbol (a variable) to all terms by the function \\Vert \\_~\\Vert _{\\#}:P\\mathbb {R} \\rightarrow \\underline{\\Upsilon } (with \\Vert r\\Vert _{\\#} =\\Vert r\\Vert ), where \\underline{\\Upsilon } is the universal databaseinstance (set of all relations for a given universe ).", "For agiven term t_R with relational symbols r_1,..,r_k \\in \\mathbb {R}, \\Vert t_R\\Vert _{\\#} is the relational table obtained fromthis expression for the given set of relations \\Vert r_1\\Vert ,...,\\Vert r_k\\Vert \\in \\underline{\\Upsilon }, with the constraint that \\\\ \\Vert t_R \\emph {UNION} t^{\\prime }_R\\Vert _{\\#} = \\Vert t_R \\Vert _{\\#} \\bigcup \\Vert t^{\\prime }_R\\Vert _{\\#} ifthe relations \\Vert t_R \\Vert _{\\#} and \\Vert t^{\\prime }_R \\Vert _{\\#} areunion compatible; \\perp = \\lbrace <>\\rbrace = \\Vert r_\\emptyset \\Vert (empty relation) otherwise.\\\\We say that two terms t_R, t^{\\prime }_R \\in P X are equivalent (orequal), denoted by t_R \\approx t^{\\prime }_R, if for all assignments~\\Vert t_R\\Vert _{\\#} = \\Vert t^{\\prime }_R\\Vert _{\\#}.\\end{definition}We saythat an extension \\Vert t_R\\Vert _{\\#}, of a term t_R \\in PX, is\\emph {vector relation} of the \\emph {vector view} denoted by\\overrightarrow{t_R} if the type of \\Vert t_R\\Vert _{\\#} is equal to thetype of the vector relation r_V.\\\\Let R = \\Vert \\overrightarrow{t_R}\\Vert _{\\#} be the relational table withthe four attributes (as r_V){\\tt r-name},{\\tt t-index}\\\\{\\tt a-name} and {\\tt value}, thenits used-defined view representation can be derived as follows:\\begin{definition} \\textsc {View Materialization}: Let t_R \\in PX be a user-defined SPJU(Select-Project-Join-Union) view over a database schema \\mathcal {A}=(S_A,\\Sigma _A) with the type (the tuple of the view columns)\\mathfrak {S} =\\langle (r_{k_1},name_{k_1}),...,(r_{k_m},name_{k_m})\\rangle , wherethe i-th column (r_{k_i},name_{k_i}) is the column with nameequal to name_{k_i} of the relation name r_{k_i}\\in S_A, 1\\le i \\le m, and \\overrightarrow{t_R} be the rewritten query overr_V.", "Let R = \\Vert \\overrightarrow{t_R}\\Vert _{\\#} be the resultingrelational table with the four attributes (as r_V){\\tt r-name},{\\tt t-index},{\\tt a-name} and {\\tt value}.", "Wedefine the operation \\emph {\\underline{VIEW}} of the transformationof R into the user defined viewrepresentation by:\\\\\\emph {\\underline{VIEW}}(\\mathfrak {S},R) = \\lbrace \\langle d_1,...,d_m\\rangle ~|~\\exists ID \\in \\pi _3(R), \\forall _{1\\le i \\le m}(\\langle r_{k_i},name_{k_i},ID,d_i \\rangle \\in R; otherwise setd_i to \\emph {NULL} \\rbrace .\\end{definition}Notice that we have \\Vert r_k\\Vert = \\underline{VIEW}(\\mathfrak {S},R) =\\underline{MATTER}(r_k,R) for each r_k \\in S_A with R =\\bigcup _{\\textbf {d} \\in \\Vert r_k\\Vert }\\underline{PARSE}(r_k,\\textbf {d}),and \\mathfrak {S} =\\langle (r_{k},nr_{r_k}(1)),...,(r_{k},nr_{r_k}(ar(r_k)))\\rangle ,and hence the nonSQL operation \\underline{MATTER} is a special caseof the operation \\underline{VIEW}.\\\\For any original user-defined query (term)t_R over a user-defined database schema \\mathcal {A}, by\\overrightarrow{t_R} we denote the equivalent (rewritten) queryover the vector relation r_V.", "We have the following importantresult for the IRDBs:\\begin{propo} There exists a complete algorithm for theterm rewriting of any user-defined SQL term t_R over a schema \\mathcal {A}, of thefull relational algebra \\Sigma _{RE} in Definition\\ref {def:relAlg}, into an equivalent vector query\\overrightarrow{t_R} over the vector relation r_V.", "\\\\If ~t_Ris a SPJU term (in Definition \\ref {def:view-mater}) of the type\\mathfrak {S} then \\Vert t_R\\Vert _{\\#} =\\emph {\\underline{VIEW}}(\\mathfrak {S},\\Vert \\overrightarrow{t_R}\\Vert _{\\#}).\\end{propo}\\textbf {Proof:} In this proof we will use the convention that twoNULL values can not be compared as equal, because their meaning isthat the value is missing, and two missing values not necessarily arequal.", "In fact in r_V we do not store the null values but considerthem as unknown missing values.", "Thus, when there are null values inthe columns of the user-defined tables being joined, the null valuesdo not match each other.", "\\\\Let us show that there is such a query(relation algebra term) rewriting for each basic relational operatorpreviously described, recursively (in what follows, if ~t_R = rthen ~\\overrightarrow{t_R} = ~r_V WHERE {\\tt r-name} = r):\\begin{enumerate}\\item (Rename).", "t^{\\prime }_R = r~ RENAME ~name_1~AS~name_2 where the result is identical to input argument (relation) r except that thecolumn i with name nr_r(i) = name_1 in all tuples is renamed to nr_r(i) =name_2.", "The rewritten vector query is\\\\\\overrightarrow{t^{\\prime }_R}= ~UPDATE ~r_V~ SET~ [{\\tt a-name} = name_2] WHERE ({\\tt r-name} = r)\\wedge ({\\tt a-name} =name_1);\\\\\\item (Projection).", "t^{\\prime }_R = ~t_R~[S], whereand S = \\langle (r_{j_1},nr_{r_{j_1}}(i_1)),...,(r_{j_k},nr_{r_{j_k}}(i_k))\\rangle \\subseteq \\mathfrak {S}, with k \\ge 1 and1 \\le i_m \\le ar(r_{j_m}) for 1 \\le m \\le k, is a subset of the type \\mathfrak {S} of the term t_R.", "We define the rewritten vectorquery\\\\\\overrightarrow{t^{\\prime }_R} = \\overrightarrow{~t_R~[S]}\\\\ =~\\overrightarrow{t_R} WHERE ((nr_{\\overrightarrow{t_R}}(1) =r_{j_1}) \\wedge (nr_{\\overrightarrow{t_R}}(2) = nr_{r_{j_1}}(i_1)))\\vee ...\\vee ((nr_{\\overrightarrow{t_R}}(1) = r_{j_k})\\\\ \\wedge (nr_{\\overrightarrow{t_R}}(2) = nr_{r_{j_k}}(i_k)));\\\\\\item (Join).", "t^{\\prime }_R = t_{R,1} \\bowtie _S t_{R,2}, where S =(((r_{l_1},nr_{r_{l_1}}(i_1)),(r_{n_1},nr_{r_{n_1}}(j_1))),...,\\\\(((r_{l_m},nr_{r_{l_m}}(i_m)),(r_{n_m},nr_{r_{n_m}}(j_m)))))with 1\\le i_k \\le |\\mathfrak {S}_1| and 1\\le j_k \\le |\\mathfrak {S}_2| for 1\\le k \\le m, where \\mathfrak {S}_1 and \\mathfrak {S}_2 are the types of t_{R,1} and t_{R,2}, respectively.", ".\\\\ Let us define the following relational algebra terms:\\\\r = \\overbrace{\\overrightarrow{t_{R,1}}\\bigotimes ...\\bigotimes \\overrightarrow{t_{R,1}}}^{m}\\bigotimes \\overbrace{\\overrightarrow{t_{R,2}}\\bigotimes ...\\bigotimes \\overrightarrow{t_{R,2}}}^{m}~;(the first m are for the attributes of \\overrightarrow{t_{R,1}} in S, and the last mare for the corresponded joined attributes of \\overrightarrow{t_{R,2}} in S).Note that each column name of \\overrightarrow{t_{R,1}} will berenamed m times in order to have for each column different namein standard way as for attributes: for example, the column{\\tt a-name} in \\overrightarrow{t_{R,1}} will be repeated by{\\tt a-name}(1),..., {\\tt a-name}(m) (and similarly for eachcolumn name of \\overrightarrow{t_{R,2}}), and the attributeat_{r}(2) (of the column {\\tt t-index} in r_V) will berepeated by its copies at_{r}(2)(1),...,at_{r}(2)(2m) (in whatfollows will be also generated the (2m+1)-th copyat_{r}(2)(2m+1) in the algebra term t_2).", "Thus,\\\\t_1 = r WHERE (\\bigwedge _{1\\le k \\le m} ((nr_r(4k-3) = r_{l_k}) \\wedge (nr_r(4k-1) = nr_{r_{l_k}}(i_k)) \\wedge (nr_r(4(m+k)-3) =r_{n_k})\\wedge (nr_r(4(m+k)-1) = nr_{r_{n_k}}(j_k)) \\wedge (nr_r(4k) = nr_r(4(m+k)+4)))\\\\\\wedge ((m =1) \\vee ((nr_r(2) = nr_r(6) = ... = nr_r(4m-2)) \\\\\\wedge (nr_r(4m+2)= nr_r(4m+6) = ... = nr_r(8m-2)))));\\\\t_2 = (EXTEND t_1 ADD ((atr_r(2))(2m+1), name_3,Hash(atr_r(1),atr_r(2),...,\\\\atr_r(8m))))[nr_{t_1}(2), nr_{t_1}(4m+2),name_3]; \\\\where name_3 is a fresh new name and \\\\\\Vert t_2\\Vert _{\\#} = \\lbrace \\langle ID_1,ID_2,ID_3 \\rangle ~|~ ID_1 is the tuple-index in \\Vert t_{R,1}\\Vert _{\\#} and ID_2 is the corresponding joined tuple-index in \\Vert t_{R,2}\\Vert _{\\#}, while ID_3 is the fresh new generated (by Hash function) tuple-index forthe tuple obtained by join operation \\rbrace ,\\\\ then for the Cartesian products t_3 = \\overrightarrow{t_{R,1}} \\bigotimes t_2 and t_4 = \\overrightarrow{t_{R,2}} \\bigotimes t_2,\\\\\\overrightarrow{t^{\\prime }_R} = \\overrightarrow{t_{R,1} \\bowtie _S t_{R,2}} \\\\=((t_3 WHERE (nr_{t_3}(2) = nr_{t_3}(5)))[nr_{t_3}(1),name_3,nr_{t_3}(3),nr_{t_3}(4)])\\\\UNION ((t_4 WHERE (nr_{t_4}(2) = nr_{t_4}(6)))[nr_{t_4}(1),name_3,nr_{t_4}(3),nr_{t_4}(4)]);\\\\\\item (Selection).", "t^{\\prime }_R = ~t_R~ WHERE ~C:\\\\4.1 When a condition C is a finite-length logical formula that consists of atoms^{\\prime }((r_{i_1},name_i) ~\\theta ~(r_{j_1},name_j))^{\\prime }~ or ~^{\\prime }(r_{i_1},name_i) ~\\theta ~\\overline{d}~^{\\prime } or ~^{\\prime }(r_{i_1},name_i) ~ NOT NULL^{\\prime } with built-in predicates \\theta \\in \\Sigma _\\theta \\supseteq \\lbrace \\doteq ,>,< \\rbrace , a constant \\overline{d}^{\\prime },and the logical operators, between the columns in the type \\mathfrak {S} of the termt_R.", "\\\\The condition C, composed by k \\ge 1 different columnsin \\mathfrak {S}, we denote byC((r_{i_1},name_{i_1}),...,(r_{i_k},name_{i_k})), k \\ge 1, and hence we define the rewritten vectorquery\\\\\\overrightarrow{t^{\\prime }_R} = \\overrightarrow{~t_R~ WHERE ~C} = ~\\overrightarrow{t_R} WHERE nr_{\\overrightarrow{t_R}}(2) IN t_1\\\\where for ~r = \\overbrace{\\overrightarrow{t_R}\\bigotimes ... \\bigotimes \\overrightarrow{t_R}}^k ~ we define the unary relation whichcontains the tuple-indexes of the relation \\Vert t_R\\Vert for itstuples which satisfy the selection condition C,by the following selection term \\\\t_1 = (r WHERE ((nr_r(1) = r_{i_1} \\wedge nr_r(3) =name_{i_1}) \\wedge ...\\wedge (nr_r(1+4(k-1)) = r_{i_k} \\wedge nr_r(3+4(k-1)) =name_{i_1})) \\wedge C(nr_r(4),...,nr_r(4k)) \\\\\\wedge ((k = 1)\\vee (nr_r(2) = nr_r(6) =... = nr_r(2 +4(k-1)))))[nr_r(2)];\\\\4.2 Case when C = ~(r_{i_1},name_i) ~ NULL,\\\\\\overrightarrow{t^{\\prime }_R} = \\overrightarrow{~t_R~WHERE~(r_{i_1},name_i) ~ NULL } = \\overrightarrow{t_R} WHEREnr_{\\overrightarrow{t_R}}(2) NOT IN t_2,\\\\where t_2 = (\\overrightarrow{~t_R~} WHERE((nr_{\\overrightarrow{t_R}}(1) = r_{i_1}) \\wedge (nr_{\\overrightarrow{t_R}}(3) =name_i)))[(nr_{\\overrightarrow{t_R}}(2)].\\\\From the fact that ~t_R~ WHERE ~C_1 \\wedge C_2 = ~(t_R~ WHERE~C_1) WHERE C_2 and ~t_R~ WHERE ~C_1 \\vee C_2 = ~(t_R~ WHERE~C_1) UNION ~(t_R~ WHERE ~C_2), and De Morgan laws, \\lnot (C_1\\wedge C_2) = \\lnot C_1 \\vee \\lnot C_2, \\lnot (C_1 \\vee C_2) = \\lnot C_1 \\wedge \\lnot C_2, we can always divide any selection in thecomponents of the two disjoint cases above;\\\\\\item (Union).", "t^{\\prime }_R = t_{R,1}~ UNION_R ~t_{R,2},where R is a table \\lbrace \\langle r_{l_k},name_{l_k},r_{n_k},name_{n_k}\\rangle ~|~\\\\1\\le k \\le m\\rbrace such that\\mathfrak {S}_1 = \\langle (r_{l_1},name_{l_1}),...,(r_{l_m},name_{l_m})\\rangle and \\mathfrak {S}_2 = \\langle (r_{n_1},name_{n_1}),..., (r_{n_m},name_{n_m})\\rangle are thetypes of t_{R,1} and t_{R,1}, respectively, with theunion-compatible columns (\\langle r_{l_k},name_{l_k}) and(\\langle r_{n_k},name_{n_k}) for every 1\\le k \\le m.\\\\ Wedefine the relational algebra term t_1 =\\overrightarrow{t_{R,2}} \\bigotimes R, so that\\\\\\overrightarrow{t^{\\prime }_R} = \\overrightarrow{t_{R,1}~ UNION_R ~t_{R,2}}\\\\= \\overrightarrow{t_{R,1}} UNION ((t_1 WHERE ((nr_{t_1}(1) =nr_{t_1}(7)) \\wedge (nr_{t_1}(3) =nr_{t_1}(8))))[nr_{t_1}(5),\\\\nr_{t_1}(2),nr_{t_1}(6),nr_{t_1}(4)]),\\\\so that the relation-column names of the union will be equal to thecolumn names of the first term in this union;\\\\\\item (Set difference).", "t^{\\prime }_R = t_{R,1}~ MINUS_R ~t_{R,2},where where R is a table \\lbrace \\langle r_{l_k},nr_{r_{l_k}}(i_k),\\\\r_{n_k},nr_{r_{n_k}}(j_k)\\rangle ~|~1\\le k \\le m\\rbrace such that\\mathfrak {S}_1 = \\langle (r_{l_1},nr_{r_{l_1}}(i_1)),...,(r_{l_m},nr_{r_{l_m}}(i_m))\\rangle and \\mathfrak {S}_2 = \\langle (r_{n_1},nr_{r_{n_1}}(j_1)),..., (r_{n_m},nr_{r_{n_m}}(j_m))\\rangle are the types of t_{R,1} and t_{R,1}, respectively, with theunion-compatible columns (\\langle r_{l_k},nr_{r_{l_k}}(i_k)) and(\\langle r_{n_k},nr_{r_{n_k}}(j_k)) for every 1\\le k \\le m.Let us define the following relational algebra terms:\\\\r = \\overbrace{\\overrightarrow{t_{R,1}}\\bigotimes ...\\bigotimes \\overrightarrow{t_{R,1}}}^{m}\\bigotimes \\overbrace{\\overrightarrow{t_{R,2}}\\bigotimes ...\\bigotimes \\overrightarrow{t_{R,2}}}^{m}~;(the first m are for the attributes of \\overrightarrow{t_{R,1}} in \\mathfrak {S}_1, and the last mare for the corresponded joined attributes of \\overrightarrow{t_{R,2}} in \\mathfrak {S}_2).Thus, \\\\ t_1 = (r WHERE (\\bigwedge _{1\\le k \\le m} ((nr_r(4k-3) = r_{l_k}) \\wedge (nr_r(4k-1) = nr_{r_{l_k}}(i_k)) \\wedge (nr_r(4(m+k)-3) =r_{n_k})\\wedge (nr_r(4(m+k)-1) = nr_{r_{n_k}}(j_k)) \\wedge (nr_r(4k) = nr_r(4(m+k)+4)))\\\\\\wedge ((m =1) \\vee ((nr_r(2) = nr_r(6) = ... = nr_r(4m-2)) \\\\\\wedge (nr_r(4m+2) = nr_r(4m+6) = ... = nr_r(8m-2))))))[nr_{t_1}(2)];\\\\where \\Vert t_1\\Vert _{\\#} = \\lbrace \\langle ID_1\\rangle ~|~ID_1 is the tuple-index of a tuple in \\Vert t_{R,1}\\Vert _{\\#} for which there exists an equal tuple in\\Vert t_{R,2}\\Vert _{\\#}\\rbrace .", "Then,\\\\\\overrightarrow{t^{\\prime }_R} = \\overrightarrow{t_{R,1}~ MINUS_R~t_{R,2}} = \\overrightarrow{t_{R,1}} WHEREnr_{\\overrightarrow{t_{R,1}}}(2) NOT IN t_1.\\end{enumerate}It is easy to show that for the cases from 2 to 6, we obtain that\\Vert t^{\\prime }_R\\Vert _{\\#} =\\emph {\\underline{VIEW}}(\\mathfrak {S},\\Vert \\overrightarrow{t^{\\prime }_R}\\Vert _{\\#}),where \\mathfrak {S} is the type of the relational algebra termt^{\\prime }_R.", "Thus, for any SPJU term t_R obtained by the composition ofthese basic relational algebra operators we have that \\Vert t_R\\Vert _{\\#}=\\emph {\\underline{VIEW}}(\\mathfrak {S},\\Vert \\overrightarrow{t_R}\\Vert _{\\#}).\\\\The update operators are rewritten as follows:\\begin{enumerate}\\item (Insert).", "INSERT INTO ~r[S]~ VALUES ~(d_1,...,d_m), where S= \\langle nr_r(i_1),...,\\\\nr_r(i_m) \\rangle , 1\\le m \\le ar(r),is the subset of mutually different attribute names of r and allv_i, 1\\le i \\le m are the values different from NULL.", "It isrewritten into the following set of terms:\\\\ \\lbrace INSERT INTO~r_V[{\\tt r-name}, {\\tt t-index}, {\\tt a-name},{\\tt value}]~VALUES ~(r,Hash(d_1,\\\\...,d_m),nr_r(i_k),d_k)~|~1\\le k \\le m \\rbrace .\\\\Note that before the execution of this set of insertion in r_V,the DBMS has to control if it satisfy all user-defined integrityconstraints in the user-defined database schema \\mathcal {A};\\\\\\item (Delete).", "DELETE FROM ~r~ WHERE ~C, is rewritten into theterm:\\\\DELETE FROM ~r_V~ WHERE {\\tt t-index} IN~\\overrightarrow{t_R}[nr_{\\overrightarrow{t_R}}(2)],\\\\where \\overrightarrow{t_R} = \\overrightarrow{r~ WHERE ~C} is the selection term as described in point 4 above;\\\\\\item (Update).", "the existence of the rewriting of this operationis obvious, from the fact that it can always be decomposed asdeletion and after that the insertion of the tuples.\\end{enumerate}\\square \\\\This proposition demonstrates that the IRDB is full SQL database, sothat each user-defined query over the used-defined RDB databaseschema \\mathcal {A} can be equivalently transformed by query-rewriting intoa query over the vector relation r_V.", "However, in the IRDBMSs wecan use more powerful and efficient algorithms in order to executeeach original user-defined query over the vector relation r_V.\\\\Notice that this proposition demonstrates that the IRDB is a kind ofGAV Data Integration System I= (\\mathcal {A},\\mathcal {S},\\mathcal {M}) in Definition\\ref {def:parsing} where we do not materialize the user-definedschema \\mathcal {A} but only the vector relation r_V \\in \\mathcal {S} and eachoriginal query q(\\textbf {x}) over the empty schema \\mathcal {A} will berewritten into a vector query \\overrightarrow{q(\\textbf {x})} ofthe type \\mathfrak {S} over the vector relation r_V, and then theresulting view\\underline{VIEW}(\\mathfrak {S},\\Vert \\overrightarrow{q(\\textbf {x})}\\Vert _{\\#})will be returned to user^{\\prime }s application.\\\\Thus, an IRDB is a member of the NewSQL, that is, a member of aclass of modern relational database management systems that seek toprovide the same scalable performance of NoSQL systems for onlinetransaction processing (read-write) workloads while stillmaintaining the ACID guarantees of a traditional database system.\\section {Conclusion}The method of parsing of a relational instance-database A with theuser-defined schema \\mathcal {A} into a vector relation\\overrightarrow{A}, used in order to represent the information ina standard and simple key/value form, today in various applicationsof Big Data, introduces the intensional concepts for theuser-defined relations of the schema \\mathcal {A}.", "In Tarskian semantics ofthe FOL used to define the semantics of the standard RDBs, onedefines what it takes for a sentence in a language to be truerelative to a model.", "This puts one in a position to define what ittakes for a sentence in a language to be valid.", "Tarskian semanticsoften proves quite useful in logic.", "Despite this, Tarskian semanticsneglects meaning, as if truth in language were autonomous.", "Becauseof that the Tarskian theory of truth becomes inessential to thesemantics for more expressive logics, or more ^{\\prime }natural^{\\prime } languages.\\\\Both, Montague^{\\prime }s and Bealer^{\\prime }s approaches were useful for thisinvestigation of the intensional FOL with intensional abstractionoperator, but the first is not adequate and explains why we adoptedtwo-step intensional semantics (intensional interpretation with theset of extensionalization functions).", "Based on this intensionalextension of the FOL, we defined a new family of IRDBs.", "We haveshown that also with this extended intensional semantics we maycontinue to use the same SQL used for the RDBs.\\\\This new family of IRDBs extends the traditional RDBS with newfeatures.", "However, it is compatible in the way how to present thedata by user-defined database schemas (as in RDBs) and with SQL formanagement of such a relational data.", "The structure of RDB isparsed into a vector key/value relation so that we obtain a columnrepresentation of data used in Big Data applications, covering thekey/value and column-based Big Data applications as well, into aunifying RDB framework.\\\\Note that the method of parsing is well suited for the migrationfrom all existent RDB applications where the data is stored in therelational tables, so that this solution gives the possibility topass easily from the actual RDBs into the new machine engines forthe IRDB.", "We preserve all metadata (RDB schema definitions) withoutmodification and only dematerialize its relational tables bytransferring their stored data into the vector relation r_V(possibly in a number of disjoint partitions over a number ofnodes).", "From the fact that we are using the query rewriting IDBMS,the current user^{\\prime }s (legacy) applications does not need anymodification and they continue to \"see\" the same user-defined RDBschema as before.", "Consequently, this IRDB solution is adequate for amassive migration from the already obsolete and slow RDBMSs into anew family of fast, NewSQL schema-flexible (with also ^{\\prime }Openschemas^{\\prime }) and Big Data scalable IRDBMSs.", "}$" ] ]
1403.0017
[ [ "Kinetics of phase separation in thermally isolated critical binary\n fluids" ], [ "Abstract Spinodal decomposition in a near-critical binary fluid is examined for experimental scenarios in which the liquid is quenched abruptly by changing the pressure and the subsequent phase separation occurs with no heat flow from the outside, i.e., adiabatically.", "Equations of motion for the system volume and effective temperature are derived.", "It is shown that for this case that the nonequilibrium decomposition process is well approximated as one of constant entropy, i.e., as thermodynamically reversible.", "Quantitative comparison, with no adjustable parameters, is made with experimental light scattering data of Bailey and Cannell [$\\rm {Phys.\\ Rev.\\ Lett.\\ }{\\bf 70}$, 2110 (1993)].", "It is found that including these adiabatic effects accounts for most of the discrepancies between these experiments and previous isothermal theory.", "The equilibrium static critical properties of the isothermal theory are also examined, this discussion serving to justify some approximations in the current theory." ], [ "Introduction", "Spinodal decomposition is the process of phase separation of a thermodynamically unstable mixture.", "[1], [2], [3], [4] It and its complement, nucleation, are two of the most common mechanisms of phase transformation for systems governed by a conserved order parameter.", "They also are exploited in many commercial processes for creating alloy materials.", "In decomposition, experiments typically measure the intensity $I(k,t)$ of X-rays, neutrons or light scattered by the mixture at various momentum transfer wavevectors $k=\\vert {\\bf k}\\vert $ and times $t$ .", "Information about the state of the mixture, e.g., metal alloy, polymer blend, composite glass or binary fluid, is obtained by relating $I(k,t)$ to the structure factor ${\\hat{S}}(k,t)$ , which is the Fourier transform of the density-density correlation function.", "To initiate decomposition, a uniform mixture in equilibrium is quenched into the middle-part of the two-phase coexistence region, usually by cooling the sample.", "After the quench a ring of scattered radiation surrounding the incident beam of the radiation probe appears.", "The peak intensity of the ring corresponds initially to a wavevector $k_m\\approx 1/\\xi $ , where $\\xi $ is the correlation length of the equilibrium coexisting phases (assumed to be the same for both) at the quenched temperature.", "Over time this ring shrinks, implying that the precipitate is coarsening.", "At late times $k_m$ has been shown to vary as a power of the time $t$ , $k_m \\sim t^{-a_q}$ , and ${\\hat{S}}(k,t)$ to reduce to a scaling form.", "This late stage scaling is thought to be due to the formation of growing domains of average size $L\\sim 1/k_m$ .", "These domains have well-defined interfaces of thickness $\\xi $ and the composition inside the domains is near the coexistence values of the mixture.", "Decomposition was studied initially for systems in which phase separation is driven by single particle diffusion, such as metal alloys.", "The first theories of the early stage of decomposition in such substances were due to Cahn[1] and Cook[5], and later Langer, Bar-on and Miller (LBM)[6].", "Experimental tests of these theories have not been entirely unambiguous though.", "For metal alloys, lattice mismatch of the two components can cause stresses to build during unmixing, which slows the rate of decomposition.", "These strains can be minimized by matching the lattice constants of the individual components [7], or avoided entirely by examining unmixing in liquids [8], [9].", "In liquids though, unmixing is greatly accelerated by advection.", "Kawasaki and Ohta (KO) extended the LBM theory to binary liquids by incorporating these hydrodynamic effects.", "[10] A careful set of experiments to test this KO theory was done by Bailey and Cannell (BC) using 3-methylpentane and nitroethane (3MP+NE) in the critical region.", "[11] The critical equilibrium properties of 3MP+NE have been well characterized.", "Further, the components of this binary liquid have very similar indices of refraction, which minimizes multiple scattering effects during decomposition.", "Since all the parameters of the KO-LBM theory can be obtained from equilibrium measurements, a clear comparison with the theory would seem to be possible.", "However, as BC have discussed, their experiments violated an almost universal theoretical assumption for this class of non-equilibrium phenomena, namely, that the temperature is a control parameter.", "Rather, the quenches occurred by rapidly decreasing the pressure and then holding it constant during the decomposition.", "On the timescale of their experiments, no heat from the container walls was able to reach the portion of the liquid being probed; thus the decomposition occurred adiabatically rather than isothermally.", "The problem with controlling only the pressure is that unmixing releases heat (being exothermic for most simple liquids), and so the temperature of the sample will be increasing over time.", "Theories of dynamic critical phenomena and the KO theory itself predict that the characteristic relaxation time of a binary liquid scales as $\\xi ^{3+z_\\eta } \\sim {\\vert \\epsilon \\vert }^{-1.94}$ , where $\\epsilon = {T/T_c -1}$ is the reduced temperature, with $T$ and $T_c$ being the absolute and critical temperature, respectively.The exponent $z_\\eta $ arises from the temperature dependence of the shear viscosity.", "See Table REF .", "The experiments of BC were done at absolute reduced temperatures around $10^{-5}$ , and so small changes in $T$ could cause large changes in the relaxation time making comparison with theory potentially troublesome.", "More fundamentally though, what does one mean by “temperature” when a system is driven so far from equilibrium?", "This paper has two purposes.", "First, it gives an answer to this and some related questions, and generalizes the KO-LBM theory to adiabatic decomposition.", "Second, it compares quantitatively this generalized theory to the BC experiments.", "The problem of decomposition under adiabatic conditions was first addressed in [13].", "The concepts and approach described in that work are further explored in the present one.", "The paper is organized as follows.", "In Sec.", "the basic theory of adiabatic decomposition is presented.", "The product of the theory is an equation of motion for the effective system temperature.", "Then, the isothermal coarse-grained free energy used in KO and LBM is generalized to describe systems for which the temperature is changing.", "In Sec.", "the hydrodynamic KO theory for the equation of motion of the structure factor is described, one reason being to show how the temperature dependence of the theory arises.", "The equilibrium form of the LBM theory is also discussed, it being used to compute initial conditions for the kinetic theory.", "In Sec.", "the temperature and structure factor kinetic equations, along with the equilibrium equations for the structure factor, are scaled.", "Then, the scheme used here to solve them numerically is discussed.", "In Sec.", "the parameters in the temperature dependent coarse-grained free energy are determined using the equilibrium LBM theory.", "The predictions of the equilibrium LBM theory in the critical region are then discussed, primarily to show in what regions of the phase diagram the theory can be used to give accurate initial conditions, and properly describe the un-mixing process in a thermodynamic sense.", "In Sec.", "general predictions of the adiabatic and isothermal kinetic theories are presented, including the behavior of the KO theory at late times, far beyond its supposed regime of validity.", "The adiabatic and isothermal theories are then compared quantitatively with the BC experiments.", "Finally, in Sec.", "the paper is summarized and directions for further work are discussed." ], [ "Adiabatic Decomposition", "In this section a theory of adiabatic decomposition in a binary substance is presented.", "The theory generalizes any isothermal, statistical theory of decomposition, such as KO or LBM.", "In what follows, the equilibrium properties of critical binary fluids will be referred to justify some theoretical approximations.", "Table REF below contains equilibrium data of 3MP+NE relevant to the theory here.", "Table REF below contains relevant critical exponent values and amplitude relations.", "In this work, the critical point for a given pressure $P$ is denoted by the concentration $c_c$ and temperature $T_c$ .", "As mentioned above, the reduced temperature $\\epsilon = T/T_c-1$ .", "For 3MP+NE at the pressures of interest, $T_c$ varies linearly with pressure, so $dT_c\\over dP$ is a constant.", "[14] In the critical region, the miscibility gap has the scaling form $\\Delta c=2B{\\vert \\epsilon \\vert }^{\\beta }$ ; the correlation length $\\xi =\\xi ^{\\pm }_0{\\vert \\epsilon \\vert }^{-\\nu }$ ; and the susceptibility $\\chi =\\Gamma ^{\\pm }{\\vert \\epsilon \\vert }^{-\\gamma }$ , with $\\beta , \\gamma $ and $\\nu $ being critical exponents and $B, \\xi _0^\\pm $ and $\\Gamma ^\\pm $ being critical amplitudes.", "Here, “+” refers to a one-phase value obtained on the critical isobar above $T_c$ , while “-” refers to a two-phase coexistence value below $T_c$ .", "The quantities $c_c,B,\\xi _0,\\Gamma $ and any other critical amplitude mentioned in this text have either been shown experimentally to be, or are assumed to be constant over the pressures of experimental interest.", "[14] Table: Equilibrium one-phase (T>T c T>T_c) data of an on-critical mixture of3MP+NE relevant to the present work.", "All units are MKS.Table: Theoretical critical exponent and amplitude relations for binary fluidsrelevant to the present work." ], [ " Basic Theory", "The essence of the theory here is to exploit how one constructs the coarse-grained free energy ${\\cal F}$ used in previous theories of decomposition.In an alternative approach, Onuki has examined the sensitivity of nucleation near the gas-liquid critical point to various types of thermodynamic constraints, including adiabatic ones, during and after a quench.", "[49] It is defined as follows.", "[24] Consider a binary mixture of $A$ and $B$ -type molecules in strong contact with an external reservoir at temperature $T$ .", "Let $c({\\bf r})$ be the concentration of $A$ -type molecules in a cell of size $a^3$ centered at position ${\\bf r}$ .", "The cell size is mesoscopic on the order of the equilibrium correlation length $\\xi $ , which for mixtures in the critical region can be hundreds or even thousands of angstroms.", "The coarse-grained free energy ${\\cal F}\\equiv {\\cal F}[c]$ , is a functional of the concentration field $c({\\bf r})$ .", "In mean-field theories a change in ${\\cal F}[c]$ due to a change in the concentration at some point ${\\bf r}$ acts as a local thermodynamic driving force or chemical potential $\\mu ({\\bf r})$ .", "Gradients in $\\mu ({\\bf r})$ in turn cause mass diffusion.", "The coarse-grained free energy is constructed by fixing the value of $c({\\bf r})$ in each cell and performing the partition sum over all states of the system consistent with the configuration $[c]$ .", "Let the microscopic Hamiltonian be $H$ , then $exp\\bigl (-{{\\cal F}[c]+ F_r\\over k_BT}\\bigr )=\\sum _{{\\rm states}\\atop {{\\rm consistent}\\atop {\\rm with}{ } [c]}}exp(-{H\\over k_BT}),$ where $k_B$ is Boltzmann's constant.", "The coarse-grained free energy ${\\cal F}$ then describes the properties of all concentration modes of wavenumber $k$ less than some cut-off $\\Lambda \\sim 1/a$ .", "The quantity $F_r$ is the part of the total equilibrium free energy that is independent of the configuration $[c]$ .", "For example, it is assumed that $F_r$ contains all vibrational degrees of freedom, which give the dominant contribution to a liquid's entropy.", "In addition, the short wavelength concentration modes ($k>\\Lambda $ ) contribute to $F_r$ , but they also contribute to ${\\cal F}$ by renormalizing its coefficients.", "Because the long wavelength concentration modes don't contribute to $F_r$ , $F_r$ is an analytic function of $T-T_c.$[25] Since the interest here is in the kinetics of phase separation, in integrating out these degrees of freedom it is assumed that they relax very quickly compared to the modes described by $[c]$ .", "That is, the modes in $F_r$ are able to equilibrate between any characteristic change in $[c]$ .", "Note that the separation of the partial free energy into two terms ${\\cal F}$ and $F_r$ implies weak coupling between those degrees of freedom that contribute to $F_r$ and the configuration $[c]$ .", "Such a separation matters little to the isothermal theory as $F_r$ is ignored.", "However, what if the system were not in strong contact with an external bath?", "Would it be possible to let the degrees of freedom that have been integrated out act as a thermal reservoir for the long wavelength modes in this case?", "To implement this idea and clarify the concepts, it is helpful to look first at a system that is closed, i.e.", "has a fixed total energy $E_t$ and fixed volume $V$ .", "First, does the heat released during decomposition cause non-uniformities in the temperature $T$ or pressure $P$ ?", "The speed of sound is clearly faster than any diffusion process being considered, so $P$ can be easily approximated as uniform.", "For decomposition it is expected that if any non-uniformities in temperature occur, they occur over a wavelength $\\lambda \\sim 1/q_m \\ge \\xi $ , where $q_m$ is the wavevector of the peak scattering intensity and is a measure of the size of the phase separating regions.", "Thus, the characteristic time $\\tau _T$ for heat to diffuse across a distance $\\lambda $ must be compared with the characteristic relaxation time $\\tau _c$ of a concentration mode of size $\\lambda $ .", "It can be shown that, for a typical binary fluid, ${\\tau _T\\over \\tau _c} = {D_c\\over D_T}\\sim 10^{-4}\\ll 1$ , and so $T$ can safely assumed to also be uniform.", "Now, for the closed system, the relevant “free energy” is the entropy $S$ .", "In analogy with the definition for ${\\cal F}$ and following Boltzmann, define $S([c],E_t)= k_B\\ {\\rm ln}\\Biggl [\\sum _{{\\rm states}\\atop {{\\rm consistent}\\atop {\\rm with}{ } [c]}}\\delta (H-E_t)\\Biggr ],$ as the entropy of a system with a fixed configuration $[c]$ and total energy $E_t$ .", "Here, $\\delta (x)$ is the Dirac “delta” function at point $x$ .", "However, for the approximations to follow the interest is still with ${\\cal F}$ .", "To construct ${\\cal F}$ , define the partition function $Z=&&\\sum _{{\\rm states}\\atop {{\\rm consistent}\\atop {\\rm with}\\ [c]}} exp(-\\beta H)\\nonumber \\\\=&&\\int \\ dE^\\prime \\ exp(-\\beta E^\\prime )\\Gamma (E^\\prime ,[c]),$ where $\\beta $ is a parameter to be determined, and $\\Gamma ([c],E^\\prime )=exp(S([c],E^\\prime )/k_B)$ is the number of accessible states of the system with an energy $E^\\prime $ and coarse-grained configuration $[c]$ .", "Expanding $S$ about $E_t$ it is found that, for the case in which $\\Gamma $ is a macroscopic number, $S$ is related to $Z$ by $S([c],E_t)-k_B\\beta E_t= k_B{\\rm ln}\\\\ Z,$ with $k_B\\beta =\\Bigl ({\\partial S\\over \\partial E_t}\\Bigr )_{V,[c]}.$ Now, as in the isothermal case, assume weak coupling so that $S$ can be written as $S([c],E_t) ={\\cal S}[c]+S_r,$ where $S_r$ is the part of the entropy independent of the configuration $[c]$ .", "Let the energy associated with $S_r$ and ${\\cal S}[c]$ be $E_r$ and ${\\cal E}[c]$ , respectively.", "Then, $E_t={\\cal E}[c]+ E_r.$ If the concentration field $[c]$ is held fixed, then so will be ${\\cal S}[c]$ and ${\\cal E}[c]$ ; thus, $\\Bigl ({\\partial S\\over \\partial E_t}\\Bigr )_{V,[c]}={\\Bigl ({\\partial S_r \\over \\partial E_r}\\Bigr )}_V\\equiv {1\\over T_r}.$ Combining Eqs.", "(REF )-(REF ) then gives $exp\\bigl (-{{\\cal F}[c]+ F_r\\over k_B T_r}\\bigr )=\\sum _{{\\rm states}\\atop {{\\rm consistent}\\atop {\\rm with}{ } [c]}}exp(-{H\\over k_B T_r}),$ where the coarse-grained free energy ${\\cal F}={\\cal E}-T_r{\\cal S},$ and $F_r=E_r-T_r S_r.$ Comparing Eqs.", "(REF ) and (REF ) it can be seen for this closed system that the degrees of freedom contributing to $F_r$ act explicitly as a reservoir for the concentration modes described by ${\\cal F}$ .", "Note though that because the total energy $E_t$ is constrained, $T_r$ through Eq.", "(REF ) is not a constant, but an implicit function of the concentration field $[c]$ .", "For the remainder of this work, the degrees of freedom that contribute to $F_r$ will be called the reservoir, and those that are described by the concentration field $[c]$ and contribute to ${\\cal F}[c]$ will be called the slow modes.", "Now, is $T_r$ related to any measurable temperature $T$ ?", "An average reservoir temperature, “$\\langle T_r\\rangle $ ”, can be obtained by computing the average coarse-grained energy $\\langle {\\cal E}[c]\\rangle $ using $\\rho ([c],E_t)$ , which is the probability density that the system is in a configuration $[c]$ with total system energy $E_t$ .", "That is, with $\\langle {\\cal E}[c]\\rangle $ , the average reservoir energy is $E_t - \\langle {\\cal E}[c]\\rangle $ , so an average $T_r$ is known.", "It can be shown that in equilibrium, $\\langle T_r\\rangle $ is equal to the average system temperature $T=\\Bigl ({\\partial E_t\\over \\partial S_t}\\Bigr )_V$ , where $S_t$ is the total equilibrium entropy.", "[26] When the system is out of equilibrium, the relation of $\\langle T_r\\rangle $ to the system temperature is unclear - assuming the latter can even be defined in a consistent manner.", "However, as will be seen below, such a relation is not necessary to determine the time evolution of the concentration field.", "For a closed system, the equilibration process is as follows.", "The system is prepared in some non-equilibrium state with reservoir energy $E_r$ and coarse-grained configuration $[c]$ (and thus energy ${\\cal E}[c]$ ).", "The system is then released and the configuration $[c]$ evolves.", "The evolution is driven by ${\\cal F}[c]$ , which is a function of the reservoir temperature $T_r$ .", "As the $[c]$ changes, the coarse-grained energy ${\\cal E}[c]$ changes, and, because the total energy is constant, the reservoir energy $E_r$ changes.", "A change in $E_r$ implies a change in $T_r$ , and this change in $T_r$ in turn affects the evolution of the $[c]$ and so on.", "The system eventually settles down into a state that maximizes the total entropy.", "With the ideas above, an equation of motion for $\\rho ([c],E_t)$ for the case in which mass transfer is dominated by single particle diffusion (the solid model[27]) can be derived, but the coupling between the reservoir and slow modes makes this equation not too useful.", "A simpler approach is as follows.", "The primary concern here is with average temperature changes associated with the decomposition process.", "The idea then is to let the average reservoir temperature $\\langle T_r\\rangle =T$ play the role of a pseudo control parameter, the thermodynamic driving force be ${\\cal F}={\\cal E}-T{\\cal S}$ , and derive a separate equation of motion for $T$ .", "In this way the same equation for $\\rho ([c],t)$ that has been used to describe isothermal decomposition will be used, but any temperature dependent parameter will now vary in time.", "The time evolution of $T$ can be obtained using energy conservation.", "For the closed system this is easily done.", "Eq.", "(REF ) obviously also holds for averages, so for a differential change $d\\langle E_t\\rangle =dE_r +d\\langle {\\cal E}[c]\\rangle =0.$ The reservoir energy is an equilibrium thermodynamic function so $dE_r=C_{Vr}dT,$ where $C_{Vr}= {\\Bigl ({\\partial E_r \\over \\partial T}\\Bigr )}_V$ is the reservoir heat capacity at constant volume.", "Also, $d\\langle {\\cal E}[c]\\rangle ={\\partial \\langle {\\cal E}[c]\\rangle \\over \\partial t}dt,$ where the partial time derivative of the average is defined by ${\\partial \\langle {\\cal E}[c]\\rangle \\over \\partial t}\\equiv \\int d[c]{\\cal E}[c]{\\partial \\rho ([c],t)\\over \\partial t},$ with $\\int d[c]$ denoting an integral over the space of possible concentration fields.", "Note that the term $\\bigl \\langle {\\partial {\\cal E}[c]\\over \\partial T}\\bigr \\rangle dT$ does not appear in Eq.", "(REF ) since in the above construction of ${\\cal F}[c]$ , ${\\cal E}[c]$ is independent of temperature.", "Combining Eqs.", "(REF -REF ) gives: ${dT\\over dt}=-{1\\over C_{Vr}}{\\partial \\langle {\\cal E}[c]\\rangle \\over \\partial t}.$ Since the free energies are additive, $C_{Vr}=C_V-C_{Vcg},$ where $C_V$ is the equilibrium heat capacity of the total system, and $C_{Vcg}$ is the contribution to $C_V$ from the slow modes.", "$C_V$ can be obtained from experiment and $C_{Vcg}$ can be calculated once ${\\cal F}[c]$ is defined: $C_{Vcg}= -T\\biggl ({\\partial ^2 F_{cg}\\over \\partial T^2}\\biggr )_V,$ where $F_{cg} = -k_BT {\\rm ln} Z$ is the portion of the total system Helmholtz free energy from the slow modes, with the partition function $Z = \\int d[c]exp(-{{\\cal F}[c]\\over k_BT})$ ." ], [ " Adiabatic System", "These same ideas will now be applied to adiabatic decomposition.", "In this case there is of course no heat flow between the system and the outside world, and the pressure $P$ instead of $T$ will be a control parameter.", "Under these conditions a system undergoing phase separation will reach an equilibrium state that minimizes its enthalpy $H_e=E+PV$ .", "However, as shown above, the Helmholtz free energy $F=E-TS$ seems to be the natural one to describe the decomposition process theoretically.", "What will be done here then is determine how the temperature and volume change in time during a pressure quench and subsequent decomposition.", "These time dependent values of $T$ and $V$ will then be inserted into the coefficients of the coarse-grained free energy ${\\cal F}$ to determine the evolution of the slow modes.", "One difficulty with this scheme though is that the above construction of ${\\cal F}$ does not handle well changes in volume.", "With the coarse-grained cell size $a$ fixed, the number of cells changes as the volume is changed.", "However, for a typical pressure change $\\Delta P$ for the quenches in the binary liquids of interest, the fractional volume change ${\\Delta V\\over V}\\le 10^{-5}\\ll 1$ .", "So, given the level of approximation of this theory, this ambiguity in the definition of the concentration field will be ignored.", "Since both the temperature and volume will change if the pressure changes, two independent relations are needed to determined their time evolution.", "The first relation is as follows.", "To move the near-critical binary liquid from the one-phase region toward the unstable portion of the two-phase region, the external pressure is dropped by a differential amount $dP$ , increasing the system (average) volume by $dV$ .", "No heat is allowed to flow between the system and the external world, so the average work done by the system on the external world (via a piston, say) equals the average change in the total system energy.", "Thus, $d\\langle E_t\\rangle = -PdV,$ where $d\\langle E_t\\rangle $ is given by Eq.", "(REF ) (though it is not zero in this case obviously).", "What, though, is the pressure $P$ ?", "In equilibrium, $P = P_r + P_{cg},$ where $P_r = -\\bigl ({\\partial F_r\\over \\partial V}\\bigr )_T$ and $P_{cg} = -\\bigl ({\\partial F_{cg}\\over \\partial V}\\bigr )_T$ are the partial pressures of the reservoir and slow modes, respectively, with $F_{cg}$ being defined below Eq.", "(REF ).", "However, spinodal decomposition is a non-equilibrium process.", "It necessarily does not allow the slow modes to relax completely during the quench.", "In the extreme case that the quench is so fast that the slow modes are frozen, the contribution of these modes to the pressure would be zero.", "Thus, the actual pressure of the liquid on the container walls should be less than that given by Eq.", "(REF ).", "On the other hand, it is assumed that the movement of the piston that causes the drop in external pressure is slow enough so that the degrees of freedom in the reservoir are able to remain in equilibrium.", "For example, any momentary density drop near the piston wall is rapidly distributed throughout the liquid so no turbulence or other inhomogeneous flow results.", "[28] Thus, $P\\ge P_r$ .", "Estimates of $P_{cg}$ , and the change of it, $\\Delta P_{cg}$ , during the quench would be helpful here.", "As discussed above, upper bounds will be their equilibrium values.", "In equilibrium, $P_{cg}$ is a finite function of $\\epsilon $ and so will change little for a near-critical quench.", "So, an estimate of it at any point during the quench should be sufficient.", "Now, the number of of slow modes is $N= {V\\over a^3}\\sim {V\\over \\xi _f^3}$ , where $\\xi _f$ is the equilibrium correlation length at the final temperature $T_f$ .", "Each slow mode will have an energy of order $k_B T_f\\sim k_BT_c$ , and so the equilibrium free energy of the slow modes is $F_{cg}\\sim -{V\\over \\xi _f^3}k_B T_c$ .", "In equilibrium then, $P_{cg}\\sim {k_B T_c\\over \\xi _f^3}.$ The short wavelength concentration modes also contribute to ${\\cal F}$ , so Eq.", "(REF ) may be an underestimate, but it should be accurate within an order of magnitude.", "With ${\\vert \\epsilon _f\\vert }\\sim 10^{-5}$ for the quenches of 3MP+NE of BC [11], and using data from Table REF , it is found that $P_{cg} \\sim 1$ Pa in equilibrium.", "The experiments of BC were done near standard pressure at sea level, which is around $10^5$ Pa.", "Thus, ${P_{cg}\\over P}\\sim 10^{-5} \\ll 1.$ Also, the absolute change in pressure during a typical quench for the BC experiments (e.g., $\\epsilon _i = 10^{-5}$ to $\\epsilon _f = -\\epsilon _i$ ) was $\\vert \\Delta P\\vert \\simeq 10^4$ Pa. An upper bound on $\\vert \\Delta P_{cg}\\vert $ is $P_{cg}$ , so ${\\vert \\Delta P_{cg}\\vert \\over \\vert \\Delta P\\vert } \\le 10^{-4} \\ll 1.$ Thus, $P\\simeq P_r$ throughout the quench and decomposition process.", "Eq.", "(REF ) is completed by obtaining expressions for $dE_r$ and $d\\langle {\\cal E}[c]\\rangle $ .", "For this adiabatic case both the temperature and volume change, so $dE_r={\\Bigl ({\\partial E_r \\over \\partial T}\\Bigr )}_VdT+{\\Bigl ({\\partial E_r \\over \\partial V}\\Bigr )}_TdV,$ Likewise, the change in the average coarse-grained energy is $d\\langle {\\cal E}[c]\\rangle = \\bigl \\langle {\\partial {\\cal E}[c]\\over \\partial V}\\bigr \\rangle dV +{\\partial \\langle {\\cal E}[c]\\rangle \\over \\partial t} dt.$ It will be seen in the next section that $\\bigl \\langle {\\partial \\langle {\\cal E}[c]\\rangle \\over \\partial V}\\bigr \\rangle \\simeq \\langle {\\cal E}[c]\\rangle /V$ .", "As discussed above, the relative volume changes for the near-critical quenches with 3MP+NE were very small, so this term can be ignored.", "Combining eqs.", "(REF ), (REF ) and (REF ) gives a single equation relating $dT$ and $dV$ to $dP$ (i.e., $dP_r$ ).", "A second relation is that of the differential change in the reservoir pressure $P_r$ to changes in temperature and volume: $dP_r={\\Bigl ({\\partial P_r\\over \\partial T}\\Bigr )}_VdT+{\\Bigl ({\\partial P_r\\over \\partial V}\\Bigr )}_TdV.$ Combining Eqs.", "(REF ) and (REF )-(REF ), and using standard thermodynamic relations [29], [28] gives ${dT\\over dt}={\\Bigl ({\\partial T \\over \\partial P_r}\\Bigr )}_{S_r}{dP\\over dt}-{1\\over C_{Pr}}{\\partial \\langle {\\cal E}[c]\\rangle \\over \\partial t},$ and ${dV\\over dt}= -V K_{Sr}{dP\\over dt}- {1\\over T}{\\Bigl ({\\partial T \\over \\partial P_r}\\Bigr )}_{S_r}{\\partial \\langle {\\cal E}[c]\\rangle \\over \\partial t},$ where ${\\Bigl ({\\partial T \\over \\partial P_r}\\Bigr )}_{S_r}= {VT\\alpha _{pr}\\over C_{Pr}}$ .", "Here, $\\alpha _{pr}=K_{Tr}\\Bigl ({\\partial P_r\\over \\partial T}\\Bigr )_V={1\\over V}\\Bigl ({\\partial V\\over \\partial T}\\Bigr )_{P_r}$ is the reservoir isobaric thermal expansion coefficient, $C_{Pr}=C_{Vr}{K_{Tr}\\over K_{Sr}}$ is the reservoir isobaric heat capacity, $K_{Tr}=-{1\\over V}\\Bigl ({\\partial V\\over \\partial P_r}\\Bigr )_{T_r}$ is the reservoir isothermal compressibility, and $K_{Sr}=-{1\\over V}\\Bigl ({\\partial V\\over \\partial P_r}\\Bigr )_{S_r}$ is the reservoir adiabatic compressibility.", "The meaning of the above equations is this: The pressure is changed at a known rate ${dP\\over dt}$ and work is done on the system.", "In the above approximation all the work is done on the reservoir ( ${\\Bigl ({\\partial T \\over \\partial P_r}\\Bigr )}_{S_r}$ and $K_{Sr}$ are reservoir functions).", "The reservoir reacts instantaneously and the temperature $T$ and volume $V$ change at rates given by the first terms in Eqs.", "(REF ) and REF ).", "The coupling between the work source and the slow modes is indirect.", "As the change in the reservoir causes $T$ and $V$ to change, the change in $T$ and $V$ causes the coefficients in ${\\cal F}[c]$ to change.", "A change in ${\\cal F}[c]$ causes the slow modes to be out of equilibrium.", "These modes then relax by exchanging energy with the reservoir at constrained pressure, causing $T$ and $V$ to change at a rate given by the second terms in Eqs.", "(REF and REF ).", "Looking more closely, consider a quench from the one-phase to the two-phase region.", "For simplicity, assume that the quench is very fast so that all the slow modes will be frozen during it.", "Then during the quench ${dT\\over dt}\\simeq {\\Bigl ({\\partial T \\over \\partial P_r}\\Bigr )}_{S_r}{dP\\over dt}$ .", "The final temperature $T_f$ is estimated (perhaps roughly) using the full system thermodynamic function $\\Bigl ({\\partial T\\over \\partial P}\\Bigr )_S$ , where $S$ is the total entropy.", "[14] Now, the total isobaric heat capacity, $C_P=C_V{K_{T}\\over K_{S}}$ , where $K_{T}$ and $K_{S}$ are the total isothermal and adiabatic compressibility, respectively.", "Substituting this relation and that of $C_{Pr}$ above into Eq.", "(REF ) gives: $C_{Pr}= {K_{Tr}\\over K_{Sr}}{K_{S}\\over K_{T}}C_p -{K_{Tr}\\over K_{Sr}}C_{Vcg}.$ However, since the contribution to $P$ from the slow modes has been neglected, $K_{S}\\simeq K_{Sr}$ and $K_{T}\\simeq K_{Tr}$ .", "Further, $K_{T}\\simeq K_{S}$ at the temperatures of experimental interest.", "Thus, $C_{Pr}\\simeq C_P- C_{cg},$ where $C_{cg}$ means either $C_{Vcg}$ or $C_{Pcg}$ .", "Further, from Tables REF and REF it can be seen that the singular part of the thermal expansion coefficient $\\alpha _{p}$ is much smaller than the background part for the quenches we are considering.", "The slow modes contribute only (or almost only) to the singular part of $\\alpha _{p}$ , which is much smaller than the background part for 3MP+NE; thus, $\\alpha _{pr}\\simeq \\alpha _{p}$ .", "So, since ${\\Bigl ({\\partial T \\over \\partial P_r}\\Bigr )}_{S_r}\\simeq {VT\\alpha _{p}\\over C_P-C_{cg}}> {VT\\alpha _{p}\\over C_P}=\\Bigl ({\\partial T\\over \\partial P}\\Bigr )_S$ , the temperature undershoots $T_f$ and reaches a value $T_{min}$ .", "Now after the quench the slow modes will relax and the temperature will change at a rate ${dT\\over dt}=-{1\\over C_P-C_{cg}}{\\partial \\langle {\\cal E}[c]\\rangle \\over \\partial t}$ .", "As the system phase separates ${\\partial \\langle {\\cal E}[c]\\rangle \\over \\partial t}<0$ and so the temperature will increase, reaching the equilibrium temperature $T_f$ over time.", "On the other hand, since $K_{Sr}$ and ${\\Bigl ({\\partial T \\over \\partial P_r}\\Bigr )}_{S_r}$ are both positive quantities, $V$ increases monotonically to its final value $V_f$ .", "As $t\\rightarrow \\infty $ the system reaches a state of minimum enthalpy.", "To complete the theory, useful expressions for $\\alpha _{pr},C_{Pr}$ , ${\\cal F}$ and ${\\cal E}$ are needed.", "First, consider $C_{Pr}$ which is given by Eq.", "(REF ).", "Since $C_P$ is known, one could presumably determine $C_{Pr}$ by calculating $C_{cg}\\simeq C_{Vcg}$ using Eq.", "(REF ).", "However, a sufficiently accurate form of $C_{cg}$ by this method possibly would require substantial calculation.", "Rather, $C_{Pr}$ will be approximated by its average over the interval $\\lbrace \\epsilon _i,\\epsilon _f\\rbrace $ , where $\\epsilon _i$ is the initial value of $\\epsilon $ .", "The justification for this approximation is that, as stated above, $F_r$ is an analytic function of $T-T_c$ , and so $C_{Pr}$ is a non-singular function of $\\epsilon $ .", "Then, presumably, $C_{Pr}$ varies slowly around $\\epsilon =0$ .", "It will be shown in the next section that the only important temperature and volume (actually pressure) dependent parameter to appear in the decomposition theory will be $\\epsilon $ .", "As such, the average coarse-grained energy will also only be a function of $\\epsilon $ .", "So, $C_{cg}\\approx {1\\over T_c}\\Bigl ({\\partial \\langle {\\cal E}[c]\\rangle \\over \\partial \\epsilon }\\Bigr ),$ and the partial derivative implies holding all parameters but $\\epsilon $ fixed.", "With Eq.", "(REF ), averaging Eq.", "(REF ) over $\\epsilon $ from the initial to the final temperatures, $\\epsilon _i$ and $\\epsilon _f$ , respectively, gives: $C_{Pr}\\approx && {1\\over \\Delta \\epsilon }\\int _{\\epsilon _i}^{\\epsilon _f}d\\epsilon \\bigl [ C_P- C_{cg}\\bigr ]\\\\=&&{1\\over \\Delta \\epsilon }\\Bigl [\\int _{\\epsilon _i}^{\\epsilon _f}d\\epsilon \\ C_P-{1\\over T_c}\\bigl (\\langle {\\cal E}[c]\\rangle _f-\\langle {\\cal E}[c]\\rangle _i\\bigr ) \\Bigr ],\\nonumber $ where $\\Delta \\epsilon = \\epsilon _f - \\epsilon _i$ .", "This approximation should be sufficient as long as $\\epsilon _i\\sim -\\epsilon _f$ , that is, the quenches are neither too deep nor too shallow.", "To calculate the equilibrium function $\\alpha _{pr}$ , consider a slow, differential change in the pressure $P$ that allows the system to remain in equilibrium.", "Since this process is reversible, the entropy will remain constant.", "Under these conditions Eq.", "(REF ) can be written as $1= \\Bigl \\lbrace {V\\alpha _{pr}\\over C_{Pr}}-{1\\over T_c}{dT_c\\over dP}\\Bigr \\rbrace \\biggl ({\\partial P\\over \\partial \\epsilon }\\biggr )_S-{1\\over T_c C_{Pr}}\\Bigl ({\\partial \\langle {\\cal E}[c]\\rangle \\over \\partial \\epsilon }\\Bigr ),$ where $\\biggl ({\\partial P\\over \\partial \\epsilon }\\biggr )_S=T_c\\Bigl [ \\Bigl ({\\partial T\\over \\partial P}\\Bigr )_S-{d T_c\\over dP}\\Bigr ]^{-1}.$ is an equilibrium function that relates changes in pressure to changes in the scaled temperature at constant entropy.", "In terms of its components, $\\Bigl ({\\partial T\\over \\partial P}\\Bigr )_S={VT\\alpha _{p}\\over C_P}$ .", "Combining Eqs.", "(REF ), (REF ), (REF ) and (REF ) give: $\\alpha _{pr}=\\alpha _{p}- {{d T_c\\over dP}\\over VT_c} (C_P-C_{Pr}).$ Using Tables REF and REF it can be shown that the singular parts of $\\alpha _{p}$ and $C_P$ cancel in this equation.", "Thus, with the approximation above for $C_{Pr}$ , $\\alpha _{pr}$ is a constant.", "With the above equations, it is now possible to compute the final temperature $T_f$ for a quench knowing the initial temperature $T_i$ and pressure change $\\Delta P$ .", "Since decomposition is a non-equilibrium process, it is not expected that $T_f$ will equal that computed using $\\Bigl ({\\partial T\\over \\partial P}\\Bigr )_S$ , which assumes constant entropy.", "As the experiments are near $T_c$ it is useful to work with changes in $\\epsilon $ rather than $T$ .", "In terms of $\\epsilon $ , Eq.", "(REF ) is: ${C_{Pr}\\over V}d\\epsilon = \\bigl (\\alpha _{pr}- {C_{Pr}\\over VT_c}{dT_c\\over dP}\\bigr )dP -{1\\over T_cV}d\\langle {\\cal E}[c]\\rangle .$ Substituting Eq.", "(REF ) for $C_{Pr}$ , Eq.", "(REF ) for $\\alpha _{pr}$ , and then integrating from the initial to final state gives $\\int _{\\epsilon _i}^{\\epsilon _f}d\\epsilon \\ {C_P\\over V} =\\bigl (A_b - {C_b\\over VT_c}{dT_c\\over dP}\\bigr )(P_f-P_i),$ where $P_i$ and $P_f$ are the initial and final pressures, respectively.", "Also, $A_b$ and $C_b$ are the background contributions to the total isobaric thermal expansion coefficient and isobaric heat capacity, respectively, which are the same above and below $T_c$ (see Table REF , but recognize that there the heat capacity is per unit mass).", "Note that the average energy of the slow modes does not appear in Eq.", "(REF ).", "That is, $\\langle {\\cal E}[c]\\rangle $ determines the time evolution, including the undershoot temperature $T_{min}$ , but not the final temperature $T_f$ .", "Also, Eq.", "(REF ) is the same equation that would be obtained by assuming a reversible, constant entropy process.", "In other words, the temperature rise produced during the phase separation exactly compensates for the temperature undershoot caused by the lack of equilibration of the slow modes during the quench.", "It can be shown that this expression for the final temperature $\\epsilon _f$ , Eq.", "(REF ), doesn't depend on the specific approximation, Eq.", "(REF ), for $C_{Pr}$ .", "At the least, Eq.", "(REF ) will hold as long as $C_{Pr}$ is given by Eq.", "(REF ) and the approximation for $C_{cg}$ is consistent with the value of $\\langle {\\cal E}[c]\\rangle $ , which depends only on $\\epsilon $ in equilibrium.", "Rather, one reason for this constant entropy result is the use here of properties of a binary liquid in the critical region, which includes the neglect of the coarse-grained pressure $P_{cg}$ in and out of equilibrium.", "If $P_{cg}$ were not small, then the pressure response of the slow modes and thus the liquid would depend on the quench rate, so that if the quench were fast the fluid entropy would increase in a manner similar to that of a gas expanding into a vacuum.", "A second reason is that, while the entropy of the slow modes is not necessarily small (thus the reason for accounting for $C_{cg}$ ), the dominant proportion of its change during phase separation is already accounted for in an equilibrium, constant entropy, process to get to the final state.", "For example, the heat of unmixing is included in the equilibrium function $\\Bigl ({\\partial T\\over \\partial P}\\Bigr )_S$ in spite of that being one of constant entropy.", "Thus, the overall transfer of energy between degrees of freedom in this non-equilibrium process is not much different than if the process had been an equilibrium one, so whatever entropy increase that does occur is small enough so that it can safely be approximated as zero.", "Contrary to the original expectation then, the final temperature can be computed accurately by just integrating $\\Bigl ({\\partial T\\over \\partial P}\\Bigr )_S$ .", "The last elements of the adiabatic theory are expressions for the temperature dependent coarse-grained free energy and energy.", "In isothermal decomposition, ${\\cal F}$ is taken to have the usual Ginzburg-Landau form: ${\\cal F}[u]= \\int _\\Lambda d{\\bf r}\\Bigl [{K\\over 2}\\bigl (\\nabla u({\\bf r})\\bigr )^2 + f(u({\\bf r})+c_0)\\Bigr ],$ where $u({\\bf r})\\equiv c({\\bf r}) - c_0$ is the deviation of the local concentration from its average $c_0$ .", "Also, $f(c)$ is the free energy density of a uniform system at concentration $c$ , and the gradient term is the lowest order correction to the free energy from deviations of $u({\\bf r})$ from zero.", "[30] In LBM and KO, the implicit cut-off is set to be inversely proportional to the correlation length at the quenched temperature, $T_f$ , i.e., $\\Lambda \\sim 1/\\xi _f$ .", "Further, these theories also choose $f(c)$ to have the standard “$\\varphi ^4$ ” form, it being the dominant correction to the quadratic term in the critical region.", "[25] Given this, follow LBM and let $f(c)={k_BT_f f_1\\over \\xi _f^3}\\phi (x),$ where $\\phi (x)={\\zeta \\over 2}x^2 + {\\lambda _4\\over 4}x^4.$ Here, $\\zeta $ and $\\lambda _4$ are constants to be determined.", "Also, $x = {c-c_c\\over u_{sf}}$ is a reduced concentration, with $u_{sf} = B{\\vert \\epsilon _f\\vert }^\\beta $ being half the miscibility gap at the quenched temperature $T_f$ , so that the scaled free energy density $\\phi (x)$ is symmetric about the critical concentration.", "Last, $f_1 = {\\xi _f^3 u_{sf}^2\\over \\chi _f}$ , where $\\xi _f$ and $\\chi _f$ are the correlation length and susceptibility, respectively, at $T_f$ .", "In the critical region hyperscaling holds,[29] so $f_1 = {(\\xi _0^-)^3B^2\\over \\Gamma ^-}$ , which is a temperature independent, dimensionless ratio of two-phase amplitudes.", "Last, the gradient energy coefficient $K=\\lambda _K {k_BT_f\\xi _f^2\\over \\chi _f},$ where $\\chi _f = \\Gamma ^-{\\vert \\epsilon _f\\vert }^{-\\gamma }$ is the susceptibility at $T_f$ , and $\\lambda _K$ is a dimensionless number very close to unity.", "How then should ${\\cal F}$ be generalized to describe kinetics in which the temperature is not constant?", "While the early-stage theories of KO and LBM can be used for computing equilibrium states, they are not intended to describe properly static critical phenomena.", "In spite of this, they do incorporate fluctuations to some degree.", "Thus, it can be expected that these fluctuations will at least shift the apparent distance from the critical point, in a manner similar to how they shift the coexistence concentrations away from the minima of $f(c)$ .", "So a correction for this shift in $T_c$ must be made in ${\\cal F}$ .", "What will be done here is just assume a simple temperature dependent form for ${\\cal F}$ and compute its coefficients.", "Then, the free energy will be examined to determine how well it predicts some equilibrium properties of a critical binary mixture such as the equation of state and susceptibility.", "If the free energy gives satisfactory results in regions important to the adiabatic decomposition theory, then its form and the scheme used to compute it will be considered adequate.", "In that spirit, and given the arguments above (including those leading to Eq.", "(REF )), assume that the dominant temperature dependence in the theory is in $\\zeta $ and let that parameter be linear in $\\epsilon $ : $\\zeta \\rightarrow \\zeta (\\epsilon )\\approx (\\lambda _2 - \\lambda _0){\\epsilon \\over \\vert \\epsilon _f\\vert }- \\lambda _0.$ So, $\\zeta (0) = -\\lambda _0$ and $\\zeta (\\epsilon _f) = -\\lambda _2$ , assuming $\\epsilon _f < 0$ .", "Since the experiments are in the critical region, any other temperature or volume dependence of ${\\cal F}$ will be ignored.", "As $T_c$ itself changes if the density changes, $\\epsilon $ will be a function of both $T$ and $V$ .", "However, for a system at constrained pressure, $T_c$ is a function of pressure only, so $\\epsilon $ will be considered a function of $T$ and $P$ rather than $T$ and $V$ .", "In that manner, the equation of motion for $V$ , Eq.", "(REF ), will not be used.", "With Eqs.", "(REF ) and (REF )-(REF ), the average coarse-grained free energy $\\langle {\\cal E}[c]\\rangle \\simeq -(\\lambda _2-\\lambda _0) {1 \\over 2{\\vert \\epsilon _f\\vert }}{k_BT_f f_1 V\\over \\xi _f^3} \\langle x^2\\rangle .$ The one-point average $\\langle x^2\\rangle $ can be obtained from the structure factor or one-point probability density, these quantities being the subject of the next section.", "The computation of the $\\lambda _i$ parameters will be described in Sec.", "below." ], [ "Hydrodynamic Theory", "In this section, the hydrodynamic KO theory of early-stage decomposition is described briefly.", "It is considered to be the most successful numerical theory of decomposition in critical binary fluids.", "It, like LBM, is built upon the Master equation vein of the theory of stochastic processes.", "[31] The KO theory consists of a set of equations that describe the time evolution of the structure factor ${\\hat{S}}(k,t)$ .", "Contact with experiment is made by relating ${\\hat{S}}(k,t)$ to the scattered radiation intensity $I(k,t)$ .", "[32] Now, let $u_{\\bf k}$ be the Fourier transform of the concentration deviation $u({{\\bf r}})$ .", "The structure factor ${\\hat{S}}(k)$ is defined as ${\\hat{S}}(k)=\\langle \\vert u_{\\bf k}\\vert ^2\\rangle ,$ and is the Fourier transform of the concentration-concentration correlation function $S({{\\bf r}}-{{\\bf r}}_0)=\\langle u({{\\bf r}})u({{\\bf r}}_0)\\rangle .$ This function can be obtained from theory by taking moments of $\\rho ([u],t)$ , which, as mentioned above, is the probability density that the system is in a coarse-grained configuration $[u]$ at time $t$ .", "In KO theory, the time evolution of the probability density $\\rho ([u],t)$ is determined by a Fokker-Planck equation [10]: ${\\partial \\rho \\over \\partial t}=\\Bigl [ {\\cal L}_1+{\\cal L}_2\\Bigr ]\\rho ,$ where the operators are given by ${\\cal L}_1=-\\int d{{\\bf r}}_1 d{{\\bf r}}_2{\\delta \\over \\delta u({{\\bf r}}_1)}\\nabla _1^2L_\\Lambda ({{\\bf r}}_1-{{\\bf r}}_2)\\Bigl [{\\delta {\\cal F}\\over \\delta u({{\\bf r}}_2)}+k_B T{\\delta \\over \\delta u({{\\bf r}}_2)}\\Bigr ],$ and ${\\cal L}_2=\\int d{{\\bf r}}_1 d{{\\bf r}}_2{\\delta \\over \\delta u({{\\bf r}}_1)}\\nabla _1 u({{\\bf r}}_1)\\cdot {\\bf T}({{\\bf r}}_1-{{\\bf r}}_2)\\cdot \\nabla _2 u({{\\bf r}}_2)\\Bigl [{\\delta {\\cal F}\\over \\delta u({{\\bf r}}_2)}+k_B T{\\delta \\over \\delta u({{\\bf r}}_2)}\\Bigr ].$ Here, ${\\delta \\over \\delta u({\\bf r})}$ is a functional derivative with respect to the concentration field at the point ${\\bf r}$ , and ${\\bf T}({\\bf r})$ is the Oseen tensor with components $T_{\\alpha \\beta }={1\\over 8\\pi \\eta _s r}\\Bigl [\\delta _{\\alpha \\beta }+{\\hat{r}}_\\alpha {\\hat{r}}_\\beta \\Bigr ]$ , with $\\eta _s$ being the hydrodynamic shear viscosity and ${\\hat{r}}\\equiv {{\\bf r}/r}$ .", "Eqs.", "(REF -REF ) describe phase separation driven by overdamped fluid flow, the flow in turn caused by gradients in the local chemical potential, $\\mu ({\\bf r}) = {\\delta {\\cal F}[u]\\over \\delta u({\\bf r})}$ .", "These equations are renormalized versions[33] of bare stochastic equations[34], which are formally equivalent to the Langevin equations of Model H of critical dynamics[35] in the overdamped approximation.", "The operator ${\\cal L}_1$ results from integrating out concentration fluctuations of wavenumber $k>\\Lambda $ .", "The Onsager function $L_\\Lambda (r)$ that appears in ${\\cal L}_1$ is weakly non-local and couples these short-wavelength concentration modes, $\\it via$ the fluid velocity field, to the long-wavelength modes $k<\\Lambda $ .", "$L_\\Lambda (r)$ is the inverse Fourier transform of [33]: ${\\hat{L}_\\Lambda }(k_1)={1\\over k_1^2}\\int {d{\\bf k}_2\\over (2\\pi )^3}{\\bf k}_1\\cdot {\\bf \\hat{T}}({\\bf k}_1-{\\bf k}_2)\\cdot {\\bf k}_1{\\hat{S}}_{eq}(k_2).$ Here, the integral over $k_2$ runs from $\\Lambda $ to an upper cut-off which is approximated as $\\infty $ .", "${\\bf \\hat{T}}({\\bf k})$ is the Fourier transform of the Oseen tensor with components ${1\\over \\eta _s k^2}(\\delta _{\\alpha \\beta } - {\\hat{k}}_\\alpha {\\hat{k}}_\\beta )$ .", "${\\hat{S}}_{eq}(k)$ is the equilibrium structure factor of mode $k\\ (>\\Lambda )$ , and is taken to have a Lorentzian form: ${\\hat{S}}_{eq}(k)={\\chi \\over 1+(k\\xi )^2},$ where $\\chi $ and $\\xi $ are the susceptibility and correlation length, respectively, at temperature $\\epsilon $ and concentration $c_0$ .", "Clearly, as the temperature changes, ${\\hat{S}}_{eq}(k)$ , and therefore ${\\hat{L}_\\Lambda }(k)$ , will be changing also.", "However, it can be shown [33] that a reasonable approximation to Eq.", "(REF ) is ${\\hat{L}_\\Lambda }\\simeq {\\chi \\over 3\\pi ^2\\xi ^2 \\eta _s\\Lambda }.$ Making use of Tables REF and REF for $\\chi ,\\xi $ and $\\eta _s$ , it is found that ${\\hat{L}_\\Lambda }$ has a weak temperature dependence.", "Thus, if the quenches are relatively fast, ${\\hat{S}}_{eq} (k)$ can be set to its value at the final equilibrium temperature $\\epsilon _f$ .", "Further, the weak temperature dependence of the viscosity $\\eta _s$ will also be ignored.", "Then with these approximations, the only temperature dependence in (REF ) appears in the coarse-grained free energy, ${\\cal F}[u]$ .", "An equation of motion for the structure factor ${\\hat{S}}(k,t)$ was derived by KO using Eqs.", "(REF ) and (REF -REF ).", "To evaluate two-point correlation functions in ${\\cal L}_1$ other than $S(r)$ , they used the LBM ansatz for the two-point probability density: $\\rho _2^{lbm}(u_1,u_2) = \\rho _1(u_1)\\rho _1(u_2)\\Bigl [1 + {u_1 u_2\\over \\langle u^2\\rangle ^2}S(r_{12})\\Bigr ],$ where $r_{12}\\equiv \\vert {\\bf r}_1-{\\bf r}_2\\vert $ , $u_1 \\equiv u({\\bf r}_1)$ , etc.", "This self-consistent linear approximation is expected to work best during the early stage of decomposition when the growing domains are not much larger than a few equilibrium correlation lengths and sharp interfaces have not yet formed.", "[6] Implicit in the LBM derivation is a constraint that averages taken with respect to $\\rho _2$ must reduce to their exact form in the limit $r_{12}\\rightarrow 0$ .", "That is, for arbitrary functions $h(u)$ and $g(u)$ , $\\langle h(u_1)g(u_2)\\rangle \\rightarrow \\langle h(u)g(u)\\rangle $ as $r_{12}\\rightarrow 0$ , where the latter average is taken with respect to the one-point probability density $\\rho _1(u)$ .", "Implementing this constraint in a simple way, and using the ansatz above, gives an equation for $\\rho _2$ : $\\rho _2(u_1,u_2)&& \\approx \\rho _2^{lbm}(u_1,u_2) +a^3\\delta ({\\bf r}_1-{\\bf r}_2)\\Bigl [ \\\\&&\\rho _1(u_1)\\delta (u_1-u_2) - \\rho _2^{lbm}(u_1,u_2)\\bigr \\vert _{r_{12}\\rightarrow 0}\\Bigr ].\\nonumber $ The ${\\cal L}_2$ contribution to ${\\hat{S}}(k,t)$ contains a four-point correlation function.", "KO argued that during the early stage of decomposition the coupling between modes in this correlation function would be close to gaussian.", "In this approximation the four-point correlation function reduces to a product of two-point ones.", "[10] The result is: ${\\partial {\\hat{S}}(k_1)\\over \\partial t}=&&-2{\\hat{L}_\\Lambda }(k_1)k_1^2\\Bigl [\\bigl (Kk_1^2+A\\bigr ){\\hat{S}}(k_1)-k_B T\\Bigr ]\\\\+&&2\\int ^\\Lambda {d{\\bf k}_2\\over (2\\pi )^3}{{\\bf k}_1}\\cdot {\\bf \\hat{T}}({\\bf k}_1-{\\bf k}_2)\\cdot {{\\bf k}_1}\\Bigl [K\\bigl (k_2^2-k_1^2\\bigr ){\\hat{S}}(k_2){\\hat{S}}(k_1)+k_B T{\\hat{S}}(k_2)-k_B T{\\hat{S}}(k_1)\\Bigr ].\\nonumber $ Here, $A = {1\\over \\langle u^2\\rangle }\\bigl \\langle u{\\partial f(u+c_0)\\over \\partial u}\\bigr \\rangle ,$ where the averages are taken with respect to $\\rho _1(u)$ .", "It can be shown[10] that the operator ${\\cal L}_2$ doesn't contribute directly to the equation of motion for $\\rho _1(u)$ .", "Given this, the derivation of the equation of motion for $\\rho _1(u,t)$ from Eq.", "(REF ) is almost identical to the one in LBM.", "It is found: ${\\partial \\rho _1(u)\\over \\partial t}={\\partial \\over \\partial u}\\Bigl [G(u)\\rho _1(u)+k_B T{L\\over a^3}{\\partial \\rho _1(u)\\over \\partial u}\\Bigr ],$ where $G(u)=W{u\\over \\langle u^2\\rangle }+L\\Bigl [{\\partial f\\over \\partial u}-\\bigl \\langle {\\partial f\\over \\partial u}\\bigr \\rangle -{u\\over \\langle u^2\\rangle }\\bigl \\langle u{\\partial f\\over \\partial u}\\bigr \\rangle \\Bigr ],$ $W=\\int _0^\\Lambda {dk\\over 2\\pi ^2} k^4 {\\hat{L}_\\Lambda }(k)\\bigl (Kk^2+A){\\hat{S}}(k),$ and $L=a^3\\int _0^\\Lambda {dk\\over 2\\pi ^2} k^4 {\\hat{L}_\\Lambda }(k).$ For the initial conditions of these equations, the equilibrium solution of them will be used.", "Since the equilibrium form should be independent of the mechanism of equilibration, to simplify the result, the term $\\nabla _1^2L_\\Lambda ({\\bf r}_1-{\\bf r}_2)$ can be replaced with $\\delta ({\\bf r}_1-{\\bf r}_2)$ in Eq.", "(REF ) above.", "With this, and setting the RHS of Eq.", "(REF ) to zero yields $\\rho _{1eq}(u) = exp \\Bigl [- &&{a^3\\over k_BT}\\bigl (f(u+c_0) -u\\langle {\\partial f\\over \\partial u}\\rangle \\bigr ) \\\\-&&{a^3K\\over 2k_BT} {u^2\\over \\langle u^2\\rangle }\\int _0^\\Lambda {dk\\over 2\\pi ^2}k^4{\\hat{S}}_{eq}(k)+ b_0\\Bigr ]\\nonumber $ where $b_0$ is a normalization constant.", "The equilibrium structure factor is obtained by setting the RHS of Eq.", "(REF ) to zero, giving ${\\hat{S}}_{eq}(k) = {k_BT\\over Kk^2 + A},$ with $A$ being given by Eq.", "(REF ) above, but now computed using Eq.", "(REF ).", "These kinetic and equilibrium equations were solved numerically." ], [ "Scaling of the Equations", "For numerical computation, it is helpful to scale the above equations.", "While in adiabatic decomposition the temperature will necessarily be changing with time after the quench, the final equilibrium temperature will still be the relevant one.", "So, as in KO and LBM, the scaling will be done with respect to system properties at $\\epsilon _f$ .", "Define the scaled wavevector cut-off $\\alpha ^\\ast =\\Lambda \\xi _f$ , where $\\alpha ^\\ast $ is a number close to 1.", "Define also the dimensionless wavevector, $q=k\\xi _f$ ; distance, ${\\tilde{r}} = r/\\xi _f$ ; structure factor, ${\\tilde{S}}(q) = {\\hat{S}}(k)/\\chi _f$ ; relative concentration, $y = u/u_{sf}$ ; average concentration, $x_0 = {c_0-c_c\\over u_{sf}}$ ; and time, $\\tau = {k_BT\\over 6\\pi \\eta _s\\xi _f^3}t$ .", "As for LBM, the cell volume $a^3 = \\bigl (\\int {d{\\bf k}\\over (2\\pi )^3}\\bigr )^{-1} = {6\\pi ^2\\over (\\alpha ^\\ast )^3}\\xi _f^3$ .", "It is convenient to scale the Onsager function, Eq.", "(REF ), as $\\sigma (q)=&&{6\\pi \\xi _f\\eta _s\\over \\chi _f }{\\hat{L}_\\Lambda }(q/\\xi _f)\\\\=&&K(q)-{3\\over 2\\pi }\\int _0^{\\alpha ^\\ast }dm\\ Q(q/m){1\\over 1+m^2}\\nonumber $ where $K(q)$ is a Kawasaki function[10]: $K(q)={3\\over 4}\\Bigl [({1\\over q}-{1\\over q^3})\\arctan (q)+{1\\over q^2}\\Bigr ],$ and $Q(x)={1\\over 2}\\Bigl [{1\\over x}+ {1\\over x^3}\\Bigr ]{\\rm ln}\\bigl \\vert {1+x\\over 1-x}\\bigr \\vert -{1\\over x^2}.$ Changing to the new scaled variables and performing any angular integration, the equation of motion for the structure factor becomes: ${\\partial {\\tilde{S}}(q)\\over \\partial \\tau }=&&-2\\sigma (q)q^2\\Bigl [(\\lambda _Kq^2+{\\tilde{A}}){\\tilde{S}}(q)-1\\Bigr ]\\\\&&+ {3\\over \\pi }q^2\\int _0^{\\alpha ^\\ast }dm\\ Q(q/m)\\Bigl [\\lambda _K(m^2-q^2){\\tilde{S}}(q){\\tilde{S}}(m)+{\\tilde{S}}(m)-{\\tilde{S}}(q)\\Bigr ],\\nonumber $ where ${\\tilde{A}}={1\\over \\langle y^2\\rangle }\\bigl \\langle y{\\partial \\phi (y+x_0)\\over \\partial y}\\bigr \\rangle .$ The kinetic equation for the one-point probability density $\\rho _1$ is now ${\\partial \\rho _1(y)\\over \\partial \\tau }=\\omega {\\partial \\over \\partial y}\\Bigl [g(y)\\rho _1(y)+{\\partial \\rho _1(y)\\over \\partial y}\\Bigr ],$ where $\\omega = {1\\over f_1}\\int _0^{\\alpha ^\\ast }{dq\\over 2\\pi ^2}q^4\\sigma (q),$ and $g(y)={{\\tilde{W}}y\\over \\langle y^2\\rangle }+f_0\\Bigl ({\\partial \\phi \\over \\partial y}-\\langle {\\partial \\phi \\over \\partial y}\\rangle -y{\\tilde{A}}\\Bigr ),$ with ${\\tilde{W}}={1\\over f_1\\omega }\\int _0^{\\alpha ^\\ast }{dq\\over 2\\pi ^2}\\ q^4\\sigma (q)(\\lambda _Kq^2+{\\tilde{A}}){\\tilde{S}}(q).$ Here, $f_0 = {6\\pi ^2 f_1/(\\alpha ^\\ast )^3}$ .", "Last, using Eq.", "(REF ), it can be found that the equation of motion for the reduced temperature is: ${d\\epsilon \\over d\\tau } = \\Bigl [ {\\alpha _{pr}\\over \\rho _cC_{Pr}} - {1\\over T_c}{d T_c\\over dP}\\Bigr ] {dP\\over d\\tau } - {k_B\\over \\xi _f^3\\rho _cC_{Pr}}{d{\\tilde{e}}\\over d\\tau },$ where $\\rho _c$ is the critical mass density and $C_{Pr}$ is now the reservoir heat capacity per unit mass, and the average coarse-grained energy, Eq.", "(REF ), properly scaled, is: ${\\tilde{e}} =&&{\\xi _f^3\\over Vk_BT_f}\\langle {\\cal E}[c]\\rangle \\nonumber \\\\=&&-(\\lambda _2-\\lambda _0){f_1\\over 2{\\vert \\epsilon _f\\vert }}\\langle x^2\\rangle .$ The scaled form of the equilibrium equations, (REF ) and (REF ), are, respectively, $\\rho _{1eq}(y) = exp\\Bigl [-&&f_0\\bigl (\\phi (y+x_0)- y\\langle {\\partial \\phi \\over \\partial y}\\rangle \\bigr )\\\\-&&{y^2\\over \\langle y^2\\rangle }{f_0\\lambda _K\\over 2f_1}\\int _0^{\\alpha ^\\ast } {dq\\over 2\\pi ^2} q^4{\\tilde{S}}_{eq}(q) + b_0\\Bigr ],\\nonumber $ and ${\\tilde{S}}_{eq}(q) = {1\\over \\lambda _Kq^2 + {\\tilde{A}}},$ where again $b_0$ is a normalization constant.", "From these equations it can be seen that an (linear) isothermal quench and subsequent decomposition is completely specified by the quench time $\\tau _{quench}$ , the ratio of the initial to the final scaled temperature, $\\epsilon _i\\over \\epsilon _f$ , and the average concentration $x_0$ .", "In addition to the properties of the particular fluid one wants to study, an adiabatic quench is completely specified by these same quantities plus the change in pressure, $\\Delta P$ .", "The predictions of the theory are also somewhat dependent on the value of the scaled cut-off $\\alpha ^\\ast $ .", "However, the degree of this dependence will be minimized by the method of computing the $\\lambda _i$ parameters in the coarse-grained free energy, discussed below." ], [ "Numerical Solution", "The scaled adiabatic equations were solved numerically as follows.", "${\\tilde{S}}(q)$ was solved on a grid of $N_q$ points $q_i = i\\Delta q$ , $i=1,...,N_q$ in $q$ -space, with spacing $\\Delta q = 2\\alpha ^\\ast /N_q$ .", "${\\tilde{S}}(q)$ was set to zero for all grid points $q_i > \\alpha ^\\ast $ .", "The reason the grid was extended in this manner was to be able to inverse Fourier transform ${\\tilde{S}}(q)$ if need be.", "The $q$ grid point number $N_q$ was set to $2^9=512$ for any run with a maximum time $\\tau _{max}\\le 10^3$ , and was set to $2^{10}=1024$ for longer runs of $10^3<\\tau _{max}\\le 10^4$ .", "Similarly, $\\rho _1(y)$ was solved on a grid of $N_y$ equally spaced points $y_i$ , $i=1,..,N_y$ in $y$ -space, with $y_1 = y_{min}+\\Delta y/2$ , $y_2 = y_{min}+3/2\\Delta y$ , and $y_{N_y} = y_{max}-\\Delta y/2$ , where $\\Delta y=N_y/(y_{max}-y_{min})$ and $y_{min} = -y_{max}$ .", "To ensure that $\\rho _1(y)$ could model properly behavior near the coexistence curve at $\\epsilon _f$ , $y_{max}$ was set to 2.5.", "Also, $N_y=120$ for all results shown in this work.", "It was found that no result shown here changed appreciably if $N_q$ and $N_y$ were increased beyond the above values.", "Now, the adiabatic theory consists of ODE's for ${\\tilde{S}}(q,\\tau )$ and $\\epsilon (\\tau )$ , and a PDE for $\\rho _1(y,\\tau )$ .", "To simplify the computation, the PDE for $\\rho _1(y,\\tau )$ was converted into a set of coupled ODE's, using a simple finite difference scheme.", "Let $\\rho _{1i}$ and $g_i$ be the values of $\\rho _1(y)$ and $g(y)$ at the $i$ th grid point $y_i$ .", "Then, Eq.", "(REF ) becomes ${d\\rho _{1i}\\over d\\tau }&& =\\omega \\Bigl [g_{i+1}\\rho _{1(i+1)} - g_{i-1}\\rho _{1(i-1)}\\Bigr ]/(2\\Delta y) \\nonumber \\\\+&& \\omega \\Bigl [\\rho _{1(i+1)}+ \\rho _{1(i-1)} - 2\\rho _{1i}\\Bigr ]/\\Delta y^2,$ with the boundary conditions $\\rho _{11} = \\rho _{1N_y}= 0$ .", "A total of $N = N_q/2 + N_y-2 + 1$ ODE's result.", "The integration of these in time was done using the Bulirsch-Stoer method.", "[36] The structure factor equations are stiff in the sense that the relaxation of the high-$q$ modes is much faster than the low-$q$ ones.", "The Bulirsch-Stoer method is not usually used for solving such an ODE type.", "Given that, initially the equations were also solved using a commercial package built for solving stiff ODE's.", "[37] It was found that both methods yielded the same results.", "In past work, the PDE for $\\rho _1(y)$ was solved instead using the much faster “double gaussian” method.", "[6] This method was analyzed for on-critical quenches, $x_0=0$ , and found to give essentially identical results to the finite difference method at early times.", "However, it over-estimated the phase separation at later times when the wavevector $q_m$ of the peak of the structure factor was less than $0.3$ .", "For the BC experiments, data was available out to times such that $q_m < 0.2$ .", "As a consequence, this approximation was not used here.", "For each timestep, $\\rho _1$ was normalized to prevent accumulation of round-off errors.", "The first moment, $\\langle y\\rangle $ of $\\rho _1$ was monitored to ensure that it remained zero.", "The second moment of $\\rho _1$ and the integral of ${\\tilde{S}}(q)$ were also monitored to ensure they both gave the same result for $\\langle y^2\\rangle $ .", "The equilibrium equations, (REF ) and (REF ), were solved using the same grids for $q$ and $y$ , though here $N_q$ was set to $2^{12}=4096$ .", "Simple iteration was used for them.", "The initial guess for ${\\tilde{S}}_{eq}(q)$ was a scaled version of Eq.", "(REF ) at the relevant initial $\\epsilon $ and $x_0$ .", "However, if the LBM solution of ${\\tilde{S}}_{eq}(q)$ differed appreciably from the known value, bootstrap, i.e., a previous guess at a nearby temperature and concentration, was used instead." ], [ "Computing the Coarse-Grained Free Energy", "The last ingredients of the theory are values for the $\\lambda _i$ parameters in the coarse-grained free energy, defined in Sec.", "REF above.", "Similar to LBM, these parameters were determined by using ${\\cal F}$ to compute the equilibrium structure factor and chemical potential on coexistence at the final temperature $\\epsilon _f$ , and at the critical point $\\epsilon =0$ .", "There are a number of ways to accomplish this task.", "The one probably most accurate for the least amount of effort is to use the LBM equilibrium solution for ${\\tilde{S}}(q)$ and $\\rho _1(y)$ .", "[38] This scheme was used here.", "It amounts to applying $N$ equilibrium conditions to the LBM equations for the $N$ unknowns, and finding the solution of them using the Newton-Raphson method.", "[36] Here, all derivatives required by this method were computed numerically.", "For here and elsewhere in this work, the free energy amplitude $f_1$ was set to 0.210, consistent with the critical amplitude values in Table REF .", "On coexistence, $\\epsilon =\\epsilon _f$ and $x_0=1$ , the scaled equilibrium structure factor ${\\tilde{S}}(q) = 1/(1 + q^2)$ .", "Two relations obtained from this equation were ${\\tilde{S}}(0)=1$ and $\\langle y^2\\rangle =&&{1\\over 2\\pi ^2f_1}\\int _0^{\\alpha ^\\ast } dq q^2{\\tilde{S}}(q)\\nonumber \\\\=&&{1\\over 2\\pi ^2f_1}\\bigl [\\alpha ^\\ast - \\arctan (\\alpha ^\\ast )\\bigr ].$ A third relation is that the exchange chemical potential must vanish on coexistence: ${\\tilde{\\mu }} = \\langle {\\partial \\phi \\over \\partial x}\\rangle = 0$ .", "These three equations are sufficient to determine $\\lambda _K$ , $\\lambda _2$ and $\\lambda _4$ for any cut-off $\\alpha ^\\ast $ .", "The last parameter $\\lambda _0$ was determined by requiring that ${\\tilde{S}}(0) = \\infty $ , i.e., $\\tilde{A} = 0$ , at the critical point, $\\epsilon =0$ and $x_0=0$ .", "Values for these $\\lambda _i^{\\prime }s$ for various cut-offs are shown in Table REF below.", "Note that the value of $\\lambda _0$ does not depend on the form of the temperature dependence of $\\zeta $ , Eq.", "(REF ), only that it reduce to $-\\lambda _0$ at $\\epsilon = 0$ .", "Table: Coarse-grained free energy coefficients λ i \\lambda _i for variouswavevector cut-offs α * \\alpha ^\\ast .To justify some approximations made previously, it is helpful to examine the predictions of the equilibrium LBM theory on-critical above $T_c$ and on-coexistence below it.", "Figure REF shows LBM predictions for the equilibrium, on-critical inverse susceptibility ${\\tilde{S}}^{-1}(0)$ as a function of ${\\epsilon \\over \\vert \\epsilon _f\\vert }>0$ for two cutoff values $\\alpha ^\\ast =1$ and $\\pi /2$ .", "Also shown are the expected scaling predictions for a simple binary fluid (3-D Ising universality class): ${\\tilde{S}}(0)^{-1}=\\Gamma ^-/\\Gamma ^+\\vert {\\epsilon \\over \\epsilon _f}\\vert ^{\\gamma }$ where the amplitude ratio and exponent are obtained from Table REF .", "For ${\\epsilon \\over \\vert \\epsilon _f\\vert }\\ll 1$ , it is found that LBM also predicts scaling behavior, with the exponent $\\gamma $ approximately equal to the mean spherical model value of 2.", "[29] Since the accepted 3-D Ising value of $\\gamma \\approx 1.240$ , the LBM theory does not perform well in this limit as expected.", "However, it can seen for higher temperatures the theory performs much better.", "For $0.5\\le {\\epsilon \\over \\vert \\epsilon _f\\vert }\\le 10$ , the LBM predictions for ${\\tilde{S}}(0)$ are within 10% and 20% of the exact values for $\\alpha ^\\ast =1$ and $\\alpha ^\\ast =\\pi /2$ , respectively.", "For higher temperatures, the agreement lessens, but ${\\tilde{S}}(0)$ is small there anyways.", "It was found that the predictions of the hydrodynamic theory are pretty much insensitive to such small variations in the initial conditions.", "(More important is that ${\\tilde{S}}(q)$ and $\\rho _1(y)$ be consistent with each other.)", "Define an effective susceptibility exponent $\\gamma _{eff} = -ln\\bigl [{\\tilde{S}}(0,\\epsilon _1)/{\\tilde{S}}(0,\\epsilon _2)\\bigr ]/ln\\bigl [\\epsilon _1/\\epsilon _2\\bigr ],$ where the temperatures $\\epsilon _1$ and $\\epsilon _2$ are close to each other in some sense.", "Then, over this temperature range, $0.5\\le {\\epsilon \\over \\vert \\epsilon _f\\vert }\\le 10$ , LBM predicts that $\\gamma _{eff}$ varies from 1.39 down to 1.10 for $\\alpha ^\\ast =1$ , and 1.47 down to 1.15 for $\\alpha ^\\ast =\\pi /2$ .", "Also, for $\\alpha ^\\ast =1$ and ${\\epsilon \\over \\vert \\epsilon _f\\vert }=1.5$ , and $\\alpha ^\\ast =\\pi /2$ and ${\\epsilon \\over \\vert \\epsilon _f\\vert }=2.3$ , $\\gamma _{eff} = 1.24$ , i.e., is exact.", "Thus, the LBM theory seems to describe properly the temperature dependence of critical fluctuations within a window near ${\\epsilon \\over \\vert \\epsilon _f\\vert }=1$ .", "It is concluded then that using the equilibrium LBM theory, along with ${\\cal F}$ defined in Sec.", "REF above, to give initial conditions for ${\\tilde{S}}(q)$ and $\\rho _1(y)$ is acceptable as long as the quenches are not too deep, ${\\epsilon _i\\over \\vert \\epsilon _f\\vert } > 0.5$ .", "Figure: LBM predictions, using the temperature dependent coarse-grained free energydefined in Sec.", ", for the equilibrium, one-phase,on-critical, inverse susceptibility S ˜(0) -1 {\\tilde{S}}(0)^{-1} as afunction of the scaled temperature ϵ |ϵ f |>0{\\epsilon \\over \\vert \\epsilon _f\\vert }>0.Results for two cut-offs α * \\alpha ^\\ast are shown, along with “exact” scalingvalues.Figure: LBM predictions, using the temperature dependent coarse-grained free energydefined in Sec.", ",for the coexistence curve ϵ(x 0 )\\epsilon (x_0)for two cut-offs α * =1\\alpha ^\\ast =1 and π/2\\pi /2.", "Also shown is the “exact”scaling form in the critical region, ϵ |ϵ f |=-x 0 1/β {\\epsilon \\over \\vert \\epsilon _f\\vert }= -x_0^{1/\\beta },where β≈0.33\\beta \\approx 0.33.Examining the LBM predictions for the two-phase coexistence curve $\\epsilon (x_0)$ is also illuminating.", "Figure REF shows $\\epsilon (x_0)$ for positive $x_0$ (the curve is symmetrical about $x_0=0$ ) for cutoffs $\\alpha ^\\ast = 1$ and $\\pi /2$ .", "Also shown is the accepted scaling form for a system in the 3-D Ising universality class: $\\epsilon (x_0)\\sim -\\vert x_0\\vert ^{1/\\beta }$ , with $\\beta \\approx 0.33$ .", "Thus, $\\epsilon (x_0)$ should have a maximum at $x_0=0$ .", "However, as can be seen, instead of a maximum at $x_0=0$ , the LBM predictions overshoot $\\epsilon =0$ and have a maximum at around $x_0=0.2$ and $0.25$ for $\\alpha ^\\ast =1$ and $\\pi /2$ , respectively.The cause of this overshoot was investigated.", "It was found that it could be corrected by making the theory self-consistent, not on the second moment of the one-point probability density, but on the susceptibility.", "While it is possible that the kinetic version of the theory could be corrected in a similar manner, it was found not to be necessary in order to make a proper comparison with the experiments considered in this work.", "Thus, the LBM theory should not be used to describe the initial state for quenches that are deep, ${\\epsilon \\over \\vert \\epsilon _f\\vert }\\ll 1$ , and off-critical.", "Also, contrary to its behavior above $T_c$ , the LBM predictions for $\\gamma _{eff}$ below $T_c$ are always below the accepted 3-D Ising scaling value of $1.240$ , at best reaching $0.9$ at ${\\epsilon \\over \\vert \\epsilon _f\\vert }=-2.0$ .", "On the other hand, the theory's predictions for an effective $\\beta $ exponent, $\\beta _{eff}$ , defined analogously to $\\gamma _{eff}$ in Eq.", "(REF ) above, are near the accepted 3-D Ising value for temperatures near $\\epsilon _f$ .", "For $\\alpha ^\\ast =1$ and ${\\epsilon \\over \\vert \\epsilon _f\\vert }=-0.85$ , and $\\alpha ^\\ast =\\pi /2$ and ${\\epsilon \\over \\vert \\epsilon _f\\vert }=-1$ , $\\beta _{eff}$ equals the accepted $\\beta $ value of $0.328$ .", "Also, for temperatures near $\\epsilon _f$ , the predictions of LBM for $\\epsilon (x_0)$ are in good agreement with the accepted values in the range $-2.0\\le {\\epsilon \\over \\vert \\epsilon _f\\vert }\\le -0.5$ .", "As will be seen below, the adiabatic theory predicts that the temperature undershoot after an on-critical quench of 3MP+NE does not go below $-2\\epsilon _f$ .", "Thus, if the quenches are fast, the on-coexistence equilibrium predictions of the LBM theory should be acceptable.", "It is concluded that if the quenches are not too deep, and are fast enough so that the fluid spends little time exploring the region near $T_c$ , then the temperature dependent coarse-grained free energy defined in Sec.", "REF is adequate." ], [ "Results", "In this section, general predictions of the adiabatic and isothermal decomposition theories are discussed, and then the theories are compared with experiment.", "In an unpublished work, Schwartz showed that the original numerical scheme of KO was not quite right,[40] leading to erroneous results for the structure factor at intermediate and late times.", "Given this, aspects of the isothermal KO theory by itself will also be discussed." ], [ "General Predictions", "In the isothermal decomposition theory, the equations scale completely.", "That is, no system or temperature dependent parameter appears in the theory when the equations are scaled using parameters appropriate to the critical region.", "Thus, if initial (and quench) conditions are ignored, the theory predicts that if the experimental data is appropriately rescaled then the data should superimpose for any binary fluid and any quench temperature.", "However, the adiabatic theory does not scale.", "The parameters appearing in the equation for $\\epsilon $ , (REF ), are strongly system dependent and some, $\\alpha _{p}$ and $C_P$ , are temperature dependent.", "To illustrate the adiabatic results in this section then, data for 3MP+NE will just be used.", "The equation for $\\epsilon $ requires $\\alpha _{p},C_P,\\rho _c,T_c,{d T_c\\over dP}$ and $\\xi _0^\\pm $ , which, for 3MP+NE, can be obtained from Tables REF and REF .", "The other parameters appearing in the theory are the universal amplitude $f_1$ , the cut-off $\\alpha ^\\ast $ , and the cut-off dependent free energy parameters $\\lambda _i$ .", "As mentioned above, $f_1$ was set to 0.210, and in Sec.", "values for the $\\lambda _i$ were computed for various cut-off values.", "What remains then is to determine an appropriate cut-off.", "In isothermal decomposition, $\\alpha ^\\ast $ is determined by requiring that it be large enough so that no unstable modes are integrated out.", "In the mean-field theory of Cahn,[1] the dominant unstable wavevector is at $q={1\\over {\\sqrt{2}}}$ , with the largest unstable mode occurring at $q=1$ .", "Thus, $\\alpha ^\\ast \\ge 1$ .", "While the statistical theory here gives free energy parameters that are cut-off dependent, this relation roughly holds here too.", "On the other hand, the time dependent inverse susceptibility, $A(t)$ , appearing in the equation of motion for the structure factor, Eq.", "(REF ) does not vary with the wavevector $k$ of the mode.", "In other words, the LBM ansatz produces a mean-field form for this time dependent inverse susceptibility.", "[6] The goal then should be to include as few concentration modes as possible into this mean-field approximation, that is, to make $\\alpha ^\\ast $ as small as possible.", "A good compromise between these opposing needs is to follow LBM and just let $\\alpha ^\\ast =1$ for isothermal decomposition.", "In adiabatic decomposition, the wavevector of the largest unstable mode will depend upon the degree of temperature undershoot.", "What has been done here is set $\\alpha ^\\ast $ and then examine the temperature undershoot at a short time after the end of the quench, say, $\\tau =0.1$ .", "The inverse of the equilibrium correlation length for that temperature at that time was then identified to be the minimum cut-off value, in analogy with the isothermal case.", "For the quenches considered here, it was found that setting $\\alpha ^\\ast =1.4$ was reasonable.", "For consistency, this cut-off was also used for the isothermal runs.", "With the parameters in the equations determined, a quench is specified by the initial scaled temperature $\\epsilon _i$ , the pressure change $\\Delta P$ , and the quench time $\\tau _{quench}$ .", "The final temperature was determined by integrating the thermodynamic function $\\Bigl ({\\partial \\epsilon \\over \\partial P}\\Bigr )_S$ given in Eq.", "(REF ).", "The quench time varied with experiment, but was either known or could be deduced.", "Figure REF shows the scaled temperature, $\\epsilon /\\vert \\epsilon _f\\vert $ as a function of the scaled time $\\tau $ for three adiabatic runs ending at the temperatures $T_c\\epsilon _f= -0.04$ mK, $-0.4$ mK and $-4.0$ mK, with initial temperatures $\\epsilon _i = -10\\epsilon _f$ .", "The quenches were on-critical so $x_0=0$ .", "The scaled quench time $\\tau _{quench}$ has been set to be $0.01$ ; thus the initial temperature drop does not appear on the graph.", "Clearly, the temperature undershoot is large; the temperature reached immediately after the quench is roughly $-1.8 |\\epsilon _f|$ , with the smaller final temperatures giving the greater undershoot.", "Note also that there is not much difference in the scaled temperature trajectories even though the final scaled temperatures differ by a factor of 100.", "Figure: Scaled temperature ϵ/|ϵ f |\\epsilon /|\\epsilon _f| as afunction of scaled time τ\\tau for three adiabatic runs.The final temperatures ΔT f \\Delta T_f are indicated in the figure and theinitial temperatures are ϵ i =-10ϵ f \\epsilon _i= -10\\epsilon _f.", "The straight linedenotes the final equilibrium temperature.Figures REF and REF show results for the scaled peak intensity, ${\\tilde{S}}(q_m)$ , and scaled peak wavevector, $q_m$ as functions of the scaled time $\\tau $ .", "Results of the middle adiabatic quench in Figure REF , $T_c\\epsilon _f=-0.4$ mK, are shown along with results from an isothermal run with the same ratio of $\\epsilon _i/\\epsilon _f = -10$ .", "Also shown are results from an “LBM” version of the theory in which $\\sigma (q)$ , Eq.", "(REF ), is set to 1 and the second term in Eq.", "(REF ) due to the hydrodynamic operator ${\\cal L}_2$ is dropped.", "Setting $\\sigma (q)=1$ assumes all modes have equilibrated, which clearly is not the case, so that value should be considered an upper bound.", "In Figure REF it can be seen that the temperature undershoot causes ${\\tilde{S}}(q_m)$ for the adiabatic quench to grow initially more rapidly than the isothermal quench.", "Interestingly, ${\\tilde{S}}(q_m)$ for the adiabatic quench in Figure REF never differed from the peak height of the other two adiabatic quenches in Figure REF (not shown) by more than about $5\\%$ at late times even though there is a spread of two orders of magnitude in their final temperatures.", "This weak violation of scaling is caused by the weak divergence ($\\alpha =0.105$ ) of $\\alpha _{p}$ and $C_P$ .", "On the other hand, ${\\tilde{S}}(q_m)$ for the isothermal quench differs by at least a factor of 2 from the adiabatic runs.", "At very early times, the LBM prediction is greater than either version of the KO theory, due presumably to the overestimation of the transport function $\\sigma (q)$ .", "At later times, LBM lags appreciably behind KO as expected.", "Figure: Scaled structure factor peak S ˜(q m ){\\tilde{S}}(q_m)as a function of scaled time τ\\tau for an adiabatic and isothermal runof the KO theory.", "Also shown are predictions of an isothermal run from anLBM version of the theory.The adiabatic run is the same as the middle one in Fig.", ": ΔT f =-0.4\\Delta T_f = -0.4 mK and ϵ i /ϵ f =-10\\epsilon _i/\\epsilon _f=-10; the isothermalruns have the same ratio of initial to final temperature.", "Results foradiabatic runs for other temperature shown in Fig.", "gavepeak values that differed at most by 5%5\\% from the ΔT f =-0.4\\Delta T_f=-0.4 mK runhere.Figure: Scaled peak wavevector q m q_m as a function of scaled time τ\\tau for an adiabatic and isothermal run of the KO theory.", "Also shown arepredictions of an isothermal run from an LBM version of the theory.The conditions are the same as for the results shown in Fig..Results for the other adiabatic runs in Fig.", "were essentiallyidentical to the one shown here.In Figure REF it can bee seen that, surprisingly, the adiabatic and isothermal quenches give about the same value for the initial peak wavevector.", "This is because quenching to a lower temperature causes the fluid to coarsen at a faster rate (the characteristic decomposition time $\\tau \\sim \\xi ^3$ ), with eventually the adiabatic runs giving a slightly smaller $q_m$ .", "Thus, one may not be able to determine any great discrepancy between the isothermal theory and experiment if one looks only at the peak wavevector.", "While KO, like LBM, was created to describe the early stage of decomposition, it is interesting to examine its behavior at later times.", "Define time dependent exponents, $a_q$ and $a_s$ , so that $q_m\\sim \\tau ^{-a_q}$ and $S(q_m)\\sim \\tau ^{a_s}$ at any $\\tau $ .", "For $q_m$ , $a_q$ increases monotonically at early times, but appears to approach a constant for $\\tau > 100$ .", "It was found that in the isothermal case KO and LBM predict that $a_q \\approx 0.47$ and 0.22, respectively, for $100<\\tau \\le 1000$ .", "At larger times, $10^3< \\tau < 10^4$ , the KO value decreases slightly to $0.46$ .", "The exponent changed only slightly with cut-off, $a_q\\approx 0.47$ and 0.46, for $\\alpha ^\\ast =1$ and $\\alpha ^\\ast =\\pi /2$ , respectively, for the largest times examined, $10^3< \\tau \\le 10^4$ .", "Note that at $\\tau \\approx 10^4$ , $q_m\\approx 0.02$ .", "In absolute terms, $q_m$ varied less than 5% with cut-off out to $\\tau =100$ .", "As mentioned above, the KO theory applies to fluid flow at low Reynolds number, that is, when the viscous term in the Navier-Stokes equation is much larger than the inertial one.", "In this limit, it is expected that the dominant mechanism in the very late stages of coarsening yields $a_q = 1$ .", "[41] As a consequence, the KO value for $a_q$ can then at most be considered valid for an intermediate stage of phase separation.", "Interestingly though, a value of $a_q=0.5$ has been predicted for the late stages for the opposite case of the inertial term dominating.", "[42] But whether the agreement between this prediction and KO's is more than just happenstance will require more analysis.", "The effective time exponent of ${\\tilde{S}}(q_m)$ , $a_s$ , also increases at early times; however at $\\tau \\approx 100$ for KO it reached a maximum of 1.8 and then dropped slowly, reaching a value of 1.46 at the largest times examined, $\\tau \\approx 10^4$ .", "On the other hand, experiments have shown that $a_s$ increases monotonically with time, eventually approaching a constant.", "[8], [9] So this slowing in the growth of ${\\tilde{S}}(q_m)$ seems to indicate a gradual breaking down of the theory.", "The peak height was more sensitive to the cut-off with the maximum of $a_s$ for KO being 2.1 and 1.7 for $\\alpha ^\\ast =1$ and $\\pi /2$ , respectively.", "On the other hand, this maximum for KO always occurred when $q_m\\approx 0.2$ .", "(For LBM it occurs for $q_m\\approx 0.35$ .)", "This value of $q_m$ corresponds to an average fluctuation size of $\\pi \\xi _f/q_m\\approx 16\\xi _f$ .", "In Cahn-Hilliard theory [30], the equilibrium interface separating two phases has a width of around $4\\xi _f$ .", "Thus, at this time sharp interfaces will be forming, which the LBM and thus KO theories cannot describe.", "[6] At $\\tau = 100$ , ${\\tilde{S}}(q_m)$ for the isothermal KO theory was 458, 285 and 239 for $\\alpha ^\\ast = 1$ , 1.4 and $\\pi /2$ , respectively, so the cut-off dependence of the theory seems to decrease as the cut-off is increased.", "Scaling theory[43], [44] predicts at late times that the function $F(x) = q_m^3{\\tilde{S}}(q = x q_m)$ becomes constant.", "Interestingly, the peak of this function is almost a constant within the KO theory: $F(1)\\sim \\tau ^{\\zeta _F}$ , with $\\zeta _F \\approx 0.07$ at late times.", "This trend of the theory persists at least out to $\\tau \\approx 10^4$ ." ], [ "Comparison With Experiment", "In this section the adiabatic and isothermal theories will be compared with light scattering data of BC[11].", "Figure: Scaled structure factor S ˜(q){\\tilde{S}}(q) as a functionof scaled wavevector qq at scaled times τ=1,5\\tau = 1,5 and 20.The initial temperature for allquenches was T c ϵ i T_c\\epsilon _i= 10 mK.", "The symbols denote theexperimental results of Bailey and Cannell as follows:T c ϵ f T_c\\epsilon _f = (△)(\\bigtriangleup ) -0.116 mK,(◯)(\\bigcirc ) -0.210 mK, (□)(\\Box ) -0.538 mK, (×)(\\times )-1.036 mK, (•)(\\bullet ) -2.079 mK,(▽)(\\bigtriangledown ) -5.156 mK, and (⋄)(\\diamond ) -10.37 mK.The solid and dashed curves denote results of the adiabaticand isothermal theories, respectively.", "For the isothermal runs for each time,the upper curve denotes results for the deepest quench shown inthat time frame, while the lower one is for the shallowest.", "This meaningalso holds for the adiabatic runs for τ=1\\tau =1 and 5, but the reverseis true for the largest time, τ=20\\tau =20.Figure: Scaled structure factor S ˜(q){\\tilde{S}}(q) as a functionof scaled wavevector qq at scaled times τ=50\\tau = 50 and 100.The meaning of the symbols and curves and conditions are thesame as in Fig.", "for τ=20\\tau =20.As stated above, in the experiments of BC the decomposition occurs adiabatically.", "The quenches were for on-critical mixtures so $x_0=0$ .", "To compare the adiabatic theory with experiment the thermodynamic quantities, $T_c,{d T_c\\over dP},\\rho _c,\\alpha _{p},C_P$ and the two-phase values for $\\xi _0$ and $\\eta _s$ , are needed.", "$T_c,{d T_c\\over dP},\\rho _c,\\alpha _{p}$ and $C_P$ can be found using Tables REF and REF .", "With these tables it can be deduced that $\\xi _0^-=1.13Å$ .", "As mentioned above, the hydrodynamic shear viscosity, $\\eta _s$ , is not constant but is a singular function of $\\epsilon $ .", "In addition, the two-phase value of $\\eta _s$ has not been determined, and at present there is no definite relation between the one and two-phase amplitudes.", "However, since the scaling form for $\\eta _s$ is so weakly singular and the quenches are expected to be fast, $\\eta _s$ was simply set equal to its one-phase value at $\\xi =\\xi _f$ , i.e., $\\eta _s\\simeq {\\bar{\\eta }}(Q_0\\xi _f)^{z_\\eta }$ , where $\\bar{\\eta }$ , $Q_0$ and $z_\\eta $ are given in Table REF .", "For each quench, the initial temperature, $\\epsilon _i$ , and the pressure change, $\\Delta P$ , are known.", "The final temperature, $\\epsilon _f$ , is determined by integrating $\\Bigl ({\\partial \\epsilon \\over \\partial P}\\Bigr )_S$ , which is given by Eq.", "(REF ).", "Re-evaluation of the critical properties of 3MP+NE by BC allows us to ignore any uncertainty in $\\epsilon _f$ .", "The experimental intensity data was scaled by BC[15].", "In Figures REF and REF , the scaled structure factor ${\\tilde{S}}(q)$ is shown as a function of the scaled wavevector $q$ for various scaled times $\\tau $ .", "The quenches shown all begin at $T_c\\epsilon _i\\approx 10$ mK and have final temperatures that range from $T_c\\epsilon _f= -0.116$ mK to $-10.37$ mK.", "The solid and dashed curves denote results for the adiabatic and isothermal theories, respectively, while the points represent data of BC.", "For the isothermal runs for each time, the uppermost curve denotes the result for the deepest quench shown and the lower one denotes the shallowest.", "This meaning also holds for the adiabatics runs for $\\tau =1$ and 5, but the reverse becomes true for larger times.", "The effect of a finite quench time is included in these results; the scaled quench time $\\tau _{quench}$ ranged from $2\\times 10^{-3}$ for the shallowest quench to 11 for the deepest.", "At a very early time, $\\tau =1$ , the prediction of the adiabatic theory for the peak height ${\\tilde{S}}(q_m)$ lags behind the data by a factor of 2.", "However, at later times, $\\tau =20$ and 50, the agreement with experiment is very good, within $20\\%$ .", "At the largest time for which data is available, $\\tau =100$ , the adiabatic theory appears to start lagging behind the data again, with the difference being $30\\%$ .", "On the other hand, the predictions of the adiabatic theory for the peak wavevector $q_m$ are within a few percent of the data at all times.", "The LBM and thus KO theories are expected to work best at early times.", "For example, Mainville et al.", "obtained good agreement throughout the early stage, albeit with some fitting, between their experimental scattering data and the LBM theory.", "[7] Therefore, the disagreement between the adiabatic theory and experiment for the peak height ${\\tilde{S}}(q_m)$ at very early times is perplexing.", "One possibility is that setting the cut-off to a finite value removes the relaxation of high wavevector modes, $q>\\alpha ^\\ast $ , right after the quench.", "The relaxation of these modes then couple to lower $q$ ones, increasing their relaxation, like a wave moving through $q$ -space.", "To examine this hypothesis, the cut-off of the theory was varied from $\\alpha ^\\ast =1$ to $\\alpha ^\\ast = \\pi /2$ .", "It was found that ${\\tilde{S}}(q_m)$ at $\\tau =1$ varied by only 9% and 3% for the adiabatic and isothermal theories, respectively, for this cut-off range.", "Another possible explanation is that the adiabatic theory has underestimated the temperature undershoot.", "Though whatever the cause, further research is needed.", "The disagreement between theory and experiment at late times, is that at $\\tau =50$ , $q_m\\simeq 0.2$ , which, as has been discussed above, appears to be where the KO theory begins to break down.", "The isothermal theory predicts a peak height and peak wavevector that lags behind experiment at all times, the difference in ${\\tilde{S}}(q_m)$ becoming over a factor of three at the latest times.", "Note that the theory results shown in Figures REF and REF are not quite the same as those in a previous description of the adiabatic theory, Ref.[13].", "One reason is that in Ref.", "[13] the “double gaussian\" approximation was used to solve for the time evolution of $\\rho _1(y)$ .", "As mentioned above, while this approximation is more computationally efficient than solving the full PDE for $\\rho _1(y)$ , Eq.", "(REF ), it tends to overestimate the growth of ${\\tilde{S}}(q)$ for times such that $q_m\\ge 0.3$ .", "Advances in computer power in the years since Ref.", "[13] was published have made this approximation unnecessary.", "A second reason is that the form for the scaled free energy density $\\phi (x)$ was different in the previous work.", "This previous form was constructed to satisfy a constraint in the limit of the cut-off $\\alpha ^\\ast \\rightarrow 0$ (see Ref.", "[26] for a detailed description).", "It was subsequently concluded that the added complexity to $\\phi (x)$ needed for this constraint outweighed any improved accuracy of the theory, and so here the standard “$\\varphi ^4$ \" form for $\\phi (x)$ was used instead.", "As mentioned above, the isothermal theory predicts that if the fluid is in the critical region, the experimental data is scaled properly, and the scaled initial conditions and quench times are the same, then the scaled time evolution of any experimental run should be identical.", "It is interesting then whether the experimental data of BC show any violation of this scaling.", "Consider two experimental runs of BC with final temperatures $T_c\\epsilon _f=-2.079$ mK and -0.202 mK.", "[15] The initial temperatures were both at $\\epsilon _i \\simeq 5{\\vert \\epsilon _f\\vert }$ to eliminate the effect of initial conditions.The times for these quenches were estimated from other data by assuming that the rate of change of the pressure was the same for all quenches.", "At $\\tau =10$ , ${\\tilde{S}}(q_m)$ was measured to be 16.6 and 18.8 for the first (-2.079 mK) and second (-0.202 mK) quench, respectively.", "The adiabatic theory predicts that ${\\tilde{S}}(q_m)=11.4$ and 12.3, for the first and second quench, respectively, while the isothermal theory predicts that ${\\tilde{S}}(q_m)=7.6$ and 7.7 for those quenches.", "Both the adiabatic and isothermal theories predict that $q_m\\approx 0.47$ in agreement with both experimental runs.", "While the experimental violation of scaling is not large, the difference in ${\\tilde{S}}(q_m)$ for the two runs being 12%, the trend is in agreement with the adiabatic theory, which predicts a difference of 8%.", "So though there is certainly scatter in the data, this agreement at the least is suggestive evidence that the temperature change during decomposition is appreciable for this fluid." ], [ "Summary and Discussion", "In summary, the KO-LBM theory of spinodal decomposition in binary fluids was generalized to model experimental scenarios in which the fluid is quenched by changing the pressure and the subsequent phase separation occurs adiabatically.", "The central idea of the approach here was that the coarse-grained free energy, ${\\cal F}[c]$ , which governs the time evolution of the slowest modes, is constructed in a manner that creates a natural split in the degrees of freedom of the system.", "Those fast degrees of freedom that have been integrated out contribute to ${\\cal F}[c]$ , but also to a free energy, $F_r$ , that is independent of the configuration $[c]$ of the slow modes.", "It was shown that the fast degrees of freedom, through $F_r$ , are able to act as a thermal reservoir for the slow modes.", "Any global constraint though, such as constant energy or entropy, indirectly relates the state of the reservoir, and thus its temperature, to the particular state $[c]$ of the slow modes.", "However, it was argued that these states need be related only in an average sense.", "With that approximation, an equation of motion for the average reservoir temperature was derived, it playing the same role as the assumed constant global system temperature in previous isothermal theories of decomposition.", "The extension of the isothermal theories of KO and LBM to adiabatic conditions then consisted of: this equation of motion for the reservoir temperature; estimates for various reservoir thermodynamic derivatives, such as the heat capacity, which appear in the temperature equation; and a specification of a temperature dependent coarse-grained free energy.", "This “adiabatic” theory was then applied to an on-critical mixture of 3MP+NE.", "It was shown that the temperature change during decomposition is appreciable and accelerates the coarsening.", "The adiabatic and previous isothermal theories were then compared quantitatively, with no adjustable parameters, with data of Bailey and Cannell on 3MP+NE for the structure factor at various times during the early stage of decomposition.", "It was shown that there is a definite lack of agreement between the data and previous theory for the structure factor peak height, and that the adiabatic theory accounts for a substantial amount of this difference.", "The adiabatic theory also improves the agreement with experiment for the wavevector, $q_m$ , of the structure factor peak.", "Differences between theory and experiment though indicate that the adiabatic theory may still be underestimating the effects of temperature changes during decomposition for 3MP+NE.", "Further research is needed to determine the cause.", "The large temperature change during decomposition predicted for 3MP+NE is due partly to the size of the singular term in the isobaric heat capacity compared to the background term (see Table REF ).", "Another binary fluid, isobutyric acid and water, has a much smaller singular term, and for it, at the same reduced temperatures as the Bailey and Cannell experiments, the predictions of the adiabatic and isothermal decomposition theories are essentially the same.", "[26] It is possible though that there are other binary fluids with even larger relative singular contributions to their heat capacity than 3MP+NE, making the adiabatic effect even more pronounced.", "The adiabatic theory could possibly be extended to temperatures outside the critical region, e.g., for mixtures with longer range interactions such as polymers.", "However, if 3-D Ising critical scaling no longer holds, the isothermal KO/LBM theory itself becomes temperature dependent through at least the parameter $f_1$ (see Sec.", "REF ).", "So it is unclear how the predictions of this extended adiabatic theory would differ from the near-critical one developed here, especially as thermodynamic quantities such as the heat capacity are system dependent.", "The behavior of the isothermal KO theory at later times was also analyzed.", "It was found for times $10^2 < \\tau \\le 10^4$ , that $q_m$ scaled as $\\tau ^{-a_q}$ , with $a_q\\approx 0.46$ .", "While off-critical quenches were not examined here, it is expected that the adiabatic effect to be less for them since the heat released during phase separation should be largest for an on-critical mixture, it being roughly proportional to $1-x_0^2$ , using Eq.", "(REF ).", "Interestingly, it was shown in Section REF that the entropy increase during this adiabatic decomposition is well approximated as zero.", "That is, in the model here, the temperature rise from phase separation exactly compensates for the temperature undershoot caused by the incomplete relaxation of the slow modes during the quench, so that the final temperature reached is as if the whole process had been reversible.", "In that manner, if a fluid were quenched, allowed to phase separate at least partially, and then the pressure were reversed, the fluid upon re-mixing should reach a temperature very near its initial value.", "A similar two-step experiment was done by Siebert and Knobler in their study of nucleation.", "[46], [3] While the arguments leading to this prediction of a (almost) constant entropy decomposition relied partly on the system being a near-critical binary fluid, it might be more general.", "Answers are left to future research.", "Last, it is noted that more recent explorations of the effective temperature concept have focused on amorphous substances such as foams and glasses driven mechanically and continuously out of equilibrium.", "[47], [48].", "I thank James Langer for many helpful discussions, and Arthur Bailey and David Cannell for suggesting this problem and many discussions.", "I also thank John McCoy for some discussions of thermodynamic fundamentals.", "This work was done primarily at U.C.", "Santa Barbara." ] ]
1403.0051
[ [ "On Flux Rope Stability and Atmospheric Stratification in Models of\n Coronal Mass Ejections Triggered by Flux Emergence" ], [ "Abstract Flux emergence is widely recognized to play an important role in the initiation of coronal mass ejections.", "The Chen-Shibata (2000) model, which addresses the connection between emerging flux and flux rope eruptions, can be implemented numerically to study how emerging flux through the photosphere can impact the eruption of a pre-existing coronal flux rope.", "The model's sensitivity to the initial conditions and reconnection micro-physics is investigated with a parameter study.", "In particular, we aim to understand the stability of the coronal flux rope in the context of X-point collapse and the effects of boundary driving in both unstratified and stratified atmospheres.", "In the absence of driving, we assess the behavior of waves in the vicinity of the X-point.", "With boundary driving applied, we study the effects of reconnection micro-physics and atmospheric stratification on the eruption.", "We find that the Chen-Shibata equilibrium can be unstable to an X-point collapse even in the absence of driving due to wave accumulation at the X-point.", "However, the equilibrium can be stabilized by reducing the compressibility of the plasma, which allows small-amplitude waves to pass through the X-point without accumulation.", "Simulations with the photospheric boundary driving evaluate the impact of reconnection micro-physics and atmospheric stratification on the resulting dynamics: we show the evolution of the system to be determined primarily by the structure of the global magnetic fields with little sensitivity to the micro-physics of magnetic reconnection; and in a stratified atmosphere, we identify a novel mechanism for producing quasi-periodic behavior at the reconnection site behind a rising flux rope as a possible explanation of similar phenomena observed in solar and stellar flares." ], [ "Introduction", "Coronal mass ejections (CMEs) are a common occurrence in the Sun's atmosphere that are known to release giga-tons of plasma into interplanetary space.", "Some of the ejected plasma can reach the space environment of the Earth and have a strong and complex influence on space activity by inducing geospace disruptions that can severely impact spacecraft, power grids, and communication [1].", "While CMEs are quite commonly observed [11], especially during the peak of the solar cycle, they are still poorly understood.", "Some of the biggest CME mysteries pertain to their origin, propagation, and relation to flares.", "The initiation of CMEs has been widely studied and yet remains largely unexplained [14], [4].", "However, many observational studies of associated features have led to clues about how they occur and what factors contribute to their destabilization [15].", "Prior to an eruption, large-scale shear motions are often observed in photospheric images, especially about the magnetic neutral line [18] and in the form of sunspot rotations [44].", "In addition, patches of magnetic flux are found to emerge, expand, move, fragment, coalesce, and cancel over a wide range of length and time scales [40], [46], [2], [38].", "It is believed that shear motions, sunspot rotation, and the emergence of new flux are all related to the injection of magnetic helicity into coronal magnetic structures that could be directly involved in the eruption [3], [19], [8], [37], [29].", "In addition to the growing body of observational studies that have improved our understanding of CMEs, many new insights have also emerged from theoretical and numerical efforts.", "CMEs have been modeled in two and three dimensions using both simple analytical methods and sophisticated magnetohydrodynamic simulations [16].", "These models differ widely in physical and numerical details, each making its own choice of how to address the trade-off between complexity and computational feasibility.", "Early theoretical models explained CMEs as a loss of equilibrium, due to magnetic buoyant instabilities [45], [25], [9], as well as MHD flows [26] and reconnection [12].", "[13] proposed a CME model based on the movement of magnetic footpoints (sources) below a flux rope and the subsequent development of a singular current sheet, through which a large magnetic energy release should take place as the flux rope moves continually outwards.", "[22] refined their model and computed exact solutions for the energy release, flux rope height, current sheet length, and reconnection rate.", "The Lin & Forbes (hereafter, “LF”) model, while simplistic, provides an important step forward in CME modeling because it offers exact solutions to the time-dependent nonlinear problem of a flux rope eruption and includes more than a heuristic treatment of magnetic reconnection.", "Furthermore, it predicts many features (e.g., morphology, current sheet, post-flare loops, flows, energetics) confirmed by observations [7], [17], [23].", "A similar two-dimensional flux rope model was proposed by [5].", "Like LF, the Chen & Shibata (“CS”) model consists of a two-dimensional configuration in which a flux rope sits above the photosphere, surrounded by a line-tied coronal arcade.", "In both models, the magnetic equilibrium is destabilized by photospheric driving, causing a current sheet to form in the flux rope's wake as it moves outwards.", "However, whereas the LF model calls for a somewhat manufactured mechanism for destabilization via large-scale convergence of the sources, the CS model improves upon the LF model by incorporating flux emergence as the driver.", "While it does not lend itself to a purely analytical treatment, the CS model is suitable for numerical simulation.", "The authors report four very different outcomes based on the position and direction of the driving, showing that the location of the emergence per se is not a critical factor for destabilizing the coronal flux rope but rather that the relative orientation of the emerging flux determines whether the flux rope moves outwards/upwards (CME-like) or inwards/downwards (failed eruption).", "Several subsequent studies have built upon the CS model.", "For example, [6], [43], and [41] produced synthetic emission images from CS simulations to compare morphological features, reconnection in-flows, and coronal dimmings found in actual CME observations.", "Moreover, [43] and [42] were able to identify the formation, structure, and location of slow and fast shocks in the CMEs produced in these simulations.", "Gravitational density stratification in an isothermal atmosphere was considered by [6], [41] and also in a later study by [10] in spherical coordinates with axisymmetry.", "In this study, we re-examine the CS model using a more sophisticated numerical tool, a more realistic atmosphere, and higher spatial resolution than previous studies.", "Simulations are performed using a high-order spectral element method with numerically accurate, self-consistent treatments of diffusive transport (i.e., resistivity, viscosity, and thermal conduction).", "In addition, we reformulate the initial conditions to have magnetic fields that are everywhere continuous and differentiable, and to include a solar-like temperature profile with a sharp transition region and density stratification.", "Through an exploration of physical parameters, we find that the CS magnetic equilibrium can be unstable even without a flux emergence driver.", "Linear theory has shown that sufficient perturbation of the field lines near an X-point by waves or motion can disrupt the balance between magnetic pressure and magnetic tension, causing the X-point to collapse and form a reconnecting current sheet [39].", "Our simulations demonstrate that under a wide range of conditions the CS equilibrium is susceptible to such a collapse via nonlinear accumulation of fast magnetosonic waves at the X-point [31].", "However, we also show that in a sufficiently incompressible plasma due, for example, to the presence of a background \"guide\" magnetic field co-aligned with the axis of the flux rope, the X-point collapse does not take place and the CS magnetic equilibrium can be stabilized.", "For both stable and unstable configurations, we investigate the impact of the resistivity model enabling magnetic reconnection below the flux rope, as well as the plasma parameters in the low solar atmosphere, on the flux rope's response to the flux emergence driver.", "We show that flux emergence can produce a rising flux rope both in a stratified and an unstratified atmosphere, though the resulting ejection speed, as well as the plasma dynamics around the X-point, can be strongly effected by the magnitude of the guide field and the atmospheric stratification." ], [ "Model", "The CS model has a two-dimensional domain with motion and magnetic field allowed perpendicular (as well as parallel) to the plane of the domain ($\\mathbf {v}, \\mathbf {B} \\in \\mathbb {R}^3$ ).", "Therefore, we can write the magnetic field, normalized to some value $B_0$ , in terms of a scalar potential $\\psi $ representing the in-plane flux, and an out-of-plane scalar field: $\\mathbf {B}(x,y;t) = \\nabla (-\\psi ) \\times \\hat{\\mathbf {e}}_z + b_z \\hat{\\mathbf {e}}_z$ All quantities are normalized in terms of the first three constants found in Table 1: $L_0 = 5$  Mm, which is the unit of length; $B_0=10$  G, the unit of magnetic field strength; and $N_0 = 10^9$  cm$^{-3}$ , the unit of number density.", "Given the Alfén velocity $v_A \\equiv B_0/\\sqrt{\\mu _0 m_p N_0}$ , where $m_p$ is the proton mass, we define the unit time as $\\tau \\equiv L_0/v_A$ , unit temperature as $T_0 \\equiv B_0^2/(\\mu _0 k_B N_0)$ , and unit pressure as $P_0 \\equiv B_0^2/\\mu _0$ .", "The solar surface gravity $g_S = 274$  m/s$^2$ is similarly normalized as $g \\equiv g_S (\\tau /v_A) = 2.88 \\cdot 10^{-3}$ .", "Due to symmetries intrinsic to the model, only half of the domain in the horizontal direction has to be resolved ($x>0$ ).", "Thus, simulations are performed in a computational domain $(x,y)\\in [0,L_x]\\times [0,L_y]$ , with the solar convection zone assumed to be located below the domain ($y<0$ ).", "Table: Normalization constants" ], [ "Magnetic configuration", "The initial magnetic configuration prescribed in the CS model consists of a coronal flux rope of radius $r_0$ surrounded by an arcade of “loops” that are line-tied in the photosphere.", "The flux rope contains a current channel that is mirrored by an image current far below the photosphere (outside the computational domain), and four line currents produce a potential quadrupolar field just below the photosphere.", "Since the bottom boundary of the numerical domain coincides with the photosphere, the only visible current initially is that within the flux rope.", "In the original CS study, the coronal flux rope is given by a flux function $\\psi _l$ that results in a discontinuous current density at the edge of the flux rope.", "Therefore, we propose the following alternative: l =   r22 r0 - (r2-r02)24 r03 , $r\\le r_0$   r02 - r0 r0 + r0 r , $r > r_0$ where $r^2 = x^2 + y^2$ , and the center of the flux rope lies at ($x=0$ , $y=h$ ).", "Our formulation for $\\psi _l$ lends itself to a continuous current density: $j_l = - \\nabla ^2 \\psi _l = \\left\\lbrace \\begin{array}{l @{\\hspace{17.07164pt}} c}\\dfrac{4 r^2}{r_0^3} - \\dfrac{4}{r_0} \\ , & r\\le r_0 \\\\[4mm]0 \\ , & r > r_0\\end{array}\\right.$ The other flux components of the initial configuration, representing the image current and line currents, respectively, are kept as originally defined: $\\vphantom{\\int } \\psi _i = -\\frac{r_0}{2} \\ln \\left[x^2 + (y+h)^2 \\right]\\\\[2pt]\\psi _b = c\\,\\ln \\frac{\\left[(x+0.3)^2 + (y+0.3)^2\\right]\\left[(x-0.3)^2 + (y+0.3)^2\\right]}{\\left[(x+1.5)^2 + (y+0.3)^2\\right]\\left[(x-1.5)^2 + (y+0.3)^2\\right]}$ with $r_0=0.5$ , $h=2$ and $c=0.25628$ .", "All three flux functions are summed to produce the initial magnetic equilibrium, shown in Fig.", "REF : $\\psi = \\psi _l + \\psi _i + \\psi _b \\ .$ Figure: Contours of ψ\\psi in the initial conditions.In addition to the line currents, which produce the in-plane magnetic field, we also allow for a uniform background magnetic field out of the plane $b_{z0}\\,\\hat{\\mathbf {e}}_z$ .", "This “guide” field contributes magnetic pressure but no current.", "Figure: Contours of ψ\\psi (black) and b z b_z (magenta) at t=0t=0 for b z b_z as a function of rr (left) and as a function of ψ\\psi (right).In the original CS study, a density spike is applied to support the flux rope against radial compression: an outward-acting pressure gradient force offsets the inward-acting Lorentz force due to the flux rope's poloidal field.", "Equivalently, the flux rope can be supported against the radial Lorentz force by magnetic pressure, as in Shiota et al.", "(2005): in addition to the background guide field $b_{z0}$ , we apply an additional axial field in the flux rope which is highest in the center and diminishes over a radius of $r_0$ .", "We note that if the axial field $b_z$ is specified as a function of the flux rope radius alone the flux rope will not be force-free, as the contours of $\\psi $ are not perfectly circular due to the small but finite contributions of $\\psi _b$ and, to a lesser extent, of $\\psi _i$ to the total flux in the coronal flux rope.", "It can be seen from the left panel of Fig.", "REF that, given such a function $b_z(r)$ , the contours of $\\psi $ and $b_z$ would not be well-aligned.", "To avoid the misalignment and minimize unbalanced Lorentz forces in the initial condition, we instead choose to specify $b_z$ as a function of $\\psi $ , as follows: $& b_z = \\left\\lbrace \\begin{array}{l @{\\hspace{17.07164pt}} c}\\sqrt{b_{z0}^2 + \\dfrac{10}{3} - 8 \\left(\\dfrac{\\zeta }{r_0}\\right)^2 + 6 \\left(\\dfrac{\\zeta }{r_0}\\right)^4 -\\dfrac{4}{3} \\left(\\dfrac{\\zeta }{r_0}\\right)^6} \\, & \\zeta \\le r_0 \\\\[3mm]b_{z0} \\ , & \\zeta > r_0\\end{array}\\right.\\\\& \\zeta ^2(\\psi ) = 2r_0^2 - \\sqrt{3 r_0^4 - 4 r_0^3 (\\psi - \\psi _0)} \\ .$ (The derivation of the above equations can be found in Appendix A.)", "Note that $\\zeta =0$ when $\\psi = \\psi _0 - r_0/4$ , and $\\zeta =r_0$ when $\\psi = \\psi _0 + r_0/2$ , where $\\psi _0 \\equiv \\left[\\psi _i + \\psi _b\\right] |_{(x,y)=(0,h)}$ ." ], [ "Unstratified atmosphere", "In the case of an unstratified atmosphere, the number density field is initialized to a uniform value of $n = n_0 = 1 (N_0)$ .", "The pressure field of an electron-proton plasma can be determined by the following equation of state: $p = 2 n T$ We choose a uniform initial temperature, so the initial pressure $p=p_0$ is also uniform.", "The free parameter $p_0$ is chosen variably in the simulations to yield temperatures close to coronal values, as well as low plasma $\\beta \\equiv 2p/B^2$ .", "An unstratified atmosphere has the advantage of isolating the flux rope dynamics from the thermodynamics.", "By controlling $p_0$ , one essentially explores different regimes of the plasma $\\beta $ ." ], [ "Stratified atmosphere", "We also attempt to simulate a solar-like atmosphere by modeling the average vertical temperature profile as a hyperbolic tangent function, as in [20], [21]: $T(y) = \\frac{T_p}{T_0} + \\left(\\frac{T_c-T_p}{2 T_0}\\right)\\left[1+\\tanh \\left(\\frac{y-y_\\text{\\tiny TR}/L_0}{\\Delta y/L_0}\\right)\\right]$ with photospheric temperature $T_p=5000$ K, coronal temperature $T_c=10^6$ K, transition region height $y_\\text{\\tiny TR}=2.5$ Mm, and transition region width $\\Delta y = 0.5$ Mm.", "Given this temperature profile, we seek compatible density and pressure profiles such that the plasma is in hydrostatic equilibrium: $\\frac{dp}{dy} + n g = 0$ with the constant gravitational acceleration $g$ pointed in the $-\\hat{\\mathbf {e}}_y$ direction.", "We solve (REF ) using (REF ) and (REF ) (see the derivation in Appendix B).", "The resulting pressure profile is: $\\begin{split}p(y) = &p_0 \\exp \\left\\lbrace \\frac{g\\Delta y/L_0}{2 T_c/T_0} \\left[ - \\frac{y-y_\\text{\\tiny TR}/L_0}{\\Delta y/L_0}\\right.\\right.\\\\&\\left.\\left.+ \\frac{T_c - T_p}{2 T_p}\\ln \\left(\\frac{T_p}{T_0} \\exp \\left[- \\frac{2(y-y_\\text{\\tiny TR}/L_0)}{\\Delta y/L_0}\\right] + \\frac{T_c}{T_0} \\right) \\right] \\right\\rbrace .\\end{split}$ Here the parameter $p_0$ corresponds to a constant of integration that shifts the entire pressure profile of the atmosphere.", "As in the unstratified case, this affects the values of $\\beta $ , which should be high ($\\sim $  10) in the photosphere and low ($\\sim $  $10^{-2}$ ) in the corona.", "Therefore, we do not vary $p_0$ for the stratified simulations.", "Figure: Logarithm (base 10) of normalized number density nn (green), normalized pressure pp (blue), and temperature TT (red) as a function of height in the initial conditions for a stratified atmosphere.Profiles of the initial density, pressure, and temperature in a stratified atmosphere are plotted in Fig.", "REF .", "It is evident that all three quantities vary smoothly and by multiple orders of magnitude.", "Furthermore, this transition occurs well below the height of the flux rope core ($y=2$ )." ], [ "Numerical method", "We implement this initial configuration in the high-fidelity numerical simulation framework, HiFi, which makes use of high-order spectral elements and implicit time-stepping [27], [28].", "As a strong condition, HiFi requires all variables to be represented by continuous functions in the initial and boundary conditions.", "Therefore, the magnetic field must be everywhere differentiable, implying $\\psi \\in \\mathcal {C}^2$ .", "In addition, the boundary driving of flux needs to be differentiable in both space and time, in order for the electric and magnetic fields to be smooth.", "These conditions are well satisfied by the initial conditions described above.", "In this work, HiFi is used to integrate in time the following equations of visco-resistive MHD: $\\frac{\\partial n}{\\partial t} + \\nabla \\cdot \\left( n \\mathbf {v} \\right) = 0\\\\\\frac{\\partial (-\\psi )}{\\partial t} = - \\mathbf {v} \\times \\mathbf {B} + \\eta j_z\\\\\\frac{\\partial b_z}{\\partial t} + \\nabla \\cdot \\left( b_z \\mathbf {v} - v_z \\mathbf {B} \\right) = \\nabla \\cdot \\left( \\eta \\nabla b_z \\right)\\\\\\frac{\\partial n \\mathbf {v}}{\\partial t} + \\nabla \\cdot \\left\\lbrace n \\mathbf {v} \\mathbf {v} + p \\mathbf {I} - \\mu n \\left[\\nabla \\mathbf {v} + \\left(\\nabla \\mathbf {v} \\right)^T\\right]\\right\\rbrace = \\mathbf {j} \\times \\mathbf {B}\\\\\\frac{3}{2} \\frac{\\partial p}{\\partial t} + \\nabla \\cdot \\left( \\frac{5}{2} p \\mathbf {v} - \\kappa \\nabla T \\right) = \\mathbf {v} \\cdot \\nabla p + \\eta j^2 + \\mu n \\left[\\nabla \\mathbf {v} + \\left(\\nabla \\mathbf {v} \\right)^T\\right]:\\nabla \\mathbf {v}$ with an auxiliary equation (Ampère's law): $\\nabla \\times \\mathbf {B} = \\mathbf {j}$ The normalized transport coefficients found in Eqs.", "(REF ) – namely $\\kappa $ , $\\mu $ , $\\eta $ – control the level of dissipation of the MHD fluid quantities through molecular diffusion: temperature, velocity, and current, respectively.", "Each of these three transport parameters is chosen to be compatible with the resolution and objective of each simulation.", "Further, for some of the simulations (see below), we allow the resistivity $\\eta $ to be a function of local current density, $\\eta = \\eta _{bg} + \\eta _{anom}(\\mathbf {j})$ , where anom(j) =   0 $|\\mathbf {j}| < j_c$   anom{1 - [(|j|/jc - 1)]}2 $j_c \\le |\\mathbf {j}| \\le 2j_c$ ,   anom $ |\\mathbf {j}| > 2j_c$ $\\eta _{bg}$ is the uniform and time-independent background resistivity, and $\\eta _{anom}$ is some “anomalously enhanced\" effective resistivity, $\\bar{\\eta }_{anom} \\gg \\eta _{bg}$ , occurring due to micro-physics not captured by the MHD model whenever the current density rises above the critical current density $j_c$ ." ], [ "Boundary conditions", "The bottom boundary of the simulation domain, representing the photosphere, is perhaps the most important boundary condition affecting the outcome of a simulation.", "Flux emergence is achieved by varying the flux function at this boundary in time, which is equivalent to applying an electric field.", "This electric field determines the evolution of the magnetic field, which can be advected in or out of the domain or resistively dissipated, as described by Ohm's Law: $\\frac{\\partial \\psi }{\\partial t} = E_z = - \\hat{\\mathbf {e}}_z \\cdot \\mathbf {v} \\times \\mathbf {B} + \\eta j_z$ The resistive component, the second term on the right-hand side of (REF ), is determined by the geometry of the magnetic field at any given time.", "Therefore, by varying the flux at the boundary ($\\partial \\psi / \\partial t$ ) in a prescribed way, we also induce cross-field plasma motions ($\\mathbf {v} \\times \\mathbf {B}$ ).", "[5] prescribe two cases of localized boundary driving, namely, over a region $|x-x_0| \\le 0.3$ centered at $x_0=0$ (case A) and at $x_0=3.9$ (case B).", "We apply the same method only for the case of $x_0=0$ , but use a formulation that is smoother in time and in space: $\\begin{array}{c c}\\psi (x,0;t) = \\psi (x,0;0) + \\dfrac{\\psi _e(x)}{2} \\left[\\dfrac{t}{t_e} - \\dfrac{\\sin \\left(2 \\pi t/t_e\\right) }{2 \\pi } \\right] \\ , & t \\le t_e \\\\[5mm]\\psi _e(x) = \\dfrac{c_e}{2}\\left[ 1 + \\cos \\dfrac{\\pi (x-x_0)}{0.3}\\right] \\ , & |x|\\le 0.3\\end{array}$ where $t_e$ is the duration over which the electric field drive is applied at the boundary.", "For $t > t_e$ , the photospheric boundary is treated as a perfect conductor.", "We do not allow any in-plane flow on the bottom photospheric boundary ($v_x = 0$ , $v_y=0$ ) and force the normal gradients of $b_z, j_z$ , and temperature to be zero: i.e., $\\nabla _{\\hat{n}} \\equiv \\hat{n} \\cdot \\nabla = 0$ .", "The unstratified atmosphere also has $\\nabla _{\\hat{n}} n = 0$ , while the stratified case imposes a fixed value of the density scale height $\\partial _t\\left\\lbrace [\\nabla _{\\hat{n}} n]/n\\right\\rbrace = \\partial _t\\left\\lbrace \\nabla _{\\hat{n}}[\\ln (n)]\\right\\rbrace = 0$ .", "The left boundary is a symmetry boundary, with odd symmetry required for the horizontal and out-of-plane components of flow, $v_x$ and $v_z$ , and even symmetry imposed on all other dependent variables.", "At the outer boundaries (top and right), the gradients of density, $b_z$ , $j_z$ , and pressure are zero, and flow is only allowed in the vertical direction.", "Table REF provides a simple reference for the various boundary conditions applied in the simulations.", "The flux emergence represented by (REF ) changes the flux function just at the boundary but has no direct effect on $\\psi $ anywhere else, including just above it.", "While Ohm's Law (REF ) does relate flux evolution to fluid transport, it does not guarantee that the flux function and other quantities will be well-behaved for all time, particularly in the $y$ -direction.", "Therefore, to allow flux to slip more easily through the region just above the photospheric driving (effectively, the “chromosphere”), we apply a resistive boundary layer by enhancing $\\eta $ locally according to the following function: $\\eta = \\eta _{bg} + \\eta _{anom} + \\eta _{ph} \\exp \\left(-\\frac{y^2}{y_0^2}\\right)$ where $\\eta _{ph}$ is the photospheric value of resistivity, and $y_0=0.2$ corresponds to the height of 1 Mm above the photosphere.", "Conceptually this boundary layer emulates the enhanced resistivity of the chromosphere due to collisional impedance by neutrals [20]." ], [ "Outer corona.", "Separately, to mimic an open domain boundary with no wave reflections in the corona, a viscous boundary layer is prescribed close to the outer coronal (top and right) boundaries.", "Starting at a distance of $d=0.5$ inward from the boundary of the computational domain, $\\mu $ is increased gradually towards the domain boundary (according to a cosine profile) from a background value of $\\mu _{bg}$ up to the outer boundary value of $\\mu _{out}=1$ .", "In the same fashion, near the outer boundaries with $d=0.5$ , resistivity is ramped up from $\\eta _{bg}$ to a boundary value of $\\eta _{out}=10^{-2}$ ." ], [ "Results", "In this section, we discuss simulations of the undriven equilibrium, as well as driven simulations of flux emergence in both the stratified and unstratified cases.", "While the CS initial conditions describe an approximate equilibrium, we find that this equilibrium can be unstable to small perturbations.", "We discuss the role of MHD waves in destabilizing the flux rope via X-point collapse and the role played by plasma compressibility in stabilizing the X-point and the flux rope.", "Finally, we discuss the driven simulations of stable and unstable equilibria with different resistivity models, as well as details of the resulting eruption process.", "Figure: (No driving.)", "Four snapshots of current density j z j_z showing outward propagating MHD waves.", "Notice the trapping and interference of waves at the X-point; compounding of these waves here precipitates an X-point collapse, leading to the formation of a current sheet and initiation of magnetic reconnection." ], [ "Flux rope stability", "In [5], the coefficient $c$ (see Eq.", "above) was determined by trial and error to yield a magnetic equilibrium such that the center of the flux rope did not move “for long enough time.” (It can also be derived by requiring that the vertical component of the Lorentz force is zero at the center of the flux rope, $\\hat{y}\\cdot [\\mathbf {j}\\times \\mathbf {B}]|_{(x,y)=(0,h)}=0$ .)", "Our numerical experiments confirm that this value for $c$ is indeed appropriate for equilibrium, but we find that the equilibrium itself is tentative and unstable to perturbations.", "At the beginning of each simulation, the flux rope—which is approximately force-free—goes through a small adjustment to settle into an actual force-free equilibrium.", "In Sec.", ", we explained how formulating $b_z$ as a function of $\\psi $ forces the contours of $b_z$ and $\\psi $ to be aligned.", "Although this is an improvement on the original setup, the configuration is still not perfectly force-free because the contours of $j_z = -\\nabla ^2 \\psi = -\\nabla ^2 \\psi _l$ are still circular and therefore not aligned with the contours of $\\psi $ , producing a small but finite Lorentz force.", "The adjustment to correct the misalignment, however small it may be, is sufficient to generate waves that may destabilize the flux rope.", "Fig.", "REF illustrates the oscillation of current density induced by the fast magnetosonic waves emanating from the flux rope as it adjusts to the initial conditions.", "The distribution of the waves is not uniform because the initial adjustment is one where the flux rope is squeezed in one direction (horizontally) while expanding in the other direction (vertically).", "Therefore, the waves propagating horizontally are out of phase with those propagating vertically.", "It is interesting to note that the propagation of both types of waves is initially radial but eventually becomes oblique at the flanks of the active region due to the inhomogeneity of the magnetic pressure in the corona.", "The fast waves themselves do not directly destabilize the flux rope.", "However, the entire equilibrium can be destabilized when the waves reach the X-point below the flux rope and cause it to collapse.", "The role of fast waves in X-point collapse for a zero-$\\beta $ plasma has been studied by [31] and their behavior in the neighborhood of X-points has been investigated by similar earlier studies [32], [33], [34].", "As described in these studies, we find that the fast waves approach the X-point but tend to get trapped there if their phase speed becomes too low at the X-point.", "The trapping occurs because the waves are refracted towards the null and then wrap around it if they cannot pass through it.", "As a result, the waves push current towards the null where it accumulates exponentially in the linear regime.", "The buildup of waves at the null, however, quickly leads to nonlinear behavior, forming shocks and jets, which deform the X-point into a cusp-like geometry, which flattens and forms a current sheet [31].", "This fast-wave accumulation and resulting collapse of the X-point are evident in the sequence of figures in Fig.", "REF .", "The consequence of X-point collapse occurring as a result of fast wave accumulation at the null is that it forms a current sheet separating anti-parallel fields.", "The formation of the current sheet, in turn, kicks off magnetic reconnection that drives itself for as long as there is free magnetic energy available in the system.", "When the collapse forms a horizontal current sheet, the reconnection process draws in the flux rope from above, and it pulls itself down towards the photosphere to draw in flux from below, destroying the original configuration.", "Formation of a vertical current sheet similarly leads to a CME eruption.", "Other factors that may contribute to a collapse of the X-point in the absence of driving include boundary flows, likely related to the reflection of waves, and the asymmetric resistivity model, which intentionally biases $\\eta $ in the $y$ -direction in order to allow magnetic flux to slip through the photosphere (see previous section).", "However, we found these effects to be sub-dominant to the fast wave accumulation at the X-point in destabilizing the flux rope.", "Figure: Dependence of flux rope stability on the free parameters p 0 p_0 and b z0 b_{z0} (left), as well as on the plasma β\\beta and compressibility measure Γ\\Gamma (right).", "All simulations are performed without driving.", "The different symbols signify that the flux rope was stable (black dots); was wobbly but on average did not rise or sink (black stars); moved upwards (blue triangles); or moved downwards (red triangles)." ], [ "Sensitivity to Compressibility", "We find that sufficient magnetic pressure and/or gas pressure at the X-point can suppress the X-point collapse.", "In the absence of a guide field ($b_{z0}=0$ ), the magnetic pressure drops to zero at the X-point.", "Then, with low gas pressure, the fast wave speed is reduced drastically at the X-point causing wave refraction and accumulation.", "However, a parametric exploration of $p_0$ and $b_{z0}$ in an unstratified atmosphere revealed that increasing either of these parameters helps to stabilize the flux rope.", "The left panel of Fig.", "REF is a graphical chart of the many simulations that were performed scanning the parameter space of $p_0$ and $b_{z0}$ .", "Black dots represent simulations in which the flux rope was stable over many hundreds of Alfvén times (no X-point collapse); blue triangles represent those in which the flux rope experienced a slow rise (vertical collapse); red triangles represent those in which the flux rope descended (horizontal collapse); and black stars represent those in which the flux rope moved up and down but on average maintained the same height in the atmosphere (oscillatory X-point collapse).", "One could argue heuristically that increasing either $p$ or $b_z$ effectively decreases the compressibility of the plasma (or increases the stiffness of the medium), so any motions at the X-point need to do more work against the gas pressure or magnetic pressure to force a collapse of the magnetic topology.", "Therefore, we propose a generalized measure of two-dimensional plasma compressibility: $\\Gamma = \\frac{b_\\perp ^2}{2 p + b_{z}^2}$ and relate the free parameters of the simulations, $p_0$ and $b_{z0}$ , to the magnitude of the in-plane field $b_\\perp \\approx 1$ ($B_0$ ), as well as to the initial background plasma $\\beta $ : $\\beta = \\frac{2 p_0}{b_\\perp ^2 + b_{z0}^2}$ The right panel of Fig.", "REF provides an alternative way to assess the effect of the initial parameters on the stability of the flux rope and the X-point in terms of dimensionless quantities.", "Note that $\\beta $ and $\\Gamma $ are related to $b_{z}$ such that some combinations of the two are impossible (denoted by gray shading in the figure): $b_z^2 = \\frac{(1-\\Gamma \\beta ) b_\\perp ^2}{\\Gamma (1 + \\beta )} \\ ,$ which implies $\\Gamma \\beta \\le 1$ .", "Within the accessible parameter space we observe that above a certain level of compressibility (approximately 8, determined empirically), the X-point tends to collapse horizontally and causes the flux rope to descend.", "Within the range $4.5 < \\Gamma < 8$ , the X-point collapses vertically, causing the flux rope to move upwards out of equilibrium (though much more slowly than in a driven eruption).", "However, if the plasma is “stiffened” beyond a threshold, $\\Gamma \\lesssim 3$ , the fast waves are able to pass through the X-point as their phase speed is no longer close to zero.", "Since the waves no longer accumulate at the X-point, they do not cause it to collapse and the equilibrium is preserved.", "While it is possible that different perturbations might produce different empirical thresholds of stability, it has not been the goal of the present study to determine particular values but rather to show that the X-point stability can be fundamentally related to the accumulation of fast waves at the X-point, which can be moderated by changing the background compressibility of the plasma.", "Similarly, while the magnitude of the dissipative transport coefficients within the visco-resistive MHD simulations can have some impact on the specific stability thresholds via damping of the fast waves emanating from the flux rope, such damping does not qualitatively change the conclusion of this parameter study.", "Figure: Out-of-plane current density j z j_z (color, saturated high and low values), with magnetic flux contours (solid black), in a simulation of flux emergence into an unstratified atmosphere.", "The four panels show snapshots of the simulation at 100τ=12100\\tau = 12 min, 240τ=29240\\tau = 29 min, 480τ=58480\\tau = 58 min, and 1800τ=217.51800\\tau = 217.5 min.The premise of the CS model is that a stable pre-existing flux rope can be driven to eruption by magnetic flux emergence.", "Flux emergence is achieved through photospheric boundary driving (see Eqs.", "REF ): a small amount of flux is effectively emerged through the photospheric boundary by applying a time-dependent electric field.", "Emerging flux can cause the flux rope to move in either direction by forcing a destabilization of the X-point, similarly to the fast waves but more predictably.", "Within the underlying arcade, when the emergent flux is oppositely oriented to the local flux, it causes a vertical collapse of the X-point, leading to a rising flux rope.", "Oriented in the same sense as the local flux rope, it causes a horizontal collapse of the X-point, which forces the flux rope downwards.", "For emergence outside the arcade, likewise, it is possible to choose values for the coefficient $c_e$ in Eqs.", "REF such that the simulation results in a vertical X-point collapse, and when the sign of $c_e$ is reversed, the X-point collapses horizontally.", "However, the sign of $c_e$ must be carefully chosen based on topological and geometric considerations, including the sign of the local overlying flux.", "In this study, we restrict ourselves to discussing emergence at $x_0=0$ alone, with $c_e=1.1$ as in the original CS model.We note that the original CS reference [5] quotes $c_e=11$ and $c=2.5628$ , but these should have been quoted as a factor of 10 lower, as per personal communication with P.F. Chen.", "To evaluate the impact of reconnection micro-physics, stability of the initial condition and atmospheric stratification on the system's response to flux emergence in the CS model, the simulation study described below has been performed by changing one model parameter at a time with otherwise identical numerical and dissipation parameters.", "In the reference simulation with an unstratified atmosphere, the background magnetic “guide\" field is set to $b_{z0}=1$ , equivalent to 10 G and of the same order as $b_\\perp $ , such that the plasma compressibility measure $\\Gamma $ is less than unity and the initial configuration is stable for any plasma pressure profile.", "To minimize the impact of the size of the computational domain or the dissipative boundary layers on the results of the simulations, the domain boundaries are placed at $L_x=4$ and $L_y=10$ .", "The computational grid spanning the $(x,y)\\in [0,L_x]\\times [0,L_y]$ domain has 864 and 1536 spatial degrees of freedom in the $x$ and $y$ directions, respectively, distributed non-uniformly in such a way that the vertically elongated X-point reconnection current sheet is well-resolved in the $x$ -direction, while both magnetic and thermodynamic structures associated with flux emergence through the chromosphere can be well resolved in the $y$ -direction.", "The background resistivity throughout the domain is set to $\\eta _{bg}=10^{-5}$ , the photospheric resistivity is set to $\\eta _{ph}=10^{-2}$ , there is no anomalous resistivity $\\bar{\\eta }_{anom}=0$ , the background kinematic viscosity coefficient is set to $\\mu _{bg}=10^{-4}$ , and the heat conduction is set to $\\kappa =10^{-5}$ .", "The duration of the flux emergence is taken to be $t_e=300$ , equivalent to $36.25$  minutes.", "Figure: Unstratified atmosphere.", "Left: Height and speed of flux rope center.", "Smoothing is performed using a Hanning window of 12 points.", "The speed is computed by finite-differencing the smoothed (blue) curve.", "Right: Temperature at the X-point (center of current sheet) below the flux rope during the same period." ], [ "Flux emergence in an unstratified atmosphere", "To approximate the coronal conditions in the unstratified simulation, the initial pressure is set to $p_0=10^{-2}$ , such that the initial $\\beta $ is $\\sim 1\\%$ throughout the domain.", "To produce an eruption, the photospheric electric field drive is applied within the arcade below the X-point to generate $B_x$ opposite to the magnetic field of the arcade.", "As a result, the magnetic pressure above the photospheric boundary is reduced causing a local downflow towards the photosphere.", "This in turn reduces the plasma pressure below the X-point, which forces an in-flow at the sides of the X-point, bringing about its collapse and formation of a reconnection current sheet (e.g., see Fig.", "REF ).", "As shown in Fig.", "REF , the X-point collapse in this simulation is observed to occur at $t\\approx 100$ , forming a current sheet that reaches its maximum length and strength near $t=200$ .", "Current density then also increases along the separatrices and the field lines connected to the current sheet.", "When the new flux stops emerging ($t > t_e$ ), the current sheet persists at approximately half to a third of its peak magnitude, slowly diminishing over time for the duration of the simulation.", "As reconnection ensues, the flux rope is nudged out of equilibrium (in the $+ \\hat{\\mathbf {e}}_y$ direction) by the reconnection outflow and continues to move outwards as reconnection proceeds.", "The left panel of Fig.", "REF tracks the height of the flux rope center during the eruption by measuring the position of the magnetic O-point (black dots).", "The height measurements are smoothed (blue curve) using a Hanning window convolution over 12-point windows, and the speed (red curve) is computed by finite-differencing the smoothed height.", "The maximum speed of the flux rope is observed to be only about $0.7$  km/s, quickly slowing down further as the reconnection loses steam.", "In the right panel of Fig.", "REF , the temperature at the X-point, or the current sheet center, is plotted in mega-Kelvin showing rapid heating early in the eruption due to Joule heating at the current sheet.", "We note that this reference simulation results in a very slowly rising flux rope which is inconsistent with the original [5] simulation where the flux rope rise speed of approximately 70 km/s was observed.", "To study the sensitivity of this result to the magnitude of the background magnetic guide field and the micro-physics of reconnection at the X-point, represented here by the anomalous resistivity model similar to that of [5], a series of further simulations has been performed.", "Figure REF shows traces of the height of the flux rope center for a set of five simulations with three different values of the guide magnetic field $b_{z0}=\\lbrace 1.0,0.5,0.25\\rbrace $ and two resistivity models, one with $\\bar{\\eta }_{anom}=0$ and another with $\\bar{\\eta }_{anom}=10^{-2}$ and $j_c=10$ , both using the constant background resistivity value $\\eta _{bg}=10^{-5}$ .", "Figure: Unstratified atmosphere.", "Height of flux rope center for a set of five simulations with varying magnitude of initial background magnetic guide field and resistivity models.", "Three values of the guide magnetic field b z0 ={1.0,0.5,0.25}b_{z0}=\\lbrace 1.0,0.5,0.25\\rbrace and two resistivity models, one with η ¯ anom =0\\bar{\\eta }_{anom}=0 labeled as “eta const\", and one with η ¯ anom =10 -2 \\bar{\\eta }_{anom}=10^{-2} and j c =10j_c=10 labeled as “eta anom\", both using the constant background resistivity value η bg =10 -5 \\eta _{bg}=10^{-5}, are considered.The comparison of the five traces clearly demonstrates that the outcome of the simulations is much more sensitive to the magnitude of the background guide field, i.e.", "the global structure and stability of the magnetic configuration, than to the resistivity model.", "The two traces with $b_{z0}=1.0$ , the initially stable magnetic configuration, are virtually indistinguishable from each other despite very different resistivity models.", "The two traces with $b_{z0}=0.5$ initialized from a marginally stable configuration (see Fig.", "REF ) do show small differences during the acceleration phase.", "Here the simulation with anomalous resistivity allows for slightly faster rise, but both rise much faster than the $b_{z0}=1.0$ cases.", "And the initially unstable $b_{z0}=0.25$ case demonstrates yet faster rise of the flux rope that is comparable to the rise speed observed in the [5] simulation.", "(Only the anomalous resistivity $b_{z0}=0.25$ simulation trace is shown in Fig.", "REF because the corresponding uniform resistivity simulation produces a very intense X-point current sheet that breaks up due to secondary instabilities [24], leading to formation of further spatial sub-structure which we have chosen not to attempt to resolve.", "Detailed investigation of such multi-scale reconnection cases is left for future work.)", "We note that the choice of critical current density $j_c=10$ for onset of anomalous resistivity is such that all five simulations achieve $|{\\bf j}| > j_c$ at the X-point during the acceleration phase of the flux rope, yet that does not result in significant acceleration of the flux rope for the $b_{z0}=1.0$ and $b_{z0}=0.5$ cases.", "It is also of interest that the rapid rise of the flux rope in the $b_{z0}=0.25$ case is followed by stagnation at the height of approximately 19Mm.", "Such stagnation is indicative of the system finding a new stable magnetic equilibrium where the upward force on the flux rope is balanced by the magnetic tension distributed throughout the overlying magnetic arcade." ], [ "Flux emergence in a stratified atmosphere", "Introduction of atmospheric stratification, as described in Sec.", "REF , leads to a more realistic equilibrium plasma configuration that is much denser at the photosphere than in the unstratified corona-like case.", "The impact of the flux emergence at the bottom boundary, with and without the atmospheric stratification, is reflected in the traces of height and speed of the respective flux rope eruptions.", "For the stratified atmosphere, the height and speed of the flux rope as functions of time are shown in the left panel of Fig.", "REF and can be compared to the equivalent traces for the reference simulation in the left panel of Fig.", "REF .", "(Note the different ranges of the time axes of the two panels.)", "The two time histories are qualitatively similar, both showing rapid acceleration of the flux-rope during flux emergence, with a reduction of the ejection speed by approximately a factor of two once the driving is turned off.", "However, both the peak and the post-driving ejection speed of the CME in the stratified atmosphere are less than half of that obtained in the unstratified case.", "Figure: Stratified atmosphere.", "Left: Height and speed of flux rope center.", "Smoothing is performed using a Hanning window of 12 points.", "The speed is computed by finite-differencing the smoothed (blue) curve.", "Right: Temperature at the X-point (center of current sheet) below the flux rope during the same period.Another significant difference between the two cases of flux emergence is observed by comparing the time traces of the X-point plasma temperature, shown in the right panels of Fig.", "REF and Fig.", "REF .", "While in the unstratified atmosphere there is a notable temperature increase at the X-point at the time of eruption, in the stratified simulation the temperature decreases instead.", "Furthermore, as the flux rope begins to rise between 35 and 110 minutes into the simulation ($300 \\lesssim t \\lesssim 900$ ) the stratified simulation shows an oscillatory X-point temperature as long as the flux rope is within $\\approx \\:1$  Mm of its original position.", "The root cause of the overall X-point cooling can easily be explained as upflows of cold chromospheric plasma being advected into the coronal reconnection region.", "Nevertheless, the observed self-induced quasi-periodic oscillatory behavior of the X-point temperature is somewhat unexpected.", "Figure: Evolution in time of the reconnection site behind the CME flux rope in a stratified atmosphere.", "Each panel shows a snapshot of the temperature structure on the left, the density structure on the right, select contours of the magnetic flux ψ\\psi (the same contour values, denoting the same magnetic field lines, have been chosen for each panel), and arrows denoting the in-plane plasma flow.", "The snapshots are made 348τ=42348\\tau = 42 min, 528τ=64528\\tau = 64 min, 708τ=86708\\tau = 86 min, and 978τ=118978\\tau = 118 min into the simulation.", "Note that for illustration purposes both plasma temperature and number density are plotted using logarithmic color scales with saturated high and low values.", "Arrows showing the plasma flow have been scaled by a factor of 25 with respect to the linear dimensions of the domain so that an arrow of unit length corresponds to flow of 1.7×10 4 1.7\\times 10^4 km/s.Fig.", "REF shows the evolution in time of plasma temperature, density, and flows around the X-point during the period of quasi-periodic temperature oscillations.", "Continuous upflows of dense cool plasma convected along the magnetic field lines and into the reconnection region around the X-point are apparent throughout the evolution.", "The lower-right panel of this figure makes clear that this continuous chromospheric upflow results in quasi-periodic striations of cool dense material alternating with hotter, lower-density plasma on the recently reconnected field lines rising towards (and with) the flux rope located above.", "These striations are the signatures of the same oscillatory behavior observed on the X-point temperature trace in Fig.", "REF .", "While the origins and parametric robustness of the observed quasi-periodic phenomenon require further in-depth study that is outside of the scope of this article, a heuristic explanation of the basic physical mechanism is straightforward.", "It results from the competition between the upward directed tension force in newly reconnected magnetic field lines and the downward directed gravity acting on the dense, cold plasma deposited onto these same field-lines by the chromospheric upflows.", "As in the formation of water droplets, whenever sufficient amount of plasma accumulates in a small enough volume in the V-shaped dip of a set of recently reconnected field lines, the gravitational pull on that plasma overcomes the field's tension force and a droplet of plasma forms and falls vertically through the reconnection site itself.", "As a result, those flux-rope destined field lines that produce the droplets end up with lower density hotter plasma, while the field lines that pass through in between the droplets contain colder and heavier plasma.", "The temperature at the X-point, where the reconnection is regularly disrupted by the droplets, is similarly modulated when the plasma that has been heated by the reconnection process is periodically replaced by the cold plasma of the droplets.", "Below the reconnection site, the pattern of chromospheric upflows along the magnetic separatrices and vertical downflows through the X-point creates a circulation of plasma between reconnection's outflow and inflow.", "How, and whether or not, this circulation pattern contributes to the formation of the quasi-periodic temperature and density structure described above is left as a topic for future study." ], [ "Discussion & Conclusions", "Coronal mass ejections are eruptive solar events of enormous proportions that shed plasma and magnetic flux into interplanetary space.", "The Chen & Shibata model is a good starting point for understanding how such an eruption can originate from the destabilization of a global magnetic configuration by local flux emergence.", "It helps us to see a connection between flux emergence, a phenomenon at the solar surface, and flux rope ejection, a phenomenon in the corona.", "Many observational studies have shown spatio-temporal correlations between flux emergence and eruptive events, but few theoretical models to date have identified a precise single mechanism or sequence of processes whereby producing magnetic flux at the photosphere dynamically triggers an eruption.", "The CS model may assume an oversimplified solar atmosphere and a somewhat manufactured magnetic topology, but it does proffer a complete story.", "To determine the effects of a more realistic solar atmosphere, we have undertaken an effort to repeat the study using a different numerical suite and allowing for a stratified atmosphere with the density variation of over four orders of magnitude, as well as a sharp temperature transition between the chromosphere and the corona.", "We have found that even in the absence of stratification the initial equilibrium can be unstable to small perturbations.", "The initial adjustment of the magnetic equilibrium to slight force imbalances can generate fast waves that may not be able to propagate through the X-point below the flux rope.", "In these cases, the fast waves accumulate in such a way as to collapse the X-point and initiate reconnection.", "Thus, the equilibrium can be destabilized before any photospheric driving is applied.", "However, we also found that the stability of the CS equilibrium can be controlled by varying the compressibility of the plasma, which in a two-dimensional system is determined by the combination of thermal pressure and the magnitude of the out-of-plane component of the magnetic field.", "To quantify this effect, we defined a generalized measure of compressibility $\\Gamma $ and have empirically determined the equilibrium's stability boundaries in terms of $\\Gamma $ .", "When emulating flux emergence by applying an electric field at the photospheric boundary, in the unstratified atmosphere, the results of our simulations are qualitatively similar to those of the original study.", "However, there are also important differences and new findings.", "As opposed to the original study, when initialized in a stable configuration, our simulations show little evidence of significant flux rope acceleration or Joule heating associated with the reconnection current sheet.", "Notably, this result appears to be insensitive to the micro-physics of the reconnection region.", "By varying the magnitude of the background out-of-plane magnetic field component and thus changing the stability of the global magnetic configuration, we also show that flux rope rise speeds comparable to the original result are possible but require an unstable magnetic configuration as the initial condition.", "We further show that the micro-physics of reconnection is more likely to slow down than to accelerate the flux rope by comparing simulations with and without anomalous resistivity.", "It is well known that current-dependent anomalous resistivity allows for “fast\" magnetic reconnection with only weak dependence on the magnitude of resistivity itself [30].", "Yet, for both initially stable and quasi-stable magnetic configurations, allowing for anomalous resistivity did not result in a substantial increase of the flux rope rise speed.", "That is, merely allowing for faster reconnection did not lead to faster reconnection and faster flux rope ejection.", "On the other hand, in magnetic configurations where fast flux-rope ejection is possible, the simulations with low guide field indicate that the inability of the magnetic reconnection process to occur sufficiently fast could limit the rise speed of the flux rope.", "In the flux emergence simulations with stable magnetic configuration and realistic atmospheric stratification, the weakness of the X-point heating and the slowness of the ejected flux rope are reproduced, and amplified.", "In these simulations, changes in the magnetic field structure due to flux emergence generate persistent chromospheric upflows of cold, dense material that is convected into and dramatically cools the reconnection current sheet.", "In addition to the steady state upflows and cooling, the stratified simulations also produce another type of behavior: self-induced quasi-periodic oscillations in the X-point temperature, density, and other fluid quantities.", "The quasi-periodic oscillations observed in the stratified simulation are of transient nature, appearing after the flux emergence drive has been completed and lasting for just over an hour while the flux rope is within $\\approx 1$  Mm of its initial location.", "The robustness of this phenomenon will be a subject of future research, but our initial investigation indicates that a critical balance between the upward tension force of the reconnected magnetic field and the downward gravitational pull on the dense chromospheric plasma convected into the reconnection region has to be achieved in order for the quasi-periodic oscillations to appear in a simulation.", "While that may seem to be a prohibitive constraint, we speculate that in the three-dimensional parameter space spanned by (1) the height of the X-point, (2) the strength of the magnetic fields and (3) the horizontal location of the emerging flux relative to the separatrices of the pre-existing magnetic configuration, all quantities that can vary greatly throughout the lower solar atmosphere, there is likely embedded a two-dimensional parameter space where such balance can, indeed, be achieved.", "We note that there is also extensive observational evidence for what has been called quasi-periodic pulsations (QPP) in solar and stellar flares [36], [35] with the QPP periodicity time scale varying from fractions of a second to several minutes, comparable to the period of the oscillations produced in our simulation.", "In fact, [36] have previously resorted to the water drop formation analogy in describing what they refer to as a class of “load/unload” models of long multi-minute period QPPs.", "The plasma droplet mechanism described in Sec.", "REF above is a much more direct, and novel, analogy to the same physical process with the potential to provide a new alternative explanation for the long-duration QPPs.", "Finally, we point out that the limitations of the two-dimensional MHD model used here for modeling a region of potential flaring activity embedded into a stratified solar atmosphere are many.", "It is well known that laminar resistive reconnection cannot account for the observed rates of magnetic energy release, particle acceleration, or radiation from solar flares, while three-dimensional effects can substantially alter both the flux-rope stability properties and the micro-physics of reconnection.", "Nevertheless, we believe that the careful and systematic study described in this article is a prerequisite for performing more complete, and also substantially more challenging and complicated, studies of CME initiation by flux emergence in the future.", "E.L. thanks Neil Sheeley for his valuable insights into solving the hyperbolic function integrals.", "This work was supported by the NASA SR&T and LWS programs, as well as ONR 6.1 program.", "Simulations were performed under grants of computer time from the US DOD HPC program and the National Energy Research Scientific Computing Center, which is supported by the US DOE Office of Science." ], [ "Appendix A", "We derive the uniform pressure magnetic flux rope equilibrium with axial field from a familiar form of the Grad–Shafranov equation: $\\frac{d}{d\\psi }\\left(\\frac{b_z^2}{2}\\right) = -\\nabla ^2 \\psi = j_z$ In particular, assuming $\\psi (r)=\\psi _l(r)$ given by Eq.", "(REF ): $\\frac{b_z^2}{2} &= \\int j_z \\;d\\psi = \\int j_z \\frac{d\\psi }{dr} dr \\nonumber \\\\&= \\int \\left[ \\frac{4r^2}{r_0^3} - \\frac{4}{r_0} \\right] \\left[\\frac{r}{r_0} - \\frac{r(r^2-r_0^2)}{r_0^3}\\right] dr \\\\&= -\\frac{2}{3}\\left(\\frac{r}{r_0}\\right)^6 + 3\\left(\\frac{r}{r_0}\\right)^4 - 4 \\left(\\frac{r}{r_0}\\right)^2 + \\text{constant,}\\,\\text{for}\\ r \\le r_0 .\\nonumber $ Requiring that $b_z = b_{z0}$ for $r \\ge r_0$ , we can determine the constant of integration such that $b_z$ is continuous: $b_z(r) = \\left\\lbrace \\begin{array}{l @{\\hspace{17.07164pt}} c}\\sqrt{b_{z0}^2 + \\dfrac{10}{3} - 8 \\left(\\dfrac{r}{r_0}\\right)^2 + 6 \\left(\\dfrac{r}{r_0}\\right)^4 -\\dfrac{4}{3} \\left(\\dfrac{r}{r_0}\\right)^6} \\, & r \\le r_0 \\\\[3mm]b_{z0} \\ .", "& r > r_0\\end{array}\\right.$ Suppose, however, we wish to find $b_z$ as a function of $\\psi $ , rather than of $r$ .", "We then solve for the inverse function $\\zeta \\equiv \\psi _l^{-1}(r)$ by replacing $r$ with $\\zeta $ in Eq.", "(REF ) and rearranging terms: $\\zeta ^4 - 4r_0^2\\zeta ^2 + r_0^4 + 4 r_0^3 \\psi _l = 0 \\ .$ Solving this quadratic equation for $\\zeta ^2$ , we find: $\\zeta ^2 = 2 r_0^2 \\pm \\sqrt{3r_0^4 - 4 r_0^3 \\psi _l } \\ .$ We recover the form of Eq.", "() by rejecting the positive root (to permit small values of $\\zeta $ ), replacing $\\psi _l$ by the full functional form of $\\psi = \\psi _l + \\psi _i + \\psi _b$ to approximate a force-free initial condition with well-aligned contours of constant $\\psi $ and $b_z$ , and allowing for gauge freedom.", "Here we present a derivation of the pressure profile used in simulations of a stratified solar atmosphere, given a particular temperature profile (REF ).", "To be physically relevant, we use here dimensional quantities, rather the normalized code variables.", "We begin with the first-order differential equation governing hydrostatic equilibrium: $\\frac{dp}{dy} + m_p n g_S = 0 \\ ,$ which we divide by $p=2 n k_B T$ : $\\frac{d \\ln p}{dy} + \\frac{m_p g_S}{2 k_B T} = 0.$ Then $\\ln \\frac{p}{p_0} = -\\frac{m_p g_S}{2 k_B} \\int \\frac{dy}{T}.$ We use the profile for temperature $T$ given by (REF ), but with the following variable substitution: $u \\equiv \\frac{y-y_\\text{\\tiny TR}}{\\Delta y} \\ ,$ leading to $\\begin{split}T(u) &= T_p + \\frac{T_c-T_p}{2} \\left(1 + \\tanh u\\right) \\\\&= T_p + \\frac{T_c-T_p}{2} \\left(1 + \\frac{e^u-e^{-u}}{e^u+e^{-u}} \\right) \\\\&= \\frac{T_p\\, e^{-u} + T_c\\, e^u}{e^u + e^{-u}}.\\end{split}$ With algebraic manipulations, we can rewrite (REF ) as: $\\frac{1}{T} = \\frac{1}{T_c} + \\frac{e^{-2u}(1-T_p/T_c)}{T_p\\,e^{-2u} + T_c}.$ Then the integral in (REF ) can be evaluated: $\\begin{split}\\int \\frac{dy}{T} &= \\Delta y \\int \\frac{du}{T} = \\Delta y \\left[ \\int \\frac{du}{T_c} + \\left( 1-\\frac{T_p}{T_c} \\right) \\int \\frac{e^{-2u}\\,du}{T_p\\,e^{-2u}+T_c} \\right] \\\\&= \\Delta y \\left[ \\frac{u}{T_c} - \\frac{1}{2}\\left(\\frac{1}{T_p}-\\frac{1}{T_c}\\right) \\ln \\left(T_p e^{-2u} + T_c\\right) \\right].\\end{split}$ Finally, substituting (REF ) into (REF ) yields an expression for $p$ , in terms of $u$ : $p(u) = p_0 \\exp \\left\\lbrace \\frac{m_p g_S\\Delta y}{2 k_B T_c} \\left[\\frac{T_c-T_p}{2 T_p} \\,\\ln \\left(T_p e^{-2u} + T_c\\right) - u \\right] \\right\\rbrace .$" ] ]
1403.0231
[ [ "A rotating universe outside a Schwarzschild black hole where spacetime\n itself non-uniformly rotates" ], [ "Abstract We study a non-uniformly rotating universe outside a Schwarzschild black hole by generating a time-dependent manifold of revolution around a straight line.", "In this simple model where layers of spherical shells of the universe non-uniformly rotate, the Einstein field equations require this phenomenon to be caused by a static mass-energy distribution with time-dependent $T^{\\phi\\phi}$ (quadratic with time) and $T^{r\\phi}=T^{\\phi r}$ (linear with time).", "This indicates that a time-dependent stress along a certain direction results in a spacetime shift in that direction.", "For this model however, such material violates the null energy condition.", "Incidentally, the various coordinate systems describing the Schwarzschild solution can be viewed as arising from the freedom in parametrising the straight line and the radial function in the general method of constructing spacetime by generating manifolds of revolution around a given curve." ], [ "Introduction", "The general method of constructing spacetime by generating manifolds of revolution around a given curve was recently formulated to study curved traversable wormholes [1], [2] The original motivation that led to this formulation came from trying to helicalise a given curve [32], i.e.", "replacing the given curve by another curve which winds around it..", "This follows from the ideas of Morris and Thorne [4] that in finding solutions to general relativity, a spacetime geometry is first constructed and the Einstein field equations are subsequently used to determine the required materials to support it.", "Following the publication of Morris and Thorne's work, there has been intense research in the area of traversable wormholes, with a major result being the necessity of exotic matter to be present [5].", "A useful application of the general method in [1], [2] is the construction of curved traversable wormholes which does not assume spherical symmetry.", "This led to the finding that by carefully engineering the shape and curvature of curved wormholes, it is possible for such wormholes to admit safe geodesics through them, i.e.", "freely-falling trajectories which are locally supported by ordinary matter, thereby avoiding the need for travellers to get into direct contact with exotic matter.", "Here is how (3+1)-d spacetimes are constructed using that method.", "Given a smooth curve $\\psi (v)$ embedded in a 4-d Euclidean space, a 3-manifold of revolution is: $\\vec{\\sigma }(u,v,w)=\\psi (v)+Z(v)\\cos {u}\\ \\vec{n}_1(v)+Z(v)\\sin {u}\\cos {w}\\ \\vec{n}_2(v)+Z(v)\\sin {u}\\sin {w}\\ \\vec{n}_3(v),$ where $Z(v)$ is the radial function and $\\vec{n}_1(v),\\vec{n}_2(v),\\vec{n}_3(v)$ are three orthonormal vectors.", "The metric of this 3-manifold can be calculated, and then extended to a (3+1)-d spacetime metric.", "This method can also be used to build dynamical spacetimes, as [1] explicitly illustrated how an inflating wormhole can be constructed by letting the given curve $\\psi (v)$ and the radial function $Z(v)$ depend on time.", "In this paper, we would like to explore how the general method can be used to construct a rotating spacetime, and identify the essential attributes of the matter which produce this phenomenon.", "To investigate this in a simple model, extending a static traversable wormhole to a rotating one would not be ideal since it requires matter (exotic in some region) to already be present.", "It is instead advantageous to extend from an originally vacuum spacetime, since the matter that would be present is solely responsible for the rotational effects of the spacetime.", "Apart from the trivially flat Minkowski geometry, a Schwarzschild geometry is also vacuum (excluding the black hole, of course).", "We would hereby construct a rotating universe outside (the event horizon of) a black hole, where the spacetime itself rotates, and obtain the Minkowski version as the special case of zero black hole mass.", "To take full advantage of the spherical symmetry of the Schwarzschild geometry, the universe is prescribed to rotate in layers of rigid spherical shells.", "However, these shells of various radii need not be rotating with the same angular velocity (see Fig.", "1), so the universe as a whole would not be rigidly rotating.", "This would be interesting, as we can compare such a non-uniformly rotating universe to an expanding universe which is isotropic and homogeneous (the FLRW solution), expanding in all directions whilst carrying the matter (or galaxies) along with it [6].", "Note also that we are going to assume that the mass-energy is static, unlike the van Stockum [7], Tipler [8] or Kerr [9] solutions which describe rotating matter.", "This helps to simplify the metric, since those examples necessarily contain a non-zero $g_{t\\phi }$ cross-term which would lead to significantly arduous calculations and possibly obfuscate the interpretations of the matter properties, if incorporated into our model As seen in the next section, the spatial 3-manifold representing a non-uniformly rotating universe would itself contain a $g_{v\\phi }$ spatial cross-term..", "Figure: Each spherical shell rigidly rotates, but spherical shells of different radii may be rotating with different angular velocities.In the next section, we construct the metric that describes a non-uniformly rotating spacetime around a Schwarzschild black hole by generating a 3-manifold of revolution around a straight line.", "In section 3, we decide on the choice of parametrisation for the straight line and the radial function, showing how the freedom in parametrisation leads to various coordinate systems that describe the Schwarzschild metric.", "Section 4 is devoted to the physical properties of the mass-energy fluid that give rise to such a rotating spacetime, with a discussion section following after.", "Section 6 concludes this paper.", "We shall be working in units where $G=c=1$ ." ], [ "The metric construction", "Consider the following time-dependent 3-manifold of revolution around a straight line embedded into a 4-d Euclidean space: $\\vec{\\sigma }(t,v,\\theta ,\\phi )=\\left(\\begin{array}{c}r(v)\\cos {\\theta }\\\\r(v)\\sin {\\theta }\\cos {(\\phi +\\chi (v)\\omega t)}\\\\r(v)\\sin {\\theta }\\sin {(\\phi +\\chi (v)\\omega t)}\\\\z(v)\\end{array}\\right),$ where $t$ is the time coordinate as measured by a faraway observer, $r$ , $\\theta $ and $\\phi $ being the usual spherical coordinates for 3-d Euclidean space, and $z$ is the fourth spatial coordinate A useful lower-dimensional analogue for visualisation is to think of embedding a 2-d surface into ordinary 3-d Euclidean space using cylindrical coordinates $r$ , $\\phi $ and $z$ so that $r$ and $\\phi $ are the usual plane polar coordinates with $z$ being the third spatial coordinate.", "The general framework for generating surfaces of revolution by adding circles or 1-spheres to a given curve is described in [2]..", "The symbol $\\omega $ is a constant, with $v$ parametrising the radial coordinate $r(v)$ and the fourth spatial coordinate $z(v)$ , so it can be thought of that one is a function of the other, viz.", "$z(r)$ or $r(z)$ .", "This effectively determines the shape of the manifold of revolution around the straight line.", "Disregarding the $\\chi (v)\\omega t$ term, this 3-manifold of revolution obtained by adding spheres of radii $r(v)$ along the straight line $\\vec{L}(v)=(0,0,0,z(v))$ is a special case of the general method used in [1] to construct static curved traversable wormholes Note that in [1], the coordinates $u$ and $w$ were used in place of $\\theta $ and $\\phi $ , with the radial function denoted as $Z(v)$ (see Eq.", "(REF )).", "The reason such general symbols were used was because they were not the usual spherical coordinates as the given curve (which was not necessarily a straight line since curved wormholes were to be constructed) would naturally induce a more general coordinate system.", "Here, the focus is on manifolds of revolution around a straight line (in fact rotating about that line) so we are indeed working with the three usual spherical coordinates $r,\\theta ,\\phi $ plus the fourth spatial coordinate $z$ ..", "The time-dependence built in here by replacing $\\phi \\rightarrow \\phi +\\chi (v)\\omega t$ from the static version in Eq.", "(REF ) to yield Eq.", "(REF ) represents the fact that a spherical shell of radius $r(v)$ is rotating about the fourth coordinate axis $z$ (or the line $\\vec{L}$ ) with constant angular velocity $\\chi (v)\\omega $ .", "This can be seen by choosing any particular point on the 3-manifold i.e.", "fixing some values of $v,\\theta ,\\phi $ , and noting that as $t$ evolves this point would be rotated by an angle of $\\chi (v)\\omega t$ about $\\vec{L}$ .", "The $v$ -dependence on $\\chi $ implies that spheres of different radii $r(v)$ which are added at different points on $\\vec{L}(v)$ may in general be rotating with different angular velocities, although we would require that this variation is smooth with $v$ .", "For physical interest that we would like to consider here in this study, the following conditions are imposed: $\\chi \\rightarrow 0$ as $v\\rightarrow \\infty $ so that when extended to a (3+1)-d spacetime manifold, a faraway observer would be sitting on an asymptotically flat manifold which is non-rotating.", "Physical observations shall be discussed with respect to this inertial frame.", "$\\chi (v_0)$ is normalised to 1 at some reference point where the parameter is $v_0$ .", "This reference spherical shell of radius $r(v_0)$ would then be rotating with angular velocity $\\omega $ .", "$\\chi (v)$ monotonically decreases as $v$ goes from $v_0$ to $\\infty $ .", "The first two are boundary conditions that prohibit $\\chi $ from being a constant (since it has to be 1 at $v_0$ and 0 at $\\infty $ ), so the universe is rotating non-uniformly in contrast to being in a rigid rotation.", "This means that there is no observer that would see the universe as being globally static.", "Any observer that may be locally static would necessarily see at least one other spherical shell rotating.", "The third condition would demand that the bulk of the mass-energy be concentrated near the axis of rotation, diminishing away from it.", "Some kind of mass-energy distribution in the universe whose properties are to be found through the Einstein field equations would cause the universe itself to behave like a swirling fluid with maximum rate of swirling at the centre, and dissipating with essentially little or no swirling towards the outer edge.", "Note that this is not the same as the mass-energy being the said fluid that is rotating, rather it is the universe itself that is non-uniformly rotating Like an expanding universe, the spacetime itself is expanding as opposed to the galaxies themselves moving away from each other..", "The spatial metric $ds^2_{\\textrm {space}}$ for the 3-manifold given by Eq.", "(REF ) can be computed as follows: the components are $g_{ij}=\\vec{\\sigma }_i\\cdot \\vec{\\sigma }_j$ , where $i,j\\in \\lbrace v,\\theta ,\\phi \\rbrace $ , and $\\vec{\\sigma }_i$ denotes partial derivative with respect to $i$ .", "This gives $ds^2_{\\textrm {space}}&=&(z^{\\prime 2}+r^{\\prime 2}+\\chi ^{\\prime 2}\\omega ^2t^2r^2\\sin ^2{\\theta })dv^2+r^2d\\theta ^2+r^2\\sin ^2{\\theta }\\ d\\phi ^2\\nonumber \\\\&\\ &+2\\chi ^{\\prime }\\omega tr^2\\sin ^2{\\theta }\\ dvd\\phi ,$ where explicit dependence on $v$ for $z(v),r(v),\\chi (v)$ are suppressed for conciseness.", "This spatial metric can also be written as, $ds^2_{\\textrm {space}}&=&(z^{\\prime 2}+r^{\\prime 2})dv^2+r^2d\\theta ^2+r^2\\sin ^2{\\theta }(\\chi ^{\\prime }\\omega t\\ dv+d\\phi )^2,$ indicating how the time-dependence term leads to the $dvd\\phi $ cross-term, viz.", "$\\phi \\rightarrow \\phi +\\chi \\omega t$ for the static to rotating manifold's parametric equations corresponds to $d\\phi \\rightarrow d(\\phi +\\chi \\omega t)=d\\phi +\\chi ^{\\prime }\\omega t\\ dv$ for their metrics.", "It is clear that a constant $\\chi $ just gives the usual spherically symmetric metric (see section 7.2 in [2]).", "Our boundary conditions however, forbid this for the non-uniform rotation case.", "We can extend this to a spacetime metric of the form: $ds^2&=&g_{tt}dt^2+(z^{\\prime 2}+r^{\\prime 2})dv^2+r^2d\\theta ^2+r^2\\sin ^2{\\theta }(\\chi ^{\\prime }\\omega t\\ dv+d\\phi )^2,$ where $g_{tt}<0$ .", "Since we are chiefly concerned with the geometry outside a Schwarzschild black hole, $g_{tt}$ is a function of only $v$ .", "We would not bother with the $g_{tj}$ ($j$ being any of the spatial coordinates $v,\\theta ,\\phi $ ) cross-terms, taking them to be zero and impose the mass-energy to be static.", "This would greatly reduce the algebraic technicalities in calculating the Einstein tensor especially as the spatial metric itself already contains the $g_{v\\phi }$ cross-term." ], [ "Parametrisation of $z(v)$ and {{formula:22e52d4e-c729-44a6-92b9-e7c133cffe14}}", "We should decide on the choice of parametrisation of $z(v)$ and $r(v)$ , before proceeding with further computations.", "For the sake of discussion, let us consider the static spacetime so that $\\omega =0$ : $ds^2&=&g_{tt}dt^2+(z^{\\prime 2}+r^{\\prime 2})dv^2+r^2d\\theta ^2+r^2\\sin ^2{\\theta }\\ d\\phi ^2.$ Amongst many possible parametrisations, two simple ones are linearly parametrising $r(v)=v$ , or to linearly parametrise $z(v)=v$ .", "The former is to treat the actual radial coordinate $r$ as the parameter itself, so that $z$ becomes a function of $r$ and the metric in Eq.", "(REF ) becomes $ds^2&=&g_{tt}(r)dt^2+(z^{\\prime }(r)^2+1)dr^2+r^2d\\theta ^2+r^2\\sin ^2{\\theta }\\ d\\phi ^2.$ If one goes on to calculate the Einstein tensor and solve the vacuum field equations, one would find that $z(r)=2\\sqrt{R_s(r-R_s)}$ and $g_{tt}(r)=-(1-R_s/r)$ , where $R_s$ is the Schwarzschild radius.", "This is the usual static spherically symmetric vacuum solution expressed in Schwarzschild coordinates and $z(r)=2\\sqrt{R_s(r-R_s)}$ is recognised as Flamm's paraboloid.", "If the latter parametrisation is used instead (which can also be thought of as linearly parametrising the line $\\vec{L}$ ), then the metric in Eq.", "(REF ) becomes $ds^2&=&g_{tt}(z)dt^2+(1+r^{\\prime }(z)^2)dz^2+r(z)^2d\\theta ^2+r(z)^2\\sin ^2{\\theta }\\ d\\phi ^2.$ Solving the vacuum field equations gives $r(z)=z^2/4R_s+R_s$ and $g_{tt}(z)=-z^2/(z^2+4R_s^2)$ .", "This is actually equivalent to the Einstein-Rosen coordinates if one rescales $z=2u\\sqrt{R_s}$ Matt Visser provides an excellent description on these various coordinate systems that cover only certain regions of the Schwarzschild geometry, which played a part in leading Einstein and Rosen to deduce the Einstein-Rosen bridge from their coordinates in [33].", "The Einstein-Rosen coordinates are $ds^2=-u^2/(u^2+R_s)dt^2+4(u^2+R_s)du^2+(u^2+R_s)^2(d\\theta ^2+\\sin ^2{\\theta }\\ d\\phi ^2)$ .", "See for instance chapter five in [5]..", "It is hereby obvious that they would have naturally interpreted from such coordinates that this represents a wormhole, since spheres of radii $r(z)=z^2/4R_s+R_s$ are added to the line $\\vec{L}(z)=(0,0,0,z)$ .", "With the radial function $r(z)>0$ for all $z\\in {\\mathbb {R}}$ and having a minimum value of $R_s$ at $z=0$ , the geometrical picture of the spatial 3-manifold is a “3-d straight tube” with minimum radius $R_s$ at $z=0$ that grows into two asymptotically flat ends.", "Those two parametrisations were rather effortless, i.e.", "setting either $r(v)=v$ or $z(v)=v$ .", "As a third and perhaps not so straightforward example, isotropic coordinates can be obtained by the parametrisation $r(v)=v(1+R_s/4v)^2$ and $z(v)=(4v-R_s)\\sqrt{R_s/4v}$ The logical flow to arrive at this choice of parametrisation would be as follows: In isotropic coordinates, a spacetime metric takes the general form $ds^2=g_{tt}(v)dt^2+A(v)(dv^2+v^2d\\theta ^2+v^2\\sin ^2{\\theta }\\ d\\phi ^2)$ .", "By comparing with our generic form in Eq.", "(REF ), we see that $z$ and $r$ has to satisfy $z^{\\prime 2}=r^2/v^2-r^{\\prime 2}$ .", "By eliminating $z^{\\prime 2}$ in place of $r$ and $r^{\\prime }$ , the vacuum field equation $G_{tt}=0$ reduces to $2v^2rr^{\\prime \\prime }=(r-vr^{\\prime })^2$ , where $r(v)=v(1+R_s/4v)^2$ is a solution.", "It then follows that $z(v)=(4v-R_s)\\sqrt{R_s/4v}$ .", "Finally $g_{tt}(v)$ can be solved from other components of the vacuum field equations to give $g_{tt}(v)=-[(1-R_s/4v)/(1+R_s/4v)]^2$ .", "The metric is hence $ds^2=-[(1-R_s/4v)/(1+R_s/4v)]^2dt^2+(1+R_s/4v)^4(dv^2+v^2d\\theta ^2+v^2\\sin ^2{\\theta }\\ d\\phi ^2)$ , as anticipated.", "(See chapter 2.3.9 in [5], for instance.", "Look out for the typo in Eq.", "(2.42) where $(1+r_s/4\\rho )^2$ should be $(1+r_s/4\\rho )^4$ .", ").", "Fig.", "2 shows how the same shape function is described by the three different parametrisations of $z(v)$ and $r(v)$ that are discussed here.", "It is thus intriguing that the various coordinate systems for the Schwarzschild geometry can be seen as arising from the freedom in parametrising $z(v)$ and $r(v)$ in the general method of constructing spacetime by generating manifolds of revolution around a given curve (in this case a straight line), as formulated in [2], [1] In [2], [1] where static curved traversable wormholes were constructed, the parametrisations were essentially chosen from the beginning when the curve $\\vec{\\psi }(v)$ and the radial function $Z(v)$ were defined..", "In our subsequent analysis of the rotating universe outside a Schwarzschild black hole, we shall adopt the parametrisation $r(v)=v$ which is the Schwarzschild coordinates.", "Figure: The same curve (with R s =1R_s=1) as described by (from left) Schwarzschild coordinates (rr is linearly parametrised), Einstein-Rosen coordinates (zz is linearly parametrised), and isotropic coordinates.", "Each dot on the curve represents an increment of vv by 1, beginning from (r,z)=(R s ,0)(r,z)=(R_s,0)." ], [ "Physical properties of a non-uniformly rotating universe outside a Schwarzschild black hole", "Let us return to describing a non-uniformly rotating universe outside a Schwarzschild black hole with Schwarzschild radius $R_s$ .", "In Schwarzschild coordinates, the spacetime metric would be $ds^2&=&-\\left(1-\\frac{R_s}{r}\\right)dt^2+\\frac{1}{1-R_s/r}dr^2+r^2d\\theta ^2+r^2\\sin ^2{\\theta }(\\chi ^{\\prime }(r)\\omega t\\ dr+d\\phi )^2\\\\&=&-\\left(1-\\frac{R_s}{r}\\right)dt^2+\\left(\\frac{1}{1-R_s/r}+\\chi ^{\\prime }(r)^2\\omega ^2t^2r^2\\sin ^2{\\theta }\\right)dr^2+r^2d\\theta ^2+r^2\\sin ^2{\\theta }\\ d\\phi ^2\\nonumber \\\\&\\ &+2\\chi ^{\\prime }(r)\\omega tr^2\\sin ^2{\\theta }\\ drd\\phi ,$ where $r\\ge R_s$ is the region of interest.", "The Einstein tensor $G^{\\mu \\nu }=R^{\\mu \\nu }-Rg^{\\mu \\nu }/2$ can be calculated, with the following non-zero terms It is obvious that $\\omega =0$ gives $G^{\\mu \\nu }=0$ , so that a non-rotating Schwarzschild geometry is Ricci flat.", ": $G^{tt}&=&-\\frac{\\chi ^{\\prime 2}\\omega ^2r^3\\sin ^2{\\theta }}{4(r-R_s)}\\\\G^{rr}&=&\\frac{1}{4}\\chi ^{\\prime 2}\\omega ^2r(r-R_s)\\sin ^2{\\theta }\\\\G^{\\theta \\theta }&=&-\\frac{1}{4}\\chi ^{\\prime 2}\\omega ^2\\sin ^2{\\theta }\\\\G^{\\phi \\phi }&=&\\frac{1}{4}\\chi ^{\\prime 2}\\omega ^2\\left(\\chi ^{\\prime 2}\\omega ^2t^2r(r-R_s)\\sin ^2{\\theta }-3\\right)\\\\G^{r\\phi }&=&G^{\\phi r}=-\\frac{1}{4}\\chi ^{\\prime 3}\\omega ^3tr(r-R_s)\\sin ^2{\\theta }\\\\G^{t\\phi }&=&G^{\\phi t}=-\\frac{\\omega }{2r}(4\\chi ^{\\prime }+r\\chi ^{\\prime \\prime })$ A particularly interesting quick observation is that $G^{t\\phi }=G^{\\phi t}$ can be made to be identically zero if $\\chi $ satisfies $4\\chi ^{\\prime }+r\\chi ^{\\prime \\prime }=0$ .", "A solution to this is $\\chi (r)=P/r^3+Q$ where $P,Q$ are arbitrary constants.", "The two boundary conditions for $\\chi $ (see section 2) would be met if $P=R_s^3$ and $Q=0$ , with the reference spherical shell being the horizon of the black hole $\\chi (R_s)=1$ since the region observable by a faraway observer is $r\\ge R_s$ .", "The third condition is also met by $\\chi (r)=(R_s/r)^3$ .", "This leaves the non-zero components of $G^{\\mu \\nu }$ as $G^{tt}&=&-\\frac{9R_s^6\\omega ^2\\sin ^2{\\theta }}{4r^5(r-R_s)}\\\\G^{rr}&=&\\frac{9R_s^6\\omega ^2(r-R_s)\\sin ^2{\\theta }}{4r^7}\\\\G^{\\theta \\theta }&=&-\\frac{9R_s^6\\omega ^2\\sin ^2{\\theta }}{4r^8}\\\\G^{\\phi \\phi }&=&\\frac{27}{4}R_s^6\\omega ^2\\left(\\frac{3R_s^6\\omega ^2t^2(r-R_s)\\sin ^2{\\theta }-r^7}{r^{15}}\\right)\\\\G^{r\\phi }&=&G^{\\phi r}=\\frac{27R_s^9\\omega ^3t(r-R_s)\\sin ^2{\\theta }}{4r^{11}}.$ The physics of the mass-energy that would result in such a non-uniformly rotating universe is given by the Einstein field equations $G^{\\mu \\nu }=8\\pi T^{\\mu \\nu }$ .", "With $\\chi =(R_s/r)^3$ , the $T^{t\\phi }=T^{\\phi t}$ terms are zero, so the simplest kind of mass-energy does not involve any heat transfer.", "In full, the non-zero components of $T^{\\mu \\nu }$ are: $T^{tt}&=&-\\frac{9R_s^6\\omega ^2\\sin ^2{\\theta }}{32\\pi r^5(r-R_s)}\\le 0\\\\T^{rr}&=&\\frac{9R_s^6\\omega ^2(r-R_s)\\sin ^2{\\theta }}{32\\pi r^7}\\ge 0\\\\T^{\\theta \\theta }&=&-\\frac{9R_s^6\\omega ^2\\sin ^2{\\theta }}{32\\pi r^8}\\le 0\\\\T^{\\phi \\phi }&=&\\frac{27}{32\\pi }R_s^6\\omega ^2\\left(\\frac{3R_s^6\\omega ^2t^2(r-R_s)\\sin ^2{\\theta }-r^7}{r^{15}}\\right)\\\\T^{r\\phi }&=&T^{\\phi r}=\\frac{27R_s^9\\omega ^3t(r-R_s)\\sin ^2{\\theta }}{32\\pi r^{11}}\\ge 0\\textrm {\\ for\\ }t\\ge 0.$ A reassuring fact that can be inferred regarding the nature of the mass-energy density $T^{tt}$ is its time-independence.", "Note the contrast when compared to an expanding universe which dilutes the static material as it carries it along in the expansion.", "Here, there is no increase in volume since spherical shells of the universe are rotating but not expanding.", "The material density is naturally expected to be constant with time, lest mass-energy conservation be violated to produce the rotation.", "As seen from the frame of a faraway observer, the mass-energy density $T^{tt}$ is negative and gets enormously large towards the horizon as $\\displaystyle \\lim _{r^+\\rightarrow R_s}T^{tt}=-\\infty $ , though the $T^{rr}$ and $T^{r\\phi }=T^{\\phi r}$ stresses vanish at $r=R_s$ .", "There are two (independent) components of the stress-energy tensor which are time-dependent, viz.", "$T^{\\phi \\phi }$ and $T^{r\\phi }=T^{\\phi r}$ , where the former depends quadratically with time and linearly for the latter.", "The other two stresses $T^{rr}$ and $T^{\\theta \\theta }$ are time-independent.", "The field equations therefore reveal the astonishing effect of a mass-energy fluid whose stresses possess this directional dependence on time: Such static fluid, with time-dependent properties of $T^{\\phi \\phi }$ and $T^{r\\phi }=T^{\\phi r}$ would remarkably cause the universe to rotate, carrying the fluid along with it.", "Figure: The time-dependent stresses on a fluid element T φφ T^{\\phi \\phi } (quadratic with time) and T rφ =T φr T^{r\\phi }=T^{\\phi r} (linear with time) in the θ=π/2\\theta =\\pi /2 plane.", "The frame of a faraway observer would see the dotted circle trajectory of the static fluid element being carried along by the universe which rotates about a black hole.", "The time-independent stresses along the radial and θ\\theta -directions are not shown.Although a free particle cannot remain at rest outside a black hole (since the gravity of the black hole would attract the particle towards it), it is not difficult to imagine some kind of cosmological event where perhaps a red giant exploded and began collapsing into a black hole.", "During the explosion, the red giant would expel material outwards, eventually leading to a transient equilibrium state where the material is static outside the resulting black hole's event horizon for a period of time.", "The sign of $T^{\\phi \\phi }$ is negative for small $t$ , indicating that the material is initially under tension along the $\\phi $ -direction.", "Over time, the sign of $T^{\\phi \\phi }$ changes to positive such that the material would be under increasing pressure.", "An example of a material whose pressure increases would be a nuclear process, whereby mass is converted into thermal energy which builds up the pressure over time, though in this situation the pressure increase only takes place along the $\\phi $ -direction and the $r\\phi $ -shears.", "A realistic scenerio of this non-uniformly rotating phenomenon would only be temporary, since the increase cannot go unbounded forever.", "Fig.", "3 depicts the time-dependent stresses on a fluid element with increasing pressure along the $\\phi $ -direction and $r\\phi $ -shear forces.", "Finally, consider the covariant null vector $k_\\mu =((\\sqrt{-g^{tt}})^{-1},0,0,(\\sqrt{g^{\\phi \\phi }})^{-1})$ .", "For $\\theta \\ne \\pi /2$ , $T^{\\mu \\nu }k_\\mu k_\\nu =T^{tt}(k_t)^2+T^{\\phi \\phi }(k_\\phi )^2=-\\frac{9R_s^6\\omega ^2r\\sin ^2{\\theta }}{8\\pi [r^7+9R_s^6\\omega ^2(r-R_s)t^2\\sin ^2{\\theta }]}<0,$ implying that the null energy condition is violated The null energy condition asserts that for any null vector, $T^{\\mu \\nu }k_\\mu k_\\nu \\ge 0$ .", "This can be found in page 115 of [5]..", "This kind of mass-energy is therefore exotic in order to produce this non-uniformly rotating universe.", "The negativity of $T^{\\mu \\nu }k_\\mu k_\\nu $ for this null vector however, decreases with time." ], [ "Minkowski spacetime ($R_s=0$ )", "A non-uniformly rotating Minkowski spacetime can be thought of as the special case when the mass of the black hole is zero, or equivalently $R_s=0$ .", "We cannot however, directly substitute $R_s=0$ into the stress-energy tensor because we chose $P=R_s^3$ to satisfy our two boundary conditions for $\\chi $ .", "To obtain the correct $T^{\\mu \\nu }$ , we let $\\chi =(R_0/r)^3$ with $R_0$ being a positive constant and only consider the region where $r\\ge R_0$ .", "The non-zero components of $T^{\\mu \\nu }$ corresponding to Eqs.", "(REF -) are $T^{tt}&=&-\\frac{9R_0^6\\omega ^2\\sin ^2{\\theta }}{32\\pi r^6}\\le 0\\\\T^{rr}&=&\\frac{9R_0^6\\omega ^2\\sin ^2{\\theta }}{32\\pi r^6}\\ge 0\\\\T^{\\theta \\theta }&=&-\\frac{9R_0^6\\omega ^2\\sin ^2{\\theta }}{32\\pi r^8}\\le 0\\\\T^{\\phi \\phi }&=&\\frac{27}{32\\pi }R_0^6\\omega ^2\\left(\\frac{3R_0^6\\omega ^2t^2\\sin ^2{\\theta }-r^6}{r^{14}}\\right)\\\\T^{r\\phi }&=&T^{\\phi r}=\\frac{27R_0^9\\omega ^3t\\sin ^2{\\theta }}{32\\pi r^{10}}\\ge 0\\textrm {\\ for\\ }t\\ge 0.$ As in the universe outside a Schwarzschild black hole, the null energy condition is violated since a null vector $k_\\mu =((\\sqrt{-g^{tt}})^{-1},0,0,(\\sqrt{g^{\\phi \\phi }})^{-1})$ gives (for $\\theta \\ne \\pi /2$ ) $T^{\\mu \\nu }k_\\mu k_\\nu =T^{tt}(k_t)^2+T^{\\phi \\phi }(k_\\phi )^2=-\\frac{9R_0^6\\omega ^2\\sin ^2{\\theta }}{8\\pi (r^6+9R_0^6\\omega ^2t^2\\sin ^2{\\theta })}<0.$ The results for the non-uniformly rotating Minkowski spacetime are therefore similar to that for Schwarzschild.", "Note however that unlike Schwarzschild where $r\\ge R_s$ is the observable universe, here the entire spacetime should be observable.", "This ostensibly leads to a problem, because $r\\ge R_0\\ne 0$ otherwise $\\chi =(R_0/r)^3$ would then be identically zero.", "Nevertheless the form of $\\chi =P/r^3+Q$ was a solution to $G^{t\\phi }=G^{\\phi t}=0$ such that $T^{t\\phi }=T^{\\phi t}=0$ which gives a stress-energy tensor that does not involve heat conduction.", "It is certainly possible to choose a different $\\chi $ which is 1 at $r=0$ and monotonically decreases to 0 as $r\\rightarrow \\infty $ , like $\\chi =1/(1+r^3)$ but requires that $T^{t\\phi }=T^{\\phi t}\\ne 0$ .", "This is thus a difference between a non-uniformly rotating Minkowski and a non-uniformly rotating Schwarzschild universe.", "Furthermore, the mass-energy for the Minkowski one is naturally static since there is no black hole to gravitationally attract it towards the centre If $R_s=0$ , then from Eqs.", "(REF ) or () the metric for a non-uniformly rotating Minkowski spacetime has $g_{tt}=-1$ , which implies no tidal force.." ], [ "Further Discussion", "We began by constructing the geometry of a non-uniformly rotating universe around a Schwarzschild black hole in section 2, and subsequently showed that this is caused by a static mass-energy with time-dependent stress along the $\\phi $ -direction and $r\\phi $ -shear forces.", "The frame of a faraway observer sees that the mass-energy is negative, and violates the null energy condition.", "The exotic nature of the material may render this phenomenon as unphysical, notwithstanding the fact that there are known solutions in general relativity (like traversable wormholes [5], and warp drive [20]) which demand such physics.", "Whilst it is arguable that the Casimir effect [21], [22], [23] is a well-regarded example of exotic matter to support these kind of so-called unphysical solutions, it is certainly important to be critical and discrimate any artificially thought up spacetime with highly obscure and inordinate physical requirements.", "In spite of the possibility of being classified into the undesirable category, the purpose of our study here is not to propose an arbitrary metric and just accept whatever stress-energy tensor that follows from the field equations.", "It is actually quite the contrary as we do not demand that there must exist a particular kind of mass-energy in nature to produce our desired spacetime.", "Instead, our motivation lies in figuring out the properties of such material and uncover their key characteristics.", "If such properties are deemed drastically preposterous, then it may perhaps be interpreted as an explanation to why we do not observe such rotational effects in our universe.", "The Einstein field equations are notoriously complicated non-linear partial differential equations, and a repercussion is the difficulty to solve it exactly.", "They are nevertheless meant to be read both ways: Given the physics, here is the resulting spacetime geometry.", "Given a spacetime geometry, here is the necessary physics.", "Even though our approach is based on the unorthodox direction of specifying the spacetime geometry to figure out the necessary physics, our results can be read from the more conventional direction to reveal the effect of a static mass-energy with time-dependent $\\phi $ -stress (quadratic with time) and $r\\phi $ -shear forces (linear with time): This causes the universe to rotate, i.e.", "shift along the $\\phi $ -direction.", "Putting it in another way, the stress-energy time-dependence for a particular direction ($T^{\\phi \\phi }$ , $T^{r\\phi }=T^{\\phi r}$ ) results in a spacetime translation along that direction ($\\phi $ ).", "This may indicate that more complicated dynamical evolution of the universe can be decomposed into the respective directional time-dependence of the stress-energy tensor ($T^{rr}(t),T^{\\theta \\theta }(t)$ , etc.).", "Our success in pinning down the precise conditions for this particular case owes to the fact that we constructed such a rotating spacetime first and then use the field equations to decipher the physics.", "We therefore already have an exact solution, for what may appear to be a strange (or highly fine-tuned) specification of the stress-energy tensor.", "Surely, one may begin instead with a physically constructed stress-energy tensor with the time-dependent properties of $T^{\\phi \\phi }$ and $T^{r\\phi }=T^{\\phi r}$ .", "The major weakness in this usual approach is that unless the stress-energy is of a particularly nice form, it may be nearly impossible to analytically solve the field equations.", "An important lesson is thus had we remained obdurate and refused to be avant-garde with the field equations, we might not have discovered such properties that lead to a rotating universe.", "Our results also show the difference between a non-uniformly rotating universe and an expanding universe as described by the FLRW solution.", "The latter assumes homogeneity and isotropicity, so it does not pick out any preferred spatial direction.", "The resulting Friedmann equations govern the change in the material density with time as it causes the universe to expand.", "Our non-uniformly rotating universe on the other hand has a fixed density and it is the increase in a directional stress with time that produces the rotation.", "For future research, it would be interesting to investigate if the purported exotic nature of the material to produce the rotation is mandatory.", "One may attempt to adapt the key time-dependent features that we have found here to specify certain stress-energy tensors (perhaps with no or less severe violation of the null energy condition), and solve the field equations numerically to simulate the rotational evolution of the universe.", "Our model assumes a constant angular velocity for simplicity to glean the crucial physical insights.", "It would be desirable to study more exhaustively how the rotation may originate, and how it would end via numerical computations.", "Another possible extension would be to let the material orbit the black hole on a circular geodesic, although care is to be taken in distinguishing between the particle's own orbital motion with the rotation of the universe itself.", "This simple model presented here is based on Einstein's theory of general relativity, without any quantum effects involved.", "Recent frontier research in quantum gravity has pushed the debate on what happens near the horizon of a black hole to unprecedented heights, following the black hole information paradox [24], [25], [26], [27], [28] to firewalls near a black hole horizon [29], [30], with the latest update from Stephen Hawking suggesting that ADS-CFT supports the notion that black holes do not have event horizons [31].", "It would be exciting to extend the formulation of a non-uniformly rotating universe around a black hole to the realm of such theories where quantum mechanics plays its part as well." ], [ "Concluding remarks", "This study of a non-uniformly rotating universe around a Schwarzschild black hole is perhaps a paramount example of how the approach of constructing the spacetime metric based on its desired geometrical properties and then using the field equations to find out the physics of it has led to the discovery of a new kind of solution in general relativity.", "The key properties of the mass-energy for this spacetime have been carefully examined and discussed.", "We have also illustrated how our general method of constructing spacetime by generating manifolds of revolution around a curve [2], [1] leads to the various coordinate systems for describing the Schwarzschild solution, when the given curve is a straight line.", "These different coordinate systems can be attributed to the freedom in parametrising the straight line and the radial function.", "The author is grateful for the efforts by Meng Lee Leek from Nanyang Technological University in participating in useful discussions that aided towards the coherent formulation of this work, as well as reviewing the final draft of this manuscript." ] ]
1403.0337
[ [ "A (varying power)-law modified gravity" ], [ "Abstract In the present paper we analyze a toy model for an $f(\\phi,R)$ gravity which has the form of a power-law modified gravity in which the exponent is space-time dependent.", "Namely, we investigate the effects of adding to the Hilbert-Einstein action an $R^{\\phi}$-term.", "We present possible equivalences of the model with known models of modified gravity theories and examine the problem of matter stability in this model.", "Like $f(R)$-gravity toy models, the present one offers the possibility of unifying the early and the late-time evolution of the Universe.", "We show that the behavior of the scalar field depends globally on the size of the Universe and locally on the surrounding environment.", "For the early Universe it lets appear a huge cosmological constant that might drive inflation.", "For the late-times it lets appear globally a tiny cosmological constant." ], [ "Introduction", "Modified gravity toy models play an essential role both in providing us with a better understanding of General Relativity and in the investigation of alternative ways of extending the latter to explain the observed Universe.", "Early on [1], it has been shown that a rapid expansion of the early Universe occurs if one just added an $R^{2}$ -term to the Hilbert action.", "Recently, it was realized that this theory belongs in fact to a family of modified gravity models known as $f(R)$ -gravity theories which have the capacity of explaining also the currently accelerated expansion of the Universe, and hence, provide a unifying description of the early and the late-time evolution of the Universe (see for example [2].)", "Motivated by more fundamental theories like string theories [3] or the study of renormalization in curved space-times [4], quadratic invariants such as $R^{2}$ , $R_{\\mu \\nu }R^{\\mu \\nu }$ or $R_{\\mu \\nu \\lambda \\sigma }R^{\\mu \\nu \\lambda \\sigma }$ (as well as $\\Box R$ ) were introduced.", "Recently, however, research on modified gravity theories has extended to include more general and arbitrary functions of the curvature $R$ and the other invariants.", "Many models are proposed that range from power-laws of the form $R^{n}$ , with $n$ positive or negative, or a combination of terms with different powers, to models with more elaborated functionals of the above invariants (see the recent reviews [5], [6].)", "Power-law modified gravity models are mathematically simpler.", "The fixed powers of the curvature in these models are estimated individually by applying the model to study the evolution of the Universe as a whole [7] or to study isolated systems [8], [9].", "The required powers, however, usually do not take integral values and are determined within an interval of possible values.", "Moreover, arguments for the cosmological non-viability of power-law $f(R)$ -gravity are elaborated recently in [10], [11], [12].", "On the other hand, modifying General Relativity by introducing a scalar field in the gravitational sector also brings new possibilities.", "The prototype of such models is the Brans-Dicke scalar-tensor theory [13] that produces a variable gravitational constant using a positive-valued scalar field that does not couple directly with matter but only through geometry thanks to its non-minimal coupling with gravity.", "Therefore, in the wider class of $f(\\phi ,R)$ -modified gravity theories [2], [14] one may combine the advantages brought by the scalar field with those brought by the higher-order geometric invariants.", "In the present paper we shall analyze a $f(\\phi ,R)$ toy model that still belongs to the family of power-law models in which, however, the exponent of the curvature is not fixed but is taken to be space-time dependent by promoting it to the rank of an independent scalar field.", "Namely, we shall examine the possibility of adding to the Hilbert-Einstein action a term of the form $R^{\\phi }$ .", "We apply it to the description of the early and the late-time Universe and investigate its stability with respect to matter." ], [ "The model and its equations of motion", "In this section we shall introduce the model, examine its possible equivalences and expose its qualitative features, and then derive its field equations.", "Our model belongs to the generalized scalar-tensor theories of gravity [14]: $S=\\frac{1}{2}\\int \\mathrm {d}^{4}x\\sqrt{-g}\\left[f(\\phi ,R)-\\eta \\partial _{\\mu }\\phi \\partial ^{\\mu }\\phi \\right]+S_{M}.$ We shall work in units where $8\\pi G=c=1$ throughout the present paper.", "The functions $f(\\phi ,R)$ of the scalar field $\\phi $ and the Ricci curvature $R$ in this class of modified gravity theories are required to be regular but are otherwise arbitrary.", "The contribution of the kinetic term of the scalar field is quantified by the dimensionless parameter $\\eta $ .", "The contributions of ordinary matter are contained in the action $S_{M}=\\int \\mathrm {d}^{4}x\\sqrt{-g}\\mathcal {L}(g_{\\mu \\nu },\\psi )$ of the matter fields $\\psi $ , with possible coupling with $\\phi $ that will not be discussed here.", "The specific choice we make in this paper for the function $f(\\phi ,R)$ is the following $f(\\phi ,R)=R-\\mu ^{2}\\left(\\frac{R}{R_{0}}\\right)^{\\phi }-m^{2}\\phi ^{2}.$ Here $\\mu $ is a parameter, with a mass dimension, whose order of magnitude will be discussed in Sec. .", "$R_{0}$ is a constant, with the dimensions of (length)$^{-2}$ , that is assumed to be very big in order for the ratio to be small at low curvatures and of order unity at high curvatures that reign at the beginning of inflation.", "We shall thus identify $R_{0}$ with the Planck curvature $M^{2}_{Pl}\\sim (10^{19}\\mathrm {GeV})^{2}$ .", "The mass of the scalar field $\\phi $ is $m$ whose order of magnitude will be discussed in Sec. .", "Before deriving the field equations we shall first expose some qualitative features of the model.", "We begin by discussing the possible equivalences of the model with other known families of modified gravity models, then we discuss the qualitative behavior of the effective potential of the scalar field." ], [ "Possible equivalences", "It is well-known that when the scalar field is non-dynamical, that is if we choose $\\eta =0$ in (REF ), the scalar in a $f(\\phi ,R)$ gravity becomes an auxiliary field and it is always possible to turn the model into a pure $f(R)$ model by substituting the equation of motion of the scalar field [2], [14].", "However, for the model (REF ) one does not obtain from the equation of motion of the auxiliary field a simple expansion in terms of $R$ as can be seen from the identity (REF ) below obtained by varying $\\phi $ in (REF ).", "In fact, identity (REF ) can only be solved numerically.", "Thus even if the scalar field in the model (REF ) were non-dynamical it would not be equivalent to a simple $f(R)$ gravity, and hence it would be more convenient to treat the scalar field as an independent field.", "In fact, the scalar field may be viewed as a real parameter that defines a continuous family of gravitational Lagrangians.", "The family contains a Lagrangian suited for high curvatures and another for low curvatures.", "When $R\\gg R_{0}$ , as we shall see in Sec.", ", $\\phi \\rightarrow 0$ and the functional (REF ) becomes $f(\\phi ,R)\\rightarrow R-\\mu ^{2}$ , thus reproducing the Hilbert action with a huge cosmological constant that may serve during inflation provided that $\\mu $ is sufficiently big.", "When $R\\ll R_{0}$ one may, as we shall see in Sec.", ", have a very small but finite value $\\phi _{0}$ whence $f(\\phi ,R)\\rightarrow R-m^{2}\\phi _{0}^{2}$ , reproducing again the Hilbert action but with a tiny cosmological constant suited for the late-times of the expansion of the Universe.", "There is actually a conformal transformation that permits to simplify the model by rendering it linear in the scalar curvature when $\\phi >0$ .", "Indeed, under the conformal transformation $g_{\\mu \\nu }(x)\\rightarrow \\Omega ^{2}(x)g_{\\mu \\nu }(x),$ and provided that the scalar transforms as $\\phi (x)\\rightarrow \\Omega ^{-1}(x)\\phi (x)$ , the Lagrangian density of the gravitational sector $\\sqrt{-g}f(\\phi ,R)$ becomes, when choosing $\\Omega (x)=\\phi (x)$ , $\\sqrt{-g}\\left[\\frac{1-\\xi }{\\phi ^{2}}R+6\\frac{1-\\xi }{\\phi ^{4}}\\partial _{\\mu }\\phi \\partial ^{\\mu }\\phi -\\frac{m^{2}}{\\phi ^{4}}\\right],$ where $\\xi =\\mu ^{2}/R_{0}$ .", "The field redefinition $\\phi ^{2}=(1-\\xi )/\\sigma $ transforms (REF ) into the Lagrangian density of a general scalar-tensor theory of the Brans-Dicke type when the latter is written in the Jordan frame: $\\sqrt{-g}\\left[\\sigma R-\\frac{\\omega (\\sigma )}{\\sigma }\\partial _{\\mu }{\\sigma }\\partial ^{\\mu }\\sigma -V(\\sigma )\\right].$ The Brans-Dicke dimensionless parameter $\\omega $ takes here the value $\\omega =-3/2$ and the potential is $V(\\sigma )=m^{2}\\sigma ^{2}/(1-\\xi )^{2}$ .", "Although the gravitational sector obtained after this transformation is of the Brans-Dicke type, the conformally transformed action with its matter part is not of a Brans-Dicke type theory since the matter Lagrangian density $\\sqrt{-g}\\mathcal {L}(g_{\\mu \\nu },\\psi )$ , even without direct coupling with $\\phi $ , takes after the conformal transformation and the scalar field redefinition the form $\\sqrt{-g}\\sigma ^{2}\\mathcal {L}(\\sigma g_{\\mu \\nu },\\psi )$ and hence the scalar $\\sigma $ couples with matter and is not really the Brans-Dicke scalar of the Jordan frame." ], [ "An $R$ -dependent scalar potential", "Another important qualitative feature of this model resides in the induced potential of the scalar field due to its appearance in the exponent of the Ricci scalar.", "The shape of the potential being dependent on $R$ (see Fig.", "REF ) an effective mass for the scalar field, different from the original mass $m$ , then results.", "Indeed, the induced $R$ -dependent potential reads $2V(\\phi )=m^{2}\\phi ^{2}+\\mu ^{2}\\left(\\frac{R}{R_{0}}\\right)^{\\phi }.$ Its minimum at $\\phi _{0}$ is given by $V^{\\prime }(\\phi _{0})=0$ where the prime denotes a derivative with respect to $\\phi $ .", "Hence we find $\\phi _{0}=-\\frac{\\mu ^{2}}{2m^{2}}\\left(\\frac{R}{R_{0}}\\right)^{\\phi _{0}}\\ln \\frac{R}{R_{0}}.$ The effective mass is then obtained by writing $m^{2}_{eff}=V^{\\prime \\prime }(\\phi _{0})$ .", "The result is $m^{2}_{eff}=m^{2}\\left(1-\\phi _{0}\\ln \\frac{R}{R_{0}}\\right).$ Thus, the effective mass of the scalar field depends on the size of the Universe and, more importantly, depends also on the environment through its dependence on the curvature scalar.", "This latter property is attractive since it reminds us of the so-called Chameleon mechanism [15], [16], [17] that helps avoid positive fifth force tests on the solar system scales by providing a huge mass in the Yukawa coupling with matter.", "Indeed, using sensible estimates for $R$ [6] inside the Earth, at the Earth's atmosphere, or amongst the interstellar gas of the solar system, we have, respectively, the following approximations for the ratio $R/R_{0}$ : $\\sim 10^{-94}$ , $\\sim 10^{-106}$ , and $\\sim 10^{-117}$ .", "However, the fact that the ratio $R/R_{0}$ appears only inside a logarithm in the above expression, the effective mass is at best two orders of magnitude bigger than the original $m$ when the scalar curvature scalar satisfies $R\\ll R_{0}$ (or $R\\gg R_{0}$ .)", "As we shall see in Sec.", ", however, the possible order of magnitude of the original mass $m$ is sufficient to avoid detectable corrections to Newton's law.", "Still, a more interesting application for this varying effective mass of the scalar field may arise when the latter is used as a candidate for dark matter.", "In the present paper, however, we shall not apply the model to a detailed study of the problem of dark matter, restraining ourselves mainly to the phenomenological features of the model due to the high degree of nonlinearity of the equations involved.", "Figure: In this figure we have plotted, for φ>0\\phi >0, the function V(φ)=φ 2 +(R/R 0 ) φ V(\\phi )=\\phi ^{2}+(R/R_{0})^{\\phi } to show how the shape of the scalar field's potential changes with the curvature RR." ], [ "The field equations", "The field equations one obtains when varying the action (REF ) with respect to the scalar field and then with respect to the metric are, respectively, $\\eta \\,\\Box \\,\\phi -m^{2}\\phi -\\frac{\\mu ^{2}}{2}\\left(\\frac{R}{R_{0}}\\right)^{\\phi }\\ln \\frac{R}{R_{0}}=0,$ $G_{\\mu \\nu }&=&T^{M}_{\\mu \\nu }+T^{\\phi }_{\\mu \\nu }-\\frac{\\mu ^{2}}{2}g_{\\mu \\nu }\\left(\\frac{R}{R_{0}}\\right)^{\\phi }\\nonumber \\\\&+&\\mu ^{2}\\left(R_{\\mu \\nu }+g_{\\mu \\nu }\\Box -\\nabla _{\\mu }\\nabla _{\\nu }\\right)\\left[\\frac{\\phi }{R}\\left(\\frac{R}{R_{0}}\\right)^{\\phi }\\right].$ Here, $G_{\\mu \\nu }=R_{\\mu \\nu }-\\frac{1}{2}g_{\\mu \\nu }R$ is the Einstein tensor, $T^{M}_{\\mu \\nu }$ is the energy-momentum tensor of ordinary matter, $T^{\\phi }_{\\mu \\nu }=\\eta (\\partial _{\\mu }\\phi \\partial _{\\nu }\\phi -\\frac{1}{2}g_{\\mu \\nu }\\partial _{\\alpha }\\phi \\partial ^{\\alpha }\\phi )-\\frac{1}{2}g_{\\mu \\nu }m^{2}\\phi ^{2}$ is the energy-momentum tensor of the scalar field $\\phi $ , and $\\Box $ is the D'Alembertian operator.", "Thus, we see that in addition to the energy-momentum tensors of ordinary matter and that of the scalar field, we have a third energy-momentum tensor coming from the interaction of the scalar field with curvature.", "It is this third ingredient that makes it possible to have different sources in the Einstein equations at different curvatures.", "In the next two sections we use these equations to analyze the very early as well as the late-time expansion of the Universe.", "In this paper we shall analyze the spatially flat Friedmann-Lemaître-Robertson-Walker Universe.", "Adopting the spatially flat FLRW metric in the co-moving coordinates $(t,\\textbf {x})$ $\\mathrm {d}s^{2}=-\\mathrm {d}t^{2}+a^{2}(t)\\mathrm {d}\\textbf {x}^{2},$ where $a(t)$ is the positive time-dependent scale factor, and neglecting the spatial dependence of the scalar field $\\phi $ , equation (REF ) reads $\\eta \\,\\ddot{\\phi }+3\\eta \\,H\\dot{\\phi }+m^{2}\\phi +\\frac{\\mu ^{2}}{2}\\left(\\frac{R}{R_{0}}\\right)^{\\phi }\\ln \\frac{R}{R_{0}}=0.$ Neglecting also the matter sources, the equations one obtains when taking the 00-components of (REF ) and its trace are, respectively, $3H^{2}&=&\\frac{\\eta }{2}\\dot{\\phi }^{2}+\\frac{m^{2}}{2}\\phi ^{2}-\\left[3\\mu ^{2}(\\dot{H}+H^{2})\\frac{\\phi }{R}-\\frac{\\mu ^{2}}{2}\\right]\\left(\\frac{R}{R_{0}}\\right)^{\\phi }\\nonumber \\\\&+&3H\\mu ^{2}\\dot{\\left[\\frac{\\phi }{R}\\left(\\frac{R}{R_{0}}\\right)^{\\phi }\\right]}$ $R&=&-\\eta \\,\\dot{\\phi }^{2}+2m^{2}\\phi ^{2}-\\mu ^{2}(\\phi -2)\\left(\\frac{R}{R_{0}}\\right)^{\\phi }\\nonumber \\\\&+&3\\mu ^{2}\\ddot{\\left[\\frac{\\phi }{R}\\left(\\frac{R}{R_{0}}\\right)^{\\phi }\\right]}+9H\\mu ^{2}\\dot{\\left[\\frac{\\phi }{R}\\left(\\frac{R}{R_{0}}\\right)^{\\phi }\\right]},$ where $H=\\dot{a}/a$ is the Hubble expansion rate and an over-dot stands for a cosmic time $t$ -derivative.", "In order to analyze the cosmic evolution during the early times of the Universe we shall rewrite (REF ) and (REF ) assuming $R\\sim R_{0}$ that is $|\\ln \\frac{R}{R_{0}}|\\ll 1$ .", "In order to simplify the subsequent analysis we shall choose $\\eta =0$ in the equations (REF ) to (REF ).", "This allows us to discard the contribution of the kinetic term of $\\phi $ .", "This choice is amply justified near the origin where the equilibrium potential becomes locally flat as it is shown in Fig.", "REF .", "Hence, we can expand $\\phi $ in terms of $\\ln \\frac{R}{R_{0}}$ using equation (REF ) and then relate its time derivative to that of the Ricci scalar $R$ as follows $\\phi &=&-\\frac{\\mu ^{2}}{2m^{2}}\\ln \\frac{R}{R_{0}}+\\mathcal {O}\\left[\\left(\\ln \\frac{R}{R_{0}}\\right)^{3}\\right]\\Rightarrow \\nonumber \\\\\\dot{\\phi }&=&-\\frac{\\mu ^{2}}{2m^{2}}\\frac{\\dot{R}}{R}+\\mathcal {O}\\left[\\left(\\ln \\frac{R}{R_{0}}\\right)^{2}\\right].$ Substituting these approximations in (REF ) and (REF ) we obtain at the zeroth order approximation in $\\ln \\frac{R}{R_{0}}$ the following differential equations $3H^{2}=\\frac{\\mu ^{2}}{2}-\\frac{3\\beta \\mu ^{2}}{2}\\frac{H\\dot{R}}{R^{2}},$ $R=2\\mu ^{2}-\\frac{9\\beta \\mu ^{2}}{2}\\left(\\frac{H\\dot{R}}{R^{2}}+\\frac{\\ddot{R}}{3R^{2}}-\\frac{\\dot{R}^{2}}{R^{3}}\\right),$ where we have introduced the dimensionless ratio $\\beta =\\mu ^{2}/m^{2}$ .", "These final equations both admit a de Sitter solution with a constant $H$ constrained by the parameter $\\mu ^{2}$ to be $H=\\mu /\\sqrt{6}$ .", "This solution may be assigned to the minimum of the first curve depicted in Fig.", "REF near the origin.", "Indeed, on the one hand, we see from (REF ) that any infinitesimal increase in $\\phi $ , giving a positive $\\dot{\\phi }$ , induces a decrease in the curvature, $\\dot{R}<0$ , that sets the system rolling down the successive potentials depicted in Fig.", "REF all the way to the bottom where $R\\ll R_{0}$ .", "On the other hand, combining (REF ) and (REF ) yields, when neglecting $\\ddot{R}/R^{2}$ and $\\dot{R}^{2}/R^{3}$ , the following approximate differential equation $\\frac{\\dot{H}}{H}=\\frac{\\beta \\mu ^{2}}{4}\\frac{\\dot{R}}{R^{2}},$ where we have used $R=6\\dot{H}+12H^{2}$ in the left-hand side of (REF ).", "This indeed shows that the Hubble parameter $H$ decreases with the potential from the maximum value $H=\\mu /\\sqrt{6}$ it takes at the origin $\\phi =0$ .", "Thus, with an estimate of about $H^{2}_{I}\\sim 10^{20\\sim 38}(\\mathrm {eV})^{2}$ for the Hubble flow at inflation [6], the huge order of magnitude that must be imposed on the mass parameter $\\mu ^{2}$ follows.", "In the next section we will see that this decrease of $H$ continues even though on the lower curves of Fig.", "REF the scalar $\\phi $ becomes decreasing towards the origin again when the scalar curvature decreases below a given value of the curvature $R$ ." ], [ "The late-time expansion", "In order to analyze the cosmic evolution during the late-times of the Universe we shall use (REF ) and (REF ) assuming this time that $R\\ll R_{0}$ , that is, $|\\ln \\frac{R}{R_{0}}|\\gg 1$ .", "Furthermore, we shall assume that during this cosmic expansion the field remains constantly in equilibrium at the bottom of each of the successive curves in Fig.", "REF .", "That is, given the smallness of the Hubble flow during the late-times, the scalar field evolves adiabatically with cosmic expansion, acquiring a very small non-vanishing variation $\\phi $ only due to the continuous deformation of its effective potential as a result of the changing in the size of the Universe.", "We can thus neglect the kinetic terms by setting $\\eta =0$ .", "Therefore, equation (REF ) reads $\\left(\\frac{R}{R_{0}}\\right)^{\\phi }=-\\frac{2m^{2}\\phi }{\\mu ^{2}\\ln \\frac{R}{R_{0}}}\\ll 1.$ Using this approximation equations (REF ) and (REF ), respectively, read at the zeroth order approximation in $(\\ln \\frac{R}{R_{0}})^{-1}$ as follows $3H^{2}=\\frac{m^{2}\\phi ^{2}}{2}-6m^{2}\\phi ^{2}\\frac{H\\dot{\\phi }}{R},$ $R=2m^{2}\\phi ^{2}-18m^{2}\\phi ^{2}\\frac{H\\dot{\\phi }}{R}.$ We see that the first possibility is to have again a de Sitter solution whenever the scalar field settles down and takes on a constant value $\\phi _{0}$ .", "The constant Hubble flow then would be $H^{2}_{0}=m^{2}\\phi ^{2}_{0}/6$ .", "From the currently observed Hubble parameter $H^{2}_{0}\\sim (10^{-33}\\mathrm {eV})^{2}$ the order of magnitude of $m^{2}\\phi ^{2}_{0}$ follows but this does not imply that the mass $m$ of the scalar field would have to be fine-tuned because it is the value of the scalar field $\\phi _{0}$ that becomes very small at low curvature.", "This steams from identity (REF ) which, after multiplying the two sides of the identity by $\\phi $ and taking $R_{0}\\sim 10^{38}(\\mathrm {GeV})^{2}$ , $R\\sim H_{0}^{2}$ and $\\mu ^{2}\\sim 10^{38}(\\mathrm {eV})^{2}$ , yields the value $\\phi _{0}\\sim 10^{-105}$ in Planck units.", "Therefore, the mass $m$ can be as high as $\\sim 10^{72}\\mathrm {eV}$ .", "As indicated in Sec.", ", this order of magnitude is high enough to avoid detectable corrections to Newton's law.", "As mentioned at the end of section , the Hubble parameter actually continues to decrease due to the following fact.", "First, combining (REF ) and (REF ) yields, $\\frac{\\dot{H}}{H}=\\frac{m^{2}\\phi ^{2}\\dot{\\phi }}{R}.$ Next, by differentiating identity (REF ) once with respect to time we find $\\dot{\\phi }\\left(\\phi \\ln \\frac{R}{R_{0}}-1\\right)=-\\frac{\\dot{R}\\phi }{R}\\left[\\phi +\\left(\\ln \\frac{R}{R_{0}}\\right)^{-1}\\right],$ showing that $\\dot{\\phi }$ vanishes at $\\phi _{*}=-(\\ln \\frac{R_{*}}{R_{0}})^{-1}$ and changes sign to become negative.", "Therefore, below the scalar curvature $R_{*}$ (from (REF ), we have $R_{*}=R_{0}\\exp (-\\sqrt{2e/\\beta })$ ) the scalar field decreases and approaches the origin again.", "Therefore, the term in the right-hand side of (REF ) is in fact negative.", "This shows that at low curvature the scalar decreases, i.e.", "$\\dot{\\phi }<0$ , and the Hubble parameter decreases too." ], [ "Matter stability", "The criterion of matter stability [18] constitutes a crucial test for any modified gravity model in order to become a realistic candidate for a theory of gravity.", "In order to examine the matter stability one assumes [18], [6] that the curvature scalar decomposes as $R=R_{M}+R_{p}$ where $R_{p}$ represents a very small perturbation brought by the modified gravity terms to the scalar curvature $R_{M}={T^{M}}_{\\mu }^{\\,\\mu }$ created by matter through General Relativity, such that $R_{p}\\ll R_{M}$ .", "One then neglects spatial dependence and approximates the D'Alembertian by $\\Box \\approx -\\partial _{t}^{2}$ .", "The model is said to be stable in the presence of matter if the differential equation obtained for the perturbation $R_{p}$ , keeping only the linear terms in $R_{p}$ and its derivatives, is of the form $\\ddot{R}_{p}+\\alpha R_{p}+\\mathrm {const}.=0$ where $\\alpha $ is a positive-valued function of $R_{M}$ .", "Given that our model contains also an independent dynamical scalar field we shall treat the stability problem in two steps.", "First we shall check the stability of the scalar field under perturbations for a given curvature of the background.", "That is, we shall examine the oscillations of the scalar field about its minimum $\\phi _{0}$ when put on one of the fixed curves at the bottom of Fig.", "REF .", "Denoting by $\\phi _{p}$ the small deviation of $\\phi $ from its equilibrium at $\\phi _{0}$ , equation (REF ) implies $\\eta \\ddot{\\phi _{p}}+m^{2}\\left(\\phi _{p}+\\phi _{0}\\right)+\\frac{\\mu ^{2}}{2}\\left(\\frac{R_{M}}{R_{0}}\\right)^{\\phi _{0}+\\phi _{p}}\\ln \\frac{R_{M}}{R_{0}}=0.$ Since $\\phi _{0}$ is the value of the scalar field at equilibrium, we also have from (REF ) that $(R_{M}/R_{0})^{\\phi _{0}}=-2m^{2}\\phi _{0}/(\\mu ^{2}\\ln \\frac{R_{M}}{R_{0}})$ .", "Substituting this in (REF ), the latter becomes $\\eta \\ddot{\\phi _{p}}+m^{2}\\phi _{p}+m^{2}\\phi _{0}\\left[1-\\left(\\frac{R_{M}}{R_{0}}\\right)^{\\phi _{p}}\\right]=0.$ Given the smallness of the second term in square brackets, the differential equation satisfied by the perturbation is thus, to a good approximation, of the form $\\ddot{\\phi }+(m^{2}/\\eta )\\phi _{p}+\\mathrm {const}.=0$ .", "Therefore, provided only that $\\eta >0$ , i.e.", "that the scalar field is non-phantom, the latter is stable against perturbations $\\phi _{p}$ that come from its kinetic term.", "Now that we explicitly verified that on each potential curve the local deviations of the scalar field from its corresponding equilibrium positions are stable, we may proceed to the analysis of the stability of curvature under perturbations $R_{p}$ caused by the presence of the scalar field.", "That analysis may now be carried out by safely neglecting the kinetic term of the scalar field.", "That is, we set $\\eta =0$ and treat $\\phi $ as a non-dynamical field that remains at the bottom of each of its potential curves.", "Taking the trace of equation (REF ) we find $3\\mu ^{2}\\ddot{\\left[\\frac{\\phi }{R}\\left(\\frac{R}{R_{0}}\\right)^{\\phi }\\right]}-{R}_{p}-\\mu ^{2}\\left(\\phi -2\\right)\\left(\\frac{R}{R_{0}}\\right)^{\\phi }+2m^{2}\\phi ^{2}=0.$ Substituting $(R/R_{0})^{\\phi }$ using identity (REF ) valid for $\\eta =0$ , and then performing the second time-derivative in the above equation, the latter reads, at the first order in $(\\ln \\frac{R}{R_{0}})^{-1}$ , $\\frac{12m^{2}\\phi \\ddot{\\phi }}{R\\ln \\frac{R}{R_{0}}}-\\frac{6m^{2}\\phi ^{2}\\ddot{R}}{R^{2}\\ln \\frac{R}{R_{0}}}+R_{p}-\\frac{2m^{2}(\\phi ^{2}-2\\phi )}{\\ln \\frac{R}{R_{0}}}-2m^{2}\\phi ^{2}=0.$ Identity (REF ), however, implies that the first term in the above equation is actually of the order $(\\ln \\frac{R}{R_{0}})^{-2}$ and must accordingly be dropped out from the equation.", "Indeed, differentiating once with respect to time the two sides of identity (REF ), we learn that at the leading order, $\\ddot{\\phi }\\sim -\\phi \\ddot{R}/(R\\ln \\frac{R}{R_{0}})$ , and the first term in (REF ) is thus irrelevant at the displayed order.", "By keeping only the leading terms from each category in (REF ) we obtain the following differential equation for $R_{p}$ $\\frac{-6m^{2}\\phi ^{2}}{R^{2}\\ln \\frac{R}{R_{0}}}\\ddot{R}_{p}+R_{p}-2m^{2}\\phi ^{2}=0.$ Now from equation (REF ) we deduce that $2m^{2}\\phi ^{2}$ constitutes only a positive fraction $\\zeta $ of the perturbation $R_{p}$ .", "Substituting this in (REF ), together with the approximations $R^{2}\\approx R_{M}^{2}(1+2\\frac{R_{p}}{R_{M}})$ and $\\ln \\frac{R}{R_{0}}\\approx \\ln \\frac{R_{M}}{R_{0}}$ , then yields $\\ddot{R}_{p}+\\left[\\frac{2(1-\\zeta )}{3\\zeta }R_{M}\\ln \\frac{R_{0}}{R_{M}}\\right]R_{p}+\\mathrm {const}.=0,$ where in $\\mathrm {const}.$ we have collected all the terms depending only on $R_{M}$ and $R_{0}$ .", "The coefficient that multiplies $R_{p}$ in this second order differential equation being positive demonstrates the stability of this toy model in the presence of matter." ], [ "Summary and discussion", "We have studied in the present paper a toy model for a $f(\\phi ,R)$ gravity in which the scalar field plays the role of a parameter that permits to continuously switch from different gravitational Lagrangians according to the curvature of the environment.", "The parameter is not free but constrained by the dynamics of the model itself.", "Hence, in a sense, the model exhibits a chameleon behavior vis-à-vis the structure of its Lagrangian.", "We saw that during the expansion of the Universe, the action becomes at high curvatures the Hilbert-Einstein action augmented with a huge cosmological constant while at low curvatures it becomes the Hilbert-Einstein action with a tiny cosmological constant.", "We have seen that by identifying the constant parameter $R_{0}$ with the curvature at the beginning of inflation, any positive increase in $\\phi $ induces a decrease in the curvature all the way to its present very low value.", "Since the potential of the scalar field begins to decrease at the very instant when the field $\\phi $ leaves the origin towards the positive values, we may assign this behavior to the process of reheating during which the huge potential energy at the beginning is transformed into radiation that fills the Universe at the end of inflation.", "However, the detailed process of reheating in this model still remains to be examined more precisely.", "Finally, we would like to end this paper by describing what happens if the field $\\phi $ leaves the origin from its equilibrium position on the highest curve in Fig.", "REF towards the negative values.", "In that case each of the different curves displayed in Fig.", "REF gets a symmetric image at the left of the vertical axis.", "Therefore, when departing from the curve situated at the top of the figure and infinitesimally going to the left, the shape of the potential of the scalar field changes continuously until it reaches the lower curves as in the case of a positive scalar field.", "However, during this decrease of the potential the curvature scalar $R$ increases instead and goes way beyond its initial value $R_{0}$ .", "Hence, in this toy model there is no classical mechanism that prevents $R$ from reaching infinite values." ] ]
1403.0261
[ [ "Bootstrapping and Askey-Wilson polynomials" ], [ "Abstract The mixed moments for the Askey-Wilson polynomials are found using a bootstrapping method and connection coefficients.", "A similar bootstrapping idea on generating functions gives a new Askey-Wilson generating function.", "An important special case of this hierarchy is a polynomial which satisfies a four term recurrence, and its combinatorics is studied." ], [ "Introduction", "The Askey-Wilson polynomials [1] $p_n(x;a,b,c,d|q)$ are orthogonal polynomials in $x$ which depend upon five parameters: $a$ , $b$ , $c$ , $d$ and $q$ .", "In [2] Berg and Ismail use a bootstrapping method to prove orthogonality of Askey-Wilson polynomials by initially starting with the orthogonality of the $a=b=c=d=0$ case, the continuous $q$ -Hermite polynomials, and successively proving more general orthogonality relations, adding parameters along the way.", "In this paper we implement this idea in two different ways.", "First, using successive connection coefficients for two sets of orthogonal polynomials, we will find explicit formulas for generalized moments of Askey-Wilson polynomials, see Theorem REF .", "This method also gives a heuristic for a relation between the two measures of the two polynomial sets, see Remark REF , which is correct for the Askey-Wilson hierarchy.", "Using this idea we give a new generating function (Theorem REF ) for Askey-Wilson polynomials when $d=0.$ The second approach is to assume the two sets of polynomials have generating functions which are closely related, up to a $q$ -exponential factor.", "We prove in Theorem REF that if one set is an orthogonal set, the second set has a recurrence relation of predictable order, which may be greater than three.", "We give several examples using the Askey-Wilson hierarchy.", "Finally we consider a more detailed example of the second approach, using a generating function to define a set of polynomials called the discrete big $q$ -Hermite polynomials.", "These polynomials satisfy a 4-term recurrence relation.", "We give the moments for the pair of measures for their orthogonality relations.", "Some of the combinatorics for these polynomials is given in § .", "Finally we record in Proposition REF a possible $q$ -analogue of the Hermite polynomial addition theorem.", "We shall use basic hypergeometric notation, which is in Gasper-Rahman [6] and Ismail [7]." ], [ "Askey-Wilson polynomials and connection coefficients", "The connection coefficients are defined as the constants obtained when one expands one set of polynomials in terms of another set of polynomials.", "For the Askey-Wilson polynomials [7] $p_n(x;a,b,c,d|q)= \\frac{(ab,ac,ad)_n}{a^n}{}_{4}\\phi _{3} \\left( \\left.\\begin{matrix}q^{-n},abcdq^{n-1},ae^{i\\theta },ae^{-i\\theta }\\\\ab,ac,ad\\\\\\end{matrix}\\right| q;q\\right), \\quad x=\\cos \\theta $ we shall use the connection coefficients obtained by successively adding a parameter $(a,b,c,d)=(0,0,0,0)\\rightarrow (a,0,0,0)\\rightarrow (a,b,0,0)\\rightarrow (a,b,0,0)\\rightarrow (a,b,c,0)\\rightarrow (a,b,c,d).$ Using a simple general result on orthogonal polynomials, we derive an almost immediate proof of an explicit formula for the mixed moments of Askey-Wilson polynomials.", "First we set the notation for an orthogonal polynomial set $p_n(x).$ Let $\\mathcal {L}_p$ be the linear functional on polynomials for which orthogonality holds $\\mathcal {L}_p(p_m(x)p_n (x)) =h_n \\delta _{mn}, \\quad 0\\le m,n.$ Definition 2.1 The mixed moments of $\\mathcal {L}_p$ are $\\mathcal {L}_p(x^np_m(x)),\\quad 0\\le m,n.$ The main tool is the following Proposition, which allows the computation of mixed moments of one set of orthogonal polynomials from another set if the connection coefficients are known.", "Proposition 2.2 Let $R_n(x)$ and $S_n(x)$ be orthogonal polynomials with linear functionals $\\mathcal {L}_R$ and $\\mathcal {L}_S$ , respectively, such that $\\mathcal {L}_R(1)=\\mathcal {L}_S(1) = 1$ .", "Suppose that the connection coefficients are $R_k(x) = \\sum _{i=0}^k c_{k,i} S_i(x).$ Then $\\mathcal {L}_S(x^n S_m(x)) = \\sum _{k=0}^n \\frac{\\mathcal {L}_R(x^n R_k(x))}{\\mathcal {L}_R(R_k(x)^2)}c_{k,m} \\mathcal {L}_S(S_m(x)^2).$ If we multiply both sides of (REF ) by $S_m(x)$ and apply $\\mathcal {L}_S$ , we have $\\mathcal {L}_S(R_k(x)S_m(x)) = c_{k,m} \\mathcal {L}_S(S_m(x)^2).$ Then by expanding $x^n$ in terms of $R_k(x)$ $x^n=\\sum _{k=0}^n \\frac{\\mathcal {L}_R(x^n R_k(x))}{\\mathcal {L}_R(R_k(x)^2)} R_k(x)$ we find $\\mathcal {L}_S(x^n S_m(x)) = \\mathcal {L}_S\\left( \\sum _{k=0}^n\\frac{\\mathcal {L}_R(x^n R_k(x))}{\\mathcal {L}_R(R_k(x)^2)} R_k(x) S_m(x)\\right)= \\sum _{k=0}^n \\frac{\\mathcal {L}_R(x^n R_k(x))}{\\mathcal {L}_R(R_k(x)^2)}c_{k,m} \\mathcal {L}_S(S_m(x)^2).$ Remark 2.3 One may also use the idea of Proposition REF to give a heuristic for representing measures of the linear functionals.", "Putting $m=0,$ if representing measures were absolutely continuous, say $w_R(x)dx$ for $R_n(x)$ , and $w_S(x)dx$ for $S_n(x)$ then one might guess that $w_S(x) = w_R(x) \\sum _{k=0}^\\infty \\frac{R_k(x)}{\\mathcal {L}_R(R_k(x)^2)} c_{k,0}.$ For the rest of this section we will compute the mixed moments $\\mathcal {L}_p(x^n p_m(x))$ for the Askey-Wilson polynomials using Proposition REF starting from the $q$ -Hermite polynomials.", "Let $\\mathcal {L}_{a,b,c,d}$ be the linear functional for $p_n(x;a,b,c,d|q)$ satisfying $\\mathcal {L}_{a,b,c,d}(1)=1$ .", "Then $\\mathcal {L}=\\mathcal {L}_{0,0,0,0}$ , $\\mathcal {L}_{a}=\\mathcal {L}_{a,0,0,0}$ , $\\mathcal {L}_{a,b}=\\mathcal {L}_{a,b,0,0}$ , and $\\mathcal {L}_{a,b,c}=\\mathcal {L}_{a,b,c,0}$ are the linear functionals for these polynomials: $q$ -Hermite, $H_n(x|q)=p_n(x;0,0,0,0|q)$ , the big $q$ -Hermite $H_n(x;a|q)=p_n(x;a,0,0,0|q)$ , the Al-Salam-Chihara $Q_n(x;a,b|q)=p_n(x;a,b,0,0|q)$ , and the dual $q$ -Hahn $p_n(x;a,b,c|q)=p_n(x;a,b,c,0|q)$ .", "The $L^2$ -norms are given by [7] $\\mathcal {L}(H_n(x|q) H_m(x|q)) &= (q)_n\\delta _{mn},\\\\\\mathcal {L}_a(H_n(x;a|q) H_m(x;a|q)) &= (q)_n\\delta _{mn},\\\\\\mathcal {L}_{a,b}(Q_n(x;a,b|q) Q_m(x;a,b|q)) &= (q,ab)_n\\delta _{mn},\\\\\\mathcal {L}_{a,b,c}(p_n(x;a,b,c|q) p_m(x;a,b,c|q)) &= (q,ab,ac,bc)_n\\delta _{mn},\\\\\\mathcal {L}_{a,b,c,d}(p_n(x;a,b,c,d|q) p_m(x;a,b,c,d|q)) &=\\frac{(q,ab,ac,ad,bc,bd,cd,abcdq^{n-1})_n}{(abcd)_{2n}}\\delta _{mn}.$ To apply Proposition REF , we need the following connection coefficient formula for the Askey-Wilson polynomials given in [1] $\\frac{p_n(x;A,b,c,d|q)}{(q,bc,bd,cd)_n}=\\sum _{k=0}^n \\frac{p_k(x;a,b,c,d|q)}{(q,bc,bd,cd)_k}\\times \\frac{a^{n-k}(A/a)_{n-k}(A bcdq^{n-1})_k}{(abcdq^{k-1})_k (q,abcdq^{2k})_{n-k}}.$ The following four identities are special cases of (REF ): $H_n(x|q) &= \\sum _{k=0}^n \\genfrac[]{0.0pt}{}{n}{k}_{q} H_k(x;a|q) a^{n-k},\\\\H_n(x;a|q) &=\\sum _{k=0}^n \\genfrac[]{0.0pt}{}{n}{k}_{q} Q_k(x;a,b|q) b^{n-k},\\\\Q_n(x;a,b|q) &=(ab)_n \\sum _{k=0}^n \\genfrac[]{0.0pt}{}{n}{k}_{q} \\frac{p_k(x;a,b,c|q)}{(ab)_k} c^{n-k},\\\\\\frac{p_n(x;b,c,d|q)}{(q,bc,bd,cd)_n}&=\\sum _{k=0}^n \\frac{p_k(x;a,b,c,d|q)}{(q,bc,bd,cd)_k}\\cdot \\frac{a^{n-k}}{(abcdq^{k-1})_k (q,abcdq^{2k})_{n-k}}.$ For the initial mixed moment we need the following result proved independently by Josuat-Vergès [11] and Cigler [3] $\\mathcal {L}(x^n H_m(x;q)) = \\frac{(q)_m}{2^n} \\overline{P}(n,m),$ where $\\overline{P}(n,m) = \\sum _{k=m}^n \\left( \\binom{n}{\\frac{n-k}{2}} - \\binom{n}{\\frac{n-k}{2}-1}\\right)(-1)^{(k-m)/2} q^{\\binom{(k-m)/2+1}{2}}\\genfrac[]{0.0pt}{}{\\frac{k+m}{2}}{\\frac{k-m}{2}}_{q}.$ We shall use the convention $\\binom{n}{k} = \\genfrac[]{0.0pt}{}{n}{k}_{q} = 0$ if $k<0$ , $k>n$ , or $k$ is not an integer.", "Thus $\\overline{P}(n,m)=0$ if $n\\lnot \\equiv m \\mod {2}$ .", "Theorem 2.4 We have $\\mathcal {L}_a(x^n H_m(x;a|q)) &= \\frac{(q)_m}{2^n}\\sum _{\\alpha \\ge 0} \\overline{P}(n,\\alpha +m)\\genfrac[]{0.0pt}{}{\\alpha +m}{m}_{q} a^{\\alpha }, \\\\\\mathcal {L}_{a,b}(x^n Q_m(x;a,b|q)) &= \\frac{(q,ab)_m}{2^n}\\sum _{\\alpha ,\\beta \\ge 0} \\overline{P}(n,\\alpha +\\beta +m)\\genfrac[]{0.0pt}{}{\\alpha +\\beta +m}{\\alpha ,\\beta ,m}_{q} a^{\\alpha } b^{\\beta }, \\\\\\mathcal {L}_{a,b,c}(x^n p_m(x;a,b,c|q))&= \\frac{(q,ac,bc)_m}{2^n} \\sum _{\\alpha ,\\beta ,\\gamma \\ge 0}\\overline{P}(n,\\alpha +\\beta +\\gamma +m)\\genfrac[]{0.0pt}{}{\\alpha +\\beta +\\gamma +m}{\\alpha ,\\beta ,\\gamma ,m}_{q} \\\\&\\quad \\times a^{\\alpha } b^{\\beta } c^{\\gamma } (ab)_{\\gamma +m},\\\\\\mathcal {L}_{a,b,c,d}(x^n p_m(x;a,b,c,d|q)) &=\\frac{1}{2^n}\\sum _{\\alpha ,\\beta ,\\gamma ,\\delta \\ge 0}a^\\alpha b^\\beta c^\\gamma d^\\delta \\overline{P}(n,\\alpha +\\beta +\\gamma +\\delta ) \\genfrac[]{0.0pt}{}{\\alpha +\\beta +\\gamma +\\delta }{\\alpha ,\\beta ,\\gamma ,\\delta }_{q} \\\\&\\quad \\times \\frac{(bd)_{\\alpha }(cd)_{\\alpha }(bc)_{\\alpha +\\delta }}{(abcd)_\\alpha }\\cdot \\frac{(ab,ac,ad)_m (q^{\\alpha };q^{-1})_m}{a^m (abcdq^{\\alpha })_m}.$ By (REF ), Proposition REF and (REF ), $\\mathcal {L}_a(x^n H_m(x;a|q)) &=\\sum _{k=0}^n \\frac{\\mathcal {L}(x^n H_k(x|q))}{\\mathcal {L}(H_k(x)^2)}\\genfrac[]{0.0pt}{}{k}{m}_{q} a^{k-m}\\mathcal {L}_a(H_m(x;a|q)^2)\\\\&= \\frac{(q)_m}{2^n}\\sum _{k=0}^n \\overline{P}(n,k)\\genfrac[]{0.0pt}{}{k}{m}_{q} a^{k-m}.$ Equations (), (), and () can be proved similarly using the connection coefficient formulas (), (), and ().", "Letting $m=0$ in () we obtain a formula for the $n$ th moment of the Askey-Wilson polynomials.", "Corollary 2.5 We have $\\mathcal {L}_{a,b,c,d}(x^n) =\\frac{1}{2^n}\\sum _{\\alpha ,\\beta ,\\gamma ,\\delta \\ge 0}a^\\alpha b^\\beta c^\\gamma d^\\delta \\overline{P}(n,\\alpha +\\beta +\\gamma +\\delta ) \\genfrac[]{0.0pt}{}{\\alpha +\\beta +\\gamma +\\delta }{\\alpha ,\\beta ,\\gamma ,\\delta }_{q}\\frac{(bd)_{\\alpha }(cd)_{\\alpha }(bc)_{\\alpha +\\delta }}{(abcd)_\\alpha }.$ In [13] the authors found a slightly different formula $\\mathcal {L}_{a,b,c,d}(x^n)=\\frac{1}{2^n}\\sum _{\\alpha ,\\beta ,\\gamma ,\\delta \\ge 0}a^\\alpha b^\\beta c^\\gamma d^\\delta \\overline{P}(n,\\alpha +\\beta +\\gamma +\\delta ) \\genfrac[]{0.0pt}{}{\\alpha +\\beta +\\gamma +\\delta }{\\alpha ,\\beta ,\\gamma ,\\delta }_{q}\\frac{(ad)_{\\beta +\\gamma }(ac)_{\\beta }(bd)_{\\gamma }}{(abcd)_{\\beta +\\gamma }},$ which can be rewritten using the symmetry in $a,b,c,d$ as $\\mathcal {L}_{a,b,c,d}(x^n)=\\frac{1}{2^n}\\sum _{\\alpha ,\\beta ,\\gamma ,\\delta \\ge 0}a^\\alpha b^\\beta c^\\gamma d^\\delta \\overline{P}(n,\\alpha +\\beta +\\gamma +\\delta ) \\genfrac[]{0.0pt}{}{\\alpha +\\beta +\\gamma +\\delta }{\\alpha ,\\beta ,\\gamma ,\\delta }_{q}\\frac{(bc)_{\\alpha +\\delta }(bd)_{\\alpha }(ac)_{\\delta }}{(abcd)_{\\alpha +\\delta }}.$ One can obtain (REF ) from (REF ) by applying the $_3\\phi _1$ -transformation [6] to the $\\alpha $ -sum after fixing $\\gamma $ , $\\delta $ , and $N=\\alpha +\\beta $ .", "We next check if the heuristic in Remark REF leads to correct results in these cases.", "The absolutely continuous Askey-Wilson measure $w(x;a,b,c,d|q)$ with total mass 1 for $0<q<1$ , $\\max (|a|,|b|,|b|,|d|)<1$ is, if $x=\\cos \\theta $ , $\\theta \\in [0,\\pi ]$ , $w(\\cos \\theta ;a,b,c,d|q) &= \\frac{(q,ab,ac,ad,bc,bd,cd)_\\infty }{2\\pi (abcd)_\\infty } \\\\&\\quad \\times \\frac{(e^{2i\\theta },e^{-2i\\theta })_\\infty }{(ae^{i\\theta },ae^{-i\\theta },be^{i\\theta },be^{-i\\theta },ce^{i\\theta },ce^{-i\\theta },de^{i\\theta },de^{-i\\theta })_\\infty }.$ Then the measures for the $q$ -Hermite $H_n(x|q)$ , the big $q$ -Hermite $H_n(x;a|q)$ , the Al-Salam-Chihara $Q_n(x;a,b|q)$ , and the dual $q$ -Hahn $p_n(x;a,b,c|q)$ are, respectively, $w(\\cos \\theta ;0,0,0,0|q)$ , $w(\\cos \\theta ;a,0,0,0|q)$ , $w(\\cos \\theta ;a,b,0,0|q)$ , and $w(\\cos \\theta ;a,b,c,0|q)$ .", "Notice that each successive measure comes from the previous measure by inserting infinite products.", "Example 2.6 Let $R_k(x) = H_k(x|q)$ and $S_k(x)=H_k(x;a|q)$ so that $w_S(\\cos \\theta )=w_R(\\cos \\theta )\\frac{1}{(ae^{i\\theta },ae^{-i\\theta })_\\infty }.$ In this case, we have $\\mathcal {L}_R(R_k(x)^2) =(q)_k$ and $R_k(x) = \\sum _{i=0}^k c_{k,i} S_i(x),$ where $c_{k,i} = \\genfrac[]{0.0pt}{}{k}{i}_{q}a^{k-i}$ .", "By the heuristic in Remark REF , $w_S(x) = w_R(x) \\sum _{k=0}^\\infty \\frac{R_k(x)}{(q)_k} a^k=w_R(x)\\frac{1}{(ae^{i\\theta } , ae^{-i\\theta })_\\infty },$ where we have used the $q$ -Hermite generating function [12].", "Example 2.7 Let $R_k(x) = H_k(x;a|q)$ and $S_k(x)=Q_k(x;a,b|q)$ so that $w_S(\\cos \\theta )=w_R(\\cos \\theta )\\frac{(ab)_\\infty }{(be^{i\\theta },be^{-i\\theta })_\\infty }.$ In this case, we have $\\mathcal {L}_R(R_k(x)^2) =(q)_k$ and $R_k(x) = \\sum _{i=0}^k c_{k,i} S_i(x),$ where $c_{k,i} = \\genfrac[]{0.0pt}{}{k}{i}_{q}b^{k-i}$ .", "By the heuristic in Remark REF , $w_S(x) = w_R(x) \\sum _{k=0}^\\infty \\frac{R_k(x)}{(q)_k} c^k=w_R(x)\\frac{(ab)_\\infty }{(be^{i\\theta } , be^{-i\\theta })_\\infty },$ where we have used the big $q$ -Hermite generating function [12].", "Example 2.8 Let $R_k(x) = Q_k(x;a,b|q)$ and $S_k(x)=p_k(x;a,b,c|q)$ so that $w_S(\\cos \\theta )=w_R(\\cos \\theta )\\frac{(ac,bc)_\\infty }{(ce^{i\\theta },ce^{-i\\theta })_\\infty }.$ In this case, we have $\\mathcal {L}_R(R_k(x)^2) =(q,ab)_k$ and $R_k(x) = \\sum _{i=0}^k c_{k,i} S_i(x),$ where $c_{k,i} = \\genfrac[]{0.0pt}{}{k}{i}_{q}\\frac{(ab)_k}{(ab)_i}c^{k-i}$ .", "By the heuristic in Remark REF , $w_S(x) = w_R(x) \\sum _{k=0}^\\infty \\frac{R_k(x)}{(q,ab)_k} (ab)_k c^k=w_R(x)\\frac{(ac,bc)_\\infty }{(ce^{i\\theta } , ce^{-i\\theta })_\\infty },$ where we have used the Al-Salam-Chihara generating function [12].", "Notice that in the above example we used the known generating function for the Al-Salam-Chihara polynomials $Q_n(x;a,b|q)$ .", "If we apply the same steps to $R_k(x)=p_k(x;a,b,c,0|q)$ and $S_k(x)=p_k(x;a,b,c,d|q)$ , a new generating function appears.", "Theorem 2.9 We have $(abct)_\\infty \\sum _{k=0}^\\infty \\frac{p_k(x;a,b,c,0|q)}{(q,abct)_k} t^k= \\frac{(at,bt,ct)_\\infty }{(te^{i\\theta } , te^{-i\\theta })_\\infty }.$ We must show $(abct)_\\infty \\sum _{n=0}^\\infty \\frac{t^n}{(q,abct)_n} p_n(x;a,b,c,0|q)=\\frac{(bt,ct)_\\infty }{(te^{i\\theta },te^{-i\\theta })_\\infty }(at)_\\infty .$ Using the Al-Salam-Chihara generating function and the $q$ -binomial theorem [6], (REF ) is equivalent to $\\sum _{n=0}^N \\frac{p_n(x;b,c,0,0|q)}{(q)_n} \\frac{(-a)^{N-n}q^{\\binom{N-n}{2}}}{(q)_{N-n}}=\\sum _{n=0}^N \\frac{p_n(x;a,b,c,0|q)}{(q)_n} \\frac{(-abcq^n)^{N-n}q^{\\binom{N-n}{2}}}{(q)_{N-n}}.$ Now use the connection coefficients $p_n(x;b,c,0,0|q)= (bc)_n \\sum _{k=0}^n \\genfrac[]{0.0pt}{}{n}{k}_{q}p_k(x;a,b,c,0|q)\\frac{a^{n-k}}{(bc)_{k}},$ to show that (REF ) follows from $\\sum _{n=k}^N \\frac{(bc)_n}{(q)_n} \\genfrac[]{0.0pt}{}{n}{k}_{q}\\frac{a^{N-k}}{(bc)_k}\\frac{(-1)^{N-n}q^{\\binom{N-n}{2}}}{(q)_{N-n}}=\\frac{1}{(q)_k}\\frac{(-abcq^k)^{N-k}}{(q)_{N-k}} q^{\\binom{N-k}{2}}.$ This summation is a special case of the $q$ -Vandermonde theorem [6].", "A generalization of Theorem REF to Askey-Wilson polynomials is given in [10].", "A natural generalization of the mixed moments in () is $\\mathcal {L}_{a,b,c,d}(x^n p_m(x;a,b,c,d|q)p_\\ell (x;a,b,c,d|q)).$ For general orthogonal polynomials Viennot has given a combinatorial interpretation for $\\mathcal {L}(x^np_mp_\\ell )$ in terms of weighted Motzkin paths.", "An explicit formula when $p_n= p_n(x;a,b,c,d|q)$ may be given using (REF ) and a $q$ -Taylor expansion [9], but we do not state the result here." ], [ "Generating functions", "In §  we noted the following generating functions for our bootstrapping polynomials: continuous $q$ -Hermite $H_n(x|q)$ , continuous big $q$ -Hermite $H_n(x;a|q)$ , and Al-Salam-Chihara $Q_n(x;a,b|q)$ $\\sum _{n=0}^\\infty \\frac{H_n(x|q)}{(q)_n} t^n =\\frac{1}{(te^{i\\theta },te^{-i\\theta })_\\infty },$ $\\sum _{n=0}^\\infty \\frac{H_n(x;a|q)}{(q)_n} t^n= \\frac{(at)_\\infty }{(te^{i\\theta } , te^{-i\\theta })_\\infty },$ $\\sum _{n=0}^\\infty \\frac{Q_n(x;a,b|q)}{(q)_n}t^n= \\frac{(at,bt)_\\infty }{(te^{i\\theta } , te^{-i\\theta })_\\infty }.$ Note that (REF ) is obtained from (REF ) by multiplying by $(at)_\\infty $ and (REF ) is obtained from (REF ) by multiplying by $(bt)_\\infty $ .", "However, if we multiply (REF ) by $(ct)_\\infty ,$ we no longer have a generating function for orthogonal polynomials.", "It is the generating function for polynomials which satisfy a recurrence relation of finite order, but longer than order three, which orthogonal polynomials have.", "The purpose of this section is to explain this phenomenon.", "We consider polynomials whose generating function are obtained by multiplying the generating function of orthogonal polynomials by $(yt)_\\infty $ or $1/(-yt)_\\infty .$ We say that polynomials $p_n(x)$ satisfy a $d$ -term recurrence relation if there exist a real number $A$ and sequences $\\lbrace b_{n}^{(0)}\\rbrace _{n\\ge 0}, \\lbrace b_{n}^{(1)}\\rbrace _{n\\ge 1},\\dots ,\\lbrace b_{n}^{(d-2)}\\rbrace _{n\\ge d-2}$ such that, for $n\\ge 0$ , $p_{n+1}(x) = (Ax - b_n^{(0)})p_n(x) - b_n^{(1)}p_{n-1}(x)-\\dots -b_n^{(d-2)}p_{n-d+2}(x),$ where $p_{i}(x)=0$ for $i<0$ .", "Theorem 3.1 Let $p_n(x)$ be polynomials satisfying $p_{n+1}(x) = (Ax-b_n)p_n(x)-\\lambda _np_{n-1}(x)$ for $n\\ge 0$ , where $p_{-1}(x)=0$ and $p_0(x)=1$ .", "If $b_{k}$ and $\\frac{\\lambda _{k}}{1-q^{k}}$ are polynomials in $q^k$ of degree $r$ and $s$ , respectively, which are independent of $y$ , then the polynomials $P^{(1)}_n(x,y)$ in $x$ defined by $\\sum _{n=0}^\\infty P^{(1)}_{n} (x,y) \\frac{t^n}{(q)_n}=(yt)_\\infty \\sum _{n=0}^\\infty p_{n} (x) \\frac{t^n}{(q)_n}$ satisfy a $d$ -term recurrence relation for $d=\\max (r+2,s+3)$ .", "We use two lemmas to prove Theorem REF .", "In the following lemmas we use the same notations as in Theorem REF .", "Lemma 3.2 We have $P^{(1)}_n(x,y) = P^{(1)}_n(x,yq) - y(1-q^n) P^{(1)}_{n-1}(x,yq).$ This is obtained by equating the coefficients of $t^n$ in $\\sum _{n=0}^\\infty P^{(1)}_n(x,y) \\frac{t^n}{(q)_n}= (1-yt) \\sum _{n=0}^\\infty P^{(1)}_n(x,yq) \\frac{t^n}{(q)_n}.$ Lemma 3.3 Suppose that $b_{k}$ and $\\frac{\\lambda _{k}}{1-q^{k}}$ are polynomials in $q^k$ of degree $r$ and $s$ , respectively, i.e., $b_k = \\sum _{j=0}^r c_j (q^k)^j,\\qquad \\frac{\\lambda _{k}}{1-q^{k}} = \\sum _{j=0}^s d_j (q^k)^j.$ Then $P_{n+1}^{(1)}(x,y) = (Ax-y) P_{n}^{(1)}(x,yq)-\\sum _{j=0}^r c_j q^{nj} P^{(1)}_n (x,yq^{1-j})-(1-q^n)\\sum _{j=0}^s d_j q^{nj} P^{(1)}_{n-1} (x,yq^{1-j}).$ Expanding $(yt)_\\infty $ using the $q$ -binomial theorem, we have $P^{(1)}_n(x,y) = \\sum _{k=0}^n \\genfrac[]{0.0pt}{}{n}{k}_{q} (-1)^k y^k q^{\\binom{k}{2}}p_{n-k}(x).$ Using the relation $\\genfrac[]{0.0pt}{}{n+1}{k}_{q} = \\genfrac[]{0.0pt}{}{n}{k-1}_{q}+q^k\\genfrac[]{0.0pt}{}{n}{k}_{q}$ , we have $P^{(1)}_{n+1}(x,y) &= \\sum _{k=0}^{n+1} \\left(\\genfrac[]{0.0pt}{}{n}{k-1}_{q}+q^k\\genfrac[]{0.0pt}{}{n}{k}_{q}\\right)(-1)^k y^k q^{\\binom{k}{2}} p_{n+1-k}(x)\\\\&= -y P^{(1)}_n(x,yq) +\\sum _{k=0}^n\\genfrac[]{0.0pt}{}{n}{k}_{q} (-1)^k (yq)^k q^{\\binom{k}{2}} p_{n+1-k}(x).$ By $\\genfrac[]{0.0pt}{}{n}{k}_{q} = \\frac{1-q^n}{1-q^{n-k}}\\genfrac[]{0.0pt}{}{n-1}{k}_{q}$ and the 3-term recurrence $p_{n+1-k}(x)=(Ax-b_{n-k})p_{n-k}(x)-\\lambda _{n-k}p_{n-1-k}(x),$ we get $P^{(1)}_{n+1}(x,y) = (Ax-y) P^{(1)}_n(x,yq) -\\sum _{k=0}^n\\genfrac[]{0.0pt}{}{n}{k}_{q} (-1)^k (yq)^k q^{\\binom{k}{2}} p_{n-k}(x) b_{n-k}\\\\-(1-q^n) \\sum _{k=0}^{n-1} \\genfrac[]{0.0pt}{}{n-1}{k}_{q} (-1)^k (yq)^k q^{\\binom{k}{2}}p_{n-1-k}(x) \\frac{\\lambda _{n-k}}{1-q^{n-k}}.$ Since $b_{n-k} = \\sum _{j=0}^r c_j q^{nj} (q^k)^{-j},\\qquad \\frac{\\lambda _{n-k}}{1-q^{n-k}} = \\sum _{j=0}^s q^{nj} d_j (q^k)^{-j},$ and $P^{(1)}_n(x,yq^{1-j}) = \\sum _{k=0}^n \\genfrac[]{0.0pt}{}{n}{k}_{q} (-1)^k (yq)^kq^{\\binom{k}{2}} p_{n-k}(x) (q^{k})^{-j},$ we obtain the desired recurrence relation.", "Now we can prove Theorem REF .", "By Lemma REF , we can write $P_{n+1}^{(1)}(x,y) = (Ax-y) P_{n}^{(1)}(x,yq)-\\sum _{j=0}^r c_j q^{nj} P^{(1)}_n (x,yq^{1-j})-(1-q^n)\\sum _{j=0}^s d_j q^{nj} P^{(1)}_{n-1} (x,yq^{1-j}).$ Using Lemma REF we can express $P^{(1)}_k (x,yq^{1-j})$ as a linear combination of $P^{(1)}_k (x,yq), P^{(1)}_{k-1} (x,yq), \\dots , P^{(1)}_{k-j} (x,yq).$ Replacing $y$ by $y/q$ , we obtain a $\\max (r+2,s+3)$ -term recurrence relation for $P^{(1)}_n(x,y)$ .", "Remark 3.4 One may verify that the order of recurrence for $P_{n}^{(1)}(x,y)$ is exactly $\\max (2+r,3+s)$ in the following way.", "Lemma REF is applied $s$ times to the term $P^{(1)}_{n-1} (x,yq^{1-s})$ to obtain a linear combination of $ P^{(1)}_{n-1} (x,yq) ,P^{(1)}_{n-2} (x,yq), \\cdots , P^{(1)}_{n-s-1} (x,yq).$ The coefficient of $P^{(1)}_{n-s-1} (x,yq)$ in this expansion is $(-1)^s (q^{n-1};q^{-1})_s y^s q^{\\binom{s}{2}}.$ Similarly, considering $P^{(1)}_{n} (x,yq^{1-r})$ , the coefficient of $P^{(1)}_{n-r} (x,yq)$ in the expansion is $(-1)^r (q^{n};q^{-1})_r y^r q^{\\binom{r}{2}}.$ These terms are non-zero, give a recurrence of order $\\max (r+2,s+3)$ , and could only cancel if $r=s+1.$ In this case, the coefficient of $P^{(1)}_{n-s-1} (x,yq)$ is $(q^n;q^{-1})_{s+1} (-1)^{s+1}y^s q^{\\binom{s}{2}}q^{ns}\\left( d_s-yc_{r}q^{r+s}\\right).$ Since $d_s$ and $c_r$ are non-zero and independent of $y$ , this is non-zero.", "Remark 3.5 Theorem REF can be generalized for polynomials $p_n(x)$ satisfying a finite term recurrence relation of order greater than 3.", "For instance, if $p_{n+1}(x) = (Ax-b_n)p_n(x)-\\lambda _np_{n-1}(x) - \\nu _n p_{n-2}(x)$ , then using $\\genfrac[]{0.0pt}{}{n}{k}_{q} =\\frac{1-q^n}{1-q^{n-k}}\\genfrac[]{0.0pt}{}{n-1}{k}_{q}$ twice one can see that Equation (REF ) has the following extra sum in the right hand side: $-(1-q^n)(1-q^{n-1}) \\sum _{k=0}^{n-1} \\genfrac[]{0.0pt}{}{n-1}{k}_{q} (-1)^k (yq)^k q^{\\binom{k}{2}}p_{n-2-k}(x) \\frac{\\nu _{n-k}}{(1-q^{n-k})(1-q^{n-k-1})}.$ Thus if $\\frac{\\nu _k}{(1-q^k)(1-q^{k-1})}$ is a polynomial in $q^k$ then $P_n^{(1)}(x,y)$ satisfy a finite term recurrence relation.", "Note that by using Lemmas REF and REF , one can find a recurrence relation for $P_n^{(1)}(x,y)$ in Theorem REF .", "An analogous theorem holds for polynomial in $q^{-k}.$ We state the result without proof.", "Theorem 3.6 Let $p_n(x)$ be polynomials satisfying $p_{n+1}(x) =(Ax-b_n)p_n(x)-\\lambda _n p_{n-1}(x)$ for $n\\ge 0$ , where $p_{-1}(x)=0$ and $p_0(x)=1$ .", "If $b_{k}$ and $\\frac{\\lambda _{k}}{1-q^{k}}$ are polynomials in $q^{-k}$ of degree $r$ and $s$ , respectively, which are independent of $y$ , and the constant term of $\\frac{\\lambda _{k}}{1-q^{k}}$ is zero, then the polynomials $P^{(2)}_n(x,y)$ defined by $\\sum _{n=0}^\\infty P^{(2)}_{n} (x,y) \\frac{q^{\\binom{n}{2}}t^n}{(q)_n}=\\frac{1}{(-yt)_\\infty }\\sum _{n=0}^\\infty p_{n} (x) \\frac{q^{\\binom{n}{2}}t^n}{(q)_n}$ satisfy a $d$ -term recurrence relation for $d= \\max (r+1,s+2)$ .", "We now give several applications of Theorem REF and Theorem REF .", "In the following examples, we use the notation in these theorems.", "Example 3.7 Let $p_n(x)$ be the continuous $q$ -Hermite polynomial $H_n(x|q)$ .", "Then $A=2,b_n =0$ , and $\\lambda _n=1-q^n$ .", "Since $r=-\\infty $ and $s=0$ , $P^{(1)}_n(x,y)$ satisfies a 3-term recurrence relation.", "By Lemma REF , we have $P^{(1)}_{n+1}(x,y) = (2x-y) P^{(1)}_{n}(x,yq) - (1-q^n) P^{(1)}_{n-1}(x,yq).$ By Lemma REF we have $P^{(1)}_{n+1}(x,y) = P^{(1)}_{n+1}(x,yq) - y(1-q^n) P^{(1)}_{n}(x,yq).$ Thus $P^{(1)}_{n+1}(x,yq) =(2x-yq^n) P^{(1)}_{n}(x,yq) - (1-q^n) P^{(1)}_{n-1}(x,yq).$ Replacing $y$ by $y/q$ we obtain $P^{(1)}_{n+1}(x,y) =(2x-yq^{n-1}) P^{(1)}_{n}(x,y) - (1-q^n) P^{(1)}_{n-1}(x,y).$ Thus $P_n(x,y)$ are orthogonal polynomials, which are the continuous big $q$ -Hermite polynomials $H_n(x;y|q)$ .", "Example 3.8 Let $p_n(x)$ be the continuous big $q$ -Hermite polynomials $H_n(x;a|q)$ .", "Then $A=2, b_n = aq^n$ , and $\\lambda _n = 1-q^n$ .", "Since $r=1$ and $s=0$ , $P^{(1)}_n(x,y)$ satisfies a 3-term recurrence relation.", "Using the same method as in the previous example, we obtain $P^{(1)}_{n+1}(x,y) =(2x-(a+y)q^{n}) P^{(1)}_{n}(x,y) - (1-q^n)(1-ayq^{n-1})P^{(1)}_{n-1}(x,y).$ Thus $P^{(1)}_n(x,y)$ are orthogonal polynomials, which are the Al-Salam-Chihara polynomials $Q_n(x;a,y|q)$ .", "Example 3.9 Let $p_n(x)$ be the Al-Salam-Chihara polynomials $Q_n(x;a,b|q)$ .", "Then $A=2,b_n= (a+b)q^n$ , and $\\lambda _n = (1-q^n) (1-abq^{n-1})$ .", "Since $r=1$ and $s=1$ , $P_n(x,y)$ satisfies a 4-term recurrence relation.", "By Lemma REF , we have $P^{(1)}_{n+1}(x,y) = (2x-y)P^{(1)}_n(x,yq) - (a+b)q^n P^{(1)}_n(x,y)-(1-q^n)(-abq^{n-1}P^{(1)}_{n-1}(x,y) + P^{(1)}_{n-1}(x,yq)).$ Using Lemma REF we get $P^{(1)}_{n+1} = (2x-(a+b+y)q^n)P^{(1)}_n-(1-q^n)(1-(ab+ay+by)q^{n-1}) P^{(1)}_{n-1}-abyq^{n-2}(1-q^n)(1-q^{n-1}) P^{(1)}_{n-2}.$ Example 3.10 Let $p_n(x)$ be the continuous dual $q$ -Hahn polynomials $p_n(x;a,b,c|q)$ .", "Then $A=2$ and $b_n &= (a+b+c)q^n -abcq^{2n}-abcq^{2n-1}, \\\\\\lambda _n & = (1-q^n) (1-abq^{n-1}) (1-bcq^{n-1}) (1-caq^{n-1}).$ Since $r=2$ and $s=3$ , $P_n(x,y)$ satisfies a 6-term recurrence relation.", "It is possible to find an explicit recurrence relation using the same idea as in the previous example.", "Example 3.11 Let $p_n(x)$ be the discrete $q$ -Hermite I polynomial $h_n(x;q)$ .", "Then $A=1,b_n =0$ , and $\\lambda _n=q^{n-1}(1-q^n)$ .", "Since $r=-\\infty $ and $s=1$ , $P^{(1)}_n(x,y)$ satisfies a 4-term recurrence relation which is $P^{(1)}_{n+1}(x,y) = (x-yq^{n}) P^{(1)}_n (x,y) -q^{n-1}(1-q^n) P^{(1)}_{n-1}(x,y)+yq^{n-2}(1-q^n)(1-q^{n-1}) P^{(1)}_{n-2}(x,y).$ In §4 we will study $P^{(1)}_n(x,y)=h_n(x,y;q)$ , the discrete big $q$ -Hermite I polynomials $h_n(x,y;q)$ .", "This is a proof of Theorem REF .", "Example 3.12 Let $p_n(x)$ be the discrete $q$ -Hermite II polynomial $\\tilde{h}_n(x;q)$ .", "Then $A=1, b_n =0$ , and $\\lambda _n=q^{-2n+1}(1-q^n)$ .", "Since $b_n$ and $\\lambda _n/(1-q^n)$ are polynomials in $q^{-n}$ of degrees $-\\infty $ and 2, respectively, and the constant term of $\\lambda _n/(1-q^n)$ is 0, so $P^{(2)}_n(x,y)$ satisfies a 4-term recurrence relation.", "It is $P^{(2)}_{n+1}(x,y) = (x-yq^{-n}) P^{(2)}_n (x,y) -q^{-2n+1}(1-q^n) P^{(2)}_{n-1}(x,y)-yq^{3-3n}(1-q^n)(1-q^{n-1}) P^{(2)}_{n-2}(x,y).$ $P^{(2)}_n(x,y)$ are the discrete big $q$ -Hermite II polynomials $\\tilde{h}_n(x,y;q)$ of § .", "Example 3.13 The Al-Salam–Carlitz I polynomials $U_n^{(a)}(x;q)$ are defined by $\\sum _{n=0}^\\infty \\frac{U_n^{(a)}(x;q)}{(q)_n}t^n=\\frac{(t)_\\infty (at)_\\infty }{(xt)_\\infty }.$ They have the 3-term recurrence relation $U_{n+1}^{(a)}(x;q) = (x-(1+a)q^{n}) U_{n}^{(a)}(x;q)+aq^{n-1}(1-q^n) U_{n-1}^{(a)}(x;q).$ Let $p_n(x)$ be the polynomials with generating function $\\sum _{n=0}^\\infty \\frac{p_n(x)}{(q)_n} t^n=\\frac{(t)_\\infty }{(xt)_\\infty }=\\sum _{n=0}^\\infty \\frac{x^n(1/x)_n}{(q)_n} t^n.$ Then $p_n(x) = x^n (1/x)_n$ .", "Thus $p_{n+1}(x) = (x-q^{n}) p_n(x)$ , and we have $A=1, b_n = q^{n}$ , and $\\lambda _n =0$ , and $U_n^{(a)}(x;q) = P^{(1)}_n(x,a)$ .", "Example 3.14 The Al-Salam–Carlitz II polynomials $V_n^{(a)}(x;q)$ are defined by $\\sum _{n=0}^\\infty \\frac{(-1)^n q^{\\binom{n}{2}}}{(q)_n} V_n^{(a)}(x;q) t^n=\\frac{(xt)_\\infty }{(t)_\\infty (at)_\\infty }.$ They have the 3-term recurrence relation $V_{n+1}^{(a)}(x;q) = (x-(1+a)q^{-n}) V_{n}^{(a)}(x;q)-aq^{-2n+1}(1-q^n) V_{n-1}^{(a)}(x;q).$ Let $p_n(x)$ be the polynomials with generating function $\\sum _{n=0}^\\infty \\frac{q^{\\binom{n}{2}}}{(q)_n}p_n(x) t^n=\\frac{(xt)_\\infty }{(t)_\\infty }=\\sum _{n=0}^\\infty \\frac{(x)_n}{(q)_n} t^n=\\sum _{n=0}^\\infty \\frac{(-1)^nq^{\\binom{n}{2}}x^n(1/x)_n}{(q)_n} t^n.$ Then $p_n(x) = (-1)^n x^n (1/x)_n$ .", "Thus $p_{n+1}(x) = (-x+q^{-n}) p_n(x)$ , and we have $A=-1, b_n = -q^{-n}$ , and $\\lambda _n =0$ and we obtain $V_n^{(a)}(x;q) = (-1)^n P^{(2)}_n(-x,-a)$ and (REF ).", "Garrett, Ismail, and Stanton [5] considered the polynomials $\\hat{H}_n(x|q)$ defined by the generating function $\\sum _{n=0}^\\infty \\hat{H}_n(x|q) \\frac{t^n}{(q)_n}=\\frac{(t^2;q)_\\infty }{(te^{i\\theta },te^{-i\\theta };q)_\\infty }=(t^2;q)_\\infty \\sum _{n=0}^\\infty H_n(x|q) \\frac{t^n}{(q)_n}.$ It turns out that $p_n=\\hat{H}_n(x|q)$ satisfies the 5-term recurrence relation $p_{n+1}= 2xp_n +(q^{2n}+q^{2n-1}-q^{n-1}-1)p_{n-1}+q^{n-2}(1-q^n)(1-q^{n-1})(1-q^{n-2})p_{n-3}.$ The following generalization of Theorem REF explains this phenomenon for $m=2$ , $r=0$ , and $s=0$ .", "We omit the proof, which is similar to that of Theorem REF .", "Theorem 3.15 Let $m$ be a positive integer.", "Let $p_n(x)$ be polynomials satisfying $p_{n+1}(x) =(Ax-b_n)p_n(x)-\\lambda _n p_{n-1}(x)$ for $n\\ge 0$ , where $p_{-1}(x)=0$ and $p_0(x)=1$ .", "If $b_{k}$ and $\\frac{\\lambda _{k}}{1-q^{k}}$ are polynomials in $q^k$ of degree $r$ and $s$ , respectively, which are independent of $y$ , then the polynomials $P_n(x,y)$ in $x$ defined by $\\sum _{n=0}^\\infty P_{n} (x,y) \\frac{t^n}{(q)_n}=(yt^m)_\\infty \\sum _{n=0}^\\infty p_{n} (x) \\frac{t^n}{(q)_n}$ satisfy a $d$ -term recurrence relation for $d= \\max (rm^2+2,sm^2+3,m^2+1)$ ." ], [ "Discrete big $q$ -Hermite polynomials", "In this section we study a set of polynomials which satisfy a 4-term recurrence relation, called the discrete big $q$ -Hermite polynomials (see Definition REF ).", "These polynomials generalize the discrete $q$ -Hermite polynomials and appear in Example REF .", "Recall [7] that the continuous $q$ -Hermite polynomials $H_n(x|q)$ are defined by $\\sum _{n=0}^\\infty \\frac{H_n(x|q)}{(q)_n} t^n =\\frac{1}{(te^{i\\theta },te^{-i\\theta })_\\infty },$ and the continuous big $q$ -Hermite polynomials $H_n(x;a|q)$ are defined by $\\sum _{n=0}^\\infty \\frac{H_n(x;a|q)}{(q)_n} t^n =\\frac{(at)_\\infty }{(te^{i\\theta },te^{-i\\theta })_\\infty }.$ Observe that the generating function for $H_n(x;a|q)$ is the generating function for $H_n(x|q)$ multiplied by $(at)_\\infty $ .", "In this section we introduce discrete big $q$ -Hermite polynomials in an analogous way.", "The discrete $q$ -Hermite I polynomials $h_n(x;q)$ have generating function $\\sum _{n=0}^\\infty \\frac{h_n(x;q)}{(q;q)_n} t^n= \\frac{(t^2;q^2)_\\infty }{(xt)_\\infty }.$ Definition 4.1 The discrete big $q$ -Hermite I polynomials $h_n(x,y;q)$ are given by $\\sum _{n=0}^\\infty h_n(x,y;q) \\frac{t^n}{(q;q)_n}= \\frac{(t^2;q^2)_\\infty (yt)_\\infty }{(xt)_\\infty }.$ Expanding the right hand side of (REF ) using the $q$ -binomial theorem, we find the following expression for $h_n(x,y;q)$ .", "Proposition 4.2 For $n\\ge 0,$ $h_n(x,y;q) = \\sum _{k=0}^{\\left\\lfloor n/2\\right\\rfloor }\\genfrac[]{0.0pt}{}{n}{2k}_{q} (q;q^2)_k q^{2\\binom{k}{2}}(-1)^k x^{n-2k} (y/x;q)_{n-2k}.$ The polynomials $h_n(x,y;q)$ are orthogonal polynomials in neither $x$ nor $y$ .", "However they satisfy the following simple 4-term recurrence relation which was established in Example REF .", "Theorem 4.3 For $n\\ge 0,$ $h_{n+1}(x,y;q) = (x-yq^{n}) h_n (x,y;q) -q^{n-1}(1-q^n)h_{n-1}(x,y;q)+yq^{n-2}(1-q^n)(1-q^{n-1})h_{n-2}(x,y;q).$ Note that when $y=0$ , the 4-term recurrence relation reduces to the 3-term recurrence relation for the discrete $q$ -Hermite I polynomials.", "The polynomials $h_n(x,y;q)$ are not symmetric in $x$ and $y$ .", "If we consider $h_n(x,y;q)$ as a polynomial in $y$ , then it does not satisfy a finite term recurrence relation, see Proposition REF .", "Since $h_n(x,y;q)$ satisfies a 4-term recurrence, it is a multiple orthogonal polynomial in $x.$ Thus there are two linear functionals $\\mathcal {L}^{(0)}$ and $\\mathcal {L}^{(1)}$ such that, for $i\\in \\lbrace 0,1\\rbrace $ , $\\mathcal {L}^{(i)}(h_m)=\\delta _{mi}, \\quad m \\ge 0,$ $\\mathcal {L}^{(i)}(h_m (x,y;q) h_n (x,y;q)) = 0 \\quad \\mbox{if $m>2n+i$, and} \\quad \\mathcal {L}^{(i)}(h_{2n+i}(x,y;q) h_n (x,y;q)) \\ne 0.$ We have explicit formulas for the moments for $\\mathcal {L}^{(0)}$ and $\\mathcal {L}^{(1)}$ .", "Theorem 4.4 The moments for the discrete big $q$ -Hermite polynomials are $\\mathcal {L}^{(0)}(x^n) = \\sum _{k=0}^{\\left\\lfloor n/2\\right\\rfloor } \\genfrac[]{0.0pt}{}{n}{2k}_{q} (q;q^2)_k y^{n-2k},$ $\\mathcal {L}^{(1)}(x^n) = (1-q^n) \\sum _{k=0}^{\\left\\lfloor n/2\\right\\rfloor } \\genfrac[]{0.0pt}{}{n-1}{2k}_{q} (q;q^2)_k y^{n-2k-1}.$ Before proving Theorem REF we show that in general there is a way to find the linear functionals of $d$ -orthogonal polynomials if we know how to expand certain orthogonal polynomials in terms of these $d$ -orthogonal polynomials.", "This is similar to Proposition REF .", "Theorem 4.5 Let $R_n(x)$ be orthogonal polynomials with linear functionals $\\mathcal {L}_R$ such that $\\mathcal {L}_R(1) = 1$ .", "Let $S_n(x)$ be $d$ -orthogonal polynomials with linear functionals $\\lbrace \\mathcal {L}_S^{(i)}\\rbrace _{i=0}^{d-1}$ such that $\\mathcal {L}_S^{(i)}(S_n(x)) =\\delta _{n,i}$ .", "Suppose $R_k(x) = \\sum _{m=0}^k c_{km} S_m(x).$ Then $\\mathcal {L}_S^{(i)}(x^n) = \\sum _{k=0}^n \\frac{\\mathcal {L}_R(x^n R_k(x))}{\\mathcal {L}_R(R_k(x)^2)}d_{k,i},$ where $d_{k,i}={\\left\\lbrace \\begin{array}{ll}c_{k,i} {\\text{ if }} k\\ge i,\\\\0 {\\text{ \\quad if }} k<i.\\end{array}\\right.", "}$ If we apply $\\mathcal {L}_S^{(i)}$ to both sides of (REF ), we have $\\mathcal {L}_S^{(i)}(R_k(x)) = d_{k,i}.$ Then by expanding $x^n$ in terms of $R_k(x)$ we get $\\mathcal {L}_S^{(i)}(x^n) = \\mathcal {L}_S^{(i)}\\left( \\sum _{k=0}^n\\frac{\\mathcal {L}_R(x^n R_k(x))}{\\mathcal {L}_R(R_k(x)^2)} R_k(x)\\right)= \\sum _{k=0}^n \\frac{\\mathcal {L}_R(x^n R_k(x))}{\\mathcal {L}_R(R_k(x)^2)}d_{k,i}.$ We will apply Theorem REF with $R_n(x) = h_n(x;q)$ and $S_n(x) = h_n(x,y;q)$ to prove Theorem REF .", "The first ingredient is (REF ), which follows from the generating function (REF ) $h_k(x;q) = \\sum _{m=0}^k \\genfrac[]{0.0pt}{}{k}{m}_{q} y^{k-m} h_m(x,y;q).$ The second ingredient is the value of $\\mathcal {L}_h(x^nh_k).$ Proposition 4.6 Let $\\mathcal {L}_h$ be the linear functional for $h_n(x;q)$ with $\\mathcal {L}_h(1)=1$ .", "Then $\\mathcal {L}_h(x^n h_{m}(x;q)) ={\\left\\lbrace \\begin{array}{ll} 0 {\\text{ if }} m>n {\\text{ or }}n\\lnot \\equiv m\\mod {2},\\\\\\frac{q^{\\binom{m}{2}}(q)_n}{(q^2;q^2)_{\\frac{n-m}{2}}} {\\text{ if }}n\\ge m, n\\equiv m\\mod {2}.\\end{array}\\right.", "}$ Clearly we may assume that $n\\ge m$ and $n\\equiv m\\mod {2}.$ Using the explicit formula $h_m(x;q) =x^m {}_{2}\\phi _{0} \\left( \\left.\\begin{matrix}q^{-m},q^{-m+1}\\\\-\\\\\\end{matrix}\\right| q^2, \\frac{q^{2m-1}}{x^2}\\right),$ and the fact $\\mathcal {L}_h(x^k)={\\left\\lbrace \\begin{array}{ll}0 {\\text{\\qquad \\qquad if $k$ is odd,}}\\\\(q;q^2)_{k/2} {\\text{ if $k$ is even,}}\\end{array}\\right.", "}$ we obtain $\\mathcal {L}_h(x^n h_{m}(x;q)) = (q;q^2)_{\\frac{n+m}{2}}{}_{2}\\phi _{1} \\left( \\left.\\begin{matrix}q^{-m},q^{-m+1}\\\\q^{-n-m+1}\\\\\\end{matrix}\\right| q^2,q^{m-n}\\right),$ $\\mathcal {L}_h(x^n h_{m}(x;q)) = (q;q^2)_{\\frac{n+m}{2}}$ which is evaluable by the $q$ -Vandermonde theorem [6].", "The discrete $q$ -Hermite polynomials have the following orthogonality: $\\mathcal {L}_h(h_m(x;q) h_n(x;q))= q^{\\binom{n}{2}} (q)_n \\delta _{mn}.$ Using Theorem REF , Proposition REF , and (REF ) we have proven Theorem REF .", "We do not know representing measures for the moments in Theorem REF .", "One may also find a recurrence relation for $h_n(x,y;q)$ as a polynomial in $y$ , whose proof is routine.", "Proposition 4.7 For $n\\ge 0$ , we have $yq^nh_n(x,y;q)=-h_{n+1}(x,y;q)+\\sum _{k=0}^n (q^n;q^{-1})_k (-1)^k h_{n-k}(x,y,;q)\\times {\\left\\lbrace \\begin{array}{ll} x {\\text{ if $k$ is even}}\\\\1 {\\text{ if $k$ is odd.}}\\end{array}\\right.", "}$ We can also consider discrete $q$ -Hermite II polynomials.", "The discrete $q$ -Hermite II polynomials $\\tilde{h}_n(x,y;q)$ have the generating function $\\sum _{n=0}^\\infty \\frac{q^{\\binom{n}{2}} \\tilde{h}_n(x;q)}{(q)_n} t^n =\\frac{(-xt)_\\infty }{(-t^2;q^2)_\\infty }.$ We define the discrete big $q$ -Hermite II polynomials $\\tilde{h}_n(x,y;q)$ by $\\sum _{n=0}^\\infty \\tilde{h}_n(x,y;q) \\frac{q^{\\binom{n}{2}} t^n}{(q;q)_n}= \\frac{1}{(-t^2;q^2)_\\infty } \\frac{(-xt;q)_\\infty }{(-yt;q)_\\infty }.$ Then $\\tilde{h}_n(x,0|q)$ is the discrete $q$ -Hermite II polynomial.", "The following proposition is straightforward to check.", "Proposition 4.8 For $n\\ge 0$ , we have $\\tilde{h}_n(x,y;q) = i^{-n} h_n(ix, iy;q^{-1}).$" ], [ "Combinatorics of the discrete big $q$ -Hermite polynomials", "In this section we give some combinatorial information about the discrete big $q$ -Hermite polynomials.", "This includes a combinatorial interpretation of the polynomials (Theorem REF ), and a combinatorial proof of the 4-term recurrence relation.", "Viennot's interpretation of the moments as weighted generalized Motzkin paths is also considered.", "For the purpose of studying $h_n(x,y;q)$ combinatorially we will consider the following rescaled continuous big $q$ -Hermite polynomials $h^*_n(x,y;q)$ : $h^*_n(x,y;q) = (1-q)^{-n/2} h_n(x\\sqrt{1-q} ,y\\sqrt{1-q}|q).$ By (REF ) we have $h^*_n(x,y;q) = \\sum _{k=0}^{\\left\\lfloor n/2\\right\\rfloor }(-1)^k q^{2\\binom{k}{2}} [2k-1]_q!!", "\\genfrac[]{0.0pt}{}{n}{2k}_{q}x^{n-2k} (y/x;q)_{n-2k}.$ Because $h^*_n(x,y;1) = H_n(x-y),$ which is a generating function for bicolored matchings of $[n]:=\\lbrace 1,2,\\dots ,n\\rbrace ,$ we need to consider $q$ -statistics on matchings.", "A matching of $[n]=\\lbrace 1,2,\\dots ,n\\rbrace $ is a set partition of $[n]$ in which every block is of size 1 or 2.", "A block of a matching is called a fixed point if its size is 1, and an edge if its size is 2.", "When we write an edge $\\lbrace u,v\\rbrace $ we will always assume that $u<v$ .", "A fixed point bi-colored matching or FB-matching is a matching for which every fixed point is colored with $x$ or $y$ .", "Let $\\mathcal {FBM}(n)$ be the set of FB-matchings of $[n]$ .", "Let $\\pi \\in \\mathcal {FBM}(n)$ .", "A crossing of $\\pi $ is a pair of two edges $\\lbrace a,b\\rbrace $ and $\\lbrace c,d\\rbrace $ such that $a<c<b<d$ .", "A nesting of $\\pi $ is a pair of two edges $\\lbrace a,b\\rbrace $ and $\\lbrace c,d\\rbrace $ such that $a<c<d<b$ .", "An alignment of $\\pi $ is a pair of two edges $\\lbrace a,b\\rbrace $ and $\\lbrace c,d\\rbrace $ such that $a<b<c<d$ .", "The block-word $\\mathrm {bw}(\\pi )$ of $\\pi $ is the word $w_1w_2\\dots w_n$ such that $w_i = 1$ if $i$ is a fixed point and $w_i=0$ otherwise.", "An inversion of a word $w_1w_2\\dots w_n$ is a pair of integers $i<j$ such that $w_i>w_j$ .", "The number of inversions of $w$ is denoted by $\\operatorname{inv}(w)$ .", "Suppose that $\\pi $ has $k$ edges and $n-2k$ fixed points.", "The weight $\\operatorname{wt}(\\pi )$ of $\\pi $ is defined by $\\operatorname{wt}(\\pi ) = (-1)^k q^{2\\binom{k}{2}+2\\operatorname{al}(\\pi )+\\operatorname{cr}(\\pi ) + \\operatorname{inv}(\\mathrm {bw}(\\pi ))}z_1z_2\\dots z_{n-2k},$ where $z_i=x$ if the $i$ th fixed point is colored with $x$ , and $z_i=-yq^{i-1}$ if the $i$ th fixed point is colored with $y$ .", "A complete matching is a matching without fixed points.", "Let $\\mathcal {CM}(2n)$ denote the set of complete matchings of $[2n]$ .", "Proposition 5.1 We have $\\sum _{\\pi \\in \\mathcal {CM}(2n)} q^{2\\operatorname{al}(\\pi )+\\operatorname{cr}(\\pi )} = [2n-1]_q!", "!.$ It is known that $\\sum _{\\pi \\in \\mathcal {CM}(2n)} q^{\\operatorname{cr}(\\pi )+2\\operatorname{ne}(\\pi )} =\\sum _{\\pi \\in \\mathcal {CM}(2n)} q^{2\\operatorname{cr}(\\pi )+\\operatorname{ne}(\\pi )} = [2n-1]_q!", "!.$ Since a pair of two edges is either an alignment, a crossing, or a nesting we have $\\operatorname{al}(\\pi )+\\operatorname{ne}(\\pi )+\\operatorname{cr}(\\pi )=\\binom{n}{2}$ .", "Thus $\\sum _{\\pi \\in \\mathcal {CM}(2n)} q^{2\\operatorname{al}(\\pi )+\\operatorname{cr}(\\pi )} =q^{2\\binom{n}{2}}\\sum _{\\pi \\in \\mathcal {CM}(2n)} q^{-2\\operatorname{ne}(\\pi )-\\operatorname{cr}(\\pi )} =q^{2\\binom{n}{2}} [2n-1]_{q^{-1}}!!", "= [2n-1]_q!", "!.$ Theorem 5.2 We have $h^*_n(x,y;q) = \\sum _{\\pi \\in \\mathcal {FBM}(n)} \\operatorname{wt}(\\pi ).$ Let $M(n)$ be the set of 4-tuples $(k,w,\\sigma ,X)$ such that $0\\le k\\le \\left\\lfloor n/2\\right\\rfloor $ , $w$ is a word of length $n$ consisting of $k$ 0's and $n-2k$ 1's, $\\sigma \\in \\mathcal {CM}(2k)$ , and $Z=(z_1,z_2,\\dots ,z_{n-2k})$ is a sequence such that $z_i$ is either $x$ or $-yq^{i-1}$ for each $i$ .", "For $\\pi \\in \\mathcal {FBM}(n)$ we define $g(\\pi )$ to be the 4-tuple $(k,w,\\sigma ,Z)\\in M(n)$ , where $k$ is the number of edges of $\\pi $ , $w=\\mathrm {bw}(\\pi )$ , $\\sigma $ is the induced complete matching of $\\pi $ , and $Z=(z_1,z_2,\\dots , z_{n-2k})$ is the sequence such that $z_i=x$ if the $i$ th fixed point is colored with $x$ , and $z_i=-yq^{i-1}$ if the $i$ th fixed point is colored with $y$ .", "Here, the induced complete matching of $\\pi $ is the complete matching of $[2k]$ for which $i$ and $j$ form an edge if and only if the $i$ th non-fixed point and the $j$ th non-fixed point of $\\pi $ form an edge.", "It is easy to see that $g$ is a bijection from $\\mathcal {FBM}(n)$ to $M(n)$ such that if $g(\\pi )=(k,w,\\sigma ,Z)$ with $Z=(z_1,z_2,\\cdots ,z_{n-2k})$ then $\\operatorname{wt}(\\pi ) = (-1)^k q^{2\\binom{k}{2}} q^{2\\operatorname{al}(\\sigma )+\\operatorname{cr}(\\sigma )} q^{\\operatorname{inv}(w)}z_1z_2\\cdots z_{n-2k}.$ Thus $\\sum _{\\pi \\in \\mathcal {FBM}(n)} \\operatorname{wt}(\\pi ) &=\\sum _{(k,w,\\sigma ,Z)\\in M(n)} (-1)^k q^{\\binom{k}{2}}q^{2\\operatorname{al}(\\sigma )+\\operatorname{cr}(\\sigma )} q^{\\operatorname{inv}(w)}z_1z_2\\cdots z_{n-2k}.$ Here once $k$ is fixed $\\sigma $ can be any complete matching of $[2k]$ , $w$ can be any word consisting of $k$ 0's and $n-2k$ 1's, and for $Z=(z_1,z_2,\\cdots ,z_{n-2k})$ each $z_i$ can be either $x$ or $-yq^{i-1}$ .", "Thus the sum of $q^{2\\operatorname{al}(\\sigma )+\\operatorname{cr}(\\sigma )}$ for all such $\\sigma $ 's gives $[2k-1]_q!", "!$ , the sum of $\\operatorname{inv}(w)$ for all such $w$ gives $\\genfrac[]{0.0pt}{}{n}{2k}_{q}$ , the sum of $z_1z_2\\cdots z_{n-2k}$ for all such $Z$ gives $(x/y)_{n-2k}$ .", "This finishes the proof.", "Proposition 5.3 For $n\\ge 0$ , we have $h^*_{n+1} = (x-yq^n) h^*_n - q^{n-1}[n]_q h^*_{n-1}+y q^{n-2}[n-1]_q(1-q^n) h^*_{n-2}.$ Let $W_-(n)$ be the sum of $\\operatorname{wt}(\\pi )$ for all $\\pi \\in \\mathcal {FBM}(n)$ such that $n$ is not a fixed point.", "Let $W_x(n)$ (respectively $W_y(n)$ ) be the sum of $\\operatorname{wt}(\\pi )$ for all $\\pi \\in \\mathcal {FBM}(n)$ such that $n$ is a fixed point colored with $x$ (respectively $y$ ).", "Then $h^*_{n+1}(x,y;q) =\\sum _{\\pi \\in \\mathcal {FBM}(n)} \\operatorname{wt}(\\pi ) = W_-(n+1) + W_x(n+1) + W_y(n+1).$ We claim that $W_x(n+1) &= x h^*_{n}(x,y;q), \\\\W_y(n+1) &= -yq^n (W_x(n)+W_y(n)) - yW_-(n),\\\\W_-(n+1) &= -q^{n-1}[n]_q h^*_{n-1}(x,y;q).$ From (REF ) we easily get (REF ).", "For (), consider a matching $\\pi \\in \\mathcal {FBM}(n+1)$ such that $n+1$ is connected with $i$ where $1\\le i\\le n$ .", "Suppose that $\\pi $ has $k$ edges and $n+1-2k$ fixed points.", "Let us compute the contribution of an edge of a fixed point together with the edge $\\lbrace i,n+1\\rbrace $ to $2\\operatorname{al}(\\pi )+\\operatorname{cr}(\\pi ) +\\operatorname{inv}(\\mathrm {bw}(\\pi ))$ .", "An edge with two integers less than $i$ contributes 2 to $2\\operatorname{al}(\\pi )$ .", "An edge with exactly one integer less than $i$ contributes 1 to $\\operatorname{cr}(\\pi )$ .", "An edge with two integers greater than $i$ contributes nothing.", "Each fixed point of $\\pi $ less than $i$ contributes 2 to $\\operatorname{inv}(\\mathrm {bw}(\\pi ))$ together with the edge $\\lbrace i,n+1\\rbrace $ .", "Each fixed point of $\\pi $ greater than $i$ contributes 1 to $\\operatorname{inv}(\\mathrm {bw}(\\pi ))$ together with the edge $\\lbrace i,n+1\\rbrace $ .", "Thus the contribution of the edge $\\lbrace i,n+1\\rbrace $ to $2\\operatorname{al}(\\pi )+\\operatorname{cr}(\\pi ) +\\operatorname{inv}(\\mathrm {bw}(\\pi ))$ is equal to $i-1 + (n+1-2k)$ .", "Let $\\sigma $ be the matching obtained from $\\pi $ by removing the edge $\\lbrace i,n+1\\rbrace $ .", "Then $2\\operatorname{al}(\\pi )+\\operatorname{cr}(\\pi ) + \\operatorname{inv}(\\mathrm {bw}(\\pi )) =2\\operatorname{al}(\\sigma )+\\operatorname{cr}(\\sigma ) + \\operatorname{inv}(\\mathrm {bw}(\\sigma ))+i-1 + (n+1-2k).$ Thus, using (REF ), the above identity and $2\\binom{k}{2} =2\\binom{k-1}{2}+2k-2$ , we have $\\operatorname{wt}(\\pi ) = -q^{n-1} q^{i-1} \\operatorname{wt}(\\sigma )$ .", "Since $i$ can be any integer from 1 to $n$ and $\\sigma \\in \\mathcal {FBM}(n-1)$ we get ().", "Now we prove ().", "Consider a matching $\\pi \\in \\mathcal {FBM}(n+1)$ such that $n+1$ is a fixed point colored with $y$ .", "Suppose that $\\pi $ has $k$ edges with $2k$ non-fixed points $b_1<b_2<\\dots <b_{2k}$ .", "For $0\\le i\\le 2k+1$ , let $a_i =b_i-b_{i-1}-1$ , where $b_0=0$ and $b_{2k+1}=n$ .", "Then $a_0+a_1+\\cdots +a_{2k+1}=n-2k$ .", "Let $\\sigma $ be the matching obtained from $\\pi $ by removing $n+1$ .", "Then we have $\\operatorname{wt}(\\pi ) = -yq^{n-2k}\\operatorname{wt}(\\sigma )$ .", "We consider two cases.", "Case 1: $a_0\\ne 0$ .", "Let $\\tau $ be the matching obtained from $\\sigma $ by changing 1 into $n$ and decreasing the other integers by 1.", "We color the $i$ th fixed point of $\\tau $ with the same color of the $i$ th fixed point of $\\sigma $ .", "Then $\\operatorname{wt}(\\sigma ) = q^{2k} \\operatorname{wt}(\\tau )$ and $\\operatorname{wt}(\\pi )=-yq^n(\\tau )$ .", "Since $n$ is a fixed point in $\\tau $ the sum of $\\operatorname{wt}(\\pi )$ in this case gives $-yq^n (W_x(n)+W_y(n))$ .", "Case 2: $a_0=0$ .", "Note that $\\mathrm {bw}(\\sigma )=0\\overbrace{1\\cdots 1}^{a_1}0\\overbrace{1\\cdots 1}^{a_2}0 \\cdots 0\\overbrace{1\\dots 1}^{a_{2k}}0\\overbrace{1\\dots 1}^{a_{2k+1}}.$ We define $\\tau $ to be the matching with $\\mathrm {bw}(\\tau )=\\overbrace{1\\cdots 1}^{a_1}0\\overbrace{1\\cdots 1}^{a_2}0\\overbrace{1\\cdots 1}^{a_3}1 \\cdots 0\\overbrace{1\\dots 1}^{a_{2k+1}}0$ and the $i$ th fixed point of $\\tau $ is colored with the same color of the $i$ th fixed point of $\\sigma $ .", "Then $\\operatorname{wt}(\\sigma ) = q^{-n+2k}\\operatorname{wt}(\\tau )$ and $\\operatorname{wt}(\\pi ) =-y\\operatorname{wt}(\\tau )$ .", "Since $n$ is a non-fixed point in $\\tau $ , the sum of $\\operatorname{wt}(\\pi )$ in this case gives $- yW_-(n)$ .", "It is easy to see that (REF ), (), and () implies the 4-term recurrence relation.", "Since the polynomials $h_n(x,y;q)$ satisfy a 4-term recurrence relation, they are 2-fold multiple orthogonal polynomials in $x$ .", "By Viennot's theory, we can express the two moments $\\mathcal {L}^{(0)}(x^n)$ and $\\mathcal {L}^{(1)}(x^n)$ as a sum of weights of certain lattice paths.", "A 2-Motzkin path is a lattice path consisting of an up step $(1,1)$ , a horizontal step $(1,0)$ , a down step $(1,-1)$ , and a double down step $(1,-2)$ , which starts at the origin and never goes below the $x$ -axis.", "For $i=0,1$ let $\\operatorname{Mot}_i(n)$ denote the set of 2-Motzkin paths of length $n$ with final height $i$ .", "The weight of $M\\in \\operatorname{Mot}_i(n)$ is the product of weights of all steps, where the weight of each step is defined as follows.", "An up step has weight 1.", "A horizontal step starting at level $i$ has weight $yq^i$ .", "A down step starting at level $i$ has weight $q^{i-1}(1-q^i)$ .", "A double down step starting at level $i$ has weight $-yq^{i-2}(1-q^i) (1-q^{i-1})$ .", "Then by Viennot's theory we have $\\mathcal {L}_i(y^n) = \\sum _{M\\in \\operatorname{Mot}_i(n)} \\operatorname{wt}(M).$ Thus we obtain the following corollary from Theorem REF .", "Corollary 5.4 For $n\\ge 0$ , we have $\\sum _{M\\in \\operatorname{Mot}_0(n)} \\operatorname{wt}(M)&= \\sum _{k=0}^{\\left\\lfloor n/2\\right\\rfloor } \\genfrac[]{0.0pt}{}{n}{2k}_{q} (q;q^2)_k y^{n-2k},\\\\\\sum _{M\\in \\operatorname{Mot}_1(n)} \\operatorname{wt}(M)&= (1-q^n) \\sum _{k=0}^{\\left\\lfloor n/2\\right\\rfloor } \\genfrac[]{0.0pt}{}{n-1}{2k}_{q} (q;q^2)_k y^{n-2k-1}.$ It would be interesting to prove the above corollary combinatorially." ], [ "An addition theorem", "A Hermite polynomial addition theorem is $H_n(x+y)=\\sum _{k=0}^n \\binom{n}{k} H_k(x/a)a^kH_{n-k}(y/b)b^{n-k}$ where $a^2+b^2=1$ .", "We give a $q$ -analogue of this result (Proposition REF ) using the discrete big $q$ -Hermite polynomials.", "We will use $h_n(x,y;q)$ as our $q$ -version of $H_n(x-y)$ , $\\lim _{q\\rightarrow 1} h^*_n(x,y;q)=\\lim _{q\\rightarrow 1}\\frac{h_n(x\\sqrt{1-q},y\\sqrt{1-q};q)}{(1-q)^{n/2}}=H_n(x-y).$ and $h_n(x/a,0;q),$ the discrete $q$ -Hermite, as our version of $H_n(x/a)$ $\\lim _{q\\rightarrow 1} h^*_n(x,0;q)=\\lim _{q\\rightarrow 1}\\frac{h_n(x\\sqrt{1-q},0;q)}{(1-q)^{n/2}}=H_n(x).$ Another $q$ -version of $b^{n-k}H_{n-k}(y/b)$ , $a^2+b^2=1$ is given by $p_{n-k}(y,a;q)$ , where $p_{t}(y,a;q)=\\sum _{m=0}^{[t/2]} \\genfrac[]{0.0pt}{}{t}{2m}_q(q;q^2)_m a^{2m}(1/a^2;q^2)_my^{t-2m} q^{\\binom{t-2m}{2}}.$ $\\lim _{q\\rightarrow 1} \\frac{p_t(y\\sqrt{1-q},a;q)}{(1-q)^{t/2}}=b^tH_n(x/b).$ The result is Proposition 6.1 For $n\\ge 0$ , $h_n(x,y;q)=(-1)^n\\sum _{k=0}^n \\genfrac[]{0.0pt}{}{n}{k}_q h_k(x/a,0|q) (-a)^k p_{n-k}(y,a|q).$ The generating function of $p_n$ is $F(y,a,w)=\\sum _{n=0}^\\infty \\frac{p_n(y,a;q)}{(q)_n} w^n=\\frac{(w^2;q^2)_\\infty (-yw)_\\infty }{(a^2w^2;q^2)_\\infty }.$ If $G(x,y,t)=\\frac{(t^2;q^2)_\\infty (yt)_\\infty }{(xt)_\\infty }$ is the discrete big $q$ -Hermite generating function, then $G(x,y,-t)= G(x/a,0,-at) F(y,a,t),$ which gives Proposition REF ." ] ]
1403.0053
[ [ "Electron quantum optics in ballistic chiral conductors" ], [ "Abstract The edge channels of the quantum Hall effect provide one dimensional chiral and ballistic wires along which electrons can be guided in optics like setup.", "Electronic propagation can then be analyzed using concepts and tools derived from optics.", "After a brief review of electron optics experiments performed using stationary current sources which continuously emit electrons in the conductor, this paper focuses on triggered sources, which can generate on-demand a single particle state.", "It first outlines the electron optics formalism and its analogies and differences with photon optics and then turns to the presentation of single electron emitters and their characterization through the measurements of the average electrical current and its correlations.", "This is followed by a discussion of electron quantum optics experiments in the Hanbury-Brown and Twiss geometry where two-particle interferences occur.", "Finally, Coulomb interactions effects and their influence on single electron states are considered." ], [ "Introduction", "Mesoscopic electronic transport aims at revealing and studying the quantum mechanical effects that take place in micronic samples, whose size becomes shorter than the coherence length on which the phase of the electronic wavefunction is preserved at very low temperatures.", "In particular, such effects can be emphasized when the electronic propagation in the sample is not only coherent but also ballistic and one-dimensional.", "The wave nature of electronic propagation then bears strong analogies with the propagation of photons in vacuum.", "Using analogs of beam-splitters and optical fibers, the electronic equivalents of optical setups can be implemented in a solid state system and used to characterize electronic sources.", "These optical experiments provide a powerful tool to improve the understanding of electron propagation in quantum conductors.", "Inspired by the controlled manipulations of the quantum state of light, the recent development of single electron emitters has opened the way to the controlled preparation, manipulation and characterization of single to few electronic excitations that propagate in optics-like setups.", "These electron quantum optics experiments enable to bring quantum coherent electronics down to the single particle scale.", "However, these experiments go beyond the simple transposition of optics concepts in electronics as several major differences occur between electron and photons.", "Firstly statistics differ, electrons being fermions while photons are bosons.", "The other major differences come from the presence of the Fermi sea and the Coulomb interaction.", "While photon propagation is interaction free in vacuum, electrons propagate in the sea of the surrounding electrons interacting with each others through the long range Coulomb interaction turning electron quantum optics into a complex many body problem.", "This article will be restricted to the implementation of such experiments in Gallium Arsenide two-dimensional electron gases.", "These samples provide the high mobility necessary to reach the ballistic regime and by applying a high magnetic field perpendicular to the sample enable to reach the quantum Hall effect in which electronic propagation occurs along one dimensional chiral edge channels.", "The latter situation is the most suitable to implement electron optics experiments.", "Firstly because electrons can be guided along one dimensional quantum rails, secondly because chirality prevents interferences between the electron sources and the optics-like setup used to characterize it.", "After briefly recalling the main analogies between electron propagation along the one dimensional chiral edge channels and photon propagation in optics setups, we will review the pioneer experiments that have been realized in these systems and that demonstrate the relevance of these analogies.", "Most of these experiments have been realized with DC sources that generate a continuous flow of electrons in the system and thus do not reach the single particle scale.", "The core of this review will then deal with the generation and characterization of single particle states using single electron emitters." ], [ "Optics-like setups for electrons propagating along one dimensional chiral edge channels", "The first ingredient to implement quantum optics experiment with electrons is a medium in which ballistic and coherent propagation is ensured on a large scale.", "In condensed matter, this is provided by two-dimensional electron gases: these semiconductor hetero-structures (in our case and most frequently GaAs-AlGaAs) are grown by molecular beam epitaxy, which supplies crystalline structures with an extreme degree of purity.", "Thus mobilities up to about $10-30\\times 10^6\\ \\rm {cm^2.V^{-1}.s^{-1}}$ have been reported [1], [2], [3], and mean-free path $l_e$ as well as phase coherence lengths $l_\\phi $ can be on the order of $10-20\\,\\mu $ m. These properties enable to pattern samples with e-beam lithography in such a way that the phase coherence of the wavefunction is preserved over the whole structure, thus fulfilling a first requirement to build an electron optics experiment in a condensed matter system.", "The simplest interference pattern can be produced for example in Young's double-slit experiment [4] where the phase difference between paths is tuned via the enclosed Aharonov-Bohm flux, leading to the observation of an interference pattern in the current.", "Figure: a) Schematics of a 2DEG in the integer quantum Hall regime : when a strong perpendicular magnetic field is applied, electronic transport is governed by chiral edge channels.", "b) Schematics of a quantum point contact (QPC): when a negative voltage V qpc V_{qpc} is applied on split gates deposited above the 2DEG, a tunnel barrier is created and enables to realize the electronic analog of a beamsplitter.Besides, electrons have to be guided from their emission to their detection through all the optical elements.", "A powerful implementation of phase coherent quantum rails is provided by (integer) quantum Hall effect.", "Under a strong perpendicular magnetic field, electronic transport in the 2DEG is governed by chiral one-dimensional conduction channels appearing on the edges while the bulk remains insulating (see Fig.REF a).", "The appearance of such edge channels results from the bending of Zeeman-split Landau levels near the edges of the samples [5].", "Importantly, these edge channels are chiral: electrons flow with opposite velocities on opposite edges.", "The number of filled landau levels called the filling factor $\\nu $ is the number of one dimensional channels flowing on one edge.", "It depends on the magnetic field: as $B$ increases, the Landau levels are shifted upward with respect to the Fermi energy, so that the number of Zeeman-split Landau levels crossing the Fermi level (that is, the number of filled Landau levels) decreases.", "The conductance $G$ of the 2DEG is quantized in units of the inverse of the Klitzing resistance $e^2/h=R_K^{-1}$ (where $R_K=25.8\\, \\rm {k\\Omega }$ ) and given by the number of edge channels: $G=\\nu R_K^{-1}$ .", "Many experiments are performed at filling factor $\\nu =2$ , where electronic transport occurs on two edge channels, which are spin-polarized, corresponding to two Zeeman-split levels.", "In the quantum Hall regime, the mean free path of electrons is considerably increased, up to $l_e\\sim 100\\,\\mu $ m: the chirality imposed by the magnetic field reduces backscattering drastically, as an electron has to scatter from one edge to the counter-propagating one to backscatter, which can only be done when Landau levels are partially filled in the bulk.", "Beside the absence of backscattering in the edge channels [6], large phase coherence lengths have also been measured ($l_\\phi \\sim 20\\,\\mu $ m at 20 mK [7]).", "However, backscattering can be induced locally on a controlled way using a quantum point contact (QPC) which consists of a pair of electrostatic gates deposited on the surface of the sample with a typical distance between the gates of a few hundreds of nanometers.", "The typical geometry of QPC gates is shown in Fig.REF b): when a negative gate voltage $V_{qpc}$ is applied on the gates, a constriction is created in the 2DEG between the gates because of electrostatic repulsion.", "This constriction gives rise to a potential barrier, the shape of which can be determined from the geometry of the gates [8].", "At high magnetic field, the transmission through the QPC is described in terms of edge channels following equipotential lines, which are reflected one by one as the QPC gate voltage is swept towards large negative values [9].", "The conductance at low magnetic fields presents steps in units of $2e^2/h$ as Landau levels are spin-degenerate.", "At higher magnetic field, the height of the conductance steps is equal to $e^2/h$ , reflecting Zeeman-split Landau levels and spin-polarized edge channels.", "Between two conductance plateaus, one of the edge channels is partially transmitted and accounts for a contribution $T\\frac{e^2}{h}$ to the conductance, proportional to the transmission probability $T$ .", "In particular, when set at the exact half of the opening of the first conductance channel, the outer edge channel is partially transmitted with a probability $T=0.5$ , while all other edge channels are fully reflected.", "The quantum point contact therefore acts as a tunable, channel-selective electronic beamsplitter in full analogy with the beamsplitters used in optical setups.", "In the quantum Hall effect regime, electrons thus propagate along one-dimensional ballistic and phase coherent chiral edge channels which can be partitioned by electronic beamsplitters.", "These are the key ingredients to implement optics-like setups in electronics.", "The last missing elements are the electronic source that emits electrons and the detection apparatus.", "The measurement of light intensity and its correlations in usual quantum optics experiments is replaced by the measurement of the electrical current and its fluctuations (noise) for electrons.", "Concerning the electron emitter, this review will focus on triggered emitters that can emit particles on demand in the conductor.", "However, most of electron optics experiments and in particular the first ones have been performed using stationary dc sources that generate a continuous flow of charges in the system.", "Such a source can be implemented by applying a voltage bias $V$ to the edge channel, hence shifting the chemical potential of the edge by $-e V$ .", "As a result, electrons generated in the edge channel are naturally regularly ordered, with an average time $h/eV$ between charges [10].", "The origin of this behavior is Pauli's exclusion principle, that prevents the presence of two electrons at the same position in the electron beam.", "As a consequence of Fermi statistics, a voltage biased ballistic conductor naturally produces a noiseless current [11], [12].", "Starting from the late nineties, many electron optics experiments have been performed to investigate the coherence and statistical properties of such sources." ], [ "Electron optics experiments", "The coherence properties of stationary electron sources have been studied in electronic Mach-Zehnder interferometers [13], [14], [7], [15].", "Using two QPC's as electronic beamsplitters and benefiting from the ballistic propagation of electrons along the edges, single electron interferences can be observed in the current flowing at the output of the interferometer.", "The phase difference between both arms can be varied by electrostatic influence of an additional gate or by changing the magnetic field, thus changing the magnetic flux in the closed loop of the interferometer.", "This constitutes a very striking demonstration of the phase coherence of the electronic waves as the modulation of the current can be close to 100%.", "It is important to stress the role of chirality in these experiments, as a way to decouple source and interferometer.", "Indeed, backscattering of electrons towards the source in non-chiral systems can lead to the modification of the source properties by the presence of the interferometer itself.", "An important difference between electrons and photons is also revealed in these experiments.", "Indeed, electrons interact with each others and this interaction tends to reduce the coherence of the electronic wavepacket which induces a reduction of the contrast [16], [17], [18] when varying the length difference between the interferometer arms.", "The statistical properties of stationary sources have also been studied in the electronic analog of the Hanbury-Brown & Twiss geometry [19], [20], [21].", "In this setup, a beam of electrons is partitioned on an electronic beamsplitter and the correlations $\\langle I_t(t)I_r(t^{\\prime })\\rangle $ between both transmitted $I_t(t)$ and reflected $I_r(t^{\\prime })$ intensities are recorded.", "The random partitioning on the beamsplitter is a discrete process at the scale of individual particles: an electron (or a photon in optics) is either transmitted or reflected, so that the intensity correlations encode detailed information on the emission statistics of the source by comparing it with the reference of a poissonian process.", "In current experiments, the $t, t^{\\prime }$ time information is lost and the current fluctuations on long times are measured.", "For a dc biased ohmic contact, the regular and noiseless flow of electrons at the input of the splitter is reflected in the perfect anticorrelations of the output currents, $ \\langle I_t I_r\\rangle = 0$ .", "The nature of the physical effects probed in these two types of experiments is quite different.", "Indeed, Mach-Zehnder interferometers probe the wave properties of the source, and interference patterns arise from a collection of many single-particle events.", "For light, classical analysis in terms of wave physics started during the 17th century (e.g.", "by Hooke, Huyghens) to be further developed during the 18th and 19th centuries (e.g.", "by Young and Maxwell) and is associated with first order coherence function $\\mathcal {G}^{(1)}(\\mathbf {r},t;\\mathbf {r^{\\prime }},t^{\\prime })=\\langle E^{*}(\\mathbf {r},t)E(\\mathbf {r^{\\prime }},t^{\\prime })\\rangle $ , that encodes the coherence properties of the electric field $E(\\mathbf {r},t)$ at position $\\mathbf {r}$ and time $t$ .", "The information obtained through Hanbury-Brown & Twiss interferometry differ from a wave picture, as random partitioning on the beamsplitter is a discrete process, thus encoding information on the discrete nature of the involved particles.", "A classical model in terms of corpuscles can explain the features observed, and are described in optics using second order coherence function $\\mathcal {G}^{(2)}(\\mathbf {r},t;\\mathbf {r^{\\prime }},t^{\\prime })=\\langle E^{*}(\\mathbf {r^{\\prime }},t^{\\prime })E^{*}(\\mathbf {r},t)E(\\mathbf {r},t)E(\\mathbf {r^{\\prime }},t^{\\prime })\\rangle $ .", "The classical definitions of first and second order coherence of the electromagnetic field were extended by Glauber [22] to describe non-classical states of light by introducing the quantized electromagnetic field $\\hat{E}(\\mathbf {r},t)$ .", "This description is currently the basic tool to characterize light sources in quantum optics experiments.", "It can be adapted to electrons in quantum conductors, and as in photon optics, both aspects of wave and particle nature of the carriers can be reconciled into a unified theory of coherence \"à la Glauber\".", "Still, a few experiments cannot be understood within the wave nor the corpuscular description: this is the case when two-particle interferences effects related to the exchange between two indistinguishable particles take place.", "The collision of two particles emitted at two different inputs of a beamsplitter can be used to measure their degree of indistinguishability.", "In the case of bosons, indistinguishable partners always exit in the same output.", "This results in a dip in the coincidence counts between two detectors placed at the output of the splitter when both photons arrive simultaneously on the splitter as observed by Hong-Ou-Mandel (HOM) [23] in the late eighties.", "Fermionic statistics leads to the opposite behavior: particles exit in different outputs.", "This two particle interference effect has been observed using two stationary sources (dc biased contacts) and recording the reduction in the current fluctuations at the output of the splitter [24].", "The interference term could also be fully controlled [25], [26] by varying the Aharonov-Bohm flux through a two-particle interferometer of geometry close to the Mach-Zehnder interferometer described above.", "In the latter case two-particle interferences can be used to post-select entangled electron pairs at the output of the interferometer.", "The production of a continuous flow of entangled electron-hole pairs has also been proposed using a beam splitter partitioning two edge channels [27].", "All these experiments emphasize the analogies between electron and photon propagation and provide important quantitative information on the electron source.", "They also show the differences between electron and photon optics, regarding the effect of Coulomb interaction or Fermi statistics.", "However, as particles are emitted continuously in the conductor, they miss the single particle resolution necessary to manipulate single particle states.", "In optics, the development of triggered single photon sources has enabled the manipulation and characterization of quantum states of light, opening the way towards the all-optical quantum computation [28].", "In electronics as well, several types of sources have been recently developed in quantum Hall edge channels, so that the field of electron quantum optics is now accessible.", "In the first section, we introduce the formalism of electronic coherence functions as inspired by Glauber theory of light.", "It appears particularly suitable to describe the single electrons generated by triggered sources that we briefly review in the second section, focusing on the mesoscopic capacitor used as a single electron source.", "The use and study of short time current correlations to unveil the statistical properties of a triggered emitter are presented in the third section.", "We then discuss the two particle exchange interferences that take place in the Hanbury-Brown & Twiss interferometer and analyze how these effects can be revealed in the partitioning of a single source as well as in a controlled two-electron collision.", "Finally the crucial issue of interactions between electrons and their impact on electron quantum optics experiments is discussed in the last section." ], [ "Electron optics formalism", "A single edge channel is modeled as a one dimensional wire along which the electronic propagation is chiral, ballistic and spin polarized.", "The electronic degrees of freedom are described by the fermion field operator $\\hat{\\Psi }(x,t)$ that annihilates one electron at time $t$ and position $x$ of the edge channel, or equivalently, in the Fourier representation, by the operator $\\hat{a}(\\epsilon )$ that annihilates one electron of energy $\\epsilon $ in the channel.", "Neglecting here Coulomb interactions which effects will be discussed in section , the free propagation of the fermionic field simply corresponds to the forward propagation of electronic waves at constant velocity $v$ : $\\hat{\\Psi }(x,t) = \\frac{1}{\\sqrt{h v}} \\int d\\epsilon \\; \\hat{a} (\\epsilon ) e^{i \\frac{\\epsilon }{\\hbar } (x/v- t)} $ This time evolution is particularly simple as the fermion field operator $\\hat{\\Psi }(x,t)$ only depends on $x$ and $t$ through the difference $x-vt$ .", "The ballistic propagation of electrons along quantum Hall edge channels bears strong similarities with the propagation of photons in vacuum.", "These profound analogies can be noticed in the formalism describing the dynamics of the fermion field operator $\\hat{\\Psi }(x,t)$ on the one hand and the electric field operator, $\\hat{E}(x,t) = \\hat{E}^{+}(x,t) + \\hat{E}^{-}(x,t)$ in quantum optics on the other hand [29]: $\\hat{E}^{+}(x,t)& = & i \\int d\\epsilon \\sqrt{\\frac{\\epsilon }{2 h c \\varepsilon _0 S}}\\; \\hat{a}(\\epsilon ) \\; e^{i \\frac{\\epsilon }{\\hbar }(x/c - t)} \\\\\\hat{E}^{-}(x,t)& = & \\big (\\hat{E}^{+}(x,t) \\big )^{\\dag }$ Where $S$ is the transverse section perpendicular to the one dimensional propagation along the $x$ direction and $c$ the celerity of light propagation.", "For simplicity the polarization of the electric field has been omitted.", "From Eq.", "(REF ) one can see that the fermion field operator $\\hat{\\Psi }(x,t)$ is very similar to $\\hat{E}^{+}(x,t)$ , the part of the electric field that annihilates photons, where the complex conjugate $\\hat{\\Psi }^{\\dag }(x,t)$ is similar to $\\hat{E}^{-}(x,t)$ that creates photons.", "The electrical current $\\hat{I}(x,t)$ in electron optics will then be the analog to the light intensity $\\hat{I}_{ph}(x,t)$ in usual photon optics: $\\hat{I}(x,t) = -e v \\hat{\\Psi }^{\\dag }(x,t) \\hat{\\Psi }(x,t) \\quad \\hat{I}_{ph}(x,t) = \\hat{E}^{-}(x,t) \\hat{E}^{+}(x,t)$ More generally, the coherence properties of electron sources can be studied by characterizing the first order coherence $\\mathcal {G}^{(1,e)}(x,t;x^{\\prime },t^{\\prime })$ [30], [31] defined in full analogy with Glauber's theory of optical coherences [22] with $\\hat{\\Psi }(x,t)$ replacing $\\hat{E}^{+}(x,t)$ .", "However, as $\\hat{\\Psi }(x,t)$ only depends on $x$ and $t$ through the difference $x -vt$ , we will only retain the time dependence of $\\mathcal {G}^{(1,e)}$ and set $x=x^{\\prime }=0$ in the rest of the manuscript: $\\mathcal {G}^{(1,e)}(t,t^{\\prime }) & =& \\langle \\hat{\\Psi }^{\\dag }(t^{\\prime })\\hat{\\Psi }(t) \\rangle $ The first order coherence can also be defined for holes, $\\mathcal {G}^{(1,h)}(t,t^{\\prime }) = \\langle \\hat{\\Psi }(t^{\\prime })\\hat{\\Psi }^{\\dag }(t) \\rangle $ and is directly related to the electron coherence, $\\mathcal {G}^{(1,h)}(t,t^{\\prime })= \\frac{\\delta (t-t^{\\prime })}{v} - \\mathcal {G}^{(1,e)}(t,t^{\\prime })^{*}$ .", "We will thus use mainly the electron coherence, the expression for holes will be used when it simplifies the notations.", "The diagonal part, $t=t^{\\prime }$ , of the first order coherence represent the 'populations' of the electronic source per unit of length, that is the electronic density which is proportional (with a factor $-ev$ ) to the electrical current at time $t$ .", "The off-diagonal parts represent the coherences that are probed in an electronic interference experiment.", "In an equivalent way, coherence properties can also be defined in Fourier space: $\\tilde{\\mathcal {G}}^{(1,e)}(\\epsilon ,\\epsilon ^{\\prime }) & =& \\frac{h}{v} \\langle \\hat{a}^{\\dag }(\\epsilon ^{\\prime })\\hat{a}(\\epsilon ) \\rangle $ The diagonal elements, or populations, are then proportional to the number of electrons per unit energy while the off diagonal terms represent the coherences in energy space.", "It is worth noticing that in the case of a stationary emitter ($\\mathcal {G}^{(1,e)}(t,t^{\\prime }) = \\mathcal {G}^{(1,e)}(t-t^{\\prime })$ ), these off diagonal terms in energy space vanish and the first order coherence can be characterized by the populations in energy only: $\\tilde{\\mathcal {G}}^{(1,e)}(\\epsilon ,\\epsilon ^{\\prime }) \\propto \\delta (\\epsilon -\\epsilon ^{\\prime })$ ." ], [ "Electron-photon differences", "Despite the deep analogies between electron and photon optics, some major differences remain.", "The first and most obvious one comes from the Coulomb interaction that affects electron and hole interactions.", "Contrary to photons, the propagation of a single elementary excitation is a complex many-body problem as one should consider its interaction with the large number of surrounding electrons that build the Fermi sea.", "This interaction leads in general to the relaxation and decoherence of single electronic excitations propagating in the conductor and will be discussed in section .", "However, the free dynamics described by Eqs.", "(REF ) that neglects interaction effects already capture many interesting features of electronic propagation in ballistic conductors that will be discussed first.", "Another major difference is related to the statistics, fermions versus bosons, with important consequences on the nature of the vacuum.", "At equilibrium in a conductor, many electrons are present and occupy with unit probability states up to the Fermi energy $\\epsilon _F$ .", "The equilibrium state of the edge channel at temperature $T$ will be labeled as $| F \\rangle $ .", "As a first consequence, and contrary to optics, even at equilibrium, the first order coherence function does not vanish due to the non-zero contribution from the Fermi sea which we label $\\mathcal {G}^{(1,e)}_F(t,t^{\\prime }) = \\langle F | \\hat{\\Psi }^{\\dag }(t^{\\prime })\\hat{\\Psi }(t) | F \\rangle $ .", "It can be more easily computed in Fourier space, $\\tilde{\\mathcal {G}}^{(1,e)}_F(\\epsilon ,\\epsilon ^{\\prime })= \\frac{h}{v} f(\\epsilon ) \\delta (\\epsilon -\\epsilon ^{\\prime })$ where it is diagonal and thus characterized by the population in each energy state given by the Fermi distribution $f(\\epsilon )$ at temperature $T$ .", "We will therefore consider the deviations of the first order coherence function compared to the equilibrium situation: $\\Delta \\mathcal {G}^{(1,e)}(t,t^{\\prime }) = \\mathcal {G}^{(1,e)}(t,t^{\\prime }) - \\mathcal {G}^{(1,e)}_F(t,t^{\\prime })$ .", "The electrical current carried by the edge channel does not vanish as well at equilibrium, $I_F= \\langle F | \\hat{I}(t) | F \\rangle = -\\frac{e}{h} \\int d\\epsilon \\; f(\\epsilon ) $ .", "This equilibrium current is canceled by the opposite equilibrium current carried by the counterpropagating edge channel located on the opposite edge of the sample.", "In an experiment, the current is measured on an ohmic contact which collects the total current, difference between the incoming current carried by one edge and the outgoing current carried by the counterpropagating edge.", "The ohmic contact plays the role of a reservoir at thermal equilibrium such that the outgoing edge is at thermal equilibrium and carries the current $I_F$ .", "The total current measured is then, $:I(t): = I(t)-I_F$ .", "In the following, in order to lighten the notations, $I(t)$ will refer to the total current, the Fermi sea contribution will always be subtracted, $I(t)=:I(t):$ .", "The measurement of the electrical current on an ohmic contact thus characterizes the deviation of the state of a quantum Hall edge channel compared to its equilibrium state.", "It is proportional to the diagonal terms of the excess first order coherence of the source in time domain.", "$I(t)= -ev \\Delta \\mathcal {G}^{(1,e)}(t,t)$ In term of elementary excitations, deviations from the Fermi sea consist in the creation of electrons above the Fermi sea and the destruction of electrons below it, or equivalently, the creation of holes of positive energy.", "Contrary to optics, where all the photons contribute with a positive sign to the light intensity, two kinds of particles with opposite charge and thus opposite contributions to the electrical current are present in electron optics.", "As we will see in the following of this manuscript, the propagation of carriers of opposite charge related to the presence of the Fermi sea leads to important differences with optics.", "Excess electron $\\delta n_e(\\epsilon )$ and hole $\\delta n_h(\\epsilon )$ populations are related to the diagonal terms, the populations, of the excess first order coherence in Fourier space: $\\delta n_e(\\epsilon ) & = & \\frac{v}{h} \\Delta \\tilde{\\mathcal {G}}^{(1,e)}(\\epsilon , \\epsilon ) \\\\\\langle \\delta N_e \\rangle & =& \\int _{0}^{+\\infty } d\\epsilon \\;\\delta n_e(\\epsilon ) \\\\\\delta n_h(\\epsilon ) & =& \\frac{v}{h} \\Delta \\tilde{\\mathcal {G}}^{(1,h)}(-\\epsilon , -\\epsilon ) = -\\delta n_e(-\\epsilon ) \\\\\\langle \\delta N_h \\rangle & =& \\int _{0}^{+\\infty } d\\epsilon \\;\\delta n_h(\\epsilon )$" ], [ "Stationary source versus single particle emission", "Stationary sources are the most commonly used in electron optics experiments and are implemented by applying a stationary bias $V$ to an ohmic contact which shifts the chemical potential of the edge channel by $-eV$ .", "For such a stationary source, the first order coherence function does not depend separately on both times $t$ and $t^{\\prime }$ but only on the time difference $t-t^{\\prime }$ .", "As already mentioned, such a source is fully characterized by its diagonal components in Fourier space $\\mathcal {G}^{(1,e)}(\\epsilon ,\\epsilon ^{\\prime }) \\propto \\delta (\\epsilon - \\epsilon ^{\\prime })$ .", "In the case of the voltage biased ohmic contact, the electron population is simply given by the difference of the equilibrium Fermi distributions with and without the applied bias : $\\Delta \\tilde{\\mathcal {G}}^{(1,e)}(\\epsilon ,\\epsilon ^{\\prime }) = \\frac{h}{v} \\big [ f(\\epsilon + eV) - f(\\epsilon ) \\big ] \\delta (\\epsilon -\\epsilon ^{\\prime }) $ .", "The corresponding total number of electrons emitted per unit of energy in a long but finite measurement time $T_{meas}$ is then given by $\\delta n_e(\\epsilon ) = \\big [ f(\\epsilon + eV) - f(\\epsilon ) \\big ] \\frac{T_{meas}}{h}$ .", "As mentioned in section REF , many electron optics experiments have been performed with this source to investigate the coherence properties of these sources using electronic interferometers.", "A different route of electron optics is the study of the propagation and the manipulation of single particle (electron or hole) states.", "Such a single electron state corresponding to the creation of one additional electron in wave function $\\phi ^e(x)$ above the Fermi sea can be formally written as: $\\hat{\\Psi }^{\\dag } [\\phi ^e]|F \\rangle & =& \\int dx \\; \\phi ^{e}(x) \\hat{\\Psi }^{\\dag }(x) |F \\rangle $ where $\\phi ^{e}(x)$ is the electronic wave function which Fourier components $\\tilde{\\phi }^{e}(\\epsilon )$ are only non-zero for $\\epsilon >0$ , corresponding to the filling of electronic states above the Fermi energy (at finite temperature, the single particle state has to be separated from the thermal excitations of the Fermi sea).", "This state is fully characterized by the first order coherence function: $\\Delta \\mathcal {G}^{(1,e)}(t,t^{\\prime })& =& \\phi ^{e}(-vt) \\phi ^{e,*}(-vt^{\\prime }) \\\\\\Delta \\tilde{\\mathcal {G}}^{(1,e)}(\\epsilon ,\\epsilon ^{\\prime })& =& \\tilde{\\phi }^{e}(\\epsilon ) \\tilde{\\phi }^{e,*}(\\epsilon ^{\\prime }) $ In a two dimensional ($\\epsilon , \\epsilon ^{\\prime }$ ) representation of the first order coherence in Fourier space, such a single electron state can be represented as a spot in the $\\epsilon >0, \\epsilon ^{\\prime }>0$ quadrant (see Fig.REF , right panel).", "This quadrant thus corresponds to the electron states.", "The coherence of the wave function appears in the off diagonal components ($\\epsilon \\ne \\epsilon ^{\\prime }$ ) which clearly enunciates the fact that such single particle states cannot be generated by a stationary emitter but requires the use of a triggered ac source.", "These single electron emitters open a new route for electronic transport, where the object of study is an electronic wavefunction that evolves in time instead of the set of occupation probabilities for the electronic states.", "The study of such a source and its ability to produce single electron states will be the purpose of the next section.", "Figure: Left panel : Quadrants of the electronic coherence function in Fourier space : electron (e)(e), hole (h)(h), and mixed electron/hole (e/h)(e/h).", "Right panel : schematic representation of a single-electron state created on top of the Fermi sea : the Fermi sea is represented by the half-diagonal ϵ=ϵ ' <0\\epsilon =\\epsilon ^{\\prime }<0 with no transverse extension.", "The single-electron state is pictured by a dot in the (e)(e)-quadrant.Note that the symmetric situation of a single hole creation can be described by the following state: $\\hat{\\Psi } [\\phi ^h]|F \\rangle = \\int dx \\; \\phi ^{h,*}(x) \\hat{\\Psi } (x) |F \\rangle $ where $\\tilde{\\phi }^{h}(\\epsilon )$ has only non vanishing components for $\\epsilon <0$ corresponding to electronic states below the Fermi energy.", "Its first order coherence function, $\\Delta \\tilde{\\mathcal {G}}^{(1,e)}(\\epsilon ,\\epsilon ^{\\prime }) = - \\tilde{\\phi }^{h}(\\epsilon ) \\tilde{\\phi }^{h,*}(\\epsilon ^{\\prime }) $ corresponds to a spot in the $(\\epsilon <0,\\epsilon ^{\\prime }<0)$ -quadrant of hole states (see Fig.REF ).", "Note that the minus sign reflects the fact that a hole is an absence of electron in the Fermi sea.", "The two remaining quadrants ($\\epsilon >0, \\epsilon ^{\\prime } <0$ and $\\epsilon <0, \\epsilon ^{\\prime }>0$ ) in the $(\\epsilon ,\\epsilon ^{\\prime })$ -plane are called the electron/hole coherences.", "They can be understood as the manifestation of a non fixed number of excitations (electrons and holes) which characterizes states that are neither purely electron nor purely hole states.", "An example of such a state can be written as: $|\\Psi \\rangle = \\alpha |F \\rangle + \\beta \\int dx dx^{\\prime } \\; \\phi ^{h^,*}(x) \\phi ^{e}(x^{\\prime }) \\hat{\\Psi } (x) \\hat{\\Psi }^{\\dag } (x^{\\prime })|F \\rangle $ This state is the coherent superposition of the equilibrium state and a non-equilibrium state that corresponds to the creation of one electron and one hole (one electron/hole pair).", "The total number of particles stays fixed but the number of excitations is not, such that this state cannot be seen as a pure 'electron-hole' pair.", "By computing the first order coherence function in Fourier space, one gets (zero temperature has been assumed for simplicity): $\\Delta \\mathcal {G}^{(1,e)}(\\epsilon ,\\epsilon ^{\\prime }) & = & |\\beta |^2 \\big [\\tilde{\\phi }^{e}(\\epsilon ) \\tilde{\\phi }^{e,*}(\\epsilon ^{\\prime }) -\\tilde{\\phi }^{h}(\\epsilon ) \\tilde{\\phi }^{h,*}(\\epsilon ^{\\prime })\\big ] \\nonumber \\\\& -& \\alpha ^* \\beta \\tilde{\\phi }^{e}(\\epsilon ) \\tilde{\\phi }^{h,*}(\\epsilon ^{\\prime }) - \\alpha \\beta ^* \\tilde{\\phi }^{h}(\\epsilon ) \\tilde{\\phi }^{e,*}(\\epsilon ^{\\prime }) \\nonumber \\\\$ The first two terms correspond to the electron and hole states discussed previously.", "The last two terms correspond to spots in the electron/hole quadrants of the $(\\epsilon ,\\epsilon ^{\\prime })$ -plane.", "This kind of terms will appear when the source fails to create a well defined number of electron/hole excitations but rather a coherent superposition of states with different number of excitations." ], [ "Generation of quantized currents", "The first manipulations of electrical currents at the single charge scale have been implemented in metallic electron boxes.", "In these systems, taking advantage of the quantization of the charge, quantized currents could be generated in single electron pumps with a repetition frequency of a few tens of MHz [32], [33].", "These single electron pumps have been realized almost simultaneously in semiconducting nanostructures [34] where the operating frequency was recently extended to GHz frequencies [35], [36].", "These technologies have also been implemented under a strong magnetic field [37], [38], [39], to inject electrons in high-energy quantum Hall edge channels.", "Another route for quantized current generation is to trap a single electron in the electrostatic potential generated by a surface acoustic wave propagating [40], [41] through the sample.", "This technique has recently enabled the transfer of single charges between two distant quantum dots [42], [43].", "However, even if these devices are good candidates to generate and manipulate single electron quantum states in one dimensional conductors, their main applications concern metrology and a possible quantum representation of the ampere (for a review on single electron pumps and their metrological applications, see [44]).", "Another proposal to generate single particle states in ballistic conductors, and which relies on a much simpler device, has been proposed [45], [46], [47], [48]: the DC bias applied to an ohmic contact, and that generates a stationary current, is replaced by a pulsed time dependent excitation $V_{exc}(t)$ .", "For an arbitrary time dependence and amplitude of the excitation, such a time dependent bias generates an arbitrary state that, in general, is not an eigenstate of the particle number but is the superposition of various numbers of electron and hole excitations.", "The differences of such a many body state compared to the creation of a single electronic state above the Fermi sea can be outlined using the first order coherence function of the source, $\\Delta \\mathcal {G}^{(1,e)}(\\epsilon ,\\epsilon ^{\\prime })$ in Fourier space.", "Contrary to the single electronic excitation which has only non-zero values in the electron domain $\\epsilon ,\\epsilon ^{\\prime }>0$ , such a state has also non zero values in the hole sector ($\\epsilon ,\\epsilon ^{\\prime }<0$ ) representing the spurious hole excitations generated by the source.", "Finally, in this case, $\\Delta \\mathcal {G}^{(1,e)}(\\epsilon ,\\epsilon ^{\\prime })$ also exhibits non zero electron-hole coherences as such a state is not an eigenstate of the excitation number.", "It can be shown that by applying a specific Lorentzian shaped pulse containing a quantized number of charges: $e^2/h \\; \\int dt V(t) = n e$ , exactly $n$ electronic excitations could be generated in the electron sector without creating any hole excitation.", "In particular, the voltage $V(t)=\\frac{h \\tau _0/\\pi }{t^2 + \\tau _0^2}$ generates a single electron above the Fermi sea as recently experimentally demonstrated [49].", "We followed a different route to generate single particle states which bears more resemblance with the single electron pumps mentioned before.", "The emitter, called a mesoscopic capacitor consists in a quantum dot capacitively coupled to a metallic top gate and tunnel coupled to the conductor.", "Compared to the pumps presented above, only one tunnel barrier is necessary such that the device is easier to tune.", "This difference implies that the source is ac driven and thus generates a quantized ac current whereas pumps generate a quantized dc current (note that in a recent proposal, Battista et al.", "[50], [51] suggested a new geometry where the electron and hole streams are separated, such that a dc current is generated).", "Compared to Lorentzian pulses, the single particle emission process does not depend much on the exact shape of the excitation drive.", "The quantization of the emitted current is ensured by the charge quantization in the dot.", "Another difference comes from the possibility to tune the energy of the emitted particle, as emission comes from a single energy level of the dot which energy can be tuned to some extent." ], [ "The mesoscopic capacitor", "The mesoscopic capacitor [52], [53], [54] is depicted in Fig.REF .", "It consists of a submicron-sized cavity (or quantum dot) tunnel coupled to a two-dimensional electron gas through a quantum point contact (QPC) whose transparency $D$ is controlled by the gate voltage $V_\\mathrm {g}$ .", "The potential of the dot is controlled by a metallic top gate deposited on top of the dot and capacitively coupled to it.", "This conductor realizes the quantum version of a RC circuit, where the dot and electrode define the two plates of a capacitor while the quantum point contact plays the role of the resistor.", "As mentioned in the first section, a large perpendicular magnetic field is applied to the sample in order to reach the integer quantum Hall regime, and we consider the situation where a single edge channel is coupled to the dot.", "Electronic transport can thus be described by the propagation of spinless electronic waves in a one dimensional conductor.", "Electrons in the incoming edge channel can tunnel onto the quantum dot with the amplitude $\\sqrt{\\mathcal {D}}=\\sqrt{1-r^2}$ , perform several round-trips inside the cavity, each taking the finite time $\\tau _0 =l/v$ ($l$ is the dot circumference), before finally tunneling back out into the outgoing edge state.", "In these expressions, the reflection amplitude $r$ has for convenience been assumed to be real and energy-independent.", "For a micron size cavity, $\\tau _0$ typically equals a few tens of picoseconds.", "As a result of these coherent oscillations inside the electronic cavity, the propagation in the quantum dot can be described by a discrete energy spectrum with energy levels that are separated by a constant level spacing $\\Delta $ related to the time of one round-trip $\\Delta =h/\\tau _0$ , see Fig.", "3.", "The levels are broadened by the finite coupling between the quantum dot and the electron gas, determined by the QPC transmission $D$ .", "This discrete spectrum can be shifted compared to the Fermi energy first in a static manner, when a static potential $V_0$ is applied to the top gate, but also dynamically, when a time dependent excitation $V_{exc}(t)$ is applied.", "When a square shape excitation is applied, it causes a sudden shift of the quantum dot energy spectrum.", "We consider the optimal situation where the highest occupied energy level is initially located at energy $\\epsilon _F$ at resonance with the Fermi energy in the absence of drive (labelled by $i$ on Fig.REF , lower panel).", "Figure: The mesoscopic capacitor.", "Upper panel, sketch of the mesoscopic capacitor.", "Lower panel, sketch of single electron/hole emission process.When a square drive is applied with a peak to peak amplitude $2 e V_{exc}$ comparable to $\\Delta $ , an electron is emitted above the Fermi energy from the highest occupied energy level in the first half period (labeled as 1 in Fig.REF ), an electron is then absorbed from the electron gas (corresponding to the emission of a hole as indicated in Fig.REF ) in the second half period (labeled as 2 in Fig.REF ).", "Repeating this sequence at a drive frequency of $f\\sim 1$ GHz thus gives rise to periodic emission of a single electron followed by a single hole [55].", "Previous discussion neglects the effects of Coulomb interaction inside the dot.", "It is characterized by the charging energy $E_c = \\frac{e^2}{C_g}$ , where $C_g$ is the geometrical capacitance of the dot.", "It adds to the orbital level spacing $\\Delta $ in the addition energy of the dot $\\Delta ^{*} = \\Delta + \\frac{e^2}{C_g}$ that defines the energetic cost associated with the addition or removal of one electron in the dot.", "It is thus the relevant energy scale for charge transfers between the dot and the edge channel.", "However, the magnitude of Coulomb interaction effects has been estimated to be of the same order as the orbital level spacing [56].", "This rather low contribution of interactions explains the success of the non-interacting models used throughout this manuscript to describe the dot.", "In these non-interacting models, we take the level spacing $\\Delta $ to be equal to $\\Delta ^{*}$ which captures both orbital and interaction effects.", "This emission of a quantized number of particles by the dot can be first characterized through the current generated by the emitter averaged on a large number of emission sequences." ], [ "Average current quantization", "An important characteristic of the mesoscopic capacitor lies in its capacitive coupling, such that it cannot generate any dc current.", "This emitter is intrinsically an ac emitter and, as such, can be characterized through ac measurements of the current averaged on a large number of electron/hole emission cycles.", "This current $\\langle I(t) \\rangle $ can first be measured in time domain [57], using a fast averaging card with a sampling time of 500 ps and averaging on approximately $10^8$ single electron/hole emission sequences.", "To get a good resolution on the time dependence of the current, this card limits the drive frequency to a few tens of MHz.", "The resulting current generated by the source for a drive frequency of 32 MHz is represented on Figure REF .", "We observe an exponential decay of the current with a positive contribution that corresponds to the emission of the electron followed by its opposite counterpart that corresponds to the emission of the hole.", "This exponential decay corresponds to what one would naively expect for a RC circuit.", "At $t=0$ the square excitation triggers the charge emission by promoting an occupied discrete level above the Fermi energy which is then coupled to the continuum of empty states in the edge channel.", "The probability of charge emission, and hence the current, follows an exponential decay on an average time governed by the transmission $D$ and the level spacing, $\\tau _e = h/D \\Delta $ [58].", "On Fig.", "REF (left panel), the escape time is $\\tau _e=0.9$ ns, much smaller than the half period, such that the electron is allowed enough time to escape the dot.", "This is reflected by the measured quantization of the average transmitted charge [55], [57], $\\langle Q_t \\rangle = \\int _0 ^{T_0/2} \\langle I(t) \\rangle dt =e$ (where $T_0=1/f=2\\pi /\\Omega $ is the period of the excitation drive), which shows that one electron and hole are emitted on average by the source.", "By tuning the transmission, the escape time can be controlled and varied.", "On Fig.", "REF (right panel) $\\tau =10$ ns which is comparable with the half-period.", "In this situation, some single electron events are lost and the average emitted charge is not quantized anymore, which defines a probability of charge emission $P$ , $\\langle Q_t \\rangle =P e <e$ ($P=0.7$ for $\\tau _e=10$ ns ).", "For an exponential decay of the current, the emission probability can be easily computed, $P=\\tanh (\\frac{T_0}{4 \\tau _e})$ .", "Figure: Measurements of the average current in the time domain.", "The black traces represent the experimental points while the blue trace is an exponential fit.", "The escape times and average transmitted charges are τ e =0.9\\tau _e = 0.9 ns, 〈Q t 〉=e\\langle Q_t \\rangle = e, left panel, τ e =10\\tau _e = 10 ns, 〈Q t 〉=0.7e\\langle Q_t \\rangle = 0.7 e, right panel.", "The red dotted line represents the square excitation voltage.At higher frequencies (GHz frequencies), the dot cannot be characterized by current measurements in the time domain anymore as the limited 500 ps resolution becomes larger than the half-period.", "In that case, we measure the first harmonic of the current $I_{\\Omega }$ in modulus and phase using a homodyne detection.", "The quantization of the emitted charge is then reflected in a quantization of the current modulus $| I_{\\Omega } | =2 e f$ while the escape time can be deduced from the measurement of the phase $\\phi $ , $\\tan { \\phi } = \\Omega \\tau _e$ .", "Fig.", "REF (upper panel) presents the measurement of the modulus of the current as a function of the dot transmission (horizontal axis) and the excitation drive amplitude (vertical axis).", "The value of the current modulus is encoded in a color scale.", "White diamonds correspond to areas of quantized modulus of the ac current $| I_{\\Omega } | =2 e f$ [55].", "These diamonds are blurred at high transmissions, where the charge quantization on the dot is lost due to charge fluctuations, they also vanish at small transmission when the average emission time becomes comparable or longer than the half period.", "This quantization of the average ac current is the counterpart, in the frequency domain, of the charge quantization for time domain measurements.", "The single electron emitter can be very conveniently described by the scattering theory of electronic waves submitted to a time-dependent scatterer.", "As the scatterer is periodically driven, one can apply the Floquet scattering theory [59], [60], [61], [54].", "Any physical quantity can be numerically computed from the calculation of the Floquet scattering matrix.", "In particular, Floquet calculations can be compared with the current modulus measurements plotted on Figure REF (simulations are on the lower panel), for any excitation drive $V_{exc}(t)$ .", "The excitation is a square drive the electronic temperature is $T_\\mathrm {el}\\approx 60$ mK and the level spacing of the dot is $\\Delta =4.2$ K. The QPC gate voltage $V_g$ controls both the transmission $D(V_g)$ and the dot potential $V_0(V_g)$ .", "For the transmission $D(V_g)$ , we use a saddle-point transmission law [8] with two parameters, for the potential $V_0(V_g)$ , we use a capacitive coupling of the dot potential to the QPC gate characterized by a linear variation.", "Using these parameters, the agreement between the experimental data and numerical calculations is very good, up to small energy-dependent variations in the QPC transmission which were not included in the model.", "Figure: Two dimensional color plot of the modulus of the average current.", "The top figure represents the experimental points while the bottom figure is a simulation using Floquet scattering theory." ], [ "Second order coherence function", "Although the measurement of the quantization of the charge emitted on one period is a strong indication that the source acts as an on-demand single particle emitter, it cannot be used as a demonstration that single particle emission is achieved at each of the source's cycles.", "The emitted charge is averaged on a huge number of emission periods, and hence does not provide any information on the statistics of electron emission.", "As can be seen on Fig.REF , the absence of electron emission on one cycle could be compensated by the emission of two electrons on the second one.", "An additional electron/hole pair could also be emitted in one cycle [62], [63].", "These various processes would not affect the average emitted charge and the quantization of the average current.", "In optics, single particle emission by photonic sources is demonstrated by the use of light intensity correlations [29], [64], [65], [66].", "In electronics as well, to demonstrate that exactly a single particle is emitted, one needs to go beyond the measurement of average quantities and study the correlations of the emitted current.", "Single particle emission can be demonstrated through the measurement of second order correlations functions of the electrical current.", "Figure: Sketch of electron/hole emission sequences.", "Electrons/holes are represented by blue/white dots.", "Spurious events are emphasized by red circles.The second order correlation is usually defined by the joint probability to detect one particle at time $t$ and one particle at time $t^{\\prime }$ .", "It reveals the correlations between particles, that is, their tendency to arrive close to each others (called bunching), or on the opposite to be well separated (antibunching).", "Here, as we rely on current, or density measurements, we focus on the density-density correlation function.", "Using the fermion field operator at times $t$ and $t^{\\prime }$ , it goes like: $&&\\mathcal {C}^{(2)}_0(t,t^{\\prime }) = \\langle \\hat{\\Psi }^{\\dag }(t)\\hat{\\Psi }(t)\\hat{\\Psi }^{\\dag }(t^{\\prime })\\hat{\\Psi }(t^{\\prime }) \\rangle \\\\& & = \\frac{\\delta (t-t^{\\prime })}{v} \\langle \\hat{\\Psi }^{\\dag }(t)\\hat{\\Psi }(t) \\rangle + \\langle \\hat{\\Psi }^{\\dag }(t^{\\prime })\\hat{\\Psi }^{\\dag }(t)\\hat{\\Psi }(t) \\hat{\\Psi }(t^{\\prime }) \\rangle $ The first term merely represents the autocorrelation of the charge at equal times and is proportional to the number of particles, that is to the average density.", "It is usually referred to as the shot noise term and reflects charge granularity.", "The second term is the joint probability to detect one particle at time $t$ and one particle at time $t^{\\prime }$ and encodes the correlations between particles.", "It is called the second order coherence function $\\mathcal {G}^{(2)}(t,t^{\\prime })$ in a description \"à la Glauber\" of the electromagnetic field.", "In particular, if a single particle is present in the system (and only in this situation), this term vanishes for all times $t$ , $t^{\\prime }$ .", "It is therefore through the measurement of this term that single particle emission is asserted (in optics for example).", "Note that in many cases, and in particular, in the cases considered in this manuscript, the second order correlations can be expressed as a function of the first order ones through the use of Wick's theorem In particular, Wick theorem can be applied to single particle states resulting from the addition of one electron or one hole or to the case of a periodically driven scatterer which we treat through Floquet scattering formalism.", "However, Wick theorem would not apply in the case where electron-electron interactions are present.", "$&&\\mathcal {C}^{(2)}_0(t,t^{\\prime }) = \\frac{\\delta (t-t^{\\prime })}{v} \\mathcal {G}^{(1,e)}(t,t) + \\nonumber \\\\&& \\mathcal {G}^{(1,e)}(t,t)\\mathcal {G}^{(1,e)}(t^{\\prime },t^{\\prime }) \\Big [1 - \\frac{|\\mathcal {G}^{(1,e)}(t,t^{\\prime })|^2}{\\mathcal {G}^{(1,e)}(t,t)\\mathcal {G}^{(1,e)}(t^{\\prime },t^{\\prime })} \\Big ]$ Focusing the discussion on the second term which encodes the correlations between particles, we observe that perfect antibunching is always observed for $t=t^{\\prime }$ as two fermions cannot be detected at the same time due to Pauli exclusion principle.", "However, in general, two fermions can be detected at arbitrary times $t \\ne t^{\\prime }$ except for a single particle state where the second term vanishes for arbitrary times $t,t^{\\prime }$ .", "Indeed, ignoring first the presence of the Fermi sea, the single electron coherence of a single particle state reads, $\\mathcal {G}^{(1)}(t,t^{\\prime }) = \\phi ^{e}(-vt) \\phi ^{e,*}(-vt^{\\prime })$ such that $|\\mathcal {G}^{(1)}(t,t^{\\prime })|^2 = \\mathcal {G}^{(1)}(t,t)\\mathcal {G}^{(1)}(t^{\\prime },t^{\\prime })$ for all times $t,t^{\\prime }$ .", "In an experimental situation, the emission of a single particle state is periodically triggered with a period $T_0$ .", "Considering an emitter with an average emission time $\\tau _e$ , the expected typical resulting trace for $\\mathcal {C}^{(2)}_0(t,t^{\\prime })$ (averaged on the absolute time $t$ ) can be plotted on Fig.REF .", "The first term in Eq.", "(REF ) is a Dirac peak and is plotted in blue.", "The second term is represented in red, lateral peaks centered on $t^{\\prime }-t = n \\times T_0$ and of width $\\tau _e$ correspond to the detection of two subsequent emission events separated by time $n T_0$ .", "These peaks disappear on short times ($n=0$ ) as two different particles cannot be detected within the same emission period.", "This suppression is the hallmark of a single particle state: whenever two or more particles are emitted on the same emission period, this central peak would reappear.", "Figure: Sketch of the second order correlation.", "Single particle wavepackets of width τ e \\tau _e are emitted with period T 0 T_0.", "The blue trace represents the first term in Eq.", "() while the red trace represents the second term.", "The latter goes to zero on short time, reflecting that the source emits particles one by one.However, one must be careful in the use of these arguments, as true single particle states are not available in quantum conductors due to the presence of the Fermi sea.", "We can only produce single particle states defined by the addition of one electron (or one hole) above (or below) the Fermi sea which consists in a large number of electrons.", "It is thus not clear whether we can apply the above reasoning and use the second order correlation functions to detect states that result from the addition of a single electron above the Fermi sea (or equivalently the addition of a single hole below).", "We would also like to slightly change the definition of the second order correlation function in such a way that it can be directly expressed as a function of the natural observable of this system, that is, the electrical current.", "We thus adopt this new definition of the second order correlation function which is defined through the measurement of the excess current correlations at times $t$ and $t^{\\prime }$ : $\\mathcal {C}^{(2)}(t,t^{\\prime }) & = & \\langle \\hat{I}(t^{\\prime })\\hat{I}(t) \\rangle - \\langle \\hat{I}(t^{\\prime })\\hat{I}(t) \\rangle _F $ As seen in section REF , the second term is necessary to suppress the current correlations that already exist at equilibrium when the source is off.", "To enlighten the analogies between this expression and the previous definition that was valid in the absence of the Fermi sea, let us consider as previously a case where Wick theorem applies: $\\mathcal {C}^{(2)}(t,t^{\\prime }) & = & \\delta (t-t^{\\prime }) \\langle \\hat{I}(t) \\rangle \\nonumber \\\\& + & \\langle \\hat{I}(t^{\\prime }) \\rangle \\langle \\hat{I}(t) \\rangle \\Big [1 - \\frac{|\\Delta \\mathcal {G}^{(1,e)}(t,t^{\\prime })|^2 }{\\Delta \\mathcal {G}^{(1,e)}(t,t) \\Delta \\mathcal {G}^{(1,e)}(t^{\\prime },t^{\\prime })}\\Big ] \\nonumber \\\\& - & e^2 v^2 \\mathcal {G}^{(1,e)}_{F}(t,t^{\\prime }) \\; \\Delta \\mathcal {G}^{(1,e)}(t^{\\prime },t) \\nonumber \\\\& - & e^2 v^2 \\mathcal {G}^{(1,e)}_{F}(t^{\\prime },t) \\; \\Delta \\mathcal {G}^{(1,e)}(t,t^{\\prime }) $ This expression presents many analogies with Eq.", "(REF ), in particular, the first two terms are identical except for the replacement of $\\mathcal {G}^{(1,e)}$ by the contribution of the source only, $\\Delta \\mathcal {G}^{(1,e)}$ .", "These two terms thus provide a way to identify the single particle states generated by the source.", "However, the last two terms are not present in Eq.", "(REF ) as they represent correlations between the Fermi sea and the single particle source.", "Contrary to the first order correlation where the source and Fermi sea contributions could be separated, this is not the case in the second order correlations." ], [ "High frequency noise of a single particle emitter", "In electronics, current correlations are measured through the current noise spectrum $S(\\omega )$ .", "It is usually defined for a stationary process.", "For a non stationary process, it can be defined in analogy by performing an average on the current fluctuations on the absolute time $t$ : $S(\\omega ) & = & 2 \\int d\\tau \\overline{\\langle \\delta I(t+\\tau ) \\delta I(t) \\rangle }^t e^{-i\\omega \\tau }$ In the following, equilibrium noise contribution that can be measured when the source is off will always be subtracted from the noise spectrum in order to analyze the source contribution to the noise only.", "$S(\\omega )$ is then directly given by the Fourier transform of the second order correlation defined above by Eqs.", "(REF ) and(REF ) up to an additional contribution related to the average current: $S(\\omega ) & = & 2 \\int d\\tau \\big [ \\overline{\\mathcal {C}^{(2)}(t,t+\\tau ) \\rangle }^t + \\overline{\\langle \\hat{I}(t+\\tau ) \\rangle \\langle \\hat{I}(t) \\rangle }^t \\big ] e^{-i\\omega \\tau } \\nonumber \\\\$ The current noise spectrum provides a direct access to the second order correlation function and is thus an appropriate tool to demonstrate single particle emission.", "However, it is important to characterize the contribution to the noise spectrum of the last terms of Eq.", "(REF ), which we label $S_F(\\omega )$ as these terms did not provide information on the source only but on correlations between the source and the Fermi sea.", "$S_F(\\omega ) & =& - \\frac{2 e^2}{T_{meas}}\\int d\\epsilon f(\\epsilon ) \\Big [ \\delta n_{e}(\\epsilon -\\hbar \\omega ) +\\delta n_e(\\epsilon + \\hbar \\omega ) \\Big ] \\nonumber \\\\ $ To evaluate this contribution, let us consider a source that emits one electron at energy $\\epsilon _e \\approx \\Delta /2$ above the Fermi sea.", "As $\\delta n_{e}(\\epsilon \\pm \\hbar \\omega )$ represents the population of excitations emitted by the source at energy $\\epsilon \\pm \\hbar \\omega $ , it is non zero when the energy is of the order of $\\epsilon _e$ .", "However, from the Fermi sea contribution $f(\\epsilon )$ , we have $\\epsilon \\le 0$ which means that $S_F(\\omega )$ becomes only non negligible at high frequency $ \\hbar \\omega \\approx \\epsilon _e$ .", "Generally, this contribution can be safely neglected if the frequency is much lower than the energy of the excitations emitted by the source.", "Practically, this approximation holds for a measurement frequency $f \\ll \\frac{\\Delta }{2 h}$ , with $\\frac{\\Delta }{2 h}\\gtrsim 20$ GHz.", "In the following, measurements were performed at $f \\simeq 1$ GHz such that correlations between the source and the Fermi sea can safely be neglected in the noise measurements and the current correlations can be used to analyze the statistics of the source exactly as if the source was emitting in vacuum.", "From Eq.", "(REF ) we then directly obtain for a single particle emitter: $\\mathcal {C}^{(2)}(t,t^{\\prime }) & = & \\delta (t-t^{\\prime }) \\langle \\hat{I}(t) \\rangle \\\\\\langle \\delta \\hat{I}(t^{\\prime }) \\delta \\hat{I}(t) \\rangle & = & \\delta (t-t^{\\prime }) \\langle \\hat{I}(t) \\rangle -\\langle \\hat{I}(t^{\\prime }) \\rangle \\langle \\hat{I}(t)\\rangle $ Considering an exponential dependence of the average current, $\\hat{I} = \\frac{e}{\\tau _e} e^{-t/\\tau _e}$ , the noise spectrum can be explicitly computed [68]: $S(\\omega ) & = & 2 e^2 f \\Big [1 - \\frac{1}{1 + \\omega ^2 \\tau _e^2} \\Big ] = 2e^2 f \\frac{\\omega ^2 \\tau _e^2}{1 + \\omega ^2 \\tau _e^2} $ This result has been obtained using a semiclassical stochastic model [69] of single electron emission from a dot containing a single electron.", "This model also does not take into account correlations with the Fermi sea.", "Note also that our single electron emitter generates one electron followed by one hole in a period $T_0$ , that is one charge in time $T_0/2$ .", "The factor $2 e^2 f$ in Eq.", "(REF ) needs to be replaced by $4e^2 f$ in our case.", "A typical trace for the noise spectrum is plotted on Fig.REF .", "The noise vanishes at low frequency and grows on a scale given by the average escape time, $\\omega \\sim 1/\\tau _e$ .", "To reveal single particle emission, current correlations need to be measured on a time scale shorter than the average escape time, that is, through high frequency noise measurements (typically at GHz frequencies) [70].", "As exactly a single particle is emitted at each cycle of the source, the fluctuations cannot be attributed to fluctuations in the emitted charge but rather to fluctuations in the emission time.", "Due to the tunneling emission process, there is a random jitter between the emission trigger and the emission time.", "Following Eq.", "(REF ), the noise goes to a white noise limit at high frequency $\\omega \\tau _e \\gg 1$ where correlations are dominated by the first term proportional to $\\delta (t-t^{\\prime })$ [56], [71].", "However at these high frequencies, correlations with the Fermi sea, Eq.", "(REF ) cannot be neglected and are responsible for a high frequency cutoff of the noise when $\\omega \\ge \\Delta /(2 \\hbar )$ (correlations with the Fermi sea are plotted on blue dashed line on Fig.REF .", "Indeed this cutoff can be interpreted as the impossibility for a particle of energy $\\Delta /2$ above the Fermi sea to emit a photon of energy greater than $\\hbar \\omega = \\Delta /2$ due to Pauli blocking by the Fermi sea.", "A good choice of the measurement frequency thus lies between these two limits : $\\frac{1}{\\tau _e} \\approx \\omega \\ll \\Delta / \\hbar $ which naturally sets the GHz as the appropriate range.", "Figure: Different terms of the noise spectrum S(ω)S(\\omega ) of a single particle emitter.", "The blue dashed line represents the Fermi sea contribution responsible for the high frequency cut-off, -S F -S_{F} defined from Eq.", "(), while the red trace is the noise spectrum neglecting the Fermi sea contribution.", "The black trace is the total noise spectrum obtained from the substraction of the blue dashed line to the red trace." ], [ "High frequency noise measurements", "In the noise measurement, the output ohmic contact on Figure REF is used both for the determination of the average current and the high frequency noise (for further experimental details, see ref.", "[72]).", "The typical order of magnitude for the noise is given by $e^2f \\approx 4.", "10^{-29} A^2.Hz^{-1}$ for a drive frequency $f\\approx 1.5 GHz$ .", "We implemented a high frequency noise measurement with a 600 MHz bandwidth centered on the drive frequency and a noise sensitivity of a few $10^{-30} A^2.Hz^{-1}$ in a few hours measurement time.", "The noise was calibrated by measuring the equilibrium noise of a 50 Ohms resistor as a function of the temperature.", "In such noise measurements, it is very hard to change the measurement frequency as it would be required in order to check Eq.", "(REF ).", "However, the dependence in the measurement frequency goes like $\\omega \\tau _e$ which allows to work at fixed frequency, chosen as $\\omega =2 \\pi f$ (where $f$ is the frequency of the excitation drive) but variable average escape time to check the frequency dependence.", "Measurements of the noise [68], [54] as a function of the escape time are plotted on Fig.", "REF .", "For short escape times, the noise exactly follows the expected dependence (blue trace).", "However, when the escape time becomes comparable with the half period, the noise deviates from the limit of the perfect emitter.", "This can be understood, as in this limit of long escape times, electrons do not have enough time to escape the dot and the probability of single charge emission deviates from 1 (black dots on Fig.", "REF ).", "For an average current following an exponential dependence, the probability $P$ can be computed as a function of the average escape time, $P= \\tanh {T_0/4\\tau _e}$ .", "As can be seen on Fig.", "REF , the experimental points fall precisely on this $\\tanh {T_0/4\\tau _e}$ dependence (black trace).", "This finite probability of charge emission has been taken into account in the heuristic semiclassical model [69], [54] of single charge emission mentioned above, the perfect emitter formula is then modified in the following way: $S(\\omega ) & = & 4 e^2 f \\tanh {\\frac{T_0}{4\\tau _e}} \\frac{\\omega ^2 \\tau _e^2}{1 + \\omega ^2 \\tau _e^2} $ This dependence of the noise for an arbitrary value of the dot transmission can also be confirmed by numerical simulations within the Floquet scattering formalism [54] described above or by real time calculations of single charge emission in a tight-binding model [73].", "Our data points agree remarkably well with this dependence (red trace) which defines two limits.", "For short times, the noise follows the perfect emitter limit, there are no fluctuations in the emitted charge and the noise is governed by the random jitter in the emission time.", "In the long time limit, the fluctuations are governed by the fluctuations in the number of emitted charges.", "Taking $\\omega \\tau _e \\gg 1$ in Eq.", "(REF ), the noise becomes independent of frequency and proportional to the average current, $S(\\omega ) \\approx 2 e |\\langle \\hat{I}(t) \\rangle |$ for $\\tau _e \\gg T_0/2$ .", "In this limit single charge emission becomes a random poissonian process.", "Figure REF shows the proper conditions to operate the source as a good single particle emitter, for $\\tau _e\\le 0.3 T_0/2$ , the source follows the perfect emitter limit.", "To conclude this section, average current measurement of a triggered electron emitter show that the source emits on average a quantized number of particles.", "The measurement of second order correlations can then be used to demonstrate that a single particle is emitted at each emission cycle.", "This single electron emitter will then be used to characterize and manipulate single electron states in optics-like setups.", "In particular, the Hanbury-Brown and Twiss geometry, where the electron beams are partitioned by a beam-splitter will be thoroughly studied." ], [ "Hanbury-Brown & Twiss interferometry", "When studying the correlations between two sources using two detectors, the Hanbury-Brown & Twiss effect arises from two-particle interferences between direct and exchange paths, pictured on Fig.REF a).", "As discovered in 1956 when observing distant stars [74], intensity correlations offer a powerful way to study the emission statistics of sources.", "In particular, two particle interferences lead to different possible outcomes depending on the fermionic or bosonic character of the two indistinguishable particles that would impinge on a beamsplitter (Fig.", "REF b)).", "On one hand, indistinguishable electrons (fermions) antibunch: the only possible outcome is to measure one electron in each output arm.", "On the other hand, indistinguishable photons (bosons) bunch: two photons are then measured in one of the outputs.", "Thus, when such particles collide and bunch/antibunch on the beam-splitter, the fluctuations and correlations of output currents encode information on the single particle content of the incoming beams.", "First observed with light sources [75], the HBT effect has since then been observed for electrons propagating in a two dimensional electron gas [19], [20], [21].", "Figure: a) Direct and exchange paths, that interfere when placing two sources in inputs 1 and 2 and recording correlations between two detectors at outputs 3 and 4. b) Possible outcomes of two-particle interference experiments when two indistinguishable particles are placed in the inputs of a beamsplitter.Figure: The Hanbury-Brown & Twiss geometry consists in the measurements of intensity auto- or cross- correlations at the outputs of a beam-splitter (outputs 3 and 4).", "Depending on the sources (inputs 1 and 2), different properties can be inferred.", "The source under study (source 1) is plugged in input 1.", "Three cases corresponding to three different sources connected to the second input are considered in this article: a) source 2 is a Fermi sea (\"vacuum\") and a single source is partitioned on the splitter, b) source 2 is identical to the one in 1 and the setup is analogous to the optical Hong-Ou-Mandel experiment, c) source 2 is a reference source used in a tomography protocol of source 1.A convenient way to implement the interference between the two exchanged paths on two detectors is to use the geometry described on Fig.REF .", "The two sources are placed at the two inputs of a beam-splitter and the two detectors at the two outputs.", "A coincidence detection event on the detectors has then two exchanged contributions.", "Particles emitted by source 1 and 2 can be reflected to 3 and 4 or transmitted to 4 and 3.", "These two paths lead to two-particle interferences in the coincidence counts of the two detectors.", "Using electron sources, a quantum point contact can be used as a tunable electronic beam-splitter with energy-independent reflexion and transmission coefficients $R$ and $T$ ($R+T=1$ ) relating incoming to outgoing modes.", "As single particle detection is not available yet for electrons (at least for subnanosecond time scales), coincidence counts are replaced in electronics by current correlations.", "The output current operators $\\hat{I}_\\alpha (t),\\ (\\alpha \\in \\lbrace 3,4\\rbrace )$ and the output current correlations $S_{\\alpha \\beta }(t^{\\prime },t)=\\langle \\delta \\hat{I}_\\alpha (t^{\\prime })\\delta \\hat{I}_\\beta (t)\\rangle ,\\ (\\alpha ,\\beta \\in \\lbrace 3,4\\rbrace )$ can be expressed in terms of input currents and correlations : $S_{33}(t^{\\prime },t)&=&R^2 S_{11}(t^{\\prime },t)+T^2 S_{22}(t^{\\prime },t)+RT Q(t,t^{\\prime }) \\quad \\\\S_{44}(t^{\\prime },t)&=&T^2 S_{11}(t^{\\prime },t)+R^2 S_{22}(t^{\\prime },t)+RT Q(t,t^{\\prime }) \\quad \\\\S_{34}(t^{\\prime },t)&=&RT \\big (S_{11}(t^{\\prime },t)+S_{22}(t^{\\prime },t)- Q(t,t^{\\prime })) \\quad $ where $S_{11}(t^{\\prime },t)$ and $S_{22}(t^{\\prime },t)$ are the current fluctuations in inputs 1 and 2 and $Q(t,t^{\\prime })$ denotes the quantum Hanbury-Brown & Twiss contribution to outcoming current correlations.", "It encodes the aforementioned two-particle interferences and involves the coherence functions of incoming electrons and holes : $Q(t,t^{\\prime })&=& e^2 v^2 \\; \\mathcal {G}_1^{(1,e)}(t,t^{\\prime })\\mathcal {G}_2^{(1,h)}(t,t^{\\prime }) \\nonumber \\\\& + & e^2 v^2 \\;\\mathcal {G}_1^{(1,h)}(t,t^{\\prime })\\mathcal {G}_2^{(1,e)}(t,t^{\\prime })$ This quantum two-particle interference can be unveiled through the measurement of zero-frequency correlations.", "Namely, standard low-frequency noise measurement setup gives access to the averaged quantities $S_{\\alpha \\beta }(\\omega =0)=2\\int d\\tau \\ \\overline{S_{\\alpha \\beta }(t + \\tau ,t)^t}$ .", "Thus it is possible to access the averaged HBT contribution $\\overline{Q}&=&2 e^2 v^2\\int d\\tau \\ \\big [ \\; \\overline{\\mathcal {G}_1^{(1,e)}(t,t+\\tau )\\mathcal {G}_2^{(1,h)}(t,t+\\tau )^t} \\nonumber \\\\& &\\quad + \\; \\overline{\\mathcal {G}_1^{(1,h)}(t,t+\\tau )\\mathcal {G}_1^{(1,e)}(t,t+\\tau )^t} \\big ] $ which is nothing but the overlap between the single electron and hole coherences of channels 1 and 2, and plays a key role in the various experiments one can perform in the Hanbury-Brown & Twiss geometry.", "In the following, we will study the three situations described on Fig.REF .", "In the first one, a single source is used and partitioned on the splitter while the second input is kept 'empty'.", "Contrary to the true vacuum obtained in the optical experiment, in electronics, this second input is always connected to a Fermi sea which is a source at equilibrium.", "This leads to important differences in the electronic version of this experiment.", "In the second experiment, each input is connected to a triggered single electron emitter.", "Two single electrons collide synchronously on the splitter realizing the electronic analog of the Hong-Ou-Mandel experiment in optics [23], [76], [77], [78].", "Finally, using a reference state in one input, an unknown input state can be reconstructed and imaged by measuring its overlap with the known reference state.", "The principle of such a single electron state tomography will be described in the last section." ], [ "Single source partitioning", "Let us first consider the electronic analog of the seminal experiment performed by Hanbury-Brown & Twiss to characterize optical sources [75], in which a light source is placed in input 1 whereas the second arm is empty and described by the vacuum.", "In the electronic analog, the single electron source described previously is used, while the empty arm now consists of a Fermi sea at equilibrium, with fixed temperature and chemical potential.", "The purpose of this experiment is not here to obtain the charge statistics of the source, that is accessed via high-frequency autocorrelations described in the previous section.", "It in fact reveals the number of elementary excitations (electron/hole pairs) produced by the electron source, which has no optical counterpart and stems from the fact that particles with opposite charges contribute with opposite signs to the current.", "The total number of elementary excitations emitted from the source is hard to access through a direct measurement of the current or its correlations (that is without partitioning).", "Indeed, the emission of one additional spurious electron/hole pair in one driving period, as represented on Fig.REF (sixth period of the drive on the figure) is a neutral process and cannot be revealed in the current if the time resolution of the current measurement is longer that the temporal separation between the electron and the hole.", "This temporal resolution is estimated to be a few tens of picoseconds in the high frequency noise measurement presented previously.", "Spurious electron/hole pairs emitted by the source on a shorter time scale thus cannot be detected.", "However, the random and independent partitioning of electrons and holes on the splitter can be used to deduce their number from the low frequency current fluctuations of the output currents.", "Using Eqs.", "(-REF ), the excess output current correlations and their low frequency spectrum are given by : $\\Delta S_{33}(t^{\\prime },t) &=& \\Delta S_{44}(t^{\\prime },t) = - \\Delta S_{34}(t^{\\prime },t) = R T \\Delta Q(t,t^{\\prime }) \\nonumber \\\\\\\\S_{33}(\\omega =0) &=& RT \\Delta \\overline{Q} \\\\& =& 2 RT e^2v^2 \\int d\\tau \\Big [ \\overline{\\Delta \\mathcal {G}_1^{(1,e)}(t,t+\\tau )}^{t} \\mathcal {G}_F^{(1,h)}(\\tau ) \\nonumber \\\\& & \\quad \\quad \\quad \\quad + \\; \\overline{\\Delta \\mathcal {G}^{(1,h)}_1(t,t+\\tau )}^{t} \\mathcal {G}_F^{(1,e)}(\\tau ) \\Big ] $ Where $\\Delta Q(t,t^{\\prime })$ is the excess HBT contribution with respect to equilibrium.", "As can be seen in Eq.", "(REF ) and contrary to optics, the single source partitioning experiment involves two sources, the triggered emitter and the Fermi sea at finite temperature, through the overlap between their first order coherence $\\Delta \\mathcal {G}^{(1)}_1$ and $\\mathcal {G}_F^{(1)}$ .", "This overlap is more easily expressed in Fourier space: $\\Delta \\overline{Q}&=& 2 \\frac{e^2}{T_{meas}} \\int _{0}^{+\\infty } d\\epsilon \\ \\big [ \\delta n_e(\\epsilon ) + \\delta n_h(\\epsilon ) \\big ] \\big (1-2 f(\\epsilon )\\big ) \\nonumber \\\\ $ Where $\\delta n_e(\\epsilon )$ is the excess number of electrons (at energy $\\epsilon \\ge 0$ above the Fermi energy) emitted per unit energy in the long measurement time $T_{meas}$ .", "Similarly, $\\delta n_h(\\epsilon )$ is the energy density of the number of holes emitted at energy $\\epsilon \\ge 0$ (corresponding to a missing electron at energy $-\\epsilon $ below the Fermi energy) in the measurement time $T_{meas}$ .", "For a periodic emitter of frequency $f$ , it is more convenient to use the energy density of the number of excitations emitted in one period.", "To avoid defining too many notations, in the rest of the manuscript, $\\delta n_e(\\epsilon )$ (resp.", "$\\delta n_h(\\epsilon )$ ) will refer to the energy density of electrons (resp.", "holes) emitted in one period.", "Defining $\\delta N_{HBT}$ as the number of electron/hole pairs counted per period by the partition noise measurement, Eq.", "(REF ) then becomes: $\\Delta \\overline{Q}&=& 4 e^2 f \\delta N_{HBT} \\\\\\delta N_{HBT} & = & \\int _{0}^{+\\infty } d\\epsilon \\ \\frac{\\delta n_e(\\epsilon ) + \\delta n_h(\\epsilon )}{2} \\big (1-2 f(\\epsilon )\\big )$ Considering first the limit of zero temperature, $\\delta N_{HBT}=\\frac{\\langle \\delta N_e \\rangle + \\langle \\delta N_h \\rangle }{2}$ equals the average number of electrons/holes emitted in one period.", "This result can be understood by a simple classical reasoning: electrons and holes are independently partitioned on the beam-splitter following a binomial law.", "As a consequence, the low-frequency output noise is proportional to the number of elementary excitations arriving on the splitter.", "Consequently, measuring the HBT contribution directly gives access to the total number of excitations generated per emission cycle.", "However, large deviations to this classical result can be observed due to finite temperature.", "Indeed, input arms are populated with thermal electron/hole excitations that can interfere with the ones generated by the source, thus affecting their partitioning.", "As seen in Eq.", "REF , $\\delta N_{HBT}$ is corrected by $-\\int d\\epsilon \\big (\\delta n_e(\\epsilon )+\\delta n_h(\\epsilon )\\big )f(\\epsilon )$ , corresponding to the energy overlap of thermal excitations and the particles triggered by the source.", "The minus sign reflects the fermionic nature of particles colliding on the QPC.", "For vanishing temperatures, classical partitioning is recovered.", "For non-vanishing temperature, a fraction of the triggered excitations reaching the beamsplitter find thermal ones at the same energy.", "In virtue of Fermi-Dirac statistics, these indistinguishable excitations antibunch (see Fig.REF ): the only possible outcome consists of one excitation in each output, so that no fluctuations are expected in that case, thus reducing the amplitude of the HBT correlations.", "An experimental realization [79] confirms these findings.", "The single electron emitter described in the previous section REF is placed on input 1 of a quantum point contact (at a distance of approximately 3 microns), see Fig.REF .", "Low frequency current correlations $S_{44}$ are measured on output 4 while output 3 is used to to characterize the source through high frequency measurements of the average ac current generated by the source.", "The emitter is driven at a frequency of 1.7 GHz with different excitation drives (sine or square waves) so as to generate different wavepackets.", "For transmissions $0.2<D<0.7$ , the average emitted charge $\\langle Q_t\\rangle $ deduced from measurements of the average ac current equals the elementary charge $e$ with an accuracy of 10 %.", "For $D\\simeq 1$ , $\\langle Q_t\\rangle $ exceeds $e$ as quantization effects in the dot vanish, and $\\langle Q_t\\rangle \\rightarrow 0$ for $D\\rightarrow 0$ .", "Figure: Modified SEM picture of the sample used in the Hanbury-Brown & Twiss experiment.", "A perpendicular magnetic field B=3.2B=3.2 T is applied in order to work at filling factor ν=2\\nu =2.", "The two edge channels are represented by blue lines.", "The emitter is placed on input 1, 2.5 microns before the electronic splitter whose gate voltage V qpc V_{qpc} is set to fully reflect the inner edge while the outer edge can be partially transmitted with tuneable transmission TT.", "The emitter is tunnel coupled to the outer edge channel with a transmission DD tuned by the gate voltage V g V_g.", "Electron emission is triggered by the excitation drive V exc (t)V_{exc}(t).", "Average measurements of the AC current generated by the source are performed on output 3, whereas output 4 is dedicated to the low frequency noise measurements S 4,4 S_{4,4}.Figure: Low frequency HBT correlation S 44 S_{44} as a function of the transmission of the beamsplitter TT, in units e 2 fe^2f (left axis) and in A 2 Hz -1 \\rm A^2Hz^{-1}.", "Three different rf drives are presented : sine drive at transmission D=1D=1 (black triangles), sine drive at transmission D=0.3D=0.3 (red dots), square drive at transmission D=0.4D=0.4 (green squares).", "The plain lines represent fits with the expected T(1-T)T(1-T) dependence.", "Different amplitudes of noise are obtained, reflecting the fact that antibunching with thermal excitations strongly depends on the energy distribution of the generated wavepackets.Fig.REF presents the HBT low frequency correlations as a function of the beam-splitter transmission $T$ .", "For all three curves, the $T(1-T)$ dependence is observed, but the noise magnitude notably differ.", "In particular, $\\delta N_{HBT}<\\langle Q\\rangle $ , invalidating the classical partitioning of a single electron/hole pair.", "This discrepancy is attributed to the non-zero overlap between triggered excitations and thermal ones, whose exact value strongly depends on the driving parameters.", "An intuitive picture can be proposed.", "The highest value of $\\delta N_{HBT}$ is observed with a square drive.", "In this case, a single energy level in the dot is rapidly raised from below to above the Fermi level of the reservoir, and the quasiparticle is emitted at an energy $\\epsilon _e \\simeq \\frac{\\Delta }{2}>k_B T_{el}$ well separated from thermal excitations.", "Therefore, we expect the outcoming noise to be maximum.", "For a sine wave, the rise of the energy level in the dot is slower and the electron is emitted at lower energies and thus more prone to antibunch with thermal excitations.", "This tends to reduce $\\delta N_{HBT}$ .", "As the transmission $D$ is lowered, the escape time $\\tau _e$ increases and electron emission occurs at later times, corresponding to higher levels of the sine drive.", "The quasiparticle is then emitted at higher energies and are less sensitive to thermal excitations.", "$\\delta N_{HBT}$ is then increased, as seen by comparing the black and red traces of Fig.REF .", "This intuitive picture can be confronted to numerical calculations within the Floquet scattering theory [61], [54] which can be used to calculate $\\delta n_e(\\epsilon )$ and $\\delta n_h(\\epsilon )$ for any type of excitation drive (sine or square) and any value of the dot parameters.", "The resulting curves for the energy distributions can be found on ref [79], they confirm the intuitive picture discussed above.", "Figure: HBT contribution δN HBT \\delta N_{HBT} as a function of the dot transmission DD for sine drive (left panel) and a square drive (right panel).", "Experimental points are represented by dots (sine) and squares (square drive) and compared with numerical simulations based on Floquet scattering theory: T el,1 =T el,2 =0T_{el,1}=T_{el,2}=0 (red dashes), T el,1 =T el,2 =150T_{el,1}=T_{el,2}=150 mK (black dashes), and T el,1 =150T_{el,1}=150 mK, T el,2 =0T_{el,2}=0 and T el,1 =0T_{el,1}=0, T el,2 =150T_{el,2}=150 mK (blue plain and dashed lines).These differences in energy distributions can be revealed by the Hanbury-Brown & Twiss interferometry, as shown on Fig.REF that presents measurements of $\\delta N_{HBT}$ as a function of the dot transmission $D$ for two different drives, sine or square.", "Floquet calculations for square and sine drives at $T_{el}=0$ are presented in red dashed line: they are almost identical and reach $\\delta N_{HBT}\\simeq 1$ for $D\\in [0.2,0.7]$ , as expected for an ideal source that does not emit additional electron-hole pairs.", "For $D<0.2$ , the shot noise regime is recovered whereas quantization effects in the dot are progressively lost for $D>0.7$ .", "The effect of temperature in arm 2 ($T_{el,2}=150$ mK, $T_{el,2}=0$ ) is shown in blue line.", "As already discussed, the presence of thermal excitations reduces $\\delta N_{HBT}$ .", "This effect decreases when lowering the transmission, and is more pronounced for sine wave than for square drive.", "Remarkably, the effect of temperature in arm 1 (blue dashes) is identical to the one in arm 2.", "When a temperature of 150 mK (extracted from noise thermometry) is introduced in both arms, a good agreement is found with the experimental data (black dashes).", "This confirms the tendency to produce low energy excitations when using a sine drive, and energy-resolved excitations using a square drive.", "Note that the Floquet calculations do not take into account the energy relaxation [80] along the 3 microns propagation towards the splitter that will be discussed in the last section of this article.", "It only provides the energy distribution at the output of the source, 3 microns away from the splitter where the collision with thermal excitations occur.", "The good agreement with Floquet calculation implies that energy relaxation has a small effect on the total number of excitations and would require a direct measurement of the energy distribution (and not of its integral on all energies) to be characterized." ], [ "Hong-Ou-Mandel experiment", "The previously discussed antibunching effect bears strong analogies with the photon coalescence observed in the Hong-Ou-Mandel experiment [23].", "While quasiparticles are generated on-demand in the first input, thermal excitations are however randomly emitted in the second input.", "To recreate the electronic analog of the seminal Hong-Ou-Mandel experiment [81], [82], [83], two identical but independent single electron sources can be placed in the two input arms of the beamsplitter, as pictured in Fig.", "REF .", "As in the seminal HOM experiment, the antibunching of the on-demand quasiparticles provides a direct measurement of the overlap of the two mono-electronic wavefunctions, i.e.", "their degree of indistinguishability.", "Indeed, for two sources generating periodically (period $1/f$ ) a single electron described by the wavefunctions $\\phi _1^{e}(x)$ and $\\phi _2^{e}(x)$ above the Fermi sea (well separated from thermal excitations), as seen in section REF , the coherence function for source i reads $\\Delta \\mathcal {G}_i^{(1,e)}(t,t^{\\prime }) = \\phi _i^{e}(-vt) \\phi _i^{e,*}(-vt^{\\prime })$ such that we have: $\\Delta \\overline{Q}&=& 4e^2f\\big (1-|\\langle \\phi _1^{e}|\\phi _2^{e}\\rangle |^2\\big )$ For perfectly distinguishable electrons, $\\langle \\phi _1|\\phi _2\\rangle =0$ and the classical random partitioning of two electrons is recovered.", "However, for perfectly indistinguishable electrons, $\\langle \\phi _1|\\phi _2\\rangle =1$ and the random partitioning is fully suppressed.", "The overlap between the two particles can be modulated by varying the delay $\\tau $ between the excitations drives.", "Dividing $\\Delta \\overline{Q}$ by the total partition noise of both sources ($2 e^2f$ for each source neglecting temperature effects) one then gets the normalized HOM correlations $\\Delta \\overline{q}$ as: $\\Delta \\overline{q}&=&1-\\big |\\int dt\\ \\phi _1^{e,*}(t)\\phi _2^{e}(t+\\tau )\\big |^2$ When working at finite temperature, the partition noise in the HOM and HBT configurations is reduced from their overlap with thermal excitations (see previous section).", "However, if the generated quantum states in sources 1 and 2 remain indistinguishable, the antibunching effect remains total and numerical simulations using the Floquet scattering formalism show that $\\Delta \\overline{q}$ is only marginally modified.", "This experiment [84] was realized using similar sources (level spacings $\\Delta _1\\simeq \\Delta _2\\simeq 1.4\\pm 0.1$ K), driven at frequency $f=2.1$ GHz with square waves.", "A delay $\\tau $ between both drives can be tuned with an accuracy of 7 ps.", "For $D_1\\simeq D_2\\simeq 0.4$ , both sources are expected to produce energy-resolved excitations relatively well-separated from the Fermi sea and with charge $\\langle Q_t \\rangle \\simeq e$ , thus achieving with reasonable accuracy the ideal generation of single-electrons wavepackets.", "Figure: Modified SEM picture of the sample used in the Hong-Ou-Mandel experiment.", "The electron gas is represented in blue.", "Two single-electron emitters are located at inputs 1 and 2 of a quantum point contact used as a single electron beamsplitter.", "Transparencies D 1 D_1 and D 2 D_2 and static potentials of dots 1 and 2 are tuned by gate voltages V g,1 V_{g,1} and V g,2 V_{g,2}.", "Electron/hole emissions are triggered by excitation drives V exc,1 V_{exc,1} and V exc,2 V_{exc,2}.", "The transparency of the beamsplitter partitioning the inner edge channel (blue line) is tuned by gate voltage V qpc V_{qpc} and set at T=1/2T=1/2.", "The average ac current generated by sources 1 and 2 are measured on output 3 while the low frequency output noise S 44 S_{44} is measured on output 4.Figure: Excess noise Δq ¯\\Delta \\overline{q} as a function of time delay τ\\tau and normalized by the value on the plateau observed for long delays.", "The sum of both partition noises (in the HBT configuration) is depicted by the blue blurry line, while the red trace is obtained with a fit by Δq ¯=1-ηe -|τ-τ 0 |/τ e \\Delta \\overline{q}=1-\\eta e^{-|\\tau -\\tau _0|/\\tau _e}The resulting HOM correlations are presented in Fig.REF as a function of delay $\\tau $ .", "A dip in the correlations is clearly observed around $\\tau =0$ .", "The measured noise is normalized by its value on the plateaus observed at large delays, and matches as expected the sum of the HBT contributions of each source, that are measured independently by alternatively turning one of the sources off.", "As seen in section , for a square wave excitation, single electron emission is described by an exponentially decaying wavepacket, with decay time $\\tau _e$ and energy $\\epsilon _0$ that depends on the amplitude of the square excitation: $\\phi _1^{e}(t)=\\phi _2^{e}(t)=\\frac{\\theta (t)}{\\sqrt{\\tau _e}}e^{-t/2\\tau _e}e^{-i\\epsilon _0 t/\\hbar }$ .", "$\\Delta \\overline{q}$ then takes the following simple form : $\\Delta \\overline{q}&=&1-e^{-|\\tau |/\\tau _e}$ Taking into account a loss in the visibility $\\eta $ and an error on synchronization $\\tau _0$ , fitting with $\\Delta \\overline{q}=1-\\eta e^{-|\\tau -\\tau _0|/\\tau _e}$ then gives $\\tau _0\\simeq 11$ ps, $\\tau _e=62\\pm 10$ ps and $\\eta =0.5$ .", "The extracted value of $\\tau _e$ is consistent with independent measurements via the average current.", "Though effects of the partial indistinguishability of the generated excitations are indubitable, the visibility $\\eta $ is far from unity.", "This may be the result of parameter mismatch between the two sources, resulting in reduced overlap of the wavepackets, but also from decoherence effects due to interaction with the environment.", "Such effects will be discussed in section ." ], [ "Electron-hole correlations in the Hong-Ou-Mandel setup", "A unique property of electron optics compared to photon optics is the ability to manipulate hole excitations in addition to electron excitations.", "Performing the HOM experiment with identical single hole excitations in the two input arms of the beamsplitter will produce results similar to those of electrons (with hole wavefunctions replacing electron wavefunctions in Eq.", "(REF )).", "But performing the HOM experiment while injecting a single electron excitation in one input arm of the beam-splitter, and a single hole excitation in the other arm will produce results which have no counterpart in optics.", "[83] In order to get useful analytical formulas, we first consider theoretically states where one electron charge has been added (removed) from the Fermi sea $| \\Psi _e \\rangle & = \\int \\!", "dx \\; \\phi ^e(x) \\, \\psi ^{\\dagger }(x) \\, |F \\rangle \\nonumber \\\\| \\Psi _h \\rangle & = \\int \\!", "dx \\; \\phi ^h(x) \\, \\psi (x) \\, |F \\rangle $ where $|F \\rangle $ is the Fermi sea at temperature $T_{el}$ , and $\\phi ^e(x)$ , $\\phi ^h(x)$ the electron and the hole wavefunctions in real space.", "Taking the electron-hole symmetric case for simplicity ($\\phi ^e(\\epsilon _F + \\delta \\epsilon ) = \\phi ^h(\\epsilon _F - \\delta \\epsilon )$ ), the normalized HOM correlation $\\Delta \\bar{q}$ becomes: $\\Delta \\bar{q} = 1 + \\left|\\frac{ \\int _0^\\infty \\!", "d\\epsilon \\,\\phi ^{e}(\\epsilon ) \\phi ^{h, *}(\\epsilon ) e^{-i \\epsilon \\tau /\\hbar } f(\\epsilon ) (1-f(\\epsilon ))}{\\int _0^\\infty \\!", "d\\epsilon \\, |\\phi ^{e}(\\epsilon )|^2 (1-f(\\epsilon ))^2} \\right|^2 \\; .$ Comparing this with Eq.", "(REF ), we notice important changes.", "First, the interferences contribute now with a positive sign to the HOM correlations, that is, the opposite of the electron-electron case.", "Electron-hole interferences produce a“HOM peak” rather than a dip.", "Second, the value of this peak depends on the overlap of the electron and the hole wave packets times the Fermi product $f(\\epsilon ) (1-f(\\epsilon ))$ .", "This peak thus vanishes as $T_{el} \\rightarrow 0$ since it requires a significant overlap between electron and hole wave packets, a situation which only happens in an energy range $\\sim k_B T_{el}$ around $\\epsilon _F$ , where electronic states are neither fully occupied nor empty.", "Note that the many-body state $|\\Psi _e\\rangle $ (or $|\\Psi _h\\rangle $ ) created by the application of the electron creation (or annihilation) operator is quite complex when the wavepacket $\\phi ^e(x)$ (or $\\phi ^h(x)$ ) has an important weight close to the Fermi energy.", "Indeed, due to the changes imposed on the Fermi sea, many electron-hole pairs are created, and the state is not simply one electron (or one hole) plus the unperturbed Fermi sea.", "The appearance of a positive HOM peak can be attributed to interferences between these electron-hole pairs coming from the two branches of the setup.", "It is quite remarkable that eventually, the peak can simply be computed from the overlap of the electron and hole wavepackets (see Eq.", "(REF )).", "Figure: Upper panel: Electron (left) and hole (right) emission process, for a square voltage drive of period T 0 =400T_0=400 (in units of ℏ/Δ\\hbar /\\Delta ), for the twopositions of the dot levels (values V + V_+ and V - V_- of the drive).", "Theposition of the dot levels is parametrized by ϵ\\epsilon with respect to the Fermi energy E F E_F.Bottom panel: Theoretical prediction for the excess noise Δq ¯\\Delta \\bar{q} as a function of the time delay τ\\tau ,showing the electron-hole HOM peaks around τ=T 0 /2=200\\tau =T_0/2=200.", "The different curves are for ϵ=0.5,0.4,0.25\\epsilon = 0.5,0.4, 0.25 and 0 (in units of Δ\\Delta ).", "T el =0.1ΔT_{el}=0.1 \\Delta and the transparency D=0.2D=0.2 in both panels.To simulate the electron-hole HOM peak with the real electron emitters, we have used the Floquet scattering matrix formalism.", "We have computed the correlations $\\Delta \\bar{q}$ when the two single electron sources in the two input arms of the beam splitter are submitted to a square drive.", "As these sources periodically emit an electron and then (after half a period) a hole, the correlations obtained for a time delay close to a half-period correspond to the correlations between an electron and a hole.", "The results for $\\Delta \\bar{q}$ as a function of the time-delay $\\tau $ are shown on Fig.REF , for a drive period of 400 (in units of $\\hbar /\\Delta $ ).", "As the correlations are proportional to the overlap in energy of the electron and the hole wavefunctions (see Eq.", "(REF )), in order to observe a peak the electron emission and the hole emission need to happen at energies not too far apart.", "This can be controlled by the dot level position of the single electron source with respect to the Fermi energy: when a dot level is close to resonance with the Fermi energy ($\\epsilon =0$ on Fig.REF ), the energy overlap between the emitted electron and the emitted hole is important, and a large peak in the correlations $\\Delta \\bar{q}$ is observed.", "On the other hand, when the dot levels of the single electron sources are far from resonance ($\\epsilon =0.5$ on Fig.REF ), there is no overlap in energy between the emitted electron and the emitted hole, and no peak is visible in the correlations, as observed on the experimental data of Fig.", "REF where electron/hole correlations are below experimental resolution.", "The temperature used in these simulations is $T_{el} = 0.1 \\Delta $ , which is similar to the experimental value." ], [ "Tomography of a periodic electron source", "In the previous experiments, properties of the source can be inferred by measuring, through current correlations, the resemblance between the state in input arm 1 and its counterpart in input arm 2.", "Indeed, HBT correlations yield information on the energy distribution of the source, by taking the Fermi sea as a reference, whereas HOM correlation demonstrate the indistinguishability of two quantum states generated by two independent sources.", "In fact, the complete coherence function in energy domain $\\Delta \\tilde{\\mathcal {G}}^{(1,e)}(\\epsilon ,\\epsilon ^{\\prime })$ of a source of electrons and holes can be obtained in the HBT geometry by placing in input arm 2 different reference sources and measuring the corresponding current correlations.", "These spectroscopy [85] and tomography processes [31], inspired by the optics equivalent [86], [87], [88] could provide a direct image of electron wavepackets propagating in quantum Hall edge channels through the determination of the first order coherence in the $\\epsilon $ , $\\epsilon ^{\\prime }$ plane.", "For a periodic source, the definition of the first order coherence in the energy domain needs to be slightly modified.", "Indeed, $\\Delta \\mathcal {G}^{(1,e)}(t,t^{\\prime })$ has a T-periodicity in the time $\\bar{t}=\\frac{t+t^{\\prime }}{2}$ , and no periodicity along $\\tau =t-t^{\\prime }$ .", "Using these two variables in time, the Fourier transform is defined in the following way: $\\mathcal {G}^{(1,e)}(t,t^{\\prime }) = \\sum _{n=-\\infty }^{+ \\infty } e^{-in \\Omega \\bar{t}} \\int \\frac{d\\omega }{2 \\pi } \\tilde{\\mathcal {G}}^{(1,e)}_n(\\omega ) e^{-i \\omega \\tau }$ From the above definition, $\\tilde{\\mathcal {G}}^{(1,e)}_n(\\omega )$ and $\\tilde{\\mathcal {G}}^{(1,e)}(\\epsilon , \\epsilon ^{\\prime })$ are related through: $\\tilde{\\mathcal {G}}^{(1,e)}(\\epsilon , \\epsilon ^{\\prime }) = \\sum _{n=-\\infty }^{+ \\infty } \\frac{\\delta (\\epsilon - \\epsilon ^{\\prime } - n\\hbar \\Omega )}{h} \\tilde{\\mathcal {G}}^{(1,e)}_n \\Big (\\frac{\\epsilon + \\epsilon ^{\\prime }}{2} \\Big ) \\quad $ Due to the periodicity in time, $\\tilde{\\mathcal {G}}^{(1,e)}(\\epsilon , \\epsilon ^{\\prime })$ takes discrete values along the energy difference $\\epsilon - \\epsilon ^{\\prime } = n \\hbar \\Omega $ while the sum of energies takes continuous values, $\\frac{\\epsilon + \\epsilon ^{\\prime }}{2} = \\hbar \\omega $ .", "The population in energy domain thus corresponds to the $n=0$ component of $\\tilde{\\mathcal {G}}^{(1,e)}_n(\\omega )$ while the coherences correspond to $n \\ne 0$ .", "Figure: The spectroscopy and tomography of a periodic electron source can be achieved by modulating in a controlled way the two-particle interference, in the HBT geometry, between the source under study and reference sources.", "a) Sweeping the voltage V dc V_{dc} applied on the ohmic contact in input 2 enables to extract the diagonal part of the coherence function of the source in input 1, namely the energy distribution δn e/h \\delta n_{e/h}.", "b) A dynamical modulation of the partition noise by applying a voltage V n (t)=V dc +V ac cos(nΩt+φ)V_n(t)=V_{dc}+V_{ac}\\cos (n\\Omega t+\\phi ) similarly gives access to the harmonics Δ𝒢 n ,n≠0\\Delta \\mathcal {G}_n, n\\ne 0 of the coherence function.Figure: Examples of coherence functions in the complex plane.", "For transmissions D=1D=1, D=0.4D=0.4, D=0.1D=0.1, odd and even harmonics of coherence functions Δ𝒢 (e) \\Delta \\mathcal {G}^{(e)} are plotted as a function of energy in a 2D plots.", "In contrast with the case D=1D=1, excitations are energy resolved at rather high energies ±Δ/2\\pm \\Delta /2 for D=0.4D=0.4 and D=0.1D=0.1.", "When emission probability drops (for D=0.1D=0.1), emission of holes and electrons are correlated as the generation of an electron is subject to the generation of the preceding holeThe source contribution of the coherence function $\\Delta \\tilde{\\mathcal {G}}^{(1,e)}_n(\\omega )$ can be fully reconstructed in the $n \\hbar \\Omega = \\epsilon - \\epsilon ^{\\prime }$ , $\\hbar \\omega = \\frac{\\epsilon + \\epsilon ^{\\prime }}{2}$ plane by applying as a reference state on input 2 a voltage $V_n(t)=V_{dc}+V_{ac}\\cos (n\\Omega t+\\phi )$ sum of a dc bias and an ac excitation at angular frequency $n\\Omega $ .", "The complete description of this tomography protocol lies beyond the scope of this article and can be found in Ref.[31].", "However an intuitive understanding can be drawn, that mainly relies on the two-particle interference between the electron source under study and the reference source.", "Let us first focus on the reconstruction of the $n=0$ component of the coherence function, associated with the energy distribution, $\\delta n_{e/h}$ that is on the spectroscopy of the electron source.", "A sketch supporting this discussion is presented Fig.REF a).", "In the case $n=0$ , only the dc part of the voltage applied on input 2 is kept: $V_0(t)=V_{dc}$ that shifts the chemical potential of the connected edge by the value $-e V_{dc}$ .", "As already mentioned, a two-particle interference can only occur between states of same energy.", "An electron at a well defined energy $\\epsilon _0$ finds a symmetric partner in input 2 only if $\\epsilon _0<-eV_{dc}$ (in the limit of vanishing temperature).", "Under this threshold, antibunching occurs with unit probability and partition noise is reduced to zero.", "Otherwise, for $\\epsilon _0> -e V_{dc}$ , the random partitioning takes place, regardless of the presence of the DC bias.", "Accordingly, by sweeping the bias $V_{dc}$ , one can then reconstruct the probability of finding a particle at energy $\\epsilon $ , namely $\\delta n_{e/h}(\\epsilon ) = \\frac{h^2 f}{v} \\Delta \\tilde{\\mathcal {G}}^{(1,e)}_n(\\epsilon /\\hbar )$ from the $V_{dc}$ dependence of the partition noise due to antibunching effects.", "Due to thermal smearing effects, the resolution of such a spectroscopy is in fact limited to $kT_{el}$ in the presence of a finite temperature $T_{el}$ .", "In the same manner (Fig.REF b)), dynamical modulations of the noise with a reference voltage $V_n(t)=V_{dc}+V_{ac}\\cos (n\\Omega t+\\phi )$ enables to gain access to harmonics $\\Delta \\tilde{\\mathcal {G}}^{(1,e)}_n(\\omega )$ for $n\\ne 0$ , that is the off diagonal elements $\\epsilon -\\epsilon ^{\\prime } = n\\hbar \\Omega $ in the $\\epsilon , \\epsilon ^{\\prime }$ plane.", "Using once again the Floquet scattering formalism, simulations of the coherence function of a periodic source have been realized, in the case of the single electron/hole source: three cases with different sets of parameters illustrate the key features on Fig.REF .", "For clarity, odd and even harmonics of $\\Delta \\mathcal {G}^{(e)}(\\omega )$ are plotted on separate graphs as they have different parity with respect to $\\omega $ : $\\Delta \\mathcal {G}^{(e)}_{2p}$ is odd while $\\Delta \\mathcal {G}^{(e)}_{2p+1}$ is even.", "First, these graphs clearly highlight the four quadrants identified in Fig.REF .", "For a transmission $D=0.4$ , the parameters are close to the optimal values : every charge is emitted during the dedicated emission cycle and the excitations are highly energy-resolved, around energies $\\pm \\Delta /2$ .", "Only weak $e/h$ coherences are detected: the emission probability is very close to one, so that the emission of an electron is decorrelated from the emission of the previous hole as the emission probability is close to one.", "Going towards higher transmission ($D=1$ ) yields excitations that lie mostly at low energy, and spread over a wide range of energies.", "Since the transmission is high, the two emission events of electron and holes are once again decorrelated.", "On the opposite, for lower transmissions ($D=0.1$ ), strong $e/h$ coherences appear as the emission probability is much smaller than 1.", "Production of holes and electrons are correlated as the emission of an electron is subject to the emission of the preceding hole, which does not take place in each cycle.", "Note that, as suggested in refs [89], [90], the coherence function of the source could also be measured in time domain, $\\Delta \\mathcal {G}^{(1,e)}(t,t+\\tau )$ , measuring the current at time $t$ at the output of a Mach-Zehnder interferometer as a function of the difference $\\tau $ in the propagation time between the two arms of the interferometer.", "This method implies a simpler measurement (average current instead of current fluctuations) but a more complicated sample.", "Also, decoherence effects during the propagation in the interferometer [91], [92], [93], [94] would have to be taken care of." ], [ "Interaction mechanism in quantum Hall edge channels", "In the previous sections of this manuscript, electron-electron interactions have been neglected, regarding the presentation of the general framework of electron quantum optics (section ) as well as in the discussion of the experimental results where the propagation along the channels was assumed to be interaction free and dissipationless.", "Most results can indeed be first analyzed without taking into account the presence of interaction-induced decoherence of the mono-electronic excitations.", "However, due to their one-dimensional nature, quantum Hall edge channels are prone to emphasize interaction effects.", "In 1D systems, the motion of an electron interacting with its neighbours strongly affects the latter, so that the picture of quasi-free quasiparticles (Fermi liquid paradigm) holding for 2D and 3D systems is not adapted.", "It is replaced by the Luttinger liquid description, that relies on bosonic collective excitations [95], called edge-magnetoplasmons in quantum Hall systems.", "Moreover, inter-channel Coulomb interactions then couple neighboring edge channels (at filling factor $\\nu >1$ ), leading to the appearance of new collective propagation eigenmodes [92].", "Figure: In case of strong coupling between edge channels, a charge density wave in channel 1 is decomposed on two new propagation eigenmodes: a slow neutral mode of velocity v n v_n with antisymmetric distribution of the charge, and a fast charge mode (velocity v ρ ≫v n v_\\rho \\gg v_n) with a symmetric repartition of the charge.The simplest model of two interacting co-propagating edge channels ($\\nu =2$ ) illustrates the typical interaction mechanism.", "In the absence of both inter and intra channel interactions, currents propagate independently in each channel at the bare Fermi velocity $v$ .", "The current $i_k(x, \\omega )$ flowing in channel $k$ ($k=1, 2$ ) at position $x$ and angular frequency $\\omega $ is simply related to the current at position $x=0$ by the phase $e^{i\\frac{\\omega x}{v}}$ acquired along the propagation: $i_k(x, \\omega )=e^{i\\frac{\\omega x}{v}}i_k(0)$ .", "If only intrachannel interactions are turned on, channels 1 and 2 are not coupled such that current propagation along each channel is still described by a phase with a velocity renormalized by interactions.", "However, when including interchannel interactions, outcoming currents $i_k(x, \\omega )$ at position $x$ are related to incoming ones at position $x=0$ via a $2\\times 2$ scattering matrix $S_{emp}(\\omega ,x)$ [96].", "Note that $S_{emp}$ describes the scattering of edge magnetoplasmons and not electrons, so that $S_{emp}$ acts on the current rather than on the fermion field operator $\\hat{a}$ in usual Landauer-Büttiker scattering formalism.", "The diagonalization of the scattering matrix $S_{emp}$ then gives access to the new propagation eigenmodes, that couple both channels.", "In particular, in the limit of strong interactions the two eigenmodes consist in a slow neutral dipolar mode for which the charge is anti-symmetrically distributed between both channels, and fast charge mode with symmetric charge distribution [92] as depicted in Fig.REF .", "Due to Coulomb repulsion, the charge mode propagates much faster than the neutral one, $v_\\rho \\gg v_n$ .", "The appearance of these eigenmodes bears strong similarities with the separation of the spin and charge degrees of freedom in non-chiral quantum wires [97], [98], [99].", "Various experiments have been carried out to investigate the coupling between edge channels and their effect on the relaxation and decoherence of electronic excitations.", "This coupling has been shown to be responsible for the loss of the visibility of the interference pattern in Mach-Zehnder interferometers at filling factor $\\nu =2$ [16], [100], [7].", "In this case, the coupling of the external channel (which is the one probed in the interferometer) to the neighboring one leads to decoherence as information on the quantum state generated in the outer channel is capacitively transferred to the inner one acting as the environment.", "The influence of interchannel coupling on the energy relaxation of out of equilibrium excitations emitted in the outer edge channel has also been probed [101], [102] at filling factor $\\nu =2$ using a quantum dot as an energy filter.", "These results have shown that coherence is lost and energy relaxes on a typical length of a few microns.", "Numerous theoretical works have successfully interpreted decoherence in interferometers [91], [92], [103] and energy relaxation along propagation [104], [96], [105], [106] as stemming from interchannel Coulomb interactions.", "As a consequence, decoherence and relaxation can be controlled to some extent for example by the use of additional gates used to screen the interchannel interaction or by closing the internal edge channel which then acquires a gapped discrete spectrum such that interactions are fully frozen for energies below the gap.", "The latter technique has been shown to decrease both the energy relaxation [107] and the coherence length [18], [108].", "Coupling between channels have also been investigated through high frequency current measurements that directly probe the propagation of edge magnetoplasmons in a quantum Hall circuit.", "Numerous experimental works have investigated the propagation of charge along quantum Hall edge channels, both in the time [109], [110], [111] or in the frequency domain [112], [113], [114].", "However, in the $\\nu =2$ case for example, to access all the terms of the $2\\times 2$ scattering matrix $S_{emp}(\\omega ,x)$ and reveal the nature of the eigenmodes, one needs to selectively address each edge channel individually.", "Using a mesoscopic capacitor to selectively inject an edge magnetosplasmon in the outer edge channel and a quantum point contact to analyze the scattering of the emp to the outer and inner edge channels after a controlled interaction length, the scattering parameters of $S_{emp}(\\omega ,x)$ and their frequency dependence could be investigated, thus revealing the nature of the neutral and charge eigenmodes by a direct measurement of the current at high frequency [115].", "Recently, interchannel interactions could also be characterized using partition noise measurements [116] to measure the excitations (electron/hole pairs) induced in the inner channel when electrons were injected selectively in the outer one.", "The existence and nature of the interchannel coupling is thus now well established, however its influence on any arbitrary single electron state generated above the Fermi sea by single particle emitter is a challenging problem that still requires theoretical and experimental investigation.", "Some results can be obtained in the specific case of a single electron state emitted at a perfectly well defined energy $\\epsilon _0$ above the Fermi sea [80]." ], [ "Decoherence of an energy-resolved excitation", "A single electronic excitation created on top of the Fermi sea enters at $x=0$ in a region where it interacts, via Coulomb interaction, with an environment along a propagation length $l$ , see Fig.", "REF .", "The external environment, which can be any capacitively coupled conductor like an external gate or the adjacent edge channel in the $\\nu =2$ case is labeled as conductor 2 while the edge channel along which the excitation propagates will be labeled as conductor 1.", "As discussed previously in the context of two coupled edge channels at $\\nu =2$ , the interaction between both conductors can be encoded in the scattering matrix $S_{emp}(\\omega ,l)$ which gives the scattering coefficients for charge density waves of angular frequency $\\omega $ propagating in conductors 1 and 2 from the input $x=0$ to the output $x=l$ of the interaction region.", "During propagation in the interaction region, a single particle will emit plasmonic waves in the environment (in the following the input state of the environment will be considered to be at equilibrium at zero temperature).", "The environment and edge channel 1 are then described by a complex many-body state where the edge channel and environment are entangled.", "Tracing out the environmental degrees of freedom at the output, the state of edge channel 1 cannot be described as a pure state anymore and the off-diagonal terms of the first order coherence function can be drastically reduced.", "Figure: Schematics of interactions with the environment.", "A single particle propagating on an edge channel enters the interaction region.", "After the emission of plasmonic waves in the environment, the edge channel is entangled with the environment at the output.The single electron coherence that describes the electronic state in edge channel 1 at the input of the interaction region ($x,y \\ge 0$ ) is known, $\\Delta \\mathcal {G}^{(1,in)}(x,y) = \\phi ^{e}(x) \\phi ^{e,*}(y) \\propto e^{i \\frac{\\epsilon _0 (x-y)}{\\hbar v}}$ (we prefer here to use the $x, y$ notation than the $t, t^{\\prime }$ one to distinguish between the input and output of the interaction region).", "In Fourier space, the energy distribution consists of a Dirac peak at energy $\\epsilon _0$ above the Fermi sea, $\\delta n_{e}(\\epsilon ) = \\delta (\\epsilon -\\epsilon _0)$ (see blue curve on Fig.REF .a)).", "Note that as a consequence of the specific choice of the input wavepacket (plane wave of well defined energy), the input state is stationary in time such that the coherence function in Fourier space is fully determined by the diagonal part $\\delta n_{e}(\\epsilon )$ .", "At the output of the interaction region, one can guess the shape of the output energy distribution: the electron has lost some energy, as a consequence, the quasiparticle peak is reduced to the height $Z \\le 1$ (which eventually goes down to zero as the propagation length increases, see Fig.REF .b)) and a relaxation tail $\\delta n_{e}^{(t)}(\\epsilon )$ appears below the quasiparticle peak.", "This energy can be transferred both to the environment but also to the Fermi sea through the creation of additional electron-hole pairs.", "This can be seen by the appearance of a non-equilibrium energy distribution $\\delta n_{e}^{(r)}(\\epsilon )$ at small energies above the Fermi sea.", "At high enough energy $\\epsilon _0$ each of these two contributions can be identified and associated with a decoherence coefficient of the emitted wavepacket: $\\phi ^{e}(x) \\phi ^{e,*}(y) \\rightarrow \\phi ^{e}(x) \\phi ^{e,*}(y) \\; \\mathcal {D}(x-y)$ with $ \\mathcal {D}(x-y) =\\mathcal {D}_{FS}(x-y) \\times \\mathcal {D}_{env}(x-y) $ where $\\mathcal {D}_{FS}$ stands for a Fermi sea induced decoherence and $\\mathcal {D}_{env}$ for the decoherence induced by the external environment.", "These two decoherence coefficients can be directly expressed as a function of the plasmon scattering matrix in the interaction region [80]: $\\mathcal {D}_{env}(x-y)=\\exp {\\int ^\\infty _0 \\frac{d\\omega }{\\omega } |S_{21}(\\omega )|^2 \\big (e^{i\\frac{\\omega (x-y)}{v}}-1\\big )} \\quad \\quad \\quad \\; \\; \\\\\\mathcal {D}_{FS}(x-y)=\\exp {\\int ^\\infty _0 \\frac{d\\omega }{\\omega }|1 - S_{11}(\\omega )|^2 \\big (e^{i\\frac{\\omega (x-y)}{v}}-1\\big )} \\quad \\quad \\\\\\mathcal {D}(x-y)=\\exp {\\int ^\\infty _0 \\frac{d\\omega }{\\omega }2 \\Re \\big (1-S_{11}(\\omega )\\big )\\big (e^{i\\frac{\\omega (x-y)}{v}}-1\\big )} \\quad \\quad $ In this regime, the Fermi sea appears as an extra dissipation channel which must be taken into account into an effective environment.", "Note that here, this picture emerges in the high energy limit and is not valid when the extra-particle relaxes down to the Fermi surface.", "In this latter case, separation of the extra particle and the additional electron-hole pairs created above the Fermi sea is not possible and the decoherence coefficient $\\mathcal {D}(x-y)$ cannot be identified as easily.", "This decoherence coefficient which suppresses the off diagonal coefficients of the first order coherence ($\\mathcal {D}(x-y) \\rightarrow 0 $ for $|x-y| \\rightarrow \\infty $ ) has important consequences on a Hong-Ou-Mandel experiment which is a sensitive probe of the off-diagonal components (coherences).", "Let us assume for simplicity that the decoherence factor takes the simple form $\\mathcal {D}(t,t^{\\prime }) = e^{- \\frac{|t-t^{\\prime }|}{\\tau _c}}$ (the decoherence factor has been expressed in time instead of position using $x=-vt$ ).", "In this case, Eq.", "(REF ) for the normalized output noise in the HOM experiment which was valid in the case of two pure states at the input of the splitter (absence of decoherence) needs by the following expression which takes into account decoherence: $\\Delta \\overline{q} & =& 1 - \\int dt dt^{\\prime } \\phi ^{e}_1(t) \\phi ^{e,*}_1(t^{\\prime }) \\phi ^{e}_2(t^{\\prime }) \\phi ^{e}_2(t) \\mathcal {D}_1(t,t^{\\prime })\\mathcal {D}_2(t^{\\prime },t) \\nonumber \\\\$ Taking $\\phi ^{e}_1(t) = \\frac{\\Theta (t)}{\\sqrt{\\tau _e}} e^{-\\frac{t}{2 \\tau _e}} e^{-i \\epsilon _0 t/\\hbar }$ , $\\phi ^{e}_2(t) = \\phi ^{e}_1(t+\\tau ) $ where $\\tau $ is the tuneable time delay between the emission of the two sources, and $\\mathcal {D}_1(t,t^{\\prime }) = \\mathcal {D}_2(t,t^{\\prime })= e^{-\\frac{|t-t^{\\prime }|}{\\tau _c}}$ , one obtains: $\\Delta \\overline{q} & =& 1 - \\eta e^{-\\frac{|\\tau |}{\\tau _e}} \\\\\\eta & =& \\frac{1}{1+ 2 \\tau _e/\\tau _c}$ This model of decoherence predicts a reduction of the HOM dip at $\\tau =0$ while keeping the shape of an exponential decay when varying the delay $\\tau $ .", "This model predicts that a wavepacket with a small temporal extension and in particular much smaller than the coherence time $\\tau _c$ is not affected by decoherence, $\\eta \\approx 1$ .", "On the contrary, a wavepacket with a large temporal extension ($\\tau _e \\gg \\tau _c$ ) is drastically affected by decoherence and the HOM dip vanishes, $\\eta \\approx \\frac{\\tau _c}{2 \\tau _e}$ .", "In this limit, the electron cannot be described by a coherent wavepacket with a well defined phase relationship between its various temporal component but rather by a classical probability distribution of different emission times of typical extension given by $\\tau _e$ .", "In this case, the width $\\tau _e$ plays the role of a random delay between the two sources which explains the reduction of the HOM dip.", "In our experiment, we measure $\\eta \\approx 0.5$ for $\\tau _e \\approx 50$ ps which is consistent with $\\tau _c \\approx 100$ ps.", "Figure: a) Energy distribution before (blue curve) and after (red curve) interaction along the propagation length ll.", "b) Typical dependence of the quasiparticle peak height on the interaction length ll." ], [ "Interactions in the Hong-Ou-Mandel setup", "We now provide a quantitative description of the effects of Coulomb interactions in the Hong-Ou-Mandel setup in the case of interchannel coupling at filling factor 2.", "We consider the case of short range interchannel interactions and strong coupling such that the eigenmodes are the symmetric fast charge mode (with velocity $v_{\\rho }$ ) and the slow antisymmetric neutral mode (with velocity $v_n \\ll v_{\\rho }$ ) as described in Sec.", "REF .", "Finite temperature of the leads can also be included.", "The single electron source is modeled through the injection of single wave-packets at a given distance $l$ from the QPC (chosen symmetrically for the two incoming arms: $x=\\pm l$ ).", "As previously discussed, the wavepackets are defined as exponentials in real-space, $\\phi _2 (x) = \\frac{1}{\\sqrt{v \\tau _e}} e^{-i \\epsilon _0 x/(\\hbar v)} e^{-x/(2 v \\tau _e)} \\theta (x)$ , and for the sake of simplicity, we focus on the interference between identical wave-packets, $\\phi _1 (x)=\\phi _2 (-x)$ .", "The normalized HOM correlation then reads [117]: $\\Delta \\bar{q} (\\tau ) = 1 - \\frac{ \\mathrm {Re} \\left[ q_{HOM} \\right]}{\\mathrm {Re} \\left[ q_{HBT} \\right]} $ where $q_{HOM} &= \\int d x_1 d y_1 \\int d x_2 d y_2 \\phi _1 (x_1) \\phi _1^* (y_1) g(0,x_1-y_1) \\nonumber \\\\& \\times \\phi _2 (x_2) \\phi _2^* (y_2) g (0,y_2-x_2) \\int d t d t^{\\prime } \\mathrm {Re} \\left[ g(t^{\\prime } - t,0)^2 \\right] \\nonumber \\\\& \\times \\left[ 1 - \\frac{h (t;x_2,y_2)}{h (t^{\\prime };x_2,y_2)} \\right] \\times \\left[ 1 - \\frac{h (t^{\\prime }+\\tau ;-x_1,-y_1)}{h (t+\\tau ;-x_1,-y_1)} \\right] \\\\q_{HBT} &= \\int d x_1 d y_1 \\int d x_2 d y_2 \\phi _1 (x_1) \\phi _1^* (y_1) g(0,x_1-y_1) \\nonumber \\\\& \\times \\phi _2 (x_2) \\phi _2^* (y_2) g (0,y_2-x_2) \\int d t d t^{\\prime } \\mathrm {Re} \\left[ g(t^{\\prime } - t,0)^2 \\right] \\nonumber \\\\& \\times \\left[ 2 - \\frac{h (t;x_2,y_2)}{h (t^{\\prime };x_2,y_2)} - \\frac{h (t^{\\prime };-x_1,-y_1)}{h (t;-x_1,-y_1)} \\right]$ and the auxiliary functions introduced are given by $g(t,x)&= \\left[ \\frac{\\sinh \\left( i \\frac{\\pi a}{\\beta v_\\rho } \\right)}{\\sinh \\left( \\frac{ia + v_\\rho t - x}{\\beta v_\\rho /\\pi } \\right)} \\frac{\\sinh \\left( i \\frac{\\pi a}{\\beta v_n} \\right)}{\\sinh \\left( \\frac{ia + v_n t - x}{\\beta v_n /\\pi } \\right)} \\right]^{1/2} , \\\\h (t;x,y) &= \\left[ \\frac{\\sinh \\left( \\frac{ia - v_\\rho t + x + l}{\\beta v_\\rho / \\pi } \\right)}{\\sinh \\left(\\frac{ia + v_\\rho t - y - l}{\\beta v_\\rho /\\pi } \\right)} \\right]^{\\frac{1}{2}} \\left[ \\frac{\\sinh \\left( \\frac{ia - v_n t + x + l}{\\beta v_n / \\pi } \\right)}{\\sinh \\left( \\frac{ia + v_n t - y - l}{\\beta v_n /\\pi } \\right)} \\right]^{\\frac{1}{2}} .$ The variable $a$ is a spatial cutoff, which ultimately needs to be sent to 0, and $\\beta = 1/(k_B T_{el})$ .", "Figure: Normalized HOM correlations as a function of the time delay τ\\tau , for two different type of wave-packets: (upper) one with an escape time τ e =22\\tau _e= 22 ps and emitted energy ϵ 0 =0.175\\epsilon _0=0.175 Kand (lower) one with an escape time τ e =44\\tau _e= 44 ps and emitted energy ϵ 0 =0.7\\epsilon _0=0.7 K. In both cases, T el =0.1T_{el} = 0.1 K.Numerical evaluation of Eq.", "REF can be performed thanks to a quasi Monte Carlo algorithm using importance sampling [118], results are presented on Fig.REF .", "As we vary the time delay $\\tau $ of the right-moving electron over the left-moving one, our computations reveal the presence of three characteristic signatures in the noise (see Fig.", "REF ) : a central dip at $\\tau = 0$ , and two side structures which emerge symmetrically with respect to the central dip at $\\tau = \\pm l (v_\\rho -v_n) /v_\\rho v_n$ .", "The depth and shape of these three dips are conditioned by the energy resolution of the incoming wave-packets.", "Away from these three features, the normalized correlations saturate at a constant value, representing the Hanbury-Brown and Twiss contribution.", "This corresponds to the situation where the electrons injected on the two incoming arms scatter independently at the QPC.", "This interference pattern can be interpreted in terms of the different excitations propagating along the partitioned edge channel.", "After injection, the electron fractionalizes into two modes: a slow neutral mode with anti-symmetric distribution of the charge between the injection and the co-propagating channels and a fast charge mode with a symmetric repartition of the charge among the two channels.", "The central dip, which corresponds to the symmetric situation of synchronized injections, thus probes the interference of excitations with the same velocity and charge.", "These identical excitations interfere destructively, leading to a reduction of the noise (in absolute value), thus producing a dip in the normalized HOM correlations.", "A striking difference with the non-interacting case is that the central dip never reaches down to 0 as observed experimentally (see Sec.", "REF ).", "The depth of this dip is actually a probing tool of the degree of indistinguishability between the colliding excitations [81].", "Our present work suggests that because of the strong inter-channel coupling, some coherence is lost in the other channels, and the Coulomb-induced decoherence leads to this characteristic loss of contrast for the HOM dip.", "This effect gets more pronounced for further energy-resolved packets.", "As depicted in Fig.", "REF , while for “wide” packets in energy ($\\gamma =\\frac{2\\epsilon _0 \\tau _e}{\\hbar } \\approx 1$ ) the contrast (defined as $\\eta =1-\\Delta \\bar{q} (0)$ ) is still pretty good, $\\eta \\sim 0.8$ , the loss of contrast can be dramatic for energy-resolved packets, with $\\eta \\sim 0.4$ for $\\gamma =8$ .", "Adjusting the time delay appropriately, one can also probe interferences between excitations that have different velocities.", "This effect is responsible for the side structures appearing in the noise: at $\\tau = l (v_\\rho - v_n)/(v_\\rho v_n)$ , the fast right-moving excitation and the slow left-moving one reach the QPC at the same time while the dip at $\\tau =-l (v_\\rho - v_n)/(v_\\rho v_n)$ corresponds to the collision between a slow right-moving excitation and a fast left-moving one.", "Like the central dip, these lateral structures correspond to the collision of two excitations of the same charge, which interfere destructively.", "Their depth is however less than half the one of the central dip.", "This can be attributed to the velocity mismatch between interfering excitations, as it indicates that they are more distinguishable.", "This difference of velocity of the two colliding objects is also responsible for the asymmetry of the lateral dips.", "Typically, the slope is steeper for smaller $| \\tau |$ .", "This asymmetry is very similar to the one encountered in the non-interacting case for interfering packets with different shapes, where a broad right-moving packet in space collides onto a thin left-moving one [83]." ], [ "Conclusion", "As detailed in this manuscript, optical tools and concepts can be used in a very efficient way to understand and characterize electronic propagation in a quantum conductor.", "Within this framework, electronic transport is analyzed through a simple single particle description which captures most of the features of electron propagation but is only correct in the non-interacting photon-like case.", "In the presence of Coulomb interactions the correct description relies on the resolution of a complex many-body problem.", "The production and manipulation of single-particle states provide a direct test bench for single-particle physics.", "Using controlled emitters with tuneable parameters, a wide range of single particle wavefunctions can be engineered both in time or energy [79], [119] space.", "Coulomb interaction during propagation with the surrounding electrons of the Fermi sea and nearby conductors will strongly affect the state of a single excitation.", "Consequently, even the propagation of a single electron tends to a complex many body problem : as the electronic wavepacket propagates, it relaxes and decoheres, and additional electron-hole excitations are generated.", "This mechanism sets the limits of electron quantum optics : during propagation, a single-particle excitation is diluted in collective excitations, so that the possibility of manipulating a pure single-particle quantum state is lost.", "To get a complete understanding of the effects of Coulomb interactions, it is necessary to picture fully the electronic wavefunction in energy or time domains.", "The tomography protocol suggested in [120] provides a complete imaging of the first order coherence in energy domain from noise measurements in the Hanbury-Brown and Twiss geometry.", "In particular the energy distribution of mono-electronic excitations could be extracted from the variation of the output noise when shifting the chemical potential of a Fermi sea used as reference state in one input.", "The measured energy distribution after a tuneable propagation length could be compared with the non interacting theory in analogy with the spectroscopy of a non-equilibrium stationary electron beam performed in ref.", "[101], [51] using a quantum dot as an energy filter.", "The energy distribution is also directly related to heat transfers and heat fluctuations generated by single particle emitters [121], [122] and could thus be inferred from nano-caloritronic measurements.", "In the time domain, the first order coherence could be measured using a single electron emitter at the input of a Mach-Zehnder interferometer [89].", "Beyond the study of the propagation of a single excitation, proposals have been made to manipulate coherently single to few electronic excitations, connecting the physics of quantum conductors to quantum information processing.", "For example, the Mach-Zehnder geometry, together with two single electron emitters placed at the input, could be used to postselect entangled electron pairs [123], [124], [125] or to generate GHZ states [126].", "However, such coherent manipulations would require to reduce and circumvent the effect of Coulomb interaction in quantum Hall edge channels for example by closing the internal edge channel [107], [108].", "Energy exchanges between neighboring edge channels are then frozen for energies below the excitation gap of the internal edge.", "As pioneered in [127], coherent manipulations could also be performed on the spin degree of freedom.", "By transferring charge in a controlled manner between the two co-propagating edge channels of opposite spins at filling factor $\\nu =2$ , any coherent superpositions of spins could be achieved.", "Finally, another extremely interesting route would be to extend these concepts to other ballistic electronic systems.", "Of particular interest would be the study of triggered charge emission along the edge channels of the fractional quantum Hall regime [128].", "The question is whether one can emit and manipulate a single quasiparticle of fractional charge in the same fashion as single electronic excitations for integer values of the filling factor.", "In particular the study of two-particle interference would be of particular interest as they are sensitive to the phase associated with the exchange of two particles and could thus provide a way to measure the statistics of fractional excitations.", "Another possible implementation would be the recently discovered helical edge states of quantum spin Hall effect [129], [130] : an equivalent of the mesoscopic capacitor in such a system has already been proposed [131], [132], enabling the generation of time-bin entangled pairs of electrons.", "Acknowledgement.", "This work was supported by the ANR grant '1shot', ANR-2010-BLANC-0412.", "We warmly thank M. Albert, C. Flindt, G. Haack and M. Moskalets for fruitful discussions and Markus Büttiker for his strong support and inspiration to this work." ] ]
1403.0118
[ [ "Some consequences of thermodynamic feasibility for the multistability\n and injectivity in chemical reaction networks" ], [ "Abstract Power law dynamics is used to describe the stability behavior in metabolic networks such as chemical reaction networks (CRN's).", "These systems allow multiple steady states within a single stoichiometric class.", "On the other side thermodynamic constraints such as loop-less fluxes represented by the Gorban theorem of alternatives applied to these networks reveal considerable restrictions to their dynamics by eliminating multistability of CRN's in general.", "Thermodynamic feasible CRN's are contained in the class of injective CRN's.", "We can give an alternative proof of the detailed balance with Brewer's Fixed Point Theorem.", "Furthermore we can derive by the loop-less principle the extended detailed balance.", "This paper establishes a link between recent research in CRN theory and thermodynamic basics.", "The result has also consequences for the picture of multiple steady states as assumed for cell differentiation and regulation.", "CRN's provide from their perspective not enough means to maintain multistability without regulation or external control." ], [ "Introduction", "Summarizing the results which have been lately obtained in [1] and [2] we derive some obvious consequences.", "The loop-less and so called thermodynamicall feasible fluxes outlined and specifed in [1] obey in an almost natural way the injectivity conditions in [2].", "There is a long history of achievements analyzing injectivity and multistationarity of chemical reaction networks (CRN's) ([10], [11],[12], [15],[16]).", "There have been nomerous refinements and generalizations of prevoius resuslts in ([3],[2],[4],[6],[8],[13],[14]).", "We would like to insert thermodynamical requirements [1] into CRN's as recently manifestet in [2] to elucidate their consequences for their stability behaviour." ], [ "Thermodynamic Considerations", "In this section we give some basic explanations for the physical description of chemical reactions as occuring in chemical reaction networks of continuous stirred tank reactors (CSTR).", "Generally there is a known thermodynamic potiential that governs reaction kinetics between complexes.", "We will consider every reaction as reversible and described by the boltzmann distribution between potentials.", "Reaction dynamics derived from power law kinetics allow by that assumption flows in both directions (reversible).", "We will therefore assume that all reactions are reversible unless we explicitely mention where we can neglect full reversibility.", "An example of a reversible reaction for illustration considered here is: $A + B \\; \\; \\; \\mathop {\\stackrel{\\kappa _f}{\\rightleftarrows }}_{\\kappa _b} \\; \\; \\; C + D \\ .$ This reaction has a reaction constant for both directions.", "A reaction constant $\\kappa (T)$ is almost universally described as dependent upon activation potential $\\Delta E_a$ as in ([17], p. 9, 1-25) $\\kappa (T)=\\kappa ^{o} \\cdot e^{-\\Delta E_a/kT} \\ .$ CSTR nearly operating under sonstant tempterature $T$ have reaction constants that can be assumed to be fixed approximately.", "On the other side we have to check whether applied theorems withstand validity under perturbations of parameters.", "For a theoretical derivation of the formula for the forward and backward reaction constants for power law kinetics see [18].", "We denote here the concentrations of the species $\\lbrace A,B,C,D\\rbrace $ as $\\lbrace x_A,x_B,x_C,x_D\\rbrace $ in reaction (REF ).", "Complexes in a reaction are the union of all reactant species and all product species.", "In the case of reaction (REF ) we have the complexes $\\mathcal {C}_1=\\lbrace A+B\\rbrace $ and $\\mathcal {C}_2=\\lbrace C+D\\rbrace $ with $\\mathcal {C}=\\lbrace \\mathcal {C}_1,\\mathcal {C}_2\\rbrace $ being the collection of all reactions.", "Assuming powerlaw kinetics for reaction (REF ) we obtain for the change rate of species $x_A$ : $\\dot{x}_A= \\kappa _b x_C x_D - \\kappa _f x_A x_B$ Similar relations hold for all other three species.", "A more detailed treatment of reaction constants with the example given in eqn.", "(REF ) is obtained in [18] where we have at the equilibrium steady state the following relation: $\\frac{\\kappa _f(T)}{\\kappa _b(T)} = \\left(\\frac{\\mu _{CD}}{\\mu _{AB}} \\right)^{(3/2)} \\left(\\frac{x_C x_D}{x_A x_B} \\right)_{int} \\exp (- \\Delta E_0 /k_b T)$ The energy difference $\\Delta E_0 = E_{0,\\mathcal {C}_2} - E_{0,\\mathcal {C}_1}$ is given by the zero-point energies of the reactant ($E_{0,\\mathcal {C}_1,}$ ) and product ($E_{0,\\mathcal {C}_2}$ ) complex.", "Here we do neglect the reduced masses $\\mu _{CD}$ and $\\mu _{AB}$ , since we can absorb them into the related reaction constants of the specific reaction." ], [ "Background material", "A chemical reaction as in equation (REF ) $R_i: \\mathcal {C}_1 \\rightarrow \\mathcal {C}_2$ between two complexes $\\mathcal {C}_1$ and $\\mathcal {C}_2$ is defined by the reactant complex $\\mathcal {C}_1=\\lbrace A,B\\rbrace $ and product complex $\\mathcal {C}_2=\\lbrace C,D\\rbrace $ with stoichiometric vectors $y_1=y_{AB}=(1,1,0,0)$ and $y_2=y_{CD}=(0,0,1,1)$ .", "Furthermore we have the associated forward and backward reaction constants $\\kappa _{\\mathcal {C}_1\\rightarrow \\mathcal {C}_2} $ and $ \\kappa _{\\mathcal {C}_2\\rightarrow \\mathcal {C}_1}$ .", "We can also denote the difference stoichiometric vector $[y_2-y_1]=(-1,-1,1,1)$ .", "and by enumerating the species by $x=(x_1,x_2,x_3,x_4)=(x_A,x_B,x_C,x_D)$ we can rewrite equation (REF ) by: $\\dot{x}_1= \\kappa _{\\mathcal {C}_2\\rightarrow \\mathcal {C}_1} x^{y_2} - \\kappa _{\\mathcal {C}_1\\rightarrow \\mathcal {C}_2} x^{y_1}$ where $x^{y}=\\prod _{i\\in [4]}x_i^{(y)_i}$ .", "Following the notation given in [3] and [2] for a CRN we can form the stoichiometric difference matrix $A=\\lbrace [y_{CD}-y_{AB}],[y_{AB}-y_{CD}]\\rbrace \\in \\mathbb {R}^{4 \\times 2}$ , the diagonal reaction constant matrix $diag(\\kappa )=diag(\\kappa _{\\mathcal {C}_1\\rightarrow \\mathcal {C}_2} ,\\kappa _{\\mathcal {C}_2\\rightarrow \\mathcal {C}_1} ) \\in \\mathbb {R}^{2 \\times 2}$ and the complex matrix $B=\\lbrace y_{AB},y_{CD}\\rbrace \\in \\mathbb {R}^{2 \\times 4}$ and rewrite the change rate of all species $x$ as: $\\dot{x}=A\\ diag(\\kappa ) x^B \\ ,$ where $x^B\\in \\mathbb {R}^2$ is calculated for each row-vector in $B$ .", "Generally we define for the case of $n$ species $x \\in {\\mathbb {R}^n_{+}}$ involved in $r$ reactions $\\mathcal {R}$ (possibly reversible or not) and corresponding stoichiometric difference matrix $A \\in \\mathbb {R}^{n\\times r}$ and complex matrix $B \\in \\mathbb {R}^{r \\times n}$ with associated reaction rates $\\kappa \\in \\mathbb {R}^r_{+}$ the generalized polynomial map $f_{\\kappa } (x): \\mathbb {R}^{n}_{+} \\rightarrow \\mathbb {R}^n$ , where we have $A_{\\kappa }= A \\ diag (\\kappa )$ , by: $\\frac{ {\\sf d} x}{ {\\sf d} t} =f_{\\kappa }(x) = A_{\\kappa } x^B$ In the case of a fully reversible network we have for each reaction the forward $\\kappa ^{i}_f$ and backward $\\kappa ^{i}_b$ reaction constant where $i \\in [r] $ with $r=2r^{\\prime }$ reactions in total.", "We will develop the subject for the fully reversible case even when we can admit less restrictive conditions for the validity of the result.", "We will first state the result from [1] here.", "We will consider $r$ reactions $\\mathcal {R}$ with positive reaction constants $\\kappa _j$ , $j \\in [r]$ over $n$ different species.", "The number $p$ of complexes $y_i \\in \\mathbb {R}^n_{+}$ , $i \\in [p]$ are reduced to these taking place in one of the $r$ unidirectional reactions.", "The difference stoichiometry vectors of each reaction $j \\in [r]$ denoted by $[y-y^{\\prime }]^{(j)}$ form the colums of the matrix $A$ .", "The notation here is the same for a matrix $A$ representing the internal reaction of a CRN.", "We exclude here external reactions first and analyse the internal system of reactions.", "At the end of the text we will insert an external flux representing the inflow of a chemostat reactor (CFSTR).", "In order to consider thermodynamic aspects in a flux distribution we have to assign potential differences $\\Delta {G}$ between the complexes of each reaction of the CRN in form of a vector of potentials for the complexes.", "The Gibbs potential for example (REF ) is related to equation (REF ) by $\\Delta {G} = y_C G_C^0 + y_D G_D^0 - y_A G_A^0 - y_B G_B^0 + RT \\, \\ln (K_a)$ over the constant $R=N_A \\cdot k_b$ , the activities $K_a=\\prod _{i} x^{[y_2-y_1]}$ from equation (REF ) and the zero point Energies $G_x^0$ (see also [5] eqn.", "(1)).", "Through that notation we can find a vector $\\gamma \\in \\mathbb {R}^n$ for the potentials of the individual spezies depending on their concentrations and stoichiometric coefficient, such that we obtain $\\Delta G = \\gamma ^T A$ as the differential energy between the complexes for the current temperature and spezies concentrations.", "The following classification of fluxes can be traced back to the Gordan theorem of alternatives [1] which we will state here: Theorem 1 (Gordan's theorem) $\\forall A\\in \\mathbb {R}^{ n\\times m}$ exactly one of the following two statements is true: (a) $\\exists z \\in \\mathbb {R}^m_{+} \\setminus \\lbrace 0\\rbrace $ , s.t.", "$Az=0$ (b) $\\exists y \\in \\mathbb {R}^n$ s.t.", "$ A^{\\top } y>0$ In [1] a transformation of the Gordan theorem for the case of reversible fluxes of a chemical reaction nework is given.", "A reaction system fully reversible will be called loop-free/thermodynamically feasible (b) or thermodynamically not feasible with loops (a) if the following holds: Corollary 3.1 For all $\\hat{A} \\in \\mathbb {R}^{n \\times r}$ where $n$ is the number of species and $r$ the number of (bidirectional/reversible) reactions and every $\\nu \\in \\mathbb {R}^r$ one of the following cases is true: (a) $\\exists \\hat{z} \\in \\mathbb {R}^r \\setminus \\lbrace 0\\rbrace $ , s.t.", "$(\\forall i \\ sign(\\hat{z}_i) \\in \\lbrace sign(\\nu _i),0\\rbrace )\\wedge \\hat{A} \\hat{z}=0$ (b) $\\exists \\gamma \\in \\mathbb {R}^n$ s.t.", "$(\\forall i \\ sign(\\hat{A}^{\\top }\\gamma )_i = - sign(\\nu _i) \\vee \\nu _i=0)$ See [1].", "The idea behind that alternative is that we cannot have a flux keeping the concentrations of the species constant when there are differences between the potential of the complexes.", "The net energy consumption would be zero and the turnover would be non-zero which would be impossible due to the conservation of energy.", "It is more important to know that there is a potential distribution behind that which does not allow thermodynamically infeasible fluxes.", "In Corollary REF we were choosing $\\gamma $ instead of $y$ in order to avoid an overlap with the stoichiometry vector $y_i$ and also to give the link to the chemical potential introduced in equations (REF ) and (REF ) since $\\gamma ^{\\top }A$ is equivalent to $A^{\\top }\\gamma $ .", "(b) in Corollary REF reflects the fact that the flux $\\nu _i$ is in opposite direction to the increasing potential $(A^{\\top }\\gamma )_i$ between complexes.", "We can link that relation to our reversible system.", "We set $m=2r$ the number of all unidirectional reaction in a fully reversible chemical reaction network and order the signs of the flux $\\nu \\in \\mathbb {R}^r$ with $sign(\\nu _i)=d_i$ for $i \\in [r]$ according to the first $r$ forward and $r$ backward fluxes or each reversible reaction where we have $d_i=-d_{i+r}$ and the total flux results as the sum of the forward and backward flux: $\\nu _i = z_i-z_{i+r}$ for $z \\in \\mathbb {R}^m_{+}$ .", "We can set up the following result which is an equivalent formulation of loop-free fluxes from Corollary REF for unidirectional fully reversible CRN's.", "Corollary 3.2 For all $A \\in \\mathbb {R}^{n \\times m}$ where $n$ is the number of species and $m=2r$ the number of reactions and every $\\nu \\in \\mathbb {R}^r$ one of the following cases is true: (a) $\\exists z \\in \\mathbb {R}^m_{+} \\setminus \\lbrace 0\\rbrace $ $\\wedge $ $(\\exists j \\in [r]$ with $ z_j \\ne z_{j+r} )$ , s.t.", "$(\\forall i \\in [r] \\ sign(z_i-z_{i+r}) \\in \\lbrace sign(\\nu _i),0\\rbrace )\\wedge A z=0$ (b) $\\exists \\gamma \\in \\mathbb {R}^n$ s.t.", "$(\\forall i (\\ sign(A^{\\top }\\gamma )_i = - \\ sign(A^{\\top }\\gamma )_{i+r} = - sign(\\nu _i) ) \\vee \\nu _i=0)$ Equivalence between Corollary REF and REF concerning (a) can be seen by doubling the matrix $\\hat{A}$ for the bidirectional case by setting $A=(\\hat{A},-\\hat{A})$ and also doubling the vector $\\hat{z}$ by setting $z_i=\\max {(\\hat{z}_i,0)}$ and $z_{i+r}=-\\min {(\\hat{z}_i,0)}$ for $i \\in [r]$ .", "The reverse can be done by halving $A$ to form $\\hat{A}$ and by taking differences $\\hat{z}_i=z_i-z_{i+r}$ for $i \\in [r]$ .", "(b) is equivalent in both Corollaries.", "Remark 3.3 Corollary REF can be extended to the case where reaction $\\mathcal {R}_i$ , $i \\in [r]$ are not reversible by choosing $\\nu \\in \\mathbb {R}^r$ such that the sign of $\\nu _i$ is in accordance with the direction of the reaction $\\mathcal {R}_i$ .", "Remark 3.4 The exclusion of the case (a) comes as the assumption that there is no component $x$ of $\\nu $ that is in the nullspace of $A$ .", "The process of elimination of components $x \\in \\ker (A)$ implies that $\\nu $ is orthogonal to the nullspace of $A$ : $\\nu \\perp \\ker (A) \\ .$ We can now use that fact from equation (REF ) to derive conditions for possible injectivity according to [2].", "Therefore we have to suffer some more notation.", "The sign $\\sigma (a)$ of a vector $a \\in \\mathbb {R}^n$ is given by $\\sigma (a)_i=sign(a_i)$ .", "Therefore we have $\\sigma (a) \\in \\lbrace -1,0,1\\rbrace ^n$ .", "For a subspace $K\\subset \\mathbb {R}^n$ we get consequently $\\sigma (K)=\\lbrace \\sigma (a)\\vert a \\in K\\rbrace $ .", "Furthermore we define $\\Sigma (K)=\\sigma ^{-1}(\\sigma (K))$ .", "We can now state the following theorem: Theorem 2 ([2]) Let $f_{\\kappa }: \\mathbb {R}^n_{+}\\rightarrow \\mathbb {R}^m$ be the generalized polynomial map $f_{\\kappa }(x) = A_{\\kappa } x^B $ , where $A \\in \\mathbb {R}^{n\\times r}$ , $B \\in \\mathbb {R}^{r \\times n}$ and reaction rates $\\kappa \\in \\mathbb {R}^r_{+}$ .", "Let $K\\subset \\mathbb {R}^n$ with $K^*=K \\setminus \\lbrace 0\\rbrace $ , the following statements are equivalent: (inj) $f_{\\kappa }$ is injective with respect to $K$ , for all $\\kappa \\in \\mathbb {R}^r_{+}$ (sig) $\\sigma (\\ker (A)) \\cap \\sigma (B(\\Sigma (K^*)))= \\varnothing $ .", "See [2] Theorem 1.4.", "The number of reactions $r$ in theorem REF includes both reversible and nonreversible reactions by counting reversible reactions double and irreversible reactions single.", "For further purposes we need the analysis of the second $(sig)$ property.", "We know from [2]: Lemma 3.5 Let $B \\in \\mathbb {R}^{r \\times n}$ and $K \\subset \\mathbb {R}^n$ where we set $K^*=K \\setminus \\lbrace 0\\rbrace $ .", "Further let $\\varphi _B: \\mathbb {R}^n_{+} \\rightarrow \\mathbb {R}^r_{+}$ be the generalized polynomial map $\\varphi _B(x)=x^B$ , then the following statements are equivalent: $\\varphi _B$ is injective with respect to $K$ .", "$\\sigma (ker(B)) \\cap \\sigma (K^*) = \\varnothing $ See [2] Proposition 2.5." ], [ "Injectivity relations for thermodynamic feasible fluxes", "We will now describe the system under consideration.", "We will use the CRN's as introduced in [3].", "By setting $A=SE$ we have similar to eqn.", "(REF ) the specific CRN $\\frac{ {\\sf d} x}{ {\\sf d} t} =f_{\\kappa }(x) = SE \\ diag({\\kappa }) x^B \\ .$ The columns of $S$ are the stoichiometry vectors of all $p$ complexes $ y_{j_p} \\in \\mathcal {C}$ , $j \\in [p]$ involved in the $r$ reactions $\\mathcal {R}$ .", "$E$ is the incidence matrix between the interacting complexes forming the matrix $A$ , which consists of all stoichiometric differences of the reacting complexes $[y_i -y_i^{^{\\prime }} ] \\in \\mathcal {R}$ , $i \\in [r]$ .", "The rows of $B$ are all reactant complexes of each reaction .", "We define $K={\\textit {im}}(A)$ .", "For $x,y \\in \\mathbb {R}^r$ we denote $\\sigma (x) \\subseteq \\sigma (y) $ if $\\sigma (x)_i \\in \\lbrace \\sigma (y)_i,0\\rbrace $ , $\\forall \\ i \\in [r]$ .", "We now use the relation in eqn.", "(REF ) to show the following lemma: Lemma 4.1 $\\ker (A) \\bot \\ diag(\\kappa ) x^B \\Longleftrightarrow \\ker (A_{\\kappa }) \\bot x^B \\ ,$ $diag(\\kappa )$ is orderpreserving since we have $\\kappa \\in \\mathbb {R}^r_{+}$ s.t.", "we have an equivalence between $a \\in \\ker (A_{\\kappa })$ with $\\sigma (a)\\subseteq \\sigma (x^B)$ and $diag(\\kappa ) a=b \\in \\ker (A)$ with $\\sigma (b) \\subseteq \\sigma (diag(\\kappa ) x^B) = \\sigma (\\nu )$ through $\\sigma (a)=\\sigma (b)$ .", "By the same minimization process as pointed out in Remark REF , we obtain equation (REF ).", "Lemma 4.2 Let $V,W \\subset \\mathbb {R}^n$ be two subspaces for which $v \\in V$ and $w \\in W$ implies $v \\bot w$ then $\\sigma (V) \\cap \\sigma (W^*)=\\emptyset $ .", "(The converse does not hold).", "Assume there exists $v \\in V$ and $w \\in W$ s.t.", "$\\sigma (v)=\\sigma (w)\\ne 0$ then $v \\cdot w > 0$ which contradicts $v \\perp w$ .", "We can now state our main theorem: Theorem 3 For a system as in equation (REF ) where $n$ is the number of species with concentrations $x \\in {\\mathbb {R}^n_{+}}$ involved in $r$ reactions $\\lbrace \\mathcal {R}_i\\rbrace _{i \\in [r]}$ and stoichiometric difference matrix $A \\in \\mathbb {R}^{n\\times r}$ and complex matrix $B \\in \\mathbb {R}^{r \\times n}$ with reaction rates $\\kappa \\in \\mathbb {R}^r_{+}$ and corresponding generalized polynomial map $f_{\\kappa } (x): \\mathbb {R}^{n}_{+} \\rightarrow \\mathbb {R}^n$ with $A_{\\kappa }= A \\ diag (\\kappa )$ we get under the condition that there exists a specific ${\\kappa }^{t} \\in {\\mathbb {R}^n_{+}}$ s.t.", "$diag(\\kappa ^{t}) x^B \\perp \\ker (A)$ (c.f.", "eqn.", "(REF )) holds for all $x \\in \\mathbb {R}^n_{+}$ the following sufficient conditions for injectitivity in the sense of theorem (REF ): $span(reaction \\ differences\\ in \\ A) \\subseteq span(reactant \\ complexes \\ in \\ B) \\ .$ Lemma REF provides more than we need to proove (REF ) (sig) and is part of the proof of (REF ) (sig), since we need only the disjoint sign condition.", "The relation holds for all $\\kappa \\in \\mathbb {R}^r_{+}$ .", "To see that a loop free flux system implies injectivity we have to show that $\\varphi _B$ is injective with respect to $K$ and we have to show that the image of $B$ with respect to $K$ is perpendicular/sign-disjoint to $ker(A)$ (theorem REF , (sig)).", "We derive another relation from (REF ) and (REF ) by using the fact that the differential $\\delta \\nu $ of the flux $\\nu $ does also satisfy these relations.", "$\\frac{{\\sf d} \\varphi _B(x)}{{\\sf d}x} = diag{(x^B)} B \\ diag{(x^{-1})} \\in \\mathbb {R}^{r \\times n}, \\ x \\in \\mathbb {R}^n_{+}$ We have $\\delta \\nu \\perp \\ker (A)$ too.", "Calculating $\\delta \\nu $ : $\\delta \\nu = diag(\\kappa ) \\frac{{\\sf d} \\varphi _B(x)}{{\\sf d}x} \\cdot \\frac{{\\sf d} x}{{\\sf d}t} \\cdot {\\sf d} t = diag(\\kappa ) \\ diag{(x^B)} B \\ diag{(x^{-1})} \\cdot SE \\ diag(\\kappa ) x^B \\cdot {\\sf d} t$ Lemma REF together with condition (REF ) and (REF ) shows that (sig) in theorem REF is satisfied in the case where we set $\\sigma (B(\\Sigma (K))^*)=\\sigma (B(\\Sigma (K)) \\setminus \\lbrace 0\\rbrace )$ instead of $\\sigma (B(\\Sigma (K^*)))$ for all $\\kappa $ .", "It remains to show that $\\varphi _B$ is injective with respect to $K$ in order to apply the $*$ -operator to $K$ directly.", "Remark 4.3 Relation (REF ) holds for all $x \\in \\mathbb {R}^n_{+}$ .", "This might be a too restrictive condition for CRN systems.", "We assume that there exists such a parameter system such that condition (REF ) is satisfied.", "In the theorem REF we also allow $\\kappa $ for which thermodynamic feasibility is not allowed.", "But we obtain in that case that thermodynamic feasible reaction systems from theorem REF are contained in the set of injective systems as characterized in theorem REF .", "The basis of $K=\\textit {im}(SE)=\\textit {im}(A)$ consists of the stoichiometric differences $[y_i-y_i^{\\prime }]\\in \\mathcal {R}$ .", "The basis of the rowspace of $B$ consists of all reactant complexes $y_i^{\\prime }$ .", "According to theorem REF , (sig) and lemma REF , 2. we need to show that the columnspace of $SE$ maps injectively on the rowspace of $B$ .", "By the definition of equation (REF ) this shows theorem REF .", "$\\blacksquare $ Lemma 4.4 For $[y_i-y_i^{^{\\prime }}]\\in \\mathcal {R}$ with $y_i \\ne y_i^{^{\\prime }}$ at least one of the following two cases is true: a) $ y_i \\cdot [y_i-y_i{^{\\prime }}] \\ne 0$ b) $ y_i^{\\prime } \\cdot [y_i-y_i{^{\\prime }}] \\ne 0$ Assume that both are zero then we would have $0 < [y_i-y_i{^{\\prime }}]\\cdot [y_i-y_i{^{\\prime }}]=0$ .", "Corollary 4.5 For a system of $r$ reversible reactions with thermodynamic feasible fluxes the corresponding generalized polynomial $f_{\\kappa }$ is injective.", "We can check that by selecting a subset of reactions differences $[y_{k_i}-y_{k_i}^{\\prime }] \\in \\mathcal {R}$ for $ i \\in [k]$ where $k = dim(K)$ .", "In the same way we can select a subset of maximum $k \\le k^{\\prime }\\le 2k$ row vectors $\\lbrace y_{i_{k^{\\prime }}}^{\\prime \\prime }\\rbrace _{i \\in [k^{\\prime }]}$ of $B$ out of the $\\lbrace y_{k_i},y_{k_i}^{\\prime }\\rbrace _{i \\in [k]}$ pairs for which $span {(\\lbrace [y_{k_i}-y_{k_i}^{\\prime }]\\rbrace _{i \\in [k]})} \\subseteq span{(\\lbrace y_{i_k}^{\\prime \\prime }\\rbrace _{i \\in [k^{\\prime }]})}$ holds since the columnspace of $SE=A$ is contained in the rowspace of $B$ .", "Together with lemma REF we see that $K^* $ is mapped injectively into $\\textit {im}(B)$ , which is orthogonal to $\\ker {A_{\\kappa }}$ .", "Corollary 4.6 For all weakly reversible thermodynamically feasible fluxes the generalized polynomial map $f_{\\kappa }(x)$ is injective.", "Weak reversibility inplies that every reactant and product complex is represented at least once in the rows of $B$ .", "Hence the columnspace of $SE=A$ is contained in the rowspace of $B$ .", "Deficiency as introduced in [11] is replaced by thermodynamic feasibility as represented in equation (REF ).", "The injectivity relation is reduced to $span(reaction \\ di ff erences) \\subseteq span(reactant \\ complexes) \\ .$" ], [ "Continuous flow stirred tank reactors", "We can extend the closed system of reactions as developed until now by a continuous external flow as described in the continuous flow stirred tank reactor (CFSTR).", "We introduce an artificial reaction by the inflow $y^*$ as a reactant complex and the resulting outflow $y^{\\prime *}$ as a product complex by setting $\\Delta y^*=[y^*-y^{\\prime *}]$ .", "By that reaction a stoichiometry class is fixed from external imposed conditions.", "We assume that the interior system given by the closed CRN as described until now has a thermodynamical feasible flux system and especially an interior fixed point and is hence injective by definition.", "The response to the external flux is equivalent to the fixation of the system to a starting position, which is unique by injectivity of the interior system.", "By these assumption we obtain the corollary: Corollary 5.1 A CRN as given under the same assumptions from theorem REF is injective with respect to a continuous inflow $y^*$ where $y_i^*$ are the concentrations of the species for the inflow and ${y^{\\prime }}_i^*$ are the concentrations in the outflow which is the species concentration in the system.", "We can write for the response of the internal system: $f_{\\kappa }(x)= [{y}^*-{y^{\\prime }}^*]$ $y^*_i=c_i$ is the number of species $x_i$ inflow per unit Volume and similar for ${y^{\\prime }}^*_i=x_i$ .", "The reactionrate $\\kappa ^*=1$ ." ], [ "Conclusion", "Including thermodynamic principles into CRN's leads to a restriction of the available parameter space.", "Thermodynamic feasible reaction dynamics requires injective generalized polynomial maps for the dynamics of the species concentrations.", "Reversible and weakly reversible CRN's.", "imply injectivity.", "Regarding cell differentiation we can conclude that metabolic networks are regulated by signal transduction and not by triggering intrinsic multistability.", "Therefore we can assume or predict that mutistability is governed by regulatory mechanisms, which are not subjected to powerlaw kinetics and thermodynamic energy potentials." ], [ "Acknowledgements", "The work was done during my stay at the Friedrich Alexander University in Erlangen at the Department of Mathematics.", "I would like to thank Gerhard Keller and Andreas Knauf for helpfull discussions." ] ]
1403.0241
[ [ "Fragile antiferromagnetism in the heavy-fermion compound YbBiPt" ], [ "Abstract We report results from neutron scattering experiments on single crystals of YbBiPt that demonstrate antiferromagnetic order characterized by a propagation vector, $\\tau_{\\rm{AFM}}$ = ($\\frac{1}{2} \\frac{1}{2} \\frac{1}{2}$), and ordered moments that align along the [1 1 1] direction of the cubic unit cell.", "We describe the scattering in terms of a two-Gaussian peak fit, which consists of a narrower component that appears below $T_{\\rm{N}}~\\approx 0.4$ K and corresponds to a magnetic correlation length of $\\xi_{\\rm{n}} \\approx$ 80 $\\rm{\\AA}$, and a broad component that persists up to $T^*\\approx$ 0.7 K and corresponds to antiferromagnetic correlations extending over $\\xi_{\\rm{b}} \\approx$ 20 $\\rm{\\AA}$.", "Our results illustrate the fragile magnetic order present in YbBiPt and provide a path forward for microscopic investigations of the ground states and fluctuations associated with the purported quantum critical point in this heavy-fermion compound." ], [ "Fragile antiferromagnetism in the heavy-fermion compound YbBiPt B. G. Ueland Ames Laboratory, U.S. DOE, Iowa State University, Ames, Iowa 50011, USA Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA A. Kreyssig Ames Laboratory, U.S. DOE, Iowa State University, Ames, Iowa 50011, USA Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA K. Prokeš Helmholtz-Zentrum Berlin für Materialien und Energie, Hahn-Meitner-Platz 1, 14109 Berlin, Germany J. W. Lynn NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA L. W. Harriger NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA D. K. Pratt NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA D. K. Singh NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA Department of Materials Science and Engineering, University of Maryland, College Park, MD 20742, USA T. W. Heitmann The Missouri Research Reactor, University of Missouri, Columbia, Missouri 65211, USA S. Sauerbrei Ames Laboratory, U.S. DOE, Iowa State University, Ames, Iowa 50011, USA Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA S. M. Saunders Ames Laboratory, U.S. DOE, Iowa State University, Ames, Iowa 50011, USA Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA E. D. Mun Ames Laboratory, U.S. DOE, Iowa State University, Ames, Iowa 50011, USA Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA S. L. Bud'ko Ames Laboratory, U.S. DOE, Iowa State University, Ames, Iowa 50011, USA Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA R. J. McQueeney Ames Laboratory, U.S. DOE, Iowa State University, Ames, Iowa 50011, USA Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA P. C. Canfield Ames Laboratory, U.S. DOE, Iowa State University, Ames, Iowa 50011, USA Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA A. I. Goldman Ames Laboratory, U.S. DOE, Iowa State University, Ames, Iowa 50011, USA Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA 75.30.Mb, 75.50.Ee, 75.30.Kz, 71.10.Hf We report results from neutron scattering experiments on single crystals of YbBiPt that demonstrate antiferromagnetic order characterized by a propagation vector, $\\tau _{\\rm {AFM}}$ = ($\\frac{1}{2} \\frac{1}{2} \\frac{1}{2}$ ), and ordered moments that align along the [1 1 1] direction of the cubic unit cell.", "We describe the scattering in terms of a two-Gaussian peak fit, which consists of a narrower component that appears below $T_{\\rm {N}}~\\approx ~0.4$ K and corresponds to a magnetic correlation length of $\\xi _{\\rm {n}} \\approx $ 80 $\\rm {Å}$ , and a broad component that persists up to $T^*\\approx $ 0.7 K and corresponds to antiferromagnetic correlations extending over $\\xi _{\\rm {b}} \\approx $ 20 $\\rm {Å}$ .", "Our results illustrate the fragile magnetic order present in YbBiPt and provide a path forward for microscopic investigations of the ground states and fluctuations associated with the purported quantum critical point in this heavy-fermion compound.", "Unusual magnetic behavior may occur in heavy-fermion systems [1] in close proximity to a magnetic quantum critical point (QCP) [2], [3], [4] due to the entanglement of conduction electrons and localized moments and the competition between potential ground states [6], [5].", "Quantum phase transitions occur at $T$ = 0 K and are driven by some non-thermal parameter such as magnetic field or pressure [7].", "In strongly-correlated electron systems such as heavy-fermions compounds, quantum phase transitions may be accompanied by large changes in the Fermi-surface and can lead to non-Fermi liquid behavior, enhanced quantum fluctuations, and may result in superconductivity or other novel ground states [5], [8].", "Two scenarios are often discussed in the context of heavy-fermions with QCPs [3]: (1) the conventional spin-density-wave (SDW) scenario where the quasiparticles are formed below the Kondo temperature ($T_{\\rm {K}}$ ) and survive in the vicinity of the QCP yielding critical fluctuations localized at small regions of the Fermi-surface [9], [10]; and (2) the Kondo breakdown scenario [5] where localization of the f electrons at the QCP breaks the Kondo coupling yielding large changes of the Fermi-surface accompanied by a magnetic transition.", "CeCu$_{2}$ Si$_{2}$ [11] and Ce$_{1-x}$ La$_{x}$ Ru$_{2}$ Si$_{2}$ [12] are cited as examples of materials described by the conventional SDW scheme, whereas CeCu$_{6-x}$ Au$_x$ [13] and YbRh$_{2}$ Si$_{2}$ [14] provide examples relevant to the latter scenario.", "Experiments on YbRh$_{2-x}$ Ir$_{x}$ Si$_{2}$ have shown that substituting 6% Ir for Rh detaches the Kondo-breakdown point from the QCP resulting in an extended intermediate-field range of non-Fermi liquid (NFL) behavior, characteristic of a \"spin-liquid\"-type ground state [14].", "The stoichiometric compound YbAgGe [15] also exhibits an extended region of NFL behavior with applied magnetic field [16], [17].", "However, accessing the ordered magnetic state close to the QCP and studying the evolution of the microscopic magnetic correlations in the NFL regime is complicated by either the requirement of attaining extremely low temperatures ($T_{\\rm {N}}~\\approx ~0.05$ K) for YbRh$_{2}$ Si$_{2}$ [18], [19], or by a complex series of magnetic transitions with applied field for YbAgGe [17].", "YbBiPt offers an important alternative stoichiometric system with: (1) the simplicity of a cubic lattice; (2) temperatures and fields that are low, but readily achievable for scattering measurements close to the QCP; and (3) a rather simple $H-T$ phase diagram with an extended region of NFL behavior [20], [21], [22], [23], [24], [25].", "YbBiPt belongs to the series of cubic half-Heusler (space group $F\\overline{4}3m$ ) $R$ BiPt compounds (R = rare earth) [25], [27], [28], with the magnetic Yb ions forming a face-centered-cubic magnetic sublattice.", "Its discovery generated strong interest due to its extraordinary Sommerfield coefficient ($\\gamma ~\\approx $ 8 J/mol-K$^2$ ) and classification as a heavy-fermion compound [20], [21], [22], [23].", "All of its relevant energy scales including the Kondo temperature ($T_{\\rm {K}} \\approx $ 1 K) that describes the magnetic coupling between the localized and itinerant moments, the Weiss temperature ($\\theta _{\\rm {W}} \\approx $  -2 K) that describes the mean-field magnetic exchange strength, the Néel temperature for the proposed spin-density-wave order ($T_{\\rm {N}}$ = 0.4 K), and the crystalline electric field splitting ($\\Delta E <$ 1 meV) are small and comparable, suggesting a complex interplay of competing interactions at low temperature.", "It has also been suggested that YbBiPt offers the realization of a topological Kondo insulator [29].", "Much of the recent attention on YbBiPt has focused on the possibility of a magnetic-field-tuned antiferromagnetic (AFM) QCP occurring at a low critical magnetic field of $\\mu _{0}H_{\\rm {c}}$ = 0.4 T [24].", "Thermodynamic and transport measurements in ambient field suggest that YbBiPt manifests AFM order below $T_{\\rm {N}}$ = 0.4 K. In particular, a clear anomaly is observed at $T_{\\rm {N}}$ in electrical resistance data that is consistent with spin-density-wave type AFM order that partially gaps the Fermi surface [20].", "This feature is strongly suppressed upon the application of a modest magnetic field ($\\mu _{0}H > \\mu _{0}H_{\\rm {c}}$ ), and non-Fermi liquid behavior is found for $\\mu _{0}H_{\\rm {c}} < H <$ 0.8 T, followed by Fermi-liquid behavior for $\\mu _{0}H >$ 0.8 T [24].", "Although the locations of the field-induced phase transitions and Fermi-liquid behavior have been mapped out [24], it is not yet clear whether YbBiPt is best described by the conventional SDW or the Kondo breakdown scenario.", "It also is notable that scattering measurements over the past 22 years have failed to identify magnetic ordering in powder [25] or single crystal samples, leading to uncertainty regarding the true nature of the proposed AFM transition that is somewhat reminiscent of the \"hidden order\" paradox in URu$_2$ Si$_2$ [26].", "Furthermore, muon spin-relaxation ($\\mu $ SR) measurements have found evidence of spatially inhomogeneous and disordered magnetism in powder samples [30] which suggests that any magnetic order in YbBiPt is likely quite fragile.", "Clarifying the nature of the transition at $T$ = 0.4 K in YbBiPt represents a key step towards performing microscopic investigations of magnetism close to the QCP.", "Here, we present results from neutron scattering experiments on single crystals of YbBiPt that identify and characterize the low temperature AFM order by the magnetic propagation vector $\\tau _{\\rm {AFM}}$ = ($\\frac{1}{2}~\\frac{1}{2}~\\frac{1}{2}$ ) and moments collinear with $\\tau _{\\rm {AFM}}$ .", "We further show that the observed magnetic scattering can be modeled by a two-Gaussian peak fit consisting of a narrower Gaussian component that appears below $T_{\\rm {N}}$ with a magnetic correlation length of $\\xi _{\\rm {n}} \\approx $ 80 $\\rm {Å}$ , and a broader Gaussian component that persists up to $T^* \\approx $ 0.7 K that is consistent with short-range AFM correlations occurring over $\\xi _{\\rm {b}} \\approx $ 20 $\\rm {Å}$ .", "We suggest that the narrower and broad components of the scattering illustrate the competition among the low-energy magnetic interactions and lend themselves to a picture of fragile magnetic order occurring at low temperature.", "Figure: (color online) Contour plots of diffraction data taken for points in the (HHLH H L) plane corresponding to the antiferromagnetic propagation vector τ AFM \\tau _{\\rm {AFM}} = (1 21 21 2\\frac{1}{2} \\frac{1}{2} \\frac{1}{2}).", "The intensity of the scattering is indicated by color.", "(a) Data for the (1 21 25 2\\frac{1}{2} \\frac{1}{2} \\frac{5}{2}) position for TT = 0.1 K and (b) TT = 0.75 K, and (c) after subtracting TT = 0.1 K data by the TT = 0.75 K data.", "Panels (d) and (e) show data for the (1 21 23 2\\frac{1}{2} \\frac{1}{2} \\frac{3}{2}) and (1 21 21 2\\frac{1}{2} \\frac{1}{2} \\frac{1}{2}) positions, respectively, after subtracting the TT = 0.1 K data by the corresponding TT = 0.8 K data.", "(f) Diagram of the (HHLH H L) reciprocal lattice plane for YbBiPt.", "Nuclear Bragg points are indicated by black crosses, and possible magnetic Bragg points are indicated by circles.", "Solid circles correspond to measured points, and points where the intensity of the magnetic scattering is zero are marked with ×\\times 's.", "Dashed lines indicate the magnetic Brillouin zones.Figure: (color online) Detailed scattering data for (1 21 23 2\\frac{1}{2} \\frac{1}{2} \\frac{3}{2}).", "(a) Data from a rocking scan taken at TT = 0.08 K. Blue and red lines show the broad and narrower components of the two-Gaussian peak fit, respectively, and the shaded red area corresponds to scattering contributed by the narrower Gaussian peak.", "Black crosses at the bottom show scaled data from a rocking scan at the (1 1 1) nuclear Bragg peak, which is split due to a small misalignment of the two co-aligned crystals.", "(b) Data from rocking scans taken at TT = 0.45 K (open circles) and 0.75 K (filled dark gray circles).", "The blue curve represents a fit to a single Gaussian peak, while the dark gray line depicts the background.", "(c) QQ-dependence of the scattering at (1 21 23 2\\frac{1}{2} \\frac{1}{2} \\frac{3}{2}) and (1 21 25 2\\frac{1}{2} \\frac{1}{2} \\frac{5}{2}) compared to the square of the Yb 3+ ^{3+} magnetic form factor (solid line).", "(d) Energy dependence of the scattering from constant 𝐐\\textbf {Q} scans at TT = 0.08 K for Q = (1 21 23 2\\frac{1}{2} \\frac{1}{2} \\frac{3}{2}) (upward pointing triangles), and background scans at two different Q-positions (circles and downward pointing triangles).", "Lines represent fits to Gaussian peaks, and the energy resolution (FWHM) is indicated by the horizontal line.", "The shaded area corresponds to elastic magnetic scattering at (1 21 23 2\\frac{1}{2} \\frac{1}{2} \\frac{3}{2}).", "Uncertainties represent one standard deviation.Single crystals of YbBiPt were grown out of a Bi flux as described previously [24] and ranged in mass from several hundred mg to nearly 2 g. Several samples with total masses of 1 - 3 g and total mosaic spreads of $\\approx 1^{\\circ }$ FWHM were assembled for neutron scattering experiments using either one crystal or two co-aligned crystals.", "Given the strong sensitivity of the samples to pressure and strain [22], [24], several methods and glues [an amorphous fluoropolymer (CYTOP) or dental glue (HBM X60)] were used to fix the crystals to a Cu sample holder which was then thermally anchored to the bottom of a dilution refrigerator.", "For the samples attached with the fluoropolymer, Cu wire was loosely wrapped around the crystals and anchored to the sample holder to ensure mechanical stability.", "Neutron scattering experiments were performed on the E-4 two-axis diffractometer at Helmholtz-Zentrum Berlin, and the SPINS cold-neutron and BT-7 thermal-neutron triple-axis spectrometers [31] at the NIST Center for Neutron Research.", "Incident neutrons with wavelengths of $\\lambda $ = 2.451, 5.504, and 2.359 Å, for E-4, SPINS, and BT-7, respectively, were selected by a pyrolitic graphite (PG) monochromator, and PG or liquid nitrogen cooled Be filters were inserted to reduce contamination from higher-order wavelengths.", "A 40$^{\\prime }$ or 80$^{\\prime }$ Söller collimator was used between the monochromator and sample and a 120$^{\\prime }$ (E-4 and SPINS) or 80$^{\\prime }$ (BT-7) radial collimator was placed immediately after the sample.", "BT-7 was operated in two-axis mode and both E-4 and BT-7 utilized position sensitive detectors.", "On SPINS, a PG analyzer horizontally focused to a single $^3$ He detector was used to select a fixed final neutron wavelength of $\\lambda $ = 5.504 Å.", "The LAMP and DAVE software packages were used for data reduction [32], [33].", "Comprehensive searches for magnetic scattering in the ($H 0 L$ ) and ($H H L$ ) reciprocal lattice planes were undertaken on E-4 and resulted in the discovery of additional scattering below $T^*\\approx $ 0.7 K at half-integer positions ($\\frac{h}{2} \\frac{h}{2} \\frac{l}{2}$ ) with $h$ and $l$ odd integers and $h\\ne l$ .", "Fig.", "REF shows diffraction data from rocking scans taken in the ($H H L$ ) plane using the BT-7 spectrometer.", "Figure REF (a) shows a broad peak (in both the longitudinal and transverse directions) centered at ($\\frac{1}{2} \\frac{1}{2} \\frac{5}{2}$ ) for $T$ = 0.1 K, and Fig.", "REF (b) shows that the peak is absent for $T$ = 0.75 K. Figure REF (c) shows the same region after subtracting the $T$ = 0.75 K data from the $T$ = 0.1 K data.", "Figures REF (d) and (e) show similar plots of $T$ = 0.1 K data after subtracting $T$ = 0.8 K data for the ($\\frac{1}{2} \\frac{1}{2} \\frac{3}{2}$ ) and ($\\frac{1}{2} \\frac{1}{2} \\frac{1}{2}$ ) positions, respectively.", "A broad peak centered at ($\\frac{1}{2} \\frac{1}{2} \\frac{3}{2}$ ), similar to the one at ($\\frac{1}{2} \\frac{1}{2} \\frac{5}{2}$ ), is observed in Fig.", "REF (d), whereas Fig.", "REF (e) shows that the peak is absent at the ($\\frac{1}{2} \\frac{1}{2} \\frac{1}{2}$ ) position.", "Although not shown, distinct but broad peaks similar to those in Figs.", "REF (c) and (d) were identified at the ($\\pm \\frac{1}{2}\\pm \\frac{1}{2}\\mp \\frac{3}{2}$ ) and ($\\pm \\frac{3}{2}\\pm \\frac{3}{2}\\pm \\frac{1}{2}$ ) positions, whereas no peaks were observed at the ($\\pm \\frac{3}{2}\\pm \\frac{3}{2}~\\frac{3}{2}$ ) positions.", "Figure REF (f) shows a diagram which summarizes our observations in the first quadrant of the ($H H L$ ) plane.", "Since the intensity of the magnetic scattering is proportional to the component of the moment perpendicular to the neutron momentum transfer, Q, the systematic absence of scattering at the ($\\frac{1}{2} \\frac{1}{2} \\frac{1}{2}$ ) and ($\\pm \\frac{3}{2}\\pm \\frac{3}{2}~\\frac{3}{2}$ ) positions indicates that the ordered moment is aligned along the [1 1 1] direction.", "Hence, we conclude that the AFM propagation vector is $\\tau _{\\rm {AFM}}$ = ($\\frac{1}{2} \\frac{1}{2} \\frac{1}{2}$ ), and that the ordered moments are collinear with $\\tau _{\\rm {AFM}}$ .", "To study the magnetic scattering in more detail, we performed measurements on a co-aligned sample on the SPINS spectrometer as a function of the neutron energy transfer $E$ .", "Elastic ($E$ = 0) data from rocking scans centered at the ($\\frac{1}{2} \\frac{1}{2} \\frac{3}{2}$ ) position for $T$ = 0.08 K are shown in Fig.", "REF (a), and data for $T$ = 0.45 and 0.75 K are shown in Fig.", "REF (b).", "For comparison, data from a rocking scan through the (1 1 1) nuclear Bragg peak are shown at the bottom of Fig.", "REF (a).", "The magnetic scattering at $T$ = 0.08 and 0.45 K is much broader than the nuclear peak which indicates that a finite magnetic correlation length exists for the AFM order.", "For $T$ = 0.75 K the magnetic peaks are absent.", "Upon lowering the temperature below $T^*$ , broad scattering appears that grows in intensity with decreasing temperature and is well described by a single Gaussian peak.", "For temperatures below $T_{\\rm {N}}~\\approx $ 0.4 K, Fig.", "REF (a) shows that a single Gaussian peak no longer adequately describes the observed scattering since additional intensity with a narrower distribution is evident for $T$ = 0.08 K, and we describe the scattering data for $T < T_{\\rm {N}}$ by a two-Gaussian peak fit that is the sum of a broad Gaussian component and a concentric narrower Gaussian component.", "Since the centers and the FWHM's of the broad and narrower components of the two-Gaussian peak fit do not vary significantly with temperature, they were fixed to the values obtained at $T$ = 0.08 K [$\\Delta \\theta _{\\rm {narrow}}$ = 3.2(9)$^\\circ $ and $\\Delta \\theta _{\\rm {broad}}$ = 12.5(9)$^\\circ $ ].", "A constant background determined at $T$ = 0.75 K has also been used.", "Normalizing to the integrated intensity of the (1 1 1) nuclear reflection, and assuming equally populated magnetic domains and contributions from the full volume of the sample, we calculate the average magnetic moment at $T$ = 0.08 K associated with the total measured magnetic scattering at $\\tau _{\\rm {AFM}}$ to be $\\approx ~0.8~\\mu _{\\rm {B}}$ .", "At this point we do not attempt to partition the ordered moment between the broad and narrower components but note that the ratio of their integrated intensities is approximately 12:1.", "For the magnetic structure described above, we can also compare the $Q$ -dependence of the scattering at the ($\\frac{1}{2} \\frac{1}{2} \\frac{3}{2}$ ) and ($\\frac{1}{2} \\frac{1}{2} \\frac{5}{2}$ ) magnetic Bragg positions with that expected for the Yb$^{3+}$ magnetic form factor [see Fig.", "REF (c)] and find good agreement.", "Taken together with the systematic absence of scattering at the ($\\frac{1}{2} \\frac{1}{2} \\frac{1}{2}$ ) and ($\\pm \\frac{3}{2}\\pm \\frac{3}{2}~\\frac{3}{2}$ ) positions, these data confirm the magnetic origin of the half-integer diffraction peaks.", "Figure: (color online) Temperature dependence of the integrated intensities of the narrower (left axis) and broad (right axis) components of the two-Gaussian peak fits to the magnetic scattering at (1 21 23 2\\frac{1}{2}~\\frac{1}{2}~\\frac{3}{2}).We note that the data below $T_{\\rm {N}}$ may also be described by a single Lorentzian-squared peak although the fit does not quite capture all of the low-temperature intensity at the center of the peak.", "Further measurements using significantly larger samples may be required to ultimately determine the most appropriate fitting function for the magnetic scattering.", "Nevertheless, the temperature dependence of the integrated intensities of the components of the two-Gaussian peak fit are shown in Fig.", "REF and suggest that the two-Gaussian peak fit captures the essential features of the scattering: the narrower component decreases smoothly with increasing temperature and is absent above $T_{\\rm {N}}$ $\\approx $ 0.4 K, consistent with the bulk thermodynamic and transport measurement results, while the integrated intensity of the broad component also decreases smoothly with increasing temperature but persists up to $T^* \\approx 0.7$  K. The magnetic correlation lengths associated with the components of the two-Gaussian peak fit can be derived from the FWHM of the peaks in the rocking scan data and are $\\xi _{\\rm {n}} \\approx 80$  Å and $\\xi _{\\rm {b}} \\approx 20$  Å for the narrower and broad components, respectively.", "The presence of broad magnetic scattering and finite correlation lengths appears consistent with previous $\\mu $ SR measurements on powder samples which concluded that the ordered moment in YbBiPt is spatially inhomogeneous [30].", "However, the $\\mu $ SR measurements were performed on powders raising the possibility of strain effects [24].", "The present measurements were performed on single crystals mounted to minimize or eliminate strain effects.", "We believe that this unusual magnetic behavior is intrinsic to YbBiPt and does not arise from chemical or structural inhomogeneities because: (1) The broad magnetic component has been found for all three sets of measured samples despite the crystals coming from different growth batches and despite different mounting methods.", "(2) All samples measured present resolution limited nuclear diffraction peak widths in longitudinal and transverse scans.", "(3) Resistivity measurements on samples from batches prepared in an identical manner all exhibit a single sharp transition at $T=0.4$  K. (4) The measured residual resistivity ratios are on the order of 20:1, and quantum oscillations have been observed in the thermopower [24] and magnetoresistance data [34].", "(5) Previous neutron powder diffraction measurements found no evidence of chemical disorder in identically grown YbBiPt samples.", "(6) We made measurements on two crystals of YbBiPt (selected from the batches used for our neutron scattering measurements) using high energy (232 keV) x-ray diffraction to probe the bulk of the crystals and found no second phase coherent with the YbBiPt chemical lattice.", "Nevertheless, the apparent onset of short-range magnetic correlations at $T^*$ is surprising since it is well above $T_{\\rm {N}}$ and, to the best of our knowledge, no distinct signature of this feature has been previously reported.", "Given the relatively small ordered moment and the sizable broadening of the magnetic peaks, it is also now clear why previous neutron powder diffraction measurements failed to detect the magnetic order [25].", "To check whether the scattering at $\\tau _{\\rm {AFM}}$ arises from low-energy magnetic fluctuations rather than static order we performed constant-Q energy scans for $T$ = 0.08 K on SPINS at Q = ($\\frac{1}{2}~\\frac{1}{2}~\\frac{3}{2}$ ), and at positions well separated from the AFM Bragg position to capture the incoherent scattering background.", "These data are shown in Fig.", "REF (d), where the shaded area corresponds to the additional magnetic scattering at the AFM position.", "The lines in Fig.", "REF (d) represent Gaussian fits with measured values for the FWHM of $\\Delta E$ = 0.088(5), 0.090(7), and 0.089(6) meV, for Q = ($\\frac{1}{2} \\frac{1}{2} \\frac{3}{2}$ ), (0.4 0.4 1.3), and (0.31 0.31 1.5), respectively.", "The instrumental energy resolution was determined from the FWHM of the elastic incoherent scattering from plastic to be $\\Delta E$ = 0.087(1) meV and is indicated by the horizontal bar.", "We conclude that the peaks shown in Figs.", "REF (a) and (b) are elastic within our current experimental resolution, although we can not exclude that the scattering is quasielastic on an energy scale much smaller than 0.09 meV.", "We note this possibility because all of the relevant energy scales in YbBiPt are on the order of the present energy resolution.", "However, the systematic absence of scattering at the ($\\frac{h}{2} \\frac{h}{2} \\frac{h}{2}$ ) points would require that any quasielastic fluctuations be longitudinal (e.g.", "in the magnitude of the moment), and the absence of any change in the magnetic correlation lengths with temperature would be puzzling.", "Evidence for unusual magnetic order in close proximity to a QCP has been found for other strongly-correlated materials fitting either the conventional SDW or Kondo-breakdown scenarios.", "For example, CeCu$_{6-x}$ Au$_x$ exhibits dynamic short-range magnetic correlations for $x = x_{\\rm {c}}$ = 0.1 (the critical concentration where non-Fermi-liquid behavior is clearly observed) [35].", "For $x$ = 0.2, static short-range AFM order coexists with long-range AFM order at a different propagation vector [36], and persists well above $T_{\\rm {N}}$ derived from specific-heat and AC-susceptibility measurements [37].", "This is similar to the temperature dependence of the broad scattering component in YbBiPt.", "The presence of both broad and narrower components of the magnetic scattering may arise from a number of sources including the competition between magnetic and nonmagnetic ground states, the possible frustration inherent to the sublattice of side-sharing tetrahedra of Yb moments, as well as fluctuations associated with the nearby QCP.", "We thank R. Flint for useful discussions and gratefully acknowledge the Missouri University Research Reactor, Helmholtz-Zentrum Berlin für Materialien und Energie, the National Institute of Standards and Technology, and the U.S. Dept.", "of Commerce for the allocated beamtime and support during the experiments.", "Work at the Ames Laboratory was supported by the Department of Energy, Basic Energy Sciences, Division of Materials Sciences & Engineering, under Contract No.", "DE-AC02-07CH11358." ] ]
1403.0526
[ [ "New Recurrence Relationships between Orthogonal Polynomials which Lead\n to New Lanczos-type Algorithms" ], [ "Abstract Lanczos methods for solving $\\textit{A}\\textbf{x}=\\textbf{b}$ consist in constructing a sequence of vectors $(\\textbf{x}_k), k=1,...$ such that $\\textbf{r}_{k}=\\textbf{b}-\\textit{A}\\textbf{x}_{k}=\\textit{P}_{k}(\\textit{A})\\textbf{r}_{0}$,, where $\\textit{P}_{k}$ is the orthogonal polynomial of degree at most $k$ with respect to the linear functional $c$ defined as $c(\\xi^i)=(\\textbf{y},\\textit{A}^i\\textbf{r}_{0})$.", "Let $\\textit{P}^{(1)}_{k}$ be the regular monic polynomial of degree $k$ belonging to the family of formal orthogonal polynomials (FOP) with respect to $c^{(1)}$ defined as $c^{(1)}(\\xi^{i})=c(\\xi^{i+1})$.", "All Lanczos-type algorithms are characterized by the choice of one or two recurrence relationships, one for $\\textit{P}_{k}$ and one for $\\textit{P}^{(1)}_{k}$.", "We shall study some new recurrence relations involving $\\textit{P}_{k}$ and $\\textit{P}^{(1)}_{k}$ and their possible combination to obtain new Lanczos-type algorithms.", "We will show that some recurrence relations exist, but cannot be used to derive Lanczos-type algorithms, while others do not exist at all." ], [ "Introduction", "Let In 1950, C. Lanczos proposed a method for transforming a matrix into a similar tridiagonal matrix.", "We know, by Cayley-Hamilton theorem, that the computation of the characteristic polynomial of a matrix and the solution of linear equations are equivalent, Lanczos, , in 1952 used his method for solving systems of Linear equations.", "Since then, several Lanczos-type algorithms have been obtained and among them, the famous conjugate gradient algorithm of Hestenes and Stiefel, , when the matrix is Hermitian, and the bi-conjugate gradient algorithm of Fletcher, , for the general case.", "In the last three decades, Lanczos algorithms have evolved and different variants have been derived, , , , , , , , , , , , , , , , , .", "Although Lanczos-type algorithms can be derived by using linear algebra techniques, the formal orthogonal polynomials (FOP) approach is perhaps the most common for deriving them.", "In fact, all recursive algorithms implementing the Lanczos method can be derived using the theory of FOP, .", "A drawback of these algorithms is their inherent fragility due to the nonexistence of some orthogonal polynomials.", "This causes them to breakdown well before convergence.", "To avoid these breakdowns, variants that jump over the nonexisting polynomials have been developed; they are referred to as breakdown-free algorithms, , , , , , .", "Note that it is not the purpose of this paper to discuss this breakdown issue although it will be highlighted when necessary.", "Two types of recurrence relations are needed: one for $P_{k}(x)$ and one for $\\textit {P}^{(1)}_{k}(x)$ , .", "In , , recurrence relations for the computation of polynomials $\\textit {P}_{k}(x)$ are represented by $A_{i}$ and those for polynomials $\\textit {P}^{(1)}_{k}(x)$ , by $B_{j}$ .", "Table 1 and Table 2 below give a comprehensive list.", "C. Baheux and C. Brezinski have exploited some of the polynomial relations which involve few matrix-vector multiplications.", "In their work, the only relations that they studied were those where the degrees of the polynomials in the right and left hand sides of the relation differ by ONE or TWO at most.", "We are studying relations where the difference in degrees is TWO or THREE.", "For full details of these relations, see .", "The following notation has been introduced in , .", "We will adopt it here and extend the list accordingly.", "The paper is organized as follow.", "Section 2, briefly recalls the Lanczos algorithm.", "Section 3, derives some of the possible recurrence relations $A_{i}$ and $B_{j}$ given in Table 2 and their combination to obtain Lanczos-type algorithms.", "It also discusses two recurrence relations which although exist and satisfy the normalization and orthogonality conditions, cannot be used for the computation of $\\textbf {r}_{k}$ and hence $\\textbf {x}_{k}$ .", "Section 4 is the conclusion." ], [ "The Lanczos algorithm", "Consider a linear system in $\\textit {R}^n$ with $n$ unknowns $\\textit {A}\\textbf {x}=\\textbf {b}.$ Lanczos method, , , , , for solving (1), consists in constructing a sequence of vectors $\\textbf {x}_{k}$ as follows.", "$\\bullet $ choose two arbitrary vectors $\\textbf {x}_{0}$ and $\\textbf {y}\\ne 0$ in $\\textit {R}^n$ , $\\bullet $ set $\\textbf {r}_{0}=\\textbf {b}-\\textit {A}\\textbf {x}_{0}$ , $\\bullet $ determine $\\textbf {x}_{k}$ such that $\\textbf {x}_{k}-\\textrm {x}_{0}\\in \\textit {E}_{k}=span(\\textbf {r}_{0},\\textit {A}\\textbf {r}_{0},\\dots ,\\textit {A}^{k-1}\\textbf {r}_{0})$ $\\textbf {r}_{k}=\\textbf {b}-\\textit {A}\\textbf {x}_{k}\\bot \\textit {F}_{k}=span(\\textbf {y},\\textit {A}^T\\textbf {y},\\dots ,\\textit {A}^{T^{k-1}}\\textbf {y})$ where $A^T$ is transpose of $A$ .", "$\\textbf {x}_k-\\textbf {x}_0$ can be written as $\\textbf {x}_{k}-\\textbf {x}_{0}=-\\alpha _{1}\\textbf {r}_{0}- \\dots -\\alpha _{k}\\textit {A}^{k-1}\\textbf {r}_{0}.$ Multiplying both sides by $A$ , adding and subtracting b and simplifying, we get $\\textbf {r}_{k}=\\textbf {r}_{0}+\\alpha _{1}\\textit {A}\\textbf {r}_{0}+\\dots +\\alpha _{k}\\textit {A}^{k}\\textbf {r}_{0}$ and the orthogonality condition above give $(\\textit {A}^{T^{i}}\\textbf {y},\\textbf {r}_{k})=0 \\mbox{ for i = 0,\\dots , k-1},$ which is a system of $k$ linear equations in the $k$ unknowns $\\alpha _{1},\\dots ,\\alpha _{k}$ .", "This system is nonsingular only if $\\textbf {r}_{0},\\textit {A}\\textbf {r}_{0},\\dots ,\\textit {A}^{k-1}\\textbf {r}_{0}$ and $\\textbf {y},\\textit {A}^T\\textbf {y},\\dots ,\\textit {A}^{T^{k-1}}\\textbf {y}$ are linearly independent.", "If we set $\\textit {P}_{k}(\\xi )=1+\\alpha _{1}\\xi +\\dots +\\alpha _{k}\\xi ^{k}$ then we have $\\textbf {r}_{k}=\\textit {P}_{k}(\\textit {A})\\textbf {r}_{0}.$ Moreover, if we set $c_{i}=(\\textbf {y},\\textit {A}^i\\textbf {r}_{0}) \\mbox{ for i=0,1,\\dots }$ and we define the linear functional $c$ on the space of polynomials by $c(\\xi ^i)=c_{i} \\mbox{ for i=0,1,\\dots }$ then the preceding orthogonality conditions can be written as $c(\\xi ^i\\textit {P}_{k})=0 \\mbox{ for i=0,\\dots },k-1$ These relations show that $\\textit {P}_{k}$ is the polynomial of degree at most $k$ belonging to the family of orthogonal polynomials with respect to c, .", "This polynomial is defined apart from a multiplying factor which was chosen, in our case, such that $\\textit {P}_{k}(0)=1$ .", "With this normalization condition, $\\textit {P}_{k}$ exists and is unique if and only if the following Hankel determinant $\\textbf {H}^{(1)}_{k}=\\left|\\begin{array}{cccc}c_{1} & c_{2} & \\cdots & c_{k}\\\\c_{2} & c_{3} & \\cdots & c_{k+1}\\\\\\vdots & \\vdots & & \\vdots \\\\c_{k} & c_{k+1} & \\cdots & c_{2k-1}\\\\\\end{array}\\right|$ is different from zero.", "Let us now consider the monic polynomial $\\textit {P}^{(1)}_{k}$ of degree $k$ belonging to the FOP with respect to the functional $c^{(1)}$ defined by $c^{(1)}(\\xi ^i)=c(\\xi ^{i+1})$ .", "$\\textit {P}^{(1)}_{k}$ exists and is unique, if and only if the Hankel determinant $\\textbf {H}^{(1)}_{k}\\ne 0$ , which is the same condition as for the existence and uniqueness of $\\textit {P}_{k}$ .", "A Lanczos-type algorithm consists in computing $\\textit {P}_{k}$ recursively, then $\\textbf {r}_{k}$ and finally $\\textbf {x}_{k}$ such that $\\textbf {r}_{k}=\\textbf {b}-\\textit {A}\\textbf {x}_{k}$ , without inverting $A$ , which gives the solution of the system $(1)$ in at most $n$ steps, in exact arithmetic where $n$ is the dimension of the system of linear equations, , ." ], [ "Recursive computation of $\\textit {P}_{k}$ and {{formula:a8c3e5f3-06a8-436b-b040-fa31e804fbc8}}", "The recursive computation of the polynomials $\\textit {P}_{k}$ , needed in the Lanczos method, can be achieved in many ways.", "We can use the usual three-term recurrence relation, or the relation involving the polynomials of the form $\\textit {P}^{(1)}_{k}$ .", "Such recurrence relations lead to all known algorithms for implementing Lanczos-type algorithms.", "For a unified presentation of all these methods based on the theory of FOP, see , , .", "We need two recurrence relations, one for $\\textit {P}_{k}$ and one for $\\textit {P}^{(1)}_{k}$ .", "All Lanczos-type algorithms are characterized by the choice of these recurrence relationships.", "In the following we will discuss some of these recurrence relations for $\\textit {P}_{k}$ , , and derive new recurrence relationships between adjacent orthogonal polynomials, , , , , , , , which can be used to design new Lanczos-type algorithms, as has been shown in , , .", "Note that the term “Lanczos process” and “Lanczos-type algorithm” are used interchangeably throughout the paper." ], [ "Relations $A_{i}$", "We will follow the notation explained in Section 1.", "First we will derive relations $A_{i}$ ( $i>10$ ) for $P_{k}$ which can be used to find $\\textbf {r}_{k}$ and then $\\textbf {x}_{k}$ without inverting $A$ , the matrix of the system to be solved.", "We will only try to find the constant coefficients of recurrence relations which can be used for the implementation of Lanczos-type algorithms.", "If a recurrence relation exists but cannot be used for such an implementation, then there is no need to calculate its coefficients.", "The reason for that will be given.", "Note, however, that when a recurrence relation exists and can be computed, leading therefore to a Lanczos-type algorithm, the algorithm may still break down for two reasons: 1.", "There is a loss of orthogonality as in most known Lanczos-type algorithms, , .", "2.", "When coefficients involve determinants in their denominators, these determinants may be zero (or rather, in practice, just close to zero).", "In this case, a check on the value of the determinants will determine whether the process has to be stopped or not.", "This check appears in the relevant algorithms given in .", "This kind of breakdown is called ghost breakdown, , .", "It may be cured by conditioning methods, , , .", "This is not considered here.", "However, it can also be cured using the restarting and switching strategies, presented in , , .", "In the following $P_{k}$ stands for $P_{k}(x)$ and $P^{(1)}_{k}$ for $P^{(1)}_{k}(x)$ .", "But, before we derive the relations $A_i$ and $B_j$ , we first define the notion of“orthogonal polynomials sequence\", .", "Definition 1.", "A sequence $\\lbrace P_n\\rbrace $ is called an orthogonal polynomial sequence with respect to the linear functional $c$ if, for all nonnegative integers $m$ and $n$ , $(i)$ $P_n$ is polynomial of degree $n$ , $(ii)$ $c(x^mP_n)=0$ , for $m\\ne n$ , $(iii)$ $c(x^nP_n)\\ne 0$ ." ], [ "Relation $A_{11}$", "As explained earlier, we follow up from what is already known up to $A_{10}$ , .", "$A_{11}$ is therefore the natural follow up.", "Consider the following recurrence relationship $P_{k}(x)=(A_{k}x^{3}+B_{k}x^2+C_{k}x+D_{k})P_{k-3}(x)+(E_{k}x+F_{k})P^{(1)}_{k-1}(x),$ where $P_{k}$ , $P^{(1)}_{k-1}$ and $P_{k-3}$ are polynomials of degree $k$ , $k-1$ and $k-3$ respectively.", "Proposition 3.1: Relation of the form $A_{11}$ does not exist.", "Proof: Let us see if all coefficients can be identified.", "If $x^{i}$ is a polynomial of exact degree $i$ then $\\forall i=0, \\dots ,k-1$ , $c(x^{i}P_{k})=0.", "\\longrightarrow (C_{1})$ $\\forall i=0, \\dots ,k-2$ , $c^{(1)}(x^{i}P^{(1)}_{k-1})=0.", "\\longrightarrow (C_{2})$ $\\forall i=0, \\dots ,k-4$ , $c(x^{i}P_{k-3})=0.", "\\longrightarrow (C_{3})$ where $c$ and $c^{(1)}$ are defined respectively as follows.", "$c(x^i)=c_{i}.", "\\longrightarrow (C_{4})$ $c^{(1)}(x^i)=c(x^{i+1}).", "\\longrightarrow (C_{5})$ Since $P_k(0)=1$ , $\\forall k$ , then for $x=0$ , equation (2) becomes $1=D_{k}+F_{k}P^{(1)}_{k-1}(0).$ Multiplying both sides of equation (2) by $x^{i}$ and applying $c$ using the condition $(C_{5})$ where necessary, it can be written as $&c(x^{i}P_{k})=A_{k}c(x^{i+3}P_{k-3})+B_{k}c(x^{i+2}P_{k-3}) +C_{k}c(x^{i+1}P_{k-3})\\\\ \\nonumber & \\quad \\quad \\qquad +D_{k}c(x^{i}P_{k-3}) +E_{k}c^{(1)}(x^{i}P^{(1)}_{k-1})+F_{k}c(x^{i}P^{(1)}_{k-1}).$ For $i=0$ , equation (4) gives $0=F_{k}c(P^{(1)}_{k-1})$ .", "Since $c(P^{(1)}_{k-1})\\ne 0$ , , then $F_{k}=0$ .", "Hence from (3), we have $D_{k}=1$ .", "Equation (4) is always true for $i=1,...,k-7$ by $(C_{2})$ and $(C_{3})$ .", "For $i=k-6$ , (4) becomes $A_{k}c(x^{k-3}P_{k-3})=0$ , as all other terms vanish due to conditions $(C_{1})$ , $(C_{2})$ and $(C_{3})$ .", "But, according to condition $(iii)$ of definition 1 in Section $3.1$ , $c(x^{k-3}P_{k-3})\\ne 0$ , therefore, $A_{k}=0$ .", "For $i=k-5$ , (4) becomes $B_{k}c(x^{k-3}P_{k-3})=0$ .", "Since $c(x^{k-3}P_{k-3})\\ne 0$ , $B_{k}=0$ .", "For $i=k-4$ , (4) becomes $C_{k}c(x^{k-3}P_{k-3})=0$ .", "Since $c(x^{k-3}P_{k-3})\\ne 0$ , $C_{k}=0$ .", "Putting values of $A_{k}$ , $B_{k}$ , $C_{k}$ , $D_{k}$ and $F_{k}$ in equation (4), we get $c(x^{i}P_{k})=c(x^{i}P_{k-3})+E_{k}c^{(1)}(x^{i}P^{(1)}_{k-1}).$ For $i=k-3$ , this equation becomes $c(x^{k-3}P_{k})=c(x^{k-3}P_{k-3})+E_{k}c^{(1)}(x^{k-3}P^{(1)}_{k-1}).$ Using $(C_{1})$ and $(C_{2})$ , both $c(x^{k-3}P_{k})=0$ and $c^{(1)}(x^{k-3}P^{(1)}_{k-1})=0$ , therefore we get $c(x^{k-3}P_{k-3})=0,$ which is impossible due to condition $(iii)$ of definition 1 given in Section $3.1$ .", "Therefore, Proposition $3.1$ holds." ], [ "Relation $A_{12}$", "Consider the following recurrence relationship for $k\\ge 3$ , $P_{k}(x)=A_{k}[(x^{2}+B_{k}x+C_{k})P_{k-2}(x)+(D_{k}x^3+E_{k}x^2+F_{k}x+G_{k})P_{k-3}(x)],$ This recurrence relation has been considered in , ." ], [ "Relation $A_{13}$", "Consider the following recurrence relationship $P_{k}(x)=A_{k}[(x^{2}+B_{k}x+C_{k})P_{k-2}(x)+(D_{k}x^3+E_{k}x^2+F_{k}x+G_{k})P^{(1)}_{k-3}(x)],$ where $P_{k}$ , $P_{k-2}$ and $P^{(1)}_{k-3}$ are orthogonal polynomials of degree $k$ , $k-2$ and $k-3$ respectively and $A_{k}$ , $B_{k}$ , $C_{k}$ , $D_{k}$ , $E_{k}$ , $F_{k}$ and $G_{k}$ are constants to be determined using the normalization condition $P_{k}(0)=1$ and the orthogonality condition $(C_{1})$ .", "Proposition 3.2: Relation of the form $A_{13}$ exists.", "Proof: We know that $\\forall i=0, \\dots ,k-1$ , $c^{(1)}(x^{i}P^{(1)}_{k})=0$ .", "$\\longrightarrow $ $(C_{6})$ Since $\\forall k, P_{k}(0)=1$ , equation (REF ) gives $1=A_{k}[C_{k}+G_{k}P^{(1)}_{k-3}(0)].$ Multiplying both sides of (REF ) by $x^{i}$ and then applying the linear functional $c$ , we get $\\begin{array}{l}c(x^{i}P_{k})=A_{k}[c(x^{i+2}P_{k-2})+B_{k}c(x^{i+1}P_{k-2})+C_{k}c(x^{i}P_{k-2})\\\\+D_{k}c^{(1)}(x^{i+2}P^{(1)}_{k-3})+E_{k}c^{(1)}(x^{i+1}P^{(1)}_{k-3})+F_{k}c^{(1)}(x^{i}P^{(1)}_{k-3})+G_{k}c(x^{i}P^{(1)}_{k-3})].\\end{array}$ For $i=0$ , equation (REF ) becomes, $0=G_{k}c(P^{(1)}_{k-3})$ .", "Since $c(P^{(1)}_{k-3})\\ne 0$ , this implies that $G_{k}=0.$ Therefore, from equation (REF ) we have $A_{k}=\\frac{1}{C_{k}}$ .", "The orthogonality condition $(C_1)$ is always true for $i=1,\\dots ,k-6$ .", "For $i=k-5$ , we have $D_{k}c^{(1)}(x^{k-3}P^{(1)}_{k-3})=0$ , which implies that $D_{k}=0$ .", "For $i=k-4$ , (REF ) becomes $c(x^{k-2}P_{k-2})+E_{k}c^{(1)}(x^{k-3}P^{(1)}_{k-3})=0$ , $E_{k}=-\\frac{c(x^{k-2}P_{k-2})}{c^{(1)}(x^{k-3}P^{(1)}_{k-3})}.$ For $i=k-3$ , (REF ) gives $\\begin{array}{l}B_{k}c(x^{k-2}P_{k-2})+F_{k}c^{(1)}(x^{k-3}P^{(1)}_{k-3})\\\\=-c(x^{k-1}P_{k-2})-E_{k}c^{(1)}(x^{k-2}P^{(1)}_{k-3}).\\end{array}$ For $i=k-2$ , (REF ) becomes $\\begin{array}{l}B_{k}c(x^{k-1}P_{k-2})+C_{k}c(x^{k-2}P_{k-2})+F_{k}c^{(1)}(x^{k-2}P^{(1)}_{k-3})\\\\=-c(x^{k}P_{k-2})-E_{k}c^{(1)}(x^{k-1}P^{(1)}_{k-3}).\\end{array}$ For $i=k-1$ , we get $\\begin{array}{l}B_{k}c(x^{k}P_{k-2})+C_{k}c(x^{k-1}P_{k-2})+F_{k}c^{(1)}(x^{k-1}P^{(1)}_{k-3})\\\\=-c(x^{k+1}P_{k-2})-E_{k}c^{(1)}(x^{k}P^{(1)}_{k-3}).\\end{array}$ Let $a_{11}$ , $a_{21}$ , $a_{31}$ , $a_{12}$ , $a_{22}$ , $a_{32}$ and $a_{13}$ , $a_{23}$ , $a_{33}$ be the coefficients of $B_{k}$ , $C_{k}$ and $G_{k}$ in equations (REF ), (REF ) and (REF ) respectively and suppose $b_{1}$ , $b_{2}$ and $b_{3}$ are the corresponding right hand side terms of these equations.", "Then we have $a_{11}=c(x^{k-2}P_{k-2})$ , $a_{12}=0$ , $a_{13}=c^{(1)}(x^{k-3}P^{(1)}_{k-3})$ , $a_{21}=c(x^{k-1}P_{k-2})$ , $a_{22}=c(x^{k-2}P_{k-2})$ , $a_{23}=c^{(1)}(x^{k-2}P^{(1)}_{k-3})$ , $a_{31}=c(x^{k}P_{k-2})$ , $a_{32}=c(x^{k-1}P_{k-2})$ , $a_{33}=c^{(1)}(x^{k-1}P^{(1)}_{k-3})$ , $b_{1}=-c(x^{k-1}P_{k-2})-E_{k}c^{(1)}(x^{k-2}P^{(1)}_{k-3})$ , $b_{2}=-c(x^{k}P_{k-2})-E_{k}c^{(1)}(x^{k-1}P^{(1)}_{k-3})$ , $b_{3}=-c(x^{k+1}P_{k-2})-E_{k}c^{(1)}(x^{k}P^{(1)}_{k-3})$ , $a_{11}B_{k}+0C_{k}+a_{13}F_{k}=b_{1},$ $a_{21}B_{k}+a_{22}C_{k}+a_{23}F_{k}=b_{2},$ $a_{31}B_{k}+a_{32}C_{k}+a_{33}F_{k}=b_{3}.$ If $\\Delta _{k}$ represents the determinant of the coefficients matrix of the above system of equations then we have $\\Delta _{k}=a_{11}(a_{22}a_{33}-a_{32}a_{23})+a_{13}(a_{21}a_{32}-a_{31}a_{22}).$ If $\\Delta _{k}\\ne 0$ , then $B_{k}=\\frac{[b_{1}(a_{22}a_{33}-a_{32}a_{23})+a_{13}(b_{2}a_{32}-b_{3}a_{22})]}{\\Delta _{k}},$ $F_{k}=\\frac{b_{1}-a_{11}B_{k}}{a_{13}},$ $C_{k}=\\frac{b_{2}-a_{21}B_{k}-a_{23}F_{k}}{a_{22}}$ and $A_{k}=\\frac{1}{C_{k}}.$ Hence, $P_{k}(x)=A_{k}[(x^{2}+B_{k}x+C_{k})P_{k-2}(x)+(E_{k}x^2+F_{k}x)P^{(1)}_{k-3}(x)].$ Here again, the $A_{13}$ recurrence relation exists.", "It therefore, can be used to implement the Lanczos process.", "Remark 1: If $\\Delta _k = 0$ then we cannot estimate coefficient $B_k$ , which means relation $A_{13}$ may not exit.", "However, it may exist, but for one or more of its coefficients which cannot be estimated for numerical reasons.", "This would be a case of ghost breakdown." ], [ "Relation $A_{14}$", "Consider the following recurrence relationship $P_{k}(x)=A_{k}[(x^{2}+B_{k}x+C_{k})P^{(1)}_{k-2}(x)+(D_{k}x^3+E_{k}x^2+F_{k}x+G_{k})P^{(1)}_{k-3}(x)],$ where $P_{k}$ , $P^{(1)}_{k-2}$ and $P^{(1)}_{k-3}$ are orthogonal polynomials of degree $k$ , $k-2$ and $k-3$ respectively and $A_{k}$ , $B_{k}$ , $C_{k}$ , $D_{k}$ , $E_{k}$ , $F_{k}$ and $G_{k}$ are constants to be determined using the normalization condition $P_{k}(0)=1$ and the orthogonality conditions $(C_{1})$ and $(C_{6})$ .", "Proposition 3.3: Relation of the form $A_{14}$ exists.", "Proof: Since $\\forall k, P_{k}(0)=1$ , equation (REF ) gives $1=A_{k}[C_{k}P^{(1)}_{k-2}(0)+G_{k}P^{(1)}_{k-3}(0)].$ Multiplying both sides of equation (REF ) by $x^{i}$ and then applying the linear functional $c$ and using condition $(C_{5})$ , we get $\\begin{array}{l}c(x^{i}P_{k})=A_{k}[c^{(1)}(x^{i+1}P^{(1)}_{k-2})+B_{k}c^{(1)}(x^{i}P^{(1)}_{k-2})+C_{k}c(x^{i}P^{(1)}_{k-2})\\\\+D_{k}c^{(1)}(x^{i+2}P^{(1)}_{k-3})+E_{k}c^{(1)}(x^{i+1}P^{(1)}_{k-3})+F_{k}c^{(1)}(x^{i}P^{(1)}_{k-3})+G_{k}c(x^{i}P^{(1)}_{k-3})].\\end{array}$ For $i=0$ , equation (REF ) becomes $C_{k}c(P^{(1)}_{k-2})+G_{k}c(P^{(1)}_{k-3})=0.$ The orthogonality condition for (REF ) is always true for $i=1, \\dots ,k-6$ .", "For $i=k-5$ , we get $D_{k}c^{(1)}(x^{k-3}P^{(1)}_{k-3})=0$ .", "Since $c^{(1)}(x^{k-3}P^{(1)}_{k-3})\\ne 0$ , $D_{k}=0$ .", "For $i=k-4$ , $E_{k}c^{(1)}(x^{k-3}P^{(1)}_{k-3})=0$ .", "Since $c^{(1)}(x^{k-3}P^{(1)}_{k-3})\\ne 0$ , we have $E_{k}=0$ .", "For $i=k-3$ , $F_{k}c^{(1)}(x^{k-3}P^{(1)}_{k-3})+G_{k}c(x^{k-3}P^{(1)}_{k-3})=-c^{(1)}(x^{k-2}P^{(1)}_{k-2}).", "$ For $i=k-2$ , $\\begin{array}{l}B_{k}c^{(1)}(x^{k-2}P^{(1)}_{k-2})+C_{k}c(x^{k-2}P^{(1)}_{k-2})+F_{k}c^{(1)}(x^{k-2}P^{(1)}_{k-3})\\\\+G_{k}c(x^{k-2}P^{(1)}_{k-3})=-c^{(1)}(x^{k-1}P^{(1)}_{k-2}).", "\\end{array}$ For $i=k-1$ , $\\begin{array}{l}B_{k}c^{(1)}(x^{k-1}P^{(1)}_{k-2})+C_{k}c(x^{k-1}P^{(1)}_{k-2})+F_{k}c^{(1)}(x^{k-1}P^{(1)}_{k-3})\\\\+G_{k}c(x^{k-1}P^{(1)}_{k-3})=-c^{(1)}(x^{k}P^{(1)}_{k-2}).", "\\end{array}$ The values of $A_{k}$ , $B_{k}$ , $C_{k}$ , $F_{k}$ and $G_{k}$ can be obtained by solving equations (REF ), (REF ), (REF ), (REF ) and (REF ).", "Hence $P_{k}(x)=A_{k}[(x^{2}+B_{k}x+C_{k})P^{(1)}_{k-2}(x)+(F_{k}x+G_{k})P^{(1)}_{k-3}(x)].$ Now multiplying both sides of the above relation by $\\textbf {r}_{0}$ , replacing $x$ by $A$ and using the relations $\\textbf {r}_{k}=P_{k}(A)\\textbf {r}_{0}$ and $\\textbf {z}_{k}=P^{(1)}_{k}(A)\\textbf {r}_{0}$ , we get $\\textbf {r}_{k}=A_{k}[(\\textit {A}^{2}+B_{k}\\textit {A}+C_{k}\\textit {I})\\textbf {z}_{k-2}+(F_{k}\\textit {A}+G_{k}\\textit {I})\\textbf {z}_{k-3}].$ Using $\\textbf {r}_{k}=\\textbf {b}-\\textit {A}\\textbf {x}_{k}$ , we get from the last equation $\\textit {A}\\textbf {x}_{k}=\\textbf {b}-A_{k}[(\\textit {A}^{2}+B_{k}\\textit {A}+C_{k}\\textit {I})\\textbf {z}_{k-2}+(F_{k}\\textit {A}+G_{k}\\textit {I})\\textbf {z}_{k-3}].$ From this relation it is clear that we cannot find $\\textbf {x}_{k}$ from $\\textbf {r}_{k}$ without inverting A.", "Hence, this recurrence relation exist as stipulated by Proposition 3.3.", "But, it is not desirable to implement a Lanczos-type algorithm.", "For recurrence relations $A_{15}$ , $A_{16}$ , $A_{17}$ , $A_{18}$ and $A_{19}$ and their corresponding coefficients, consult ." ], [ "Relations $B_{j}$", "Now we consider relations of the type $B_{j}$ which have not been considered before, , , i.e $B_{j}$ with $j>10$ .", "These relations, when they exist will be used in combination with relations $A_{i}$ to derive further Lanczos-type algorithms as explained in ." ], [ "Relation $B_{11}$", "Consider the following recurrence relationship $P^{(1)}_{k}(x)=(A_{k}x^3+B_{k}x^2+C_{k}x+D_{k})P_{k-3}(x)+(E_{k}x+F_{k})P_{k-1}(x),$ where $P^{(1)}_{k}(x)$ , $P_{k-1}(x)$ and $P_{k-3}(x)$ are orthogonal polynomials of degree $k$ , $k-1$ and $k-3$ respectively.", "Proposition 3.4: Relation of the form $B_{11}$ does not exist.", "Proof: Let $x^{i}$ be a polynomial of exact degree $i$ then $\\forall i=0, \\dots ,k-4$ , $c(x^{i}P_{k-3})=0.", "\\longrightarrow (C_{10})$ Multiply both sides of equation (REF ) by $x^{i}$ and applying $c^{(1)}$ and also using condition $(C_{5})$ where necessary, we get $\\begin{array}{l}c^{(1)}(x^{i}P^{(1)}_{k})=A_{k}c(x^{i+4}P_{k-3})+B_{k}c(x^{i+3}P_{k-3})+C_{k}c(x^{i+2}P_{k-3})\\\\+D_{k}c(x^{i+1}P_{k-3})+E_{k}c(x^{i+2}P_{k-1})+F_{k}c(x^{i+1}P_{k-1}).\\end{array}$ The relation (REF ) is always true for $i=0,\\dots ,k-8.$ For $i=k-7$ , we have $A_{k}c(x^{k-3}P_{k-3})=0$ .", "Which implies $A_{k}=0$ , because $c(x^{k-3}P_{k-3})\\ne 0.$ Similarly for $i=k-6$ , $i=k-5$ , $i=k-4$ and $i=k-3$ , we get respectively $B_{k}=0$ , $C_{k}=0$ , $D_{k}=0$ and $E_{k}=0$ .", "But $P^{(1)}_{k}(x)$ is a monic polynomial of degree $k$ and we see that $A_{k}=0$ .", "Therefore $E_{k}=1$ .", "If $E_{k}=0$ then $P^{(1)}_{k}(x)$ is no more of degree $k$ as the degree of the $P^{(1)}_{k}$ depends on $A_{k}$ and $E_{k}$ and we know that $A_{k}=0$ and if $E_{k}=1$ then $c(x^{k-1}P_{k-1})=0$ which is also impossible.", "Similarly for $i=k-2$ , we get $F_{k}=0$ and if $E_{k}=0$ , then $P^{(1)}_{k}=0.$ Hence the relation $B_{11}$ does not exist, therefore, Proposition $3.4$ holds.", "Recurrence relation $B_{12}$ has been explored in ." ], [ "Relation $B_{13}$", "Consider the following recurrence relationship $P^{(1)}_{k}(x)=(A_{k}x^3+B_{k}x^2+C_{k}x+D_{k})P^{(1)}_{k-3}(x)+(E_{k}x^2+F_{k}x+G_{k})P^{(1)}_{k-2}(x),$ where $P^{(1)}_{k}(x)$ , $P^{(1)}_{k-2}(x)$ and $P^{(1)}_{k-3}(x)$ are orthogonal polynomials of degree $k$ , $k-2$ and $k-3$ respectively.", "Proposition 3.5: Relation of the form $B_{13}$ exists.", "Proof: Let $x^{i}$ be a polynomial of exact degree $i$ then $\\forall i=0, \\dots ,k-3$ , $c^{(1)}(x^{i}P^{(1)}_{k-2})=0.", "\\longrightarrow (C_{11})$ Multiply both sides of equation (REF ) by $x^{i}$ and applying $c^{(1)}$ , we get $\\begin{array}{l}c^{(1)}(x^{i}P^{(1)}_{k})=A_{k}c^{(1)}(x^{i+3}P^{(1)}_{k-3})+B_{k}c^{(1)}(x^{i+2}P^{(1)}_{k-3})+C_{k}c^{(1)}(x^{i+1}P^{(1)}_{k-3})\\\\+D_{k}c^{(1)}(x^{i}P^{(1)}_{k-3})+E_{k}c^{(1)}(x^{i+2}P^{(1)}_{k-2})+F_{k}c^{(1)}(x^{i+1}P^{(1)}_{k-2})+G_{k}c^{(1)}(x^{i}P^{(1)}_{k-2}).\\end{array}$ The orthogonality condition is always true for $i=0,...,k-7$ .", "For $i=k-6$ , we get $A_{k}c^{(1)}(x^{k-3}P^{(1)}_{k-3})=0$ , which implies that $A_{k}=0$ as $c^{(1)}(x^{k-3}P^{(1)}_{k-3})\\ne 0.$ But $P^{(1)}_{k}(x)$ is a monic polynomial of degree $k$ .", "Therefore $E_{k}=1$ .", "For $i=k-5$ , we get $B_{k}c^{(1)}(x^{k-3}P^{(1)}_{k-3})=0$ .", "Since $c^{(1)}(x^{k-3}P^{(1)}_{k-3})\\ne 0$ , $B_{k}=0$ .", "For $i=k-4$ , we have $C_{k}=-\\frac{c^{(1)}(x^{k-2}P^{(1)}_{k-2})}{c^{(1)}(x^{k-3}P^{(1)}_{k-3})}.$ For $i=k-3$ , we get $D_{k}c^{(1)}(x^{k-3}P^{(1)}_{k-3})+F_{k}c^{(1)}(x^{k-2}P^{(1)}_{k-2})=-c^{(1)}(x^{k-1}P^{(1)}_{k-2})-C_{k}c^{(1)}(x^{k-2}P^{(1)}_{k-3}).$ For $i=k-2$ , (REF ) becomes $\\begin{array}{r}D_{k}c^{(1)}(x^{k-2}P^{(1)}_{k-3})+F_{k}c^{(1)}(x^{k-1}P^{(1)}_{k-2})+G_{k}c^{(1)}(x^{k-2}P^{(1)}_{k-2})\\\\=-c^{(1)}(x^{k}P^{(1)}_{k-2})-C_{k}c^{(1)}(x^{k-1}P^{(1)}_{k-3}).\\end{array}$ For $i=k-1$ , (REF ) gives $\\begin{array}{r}D_{k}c^{(1)}(x^{k-1}P^{(1)}_{k-3})+F_{k}c^{(1)}(x^kP^{(1)}_{k-2})+G_{k}c^{(1)}(x^{k-1}P^{(1)}_{k-2})\\\\=-c^{(1)}(x^{k+1}P^{(1)}_{k-2})-C_{k}c^{(1)}(x^{k}P^{(1)}_{k-3}).\\end{array}$ Let $a^{\\prime }_{11}=c^{(1)}(x^{k-3}P^{(1)}_{k-3})$ , using $(C_{5})$ , $a^{\\prime }_{11} =c(x^{k-2}P^{(1)}_{k-3})$ .", "By the same condition we can write, $a^{\\prime }_{12}=c^{(1)}(x^{k-2}P^{(1)}_{k-2})=c(x^{k-1}P^{(1)}_{k-2})$ , $a^{\\prime }_{13}=0$ , $a^{\\prime }_{21}=c^{(1)}(x^{k-2}P^{(1)}_{k-3})=c(x^{k-1}P^{(1)}_{k-3})$ , $a^{\\prime }_{22}=c^{(1)}(x^{k-1}P^{(1)}_{k-2})=c(x^{k}P^{(1)}_{k-2})$ , $a^{\\prime }_{23}=c^{(1)}(x^{k-2}P^{(1)}_{k-2})=a^{\\prime }_{12}$ , $a^{\\prime }_{31}=c^{(1)}(x^{k-1}P^{(1)}_{k-3})=c(x^{k}P^{(1)}_{k-3})$ , $a^{\\prime }_{32}=c^{(1)}(x^{k}P^{(1)}_{k-2})=c(x^{k+1}P^{(1)}_{k-2})$ , $a^{\\prime }_{33}=c^{(1)}(x^{k-1}P^{(1)}_{k-2})=a^{\\prime }_{22}$ , $b^{\\prime }_{1}=-c^{(1)}(x^{k-1}P^{(1)}_{k-2})-C_{k}c^{(1)}(x^{k-2}P^{(1)}_{k-3})=-a^{\\prime }_{22}-a^{\\prime }_{21}C_{k}$ , $b^{\\prime }_{2}=-c^{(1)}(x^{k}P^{(1)}_{k-2})-C_{k}c^{(1)}(x^{k-1}P^{(1)}_{k-3})=-a^{\\prime }_{32}-a^{\\prime }_{31}C_{k}$ , $b^{\\prime }_{3}=-c^{(1)}(x^{k+1}P^{(1)}_{k-2})-C_{k}c^{(1)}(x^{k}P^{(1)}_{k-3})$ .", "Then equations (REF ), (REF ) and (REF ) become $a^{\\prime }_{11}D_{k}+a^{\\prime }_{12}F_{k}=b^{\\prime }_{1},$ $a^{\\prime }_{21}D_{k}+a^{\\prime }_{22}F_{k}+a^{\\prime }_{23}G_{k}=b^{\\prime }_{2}$ and $a^{\\prime }_{31}D_{k}+a^{\\prime }_{32}F_{k}+a^{\\prime }_{33}G_{k}=b^{\\prime }_{3}.$ If $\\Delta ^{\\prime }_{k}$ is the determinant of the coefficient matrix of the equations (REF ), (REF ) and (REF ) then $\\Delta ^{\\prime }_{k}=a^{\\prime }_{11}(a^{\\prime }_{22}a^{\\prime }_{33}-a^{\\prime }_{32}a^{\\prime }_{23})-a^{\\prime }_{12}(a^{\\prime }_{21}a^{\\prime }_{33}-a^{\\prime }_{31}a^{\\prime }_{23}).$ If $\\Delta ^{\\prime }_{k}\\ne 0$ , then $D_{k}=\\frac{b^{\\prime }_{1}(a^{\\prime }_{22}a^{\\prime }_{33}-a^{\\prime }_{32}a^{\\prime }_{23})-a^{\\prime }_{12}(b^{\\prime }_{2}a^{\\prime }_{33}-b^{\\prime }_{3}a^{\\prime }_{23})}{\\Delta ^{\\prime }_{k}},$ $F_{k}=\\frac{b^{\\prime }_{1}-a^{\\prime }_{11}D_{k}}{a^{\\prime }_{12}}$ and $G_{k}=\\frac{b^{\\prime }_{2}-a^{\\prime }_{21}D_{k}-a^{\\prime }_{22}F_{k}}{a^{\\prime }_{23}}.$ Hence, relation (REF ) can be written as $P^{(1)}_{k}(x)=(C_{k}x+D_{k})P^{(1)}_{k-3}(x)+(x^2+F_{k}x+G_{k})P^{(1)}_{k-2}(x),$ and, therefore, exists as stipulated in Proposition 3.5.", "Remark 2: For the case where $\\Delta ^{\\prime }_k=0$ , please consult Remark 1 above.", "For recurrence relations $B_{14}$ , $B_{15}$ and $B_{16}$ and their corresponding coefficients, see ." ], [ "Conclusion", "In this paper, we looked in a systematic way at new recurrence relations between FOP’s which have not been considered before.", "In particular, we have shown that relations $A_{11}$ , $A_{17}$ , $B_{11}$ and $B_{12}$ do not exist; relations $A_{14}$ , $A_{15}$ , $A_{18}$ and $B_{14}$ exist but are not suitable for implementing new Lanczostype algorithms; and relations $A_{12}$ , $A_{13}$ , $A_{16}$ , $A_{19}$ , $B_{13}$ , $B_{15}$ and $B_{16}$ exist and can be used for the implementation of new Lanczos-type algorithms, .", "Relation $A_{12}$ is self-sufficient and leads to a new Lanczos-type algorithm on its own, , while the rest of the relations can lead to Lanczos-type algorithms when combined in $A_i/B_j$ fashion.", "Possible combinations, which are studied in , , are: $A_{13}/B_{13}$ , $A_{13}/B_{15}$ , $A_{13}/B_{16}$ , $A_{16}/B_{13}$ , $A_{16}/B_{15}$ , $A_{16}/B_{16}$ , $A_{19}/B_{13}$ , $A_{19}/B_{15}$ , $A_{19}/B_{16}$ ." ] ]
1403.0323
[ [ "On moments of Cantor and related distributions" ], [ "Abstract We provide several simple recursive formulae for the moment sequence of infinite Bernoulli convolution.", "We relate moments of one infinite Bernoulli convolution with others having different but related parameters.", "We give examples relating Euler numbers to the moments of infinite Bernoulli convolutions.", "One of the examples provides moment interpretation of Pell numbers as well as new identities satisfied by Pell and Lucas numbers." ], [ "Introduction", "The aim of this note is to add a few simple observations to the analysis of the distribution of the so called fatigue symmetric walk (term appearing in [12]).", "These observations are based on the reformulation of known results scattered through literature.", "We however pay more attention to the moment sequences and less to the properties of distributions that produce these moment sequences.", "It seems that the main novelty of the paper lies in the probabilistic interpretation of Pell and Lucas numbers and easy proofs of some identities satisfied by these numbers.", "However in order to place these results in the proper context we recall definition and basic properties of infinite Bernoulli convolutions.", "In deriving properties of these convolutions we recall some known, important results.", "The paper is organized as follows.", "After recalling definition and basic facts on the fatigue random walks we concentrate on the moment sequences of infinite Bernoulli convolutions.", "We formulate a corollary of the results of the paper expressed in terms of moment sequence.", "This corollary formulated in terms of number sequences provides identities of Pell and Lucas numbers of even order (Remark REF )." ], [ "Infinite Bernoulli Convolutions", "Let $\\left\\lbrace X_{n}\\right\\rbrace _{n\\ge 1}$ be the sequence of i.i.d.", "random variables such that $P(X_{1}=1)=P(X_{1}=-1)=1/2.", "$ Further let $\\left\\lbrace c_{n}\\right\\rbrace _{n\\ge 1}$ be a sequence of reals such that $\\sum _{n\\ge 1}c_{n}^{2}<\\infty .$ We define random variable: $S=\\sum _{n\\ge 1}c_{n}X_{n}.$ By Kolmogorov 3 series theorem $S$ exists and moreover it is square integrable.", "$ES^{2}=\\sum _{n\\ge 1}c_{n}^{2}.$ Obviously $ES=0.$ Let $\\varphi (t)$ denote characteristic function of $S.$ By the standard argument we have $\\varphi (t)=E\\exp (it\\sum _{s\\ge 1}c_{n}X_{n})=E\\prod _{n\\ge 1}\\exp (itc_{n}X_{n})=\\prod _{n\\ge 1}(\\exp (itc_{n})/2+\\exp (-itc_{n})/2)=\\prod _{n\\ge 1}\\cos (tc_{n}).$ We will concentrate on the special form of the sequence $c_{n}$ namely we will assume that $c_{n}=\\lambda ^{-n}$ for some $\\lambda >1.$ It is known (see [13]) that for all $\\lambda $ distribution of $S=S\\left( \\lambda \\right) $ is continuous that is $P_{S}(\\left\\lbrace x\\right\\rbrace )=0$ for all $x\\in \\mathbb {R}.$ Moreover it is also known (see [16], [17]) that if for almost $\\lambda \\in (1,2]$ this distribution is absolutely continuous and for almost all $\\lambda \\in (1,\\sqrt{2}]$ it has square integrable density.", "Garsia in [7], Theorem 1.8 showed examples of such $\\lambda $ leading to absolutely continuous distribution.", "Namely such $\\lambda \\in (1,2)$ are the roots of monic polynomials $P$ with integer coefficients such that $\\left|P(0)\\right|=2 $ and $\\lambda \\prod _{\\left|\\alpha _{i}\\right|>1}\\left|\\alpha _{i}\\right|=2,$ where $\\left\\lbrace \\alpha _{i}\\right\\rbrace $ are the remaining roots of $P$ .", "There are known (see [4], [5]) countable instances of $\\lambda \\in (1,2]$ that this distribution is singular.", "We will denote by $\\varphi _{\\lambda }$ the characteristic function of $S\\left( \\lambda \\right).$ Following [4] we know that the values $\\lambda $ such that $\\varphi _{\\lambda }(t)$ does not tend to zero as $t\\longrightarrow \\infty $ consequently related distribution is singular (by Riemann–Lebesgue Lemma) are the so called Pisot or PV- numbers i.e.", "sole roots of such monic irreducible polynomials $P$ with integer coefficients having the property that all other roots have absolute values less than $1.$ We must then have $P(0)=1.$ Examples of such numbers are the so called 'golden ratio' $\\left( 1+\\sqrt{5}\\right) /2$ or the so called 'silver ratio' $1+\\sqrt{2}.$ Moreover following [14] one knows that PV numbers are the only numbers $\\lambda \\in (1,2]$ for which $\\varphi _{\\lambda }$ does not tend to zero.", "Of course singularity of the distribution of $S\\left(\\lambda \\right) $ can occur for $\\lambda $ not being PV numbers.", "For $\\lambda >2$ it is known that the distribution of $S$ is singular [10].", "To simplify notation we will write ${supp}X,$ where $X$ is a random variable meaning ${supp}P_{X},$ where $P_{X}$ denotes distribution of $X.$ Similarly $X\\ast Y$ denotes random variable whose distribution is a convolution of distributions of $X$ and $Y.$ We have simple Lemma.", "Lemma 1 i) ${supp}(S\\left( \\lambda \\right) )\\subset \\left[ -\\frac{1}{\\lambda -1},\\frac{1}{\\lambda -1}\\right] .$ In particular: ia) if $\\lambda =2$ then $S\\sim U([-1,1])$ and ib) if $\\lambda =3$ then ${supp}(S+1/2)$ is equal to Cantor set.", "In general if $\\lambda $ is a positive integer then ${supp}(S+1/(\\lambda -1))$ consist of all numbers of the form $\\sum _{j\\ge 1}r_{j}\\lambda ^{-j}$ where $r_{j}\\in \\lbrace 0,2\\rbrace $ .", "Moreover the distribution of $(S(\\lambda )+1/(\\lambda -1))$ is 'uniform' on this a set.", "ii) $\\forall k\\ge 1:$ $\\varphi _{\\lambda }(\\lambda ^{k}t)=\\varphi _{\\lambda }(t)\\prod _{j=0}^{k-1}\\cos \\left( \\lambda ^{j}t\\right) .", "$ iii) $\\forall k\\ge 1:S\\left( \\lambda \\right) \\sim \\sum _{i=1}^{k}\\lambda ^{i-1}S_{i}(\\lambda ^{k}),$ where $S_{i}\\left( \\tau \\right) $ ( $i=1,\\ldots ,k)$ are i.i.d.", "random variables each having distribution $S\\left( \\tau \\right) .$ Consequently $\\varphi _{\\lambda }(t)=\\prod _{j=1}^{k}\\varphi _{\\lambda ^{k}}\\left(\\lambda ^{j-1}t\\right) $ iv) Let us denote $m_{n}(\\lambda )=ES(\\lambda )^{n}.$ Then $\\forall n\\ge 1:m_{2n-1}(\\lambda )=0$ and $m_{2n}(\\lambda )=\\frac{1}{\\lambda ^{2kn}-1}\\sum _{j=0}^{n-1}\\binom{2n}{2j}m_{2j}(\\lambda )W_{2(n-j)}^{(k)}(\\lambda ),$ with $m_{0}=1$ , where $W_{n}^{(1)}=1,$ $W_{n}^{(k)}\\left( \\lambda \\right) =\\left.", "\\frac{d^{n}}{dt^{n}}(\\prod _{j=1}^{k}\\cosh (\\lambda ^{j-1}t))\\right|_{t=0}$ $=\\frac{1}{2^{k-1}}\\sum _{i_{1}=0,\\ldots ,i_{k-1}=0}^{1}(1+\\sum _{j=1}^{k}(2i_{j}-1)\\lambda ^{j})^{2n}.$ In particular we have: $m_{2n}(\\lambda ) &=&\\frac{1}{\\lambda ^{2n}-1}\\sum _{j=0}^{n-1}m_{2j}(\\lambda )\\binom{2n}{2j}, \\\\m_{2n}\\left( \\lambda \\right) &=&\\frac{1}{\\lambda ^{4n}-1}\\sum _{j=0}^{n-1}\\binom{2n}{2j}m_{2j}(\\lambda )\\sum _{l=0}^{2(n-j)}\\binom{2(n-j)}{2l}\\lambda ^{2l}.", "$ v) $\\forall k\\ge 1:m_{2k}(\\lambda )\\mathbb {=}\\frac{-1}{\\lambda ^{2k}-1}\\sum _{j=0}^{k-1}\\binom{2k}{2j}\\lambda ^{2j}E_{2(k-j)}m_{2j}(\\lambda ),$ where $E_{k}$ denotes $k-$ th Euler number.", "i) First of all notice that $\\frac{1}{\\lambda -1}=\\sum _{n\\ge 1}1/\\lambda ^{n},$ hence $S+\\frac{1}{\\lambda -1}=\\sum _{n\\ge 1}\\frac{1}{\\lambda ^{n}}(X_{n}+1).$ Now since $P(X_{n}+1=0)=P(X_{n}+1=2)=1/2$ we see that ${supp}(S+\\frac{1}{\\lambda -1})\\subset [0,\\frac{2}{\\lambda -1}].$ Notice also that if $\\lambda =2$ then $S+1=2\\sum _{n\\ge 1}\\frac{1}{2^{n}}Y_{n},$ where $P(Y_{n}=0)=P(Y_{n}=1)=1/2.$ In other words $(S+1)/2$ is any number chosen from $[0,1]$ with equal chances that is $(S+1)/2$ has uniform distribution on $[0,1].$ When $\\lambda =3$ we see that $S+1/2$ is a number that can be written with the help of $^{\\prime }0^{\\prime }$ and $^{\\prime }2^{\\prime }$ in ternary expansion.", "In other words $S+1/2$ is number drawn from Cantor set with equal chances.", "For $\\lambda $ integer we argue in the similar way.", "ii) We have $\\varphi _{S}(\\lambda ^{k}t)=\\prod _{n\\ge 1}\\cos (\\lambda ^{k}t\\frac{1}{\\lambda ^{n}})=\\varphi _{S}\\left( t\\right) \\prod _{j=0}^{k-1}\\cos \\left(\\lambda ^{j}t\\right) $ .", "iii) Fix integer $k$ .", "Notice that we have: $S\\left( \\lambda \\right) =\\sum _{n\\ge 1}\\lambda ^{-n}X_{n}=\\\\\\sum _{j\\ge 1}\\lambda ^{-kj}X_{kj}+\\newline \\sum _{j\\ge 1}\\lambda ^{-kj+1}X_{kj-1}+\\ldots +\\sum _{j\\ge 1}\\lambda ^{-kj+k-1}X_{kj-k+1}= \\\\\\sum _{m=1}^{k}\\lambda ^{m-1}\\sum _{j\\ge 1}\\left( \\lambda ^{k}\\right) ^{-j}X_{kj-m+1}.$ Now since by assumption all $X_{i}$ are i.i.d.", "we deduce that $S_{i}(\\lambda ^{k})$ are i.i.d.", "random variables with distribution defined by $\\varphi _{\\lambda ^{k}}(t).$ Hence we have $\\varphi _{\\lambda }(t)=\\prod _{j=1}^{k}\\varphi _{\\lambda ^{k}}(\\lambda ^{j-1}t).$ iv) First of all we notice that $\\varphi _{S}(t)$ is an even function hence all derivatives of odd order at zero are equal 0.", "Secondly let $\\psi _{\\lambda }(t)$ denote moment generating function of $S(\\lambda ).$ It is easy to notice that $\\psi _{\\lambda }(s)=\\varphi _{S\\left( \\lambda \\right) }(-is).$ Let us denote $m_{2n}\\left( \\lambda \\right) =\\psi _{\\lambda }^{(2n)}(0).$ Basing on the elementary formula $\\cosh (\\alpha )\\cosh (\\beta )=\\frac{1}{2}(\\cosh (\\alpha +\\beta )+\\cosh \\left( \\alpha -\\beta \\right) ),$ we can easily obtain by induction the following identity: $\\prod _{j=0}^{k-1}\\cosh (\\lambda ^{j}t)=\\frac{1}{2^{k-1}}\\sum _{i_{1}=0,\\ldots ,i_{k-1}=0}^{1}\\cosh (t(1+\\sum _{j=1}^{k}(2i_{j}-1)\\lambda ^{j})).$ Since $\\left.", "(\\cosh \\alpha t)^{(2n)}\\right|_{t=0}=\\alpha ^{2n}$ , we have $\\left.", "\\left( \\prod _{j=0}^{k-1}\\cosh (\\lambda ^{j}t)\\right)^{(2n)}\\right|_{t=0}= \\\\\\frac{1}{2^{k-1}}\\sum _{i_{1}=0,\\ldots ,i_{k-1}=0}^{1}\\left.\\left( \\cosh (t(1+\\sum _{j=1}^{k}(2i_{j}-1)\\lambda ^{j}))\\right)^{(2n)}\\right|_{t=0}=W_{2n}^{(k)}\\left( \\lambda \\right) .$ $$ Now using Leibnitz formula for differentiation applied to (REF ) we get $f^{(2n)}(\\lambda ^{k}t)\\lambda ^{2kn}=\\sum _{j=0}^{2n}\\binom{2n}{j}(\\prod _{i=0}^{k-1}\\cosh \\lambda ^{i}t)^{\\left( j\\right) }f^{\\left(2n-j\\right) }\\left( t\\right) .$ Setting $t=0$ and using the fact that all derivatives of both $f$ and $\\cosh t$ of odd order at zero are zeros we get the desired formula.", "v) We use result of [18] that states that for each $N$ inverse of lower triangular matrix of degree $N\\times N$ with $(i,j)$ entry $\\binom{2i}{2j}$ is the lower triangular matrix with $(i,j)-th$ entry equal to $\\binom{2i}{2j}E_{2(i-j)}.$ Remark 1 Formula (REF ) is known in a slightly different form it appeared in [8], [6] and [1].", "Remark 2 Notice that polynomials $\\left\\lbrace W_{n}^{(k)}(\\lambda )\\right\\rbrace _{k,n\\ge 1}$ satisfy the following recursive relationship for $k>1:$ $W_{n}^{(k)}\\left( \\lambda \\right) =\\sum _{j=0}^{n}\\binom{2n}{2j}\\lambda ^{2j}W_{j}^{\\left( k-1\\right) }\\left( \\lambda \\right) ,$ with $W_{n}^{(1)}(\\lambda )=1.$ Hence its generating functions satisfy $\\Theta _{k}(t)$ the following relationship $\\Theta _{k}(t,\\lambda )=\\Theta _{k-1}(\\lambda t,\\lambda )\\cosh t,$ where we have have denoted: $\\Theta _{k}(t,\\lambda )=\\sum _{n\\ge 0}\\frac{t^{2n}}{\\left( 2n\\right) !", "}W_{n}^{(k)}(\\lambda ).$ Remark 3 Notice that the above mentioned lemma provides an example of two singular distributions whose convolution is a uniform distribution.", "Namely we have $S(4)\\ast 2S(4)=S(2).$ Similarly we have $S(2)=S\\left( 8\\right) \\ast 2S(8)\\ast 4S(8)$ or $S(2)=S(2^{k})\\ast \\ldots \\ast 2^{k-1}S(2^{k}).$ The first example was already noticed by Kersher and Wintner in [10],(22a).", "Remark 4 We can deduce even more from theses examples namely following the result of Kersher [9], p.451 that characteristic functions $\\varphi _{n}(t)$ of $S\\left( n\\right) $ (where $n$ is an integer $>2$ ) do not tend to zero as $t\\longrightarrow \\infty .$ Thus since we have $\\varphi _{4}(t)\\varphi _{4}(2t)=\\sin t/t$ and $\\varphi _{8}(t)\\varphi _{8}(2t)\\varphi _{8}(4t)=\\sin t/t$ we deduce that if $t_{k}\\longrightarrow \\infty $ is a sequence such that $\\left|\\varphi _{4}(t_{k})\\right|>\\varepsilon >0$ for suitable $\\varepsilon $ then $\\varphi _{4}(2t_{k})\\longrightarrow 0.$ Similarly if $t_{k}\\longrightarrow \\infty $ such that $\\left|\\varphi _{8}(t_{k})\\right|>\\varepsilon >0$ then $\\varphi _{8}(2t_{k})\\varphi _{8}(4t_{k})\\longrightarrow 0$ .", "Similar observations can be made can be made in more general situation.", "As it is known from the papers of Erdős [4], [5] the situation that $\\left|\\varphi _{\\lambda }(t_{k})\\right|>\\varepsilon >0$ for some sequence $t_{k}\\longrightarrow \\infty $ occurs when $\\lambda $ is a Pisot number (briefly PV-number).", "On the other hand as it is known roots of Pisot numbers are not Pisot, hence using above mentioned result of Salem $\\left|\\varphi _{\\lambda ^{1/k}}(t)\\right|\\longrightarrow 0$ as $t\\longrightarrow \\infty ,$ where $\\lambda $ is some PV number and $k>1$ any integer.", "But we have $\\varphi _{\\lambda ^{1/k}}(t_{n})=\\prod _{j=1}^{k}\\varphi _{\\lambda }\\left( \\lambda ^{(j-1)/k}t_{n}\\right)\\longrightarrow 0,$ where $t_{n}\\longrightarrow \\infty $ is such a sequence that $\\left|\\varphi _{\\lambda }(t_{n})\\right|>\\varepsilon >0.$ Remark 5 One knows that if $\\lambda =q/p$ where $p$ and $q$ are relatively prime integers and $p>1$ then $\\varphi _{\\lambda }(t)=O((\\log |t|)^{-\\gamma })$ where $\\gamma =\\gamma \\left( p,q\\right) >0$ as $t\\longrightarrow \\infty $ (see [9],(3)).", "Besides we know that then distribution of $S\\left( \\lambda \\right) $ is singular.", "Hence from our considerations it follows that if $\\lambda =(q/p)^{1/k}$ for some integer $k$ then $\\varphi _{\\lambda }(t)=O((\\log |t|)^{-k\\gamma }).$ Is it also singular?" ], [ "Moment sequences", "To give connection of certain moment sequences with some known integer sequences let us remark the following: Remark 6 i) $9^{n}m_{2n}(3)&=&\\sum _{j=0}^{n}\\binom{2n}{2j}m_{2j}(3), \\\\81^{n}m_{2n}(3) &=&\\sum _{j=0}^{n}\\binom{2n}{2j}m_{2j}(3)(2^{4(n-j)-1}+2^{2\\left( n-j\\right) -1}).$ ii) $5^{n}m_{2n}\\left( \\sqrt{5}\\right) &=&\\sum _{j=0}^{n}\\binom{2n}{2j}m_{2j}(\\sqrt{5}), \\\\25^{n}m_{2n}\\left( \\sqrt{5}\\right) &=&\\sum _{j=0}^{n}\\binom{2n}{2j}m_{2j}(\\sqrt{5})4^{n-j}L_{2(n-j)}/2,$ where $L_{n}$ denotes $n-$ th Lucas number defined below.", "i) The first assertion is a direct application of (REF ) while in proving the second one we use () and the fact that $\\sum _{j=0}^{n}\\binom{2n}{2j}9^{j}=4^{n}(4^{n}+1)/2$ as shown by [21], (seq.", "no.", "A026244).", "ii) Again the first statement follows (REF ) while the second follows () and the fact that $\\sum _{j=0}^{n}\\binom{2n}{2j}5^{j}=4^{n}T_{n}(3/2),$ where $T_{n}$ denotes Chebyshev polynomial of the first kind.", "([21], seq.", "no.", "A099140).", "Further we use the fact that $T_{n}(3/2)=L_{2n}/2.$ ([21], seq.", "no.", "A005248).", "We also have the following Lemma.", "Lemma 2 $\\forall n\\ge 1,k\\ge 2:$ $m_{2n}(\\lambda )=\\sum _{i_{1},\\ldots ,i_{k}=0}^{n}\\frac{(2n)!", "}{(2i_{1})!\\ldots (2i_{k})!", "}\\lambda ^{2(i_{2}+2i_{3}\\ldots (k-1)i_{k})}\\prod _{j=1}^{k}m_{2i_{j}}\\left( \\lambda ^{k}\\right) .", "$ In particular: $m_{2n}(\\lambda )&=&\\sum _{j=0}^{n}\\binom{2n}{2j}\\lambda ^{2j}m_{2j}(\\lambda ^{2})m_{2n-2j}(\\lambda ^{2}), \\\\m_{2n}(\\lambda )&=&\\sum _{\\begin{array}{c} i,j=0 \\\\ i+j\\le n\\end{array}}\\frac{(2n)!}{(2i)!(2j)!(2(n-i-j))!", "}\\times \\\\&&\\lambda ^{2i}\\lambda ^{4j}m_{2j}(\\lambda ^{3})m_{2i}(\\lambda ^{3})m_{2n-2i-2j}(\\lambda ^{3}).", "$ (REF ) follows directly Lemma REF , iii).", "As a corollary we get the following four observations: Corollary 1 i) $\\forall n\\ge 1:4^{n}=\\sum _{j=0}^{n}\\binom{2n+1}{2j+1}$ and $1=\\sum _{j=0}^{n}\\binom{2n+1}{2j+1}4^{j}E_{2(n-j)}.$ ii) $S\\left( \\sqrt{2}\\right) $ has density $g(x)=\\left\\lbrace \\begin{array}{ccc}\\sqrt{2}/4 & if & \\left|x\\right|\\le \\sqrt{2}-1, \\\\\\sqrt{2}(\\sqrt{2}+1-\\left|x\\right|)/8 & if & \\sqrt{2}-1\\le \\left|x\\right|\\le \\sqrt{2}+1, \\\\0 & if & \\left|x\\right|>1+\\sqrt{2}.\\end{array}\\right.$ iii) Let us denote $\\delta _{n}=(\\sqrt{2}+1)^{n}$ , then $m_{2n}(\\sqrt{2})=\\left( \\delta _{2n+2}-\\delta _{2n+2}^{-1}\\right) /(4\\sqrt{2}(n+1)(2n+1)).$ Since for $\\lambda =2$ random variable $S\\sim U[-1,1]$ its moments are equal to $ES^{2n}=\\frac{1}{2n+1}.$ Now we use Lemma REF iii) and iv).", "ii) From the proof of Lemma REF it follows that $S\\left( \\sqrt{2}\\right) \\sim S\\left( 2\\right) +\\sqrt{2}S\\left( 2\\right) .$ Now keeping in mind that $S\\left(2\\right) \\sim U(-1,1)$ we deduce that $g(x)=\\frac{\\sqrt{2}}{8}\\int _{-1}^{1}h(x-t)dt,$ where $h\\left( x\\right) =\\left\\lbrace \\begin{array}{ccc}\\frac{\\sqrt{2}}{4} & if & \\left|x\\right|\\le \\sqrt{2}, \\\\0 & if & otherwise.\\end{array}\\right.", ".$ iii) By straightforward calculations we get $m_{2n}(\\sqrt{2})=2\\int _{0}^{\\sqrt{2}+1}x^{2n}g(x)dx=\\frac{\\sqrt{2}}{2}\\int _{0}^{\\sqrt{2}-1}x^{2n}dx+\\frac{\\sqrt{2}}{4}\\int _{\\sqrt{2}-1}^{\\sqrt{2}+1}x^{2n}(\\sqrt{2}+1-x)dx.$ Remark 7 Let us apply formulae: (REF ), (REF ), () and observe by direct calculation that $2\\sum _{j=0}^{n}\\binom{2n}{2j}2^{j}=(1+\\sqrt{2})^{2n}+(1-\\sqrt{2})^{2n}$ .", "We get the following identities: $\\forall n\\ge 1:$ $m_{2n}(\\sqrt{2})&=&\\sum _{j=0}^{n}\\binom{2n}{2j}\\frac{2^{j}}{(2n-2j+1)(2j+1)} \\\\&=&\\frac{1}{2^{n}-1}\\sum _{j=0}^{n-1}\\binom{2n}{2j}m_{2j}(\\sqrt{2})\\\\&=&\\frac{1}{4^{n}-1}\\sum _{j=0}^{n-1}\\binom{2n}{2j}m_{2j}(\\sqrt{2})\\tau _{2(n-j)},$ $$ where $\\tau _{n}=(1+(-1)^{n})(\\delta _{n}+\\delta _{n}^{-1})/4.$ Now let us recall the definition of the so called Pell and Pell–Lucas numbers.", "Using sequence $\\delta _{n}$ Pell numbers $\\left\\lbrace P_{n}\\right\\rbrace $ and Pell–Lucas numbers $\\left\\lbrace Q_{n}\\right\\rbrace $ are defined $P_{n}&=&(\\delta _{n}+(-1)^{n+1}\\delta _{n}^{-1})/(2\\sqrt{2}), \\\\Q_{n}&=&\\delta _{n}+\\left( -1\\right) ^{n}\\delta _{n}^{-1}, $ where $\\delta _{n}$ is defined in REF ,iii).", "Using these definitions we can rephrase assertions of Corollary REF and Remark REF adding to recently discovered ([15], [2]) new identities satisfied by Pell and Pell-Lucas numbers and of course probabilistic interpretation of Pell numbers.", "Remark 8 i) $m_{2n}(\\sqrt{2})=\\frac{P_{2n+2}}{(2n+2)(2n+1)},$ $\\tau _{2n}=Q_{2n}/2.$ ii) $\\forall n\\ge 1:$ $P_{2n+2}&=&\\sum _{j=0}^{n}\\binom{2n+2}{2j+1}2^{j}, \\\\Q_{2n} &=&2\\sum _{j=0}^{n}\\binom{2n}{2j}2^{j}, \\\\2^{n-1}P_{2n}&=&\\sum _{j=0}^{n}\\binom{2n}{2j}P_{2j}, \\\\2^{2n-1}P_{2n}&=&\\sum _{j=0}^{n}\\binom{2n}{2j}P_{2j}Q_{2(n-j)}, \\\\\\sum _{j=0}^{n}\\binom{2n}{2j}(1+\\sqrt{2})^{2j}&=&2^{n-1}+2^{n-2}Q_{2n}+2^{n-1}\\sqrt{2}P_{2n}.", "$ Only last statement requires justification.", "First we find that $\\sum _{j=0}^{n}\\binom{2n}{2j}Q_{2j}=2^{n}(1+Q_{2n}/2)$ using ().", "Then we use (REF ), () and ()." ] ]
1403.0386
[ [ "Generalizations of Kaplansky Theorem Related to Linear Operators" ], [ "Abstract The purpose of this paper is to generalize a very famous result on products of normal operators, due to I. Kaplansky.", "The context of generalization is that of bounded hyponormal and unbounded normal operators on complex separable Hilbert spaces.", "Some examples \"spice up\" the paper." ], [ "Introduction", "Normal operators are a major class of bounded and unbounded operators.", "Among their virtues, they are the largest class of single operators for which the spectral theorem is proved (cf.", "[15]).", "There are other classes of interesting non-normal operators such as hyponormal and subnormal operators (among others).", "They have been of interest to many mathematicians and have been extensively investigated enough so that even monographs have been devoted to them.", "See for instance [3] and .", "In this paper we are mainly interested in generalizing the following result to unbounded normal and bounded hyponormal operators: Theorem 1.1 (Kaplansky, [8]) Let $A$ and $B$ be two bounded operators on a Hilbert space such that $AB$ and $A$ are normal.", "Then $B$ commutes with $AA^*$ iff $BA$ is normal.", "Before recalling some essential background, we make the following observation: All operators are linear and are defined on a separable complex Hilbert space, which we will denote henceforth by $H$ .", "A bounded operator $A$ on $H$ is said to be normal if $AA^*=A^*A$ .", "$A$ is called hyponormal if $AA^*\\le A^*A$ , that is iff $\\Vert A^*x\\Vert \\le \\Vert Ax\\Vert $ for all $x\\in H$ .", "Hence a normal operator is always hyponormal.", "Obviously, a hyponormal operator need not be normal.", "However, and in a finite-dimensional setting, a hyponormal operator is normal too.", "This is proved via a nice and simple trace argument (see e.g.", "[7]).", "Since the paper is also concerned with unbounded operators, and for the readers convenience, we recall some known notions and results about unbounded operators.", "If $A$ and $B$ are two unbounded operators with domains $D(A)$ and $D(B)$ respectively, then $B$ is said to be an extension of $A$ , and we denote it by $A\\subset B$ , if $D(A)\\subset D(B)$ and $A$ and $B$ coincide on each element of $D(A)$ .", "An operator $A$ is said to be densely defined if $D(A)$ is dense in $H$ .", "The (Hilbert) adjoint of $A$ is denoted by $A^*$ and it is known to be unique if $A$ is densely defined.", "An operator $A$ is said to be closed if its graph is closed in $H\\times H$ .", "We say that the unbounded $A$ is self-adjoint if $A=A^*$ , and we say that it is normal if $A$ is closed and $AA^*=A^*A$ .", "Recall also that the product $BA$ is closed if for instance $B$ is closed and $A$ is bounded, and that if $A$ , $B$ and $AB$ are densely-defined, then only $B^*A^*\\subset (AB)^*$ holds; and if further $A$ is assumed to be bounded, then $B^*A^*=(AB)^*$ .", "The notion of hyponormality extends naturally to unbounded operators.", "an unbounded $A$ is called hyponormal if: $D(A)\\subset D(A^*)$ , $\\Vert A^*x\\Vert \\le \\Vert Ax\\Vert $ for all $x\\in D(A)$ .", "It is also convenient to recall the following theorem which appeared in [16], but we state it in the form we need.", "Theorem 1.2 (Stochel) If $T$ is a closed subnormal (resp.", "closed hyponormal) operator and $S$ is a closed hyponormal (resp.", "closed subnormal) operator verifying $XT^*\\subset SX$ where $X$ is a bounded operator, then both $S$ and $T^*$ are normal once $\\ker X=\\ker X^*=\\lbrace 0\\rbrace $ .", "Any other result or notion (such as the classical Fuglede-Putnam theorem, the polar decomposition, subnormality etc...) will be assumed to be known by readers.", "For more details, the interested reader is referred to [1], [2], [6], [14] and [15].", "For other works related to products of normal (bounded and unbounded) operators, the reader may consult [5], [9], [11], [12] and [13], and the references therein." ], [ "Main Results: The Bounded Case", "The following known lemma is essential (we include a proof): Lemma 2.1 Let $S$ and $T$ be two bounded self-adjoint operators on a Hilbert space $H$ .", "If $U$ is any operator, then $S\\ge T \\Longrightarrow USU^*\\ge UTU^*.$ Let $x\\in H$ .", "We have $<USU^*x,x>= & <SU^*x,U^*x>\\\\\\ge & <TU^*x,U^*x>\\\\=&<UTU^*x,x>.$ As a direct application of the previous result we have the following Kaplansky-like theorem: Proposition 2.1 Let $A$ and $B$ be two bounded operators on a Hilbert space such that $A$ is normal and $AB$ is hyponormal.", "Then $AA^*B=BAA^*\\Longrightarrow \\text{ $BA$ is hyponormal.", "}$ Since $A$ is normal, we know that $A=PU=UP$ where $P$ is positive and $U$ is unitary.", "Hence $AA^*B=BAA^*\\Longrightarrow P^2B=BP^2\\Longrightarrow PB=BB$ so that $U^*ABU=U^*UPBU=PBU=BA.$ Finally, we have $BA(BA)^*&=(U^*(AB)U)(U^*ABU)^*\\\\&=U^*ABUU^*(AB)^*U\\\\&=U^*(AB)(AB)^*U\\\\&\\le U^*(AB)^*ABU\\\\&=(BA)^*BA.$ The reverse implication does not hold in the previous result (even if $A$ is self-adjoint) as shown in the following example: Example 1 Let $A$ and $B$ be acting on the standard basis $(e_n)$ of $\\ell ^2(\\mathbb {N})$ by: $Ae_n=\\alpha _ne_n \\text{ and } Be_n=e_{n+1},~\\forall n\\ge 1$ respectively.", "Assume further that $\\alpha _n$ is bounded, real-valued and positive, for all $n$ .", "Hence $A$ is self-adjoint (hence normal!)", "and positive.", "Then $ABe_n=\\alpha _{n}e_{n+1},~\\forall n\\ge 1.$ For convenience, let us carry out the calculations as infinite matrices.", "Then $ AB=\\begin{bmatrix} 0 & 0 & & & &\\text{\\Large {0}} \\\\ \\alpha _1 & 0 &0 & &\\\\ 0 & \\alpha _2 & 0 & 0 &\\\\& 0 & \\alpha _3 & 0 & \\ddots \\\\& & 0 & \\ddots & 0 & \\\\\\text{\\Large {0}} & & & \\ddots & \\ddots & \\ddots \\end{bmatrix} \\text{ so that } (AB)^*=\\begin{bmatrix} 0 & \\alpha _1 & & & &\\text{\\Large {0}} \\\\ 0 & 0 &\\alpha _2 & &\\\\ 0 & 0 & 0 & \\alpha _3 &\\\\& 0 & 0 & 0 & \\ddots \\\\& & 0 & \\ddots & 0 & \\\\\\text{\\Large {0}} & & & \\ddots & \\ddots & \\ddots \\end{bmatrix}.$ Hence $ AB(AB)^*=\\begin{bmatrix} 0 & 0 & & & &\\text{\\Large {0}} \\\\ 0 &\\alpha _1^2 &0 & &\\\\ 0 & 0 & \\alpha _2^2 & 0 &\\\\& 0 & 0 & \\alpha _3^2 & \\ddots \\\\& & 0 & \\ddots & \\ddots & \\\\\\text{\\Large {0}} & & & \\ddots & \\ddots & \\ddots \\end{bmatrix}$ and $(AB)^*AB=\\begin{bmatrix} \\alpha _1^2 & 0 & & & &\\text{\\Large {0}} \\\\ 0&\\alpha _2^2 & 0 & &\\\\ 0 & 0 & \\alpha _3^2 & 0 &\\\\& 0 & 0 & \\ddots & \\ddots \\\\& & 0 & \\ddots & \\ddots & \\\\\\text{\\Large {0}} & & & \\ddots & \\ddots & \\ddots \\end{bmatrix}.$ It thus becomes clear that $AB$ is hyponormal iff $\\alpha _n\\le \\alpha _{n+1}$ .", "Similarly $BAe_n=\\alpha _{n+1}e_{n+1},~\\forall n\\ge 1.$ Whence the matrix representing $BA$ is given by: $BA=\\begin{bmatrix} 0 & 0 & & & &\\text{\\Large {0}} \\\\ \\alpha _2 & 0 &0 & &\\\\ 0 & \\alpha _3 & 0 & 0 &\\\\& 0 & \\alpha _4 & 0 & \\ddots \\\\& & 0 & \\ddots & 0 & \\\\\\text{\\Large {0}} & & & \\ddots & \\ddots & \\ddots \\end{bmatrix} \\text{ so that } (BA)^*=\\begin{bmatrix} 0 & \\alpha _2 & & & &\\text{\\Large {0}} \\\\ 0 & 0 &\\alpha _3 & &\\\\ 0 & 0 & 0 & \\alpha _4 &\\\\& 0 & 0 & 0 & \\ddots \\\\& & 0 & \\ddots & 0 & \\\\\\text{\\Large {0}} & & & \\ddots & \\ddots & \\ddots \\end{bmatrix}.$ Therefore, $BA(BA)^*=\\begin{bmatrix} 0 & 0 & & & &\\text{\\Large {0}} \\\\ 0 &\\alpha _2^2 &0 & &\\\\ 0 & 0 & \\alpha _3^2 & 0 &\\\\& 0 & 0 & \\alpha _4^2 & \\ddots \\\\& & 0 & \\ddots & \\ddots & \\\\\\text{\\Large {0}} & & & \\ddots & \\ddots & \\ddots \\end{bmatrix}$ and $(BA)^*BA=\\begin{bmatrix} \\alpha _2^2 & 0 & & & &\\text{\\Large {0}} \\\\ 0&\\alpha _3^2 & 0 & &\\\\ 0 & 0 & \\alpha _4^2 & 0 &\\\\& 0 & 0 & \\ddots & \\ddots \\\\& & 0 & \\ddots & \\ddots & \\\\\\text{\\Large {0}} & & & \\ddots & \\ddots & \\ddots \\end{bmatrix}.$ Accordingly, $BA$ is hyponormal iff $\\alpha _n\\le \\alpha _{n+1}$ (thankfully, this is the same condition for the hyponormality of $AB$ ).", "Finally, $BA^2=\\begin{bmatrix} 0 & 0 & & & &\\text{\\Large {0}} \\\\ \\alpha _1^2 & 0 &0 & &\\\\ 0 & \\alpha _2^2 & 0 & 0 &\\\\& 0 & \\alpha _3^2 & 0 & \\ddots \\\\& & 0 & \\ddots & 0 & \\\\\\text{\\Large {0}} & & & \\ddots & \\ddots & \\ddots \\end{bmatrix}\\ne A^2B=\\begin{bmatrix} 0 & 0 & & & &\\text{\\Large {0}} \\\\ \\alpha _2^2 & 0 &0 & &\\\\ 0 & \\alpha _3^2 & 0 & 0 &\\\\& 0 & \\alpha _4^2 & 0 & \\ddots \\\\& & 0 & \\ddots & 0 & \\\\\\text{\\Large {0}} & & & \\ddots & \\ddots & \\ddots \\end{bmatrix}$ Remark An explicit example of such an $(\\alpha _n)$ verifying the required hypotheses would be to take: $\\left\\lbrace \\begin{array}{c}\\alpha _1=0 \\\\\\alpha _{n+1}=\\sqrt{2+\\alpha _n}\\end{array}\\right.$ Then $(\\alpha _n)$ is bounded (in fact, $0\\le \\alpha _n<2$ , for all $n$ ), increasing and such that $\\alpha _1=0\\ne \\alpha _2=\\sqrt{2}$ .", "Going back to Proposition , we observe that the result obviously holds by replacing \"hyponormal\" by \"co-hyponormal\".", "Thus we have Proposition 2.2 Let $A$ and $B$ be two bounded operators on a Hilbert space such that $A$ is normal and $AB$ is co-hyponormal.", "Then $AA^*B=BAA^*\\Longrightarrow \\text{ $BA$ is co-hyponormal.", "}$ Remark The same previous example, mutatis mutandis, works as a counterexample to show that \"$BA$ co-hyponormal $\\Rightarrow AA^*B=BAA^*$ \" need not hold.", "We now come to a very important result of the paper: Theorem 2.1 Let $A,B$ be two bounded operators such that $A$ is also normal.", "Assume that $AB$ is hyponormal and that $BA$ is co-hyponormal.", "Then $AA^*B=BA^*A \\Longleftrightarrow BA \\text{ and } AB \\text{ are normal.", "}$ \"$\\Longrightarrow $ \": Since $A$ is normal, we have $A=UP=PU$ where $U$ is unitary and $P$ is positive.", "Since $AA^*B=BA^*A$ , we obtain $P^2B=BP^2$ or just $BP=PB$ by the positivity of $P$ .", "Therefore, we may write $(AB)^*AB=&(U(BA)U^*)^*(U(BA)U^*)\\\\=&U(BA)^*U^*U(BA)U^*\\\\=&U(BA)^*(BA)U^*\\\\\\le &U(BA)(BA)^*U^* \\text{ (since $BA$ is co-hyponormal and by Lemma \\ref {Lemma A>B UAU*>UBU*})}\\\\=&U(BA)U^*U(BA)^*U^*\\\\=&(AB)(AB)^*,$ that is $AB$ is co-hyponormal.", "Since it is already hyponormal, it immediately follows that $AB$ is normal.", "To prove that $BA$ is normal we apply a similar idea and we have $(BA)(BA)^*=&(U^*(AB)U)(U^*(AB)U)^*\\\\=&U^*(AB)^*UU^*(AB)^*U\\\\=&U^*(AB)(AB)^*U\\\\\\le &U^*(AB)^*(AB)U \\text{ (since $AB$ is hyponormal and by Lemma \\ref {Lemma A>B UAU*>UBU*})}\\\\=&U^*(AB)^*UU^*(AB)U\\\\=&(BA)^*(BA),$ that is $BA$ is hyponormal, and since it is also co-hyponormal, we conclude that $BA$ is normal.", "\"$\\Longleftarrow $ \": To prove the the reverse implication, we use the celebrated Fuglede-Putnam theorem (see e.g.", "[2]) and we have: $ABA=ABA&\\Longrightarrow A(BA)=(AB)A\\\\&\\Longrightarrow A(BA)^*=(AB)^*A\\\\&\\Longrightarrow AA^*B^*=B^*A^*A\\\\&\\Longrightarrow BAA^*=A^*AB.$ This completes the proof." ], [ "Main Results: The Unbounded Case", "We start this section by giving a counterexample that shows that the same assumptions, as in Theorem REF , would not yield the same results if $B$ is an unbounded operator, let alone the case where both $A$ and $B$ are unbounded.", "What we want is a normal bounded operator $A$ and an unbounded (and closed) operator $B$ such that $BA$ is normal, $A^*AB\\subset BA^*A$ but $AB$ is not normal.", "Example 2 Let $Bf(x)=e^{x^2}f(x) \\text{ and } Af(x)=e^{-x^2}f(x)$ on their respective domains $D(B)=\\lbrace f\\in L^2(\\mathbb {R}):~e^{x^2}f\\in L^2(\\mathbb {R})\\rbrace \\text{ and } D(A)=L^2(\\mathbb {R}).$ Then it is well known that $A$ is bounded and self-adjoint (hence normal), and that $B$ is self-adjoint (hence closed).", "Now $AB$ is not normal for it is not closed as $AB\\subset I$ .", "$BA$ is normal as $BA=I$ (on $L^2(\\mathbb {R})$ ).", "Hence $AB\\subset BA$ which implies that $AAB\\subset ABA\\Longrightarrow AAB\\subset ABA\\subset BAA.$ Now, we state and prove the generalization of Theorem REF to unbounded operators.", "We have Theorem 3.1 Let $B$ be an unbounded closed operator and $A$ a bounded one such that $AB$ and $A$ are normal.", "Then $BA \\text{ normal } \\Longrightarrow A^*AB\\subset BA^*A.$ If further $BA$ is hyponormal (resp.", "subnormal), then $BA \\text{ normal} \\Longleftarrow A^*AB\\subset BA^*A.$ \"$\\Longrightarrow $ \": Since $AB$ and $BA$ are normal, the equation $A(BA)=(AB)A$ implies that $A(BA)^*=(AB)^*A$ by the Fuglede-Putnam theorem (see e.g.", "[2]).", "Hence $AA^*B^*\\subset B^*A^*A \\text{ or } A^*AB\\subset BA^*A.$ \"$\\Longleftarrow $ \": The idea of proof in this case is similar in core to Kaplansky's (cf.", "[8]).", "Let $A=UP$ be the polar decomposition of $A$ , where $U$ is unitary and $P$ is positive (remember that they also commute and that $P=\\sqrt{A^*A}$ ), then one may write $U^*ABU=U^*UPBU=PBU\\subset BP U=BA$ or $U^*AB=U^*\\overline{AB}=U^*((AB)^*)^*\\subset BA U^*$ (by the closedness of $AB$ ).", "Since $(AB)^*$ is normal, it is closed and subnormal.", "Since $B$ is closed and $A$ is bounded, $BA$ is closed.", "Since it is hyponormal, Theorem REF applies and yields the normality of $BA$ as $U$ is invertible.", "The proof is very much alike in the case of subnormality.", "Imposing another commutativity condition allows us to generalize Theorem REF to unbounded normal operators by bypassing hyponormality and subnormality: Theorem 3.2 Let $B$ be an unbounded closed operator and $A$ a bounded one such that all of $AB$ , $A$ and $B$ are all normal.", "Then $A^*AB\\subset BA^*A \\text{ and } AB^*B\\subset B^*BA \\Longrightarrow BA \\text{ normal } .$ The proof is partly based on the following interesting result of maximality of self-adjoint operators: Proposition 3.1 (Devinatz-Nussbaum-von Neumann, [4]) Let $A$ , $B$ and $C$ be unbounded self-adjoint operators.", "Then $A\\subseteq BC \\Longrightarrow A=BC.$ Now we give the proof of Theorem : First, $BA$ is closed as $A$ is bounded and $B$ is closed.", "So $BA(BA)^*$ (and $(BA)^*BA$ ) is self-adjoint.", "Then we have $A^*ABB^*\\subset BA^*AB^*=BAA^*B^*\\subset BA(BA)^*$ and hence $BA(BA)^*\\subset (A^*ABB^*)^*=BB^*A^*A$ so that Proposition REF gives us $BA(BA)^*=BB^*A^*A$ for both $BB^*$ and $A^*A$ are self-adjoint since $B$ is closed and $A$ is bounded respectively.", "Similarly $A^*AB^*B\\subset A^*B^*BA\\subset (BA)^*BA.$ Adjointing the previous \"inclusion\" and applying again Proposition REF yield $(BA)^*BA=B^*BA^*A=BB^*A^*A,$ establishing the normality of $BA$ ." ] ]
1403.0523
[ [ "A weak-value model for virtual particles supplying the electric current\n in graphene: the minimal conductivity and the Schwinger mechanism" ], [ "Abstract We propose a model for the electric current in graphene in which electric carriers are supplied by virtual particles allowed by the uncertainty relations.", "The process to make a virtual particle real is described by a weak value of a group velocity: the velocity is requisite for the electric field to give the virtual particle the appropriate changes of both energy and momentum.", "With the weak value, we approximately estimate the electric current, considering the ballistic transport of the electric carriers.", "The current shows the quasi-Ohimic with the minimal conductivity of the order of e^2/h per channel.", "Crossing a certain ballistic time scale, it is brought to obey the Schwinger mechanism." ], [ "Introduction", "Graphene is fascinating material due to its applicability for electronic devices and its physical properties are also attractive in fundamental physics[1].", "In a single layer graphene, the low energy excitation of a quasi particle can be well described by the 2+1 dimensional massless Dirac equation.", "With Pauli matrices $\\hat{\\sigma }_i$ , the Hamiltonian can be represented by $\\hat{H}=v_f(\\hat{\\sigma }_x \\hat{p}_x +\\hat{\\sigma }_y \\hat{p}_y), $ where $v_f$ is the Fermi velocity, which corresponds to the velocity of light $c$ .", "The absolute velocity of a particle always takes $v_f$ like a photon.", "Consequently, graphene can be a tool for demonstrating relativistic phenomena like Klein's paradox[2] and Schwinger mechanism[3], [4], which must be confirmed in the electrodynamics.", "On the electric current in graphene, when the chemical potential and the temperature were zero, the minimal conductivity was experimentally found, of which the order was $e^2/h$ per channel (per valley and per spin)[5]: the electric current $j$ shows the linear response on the electric field $\\varepsilon $ as $j\\sim (e^2/h)\\varepsilon $ , which is called the quasi-Ohmic.", "Theoretical works have succeeded in obtaining the minimal conductivity, using the linear response theory[6], [7], [8], [9], [10], [11], Landauer formula[12] and the dynamical approach[13], [14].", "Although their results show subtle different values like $e^2/(\\pi h)$ , there is a consensus on the minimal conductivity of the order of $e^2/h$ per channel.", "Furthermore, it was also predicted that, as the electric field is stronger, so the electric current is beyond the linear response to the electric field as $j\\propto \\varepsilon ^{3/2}$ , which is owing to the Schwinger mechanism[3], [4], [13], [14].", "Schwinger mechanism originally represents a particle-antiparticle creation from a vacuum in a uniform electric field[15], while a hole substitutes for an antiparticle in graphene.", "The electric current can be considered as the ballistic transport of charges, since the ballistic time is long in graphene: the physical behavior can be assigned by the ballistic time.", "In fact, with the ballistic time $t_{bal}$ , the electric current by the Schwinger mechanism is approximately given by $j\\sim en(t_{bal})v_f$ , where $n(t_{bal})$ represents the density of the electric carriers (charges).", "$n(t_{bal})$ can be derived from the 2+1 dimensional massless ($m=0$ ) pair creation rate of the Schwinger mechanism[3], [16], $\\frac{dn}{dt} &=& \\frac{e^{3/2}\\varepsilon ^{3/2}}{4\\pi ^2\\hbar ^{3/2}c^{1/2}}{\\rm exp}\\left(-\\frac{\\pi m^2c^3 }{e\\varepsilon \\hbar }\\right) \\\\&=& \\frac{e^{3/2}\\varepsilon ^{3/2}}{4\\pi ^2\\hbar ^{3/2}v_f^{1/2}} \\ \\ \\ (m=0, c=v_f).", "$ Whether the electric current shows the quasi-Ohmic or the Schwinger mechanism, on first glance, it is surprising that graphene is capable of leading a current.", "There is no electric carrier when the chemical potential and the temperature are zero: the density of states is proportional to the absolute value of the energy, $|E|$ [1].", "Consequently, there must be two processes for the electric current: creation and acceleration (or reorientation Note that the absolute velocity of a particle must be $v_f$ .", "Then, `reorientation' will be more precise.", ").", "If the ballistic time $t_{bal}$ is long enough, an electric carrier can be accelerated to the direction of the electric field after the creation.", "When the electric current is mostly composed of the creation processes, it behaves as the quasi-Ohmic.", "On the other hand, as the contribution of the acceleration processes surpasses the previous one, it shows the Schwinger mechanism, in which all the electric carriers are effectively in the direction of the electric field with the velocity of $v_f$ , i.e.", "$j\\sim en(t_{bal})v_f$ .", "The time scale of the ballistic time for their crossover is given by $t_{c}=\\sqrt{\\frac{\\hbar }{e\\varepsilon v_f}}.", "$ As the electric field is stronger, this crossover time becomes smaller and the Schwinger mechanism will appear.", "In earlier studies[4], [13], [14], it was found that, while the quasi-Ohmic current could be obtained with the perturbation on the electric field, the electric current showed the Schwinger mechanism at last in which the perturbative treatment failed beyond the time scale (REF ).", "In [17], we showed the case when a group velocity of a quantum particle was given by a weak value in the 1+1 dimensional Dirac equation, which was applied to a transmission through a supercritical step potential.", "In this paper, we show that this weak-value formalism is also valid for describing the creation process of an electric carrier in graphene.", "In fact, it has been pointed out that a weak value is useful for a description of a localized event like a pair creation[18].", "To begin with, a weak value is introduced as a result of weak measurements: using weak measurements, we can extract a physical value on an observable without disturbing a quantum system to be measured[19], [20].", "Actually, direct observations of quantum systems have been performed[21], [22], [23].", "In optical physics, the signal amplification effect of weak measurements has also been studied for high sensitive measurements like observations of the Hall effect[24], a beam deflection[25], a phase shift[26], [27], and the Kerr nonlinearities[28], including the theoretical researches[29], [30], [31], [32].", "In solid systems, such effect has been used for a charge sensing[33] and an atomic spontaneous emission[34].", "In addition to the applications as mentioned above, weak measurements have offered interesting approaches for the foundation of quantum mechanics like quantum paradoxes[35], [36], [37], [38], [39] and the violation of the Leggett-Garg inequality[40], [41], [42], [43].", "Apart from weak measurements, a weak value has been useful for explaining quantum phenomena[17], [18], [44], [45], [46], [47].", "Then, the significance of a weak value itself has also been discussed in the context of a measurement[48], [49], [50], [51] and the validity for a description of quantum mechanics[43], [52], [53], [54], [55], [56].", "Our result shows one of the interesting cases in which a weak value emerges as a real value of a physical quantity like [17].", "In addition, aside from an issue of a weak value, it also gives a new insight into graphene in the sense that the creation process is related to virtual particles allowed by the uncertainty relations.", "We focus on just the creation process and do not care the acceleration one.", "Nevertheless, it is enough for approximating the electric current for each mechanism, the quasi-Ohmic and the Schwinger mechanism.", "We show that a weak value may also appear as a group velocity even in the 2+1 dimensional massless Dirac equation here[17].", "According to equation (REF ), a plane wave with a (positive or negative) energy $\\pm E$ and a momentum $p=(p_x,p_y)$ can be described as follows, $\\frac{1}{\\sqrt{2}}\\left[\\begin{array}{c}e^{-i\\theta /2} \\\\\\pm e^{i\\theta /2}\\\\\\end{array}\\right]e^{\\frac{i}{\\hbar }(p_x x+p_y y)} = |\\pm E\\rangle \\psi _{p_x,p_y}(x,y), $ where $\\theta ={\\rm Arctan}(p_y/p_x)$ shows the direction of the momentum.", "They satisfy the energy-momentum relation, $E^2=v_f^2p^2=v_f^2p_x^2+v_f^2p_y^2$ .", "$|\\pm E\\rangle $ is independent of $x$ and $y$ , which is called the chirality.", "The dependent part $\\psi _{p_x,p_y}(x,y)$ is called the space part.", "Consider the case that a chirality prepared in the initial state $|E\\rangle $ is finally found in $|E^{\\prime }\\rangle $ , which is referred to as the preselection in $|E\\rangle $ and the postselection in $|E^{\\prime }\\rangle $ .", "When $t$ is small enough, the time evolution of the space part is given as follows, $& &\\langle E^{\\prime }| e^{-\\frac{i}{\\hbar }v_f(\\hat{\\sigma }_x\\hat{p}_x+\\hat{\\sigma }_y\\hat{p}_y)t}|E\\rangle \\psi _{p_x,p_y}(x,y) \\\\&\\sim & \\langle E^{\\prime }|E\\rangle e^{-\\frac{i}{\\hbar }v_f\\langle \\hat{\\sigma }_x\\rangle _{\\bf w}\\hat{p}_xt}e^{-\\frac{i}{\\hbar }v_f\\langle \\hat{\\sigma }_y\\rangle _{\\bf w}\\hat{p}_yt}\\psi _{p_x,p_y}(x,y) \\ \\ \\ (t \\ \\sim \\ 0) \\\\&\\sim & \\langle E^{\\prime }|E\\rangle \\psi _{p_x,p_y}(x-v_f\\langle \\hat{\\sigma }_x\\rangle _{\\bf w}t,y-v_f\\langle \\hat{\\sigma }_y\\rangle _{\\bf w} t)\\ \\ \\ (t \\ \\sim \\ 0), $ where $\\langle \\hat{\\sigma }_x\\rangle _{\\bf w}$ is a weak value, $\\langle \\hat{\\sigma }_x\\rangle _{\\bf w}=\\frac{\\langle E^{\\prime }|\\hat{\\sigma }_x|E\\rangle }{\\langle E^{\\prime }|E\\rangle }, $ and the weak value of $\\hat{\\sigma }_y$ is given in a similar way.", "$v_f\\langle \\hat{\\sigma }_x\\rangle _{\\bf w}$ and $v_f\\langle \\hat{\\sigma }_y\\rangle _{\\bf w}$ correspond to the (group) velocities in $x$ and $y$ directions respectively.", "In fact, without the postselection, they give $v_f\\langle \\hat{\\sigma }_x\\rangle =p_xv_f^2/E=v_f{\\rm cos}\\theta $ and $v_f\\langle \\hat{\\sigma }_y\\rangle =p_yv_f^2/E=v_f{\\rm sin}\\theta $ .", "If $|E^{\\prime }\\rangle $ represents the chirality of the eigenstate of the energy $E^{\\prime }$ and the momentum $p^{\\prime }=(p^{\\prime }_x,p^{\\prime }_y)$ , the weak values are given as follows, $\\langle \\hat{\\sigma }_x\\rangle _{\\bf w} &=& \\frac{{\\rm sin}[(\\theta +\\theta ^{\\prime })/2]}{{\\rm sin}[(\\theta -\\theta ^{\\prime })/2]} \\\\\\langle \\hat{\\sigma }_y\\rangle _{\\bf w} &=& -\\frac{{\\rm cos}[(\\theta +\\theta ^{\\prime })/2]}{{\\rm sin}[(\\theta -\\theta ^{\\prime })/2]} , $ where $\\theta ^{\\prime }={\\rm Arctan}(p^{\\prime }_y/p^{\\prime }_x)$ .", "Although a weak value is generally a complex number as shown in equation (REF ), it is always real number as far as considering energy eigenstates in our case.", "That is why we treat a weak value as a real number hereafter.", "In the next section, considering a transition between energy eigenstates, we try to describe a creation process in graphene by a pre-postselection of a chirality.", "We show that the weal value of a group velocity (REF ) is requisite for the electric field to yield the changes of both the energy and the momentum appropriately for such transition.", "In section , we assume that the creation process for an electric carrier is triggered off by a virtual particle, which is allowed by the uncertainty relations.", "The weak-value formalism for describing a time evolution like () is justified for such virtual particles, although $t$ is not always $\\sim 0$ .", "In section , we approximately estimate the electric current in graphene, using a weak value of a group velocity.", "It is shown that, crossing the time scale (REF ), the current flows in the different manners, namely, the quasi-Ohmic and the Schwinger mechanism.", "Section is devoted to our conclusion." ], [ "A transition for creating an electric carrier in graphene", "First of all, there must be a creation process of an electric carrier in graphene so as to be capable of leading an electric current.", "An electric carrier will be supplied by creating a particle-hole pair, which is represented by a transition of a particle in the Dirac sea (valence band) to the vacuum (conduction band).", "For this purpose, we consider a transition as shown in figure REF , supposing that the electric field $\\varepsilon $ is in $x$ direction.", "Initially, a particle in the Dirac sea has a negative (kinetic) energy $-E$ and a momentum $(-p_x,p_y)$ , where $E, p_x\\ge 0$ .", "By the electric field, the particle might change to the one with a positive (kinetic) energy $E^{\\prime }$ and a momentum $(p^{\\prime }_x,p_y)$ , where $E^{\\prime }, p^{\\prime }_x\\ge 0$ .", "The momentum in $y$ direction does not change, because the electric field is zero in this direction.", "The signs of $p_x$ and $p^{\\prime }_x$ provide $+x$ velocities, which is in the direction of the electric field, because they give the group velocities $(-p_xv_f^2)/(-E)\\ge 0$ and $p^{\\prime }_xv_f^2/E^{\\prime }\\ge 0$ respectively.", "As an energy eigenstate can be specified by a chirality as shown in equation (REF ), this transition process can be described by the pre-postselection on the chirality, $|E\\rangle $ and $|E^{\\prime }\\rangle $ .", "Like equation (), when the time $t$ is very small, the time evolution of the space part can be approximately expressed as follows, $& &\\langle E^{\\prime }| e^{-\\frac{i}{\\hbar }(v_f(\\hat{\\sigma }_x\\hat{p}_x+\\hat{\\sigma }_y\\hat{p}_y)-e\\varepsilon x)t}|E\\rangle \\psi _{p_x,p_y}(x,y) \\\\&\\sim & \\langle E^{\\prime }|E\\rangle e^{-\\frac{i}{\\hbar }v_f\\langle \\hat{\\sigma }_x\\rangle _{\\bf w}\\hat{p}_xt}e^{-\\frac{i}{\\hbar }v_f\\langle \\hat{\\sigma }_y\\rangle _{\\bf w}\\hat{p}_yt}e^{\\frac{i}{\\hbar }e\\varepsilon x t}\\psi _{p_x,p_y}(x,y) \\ \\ \\ (t \\ \\sim \\ 0) \\\\&\\sim & \\langle E^{\\prime }|E\\rangle \\psi _{p_x+e\\varepsilon t,p_y}(x-v_f\\langle \\hat{\\sigma }_x\\rangle _{\\bf w}t,y-v_f\\langle \\hat{\\sigma }_y\\rangle _{\\bf w} t) \\ \\ \\ (t \\sim \\ 0).", "$ This is different from equation () in the respect that there is a momentum shift for $p_x$ due to the electric field [17].", "At this stage, however, it is not clear whether this weak-value formalism is valid, as we have not yet verified that the time is small enough for this approximation.", "The validity of the approximation will be discussed in the next section.", "At any rate, the weak values of the group velocities in $x$ and $y$ directions can be respectively defined by equation (REF ) and equation (), with $\\theta ={\\rm Arctan}(p_y/(-p_x))$ and $\\theta ^{\\prime }={\\rm Arctan}(p_y/p^{\\prime }_x)$ .", "These group velocities represent the velocities driven by the transitions.", "Because the electric field is zero in $y$ direction, the group velocity in this direction should be zero, namely, $\\langle \\hat{\\sigma }_y\\rangle _{\\bf w}=0,$ equivalently, $\\theta ^{\\prime }+\\theta =\\pm \\pi \\ \\ {\\rm i.e.", "}, \\ \\ p_x^{\\prime }=p_x \\ {\\rm and} \\ E^{\\prime }=E.", "$ If not, particles seem to accomplish transitions with zero electric field and the velocity, which causes the current, emerges in $y$ direction.", "Equation (REF ) shows that a transition is selective[57]: a particle with a negative energy $-E$ and a momentum $(-p_x, p_y)$ may turn out one with $E$ and $(p_x, p_y)$ .", "In this case, $\\langle \\hat{\\sigma }_x\\rangle _{\\bf w}$ is given by $\\langle \\hat{\\sigma }_x\\rangle _{\\bf w} = \\frac{\\sqrt{p_x^2+p_y^2}}{p_x}=\\frac{1}{{\\rm cos}\\theta } >1, $ which is a strange value, that is, the corresponding group velocity $v_f\\langle \\hat{\\sigma }_x\\rangle _{\\bf w}$ is more than $v_f$[47], [58], [59].", "The appearance of such strange weak value agrees with the result of [17] due to a transition from a negative energy state to a positive one.", "To clarify the meaning of this strange velocity, we consider the inside details of the transition process.", "During the transition, the changes of the energy and the momentum are $2E(=E-(-E))\\equiv \\Delta E$ and $2p_x(=p_x-(-p_x))\\equiv \\Delta p_x$ respectively.", "The force $e\\varepsilon $ by the electric field acts on a particle in $x$ direction.", "Define $\\Delta x$ as the moving distance for the duration of the transition.", "As the energy change is equivalent to the work done by the electric field, it satisfies, $\\Delta E=e\\varepsilon \\Delta x.", "$ With the time needed for the transition $\\Delta t$ , we also obtain, $\\Delta p_x = e\\varepsilon \\Delta t, $ because of the equivalence between the momentum change and the impulse.", "Then, we can define the average (group) velocity $v_g$ during the transition process as follows, $v_g &\\equiv & \\frac{\\Delta x}{\\Delta t} \\\\&=&\\frac{\\Delta E}{\\Delta p_x}=\\frac{2E}{2p_x}=\\frac{v_f\\sqrt{p_x^2+p_y^2}}{p_x}.", "$ From equations (REF ) and (REF ), we can find, $v_g=v_f\\langle \\hat{\\sigma }_x\\rangle _{\\bf w},$ which shows that, in fact, the weak value of the group velocity (REF ) is requisite to satisfy the energy change (REF ) and the momentum change (REF ) simultaneously.", "Using a weak value, we can also estimate a probability of occurring a transition.", "In a specific postselection, a weak value may take a strange value lying outside of the spectra of eigenvalues.", "However, the average value should be within the conventional range of value in considering all the possible postselection.", "In our case, when it succeeds in postselecting a chirality by a positive energy eigenstate $|E\\rangle $ , $v_f\\langle \\hat{\\sigma }_x\\rangle _{\\bf w}$ is more than $v_f$ as shown in equation (REF ).", "Such weak value yields a group velocity of a current due to the transition process, because a transition corresponds to a creation of a particle-hole pair, namely, a carrier.", "Note that the current does not contain the effect of the process after the creation, i.e.", "the acceleration.", "Without a transition, it brings about zero group velocity and does not contribute to generating a current, because a particle keeps in the Dirac sea as before.", "Including such non-transition particles, the average velocity should be conventionally less than $v_f$ , by which the net velocity of the current is given.", "After all, the current does not flow beyond $v_f$ like superluminal velocity: it never occurs as a strange physical phenomenon with a strange value of a physical quantity as the whole.", "For this reason, all the particle do not transmit to a positive energy, and such transition happens with some probability $T$ .", "In [17], we could actually estimate the transmission probability for a step potential by making a weal value of a group velocity at the step consistent with an average velocity of the flux outside the step.", "In a similar way, we can obtain a transition probability $T$ as shown in figure REF .", "The number of transition particles in a positive energy eigenstate $|E\\rangle $ is equivalent to the one in a negative energy eigenstate $|-E\\rangle $ , which corresponds to the holes.", "In addition, the group velocities of particles just before and just after a transition, namely, the group velocities in $|E\\rangle $ and $|-E\\rangle $ are the same as $v_f{\\rm cos}\\theta (=p_xv_f^2/E=(-p_xv_f^2)/(-E))$ .", "Consequently, the average velocity of the current driven by the transition should be also $v_f{\\rm cos}\\theta $ .", "On the other hand, as we have mentioned, the transition itself generates the group velocity $v_f\\langle \\hat{\\sigma }_x\\rangle _{\\bf w}$ more than $v_f{\\rm cos}\\theta $ .", "If the transition probability is given by $T$ , the average velocity $Tv_f\\langle \\hat{\\sigma }_x\\rangle _{\\bf w}$ has to satisfy $Tv_f\\langle \\hat{\\sigma }_x\\rangle _{\\bf w}=v_f{\\rm cos}\\theta .", "$ From this equation, we can find the transition probability as follows, $T(p_x,p_y)={\\rm cos}^2\\theta =\\frac{p_x^2}{p_x^2+p_y^2}.", "$ This transition probability is the same as the transmission probability for the step potential shown in [57], in which n-p junction in graphene is treated.", "This agreement is plausible, as we have derived the `transition' probability in the same manner of the `transmission' probability for the step potential like [17].", "They have a common point that the electric field brings about the process (transition or transmission), which generates and determines the velocity of the current.", "We have discussed a transition between energy eigenstates by an electric field.", "A creation of an electric carrier in graphene should be described by this picture.", "In the next section, we consider how much energy states can participate in such transitions for creating electric carriers.", "Figure: The current due to transitions from a negative energy level -E-E to the positive one EE, except for the accelerations after the transitions.A particle in -E-E (EE) has the group velocity v f cos θv_f{\\rm cos}\\theta just before (after) a transition.The current driven by the creations is composed of such particles with a homogeneous density.Note that the current in the negative energy states corresponds to the flux of holes in the opposite direction.The entire flux shows the current of the charges in +x+x direction.As a particle has the group velocity v f 〈σ ^ x 〉 𝐰 v_f\\langle \\hat{\\sigma }_x\\rangle _{\\bf w} during a transition,a transition should occur with a probability TT to agree with the average velocity of the flux as shown in equation ()." ], [ "An electric carrier and the uncertainty relations", "We assume that a transition for creating an electric carrier is triggered off by fluctuations allowed by the uncertainty relations.", "According to the uncertainty relation between energy and time, $\\delta E \\delta t \\sim \\hbar , $ an energy fluctuation $\\delta E$ can occur during a time $\\delta t$ , which means that a virtual particle with the energy $\\delta E$ can exist during the lifetime $\\delta t$ .", "In a similar way, a virtual particle with a momentum $\\delta p_x$ can be considered within a space $\\delta x$ in $x$ direction, and they satisfy the uncertainty relation, $\\delta x\\delta p_x \\sim \\hbar .", "$ In our case, the virtual particle can correspond to a virtual transition like figure REF , and has the energy $2E\\equiv \\delta E$ and the momentum $2p_x\\equiv \\delta p_x$ .", "Note that we are concerned about the case in which the momentum change in $y$ direction is zero due to zero electric field and do not have to care the fluctuation of the momentum in this direction.", "Such virtual transition with the energy $\\delta E$ and the momentum $\\delta p_x$ , however, is not always consistent with a real particle, because it does not always satisfy the appropriate energy-momentum relation, $\\delta E^2=v_f^2(\\delta p_x^2+p_y^2)$ : a virtual particle satisfy the uncertainty relations (REF ) and (REF ) independently.", "In fact, to make a virtual particle contribute to the electric current as a real one would, the electric field must satisfy (REF ) and (REF ) simultaneously, which is, as we mentioned before, accomplished by the weak value of the group velocity (REF ).", "In other words, a virtual particle gives the electric field a chance to do the work and the impulse, by which we mean the electric current is able to pass in graphene.", "Within the lifetime $\\delta t=\\hbar /2E=\\hbar /(2v_f\\sqrt{p_x^2+p_y^2})$ , the electric field must achieve the impulse $2p_x=\\Delta p_x$ , which takes the time $\\Delta t$ (see equation (REF )), namely, $\\Delta t \\le \\delta t, $ from which we can obtain $p_x^2(p_x^2+p_y^2)\\le \\frac{e^2\\varepsilon ^2\\hbar ^2}{16v_f^2}.", "$ (REF ) assigns the energy states which may contribute to electric carriers via virtual particles.", "The same result can be derived from the relation between the work and the space instead of the impulse and the time: according to equation (REF ), the work $2E=\\Delta E$ needs the space $\\Delta x$ , which should be smaller than the fluctuation $\\delta x=\\hbar /2p_x$ as follows, $\\Delta x \\le \\delta x.", "$ Satisfying (REF ) and (REF ) simultaneously, the weak value makes the uncertainty relations (REF ) and (REF ) equivalent in the sense that it selects a real particle from virtual particles in the independent uncertainty relations (REF ) and (REF ).", "As a result, it is plausible that (REF ) and (REF ) derive the same result (REF ), because the weak value satisfies the appropriate changes of both the energy and the momentum.", "So far, we have proceeded to a discussion as if the approximation of the weak-value formalism, equation (), is valid and a velocity of a particle during a transition is given by a weak value.", "As follows, we verify that this approximation is adequate as far as the above-mentioned transition stemming from a virtual particle.", "Expanding on $t$ , we can describe equation (REF ) as follows, $& &\\langle E^{\\prime }| e^{-\\frac{i}{\\hbar }(v_f(\\hat{\\sigma }_x\\hat{p}_x+\\hat{\\sigma }_y\\hat{p}_y)-e\\varepsilon x)t}|E\\rangle \\psi _{p_x,p_y}(x,y) \\nonumber \\\\&=& \\langle E^{\\prime }|E\\rangle \\sum _{k=0}^{\\infty }\\frac{1}{k!", "}\\Bigl (-\\frac{i}{\\hbar }t\\Bigr )^k\\frac{\\langle E^{\\prime }|(v_f(\\hat{\\sigma }_x\\hat{p}_x+\\hat{\\sigma }_y\\hat{p}_y)-e\\varepsilon x)^k|E\\rangle }{\\langle E^{\\prime }|E\\rangle } \\psi _{p_x,p_y}(x,y)$ This equation coincides with equation () by the first order of $t$ , which is given by $& & \\langle E^{\\prime }|E\\rangle e^{-\\frac{i}{\\hbar }v_f\\langle \\hat{\\sigma }_x\\rangle _{\\bf w}\\hat{p}_xt}e^{-\\frac{i}{\\hbar }v_f\\langle \\hat{\\sigma }_y\\rangle _{\\bf w}\\hat{p}_yt}e^{\\frac{i}{\\hbar }e\\varepsilon x t}\\psi _{p_x,p_y}(x,y) \\nonumber \\\\&=& \\langle E^{\\prime }|E\\rangle \\sum _{k=0}^{\\infty }\\frac{1}{k!", "}\\Bigl (-\\frac{i}{\\hbar }v_f\\langle \\hat{\\sigma }_x\\rangle _{\\bf w}\\hat{p}_xt\\Bigr )^k \\nonumber \\\\& & \\ \\ \\ \\ \\ \\ \\sum _{k^{\\prime }=0}^{\\infty }\\frac{1}{k^{\\prime }!", "}\\Bigl (-\\frac{i}{\\hbar }v_f\\langle \\hat{\\sigma }_y\\rangle _{\\bf w}\\hat{p}_yt\\Bigr )^{k^{\\prime }}\\sum _{k^{\\prime \\prime }=0}^{\\infty }\\frac{1}{k^{\\prime \\prime }!", "}\\Bigl (\\frac{i}{\\hbar }e\\varepsilon xt\\Bigr )^{k^{\\prime \\prime }}\\psi _{p_x,p_y}(x,y).", "$ Consequently, equation (REF ) can be approximated to equation (), when the terms of $O(t^k)$ $(k\\ge 2)$ can be neglected, which should be properly satisfied in $t\\sim 0$ .", "If it satisfies not $t\\sim 0$ but that the higher terms of $O(t^k)$ $(k\\ge 2)$ are smaller than the first one, however, this approximation will also stand for rough estimations.", "In our case, we can obtain all what we need to valuate the higher terms: $\\langle \\hat{\\sigma }_x\\rangle _{\\bf w} = O(\\Delta E/\\Delta p_x)/v_f$ , $\\langle \\hat{\\sigma }_y\\rangle _{\\bf w} = 0$ , $\\langle \\hat{\\sigma }_z\\rangle _{\\bf w} = O(\\langle \\hat{\\sigma }_x\\rangle _{\\bf w})$ , $e\\varepsilon = O(\\Delta p_x/\\Delta t)$ , $2p_x=O(\\Delta p_x)$ , and $t=O(\\Delta t)$ .", "For example, one of the second terms in equation (REF ) can be estimated as follows, $& & \\frac{1}{2!", "}\\left(\\frac{i}{\\hbar }\\right)^2v_f^2\\langle \\hat{\\sigma }_x\\rangle _{\\bf w}^2\\hat{p}_x^2t^2\\psi _{p_x,p_y}(x,y) \\nonumber \\\\&=& \\frac{1}{2!", "}\\left(\\frac{i}{\\hbar }\\right)^2O\\left(\\frac{\\Delta E^2}{\\Delta p_x^2}\\right)O(\\Delta p_x^2) O(\\Delta t^2)\\psi _{p_x,p_y}(x,y) \\nonumber \\\\&=& \\frac{1}{2!", "}\\left(\\frac{i}{\\hbar }\\right)^2O(\\Delta E^2\\Delta t^2)\\psi _{p_x,p_y}(x,y).$ Because of $\\Delta E\\Delta t<\\hbar $ , this term is smaller than the first one.", "In a similar way, we can verify that the other higher terms are also smaller.", "As a result, it is reasonable to describe the weak-value formalism for the transition starting from a virtual particle in the uncertainty relations: as far as the rough estimation of the electric current, we can regard the velocity of a particle as the weak value during the transition process." ], [ "The ballistic transport in graphene with a weak value", "In graphene, the ballistic time $t_{bal}$ is long, within which we can pay no attention to the interactions with phonons, electrons, and so on.", "The effect of the disorder can also be ignored.", "In the spirit of Drude model, the electric current in graphene can be explained with such ballistic transport[13], [14]: the ballistic time $t_{bal}$ , which is mostly given by $t_{bal}=L/v_f$ with the size of the graphene sample $L$ , corresponds to the mean free time.", "Moving the Dirac point, the net velocity appears along the electric field and brings about the current[1], the behavior of which is assigned by $t_{bal}$ .", "In our case, $t_{bal}$ should similarly participate in the electric current.", "The transition time for creating an electric carrier $\\Delta t$ must be smaller than $t_{bal}$ , namely, $\\Delta t\\le t_{bal}$ , from which we can obtain, $0\\le p_x\\le \\frac{1}{2}e\\varepsilon t_{bal}.", "$ The distance $\\Delta x$ to achieve the transition is also smaller than $L$ , i.e.", "$\\Delta x\\le L$ .", "Then, we can find, $\\sqrt{p_x^2+p_y^2}\\le \\frac{1}{2}e\\varepsilon t_{bal}, $ which includes (REF ): the energy states satisfying (REF ) can actually participate in the electric current, given $t_{bal}$ .", "As a result, the actual electric current should consist of the energy states in both (REF ) and (REF ).", "Note that (REF ) represents candidates for electric carriers via virtual particles.", "We have not been concerned about the amount of $t_{bal}$ itself.", "However, if $t_{bal}$ is small enough, the energy fluctuation $\\delta E_{bal}$ may be effective on the current where $\\delta E_{bal}$ is given by the uncertainty relation of $\\delta E_{bal} \\ t_{bal} \\sim \\hbar $ .", "With the energy fluctuation $2E\\le \\delta E_{bal}$ ($E=v_f\\sqrt{p_x^2+p_y^2}$ ), it gives $\\sqrt{p_x^2+p_y^2}\\le \\frac{\\hbar }{2v_ft_{bal}}, $ within which the energy states are involved in the energy fluctuation $\\delta E_{bal}$ .", "Figure REF represents (REF ), (REF ), and (REF ) for the various amounts of $t_{bal}$ .", "Clearly, they show the different features, crossing the time scale $t_{c}$ given by (REF ).", "In $t_{bal}<t_{c}$ , all states to be considered are included in the energy fluctuation by the ballistic time (REF ) as denoted by O, while they are divided into two regions, namely, the inside and the outside of the fluctuation (O and S) in $t_{bal}>t_{c}$ .", "After all, when $t_{bal}>>t_c$ , the most states are out of the energy fluctuation and belong to S. It follows that each case of $t_{bal}$ shows a different mechanism of the electric current.", "Figure: The figures of (), (), and () in the various cases of t bal t_{bal}: (a)t bal <t c t_{bal}<t_c, (b)t bal =t c t_{bal}=t_c, and t bal >t c t_{bal}>t_c.Their values of t bal t_{bal} are chosen appropriately.", "p x p_x and p y p_y are also normalized suitably.The regions surrounded by the blue curves represent (), which provide candidates for electric carriers via virtual particles.The regions assigned by the ballistic times, i.e.", "() are within the red solid circles.The energy fluctuations due to the ballistic times, which are given by (), are indicated by the dashed circles.The regions satisfying both () and (), which are within both the blue curves and the red solid circles, are colored.According as the inside or the outside of the dashed circle, they are color-coded by the dark green or the pale gray, which are referred to as O and S respectively.Note that p x ≥0p_x\\ge 0 are concerned, because the initial momentum -p x -p_x should be negative.These colored regions provide the energy states contributing to the electric current: they assign the regions of the integrations, () and (), for counting the energy states.In (a), the colored region is utterly within the dashed circle as shown by O.Getting the larger t bal t_{bal}, the size of the red circle overtakes the dashed one in (b).In (c), the colored regions are divided into O and S by the dashed circle, namely, the energy fluctuation by the ballistic time.The time scale of $t_{bal}<t_{c}$ corresponds to the quasi-Ohmic.", "In this case, the electric current is significantly composed of the creation processes, because $t_{bal}$ is not long enough for an acceleration after a creation[13], [14].", "For this reason, although our model describes just the creation process, we can actually try to attain the result of the quasi-Ohmic.", "As shown in figure REF (a), all the energy state contributing to the creation process is within the energy fluctuation by the ballistic time, (REF ).", "Then, for each state of an energy $E$ , we can consider the number of virtual particles as $\\delta E_{bal}/2E$ : one state may supply more than one virtual particle.", "This means that, in addition to approximating the number of contributing energy states by the uncertainty relations (REF ) and (REF ), we are also trying to approximate the contribution per state, using the uncertainly relation on the ballistic time scale.", "Each virtual particle provides an opportunity for the electric field to work of $2E$ , which is accomplished with the time needed $\\Delta t$ .", "Then, the work per unit time for each state is given by $2E/\\Delta t$ .", "With the transition probability $T(p_x,p_y)$ , the whole work per unit time done by the electric field can be estimated as follows, $& & \\frac{1}{(2\\pi \\hbar )^2}\\int \\int _{\\rm O}dp_xdp_yT(p_x,p_y)\\frac{2E}{\\Delta t}\\frac{\\delta E_{bal}}{2E} \\\\&=& \\frac{1}{(2\\pi \\hbar )^2}\\frac{e\\varepsilon \\hbar }{2t_{bal}}\\int \\int _{\\rm O}dp_xdp_y\\frac{p_x}{p_x^2+p_y^2} \\nonumber \\\\&=& \\frac{1}{(2\\pi \\hbar )^2}\\frac{e\\varepsilon \\hbar }{2t_{bal}}\\int _0^{\\frac{1}{2}e\\varepsilon t_{bal}}dr\\int _{-\\pi /2}^{\\pi /2}d\\theta {\\rm cos}\\theta \\ \\ \\ (p_x=r{\\rm cos}\\theta , \\ p_y=r{\\rm sin}\\theta ) \\nonumber \\\\&=& \\frac{e^2\\varepsilon ^2}{4\\pi h}.", "$ When the electric current is proportional to the electric field as $j=\\sigma \\varepsilon $ with the conductivity $\\sigma $ , the work per unit time is given by $j\\varepsilon =\\sigma \\varepsilon ^2$ .", "Comparing equation () to $j\\varepsilon $ , we can find the electric current, $j=\\frac{e^2}{4\\pi h}\\varepsilon , $ and the conductivity, $\\sigma =\\frac{e^2}{4\\pi h}.", "$ We have obtained the linearity of the electric current, namely, the quasi-Ohmic.", "The estimated conductivity (REF ) almost accords with the minimal conductivity $\\sim e^2/h$ per channel, especially the theoretical value like $e^2/(\\pi h)$ .", "In this case, $\\delta E_{bal}/2E$ -fold virtual particles play roles of the electric carriers per state.", "Of course, the electric field actually performs the corresponding work and impulse, which is the cause of the conductivity or the resistivity unlike Joule heat in the Ohmic.", "It resembles a pi-meson taking on a nuclear force between nuclear particles, as we can approximate the mass of the pi-meson $m_\\pi $ with the energy-time uncertainty relation, $\\delta E_\\pi \\delta t_\\pi =(m_\\pi c^2)\\delta t_\\pi \\sim \\hbar $ .", "$\\delta t_\\pi \\sim \\hbar /m_\\hbar c^2$ corresponds to the lifetime of the pi-meson.", "Such pi-meson is effective within $c\\delta t_\\pi $ , namely, the Compton wave length, $\\hbar /m_\\pi c$ .", "Estimating this length as the size of the atomic nuclei, $d\\sim 10^{-15}{\\rm m}$ , we can find $m_\\pi c^2\\sim 200{\\rm MeV}$ , which agrees with $m_\\pi c^2\\sim 140{\\rm MeV}$ .", "It is the fact that the atomic nuclei is stable due to the nuclear force with a mediation of a virtual particle of a pi-meson, which does not emerge from the nuclei.", "In this sense, the virtual particle is real as far as no violation of the energy conservation, which is the same as an electric carrier for the quasi-Ohmic in graphene.", "When $t_{bal}>>t_{c}$ , the energy fluctuation due to the ballistic time is very small for the most states, which belong to the region S as shown in figure REF (c).", "For such states, the effect of the fluctuation can be neglected unlike the quasi-Ohmic case: one state supplies one particle.", "While the region O provides the quasi-Ohmic current as mentioned before, the contribution of the region S to the electric current can be also valuated.", "The ballistic time is enough to accelerate a created carrier to $v_f$ effectively in $x$ direction.", "Then, all what we need is the density of electric carriers $n(t_{bal})$ , with which we can estimate the electric current as $j\\sim en(t_{bal})v_f$ approximately[13], [14].", "$n(t_{bal})$ can be derived from the creation rate of the electric carriers $dn/dt$ , which should correspond to the pair creation rate by the Schwinger mechanism, equation (REF ).", "As we mentioned before, in a state of an energy $E$ and a momentum $(p_x,p_y)$ , it takes the time $\\Delta t$ for accomplishing the transition of the creation.", "With the transition probability $T(p_x,p_y)$ , the number of created particles per unit time is given by $T(p_x,p_y)/\\Delta t$ for the state.", "Consequently, the rate $dn/dt$ is given as follows, $\\frac{dn}{dt} &=& \\frac{1}{(2\\pi \\hbar )^2}\\int \\int _{\\rm S}dp_xdp_y\\frac{T}{\\Delta t} \\\\&=& \\frac{e\\varepsilon }{2(2\\pi \\hbar )^2}\\int \\int _{\\rm S}dp_xdp_y\\frac{p_x}{p_x^2+p_y^2} \\nonumber \\\\&\\sim & \\frac{e\\varepsilon }{2(2\\pi \\hbar )^2}\\int _0^{\\frac{1}{2}\\sqrt{\\frac{e\\varepsilon \\hbar }{v_f}}}dp_x\\int _{-\\sqrt{\\frac{e^2\\varepsilon ^2\\hbar ^2}{16v_f^2p_x^2}-p_x^2}}^{\\sqrt{\\frac{e^2\\varepsilon ^2\\hbar ^2}{16v_f^2p_x^2}-p_x^2}}dp_y\\frac{p_x}{p_x^2+p_y^2} \\ \\ \\ (t_{bal}>>t_c)\\nonumber \\\\&=& \\frac{e^{3/2}\\varepsilon ^{3/2}}{4\\pi ^2\\hbar ^{3/2}v_f^{1/2}} \\int _0^1ds {\\rm Arctan}\\sqrt{\\frac{1}{s^4}-1}= \\frac{e^{3/2}\\varepsilon ^{3/2}}{4\\pi ^2\\hbar ^{3/2}v_f^{1/2}} \\frac{B(\\frac{1}{2}, \\frac{3}{4})}{4},$ where $B(m,n)$ represents the beta function $B(m,n)=2\\int _0^{\\pi /2}({\\rm sin}\\theta )^{2m-1}({\\rm cos}\\theta )^{2n-1} d\\theta $.", "In the above approximation, the higher-order terms above $O(t_c/t_{bal})$ have been neglected because of $t_{bal}>>t_c$ .", "As $B(1/2,3/4)/4$ is about 0.6, this result roughly corresponds with the rate of the Schwinger mechanism, equation (REF )[60].", "Note that, compared to this current by the Schwinger mechanism, the quasi-Ohmic current can be ignored in this case." ], [ "Conclusion", "We have shown that a creation process of an electric carrier in graphene can be described by a weak-value formalism.", "Although our weak-value model did not cover the entire physics, namely, the acceleration process, it was enough to show the feature of the electric currents for the cases of the quasi-Ohmic ($\\propto \\varepsilon $ ) and the Schwinger mechanism ($\\propto \\varepsilon ^{3/2}$ ).", "Our goal was not to make a strict estimation: not to settle in the value of the minimal conductivity.", "However, our estimation of the currents approximately agrees with the earlier studies.", "At the cost of a rigorous discussion, we have clarified the process to supply electric carriers in graphene related to virtual particles by the uncertainty relations.", "In this sense, our approach is different from the earlier studies.", "In the dynamical approach[13], [14], the crossing time scale (REF ) was derived, beyond which the perturbative treatment failed.", "There, the Schwinger mechanism can be understood as a transmission picture with WKB approximation or be comprehensible in the context of the Landau-Zener transition[16], [61], [62]: considering the infinite past and future, the entire physics can be determined.", "In contrast, our model describes the physics of the turning point of transition from a negative energy state to a positive one, i.e.", "the creation process itself, by which we have tried to valuate the entire current in graphene.", "Note that the transition probability $T$ of equation (REF ) is irrelevant to the Landau-Zener transition probability.", "A weak value was originally introduced by Aharonov, Albert and Vaidman as a result of weak measurements[19].", "According to a weak value, a pointer of a measurement apparatus is surely moved, although the weak value may take a strange value lying outside the eigenvalue spectrum.", "While such physical effect to the pointer is one of the actual phenomena of a weak value, our model shows the case that a weak value emerges as an actual value of a physical quantity, following [17].", "In addition, we have found the new insight in that a weak value of a group velocity makes a virtual particle in the uncertainty relations real.", "Irrespective of a strange value, a weak value has often allowed us to treat a quantum particle as a classical one.", "Interestingly, the figure depicted by (quantum) weak values, however, does not alway accord with the one of classical physics as shown in [63].", "A weak value seems to be simple, but not to be superficial.", "Then, as we have seen, it needs to clarify how a weak value becomes effective in physics for understanding the meaning of the value." ], [ "Acknowledgements", "This work was supported by the Funding Program for World-Leading Innovative R & D on Science and Technology (FIRST), and JSPS Grant-in-Aid for Scientific Research(A) 25247068." ] ]
1403.0082
[ [ "Emergent Lorentz Signature, Fermions, and the Standard Model" ], [ "Abstract This article investigates the construction of fermions and the formulation of the Standard Model of particle physics in a theory in which the Lorentz signature emerges from an underlying microscopic purely Euclidean $SO(4)$ theory.", "Couplings to a clock field are responsible for triggering the change of signature of the effective metric in which the standard fields propagate.", "We demonstrate that Weyl and Majorana fermions can be constructed in this framework.", "This construction differs from other studies of Euclidean fermions, as the coupling to the clock field allows us to write down an action which flows to the usual action in Minkowski spacetime.", "We then show how the Standard Model can be obtained in this theory and consider the constraints on non-Standard-Model operators which can appear in the QED sector due to CPT and Lorentz violation." ], [ "Introduction", "Part of the art of theoretical physics is to find the mathematical structures that allow us to formalize and simplify the laws of nature.", "These structures include the description of spacetime (dimension, topology, ...) and matter and their interactions (fields, symmetries, ...).", "While there is a large amount of freedom in the choice of these mathematical structures, the developments of theoretical physics have taught us that some of them are better suited to describe certain classes of phenomena.", "However, these choices are only validated by the mathematical consistency of the theory and, in the end, by the agreement of their predictions with experiments.", "Among all of these structures, and in the framework of metric theories of gravitation, the signature of the metric is in principle arbitrary.", "It seems that on the scales that have been probed so far there is the need for only one time dimension and three spatial dimensions.", "It is also now universally accepted that the relativistic structure is a central ingredient of the construction of any realistic field theory, in particular as the cleanest way to implement the notion of causality.", "Spacetime enjoys a locally Minkowski structure and, when gravity is included, the equivalence principle implies (this is not a theoretical requirement, but an experimental fact, required at a given accuracy) that all the fields are universally coupled to the same Lorentzian metric.", "Thus, we usually take for granted that spacetime is 4-dimensional manifold endowed with a metric of mixed signature, e.g.", "$(-,+,+,+)$ .", "While the existence of two time directions may lead to confusion [1], [2], several models for the birth of the universe [3], [4], [5], [6] are based on a change of signature via an instanton in which a Riemannian and a Lorentzian manifold are joined across a hypersurface.", "While there is no time in the Euclidean region, with signature $(+,+,+,+)$ , it flips to $(-,+,+,+)$ across this hypersurface, which may be thought of as the origin of time from the Lorentzian point of view.", "Eddington even suggested [7] that it can flip across some surface to $(-,-,+,+)$ and signature flips also arise in brane or loop quantum cosmology [8], [9], [10].", "It is legitimate to investigate whether the signature of the metric is only a convenient way to implement causality or whether it is just a property of an effective description of a microscopic theory in which there is no such notion.", "In Ref.", "[11], , two of us have proposed that at the microscopic level the metric is Riemannian and that the Lorentzian structure, usually thought of as fundamental, is in fact an effective property that emerges in some regions of a 4-dimensional space with a positive definite metric.", "There has been some related work in the past — for instance, the work by Barbero [13] (with more than second-order derivatives in the equations of motion, however), or in Einstein-Aether theory [14], (although without an order parameter connecting the Euclidean and Lorentzian theories) and scalar gravity [16].", "We argued that a decent classical field theory for scalars, vectors, and spinors in flat spacetime can be constructed, and that gravity can be included under the form of a covariant Galileon theory instead of general relativity.", "This mechanism of emergent Lorentz signature may also serve as a new way to circumvent the issue of non-unitarity in some higher-derivative quantum gravity theories [17], [18].", "Among the gaps emphasized in this work, we have pointed out that (1) the construction is restricted to classical field theory and the spinor sector suffers from a severe fine-tuning to ensure CPT invariance (see e.g. Ref.", "[19] and references therein for recent constraints on CPT violation), and that (2) it requires the construction of Majorana and Weyl spinors in order to formulate the Standard Model (SM) and its extensions.", "It is well known that Majorana fermions are technically impossible to construct in a 4d Euclidean theory, but several authors have found alternative constructions [20], [21], [22], [23], [24].", "However, these techniques are often aimed at a Wick rotation to or from a Lorentzian theory and can involve doubling the fermion degrees of freedom or other aspects which are ill suited to our application.", "With the aim of developing a theory which flows to the usual actions in Minkowski space (which may look very different in Euclidean space) and the available couplings to the clock field, we arrive at a new formulation for Weyl and Majorana spinors.", "As our goals and setting are different than in previous studies, we do not need to use the techniques employed there, such as fermion doubling or the ad hoc construction of different spinors.", "The Weyl spinors and coupling to the clock field allow us to directly construct an emergent version of the SM, with its chiral and metric structure inherited from an originally Euclidean theory.", "This article is organized as follows.", "In Section we briefly review the construction given in Ref.", "[11], .", "Following that, in Section we extend the fermion sector to include Weyl and Majorana fermions, which is quite distinct from the usual considerations in Euclidean space.", "For fermions, an alternative “derivation” of several of the choices in this construction are detailed in the Appendix.", "In Section we then show how to construct the Standard Model in this framework of an emergent Lorentzian metric.", "There are additional operators which can arise in this theory and, in Section , we categorize and analyze the constraints on such operators in the QED sector of the SM.", "Finally, we gather further comments, conclusions, and future directions in Section ." ], [ "Emergent Lorentz Signature", "In this section we briefly lay out our conventions and review the construction given in Ref.", "[11], for a theory with an effective Lorentz signature emerging from a locally Euclidean metric.", "The Minkowski metric, $\\eta _{\\mu \\nu }$ , is mostly positive with signature $(-, +, +, +)$ , while the Euclidean metric has positive signature and is denoted $\\delta _{\\mu \\nu }$ ." ], [ "Basics of the mechanism", "From the point of view of the Euclidean theory, at the fundamental level, there is no concept of time (one cannot single out a privileged direction) until the clock field, $\\phi $ , picks out a direction through its derivative having a nonzero vacuum expectation value (vev).", "We will always work in some patch $\\mathcal {M}_0$ where this vev can be considered a constant, $\\partial _\\mu \\phi = M^2n_\\mu ,$ with $M$ a mass scale for units and $n_\\mu $ a constant unit vector which now defines a particular direction, related to the notion of time (the direction which will change signature in the effective metric).", "Thus, we have $\\mathrm {d}t = n_\\mu \\mathrm {d}x^\\mu $ and choose $t \\equiv \\frac{\\phi }{M^2}.$ The other three coordinates (with positive signature) are the coordinates of a hypersurface normal to $n_\\mu $ .", "We can now write down actions in the Euclidean theory, which will flow to a Minkowski theory (in the sense that the fields propagate in an effective Minkowski metric) when restricted to $\\mathcal {M}_0$ after the gradient of the clock field has a vev.", "Here we just summarize the results obtained in Ref.", "[11], , which has further details.", "For a scalar field $\\chi $ with potential $V(\\chi )$ , we consider a Euclidean action of the form S= d4x[ -12- V() .", ".", "+ 1M4( )2 ].", "In $\\mathcal {M}_0$ the last term becomes $M^4(\\partial _t\\chi )^2$ so that the above action leads to the usual action for a scalar field in Minkowski space, $S_\\chi = \\int \\mathrm {d}t\\mathrm {d}^3x \\left[ -\\frac{1}{2}\\eta ^{\\mu \\nu }\\partial _\\mu \\chi \\partial _\\nu \\chi - V\\left(\\chi \\right) \\right].$ The case of a vector field, $A_\\mu $ , with field strength tensor $F_{\\mu \\nu }^E$ (the $E$ denotes that indices are raised/lowered with the Euclidean metric) is also straightforward.", "The action $S_A = \\frac{1}{4}\\int \\mathrm {d}^4x\\left[ -F_{\\mu \\nu }^EF^{\\mu \\nu }_E + \\frac{4}{M^4}F^{\\mu \\rho }_EF^\\nu _{E\\rho }\\partial _\\mu \\phi \\partial _\\nu \\phi \\right],$ with the second term equaling $4\\delta ^{ij}F_{0i}F_{0j}$ in $\\mathcal {M}_0$ , becomes the standard Maxwell action for a vector field in Minkowski spacetime, $S_A = -\\frac{1}{4}\\int \\mathrm {d}t\\mathrm {d}^3x\\eta ^{\\mu \\alpha }\\eta ^{\\nu \\beta }F_{\\alpha \\beta }F_{\\mu \\nu }.$" ], [ "Dirac Fermions", "We will now consider Dirac fermions, which will require a bit more detail and care, as we have to be careful with the Clifford algebra and the gamma matrices to build a proper action.", "This will be extended to Weyl and Majorana fermions in the following section, while a more (Clifford) basis agnostic derivation of these conventions can be found in the Appendix.", "In general, the gamma matrices $\\gamma ^\\mu $ satisfyThe overall sign can be changed by a factor of $i$ in the gamma matrices, changing the Hermiticity of the matrices.", "$\\left\\lbrace \\gamma ^\\mu , \\gamma ^\\nu \\right\\rbrace = -2g^{\\mu \\nu },$ for a metric $g^{\\mu \\nu }$ .", "These matrices generate the group ($SO(4)$ or $SO(3,1)$ in our case) generators $S^{\\mu \\nu } \\equiv \\frac{i}{4}[\\gamma ^\\mu , \\gamma ^\\nu ].$ In Minkowski space we will use the common Weyl or chiral representation with the Pauli matrices $\\sigma ^\\mu \\equiv (1,\\sigma ^i), \\bar{\\sigma }^\\mu \\equiv (1, -\\sigma ^i)$ , and the gamma matrices $\\gamma ^\\mu _M \\equiv \\left(\\begin{matrix}0 & \\sigma ^\\mu \\\\\\bar{\\sigma }^\\mu & 0\\end{matrix}\\right).$ We defineNote that this definition includes a minus sign.", "$\\gamma ^5_M$ as $\\gamma ^5_M \\equiv -i\\gamma ^0_M \\gamma ^1_M \\gamma ^2_M \\gamma ^3_M = \\mathrm {diag} (1,1,-1,-1),$ which is Hermitian, squares to 1, and anticommutes with all $\\gamma ^\\mu _M$ .", "A 4-component Dirac spinor, $\\psi _M$ , transforms as $\\psi _M \\rightarrow \\Lambda _{M,\\frac{1}{2}}\\psi _M, \\quad \\Lambda _{M,\\frac{1}{2}} = \\exp \\left[-\\frac{i}{2}\\omega _{\\mu \\nu }S^{\\mu \\nu }_M\\right]$ with $\\omega $ an antisymmetric tensor and $\\Lambda _{M,\\frac{1}{2}}$ not unitary in general.", "In order to form Lorentz invariants for an action, we define the usual barred spinor, $\\bar{\\psi }_M \\equiv \\psi _M^\\dagger \\gamma ^0_M, \\quad \\bar{\\psi }_M \\rightarrow \\bar{\\psi }_M\\Lambda ^{-1}_{M,\\frac{1}{2}}.$ The standard action for the Dirac field in Minkowski space is given by $S^M_\\psi = \\int \\mathrm {d}^4x~\\bar{\\psi }_M\\left( \\frac{i}{2}\\gamma ^\\mu _M\\overleftrightarrow{\\partial }_\\mu - m \\right)\\psi _M.$ In Euclidean space the gamma matrices are chosen as $\\gamma ^0_E \\equiv i\\gamma ^5_M, \\quad \\gamma ^i_E \\equiv \\gamma ^i_M,$ and $\\gamma ^5_E$ satisfies the same properties, now defined as $\\gamma ^5_E \\equiv \\gamma ^0_E \\gamma ^1_E \\gamma ^2_E \\gamma ^3_E = \\gamma ^0_M.$ The generators of $SO(4)$ , $S^{\\mu \\nu }_E$ , are now Hermitian and so $\\Lambda _{E,\\frac{1}{2}}$ is a unitary transformation of the 4-component spinor $\\psi _E$ , $\\psi _E \\rightarrow \\Lambda _{E,\\frac{1}{2}}\\psi _E, \\quad \\Lambda _{E,\\frac{1}{2}} = \\exp \\left[-\\frac{i}{2}\\omega _{\\mu \\nu }S^{\\mu \\nu }_E\\right].$ Both $\\bar{\\psi }_E (=\\psi ^\\dagger _E\\gamma ^0_M = \\psi ^\\dagger \\gamma ^5_E)$ and $\\psi ^\\dagger $ transform the same way, $\\bar{\\psi }_E \\rightarrow \\bar{\\psi }_E\\Lambda ^{-1}_{E,\\frac{1}{2}}, \\quad \\psi ^\\dagger _E \\rightarrow \\psi ^\\dagger _E\\Lambda ^{-1}_{E,\\frac{1}{2}},$ and can form $SO(4)$ invariants with $\\psi _E$ .", "We will favor the bar notation to make the connection to the Lorentzian theory explicit.", "The following Euclidean action, S= d4x{E( i2E - m )E. .", "+ 12M2[ (i5E) -(iE)]}, becomes the Minkowski Dirac action, Eq.", "(REF ), after the clock field picks out a direction in $\\mathcal {M}_0$ ." ], [ "Weyl and Majorana Fermions", "We now extend the above procedure for Weyl and Majorana fermions.", "By a Weyl fermion, we mean a 4-component spinor that is an eigenstate of $\\gamma ^5$ , $\\gamma ^5_{E,M}\\psi _\\pm ^{E,M} = \\pm \\psi _\\pm ^{E,M},$ and we recall that in the representation used above, $\\gamma ^5_E \\ne \\gamma ^5_M$ .", "It is important to note that the $\\gamma ^\\mu _E$ representation we have used is not the same as the Weyl or chiral representation: it does not make manifest the algebra isomorphismIt is important to note that unlike in the Lorentzian case, the representations of these $SU(2)$ s are not related by complex conjugation.", "In other words, a 2-component spinor and its complex conjugate transform under the same $SU(2)$ and can make an $SO(4)$ invariant.", "For a review of 2-component spinors, see Ref.", "[25] and references therein, as well as Ref.", "[26] for Euclidean space.", "$SO(4) = SU(2)_-\\times SU(2)_+$ .", "In other words, a general $SO(4)$ transformation of a 4-component spinor, $\\psi ^E$ , in this description does not separate into two 2-component spinors (the top/bottom of the 4-component spinor) transforming in separate $SU(2)$ s. This is also why we have suppressed all spinor indices, as there is not the usual separation into dotted and undotted indices labeling the different $SU(2)$ s. However, the eigenstates of $\\gamma ^5_E$ take the following form, $\\psi ^E_\\pm = \\left(\\begin{matrix}\\xi _\\pm \\\\\\pm \\xi _\\pm \\end{matrix}\\right),$ with $\\xi _\\pm $ transforming as a 2-component spinor under $SU(2)_\\pm $ .", "We can also form the usual projection matrices with $(1\\pm \\gamma ^5_E)/2$ .", "For $\\psi ^E_\\pm $ then, we can make a direct connection with 2-component spinors in this formalism.", "It should be noted, however, that it is best to work in one form or the other, as the decomposition between 4- and 2-component spinors is completely different in our Euclidean and Lorentzian theories.See Ref.", "[25] and references therein for details in translating between 2- and 4-component spinors, in 4d Minkowski in particular.", "In the Lorentzian theory, the Weyl spinors are of the form $\\psi ^M_{L(-)} = \\left(\\begin{matrix}\\xi _-\\\\0\\end{matrix}\\right), \\qquad \\psi ^M_{R(+)} = \\left(\\begin{matrix}0\\\\\\xi _+\\end{matrix}\\right).$ To construct an action in the Euclidean theory which will become the appropriate action in the Lorentzian theory we might try using the terms $i\\overline{\\psi ^E_\\pm }\\gamma ^\\mu _E\\partial _\\mu \\psi ^E_\\pm ~, \\qquad \\delta ^{\\mu \\nu }\\left(i\\overline{\\psi ^E_\\pm }\\gamma ^\\rho _E\\partial _\\mu \\psi ^E_\\pm \\right)\\partial _\\rho \\phi \\partial _\\nu \\phi ,$ but unfortunately they vanish identically.", "Instead, we can construct an appropriate action as S= 14M2dtd3x[ E5E( i - EE )E. .", "+  h.c.  E].", "After the gradient of the clock field has a vev on $\\mathcal {M}_0$ , this becomes the standard Lorentzian action for 2-component Weyl spinors, $S_\\pm = \\int \\mathrm {d}t\\mathrm {d}^3x{\\left\\lbrace \\begin{array}{ll}i\\xi _{-,L}^\\dag \\bar{\\sigma }^\\mu \\partial _\\mu \\xi _{-,L}\\\\i\\xi _{+,R}^\\dag \\sigma ^\\mu \\partial _\\mu \\xi _{+,R}\\end{array}\\right.", "},$ where the subscript indicates the $SU(2)$ representation from the Euclidean ($_\\mp $ ) to Lorentzian ($_{L,R}$ ).", "To connect to the 4-component spinors, we recognize that, once the gradient of the clock field has a vev, we want eigenstates of $\\gamma ^5_M$ , $\\gamma ^5_E\\psi ^E_\\pm = \\pm \\psi ^E_\\pm \\quad \\rightarrow \\quad \\gamma ^5_M\\psi ^M_\\pm = \\pm \\psi ^M_\\pm .$ This naturally comes out of the action of Eq. ().", "By inserting $(-i*i)$ in the second term and using the properties of the gamma matrices the action becomes $S = \\int \\mathrm {d}t\\mathrm {d}^3x~i\\overline{\\psi }^M_\\pm \\gamma ^\\mu \\partial _\\mu \\psi ^M_\\pm .$ Finally, we also want to incorporate Majorana spinors (representing fermions which are their own antiparticles).", "As is well known, we cannot directly have a Majorana spinor in 4d Euclidean space: $\\psi ^\\mathcal {C}_E = \\psi _E$ is only satisfied for the zero spinor, where $\\psi ^\\mathcal {C}_E \\equiv C_E\\overline{\\psi }^T$ is the Euclidean charge conjugate spinor and $C_E$ will be defined below (see Eq.", "(REF )).", "However, using the above formulation of Weyl spinors, we can write down a Lagrangian for a single Weyl fermion with a mass term.", "This captures the physical properties of a Majorana spinor, and in the Lorentzian theory this will correspond to the usual Majorana spinor (a self-conjugate 4-spinor).", "From our form of Weyl spinors, Eq.", "(REF ), we write a Majorana mass term (the right-hand side is exactly the Lorentzian 2-component form as we have rotated $\\psi $ to change the signs) as $\\frac{1}{4} m(\\psi ^E_\\pm )^TC_E\\psi ^E_\\pm ~+~\\mathrm {h.c.} = \\frac{1}{2}m\\xi _\\pm \\xi _\\pm ~+~\\mathrm {h.c.}$ with $T$ denoting the transpose, and where we need the matrix $C_E$ to have the term be $SO(4)$ invariant (i.e.", "to provide the (suppressed) invariant to combine the $\\xi _\\pm $ spinors as in the usual 2-component formalism).", "In other words, we require $\\Lambda ^T_{E,\\frac{1}{2}}C_E = C_E \\Lambda ^{-1}_{E,\\frac{1}{2}},$ which is satisfied by the matrixWe are not using explicit spinor indices, so we consider this as a numerical identification.", "$C_E = \\gamma ^1_E\\gamma ^3_E,$ with properties $C_E^T = C_E^\\dag = C_E^{-1} = -C_E.$ This is similar to the numerical structure of a charge conjugation matrix,We have not shown how it operates directly on (anti)particles.", "Also, the matrix satisfies $C_E^{-1}\\gamma ^\\mu _EC_E= (\\gamma ^\\mu _E)^T$ rather than giving $-(\\gamma ^\\mu _E)^T$ as in the usual Minkowski space definition.", "but again, we cannot enforce that a spinor is self-conjugate and nontrivial in the 4d Euclidean theory (as one can see directly given $C_E$ above).", "When we move to the Lorentzian theory, this matrix becomes $\\widetilde{C}_M = \\gamma ^1_M\\gamma ^3_M,$ which is almost the Lorentzian charge conjugation matrix.", "If we use a factorThe sign is automatic from the left-handed field, or through a field rotation for the right-handed field (the sign of the Majorana mass term can be changed freely).", "of $-\\gamma ^5_M$ in the mass term from the property of the (now Lorentzian) Weyl spinor, we can now identify (again, as a numerical identity through direct computation of the necessary properties) this with the Lorentzian charge conjugation matrix $C_M$ , $C_M = -\\gamma ^5_M\\gamma ^1_M\\gamma ^3_M.$ The structure of this mass term, $\\frac{1}{2}m\\psi ^T_{\\pm ,M}C_M\\psi _{\\pm ,M},$ is exactly a Majorana mass term with the identification of the Majorana condition, $\\psi ^C_M \\equiv C_M\\overline{\\psi }^T_M = \\psi _M ~~\\mathrm {or}~~ \\overline{\\psi }_M = \\psi ^T_MC_M.$ Note that the degrees of freedom match, as we have moved from a Weyl spinor in Euclidean space to a Majorana spinor (or equivalently a single Weyl spinor with a (Majorana) mass term) in Lorentzian space, each with two complex degrees of freedom off shell.", "In Minkowski space the Majorana spinors take the following form in terms of 2-component spinors (either a single left- or right-handed spinor), $\\psi _{M(-)} = \\left(\\begin{matrix}\\xi _-\\\\\\xi _-^\\dagger \\end{matrix}\\right), \\qquad \\psi _{M(+)} = \\left(\\begin{matrix}\\xi _+^\\dagger \\\\\\xi _+\\end{matrix}\\right),$ again with the caveat that one should be careful in mixing the 2- and 4-component languages between the Euclidean and Lorentzian theories." ], [ "The Standard Model", "We have all the ingredients we need to construct the SM in flat spacetime from an originally $SO(4)$ Euclidean theory.", "The SM contains the gauge field strength terms for each group, kinetic terms for each matter field, and Yukawa terms coupling the Higgs field to the matter fields to give mass terms from the Higgs mechanism.", "A key structure is that the weak gauge group, $SU(2)_L$ , acts only on left-handed fields.", "It is this chiral structure of the weak force which requires the Yukawa interactions with the Higgs field (or some other mechanism entirely) in order for the fermions to have mass.", "We have already seen how to construct kinetic terms (and gauge field strengths) which flow from the Euclidean theory to the proper terms with a Lorentzian signature, for all of the fields we need.", "Let us now consider the necessary Yukawa interaction terms between the Higgs and fermion matter fields.", "These terms do not change form as the background metric changes, and we can use the usual terms in the $SO(4)$ theory.", "As we must treat left- and right-handed fields differently under the weak force, we rely on the Weyl spinors (or projections) we constructed earlier.", "A common simplification is to write the SM Lagrangian purely in terms of left-handed fields.", "In this form, the right-handed fields which do not couple to the weak force are written as antifermions of a new species of left-handed fermions.", "For instance, for the up and down quarks, the left-handed $SU(2)_L$ doublet is $Q$ , and the right-handed $SU(2)_L$ singlets are $\\bar{u}_R$ and $\\bar{d}_R$ with the bar purely part of the name.", "We then use their left-handed antiparticles, $\\bar{u}, \\bar{d}$ , in writing a Lagrangian.", "Yukawa terms in the Euclidean theory then look just like in the SM.", "For example, for the first generation of quarks (with $H$ the Higgs scalar $SU(2)_L$ doublet), $Q_EH_E\\bar{d}_E + Q_E\\epsilon H^\\dag \\bar{u}_E + (\\mathrm {h.c.}),$ where $Q_E, H_E, \\bar{u}_E,$ and $\\bar{d}_E$ are all Euclidean Weyl spinors with $\\gamma ^5_E$ eigenvalue $-1$ and all indices are suppressed (the $\\epsilon $ tensor combines $Q_E$ and $H_E^\\dag $ antisymmetrically in $SU(2)_L$ indices).", "Once we go to the Lorentzian theory, the Euclidean Weyl spinors become the left-handed projections of the SM fields, and we have exactly the SM.", "The leptons and other generations all follow in the same way.", "One thing to note is how the right-handed terms are generated in terms of these left-handed fields.", "In the Lorentzian theory, the conjugate of a left-handed field is right-handed, and vice versa.", "We do not have this group structure in the Euclidean theory.", "Thus, when we write the Hermitian conjugate terms in the $SO(4)$ theory, they are still fields with $\\gamma ^5_E$ eigenvalue $-1$ .", "Once we are in the Lorentzian theory, however, Weyl spinors are not self-conjugate, and the Hermitian conjugate terms are right-handed.", "After the Higgs mechanism the fermions are all (except for the neutrino) paired up into Dirac mass terms, which mix the left- and right-handed components." ], [ "Constraints in QED", "As was remarked in Ref.", "[11], , there is a tuning necessary in the couplings to the clock field to reach the standard Lorentzian theory.", "In this section we will restrict ourselves to the QED sector and examine the constraints on these terms by using the work summarized in Refs.", "[27], [28] (see references therein for details on the parameterization of operators and the relevant experimental results).", "We will work in flat (Minkowski) space with a single fermion flavor (the electron/positron); some constraints may change in more general settings.", "We will make a connection from our model to the parameterization of Lorentz violating operators in the Standard Model Extension (SME) used in Refs.", "[27], [28].", "The SME encapsulates the minimal set of dimension 3 and 4 CPT and Lorentz violating operators, and constructs observables which can be constrained by experiment.", "The general Minkowski space QED Lagrangian (with electromagnetic tensor $F_{\\mu \\nu }$ and fermion $\\psi $ ) in the SME is $\\mathcal {L} = \\frac{i}{2}\\overline{\\psi }\\Gamma _\\nu \\stackrel{\\leftrightarrow }{\\partial ^\\nu }\\psi - \\overline{\\psi }M\\psi - \\frac{1}{4}K_{\\mu \\nu } F^{\\mu \\nu },$ with + c+ d5+ e   + if5 + 12g, M m + a+ b5+ 12H, K F -2(kAF)A+ (kF)F, where $\\frac{1}{2}\\Sigma ^{\\mu \\nu } \\equiv \\frac{i}{4}[\\gamma ^\\mu , \\psi ^\\nu ]$ and all $\\gamma ^\\mu $ are in Minkowski space.", "The observables are combinations of the free parameters $a_\\mu , b_\\mu , c_{\\mu \\nu }, d_{\\mu \\nu }, e_\\nu , f_\\nu ,g_{\\lambda \\mu \\nu }, H_{\\mu \\nu }, (k_{AF})^\\kappa ,$ and $(k_F)_{\\kappa \\lambda \\mu \\nu }$ (see Refs.", "[27], [28] for the precise definitions and counting of independent parameters and observable combinations).", "Let us start with the photon.", "We can parameterize any deviation from the interaction term with the clock field which leads to the Lorentzian theory as $\\frac{4}{M^4}(1 + \\epsilon _A)F^{\\mu \\rho }_E F^\\nu _{E\\rho }\\partial _\\mu \\phi \\partial _\\nu \\phi ,$ with $\\epsilon _A$ the deviation from Eq.", "(REF ).", "In the Lorentzian theory then, we end up with the additional term $\\epsilon _A \\delta _{ij} F^{0i} F^{0j}.$ This is a CPT even operator, which violates Lorentz invariance, corresponding to the SME parameter $(k_F)_{\\kappa \\lambda \\mu \\nu }$ in Eq.", "(): it is constrained to have $|\\epsilon _A| <\\mathcal {O}(10^{-32})$ (cf.", "the observables $\\tilde{\\kappa }$ , in particular the component $\\tilde{\\kappa }_{e+}^{ZZ}$ in Ref. [27]).", "In the matter sector, we parametrize a deviation from Eq.", "(REF ), which gives the proper Minkowski Lagrangian in $\\mathcal {M}_0$ , with the parameters $\\epsilon _{\\psi _{1,2}}$ as 12M2 [ (1 + 1) (iE5E) .", ".", "- (1 + 2)(iE)].", "We then have the following Lorentz violating operators in the theory in Minkowski space, the first of which is CPT even, the second CPT odd: $\\frac{i}{2}\\epsilon _{\\psi _1}\\left(\\overline{\\psi }\\gamma ^0_M\\stackrel{\\leftrightarrow }{\\partial _0}\\psi \\right)+ \\frac{1}{2}\\epsilon _{\\psi _2}\\left(\\overline{\\psi }\\gamma ^5_M\\stackrel{\\leftrightarrow }{\\partial _0}\\psi \\right).$ However, through a field redefinition this second operator ($f_0$ in the SME above) can actually be removed at leading order (in $\\epsilon _{\\psi _2}$ ) and absorbed into $\\epsilon _{\\psi _1}$ at second order (see the discussion in Refs.", "[27], [28] and references therein).", "Thus we do not have CPT violation, regardless of the precise tuning, contrary to what was stated originally in [11], .", "The CPT even operator must have coefficient $|\\epsilon _{\\psi _1}| < \\mathcal {O}(10^{-15})$ (corresponding to $\\tilde{c}_{TT}/m_e$ in Ref.", "[27]) and this gives a constraint, through field redefinition, of $|\\epsilon _{\\psi _2}| < \\mathcal {O}(10^{-7})$ .", "We have seen that there is a precise tuning in the couplings of the SM fields to the clock field needed to avoid Lorentz violation constraints.", "Besides the tuning in these coefficients, there are other possible terms which can be dangerous, as noted in Ref.", "[11], .", "Of the 10 terms which are scalars under $SO(4)$ , Hermitian, and include at most one derivative acting on spinors, we have the usual mass and kinetic terms, and the 2 terms we have already included.", "There are 4 additional terms with couplings to the clock field, (E),    (i5EE), (i),    (i5EE).", "The first term corresponds, in the Lorentzian theory, to a $\\gamma ^5$ mass term, which can be removed through a chiral transformation.", "The third term (corresponding to $e_\\mu $ in the SME) is CPT and Lorentz violating, but can also be removed by transformations and field redefinitions (it can be absorbed into $a_\\mu $ and is not observable with a single flavor in flat space; see the summary in Ref.", "[28] and references therein).", "The second term ($b_T$ in the SME) is CPT odd and Lorentz violating, constrained to be less than $\\mathcal {O}(10^{-27}~\\mathrm {GeV})$ (see the combinations $\\tilde{b}_T$ and $\\tilde{g}_T$ in Ref. [27]).", "This is problematic as the generated mass scale is presumably $\\sim M$ , and there does not appear to be a simple way to forbid such a term.", "Finally, the fourth term, which is CPT even and Lorentz violating, is constrained to be less than $\\mathcal {O}(10^{-24})$ (constrained via the tracelessness of $d_{\\mu \\nu }$ ; see the observable $\\tilde{d}_+$ in Ref. [27]).", "Again, there is not an obvious way to forbid such a term, and its pure number coefficient is a free parameter.", "Finally, we also have two terms which do not involve interactions with the clock field.", "One is the standard $\\gamma ^5_E$ mass term, $\\overline{\\psi }\\gamma ^5_E\\psi $ , which we will transform away in the Euclidean theory (or it corresponds to the unobservable parameter $a_0$ in the SME).", "The second term is $\\overline{\\psi }\\gamma ^5_E\\gamma ^\\mu _E\\stackrel{\\leftrightarrow }{\\partial _\\mu }\\psi $ , which we can write in the Lorentzian theory as (i0M5M0 + 0MiMi) = -i(5M0M0 + 0ii), where we used that $\\gamma ^0_M\\gamma ^i_M = \\frac{1}{2}[\\gamma ^0_M,\\gamma ^i_M] +\\frac{1}{2}\\lbrace \\gamma ^0_M,\\gamma ^i_M\\rbrace = \\frac{1}{2}[\\gamma ^0_M,\\gamma ^i_M]$ .", "The first term in Eq.", "() is the same as the last term discussed in the previous paragraph, and thus has the same constraint.", "In the SME, the second term is a component of the trace part of the parameter $g_{\\mu \\nu \\lambda }$ (the coefficient of a CPT odd and Lorentz violating operator), $g^{(T)}_\\mu \\equiv g_{\\mu \\nu }^{~~~\\nu }$ (note that $g_{000}$ does not contribute since $\\Sigma ^{00} = 0$ ).", "This is not an observable component of $g$ as it can be removed through a field redefinition (see Ref.", "[28] and references therein)." ], [ "Discussion, Conclusion, and Outlook", "This article follows the idea that the apparent Lorentzian dynamics of usual field theories is an emergent property and that the underlying field theory is in fact strictly Riemannian.", "This requires the introduction of the clock field, a scalar field playing the role of the physical time.", "The microscopic theory is Euclidean, and time evolution is just an effective and emergent property, which holds on some energy scales, and in some regions of the Euclidean space.", "Through interactions with the clock field the effective theory flows to the standard Lorentzian picture.", "In Ref.", "[11], , we were able to perform a construction in flat spacetime for scalar, vector, and Dirac spinors restricted to classical fields.", "In order for all the fields to propagate in the same emergent Lorentzian metric, the couplings to the clock field needed to be adjusted with care.", "This work was a proof of concept in constructing a model with the Lorentzian metric only emerging at energies below the vev of the gradient of the clock field, with many open and interesting questions.", "In this work we have addressed several of these questions.", "The present analysis has shown that it is possible to construct a Euclidean theory with fermions that reduce, once the gradient of the clock field has a vev on ${\\cal M}_0$ , to Lorentzian Weyl and Majorana fermions.", "This completes the basic fields needed in the Standard Model and common extensions.", "The clock field allows us to avoid the typical difficulties in constructing Euclidean theories of these types of fermions.", "We then showed that it is possible to construct a Euclidean theory leading to an emergent version of the Standard Model by adding the Standard Model structure to the dynamics necessary for the emergence of a Lorentzian metric.", "To finish, we have analyzed with care the fine-tuning required to ensure CPT and Lorentz invariance.", "One crucial point is that the terms necessary in our model do not induce CPT violation.", "Bounds on the deviations from the adjusted couplings to the clock field, as well as other possible interaction terms in this framework, can be obtained from experimental QED constraints.", "Forbidding additional operators and ensuring the value of the necessary coupling constants is an open question.", "There are still many interesting future directions to pursue in this framework for emergent Lorentz symmetry.", "One would like to move beyond the classical level and quantize the theory, as well as understand the mechanism which leads to the vev of the clock field.", "The possible violation of CPT and Lorentz symmetry also needs to be investigated further.", "Even with these and other open questions, we now have a basic model which can reproduce the Standard Model and its Lorentzian background with time evolution from a purely Riemannian theory with no concept of time.", "This work was supported by the World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan.", "This work was made in the ILP LABEX (under reference ANR-10-LABX-63) and was supported by French state funds managed by the ANR within the Investissements d'Avenir programme under reference ANR-11-IDEX-0004-02 and by the ANR VACOUL, ANR-10-BLAN-0510.", "One of us (SM) acknowledges the support by Grant-in-Aid for Scientific Research 24540256 and 21111006.", "*" ], [ "Gamma Matrices and Fermions", "In this appendix we will try to motivate some of the choices made in our Euclidean formulation of fermions.", "Our procedure will be to take the action proposed in the Euclidean theory as an ansatz, and require that we end with a proper Lorentzian theory.", "This will then define the relationship between the representations of the gamma matrices (which will not be chosen a priori) and identifications between quantities in the two theories.", "Let us start with a massless Dirac fermion in the $SO(4)$ theory, coupled to the clock field as in the action of Eq.", "(REF ), S= d4x{E( i2E - m )E .", ".", "+ 12M2[ (iE5E) -(iE)]}, but without assuming the form of $\\overline{\\psi }_E$ or $\\gamma ^\\mu _E$ .", "Although we can form an $SO(4)$ invariant with $\\psi _E^\\dag \\psi _E$ , we wish to mirror the usual Lorentzian construction, so we have used $\\overline{\\psi }$ .", "In order for this to transform as $\\psi _E^\\dag $ (i.e.", "in the opposite way of $\\psi _E$ ), any matrix we attach to $\\psi _E^\\dag $ to form $\\overline{\\psi }_E$ must commute with the $SO(4)$ generators.", "Thus we have $\\overline{\\psi }_E \\equiv \\psi _E^\\dag \\gamma ^5_E.$ After the clock field's derivative has a vev $M^2$ , chosen to define the $t$ -direction, the action becomes $\\mathcal {S}_\\psi \\rightarrow \\int \\mathrm {d}t\\mathrm {d}^3x~\\left(\\overline{\\psi }i\\gamma ^i_E\\partial _i\\psi \\right) + i\\left(\\overline{\\psi }\\gamma ^5_E\\partial _0\\psi \\right).$ Since the clock field has now picked out a direction, morphing $SO(4)$ to $SO(3,1)$ , we expect that we should now have a free fermion propagating in Minkowski space.", "We recover the usual action, $\\mathcal {S}_M = \\int \\mathrm {d}t\\mathrm {d}^3x~i\\overline{\\psi }\\gamma ^\\mu _M\\partial _\\mu \\psi = i\\overline{\\psi }\\left(\\gamma ^0_M\\partial _0 + \\gamma ^i_M\\partial _i\\right)\\psi $ by identifyingNote that usually $\\beta $ and $\\gamma ^0$ are used interchangeably because they are numerically the same.", "However, at least in the chiral representation, the spin structure is different.", "5E      M M, iE iM,       5E 0M.", "From these identifications and the definition of $\\gamma ^5_E$ we know that $\\lbrace \\gamma ^5_E, \\gamma ^i_M\\rbrace = 0$ and $[\\gamma ^5_E, \\gamma ^0_M] =0$ .", "Therefore $\\gamma ^5_E\\gamma ^\\mu _M(\\gamma ^5_E)^{-1} =(\\gamma ^\\mu _M)^\\dagger $ , with $\\gamma ^0_M$ Hermitian and $\\gamma ^i_M$ anti-Hermitian (from the definition of the Clifford algebra as $\\lbrace \\gamma ^\\mu _M, \\gamma ^\\nu _M\\rbrace = -2\\eta ^{\\mu \\nu }$ ).", "Combined with $\\gamma ^5_E$ being Hermitian, or by direct computation, we find that it has the right properties (see, e.g., Appendix G of Ref.", "[25]) to be the matrix $\\beta $ : $\\overline{\\psi }_M$ transforms oppositely of $\\psi _M$ such that $\\overline{\\psi }_M\\psi _M$ is a (Hermitian) Lorentz scalar.", "Furthermore, the Clifford algebra for $SO(4)$ tells us that $(\\gamma ^0_E)^2 = -1$ and we chose an anti-Hermitian representation, $(\\gamma ^0_E)^\\dagger = -\\gamma ^0_E$ , such that the $SO(4)$ generators we defined were Hermitian.", "Since $\\gamma ^0_E$ anticommutes with all the $\\gamma ^\\mu _M$ this implies $\\gamma ^0_E = i\\gamma ^5_M.$ All the above requirements are then consistent, coming from the proposed $SO(4)$ action.", "The gamma matrices all match what was given in Sec.", "REF .", "For Weyl spinors (Euclidean spinor eigenstates of $\\gamma ^5_E$ ) we can follow the same procedure, and we find that we reach the Minkowski Weyl action with the same identification of gamma matrices, including that the spinor is now an eigenstate of $\\gamma ^5_M$ ." ] ]
1403.0580
[ [ "Algebraic Properties of Valued Constraint Satisfaction Problem" ], [ "Abstract The paper presents an algebraic framework for optimization problems expressible as Valued Constraint Satisfaction Problems.", "Our results generalize the algebraic framework for the decision version (CSPs) provided by Bulatov et al.", "[SICOMP 2005].", "We introduce the notions of weighted algebras and varieties and use the Galois connection due to Cohen et al.", "[SICOMP 2013] to link VCSP languages to weighted algebras.", "We show that the difficulty of VCSP depends only on the weighted variety generated by the associated weighted algebra.", "Paralleling the results for CSPs we exhibit a reduction to cores and rigid cores which allows us to focus on idempotent weighted varieties.", "Further, we propose an analogue of the Algebraic CSP Dichotomy Conjecture; prove the hardness direction and verify that it agrees with known results for VCSPs on two-element sets [Cohen et al.", "2006], finite-valued VCSPs [Thapper and Zivny 2013] and conservative VCSPs [Kolmogorov and Zivny 2013]." ], [ "Introduction", "An instance of the Constraint Satisfaction Problem (CSP) consists of variables (to be evaluated in a domain) and constraints restricting the evaluations.", "The aim is to find an evaluation satisfying all the constraints or satisfying the maximal possible number of constraints or approximating the maximal possible number of satisfied constraints etc.", "depending on the version of the problem.", "Further one can divide constraint satisfaction problems with respect to the size of the domain, the allowed constraints or the shape of the instances.", "A particularly interesting version of CSP was proposed in a seminal paper of Feder and Vardi [12].", "In this version a CSP is defined by a language which consists of a finite number of relations over a finite set.", "An instance of such a CSP is allowed if all the constraint relations are from this set.", "The goal is to determine whether an instance has a solution satisfying all the constraints.", "Each language clearly defines a problem in NP; the whole family of problems is interesting for another reason: it is robust enough to include some well studied computational problems, e.g.", "2-colorability, 3-SAT, solving systems of linear equations over ${\\mathbb {Z}}_p$ , and still is conjectured [12] not to contain problems of intermediate complexity.", "This conjecture is known as the Constraint Satisfaction Dichotomy Conjecture of Feder and Vardi.", "Confirming this conjecture would establish CSPs as one of the largest natural subclasses of NP without problems of intermediate complexity.", "The conjecture always attracted a lot of attention, but the first results, even very interesting ones, were usually very specialized (e.g. [14]).", "A major breakthrough appeared with a series of papers establishing the algebraic approach to CSP [16], [3], [7].", "This deep connection with an independently developed branch of mathematics introduced a new viewpoint and provided tools necessary to tackle wide classes of CSP languages at once.", "At the heart of this approach lies a Galois connection between languages and clones of operations called polymorphisms (which completely determine the complexity of the language).", "Results obtained using this new methods include a full complexity classifications for CSPs on three-element sets [5] and containing all unary relations [4], [6].", "Moreover, the algebraic approach to CSP allowed to propose a boundary between the tractable and NP-complete problems: this conjecture is known as the Algebraic Dichotomy Conjecture.", "Unfortunately, despite many efforts (e.g.", "[5]), both conjectures remain open.", "The Valued Constraint Satisfaction Problem (VCSP) further extends the approach proposed by Feder and Vardi.", "The role of constraints is played by cost functions describing the price of choosing particular values for variables as a part of the solution.", "This generalization allows to construct languages modeling standard optimization problems, for example MAX-CUT.", "Moreover, by allowing $\\infty $ as a cost of a tuple, a VCSP language can additionally model every problem that CSP can model, as well as hybrid problems like MIN-VERTEX-COVER.", "This makes the extended framework even more general (compare the survey [17]).", "A number of classes of VCSPs have been thoroughly investigated.", "The underlying structure suggested capturing the properties of languages of cost functions using an amalgamation of algebraic and numerical techniques [25], [10].", "The first approach which provides a Galois correspondence (mirroring the Galois correspondence for CSPs) was proposed by Cohen et al. [9].", "A weighted clone defined in this paper fully captures the complexity of a VCSP language.", "The present paper builds on that correspondence imitating the line of research for CSPs [7].", "It is organized in the following way: Section 2 contains preliminaries and basic definitions.", "In Section 3 we present a reduction to cores and rigid cores.", "Section 4 introduces a concept of a weighted algebra and a weighted variety, and shows that those notions are well behaved in the context of the Galois connection for VCSP.", "Reductions developed in Section 3 together with definitions from Section 4 allow us to focus on idempotent varieties.", "Section 5 states a conjecture postulating (for idempotent varieties) the division between the tractable and NP-hard cases of VCSP.", "The conjecture is clearly a strengthening of the Algebraic Dichotomy Conjecture [7].", "Section 5 contains additionally the proof of the hardness direction of the conjecture as well as the reasoning showing that the conjecture agrees with complexity classifications for VCSPs on two-element sets [10], with finite-valued cost functions [25], and with conservative cost functions [19].", "Throughout the paper, let $\\overline{\\mathbb {Q}} = \\mathbb {Q}\\cup \\lbrace \\infty \\rbrace $ denote the set of rational numbers with (positive) infinity.", "We assume that $x + \\infty = \\infty $ and $y \\cdot \\infty = \\infty $ for $y \\ge 0$ .", "An $r$ -ary relation on a set $D$ is a subset of $D^r$ , a cost function on $D$ of arity $r$ is a function from $D^r$ to $\\overline{\\mathbb {Q}}$ .", "We denote by $\\Phi _{D}$ the set of all cost functions on $D$ .", "A cost function which takes only finite values is called finite-valued.", "A $\\lbrace 0, \\infty \\rbrace $ -valued cost function is called crisp and can be viewed as a relation.", "An instance of the valued constraint satisfaction problem (VCSP) is a triple $I= ( V, D,$ with $V$ a finite set of variables, $D$ a finite domain and $ a finite multi-set of \\emph {constraints}.", "Each constraint is a pair $ C = (, )$ with $$ a tuple of variables of length $ r$ and $$ a cost function on $ D$ of arity $ r$.$ An assignment for $I$ is a mapping $s \\colon V \\rightarrow D$ .", "The cost of an assignment $s$ is given by $Cost_{I}(s) = \\sum _{(\\sigma ,\\varrho ) \\in \\varrho (s(\\sigma )) (where s is applied component-wise).", "To solve I is to find an assignment with a minimal cost, called an \\emph {optimal} assignment.", "}$ In the Max-Cut problem, one needs to find a partition of the vertices of a given graph into two sets, such that the number of edges with ends in different sets is maximal.", "This problem is NP-hard.", "The Max-Cut problem can be expressed as an instance of VCSP.", "The domain has two elements 0 and 1.", "Variables in the instance are vertices of the graph and for each edge $e$ there is a constraint of a form $(e, \\varrho _{XOR})$ , where $\\varrho _{XOR}$ is a binary cost function defined by $\\varrho _{XOR}(x,y) = {\\left\\lbrace \\begin{array}{ll} 1 &\\mbox{if } x=y, \\\\0 & \\mbox{otherwise.}", "\\end{array}\\right.", "}$ Any assignment of the values 0 and 1 to the variables corresponds to a partition of the graph.", "The cost of an assignment is equal to the number of edges of the graph minus the number of cut edges.", "Any set $\\Gamma \\subseteq \\Phi _{D}$ is called a valued constraint language over $D$ , or simply a language.", "If all cost functions from $\\Gamma $ are $\\lbrace 0, \\infty \\rbrace $ -valued or finite-valued, we call it a crisp or finite-valued language, respectively.", "If $\\Gamma $ is a language, but not necessarily finite-valued or crisp, we sometimes stress this fact by saying that $\\Gamma $ is a general-valued language.", "By $\\operatorname{VCSP}(\\Gamma )$ we denote the class of all VSCP instances in which all cost functions in all constraints belong to $\\Gamma $ .", "$\\operatorname{VCSP}(\\Gamma _{crisp})$ , where $\\Gamma _{crisp}$ is the language consisting of all crisp cost functions on some fixed set $D$ , is equivalent to the classical CSP.", "For an instance $I\\in \\operatorname{VCSP}(\\Gamma )$ we denote by $\\operatorname{Opt}_{\\Gamma }(I)$ the cost of an optimal assignment.", "We say that a language $\\Gamma $ is tractable if, for every finite subset $\\Gamma ^{\\prime } \\subseteq \\Gamma $ , there exists an algorithm solving any instance $I\\in \\operatorname{VCSP}(\\Gamma ^{\\prime })$ in polynomial time, and we say that $\\Gamma $ is NP-hard if $\\operatorname{VCSP}(\\Gamma ^{\\prime })$ is NP-hard for some finite $\\Gamma ^{\\prime } \\subseteq \\Gamma $ .", "Example REF shows that the language $\\lbrace \\varrho _{XOR} \\rbrace $ is NP-hard.", "Weighted Relational Clones.", "We follow the exposition from [9] and define a closure operator on valued constraint languages that preserves tractability.", "A cost function $\\varrho $ is expressible over a valued constraint language $\\Gamma \\subseteq \\Phi _{D}$ if there exists an instance $I_{\\varrho } \\in \\operatorname{VCSP}(\\Gamma )$ and a list $(v_1, \\dots , v_r)$ of variables of $I_{\\varrho }$ , such that $\\varrho (x_{1}, \\dots , x_{r}) = \\min _{\\lbrace s \\colon V \\rightarrow D \\ | \\ s(v_i)=x_i \\rbrace } Cost_{I_{\\varrho }}(s).$ Note that the list of variables $(v_1, \\dots , v_r)$ in the definition above might contain repeated entries.", "Hence, it is possible that there are no assignments $s$ such that $s(v_i)=x_i$ for all $i$ .", "We define the minimum over the empty set to be $\\infty $ .", "A set $\\Gamma \\subseteq \\Phi _{D}$ is a weighted relational clone if it is closed under, expressibility, scaling by non-negative rational constants, and addition of rational constants.", "We define $\\operatorname{wRelClo}(\\Gamma )$ to be the smallest weighted relational clone containing $\\Gamma $ .", "If $\\varrho (x_1, \\dots , x_{r}) = \\varrho _1(y_1, \\dots , y_s) + \\varrho _2(z_1, \\dots , z_t)$ for some fixed choice of arguments $y_1, \\dots , y_s, z_1, \\dots , z_t$ from amongst $x_1, \\dots , x_{r}$ then the cost function $\\varrho $ is said to be obtained by addition from the cost functions $\\varrho _1$ and $\\varrho _2$ .", "It is easy to see that a weighted relational clone is closed under addition, and minimisation over arbitrary arguments.", "The following result shows that we can restrict our attention to languages which are weighted relational clones.", "[Cohen et al.", "[9]] A valued constraint language $\\Gamma $ is tractable if and only if $\\operatorname{wRelClo}(\\Gamma )$ is tractable, and it is NP-hard if and only if $\\operatorname{wRelClo}(\\Gamma )$ is NP-hard.", "Weighted polymorphisms.", "A $k$ -ary operation on $D$ is a function $f \\colon D^k \\rightarrow D$ .", "We denote by $\\mathcal {O}_D$ the set of all finitary operations on $D$ and by $\\mathcal {O}_D^{(k)}$ the set of all $k$ -ary operations on $D$ .", "The $k$ -ary projections, defined for all $i \\in \\lbrace 1, \\dots , k\\rbrace $ , are the operations $\\pi ^{(k)}_i$ such that $\\pi ^{(k)}_i(x_1, \\dots , x_k) = x_i$ .", "Let $f \\in \\mathcal {O}_D^{(k)}$ and $g_1,\\dots , g_k \\in \\mathcal {O}_D^{(l)}$ .", "The $l$ -ary operation $f[g_1,\\dots , g_k]$ defined by $f[g_1,\\dots , g_k](x_1,\\dots , x_l)=f(g_1(x_1,\\dots , x_l),\\dots , g_k(x_1,\\dots , x_l))$ is called the superposition of $f$ and $g_1,\\dots , g_k$ .", "A set $C \\subseteq \\mathcal {O}_{D}$ is a clone of operations (or simply a clone) if it contains all projections on $D$ and is closed under superposition.", "The set of $k$ -ary operations in a clone $C$ is denoted $C^{(k)}$ .", "The smallest possible clone of operations over a fixed set $D$ is the set of all projections on $D$ , which we denote $\\Pi _{D}$ .", "Following [9] we define a $k$ -ary weighting of a clone $C$ to be a function $\\omega \\colon C^{(k)} \\rightarrow \\mathbb {Q}$ such that $\\sum _{f \\in C^{(k)}} \\ \\omega (f) = 0$ , and if $\\omega (f) < 0$ then $f$ is a projection.", "The set of operations to which a weighting $\\omega $ assigns positive weights is called the support of $\\omega $ and denoted $\\operatorname{supp}(\\omega )$ .", "A new weighting of the same clone can be obtained by scaling a weighting by a non-negative rational, adding two weightings of the same arity and by the following operation called superposition.", "Let $\\omega $ be a $k$ -ary weighting of a clone $C$ and let $g_1, \\dots , g_k \\in C^{(l)}$ .", "A superposition of $\\omega $ and $g_1, \\dots , g_k$ is a function $\\omega [g_1, \\dots , g_k] \\colon C^{(l)} \\rightarrow \\mathbb {Q}$ defined by $\\omega [g_1, \\dots , g_k](f^{\\prime }) \\ = \\sum _{\\lbrace f \\in C^{(k)} \\ | \\ f[g_1, \\dots , g_k]=f^{\\prime } \\rbrace } \\omega (f).$ The sum of weights that any superposition $\\omega [g_1, \\dots , g_k]$ assigns to the operations in $C^{(l)}$ is equal to the sum of weights in $\\omega $ , which is 0.", "However, it may happen that a superposition assigns a negative value to an operation that is not a projection.", "A superposition is said to be proper if the result is a valid weighting.", "A non-empty set of weightings over a fixed clone $C$ is called a weighted clone if it is closed under non-negative scaling, addition of weightings of equal arity and proper superposition with operations from $C$ .", "For any clone of operations $C$ , the set of all weightings over $C$ and the set of all zero-valued weightings of $C$ are weighted clones.", "We say that an $r$ -ary relation $R$ on $D$ is compatible with an operation $f \\colon D^k \\rightarrow D$ if, for any list of $r$ -tuples ${\\bf x_1, \\dots , x_k} \\in R$ we have $f({\\bf {x_1, \\dots , x_k}}) \\in R$ (where $f$ is applied coordinate-wise).", "Let $\\varrho \\colon D^r \\rightarrow \\overline{\\mathbb {Q}}$ be a cost function.", "We define $\\operatorname{Feas}(\\varrho ) = \\lbrace {\\bf x} \\in D^r \\ | \\ \\varrho (\\bf {x}) \\mbox{ is finite} \\rbrace $ to be the feasibility relation of $\\varrho $ .", "We call an operation $f \\colon D^k \\rightarrow D$ a polymorphism of $\\varrho $ if the relation $\\operatorname{Feas}(\\varrho )$ is compatible with it.", "For a valued constraint language $\\Gamma $ we denote by $\\operatorname{Pol}(\\Gamma )$ the set of operations which are polymorphisms of all cost functions $\\varrho \\in \\Gamma $ .", "It is easy to verify that $\\operatorname{Pol}(\\Gamma )$ is a clone.", "The set of $m$ -ary operations in $\\operatorname{Pol}(\\Gamma )$ is denoted $\\operatorname{Pol}_{m}(\\Gamma )$ .", "For crisp cost functions (relations) this notion of polymorphism corresponds precisely to the standard notion of polymorphism which has played a crucial role in the complexity analysis for the CSP [16], [3].", "Take $\\varrho $ to be a cost function of arity $r$ on $D$ , and let $C \\subseteq \\operatorname{Pol}(\\lbrace \\varrho \\rbrace )$ be a clone of operations.", "A weighting $\\omega \\colon C^{(k)} \\rightarrow \\mathbb {Q}$ is called a weighted polymorphism of $\\varrho $ if, for any list of $r$ -tuples $\\bf x_1, \\dots , x_k \\in \\operatorname{Feas}(\\varrho )$ , we have $\\sum _{f \\in C^{(k)}} \\omega (f) \\cdot \\varrho (f(\\mathbf {x_1, \\dots , x_k})) \\le 0.$ For a valued constraint language $\\Gamma $ we denote by $\\operatorname{wPol}(\\Gamma )$ the set of those weightings of the clone $\\operatorname{Pol}(\\Gamma )$ that are weighted polymorphisms of all cost functions $\\varrho \\in \\Gamma $ .", "The set of weightings $\\operatorname{wPol}(\\Gamma )$ is a weighted clone [9].", "For any lattice-ordered set $D$ , a function $\\varrho \\colon D^{r} \\rightarrow \\mathbb {Q}$ is called submodular if for all $\\mathbf {x_1, x_2} \\in D^{r}$ $\\varrho (\\min (\\mathbf {x_1,x_2})) + \\varrho (\\max (\\mathbf {x_1,x_2})) - \\varrho (\\mathbf {x_1}) - \\varrho (\\mathbf {x_2}) \\le 0.$ The above condition can be equivalently expressed by saying that the set of submodular functions on $D$ is the set of cost functions with a binary weighted polymorphism $\\omega $ , defined as follows: $\\omega (f) = {\\left\\lbrace \\begin{array}{ll} -1 &\\mbox{if } f \\mbox{ is a projection,} \\\\\\ \\ 1 & \\mbox{if } f \\mbox{ is one of the operations } \\min \\mbox{ or } \\max , \\\\\\ \\ 0 & \\mbox{otherwise.}", "\\end{array}\\right.", "}$ An operation $f$ is idempotent if $f(x,...,x) = x$ .", "A weighted polymorphism is called idempotent if all operations in its support are idempotent.", "An operation $f \\in \\mathcal {O}_D^{(k)}$ is cyclic if for every $x_1, \\dots , x_k \\in D$ we have that $f(x_1,x_2, \\dots ,x_k) = f(x_2, \\dots , x_k ,x_1)$ .", "A weighted polymorphism is called cyclic if its support is non-empty and contains cyclic operations only.", "A cost function $\\varrho $ is said to be improved by a weighting $\\omega $ if $\\omega $ is a weighted polymorphism of $\\varrho $ .", "For any set $W$ of weightings over a fixed clone $C \\subseteq \\mathcal {O}_{D}$ we denote by $\\operatorname{Imp}(W)$ the set of cost functions on $D$ which are improved by all weightings $\\omega \\in W$ .", "The following result, together with Theorem REF , implies that tractable valued constraint languages can be characterised by weighted polymorphisms.", "[Cohen et al.", "[9]] For any finite valued constraint language $\\Gamma $ , we have $\\operatorname{Imp}(\\operatorname{wPol}(\\Gamma )) = \\operatorname{wRelClo}(\\Gamma )$ .", "For more information on the valued constraint satisfaction problem see the recent survey [17]." ], [ "Algebras and varieties", "In this subsection we introduce the basic concepts of universal algebra that serve us as tools later on in this paper.", "An algebraic signature is a set of function symbols together with (finite) arities.", "An algebra $\\textbf {A}$ over a fixed signature $\\Sigma $ consists of a set $A$ , called the universe of $\\textbf {A}$ , and a set of basic operations that correspond to the symbols in the signature, i.e., if the signature contains a $k$ -ary symbol $f$ then the algebra has a basic operation $f^{\\textbf {A}}$ , which is a function $f^{\\textbf {A}} \\colon A^{k} \\rightarrow A$ .", "A subset $B$ of the universe of an algebra $\\textbf {A}$ is a subuniverse of $\\textbf {A}$ if it is closed under all operations of $\\textbf {A}$ .", "An algebra $\\textbf {B}$ is a subalgebra of $\\textbf {A}$ if $B$ is a subuniverse of $\\textbf {A}$ and the operations of $\\textbf {B}$ are restrictions of all the operations of $\\textbf {A}$ to $B$ .", "Let $(\\textbf {A}_{i})_{i \\in I}$ be a family of algebras (over the same signature).", "Their product $\\Pi _{i \\in I} \\textbf {A}_{i}$ is an algebra with the universe equal to the cartesian product of the $A_{i}$ 's and operations computed coordinate-wise.", "For two algebras $\\textbf {A}$ and $\\textbf {B}$ (over the same signature), a homomorphism from $\\textbf {A}$ to $\\textbf {B}$ is a function $h \\colon A \\rightarrow B$ that preserves all operations.", "It is easy to see, that an image of a homomorphism $h \\colon A \\rightarrow B$ is a subalgebra of $\\textbf {B}$ .", "Let $\\mathcal {K}$ be a class of algebras over a fixed signature $\\Sigma $ .", "We denote by $\\operatorname{S}(\\mathcal {K})$ the class of all subalgebras of algebras in $\\mathcal {K}$ , by $\\operatorname{P}(\\mathcal {K})$ the class of all products of algebras in $\\mathcal {K}$ , by $\\operatorname{P}_{fin}(\\mathcal {K})$ the class of all finite products, and by $\\operatorname{H}(\\mathcal {K})$ the class of all homomorphic images of algebras in $\\mathcal {K}$ .", "If $\\mathcal {K}=\\lbrace \\textbf {A}\\rbrace $ we write $\\operatorname{S}(\\textbf {A})$ , $\\operatorname{P}(\\textbf {A})$ , and $\\operatorname{H}(\\textbf {A})$ instead of $\\operatorname{S}(\\lbrace \\textbf {A}\\rbrace )$ , $\\operatorname{P}(\\lbrace \\textbf {A}\\rbrace )$ , and $\\operatorname{H}(\\lbrace \\textbf {A}\\rbrace )$ , respectively.", "Similarly $\\mathcal {K})$ is the smallest class of algebras closed under all three operations.", "For an algebra $\\textbf {A}$ the variety $\\lbrace \\textbf {A}\\rbrace )$  (denoted $\\textbf {A})$ ) is the variety generated by $\\textbf {A}$ , and ${fin}(\\textbf {A})$ is the class of finite algebras in $\\textbf {A})$ .", "The variety $\\textbf {A})$ can be characterised as follows: [Tarski [23]] For any finite algebra $\\textbf {A}$ , we have $\\textbf {A}) = \\operatorname{HSP}(\\textbf {A}) \\text{ \\ \\ and \\ \\ } {fin}(\\textbf {A}) = \\operatorname{HSP}_{fin}(\\textbf {A}).$ We say that an equivalence relation $\\sim $ on $A$ is a congruence of $\\textbf {A}$ if the following condition is satisfied for all operations $f$ of $\\textbf {A}$ : if for all $i \\in \\lbrace 1, \\dots , k\\rbrace $ , we have $a_i \\sim b_i$ , then $f(a_1, \\dots , a_k) \\sim f(b_1, \\dots , b_k),$ where $k$ is the arity of $f$ .", "Every congruence $\\sim $ of $\\textbf {A}$ determines a quotient algebra $\\textbf {A}/{\\sim }$ .", "Its universe is the set of the equivalence classes $A /{\\sim }$ and operations are defined using their arbitrarily chosen representatives.", "A term $t$ in a signature $\\Sigma $ is a formal expression built from variables and symbols in $\\Sigma $ that syntactically describes the composition of basic operations.", "For an algebra $\\textbf {A}$ over $\\Sigma $ a term operation $t^{\\textbf {A}}$ is an operation obtained by composing the basic operations of $\\textbf {A}$ according to $t$ .", "Let $s$ and $t$ be a pair of terms in a signature $\\Sigma $ .", "We say that $\\textbf {A}$ satisfies the identity $s \\approx t$ if the term operations $s^{\\textbf {A}}$ and $t^{\\textbf {A}}$ are equal.", "We say that a class of algebras $ over $$ satisfies the identity $ s t$ if every algebra in $ does.", "[Birkhoff [2]] A class of algebras $ is a variety if and only if there exists a set of identities such that $ contains precisely those algebras that satisfy all the identities from this set.", "It follows from Birkhoff's theorem that the variety $\\textbf {A})$ is the class of algebras that satisfy all the identities satisfied by $\\textbf {A}$ .", "Moreover, if $\\textbf {A}$ is finite then $\\textbf {A})$ is locally finite, i.e., every finitely generated algebra in $\\textbf {A})$ is finite." ], [ "Core Valued Constraint Languages", "For each valued constraint language $\\Gamma $ there is an associated algebra.", "It has universe $D$ and the set of operations $\\operatorname{Pol}(\\Gamma )$ .", "If all polymorphisms of $\\Gamma $ are idempotent it means that the algebra $(D, \\operatorname{Pol}(\\Gamma ))$ satisfies the identity $f(x,\\ldots , x) \\approx x$ for every operation $f$ .", "Such algebras are called idempotent.", "In this section we prove that every finite valued constraint language has a computationally equivalent valued constraint language whose associated algebra is idempotent." ], [ "Positive Clone.", "Those polymorphisms of a given language $\\Gamma $ which are assigned a positive weight by some weighted polymorphisms $\\omega \\in \\operatorname{wPol}(\\Gamma )$ are of special interest in the rest of the paper.", "We begin this section by proving that they form a clone.", "Let $\\mathcal {C}$ be a weighted clone over a set $D$ .", "The following proposition shows that the set $\\bigcup _{\\omega \\in \\mathcal {C}} \\operatorname{supp}(\\omega )$ , together with the set of projections $\\Pi _{D}$ , is a clone.", "We call it the positive clone of $\\mathcal {C}$ and denote by $C^{+}$  (if $\\mathcal {C}$ is $\\operatorname{wPol}(\\Gamma )$ then $C^+$ is denoted by $\\operatorname{Pol}^+(\\Gamma )$ ).", "If $\\mathcal {C}$ is a weighted clone then $C^+$ is a clone.", "We will use the following technical lemma (Lemma 6.5 from [9]).", "It implies that any weighting that can be expressed as a weighted sum of arbitrary superpositions can also be expressed as a superposition of a weighted sum of proper superpositions.", "Let $\\mathcal {C}$ be a weighted clone, and let $\\omega _1$ and $\\omega _2$ be weightings in $\\mathcal {C}$ , of arity $k$ and $l$ respectively.", "For any $m$ -ary operations $f_1, \\dots ,f_k, g_1, \\dots ,g_l$ of $C$ : $c_1 \\omega _1[f_1, \\dots ,f_k] + c_2 \\omega _2[g_1, \\dots ,g_l] =\\omega [f_1, \\dots ,f_k, g_1, \\dots ,g_l],$ where $\\omega = c_1 \\omega _1[\\pi _1^{(k+l)}, \\ldots , \\pi _k^{(k+l)}] + c_2 \\omega _2[\\pi _{k+1}^{(k+l)}, \\ldots , \\pi _{k+l}^{(k+l)}].$ We need to show that the set $C^{+}$ is closed under superposition.", "Take a $k$ -ary operation $f$ and a list of $l$ -ary operations $g_1, \\dots , g_k$ that all belong to $C^+$ .", "If $f$ is a projection there is nothing to prove.", "Otherwise there is a weighting $\\omega \\in \\mathcal {C}$ such that $\\omega (f)>0$ .", "Similarly for each $g_i$ which is not a projection we find $\\omega _i$ such that $\\omega _i(g_i)>0$  (if $g_i$ is a projection we put $\\omega _i$ to be the zero-valued $l$ -ary weighting).", "Now, there exist non-negative rational numbers $i_j$ such that the sum $\\omega [g_1,\\cdots ,g_k] + i_1\\omega _1[\\pi _1^l,\\cdots ,\\pi _l^l]+\\cdots + i_k\\omega _k[\\pi _1^l,\\cdots ,\\pi _l^l]$ is a valid weighting.", "By Lemma REF this weighting can be obtained as a superposition of a sum of proper superpositions and therefore belongs to $\\mathcal {C}$ which finishes the proof." ], [ "Cores.", "Let $\\Gamma $ be a valued constraint language with a domain $D$ .", "For $S \\subseteq D$ we denote by $\\Gamma [S]$ the valued constraint language defined on a domain $S$ and containing the restriction of every cost function $\\varrho \\in \\Gamma $ to $S$ .", "We show that $\\Gamma $ has a computationally equivalent valued constraint language $\\Gamma ^{\\prime }$ such that $\\operatorname{Pol}_1^+(\\Gamma ^{\\prime })$ contains only bijective operations.", "Such a language is called a core.", "Moreover, $\\Gamma ^{\\prime }$ can be chosen to be equal $\\Gamma [S]$ for some $S \\subseteq D$ .", "For every valued constraint language $\\Gamma $ there exists a core language $\\Gamma ^{\\prime }$ , such that the valued constraint language $\\Gamma $ is tractable if and only if $\\Gamma ^{\\prime }$ is tractable, and it is NP-hard if and only if $ \\Gamma ^{\\prime }$ is NP-hard.", "We prove the above result by generalizing the arguments for finite-valued languages given in [15], [25].", "We need an auxiliary lemma.", "For a valued constraint language $\\Gamma $ , let $f \\in \\operatorname{Pol}_1^{+}(\\Gamma )$ and let $I\\in \\operatorname{VCSP}(\\Gamma )$ .", "If $s$ is an optimal assignment for $I$ , then $f(s)$ is also optimal.", "Let $f$ , $I$ and $s$ be like in the statement of the lemma.", "Observe that $Cost_{I}$ (see Definition REF ) can be seen as a cost function whose arity is equal to the number of variables in $I$ .", "Moreover, $Cost_{I}$ belongs to $\\operatorname{wRelClo}(\\Gamma )$ as it is clearly expressible over $\\Gamma $ .", "If $Cost_{I}(s) = \\infty $ then there is no assignment with a finite cost and we are done.", "Assume that $Cost_{I}(s) < \\infty $ , which means that $s \\in \\operatorname{Feas}(Cost_{I})$ .", "If $f \\ne \\operatorname{id}$ , then there exists a weighted polymorphism $\\omega $ with $\\omega (f) > 0$ .", "By definition the following inequality is satisfied: $\\sum _{g \\in \\operatorname{Pol}_{1}(\\Gamma )} \\omega (g) \\cdot Cost_{I}(g(s)) \\le 0.$ Without loss of generality we can assume that $\\omega (\\operatorname{id}) = -1$ .", "Then we have that $\\sum _{g \\in \\operatorname{supp}(\\omega )} \\omega (g) = 1$ and the inequality above can be rewritten as $\\sum _{g \\in \\operatorname{supp}(\\omega )} \\omega (g) \\cdot Cost_{I}(g(s)) \\le Cost_{I}(s).$ On the other hand, $\\sum _{g \\in \\operatorname{supp}(\\omega )} \\omega (g) \\cdot Cost_{I}(g(s)) \\ge \\sum _{g \\in \\operatorname{supp}(\\omega )} \\omega (g) \\cdot Cost_{I}(s) = Cost_{I}(s).$ Therefore $Cost_{I}(g(s)) = Cost_{I}(s)$ for each operation $g \\in \\operatorname{supp}(\\omega )$ .", "Since $f \\in \\operatorname{supp}(\\omega )$ and $s$ is optimal, $f(s)$ is also optimal.", "(of Proposition REF ) Let $\\Gamma $ be a valued constraint language over a domain $D$ .", "Suppose that there is a unary polymorphism $f \\in \\operatorname{Pol}^{+}(\\Gamma )$ that is not bijective.", "Let $\\Gamma ^{\\prime } = \\Gamma [f(D)]$ , where $f(D)D$ denotes the range of $f$ .", "There is a natural correspondence between instances of $\\operatorname{VCSP}(\\Gamma ^{\\prime })$ and instances of $\\operatorname{VCSP}(\\Gamma )$ , induced by the correspondence between functions in $\\Gamma $ and their restrictions in $\\Gamma ^{\\prime }$ .", "For any instance $I^{\\prime }$ of $\\operatorname{VCSP}(\\Gamma ^{\\prime })$ the corresponding instance $I$ of $\\operatorname{VCSP}(\\Gamma )$ has the same variables.", "The cost function $\\varrho ^{\\prime }$ in each constraint is replaced by any cost function $\\varrho $ from $\\Gamma $ , which is equal to $\\varrho ^{\\prime }$ when restricted to $f(D)$ .", "We show that $\\operatorname{Opt}_{\\Gamma }(I) = \\operatorname{Opt}_{\\Gamma ^{\\prime }}(I^{\\prime })$ .", "Any assignment for $I^{\\prime }$ is also an assignment for $I$ , and hence $\\operatorname{Opt}_{\\Gamma }(I) \\le \\operatorname{Opt}_{\\Gamma ^{\\prime }}(I^{\\prime })$ .", "Furthermore, by Lemma REF for each $s$ that is an optimal assignment for $I$ , we have $Cost_{I}(s) = Cost_{I}(f(s)) = Cost_{I^{\\prime }}(f(s)).$ Therefore, $\\operatorname{Opt}_{\\Gamma }(I) \\ge \\operatorname{Opt}_{\\Gamma ^{\\prime }}(I^{\\prime })$ .", "It follows that $\\operatorname{VCSP}(\\Gamma )$ is tractable if and only if $\\operatorname{VCSP}(\\Gamma ^{\\prime })$ is tractable, and it is NP-hard if and only if $\\operatorname{VCSP}(\\Gamma ^{\\prime })$ is NP-hard.", "Moreover, the valued constraint language $\\Gamma ^{\\prime }$ is defined over a smaller domain.", "We replace $\\Gamma $ with $\\Gamma ^{\\prime }$ and repeat this procedure, until we obtain a language $\\Gamma ^{\\prime }$ that is a core.", "For core languages we characterize the set of unary weighted polymorphisms.", "Let $\\Gamma $ be a core valued constraint language.", "A unary weighting $\\omega $ is a weighted polymorphism of $\\Gamma $ if and only if it assigns positive weights only to such bijective operations $f \\in \\operatorname{Pol}_{1}(\\Gamma )$ that, for all cost functions $\\varrho \\in \\Gamma $ , satisfy $\\varrho \\circ f = \\varrho $ .", "If a valid unary weighting $\\omega $ assigns positive weights only to such operations $f \\in \\operatorname{Pol}_{1}(\\Gamma )$ that, for all cost functions $\\varrho \\in \\Gamma $ , satisfy $\\varrho \\circ f = \\varrho $ , then for each $\\varrho \\in \\Gamma $ and a tuple $\\mathbf {x} \\in \\operatorname{Feas}(\\varrho )$ $\\sum _{f \\in \\operatorname{Pol}_{1}(\\Gamma )} \\omega (f) \\cdot \\varrho (f(\\mathbf {x})) = \\sum _{f \\in \\operatorname{Pol}_{1}(\\Gamma )} \\omega (f) \\cdot \\varrho (\\mathbf {x}) = 0,$ and $\\omega $ is clearly a weighted polymorphism of $\\Gamma $ .", "For the other direction, let $\\omega $ be a unary weighted polymorphism of $\\Gamma $ , such that $\\operatorname{supp}(\\omega ) \\ne \\emptyset $ .", "Without loss of generality assume that $\\omega (\\operatorname{id})=-1$ .", "Since $\\Gamma $ is a core language, the operations $g \\in \\operatorname{supp}(\\omega )$ are bijective.", "For $\\varrho \\in \\Gamma $ and a tuple $\\mathbf {x} \\in \\operatorname{Feas}(\\varrho )$ for which $\\varrho $ takes the minimal value, we have $\\sum _{g \\in \\operatorname{supp}(\\omega )} \\omega (g) \\cdot \\varrho (g(\\mathbf {x})) + \\omega (\\operatorname{id}) \\cdot \\varrho (\\mathbf {x}) \\le 0, \\text{ hence}$ $\\varrho (\\mathbf {x}) \\ge \\sum _{g \\in \\operatorname{supp}(\\omega )} \\omega (g) \\cdot \\varrho (g(\\mathbf {x})) \\ge \\sum _{g \\in \\operatorname{supp}(\\omega )} \\omega (g) \\cdot \\varrho (\\mathbf {x}) = \\varrho (\\mathbf {x}).$ Therefore $\\varrho (g(\\mathbf {x})) = \\varrho (\\mathbf {x})$ for each $g \\in \\operatorname{supp}(\\omega )$ , which means that the operations in the support preserve the minimal weight.", "Note that, since each $g \\in \\operatorname{supp}(\\omega )$ is bijective, it determines a bijection of the set $\\operatorname{Feas}(\\varrho )$ .", "We have shown that this bijection preserves the set of tuples with minimal weight.", "It can be similarly shown by induction that it preserves the set of tuples with any other fixed weight.", "Hence, we have proved that $\\varrho \\circ g = \\varrho $ for all $g \\in \\operatorname{supp}(\\omega )$ .", "This implies that, for any core language $\\Gamma $ , a unary polymorphism belongs to $\\operatorname{Pol}^+(\\Gamma )$ if and only if it is bijective and preserves all cost functions in $\\Gamma $ .", "Let $\\Gamma $ be a finite core valued constraint language over a domain $D$ .", "For each arity $m$ we fix an enumeration of all the elements of $D^m$ .", "This allows us to treat every $m$ -ary operation $f \\in \\mathcal {O}_D^{(m)}$ as a $|D^m|$ -tuple.", "We define a $|D^m|$ -ary cost function in $\\operatorname{wRelClo}(\\Gamma )$ that precisely distinguishes the $m$ -ary operations in the positive clone from all the other $m$ -ary polymorphisms.", "To do this we need the following technical lemma, which is a variant of the well known Farkas' Lemma used in linear programming: [Farkas [11]] Let $S$ and $T$ be finite sets of indices, where $T$ is a disjoint union of two subsets, $T_{\\ge }$ and $T_{=}$ .", "For all $i \\in S$ , and all $j \\in T$ , let $a_{i,j}$ and $b_{j}$ be rational numbers.", "Exactly one of the following holds: Either there exists a set of non-negative rational numbers $\\lbrace z_i \\ | \\ i \\in S \\rbrace $ and a rational number $C$ such that $\\text{for each } j \\in R_{\\ge }, \\ \\ \\sum _{i \\in S} a_{i,j} z_{i} \\ge b_{j} + C,$ $\\text{for each } j \\in R_{=}, \\ \\ \\sum _{i \\in S} a_{i,j} z_{i} = b_{j} + C.$ Or else there exists a set of rational numbers $\\lbrace y_{j} \\ | \\ j \\in T \\rbrace $ such that $\\sum _{j \\in T} y_j = 0$ and $\\text{for each } j \\in T_\\ge , \\ \\ y_j \\ge 0,$ $\\text{for each } i \\in S, \\ \\ \\sum _{j \\in T} y_j a_{i,j} \\le 0,$ $\\text{and } \\sum _{j \\in T} y_j b_j > 0.$ The set $\\lbrace y_j \\ | \\ j \\in T \\rbrace $ defined in the lemma is called a certificate of unsolvability.", "Let $\\Gamma $ be a finite core valued constraint language over a domain $D$ .", "For every $m$ there exists a cost function $\\varrho \\colon \\mathcal {O}_D^{(m)} \\rightarrow \\overline{Q}$ in $\\operatorname{wRelClo}(\\Gamma )$ , and a rational number $P$ , such that for every $f \\in \\mathcal {O}_D^{(m)}$ the following conditions are satisfied: $\\varrho (f) \\ge P$ , $\\varrho (f) < \\infty $ if and only if $f \\in \\operatorname{Pol}(\\Gamma )$ , $\\varrho (f) = P $ if and only if $f \\in \\operatorname{Pol}^{+}(\\Gamma )$ .", "The cost function $\\varrho $ is given by a sum of all cost functions in $\\Gamma $ with positive coefficients that we define later on.", "Like in the classical CSP, a cost function whose feasibility relation contains exactly those $|D^m|$ -tuples which are $m$ -ary polymorphisms of $\\Gamma $ is defined by: $\\sum _{\\begin{array}{c}\\varrho \\in \\Gamma \\\\ ({\\bf a_1},\\cdots , {\\bf a_m})\\in (\\operatorname{Feas}(\\varrho ))^m\\end{array}} \\varrho (x_{{\\bf b_1}},\\cdots , x_{{\\bf b_{r}}}),$ where ${\\bf b_i}(j) = {\\bf a_j}(i)$ , and $r$ is the arity of $\\varrho $ .", "For each summand we introduce a variable $z_{\\varrho ,{\\bf a_1},\\cdots , {\\bf a_m}}$ and, for each $f\\in \\operatorname{Pol}^+_m(\\Gamma )$ we write: $\\sum _{\\begin{array}{c}\\varrho \\in \\Gamma \\\\ ({\\bf a_1},\\cdots , {\\bf a_m}) \\in (\\operatorname{Feas}(\\varrho ))^m\\end{array}}z_{\\varrho ,{\\bf a_1},\\cdots , {\\bf a_m}} \\varrho (f({\\bf b_1}),\\cdots ,f({\\bf b_{r}}) ) = 0 + C,$ while for each $f\\in \\operatorname{Pol}_m(\\Gamma )\\setminus \\operatorname{Pol}^+_m(\\Gamma )$ : $\\sum _{\\begin{array}{c}\\varrho \\in \\Gamma \\\\ ({\\bf a_1},\\cdots , {\\bf a_m}) \\in (\\operatorname{Feas}(\\varrho ))^m\\end{array}}z_{\\varrho ,{\\bf a_1},\\cdots , {\\bf a_m}} \\varrho (f({\\bf b_1}),\\cdots ,f({\\bf b_{r}}) ) \\ge 1 + C,$ where ${\\bf b_i}(j) = {\\bf a_j}(i)$ , and $r$ is the arity of $\\varrho $ .", "By putting the above equalities and inequalities together we obtain a system of linear inequalities and equations.", "By Lemma REF there are two mutually exclusive possibilities.", "First, there may exist a set of non-negative rational numbers $z_{\\varrho ,{\\bf a_1},\\cdots , {\\bf a_m}}$ and a rational number $C$ , such that this system is satisfied.", "Then the proposition is proved: items 1. and 3. follow trivially from construction.", "Item 2. follows by definition of the cost function.", "Otherwise, there exists a set $\\lbrace y_f \\ | \\ f \\in \\operatorname{Pol}_m(\\Gamma ) \\rbrace $ which forms the certificate of unsolvability.", "Then let us consider a weighting defined by $\\omega (f) = y_f$ .", "If $\\omega $ is a valid weighting, then it is an $m$ -ary weighted polymorphism of $\\Gamma $ .", "Moreover, $\\omega $ assigns to all operations in $\\operatorname{Pol}_m(\\Gamma ) \\setminus \\operatorname{Pol}^{+}_m(\\Gamma )$ non-negative weights that sum up to a positive number.", "Hence, for some $h \\in \\operatorname{Pol}_m(\\Gamma ) \\setminus \\operatorname{Pol}^{+}_m(\\Gamma )$ , we have $\\omega (h) > 0$ , which contradicts $h \\notin \\operatorname{Pol}^{+}_m(\\Gamma )$ .", "If it happens that $y_g < 0$ for some operation $g \\in \\operatorname{Pol}^{+}_m(\\Gamma )$ that is not a projection, then there exists an $m$ -ary weighted polymorphism of $\\Gamma $ which assigns a positive weight to $g$ .", "By scaling it and adding to $\\omega $  (as in the proof of Proposition REF ), we obtain the weighted polymorphism needed for the contradiction." ], [ "Rigid cores.", "We further reduce the class of languages that we need to consider.", "Let $\\Gamma $ be a core valued constraint language over an $n$ -element domain $D = \\lbrace d_1, \\dots , d_n \\rbrace $ .", "For each $i \\in \\lbrace 1, \\dots , n \\rbrace $ , let $N_i(x) = {\\left\\lbrace \\begin{array}{ll} 0 &\\mbox{if } x=d_i, \\\\\\infty & \\mbox{otherwise.}", "\\end{array}\\right.", "}$ and let $\\Gamma _c$ denote the valued constraint language obtained from $\\Gamma $ by adding all cost functions $N_i$ .", "Observe that $\\operatorname{Pol}(\\Gamma _c) = \\operatorname{IdPol}(\\Gamma )$ , where by $\\operatorname{IdPol}(\\Gamma )$ we denote the set of idempotent polymorphisms of the language $\\Gamma $ .", "Hence, the only unary polymorphism of $\\Gamma _c$ is the identity, which also means that there is only one unary weighted polymorphism of $\\Gamma _c$ – the zero-valued polymorphism.", "A valued constraint language $\\Gamma $ is a rigid core if there is exactly one unary polymorphism of $\\Gamma $ , which is the identity.", "The notion of rigid core corresponds to the classical notion of rigid core considered in CSP [7].", "A valued constraint language $\\Gamma $ is a rigid core if the set of feasibility relations of all cost functions from $\\Gamma $ is a rigid core in the standard sense, which is also equivalent to all polymorphisms of $\\Gamma $ being idempotent.", "We now prove a result which, together with Proposition REF , implies that for each finite language $\\Gamma $ , there is a computationally equivalent language that is a rigid core.", "Let $\\Gamma $ be a valued constraint language which is finite and a core.", "The valued constraint language $\\Gamma _c$ is a rigid core.", "Moreover, $\\Gamma $ is tractable if and only if $\\Gamma _c$ is tractable, and $\\Gamma $ is NP-hard if and only if $\\Gamma _c$ is NP-hard.", "Let $\\Gamma $ be a finite core valued constraint language over a domain $D = \\lbrace d_1, \\dots , d_n \\rbrace $ .", "It follows from Proposition REF that there exist an $n$ -ary cost function $N \\in \\operatorname{wRelClo}(\\Gamma )$ , and positive rational numbers $P < Q$ , such that the following conditions are satisfied: $N(x_1, \\dots , x_n) = P $ if and only if the unary operation $g$ defined by $d_i \\mapsto x_i$ belongs to $\\operatorname{Pol}^{+}(\\Gamma )$ , $N(x_1, \\dots , x_n) > Q $ if and only if the unary operation $g$ defined by $d_i \\mapsto x_i$ belongs to $\\operatorname{Pol}(\\Gamma ) \\setminus \\operatorname{Pol}^{+}(\\Gamma )$ , otherwise $N(x_1, \\dots , x_n) = \\infty $ .", "Assume without loss of generality that $N \\in \\Gamma $ .", "We show a polynomial-time Turing reduction from $\\operatorname{VCSP}(\\Gamma _c)$ to $\\operatorname{VCSP}(\\Gamma )$ .", "Let $I_c = (V_c,D, c)$ be an instance of $\\operatorname{VCSP}(\\Gamma _c)$ .", "The set of variables $V$ in the new instance $I$ is a disjoint union of $V_c$ and $\\lbrace v_1, \\dots , v_n \\rbrace $ .", "For every constraint of the form $((v),N_i)$ in $c$ we: add a constraint $((v,v_i),\\varrho _=)$ , where $\\varrho _=(x,y) = {\\left\\lbrace \\begin{array}{ll} 0 &\\mbox{if } x=y, \\\\\\infty & \\mbox{otherwise} \\end{array}\\right.", "}$ (this cost function is expressible over every valued constraint language, so without loss of generality we can assume that $\\varrho _= \\in \\Gamma $ ), remove the constraint $((v),N_i)$ from $c$ .", "We obtain a new set of constraints 1, where all cost functions are already from $\\Gamma $ .", "Let $C$ be the sum of weights that all cost functions in all constraints in 1 assign to all tuples in their feasibility relations.", "The final set of constraints $ additionally contains $ m$ constraints of the form $ ((v1, ..., vn), N)$, where $ m$ is big enough to ensure that $ m (Q-P) > C$.$ There are three possibilities: If $\\operatorname{Opt}_{\\Gamma }(I) = \\infty $ then no assignment for $I_c$ has a finite cost.", "Suppose otherwise and let $s_c$ be an assignment for $I_c$ with a finite cost.", "Then $s_c$ gives rise to an assignment $s$ for $I$ with a finite cost.", "It coincides with $s_c$ on $V_c$ and for each $i \\in \\lbrace 1, \\dots , n\\rbrace $ , we set $s(v_i) = d_i$ .", "The optimal assignment $s$ for $I$ satisfies $N(s(v_1, \\dots , v_n))=P$ .", "Then the tuple $s(v_1, \\dots , v_n)$ determines a unary operation $g$ , defined by $d_i \\mapsto s(v_i)$ .", "The operation $g$ , by the definition of the cost function $N$ , belongs to the positive clone $\\operatorname{Pol}^{+}(\\Gamma )$ .", "Hence, $g^{-1}$ also belongs to the positive clone.", "Since $\\Gamma $ is a core, the assignment $g^{-1}(s)$ is optimal for $I$ .", "Its restriction onto $V_c$ is an optimal assignment for $I_c$ .", "The optimal assignment $s$ for $I$ satisfies $N(s(v_1, \\dots , v_n)) > Q$ .", "While there are $m$ constraints of the form $((v_1, \\dots , v_n), N)$ , we have $Cost_{I}(s) \\ge m \\cdot Q > m \\cdot P + C.$ If there was any assignment $s_c$ for $I_c$ with a finite cost, the corresponding assignment $s$ for $I$ would satisfy $Cost_{I}(s) < m \\cdot P +C$ , which gives a contradiction, and implies that $\\operatorname{Opt}_{\\Gamma _c}(I_c) = \\infty $ .", "If $\\Gamma $ is a core language then the positive clone of $\\Gamma _c$ contains precisely the idempotent operations from the positive clone of $\\Gamma $ .", "To show this, we first prove the following lemma: Let $\\Gamma $ be a core valued constraint language.", "For every weighted polymorphism $\\omega \\in \\operatorname{wPol}(\\Gamma )$ there exists an idempotent weighted polymorphism $\\omega ^{\\prime } \\in \\operatorname{wPol}(\\Gamma )$ such that $\\operatorname{supp}(\\omega ) \\cap \\operatorname{IdPol}(\\Gamma ) \\subseteq \\operatorname{supp}(\\omega ^{\\prime })$ .", "Moreover, if $\\omega $ is cyclic then $\\omega ^{\\prime }$ can be chosen to be cyclic.", "Consider a weighted polymorphism $\\omega \\in \\operatorname{wPol}(\\Gamma )$ .", "Take a non-idempotent operation $g \\in supp(\\omega )$ and let $h$ be a unary operation defined by $h(x) = g(x, \\ldots , x)$ .", "Since $\\operatorname{Pol}^+(\\Gamma )$ is a clone of operations, $h \\in \\operatorname{Pol}^+(\\Gamma )$ .", "Then by Proposition REF the operation $h$ is bijective and preserves all cost functions in $\\Gamma $ .", "We modify the weighted polymorphism $\\omega $ by adding $\\omega (g)$ to the weight of the idempotent operation $h^{-1} \\circ g$ and then assigning weight 0 to the operation $g$ .", "It is straightforward to check that the new weighting is a weighted polymorphism of $\\Gamma $ .", "If $g$ is cyclic then so is $h^{-1} \\circ g$ .", "Hence if $\\omega $ is cyclic then so is the new weighting.", "We repeat this construction for every non-idempotent operation in $\\operatorname{supp}(\\omega )$ .", "Finally, we obtain an idempotent weighted polymorphism $\\omega ^{\\prime }$ which satisfies the conditions of the lemma.", "Let $\\Gamma $ be a valued constraint language which is a core.", "Then $\\operatorname{IdPol}^+(\\Gamma ) = \\operatorname{Pol}^+(\\Gamma _c)$ .", "Clearly both sets contain all the projections.", "Let us take $f \\in \\operatorname{Pol}^+(\\Gamma _c)$ that is not a projection and let $\\omega $ be a weighted polymorphism of $\\Gamma _c$ such that $f \\in \\operatorname{supp}(\\omega )$ .", "There is a corresponding weighted polymorphism $\\omega ^{\\prime }$ of $\\Gamma $ , which is equal to $\\omega $ on the idempotent operations and equal 0 otherwise.", "Then we have $f \\in \\operatorname{supp}(\\omega ^{\\prime })$ .", "Since $f$ is idempotent it follows that $f \\in \\operatorname{IdPol}^+(\\Gamma )$ .", "To prove the reverse inclusion consider $f \\in \\operatorname{IdPol}^+(\\Gamma )$ that is not a projection.", "Let $\\omega $ be a weighted polymorphism of $\\Gamma $ such that $f \\in \\operatorname{supp}(\\omega )$ .", "By Lemma REF there exists an idempotent weighted polymorphism $\\omega ^{\\prime }$ of $\\Gamma $ such that $\\operatorname{supp}(\\omega ) \\cap \\operatorname{IdPol}(\\Gamma ) \\subseteq \\operatorname{supp}(\\omega ^{\\prime })$ .", "The weighting $\\omega ^{\\prime \\prime }$ , defined as a restriction of $\\omega ^{\\prime }$ to the idempotent operations, is a weighted polymorphism of $\\Gamma _c$ with $f \\in \\operatorname{supp}(\\omega ^{\\prime \\prime })$ .", "Hence $f \\in \\operatorname{Pol}^+(\\Gamma _c)$ ." ], [ "Weighted varieties", "One of the fundamental results of the algebraic approach to CSP [7], [3], [20] says that the complexity of a crisp language $\\Gamma $ depends only on the variety generated by the algebra $(D, \\operatorname{Pol}(\\Gamma ))$ .", "We generalize this fact to VCSP.", "A $k$ -ary weighting $\\omega $ of an algebra $\\textbf {A}$ is a function that assigns rational weights to all $k$ -ary term operations of $\\textbf {A}$ in such a way, that the sum of all weights is 0, and if $\\omega (f) < 0$ then $f$ is a projection.", "A (proper) superposition $\\omega [g_1, \\dots , g_k]$ of a weighting $\\omega $ with a list of $l$ -ary term operations $g_1, \\dots , g_k$ from $\\textbf {A}$ is defined the same way as for clones (see Definition REF ).", "An algebra $\\textbf {A}$ together with a set of weightings closed under non-negative scaling, addition of weightings of equal arity and proper superposition with operations from $\\textbf {A}$ is called a weighted algebra.", "For a variety $ over a signature $$ and a term $ t$ we denote by $ [t]$ the equivalence class of $ t$ under the relation $$ such that $ t s$ if and only if the variety $ satisfies the identity $t \\approx s$ (we skip the subscript, writing $[t]$ instead $[t]_{, whenever the variety is clear from the context).", "Observe that if the variety is locally finite then there are finitely many equivalence classes of terms of a fixed arity~\\cite {BS}.", "}\\begin{definition}Let be a locally finite variety over a signature \\Sigma .", "A k-ary \\emph {weighting} \\omega of is a function that assigns rational weights to all equivalence classes of k-ary terms over \\Sigma in such a way, that the sum of all weights is 0, and if \\omega ([t]) < 0 then satisfies the identity t(x_1, \\dots , x_k) \\approx x_i for some i \\in \\lbrace 1, \\dots , k\\rbrace .The variety together with a nonempty set of weightings is called a \\emph {weighted variety}.\\end{definition}$ Take any finite algebra $\\textbf {B}\\in .", "A $ k$-ary weighting $$ of $ induces a weighting $\\omega ^\\textbf {B}$ of $\\textbf {B}$ in a natural way: $\\omega ^{\\textbf {B}}(f) = \\sum _{\\lbrace [t] \\ | \\ t^\\textbf {B}= f \\rbrace } \\omega ([t]).$ If $\\omega ([t]) < 0$ then the term operation $t^\\textbf {B}$ is a projection, and hence the weighting $\\omega ^{\\textbf {B}}$ is proper.", "For a weighted variety $, by $B we mean the algebra $\\textbf {B}$ together with the set of weightings induced by $.$ For every weighting $\\omega $ of a finite weighted algebra $\\textbf {A}$ there is a corresponding weighting $\\omega $ of the variety $\\textbf {A})$ defined by $\\omega ([t]) = \\omega (t^{\\textbf {A}})$ .", "It follows from Birkhoff's theorem (see Theorem REF ) that it is well defined.", "A weighted variety $\\textbf {A})$ generated by a weighted algebra $\\textbf {A}$ is the variety $\\textbf {A})$ together with the set of weightings corresponding to the weightings of $\\textbf {A}$ .", "The correspondence is one-to-one so for simplicity we often identify the weightings of $\\textbf {A})$ with the weightings of $\\textbf {A}$ .", "We prove that every finite algebra $\\textbf {B}\\in \\textbf {A})$ together with the set of weightings induced by $\\textbf {A})$ is a weighted algebra.", "It is straightforward to check its closure under non-negative scaling and addition of weightings of equal arity.", "We only need to show that $\\textbf {B}$ is closed under proper superpositions.", "For a finite weighted algebra $\\textbf {A}$ over a fixed signature $\\Sigma $ and a finite algebra $\\textbf {B}\\in \\textbf {A})$ let $\\omega ^{\\textbf {B}}$ be a $k$ -ary weighting of $\\textbf {B}$ induced by the weighted variety $\\textbf {A})$ .", "If for some list $f_1^{\\textbf {B}}, \\ldots , f_k^{\\textbf {B}}$ of $l$ -ary term operations from $\\textbf {B}$ the weighting $\\omega ^{\\textbf {B}}[f_1^{\\textbf {B}}, \\ldots , f_k^{\\textbf {B}}]$ is proper then it is induced by some weighting of $\\textbf {A})$ .", "In the proof we use Gordan's Theorem (which is a straightforward consequence of Lemma REF ).", "[Gordan [13]] Let $S$ and $T$ be finite sets of indices.", "For all $i \\in S$ , and all $j \\in T$ , let $a_{i,j}$ be rational numbers.", "Exactly one of the following holds: Either there exists a set of non-negative rational numbers $\\lbrace z_i \\ | \\ i \\in S \\rbrace $ such that $\\text{for some } i \\in S, \\ \\ z_i >0,$ $\\text{for each } j \\in T, \\ \\ \\sum _{i \\in S} a_{i,j} z_{i} =0.$ Or else there exists a set of rational numbers $\\lbrace y_{j} \\ | \\ j \\in T \\rbrace $ such that $\\text{for each } i \\in S, \\ \\ \\sum _{j \\in T} y_j a_{i,j} > 0.$ (of Proposition ) Let $\\omega ^{\\textbf {B}}$ and $f_1^{\\textbf {B}}, \\ldots , f_k^{\\textbf {B}}$ be as in the statement of the proposition.", "Assume that the weighting $\\omega ^{\\textbf {B}}[f_1^{\\textbf {B}}, \\ldots , f_k^{\\textbf {B}}]$ is proper.", "Notice that the following conditions are equivalent: the operation $f_i^\\textbf {B}$ is the projection $\\pi _j$ on the $j$ -th coordinate, there exists a term $t$ such that $t^{\\textbf {B}} = f_i^{\\textbf {B}}$ and $t^\\textbf {A}$ is the projection $\\pi _j$ on the $j$ -th coordinate.", "For each $i \\in \\lbrace 1, \\dots , k\\rbrace $ consider the set $F_i$ of equivalence classes of terms over $\\Sigma $ defined by $F_i = {\\left\\lbrace \\begin{array}{ll} \\lbrace [t] \\ | \\ t^\\textbf {A}= \\pi _j \\rbrace &\\mbox{if } f_i^\\textbf {B}= \\pi _j, \\\\\\lbrace [t] \\ | \\ t^{\\textbf {B}} = f_i^{\\textbf {B}}\\rbrace & \\mbox{otherwise} \\end{array}\\right.", "}$ (observe that if $f_i^\\textbf {B}$ is a projection then $F_i$ contains a single equivalence class).", "Take $\\omega $ to be some $k$ -ary weighting of $\\textbf {A}$ that induces $\\omega ^{\\textbf {B}}$ , and let $W = \\lbrace \\omega [t_1^\\textbf {A}, \\ldots , t_k^\\textbf {A}] \\ | \\ [t_i] \\in F_i \\rbrace .$ Suppose that for some choice of equivalence classes $[t_i] \\in F_i$ the superposition $\\omega [t_1^\\textbf {A}, \\ldots , t_k^\\textbf {A}]$ is proper.", "The weighting $\\omega [t_1^\\textbf {A}, \\ldots , t_k^\\textbf {A}]$ of $\\textbf {A}$ induces a weighting of $\\textbf {B}$ which is equal to $\\omega ^{\\textbf {B}}[f_1^{\\textbf {B}}, \\ldots , f_k^{\\textbf {B}}]$ , thus in this case the proof is concluded.", "This shows that a superposition of $\\omega ^{\\textbf {B}}$ with any list of projections is always induced by some weighting of $\\textbf {A}$ .", "Now let us deal with the case when none of the weightings in $W$ is proper.", "Without loss of generality we can assume that the operations $f_1^{\\textbf {B}}, \\ldots , f_k^{\\textbf {B}}$ are pairwise distinct (otherwise we replace $\\omega ^{\\textbf {B}}$ by its superposition with a suitable list of projections) and hence the sets $F_i$ are disjoint.", "Let $F= \\bigcup F_i$ .", "We remove from $F$ the element of $F_i$ if $f_i^\\textbf {B}$ is a projection.", "The removed elements cannot cause a problem and therefore we assume that for every $[t] \\in F$ the operation $t^\\textbf {A}$ is not a projection.", "We apply Gordan's Theorem to the following system of linear equations: $\\sum _{\\nu \\in W} \\nu (t^\\textbf {A}) \\cdot z_\\nu - z_{[t]} =0, \\mbox{ for each } [t] \\in F.$ If this system has a non-zero solution in non-negative rational numbers then $z_\\nu > 0$ for some $\\nu \\in W$ .", "Observe that the weighting $\\upsilon = \\sum _{\\nu \\in W} \\nu \\cdot z_\\nu $ is proper.", "Indeed, by the definition of a superposition the only non-projections that could be assigned negative weights by $\\upsilon $ are the operations $t^{\\textbf {A}}$ where $[t] \\in F$ .", "But each such operation $t^{\\textbf {A}}$ is assigned a non-negative weight $z_{[t]}$ .", "Hence, by Lemma REF the weighting $\\upsilon $ is equal to a proper superposition of some weighting of $\\textbf {A}$ with a list of $l$ -ary term operations of $\\textbf {A}$ .", "Finally, let $p = \\sum _{\\nu \\in W} z_\\nu > 0$ .", "The weighting ${1\\over p} \\upsilon $ of $\\textbf {A}$ induces a weighting of $\\textbf {B}$ which is equal to $\\omega ^{\\textbf {B}}[f_1^{\\textbf {B}}, \\ldots , f_k^{\\textbf {B}}]$ .", "Otherwise, there exists a set $\\lbrace y_{[t]} \\ | \\ [t] \\in F \\rbrace $ of rational numbers, such that $\\mbox{for each } \\nu \\in W, \\ \\ \\sum _{[t] \\in F} y_{[t]} \\cdot \\nu (t^\\textbf {A}) > 0,$ and $y_{[t]} <0$ for each $[t] \\in F$ .", "For every $i \\in \\lbrace 1, \\dots , k\\rbrace $ let us choose $[t_i] \\in F_i$ satisfying $y_{[t_i]} = \\max \\lbrace y_{[t]} \\ | \\ [t] \\in F_i \\rbrace $ (if $f_i^\\textbf {B}$ is a projection then we choose $[t_i] \\in F_i$ to be the only element of $F_i$ and put $y_{[t_i]}=0$ ) and consider the weighting $\\upsilon = \\omega [t_1^\\textbf {A}, \\ldots , t_k^\\textbf {A}]$ .", "Notice that $\\upsilon $ may assign negative weights only to operations $t_i^\\textbf {A}$ .", "Since $\\sum _{[t] \\in F_1} y_{[t]} \\cdot \\upsilon (t^\\textbf {A}) + \\dots + \\sum _{[t] \\in F_k} y_{[t]} \\cdot \\upsilon (t^\\textbf {A}) >0,$ then $\\sum _{[t] \\in F_i} y_{[t]} \\cdot \\upsilon (t^\\textbf {A}) > 0$ for some $i \\in \\lbrace 1, \\dots , k\\rbrace $ .", "Hence $0 < \\sum _{[t] \\in F_i} y_{[t]} \\cdot \\upsilon (t^\\textbf {A}) \\le \\sum _{[t] \\in F_i} y_{[t_i]} \\cdot \\upsilon (t^\\textbf {A}) =y_{[t_i]} \\cdot \\sum _{[t] \\in F_i} \\upsilon (t^\\textbf {A}).$ It follows that $\\sum _{[t] \\in F_i} \\upsilon (t^\\textbf {A}) < 0$ , which is a contradiction, since $\\sum _{[t] \\in F_i} \\upsilon (t^\\textbf {A})$ is the weight that the proper weighting $\\omega ^{\\textbf {B}}[f_1^{\\textbf {B}}, \\ldots , f_k^{\\textbf {B}}]$ assigns to the operation $f_i^\\textbf {B}$ (which is not a projection).", "For a finite weighted algebra $\\textbf {A}$ let $\\operatorname{Imp}(\\textbf {A})$ denote the set of those cost functions on $A$ that are improved by all weightings of $\\textbf {A}$ .", "We prove that for each finite weighted algebra $\\textbf {B}\\in \\textbf {A})$ the valued constraint language $\\operatorname{Imp}(\\textbf {B})$ is not harder then $\\operatorname{Imp}(\\textbf {A})$ .", "The proof consists of a sequence of lemmas.", "Let $\\textbf {A}$ be a finite weighted algebra.", "For any $\\textbf {B}\\in P_{fin}(\\textbf {A})$ , there is a polynomial-time reduction of $\\operatorname{VCSP}(\\operatorname{Imp}(\\textbf {B}))$ to $\\operatorname{VCSP}(\\operatorname{Imp}(\\textbf {A}))$ .", "Let $A^n$ be the universe of $\\textbf {B}$ and let $\\Gamma $ be a finite subset of $\\operatorname{Imp}(\\textbf {B})$ .", "Take $\\varrho \\in \\Gamma $ to be an $r$ -ary cost function.", "There is a natural way of defining a corresponding cost function of arity $n \\cdot r$ on the set $A$ .", "We denote this cost function by $\\varrho ^{\\prime }$ .", "Let $\\omega $ be a $k$ -ary weighting of the weighted algebra $\\textbf {A}$ .", "The corresponding $k$ -ary weighting $\\omega ^{\\textbf {B}}$ of $\\textbf {B}$ is a weighted polymorphism of $\\varrho $ .", "Then it is not hard to show that $\\omega $ is a weighted polymorphism of $\\varrho ^{\\prime }$ , as the basic operations of $\\textbf {B}$ are the operations of $\\textbf {A}$ computed coordinate-wise.", "Hence, each weighting of $\\textbf {A}$ is a weighted polymorphism of $\\varrho ^{\\prime }$ , which means that $\\varrho ^{\\prime } \\in \\operatorname{Imp}(\\textbf {A})$ .", "For each $\\varrho \\in \\Gamma $ we have defined a corresponding $\\varrho ^{\\prime } \\in \\operatorname{Imp}(\\textbf {A})$ .", "Let $\\Gamma ^{\\prime } \\subseteq \\operatorname{Imp}(\\textbf {A})$ be the (finite) set of all those cost functions.", "Now take an arbitrary instance $I= (V, A^n, $ of $\\operatorname{VCSP}(\\Gamma )$ .", "Replace the domain $A^n$ by $A$ , and each variable $v_i \\in V$ by a set of $n$ variables $\\lbrace v_i^1, \\dots , v_i^n \\rbrace $ , obtaining a new set of variables $V^{\\prime }$ .", "In each constraint $(\\sigma , \\varrho ) \\in , where $$ is an $ r$-ary cost function, replace the $ r$-tuple $$ of variables from $ V$ by the corresponding $ n r$-tuple of variables from $ V'$, and the cost function $$ by the corresponding cost function $ '$ from $ '$.", "The new instance $ I'=(V',A,)$ is an instance of $ VCSP(')$.", "It is easy to see that there is a one-to-one correspondence between the optimal assignments for $ I$ and the optimal assignments for $ I'$.$ Let $\\textbf {A}$ be a finite weighted algebra.", "For any $\\textbf {B}\\in S(\\textbf {A})$ , there is a polynomial-time reduction of $\\operatorname{VCSP}(\\operatorname{Imp}(\\textbf {B}))$ to $\\operatorname{VCSP}(\\operatorname{Imp}(\\textbf {A}))$ .", "Notice that $\\operatorname{Imp}(\\textbf {B}) \\subseteq \\operatorname{Imp}(\\textbf {A})$ , so there is nothing to be proved.", "Let $\\textbf {A}$ be a finite weighted algebra.", "For any $\\textbf {B}\\in H(\\textbf {A})$ , there is a polynomial-time reduction of $\\operatorname{VCSP}(\\operatorname{Imp}(\\textbf {B}))$ to $\\operatorname{VCSP}(\\operatorname{Imp}(\\textbf {A}))$ .", "By the isomorphism theorem we can consider $\\textbf {B}$ to be a quotient algebra $\\textbf {A}/ {\\sim }$ rather than a homomorphic image of $\\textbf {A}$ .", "Let $A /{\\sim }$ be the universe of $\\textbf {B}$ and let $\\Gamma $ be a finite subset of $\\operatorname{Imp}(\\textbf {B})$ .", "Take $\\varrho \\in \\Gamma $ to be a $r$ -ary cost function.", "We define a corresponding cost function $\\varrho ^{\\prime }$ of arity $r$ on the set $A$ by $\\varrho ^{\\prime }(x_1, \\dots , x_r) = \\varrho ([x_1]_{\\sim }, \\dots [x_r]_{\\sim })$ .", "Let $\\omega $ be a $k$ -ary weighting of the weighted algebra $\\textbf {A}$ .", "The corresponding $k$ -ary weighting $\\omega ^{\\textbf {B}}$ of $\\textbf {B}$ is a weighted polymorphism of $\\varrho $ .", "It is not hard to show that $\\omega $ is a weighted polymorphism of $\\varrho ^{\\prime }$ .", "Hence, each weighting of $\\textbf {A}$ is a weighted polymorphism of $\\varrho ^{\\prime }$ , which means that $\\varrho ^{\\prime } \\in \\operatorname{Imp}(\\textbf {A})$ .", "For each $\\varrho \\in \\Gamma $ we have defined a corresponding $\\varrho ^{\\prime } \\in \\operatorname{Imp}(\\textbf {A})$ .", "Let $\\Gamma ^{\\prime } \\subseteq \\operatorname{Imp}(\\textbf {A})$ be the (finite) set of all those cost functions.", "Now take an arbitrary instance $I= (V, A /{\\sim }, $ of $\\operatorname{VCSP}(\\Gamma )$ .", "Replace the domain $A /{\\sim }$ by $A$ .", "In each constraint $(\\sigma , \\varrho ) \\in replace the cost function $$ by a corresponding cost function $ '$ from $ '$.", "The new instance $ I'=(V,A,)$ is an instance of $ VCSP(')$.$ If $s^{\\prime } \\colon V \\rightarrow A$ is an optimal assignment for $I^{\\prime }$ , then $s \\colon V \\rightarrow A /{\\sim }$ defined by $s(v) = [s^{\\prime }(v)]_{\\sim }$ is an optimal assignment for $I$ .", "On the other hand, if $s \\colon V \\rightarrow A /{\\sim }$ is an optimal assignment for $I$ , then any assignment $s^{\\prime } \\colon V \\rightarrow A$ , such that for each $v \\in V$ , we have $s^{\\prime }(v) \\in s(v)$ , is optimal for $I^{\\prime }$ .", "The above lemmas together with Proposition REF , imply the following: For any finite weighted algebra $\\textbf {A}$ , and any finite $\\textbf {B}\\in \\textbf {A})$ , there is a polynomial-time reduction of $\\operatorname{VCSP}(\\operatorname{Imp}(\\textbf {B}))$ to $\\operatorname{VCSP}(\\operatorname{Imp}(\\textbf {A}))$ .", "For a valued constraint language $\\Gamma $ the weighted algebra $(D, \\operatorname{wPol}(\\Gamma ))$ is the algebra $(D, \\operatorname{Pol}(\\Gamma ))$ together with the set of weightings $\\operatorname{wPol}(\\Gamma )$ .", "By Proposition  the complexity of $\\Gamma $ depends only on the weighted variety generated by the weighted algebra $(D, \\operatorname{wPol}(\\Gamma ))$ ." ], [ "Dichotomy conjecture", "An operation $t$ of arity $k$ is called a Taylor operation of an algebra (or a variety), if $t$ is idempotent and for every $j \\le k$ it satisfies an identity of the form $t(\\square _1,\\square _2, \\dots ,\\square _k) \\approx t(\\triangle _1,\\triangle _2, \\dots ,\\triangle _k),$ where all $\\square _i$ ’s and $\\triangle _i$ ’s are substituted with either $x$ or $y$ , but $\\square _j$ is $x$ whenever $\\triangle _j$ is $y$ .", "In this section we work towards a proof of the following theorem: Let $\\Gamma $ be a finite core valued constraint language.", "If $\\operatorname{Pol}^{+}(\\Gamma )$ does not have a Taylor operation, then $\\Gamma $ is NP-hard.", "We conjectureThe conjecture was suggested in a conversation by Libor Barto, however it might have appeared independently earlier.", "that these are the only cases of finite core languages which give rise to NP-hard VCSPs.", "Let $\\Gamma $ be a finite core valued constraint language.", "If $\\operatorname{Pol}^{+}(\\Gamma )$ does not have a Taylor operation, then $\\Gamma $ is NP-hard.", "Otherwise it is tractable.", "For crisp languages $\\operatorname{Pol}^{+}(\\Gamma ) = \\operatorname{Pol}(\\Gamma )$ .", "Therefore Theorem  generalizes the well-known result of Bulatov, Jeavons and Krokhin [7], [3] concerning crisp core languages.", "Similarly the above conjecture is a generalization of The Algebraic Dichotomy Conjecture for CSP.", "Later on we show that it is supported by all known partial results on the complexity of VCSPs.", "To prove Theorem  we use the following characterization of algebras possessing a Taylor operation: [Taylor [24]] Let $\\textbf {A}$ be a finite idempotent algebra, then the following are equivalent: $\\textbf {A}$ has a Taylor operation, $\\textbf {A})$ (equivalently $\\operatorname{HS}(\\textbf {A})$ ) does not contain a two-element algebra whose every term operation is a projection.", "First let us prove an auxiliary lemma.", "Let $\\Gamma $ be a finite core valued constraint language over a domain $D$ , and let $R$ be an $r$ -ary relation which is compatible with every polymorphism from $\\operatorname{Pol}^+(\\Gamma )$ .", "Then there exists a cost function $\\varrho _R$ in $\\operatorname{wRelClo}(\\Gamma )$ , and a rational number $P$ , such that for every $r$ -tuple $\\bf x$ the following conditions are satisfied: $\\varrho _R ({\\bf x}) \\ge P$ and $\\varrho _R ({\\bf x}) = P$ if and only if ${\\bf x} \\in R$ .", "Let $R=\\lbrace {\\bf x_1},\\cdots ,{\\bf x_m}\\rbrace $ be a relation as in the statement of the lemma.", "By Proposition REF there exists a cost function $\\varrho ^{\\prime } \\colon \\mathcal {O}_D^{(m)} \\rightarrow \\overline{Q}$ in $\\operatorname{wRelClo}(\\Gamma )$ , and a rational number $P$ , such that for every $f \\in \\mathcal {O}_D^{(m)}$ : $\\varrho ^{\\prime } (f) \\ge P$ , $\\varrho ^{\\prime } (f) < \\infty $ if and only if $f \\in \\operatorname{Pol}(\\Gamma )$ , $\\varrho ^{\\prime } (f) = P $ if and only if $f \\in \\operatorname{Pol}^{+}(\\Gamma )$ .", "Consider the coordinates ${\\bf b_1},\\cdots ,{\\bf b_r}$ , such that ${\\bf b_i}(j) = {\\bf x_j}(i)$ .", "Minimising the cost function $\\varrho ^{\\prime }$ over all the other coordinates we obtain a cost function $\\varrho $ satisfying the given conditions.", "(of Theorem ) Let $\\Gamma $ be a finite core valued constraint language over a domain $D$ , and let $\\Gamma _c$ be a rigid core of $\\Gamma $ as defined in Subsection REF .", "Suppose that $\\operatorname{Pol}^{+}(\\Gamma )$ does not have a Taylor operation.", "By Proposition REF we have that $\\operatorname{IdPol}^+(\\Gamma ) = \\operatorname{Pol}^+(\\Gamma _c)$ .", "Therefore, $\\operatorname{Pol}^+(\\Gamma _c)$ does not have a Taylor operation.", "Below we prove that $\\operatorname{VCSP}(\\Gamma _c)$ is NP-hard.", "This implies, by Proposition REF , that $\\operatorname{VCSP}(\\Gamma )$ is NP-hard which concludes the proof.", "Let $\\textbf {A}$ denote the idempotent algebra $\\operatorname{Pol}^+(\\Gamma _c)$ over the universe $D$ .", "By Taylor's theorem $\\operatorname{HS}(\\textbf {A})$ contains a two-element algebra $\\textbf {B}$ whose every term operation is a projection.", "By the isomorphism theorem we can consider $\\textbf {B}$ to be a quotient algebra rather than a homomorphic image of a subalgebra of $\\textbf {A}$ .", "In other words, there exists a binary relation $S$ compatible with $\\textbf {A}$ , which is an equivalence relation on some subuniverse $D^{\\prime }$ of $D$ and has two equivalence classes $[d_0]_{S}$ and $[d_1]_{S}$ .", "Moreover, the term operations defined on the set of equivalence classes of $S$ using their arbitrarily chosen representatives are all projections.", "Every relation is compatible with a two-element algebra whose every term operation is a projection.", "Consider the relation $R=\\lbrace (1,0,0), (0,1,0), (0,0,1)\\rbrace $ .", "It corresponds to the One-in-Three Sat problem, which is NP-complete [21].", "We define a ternary relation $S_{\\textsc {1in3}}$ on $D^{\\prime }$ by: $S_{\\textsc {1in3}} = \\lbrace (x_1, x_2, x_3) \\ \\colon \\text{exactly one of } x_1, x_2, x_3 \\text{ belongs to } [d_1]_S \\rbrace .$ This relation is compatible with $\\textbf {A}$ .", "Hence there exists a cost function $\\varrho _{S_{\\textsc {1in3}}}$ in $\\operatorname{wRelClo}(\\Gamma _c)$ satisfying the conditions given by Lemma .", "Now, for every instance of One-in-Three Sat it is easy to construct (in polynomial time) an instance of $\\operatorname{VCSP}(\\Gamma _c)$ such that if the instance of One-in-Three Sat has a solution then this solution gives rise to the minimal evaluation of the constructed instance.", "This finishes the reduction.", "As the Taylor operation is difficult to work with, in the following we use a characterization of Taylor algebras as the algebras possessing a cyclic term.", "[Barto and Kozik [1]] A finite idempotent algebra $\\textbf {A}$ has a Taylor operation if and only if it has a cyclic operation if and only if it has a cyclic operation of arity $p$ , for every prime $p > |\\textbf {A}|$ .", "Since $(D,\\operatorname{IdPol}^+(\\Gamma ))$ is a finite idempotent algebra it follows that: For any finite core valued constraint language $\\Gamma $ , $\\operatorname{Pol}^{+}(\\Gamma )$ has a Taylor operation if and only if it has an idempotent cyclic operation." ], [ "Two-element domain", "A complete complexity classification for valued constraint languages over a two-element domain was established in [10].", "All tractable languages have been defined via multimorphisms, which are a more restricted form of weighted polymorphisms.", "A $k$ -ary multimorphism of a language $\\Gamma $ , specified as a $k$ -tuple $\\langle f_1, \\ldots , f_k \\rangle $ of $k$ -ary operations on $D$ , is a $k$ -ary weighted polymorphism $\\omega $ of $\\Gamma $ such that $\\omega = {1\\over k}\\big (\\sum _i f_i - \\sum _i \\pi ^{(k)}_i\\big )$ .", "An operation $f \\in \\mathcal {O}_D^{(3)}$ is called a majority operation if for every $x,y \\in D$ we have that $f (x, x, y) = f (x, y, x) = f (y, x, x) = x$ .", "Similarly, an operation $f \\in \\mathcal {O}_D^{(3)}$ is called a minority operation if for every $x,y \\in D$ it satisfies $f(x,x,y) = f(x,y,x) = f(y,x,x) = y$ .", "Observe that on a two-element domain there is precisely one majority operation, which we denote by $\\operatorname{Mjrty}$ , and precisely one minority operation, which we denote by $\\operatorname{Mnrty}$ .", "[Cohen et al.", "[10]] Let $\\Gamma $ be a core valued constraint language on $D = \\lbrace 0, 1\\rbrace $ .", "If $\\Gamma $ admits at least one of the following six multimorphisms, then $\\Gamma $ is tractable.", "Otherwise it is NP-hard.", "$ \\langle \\min , \\min \\rangle $ , $ \\langle \\max , \\max \\rangle $ , $ \\langle \\min ,\\max \\rangle $ , $ \\langle \\operatorname{Mjrty}, \\operatorname{Mjrty}, \\operatorname{Mjrty}\\rangle $ , $ \\langle \\operatorname{Mnrty}, \\operatorname{Mnrty}, \\operatorname{Mnrty}\\rangle $ , $ \\langle \\operatorname{Mjrty}, \\operatorname{Mjrty}, \\operatorname{Mnrty}\\rangle $ .", "In [10] the complexity classification is given for languages that are not necessarily cores.", "It is not difficult to prove, though, that the general case is equivalent to the above theorem.", "This is because every language over a two-element domain which is not a core is tractable.", "We show that the dichotomy conjecture for VCSP agrees with the complexity classification for valued constraint languages over a two-element domain.", "Let $\\Gamma $ be a finite core valued constraint language on $D = \\lbrace 0, 1\\rbrace $ .", "Then $\\operatorname{Pol}^{+}(\\Gamma )$ has an idempotent cyclic operation if and only if $\\Gamma $ admits at least one of the following six multimorphisms.", "$ \\langle \\min , \\min \\rangle $ , $ \\langle \\max , \\max \\rangle $ , $ \\langle \\min ,\\max \\rangle $ , $ \\langle \\operatorname{Mjrty}, \\operatorname{Mjrty}, \\operatorname{Mjrty}\\rangle $ , $ \\langle \\operatorname{Mnrty}, \\operatorname{Mnrty}, \\operatorname{Mnrty}\\rangle $ , $ \\langle \\operatorname{Mjrty}, \\operatorname{Mjrty}, \\operatorname{Mnrty}\\rangle $ .", "On $D = \\lbrace 0,1\\rbrace $ there are precisely two constant operations, which we denote by $\\operatorname{Const}_0$ and $\\operatorname{Const}_1$ .", "By $\\operatorname{Inv}$ we denote the inversion operation defined by $\\operatorname{Inv}(0) =1$ and $\\operatorname{Inv}(1)=0$ .", "To prove Proposition REF we use the following theorem: [Cohen et al.", "[9]] Let $W$ be a weighted clone on $D = \\lbrace 0, 1\\rbrace $ that contains a weighting which assigns positive weight to at least one operation that is not a projection.", "Then $W$ contains one of the following nine weightings: $ \\langle \\operatorname{Const}_0 \\rangle $ , $ \\langle \\operatorname{Const}_1 \\rangle $ , $ \\langle \\operatorname{Inv}\\rangle $ , $ \\langle \\min , \\min \\rangle $ , $ \\langle \\max , \\max \\rangle $ , $ \\langle \\min ,\\max \\rangle $ , $ \\langle \\operatorname{Mjrty}, \\operatorname{Mjrty}, \\operatorname{Mjrty}\\rangle $ , $ \\langle \\operatorname{Mnrty}, \\operatorname{Mnrty}, \\operatorname{Mnrty}\\rangle $ , $ \\langle \\operatorname{Mjrty}, \\operatorname{Mjrty}, \\operatorname{Mnrty}\\rangle $ .", "(of Proposition REF ) Each of the operations $\\min $ , $\\max $ , $\\operatorname{Mjrty}$ and $\\operatorname{Mnrty}$ is idempotent and cyclic.", "If $\\Gamma $ admits at least one of the six multimorphisms listed in the statement of the proposition then obviously $\\operatorname{Pol}^{+}(\\Gamma )$ has an idempotent cyclic operation.", "For the other direction, let $\\Gamma $ be a finite core valued constraint language on $D = \\lbrace 0, 1\\rbrace $ such that $\\operatorname{Pol}^{+}(\\Gamma )$ has an idempotent cyclic operation.", "Let $\\Gamma _c$ be a rigid core of $\\Gamma $ as defined in Subsection REF .", "By Proposition REF we have that $\\operatorname{IdPol}^+(\\Gamma ) = \\operatorname{Pol}^+(\\Gamma _c)$ .", "Therefore, the weighted clone $\\operatorname{wPol}(\\Gamma _c)$ contains a weighting which assigns positive weight to at least one operation that is not a projection.", "Then $\\operatorname{wPol}(\\Gamma _c)$ contains one of the nine weightings listed in Theorem REF .", "Since the first three of them are not idempotent, it follows that $\\Gamma _c$ , and hence $\\Gamma $ , admits one of the six remaining multimorphisms, which finishes the proof." ], [ "Finite-valued languages", "[Thapper and Živný [25]] Let $\\Gamma $ be a finite-valued constraint language which is a core.", "If $\\Gamma $ admits an idempotent cyclic weighted polymorphism of some arity $m>1$ , then $\\Gamma $ is tractable.", "Otherwise it is NP-hard.", "To show that our conjecture agrees with the above complexity classification we prove the following result (which holds for general-valued languages): Let $\\Gamma $ be a core valued constraint language.", "Then $\\Gamma $ admits an idempotent cyclic weighted polymorphism of some arity $m>1$ if and only if $\\operatorname{Pol}^{+}(\\Gamma )$ contains an idempotent cyclic operation of the same arity.", "One implication is strightforward: if $\\Gamma $ admits an idempotent cyclic weighted polymorphism of some arity $m>1$ then $\\operatorname{Pol}^{+}(\\Gamma )$ contains an idempotent cyclic operation.", "To show the other implication we use a technique of constructing weighted polymorphism introduced in [18].", "The construction of a new weighted polymorphism of arity $m$ is based on grouping operations in $\\mathcal {O}_D^{(m)}$ into so-called collections and working with weightings that assign the same weight to every operation in a collection.", "Let $\\mathbb {G}$ be a fixed set of collections, i.e., subsets of $\\mathcal {O}_D^{(m)}$ , and let $\\mathbb {G}^* \\subseteq \\mathbb {G}$ be a set of collections satisfying some desired property.", "An expansion operator $\\operatorname{Exp}$ takes a collection ${g}\\in \\mathbb {G}$ and produces a probability distribution $\\delta $ over $\\mathbb {G}$ .", "We say that $\\operatorname{Exp}$ is valid for a language $\\Gamma $ if, for any $\\varrho \\in \\Gamma $ and any ${g}\\in \\mathbb {G}$ , the probability distribution $\\delta = \\operatorname{Exp}({g})$ satisfies $\\sum _{\\mathbb {G}} \\sum _{h \\in {\\delta ( \\over |} \\varrho (h({\\bf x_1, \\dots , x_m})) \\le \\sum _{g \\in {g}} {1\\over |{g}|} \\varrho (g({\\bf x_1, \\dots , x_m})),for any {\\bf x_1, \\dots , x_m} \\in \\operatorname{Feas}(\\varrho ).", "We say that the operator \\operatorname{Exp} is \\emph {non-vanishing} (with respect to the pair (\\mathbb {G}, \\mathbb {G}^*)) if, for any {g}\\in \\mathbb {G}, there exists a sequence of collections {g}_0, {g}_1, \\dots , {g}_r with {g}_0 = {g}, such that for each i \\in \\lbrace 0, \\dots , r-1\\rbrace the collection {g}_{i+1} is assigned a non-zero probability by \\operatorname{Exp}({g}_i), and {g}_r \\in \\mathbb {G}^*.", "}$ [“Expansion Lemma” [18]] Let $\\operatorname{Exp}$ be an expansion operator which is valid for the language $\\Gamma $ and non-vanishing with respect to $(\\mathbb {G},\\mathbb {G}^*)$ .", "If $\\Gamma $ admits a weighted polymorphism $\\omega $ with $\\operatorname{supp}(\\omega ) \\subseteq \\bigcup \\mathbb {G}$ , then it also admits a weighted polymorphism $\\omega ^*$ with $\\operatorname{supp}(\\omega ^*) \\subseteq \\bigcup \\mathbb {G}^*$ .", "(of Proposition REF ) In order to show the remaining implication assume that $\\operatorname{Pol}^{+}(\\Gamma )$ contains an idempotent cyclic operation $f$ of arity $m>1$ .", "There exists a weighted polymorphism $\\omega $ of $\\Gamma $ such that $f \\in \\operatorname{supp}(\\omega )$ .", "We define $\\sim $ to be the smalles equivalence relation on $\\mathcal {O}_D^{(m)}$ such that $g \\sim g^{\\prime }$ if $g(x_1,x_2, \\dots ,x_k) = g^{\\prime }(x_2, \\dots , x_k ,x_1)$ .", "Observe that if $g \\sim g^{\\prime }$ and $g \\in \\operatorname{Pol}(\\Gamma )$ then also $g^{\\prime } \\in \\operatorname{Pol}(\\Gamma )$ .", "Let $\\mathbb {G}$ consist of the equivalence classes of the relation $\\sim $ restricted to $\\operatorname{Pol}(\\Gamma )$ , and let $\\mathbb {G}^* \\subseteq \\mathbb {G}$ be the set of all one-element equivalence classes, i.e., each ${g}\\in \\mathbb {G}^*$ contains a single cyclic operation.", "We now define the expansion operator $\\operatorname{Exp}$ .", "Take an arbitrary ${g}\\in \\mathbb {G}\\setminus \\mathbb {G}^*$  (for ${g}\\in \\mathbb {G}^*$ we produce the probability distribution choosing ${g}$ with probability 1) and choose a single operation $g \\in {g}$ .", "Notice that ${g}= \\lbrace g_1, g_2, \\ldots , g_m\\rbrace $ , where $g_1 = g$ and $g_i(x_1, \\ldots , x_n) = g(x_i, x_{i+1}, \\ldots , x_{i-1})$ for $i \\in \\lbrace 2, \\ldots , m\\rbrace $ .", "Consider a weighting $\\nu = c \\cdot (\\omega [g_1, g_2, \\dots , g_m] + \\omega [g_2, \\dots , g_m, g_1] + \\dots + \\omega [g_m, g_1, \\dots , g_{m-1}]),$ where $c$ is a suitable positive rational, which we define later onNote that the definition of $\\nu $ does not depend on the choice of $g$ from ${g}$ ..", "The weighting $\\nu $ assigns a positive weight to a cyclic operation $f[g_1, g_2, \\dots , g_m]$ .", "This proves that $\\nu $ is not zero-valued, and hence in the above definition $c$ can be chosen so that the sum of positive weights in $\\nu $ equals 1.", "We say that a weighting $\\omega $ is weight-symmetric if $\\omega (g) = \\omega (g^{\\prime })$ whenever $g \\sim g^{\\prime }$ .", "It is easy to check that $\\nu $ is weight-symmertic.", "We define $\\operatorname{Exp}({g})$ to be a probability distribution $\\delta $ on $\\mathbb {G}$ such that $\\delta ( = {\\left\\lbrace \\begin{array}{ll} | \\cdot \\nu (h) &\\mbox{if } \\operatorname{supp}(\\nu ), \\\\0 & \\mbox{otherwise,} \\end{array}\\right.", "}$ where $h$ is any of the operations in $.We have already pointed out that $$ assigns a positive weight to a cyclic operation $ f[g1, g2, ..., gm]$.", "It follows that $ Exp(g)$ assigns non-zero probability to the one-element equivalence class $ {f[g1, g2, ..., gm]}$.", "Therefore, $ Exp$ is non-vanishing.$ It remains to show that $\\operatorname{Exp}$ is valid for $\\Gamma $ .", "Observe that $\\nu $ assigns negative weights only to the operations in ${g}$ .", "Since it is weight-symmetric, $\\nu (g) = - {1\\over |{g}|}$ for every $g \\in {g}$ .", "The weighting $\\nu $ might not be valid but it is not difficult to see that it satisfies the condition characterizing weighted polymorphisms, i.e., for any cost function $\\varrho \\in \\Gamma $ , and any list of tuples $\\bf x_1, \\dots , x_m \\in \\operatorname{Feas}(\\varrho )$ , we have $\\sum _{f \\in \\operatorname{Pol}_m(\\Gamma )} \\nu (f) \\cdot \\varrho (f(\\mathbf {x_1, \\dots , x_m})) & \\le 0, \\mbox{ hence } \\\\\\sum _{h \\in \\operatorname{supp}(\\nu )} \\nu (h) \\cdot \\varrho (h(\\mathbf {x_1, \\dots , x_m})) & \\le \\sum _{g \\in {g}} {1\\over |{g}|} \\cdot \\varrho (g({\\bf x_1, \\dots , x_m})), \\mbox{ but} \\\\\\sum _{h \\in \\operatorname{supp}(\\nu )} \\nu (h) \\cdot \\varrho (h(\\mathbf {x_1, \\dots , x_m})) & = \\sum _{\\mathbb {G}} \\sum _{h \\in {\\delta ( \\over |} \\cdot \\varrho (h({\\bf x_1, \\dots , x_m})).", "}This proves that \\operatorname{Exp} is valid, so by Lemma~\\ref {expansion} the language \\Gamma admits a weighted polymorphism \\omega ^* whose support contains only cyclic m-ary operations.", "Moreover, it follows from the proof of the Expansion Lemma in~\\cite {KolTZ} that \\omega ^* can be constructed so that f \\in \\operatorname{supp}(\\omega ^*).", "By Lemma~\\ref {idemp} there exists an idempotent weighted polymorphism \\omega ^{\\prime } of \\Gamma such that \\operatorname{supp}(\\omega ^*) \\cap \\operatorname{IdPol}(\\Gamma ) \\subseteq \\operatorname{supp}(\\omega ^{\\prime }).", "Its support is non-emply and contains cyclic operations only.", "This concludes the proof.$" ], [ "Conservative languages", "A valued constraint language $\\Gamma $ over a domain $D$ is called conservative if it contains all $\\lbrace 0,1\\rbrace $ -valued unary cost functions on $D$ .", "An operation $f \\in \\mathcal {O}_D^{(k)}$ is conservative if for every $x_1, \\dots , x_k \\in D$ we have that $f(x_1,\\ldots ,x_k)\\in \\lbrace x_1,\\ldots ,x_k\\rbrace $ , and a weighted polymorphism is conservative if its support contains conservative operations only.", "A Symmetric Tournament Pair (STP) is a conservative binary multimorphism $ \\langle \\sqcap , \\sqcup \\rangle $ , where both operations are commutative, i.e., $\\sqcap (x,y)=\\sqcap (y,x)$ and $\\sqcap (x,y)=\\sqcap (y,x)$ for all $x,y \\in D$ , and moreover $\\sqcap (x,y) \\ne \\sqcup (x,y)$ for all $x \\ne y$ .", "A MJN is a ternary conservative multimorphism $\\langle \\operatorname{Mj}_1,\\operatorname{Mj}_2,\\operatorname{Mn}_3\\rangle $ , such that $\\operatorname{Mj}_1,\\operatorname{Mj}_2$ are majority operations, and $\\operatorname{Mn}_3$ is a minority operation.", "[Kolmogorov and Živný [19]] Let $\\Gamma $ be a conservative constraint language over a domain $D$ .", "If $\\Gamma $ admits a conservative binary multimorphism $\\langle \\sqcap , \\sqcup \\rangle $ and a conservative ternary multimorphism $\\langle \\operatorname{Mj}_1,\\operatorname{Mj}_2,\\operatorname{Mn}_3\\rangle $ , and there is a family $M$ of two-element subsets of $D$ , such that: for every $\\lbrace x,y\\rbrace \\in M$ , $\\langle \\sqcap , \\sqcup \\rangle $ restricted to $\\lbrace x,y\\rbrace $ is an STP, for every $\\lbrace x,y\\rbrace \\notin M$ , $\\langle \\operatorname{Mj}_1,\\operatorname{Mj}_2,\\operatorname{Mn}_3\\rangle $ restricted to $\\lbrace x,y\\rbrace $ is an MJN, then $\\Gamma $ is tractable.", "Otherwise it is NP-hard.", "Observe that every weighted polymorphism of a conservative language $\\Gamma $ is conservative.", "Indeed, consider a $k$ -ary weighted polymorphism $\\omega \\in \\operatorname{wPol}(\\Gamma )$ and take any $x_1, \\ldots , x_k \\in D$ .", "Let $\\varrho \\in \\Gamma $ be a unary cost function such that $\\varrho (x_i)=0$ for $i \\in \\lbrace 1, \\ldots , k\\rbrace $ and $\\varrho (x)=1$ otherwise.", "Then $\\sum _{g \\in \\operatorname{supp}(\\omega )} \\omega (g) \\cdot \\varrho (g(x_1, \\ldots , x_k)) = \\sum _{g \\in \\operatorname{Pol}_1(\\Gamma )} \\omega (g) \\cdot \\varrho (g(x_1, \\ldots , x_k)) \\le 0,$ hence for each $g \\in \\operatorname{supp}(\\omega )$ we have that $\\varrho (g(x_1, \\ldots , x_k))=0$ , so $g(x_1, \\ldots , x_k) \\in \\lbrace x_1,\\ldots ,x_k\\rbrace $ .", "This implies that the positive clone of a conservative language is idempotent, and hence every conservative language is a core.", "We show that our conjecture agrees with the above complexity classification: Let $\\Gamma $ be a conservative constraint language over a domain $D$ .", "Then $\\operatorname{Pol}^{+}(\\Gamma )$ has an idempotent cyclic operation if and only if $\\Gamma $ admits a conservative binary multimorphism $\\langle \\sqcap , \\sqcup \\rangle $ and a conservative ternary multimorphism $\\langle \\operatorname{Mj}_1,\\operatorname{Mj}_2,\\operatorname{Mn}_3\\rangle $ , and there is a family $M$ of two-element subsets of $D$ , such that: for every $\\lbrace x,y\\rbrace \\in M$ , $\\langle \\sqcap , \\sqcup \\rangle $ restricted to $\\lbrace x,y\\rbrace $ is an STP, for every $\\lbrace x,y\\rbrace \\notin M$ , $\\langle \\operatorname{Mj}_1,\\operatorname{Mj}_2,\\operatorname{Mn}_3\\rangle $ restricted to $\\lbrace x,y\\rbrace $ is an MJN.", "Let $\\Gamma ^{\\prime }$ be the language $\\Gamma $ together with all $\\lbrace 0,\\infty \\rbrace $ -valued unary cost functions on $D$ .", "For every weighted polymorphism $\\omega \\in \\operatorname{wPol}(\\Gamma )$ there is a corresponding weighted polymorphism of $\\Gamma ^{\\prime }$ , which is equal to $\\omega $ on the conservative operations.", "Therefore $\\operatorname{Pol}^+(\\Gamma ^{\\prime })=\\operatorname{Pol}^+(\\Gamma )$ , and $\\Gamma $ admits a conservative multimorphism $\\langle f_1, \\ldots , f_k \\rangle $ if and only if $\\Gamma ^{\\prime }$ does.", "Now let $\\varrho : D \\rightarrow \\overline{\\mathbb {Q}}$ be any general-valued unary cost function.", "Observe that $\\varrho \\in \\operatorname{wRelClo}(\\Gamma ^{\\prime })$ .", "It follows that without loss of generality we can assume that $\\Gamma $ contains all general-valued unary cost functions.", "We do so in the rest of the proof.", "Observe that every polymorphism of such language is conservative.", "Assume that $\\Gamma $ admits the two conservative multimorphisms $\\langle \\sqcap , \\sqcup \\rangle $ and $\\langle \\operatorname{Mj}_1,\\operatorname{Mj}_2,\\operatorname{Mn}_3\\rangle $ described in the statement of the proposition.", "There are only four idempotent operations on a two-element domain $\\lbrace x,y\\rbrace $ , namely: $\\max $ , $\\min $ , $\\pi _1$ and $\\pi _2$ (we assume that $\\lbrace x,y\\rbrace =\\lbrace 0,1\\rbrace $ ).", "Each of the operations $\\sqcap $ , $\\sqcup $ restricted to any two-element subset $\\lbrace x,y\\rbrace $ of $D$ must be equal to one of those four.", "Therefore it is not difficult to prove, using $\\lbrace 0,1\\rbrace $ -valued unary cost functions, that for every two-element subset $\\lbrace x,y\\rbrace $ of $D$ : either $\\langle \\sqcap , \\sqcup \\rangle $ restricted to $\\lbrace x,y\\rbrace $ is an STP, or $\\sqcap $ restricted to $\\lbrace x,y\\rbrace $ is equal to $\\pi _1$ and $\\sqcup $ restricted to $\\lbrace x,y\\rbrace $ is equal to $\\pi _2$ (possibly the other way round).", "Let $M^{\\prime }$ be the set of those two-element subsets $\\lbrace x,y\\rbrace $ of $D$ , for which $\\langle \\sqcap , \\sqcup \\rangle $ restricted to $\\lbrace x,y\\rbrace $ is an STP.", "Obviously $M \\subseteq M^{\\prime }$ .", "Let $t(x,y,z) = ((x \\sqcap y)\\sqcup (x \\sqcap z)) \\sqcup (z \\sqcap y)$ .", "For every $\\lbrace x,y\\rbrace \\in M^{\\prime }$ we have that $t$ restricted to $\\lbrace x,y\\rbrace $ is the majority operation.", "For every other two-element subset $\\lbrace x,y\\rbrace $ of $D$ the operation $t$ restricted to $\\lbrace x,y\\rbrace $ is equal to $\\pi _2$ or $\\pi _3$ .", "Now let us define $m$ to be $m(x,y,z)=\\operatorname{Mj}_1(t(x,y,z),t(y,z,x),t(z,x,y)).$ Since the operation $\\operatorname{Mj}_1$ is idempotent, for every $\\lbrace x,y\\rbrace \\in M^{\\prime }$ the operation $m$ restricted to $\\lbrace x,y\\rbrace $ is the majority operation.", "Moreover, if $\\lbrace x,y\\rbrace $ does not belong to $M^{\\prime }$ then $m$ restricted to $\\lbrace x,y\\rbrace $ is equal to $\\operatorname{Mj}_1$ (with permuted arguments).", "But $\\operatorname{Mj}_1$ restricted to $\\lbrace x,y\\rbrace $ is the majority operation.", "Therefore, $m$ is a majority operation on the whole domain $D$ .", "If there is a majority operation in the idempotent clone $\\operatorname{Pol}^+(\\Gamma )$ then there is also an idempotent cyclic operation.", "This finishes the proof of the right-to-left implication.", "The proof of the other implication consists of a sequence of claims and heavily relies on the results of [19].", "Assume that $\\operatorname{Pol}^{+}(\\Gamma )$ has an idempotent cyclic operation.", "Let $M$ be a set of all two-element subsets $\\lbrace x,y\\rbrace $ of $D$ for which there exists no binary cost function $\\varrho \\in \\operatorname{wRelClo}(\\Gamma )$ such that $(x,y),(y,x) \\in \\operatorname{Feas}(\\varrho ), \\text{ and } \\varrho (x,x)+\\varrho (y,y)>\\varrho (x,y)+\\varrho (y,x).$ We prove that $\\Gamma $ admits a conservative binary multimorphism $\\langle \\sqcap , \\sqcup \\rangle $ and a conservative ternary multimorphism $\\langle \\operatorname{Mj}_1,\\operatorname{Mj}_2,\\operatorname{Mn}_3\\rangle $ such that: for every $\\lbrace x,y\\rbrace \\in M$ , $\\langle \\sqcap , \\sqcup \\rangle $ restricted to $\\lbrace x,y\\rbrace $ is an STP, for every $\\lbrace x,y\\rbrace \\notin M$ , $\\langle \\operatorname{Mj}_1,\\operatorname{Mj}_2,\\operatorname{Mn}_3\\rangle $ restricted to $\\lbrace x,y\\rbrace $ is an MJN.", "Consider the weighted algebra $(D, \\operatorname{wPol}(\\Gamma ))$ .", "Observe that every two-element subset $\\lbrace x,y\\rbrace \\subseteq D$ is a subuniverse of $D$ and let $\\textbf {B}$ be the subalgebra with universe $B = \\lbrace x,y\\rbrace $ .", "Every two-element weighted subalgebra $\\textbf {B}$ of $(D, \\operatorname{wPol}(\\Gamma ))$ contains the weighting $ \\langle \\min ,\\max \\rangle $ or $\\langle \\operatorname{Mjrty}, \\operatorname{Mjrty}, \\operatorname{Mnrty}\\rangle $ (we assume that $B=\\lbrace 0,1\\rbrace $ ).", "The weighted algebra $(D, \\operatorname{wPol}(\\Gamma ))$ contains a weighting which assigns a positive weight to an idempotent cyclic operation.", "Therefore its weighted subalgebra $\\textbf {B}$ contains a weighting which assigns a positive weight to at least one operation that is not a projection, and hence it contains one of the nine weightings listed in Theorem REF .", "The first three of them are not idempotent.", "Moreover, for each of the weightings: $ \\langle \\min , \\min \\rangle $ , $ \\langle \\max , \\max \\rangle $ , $ \\langle \\operatorname{Mjrty}, \\operatorname{Mjrty}, \\operatorname{Mjrty}\\rangle $ , $ \\langle \\operatorname{Mnrty}, \\operatorname{Mnrty}, \\operatorname{Mnrty}\\rangle $ it is easy to find a $\\lbrace 0,1\\rbrace $ -valued unary cost function that is not improved by it.", "We conclude that $\\textbf {B}$ contains the weighting $ \\langle \\min ,\\max \\rangle $ or $\\langle \\operatorname{Mjrty}, \\operatorname{Mjrty}, \\operatorname{Mnrty}\\rangle $ , which finishes the proof of Claim REF .", "Let $\\lbrace x,y\\rbrace $ be a two-element subset of $D$ .", "There exists no binary cost function $\\varrho \\in \\operatorname{wRelClo}(\\Gamma )$ such that $(x,y),(y,x) \\in \\operatorname{Feas}(\\varrho ), \\text{ and } \\varrho (x,x)+\\varrho (y,y)>\\varrho (x,y)+\\varrho (y,x),$ and at least one of the pairs $(x,x)$ , $(y,y)$ belong to $\\operatorname{Feas}(\\varrho )$ .", "Consider the weighted subalgebra $\\textbf {B}$ of $(D, \\operatorname{wPol}(\\Gamma )$ with the universe $B = \\lbrace x,y\\rbrace $ .", "Assume that $\\lbrace x,y\\rbrace =\\lbrace 0,1\\rbrace $ .", "By Claim REF the weighted subalgebra $\\textbf {B}$ contains the weighting $ \\langle \\min ,\\max \\rangle $ or $\\langle \\operatorname{Mjrty}, \\operatorname{Mjrty}, \\operatorname{Mnrty}\\rangle $ .", "Suppose that $\\textbf {B}$ contains the weighting $\\langle \\operatorname{Mjrty}, \\operatorname{Mjrty}, \\operatorname{Mnrty}\\rangle $ and let $\\omega $ be the weighted polymorphims of $\\Gamma $ that induces $\\langle \\operatorname{Mjrty}, \\operatorname{Mjrty}, \\operatorname{Mnrty}\\rangle $ .", "Let $\\varrho $ be a binary cost function like in the statement of the claim.", "Without loss of generality assume that $(x,x) \\in \\operatorname{Feas}(\\varrho )$ .", "Then $&\\sum _{g \\in \\operatorname{Pol}_3(\\Gamma )} \\omega (g) \\cdot \\varrho (g((x,y),(y,x),(x,x))) = {1\\over 3}\\big ( 2\\varrho (\\operatorname{Mjrty}((x,y),(y,x),(x,x))) + \\\\ &+ \\varrho (\\operatorname{Mnrty}((x,y),(y,x),(x,x))) -\\varrho (x,y) -\\varrho (y,x) - \\varrho (x,x)\\big ) = \\\\&= {1\\over 3}\\big (2\\varrho (x,x) + \\varrho (y,y) -\\varrho (x,y) -\\varrho (y,x) - \\varrho (x,x)\\big ) = \\\\ &= {1\\over 3}\\big (\\varrho (x,x) + \\varrho (y,y) -\\varrho (x,y) -\\varrho (y,x)\\big )>0.$ It follows that $\\varrho $ is not improved by $\\omega $ , so $\\varrho \\notin \\operatorname{wRelClo}(\\Gamma )$ .", "Similarly we show that if $\\textbf {B}$ contains the weighting $\\langle \\min ,\\max \\rangle $ and $\\omega $ is the weighted polymorphims of $\\Gamma $ that induces $\\langle \\min ,\\max \\rangle $ , then $\\varrho $ is not improved by $\\omega $ .", "Claim REF together with Theorem 9 of [19] implies that $\\Gamma $ admits a conservative binary multimorphism $\\langle \\sqcap , \\sqcup \\rangle $ such that: for every $\\lbrace x,y\\rbrace \\in M$ , $\\langle \\sqcap , \\sqcup \\rangle $ restricted to $\\lbrace x,y\\rbrace $ is an STP, if $\\lbrace x,y\\rbrace \\notin M$ then $\\sqcap $ restricted to $\\lbrace x,y\\rbrace $ is equal to $\\pi _1$ and $\\sqcup $ restricted to $\\lbrace x,y\\rbrace $ is equal to $\\pi _2$ .", "There exists an operation $m$ in $\\operatorname{Pol}^+(\\Gamma )$ which is a majority operation.", "First we prove that there exists an operation $\\operatorname{Mj}$ in $\\operatorname{Pol}^+(\\Gamma )$ such that for every $\\lbrace x,y\\rbrace \\notin M$ the operation $\\operatorname{Mj}$ restricted to $\\lbrace x,y\\rbrace $ is the majority operation.", "To this end, take any $\\lbrace x,y\\rbrace \\notin M$ .", "By the definition of $M$ there exists a binary cost function $\\varrho \\in \\operatorname{wRelClo}(\\Gamma )$ such that $(x,y),(y,x) \\in \\operatorname{Feas}(\\varrho ), \\text{ and } \\varrho (x,x)+\\varrho (y,y)>\\varrho (x,y)+\\varrho (y,x).$ Moreover, by Claim REF none of the pairs $(x,x)$ , $(y,y)$ belongs to $\\operatorname{Feas}(\\varrho )$ .", "Therefore: there is no operation in $\\operatorname{Pol}^+(\\Gamma )$ that restricted to $\\lbrace x,y\\rbrace $ is the $\\max $ or $\\min $ operation (such an operation would not even be a polymorphism of $\\varrho $ ), and it follows from Claim REF that there exists an operation $f$ in $\\operatorname{Pol}^+(\\Gamma )$ such that $f$ restricted to $\\lbrace x,y\\rbrace $ is the majority operation.", "By Proposition 3.1 of [6] if follows from the conditions REF and REF above that there exists an operation $\\operatorname{Mj}$ in $\\operatorname{Pol}^+(\\Gamma )$ such that for every $\\lbrace x,y\\rbrace \\notin M$ the operation $\\operatorname{Mj}$ restricted to $\\lbrace x,y\\rbrace $ is the majority operation.", "Let $t(x,y,z) = ((x \\sqcap y)\\sqcup (x \\sqcap z)) \\sqcup (z \\sqcap y)$ .", "For every $\\lbrace x,y\\rbrace \\in M$ we have that $t$ restricted to $\\lbrace x,y\\rbrace $ is the majority operation.", "For every other two-element subset $\\lbrace x,y\\rbrace $ of $D$ the operation $t$ restricted to $\\lbrace x,y\\rbrace $ is equal to $\\pi _3$ .", "Now let us define $m$ to be $m(x,y,z)=\\operatorname{Mj}(t(x,y,z),t(y,z,x),t(z,x,y)).$ Since $\\operatorname{Mj}$ is idempotent, for every $\\lbrace x,y\\rbrace \\in M$ the operation $m$ restricted to $\\lbrace x,y\\rbrace $ is the majority operation.", "Moreover, if $\\lbrace x,y\\rbrace $ does not belong to $M$ then $m$ restricted to $\\lbrace x,y\\rbrace $ is equal to $\\operatorname{Mj}$ (with permuted arguments).", "But $\\operatorname{Mj}$ restricted to $\\lbrace x,y\\rbrace $ is the majority operation.", "Therefore, $m$ is a majority operation on the whole domain $D$ .", "By [19] if follows from Claims REF and REF that $\\Gamma $ admits a conservative ternary multimorphism $\\langle \\operatorname{Mj}_1,\\operatorname{Mj}_2,\\operatorname{Mn}_3\\rangle $ such that for every $\\lbrace x,y\\rbrace \\notin M$ , $\\langle \\operatorname{Mj}_1,\\operatorname{Mj}_2,\\operatorname{Mn}_3\\rangle $ restricted to $\\lbrace x,y\\rbrace $ is an MJN, which finishes the proof of Proposition REF ." ], [ "Infinite Constraint Languages.", "We finish this section by showing that a variation of Theorem  holds for infinite constraint languages.", "For that, we need to introduce fractional polymorphisms which correspond to weighted polymorphisms that can take real values.", "An $m$ -ary fractional operation $\\omega $ on $D$ is a probability distribution on $\\mathcal {O}_D^{(m)}$ .", "As for weightings, the set of operations to which a fractional operation $\\omega $ assigns a positive probability is called the support of $\\omega $ and denoted $\\operatorname{supp}(\\omega )$ .", "An $m$ -ary fractional operation $\\omega $ on $D$ is a fractional polymorphism of a cost function $\\varrho $ if, for any list of $r$ -tuples $\\bf x_1, \\dots , x_m \\in \\operatorname{Feas}(\\varrho )$ , we have $\\sum _{g \\in \\operatorname{supp}(\\omega )} \\omega (g) \\varrho (g({\\bf x_1},\\dots ,{\\bf x_m})) \\le {1\\over m} (\\varrho ({\\bf x_1})+\\dots +\\varrho ({\\bf x_m})).$ For a constraint language $\\Gamma $ we denote by $\\operatorname{fPol}(\\Gamma )$ the set of those fractional operations that are fractional polymorphisms of all cost functions $\\varrho \\in \\Gamma $ .", "Let $\\operatorname{fPol}^+(\\Gamma ) = \\lbrace g \\in \\mathcal {O}_D \\ | \\ g \\in \\operatorname{supp}(\\omega ), \\ \\omega \\in \\operatorname{fPol}(\\Gamma )\\rbrace $ .", "It is easy to see that $\\operatorname{fPol}^+(\\Gamma )$ is a clone (the proof is similar to that of Proposition REF ).", "Let $\\Gamma $ be a finite constraint language.", "Then $\\operatorname{fPol}^+(\\Gamma ) = \\operatorname{Pol}^+(\\Gamma )$ .", "Obviously $\\operatorname{Pol}^+(\\Gamma ) \\subseteq \\operatorname{fPol}^+(\\Gamma )$ .", "Let $f \\in \\operatorname{fPol}^+(\\Gamma )$ .", "This can be equivalently expressed by saying that some LP with rational coefficients has a solution.", "It is well known that every LP with rational coefficients has an optimal solution with rational coefficients (see e.g. [22]).", "It follows that $f \\in \\operatorname{Pol}^+(\\Gamma )$ .", "There exists an infinite valued constraint language $\\Gamma $ such that $\\operatorname{Pol}^+(\\Gamma ) \\operatorname{fPol}^+(\\Gamma )$ .", "An example of such language is given in [25].", "To deal with infinite languages we slightly modify the notion of a core.", "We say that a valued constraint language $\\Gamma $ is a core if all operations in $\\operatorname{fPol}^+_1(\\Gamma )$ are bijective.", "It follows from Proposition REF that for finite languages this definition coincides with the old one.", "Moreover, the following proposition states that all positively weighted unary polymorphisms can be captured in a finite part of $\\Gamma $ .", "If a valued constraint language $\\Gamma $ is a core then there exists a finite language $\\Gamma ^{\\prime } \\subseteq \\Gamma $ such that $\\operatorname{fPol}_1^+(\\Gamma ) = \\operatorname{fPol}_1^+(\\Gamma ^{\\prime })$ .", "We will use the following lemma (Lemma 7 from [25]), which is an immediate consequence of the fact that the space of $m$ -ary fractional operations on a fixed domain $D$ is compact.", "Let $\\Gamma $ be a valued constraint language and let $O \\subseteq \\mathcal {O}_D^{(m)}$ .", "If for every finite $\\Gamma ^{\\prime } \\subseteq \\Gamma $ there exists a fractional polymorphism with support in $O$ , then $\\Gamma $ has a fractional polymorphism with support in $O$ .", "(of Proposition REF ) Let $\\Gamma $ be a core valued constraint language and let $B$ be the set of such bijective operations $f$ on $D$ that, for all cost functions $\\varrho \\in \\Gamma $ , satisfy $\\varrho \\circ f = \\varrho $ .", "A proof analogous to the one of Proposition REF shows that a unary fractional operation $\\omega $ is a fractional polymorphism of $\\Gamma $ if and only if $\\operatorname{supp}(\\omega ) \\subseteq B$ .", "Take a language $\\Gamma ^{\\prime }$ such that $\\operatorname{fPol}_1^+(\\Gamma ^{\\prime }) \\ne \\operatorname{fPol}_1^+(\\Gamma )$ .", "This means that there exists a fractional polymorphism $\\omega $ of $\\Gamma ^{\\prime }$ such that there is some operation $g$ from outside $B$ in $\\operatorname{supp}(\\omega )$ .", "Then there exists a fractional polymorphism $\\omega ^{\\prime }$ of $\\Gamma ^{\\prime }$ such that $\\operatorname{supp}(\\omega ^{\\prime }) \\subseteq \\mathcal {O}_D^{(1)} \\setminus B$ .", "We construct it from the fractional polymorphism $\\omega $ by assigning probability 0 to all operations in $B$ and scaling to get a proper probability distribution.", "It is not difficult to check that the obtained fractional operation $\\omega ^{\\prime }$ is a fractional polymorphism of $\\Gamma ^{\\prime }$ .", "Now suppose that none of the finite $\\Gamma ^{\\prime } \\subseteq \\Gamma $ satisfies $\\operatorname{fPol}_1^+(\\Gamma ) = \\operatorname{fPol}_1^+(\\Gamma ^{\\prime })$ .", "Then every finite $\\Gamma ^{\\prime } \\subseteq \\Gamma $ has a fractional polymorphism with support in $\\mathcal {O}_D^{(1)} \\setminus B$ .", "It follows from Lemma REF that $\\Gamma $ has a fractional polymorphism with support in $\\mathcal {O}_D^{(1)} \\setminus B$ , which is a contradiction.", "A proof analogous to the one of Proposition REF shows that every constraint language $\\Gamma $ has a computationally equivalent language $\\Gamma ^{\\prime }$ which is a core (in the new sense).", "For every valued constraint language $\\Gamma $ there exists a core language $\\Gamma ^{\\prime }$ , such that the valued constraint language $\\Gamma $ is tractable if and only if $\\Gamma ^{\\prime }$ is tractable, and it is NP-hard if and only if $ \\Gamma ^{\\prime }$ is NP-hard.", "Moreover, using Proposition REF , we can show that constants can be added to a core language (the proposition implies that, for a core language $\\Gamma $ , the relation characterizing $\\operatorname{fPol}_1^+(\\Gamma )$ belongs to $\\operatorname{wRelClo}(\\Gamma )$ ).", "The proof follows along the same lines as that of Proposition REF .", "Let $\\Gamma $ be a valued constraint language which is a core.", "The valued constraint language $\\Gamma _c$ is a rigid core.", "Moreover, $\\Gamma $ is tractable if and only if $\\Gamma _c$ is tractable, and $\\Gamma $ is NP-hard if and only if $\\Gamma _c$ is NP-hard.", "Let $\\Gamma $ be a core valued constraint language.", "If, for every finite $\\Gamma ^{\\prime }\\subseteq \\Gamma $ , the set $\\operatorname{fPol}^{+}(\\Gamma ^{\\prime })$ has a Taylor operation, then so does $\\operatorname{fPol}^+(\\Gamma )$ .", "Consider a finite $\\Gamma ^{\\prime } \\subseteq \\Gamma $ .", "Since $\\Gamma $ is a core, by Proposition REF there exists a finite $\\hat{\\Gamma } \\subseteq \\Gamma $ which is a core.", "By $\\overline{\\Gamma ^{\\prime }}$ we denote the language $\\Gamma ^{\\prime } \\cup \\hat{\\Gamma }$ .", "It is a finite subset of $\\Gamma $ , and hence it has a Taylor operation.", "Fix some prime number $p > |D|$ .", "It follows from Theorem  and Theorem  that $\\operatorname{Pol}^{+}(\\overline{\\Gamma ^{\\prime }})$ has an idempotent cyclic operation of arity $p$ , and therefore by Proposition REF $\\overline{\\Gamma ^{\\prime }}$ admits an idempotent cyclic weighted polymorphism $\\omega ^{\\prime }$ of arity $p$ .", "Let $O$ denote the set of $p$ -ary idempotent cyclic operations on $D$ .", "From the weighted polymorphism $\\omega ^{\\prime }$ it is easy to construct a fractional polymorphism $\\overline{\\omega ^{\\prime }}$ such that $\\operatorname{supp}({\\overline{\\omega ^{\\prime }}}) \\subseteq O$ .", "Since $\\Gamma ^{\\prime } \\subseteq \\overline{\\Gamma ^{\\prime }}$ , we have that $\\overline{\\omega ^{\\prime }}$ is a fractional polymorphism of $\\Gamma ^{\\prime }$ with support in $O$ .", "This holds for every finite $\\Gamma ^{\\prime } \\subseteq \\Gamma $ .", "Hence, by Lemma REF there exists a fractional polymorphism $\\omega $ of $\\Gamma $ with support in $O$ .", "It immediately follows that if $\\Gamma $ does not have a Taylor operation in $\\operatorname{fPol}^+(\\Gamma )$ then it is NP-hard.", "Let $\\Gamma $ be a core valued constraint language.", "If $\\operatorname{fPol}^{+}(\\Gamma )$ does not have a Taylor operation, then $\\Gamma $ is NP-hard." ] ]
1403.0476
[ [ "Topos Semantics for Higher-Order Modal Logic" ], [ "Abstract We define the notion of a model of higher-order modal logic in an arbitrary elementary topos $\\mathcal{E}$.", "In contrast to the well-known interpretation of (non-modal) higher-order logic, the type of propositions is not interpreted by the subobject classifier $\\Omega_{\\mathcal{E}}$, but rather by a suitable complete Heyting algebra $H$.", "The canonical map relating $H$ and $\\Omega_{\\mathcal{E}}$ both serves to interpret equality and provides a modal operator on $H$ in the form of a comonad.", "Examples of such structures arise from surjective geometric morphisms $f : \\mathcal{F} \\to \\mathcal{E}$, where $H = f_\\ast \\Omega_{\\mathcal{F}}$.", "The logic differs from non-modal higher-order logic in that the principles of functional and propositional extensionality are no longer valid but may be replaced by modalized versions.", "The usual Kripke, neighborhood, and sheaf semantics for propositional and first-order modal logic are subsumed by this notion." ], [ "Introduction", "In many conventional systems of semantics for quantified modal logic, models are built on presheaves.", "Given a set $K$ of “possible worlds\", Kripke's semantics [11], for instance, assigns to each world $k \\in K$ a domain of quantification $P(k)$ — regarded as the set of possible individuals that “exist” in $k$ — and then $\\exists x \\, \\varphi $ is true at $k$ iff some $a \\in P(k)$ satisfies $\\varphi $ at $k$ .", "David Lewis's counterpart theory [13] does the same (though it further assumes that $P(k)$ and $P(l)$ are disjoint for $k \\ne l \\in K$ ).", "Such an assignment $P$ of domains to worlds is a presheaf $P : K \\rightarrow \\textbf {Sets}$ over the set of worlds, thus an object of the topos $\\textbf {Sets}^K$ .", "(Due to the disjointness assumption one may take counterpart theory as using objects of the slice category $\\textbf {Sets}/K$ , which however is categorically equivalent to $\\textbf {Sets}^K$ .)", "Kripke-sheaf semantics for quantified modal logic [7], [4], [6], [22] is another example of this sort.", "Indeed, both counterpart theory and Kripke-sheaf semantics interpret unary formulas by subsets of the “total set of elements\" $\\sum _{k \\in K} P(k)$ , and, more generally, $n$ -ary formulas by relations of type: $\\mathcal {P}(\\sum _{k \\in K} P(k)^n) \\cong \\mathrm {Sub}_{\\textbf {Sets}^K}(P^n).$ In fact, counterpart theory and Kripke-sheaf semantics interpret the non-modal part of the logic in the same way.", "(Kripke's semantics differs somewhat in interpreting $n$ -ary formulas instead as subsets of $K \\times (\\bigcup _{k \\in K} P(k))^n$ .)", "Among these presheaf-based semantics, the principal difference consists in how to interpret the modal operator $\\Box $ .", "Let K be a set of worlds $K =|\\textbf {K}|$ equipped with a relation $k\\le j$ of “accessibility”.", "Kripke declares that an individual $a \\in \\bigcup _{k \\in \\textbf {K}} P(k)$ satisfies a property $\\Box \\varphi $ at world $k \\in \\textbf {K}$ iff $a$ satisfies $\\varphi $ in all $j\\ge k$ .", "Lewis instead introduces a “counterpart” relation among individuals, and deems that $a \\in P(k)$ satisfies $\\Box \\varphi $ iff all counterparts of $a$ satisfy $\\varphi $ .", "We may take Kripke-sheaf semantics as giving a special case of Lewis's interpretation: Assuming K to be a preorder, the semantics takes a presheaf $P : \\textbf {K} \\rightarrow \\textbf {Sets}$ on $\\textbf {K}^{op}$ , and not just on the underlying set $|\\textbf {K}|$ , so that a model comes with comparison maps $\\alpha _{kj} : P(k) \\rightarrow P(j)$ whenever $k \\le j$ in K. Then $\\alpha _{kj} : P(k) \\rightarrow P(j)$ gives a counterpart relation: $\\alpha _{kj}(a)$ is the counterpart in the world $j$ of the individual $a \\in P(k)$ , so that $a$ satisfies $\\Box \\varphi $ iff $\\alpha _{kj}(a)$ satisfies $\\varphi $ for all $j \\ge k$ .", "(Notable differences between Kripke-sheaf semantics and Lewis's are the following: In the former, $\\textbf {K}$ can be any preorder, whereas Lewis only considers the universal relation on $\\textbf {K}$ .", "Also, since $P$ is a presheaf, the former assumes that $a \\in P(k)$ has one and only one counterpart in every $j\\ge k$ .)", "In terms of interior operators, this gives an interpretation of $\\Box $ on the poset $\\mathrm {Sub}_{\\textbf {Sets}^{|\\textbf {K}|}}(uP)$ of sub-presheaves of $uP$ where $u : \\textbf {Sets}^{\\textbf {K}} \\rightarrow \\textbf {Sets}^{|\\textbf {K}|}$ is the evident forgetful functor.", "Note that such sub-presheaves are just subsets of $\\sum _{k \\in K} P(k)$ .", "Explicitly, given the presheaf $P$ on $\\textbf {K}$ and any subset $\\varphi \\subseteq uP$ of elements of $\\sum _{k \\in K} P(k)$ , then $\\Box \\varphi \\subseteq \\varphi \\subseteq uP$ is the largest subpresheaf contained in $\\varphi $ .", "Observe that $u$ is the inverse image part of a fundamental example of a geometric morphism between toposes, namely, $u = i^*$ for the (surjective) geometric morphism $i^*\\dashv i_* : \\textbf {Sets}^{|\\textbf {K}|} \\longrightarrow \\textbf {Sets}^{\\textbf {K}}$ induced by the “inclusion” $i : |\\textbf {K}| \\hookrightarrow \\textbf {K}$ of the underlying set $|\\textbf {K}|$ into K. In particular, $u$ is restriction along $i$ .", "This observation leads to a generalization of these various presheaf models to a general topos-theoretic semantics for first-order modal logic [1], [5], [15], [19], which gives a model based on any surjective geometric morphism $f : \\mathcal {F} \\rightarrow \\mathcal {E}$ .", "Indeed, for each $A$ in $\\mathcal {E}$ , the inverse image part $f^\\ast : \\mathcal {E} \\rightarrow \\mathcal {F}$ restricts to subobjects to give an injective complete distributive lattice homomorphism $\\Delta _A : \\mathrm {Sub}_{\\mathcal {E}}(A)\\rightarrow \\mathrm {Sub}_{\\mathcal {F}}(f^\\ast A)$ , which always has a right adjoint $\\Gamma _A$ .", "Composing these yields an endofunctor $\\Delta _A \\Gamma _A$ on the Heyting algebra $\\mathrm {Sub}_{\\mathcal {F}}(f^\\ast A)$ : $@C+0.5pc{\\mathrm {Sub}_{\\mathcal {F}}(f^\\ast A)@{}[r]|-{\\top }@<1ex>[r]^-{\\Gamma _A}!R(-.8) @(ul,dl)_{\\Delta _A \\Gamma _A} &\\mathrm {Sub}_{\\mathcal {E}}(A)@<1ex>[l]^-{\\Delta _A}}$ In the special case considered above, the interior operation on the “big algebra\" $\\mathcal {P}(\\sum _{k \\in \\textbf {K}} P(k)) \\cong \\mathrm {Sub}_{\\textbf {Sets}^{|\\textbf {K}|}}(u P)$ determines the “small algebra\" $\\mathrm {Sub}_{\\textbf {Sets}^{\\textbf {K}}}(P)$ as the Heyting algebra of upsets in $\\sum _{k \\in \\textbf {K}} P(k)$ (an upset $S \\subseteq \\sum _{k \\in \\textbf {K}} P(k)$ is a subset that is closed under the counterpart relation: $a \\in S$ for $a \\in P(k)$ implies $\\alpha _{kj}(a) \\in S$ for all $j \\ge k$ .)", "Moreover, $\\Gamma _P$ is the operation giving “the largest upset contained in ...”, and $\\Delta _P$ is the inclusion of upsets into the powerset.", "In this case, the logic is “classical\", since the powerset is a Boolean algebra.", "In the general case, the operator $\\Box $ is of course interpreted by $\\Delta _A \\Gamma _A$ , which always satisfies the axioms for an S4 modality, since $\\Delta _A \\Gamma _A$ is a left exact comonad.", "The specialist will note that both $\\Delta _A$ and $\\Gamma _A$ are natural in $A$ , in a suitable sense, so that this interpretation will satisfy the Beck-Chevalley condition required for it to behave well with respect to substitution, interpreted as pullback (see [1]).", "This, then, is how topos-theoretic semantics generalizes Kripke-style and related semantics for quantified modal logic (cf. [1]).", "Now let us further observe that, since $\\mathcal {F}$ is a topos, it in fact has enough structure to also interpret higher-order logic, and so a geometric morphism $f : \\mathcal {F} \\rightarrow \\mathcal {E}$ will interpret higher-order modal logic.", "This is the logic that the current paper investigates.", "The first step of our approach is to observe that, because higher-order logic includes a type of “propositions\", interpreted by a subobject classifier $\\Omega $ , the natural operations on the various subobject lattices in (REF ) can be internalized as operations on $\\Omega $ .", "Moreover, the relevant part of the geometric morphism $f : \\mathcal {F} \\rightarrow \\mathcal {E}$ , giving rise to the modal operator, can also be internalized, so that one really just needs the topos $\\mathcal {E}$ and a certain algebraic structure on its subobject classifier $\\Omega _\\mathcal {E}$ .", "That structure replaces the geometric morphism $f$ by the induced operations on the internal algebras $f_*\\Omega _\\mathcal {F}$ and $\\Omega _\\mathcal {E}$ inside the topos $\\mathcal {E}$ .", "More generally, the idea is to describe a notion of an “algebraic\" model inside a topos $\\mathcal {E}$ , using the fact that S4 modalities always occur as adjoint pairs between suitable algebras.", "In a bit more detail, the higher-order logical language will be interpreted w.r.t.", "a complete Heyting algebra $H$ in $\\mathcal {E}$ , extending ideas from traditional algebraic semantics for intuitionistic logic [16], [21].", "The modal operator on $H$ arises from an (internal) adjunction $@C+0.5pc{H@{}[r]|-{\\top }@<1ex>[r]^-{\\tau }!R(-.2) @(ul,dl)_{i\\tau } &\\Omega _\\mathcal {E}@<1ex>[l]^-{i}}$ where $i$ is a monic frame map and $\\tau $ classifies the top element of $H$ .", "(This, of course, is just the unique map of locales from $H$ to the terminal locale.)", "Externally, for each $A\\in \\mathcal {E}$ , we then have a natural adjunction between Heyting algebras, $@C+0.5pc{\\mathrm {Hom}_{\\mathcal {E}}(A,H)@{}[r]|-{\\top }@<1ex>[r]^-{\\tau _A}!R(-.8) @(ul,dl)_{i_A\\tau _A} &\\mathrm {Sub}_{\\mathcal {E}}(A)@<1ex>[l]^-{i_A}}$ defined by composition as indicated in the following diagram: ${& H @<1ex>[d]^\\tau @{}[d]|\\dashv \\\\A @/^.5pc/[ur] [r] &\\Omega _{\\mathcal {E}}@<1ex>[u]^i\\\\}.$ Comparing (REF ) to (REF ), we note that in the case $H=f_*\\Omega _\\mathcal {F}$ for a geometric morphism $f : \\mathcal {F}\\rightarrow \\mathcal {E}$ we have: $\\mathrm {Hom}_{\\mathcal {E}}(A,H) = \\mathrm {Hom}_{\\mathcal {E}}(A,f_*\\Omega _\\mathcal {F}) \\cong \\mathrm {Hom}_{\\mathcal {F}}(f^*A,\\Omega _\\mathcal {F}) \\cong \\mathrm {Sub}_{\\mathcal {F}}(f^*A),$ as required.", "In this way, the topos semantics formulated in terms of a geometric morphism $f : \\mathcal {F} \\rightarrow \\mathcal {E}$ gives rise to an example of the required “algebraic\" structure (REF ), with $H = f_\\ast \\Omega _\\mathcal {F}$ , and the same semantics for first-order modal logic can also be defined in terms of the latter.", "On the other hand, every algebraic model in $\\mathcal {E}$ arises in this way from a geometric morphism from a suitable topos $\\mathcal {F}$ , namely the topos of internal sheaves on $H$ .", "Thus, as far as the interpretation of first-order logic goes, the algebraic approach is equivalent to the geometric one (the latter restricted to localic morphisms, which is really all that is relevant for the interpretation).", "The advantage of the algebraic approach for higher-order logic will become evident in what follows.", "To give just one example, we shall see how the interpretation results in a key new (inherently topos-theoretic) treatment of equality which illuminates the relation between modality and intensionality.", "The goal of this paper is both to present the new idea of algebraic topos semantics for higher-order modal logic and to revisit the accounts of first-order semantics that are scattered in the literature, putting them into perspective from the point of view of the unifying framework developed here.", "The question of completeness will be addressed in a separate paper [3], extending the result in [2].", "In the remainder of this paper, we first review the well-known topos semantics for (intuitionistic) higher-order logic and describe the adjunction $i \\dashv \\tau $ in some detail.", "The second section then states the formal system of higher-order modal logic that is considered here and gives the definition of its models.", "The third section discusses in detail the failure of the standard extensionality principles and the soundness of the modalized versions thereof.", "We then show how the semantics based on geometric morphisms can be captured within the present, algebraic framework.", "The last section states the representation theorem mentioned above.", "For general background in topos theory (particularly for section ) we refer the reader to [9], [10], [14], and for background on higher-order type theory to [8], [9], [12].", "We assume some basic knowledge of category-theoretical concepts, but will recall essential definitions and proofs so as to make the paper more accessible.", "The algebraic approach pursued here was first investigated by Hans-Jörg Winkler and the first author, and some of these results were already contained in [23].", "Finally, we have benefitted from many conversations with Dana Scott, whose ideas and perspective have played an obvious role in the development of our approach." ], [ "Frame-valued logic in a topos", "Recall that a topos $\\mathcal {E}$ is a cartesian closed category with equalizers and a subobject classifier $\\Omega _\\mathcal {E}$ .", "The latter is defined as an object $\\Omega _\\mathcal {E}$ together with an isomorphism $\\mathrm {Sub}_{\\mathcal {E}}(A) \\cong \\mathrm {Hom}_{\\mathcal {E}}(A,\\Omega _\\mathcal {E}),$ natural in $A$ (w.r.t.", "pullback on the left, and precomposition on the right).", "Equivalently, there is a distinguished monomorphism $\\top : 1 \\rightarrow \\Omega _{\\mathcal {E}}$ such that for each subobject $M \\rightarrowtail A$ there is a unique map $\\mu : A \\rightarrow \\Omega _{\\mathcal {E}}$ for which $M$ arises as the pullback of $\\top $ along $\\mu $ : ${M [r] @{ >->}[d] &1 [d]^\\top \\\\A [r]_\\mu & \\Omega _{\\mathcal {E}}}$ This definition determines $\\Omega _\\mathcal {E}$ up to isomorphism.", "The map $\\mu $ is called the classifying map of $M$ .", "The category Sets is a topos with subobject classifier the two-element set $\\textbf {2}$ .", "The classifying maps are the characteristic functions of subsets of a given set $A$ .", "We list some further examples that will play a role later on.", "Example 1.1 An important example is the subobject classifier in the topos of $I$ -indexed families of set, for some fixed set $I$ ; equivalently the functor category $\\textbf {Sets}^I$ .", "It is a functor $\\Omega : I \\rightarrow \\textbf {Sets}$ with components $\\Omega (i) = \\textbf {2}$ .", "The subobject classifier in $\\textbf {Sets}^{\\textbf {C}^{op}}$ , for any small category $\\textbf {C}$ , is described as follows.", "For any object $C$ in C, $\\Omega (C)$ is the set of all sieves $\\sigma $ on $C$ , i.e.", "sets of arrows $h$ with codomain $C$ such that $h \\in \\sigma $ implies $h \\mathbin {\\circ } f \\in \\sigma $ , for all $f$ with $\\text{cod}(f) = \\text{dom}(h)$ .", "For an arrow $g : D \\rightarrow C$ in C, $\\Omega (g)(\\sigma )$ is the restriction of $\\sigma $ along $g$ : $\\Omega (g)(\\sigma ) = \\lbrace f : X \\rightarrow D \\mid g \\mathbin {\\circ } f \\in \\sigma \\rbrace ,$ which is a sieve on $D$ .", "The mono $\\top : 1 \\rightarrow \\Omega $ is the natural transformation whose components pick out the maximal sieve $\\top _C$ on $C$ , i.e.", "the set of all arrows with codomain $C$ (the terminal object 1 being pointwise the singleton).", "The classifying map $\\chi _m$ of a subfunctor $m : E \\rightarrowtail F$ has components $(\\chi _m)_C(a) = \\lbrace f : X \\rightarrow C \\mid F(f)(a) \\in E(X)\\rbrace .$ In particular, if C is a preorder, then $\\Omega (C)$ is the set of all downward closed subsets of $\\downarrow C$ .", "Since in this case there is at most one arrow $g : D \\rightarrow C$ , the function $F(g)$ may be thought of as the restriction of the set $F(C)$ to $F(D)$ along the inequality $D \\le C$ .", "$\\Box $ Each $\\Omega _{\\mathcal {E}}$ is a complete Heyting algebra, internal in $\\mathcal {E}$ .", "Generally, the notion of Heyting algebra makes sense in any category with finite limits.", "It is an object $H$ in $\\mathcal {E}$ with maps ${1 [r]^{\\top ,\\bot } & H & H \\times H [l]_-{\\wedge , \\vee , \\Rightarrow }\\\\}$ that provide the Heyting structure on $H$ .", "These maps are to make certain diagrams commute, corresponding to the usual equations defining a Heyting algebra.", "For instance, commutativity of ${H \\times 1 [r]^{1 \\times \\top } [dr]_{\\pi _1} & H \\times H [d]^\\wedge \\\\& H\\\\}$ corresponds to the axiom $x \\wedge \\top = x$ , for any $x \\in H$ .", "The correspondence between the usual equational definition and commutative diagrams in a category $\\textbf {C}$ can be made precise using the internal language of $\\textbf {C}$ [14].", "The induced partial ordering on $H$ is constructed as the equalizer ${E @{ >->}[r] & H \\times H @<0.75ex>[r]^-{\\wedge } @<-0.75ex>[r]_-{\\pi _1} & H ,} $ corresponding to the usual definition $x \\le y \\ \\text{ iff }\\ x \\wedge y = x.$ The description of arbitrary joins and meets additionally requires the existence of exponentials and is an internalization of how set-indexed joins and meets in set-structures can be expressed via a suitable adjunction.", "For any object $I$ in $\\mathcal {E}$ , there is an arrow $\\Delta _I : H \\longrightarrow H^I$ that is the result of applying the functor $H^{(-)}$ to the unique map $I \\longrightarrow 1_\\mathcal {E}$ in $\\mathcal {E}$ .", "In detail, $\\Delta _I : H \\rightarrow H^I$ is the exponential transpose of $\\pi _1 : H \\times I \\rightarrow H$ across the adjunction $(-) \\times I \\dashv (-)^I$ .", "Set-theoretically, for any $x \\in H$ , $\\Delta _I(x)(i) = x$ , for all $i \\in I$ .", "The object $H^I$ inherits a poset structure (in fact, a Heyting structure) from $H$ , which set-theoretically translates into the pointwise ordering.", "$I$ -indexed joins $\\bigvee _I$ and meets $\\bigwedge _I$ are given by internal left and right adjoints to $\\Delta _I$ , respectively.", "After all, joins and meets are coproducts and products in the Heyting algebra $H$ , and these can always be defined by adjoints in exactly that way, regarding $H$ as in internal category in $\\mathcal {E}$ .", "This is analogous to externally defining $I$ -indexed products (coproducts) of families $(A_i)_{i \\in I}$ of objects in a category $\\textbf {C}$ by right (left) adjoints to the functor $\\Delta _I : \\textbf {C} \\longrightarrow \\textbf {C}^I$ Example 1.2 In case $\\mathcal {E} = \\textbf {Sets}$ , the right adjoint $\\forall _I$ to $\\Delta _I$ is explicitly computed as $\\forall _I(f) = \\bigvee \\lbrace a \\in H \\mid \\Delta _I(a) \\le f\\rbrace ,$ following the standard description of the right adjoint to a map of complete join-semilattices, in this case $\\Delta _I$ .", "In fact, it is not hard to see that $\\forall _I(f) = \\bigwedge _{i \\in I} f(i).$ The left adjoint $\\exists _I \\dashv \\Delta _I$ is described dually.", "$\\Box $ Example 1.3 An important case that will be useful later is where the category in question is of the form $\\textbf {Sets}^{\\textbf {C}^{op}}$ , for a small category $\\textbf {C}$ .", "Products in $\\textbf {Sets}^{\\textbf {C}^{op}}$ are computed pointwise.", "In particular, a Heyting algebra $H$ in $\\textbf {Sets}^{\\textbf {C}^{op}}$ has pointwise natural structure.", "That is to say, each $H(C)$ , for $C$ in C, is a Heyting algebra in such a way that e.g.", "for all binary operations $\\star $ on $H$ , $H(f) \\mathbin {\\circ } \\star _D = \\star _C \\mathbin {\\circ } (H(f) \\times H(f))$ , for any arrow $f : C \\rightarrow D$ in C. This is because the structure maps, being arrows in $\\textbf {Sets}^{\\textbf {C}^{op}}$ , are natural transformations.", "Naturality in particular means that for each $f : C \\rightarrow D$ in $\\textbf {C}$ , the map $H(f)$ preserves the Heyting structure.", "By contrast, exponentials are not computed pointwise but by the formulas $H^I(C) = \\mathrm {Hom}_{}(\\textbf {y}C \\times I,H) \\\\H^I(f) : \\eta \\mapsto \\eta \\mathbin {\\circ } (\\textbf {y}f \\times 1_I),$ where $\\textbf {y}C$ denotes the contravariant functor $\\mathrm {Hom}_{\\textbf {C}}(-,C)$ .", "The induced Heyting structure on $H^I$ is the pointwise one at each component.", "In particular, for any $\\eta , \\mu : \\textbf {y}C \\times I \\rightarrow H$ , $\\eta \\le \\mu \\ (\\text{in } H^I(C))& \\ \\text{ iff }\\ \\eta _D \\le \\mu _D, \\text{ for each }\\ D \\in \\textbf {C} \\\\& \\ \\text{ iff }\\ \\eta _D(f,b) \\le \\mu _D(f,b)\\ (\\text{in}\\ H(D)), \\text{for each } f : D \\rightarrow C, b \\in I(D).$ Since we are mainly interested in adjoints between ordered structures, for any two order-preserving maps $\\eta : H \\leftrightarrows G : \\mu $ between internal partial orderings $H,G$ in $\\textbf {Sets}^{\\textbf {C}^{op}}$ , $\\eta \\dashv \\mu $ means that $\\eta _C \\dashv \\mu _C$ at each component $C$ .", "That is to say $\\eta _C(x) \\le y \\ \\text{ iff }\\ x \\le \\mu _C(y),$ for all $x \\in H(C)$ , $y \\in G(C)$ .", "The natural transformation $\\Delta _I : H \\rightarrow H^I$ (henceforth $\\Delta $ ) determines for each $x \\in H(C)$ a natural transformation $\\Delta _C(x) : \\textbf {y}C \\times I \\rightarrow H$ with components $\\Delta _C(x)_D(f, a) = H(f)(x).$ Its right adjoint $\\forall _I : H^I \\rightarrow H$ (henceforth $\\forall $ ) has components, for any $\\eta \\in \\mathrm {Hom}_{}(\\textbf {y}C \\times I,H)$ , $\\forall _C(\\eta ) = \\bigvee \\lbrace s \\in H(C) \\mid H(f)(s) \\le \\eta _D(f,b),\\ \\text{for all}\\ f : D \\rightarrow C, b \\in I(D)\\rbrace ,$ where the join is taken in $H(C)$ .", "Dually, the left adjoint $\\exists $ of $\\Delta $ has components $\\exists _C(\\eta ) = \\bigwedge \\lbrace s \\in H(C) \\mid \\eta _D(f,b) \\le H(f)(s),\\ \\text{for all}\\ f : D \\rightarrow C, b \\in I(D)\\rbrace .$ (Note that for instance the condition on the underlying set of the join $\\forall _C(\\eta )$ expresses that $\\Delta _C(s) \\le \\eta $ as elements in $H^I(C)$ , so these definitions are in accordance with the general definition of right adjoints to $\\Delta _C$ given in the previous example.)", "Lastly, each $H(C)$ really is a complete Heyting algebra in the usual sense of having arbitrary set-indexed meets and joins (so the previous definitions of $\\forall $ and $\\exists $ actually make sense).", "For any set $J$ , the right adjoint $\\forall _J : H(C)^J \\longrightarrow H(C)$ can be found as follows.", "Consider the constant $J$ -valued functor $\\Delta J$ on C (and constant value $1_J$ on arrows in C).", "For any $C$ in C, there is an isomorphism $\\mathrm {Hom}_{\\textbf {Sets}}(J,HC) \\cong \\mathrm {Hom}_{\\widehat{\\textbf {C}}}(\\textbf {y}C \\times \\Delta J,H)$ (natural in $J$ and $H$ ).", "Given a function $h : J \\rightarrow HC$ , define a natural transformation $\\nu h : \\textbf {y}C \\times \\Delta J \\rightarrow H$ to have components $(\\nu h)_D(g,a) = H(g)f(a)$ .", "Conversely, given a natural transformation $\\eta $ on the right, define a function $f\\eta : J \\rightarrow HC$ by $f\\eta (a) = \\eta _C(1_C,a)$ .", "These assignments are mutually inverse.", "Moreover, the map that results from composing $\\Delta _J : HC \\rightarrow H^{\\Delta J}(C)$ with that isomorphism is computed as $f(\\Delta _C(x))(a) = \\Delta _C(x)_C(1_C,a) = H(1_C)(x) = x,$ so that for any $x \\in HC$ , $\\Delta _C(x)$ is the constant $x$ -valued map on $J$ .", "This justifies taking the right adjoint to $\\Delta _J$ as the sought right adjoint of the diagonal map $HC \\rightarrow H(C)^J$ .", "Indeed, for exponents $\\Delta J$ the formula for the right adjoint to $\\Delta _C$ , for instance, takes the familiar form met in the previous example $\\forall _C(\\eta ) = \\forall _J(f\\eta ) = \\bigwedge _{a \\in J} f\\eta (a) = \\bigwedge _{a \\in \\Delta J(C)} \\eta _C(1_C,a),$ or $\\forall _J(h) = \\forall _C(\\nu h) = \\bigwedge _{a \\in \\Delta J(C)} (\\nu h)_C(1_C,a) = \\bigwedge _{a \\in J} h(a),$ respectively.$\\Box $ For convenience, let us recall the Heyting structure of $\\Omega _{\\mathcal {E}}$ in more detail, as it will be useful later on.", "It is uniquely determined by the natural isomorphism (REF ) and the Yoneda lemma which “internalizes” the (complete) Heyting structure of $\\mathrm {Hom}_{\\mathcal {E}}(-,\\Omega _{\\mathcal {E}})$ (coming from $\\mathrm {Sub}_{\\mathcal {E}}(-)$ ) to $\\Omega _{\\mathcal {E}}$ .", "Since each pullback functor $f^\\ast : \\mathrm {Sub}_{}(B) \\rightarrow \\mathrm {Sub}_{}(A)$ , for $f : A \\rightarrow B$ in $\\mathcal {E}$ , preserves the Heyting structure on $\\mathrm {Sub}_{}(B)$ , all the required diagrams that define the Heyting operations on $\\Omega _{\\mathcal {E}}$ necessarily commute.", "The top element is $\\top : 1 \\rightarrow \\Omega _{\\mathcal {E}}$ , which by the previous considerations is the classifying map of the identity on the terminal object.", "The bottom element is the characteristic map of the monomorphism $0 \\rightarrowtail 1$ , where 0 is the initial object of $\\mathcal {E}$ .", "Meets $\\wedge : \\Omega _{\\mathcal {E}} \\times \\Omega _{\\mathcal {E}} \\longrightarrow \\Omega _{\\mathcal {E}}$ are given as the classifying map of $\\langle \\top , \\top \\rangle : 1 \\longrightarrow \\Omega _{\\mathcal {E}} \\times \\Omega _{\\mathcal {E}}$ , which is the classifying map of the pullback of $\\langle 1 , \\top u \\rangle $ and $\\langle \\top u , 1 \\rangle $ ($u : \\Omega _{\\mathcal {E}} \\rightarrow 1$ is the canonical map): ${1 [rr]^\\top [d]_\\top && \\Omega _{\\mathcal {E}} [d]|{\\langle 1 , \\top u \\rangle }\\\\\\Omega _{\\mathcal {E}} [rr]_-{\\langle \\top u , 1 \\rangle } && \\Omega _{\\mathcal {E}} \\times \\Omega _{\\mathcal {E}}}$ viewed as subobject of $\\Omega _{\\mathcal {E}} \\times \\Omega _{\\mathcal {E}}$ ; while $\\langle 1 , \\top u \\rangle $ and $\\langle \\top u , 1 \\rangle $ in turn arise as the subobjects classified by $\\pi _2$ and $\\pi _1$ , respectively.", "In a similar way, joins are constructed as classifying map of the image of the map $[\\langle 1 , \\top u_{\\Omega _{\\mathcal {E}}} \\rangle , \\langle \\top u_{\\Omega _{\\mathcal {E}}} , 1 \\rangle ] : \\Omega _{\\mathcal {E}} + \\Omega _{\\mathcal {E}} \\longrightarrow \\Omega _{\\mathcal {E}} \\times \\Omega _{\\mathcal {E}}.$ Implication is given as the classifying map of the equalizer ${E @{ >->}[r] & \\Omega _{\\mathcal {E}} \\times \\Omega _{\\mathcal {E}} @<0.75ex>[r]^-{\\wedge } @<-0.75ex>[r]_-{\\pi _1} & \\Omega _{\\mathcal {E}} .", "\\\\} $ The classifying map can be factored as follows, where the two squares are pullbacks: ${E [r] @{ >.>}[d] & \\Omega _{\\mathcal {E}} [r] [d]|{\\Delta _{\\Omega _{\\mathcal {E}}}} & 1 [d]^\\top \\\\\\Omega _{\\mathcal {E}} \\times \\Omega _{\\mathcal {E}} [r]_{\\langle \\wedge , \\pi _1 \\rangle } & \\Omega _{\\mathcal {E}} \\times \\Omega _{\\mathcal {E}} [r]_-{\\delta _{\\Omega _{\\mathcal {E}}}} & \\Omega _{\\mathcal {E}}}$ following a standard description of equalizers.1 Actually, the Yoneda argument determines $\\Rightarrow $ as the classifying map of the subobject $\\forall _{\\langle \\top u_{\\Omega _{\\mathcal {E}}} , 1 \\rangle }(\\top )$ of $\\Omega _{\\mathcal {E}} \\times \\Omega _{\\mathcal {E}}$ , where the latter is precisely the said equalizer.", "Using the Yoneda principle one also obtains indexed meets and joins as adjoints to the map $\\Delta _I : \\Omega _{\\mathcal {E}} \\rightarrow \\Omega _{\\mathcal {E}}^I$ .", "They are essentially provided by the fact that, for any topos $\\mathcal {E}$ , and any arrow $f :A \\rightarrow B$ in $\\mathcal {E}$ , the pullback functor $f^\\ast : \\mathrm {Sub}_{\\mathcal {E}}(B) \\longrightarrow \\mathrm {Sub}_{\\mathcal {E}}(A)$ has both a right and a left adjoint.", "Adding a parameter $X$ yields that $(1_X \\times f)^\\ast : \\mathrm {Sub}_{\\mathcal {E}}(X \\times B) \\longrightarrow \\mathrm {Sub}_{\\mathcal {E}}(X \\times A)$ restricts to a functor $\\mathrm {Hom}_{\\mathcal {E}}(X,\\Omega _{\\mathcal {E}}^B) \\longrightarrow \\mathrm {Hom}_{\\mathcal {E}}(X,\\Omega _{\\mathcal {E}}^A)$ by the isomorphisms $\\mathrm {Sub}_{\\mathcal {E}}(X \\times Y) \\cong \\mathrm {Hom}_{\\mathcal {E}}(X \\times Y,\\Omega _{\\mathcal {E}}) \\cong \\mathrm {Hom}_{\\mathcal {E}}(X,\\Omega _{\\mathcal {E}}^Y).$ These are natural in $X$ and so by Yoneda provide a map $\\Omega _{\\mathcal {E}}^B \\longrightarrow \\Omega _{\\mathcal {E}}^A,$ which is precisely $\\Omega _{\\mathcal {E}}^f$ .", "In particular, $\\Delta _I$ arises in this way from pullback along the projection $\\pi _1 : X \\times I \\rightarrow X$ : $\\pi _1^\\ast : \\mathrm {Sub}_{\\mathcal {E}}(X) \\longrightarrow \\mathrm {Sub}_{\\mathcal {E}}(X \\times I),$ that is by applying the previous argument to the map $u_I : I \\longrightarrow 1$ , as required.", "The external adjoints of $\\pi _1^\\ast $ induce the required internal adjoints of $\\Omega _{\\mathcal {E}}^{u_I} = \\Delta _I$ .", "As is well-known, one can interpret (intuitionistic) higher-order logic w.r.t.", "this algebraic structure on $\\Omega _{\\mathcal {E}}$ [12], [14].", "In particular, each formula $\\Gamma \\mid \\varphi $ , where $\\Gamma = ({x}_1 : {A}_1 , \\dots , {x}_n : {A}_n)$ is a suitable variable context for $\\varphi $ , is recursively assigned an arrow $\\llbracket {A}_1 \\rrbracket \\times \\dots \\times \\llbracket {A}_n \\rrbracket \\xrightarrow{} \\Omega _{\\mathcal {E}}$ in $\\mathcal {E}$ .", "Connectives and quantifiers are interpreted by composing with the evident Heyting structure maps of $\\Omega _{\\mathcal {E}}$ described above.", "For instance, $\\llbracket x : A \\mid \\forall y .", "\\varphi \\rrbracket $ is the arrow $\\llbracket A \\rrbracket \\xrightarrow{} \\Omega _{\\mathcal {E}}^{\\llbracket B \\rrbracket } \\xrightarrow{} \\Omega _{\\mathcal {E}},$ where $\\lambda _{\\llbracket B \\rrbracket } \\llbracket \\varphi \\rrbracket $ is the exponential transpose of $\\llbracket \\varphi \\rrbracket : \\llbracket A \\rrbracket \\times \\llbracket B \\rrbracket \\longrightarrow \\Omega _{\\mathcal {E}}.$ In particular, the equality predicate on each type $M$ is interpreted as the classifying map $\\delta _{\\llbracket M \\rrbracket }$ of the diagonal $\\langle 1_{\\llbracket M \\rrbracket } , 1_{\\llbracket M \\rrbracket } \\rangle : {\\llbracket M \\rrbracket } \\longrightarrow {\\llbracket M \\rrbracket } \\times {\\llbracket M \\rrbracket }.$ Example 1.4 When $\\mathcal {E} = \\textbf {Sets}$ , and $\\Omega _{\\mathcal {\\textbf {Sets}}} = \\textbf {2}$ , then the right adjoint $\\forall _I$ to $\\Delta _I : \\textbf {2} \\rightarrow \\textbf {2}^I$ is by definition required to satisfy $\\Delta _I(x) \\le f \\ \\text{ iff }\\ x \\le \\forall _I(f),$ which holds just in case $\\forall _I$ satisfies $\\forall _I(f) = 1 \\ \\text{ iff }\\ f(i) = 1,\\ \\text{ for all }\\ i \\in I.$ Equivalently, $\\forall _I(S) = 1 \\ \\text{ iff }\\ S = I,$ where $S \\subseteq I$ .", "Given a formula $x : X \\mid \\varphi $ , and an interpretation $\\llbracket X \\rrbracket \\xrightarrow{} \\textbf {2}$ , then $\\lambda _{\\llbracket X \\rrbracket }\\llbracket \\varphi \\rrbracket : 1 \\rightarrow \\textbf {2}^{\\llbracket X \\rrbracket }$ picks out the subset $S$ of $\\llbracket X \\rrbracket $ whose characteristic map is $\\llbracket \\varphi \\rrbracket $ , i.e.", "the set of objects in $\\llbracket X \\rrbracket $ that satisfy $\\varphi $ .", "Thus $\\llbracket \\forall x .", "\\varphi \\rrbracket = 1$ if and only if $S = \\llbracket X \\rrbracket $ , as expected.", "$\\Box $ In principle, these definitions make sense for any Heyting algebra $H$ in $\\mathcal {E}$ in place of $\\Omega _{\\mathcal {E}}$ , except for interpreting equality, since there is no notion of classifying map available for arbitrary $H$ .", "We present below a way in general to canonically interpret equality for arbitrary $H$ , closely connected to the treatment of modal operators.", "Definition 1.1 In any topos $\\mathcal {E}$ , a frame $H$ in $\\mathcal {E}$ is a complete Heyting algebra $H$ in $\\mathcal {E}$ .", "A frame homomorphism $f : H \\rightarrow G$ is a map $f$ in $\\mathcal {E}$ that is internally $\\bigvee ,\\wedge $ -preserving.", "For instance in any topos $\\mathcal {E}$ the object $1+1$ is an internal Boolean algebra, and thus a frame.", "Here, $\\tau : 1+1 \\rightarrow \\Omega _{\\mathcal {E}}$ is the classifying map of the first coprojection.", "Example 1.5 The prototypical frame is the collection of open sets $\\mathcal {O}(X)$ of a topological space $X$ .", "The set $\\mathcal {O}(X)$ is a complete Heyting algebra, as is $\\mathcal {P}(X)$ .", "However, arbitrary meets in $\\mathcal {O}(X)$ are in general not mere intersections.", "That is to say, the inclusion $i : \\mathcal {O}(X) \\hookrightarrow \\mathcal {P}(X)$ does not preserve them.", "This exhibits $\\mathcal {O}(X)$ as a subframe of $\\mathcal {P}(X)$ rather than a sub-Heyting algebra.", "The example also illustrates why the notion frame homomorphism matters at all.", "Note also that every frame map $f : H \\rightarrow G$ has a right adjoint $f_\\ast $ , defined for any $y \\in G$ as $f_\\ast (y) = \\bigvee \\lbrace x \\in H \\mid f(x) \\le y\\rbrace .$ The right adjoint to the inclusion $i$ is the interior operation on the topological space $X$ , which determines, in accordance with the formula for $f_\\ast $ , the largest open subset (w.r.t.", "$X$ ) of an arbitrary subset of $X$ .", "A related and more elementary example is the set inclusion $\\textbf {3} \\hookrightarrow \\textbf {4}$ of the three element Heyting algebra into the four element Boolean algebra, as indicated in: ${&11& &&& 11\\\\&&10 [ul]&\\hookrightarrow &01[ur]&&10[ul] \\\\&00[ur]&&&&00[ul][ur]}$ (3 may be thought of as the open set structure of the Sierpiński space.)", "The inclusion does not preserve the implication $10 \\rightarrow 00$ : $10 \\rightarrow 00 = 00,\\ \\text{in}\\ \\textbf {3}$ while $10 \\rightarrow 00 = 01,\\ \\text{in}\\ \\textbf {4}.$ Since $i$ preserving arbitrary meets is equivalent to saying that $i$ preserves implications, $\\textbf {3}$ is included in $\\textbf {4}$ as a subframe rather than as a sub-Heyting algebra.", "$\\Box $ In a topos $\\mathcal {E}$ the frame $\\Omega _{\\mathcal {E}}$ plays a distinguished role: Lemma 1.2 In any topos $\\mathcal {E}$ , the subobject classifier $\\Omega _{\\mathcal {E}}$ is the initial frame.", "That is to say, for every frame $H$ in $\\mathcal {E}$ , there is a unique frame map $i : \\Omega _{\\mathcal {E}} \\longrightarrow H$ .", "Moreover, the right adjoint $\\tau $ of $i$ is the classifying map of the top element $\\top _H : 1 \\longrightarrow H$ of $H$ .", "We refer to [9] (C1.3) for the proof.", "We will mainly be interested in those frames $H$ for which the map $i : \\Omega _{\\mathcal {E}} \\rightarrow H$ is monic, to which we will refer as faithful.1 A frame $H$ is faithful in this sense iff the inverse image part of the canonical geometric morphism $\\text{Sh}_{\\mathcal {E}}(H) \\longrightarrow \\mathcal {E}$ is faithful (see section ).", "This map $i : \\Omega _{\\mathcal {E}} \\rightarrow H$ will play a crucial role both in modelling equality and the modal operator on $H$ .", "Looking ahead, suppose given a suitable (intuitionistic) higher-order modal theory (as in section ).", "Then we shall interpret equality on a type $A$ w.r.t.", "an $H$ -valued model in a topos $\\mathcal {E}$ as the composite of $i$ with the usual classifying map of equality: $\\llbracket A \\rrbracket \\times \\llbracket A \\rrbracket \\xrightarrow{} \\Omega _{\\mathcal {E}} \\xrightarrow{} H.$ The semantics thus obtained is not sound w.r.t.", "standard higher-order intuitionistic logic; in particular, function and propositional extensionality fail (as we shall show by providing counterexamples).", "On the other hand, one can restore soundness by taking into account the following naturally arising modal operator.", "Lemma 1.3 Given a frame $H$ in a topos $\\mathcal {E}$ , let $i\\dashv \\tau $ be the canonical adjunction described in lemma REF , $i : \\Omega _{\\mathcal {E}} \\leftrightarrows H : \\tau .$ The composite $i\\mathbin {\\circ }\\tau $ is then an S4 modality on $H$ .", "The composite $i\\mathbin {\\circ }\\tau $ preserves finite meets because both components do.", "In virtue of $i \\dashv \\tau $ , the composite is a comonad, which gives the S4 laws." ], [ "Higher-order intuitionistic S4", "The formal system of higher-order modal logic considered here is simply the union of the usual axioms for higher-order logic and S4.", "The higher-order part is a version of type theory (cf.", "[8], [9], [12]).", "Types and terms are defined recursively.", "A higher-order language $\\mathcal {L}$ consists of a collection of basic types $A,B, \\dots $ along with basic terms (constants) $a : A, b : B$ .", "To stay close to topos-theoretic formulations, we assume the following type and term forming operations that inductively specify the collection of types and terms of the language: There are basic types 1, $\\textsf {P}$ If $A$ , $B$ are types, then there is a type $A \\times B$ If $A$ , $B$ are types, then there is a type $A^B$ Terms are recursively constructed as follows.", "Here we assume, for every type $A$ , an infinite set of variables of type $A$ , written as $x:A$ , to be given.", "We follow [8] in writing $\\Gamma \\mid t : B$ , for $\\Gamma = ({x}_1 : {A}_1 , \\dots , {x}_n : {A}_n)$ , involving at least all the free variables in the term $t$ .", "A context $\\Gamma $ may also be empty.", "Formally, every term $t$ always occurs in some variable context $\\Gamma $ and is well-typed only w.r.t.", "such a context.", "This is important to understand the recursive clauses below.", "To simplify notation, however, we omit $\\Gamma $ if it is unspecified and the same throughout a recursive clause.", "There are distinguished terms $\\emptyset \\mid \\ast : 1$ and $\\emptyset \\mid \\top , \\bot : \\textsf {P}$ If $ t : A$ and $s : B$ are terms, then $ \\langle t,s \\rangle : A \\times B$ is a term If $ t : A \\times B$ is a term, then there are terms $ \\pi _1 t : A$ and $ \\pi _2 t : B$ If $\\Gamma \\mid t : A$ is a term and $y : B$ a variable in $\\Gamma $ , then there is a term $\\Gamma [y:B] \\mid \\lambda y .", "t : A^B$ ; where $\\Gamma [y:B]$ is the context that results from $\\Gamma $ by deleting $y:B$ .", "If $ t : A^B$ and $ s : B$ are terms, then $ \\textsf {app}(t,s) : A$ is a term.", "For any two terms $ t : \\textsf {P}$ , $ s : \\textsf {P}$ there are terms $ t \\wedge s : \\textsf {P}$ , $ t \\vee s : \\textsf {P}$ , $ t \\Rightarrow s : \\textsf {P}$ .", "If $\\Gamma , y : B \\mid t : \\textsf {P}$ is a term, then $\\Gamma \\mid \\forall y .", "t : \\textsf {P}$ is a term; and similarly for $\\Gamma \\mid \\exists y .", "t : \\textsf {P}$ If $ t : A$ and $ s : A$ are terms, then $ s =_A t : \\textsf {P}$ is a term.", "If $ t : \\textsf {P}$ is a term, then $ \\Box t : \\textsf {P}$ is a term.", "One also assumes the usual structural rules of weakening of the variable context (adding dummy variables), contraction, and permutation.", "We may also assume that each variable declaration occurs only once in a context.", "As usual, we define a deductive system by specifying a relation $\\vdash $ between terms of type $\\textsf {P}$ .", "The crucial difference between the standard formulation of intuitionistic higher-order logic and the present one are the modified extensionality principles marked with ($\\ast $ ).", "$ \\varphi \\vdash \\varphi $ $\\displaystyle \\frac{ \\varphi \\vdash \\psi \\ \\ \\ \\ t : A}{ \\varphi [t/x] \\vdash \\psi [t/x]}$ , for $x : A$ (similarly for simultaneous substitution) $\\displaystyle \\frac{\\varphi \\vdash \\psi \\hspace{10.0pt} \\psi \\vdash \\vartheta }{ \\varphi \\vdash \\vartheta }$ $ \\top \\vdash x =_A x$ , where $x : A$ $ \\varphi \\wedge x =_A x^{\\prime } \\vdash \\varphi [x^{\\prime }/x]$ .", "where $x:A, x^{\\prime }:A$ ($\\ast $ ) $ \\Box \\forall x (f(x) =_B g(x)) \\vdash f =_{B^A} g$ , for terms $x:A$ and $f,g : B^A$ ($\\ast $ ) $ \\Box (p \\Leftrightarrow q) \\vdash p =_{\\textsf {P}} q$ , for terms $p,q: \\textsf {P}$ $ \\top \\vdash \\ast =_1 x$ , where $x:1$ $ \\top \\vdash \\pi _1\\langle x,y \\rangle =_A x$ and $ \\top \\vdash \\pi _2\\langle x,y \\rangle =_B y$ , where $ x :A$ and $ y : B$ $ \\top \\vdash \\langle \\pi _1 w , \\pi _2 w \\rangle =_{A \\times B} w$ , for $ w : A \\times B$ $\\Gamma [x:A] \\mid \\top \\vdash \\textsf {app}(\\lambda x .", "t,x^{\\prime }) =_B t[x^{\\prime }/x]$ , for $\\Gamma \\mid t : B$ and $x^{\\prime }:A$ $ \\top \\vdash \\lambda x .\\textsf {app}(w,x) =_{B^A} w$ , for $w : B^A$ $ \\varphi \\vdash \\top $ , for any $ \\varphi : \\textsf {P}$ $ \\bot \\vdash \\varphi $ , for any $ \\varphi : \\textsf {P}$ $ \\varphi \\vdash \\psi \\wedge \\vartheta \\ \\text{ iff }\\ \\varphi \\vdash \\psi $ and $\\varphi \\vdash \\vartheta $ $ \\varphi \\vee \\psi \\vdash \\vartheta \\ \\text{ iff }\\ \\varphi \\vdash \\vartheta $ and $\\psi \\vdash \\vartheta $ $ \\varphi \\vdash \\psi \\Rightarrow \\vartheta \\ \\text{ iff }\\ \\varphi \\wedge \\psi \\vdash \\vartheta $ $ \\Gamma \\mid \\exists x.", "\\varphi \\vdash \\psi \\ \\text{ iff }\\ \\Gamma , x:A \\mid \\varphi \\vdash \\psi $ $ \\Gamma \\mid \\varphi \\vdash \\forall x.", "\\psi \\ \\text{ iff }\\ \\Gamma , x:A \\mid \\varphi \\vdash \\psi $ Definition 2.1 A theory in a language $\\mathcal {L}$ as specified above consists of a set of closed sentences $\\alpha $ , i.e.", "terms of type $\\textsf {P}$ with no free variables (well-typed in the empty context), and which may be used as axioms in the form $\\Gamma \\mid \\top \\vdash \\alpha $ .", "Remark 2.2 Adding the axiom $\\Gamma \\mid \\top \\vdash \\forall p .", "p \\vee \\lnot p$ makes the logic classical.", "As is well-known there are more concise formulations of higher-order systems.", "The particular one chosen here is very close to the definition of a topos as a cartesian closed category with subobject classifier.", "One does not really need all exponential types and their constructors, however, but only those of the form $\\textsf {P}^A$ , for every type $A$ , which we write $\\textsf {P}A$ and call powertypes.", "Along these lines one may define: $\\lbrace x : A \\mid \\varphi \\rbrace :\\equiv \\lambda x.", "\\varphi : \\textsf {P}A,$ where $x : A \\mid \\varphi : \\textsf {P}$ .", "On the other hand, for $\\sigma : \\textsf {P}A$ and $x : A$ , set $x \\in \\sigma :\\equiv \\textsf {app}(\\sigma ,x).$ According to the axioms for exponential terms, we have $x^{\\prime } : A & \\mid \\top \\vdash x^{\\prime } \\in \\lbrace x : A \\mid \\varphi \\rbrace = \\varphi [x^{\\prime }/x] \\\\& \\mid \\top \\vdash \\lbrace x : A \\mid x \\in w\\rbrace = w.$ Thus one could instead take only types of the form $\\textsf {P}A$ , and the constructors $\\lbrace \\dots \\mid - \\rbrace $ and $\\in $ as basic, along with the last two axioms.", "For further simplifications see [9], [12].", "Finally, the S4 axioms are the usual ones $\\displaystyle \\frac{\\Gamma \\mid \\varphi \\vdash \\psi }{\\Gamma \\mid \\Box \\varphi \\vdash \\Box \\psi }$ $\\Gamma \\mid \\top \\vdash \\Box \\top $ $\\Gamma \\mid \\Box \\varphi \\wedge \\Box \\psi \\vdash \\Box (\\varphi \\wedge \\psi )$ $\\Gamma \\mid \\Box \\varphi \\vdash \\varphi $ $\\displaystyle \\Gamma \\mid \\Box \\varphi \\vdash \\Box \\Box \\varphi $ The first three axioms express that $\\Box $ , viewed as an operator, is a monotone finite meet preserving operation.", "The other two axioms are the $T$ and 4 axioms, respectively.", "Further useful rules provable from the axioms are necessitation $\\frac{\\Gamma \\mid \\top \\vdash \\varphi }{\\Gamma \\mid \\top \\vdash \\Box \\varphi },$ and the axiom $K$ : $\\Gamma \\mid \\Box (\\varphi \\Rightarrow \\psi ) \\vdash \\Box \\varphi \\Rightarrow \\Box \\psi .$ Although it is essentially obvious, for the sake of completeness we provide a definition of a model of this language in a topos.", "Definition 2.3 A model of a higher-order modal type theory in a topos $\\mathcal {E}$ consists of a faithful frame $H$ in $\\mathcal {E}$ , and an assignment $\\llbracket - \\rrbracket $ that assigns to each basic type $A$ in $\\mathcal {L}$ an object $\\llbracket A \\rrbracket $ in such a way that $\\llbracket 1 \\rrbracket = 1_\\mathcal {E}$ $\\llbracket \\textsf {P} \\rrbracket = H$ $\\llbracket A \\times B \\rrbracket = \\llbracket A \\rrbracket \\times \\llbracket B \\rrbracket $ $\\llbracket A^B \\rrbracket = \\llbracket A \\rrbracket ^{\\llbracket B \\rrbracket }$ .", "Moreover, each term $\\Gamma \\mid t : B$ in $\\mathcal {L}$ , where $\\Gamma = ({x}_1 : {A}_1 , \\dots , {x}_n : {A}_n)$ is a suitable variable context for $t$ , is assigned an arrow $\\llbracket t \\rrbracket : \\llbracket \\Gamma \\rrbracket \\rightarrow \\llbracket B \\rrbracket $ recursively as follows (where $\\llbracket \\Gamma \\rrbracket $ is short for $\\llbracket {A}_1 \\rrbracket \\times \\dots \\times \\llbracket {A}_n \\rrbracket $ and $\\llbracket t \\rrbracket $ really means $\\llbracket \\Gamma \\mid t : B \\rrbracket $ ).", "Each constant $c : A$ in $\\mathcal {L}$ is assigned an arrow $\\llbracket c \\rrbracket : 1_\\mathcal {E} \\rightarrow \\llbracket A \\rrbracket .$ In particular: $\\llbracket \\top \\rrbracket = \\top _H : 1_\\mathcal {E} \\longrightarrow H$ $\\llbracket \\bot \\rrbracket = \\bot _H : 1_\\mathcal {E} \\longrightarrow H$ $\\llbracket \\ast : 1 \\rrbracket = 1_{1_\\mathcal {E}}$ (the identity arrow on the terminal object).", "This extends to arbitrary terms-in-context as follows For any constant $c : A$ , $\\llbracket \\Gamma \\mid c : A \\rrbracket $ is the arrow $\\llbracket \\Gamma \\rrbracket \\xrightarrow{} 1_\\mathcal {E} \\xrightarrow{} \\llbracket A \\rrbracket $ If $\\Gamma \\mid s : A$ and $\\Gamma \\mid t : B$ are terms, then $\\llbracket \\Gamma \\mid \\langle s,t \\rangle : A \\times B \\rrbracket $ is the map $\\langle \\llbracket s \\rrbracket , \\llbracket t \\rrbracket \\rangle : \\llbracket \\Gamma \\rrbracket \\rightarrow \\llbracket A \\rrbracket \\times \\llbracket B \\rrbracket .$ If $\\Gamma \\mid t : A \\times B$ is a term, then $\\llbracket \\Gamma \\mid \\pi _1 t : A \\rrbracket $ is $\\llbracket \\Gamma \\rrbracket \\xrightarrow{} \\llbracket A \\rrbracket \\times \\llbracket B \\rrbracket \\xrightarrow{} \\llbracket A \\rrbracket ,$ and similarly for $\\pi _2 t $ .", "If $\\Gamma \\mid t : A$ is a term and $y : B$ a variable in $\\Gamma $ , then $\\llbracket \\Gamma [y:B] \\mid \\lambda y .", "t : A^B \\rrbracket $ is ${\\lambda _{\\llbracket B \\rrbracket }\\llbracket t \\rrbracket } : \\llbracket \\Gamma [y:B] \\rrbracket \\rightarrow A^{\\llbracket B \\rrbracket }$ If $\\Gamma \\mid t : A^B$ and $\\Gamma \\mid s : B$ are terms, then $\\llbracket \\Gamma \\mid \\textsf {app}(t,s) : A \\rrbracket $ is $\\langle \\llbracket t \\rrbracket ,\\llbracket s \\rrbracket \\rangle : \\llbracket \\Gamma \\rrbracket \\rightarrow A^B \\times B \\xrightarrow{} A.$ For any two terms $\\Gamma \\mid p : \\textsf {P}$ , $\\Gamma \\mid q : \\textsf {P}$ , and $\\star $ any of the connectives $\\wedge , \\vee , \\Rightarrow $ , $\\llbracket \\Gamma \\mid p \\star q : \\textsf {P} \\rrbracket $ is $\\llbracket \\Gamma \\rrbracket \\xrightarrow{} H \\times H \\xrightarrow{} H,$ where in the last line $\\star $ is the evident algebraic operation on $H$ .", "If $\\Gamma , y : B \\mid t : \\textsf {P}$ is a term, then $\\llbracket \\Gamma \\mid \\forall y .", "t : \\textsf {P} \\rrbracket $ is $\\llbracket \\Gamma \\rrbracket \\xrightarrow{} H^{\\llbracket B \\rrbracket } \\xrightarrow{} H$ and similarly for $\\llbracket \\Gamma \\mid \\exists y .", "t : \\textsf {P} \\rrbracket $ via $\\exists _{\\llbracket B \\rrbracket }$ .", "If $\\Gamma \\mid t : A$ and $\\Gamma \\mid s : A$ are terms, then $\\llbracket \\Gamma \\mid t =_A s : \\textsf {P} \\rrbracket $ is the map $\\llbracket \\Gamma \\rrbracket \\xrightarrow{} \\llbracket A \\rrbracket \\times \\llbracket A \\rrbracket \\xrightarrow{} \\Omega _{\\mathcal {E}} \\xrightarrow{} H,$ where $i$ is the unique (monic) frame map.", "If $\\Gamma \\mid t : \\textsf {P}$ is a term, then $\\llbracket \\Gamma \\mid \\Box t : \\textsf {P} \\rrbracket $ is the map $\\llbracket \\Gamma \\rrbracket \\xrightarrow{} H \\xrightarrow{} \\Omega _{\\mathcal {E}} \\xrightarrow{} H,$ where $\\tau $ is the classifying map of $\\top _H : 1 \\rightarrow H$ , as described before.", "Before moving on, let us review some common examples Examples 2.1 A well-studied class of examples are structures induced by surjective geometric morphisms $f : \\mathcal {F} \\rightarrow \\mathcal {E}$ .", "If $\\mathcal {F}$ is Boolean, then so is $f_\\ast \\Omega _{\\mathcal {F}}$ .", "For instance, there are geometric morphisms $\\textbf {Sets}^{|\\textbf {C}|} \\longrightarrow \\textbf {Sets}^{\\textbf {C}}$ induced by the inclusion $|\\textbf {C}| \\rightarrow \\textbf {C}$ .", "When $\\textbf {C}$ is a preorder, then this yields Kripke semantics for first-order modal logic.", "This case was originally studied in [6], [22].", "Similarly, the canonical geometric morphism $\\textbf {Sets}/X \\longrightarrow \\text{Sh}_{}(X)$ induced by the continuous inclusion $|X| \\hookrightarrow X$ gives rise to sheaf models for classical first- (and higher-) order modal logic, studied in [1].", "The exact structure of these examples will be discussed in more detail in section below.", "More generally, by a well-known theorem of Barr, every Grothendieck topos $\\mathcal {G}$ can be covered by a Boolean topos $\\mathcal {B}$ in the sense that there is a surjective geometric morphism $f : \\mathcal {B} \\longrightarrow \\mathcal {G}.$ For $H = {f}_\\ast \\Omega _{\\mathcal {B}}$ , this provides models in Grothendieck topoi.1 Cf.", "e.g.", "[14], IX.9.", "Actually, the geometric morphism $f$ can be extended to a surjective geometric morphism $\\mathcal {E} \\longrightarrow \\mathcal {B} \\longrightarrow \\mathcal {G}$ , where $\\mathcal {E}$ is the topos of sheaves on a topological space, although $\\mathcal {E}$ might not be Boolean ([14], IX.11).", "Of course, in any topos $\\mathcal {E}$ the subobject classifier $\\Omega _{\\mathcal {E}}$ itself would do.", "However, as noted e.g.", "in [18], [20], the resulting modal operator will be the identity on $\\Omega _{\\mathcal {E}}$ ." ], [ "Soundness of algebraic semantics", "The given system of intuitionistic higher-order S4 modal logic is sound w.r.t.", "the semantics described in def.", "REF .", "Except for the two extensionality principles, soundness is straightforward following known topos semantics.", "The reason why plain propositional extensionality fails in our semantics is the interpretation of implication.", "In the general topos semantics based on $\\Omega _{\\mathcal {E}}$ Heyting implication on $\\Omega _{\\mathcal {E}}$ is given by the map $\\Omega _{\\mathcal {E}} \\times \\Omega _{\\mathcal {E}} \\xrightarrow{} \\Omega _{\\mathcal {E}} \\times \\Omega _{\\mathcal {E}} \\xrightarrow{} \\Omega _{\\mathcal {E}}$ that immediately implies propositional extensionality.", "By contrast, for an arbitrary frame $H$ we observe: Lemma 3.1 For an arbitrary topos $\\mathcal {E}$ , and a (faithful) frame $H$ in $\\mathcal {E}$ , it is not in general the case that ${H \\times H [r]^-\\Rightarrow [d]|{\\langle \\pi _1 , \\wedge \\rangle } & H \\\\H \\times H [r]^-{\\delta _H} & \\Omega _{\\mathcal {E}} [u]_i\\\\}$ commutes.", "A counterexample may easily be found in the topos Sets with subobject classifier 2 and $H = \\mathcal {P}(X)$ , for some set $X \\ne 1$ .", "The adjunction $i : \\textbf {2} \\leftrightarrows \\mathcal {P}(X) : \\tau $ ($i \\dashv \\tau $ ) is defined by $i(x) = {\\left\\lbrace \\begin{array}{ll}X, & \\text{if}\\ x = 1\\\\\\emptyset , &\\text{if}\\ x = 0\\\\\\end{array}\\right.", "}$ and $\\tau (U) = 1 \\ \\text{ iff }\\ U = X.$ For any $U,V \\in \\mathcal {P}(X)$ , $U \\Rightarrow V = \\bigcup \\lbrace W \\in \\mathcal {P}(X) \\mid W \\cap U \\subseteq V\\rbrace .$ If $U \\nsubseteq V$ , then $U \\ne U \\cap V$ , and so $i \\delta \\langle \\pi _1,\\wedge \\rangle (U,V) = i\\delta _{\\mathcal {P}(X)}(U, U \\cap V) = i(0) = \\emptyset .$ But $U \\nsubseteq V$ does not in general imply $U \\Rightarrow V = \\emptyset $ .", "(Consider e.g.", "$V \\subseteq U \\Rightarrow V$ , for $U \\cap V \\ne \\emptyset $ .)", "As suggested by the example, the reason for the failure of plain propositional extensionality is that failure to be true (in the sense of $\\top = X \\nsubseteq U \\Rightarrow V$ ) does not imply equality to $\\bot $ in $H$ .", "On the other hand, note that $\\tau (U \\Rightarrow V) = 0$ , because $X \\nsubseteq U \\Rightarrow V$ .", "This observation generalizes.", "Although $i\\delta \\langle \\pi _1, \\wedge \\rangle =\\ \\Rightarrow $ fails in general, we have the following.", "Lemma 3.2 In any topos $\\mathcal {E}$ , the diagram ${H \\times H [r]^-\\Rightarrow [d]|{\\langle \\pi _1 , \\wedge \\rangle } & H [d]^\\tau \\\\H \\times H [r]^-{\\delta _H} & \\Omega _{\\mathcal {E}}}$ commutes, and thus ${i \\tau } \\mathbin {\\circ } \\Rightarrow \\ =\\ i \\delta _H \\langle \\pi _1 , \\wedge \\rangle .$ Consider the pullbacks ${(\\le ) [r] [d] & 1 @{=}[r] [d]^\\top & 1 [d]^\\top \\\\H \\times H [r]_-\\Rightarrow & H [r]_\\tau & \\Omega _{\\mathcal {E}}}\\\\{(\\le ) [r] [d] & H [r] [d]^\\Delta & 1[d]^\\top \\\\H \\times H [r]_-{\\langle \\pi _1 , \\wedge \\rangle } & H \\times H [r]_-{\\delta _H} & \\Omega _{\\mathcal {E}}}$ whence the claim follows from uniqueness of classifying maps.", "The left-hand square in the first diagram is a pullback by the definition of $\\Rightarrow $ , while the second diagram is the definition of the induced partial ordering on $H$ as the equalizer of $\\pi _1$ and $\\wedge $ .", "This argument neatly exhibits the conceptual role played by the modal operator $\\tau $ (more exactly, the adjunction $i \\dashv \\tau $ ).", "The soundness proof is essentially a corollary to that.", "Corollary 3.3 Modalized propositional extensionality $p : \\textsf {P}, q : \\textsf {P}\\ |\\ \\Box (p \\Leftrightarrow q) \\vdash p =_\\textsf {P} q$ is true in any model $(\\mathcal {E},H)$ .", "In view of lemma REF , and since $\\tau , i$ commute with meets, the left-hand side of the above sequent is interpreted as the map $i \\wedge (\\delta _H \\times \\delta _H) \\langle \\langle \\wedge _H,\\pi _1 \\rangle , \\langle \\wedge _H , \\pi _2 \\rangle \\rangle ,$ with $\\wedge $ the meet on $\\Omega _{\\mathcal {E}}$ .", "The right-hand side is the internal equality on $H$ : $i \\delta _H : H \\times H \\rightarrow \\Omega _{\\mathcal {E}} \\rightarrow H.$ It is clear from the properties of $\\le _\\Omega $ as a partial ordering that $\\wedge (\\delta _H \\times \\delta _H) \\langle \\langle \\wedge _H,\\pi _1 \\rangle , \\langle \\wedge _H , \\pi _2 \\rangle \\rangle \\le _{\\Omega } \\delta _H.$ Since $i$ preserves that ordering, we have $i\\wedge (\\delta _H \\times \\delta _H) \\langle \\langle \\wedge _H,\\pi _1 \\rangle , \\langle \\wedge _H , \\pi _2 \\rangle \\rangle \\le _H i\\delta _H.$ The failure of plain function extensionality and its recovering via $\\tau $ can be analyzed in a similar fashion.", "For non-modal function extensionality in the standard $\\Omega _{\\mathcal {E}}$ -valued setting essentially holds because $\\forall _Y \\mathbin {\\circ } (\\delta _X)^Y = \\delta _{X^Y}$ .", "However, in our setting we don't in general have $\\forall _Y \\mathbin {\\circ } (i\\delta _X)^Y = i\\delta _{X^Y}$ , but rather: Lemma 3.4 For any topos $\\mathcal {E}$ , and any faithful frame in $H$ , the following diagram commutes: ${\\Omega ^Y [r]^-{i^Y} & H^Y [d]^{\\tau ^Y} [r]^-{\\forall _Y} & H [d]^\\tau \\\\X^Y \\times X^Y [u]^{(\\delta _X)^Y} [r]^-{(\\delta _X)^Y} @/_2pc/[rr]_{\\delta _{X^Y}}& \\Omega ^Y [r]^-{\\forall _Y} & \\Omega \\\\}$ Hence in particular $i\\delta _{X^Y} = i \\tau \\mathbin {\\circ } \\forall _Y \\mathbin {\\circ } (i\\delta _X)^Y.$ The right-hand square of the diagram commutes by uniqueness of classifying maps, while for the left-hand square we have $\\tau i = 1$ .", "Similarly, the bottom triangle commutes, because ${X^Y [d]_{\\Delta _{X^Y}} [r]&1 @{=}[r] [d]^{\\top ^Y}& 1 [d]^\\top \\\\X^Y \\times X^Y [r]_-{(\\delta _X)^Y} &\\Omega ^Y [r]_{\\forall _Y} & \\Omega \\\\}$ is a pullback diagram.", "(Note that the left-hand square is a pullback, because the functor $(-)^Y$ , as a right adjoint, preserves these.)", "Corollary 3.5 Modal function extensionality $f: X^Y , g : X ^Y \\mid \\Box (\\forall y : Y .", "f(y) =_X g(y)) \\vdash f =_{X^Y} g.$ is true in any interpretation $(\\mathcal {E},H)$ .", "The left-hand side of the sequent is interpreted by the arrow $X^Y \\times X^Y \\xrightarrow{}H^Y \\xrightarrow{} H \\xrightarrow{} H,$ where the projections come from $X^Y \\times X^Y \\times Y$ , and $ev : X^Y \\times Y \\rightarrow X$ is the canonical evaluation.", "The right-hand side is simply $X^Y \\times X^Y \\xrightarrow{} \\Omega _\\mathcal {E} \\xrightarrow{} H.$ We need to show that the arrow $\\langle i\\tau \\forall _Y\\lambda _Y (i\\delta _X\\langle ev \\pi _{13}, ev \\pi _{23} \\rangle ) , i \\delta _{X^Y} \\rangle : X^Y \\times X^Y \\rightarrow H \\times H$ factors through the partial ordering $(\\le ) \\rightarrowtail H \\times H$ .", "Write the left-hand component as $i\\varphi $ .", "It is enough to show that $\\varphi \\le _{\\Omega } \\delta _{X^Y} : X^Y \\times X^Y,$ whence the claim follows as before, $i$ being order-preserving.", "To show that the subobject $(Q,m)$ classified by the map $\\tau \\forall _Y\\lambda _Y (i\\delta _X\\langle ev\\pi _{13}, ev\\pi _{23} \\rangle )$ factors through $\\Delta _{X^Y}$ , as subobjects of $X^Y \\times X^Y$ , observe first that $\\lambda _Y (i\\delta _X\\langle ev\\pi _{13}, ev\\pi _{23} \\rangle )$ can be written as $X^Y \\times X^Y \\xrightarrow{} (X^Y \\times X^Y \\times Y)^Y \\xrightarrow{} (X \\times X)^Y \\xrightarrow{} \\Omega ^Y \\xrightarrow{} H^Y,$ where $\\eta $ is the unit component (at $X^Y \\times X^Y$ ) of the product-exponential adjunction $(-) \\times Y \\dashv (-)^Y$ .", "By the previous lemma $\\tau \\mathbin {\\circ } \\forall _Y \\mathbin {\\circ } i^Y \\mathbin {\\circ } (\\delta _X)^Y = \\delta _{X^Y}.$ The subobject in question thus arises from pullbacks ${Q @{ >->}[d]_m [rrr]&&& X^Y [r] [d]^{\\Delta _{X^Y}} & 1 [d]^{\\top }\\\\X^Y \\times X^Y [rrr]^{\\langle ev\\pi _{13}, ev\\pi _{23} \\rangle ^Y \\mathbin {\\circ } \\eta } &&& X^Y \\times X^Y [r]^-{\\delta _{X^Y}} & \\Omega }$ But $\\langle ev\\pi _{13}, ev\\pi _{23} \\rangle ^Y \\mathbin {\\circ } \\eta $ is the identity arrow.", "For it is the transpose (along the adjunction $(-) \\times Y \\dashv (-)^Y$ ) of $\\langle ev\\pi _{13}, ev\\pi _{23} \\rangle : X^Y \\times X^Y \\times Y \\rightarrow X \\times X.$ The latter in turn is the canonical evaluation of $X^Y \\times X^Y$ viewed as the exponential $(X \\times X)^Y$ , i.e.", "the counit of the adjunction at $X \\times X$ , transposing which yields the identity.", "As a result, $\\tau \\forall _Y\\lambda _Y (i\\delta _X\\langle ev\\pi _{13}, ev\\pi _{23} \\rangle ) \\le _{\\Omega } \\delta _{X^Y},$ and therefore $i\\tau \\forall _Y\\lambda _Y (i\\delta _X\\langle ev\\pi _{13}, ev\\pi _{23} \\rangle ) \\le _H i\\delta _{X^Y}.$ Remark 3.6 Before giving a counterexample to $i\\delta _{X^Y} = \\forall _Y \\mathbin {\\circ } (i\\delta _X)^Y$ , let us remark that the equation does actually hold in the topos Sets.", "For consider $f\\ne g \\in X^Y$ , i.e.", "$f(y) \\ne g(y)$ , for some $y \\in Y$ .", "Then for any complete Heyting algebra $H$ , the function $(i\\delta _X)^Y(f,g) \\in H^Y$ is defined as $(i\\delta _X)^Y(f,g)(y) = i\\delta _X(f(y),g(y)) = \\top , \\quad \\text{if}\\ f(y) = g(y),$ and $\\bot $ otherwise.", "Thus taking the meet (cf.", "the definition in example REF ) yields $\\bigwedge _{y \\in Y} (i\\delta _X)^Y(f,g)(y) = \\bot ,$ because $f(y) \\ne g(y)$ , for some $y \\in Y$ , by assumption.", "In turn the meet equals $\\top $ just in case $f(y) = g(y)$ , for all $y \\in Y$ , i.e.", "if and only if $f = g$ .", "Proposition 3.7 It is not in general the case that for a topos $\\mathcal {E}$ and a frame $H$ in $\\mathcal {E}$ : $i\\delta _{X^Y} = \\forall _Y \\mathbin {\\circ } (i\\delta _X)^Y.$ To find a counterexample we consider a specific presheaf topos $\\textbf {Sets}^{\\textbf {C}^{op}}$ described below.1 The counterexample, in particular the choice of C and the functor $G: \\textbf {C} \\rightarrow \\textbf {Sets}$ below, follows a slightly different, though equivalent, proof first given in [23].", "Let's first recall some general facts.", "Write $\\Omega _{|\\textbf {C}|}$ for the subobject classifier in $\\textbf {Sets}^{|\\textbf {C}|} $ and choose $H = f_\\ast \\Omega _{|\\textbf {C}|}$ (henceforth $\\Omega _\\ast $ ), where $f$ is the geometric morphism $f : \\textbf {Sets}^{|\\textbf {C}|} \\rightarrow \\textbf {Sets}^{\\textbf {C}^{op}}$ induced by the inclusion $|\\textbf {C}| \\hookrightarrow \\textbf {C}$ via right Kan extensions.", "Recall moreover from the beginning that the subobject classifier $\\Omega $ of $\\textbf {Sets}^{\\textbf {C}^{op}}$ determines for each $C$ the set of all sieves on $C$ .", "By contrast, $\\Omega _\\ast (C)$ is the set of arbitrary sets of arrows with codomain $C$ (cf.", "also the example from the next section).", "Recall that in any category of the form $\\textbf {Sets}^{\\textbf {C}^{op}}$ the evaluation maps $\\varepsilon : B^A \\times A \\rightarrow B$ have components $\\varepsilon _C(\\eta , a) = \\eta _C(1_C,a),$ where $\\eta \\in B^A(C) = \\mathrm {Hom}_{}(\\textbf {y}C \\times A,B)$ and $a \\in A(C)$ .", "The exponential transpose $\\overline{\\alpha } : Z \\rightarrow B^A$ of a map $\\alpha : Z \\times A \\rightarrow B$ has components $\\overline{\\alpha }_C(z) = \\alpha \\mathbin {\\circ } (\\zeta \\times 1_A),$ where $\\zeta : \\textbf {y}C \\rightarrow Z$ corresponds under the Yoneda lemma to the element $z \\in Z(C)$ , i.e.", "is defined as $\\zeta (f) = Z(f)(z)$ , for any $f \\in \\textbf {y}C(D)$ .", "For any object $A$ in C, the functor $(-)^A$ acts on arrows $f : C \\rightarrow D$ as $f^A = \\overline{f \\mathbin {\\circ } \\varepsilon },$ for evaluation $\\varepsilon : C^A \\times A \\rightarrow C$ .", "In particular, $(i\\delta _B)^A = \\overline{i\\delta _B \\mathbin {\\circ } \\varepsilon },$ for $\\varepsilon : (B \\times B)^A \\times A \\rightarrow B \\times B$ evaluation at $A$ .", "Thus, for any pair $\\langle \\eta ,\\mu \\rangle \\in (B \\times B)^A(C) = \\mathrm {Hom}_{}(\\textbf {y}C \\times A,B \\times B),$ we have $(\\overline{i\\delta _B \\mathbin {\\circ } \\varepsilon })_C(\\eta , \\mu ) = i \\delta _B \\varepsilon ( \\langle \\eta , \\mu \\rangle ^\\ast \\times 1_A) = i \\delta _B\\langle \\eta , \\mu \\rangle .$ Here we use that $\\langle \\eta , \\mu \\rangle ^\\ast : \\textbf {y}C \\rightarrow (B \\times B)^A$ corresponds under Yoneda to the element $\\langle \\eta , \\mu \\rangle \\in (B \\times B)^A(C) = \\mathrm {Hom}_{}(\\textbf {y}C \\times A,B \\times B)$ and that $\\langle \\eta , \\mu \\rangle ^\\ast $ is equal to the exponential transpose of $\\langle \\eta , \\mu \\rangle $ .", "Accordingly, $\\forall _C(i\\delta _B)^A_C(\\eta ,\\mu ) & = \\forall _C(\\overline{i\\delta _B \\mathbin {\\circ } \\varepsilon })_C(\\eta , \\mu )\\\\& = \\forall _C(i \\delta _B\\langle \\eta , \\mu \\rangle ) \\\\& = \\bigcup \\lbrace s \\in \\Omega _\\ast (C) \\mid \\Omega _\\ast (g)(s) \\le i_D (\\delta _B)_D (\\eta _D(g,b) , \\mu _D(g,b)), \\ \\text{for all}\\ \\\\& \\phantom{= \\bigcup \\lbrace s \\in \\Omega _\\ast (C) \\mid }\\ (g: D \\rightarrow C , b \\in A(D))\\rbrace ,$ On the other hand, the classifying map of the diagonal on a functor $B : \\textbf {C}^{op} \\rightarrow \\textbf {Sets}$ is computed as $(\\delta _B)_C(x,y) = \\lbrace f : D \\rightarrow C \\mid B(f)(x) = B(f)(y)\\rbrace ,$ for all pairs $(x,y) \\in B(C)\\times B(C)$ .", "It is the maximal sieve $\\top _C$ on $C$ just in case $x = y$ .", "Now let $\\textbf {C}$ be the finite category $C \\xrightarrow{} D,$ and define a functor $G : \\textbf {C}^{op} \\rightarrow \\textbf {Sets}$ as follows:1 Although $g: C \\rightarrow D$ may be seen as the two-element poset with resulting presheaf topos $\\textbf {Sets}^\\rightarrow $ , we will not need that description.", "The objects and arrows in C merely play the role of indices, so it seems better to use the more neutral notation $C,D,g$ .", "$G(D) = \\lbrace u\\rbrace ,\\ G(C) = \\lbrace v,w\\rbrace ,\\ G(g)(u) = v.$ Furthermore, choose $\\eta , \\mu \\in G^G(D)$ such that $\\eta \\ne \\mu $ .", "Observe that, while necessarily $\\eta _D = \\mu _D : \\textbf {y}D (D) \\times G(D) \\rightarrow G(D)$ with assignment $(1_D , u) \\mapsto u,$ we can chose $\\eta ,\\mu $ in such a way that $\\eta _C(g,x) \\ne \\mu _C(g,x)$ , for some pair $(g,x) \\in \\textbf {y}D (C) \\times G(C)$ .", "Specifically, since the first component $g$ is fixed, the choice is only about $x \\in G(C)$ which in turn must concern $w \\in G(C)$ .", "For naturality requires that $G(g)\\eta _D(1_D,u) = \\eta _C(\\textbf {y}D(g) \\times G(g))_C(1_D,u) = \\eta _C(g,v),$ so that since $G(g)\\eta _D(1_D,u) = G(g)(u) = v$ , we must have $\\eta _C(g,v) = v$ ; similarly $\\mu _C(g,v) = v$ .", "However, no constraint is put on the values $\\eta _C(g,w)$ and $\\mu _C(g,w)$ , respectively.", "Then: $(\\delta _{G^G})_D(\\eta , \\mu ) = \\lbrace x : X \\rightarrow D \\mid G^G(x)(\\eta ) = G^G(x)(\\mu )\\rbrace = \\emptyset .$ For if $x = g$ , observe $G^G(g)(\\eta ) = \\eta \\mathbin {\\circ } (\\textbf {y}g \\times 1_G) \\ne \\mu \\mathbin {\\circ } (\\textbf {y}g \\times 1_G) = G^G(g)(\\mu ),$ because $\\eta _C (\\textbf {y}g \\times 1_G)_C(1_C,w) = \\eta _C(g,w) \\ne \\mu _C(g,w) = \\mu _C(\\textbf {y}g \\times 1_G)_C(1_C,w),$ where the inequality holds by construction.", "But also, if $x=1_D$ , then $G^G (x)(\\eta ) = \\eta \\ne \\mu = G^G(x)(\\mu )$ , where the inequality holds by assumption again.", "On the other hand, $\\forall _D(i\\delta _G)^G_D(\\eta ,\\mu ) = \\bigcup \\lbrace s \\in \\Omega _\\ast (D) \\mid \\Omega _\\ast (x)(s) \\le i_X (\\delta _G)_X (\\eta _X(x,b) , \\mu _X(x,b))\\rbrace = \\lbrace 1_D\\rbrace .$ for all pairs $(x : X \\rightarrow D, b \\in G(X))$ from C. It is clear that $s = \\lbrace 1_D\\rbrace $ satisfies the condition on the underlying set of the union, since for $x = 1_D$ , $\\Omega _\\ast (1_D)(\\lbrace 1_D\\rbrace ) & = \\lbrace 1_D\\rbrace \\\\& \\subseteq \\top _D = i_D(\\delta _G)_D (\\eta _D(1_D , u) , \\mu _D(1_D , u)).$ On the other hand, for $x = g$ , it is trivially always the case that $\\Omega _\\ast (g)(\\lbrace 1_D\\rbrace ) = \\emptyset \\subseteq (\\delta _G)_C( \\eta _C(g,b) , \\mu _C(g,b)),$ for all $b \\in G(C)$ .", "Furthermore, note that if $g \\in s$ , for some $s \\in \\Omega _\\ast (D)$ , then $\\Omega _\\ast (g)(s) = \\top _C = \\lbrace 1_C\\rbrace .$ So if $g \\in s$ , for some $s$ in the underlying set of the union $(\\ref {ExFunExt1})$ , we had to have $\\top _C = \\Omega _\\ast (g)(s) \\le i_C (\\delta _G)_C (\\eta _C(g,b) , \\mu _C(g,b)),$ for all $b \\in G(C)$ .", "However, since by assumption $\\eta _C(g,w) \\ne \\mu _C(g,w)$ , $(\\delta _G)_C( \\eta _C(g,w) , \\mu _C(g,w)) = \\emptyset ,$ and so $\\Omega _\\ast (g)(s) \\nleq i_C (\\delta _G)_C (\\eta _C(g,w) , \\mu _C(g,w)).$ Thus $g \\notin s$ , for all $s \\in \\Omega _\\ast (D)$ in the underlying set of $\\forall _D(i\\delta _G)^G_D(\\eta ,\\mu )$ .", "Therefore $\\forall _D(i\\delta _G)^G_D(\\eta ,\\mu ) = \\lbrace 1_D\\rbrace ,$ as claimed, and in contrast to (REF ): $i_D(\\delta _{G^G})_D(\\eta , \\mu ) = \\emptyset .$ (Of course, $\\tau (\\lbrace 1_D\\rbrace ) = \\emptyset $ , as lemma REF predicts.)", "Remark 3.8 There is an alternative, more combinatorial way of presenting the previous proof.", "The idea is to formulate the proof in terms of loop graphs rather than presheaves.", "For presheaves on the category $\\lbrace C \\xrightarrow{} D\\rbrace $ can equivalently be regarded as labelled graphs that consist only of loops and points, for instance: ${&\\\\\\bullet _a @(ur,ul)@{-}[]_c & \\bullet _b \\\\}$ Here, $G(D)$ is the set of edges and $G(C)$ the set of vertices, while $G(g)$ assigns to an edge a point, its “source”.", "Thus every loop has a unique source but each point may admit several edges on it.", "$\\Omega $ is the following graph which is easily seen to classify subgraphs: ${&\\\\\\bullet _1 @(ur,ul)@{-}[]_{11} @(dr,dl)@{-}[]^{10} & \\bullet _0 @(ur,ul)@{-}[]_{00} \\\\}$ The labelling expresses the imposed algebraic structure of $\\Omega $ with $0 < 1$ and $xy \\le uv \\ \\text{ iff }\\ x \\le u\\ \\&\\ y \\le v$ .", "Intuitively, in presheaf terms, 1 stands for the maximal sieve on $C$ and 0 for the empty sieve; similarly pairs $xy$ encode sieves on $D$ , where $x = 1$ if and only if $g$ is the sieve and $y=1$ if and only if $1_D$ is in it.", "Then the source of an edge $xy$ is just $x$ .", "For instance, the sieve $\\lbrace g\\rbrace $ on $D$ is encoded by 10.", "Then $\\Omega (g)(\\lbrace g\\rbrace ) = \\lbrace 1_C\\rbrace $ which is encoded by 1.", "Note also that the set of edges is the three-element Heyting algebra from example REF .", "By contrast $\\Omega _\\ast $ is the graph ${&\\\\\\bullet _1 @(ur,ul)@{-}[]_{11} @(dr,dl)@{-}[]^{10} & \\bullet _0 @(ur,ul)@{-}[]_{00} @(dr,dl)@{-}[]^{01}\\\\}$ Here the additional edge 01 corresponds to the fact that $\\lbrace 1_D\\rbrace \\in \\Omega _\\ast (D)$ .", "Thus the set of edges is the four-element Boolean algebra with the source map $2^2 \\rightarrow 2$ induced by the inclusion $1 \\hookrightarrow 2$ .", "The functor $G$ from before becomes the graph ${&\\\\\\bullet _v @(ur,ul)@{-}[]_u & \\bullet _w}$ while $G^G$ is ${\\bullet _{vv} @(ur,ul)@{}[]_{\\theta _1} &\\bullet _{vw} @(ur,ul)@{}[]_{\\theta _0} & \\bullet _{wv} & \\bullet _{ww}}$ The graph $\\Omega ^G$ then looks like this: ${&&&\\\\\\bullet _{11} @(ur,ul)@{-}[]_{111} @(dr,dl)@{-}[]^{110} &\\bullet _{10} @(ur,ul)@{-}[]_{101} @(dr,dl)@{-}[]^{100} &\\bullet _{01} @(dr,dl)@{-}[]^{010} &\\bullet _{00} @(dr,dl)@{-}[]^{000} }$ again with the pointwise ordering.1 The labelling can of course systematically be translated into one such that e.g.", "edges are labelled by natural transformations $\\eta : \\textbf {y}D \\times G \\rightarrow \\Omega $ as before.", "For any such $\\eta $ is uniquely determined by the values $\\eta _D(1_D,u)$ and $\\eta _C(g,w)$ .", "Vertices are just $2^2$ , as there are exactly four natural transformations $\\textbf {y}C \\times G \\rightarrow \\Omega $ , each one defined by the pair $xy$ of values of the component at $a$ ($\\Omega (C) = 2$ ).", "Their intuitive meaning in terms of sieves on $D$ is as before.", "In turn, the notation $xyz$ is chosen in such a way that the source is $xy$ .", "Thus, $xyz$ is to be read so as to mean $\\eta _D(1_D,u) = xz$ and $\\eta _C(g,w) = y$ .", "For by definition the source of an edge $\\eta $ in $\\Omega ^G$ is $\\Omega ^G(g)(\\eta )= \\eta (\\textbf {y}g \\times 1_G)$ .", "Its component at $D$ is empty while for $C$ , and $x = v$ $\\eta _C((\\textbf {y}g)_C \\times 1_{GC})(1_C,v) = \\eta _C(g,v) = \\eta _C(\\textbf {y}D(g) \\times G(g))(1_D,u) = g^\\ast \\eta _D(1_D,u),$ where the last identity holds by naturality of $\\eta $ .", "Thus the source is the pair $(g^\\ast \\eta _D(1_D,u) , \\eta _C(g,w))$ .", "In turn, $g^\\ast \\eta _D(1_D,u)$ is the first digit of $\\eta _D(1_D,u)$ .", "Moreover, in the expression $xyz$ , $y = 1 \\ \\text{ iff }\\ 1_C \\in \\eta _C(g,w)$ .", "So the source of $xyz$ is $xy$ .", "The graph $\\Omega _\\ast ^G$ is: ${&&&\\\\\\bullet _{11} @(ur,ul)@{-}[]_{111} @(dr,dl)@{-}[]^{110} &\\bullet _{10} @(ur,ul)@{-}[]_{101} @(dr,dl)@{-}[]^{100} &\\bullet _{01} @(ur,ul)@{-}[]_{011} @(dr,dl)@{-}[]^{010} &\\bullet _{00} @(dr,dl)@{-}[]^{000} @(ur,ul)@{-}[]_{001}}$ The vertices are the four element Boolean algebra $2^2$ with the pointwise ordering, and the same for the edges $2^3$ .", "The source map $xyz \\mapsto xy$ is the map $2^3 \\rightarrow 2^2$ induced by the inclusion $2 \\hookrightarrow 3$ that projects out the first two arguments of an element of $2^3$ .", "As it turns out, for $\\delta ^G : (G \\times G)^G \\rightarrow \\Omega ^G$ : $(\\delta ^G)_D (\\theta _0,\\theta _1) = 101.$ On the other hand, $\\Delta _C(x) = xx$ and $\\Delta _D(xy) = xxy$ , and so $\\forall _D(xyz) = \\bigvee \\lbrace st \\in \\Omega _\\ast (D) \\mid sst \\le xyz\\rbrace ,$ and similarly for $\\Omega $ .", "Thus $\\forall _D(101) = \\bigvee \\lbrace 00 , 01\\rbrace = 01$ , for $\\forall _D : \\Omega _\\ast ^G(D) \\rightarrow \\Omega _\\ast (D)$ , while $\\forall _D(101) = \\bigvee \\lbrace 00\\rbrace = 00$ , for $\\forall _D : \\Omega ^G(D) \\rightarrow \\Omega (D)$ .", "Note finally that function extensionality is valid in constant domain models.", "(See next section for the connection between topos semantics and Kripke models.)", "For instance, consider a loop graph where $G(D) \\cong 2 \\cong G(C)$ .", "An element in $\\Omega ^G(D)$ , as a natural transformation $\\eta _D : \\textbf {y}D \\times G \\rightarrow \\Omega $ , is completely determined by the two values $\\eta _D(1,a), \\eta _D(1,b)$ , for $\\lbrace a,b\\rbrace = G(D)$ .", "Thus, edges in $\\Omega ^G$ can be represented by sequences $xyzw$ , where $xy$ and $zw$ are the respective edges $\\eta _D(1,a)$ and $\\eta _D(1,b)$ in $\\Omega (D)$ , using the binary notation from before.", "The source of an edge $xyzw$ is $xz$ .", "On the other hand, the map $\\Delta _D : \\Omega (D) \\rightarrow \\Omega ^G(D)$ can be computed as $\\Delta _D(st) = stst$ .", "Now note that there can be no edge in $\\Omega ^G$ of the form $xy01$ or $01zw$ , because 01 is not an edge in $\\Omega $ (moreover that's the only difference between $\\Omega ^G$ and $\\Omega _\\ast ^G$ ).", "As a result, there is no edge in $\\Omega ^G$ such that applying $\\forall $ to it is different from applying $\\forall $ to that same edge in $\\Omega _\\ast ^G$ .", "For the only reason this might happen is because 01 is in the underlying set of the join $\\forall _D(xyzw) = \\bigvee \\lbrace st \\in \\Omega _\\ast (D) \\mid stst \\le xyzw\\rbrace .$ However, if $0101 \\le xyzw$ , for any edge $xyzw$ in $\\Omega ^G$ , then $xyzw = 1111$ .", "But certainly $\\forall $ has the same value on 1111 for both $\\Omega ^G$ and $\\Omega _\\ast ^G$ .", "Although the argument is for models with domain of cardinality 2, it easily generalizes to any $n$ ." ], [ "Algebraic semantics from geometric morphisms", "The canonical example of a model in the sense of def.", "REF is the case where $H = {f}_\\ast \\Omega _{\\mathcal {F}}$ , for a surjective geometric morphism $f : \\mathcal {F} \\rightarrow \\mathcal {E}$ [5], [15], [17], [19].", "We will continue to describe it in some detail to show that the known semantics for it really coincides with the one described in section , the crucial thing to check being the equality relation.", "To ease notation, we write $A^\\ast $ for $f^\\ast A$ , $A_\\ast $ for $f_\\ast A$ and $\\Omega _\\ast $ for ${f}_\\ast \\Omega _{\\mathcal {F}}$ , if $f$ is understood.", "Proposition 4.1 For any geometric morphism $f : \\mathcal {F} \\rightarrow \\mathcal {E}$ , the object $\\Omega _\\ast $ is a complete Heyting algebra in $\\mathcal {E}$ .", "The object $\\Omega _\\ast $ is a Heyting algebra under the image of $f_\\ast $ , since $f_\\ast $ preserves products.", "The same algebraic structure is equivalently determined through Yoneda by the external Heyting operations on each $\\mathrm {Sub}_{\\mathcal {F}}(A^\\ast )$ under the natural isomorphisms $\\mathrm {Sub}_{\\mathcal {F}}(A^\\ast ) \\cong \\mathrm {Hom}_{\\mathcal {F}}(A^\\ast ,\\Omega _{\\mathcal {F}}) \\cong \\mathrm {Hom}_{\\mathcal {E}}( A,\\Omega _\\ast ).$ Completeness means that $\\Omega _\\ast $ has $I$ -indexed joins and meets, for any object $I$ in $\\mathcal {E}$ .", "One way to see this is to first note that there are isomorphisms (natural in $E$ ) $\\mathrm {Hom}_{}(E,(\\Omega _\\ast )^I) \\cong \\mathrm {Hom}_{}(E \\times I,\\Omega _\\ast ) \\cong \\mathrm {Hom}_{}(E^\\ast \\times I^\\ast ,\\Omega _{\\mathcal {F}}) \\cong \\mathrm {Hom}_{}(E^\\ast ,\\Omega _{\\mathcal {F}}^{I^\\ast }),$ where we use that $f^\\ast $ preserves finite limits.", "Composition with ${\\forall _{I^\\ast }} : \\Omega _{\\mathcal {F}}^{I^\\ast } \\longrightarrow \\Omega _{\\mathcal {F}}$ hence yields a function $\\mathrm {Hom}_{}(E,(\\Omega _\\ast )^I) \\xrightarrow{} \\mathrm {Hom}_{}(E^\\ast ,\\Omega _{\\mathcal {F}}^{I^\\ast }) \\xrightarrow{} \\mathrm {Hom}_{}(E^\\ast ,\\Omega _{\\mathcal {F}}) \\xrightarrow{} \\mathrm {Hom}_{}(E,\\Omega _\\ast ),$ all natural in $E$ .", "Thus, by the Yoneda lemma, there is a unique map $\\forall _I : (\\Omega _\\ast )^I \\longrightarrow \\Omega _\\ast $ such that the function $\\mathrm {Hom}_{}(E,(\\Omega _\\ast )^I) \\longrightarrow \\mathrm {Hom}_{}(E,\\Omega _\\ast )$ from above is induced by composition with $\\forall _I$ .", "$\\forall _I$ is indeed right adjoint to $\\Delta _I : \\Omega _\\ast \\rightarrow \\Omega _\\ast ^I$ .", "For $\\Delta _{I^\\ast }: \\Omega _{\\mathcal {F}} \\rightarrow \\Omega _{\\mathcal {E}}^{I^\\ast }$ induces, by composition, a function $\\mathrm {Hom}_{}(E,\\Omega _\\ast ) \\cong \\mathrm {Hom}_{}(E^\\ast ,\\Omega _{\\mathcal {F}}) \\ \\xrightarrow{} \\mathrm {Hom}_{}(E^\\ast ,\\Omega _{\\mathcal {F}}^{I^\\ast })$ with $\\Delta _{I^\\ast } \\mathbin {\\circ } (-) \\dashv \\forall _{}{}_{I^\\ast } \\mathbin {\\circ } (-).$ This adjunction in turn is the one that corresponds by Yoneda under the isomorphism (REF ) to the adjunction $\\pi _1^\\ast \\dashv \\forall _{\\pi _1}$ : $\\forall _{\\pi _1} : \\mathrm {Sub}_{\\mathcal {F}}(E^\\ast \\times I^\\ast ) \\leftrightarrows \\mathrm {Sub}_{\\mathcal {E}}(E^\\ast ) : \\pi _1^\\ast ,$ where $\\pi _1^\\ast $ is pulling back along $\\pi _1 : E^\\ast \\times I^\\ast \\rightarrow E^\\ast $ .", "$I$ -indexed joins are treated similarly.", "The modal operator is given by the uniquely determined structure $\\tau : \\Omega _\\ast \\leftrightarrows \\Omega _{\\mathcal {E}} : i,$ where $\\tau $ is the classifying map of $\\top = f_\\ast (\\top ) : 1 \\rightarrow \\Omega _\\ast .$ Lemma 4.2 The internal adjunction (REF ) is induced via the Yoneda lemma by an external adjunction $\\Delta _A :\\mathrm {Sub}_{\\mathcal {E}}(A) \\leftrightarrows \\mathrm {Sub}_{\\mathcal {F}}(f^\\ast A) : \\Gamma _A$ which is natural in $A$ .1 Cf.", "e.g.", "[19].", "Here, $\\Delta _A$ is $f^\\ast $ restricted to subobjects of $A$ .", "It follows that $\\Delta _A$ is an injective frame map, as $f^\\ast $ is a faithful left exact left adjoint.", "On the other hand, $\\Gamma _A(X,m)$ , for any mono $m : X \\rightarrowtail f^\\ast A$ , is by definition the left-hand map in the following pullback ${\\bullet [r] @{ >->}[d] & f_\\ast X @{ >->}[d]^{f_\\ast m} \\\\A [r]_-{\\eta _A} & f_\\ast f^\\ast A.", "}$ The resulting two functions, natural in $A$ , have the form: $\\mathrm {Hom}_{\\mathcal {E}}(A,\\Omega _{\\mathcal {E}}) \\cong \\mathrm {Sub}_{\\mathcal {E}}(A) \\leftrightarrows \\mathrm {Sub}_{\\mathcal {F}}(f^\\ast A) \\cong \\mathrm {Hom}_{\\mathcal {F}}(f^\\ast A,\\Omega _{\\mathcal {F}}) \\cong \\mathrm {Hom}_{\\mathcal {F}}(A,\\Omega _\\ast ).$ By Yoneda they determine maps $\\delta : \\Omega _{\\mathcal {E}} \\leftrightarrows \\Omega _\\ast : \\gamma ,$ internally adjoint given that $\\Delta _A \\dashv \\Gamma _A$ , for each $A$ in $\\mathcal {E}$ .", "The map $\\delta $ is monic, because each $\\Delta _A$ is injective.", "It readily follows that $\\delta = i$ and $\\gamma = \\tau $ .", "For $\\delta $ is a monic frame map and $\\delta \\dashv \\gamma $ , while the arrow $\\gamma : \\Omega _\\ast \\longrightarrow \\Omega _{\\mathcal {E}}$ obtained through the Yoneda lemma as above actually is the classifying map of the top element $f_\\ast \\top : 1 \\rightarrow \\Omega _\\ast $ .", "Lemma 4.3 The internal structure $\\Omega _\\ast $ is a faithful frame, i.e.", "the canonical frame map $i : \\Omega _{\\mathcal {E}} \\rightarrow \\Omega _\\ast $ is a monomorphism.", "Since the maps $\\Delta _A$ in lemma REF are injective, this means that $\\Delta : \\mathrm {Sub}_{\\mathcal {E}}(-) \\rightarrow \\mathrm {Sub}_{\\mathcal {F}}(f^\\ast (-))$ is a monic natural transformation.", "As $i: \\Omega _{\\mathcal {E}} \\rightarrow \\Omega _\\ast $ is obtained using the Yoneda lemma from the maps $\\Delta _A$ , it readily follows that $i$ is monic, because the Yoneda embedding reflects monomorphisms.", "Formulas $\\varphi $ (in one free variable, say) are thus interpreted equivalently in any of the following ways (let $M$ interpret the type of $x$ ): $\\llbracket \\varphi \\rrbracket \\in \\mathrm {Sub}_{\\mathcal {F}}(f^\\ast M), \\ \\ \\ \\ \\ \\ M^\\ast \\xrightarrow{} \\Omega _{\\mathcal {F}}, \\ \\ \\ \\ \\ \\ M \\xrightarrow{} \\Omega _\\ast ,$ where the third one follows from definition REF .", "Moreover, let $\\delta _{M^\\ast }$ be the classifying map of the diagonal $\\langle 1_{M^\\ast } , 1_{M^\\ast } \\rangle : M^\\ast \\rightarrow M^\\ast \\times M^\\ast $ .", "We will write its transpose along $f^\\ast \\dashv f_\\ast $ simply as $M \\times M \\xrightarrow{} \\Omega _\\ast $ when $M$ is clear.", "Then we have: Lemma 4.4 The equality predicate for $M^*$ may be interpreted by the map (REF ), obtained as the transpose along $f^\\ast \\dashv f_\\ast $ of $(M \\times M)^\\ast \\cong M^\\ast \\times M^\\ast \\xrightarrow{} \\Omega _{\\mathcal {F}}.$ The proof is immediate, given the soundness of the interpretation with respect to $\\delta _{M^\\ast }$ .", "Definition 4.5 By a geometric model we shall mean a model derived from a geometric morphism in this way; specifically, where $f : \\mathcal {F} \\rightarrow \\mathcal {E}$ and $H=f_*(\\Omega _{\\mathcal {F}})$ .", "To show, finally, that geometric models are a special case of algebraic ones, the main thing that needs to be verified is that equality is interpreted the same way in each case, i.e.", ": $\\delta _\\ast = i \\mathbin {\\circ } \\delta _M.$ First, we make the following observation: Lemma 4.6 For any map $\\alpha : D \\rightarrow \\Omega _\\ast $ , we have $i \\tau \\mathbin {\\circ } \\alpha = \\alpha $ iff the subobject classified by the transpose $\\widetilde{\\alpha } : f^\\ast D \\rightarrow \\Omega _{\\mathcal {F}}$ of $\\alpha $ is of the form $f^\\ast m : f^\\ast A \\rightarrowtail f^\\ast D$ , for some $m : A \\rightarrowtail D$ in $\\mathcal {E}$ .", "$\\Box $ Proposition 4.7 For any object $D$ in $\\mathcal {E}$ , and any geometric morphism $f : \\mathcal {F} \\rightarrow \\mathcal {E}$ : $\\delta _\\ast = i \\mathbin {\\circ } \\delta _D.$ We prove this by showing $\\tau \\mathbin {\\circ } \\delta _\\ast = \\delta _D,$ whence the statement follows from $\\delta _\\ast = i \\mathbin {\\circ } \\tau \\mathbin {\\circ } \\delta _\\ast = i \\mathbin {\\circ } \\delta _D$ , where the identity $\\delta _\\ast = i \\mathbin {\\circ } \\tau \\mathbin {\\circ } \\delta _\\ast $ holds by applying lemma REF to $\\delta _\\ast $ .", "The proof is essentially contained in the following diagram ${D [rr]^{\\eta _D} [dd]_{\\Delta _D} && (D^\\ast )_\\ast [rr] [dd]|{(\\Delta ^\\ast )_\\ast } && 1[dd]^\\top @{=}[drr]&\\\\&&&&&&1[dd]^\\top \\\\D \\times D [rr]^-{\\eta _D \\times \\eta _D} @/_1.7pc/[rrrr]_{\\delta _\\ast } @/_1.7pc/[drrrrrr]_{\\delta _D} && (D^\\ast )_\\ast \\times (D^\\ast )_\\ast [rr]^-{(\\delta _{D^\\ast })_\\ast } && \\Omega _\\ast [drr]_\\tau &\\\\&&&&&&\\Omega _{\\mathcal {E}}\\\\}$ where $\\Delta _D = \\langle 1_D , 1_D \\rangle $ , $\\eta $ is the unit of $f^\\ast \\dashv f_\\ast $ , and $\\delta , \\tau $ denote the respective classifying maps.", "The square in the middle is a pullback, since $f_\\ast $ preserves them.", "Moreover, by the definition of $\\delta _D$ , the large outer square is a pullback.", "Note further that $\\delta _\\ast = (\\delta _{D^\\ast })_\\ast \\mathbin {\\circ } \\eta _{D\\times D}$ , by the definition of $\\delta _\\ast $ as the transpose of $\\delta _{D^\\ast }$ along $f^\\ast \\dashv f_\\ast $ .", "Thus the desired equality would follow if the unit square were a pullback, for then $\\tau \\mathbin {\\circ } (\\delta _{D^\\ast })_\\ast \\mathbin {\\circ } \\eta _{D\\times D} = \\tau \\mathbin {\\circ } \\delta _\\ast $ would classify $\\Delta _D$ , and so $\\tau \\mathbin {\\circ } \\delta _\\ast = \\delta _D$ .", "This is in fact the case.", "For $f : \\mathcal {F} \\rightarrow \\mathcal {E}$ being surjective (i.e.", "$f^\\ast $ faithful) implies that the unit components, and therefore $\\eta _D \\times \\eta _D$ , are monic.", "A direct verification then shows that the square is a pullback.", "Example 4.1 Kripke Models.", "As is well known, any functor $F : \\textbf {C} \\rightarrow \\textbf {D}$ induces a geometric morphism $f^\\ast \\dashv f_\\ast : \\textbf {Sets}^{\\textbf {C}} \\rightarrow \\textbf {Sets}^{\\textbf {D}} ,$ where $f^\\ast $ is precomposition with $F$ , and $f_\\ast $ is a right Kan extension.", "Let $\\textbf {C} = |\\textbf {D}|$ and $F$ the inclusion $i: |\\textbf {D}| \\rightarrow \\textbf {D}$ .", "Then the induced geometric morphism $i^\\ast \\dashv i_\\ast : \\textbf {Sets}^{|\\textbf {D}|} \\rightarrow \\textbf {Sets}^{\\textbf {D}}$ is surjective.", "The subobject classifier $\\Omega _\\textbf {D}$ in $\\textbf {Sets}^{\\textbf {D}}$ consists, for each $D$ , of the set of cosieves on $D$ , which can be construed as the functor category $2^{D/\\textbf {D}},$ where 2 is viewed as the poset $\\lbrace 0 \\le 1\\rbrace $ ; while $\\Omega _{|\\textbf {D}|}(D) = 2$ , for each $D$ in D. On the other hand, by the definition of right Kan extension, $i_\\ast \\Omega _{|\\textbf {D}|}(D) = \\prod _{h \\in D/\\textbf {D}} 2 = 2^{|D/\\textbf {D}|}$ , as can also be seen from $i_\\ast \\Omega _{|\\textbf {D}|}(D) \\cong \\mathrm {Hom}_{\\widehat{\\textbf {D}}}(\\textbf {y}D,i_\\ast \\Omega _{|\\textbf {D}|}) \\cong \\mathrm {Hom}_{\\widehat{|\\textbf {D}|}}(i^\\ast (\\textbf {y}D),\\Omega _{|\\textbf {D}|}).$ The last set is (isomorphic to) the set of subfamilies of the functor $i^\\ast (\\textbf {y}D) : |\\textbf {D}| \\rightarrow \\textbf {Sets}$ , by the definition of the subobject classifier $\\Omega _{|\\textbf {D}|}$ : each natural transformation $i^\\ast \\textbf {y}D = \\textbf {y}D\\mathbin {\\circ } i = \\mathrm {Hom}_{\\textbf {D}}(D,-) \\longrightarrow 2$ determines, for each $D^{\\prime }$ in D, a set of arrows $D \\rightarrow D^{\\prime }$ .", "On arrows $h : D \\rightarrow D^{\\prime \\prime }$ , the functor $i_\\ast \\Omega _{|\\textbf {D}|}$ is the function $i_\\ast \\Omega _{|\\textbf {D}|}(h) : i_\\ast \\Omega _{|\\textbf {D}|}(D) \\rightarrow i_\\ast \\Omega _{|\\textbf {D}|}(D^{\\prime \\prime })$ defined as $i_\\ast \\Omega _{|\\textbf {D}|}(h)(A) = \\lbrace f : D^{\\prime \\prime } \\rightarrow X \\mid f \\mathbin {\\circ } h \\in A\\rbrace .$ The components of the (internal) adjunction $i : \\Omega _\\textbf {D} \\leftrightarrows i_\\ast \\Omega _{|\\textbf {D}|} : \\tau $ then read $i_D : 2^{D/\\textbf {D}} \\leftrightarrows 2^{|D/\\textbf {D}|} : \\tau _D,$ where $i_D \\dashv \\tau _D$ “externally”.", "It is not hard to see that $i$ is the inclusion, while $\\tau _D(A) = \\bigvee \\lbrace S \\in 2^{D/\\textbf {D}} \\mid i_D(S) \\le A\\rbrace ,$ by the definition of right adjoint to the frame map $i$ (cf.", "(REF )).", "In words, $\\tau $ maps any family of arrows with domain $D$ to the largest cosieve on $D$ contained in it.", "In particular, when $\\textbf {D}$ is a preorder, then $D/\\textbf {D} = \\uparrow (D)$ , the upward closure of $D$ ; while $2^{D/\\textbf {D}}$ is the set of all monotone maps $\\uparrow (D) \\rightarrow 2$ , i.e.", "upsets of $\\uparrow (D)$ , while $2^{|D/\\textbf {D}|}$ is the set of arbitrary subsets of $\\uparrow (D)$ .", "An arrow $\\varphi : E \\rightarrow i_\\ast \\Omega _{|\\textbf {D}|} = 2^{|-/\\textbf {D}|}$ in $\\textbf {Sets}^{\\textbf {D}}$ defines an indexed subfamily $P$ of the functor $F$ , and conversely.", "Explicitly, given such $\\varphi : E \\rightarrow i_\\ast \\Omega _{|\\textbf {D}|}$ , define subsets $P_\\varphi (D) \\subseteq E(D)$ , for each $D$ in D and $a \\in E(D)$ , by $a \\in P_\\varphi (D) \\ \\text{ iff }\\ 1_D \\in \\varphi _D(a).$ Conversely, given maps $E(D) \\rightarrow 2$ , i.e.", "components of an arrow $i^\\ast E \\rightarrow \\Omega _{|\\textbf {D}|}$ in $\\textbf {Sets}^{|\\textbf {D}|}$ , or equivalently a subfamily $P$ of $E$ , define a natural transformation $\\varphi _P : E \\rightarrow i_\\ast \\Omega _{|\\textbf {D}|}$ by $(\\varphi _P)_D(a) = \\lbrace f : D \\rightarrow C \\mid E(f)(a) \\in P(C) \\rbrace ,$ These constructions are mutually inverse and so describe the canonical isomorphism $\\mathrm {Hom}_{}(E,i_\\ast \\Omega _{|\\textbf {D}|}) \\cong \\mathrm {Hom}_{}(i^\\ast E,\\Omega _{|\\textbf {D}|}) \\cong \\mathrm {Sub}_{}(i^\\ast E).$ Note also that the transpose $\\overline{\\varphi } = \\varepsilon \\varphi ^\\ast $ of $\\varphi : E \\rightarrow \\Omega _\\ast $ along the adjunction $f^\\ast \\dashv f_\\ast $ actually is the classifying map in $\\textbf {Sets}^{|\\textbf {D}|}$ of the subobject $P_\\varphi $ of $f^\\ast E$ defined in (REF ): $\\varepsilon _C\\varphi ^\\ast _C(a) = 1 & \\ \\text{ iff }\\ 1_C \\in \\varphi ^\\ast _C(a)\\\\& \\ \\text{ iff }\\ 1_C \\in \\varphi _C(a) \\\\& \\ \\text{ iff }\\ a \\in P_\\varphi (C),$ for any $a \\in E(C)$ .", "On the other hand, considering $\\Omega _{\\textbf {D}} = 2^{D/\\textbf {D}}$ instead of $2^{|D/\\textbf {D}|}$ , the same definitions (REF ) and (REF ) establish a correspondence between subfunctors of $E$ and their classifying maps in $\\textbf {Sets}^{\\textbf {D}}$ .", "In particular, the classifying map of a subfunctor of $E$ factors through $i_\\ast \\Omega _{|K|}$ via $\\tau $ .", "Thus, when D is a preorder, algebraic models in the complete Heyting algebra $i_\\ast \\Omega _{|K|}$ are precisely Kripke models on D. The “domain” of the model is given by the functor $E$ , while each $E(D)$ is the domain of individuals at each world $D$ .", "Each formula determines, as an arrow $\\varphi : E \\rightarrow i_\\ast \\Omega _{|K|}$ , a subfamily of $E$ , that is a family $(P_\\varphi (D) \\subseteq E(D))$ .", "Then $\\tau $ determines the largest compatible subfamily of that family, i.e.", "a family closed under the action of $E$ .", "Indeed, for $x \\in E(D)$ , $x \\in P_{\\tau \\varphi }(D) \\ \\text{ iff }\\ 1_D \\in (\\tau \\varphi )_D(x).$ Now $(\\tau \\varphi )_D(x)$ is the maximal sieve on $D$ just in case $\\varphi _D(x)$ is.", "So, if satisfied, the right-hand side means that $x \\in P_\\varphi (D)$ and moreover $F(f)(x) \\in P_\\varphi (C)$ , for all $C \\ge D$ .", "Semantically speaking, $x$ satisfies $\\tau \\varphi $ (at $D$ ) just in case $x$ (or rather its “counterpart” $F_{CD}(x)$ ) satisfies $\\varphi $ in all worlds accessible from $D$ .", "Thus we recovered the natural adjunction $\\Delta _E : \\mathrm {Sub}_{}(E) \\leftrightarrows \\mathrm {Sub}_{}(i^\\ast E) : \\Gamma _E$ that succinctly describes the algebraic structure of Kripke models.", "Lastly, presheaf semantics reduces to standard Kripke semantics for propositional modal logic in the following sense.", "In the latter, propositional formulas are recursively assigned elements in $\\mathcal {P}(\\textbf {K})$ , for a preorder $\\textbf {K}$ .", "Let $\\mathcal {P}(\\downarrow (-)) = \\Omega _\\ast $ be the composite functor $\\textbf {K} \\xrightarrow{} \\textbf {Sets} \\xrightarrow{} \\textbf {Sets}^{op}.$ Observe that $\\mathcal {P}(\\textbf {K}) \\cong \\mathrm {Hom}_{\\textbf {Sets}^{\\textbf {K}^{op}}}(1,\\mathcal {P}(\\downarrow (-))),$ via assignments (where $\\varphi \\subseteq \\mathcal {P}(\\textbf {K})$ ) $\\varphi \\mapsto (\\varphi _k =\\ \\downarrow (k \\cap \\varphi ) \\mid k \\in \\textbf {K})$ and $(\\varphi _k \\mid k \\in \\textbf {K}) \\mapsto \\bigcup _k \\varphi _k.$ Thus modelling formulas (in one variable, say) by maps of presheaves $M \\longrightarrow \\mathcal {P}(\\downarrow (-)) = \\Omega _\\ast $ yields precisely the familiar Kripke model idea for propositions, i.e.", "closed formulas.", "Moreover, for constant domains: $\\mathrm {Hom}_{\\textbf {Sets}^{\\textbf {K}^{op}}}(\\Delta M,\\mathcal {P}(\\downarrow (-))) \\cong \\mathrm {Hom}_{\\textbf {Sets}}(M,\\varprojlim \\mathcal {P}(\\downarrow (-))) \\cong \\mathrm {Hom}_{\\textbf {Sets}}(M,\\mathcal {P}(K)).$ Here, $\\Delta : \\textbf {Sets} \\longrightarrow \\textbf {Sets}^{\\textbf {K}^{op}}$ is the functor $\\Delta (M)(k) = M$ , for any set $M$ and $k \\in \\textbf {K}$ .", "A function $\\varphi : M \\longrightarrow \\mathcal {P}(K)$ assigns to each individual in the domain $M$ a set of worlds for which the individual satisfies the formula represented by $\\varphi $ ." ], [ "Kripke-Joyal forcing:", "Another way of seeing the close relation between presheaf semantics and Kripke semantics is via the notion of “Kripke-Joyal forcing” [14], [12].", "For any topos $\\mathcal {E}$ one can define a forcing relation $\\Vdash $ to interpret intuitionistic higher-order logic .", "Given an arrow $\\varphi : M \\rightarrow \\Omega _{\\mathcal {E}}$ , let $S_\\varphi $ be the subobject of $M$ classified by $\\varphi $ .", "Then for any $a : X \\rightarrow M$ , define $X \\Vdash \\varphi (a) \\ \\text{ iff }\\ a\\ \\text{factors through}\\ S_\\varphi .$ This holds iff $\\varphi a = \\textsf {t}_X$ , where $\\textsf {t}_X$ is the arrow $\\top \\mathbin {\\circ } \\,{!_X} :X \\rightarrow 1 \\rightarrow \\Omega _{\\mathcal {E}}$ .", "The idea is that $\\varphi $ corresponds to a formula, while $a$ is a generalized element of $M$ , thought of as a term $x : X \\mid a : M$ .", "In fact, $\\varphi $ and $a$ are terms in the internal language of $\\mathcal {E}$ , reinterpreted into $\\mathcal {E}$ by the forcing relation.", "The relation $\\Vdash $ satisfies certain recursive clauses for all the logical connectives [14], [12].", "Conversely, starting with an interpretation of the basic symbols of a higher-order type theory in a topos $\\mathcal {E}$ (as maps into $\\Omega _{\\mathcal {E}}$ ), then these recursive clauses determine when a formula is true (“at an object $X$ ”).", "When $a$ is a closed term, i.e.", "a constant, for which one may assume $X = 1$ , then this says that the two arrows ${1 [r]^a @/_1.5pc/[rr]_\\top & M [r]^\\varphi & \\Omega _{\\mathcal {E}}}$ are equal; i.e.", "the closed sentence $\\varphi [a/x]$ is “true”.", "In general, the forcing relation thus defines when formulas are true (at $X$ ), much as in Kripke semantics, as we now illustrate.", "Consider presheaf toposes of the form $\\textbf {Sets}^{\\textbf {C}^{op}}$ .", "In this case, the forcing relation $X \\Vdash \\varphi (a) $ can be restricted to objects $X$ in $\\mathcal {E}$ forming a generating set.1 Cf.", "[12].", "One says that a set $S$ of objects from $\\mathcal {E}$ is generating, iff for any $f \\ne g : A \\rightrightarrows B$ in $\\mathcal {E}$ , there is an arrow $x : X \\rightarrow A$ , for some $X \\in S$ , such that $fx \\ne gx$ .", "For presheaf toposes $\\textbf {Sets}^{\\textbf {C}^{op}}$ the representable functors $\\textbf {y}C$ form a generating set, so one may assume that $X = \\textbf {y}C$ , for some object $C$ in C. Also, by the Yoneda lemma, generalized elements $a:\\textbf {y}C \\rightarrow M$ may be replaced by actual elements $a \\in M(C)$ .", "To say that $a : \\textbf {y}C \\rightarrow M$ factors through a subobject $S \\in \\mathrm {Sub}_{\\mathcal {E}}(M)$ is then equivalent to saying that the corresponding element $a \\in M(C)$ actually lies in $S(C)$ .", "As a result, the forcing condition becomes $\\textbf {y}C \\Vdash \\varphi (a) \\ \\text{ iff }\\ a \\in S_\\varphi (C),$ where, as before, $\\varphi $ classifies the subobject $S_\\varphi $ of $M$ .", "We shall hereafter write $C \\Vdash \\dots $ instead of $\\textbf {y}C \\Vdash \\dots $ .", "Now consider the standard $\\Omega _\\ast $ -valued model for classical higher-order modal logic in a presheaf topos $\\textbf {Sets}^{\\textbf {C}^{op}}$ , associated with the canonical geometric morphism $\\textbf {Sets}^{|\\textbf {C}|} \\rightarrow \\textbf {Sets}^{\\textbf {C}^{op}}$ .", "We define another forcing relation $C \\Vdash _* \\varphi (a)$ which takes this modal logic into account.", "Definition 4.8 For any presheaf topos $\\textbf {Sets}^{\\textbf {C}^{op}}$ , define a forcing relation $\\Vdash _*$ for arrows $\\varphi : M \\rightarrow \\Omega _\\ast $ , objects $C$ in C, and elements $a \\in M(C)$ by: $C \\Vdash _* \\varphi (a) \\ \\text{ iff }\\ C \\Vdash \\overline{\\varphi }(a),$ where $\\Vdash $ on the right-hand side is the usual forcing relation w.r.t.", "$\\textbf {Sets}^{|\\textbf {C}|}$ (as defined in (REF )), and $\\overline{(-)}$ indicates transposition along $f^\\ast \\dashv f_\\ast $ .", "Further analysing the right-hand side of (REF ) gives: $C \\Vdash \\overline{\\varphi }(a) \\ \\text{ iff }\\ a \\in S_{\\overline{\\varphi }}(C)$ where $S_{\\overline{\\varphi }}$ is the subobject of $M^*$ classified by $\\overline{\\varphi }$ in $\\textbf {Sets}^{|\\textbf {C}|}$ .", "Proposition 4.9 Let $\\Vdash _*$ be the forcing relation of Definition REF .", "Then for all $\\varphi ,\\psi : M \\rightarrow \\Omega _\\ast $ and $a \\in M(C)$ the following hold: $&C \\Vdash _* \\top \\ &&\\text{always}\\\\&C \\Vdash _* \\bot \\ &&\\text{never}\\\\&C \\Vdash _* \\varphi (a) \\wedge \\psi (a) &&\\ \\text{ iff }\\ \\quad C \\Vdash _* \\varphi (a) \\ \\text{and}\\ C \\Vdash _* \\psi (a)\\\\&C \\Vdash _* \\varphi (a) \\vee \\psi (a) &&\\ \\text{ iff }\\ \\quad C \\Vdash _* \\varphi (a) \\ \\text{or}\\ C \\Vdash _* \\psi (a)\\\\&C \\Vdash _* \\varphi (a) \\Rightarrow \\psi (a) &&\\ \\text{ iff }\\ \\quad C \\Vdash _* \\varphi (a) \\ \\text{implies}\\ C \\Vdash _* \\psi (a)\\\\&C \\Vdash _* \\forall x \\varphi (x,a) &&\\ \\text{ iff }\\ \\quad C \\Vdash _* \\varphi (b,a)\\ \\text{for all}\\ b \\in M(C)\\\\&C \\Vdash _* \\exists x \\varphi (x,a) &&\\ \\text{ iff }\\ \\quad C \\Vdash _* \\varphi (b,a)\\ \\text{for some}\\ b \\in M(C)\\\\&C \\Vdash _* \\Box \\varphi (a) &&\\ \\text{ iff }\\ \\quad D \\Vdash _* \\varphi (p^\\ast a)\\ \\text{for every}\\ p: D \\rightarrow C\\\\&C \\Vdash _* t(a) \\in u(a) &&\\ \\text{ iff }\\ \\quad (1_C, t_C(a)) \\in (u_C(a))_C,\\\\&&&\\quad \\quad \\quad \\quad \\text{for}\\ t : M \\rightarrow N \\text{and}\\ u : M \\rightarrow \\Omega _\\ast ^N$ where $\\Box = i\\tau $ , and $\\forall x \\varphi $ is the arrow $M \\xrightarrow{}\\Omega _\\ast ^M \\xrightarrow{} \\Omega _\\ast $ , with $\\widehat{\\varphi }$ the exponential transpose of $M \\times M \\xrightarrow{} \\Omega _\\ast $ , and similarly for $\\exists x \\varphi (x,a)$ .", "Remark 4.10 Although $\\Vdash _*$ is a relation between objects $C$ and arrows $\\varphi : M \\rightarrow \\Omega _\\ast $ , it also makes sense to think of the $\\varphi $ as formulas, with the clauses above holding w.r.t.", "the arrow $\\llbracket \\varphi \\rrbracket $ assigned to the formula $\\varphi $ as in section .", "For instance, interpreting a syntactic expression $\\exists x \\varphi (x,y)$ (by REF ) yields an arrow $\\exists _M\\widehat{\\llbracket \\varphi \\rrbracket }$ .", "When $\\textbf {C}$ is a preorder this is then not merely similar to, but actually is the Kripkean satisfaction relation between worlds and formulas, extended to higher-order logic.", "We shall just do a few exemplary cases for the purpose of illustration.", "Consider $C \\Vdash _* \\varphi (a) \\vee \\psi (a)$ , which by definition REF means that $a \\in S_{\\overline{\\varphi \\vee \\psi }}(C)$ .", "Here, $\\Omega _\\ast \\times \\Omega _\\ast \\xrightarrow{} \\Omega _\\ast $ is the join map.", "Recall from proposition REF that $\\vee $ actually is of the form $\\vee _\\ast $ , for the join map $\\Omega \\times \\Omega \\xrightarrow{} \\Omega $ in $\\textbf {Sets}^{|\\textbf {C}|}$ .", "Thus the following commutes, by naturality of the counit $\\varepsilon $ : ${M^\\ast \\times M^\\ast [r]^-{\\langle \\varphi ^\\ast , \\psi ^\\ast \\rangle } [dr]_{\\langle \\overline{\\varphi } , \\overline{\\psi } \\rangle } & (\\Omega _\\ast )^\\ast \\times (\\Omega _\\ast )^\\ast [r]^-{(\\vee _\\ast )^\\ast } [d]|{\\varepsilon \\times \\varepsilon } & (\\Omega _\\ast )^\\ast [d]^\\varepsilon \\\\& \\Omega \\times \\Omega [r]_\\vee & \\Omega }$ That is to say, $\\overline{\\varphi \\vee \\psi } = \\overline{\\varphi } \\vee \\overline{\\psi },$ and so $S_{\\overline{\\varphi } \\vee \\overline{\\psi }} = S_{\\overline{\\varphi \\vee \\psi }}$ .", "Since $\\textbf {Sets}^{|\\textbf {C}|}$ is a Boolean topos, by the definition of $S_{\\overline{\\varphi } \\vee \\overline{\\psi }}$ in $\\textbf {Sets}^{|\\textbf {C}|}$ we have: $a \\in S_{\\overline{\\varphi } \\vee \\overline{\\psi }}(C) \\ \\text{ iff }\\ a \\in S_{\\overline{\\varphi }}(C)\\ \\text{or}\\ a \\in S_{\\overline{\\psi }}(C),$ i.e.", "if and only if $C \\Vdash _* \\varphi (a) \\ \\text{or}\\ C \\Vdash _* \\psi (a)$ .", "The argument for the other logical connectives is similar.", "For $\\forall $ , by definition, $C \\Vdash _* \\forall x \\varphi (x,a) \\ \\text{ iff }\\ a \\in S_{\\forall _M\\widehat{\\varphi }}(C),$ with $S_{\\forall _M\\widehat{\\varphi }}(C) = \\lbrace a \\in M(C) \\mid 1_C \\in (\\forall _M\\widehat{\\varphi })_C(a)\\rbrace $ defined as in (REF ).", "By the definition of $\\forall _M$ , and because $|\\textbf {C}|$ is discrete: $1_C \\in (\\forall _M\\widehat{\\varphi })_C(a) & \\ \\text{ iff }\\ 1_C \\in \\bigcup \\lbrace s \\in \\Omega _\\ast (C) \\mid \\Omega _\\ast (f)(s) \\le \\widehat{\\varphi }_C(a)_D(f,b),\\\\&\\quad \\quad \\quad \\quad \\text{for all}\\ f : D \\rightarrow C, b \\in M(D)\\rbrace \\\\& \\ \\text{ iff }\\ 1_C \\in \\bigcup \\lbrace s \\in \\Omega _\\ast (C) \\mid s \\le \\widehat{\\varphi }_C(a)_C(1_C,b),\\ \\text{for all}\\ b \\in M(C)\\rbrace \\\\& \\ \\text{ iff }\\ 1_C \\in \\varphi _C(a,b),\\ \\text{for all}\\ b \\in M(C)\\\\&\\ \\text{ iff }\\ (a,b) \\in S_\\varphi ,\\ \\text{for all}\\ b \\in M(C) \\\\&\\ \\text{ iff }\\ C \\Vdash _* \\varphi (a,b),\\ \\text{for all}\\ b \\in M(C).$ The last two equivalences hold by the definition of $S_\\varphi $ and $\\Vdash _*$ .", "To see the third equivalence, let $\\alpha : \\textbf {y}C \\rightarrow M$ be the map that corresponds under Yoneda to $a \\in M(C)$ .", "Then, by the definition of $\\widehat{\\varphi }$ (cf.", "(REF )): $\\widehat{\\varphi }_C(a)_C(1_C,b) = \\varphi _C(\\alpha \\times 1_M)_C(1_C,b) = \\varphi _C(\\alpha _C(1_C) ,b) = \\varphi _C(a,b).$ Then, if $1_C$ is in the union, it is in one of the $s \\in \\Omega _\\ast (C)$ , and thus $1_C \\in \\varphi _C(a,b)$ , for all $b \\in M(C)$ .", "On the other hand, if $1_C \\in \\varphi _C(a,b)$ , for all $b \\in M(C)$ , then $1_C$ is in the union for $s = \\lbrace 1_C\\rbrace $ .", "The clause for $\\in $ follows from its definition: $S_{\\varepsilon \\langle s,t \\rangle } & = \\lbrace a \\in M(C) \\mid 1_C \\in \\varepsilon \\langle s,t\\rangle _C(a)\\rbrace \\\\& = \\lbrace a \\in M(C) \\mid 1_C \\in \\varepsilon _C(s_C(a),t_C(a)) \\rbrace \\\\& = \\lbrace a \\in M(C) \\mid 1_C \\in (s_C(a))_C(1_C,t_C(a))\\rbrace , $ using the definition of the evaluation map $\\varepsilon : \\Omega ^A \\times A \\rightarrow \\Omega $ .", "For $\\Box $ , as before, $i\\tau \\varphi $ determines a subfamily of $M$ with components $S_{i\\tau \\varphi }(C) = \\lbrace a \\in M(C) \\mid 1_C \\in (i\\tau \\varphi )_C(a)\\rbrace .$ But $(i\\tau \\varphi )_C(a)$ is a sieve, as it factors through $\\Omega (C)$ , and so $S_{i\\tau \\varphi }(C) = \\lbrace a \\in M(C) \\mid (i\\tau \\varphi )_C(a) = \\top _C\\rbrace ,$ for $\\top _C$ the maximal sieve on $C$ .", "However, by the defining properties of $\\tau $ and $i$ , $(i\\tau \\varphi )_C(a) = \\top _C \\ \\text{ iff }\\ \\varphi _C(a) = \\top _C.$ Therefore, $S_{i\\tau \\varphi }(C) & = \\lbrace a \\in M(C) \\mid \\varphi _C(a) = \\top _C\\rbrace \\\\&= \\lbrace a \\in M(C) \\mid (\\chi _{S_\\varphi })_C(a) = \\top _C\\rbrace \\\\&= \\lbrace a \\in M(C) \\mid \\lbrace p : D \\rightarrow C \\mid p^\\ast a \\in S_\\varphi (D)\\rbrace = \\top _C\\rbrace \\\\& = \\lbrace a \\in M(C) \\mid p^\\ast a \\in S_{\\varphi }(D), \\ \\text{for all}\\ p : D \\rightarrow C\\rbrace .$ In forcing terms: $C \\Vdash _* i\\tau \\varphi (a) & \\ \\text{ iff }\\ a \\in S_{i\\tau \\varphi }(C) \\\\& \\ \\text{ iff }\\ p^\\ast a \\in S_{\\varphi }(D), \\ \\text{for all}\\ p : D \\rightarrow C\\\\& \\ \\text{ iff }\\ D \\Vdash _* \\varphi (p^\\ast a), \\ \\text{for all}\\ p : D \\rightarrow C.$ Example 4.2 Sheaf Models.", "For a topological space $X$ the (surjective) geometric morphism $i^\\ast \\dashv i_\\ast : \\textbf {Sets}/X \\longrightarrow \\text{Sh}_{}(X)$ coming from the continuous inclusion $i : |X| \\hookrightarrow X$ gives rise to modal sheaf semantics for classical S4 modal logic as described in [1].", "This is most readily seen by viewing sheaves on $X$ as local homeomorphisms over $X$ .", "In this case, the adjunction (REF ) reads: $\\Delta _\\pi : \\mathrm {Sub}_{LH/X}(E) \\leftrightarrows \\mathrm {Sub}_{\\textbf {Sets}/X}(i^\\ast E) : \\Gamma _\\pi $ where $E \\rightarrow X$ is a local homeomorphism.", "A subobject of $i^\\ast E$ in $\\textbf {Sets}/X$ is simply a commutative triangle of functions in $\\textbf {Sets}$ ${A @{^(->}[rr] [dr] && E [dl]\\\\&X&}$ which is entirely determined by a subset $A \\subseteq E$ .", "One obtains the largest subsheaf of $E$ contained in $A$ just by applying the interior operator of $E$ to $A \\subseteq E$ : ${\\text{int} A @{^(->}[rr] [dr] && E [dl]\\\\&X&}$ The horizontal inclusion is then continuous w.r.t.", "the subspace topology on $\\text{int} A$ .", "The composite is then a local homeomorphism, because the restriction of any local homeomorphism to an open subset of the total space ($E$ ) is one.", "This is therefore just the familiar topological semantics for propositional modal logic, given by the adjunction $i : \\mathrm {Sub}_{\\text{Sh}_{}(X)}(E) \\cong \\mathcal {O}(E) \\leftrightarrows \\mathcal {P}(E) \\cong \\mathrm {Sub}_{\\textbf {Sets}/X}(i^\\ast E) : \\text{int}$ In this case the algebraic formulation via maps into the subobject classifier is perhaps less intuitive.", "The subobject classifier $\\omega : \\Omega \\rightarrow X$ in $\\text{Sh}_{}(X)$ has the fibers:1 See e.g.", "[14].", "$\\omega ^{-1}(x) = \\varinjlim _{x \\in U}\\downarrow \\!", "{U}$ where $\\downarrow \\!", "{U}$ is the set of all open subsets of $U \\in \\mathcal {O}(X)$ .", "On the other hand, viewing sheaves as a special kind of presheaves, the formulation is now more familiar.", "The subobject classifier takes the form $\\Omega _X(U) = \\downarrow \\!", "{U}$ (for $V \\subseteq U$ this acts by $V \\cap -$ , i.e.", "the inverse image along the inclusion).", "Thus $\\Omega _X(U) = \\mathcal {O}(U)$ for the subspace topology on $U$ .", "In turn, $\\Omega _\\ast (U) = \\mathcal {P}(U)$ with the evident restriction along inclusions.", "Thus propositions are modelled by natural transformations $M \\rightarrow \\mathcal {P}$ to the contravariant powerset-functor, while the map $\\tau _U : \\mathcal {P}(U) \\rightarrow \\mathcal {O}(U)$ , for any $U \\subseteq X$ , picks the largest open subset contained in a given subset of $U$ , i.e.", "the interior.", "With this description, sheaf semantics may be seen as the generalization of the familiar topological semantics for propositional modal logic to quantified languages.", "The previous case of presheaves on a preorder $\\textbf {K}$ is actually a special case of this one by taking the Alexandroff topology on $\\textbf {K}$ ." ], [ "Geometric models from algebraic ones", "The foregoing shows that every geometric model gives rise to a logically equivalent algebraic model in the sense of section .", "The following observation, obtained through general topos-theoretic considerations, states the converse.", "Fact 5.1 For any complete Heyting algebra $H$ in a topos $\\mathcal {E}$ , the canonical structure $\\tau : H \\leftrightarrows \\Omega _{\\mathcal {E}} : i$ ($i \\dashv \\tau $ ) arises from a topos $ \\mathcal {H}$ and geometric morphism $g : \\mathcal {H} \\rightarrow \\mathcal {E}$ , via $H = {g}_\\ast \\Omega _{\\mathcal {H}}$ .", "(sketch) The topos $\\mathcal {H}$ may be defined as the category $\\text{Sh}_{\\mathcal {E}}(H)$ of internal sheaves on $H$ .", "A description of $\\text{Sh}_{\\mathcal {E}}(H)$ can be given in terms of locales in $\\mathcal {E}$ (see [9] C1.3).", "A local homeomorphism over the locale $H$ is an open locale map $E \\rightarrow H$ with open diagonal $E \\rightarrow E \\times _H E$ , where the codomain is the product of locale morphisms over $H$ (in $\\mathcal {E}$ ).", "This is an internalization of the notion of local homeomorphism over the “space” $H$ , in view of the fact that a continuous map $\\pi : Y \\rightarrow X$ of topological spaces is a local homeomorphism just in case both $\\pi $ and its diagonal (over $X$ ) are open maps.", "Alternately, using the internal language of $\\mathcal {E}$ , the category $\\text{Sh}_{\\mathcal {E}}(H)$ may be described as consisting of internal presheaves on the site $H$ (with the sup-topology) that satisfy the usual sheaf property in the internal language.", "See [9], C1.3 for details.", "Next, recall that for any two frames $X,Y$ in $\\mathcal {E}$ , there is an equivalence of categories $\\textbf {Fr}_\\mathcal {E}(Y,X) \\simeq \\textbf {Top}(\\text{Sh}_{\\mathcal {E}}(X) , \\text{Sh}_{\\mathcal {E}}(Y))$ between frame homomorphisms $Y \\rightarrow X$ in $\\mathcal {E}$ and geometric morphisms $\\text{Sh}_{\\mathcal {E}}(X) \\rightarrow \\text{Sh}_{\\mathcal {E}}(Y)$ [9], [14].", "Then $g : \\text{Sh}_{\\mathcal {E}}(H) \\rightarrow \\mathcal {E}$ arises under this equivalence from the frame map $i$ , noting that $\\mathcal {E} \\simeq \\text{Sh}_{\\mathcal {E}}(\\Omega _{\\mathcal {E}}).$ Externally, the idea of (REF ) is that the inverse image part $g^\\ast $ of a geometric morphism $g : \\text{Sh}_{}(X) \\rightarrow \\text{Sh}_{}(Y)$ restricts to a frame homomorphism $g^\\ast : \\mathrm {Sub}_{\\text{Sh}_{}(Y)}(1) \\rightarrow \\mathrm {Sub}_{\\text{Sh}_{}(X)}(1),$ where 1 is the terminal object, respectively.", "Observing that for any sheaf topos $\\text{Sh}_{}(X)$ , we have $\\mathrm {Sub}_{\\text{Sh}_{}(X)}(1) \\cong \\mathcal {O}(X)$ gives the required frame map.", "On the other hand, it is also well-known that a frame map $Y \\rightarrow X$ induces a geometric morphism of the required form for the sup-topology on $X$ and $Y$ , respectively.", "These constructions are inverse and relativize to an arbitrary topos $\\mathcal {E}$ instead of the usual category of Sets [9], [10].", "Moreover, the geometric morphism $g$ is surjective if $i$ is monic.", "Lastly, $H \\cong g_\\ast \\Omega _{\\text{Sh}_{\\mathcal {E}}(H)},$ because $\\text{Sh}_{\\mathcal {E}}(H)$ coincides with the hyperconnected-localic factorization of $g$ itself, which is determined (up to equivalence of categories) [10] as the sheaf topos $\\text{Sh}_{\\mathcal {E}}({g_\\ast \\Omega _{\\text{Sh}_{\\mathcal {E}}(H)}}),$ whence it follows that $H \\cong \\mathrm {Sub}_{\\text{Sh}_{\\mathcal {E}}(H)}(1) \\cong g_\\ast \\Omega _{{Sh}_{\\mathcal {E}}(H)}.$ This last observation applies in particular in case $H = {f}_\\ast \\Omega _{\\mathcal {F}}$ is already of the required form.", "Then $Sh_\\mathcal {E}(\\Omega _\\ast )$ occurs in the hyperconnected-localic factorization of $f$ : ${\\mathcal {F} [rr] [dr]_f &&Sh_\\mathcal {E}(\\Omega _\\ast ) [dl]^g\\\\&\\mathcal {E}&\\\\}$ and ${f}_\\ast \\Omega _{\\mathcal {F}} \\cong g_\\ast \\Omega _{\\text{Sh}_{\\mathcal {E}}({f}_\\ast \\Omega _{\\mathcal {F}})}.$ Externally, we have: $\\mathrm {Sub}_{\\mathcal {F}}(f^\\ast A) & \\cong \\mathrm {Hom}_{\\mathcal {F}}(f^\\ast A,\\Omega _{\\mathcal {F}})\\\\&\\cong \\mathrm {Hom}_{\\mathcal {E}}(A,{f}_\\ast \\Omega _{\\mathcal {F}}))\\\\& \\cong \\mathrm {Hom}_{\\mathcal {E}}(A,g_\\ast \\Omega _{\\text{Sh}_{\\mathcal {E}}({f}_\\ast \\Omega _{\\mathcal {F}})})\\\\& \\cong \\mathrm {Hom}_{\\text{Sh}_{\\mathcal {E}}({f}_\\ast \\Omega _{\\mathcal {F}})}(g^\\ast A,\\Omega _{\\text{Sh}_{\\mathcal {E}}({f}_\\ast \\Omega _{\\mathcal {F}})})\\\\& \\cong \\mathrm {Sub}_{\\text{Sh}_{\\mathcal {E}}({f}_\\ast \\Omega _{\\mathcal {F}})}(g^\\ast A)$ for all $A$ in $\\mathcal {E}$ .", "This allows us to restrict attention to localic surjective geometric morphisms.", "For instance, the geometric morphism $i^\\ast \\dashv i_\\ast : \\textbf {Sets}^{|\\textbf {D}|} \\rightarrow \\textbf {Sets}^{\\textbf {D}}$ considered in the previous section is localic." ] ]
1403.0020
[ [ "Mixed, Multi-color, and Bipartite Ramsey Numbers Involving Trees of\n Small Diameter" ], [ "Abstract In this paper we study Ramsey numbers for trees of diameter 3 (bistars) vs., respectively, trees of diameter 2 (stars), complete graphs, and many complete graphs.", "In the case of bistars vs. many complete graphs, we determine this number exactly as a function of the Ramsey number for the complete graphs.", "We also determine the order of growth of the bipartite $k$-color Ramsey number for a bistar." ], [ "Background", "In this paper we investigate Ramsey numbers, both classical and bipartite, for trees vs. other graphs.", "Trees have been studied less than other graphs, although there have been a number of papers in the last few years.", "Some general results applying to all trees are known, such as the following result of Gyárfás and Tuza [4].", "Theorem 1 Let $T_n$ be a tree with $n$ edges.", "Then $R_k(T_n)\\le (n-1)(k+\\sqrt{k(k-1)})+2$ .", "More recently, various researchers have studied particular trees of small diameter.", "Burr and Roberts [3] completely determine the Ramsey number $R(S_{n_1},\\ldots ,S_{n_i})$ for any number of stars, i.e., trees of diameter 2.", "Boza et. al.", "[2] determine $R(S_{n_1},\\ldots ,S_{n_i},K_{m_1},\\ldots ,K_{m_j})$ exactly as a function of $R(K_{m_1},\\ldots ,K_{m_j})$ .", "Bahls and Spencer [1] study $R(C,C)$ , where $C$ is a caterpillar, i.e., a tree whose non-leaf vertices form a path.", "They prove a general lower bound, and prove exact results in several cases, including “regular\" caterpillars, in which all non-leaf vertices have the same degree.", "We will study bistars (i.e.", "trees of diameter 3) vs. stars and bistars vs. complete graphs in Section , bistars vs. many complete graphs in Section , and bistars vs. bistars in bipartite graphs in Section ." ], [ "Notation", "For graphs $G_1,\\ldots ,G_n$ , let $R(G_1,\\ldots , G_n)$ denote the least integer $N$ such that any edge-coloring of $K_N$ in $n$ colors must contain, for some $1\\le i\\le n$ , a monochromatic $G_i$ in the $i^{\\text{th}}$ color.", "Let $S_n$ denote the $(n+1)$ -vertex graph consisting of a vertex $v$ of degree $n$ and $n$ vertices of degree 1 (a star).", "Let $B_{k,m}$ denote the $(k+m)$ -vertex graph with a vertex $v$ of degree $k$ , a vertex $w$ incident to $v$ of degree $m$ , and $k+m-2$ vertices of degree 1 (a bistar).", "We will call the edge $vw$ the spine of $B_{k,m}$ .", "(Note that some authors refer to the set of vertices $\\lbrace v,w\\rbrace $ as the spine.)", "We will depict the spine of a bistar with a double-struck edge; see Figure REF .", "Figure: A bistar, with spine indicatedFor a graph $G$ whose edges are colored red and blue, and for vertices $v$ and $w$ , if $v$ and $w$ are incident by a red edge, we will say (for the sake of brevity) that $w$ is a “red neighbor\" of $v$ .", "Let $\\text{deg}_{\\text{red}}(v)$ denote the number of red neighbors of $v$ , and let $\\Delta _{\\text{red}}(G)=\\text{max}\\lbrace \\text{deg}_\\text{red}(v):v\\in G\\rbrace $ and $\\delta _{\\text{red}}(G)=\\text{min}\\lbrace \\text{deg}_\\text{red}(v):v\\in G\\rbrace .$ In Section we will make use of cyclic colorings.", "Let $K_N$ have vertex set $\\lbrace 0,1,2,\\ldots ,N-1\\rbrace $ , and let $R\\subseteq \\mathbb {Z}_N\\backslash 0$ such that $R=-R$ , i.e., $R$ is closed under additive inverse.", "Define a coloring of $K_N$ by $uv\\textrm { is colored \\emph {red} if }u-v\\in R\\textrm { and \\emph {blue} otherwise.", "}$ Cyclic colorings are computationally nice.", "For instance, it is not hard to show that if $R\\subseteq R+R$ , then any two vertices $v$ and $w$ incident by a red edge must share a red neighbor.", "We will need this fact in the proof of Theorem REF .", "First we consider bistars vs. stars.", "We have the following easy upper bound.", "Theorem 2 $R(B_{k,m},S_n)\\le k+m+n-1$ .", "Let $N=k+m+n-1$ , and let the edges of $K_N$ be colored in red and blue.", "Suppose this coloring contains no blue $S_n$ .", "Then every red edge is the spine of a red $B_{k,m}$ , as follows.", "If there is no blue $S_n$ , then $\\Delta _{\\text{blue}}\\le n-1$ , and hence $\\delta _{\\text{red}}\\ge (N-1)-(n-1)=k+m-1$ .", "Let the edge $uv$ be colored red.", "Then both $u$ and $v$ have $(k-1)+(m-1)$ red neighbors besides each other.", "Even if these sets of neighbors coincide, we may select $k-1$ leaves for $u$ and $m-1$ leaves for $v$ , giving a red $B_{k,m}$ .", "The following lower bound uses some cyclic colorings.", "Theorem 3 $R(B_{k,m},S_n)>\\lfloor \\frac{k+m}{2}\\rfloor +n$ for $k,m\\ge 4$ .", "Let $k+m$ be odd.", "Let $N=\\lfloor \\frac{k+m}{2}\\rfloor +n$ .", "Let $G$ be any $(n-1)$ -regular graph on $N$ vertices.", "Consider the edges of $G$ to be the blue edges, and replace all non-edges of $G$ with red edges, so that the resulting $K_N$ is $\\lfloor \\frac{k+m}{2}\\rfloor $ -regular for red.", "Clearly, this coloring admits no blue $S_n$ .", "Consider the red edge set.", "If an edge $uv$ is colored red, then $u$ and $v$ combined have at most $k+m-3$ red neighbors besides each other, which is not enough to supply the needed $k-1$ red leaves for $u$ and the $m-1$ red leaves for $v$ .", "Now let $k+m$ be even, and $N=\\frac{k+m}{2}+n$ .", "We seek a subset $R\\subseteq \\mathbb {Z}_N$ that is symmetric $(R=-R)$ and of size $\\frac{k+m}{2}$ satisfying $R\\subseteq R+R$ .", "Thus each vertex will have red degree $\\frac{k+m}{2}$ , but any red edge $uv$ cannot be the spine of a red $B_{k,m}$ , since $u$ and $v$ will have a common neighbor.", "There are two cases: Case (i.", "): $\\frac{k+m}{2}$ is even.", "Let $R^{\\prime }=\\lbrace 2\\rbrace \\cup \\lbrace 2\\ell +1:1\\le \\ell \\le \\frac{k+m-4}{4}\\rbrace $ , and let $R:=R^{\\prime }\\cup -R^{\\prime }$ .", "It is easy to check that $R\\subseteq R+R$ .", "Setting $B=\\mathbb {Z}_N\\backslash \\lbrace R\\cup 0\\rbrace $ , we have $|B|=n-1$ , and so the cyclic coloring of $K_N$ induced by $R$ and $B$ has no red $B_{k,m}$ and no blue $S_n$ .", "Case (ii.", "): $\\frac{k+m}{2}$ is odd.", "Let $R^{\\prime }=\\lbrace 2\\rbrace \\cup \\lbrace 2\\ell +1:1\\le \\ell \\le \\frac{k+m-6}{4}\\rbrace $ , and set $R:=R^{\\prime }\\cup \\lbrace \\frac{k+m}{2}\\rbrace \\cup -R^{\\prime }$ .", "Again, set $B=\\mathbb {Z}_N\\backslash \\lbrace R\\cup 0\\rbrace $ , and the cyclic coloring of $K_N$ induced by $R$ and $B$ has the desired properties.", "Corollary 4 $R(B_{n,n},S_n)>2n$ for $n\\ge 4$ .", "We conjecture that the lower bound in Corollary REF is tight; that is, that $R(B_{n,n},S_n)=2n+1$ for $n\\ge 4$ .", "We show that this result obtains for $n=4$ (but not for $n=3$ ).", "Theorem 5 $R(B_{3,3},S_3)=6$ .", "A lower bound is supplied by the classic critical coloring of $K_5$ for $R(3,3)$ .", "See Figure REF .", "Figure: Critical coloring of K 5 K_5For the upper bound, suppose there exists a 2-coloring of $K_6$ with no blue $S_3$ .", "Then $\\delta _\\text{red}\\ge 3$ .", "So consider the red subgraph $G$ .", "If $G$ has a vertex of degree 5, the existence of a $B_{3,3}$ is immediate.", "If $G$ has a vertex of degree 4, then it must have 2 such vertices $u$ and $v$ .", "If $u\\nsim v$ , then $G$ must look like Figure REF .", "Figure: Configuration of the red subgraph GG.One may use any edge incident to $u$ or $v$ as a spine.", "If $u\\sim v$ , and $u$ and $v$ do not share all three remaining neighbors, then the existence of a $B_{3,3}$ is immediate.", "So suppose $u$ and $v$ have neighbors $x,y$ , and $z$ .", "The only way for $G$ to have degree sequence $(4,4,3,3,3,3)$ is for the remaining vertex $w$ to be adjacent to $x,y$ , and $z$ .", "Then we have a $B_{3,3}$ as indicated in Figure REF .", "Figure: A red B 3,3 B_{3,3}Finally, suppose $G$ is 3-regular.", "If there is no $B_{3,3}$ , then any adjacent vertices share a neighbor.", "It is not hard to see that adjacent vertices cannot share two neighbors in a 3-regular graph on 6 vertices.", "If any two adjacent vertices share exactly one neighbor, then $G$ can be partitioned into edge-disjoint triangles.", "But any vertex in such a graph must have even degree, since its degree will be twice the number of triangles in which it participates.", "This is a contradiction, so there must be some vertices $u$ and $v$ that have no common neighbor.", "But $u$ and $v$ each have degree three, immediately yielding a $B_{3,3}$ .", "Theorem 6 $R(B_{4,4},S_4)=9$ .", "The lower bound is given by Theorem REF .", "For the upper bound, suppose a 2-coloring of $K_9$ contains no blue $S_4$ .", "Then $\\delta _\\text{red}\\ge 5$ .", "Let $G$ be the red subgraph.", "Since $G$ has odd order, there must be at least one vertex $v$ of degree $\\ge 6$ .", "Suppose $v\\sim w$ .", "It is easy to see that $v$ and $w$ must have at least two neighbors in common; call them $y$ and $z$ .", "Now $v$ is adjacent to 3 other vertices; call them $x_1,x_2$ , and $x_3$ .", "There are two remaining vertices $x_4$ and $x_5$ .", "If $w$ is adjacent to either of them, we are done.", "So suppose $w$ is adjacent to $x_1$ and $x_2$ .", "If either $x_4$ or $x_5$ is adjacent to $v$ , we are done, so suppose neither $x_4$ nor $x_5$ is adjacent to $v$ or to $w$ .", "Then $x_4$ (in order to have degree $\\ge $ 5) must be adjacent to $y_1$ or to $y_2$ .", "Suppose it's $y_1$ .", "There are two cases: $y_1\\sim x_5$ .", "Then we have a red $B_{4,4}$ as indicated in Figure REF .", "Figure: A red B 4,4 B_{4,4} $y_1\\nsim x_5$ .", "Then, since deg$(y_1)\\ge 5$ , $y_1\\sim x_i$ for some $i\\in \\lbrace 1,2,3\\rbrace $ .", "Then we have a red $B_{4,4}$ as indicated in Figure REF , where $|\\lbrace i,k,\\ell \\rbrace |=3$ .", "Figure: A red B 4,4 B_{4,4} Consideration of $R(B_{5,5},S_5)$ leads into rather unpleasant case analysis when trying to reduce the upper bound from that given by Theorem REF .", "Now we consider bistars vs. complete graphs.", "Theorem 7 $R(B_{k,m},K_3)=2(k+m-1)+1$ .", "For the lower bound, let $V_1$ and $V_2$ be two red cliques, each of size $k+m-1$ , and let every edge between $V_1$ and $V_2$ be colored blue.", "For the upper bound, let $N=2(k+m-1)+1$ , and give $K_N$ an edge-coloring in red and blue.", "Suppose there is a vertex $v$ with blue degree at least $k+m$ .", "If any edge in $N_\\text{blue}(v)$ is blue, we have a blue triangle.", "If not, then $N_\\text{blue}(v)$ is a red clique of size at least $k+m$ , so it contains a red $B_{k,m}$ .", "So then suppose that $\\Delta _\\text{blue}<k+m$ .", "It follows that $\\delta _\\text{red}\\ge k+m-1$ .", "Then every red edge is the spine of some red $B_{k,m}$ .", "To see this, let $uv$ be colored red.", "Both $u$ and $v$ each have at least $k+m-2$ other red neighbors.", "Even if these red neighborhoods coincide, there are still $k-1$ red leaves for $u$ and $m-1$ red leaves for $v$ .", "Now we extend to arbitrary $K_n$ .", "Theorem 8 $R(B_{k,m},K_n)=(k+m-1)(n-1)+1$ .", "We proceed by induction on $n$ .", "Theorem REF provides the base case $n=3$ .", "So assume $n>3$ , and let $R(B_{k,m},K_{n-1})\\le (k+m-1)(n-2)+1$ .", "Let $N=(k+m-1)(n-1)+1$ , and consider any edge-coloring of $K_n$ in red and blue.", "If $\\delta _\\text{red}\\ge k+m-1$ , then every red edge is the spine of a red $B_{k,m}$ , so suppose $\\delta _\\text{red}\\le k+m-2$ .", "Then there is a vertex $v$ with blue degree at least $(k+m-1)(n-2)+1$ .", "By the induction hypothesis, the subgraph induced by $N_\\text{blue}(v)$ contains either a red $B_{k,m}$ or a blue $K_{n-1}$ .", "In the latter case, the blue $K_{n-1}$ along with $v$ forms a blue $K_n$ .", "For the lower bound, let $V_1,\\ldots ,V_{n-1}$ be vertex-disjoint red cliques, each of size $k+m-1$ .", "Color all edges among the $V_i$ 's blue.", "Clearly there are no red $B_{k,m}$ 's.", "Since the blue subgraph forms a Turan graph, there are no blue $K_n$ 's." ], [ "Mixed Multi-color Ramsey Numbers", "In [2], the authors determine $R(S_{k_1},\\ldots ,S_{k_i},k_{n_i},\\ldots ,K_{n_\\ell })$ exactly as a function of $R(K_{n_1},\\ldots ,K_{n_\\ell })$ .", "In [6], Omidi and Raeisi give a shorter proof of this result via the following lemma, whose proof is straight from The Book.", "Lemma 9 Let $G_1,\\ldots ,G_m$ be connected graphs, let $r=R(G_1,\\ldots ,G_m)$ and $r^{\\prime }=R(K_{n_1},\\ldots ,K_{n_\\ell })$ .", "If $n\\ge 2$ and $R(G_1,\\ldots ,G_m,K_n)=(r-1)(n-1)+1$ , then $R(G_1,\\ldots ,G_m,K_{n_1},\\ldots ,K_{n_\\ell })=(r-1)(r^{\\prime }-1)+1$ .", "Let $R=R(G_1,\\ldots ,G_m,K_{n_1},\\ldots ,K_{n_\\ell })$ .", "For the lower bound, give $K_{r^{\\prime }-1}$ an edge-coloring in $\\ell $ colors $\\beta _1,\\ldots ,\\beta _\\ell $ that has no copy of $K_{n_i}$ in color $\\beta _i$ .", "Replace each vertex of $K_{r^{\\prime }-1}$ by a complete graph of order $r-1$ whose edges are colored by colors $\\alpha _1,\\ldots ,\\alpha _m$ so that no copy of $G_i$ appears in color $\\alpha _i$ .", "Each edge in the original graph $K_{r^{\\prime }-1}$ expands to a copy of $K_{r-1,r-1}$ , with each edge the same color as the original edge.", "This shows that $R>(r-1)(r^{\\prime }-1)$ .", "For the upper bound, let $N=(r-1)(r^{\\prime }-1)+1$ , and color the edges of $K_N$ in colors $\\alpha _1,\\ldots ,\\alpha _m,\\beta _1,\\ldots ,\\beta _\\ell $ .", "Recolor the edges colored $\\beta _1,\\ldots ,\\beta _\\ell $ with a new color $\\alpha $ .", "Since $R(G_1,\\ldots ,G_m,K_{r^{\\prime }})= (r^{\\prime }-1)(r-1)+1=N$ , $K_N$ contains a copy of $G_i$ in color $\\alpha _i$ or a copy of $K_{r^{\\prime }}$ in color $\\alpha $ .", "In the former case we are done, so assume the latter obtains.", "Then consider the clique $K_{r^{\\prime }}$ which is colored $\\alpha $ .", "Return to the original coloring in colors $\\beta _1,\\ldots ,\\beta _\\ell $ .", "Since $R(K_{n_1},\\ldots ,K_{n_\\ell })=r^{\\prime }$ , some color class $\\beta _i$ contains a copy of $K_{n_i}$ .", "This concludes the proof.", "We will now make use of Lemma REF to determine $R(B_{k,m},K_{n_1},\\ldots ,K_{n_\\ell })$ as a function of $R(K_{n_1},\\ldots ,K_{n_\\ell })$ .", "Theorem 10 $R(B_{k,m},K_{n_1},\\ldots ,K_{n_\\ell })=(k+m-1)[R(K_{n_1},\\ldots ,K_{n_\\ell })-1]+1$ .", "From Theorem REF we have that $R(B_{k,m}, K_n)=(k+m-1)(n-1)+1$ .", "Note that $R(B_{k,m}, K_2)=k+m$ , so that $R(B_{k,m},K_2,K_n) &=R(B_{k,m},K_n)\\\\&=[R(B_{k,m},K_2)-1](n-1)+1.$ Hence we may apply Lemma REF to get $R(B_{k,m},K_{n_1},\\ldots ,K_{n_\\ell })=(k+m-1)[R(K_{n_1},\\ldots ,K_{n_\\ell })-1]+1$ .", "The authors are unsure whether a similar result can be proved for multiple bistars; we leave this as an open problem." ], [ "Bipartite Ramsey Numbers", "Let $G_1$ and $G_2$ be bipartite graphs.", "Then $BR(G_1,G_2)$ is the least integer $N$ so that any 2-coloring of the edges of $K_{N,N}$ contains either a red $G_1$ or a blue $G_2$ .", "In [5], Hattingh and Joubert determine the bipartite Ramsey number for certain bistars: Theorem 11 Let $k,n\\ge 2$ .", "Then $BR(B_{k,k},B_{n,n})=k+n-1$ .", "We generalize this result slightly.", "Theorem 12 Let $k\\ge m\\ge 2$ , $n\\ge \\ell \\ge 2$ .", "Then $BR(B_{k,m},B_{n,\\ell })=k+n-1$ .", "The upper bound follows immediately from Theorem REF .", "The lower bound construction given in Theorem 1 of Hattingh-Joubert for $BR(B_{s,s},B_{t,t})$ does not work for us.", "We need this construction: Let $L$ and $R$ be the partite sets, and let $N=k+n-2=(k-1)+(n-1)$ .", "Let $L=\\lbrace v_0,v_1,\\ldots ,v_{N-1}\\rbrace $ and $R=\\lbrace w_0,w_1,\\ldots ,w_{N-1}\\rbrace $ .", "Color $v_iw_j$ red if $(i-j)\\mod {N}\\in \\lbrace 0,1,\\ldots ,k-2\\rbrace $ , and blue if $(i-j)\\mod {N}\\in \\lbrace k-1,\\ldots ,N-1\\rbrace $ .", "Then the red subgraph is $(k-1)$ -regular, hence no red $B_{k,m}$ , and the blue subgraph is $(n-1)$ -regular, hence no blue $B_{n,\\ell }$ .", "Corollary 13 Let $T_m$ (resp., $T_n$ ) be a tree of diameter at most 3 with maximum degree $m$ (resp., $n$ ).", "Then $BR(T_m,T_n)=m+n-1$ .", "Hattingh and Joubert also prove the following $k$ -color upper bound.", "Theorem 14 For $k\\ge 2$ and $m\\ge 3$ , we have $BR_k(B_{m,m})=BR(B_{m,m},\\ldots ,B_{m,m})\\le \\left\\lceil k(m-1) +\\sqrt{(m-1)^2(k^2-k)-k(2m-4)}\\right\\rceil $ Hence $BR_k(B_{m,n})=O(k)$ .", "We provide a lower bound to get the following result.", "Theorem 15 Fix $m\\ge 3$ .", "Then $BR_k(B_{m,m})=\\Theta (k)$ .", "We show that $BR_k(B_{m,m})>k\\cdot (m-1)$ .", "Let $N=k\\cdot (m-1)$ , and consider a $k$ -coloring of the edges of $K_{N,N}$ in colors $c_0,\\ldots ,c_{k-1}$ .", "Let the partite sets be $L=\\lbrace v_0,\\ldots ,,v_{N-1}\\rbrace $ and $R=\\lbrace w_0,\\ldots , w_{N-1}\\rbrace $ .", "Color edge $v_iw_j$ with color $c_\\ell $ if and only if $\\ell \\equiv (i-j)\\mod {k}$ .", "Then the $c_\\ell $ -subgraph is $(m-1)$ -regular, hence there can be no monochromatic $B_{m,m}$ ." ] ]
1403.0273
[ [ "Higher order derivatives of approximation polynomials on $\\mathbb{R}$" ], [ "Abstract D. Leviatan has investigated the behavior of the higher order derivatives of approximation polynomials of the differentiable function $f$ on $[-1,1]$.", "Especially, when $P_n$ is the best approximation of $f$, he estimates the differences $\\|f^{(k)}-P_n^{(k)}\\|_{L_\\infty([-1,1])}$, $k=0,1,2,...$.", "In this paper, we give the analogies for them with respect to the differentiable functions on $\\mathbb{R}$, and we apply the result to the monotone approximation." ], [ "Introduction", "Let $\\mathbb {R}=(-\\infty ,\\infty )$ and ${\\mathbb {R}}^+=[0,\\infty )$ .", "We say that $f: {\\mathbb {R}}\\rightarrow {\\mathbb {R}^+}$ is quasi-increasing if there exists $C>0$ such that $f(x)\\leqslant Cf(y)$ for $0<x<y$ .", "The notation $f(x)\\sim g(x)$ means that there are positive constants $C_1, C_2$ such that for the relevant range of $x$ , $C_1\\leqslant f(x)/g(x)\\leqslant C_2$ .", "The similar notation is used for sequences and sequences of functions.", "Throughout $C,C_1,C_2,...$ denote positive constants independent of $n,x,t$ .", "The same symbol does not necessarily denote the same constant in different occurrences.", "We denote the class of polynomials with degree $n$ by $\\mathcal {P}_n$ .", "First, we introduce some classes of weights.", "Levin and Lubinsky [8] introduced the class of weights on ${\\mathbb {R}}$ as follows.", "Definition 1.1 Let $Q: \\mathbb {R}\\rightarrow [0,\\infty )$ be a continuous even function, and satisfy the following properties:    (a) $Q^{\\prime }(x)$ is continuous in $\\mathbb {R}$ , with $Q(0)=0$ .", "(b) $Q^{\\prime \\prime }(x)$ exists and is positive in $\\mathbb {R}\\backslash \\lbrace 0\\rbrace $ .", "(c) $\\lim _{x\\rightarrow \\infty }Q(x)=\\infty .$    (d) The function $T_w(x):=\\frac{xQ^{\\prime }(x)}{Q(x)}, x\\ne 0$ is quasi-increasing in $(0,\\infty )$ , with $T_w(x)\\ge \\Lambda >1, x\\in \\mathbb {R}\\backslash \\lbrace 0\\rbrace .$    (e) There exists $C_1>0$ such that $\\frac{Q^{\\prime \\prime }(x)}{|Q^{\\prime }(x)|}\\le C_1\\frac{|Q^{\\prime }(x)|}{Q(x)}, \\,\\,\\, a.e.", "\\,\\,\\, x\\in \\mathbb {R}.$ Furthermore, if there also exists a compact subinterval $J (\\ni 0)$ of $\\mathbb {R}$ , and $C_2>0$ such that $\\frac{Q^{\\prime \\prime }(x)}{|Q^{\\prime }(x)|}\\ge C_2\\frac{|Q^{\\prime }(x)|}{Q(x)}, \\,\\,\\, a.e.", "\\,\\,\\, x\\in \\mathbb {R}\\backslash J,$ then we write $w=\\exp (-Q)\\in \\mathcal {F}(C^2+)$ .", "For convenience, we denote $T$ instead of $T_w$ , if there is no confusion.", "Next, we give some typical examples of $\\mathcal {F}(C^2+)$ .", "Example 1.2 ([5]) (1) If $T(x)$ is bounded, then we call the weight $w=\\exp (-Q(x))$ the Freud-type weight and we write $w\\in \\mathcal {F}^*\\subset \\mathcal {F}(C^2+)$ .", "(2) When $T(x)$ is unbounded, then we call the weight $w=\\exp (-Q(x))$ the Erdös-type weight: (a) For $\\alpha >1$ , $l\\ge 1$ we define $Q(x):=Q_{l,\\alpha }(x)=\\exp _l(|x|^{\\alpha })-\\exp _l(0),$ where $\\exp _l(x)=\\exp (\\exp (\\exp \\ldots \\exp x)\\ldots ) (l\\,\\, \\textrm {times})$ .", "More generally, we define $Q_{l,\\alpha ,m}(x)=|x|^m\\lbrace \\exp _l (|x|^{\\alpha }) -\\tilde{\\alpha }\\exp _l(0)\\rbrace ,\\quad \\alpha +m>1,\\,\\, m\\ge 0,\\,\\, \\alpha \\ge 0,$ where $\\tilde{\\alpha }=0$ if $\\alpha =0$ , and otherwise $\\tilde{\\alpha }=1$ .", "We note that $Q_{l,0,m}$ gives a Freud-type weight, and $Q_{l,\\alpha ,m}$ , ($\\alpha >0$ ) gives an Erdös-type weight.", "(3) For $\\alpha >1$ , $Q_{\\alpha }(x)=(1+|x|)^{|x|^{\\alpha }} -1 $ gives also an Erdös-type weight.", "For a continuous function $f : [-1,1] \\rightarrow \\mathbb {R}$ , let $E_n(f)=\\inf _{P\\in \\mathcal {P}_n}\\Vert f-P\\Vert _{L_\\infty ([-1,1])}=\\inf _{P\\in \\mathcal {P}_n}\\max _{x\\in [-1,1]}|f(x)-P(x)|.$ D. Leviatan [7] has investigated the behavior of the higher order derivatives of approximation polynomials for the differentiable function $f$ on $[-1,1]$ , as follows: Theorem (Leviatan [7]).", "For $r\\ge 0$ we let $f\\in C^{(r)}[-1,1]$ , and let $P_n\\in \\mathcal {P}_n$ denote the polynomial of best approximation of $f$ on $[-1,1]$ .", "Then for each $0\\le k\\le r$ and every $-1\\le x\\le 1$ , $\\left|f^{(k)}(x)-P_n^{(k)}(x)\\right|\\le \\frac{C_r}{n^k}\\Delta _n^{-k}(x)E_{n-k}\\left(f^{(k)}\\right),\\quad n\\ge k,$ where $\\Delta _n(x):=\\sqrt{1-x^2}/n+1/n^2$ and $C_r$ is an absolute constant which depends only on $r$ .", "In this paper, we estimate $\\left|\\left(f^{(k)}(x)-P_{n;f}^{(k)}(x)\\right)w(x)\\right|$ , $x\\in \\mathbb {R}$ , $ k=0,1,...,r$ for $f\\in C^{(r)}(\\mathbb {R})$ and for some exponential type weight $w$ in $L_p(\\mathbb {R})$ -space, $1 < p\\le \\infty $ , where $P_{n;f}\\in \\mathcal {P}_n$ is the best approximation of $f$ .", "Furthermore, we give an application for a monotone approximation with linear differential operators.", "In Section 2 we write the theorems in the space $L_\\infty (\\mathbb {R})$ , then we also denote a certain assumption and some notations which need to state the theorems.", "In Section 3 we give some lemmas and the proofs of theorems.", "In Section 4, we consider the similar problem in $L_p(\\mathbb {R})$ -space, $1<p<\\infty $ .", "In Section 5, we give a simple application of the result to the monotone approximation.", "Theorems and Preliminaries First, we introduce some well-known notations.", "If $f$ is a continuous function on $\\mathbb {R}$ , then we define $\\left\\Vert fw\\right\\Vert _{L_\\infty (\\mathbb {R})}:=\\sup _{t\\in \\mathbb {R}}|f(t)w(t)|,$ and for $1\\le p<\\infty $ we denote $\\Vert fw\\Vert _{L_p(\\mathbb {R})}:=\\left(\\int _{\\mathbb {R}}\\left|f(t)w(t)\\right|^pdt\\right)^{1/p}.$ Let $1\\le p\\le \\infty $ .", "If $\\Vert wf\\Vert _{L_p(\\mathbb {R})}<\\infty $ , then we write $wf\\in L_p(\\mathbb {R})$ , and here if $p=\\infty $ , we suppose that $f\\in C(\\mathbb {R})$ and $\\lim _{|x|\\rightarrow \\infty }|w(x)f(x)|=0$ .", "We denote the rate of approximation of $f$ by $E_{p,n}(w;f):=\\inf _{P\\in \\mathcal {P}_n}\\left\\Vert (f-P)w\\right\\Vert _{L_p(\\mathbb {R})}.$ The Mhaskar-Rakhmanov-Saff numbers $a_x$ is defined as follows: $x=\\frac{2}{\\pi }\\int _0^1\\frac{a_xuQ^{\\prime }(a_xu)}{\\sqrt{1-u^2}}du, \\quad x>0.$ To write our theorems we need some preliminaries.", "We need further assumptions.", "Definition 2.1 Let $w=\\exp (-Q)\\in \\mathcal {F}(C^2+)$ and $0< \\lambda <(r+2)/(r+1)$ .", "Let $r\\ge 1$ be an integer.", "Then we write $w\\in \\mathcal {F}_\\lambda (C^{r+2}+)$ if $Q\\in C^{(r+2)}(\\mathbb {R}\\backslash \\lbrace 0\\rbrace )$ and there exist two constants $C>1$ and $K\\ge 1$ such that for all $|x|\\ge K$ , $\\frac{|Q^{\\prime }(x)|}{Q^{\\lambda }(x)} \\le C \\quad \\mbox{and} \\quad \\left| \\frac{Q^{\\prime \\prime }(x)}{Q^{\\prime }(x)} \\right| \\sim \\left|\\frac{Q^{(k+1)}(x)}{Q^{(k)}(x)} \\right|$ for every $k=2,...,r$ and also $\\left| \\frac{Q^{(r+2)}(x)}{Q^{(r+1)}(x)} \\right| \\le C \\left|\\frac{Q^{(r+1)}(x)}{Q^{(r)}(x)} \\right|.$ In particular, $w\\in \\mathcal {F}_\\lambda (C^{3}+)$ means that $Q\\in C^{(3)}(\\mathbb {R}\\backslash \\lbrace 0\\rbrace )$ and $\\frac{|Q^{\\prime }(x)|}{Q^{\\lambda }(x)} \\le C \\quad \\mbox{and} \\quad \\left| \\frac{Q^{\\prime \\prime \\prime }(x)}{Q^{\\prime \\prime }(x)} \\right| \\le C \\left|\\frac{Q^{\\prime \\prime }(x)}{Q^{\\prime }(x)} \\right|$ hold for $|x| \\ge K$ .", "In addition, let $\\mathcal {F}_\\lambda (C^{2}+):=\\mathcal {F}(C^2+)$ .", "From [5], we know that Example REF (2),(3) satisfy all conditions of Definition REF .", "Under the same condition of Definition REF we obtain an interesting theorem as follows: Theorem 2.2 ([10]) Let $r$ be a positive integer, $0< \\lambda <(r+2)/(r+1)$ and let $w=\\exp (-Q)\\in \\mathcal {F}_\\lambda (C^{r+2}+)$ .", "Then for any $\\mu $ , $\\nu $ , $\\alpha $ , $\\beta \\in \\mathbb {R}$ , we can construct a new weight $w_{\\mu ,\\nu ,\\alpha ,\\beta }\\in \\mathcal {F}_\\lambda (C^{r+1}+)$ such that $T_w^{\\mu }(x)(1+x^2)^\\nu (1+Q(x))^{\\alpha }(1+|Q^{\\prime }(x)|)^\\beta w(x)\\sim w_{\\mu ,\\nu ,\\alpha ,\\beta }(x)$ on $\\mathbb {R}$ , and $a_{n/c}(w)\\le a_n(w_{\\mu .\\nu ,\\alpha ,\\beta })\\le a_{cn}(w), \\quad c\\ge 1,$ $T_{w_{\\mu ,\\nu ,\\alpha ,\\beta }}(x)\\sim T_w(x)$ hold on $\\mathbb {R}$ .", "For a given $\\alpha \\in \\mathbb {R}$ and $w\\in \\mathcal {F}(C^2+)$ , we let $w_{\\alpha }\\in \\mathcal {F}(C^2+)$ satisfy $w_\\alpha (x) \\sim T_{w}^{\\alpha }(x)w(x)$ , and let $P_{n;f,w_\\alpha }\\in \\mathcal {P}_n$ be the best approximation of $f$ with respect to the weight $w_\\alpha $ , that is, $\\left\\Vert (f-P_{n;f,w_\\alpha })w_{\\alpha }\\right\\Vert _{L_\\infty (\\mathbb {R})}=E_{n}(w_\\alpha ,f):=\\inf _{P\\in \\mathcal {P}_n}\\left\\Vert (f-P)w_\\alpha \\right\\Vert _{L_\\infty (\\mathbb {R})}.$ Then we have the main result as follows: Theorem 2.3 Let $r \\ge 0$ be an integer.", "Let $w=\\exp (-Q)\\in \\mathcal {F}_\\lambda (C^{r+2}+)$ and $0< \\lambda <(r+2)/(r+1)$ .", "Suppose that $f\\in C^{(r)}(\\mathbb {R})$ with $\\lim _{|x|\\rightarrow \\infty }T^{1/4}(x)f^{(r)}(x)w(x)=0.$ Then there exists an absolute constant $C_r>0$ which depends only on $r$ such that for $0\\le k\\le r$ and $x\\in \\mathbb {R}$ , $\\left|\\left(f^{(k)}(x)-P_{n;f,w}^{(k)}(x)\\right)w(x)\\right|&\\le & C_r T^{k/2}(x)E_{n-k}\\left(w_{1/4},f^{(k)}\\right)\\\\&\\le & C_r T^{k/2}(x) \\left(\\frac{a_n}{n}\\right)^{r-k}E_{n-r}\\left(w_{1/4},f^{(r)}\\right).$ When $w\\in \\mathcal {F}^*$ , we can replace $w_{1/4}$ with $w$ in the above.", "Applying Theorem REF with $w$ or $w_{-1/4}$ , we have the following corollary.", "Corollary 2.4 (1) Let $w=\\exp (-Q)\\in \\mathcal {F}_\\lambda (C^{r+2}+), 0< \\lambda <(r+2)/(r+1), r\\ge 0$ .", "We suppose that $f\\in C^{(r)}(\\mathbb {R})$ with $\\lim _{|x|\\rightarrow \\infty }T^{1/4}(x)f^{(r)}(x)w(x)=0,$ then for $0\\le k\\le r$ we have $\\left\\Vert \\left(f^{(k)}-P_{n;f,w}^{(k)}\\right)w_{-k/2}\\right\\Vert _{L_\\infty (\\mathbb {R})}&\\le & C_r E_{n-k}\\left(w_{1/4},f^{(k)}\\right) \\\\&\\le & C_r \\left(\\frac{a_n}{n}\\right)^{r-k}E_{n-r}\\left(w_{1/4},f^{(r)}\\right).$    (2) Let $w=\\exp (-Q)\\in \\mathcal {F}_\\lambda (C^{r+3}+), 0< \\lambda <(r+3)/(r+2), r\\ge 0$ .", "We suppose that $f\\in C^{(r)}(\\mathbb {R})$ with $\\lim _{|x|\\rightarrow \\infty }f^{(r)}(x)w(x)=0,$ then for $0\\le k\\le r$ we have $\\left\\Vert \\left(f^{(k)}-P_{n;f,w_{-1/4}}^{(k)}\\right)w_{-(2k+1)/4}\\right\\Vert _{L_\\infty (\\mathbb {R})}&\\le & C_r E_{n-k}\\left(w,f^{(k)}\\right) \\\\&\\le & C_r \\left(\\frac{a_n}{n}\\right)^{r-k}E_{n-r}\\left(w,f^{(r)}\\right).$ When $w\\in \\mathcal {F}^*$ , we can replace $w_\\alpha $ with $w$ in the above.", "Corollary 2.5 Let $r \\ge 0$ be an integer.", "Let $w=\\exp (-Q)\\in \\mathcal {F}_\\lambda (C^{r+3}+)$ , $0< \\lambda <(r+3)/(r+2)$ , and let $w_{(2r+1)/4}f^{(r)}\\in L_\\infty (\\mathbb {R})$ .", "Then, for each $k (0\\le k\\le r)$ and the best approximation polynomial $P_{n;f,w_{k/2}}$ ; $\\left\\Vert \\left(f-P_{n;f,w_{k/2}}\\right)w_{k/2}\\right\\Vert _{L_\\infty (\\mathbb {R})}=E_{n}\\left(w_{k/2},f\\right),$ we have $\\left\\Vert \\left(f^{(k)}-P_{n;f,w_{k/2}}^{(k)}\\right)w\\right\\Vert _{L_\\infty (\\mathbb {R})}&\\le & C_r E_{n-k}\\left(w_{(2k+1)/4},f^{(k)}\\right)\\\\&\\le & C_r \\left(\\frac{a_n}{n}\\right)^{r-k}E_{n-r}\\left(w_{(2k+1)/4},f^{(r)}\\right).$ When $w\\in \\mathcal {F}^*$ , we can replace $w_{1/4}$ with $w$ in the above.", "Proof of Theorems Throughout this section we suppose $w\\in \\mathcal {F}(C^2+)$ .", "We give the proofs of theorems.", "First, we give some lemmas to prove the theorems.", "We construct the orthonormal polynomials $p_n(x)=p_n(w^2,x)$ of degree n for $w^2(x)$ , that is, $\\int _{-\\infty }^{\\infty } p_n(w^2,x)p_m(w^2,x)w^2(x)dx=\\delta _{mn} (\\textrm {Kronecker delta}).$ Let $fw\\in L_2(\\mathbb {R})$ .", "The Fourier-type series of $f$ is defined by $\\tilde{f}(x):=\\sum _{k=0}^{\\infty } a_k(w^2,f)p_k(w^2,x),\\quad a_k(w^2,f):=\\int _{-\\infty }^{\\infty } f(t)p_k(w^2,t)w^2(t)dt.$ We denote the partial sum of $\\tilde{f}(x)$ by $s_n(f,x):=s_n(w^2,f,x):=\\sum _{k=0}^{n-1} a_k(w^2,f)p_k(w^2,x).$ Moreover, we define the de la Vall$\\acute{\\textrm {e}}$ e Poussin means by $v_n(f,x):=\\frac{1}{n}\\sum _{j=n+1}^{2n}s_j(w^2,f,x).$ Proposition 3.1 ([11]) Let $w\\in \\mathcal {F}_\\lambda (C^3+), 0<\\lambda <3/2$ , and let $1\\le p\\le \\infty $ .", "When $T^{1/4}wf\\in L_p(\\mathbb {R})$ , we have $\\left\\Vert v_n(f)w\\right\\Vert _{L_p(\\mathbb {R})}\\le C \\left\\Vert T^{1/4}wf \\right\\Vert _{L_p(\\mathbb {R})},$ and so $\\left\\Vert (f-v_n(f))w \\right\\Vert _{L_p(\\mathbb {R})}\\le C E_{p,n}\\left(T^{1/4}w,f\\right).$ So, equivalently, $\\left\\Vert v_n(f)w\\right\\Vert _{L_p(\\mathbb {R})}\\le C \\left\\Vert w_{1/4}f\\right\\Vert _{L_p(\\mathbb {R})},$ and so $\\left\\Vert (f-v_n(f))w\\right\\Vert _{L_p(\\mathbb {R})}\\le C E_{p,n} \\left(w_{1/4},f\\right).$ When $w\\in \\mathcal {F}^*$ , we can replace $w_{1/4}$ with $w$ .", "Lemma 3.2 (1) ([8]) Let $L>0$ be fixed.", "Then, uniformly for $t>0$ , $a_{Lt}\\sim a_t.$    (2) ([8]) For $x >1$ , we have $|Q^{\\prime }(a_x)| \\sim \\frac{x \\sqrt{T(a_x)}}{a_x} \\quad \\mbox{and} \\quad |Q(a_x)| \\sim \\frac{x}{ \\sqrt{T(a_x)}}.$    (3) ([8]) Let $x\\in (0, \\infty )$ .", "There exists $0<\\varepsilon <1$ such that $T\\left(x\\left[1+\\frac{\\varepsilon }{T(x)}\\right]\\right)\\sim T(x).$    (4) ([9]) If $T(x)$ is unbounded, then for any $\\eta >0$ there exists $C(\\eta )>0$ such that for $t\\ge 1$ , $a_t\\le C(\\eta )t^\\eta .$ To prove the results, we need the following notations.", "We set $\\sigma (t):=\\inf \\left\\lbrace a_u: \\,\\, \\frac{a_u}{u}\\le t \\right\\rbrace , \\quad t>0,$ and $\\Phi _t(x):=\\sqrt{\\left|1-\\frac{|x|}{\\sigma (t)}\\right|}+T^{-1/2}(\\sigma (t)), \\quad x\\in \\mathbb {R}.$ Define for $fw\\in L_p(\\mathbb {R})$ , $0<p\\le \\infty $ , $\\omega _p(f,w,t)&:=&\\sup _{0<h\\le t}\\left\\Vert w(x)\\left\\lbrace f\\left(x+\\frac{h}{2}\\Phi _t(x)\\right)-f\\left(x-\\frac{h}{2}\\Phi _t(x)\\right)\\right\\rbrace \\right\\Vert _{L_p(|x|\\le \\sigma (2t))}\\\\&& \\quad +\\inf _{c\\in \\mathbb {R}}\\left\\Vert w(x)(f-c)(x)\\right\\Vert _{L_p(|x|\\ge \\sigma (4t))}$ (see [2], [3]).", "Proposition 3.3 (cf.", "[3], [2]) Let $w\\in \\mathcal {F}(C^2+)$ .", "Let $0<p\\le \\infty $ .", "Then for $f: \\mathbb {R}\\rightarrow \\mathbb {R}$ which $fw\\in L_p(\\mathbb {R})$ (and for $p=\\infty $ , we require $f$ to be continuous, and $fw$ to vanish at $\\pm \\infty $ ), we have for $n\\ge C_3$ , $E_{p,n}\\left(f,w\\right)\\le C_1\\omega _{p}\\left(f,w,C_2\\frac{a_n}{n}\\right),$ where $C_j$ , $j=1,2,3$ , do not depend on $f$ and $n$ .", "Damelin and Lubinsky [3] or Damelin [2] have treated a certain class $\\mathcal {E}_1$ of weights containing the conditions (a)-(d) in Definition REF and $\\frac{yQ^{\\prime }(y)}{xQ^{\\prime }(x)}\\le C_1 \\left(\\frac{Q(y)}{Q(x)}\\right)^{C_2},\\quad y\\ge x\\ge C_3,$ where $C_i$ , $i=1,2,3>0$ are some constants, and they obtain this Proposition for $w\\in \\mathcal {E}_1$ .", "Therefore, we may show $\\mathcal {F}(C^2+)\\subset \\mathcal {E}_1$ .", "In fact, from Definition REF (d) and (e), we have for $y\\ge x>0$ , $\\frac{Q^{\\prime }(y)}{Q^{\\prime }(x)}=\\exp \\left(\\int _x^y\\frac{Q^{\\prime \\prime }(t)}{Q^{\\prime }(t)}dt\\right)\\le \\exp \\left(C_3\\int _x^y\\frac{Q^{\\prime }(t)}{Q(t)}dt\\right)=\\left(\\frac{Q(y)}{Q(x)}\\right)^{C_3},$ and $\\frac{y}{x}=\\exp \\left(\\int _x^y\\frac{1}{t}dt\\right)\\le \\exp \\left(\\frac{1}{\\Lambda }\\int _x^y\\frac{Q^{\\prime }(t)}{Q(t)}dt\\right)=\\left(\\frac{Q(y)}{Q(x)}\\right)^{\\frac{1}{\\Lambda }}.$ Therefore, we obtain (REF ) with $C_2=C_3+\\frac{1}{\\Lambda }$ , that is, we see $\\mathcal {F}(C^2+)\\subset \\mathcal {E}_1$ .", "Theorem 3.4 Let $w\\in \\mathcal {F}(C^2+)$ .", "(1) If $f$ is a function having bounded variation on any compact interval and if $\\int _{-\\infty }^{\\infty } w(x)|df(x)|<\\infty ,$ then there exists a constant $C>0$ such that for every $t>0$ , $\\omega _1(f,w,t)\\le C t\\int _{-\\infty }^{\\infty } w(x)|df(x)|,$ and so $E_{1,n}(f,w)\\le C\\frac{a_n}{n}\\int _{-\\infty }^{\\infty } w(x)|df(x)|.$    (2) Let us suppose that $f$ is continuous and $\\lim _{|x|\\rightarrow \\infty }|(\\sqrt{T}wf)(x)|=0$ , then we have $\\lim _{t\\rightarrow 0}\\omega _{\\infty }(f,w,t)=0.$ To prove this theorem we need the following lemma.", "Lemma 3.5 ([9]) (1) For $t>0$ there exists $a_u$ such that $t=\\frac{a_u}{u} \\quad \\textrm {and} \\quad \\sigma (t)=a_u.$    (2) If $t=a_u/u$ , $u>0$ large enough and $|x-y|\\le t\\Phi _t(x),$ then there exist $C_1,C_2>0$ such that $C_1w(x)\\le w(y)\\le C_2 w(x).$ (1) Let $g(x):=f(x)-f(0)$ .", "For $t>0$ small enough let $0<h\\le t$ and $|x|\\le \\sigma (2t)<\\sigma (t)$ .", "Hence we may consider $\\Phi _t(x)\\le 2$ .", "Then by Lemma REF , $&&\\int _{-\\infty }^{\\infty } w(x)\\left|g\\left(x+\\frac{h}{2}\\Phi _t(x)\\right)-g\\left(x-\\frac{h}{2}\\Phi _t(x)\\right)\\right|dx\\\\&&=\\int _{-\\infty }^{\\infty } w(x)\\left|\\int _{x-\\frac{h}{2}\\Phi _t(x)}^{x+\\frac{h}{2}\\Phi _t(x)}df(v)\\right| dx\\le C\\int _{-\\infty }^{\\infty } \\left|\\int _{x-\\frac{h}{2}\\Phi _t(x)}^{x+\\frac{h}{2}\\Phi _t(x)}w(v)df(v)\\right| dx\\\\&&\\le \\int _{-\\infty }^{\\infty } \\int _{x-h}^{x+h} w(v)|df(v)|dx\\le \\int _{-\\infty }^{\\infty } w(v)\\int _{v-h\\le x\\le v+h} dx|df(v)|\\\\&&\\le 2h\\int _{-\\infty }^{\\infty } w(v)|df(v)|.$ Hence we have $\\int _{-\\infty }^{\\infty } w(x)\\left|g\\left(x+\\frac{h}{2}\\Phi _t(x)\\right)-g\\left(x-\\frac{h}{2}\\Phi _t(x)\\right)\\right| dx\\le 2t \\int _{-\\infty }^{\\infty } w(x)|df(x)|.$ Moreover, we see $\\inf _{c\\in \\mathbb {R}}\\left\\Vert w(x)(f-c)(x)\\right\\Vert _{L_1(|x|\\ge \\sigma (4t))}\\le \\frac{1}{Q^{\\prime }(\\sigma (4t))}\\left\\Vert Q^{\\prime }(x)w(x)g(x)\\right\\Vert _{L_1(|x|\\ge \\sigma (4t))}.$ Here we see $\\frac{\\sqrt{T(\\sigma (t))}}{Q^{\\prime }(\\sigma (t))}\\sim t.$ In fact, from Lemma REF (2), for $t=\\frac{a_u}{u}$ $Q^{\\prime }(\\sigma (t))= Q^{\\prime }(a_u)\\sim \\frac{u\\sqrt{T(a_u)}}{a_u}\\sim \\frac{\\sqrt{T(\\sigma (t))}}{t}.$ On the other hand, we have $\\int _{0}^{\\infty } Q^{\\prime }(x)w(x)|g(x)|dx&=& \\int _{0}^{\\infty } Q^{\\prime }(x)w(x)\\left|\\int _0^xdg(u)\\right|dx\\\\&\\le & \\int _{0}^{\\infty } Q^{\\prime }(x)w(x)\\int _0^x|df(u)|dx\\\\&=& -w(x)\\int _0^x|df(u)|\\bigg |_0^{\\infty }+\\int _0^{\\infty } w(x)|df(x)|\\\\&=&\\int _0^{\\infty } w(x)|df(x)|$ because $\\lim _{x\\rightarrow \\infty } w(x) =0$ from (c) of Definition REF .", "Similarly we have $\\int _{-\\infty }^0 \\left| Q^{\\prime }(x)w(x)g(x)\\right|dx\\le \\int _{-\\infty }^0 w(x)|df(x)|.$ Hence we have $\\Vert Q^{\\prime }wg\\Vert _{L_1(\\mathbb {R})}\\le \\int _{-\\infty }^{\\infty } w(u)|df(u)|.$ Therefore, using (REF ), (REF ) and (REF ), we have $\\inf _{c\\in \\mathbb {R}}\\left\\Vert w(x)(f-c)(x)\\right\\Vert _{L_1(|x|\\ge \\sigma (4t))}= O(t) \\int _{-\\infty }^{\\infty } w(x)|df(x)|.$ Consequently, by (REF ) and (REF ) we have $\\omega _1(f,w,t)\\le C t\\int _{-\\infty }^{\\infty } w(x)|df(x)|.$ Hence, setting $t=C_2\\frac{a_n}{n}$ , if we use Proposition REF , then $E_{1,n}(f,w)\\le C\\frac{a_n}{n}\\int _{-\\infty }^{\\infty } w(x)|df(x)|.$ (2) Given $\\varepsilon >0$ , and let us take $L=L(\\varepsilon )>0$ large enough as $\\sup _{|x|\\ge L}|w(x)f(x)| \\le \\frac{1}{\\sqrt{T(L)}}\\sup _{|x|\\ge L}|\\sqrt{T(x)}w(x)f(x)|<\\varepsilon \\quad (\\textrm {by our assumption}).$ Then we have $\\inf _{c\\in \\mathbb {R}}\\sup _{|x|\\ge L} \\left|w(x)(f-c)(x)\\right|\\le \\frac{1}{\\sqrt{T(L)}}\\sup _{|x|\\ge L}\\left|\\sqrt{T(x)}w(x)f(x)\\right|<\\varepsilon .$ Now, there exists $\\varepsilon >0$ small enough such that $\\frac{h}{2}\\Phi _t(x)\\le \\varepsilon \\frac{1}{T(x)}, \\quad |x|\\le \\sigma (2t),$ because if we put $t=a_u/u$ , then we see $\\sigma (t)=a_u$ and $|x|\\le \\sigma (2t)<a_u$ .", "Hence, noting [8], that is, for some $\\varepsilon >0$ , and for large enough $t$ , $T(a_t) \\le C t^{2-\\varepsilon },$ and if $w$ is the Erdös-type weight, then from Lemma REF (4), we have $t\\Phi _t(x)\\le \\frac{a_u}{u}\\le \\varepsilon \\frac{1}{T(a_u)}\\le \\varepsilon \\frac{1}{T(x)}.$ If $w\\in \\mathcal {F}^*$ , we also have (REF ), because for some $\\delta >0$ and $u>0$ large enough, $t\\Phi _t(x)\\le \\frac{a_u}{u}\\le u^{-\\delta } \\le \\varepsilon \\frac{1}{T(x)}.$ Therefore, using Lemma REF (3), Lemma REF and the assumption $\\lim _{|x|\\rightarrow \\infty }\\sqrt{T\\left(x\\right)}w\\left(x\\right)f\\left(x\\right)=0,$ for $2L\\le |x|\\le \\sigma (2t)$ , $h>0$ , $&&\\left|w(x)\\left\\lbrace f\\left(x+\\frac{h}{2}\\Phi _t(x)\\right)-f\\left(x-\\frac{h}{2}\\Phi _t(x)\\right)\\right\\rbrace \\right|\\\\&\\le & C\\Bigg [\\frac{1}{\\sqrt{T(x)}}\\left|\\sqrt{T\\left(x+\\frac{h}{2}\\Phi _t(x)\\right)}w\\left(x+\\frac{h}{2}\\Phi _t(x)\\right)f\\left(x+\\frac{h}{2}\\Phi _t(x)\\right)\\right|\\\\&& \\qquad +\\frac{1}{\\sqrt{T(x)}}\\left|\\sqrt{T\\left(x-\\frac{h}{2}\\Phi _t(x)\\right)}w\\left(x-\\frac{h}{2}\\Phi _t(x)\\right)f\\left(x-\\frac{h}{2}\\Phi _t(x)\\right)\\right|\\Bigg ]\\\\&\\le & 2\\varepsilon .$ On the other hand, $\\lim _{t\\rightarrow 0}\\sup _{0<h\\le t}\\left\\Vert w(x) \\left\\lbrace f\\left(x+\\frac{h}{2}\\Phi _t(x)\\right)-f\\left(x-\\frac{h}{2}\\Phi _t(x)\\right)\\right\\rbrace \\right\\Vert _{L_\\infty (|x|\\le 2L)}=0.$ Therefore, we have the result.", "Lemma 3.6 (cf.", "[4]) Let $g$ be a real valued function on $\\mathbb {R}$ satisfying $\\Vert gw\\Vert _{L_\\infty (\\mathbb {R})}<\\infty $ and $\\int _{-\\infty }^{\\infty } gPw^2dt=0 \\quad P\\in \\mathcal {P}_n.$ Then we have $\\left\\Vert w(x)\\int _0^x g(t)dt \\right\\Vert _{L_\\infty (\\mathbb {R})}\\le C \\frac{a_n}{n}\\Vert gw\\Vert _{L_\\infty (\\mathbb {R})}.$ Especially, if $w\\in \\mathcal {F}_\\lambda (C^3+), 0<\\lambda <3/2$ , then we have $\\left\\Vert w(x)\\int _0^x \\left(f^{\\prime }(t)-v_n(f^{\\prime })(t)\\right)dt \\right\\Vert _{L_\\infty (\\mathbb {R})}\\le C \\frac{a_n}{n}E_{n} \\left(w_{1/4},f^{\\prime }\\right).$ When $w\\in \\mathcal {F}^*$ , we also have (REF ) replacing $w_{1/4}$ with $w$ .", "We let $\\phi _x(t)=\\left\\lbrace \\begin{array}{lr}w^{-2}(t),& 0\\le t\\le x; \\\\0,& otherwise,\\end{array}\\right.$ then we have for arbitrary $P_n\\in \\mathcal {P}_n$ , $\\left|\\int _0^x g(t)dt\\right|=\\left|\\int _{-\\infty }^{\\infty } g(t)\\phi _x(t)w^2(t)dt\\right|=\\left|\\int _{-\\infty }^{\\infty } g(t)(\\phi _x(t)-P_n(t))w^2(t)dt\\right|.$ Therefore, we have $\\left|\\int _0^x g(t)dt\\right|&\\le & \\Vert gw\\Vert _{L_\\infty (\\mathbb {R})}\\inf _{P_n\\in \\mathcal {P}_n}\\int _{-\\infty }^{\\infty } \\left|\\phi _x(t)-P_n(t)\\right|w(t)dt \\\\&=&\\Vert gw\\Vert _{L_\\infty (\\mathbb {R})}E_{1,n}(w:\\phi _x).$ Here, from Theorem REF we see that $E_{1,n}(w:\\phi _x)&\\le & C\\frac{a_n}{n}\\int _{-\\infty }^{\\infty } w(t)|d\\phi _x(t)|\\le C\\frac{a_n}{n}\\int _0^x w(t)|Q^{\\prime }(t)|w^{-2}(t)dt\\\\&=& C\\frac{a_n}{n}\\int _0^x Q^{\\prime }(t)w^{-1}(t)dt\\le C\\frac{a_n}{n}w^{-1}(x).$ So, we have $\\left|w(x)\\int _0^x g(t)dt\\right|\\le \\Vert gw\\Vert _{L_\\infty (\\mathbb {R})}w(x)E_{1,n}\\left(w:\\phi _x\\right)\\le C\\frac{a_n}{n}\\Vert gw\\Vert _{L_\\infty (\\mathbb {R})}.$ Therefore, we have (REF ).", "Next we show (REF ).", "Since $v_n(f^{\\prime })(t)=\\frac{1}{n}\\sum _{j=n+1}^{2n}s_j(f^{\\prime },t),$ and for any $P\\in \\mathcal {P}_n$ , $j\\ge n+1$ , $\\int _{-\\infty }^{\\infty } \\left(f^{\\prime }(t)-s_j(f^{\\prime };t)\\right)P(t)w^2(t)dt=0,$ we have $\\int _{-\\infty }^{\\infty } \\left(f^{\\prime }(t)-v_n(f^{\\prime })(t)\\right)P(t)w^2(t)dt=0.$ Using (REF ) and (REF ), we have (REF ).", "Lemma 3.7 Let $w=\\exp (-Q)\\in \\mathcal {F}_\\lambda (C^3+)$ , $0<\\lambda <3/2$ .", "Let $\\left\\Vert w_{1/4}f^{\\prime }\\right\\Vert _{L_\\infty (\\mathbb {R})}<\\infty $ , and let $q_{n-1}\\in \\mathcal {P}_n$ be the best approximation of $f^{\\prime }$ with respect to the weight $w$ , that is, $\\left\\Vert (f^{\\prime }-q_{n-1})w \\right\\Vert _{L_\\infty (\\mathbb {R})}=E_{n-1}(w,f^{\\prime }).$ Now we set $F(x):=f(x)-\\int _0^xq_{n-1}(t)dt,$ then there exists $S_{2n}\\in \\mathcal {P}_{2n}$ such that $\\left\\Vert w \\left(F-S_{2n}\\right) \\right\\Vert _{L_\\infty (\\mathbb {R})}\\le C \\frac{a_n}{n}E_n \\left(w_{1/4},f^{\\prime }\\right),$ and $\\left\\Vert wS_{2n}^{\\prime } \\right\\Vert _{L_\\infty (\\mathbb {R})}\\le C E_{n-1}\\left(w_{1/4},f^{\\prime }\\right).$ When $w\\in \\mathcal {F}^*$ , we also have same results replacing $w_{1/4}$ with $w$ .", "Let $S_{2n}(x)=f(0)+\\int _0^x v_n \\left(f^{\\prime }-q_{n-1}\\right)(t)dt,$ then by Lemma REF (REF ), $&&\\left\\Vert w\\left(F-S_{2n}\\right)\\right\\Vert _{L_\\infty (\\mathbb {R})} \\\\&=&\\left\\Vert w\\left(f-\\int _0^xq_{n-1}(t)dt -f(0)-\\int _0^x v_n\\left(f^{\\prime }-q_{n-1}\\right)(t)dt\\right)\\right\\Vert _{L_\\infty (\\mathbb {R})}\\\\&=&\\left\\Vert w\\left(\\int _0^x \\left[f^{\\prime }(t)-v_n(f^{\\prime })(t)\\right]dt\\right)\\right\\Vert _{L_\\infty (\\mathbb {R})}\\le C\\frac{a_n}{n}E_n\\left(w_{1/4},f^{\\prime }\\right).$ Now by Proposition REF (REF ), $\\left\\Vert wS_{2n}^{\\prime }\\right\\Vert _{L_\\infty (\\mathbb {R})}&=&\\left\\Vert w\\left(v_n(f^{\\prime }-q_{n-1})\\right)\\right\\Vert _{L_\\infty (\\mathbb {R})}\\\\&\\le &\\left\\Vert \\left(f^{\\prime }-v_n\\left(f^{\\prime }\\right)\\right)w\\right\\Vert _{L_\\infty (\\mathbb {R})}+\\left\\Vert \\left(f^{\\prime }-q_{n-1}\\right)w\\right\\Vert _{L_\\infty (\\mathbb {R})}\\\\&\\le & E_{n}\\left(w_{1/4},f^{\\prime }\\right)+E_{n-1}\\left(w,f^{\\prime }\\right)\\le E_{n-1}\\left(w_{1/4},f^{\\prime }\\right).$ To prove Theorem REF we need the following theorems.", "Theorem 3.8 ([9]) Let $w\\in \\mathcal {F}(C^2+)$ , and let $r\\ge 0$ be an integer.", "Let $1\\le p\\le \\infty $ , and let $wf^{(r)}\\in L_p(\\mathbb {R})$ .", "Then we have $E_{p,n}(f,w)\\le C \\left(\\frac{a_n}{n}\\right)^k \\left\\Vert f^{(k)}w\\right\\Vert _{L_p(\\mathbb {R})}, \\quad k=1,2,...,r,$ and equivalently, $E_{p,n}(f,w)\\le C \\left(\\frac{a_n}{n}\\right)^kE_{p,n-k}\\left(f^{(k)},w\\right).$ Theorem 3.9 ([10]) Let $r\\ge 1$ be an integer and $w\\in \\mathcal {F}_\\lambda (C^{r+2}+)$ , $0< \\lambda <(r+2)/(r+1)$ , and let $1\\le p\\le \\infty $ .", "Then there exists a constant $C>0$ such that for any $1\\le k\\le r$ , any integer $n\\ge 1$ and any polynomial $P\\in \\mathcal {P}_n$ , $\\left\\Vert P^{(k)}w\\right\\Vert _{L_p(\\mathbb {R})}\\le C\\left(\\frac{n}{a_n}\\right)^k\\left\\Vert T^{k/2}Pw\\right\\Vert _{L_p(\\mathbb {R})}.$ We prove the theorem only in case of unbounded $T(x)$ , in the case of Freud case $\\mathcal {F}^*$ we can prove it similarly.", "We show that for $k=0,1,...,r$ , $\\left|\\left(f^{(k)}(x)-P_{n;f,w}^{(k)}\\right)w(x)\\right|\\le CT^{k/2}(x)E_{n-k}\\left(w_{1/4},f^{(k)}\\right).$ If $r=0$ , then (REF ) is trivial.", "For some $r\\ge 0$ we suppose that (REF ) holds, and let $f\\in C^{(r+1)}(\\mathbb {R})$ .", "Then $f^{\\prime }\\in C^{(r)}(\\mathbb {R})$ .", "Let $q_{n-1}\\in \\mathcal {P}_{n-1}$ be the polynomial of best approximation of $f^{\\prime }$ with respect to the weight $w$ .", "Then, from our assumption we have for $0\\le k\\le r$ , $\\left|\\left(f^{(k+1)}(x)-q_{n-1}^{(k)}(x)\\right)w(x)\\right|\\le C T^{k/2}(x)E_{n-k}\\left(w_{1/4},f^{(k+1)}\\right),$ that is, for $1\\le k\\le r+1$ $\\left|\\left(f^{(k)}(x)-q_{n-1}^{(k-1)}(x)\\right)w(x)\\right|\\le C T^{\\frac{k-1}{2}}(x)E_{n-k+1}\\left(w_{1/4},f^{(k)}\\right).$ Let $F(x):=f(x)-\\int _0^x q_{n-1}(t)dt=f(x)-Q_n(x),$ then $|F^{\\prime }(x)w(x)|\\le C E_{n-1}\\left(w,f^{\\prime }\\right).$ As (REF ) we set $S_{2n}=\\int _0^x(v_n(f^{\\prime })(t)-q_{n-1}(t))dt+f(0)$ , then from Lemma REF $\\left\\Vert \\left(F-S_{2n}\\right)w\\right\\Vert _{L_\\infty (\\mathbb {R})}\\le C \\frac{a_n}{n}E_n\\left(w_{1/4},f^{\\prime }\\right),$ and $\\left\\Vert S_{2n}^{\\prime }w\\right\\Vert _{L_\\infty (\\mathbb {R})}\\le C E_{n-1}\\left(w_{1/4},f^{\\prime }\\right).$ Here we apply Theorem REF with the weight $w_{-(k-1)/2}$ .", "In fact, by Theorem REF we have $w_{-(k-1)/2}\\in \\mathcal {F}_\\lambda (C^{r+2}+)$ .", "Then, noting $a_{2n}\\sim a_n$ from Lemma REF (1), we see $|S_{2n}^{(k)}(x) w_{-(k-1)/2}(x))|&\\le & C \\left(\\frac{n}{a_n}\\right)^{k-1}\\Vert S_{2n}^{\\prime }w\\Vert _{L_\\infty (\\mathbb {R})}\\\\&\\le & C \\left(\\frac{n}{a_n}\\right)^{k-1}E_{n-1}\\left(w_{1/4},f^{\\prime }\\right),$ that is, $\\left|S_{2n}^{(k)}(x) w(x)\\right|\\le C \\left(\\frac{n\\sqrt{T(x)}}{a_n}\\right)^{k-1}E_{n-1}\\left(w_{1/4},f^{\\prime }\\right),\\quad 1\\le k\\le r+1.$ Let $R_{n}\\in \\mathcal {P}_{n}$ denote the polynomial of best approximation of $F$ with $w$ .", "By Theorem REF with $w_{-\\frac{k}{2}}$ again, for $0\\le k\\le r+1$ we have $ \\nonumber \\left|(R_{n}^{(k)}-S_{2n}^{(k)}(x)) w_{-\\frac{k}{2}}(x)\\right|&\\le & C \\left(\\frac{n}{a_n}\\right)^k\\Vert (R_{n}-S_{2n})w_{-\\frac{k}{2}}(x)T^{k/2}(x)\\Vert _{L_\\infty (\\mathbb {R})}\\\\&\\le & C \\left(\\frac{n}{a_n}\\right)^k\\Vert (R_{n}-S_{2n})w\\Vert _{L_\\infty (\\mathbb {R})}$ and by (REF ) $ \\nonumber \\Vert (R_{n}-S_{2n})w\\Vert _{L_\\infty (\\mathbb {R})}&\\le & C \\left[\\Vert (F-R_{n})w\\Vert _{L_\\infty (\\mathbb {R})}+\\Vert (F-S_{2n})w\\Vert _{L_\\infty (\\mathbb {R})}\\right]\\\\\\nonumber &\\le & C \\left[E_{n}(w,F)+\\frac{a_n}{n}E_{n}\\left(w_{1/4},f^{\\prime }\\right) \\right] \\\\ \\nonumber &\\le & C \\left[\\frac{a_n}{n}E_{n-1}(w,f^{\\prime })+\\frac{a_n}{n}E_{n-1}(w_{1/4},f^{\\prime })\\right]\\\\&\\le & C \\frac{a_n}{n}E_{n-1}\\left(w_{1/4},f^{\\prime }\\right).$ Hence, from (REF ) and (REF ) we have for $0\\le k\\le r+1$ $\\nonumber |(R_{n}^{(k)}-S_{2n}^{(k)}(x)) w(x)|&\\le & C\\left|T^{k/2}(x)\\right|\\left|(R_{n}^{(k)}-S_{2n}^{(k)}(x)) w_{-\\frac{k}{2}}(x)\\right|\\\\&\\le & C \\left(\\frac{n\\sqrt{T(x)}}{a_n}\\right)^k\\frac{a_n}{n}E_{n-1}\\left(w_{1/4},f^{\\prime }\\right).$ Therefore by (REF ), (REF ) and Theorem REF , $ \\nonumber |R_{n}^{(k)}(x) w(x))|&\\le & C T^{k/2}(x)\\left(\\frac{n}{a_n}\\right)^{k-1}E_{n-1}\\left(w_{1/4},f^{\\prime }\\right) \\\\&\\le & C T^{k/2}(x)E_{n-k}\\left(w_{1/4},f^{(k)}\\right).$ Since $E_{n}(F,w)=E_{n}(w,f)$ and $E_{n}\\left(F,w\\right)=\\left\\Vert w\\left(F-R_{n}\\right)\\right\\Vert _{L_\\infty (\\mathbb {R})}=\\left\\Vert w\\left(f-Q_n-R_{n}\\right)\\right\\Vert _{L_\\infty (\\mathbb {R})}$ (see (REF )), we know that $P_{n;f,w}:=Q_n+R_n$ is the polynomial of best approximation of $f$ with $w$ .", "Now, from (REF ), (REF ) and (REF ) we have for $1\\le k\\le r+1$ , $\\left|\\left(f^{(k)}(x)-P_{n;f.w}^{(k)}(x)\\right)w(x)\\right|&=&\\left|\\left(f^{(k)}(x)-Q_n^{(k)}(x)-R_{n}^{(k)}(x)\\right)w(x)\\right|\\\\&\\le & \\left|(f^{(k)}(x)-q_{n-1}^{(k-1)}(x))w(x)\\right|+\\left|R_{n}^{(k)}(x)w(x)\\right|\\\\&\\le & C T^{k/2}(x)E_{n-k}\\left(w_{1/4},f^{(k)}\\right).$ For $k=0$ it is trivial.", "Consequently, we have (REF ) for all $r\\ge 0$ .", "Moreover, using Theorem REF , we conclude Theorem REF .", "It follows from Theroem REF .", "Applying Theorem REF with $w_{k/2}$ , we have for $0\\le j\\le r$ $\\left\\Vert (f^{(j)}-P_{n;f,w_{k/2}}^{(j)})w\\right\\Vert _{L_\\infty (\\mathbb {R})}\\le C E_{n-k}\\left(w_{(2k+1)/4},f^{(j)}\\right).$ Especially, when $j=k$ , we obtain $\\left\\Vert \\left(f^{(k)}-P_{n;f,w_{k/2}}^{(k)}\\right)w\\right\\Vert _{L_\\infty (\\mathbb {R})}\\le CE_{p,n-k}\\left(w_{(2k+1)/4},f^{(k)}\\right).$ Theorems in $L_p(\\mathbb {R})$ $(1 \\le p \\le \\infty )$ In this section we will give an analogy of Theorem REF in $L_p(\\mathbb {R})$ -space ($1 \\le p \\le \\infty $ ) and we will prove it using the same method as the proof of Theorem REF .", "Let $1 \\le p \\le \\infty $ .", "Let $w=\\exp (-Q)\\in \\mathcal {F}_\\lambda (C^3+)$ , $0<\\lambda <3/2$ , and let $\\beta >1$ be fixed.", "Then we set $w^{\\sharp }$ and $w^{\\flat }$ as follows; $&& \\frac{w(x)}{\\left\\lbrace (1+|Q^{\\prime }(x)|)(1+|x|)^{\\beta }\\right\\rbrace ^{1/p}}\\sim w^{\\sharp }(x)\\in \\mathcal {F}(C^2+);\\\\&&w(x)\\left\\lbrace (1+|Q^{\\prime }(x)|)(1+|x|)^{\\beta }\\right\\rbrace ^{1/p} \\sim w^{\\flat }(x)\\in \\mathcal {F}(C^2+)$ (see Theorem REF ).", "Theorem 4.1 Let $r \\ge 0$ be an integer.", "Let $w=\\exp (-Q)\\in \\mathcal {F}_\\lambda (C^{r+2}+)$ , $0< \\lambda <(r+2)/(r+1)$ , and let $\\beta >1$ be fixed.", "Suppose that $T^{1/4}f^{(r)}w\\in L_p(\\mathbb {R})$ .", "Let $P_{p,n;f,w}\\in \\mathcal {P}_n$ be the best approximation of $f$ with respect to the weight $w$ in $L_p(\\mathbb {R})$ -space, that is, $E_{p,n}\\left(w,f\\right):=\\inf _{P\\in \\mathcal {P}_n}\\left\\Vert \\left(f-P\\right)w\\right\\Vert _{L_p(\\mathbb {R})}=\\left\\Vert \\left(f-P_{p,n;f,w}\\right)w\\right\\Vert _{L_p(\\mathbb {R})}.$ Then there exists an absolute constant $C_r>0$ which depends only on $r$ such that for $0\\le k\\le r$ and $x\\in \\mathbb {R}$ , $\\left\\Vert \\left(f^{(k)}-P_{p,n;f,w}^{(k)}\\right)w^{\\sharp }_{-k/2}\\right\\Vert _{L_p(\\mathbb {R})}&\\le & C_r E_{p,n-k}\\left(w_{1/4},f^{(k)}\\right)\\\\&\\le & C_r \\left(\\frac{a_n}{n}\\right)^{r-k}E_{p,n-r}\\left(w_{1/4},f^{(r)}\\right).$ When $w\\in \\mathcal {F}^*$ , we can replace $w_{1/4}$ and $w^{\\sharp }_{-k/2}$ with $w$ and $w^{\\sharp }$ , respectively in the above.", "If we apply Theorem REF with $w_{-1/4}$ , then we have the following.", "Corollary 4.2 Let $r \\ge 0$ be an integer.", "Let $w=\\exp (-Q)\\in \\mathcal {F}_\\lambda (C^{r+3}+)$ , $0< \\lambda <(r+3)/(r+2)$ , and let $\\beta >0$ be fixed.", "Suppose that $wf^{(r)}\\in L_p(\\mathbb {R})$ .", "Then for $0\\le k\\le r$ we have $\\left\\Vert \\left(f^{(k)}-P_{p,n;f,w_{-1/4}}^{(k)}\\right)w^{\\sharp }_{-(2k+1)/4}\\right\\Vert _{L_p(\\mathbb {R})}&\\le & C_r E_{p,n-k}\\left(w,f^{(k)}\\right)\\\\&\\le & C_r \\left(\\frac{a_n}{n}\\right)^{r-k}E_{p,n-r}\\left(w,f^{(r)}\\right).$ When $w\\in \\mathcal {F}^*$ , we can omit $T^{-(2k+1)/4}$ in the above.", "Corollary 4.3 Let $r \\ge 0$ be an integer.", "Let $w=\\exp (-Q)\\in \\mathcal {F}_\\lambda (C^{r+3}+)$ , $0< \\lambda <(r+3)/(r+2)$ , and let $\\beta >0$ be fixed.", "(1) Let $w_{(2r+1)/4}f^{(r)}\\in L_p(\\mathbb {R})$ .", "Then, for each $k$ $(0\\le k\\le r)$ and the best approximation polynomial $P_{p,n;f,w_{k/2}}$ , we have $\\left\\Vert \\left(f^{(k)}-P_{p,n;f,w_{k/2}}^{(k)}\\right)w^{\\sharp }\\right\\Vert _{L_p(\\mathbb {R})}&\\le & C_r E_{p,n-k}\\left(w_{(2k+1)/4},f^{(k)}\\right)\\\\&\\le & C_r \\left(\\frac{a_n}{n}\\right)^{r-k}E_{p,n-r}\\left(w_{(2k+1)/4},f^{(r)}\\right).$    (2) Let $w^{\\flat }_{(2r+1)/4}f^{(r)}\\in L_p(\\mathbb {R})$ .", "Then, for each $k$ $(0\\le k\\le r)$ and the best approximation polynomial $P_{p,n;f,w^{\\flat }_{k/2}}$ , we have $\\left\\Vert \\left(f^{(k)}-P_{p,n;f,w^{\\flat }_{k/2}}^{(k)}\\right)w\\right\\Vert _{L_p(\\mathbb {R})}&\\le & C_r E_{p,n-k}\\left(w^{\\flat }_{(2k+1)/4},f^{(k)}\\right)\\\\&\\le & C_r \\left(\\frac{a_n}{n}\\right)^{r-k}E_{p,n-r}\\left(w^{\\flat }_{(2k+1)/4},f^{(r)}\\right).$ When $w\\in \\mathcal {F}^*$ , we can replace $w_{(2k+1)/4}$ and $w^{\\flat }_{(2k+1)/4}$ with $w$ and $w^{\\flat }$ , respectively in the above.", "Especially when $p=\\infty $ , we can refer to $w^{\\sharp }$ or $w^{\\flat }$ as $w$ .", "In this case, we can note that Corollary REF and Corollary REF imply Corollary REF , and Corollary REF , respectively.", "To prove Theorem REF we need to prepare some notations and lemmas.", "Lemma 4.4 ([6]) Let $w\\in \\mathcal {F}(C^2+)$ and let $1\\le p\\le \\infty $ .", "If $g:\\mathbb {R}\\rightarrow \\mathbb {R}$ is absolutely continuous, $g(0)=0$ , and $wg^{\\prime }\\in L_p(\\mathbb {R})$ , then $\\left\\Vert Q^{\\prime }wg\\right\\Vert _{L_p(\\mathbb {R})}\\le C\\left\\Vert wg^{\\prime }\\right\\Vert _{L_p(\\mathbb {R})}.$ Lemma 4.5 (cf.", "[9]) Let $w\\in \\mathcal {F}(C^2+)$ and let $1\\le p\\le \\infty $ .", "If $wf^{\\prime }\\in L_p(\\mathbb {R})$ , then $E_{p,n}(w,f)\\le C\\omega _p\\left(f,w,\\frac{a_n}{n}\\right)\\le C \\frac{a_n}{n}\\left\\Vert wf^{\\prime }\\right\\Vert _{L_p(\\mathbb {R})}.$ The first inequality follows from Proposition REF .", "We show the second inequality.", "By [9] we have $&&\\left\\Vert w(x)\\left\\lbrace f\\left(x+\\frac{h}{2}\\Phi _t(x)\\right)-f\\left(x-\\frac{h}{2}\\Phi _t(x)\\right)\\right\\rbrace \\right\\Vert ^p_{L_p(|x|\\le \\sigma (2t))} \\\\&=&h^p\\int _{\\mathbb {R}}|w(x)\\Phi _t(x)f^{\\prime }(x_t)|^pdx\\le Ch^p\\int _{\\mathbb {R}}|w(x)f^{\\prime }(x)|^pdx.$ Hence we see $\\sup _{0<h\\le t}\\left\\Vert w(x)\\left\\lbrace f\\left(x+\\frac{h}{2}\\Phi _t(x)\\right)-f\\left(x-\\frac{h}{2}\\Phi _t(x)\\right)\\right\\rbrace \\right\\Vert _{L_p(|x|\\le \\sigma (2t))}\\le Ct\\left\\Vert wf^{\\prime }\\right\\Vert _{L_p(\\mathbb {R})}.$ Now, we estimate $\\left\\Vert w(x)(f-c)(x)\\right\\Vert _{L_p(|x|\\ge \\sigma (4t))}.$ Let $g(x):=f(x)-f(0)$ .", "$\\inf _{c\\in \\mathbb {R}}\\left\\Vert w(x)\\left(f-c\\right)(x)\\right\\Vert _{L_p(|x|\\ge \\sigma (4t))}\\le \\frac{1}{Q^{\\prime }(\\sigma (4t))} \\left\\Vert Q^{\\prime }wg\\right\\Vert _{L_p(\\mathbb {R})}.$ Then we have from (REF ), $\\inf _{c\\in \\mathbb {R}}\\left\\Vert w(x)(f-c)(x)\\right\\Vert _{L_p(|x|\\ge \\sigma (4t))}\\le Ct\\Vert Q^{\\prime }wg\\Vert _{L_p(\\mathbb {R})}.$ Here, from Lemma REF we have $\\Vert Q^{\\prime }w(f-f(0))\\Vert _{L_p(\\mathbb {R})}\\le C\\Vert wf^{\\prime }\\Vert _{L_p(\\mathbb {R})}.$ From (REF ) and (REF ) we have $\\inf _{c\\in \\mathbb {R}}\\left\\Vert w(x)(f-c)(x)\\right\\Vert _{L_p(|x|\\ge \\sigma (4t))}\\le Ct \\Vert wf^{\\prime }\\Vert _{L_p(\\mathbb {R})}.$ From (REF ) and (REF ) we have the result.", "Lemma 4.6 (cf.", "[4]) Let $1 \\le p \\le \\infty $ and $\\beta >1$ , and let us define $w^{\\sharp }$ with $p$ , $\\beta $ .", "Let $g$ be a real valued function on $\\mathbb {R}$ satisfying $\\Vert gw\\Vert _{L_p(\\mathbb {R})}<\\infty $ and (REF ), then we have $\\left\\Vert w^{\\sharp }(x)\\int _0^x g(t)dt\\right\\Vert _{L_p(\\mathbb {R})}\\le C \\frac{a_n}{n}\\Vert gw\\Vert _{L_p(\\mathbb {R})}.$ Especially, if $w\\in \\mathcal {F}_\\lambda (C^3+), 0<\\lambda <3/2$ , then we have $\\left\\Vert w^{\\sharp }(x)\\int _0^x \\left(f^{\\prime }(t)-v_n(f^{\\prime })(t)\\right)dt\\right\\Vert _{L_p(\\mathbb {R})}\\le C \\frac{a_n}{n}E_{p,n}\\left(w_{1/4},f^{\\prime }\\right).$ When $w\\in \\mathcal {F}^*$ , we also have (REF ) replacing $w_{1/4}$ with $w$ .", "For arbitrary $P_n\\in \\mathcal {P}_n$ , we have by (REF ) and Hölder inequality $\\left|\\int _0^x g(t)dt\\right|\\le \\left\\Vert g(t)w(t)\\right\\Vert _{L_p(\\mathbb {R})}E_{q,n}\\left(w,\\phi _x\\right), \\quad 1\\le p \\le \\infty , 1/p+1/q=1,$ where $\\phi $ is defined in (REF ).", "Then, we obtain by Lemma REF , $\\left|\\int _0^x g(t)dt\\right|&\\le & \\left\\Vert g(t)w(t)\\right\\Vert _{L_p(\\mathbb {R})}\\frac{a_n}{n}\\left(\\int _{\\mathbb {R}}|w(t)\\phi ^{\\prime }_x(t)|^qdt\\right)^{1/q} \\\\&\\le & \\frac{a_n}{n}\\left\\Vert g(t)w(t)\\right\\Vert _{L_p(\\mathbb {R})}|Q^{\\prime }(x)|^{1-1/q}\\left(\\int _0^x Q^{\\prime }(t)w^{-q}(t)dt\\right)^{1/q}\\\\&\\le & C\\frac{a_n}{n}\\left\\Vert g(t)w(t)\\right\\Vert _{L_p(\\mathbb {R})}|Q^{\\prime }(x)|^{1-1/q}w^{-1}(x).$ Here, for $p=1$ we may consider $\\lim _{q\\rightarrow \\infty } \\left( \\int _0^xQ^{\\prime }(t)w^{-q}(t)dt\\right)^{1/q}=\\lim _{q\\rightarrow \\infty } \\left( w^{-q}(t)\\right)^{1/q}=w^{-1}(x).$ Hence, we have $\\left\\Vert \\frac{w(x)}{\\left\\lbrace (1+|Q^{\\prime }(x)|)(1+|x|)^{\\beta }\\right\\rbrace ^{1/p}}\\int _0^x g(t)dt\\right\\Vert _{L_p(\\mathbb {R})}&\\le & C\\left(\\frac{a_n}{n}\\right)\\left\\Vert \\left(1+|x|\\right)^{-\\beta /p}\\right\\Vert _{L_p(\\mathbb {R})}\\left\\Vert gw\\right\\Vert _{L_p(\\mathbb {R})}\\\\&\\le & C\\left(\\frac{a_n}{n}\\right)\\left\\Vert gw\\right\\Vert _{L_p(\\mathbb {R})}.$ Therefore, we have (REF ).", "From (REF ), (REF ) and Proposition REF , we have (REF ).", "Lemma 4.7 Let $w=\\exp (-Q)\\in \\mathcal {F}_\\lambda (C^3+)$ , $0<\\lambda <3/2$ .", "Let $1 \\le p \\le \\infty $ , $\\Vert w_{1/4}f^{\\prime }\\Vert _{L_p(\\mathbb {R})}<\\infty $ , and let $q_{n-1}\\in \\mathcal {P}_n$ be the best approximation of $f^{\\prime }$ with respect to the weight $w$ on $L_p(\\mathbb {R})$ space, that is, $\\left\\Vert (f^{\\prime }-q_{n-1})w\\right\\Vert _{L_p(\\mathbb {R})}=E_{p,n-1}(w,f^{\\prime }).$ Using $q_{n-1}$ , define $F(x)$ and $S_{2n}$ as (REF ) and (REF ).", "Then we have $\\left\\Vert w^{\\sharp }(F-S_{2n})\\right\\Vert _{L_p(\\mathbb {R})}\\le C \\frac{a_n}{n}E_{p,n}\\left(w_{1/4},f^{\\prime }\\right),$ and $\\left\\Vert wS_{2n}^{\\prime }\\right\\Vert _{L_p(\\mathbb {R})}\\le C E_{p,n-1}\\left(w_{1/4},f^{\\prime }\\right).$ When $w\\in \\mathcal {F}^*$ , we also have same results replacing $w_{1/4}$ with $w$ .", "By Lemma REF (REF ), we have the result using the same method as the proof of Lemma REF .", "We will prove it similarly to the proof of Theorem REF .", "First, let $q_{n-1}\\in \\mathcal {P}_{n-1}$ be the polynomial of best approximation of $f^{\\prime }$ with respect to the weight $w$ on $L_p(\\mathbb {R})$ space.", "Then using $q_{n-1}$ , we define $F(x)$ and $S_{2n}$ in the same method as (REF ) and (REF ).", "Then we have using Lemma REF $\\left\\Vert F^{\\prime }w\\right\\Vert _{L_p(\\mathbb {R})}= E_{p,n-1}\\left(w,f^{\\prime }\\right),$ $\\Vert (F-S_{2n})w^{\\sharp }\\Vert _{L_p(\\mathbb {R})}\\le C \\frac{a_n}{n}E_{p,n}\\left(w_{1/4},f^{\\prime }\\right)$ and $\\left\\Vert S_{2n}^{\\prime }w \\right\\Vert _{L_p(\\mathbb {R})}\\le C E_{p,n-1}\\left(w_{1/4},f^{\\prime }\\right).$ Then we see from Theorem REF and (REF ), $\\left\\Vert S_{2n}^{(k)}w_{-(k-1)/2}\\right\\Vert _{L_p(\\mathbb {R})}\\le C \\left(\\frac{n}{a_n}\\right)^{k-1}E_{p,n-1}\\left(w_{1/4},f^{\\prime }\\right)$ and using $w^{\\sharp } \\le w$ and Theorem REF $\\left\\Vert \\left(R_{n}^{(k)}-S_{2n}^{(k)}\\right) w^{\\sharp }_{-k/2}\\right\\Vert _{L_p(\\mathbb {R})}\\le C \\left(\\frac{n}{a_n}\\right)^{k-1}E_{p,n-1}\\left(w_{1/4},f^{\\prime }\\right),$ where $R_{n}\\in \\mathcal {P}_{n}$ denotes the polynomial of best approximation of $F$ with $w$ on $L_p(\\mathbb {R})$ space(by the similar calculation as (REF ) and (REF )).", "Then, we see $w^{\\sharp }_{-k/2}(x) \\le w_{-(k-1)/2}(x)$ .", "By (REF ) and (REF ) and Theorem REF , we have $\\left\\Vert R_{n}^{(k)}w^{\\sharp }_{-k/2}\\right\\Vert _{L_p(\\mathbb {R})}\\le C \\left(\\frac{n}{a_n}\\right)^{k-1}E_{p,n-1}\\left(w_{1/4},f^{\\prime }\\right)\\le C E_{p,n-k}\\left(w_{1/4},f^{(k)}\\right).$ By the same reason to (REF ), we know that $P_{p,n;f,w}:=Q_n+R_n$ is the polynomial of best approximation of $f$ with $w$ on $L_p(\\mathbb {R})$ space.", "Therefore, using $P_{p,n;f,w}$ , (REF ) and the method of mathematical induction, we have for $1\\le k\\le r+1$ , $\\left\\Vert \\left(f^{(k)}-P_{p,n;f,w}^{(k)}\\right)w^{\\sharp }_{-k/2}\\right\\Vert _{L_p(\\mathbb {R})}\\le C E_{p,n-k}\\left(w_{1/4},f^{(k)}\\right).$ It follows from Theorem REF .", "If we apply Theorem REF with $w_{k/2}$ and $w^{\\flat }_{k/2}$ , then we can obtain the results.", "Monotone Approximation Let $r>0$ be an integer.", "Let $k$ and $\\ell $ be integers with $0\\le k\\le \\ell \\le r$ .", "In this section, we consider a real function $f$ on $\\mathbb {R}$ such that $f^{(r)}(x)$ is continuous in $\\mathbb {R}$ and we let $a_j(x)$ , $j=k,k+1,...,\\ell $ be bounded on $\\mathbb {R}$ .", "Now, we define the linear differential operator (cf.", "[1]) $L:=L_{k,\\ell }:=\\sum _{j=k}^\\ell a_j(x)[d^j/dx^j].$ G. A. Anastassiou and O. Shisha [1] consider the operator (REF ) with $a_j(x)$ under some condition on $[-1,1]$ .", "They showed that if $L(f) \\ge 0$ for $f\\in C^{(r)}[-1,1]$ , there exist $Q_n \\in \\mathcal {P}_n$ such that $L(Q_n)\\ge 0$ and for some constant $C>0$ , $\\left\\Vert f-Q_n\\right\\Vert _{L_{\\infty }([-1,1])}\\le C n^{\\ell -r}\\omega \\left(f^{(p)};\\frac{1}{n}\\right),$ where $\\omega \\left(f^{(p)};t\\right)$ is the modulus of continuity.", "In this section, we will obtain a similar result with exponential-type weighted $L_{\\infty }$ -norm as the above result.", "Our main theorem is as follows.", "Theorem 5.1 Let $k$ and $\\ell $ be integers with $0\\le k\\le \\ell \\le r$ .", "Let $w=\\exp (-Q)\\in \\mathcal {F}_\\lambda (C^{r+3}+)$ , and let $T(x)$ be continuous on $\\mathbb {R}$ .", "Suppose that $w(x)f^{(r)}(x)\\rightarrow 0$ as $|x|\\rightarrow \\infty $ .", "Let $P_{n;f,w_{-1/4}}\\in \\mathcal {P}_n$ be the best approximation for $f$ with the weight $w_{-1/4}$ on $\\mathbb {R}$ .", "Suppose that for a certain $\\delta >0$ , $L(f;x)\\ge \\delta , \\quad x\\in \\mathbb {R}.$ Then, for every integer $n\\ge 1$ and $j=0,1,...,\\ell $ , $\\left\\Vert \\left(f^{(j)}-P_{n;f,w_{-1/4}}^{(j)}\\right)wT^{-(2j+1)/4}(x)\\right\\Vert _{L_\\infty (\\mathbb {R})}\\le C_j\\left(\\frac{a_n}{n}\\right)^{r-j}E_{n-r}\\left(w,f^{(r)}\\right),$ where $C_j>0$ , $0\\le j\\le \\ell $ , are independent of $n$ or $f$ , and for any fixed number $M>0$ there exists a constant $N(M,\\ell ,\\delta )>0$ such that $L(P_{n;f,w_{-1/4}};x)\\ge \\frac{\\delta }{2}, \\quad |x|\\le M, \\quad n\\ge N(M,\\ell ,\\delta ).$ From Corollary REF , we have (REF ).", "Hence, we also have $&&\\left|\\left(L(f;x)-L\\left(P_{n;f,w_{-1/4}};x\\right)\\right)w(x)T^{-(2\\ell +1)/4}(x)\\right|\\\\&=&\\left|\\sum _{j=k}^\\ell a_j(x)\\left\\lbrace f^{(j)}(x)-P_{n;f,w_{-1/4}}^{(j)}(x)\\right\\rbrace w(x)T^{-(2\\ell +1)/4}(x)\\right|\\\\&\\le & E_{n-r}\\left(w,f^{(r)}\\right)\\sum _{j=k}^\\ell |a_j(x)|C_j\\left(\\frac{a_n}{n}\\right)^{r-j}\\\\&\\le & C_{k,\\ell }\\left(\\frac{a_n}{n}\\right)^{r-\\ell }E_{n-r}\\left(w,f^{(r)}\\right),$ where we set $C_{k,\\ell }:=\\sum _{j=k}^\\ell \\Vert a_j\\Vert _{L_\\infty (\\mathbb {R})}C_j$ .", "Then we have for $|x|\\le M$ $&&\\left|L(f;x)-L\\left(P_{n;f,w_{-1/4}};x\\right)\\right|\\\\&\\le & C_{k,\\ell }\\left\\Vert w^{-1}(x)T^{(2\\ell +1)/4}(x)\\right\\Vert _{L_\\infty (|x|\\le M)}\\left(\\frac{a_n}{n}\\right)^{r-\\ell }E_{n-r}\\left(w,f^{(r)}\\right).$ Here, for $\\delta >0$ there exists $N(M,\\ell ,\\delta )>0$ such that for $n\\ge N(M,\\ell ,\\delta )$ $C_{k,\\ell }\\left\\Vert w^{-1}(x)T^{(2\\ell +1)/4}(x)\\right\\Vert _{L_\\infty (|x|\\le M)}\\left(\\frac{a_n}{n}\\right)^{r-\\ell }E_{n-r}\\left(w,f^{(r)}\\right)\\le \\frac{\\delta }{2}.$ This follows from $w(x)f^{(r)}(x)\\rightarrow 0$ as $|x|\\rightarrow \\infty $ .", "Therefore we see $\\frac{\\delta }{2}\\le L(f;x)-\\frac{\\delta }{2}\\le L(P_{n;f,w_{-1/4}};x).$ Consequently, we have (REF )." ], [ "Theorems and Preliminaries", "First, we introduce some well-known notations.", "If $f$ is a continuous function on $\\mathbb {R}$ , then we define $\\left\\Vert fw\\right\\Vert _{L_\\infty (\\mathbb {R})}:=\\sup _{t\\in \\mathbb {R}}|f(t)w(t)|,$ and for $1\\le p<\\infty $ we denote $\\Vert fw\\Vert _{L_p(\\mathbb {R})}:=\\left(\\int _{\\mathbb {R}}\\left|f(t)w(t)\\right|^pdt\\right)^{1/p}.$ Let $1\\le p\\le \\infty $ .", "If $\\Vert wf\\Vert _{L_p(\\mathbb {R})}<\\infty $ , then we write $wf\\in L_p(\\mathbb {R})$ , and here if $p=\\infty $ , we suppose that $f\\in C(\\mathbb {R})$ and $\\lim _{|x|\\rightarrow \\infty }|w(x)f(x)|=0$ .", "We denote the rate of approximation of $f$ by $E_{p,n}(w;f):=\\inf _{P\\in \\mathcal {P}_n}\\left\\Vert (f-P)w\\right\\Vert _{L_p(\\mathbb {R})}.$ The Mhaskar-Rakhmanov-Saff numbers $a_x$ is defined as follows: $x=\\frac{2}{\\pi }\\int _0^1\\frac{a_xuQ^{\\prime }(a_xu)}{\\sqrt{1-u^2}}du, \\quad x>0.$ To write our theorems we need some preliminaries.", "We need further assumptions.", "Definition 2.1 Let $w=\\exp (-Q)\\in \\mathcal {F}(C^2+)$ and $0< \\lambda <(r+2)/(r+1)$ .", "Let $r\\ge 1$ be an integer.", "Then we write $w\\in \\mathcal {F}_\\lambda (C^{r+2}+)$ if $Q\\in C^{(r+2)}(\\mathbb {R}\\backslash \\lbrace 0\\rbrace )$ and there exist two constants $C>1$ and $K\\ge 1$ such that for all $|x|\\ge K$ , $\\frac{|Q^{\\prime }(x)|}{Q^{\\lambda }(x)} \\le C \\quad \\mbox{and} \\quad \\left| \\frac{Q^{\\prime \\prime }(x)}{Q^{\\prime }(x)} \\right| \\sim \\left|\\frac{Q^{(k+1)}(x)}{Q^{(k)}(x)} \\right|$ for every $k=2,...,r$ and also $\\left| \\frac{Q^{(r+2)}(x)}{Q^{(r+1)}(x)} \\right| \\le C \\left|\\frac{Q^{(r+1)}(x)}{Q^{(r)}(x)} \\right|.$ In particular, $w\\in \\mathcal {F}_\\lambda (C^{3}+)$ means that $Q\\in C^{(3)}(\\mathbb {R}\\backslash \\lbrace 0\\rbrace )$ and $\\frac{|Q^{\\prime }(x)|}{Q^{\\lambda }(x)} \\le C \\quad \\mbox{and} \\quad \\left| \\frac{Q^{\\prime \\prime \\prime }(x)}{Q^{\\prime \\prime }(x)} \\right| \\le C \\left|\\frac{Q^{\\prime \\prime }(x)}{Q^{\\prime }(x)} \\right|$ hold for $|x| \\ge K$ .", "In addition, let $\\mathcal {F}_\\lambda (C^{2}+):=\\mathcal {F}(C^2+)$ .", "From [5], we know that Example REF (2),(3) satisfy all conditions of Definition REF .", "Under the same condition of Definition REF we obtain an interesting theorem as follows: Theorem 2.2 ([10]) Let $r$ be a positive integer, $0< \\lambda <(r+2)/(r+1)$ and let $w=\\exp (-Q)\\in \\mathcal {F}_\\lambda (C^{r+2}+)$ .", "Then for any $\\mu $ , $\\nu $ , $\\alpha $ , $\\beta \\in \\mathbb {R}$ , we can construct a new weight $w_{\\mu ,\\nu ,\\alpha ,\\beta }\\in \\mathcal {F}_\\lambda (C^{r+1}+)$ such that $T_w^{\\mu }(x)(1+x^2)^\\nu (1+Q(x))^{\\alpha }(1+|Q^{\\prime }(x)|)^\\beta w(x)\\sim w_{\\mu ,\\nu ,\\alpha ,\\beta }(x)$ on $\\mathbb {R}$ , and $a_{n/c}(w)\\le a_n(w_{\\mu .\\nu ,\\alpha ,\\beta })\\le a_{cn}(w), \\quad c\\ge 1,$ $T_{w_{\\mu ,\\nu ,\\alpha ,\\beta }}(x)\\sim T_w(x)$ hold on $\\mathbb {R}$ .", "For a given $\\alpha \\in \\mathbb {R}$ and $w\\in \\mathcal {F}(C^2+)$ , we let $w_{\\alpha }\\in \\mathcal {F}(C^2+)$ satisfy $w_\\alpha (x) \\sim T_{w}^{\\alpha }(x)w(x)$ , and let $P_{n;f,w_\\alpha }\\in \\mathcal {P}_n$ be the best approximation of $f$ with respect to the weight $w_\\alpha $ , that is, $\\left\\Vert (f-P_{n;f,w_\\alpha })w_{\\alpha }\\right\\Vert _{L_\\infty (\\mathbb {R})}=E_{n}(w_\\alpha ,f):=\\inf _{P\\in \\mathcal {P}_n}\\left\\Vert (f-P)w_\\alpha \\right\\Vert _{L_\\infty (\\mathbb {R})}.$ Then we have the main result as follows: Theorem 2.3 Let $r \\ge 0$ be an integer.", "Let $w=\\exp (-Q)\\in \\mathcal {F}_\\lambda (C^{r+2}+)$ and $0< \\lambda <(r+2)/(r+1)$ .", "Suppose that $f\\in C^{(r)}(\\mathbb {R})$ with $\\lim _{|x|\\rightarrow \\infty }T^{1/4}(x)f^{(r)}(x)w(x)=0.$ Then there exists an absolute constant $C_r>0$ which depends only on $r$ such that for $0\\le k\\le r$ and $x\\in \\mathbb {R}$ , $\\left|\\left(f^{(k)}(x)-P_{n;f,w}^{(k)}(x)\\right)w(x)\\right|&\\le & C_r T^{k/2}(x)E_{n-k}\\left(w_{1/4},f^{(k)}\\right)\\\\&\\le & C_r T^{k/2}(x) \\left(\\frac{a_n}{n}\\right)^{r-k}E_{n-r}\\left(w_{1/4},f^{(r)}\\right).$ When $w\\in \\mathcal {F}^*$ , we can replace $w_{1/4}$ with $w$ in the above.", "Applying Theorem REF with $w$ or $w_{-1/4}$ , we have the following corollary.", "Corollary 2.4 (1) Let $w=\\exp (-Q)\\in \\mathcal {F}_\\lambda (C^{r+2}+), 0< \\lambda <(r+2)/(r+1), r\\ge 0$ .", "We suppose that $f\\in C^{(r)}(\\mathbb {R})$ with $\\lim _{|x|\\rightarrow \\infty }T^{1/4}(x)f^{(r)}(x)w(x)=0,$ then for $0\\le k\\le r$ we have $\\left\\Vert \\left(f^{(k)}-P_{n;f,w}^{(k)}\\right)w_{-k/2}\\right\\Vert _{L_\\infty (\\mathbb {R})}&\\le & C_r E_{n-k}\\left(w_{1/4},f^{(k)}\\right) \\\\&\\le & C_r \\left(\\frac{a_n}{n}\\right)^{r-k}E_{n-r}\\left(w_{1/4},f^{(r)}\\right).$    (2) Let $w=\\exp (-Q)\\in \\mathcal {F}_\\lambda (C^{r+3}+), 0< \\lambda <(r+3)/(r+2), r\\ge 0$ .", "We suppose that $f\\in C^{(r)}(\\mathbb {R})$ with $\\lim _{|x|\\rightarrow \\infty }f^{(r)}(x)w(x)=0,$ then for $0\\le k\\le r$ we have $\\left\\Vert \\left(f^{(k)}-P_{n;f,w_{-1/4}}^{(k)}\\right)w_{-(2k+1)/4}\\right\\Vert _{L_\\infty (\\mathbb {R})}&\\le & C_r E_{n-k}\\left(w,f^{(k)}\\right) \\\\&\\le & C_r \\left(\\frac{a_n}{n}\\right)^{r-k}E_{n-r}\\left(w,f^{(r)}\\right).$ When $w\\in \\mathcal {F}^*$ , we can replace $w_\\alpha $ with $w$ in the above.", "Corollary 2.5 Let $r \\ge 0$ be an integer.", "Let $w=\\exp (-Q)\\in \\mathcal {F}_\\lambda (C^{r+3}+)$ , $0< \\lambda <(r+3)/(r+2)$ , and let $w_{(2r+1)/4}f^{(r)}\\in L_\\infty (\\mathbb {R})$ .", "Then, for each $k (0\\le k\\le r)$ and the best approximation polynomial $P_{n;f,w_{k/2}}$ ; $\\left\\Vert \\left(f-P_{n;f,w_{k/2}}\\right)w_{k/2}\\right\\Vert _{L_\\infty (\\mathbb {R})}=E_{n}\\left(w_{k/2},f\\right),$ we have $\\left\\Vert \\left(f^{(k)}-P_{n;f,w_{k/2}}^{(k)}\\right)w\\right\\Vert _{L_\\infty (\\mathbb {R})}&\\le & C_r E_{n-k}\\left(w_{(2k+1)/4},f^{(k)}\\right)\\\\&\\le & C_r \\left(\\frac{a_n}{n}\\right)^{r-k}E_{n-r}\\left(w_{(2k+1)/4},f^{(r)}\\right).$ When $w\\in \\mathcal {F}^*$ , we can replace $w_{1/4}$ with $w$ in the above.", "Proof of Theorems Throughout this section we suppose $w\\in \\mathcal {F}(C^2+)$ .", "We give the proofs of theorems.", "First, we give some lemmas to prove the theorems.", "We construct the orthonormal polynomials $p_n(x)=p_n(w^2,x)$ of degree n for $w^2(x)$ , that is, $\\int _{-\\infty }^{\\infty } p_n(w^2,x)p_m(w^2,x)w^2(x)dx=\\delta _{mn} (\\textrm {Kronecker delta}).$ Let $fw\\in L_2(\\mathbb {R})$ .", "The Fourier-type series of $f$ is defined by $\\tilde{f}(x):=\\sum _{k=0}^{\\infty } a_k(w^2,f)p_k(w^2,x),\\quad a_k(w^2,f):=\\int _{-\\infty }^{\\infty } f(t)p_k(w^2,t)w^2(t)dt.$ We denote the partial sum of $\\tilde{f}(x)$ by $s_n(f,x):=s_n(w^2,f,x):=\\sum _{k=0}^{n-1} a_k(w^2,f)p_k(w^2,x).$ Moreover, we define the de la Vall$\\acute{\\textrm {e}}$ e Poussin means by $v_n(f,x):=\\frac{1}{n}\\sum _{j=n+1}^{2n}s_j(w^2,f,x).$ Proposition 3.1 ([11]) Let $w\\in \\mathcal {F}_\\lambda (C^3+), 0<\\lambda <3/2$ , and let $1\\le p\\le \\infty $ .", "When $T^{1/4}wf\\in L_p(\\mathbb {R})$ , we have $\\left\\Vert v_n(f)w\\right\\Vert _{L_p(\\mathbb {R})}\\le C \\left\\Vert T^{1/4}wf \\right\\Vert _{L_p(\\mathbb {R})},$ and so $\\left\\Vert (f-v_n(f))w \\right\\Vert _{L_p(\\mathbb {R})}\\le C E_{p,n}\\left(T^{1/4}w,f\\right).$ So, equivalently, $\\left\\Vert v_n(f)w\\right\\Vert _{L_p(\\mathbb {R})}\\le C \\left\\Vert w_{1/4}f\\right\\Vert _{L_p(\\mathbb {R})},$ and so $\\left\\Vert (f-v_n(f))w\\right\\Vert _{L_p(\\mathbb {R})}\\le C E_{p,n} \\left(w_{1/4},f\\right).$ When $w\\in \\mathcal {F}^*$ , we can replace $w_{1/4}$ with $w$ .", "Lemma 3.2 (1) ([8]) Let $L>0$ be fixed.", "Then, uniformly for $t>0$ , $a_{Lt}\\sim a_t.$    (2) ([8]) For $x >1$ , we have $|Q^{\\prime }(a_x)| \\sim \\frac{x \\sqrt{T(a_x)}}{a_x} \\quad \\mbox{and} \\quad |Q(a_x)| \\sim \\frac{x}{ \\sqrt{T(a_x)}}.$    (3) ([8]) Let $x\\in (0, \\infty )$ .", "There exists $0<\\varepsilon <1$ such that $T\\left(x\\left[1+\\frac{\\varepsilon }{T(x)}\\right]\\right)\\sim T(x).$    (4) ([9]) If $T(x)$ is unbounded, then for any $\\eta >0$ there exists $C(\\eta )>0$ such that for $t\\ge 1$ , $a_t\\le C(\\eta )t^\\eta .$ To prove the results, we need the following notations.", "We set $\\sigma (t):=\\inf \\left\\lbrace a_u: \\,\\, \\frac{a_u}{u}\\le t \\right\\rbrace , \\quad t>0,$ and $\\Phi _t(x):=\\sqrt{\\left|1-\\frac{|x|}{\\sigma (t)}\\right|}+T^{-1/2}(\\sigma (t)), \\quad x\\in \\mathbb {R}.$ Define for $fw\\in L_p(\\mathbb {R})$ , $0<p\\le \\infty $ , $\\omega _p(f,w,t)&:=&\\sup _{0<h\\le t}\\left\\Vert w(x)\\left\\lbrace f\\left(x+\\frac{h}{2}\\Phi _t(x)\\right)-f\\left(x-\\frac{h}{2}\\Phi _t(x)\\right)\\right\\rbrace \\right\\Vert _{L_p(|x|\\le \\sigma (2t))}\\\\&& \\quad +\\inf _{c\\in \\mathbb {R}}\\left\\Vert w(x)(f-c)(x)\\right\\Vert _{L_p(|x|\\ge \\sigma (4t))}$ (see [2], [3]).", "Proposition 3.3 (cf.", "[3], [2]) Let $w\\in \\mathcal {F}(C^2+)$ .", "Let $0<p\\le \\infty $ .", "Then for $f: \\mathbb {R}\\rightarrow \\mathbb {R}$ which $fw\\in L_p(\\mathbb {R})$ (and for $p=\\infty $ , we require $f$ to be continuous, and $fw$ to vanish at $\\pm \\infty $ ), we have for $n\\ge C_3$ , $E_{p,n}\\left(f,w\\right)\\le C_1\\omega _{p}\\left(f,w,C_2\\frac{a_n}{n}\\right),$ where $C_j$ , $j=1,2,3$ , do not depend on $f$ and $n$ .", "Damelin and Lubinsky [3] or Damelin [2] have treated a certain class $\\mathcal {E}_1$ of weights containing the conditions (a)-(d) in Definition REF and $\\frac{yQ^{\\prime }(y)}{xQ^{\\prime }(x)}\\le C_1 \\left(\\frac{Q(y)}{Q(x)}\\right)^{C_2},\\quad y\\ge x\\ge C_3,$ where $C_i$ , $i=1,2,3>0$ are some constants, and they obtain this Proposition for $w\\in \\mathcal {E}_1$ .", "Therefore, we may show $\\mathcal {F}(C^2+)\\subset \\mathcal {E}_1$ .", "In fact, from Definition REF (d) and (e), we have for $y\\ge x>0$ , $\\frac{Q^{\\prime }(y)}{Q^{\\prime }(x)}=\\exp \\left(\\int _x^y\\frac{Q^{\\prime \\prime }(t)}{Q^{\\prime }(t)}dt\\right)\\le \\exp \\left(C_3\\int _x^y\\frac{Q^{\\prime }(t)}{Q(t)}dt\\right)=\\left(\\frac{Q(y)}{Q(x)}\\right)^{C_3},$ and $\\frac{y}{x}=\\exp \\left(\\int _x^y\\frac{1}{t}dt\\right)\\le \\exp \\left(\\frac{1}{\\Lambda }\\int _x^y\\frac{Q^{\\prime }(t)}{Q(t)}dt\\right)=\\left(\\frac{Q(y)}{Q(x)}\\right)^{\\frac{1}{\\Lambda }}.$ Therefore, we obtain (REF ) with $C_2=C_3+\\frac{1}{\\Lambda }$ , that is, we see $\\mathcal {F}(C^2+)\\subset \\mathcal {E}_1$ .", "Theorem 3.4 Let $w\\in \\mathcal {F}(C^2+)$ .", "(1) If $f$ is a function having bounded variation on any compact interval and if $\\int _{-\\infty }^{\\infty } w(x)|df(x)|<\\infty ,$ then there exists a constant $C>0$ such that for every $t>0$ , $\\omega _1(f,w,t)\\le C t\\int _{-\\infty }^{\\infty } w(x)|df(x)|,$ and so $E_{1,n}(f,w)\\le C\\frac{a_n}{n}\\int _{-\\infty }^{\\infty } w(x)|df(x)|.$    (2) Let us suppose that $f$ is continuous and $\\lim _{|x|\\rightarrow \\infty }|(\\sqrt{T}wf)(x)|=0$ , then we have $\\lim _{t\\rightarrow 0}\\omega _{\\infty }(f,w,t)=0.$ To prove this theorem we need the following lemma.", "Lemma 3.5 ([9]) (1) For $t>0$ there exists $a_u$ such that $t=\\frac{a_u}{u} \\quad \\textrm {and} \\quad \\sigma (t)=a_u.$    (2) If $t=a_u/u$ , $u>0$ large enough and $|x-y|\\le t\\Phi _t(x),$ then there exist $C_1,C_2>0$ such that $C_1w(x)\\le w(y)\\le C_2 w(x).$ (1) Let $g(x):=f(x)-f(0)$ .", "For $t>0$ small enough let $0<h\\le t$ and $|x|\\le \\sigma (2t)<\\sigma (t)$ .", "Hence we may consider $\\Phi _t(x)\\le 2$ .", "Then by Lemma REF , $&&\\int _{-\\infty }^{\\infty } w(x)\\left|g\\left(x+\\frac{h}{2}\\Phi _t(x)\\right)-g\\left(x-\\frac{h}{2}\\Phi _t(x)\\right)\\right|dx\\\\&&=\\int _{-\\infty }^{\\infty } w(x)\\left|\\int _{x-\\frac{h}{2}\\Phi _t(x)}^{x+\\frac{h}{2}\\Phi _t(x)}df(v)\\right| dx\\le C\\int _{-\\infty }^{\\infty } \\left|\\int _{x-\\frac{h}{2}\\Phi _t(x)}^{x+\\frac{h}{2}\\Phi _t(x)}w(v)df(v)\\right| dx\\\\&&\\le \\int _{-\\infty }^{\\infty } \\int _{x-h}^{x+h} w(v)|df(v)|dx\\le \\int _{-\\infty }^{\\infty } w(v)\\int _{v-h\\le x\\le v+h} dx|df(v)|\\\\&&\\le 2h\\int _{-\\infty }^{\\infty } w(v)|df(v)|.$ Hence we have $\\int _{-\\infty }^{\\infty } w(x)\\left|g\\left(x+\\frac{h}{2}\\Phi _t(x)\\right)-g\\left(x-\\frac{h}{2}\\Phi _t(x)\\right)\\right| dx\\le 2t \\int _{-\\infty }^{\\infty } w(x)|df(x)|.$ Moreover, we see $\\inf _{c\\in \\mathbb {R}}\\left\\Vert w(x)(f-c)(x)\\right\\Vert _{L_1(|x|\\ge \\sigma (4t))}\\le \\frac{1}{Q^{\\prime }(\\sigma (4t))}\\left\\Vert Q^{\\prime }(x)w(x)g(x)\\right\\Vert _{L_1(|x|\\ge \\sigma (4t))}.$ Here we see $\\frac{\\sqrt{T(\\sigma (t))}}{Q^{\\prime }(\\sigma (t))}\\sim t.$ In fact, from Lemma REF (2), for $t=\\frac{a_u}{u}$ $Q^{\\prime }(\\sigma (t))= Q^{\\prime }(a_u)\\sim \\frac{u\\sqrt{T(a_u)}}{a_u}\\sim \\frac{\\sqrt{T(\\sigma (t))}}{t}.$ On the other hand, we have $\\int _{0}^{\\infty } Q^{\\prime }(x)w(x)|g(x)|dx&=& \\int _{0}^{\\infty } Q^{\\prime }(x)w(x)\\left|\\int _0^xdg(u)\\right|dx\\\\&\\le & \\int _{0}^{\\infty } Q^{\\prime }(x)w(x)\\int _0^x|df(u)|dx\\\\&=& -w(x)\\int _0^x|df(u)|\\bigg |_0^{\\infty }+\\int _0^{\\infty } w(x)|df(x)|\\\\&=&\\int _0^{\\infty } w(x)|df(x)|$ because $\\lim _{x\\rightarrow \\infty } w(x) =0$ from (c) of Definition REF .", "Similarly we have $\\int _{-\\infty }^0 \\left| Q^{\\prime }(x)w(x)g(x)\\right|dx\\le \\int _{-\\infty }^0 w(x)|df(x)|.$ Hence we have $\\Vert Q^{\\prime }wg\\Vert _{L_1(\\mathbb {R})}\\le \\int _{-\\infty }^{\\infty } w(u)|df(u)|.$ Therefore, using (REF ), (REF ) and (REF ), we have $\\inf _{c\\in \\mathbb {R}}\\left\\Vert w(x)(f-c)(x)\\right\\Vert _{L_1(|x|\\ge \\sigma (4t))}= O(t) \\int _{-\\infty }^{\\infty } w(x)|df(x)|.$ Consequently, by (REF ) and (REF ) we have $\\omega _1(f,w,t)\\le C t\\int _{-\\infty }^{\\infty } w(x)|df(x)|.$ Hence, setting $t=C_2\\frac{a_n}{n}$ , if we use Proposition REF , then $E_{1,n}(f,w)\\le C\\frac{a_n}{n}\\int _{-\\infty }^{\\infty } w(x)|df(x)|.$ (2) Given $\\varepsilon >0$ , and let us take $L=L(\\varepsilon )>0$ large enough as $\\sup _{|x|\\ge L}|w(x)f(x)| \\le \\frac{1}{\\sqrt{T(L)}}\\sup _{|x|\\ge L}|\\sqrt{T(x)}w(x)f(x)|<\\varepsilon \\quad (\\textrm {by our assumption}).$ Then we have $\\inf _{c\\in \\mathbb {R}}\\sup _{|x|\\ge L} \\left|w(x)(f-c)(x)\\right|\\le \\frac{1}{\\sqrt{T(L)}}\\sup _{|x|\\ge L}\\left|\\sqrt{T(x)}w(x)f(x)\\right|<\\varepsilon .$ Now, there exists $\\varepsilon >0$ small enough such that $\\frac{h}{2}\\Phi _t(x)\\le \\varepsilon \\frac{1}{T(x)}, \\quad |x|\\le \\sigma (2t),$ because if we put $t=a_u/u$ , then we see $\\sigma (t)=a_u$ and $|x|\\le \\sigma (2t)<a_u$ .", "Hence, noting [8], that is, for some $\\varepsilon >0$ , and for large enough $t$ , $T(a_t) \\le C t^{2-\\varepsilon },$ and if $w$ is the Erdös-type weight, then from Lemma REF (4), we have $t\\Phi _t(x)\\le \\frac{a_u}{u}\\le \\varepsilon \\frac{1}{T(a_u)}\\le \\varepsilon \\frac{1}{T(x)}.$ If $w\\in \\mathcal {F}^*$ , we also have (REF ), because for some $\\delta >0$ and $u>0$ large enough, $t\\Phi _t(x)\\le \\frac{a_u}{u}\\le u^{-\\delta } \\le \\varepsilon \\frac{1}{T(x)}.$ Therefore, using Lemma REF (3), Lemma REF and the assumption $\\lim _{|x|\\rightarrow \\infty }\\sqrt{T\\left(x\\right)}w\\left(x\\right)f\\left(x\\right)=0,$ for $2L\\le |x|\\le \\sigma (2t)$ , $h>0$ , $&&\\left|w(x)\\left\\lbrace f\\left(x+\\frac{h}{2}\\Phi _t(x)\\right)-f\\left(x-\\frac{h}{2}\\Phi _t(x)\\right)\\right\\rbrace \\right|\\\\&\\le & C\\Bigg [\\frac{1}{\\sqrt{T(x)}}\\left|\\sqrt{T\\left(x+\\frac{h}{2}\\Phi _t(x)\\right)}w\\left(x+\\frac{h}{2}\\Phi _t(x)\\right)f\\left(x+\\frac{h}{2}\\Phi _t(x)\\right)\\right|\\\\&& \\qquad +\\frac{1}{\\sqrt{T(x)}}\\left|\\sqrt{T\\left(x-\\frac{h}{2}\\Phi _t(x)\\right)}w\\left(x-\\frac{h}{2}\\Phi _t(x)\\right)f\\left(x-\\frac{h}{2}\\Phi _t(x)\\right)\\right|\\Bigg ]\\\\&\\le & 2\\varepsilon .$ On the other hand, $\\lim _{t\\rightarrow 0}\\sup _{0<h\\le t}\\left\\Vert w(x) \\left\\lbrace f\\left(x+\\frac{h}{2}\\Phi _t(x)\\right)-f\\left(x-\\frac{h}{2}\\Phi _t(x)\\right)\\right\\rbrace \\right\\Vert _{L_\\infty (|x|\\le 2L)}=0.$ Therefore, we have the result.", "Lemma 3.6 (cf.", "[4]) Let $g$ be a real valued function on $\\mathbb {R}$ satisfying $\\Vert gw\\Vert _{L_\\infty (\\mathbb {R})}<\\infty $ and $\\int _{-\\infty }^{\\infty } gPw^2dt=0 \\quad P\\in \\mathcal {P}_n.$ Then we have $\\left\\Vert w(x)\\int _0^x g(t)dt \\right\\Vert _{L_\\infty (\\mathbb {R})}\\le C \\frac{a_n}{n}\\Vert gw\\Vert _{L_\\infty (\\mathbb {R})}.$ Especially, if $w\\in \\mathcal {F}_\\lambda (C^3+), 0<\\lambda <3/2$ , then we have $\\left\\Vert w(x)\\int _0^x \\left(f^{\\prime }(t)-v_n(f^{\\prime })(t)\\right)dt \\right\\Vert _{L_\\infty (\\mathbb {R})}\\le C \\frac{a_n}{n}E_{n} \\left(w_{1/4},f^{\\prime }\\right).$ When $w\\in \\mathcal {F}^*$ , we also have (REF ) replacing $w_{1/4}$ with $w$ .", "We let $\\phi _x(t)=\\left\\lbrace \\begin{array}{lr}w^{-2}(t),& 0\\le t\\le x; \\\\0,& otherwise,\\end{array}\\right.$ then we have for arbitrary $P_n\\in \\mathcal {P}_n$ , $\\left|\\int _0^x g(t)dt\\right|=\\left|\\int _{-\\infty }^{\\infty } g(t)\\phi _x(t)w^2(t)dt\\right|=\\left|\\int _{-\\infty }^{\\infty } g(t)(\\phi _x(t)-P_n(t))w^2(t)dt\\right|.$ Therefore, we have $\\left|\\int _0^x g(t)dt\\right|&\\le & \\Vert gw\\Vert _{L_\\infty (\\mathbb {R})}\\inf _{P_n\\in \\mathcal {P}_n}\\int _{-\\infty }^{\\infty } \\left|\\phi _x(t)-P_n(t)\\right|w(t)dt \\\\&=&\\Vert gw\\Vert _{L_\\infty (\\mathbb {R})}E_{1,n}(w:\\phi _x).$ Here, from Theorem REF we see that $E_{1,n}(w:\\phi _x)&\\le & C\\frac{a_n}{n}\\int _{-\\infty }^{\\infty } w(t)|d\\phi _x(t)|\\le C\\frac{a_n}{n}\\int _0^x w(t)|Q^{\\prime }(t)|w^{-2}(t)dt\\\\&=& C\\frac{a_n}{n}\\int _0^x Q^{\\prime }(t)w^{-1}(t)dt\\le C\\frac{a_n}{n}w^{-1}(x).$ So, we have $\\left|w(x)\\int _0^x g(t)dt\\right|\\le \\Vert gw\\Vert _{L_\\infty (\\mathbb {R})}w(x)E_{1,n}\\left(w:\\phi _x\\right)\\le C\\frac{a_n}{n}\\Vert gw\\Vert _{L_\\infty (\\mathbb {R})}.$ Therefore, we have (REF ).", "Next we show (REF ).", "Since $v_n(f^{\\prime })(t)=\\frac{1}{n}\\sum _{j=n+1}^{2n}s_j(f^{\\prime },t),$ and for any $P\\in \\mathcal {P}_n$ , $j\\ge n+1$ , $\\int _{-\\infty }^{\\infty } \\left(f^{\\prime }(t)-s_j(f^{\\prime };t)\\right)P(t)w^2(t)dt=0,$ we have $\\int _{-\\infty }^{\\infty } \\left(f^{\\prime }(t)-v_n(f^{\\prime })(t)\\right)P(t)w^2(t)dt=0.$ Using (REF ) and (REF ), we have (REF ).", "Lemma 3.7 Let $w=\\exp (-Q)\\in \\mathcal {F}_\\lambda (C^3+)$ , $0<\\lambda <3/2$ .", "Let $\\left\\Vert w_{1/4}f^{\\prime }\\right\\Vert _{L_\\infty (\\mathbb {R})}<\\infty $ , and let $q_{n-1}\\in \\mathcal {P}_n$ be the best approximation of $f^{\\prime }$ with respect to the weight $w$ , that is, $\\left\\Vert (f^{\\prime }-q_{n-1})w \\right\\Vert _{L_\\infty (\\mathbb {R})}=E_{n-1}(w,f^{\\prime }).$ Now we set $F(x):=f(x)-\\int _0^xq_{n-1}(t)dt,$ then there exists $S_{2n}\\in \\mathcal {P}_{2n}$ such that $\\left\\Vert w \\left(F-S_{2n}\\right) \\right\\Vert _{L_\\infty (\\mathbb {R})}\\le C \\frac{a_n}{n}E_n \\left(w_{1/4},f^{\\prime }\\right),$ and $\\left\\Vert wS_{2n}^{\\prime } \\right\\Vert _{L_\\infty (\\mathbb {R})}\\le C E_{n-1}\\left(w_{1/4},f^{\\prime }\\right).$ When $w\\in \\mathcal {F}^*$ , we also have same results replacing $w_{1/4}$ with $w$ .", "Let $S_{2n}(x)=f(0)+\\int _0^x v_n \\left(f^{\\prime }-q_{n-1}\\right)(t)dt,$ then by Lemma REF (REF ), $&&\\left\\Vert w\\left(F-S_{2n}\\right)\\right\\Vert _{L_\\infty (\\mathbb {R})} \\\\&=&\\left\\Vert w\\left(f-\\int _0^xq_{n-1}(t)dt -f(0)-\\int _0^x v_n\\left(f^{\\prime }-q_{n-1}\\right)(t)dt\\right)\\right\\Vert _{L_\\infty (\\mathbb {R})}\\\\&=&\\left\\Vert w\\left(\\int _0^x \\left[f^{\\prime }(t)-v_n(f^{\\prime })(t)\\right]dt\\right)\\right\\Vert _{L_\\infty (\\mathbb {R})}\\le C\\frac{a_n}{n}E_n\\left(w_{1/4},f^{\\prime }\\right).$ Now by Proposition REF (REF ), $\\left\\Vert wS_{2n}^{\\prime }\\right\\Vert _{L_\\infty (\\mathbb {R})}&=&\\left\\Vert w\\left(v_n(f^{\\prime }-q_{n-1})\\right)\\right\\Vert _{L_\\infty (\\mathbb {R})}\\\\&\\le &\\left\\Vert \\left(f^{\\prime }-v_n\\left(f^{\\prime }\\right)\\right)w\\right\\Vert _{L_\\infty (\\mathbb {R})}+\\left\\Vert \\left(f^{\\prime }-q_{n-1}\\right)w\\right\\Vert _{L_\\infty (\\mathbb {R})}\\\\&\\le & E_{n}\\left(w_{1/4},f^{\\prime }\\right)+E_{n-1}\\left(w,f^{\\prime }\\right)\\le E_{n-1}\\left(w_{1/4},f^{\\prime }\\right).$ To prove Theorem REF we need the following theorems.", "Theorem 3.8 ([9]) Let $w\\in \\mathcal {F}(C^2+)$ , and let $r\\ge 0$ be an integer.", "Let $1\\le p\\le \\infty $ , and let $wf^{(r)}\\in L_p(\\mathbb {R})$ .", "Then we have $E_{p,n}(f,w)\\le C \\left(\\frac{a_n}{n}\\right)^k \\left\\Vert f^{(k)}w\\right\\Vert _{L_p(\\mathbb {R})}, \\quad k=1,2,...,r,$ and equivalently, $E_{p,n}(f,w)\\le C \\left(\\frac{a_n}{n}\\right)^kE_{p,n-k}\\left(f^{(k)},w\\right).$ Theorem 3.9 ([10]) Let $r\\ge 1$ be an integer and $w\\in \\mathcal {F}_\\lambda (C^{r+2}+)$ , $0< \\lambda <(r+2)/(r+1)$ , and let $1\\le p\\le \\infty $ .", "Then there exists a constant $C>0$ such that for any $1\\le k\\le r$ , any integer $n\\ge 1$ and any polynomial $P\\in \\mathcal {P}_n$ , $\\left\\Vert P^{(k)}w\\right\\Vert _{L_p(\\mathbb {R})}\\le C\\left(\\frac{n}{a_n}\\right)^k\\left\\Vert T^{k/2}Pw\\right\\Vert _{L_p(\\mathbb {R})}.$ We prove the theorem only in case of unbounded $T(x)$ , in the case of Freud case $\\mathcal {F}^*$ we can prove it similarly.", "We show that for $k=0,1,...,r$ , $\\left|\\left(f^{(k)}(x)-P_{n;f,w}^{(k)}\\right)w(x)\\right|\\le CT^{k/2}(x)E_{n-k}\\left(w_{1/4},f^{(k)}\\right).$ If $r=0$ , then (REF ) is trivial.", "For some $r\\ge 0$ we suppose that (REF ) holds, and let $f\\in C^{(r+1)}(\\mathbb {R})$ .", "Then $f^{\\prime }\\in C^{(r)}(\\mathbb {R})$ .", "Let $q_{n-1}\\in \\mathcal {P}_{n-1}$ be the polynomial of best approximation of $f^{\\prime }$ with respect to the weight $w$ .", "Then, from our assumption we have for $0\\le k\\le r$ , $\\left|\\left(f^{(k+1)}(x)-q_{n-1}^{(k)}(x)\\right)w(x)\\right|\\le C T^{k/2}(x)E_{n-k}\\left(w_{1/4},f^{(k+1)}\\right),$ that is, for $1\\le k\\le r+1$ $\\left|\\left(f^{(k)}(x)-q_{n-1}^{(k-1)}(x)\\right)w(x)\\right|\\le C T^{\\frac{k-1}{2}}(x)E_{n-k+1}\\left(w_{1/4},f^{(k)}\\right).$ Let $F(x):=f(x)-\\int _0^x q_{n-1}(t)dt=f(x)-Q_n(x),$ then $|F^{\\prime }(x)w(x)|\\le C E_{n-1}\\left(w,f^{\\prime }\\right).$ As (REF ) we set $S_{2n}=\\int _0^x(v_n(f^{\\prime })(t)-q_{n-1}(t))dt+f(0)$ , then from Lemma REF $\\left\\Vert \\left(F-S_{2n}\\right)w\\right\\Vert _{L_\\infty (\\mathbb {R})}\\le C \\frac{a_n}{n}E_n\\left(w_{1/4},f^{\\prime }\\right),$ and $\\left\\Vert S_{2n}^{\\prime }w\\right\\Vert _{L_\\infty (\\mathbb {R})}\\le C E_{n-1}\\left(w_{1/4},f^{\\prime }\\right).$ Here we apply Theorem REF with the weight $w_{-(k-1)/2}$ .", "In fact, by Theorem REF we have $w_{-(k-1)/2}\\in \\mathcal {F}_\\lambda (C^{r+2}+)$ .", "Then, noting $a_{2n}\\sim a_n$ from Lemma REF (1), we see $|S_{2n}^{(k)}(x) w_{-(k-1)/2}(x))|&\\le & C \\left(\\frac{n}{a_n}\\right)^{k-1}\\Vert S_{2n}^{\\prime }w\\Vert _{L_\\infty (\\mathbb {R})}\\\\&\\le & C \\left(\\frac{n}{a_n}\\right)^{k-1}E_{n-1}\\left(w_{1/4},f^{\\prime }\\right),$ that is, $\\left|S_{2n}^{(k)}(x) w(x)\\right|\\le C \\left(\\frac{n\\sqrt{T(x)}}{a_n}\\right)^{k-1}E_{n-1}\\left(w_{1/4},f^{\\prime }\\right),\\quad 1\\le k\\le r+1.$ Let $R_{n}\\in \\mathcal {P}_{n}$ denote the polynomial of best approximation of $F$ with $w$ .", "By Theorem REF with $w_{-\\frac{k}{2}}$ again, for $0\\le k\\le r+1$ we have $ \\nonumber \\left|(R_{n}^{(k)}-S_{2n}^{(k)}(x)) w_{-\\frac{k}{2}}(x)\\right|&\\le & C \\left(\\frac{n}{a_n}\\right)^k\\Vert (R_{n}-S_{2n})w_{-\\frac{k}{2}}(x)T^{k/2}(x)\\Vert _{L_\\infty (\\mathbb {R})}\\\\&\\le & C \\left(\\frac{n}{a_n}\\right)^k\\Vert (R_{n}-S_{2n})w\\Vert _{L_\\infty (\\mathbb {R})}$ and by (REF ) $ \\nonumber \\Vert (R_{n}-S_{2n})w\\Vert _{L_\\infty (\\mathbb {R})}&\\le & C \\left[\\Vert (F-R_{n})w\\Vert _{L_\\infty (\\mathbb {R})}+\\Vert (F-S_{2n})w\\Vert _{L_\\infty (\\mathbb {R})}\\right]\\\\\\nonumber &\\le & C \\left[E_{n}(w,F)+\\frac{a_n}{n}E_{n}\\left(w_{1/4},f^{\\prime }\\right) \\right] \\\\ \\nonumber &\\le & C \\left[\\frac{a_n}{n}E_{n-1}(w,f^{\\prime })+\\frac{a_n}{n}E_{n-1}(w_{1/4},f^{\\prime })\\right]\\\\&\\le & C \\frac{a_n}{n}E_{n-1}\\left(w_{1/4},f^{\\prime }\\right).$ Hence, from (REF ) and (REF ) we have for $0\\le k\\le r+1$ $\\nonumber |(R_{n}^{(k)}-S_{2n}^{(k)}(x)) w(x)|&\\le & C\\left|T^{k/2}(x)\\right|\\left|(R_{n}^{(k)}-S_{2n}^{(k)}(x)) w_{-\\frac{k}{2}}(x)\\right|\\\\&\\le & C \\left(\\frac{n\\sqrt{T(x)}}{a_n}\\right)^k\\frac{a_n}{n}E_{n-1}\\left(w_{1/4},f^{\\prime }\\right).$ Therefore by (REF ), (REF ) and Theorem REF , $ \\nonumber |R_{n}^{(k)}(x) w(x))|&\\le & C T^{k/2}(x)\\left(\\frac{n}{a_n}\\right)^{k-1}E_{n-1}\\left(w_{1/4},f^{\\prime }\\right) \\\\&\\le & C T^{k/2}(x)E_{n-k}\\left(w_{1/4},f^{(k)}\\right).$ Since $E_{n}(F,w)=E_{n}(w,f)$ and $E_{n}\\left(F,w\\right)=\\left\\Vert w\\left(F-R_{n}\\right)\\right\\Vert _{L_\\infty (\\mathbb {R})}=\\left\\Vert w\\left(f-Q_n-R_{n}\\right)\\right\\Vert _{L_\\infty (\\mathbb {R})}$ (see (REF )), we know that $P_{n;f,w}:=Q_n+R_n$ is the polynomial of best approximation of $f$ with $w$ .", "Now, from (REF ), (REF ) and (REF ) we have for $1\\le k\\le r+1$ , $\\left|\\left(f^{(k)}(x)-P_{n;f.w}^{(k)}(x)\\right)w(x)\\right|&=&\\left|\\left(f^{(k)}(x)-Q_n^{(k)}(x)-R_{n}^{(k)}(x)\\right)w(x)\\right|\\\\&\\le & \\left|(f^{(k)}(x)-q_{n-1}^{(k-1)}(x))w(x)\\right|+\\left|R_{n}^{(k)}(x)w(x)\\right|\\\\&\\le & C T^{k/2}(x)E_{n-k}\\left(w_{1/4},f^{(k)}\\right).$ For $k=0$ it is trivial.", "Consequently, we have (REF ) for all $r\\ge 0$ .", "Moreover, using Theorem REF , we conclude Theorem REF .", "It follows from Theroem REF .", "Applying Theorem REF with $w_{k/2}$ , we have for $0\\le j\\le r$ $\\left\\Vert (f^{(j)}-P_{n;f,w_{k/2}}^{(j)})w\\right\\Vert _{L_\\infty (\\mathbb {R})}\\le C E_{n-k}\\left(w_{(2k+1)/4},f^{(j)}\\right).$ Especially, when $j=k$ , we obtain $\\left\\Vert \\left(f^{(k)}-P_{n;f,w_{k/2}}^{(k)}\\right)w\\right\\Vert _{L_\\infty (\\mathbb {R})}\\le CE_{p,n-k}\\left(w_{(2k+1)/4},f^{(k)}\\right).$ Theorems in $L_p(\\mathbb {R})$ $(1 \\le p \\le \\infty )$ In this section we will give an analogy of Theorem REF in $L_p(\\mathbb {R})$ -space ($1 \\le p \\le \\infty $ ) and we will prove it using the same method as the proof of Theorem REF .", "Let $1 \\le p \\le \\infty $ .", "Let $w=\\exp (-Q)\\in \\mathcal {F}_\\lambda (C^3+)$ , $0<\\lambda <3/2$ , and let $\\beta >1$ be fixed.", "Then we set $w^{\\sharp }$ and $w^{\\flat }$ as follows; $&& \\frac{w(x)}{\\left\\lbrace (1+|Q^{\\prime }(x)|)(1+|x|)^{\\beta }\\right\\rbrace ^{1/p}}\\sim w^{\\sharp }(x)\\in \\mathcal {F}(C^2+);\\\\&&w(x)\\left\\lbrace (1+|Q^{\\prime }(x)|)(1+|x|)^{\\beta }\\right\\rbrace ^{1/p} \\sim w^{\\flat }(x)\\in \\mathcal {F}(C^2+)$ (see Theorem REF ).", "Theorem 4.1 Let $r \\ge 0$ be an integer.", "Let $w=\\exp (-Q)\\in \\mathcal {F}_\\lambda (C^{r+2}+)$ , $0< \\lambda <(r+2)/(r+1)$ , and let $\\beta >1$ be fixed.", "Suppose that $T^{1/4}f^{(r)}w\\in L_p(\\mathbb {R})$ .", "Let $P_{p,n;f,w}\\in \\mathcal {P}_n$ be the best approximation of $f$ with respect to the weight $w$ in $L_p(\\mathbb {R})$ -space, that is, $E_{p,n}\\left(w,f\\right):=\\inf _{P\\in \\mathcal {P}_n}\\left\\Vert \\left(f-P\\right)w\\right\\Vert _{L_p(\\mathbb {R})}=\\left\\Vert \\left(f-P_{p,n;f,w}\\right)w\\right\\Vert _{L_p(\\mathbb {R})}.$ Then there exists an absolute constant $C_r>0$ which depends only on $r$ such that for $0\\le k\\le r$ and $x\\in \\mathbb {R}$ , $\\left\\Vert \\left(f^{(k)}-P_{p,n;f,w}^{(k)}\\right)w^{\\sharp }_{-k/2}\\right\\Vert _{L_p(\\mathbb {R})}&\\le & C_r E_{p,n-k}\\left(w_{1/4},f^{(k)}\\right)\\\\&\\le & C_r \\left(\\frac{a_n}{n}\\right)^{r-k}E_{p,n-r}\\left(w_{1/4},f^{(r)}\\right).$ When $w\\in \\mathcal {F}^*$ , we can replace $w_{1/4}$ and $w^{\\sharp }_{-k/2}$ with $w$ and $w^{\\sharp }$ , respectively in the above.", "If we apply Theorem REF with $w_{-1/4}$ , then we have the following.", "Corollary 4.2 Let $r \\ge 0$ be an integer.", "Let $w=\\exp (-Q)\\in \\mathcal {F}_\\lambda (C^{r+3}+)$ , $0< \\lambda <(r+3)/(r+2)$ , and let $\\beta >0$ be fixed.", "Suppose that $wf^{(r)}\\in L_p(\\mathbb {R})$ .", "Then for $0\\le k\\le r$ we have $\\left\\Vert \\left(f^{(k)}-P_{p,n;f,w_{-1/4}}^{(k)}\\right)w^{\\sharp }_{-(2k+1)/4}\\right\\Vert _{L_p(\\mathbb {R})}&\\le & C_r E_{p,n-k}\\left(w,f^{(k)}\\right)\\\\&\\le & C_r \\left(\\frac{a_n}{n}\\right)^{r-k}E_{p,n-r}\\left(w,f^{(r)}\\right).$ When $w\\in \\mathcal {F}^*$ , we can omit $T^{-(2k+1)/4}$ in the above.", "Corollary 4.3 Let $r \\ge 0$ be an integer.", "Let $w=\\exp (-Q)\\in \\mathcal {F}_\\lambda (C^{r+3}+)$ , $0< \\lambda <(r+3)/(r+2)$ , and let $\\beta >0$ be fixed.", "(1) Let $w_{(2r+1)/4}f^{(r)}\\in L_p(\\mathbb {R})$ .", "Then, for each $k$ $(0\\le k\\le r)$ and the best approximation polynomial $P_{p,n;f,w_{k/2}}$ , we have $\\left\\Vert \\left(f^{(k)}-P_{p,n;f,w_{k/2}}^{(k)}\\right)w^{\\sharp }\\right\\Vert _{L_p(\\mathbb {R})}&\\le & C_r E_{p,n-k}\\left(w_{(2k+1)/4},f^{(k)}\\right)\\\\&\\le & C_r \\left(\\frac{a_n}{n}\\right)^{r-k}E_{p,n-r}\\left(w_{(2k+1)/4},f^{(r)}\\right).$    (2) Let $w^{\\flat }_{(2r+1)/4}f^{(r)}\\in L_p(\\mathbb {R})$ .", "Then, for each $k$ $(0\\le k\\le r)$ and the best approximation polynomial $P_{p,n;f,w^{\\flat }_{k/2}}$ , we have $\\left\\Vert \\left(f^{(k)}-P_{p,n;f,w^{\\flat }_{k/2}}^{(k)}\\right)w\\right\\Vert _{L_p(\\mathbb {R})}&\\le & C_r E_{p,n-k}\\left(w^{\\flat }_{(2k+1)/4},f^{(k)}\\right)\\\\&\\le & C_r \\left(\\frac{a_n}{n}\\right)^{r-k}E_{p,n-r}\\left(w^{\\flat }_{(2k+1)/4},f^{(r)}\\right).$ When $w\\in \\mathcal {F}^*$ , we can replace $w_{(2k+1)/4}$ and $w^{\\flat }_{(2k+1)/4}$ with $w$ and $w^{\\flat }$ , respectively in the above.", "Especially when $p=\\infty $ , we can refer to $w^{\\sharp }$ or $w^{\\flat }$ as $w$ .", "In this case, we can note that Corollary REF and Corollary REF imply Corollary REF , and Corollary REF , respectively.", "To prove Theorem REF we need to prepare some notations and lemmas.", "Lemma 4.4 ([6]) Let $w\\in \\mathcal {F}(C^2+)$ and let $1\\le p\\le \\infty $ .", "If $g:\\mathbb {R}\\rightarrow \\mathbb {R}$ is absolutely continuous, $g(0)=0$ , and $wg^{\\prime }\\in L_p(\\mathbb {R})$ , then $\\left\\Vert Q^{\\prime }wg\\right\\Vert _{L_p(\\mathbb {R})}\\le C\\left\\Vert wg^{\\prime }\\right\\Vert _{L_p(\\mathbb {R})}.$ Lemma 4.5 (cf.", "[9]) Let $w\\in \\mathcal {F}(C^2+)$ and let $1\\le p\\le \\infty $ .", "If $wf^{\\prime }\\in L_p(\\mathbb {R})$ , then $E_{p,n}(w,f)\\le C\\omega _p\\left(f,w,\\frac{a_n}{n}\\right)\\le C \\frac{a_n}{n}\\left\\Vert wf^{\\prime }\\right\\Vert _{L_p(\\mathbb {R})}.$ The first inequality follows from Proposition REF .", "We show the second inequality.", "By [9] we have $&&\\left\\Vert w(x)\\left\\lbrace f\\left(x+\\frac{h}{2}\\Phi _t(x)\\right)-f\\left(x-\\frac{h}{2}\\Phi _t(x)\\right)\\right\\rbrace \\right\\Vert ^p_{L_p(|x|\\le \\sigma (2t))} \\\\&=&h^p\\int _{\\mathbb {R}}|w(x)\\Phi _t(x)f^{\\prime }(x_t)|^pdx\\le Ch^p\\int _{\\mathbb {R}}|w(x)f^{\\prime }(x)|^pdx.$ Hence we see $\\sup _{0<h\\le t}\\left\\Vert w(x)\\left\\lbrace f\\left(x+\\frac{h}{2}\\Phi _t(x)\\right)-f\\left(x-\\frac{h}{2}\\Phi _t(x)\\right)\\right\\rbrace \\right\\Vert _{L_p(|x|\\le \\sigma (2t))}\\le Ct\\left\\Vert wf^{\\prime }\\right\\Vert _{L_p(\\mathbb {R})}.$ Now, we estimate $\\left\\Vert w(x)(f-c)(x)\\right\\Vert _{L_p(|x|\\ge \\sigma (4t))}.$ Let $g(x):=f(x)-f(0)$ .", "$\\inf _{c\\in \\mathbb {R}}\\left\\Vert w(x)\\left(f-c\\right)(x)\\right\\Vert _{L_p(|x|\\ge \\sigma (4t))}\\le \\frac{1}{Q^{\\prime }(\\sigma (4t))} \\left\\Vert Q^{\\prime }wg\\right\\Vert _{L_p(\\mathbb {R})}.$ Then we have from (REF ), $\\inf _{c\\in \\mathbb {R}}\\left\\Vert w(x)(f-c)(x)\\right\\Vert _{L_p(|x|\\ge \\sigma (4t))}\\le Ct\\Vert Q^{\\prime }wg\\Vert _{L_p(\\mathbb {R})}.$ Here, from Lemma REF we have $\\Vert Q^{\\prime }w(f-f(0))\\Vert _{L_p(\\mathbb {R})}\\le C\\Vert wf^{\\prime }\\Vert _{L_p(\\mathbb {R})}.$ From (REF ) and (REF ) we have $\\inf _{c\\in \\mathbb {R}}\\left\\Vert w(x)(f-c)(x)\\right\\Vert _{L_p(|x|\\ge \\sigma (4t))}\\le Ct \\Vert wf^{\\prime }\\Vert _{L_p(\\mathbb {R})}.$ From (REF ) and (REF ) we have the result.", "Lemma 4.6 (cf.", "[4]) Let $1 \\le p \\le \\infty $ and $\\beta >1$ , and let us define $w^{\\sharp }$ with $p$ , $\\beta $ .", "Let $g$ be a real valued function on $\\mathbb {R}$ satisfying $\\Vert gw\\Vert _{L_p(\\mathbb {R})}<\\infty $ and (REF ), then we have $\\left\\Vert w^{\\sharp }(x)\\int _0^x g(t)dt\\right\\Vert _{L_p(\\mathbb {R})}\\le C \\frac{a_n}{n}\\Vert gw\\Vert _{L_p(\\mathbb {R})}.$ Especially, if $w\\in \\mathcal {F}_\\lambda (C^3+), 0<\\lambda <3/2$ , then we have $\\left\\Vert w^{\\sharp }(x)\\int _0^x \\left(f^{\\prime }(t)-v_n(f^{\\prime })(t)\\right)dt\\right\\Vert _{L_p(\\mathbb {R})}\\le C \\frac{a_n}{n}E_{p,n}\\left(w_{1/4},f^{\\prime }\\right).$ When $w\\in \\mathcal {F}^*$ , we also have (REF ) replacing $w_{1/4}$ with $w$ .", "For arbitrary $P_n\\in \\mathcal {P}_n$ , we have by (REF ) and Hölder inequality $\\left|\\int _0^x g(t)dt\\right|\\le \\left\\Vert g(t)w(t)\\right\\Vert _{L_p(\\mathbb {R})}E_{q,n}\\left(w,\\phi _x\\right), \\quad 1\\le p \\le \\infty , 1/p+1/q=1,$ where $\\phi $ is defined in (REF ).", "Then, we obtain by Lemma REF , $\\left|\\int _0^x g(t)dt\\right|&\\le & \\left\\Vert g(t)w(t)\\right\\Vert _{L_p(\\mathbb {R})}\\frac{a_n}{n}\\left(\\int _{\\mathbb {R}}|w(t)\\phi ^{\\prime }_x(t)|^qdt\\right)^{1/q} \\\\&\\le & \\frac{a_n}{n}\\left\\Vert g(t)w(t)\\right\\Vert _{L_p(\\mathbb {R})}|Q^{\\prime }(x)|^{1-1/q}\\left(\\int _0^x Q^{\\prime }(t)w^{-q}(t)dt\\right)^{1/q}\\\\&\\le & C\\frac{a_n}{n}\\left\\Vert g(t)w(t)\\right\\Vert _{L_p(\\mathbb {R})}|Q^{\\prime }(x)|^{1-1/q}w^{-1}(x).$ Here, for $p=1$ we may consider $\\lim _{q\\rightarrow \\infty } \\left( \\int _0^xQ^{\\prime }(t)w^{-q}(t)dt\\right)^{1/q}=\\lim _{q\\rightarrow \\infty } \\left( w^{-q}(t)\\right)^{1/q}=w^{-1}(x).$ Hence, we have $\\left\\Vert \\frac{w(x)}{\\left\\lbrace (1+|Q^{\\prime }(x)|)(1+|x|)^{\\beta }\\right\\rbrace ^{1/p}}\\int _0^x g(t)dt\\right\\Vert _{L_p(\\mathbb {R})}&\\le & C\\left(\\frac{a_n}{n}\\right)\\left\\Vert \\left(1+|x|\\right)^{-\\beta /p}\\right\\Vert _{L_p(\\mathbb {R})}\\left\\Vert gw\\right\\Vert _{L_p(\\mathbb {R})}\\\\&\\le & C\\left(\\frac{a_n}{n}\\right)\\left\\Vert gw\\right\\Vert _{L_p(\\mathbb {R})}.$ Therefore, we have (REF ).", "From (REF ), (REF ) and Proposition REF , we have (REF ).", "Lemma 4.7 Let $w=\\exp (-Q)\\in \\mathcal {F}_\\lambda (C^3+)$ , $0<\\lambda <3/2$ .", "Let $1 \\le p \\le \\infty $ , $\\Vert w_{1/4}f^{\\prime }\\Vert _{L_p(\\mathbb {R})}<\\infty $ , and let $q_{n-1}\\in \\mathcal {P}_n$ be the best approximation of $f^{\\prime }$ with respect to the weight $w$ on $L_p(\\mathbb {R})$ space, that is, $\\left\\Vert (f^{\\prime }-q_{n-1})w\\right\\Vert _{L_p(\\mathbb {R})}=E_{p,n-1}(w,f^{\\prime }).$ Using $q_{n-1}$ , define $F(x)$ and $S_{2n}$ as (REF ) and (REF ).", "Then we have $\\left\\Vert w^{\\sharp }(F-S_{2n})\\right\\Vert _{L_p(\\mathbb {R})}\\le C \\frac{a_n}{n}E_{p,n}\\left(w_{1/4},f^{\\prime }\\right),$ and $\\left\\Vert wS_{2n}^{\\prime }\\right\\Vert _{L_p(\\mathbb {R})}\\le C E_{p,n-1}\\left(w_{1/4},f^{\\prime }\\right).$ When $w\\in \\mathcal {F}^*$ , we also have same results replacing $w_{1/4}$ with $w$ .", "By Lemma REF (REF ), we have the result using the same method as the proof of Lemma REF .", "We will prove it similarly to the proof of Theorem REF .", "First, let $q_{n-1}\\in \\mathcal {P}_{n-1}$ be the polynomial of best approximation of $f^{\\prime }$ with respect to the weight $w$ on $L_p(\\mathbb {R})$ space.", "Then using $q_{n-1}$ , we define $F(x)$ and $S_{2n}$ in the same method as (REF ) and (REF ).", "Then we have using Lemma REF $\\left\\Vert F^{\\prime }w\\right\\Vert _{L_p(\\mathbb {R})}= E_{p,n-1}\\left(w,f^{\\prime }\\right),$ $\\Vert (F-S_{2n})w^{\\sharp }\\Vert _{L_p(\\mathbb {R})}\\le C \\frac{a_n}{n}E_{p,n}\\left(w_{1/4},f^{\\prime }\\right)$ and $\\left\\Vert S_{2n}^{\\prime }w \\right\\Vert _{L_p(\\mathbb {R})}\\le C E_{p,n-1}\\left(w_{1/4},f^{\\prime }\\right).$ Then we see from Theorem REF and (REF ), $\\left\\Vert S_{2n}^{(k)}w_{-(k-1)/2}\\right\\Vert _{L_p(\\mathbb {R})}\\le C \\left(\\frac{n}{a_n}\\right)^{k-1}E_{p,n-1}\\left(w_{1/4},f^{\\prime }\\right)$ and using $w^{\\sharp } \\le w$ and Theorem REF $\\left\\Vert \\left(R_{n}^{(k)}-S_{2n}^{(k)}\\right) w^{\\sharp }_{-k/2}\\right\\Vert _{L_p(\\mathbb {R})}\\le C \\left(\\frac{n}{a_n}\\right)^{k-1}E_{p,n-1}\\left(w_{1/4},f^{\\prime }\\right),$ where $R_{n}\\in \\mathcal {P}_{n}$ denotes the polynomial of best approximation of $F$ with $w$ on $L_p(\\mathbb {R})$ space(by the similar calculation as (REF ) and (REF )).", "Then, we see $w^{\\sharp }_{-k/2}(x) \\le w_{-(k-1)/2}(x)$ .", "By (REF ) and (REF ) and Theorem REF , we have $\\left\\Vert R_{n}^{(k)}w^{\\sharp }_{-k/2}\\right\\Vert _{L_p(\\mathbb {R})}\\le C \\left(\\frac{n}{a_n}\\right)^{k-1}E_{p,n-1}\\left(w_{1/4},f^{\\prime }\\right)\\le C E_{p,n-k}\\left(w_{1/4},f^{(k)}\\right).$ By the same reason to (REF ), we know that $P_{p,n;f,w}:=Q_n+R_n$ is the polynomial of best approximation of $f$ with $w$ on $L_p(\\mathbb {R})$ space.", "Therefore, using $P_{p,n;f,w}$ , (REF ) and the method of mathematical induction, we have for $1\\le k\\le r+1$ , $\\left\\Vert \\left(f^{(k)}-P_{p,n;f,w}^{(k)}\\right)w^{\\sharp }_{-k/2}\\right\\Vert _{L_p(\\mathbb {R})}\\le C E_{p,n-k}\\left(w_{1/4},f^{(k)}\\right).$ It follows from Theorem REF .", "If we apply Theorem REF with $w_{k/2}$ and $w^{\\flat }_{k/2}$ , then we can obtain the results.", "Monotone Approximation Let $r>0$ be an integer.", "Let $k$ and $\\ell $ be integers with $0\\le k\\le \\ell \\le r$ .", "In this section, we consider a real function $f$ on $\\mathbb {R}$ such that $f^{(r)}(x)$ is continuous in $\\mathbb {R}$ and we let $a_j(x)$ , $j=k,k+1,...,\\ell $ be bounded on $\\mathbb {R}$ .", "Now, we define the linear differential operator (cf.", "[1]) $L:=L_{k,\\ell }:=\\sum _{j=k}^\\ell a_j(x)[d^j/dx^j].$ G. A. Anastassiou and O. Shisha [1] consider the operator (REF ) with $a_j(x)$ under some condition on $[-1,1]$ .", "They showed that if $L(f) \\ge 0$ for $f\\in C^{(r)}[-1,1]$ , there exist $Q_n \\in \\mathcal {P}_n$ such that $L(Q_n)\\ge 0$ and for some constant $C>0$ , $\\left\\Vert f-Q_n\\right\\Vert _{L_{\\infty }([-1,1])}\\le C n^{\\ell -r}\\omega \\left(f^{(p)};\\frac{1}{n}\\right),$ where $\\omega \\left(f^{(p)};t\\right)$ is the modulus of continuity.", "In this section, we will obtain a similar result with exponential-type weighted $L_{\\infty }$ -norm as the above result.", "Our main theorem is as follows.", "Theorem 5.1 Let $k$ and $\\ell $ be integers with $0\\le k\\le \\ell \\le r$ .", "Let $w=\\exp (-Q)\\in \\mathcal {F}_\\lambda (C^{r+3}+)$ , and let $T(x)$ be continuous on $\\mathbb {R}$ .", "Suppose that $w(x)f^{(r)}(x)\\rightarrow 0$ as $|x|\\rightarrow \\infty $ .", "Let $P_{n;f,w_{-1/4}}\\in \\mathcal {P}_n$ be the best approximation for $f$ with the weight $w_{-1/4}$ on $\\mathbb {R}$ .", "Suppose that for a certain $\\delta >0$ , $L(f;x)\\ge \\delta , \\quad x\\in \\mathbb {R}.$ Then, for every integer $n\\ge 1$ and $j=0,1,...,\\ell $ , $\\left\\Vert \\left(f^{(j)}-P_{n;f,w_{-1/4}}^{(j)}\\right)wT^{-(2j+1)/4}(x)\\right\\Vert _{L_\\infty (\\mathbb {R})}\\le C_j\\left(\\frac{a_n}{n}\\right)^{r-j}E_{n-r}\\left(w,f^{(r)}\\right),$ where $C_j>0$ , $0\\le j\\le \\ell $ , are independent of $n$ or $f$ , and for any fixed number $M>0$ there exists a constant $N(M,\\ell ,\\delta )>0$ such that $L(P_{n;f,w_{-1/4}};x)\\ge \\frac{\\delta }{2}, \\quad |x|\\le M, \\quad n\\ge N(M,\\ell ,\\delta ).$ From Corollary REF , we have (REF ).", "Hence, we also have $&&\\left|\\left(L(f;x)-L\\left(P_{n;f,w_{-1/4}};x\\right)\\right)w(x)T^{-(2\\ell +1)/4}(x)\\right|\\\\&=&\\left|\\sum _{j=k}^\\ell a_j(x)\\left\\lbrace f^{(j)}(x)-P_{n;f,w_{-1/4}}^{(j)}(x)\\right\\rbrace w(x)T^{-(2\\ell +1)/4}(x)\\right|\\\\&\\le & E_{n-r}\\left(w,f^{(r)}\\right)\\sum _{j=k}^\\ell |a_j(x)|C_j\\left(\\frac{a_n}{n}\\right)^{r-j}\\\\&\\le & C_{k,\\ell }\\left(\\frac{a_n}{n}\\right)^{r-\\ell }E_{n-r}\\left(w,f^{(r)}\\right),$ where we set $C_{k,\\ell }:=\\sum _{j=k}^\\ell \\Vert a_j\\Vert _{L_\\infty (\\mathbb {R})}C_j$ .", "Then we have for $|x|\\le M$ $&&\\left|L(f;x)-L\\left(P_{n;f,w_{-1/4}};x\\right)\\right|\\\\&\\le & C_{k,\\ell }\\left\\Vert w^{-1}(x)T^{(2\\ell +1)/4}(x)\\right\\Vert _{L_\\infty (|x|\\le M)}\\left(\\frac{a_n}{n}\\right)^{r-\\ell }E_{n-r}\\left(w,f^{(r)}\\right).$ Here, for $\\delta >0$ there exists $N(M,\\ell ,\\delta )>0$ such that for $n\\ge N(M,\\ell ,\\delta )$ $C_{k,\\ell }\\left\\Vert w^{-1}(x)T^{(2\\ell +1)/4}(x)\\right\\Vert _{L_\\infty (|x|\\le M)}\\left(\\frac{a_n}{n}\\right)^{r-\\ell }E_{n-r}\\left(w,f^{(r)}\\right)\\le \\frac{\\delta }{2}.$ This follows from $w(x)f^{(r)}(x)\\rightarrow 0$ as $|x|\\rightarrow \\infty $ .", "Therefore we see $\\frac{\\delta }{2}\\le L(f;x)-\\frac{\\delta }{2}\\le L(P_{n;f,w_{-1/4}};x).$ Consequently, we have (REF )." ], [ "Proof of Theorems", "Throughout this section we suppose $w\\in \\mathcal {F}(C^2+)$ .", "We give the proofs of theorems.", "First, we give some lemmas to prove the theorems.", "We construct the orthonormal polynomials $p_n(x)=p_n(w^2,x)$ of degree n for $w^2(x)$ , that is, $\\int _{-\\infty }^{\\infty } p_n(w^2,x)p_m(w^2,x)w^2(x)dx=\\delta _{mn} (\\textrm {Kronecker delta}).$ Let $fw\\in L_2(\\mathbb {R})$ .", "The Fourier-type series of $f$ is defined by $\\tilde{f}(x):=\\sum _{k=0}^{\\infty } a_k(w^2,f)p_k(w^2,x),\\quad a_k(w^2,f):=\\int _{-\\infty }^{\\infty } f(t)p_k(w^2,t)w^2(t)dt.$ We denote the partial sum of $\\tilde{f}(x)$ by $s_n(f,x):=s_n(w^2,f,x):=\\sum _{k=0}^{n-1} a_k(w^2,f)p_k(w^2,x).$ Moreover, we define the de la Vall$\\acute{\\textrm {e}}$ e Poussin means by $v_n(f,x):=\\frac{1}{n}\\sum _{j=n+1}^{2n}s_j(w^2,f,x).$ Proposition 3.1 ([11]) Let $w\\in \\mathcal {F}_\\lambda (C^3+), 0<\\lambda <3/2$ , and let $1\\le p\\le \\infty $ .", "When $T^{1/4}wf\\in L_p(\\mathbb {R})$ , we have $\\left\\Vert v_n(f)w\\right\\Vert _{L_p(\\mathbb {R})}\\le C \\left\\Vert T^{1/4}wf \\right\\Vert _{L_p(\\mathbb {R})},$ and so $\\left\\Vert (f-v_n(f))w \\right\\Vert _{L_p(\\mathbb {R})}\\le C E_{p,n}\\left(T^{1/4}w,f\\right).$ So, equivalently, $\\left\\Vert v_n(f)w\\right\\Vert _{L_p(\\mathbb {R})}\\le C \\left\\Vert w_{1/4}f\\right\\Vert _{L_p(\\mathbb {R})},$ and so $\\left\\Vert (f-v_n(f))w\\right\\Vert _{L_p(\\mathbb {R})}\\le C E_{p,n} \\left(w_{1/4},f\\right).$ When $w\\in \\mathcal {F}^*$ , we can replace $w_{1/4}$ with $w$ .", "Lemma 3.2 (1) ([8]) Let $L>0$ be fixed.", "Then, uniformly for $t>0$ , $a_{Lt}\\sim a_t.$    (2) ([8]) For $x >1$ , we have $|Q^{\\prime }(a_x)| \\sim \\frac{x \\sqrt{T(a_x)}}{a_x} \\quad \\mbox{and} \\quad |Q(a_x)| \\sim \\frac{x}{ \\sqrt{T(a_x)}}.$    (3) ([8]) Let $x\\in (0, \\infty )$ .", "There exists $0<\\varepsilon <1$ such that $T\\left(x\\left[1+\\frac{\\varepsilon }{T(x)}\\right]\\right)\\sim T(x).$    (4) ([9]) If $T(x)$ is unbounded, then for any $\\eta >0$ there exists $C(\\eta )>0$ such that for $t\\ge 1$ , $a_t\\le C(\\eta )t^\\eta .$ To prove the results, we need the following notations.", "We set $\\sigma (t):=\\inf \\left\\lbrace a_u: \\,\\, \\frac{a_u}{u}\\le t \\right\\rbrace , \\quad t>0,$ and $\\Phi _t(x):=\\sqrt{\\left|1-\\frac{|x|}{\\sigma (t)}\\right|}+T^{-1/2}(\\sigma (t)), \\quad x\\in \\mathbb {R}.$ Define for $fw\\in L_p(\\mathbb {R})$ , $0<p\\le \\infty $ , $\\omega _p(f,w,t)&:=&\\sup _{0<h\\le t}\\left\\Vert w(x)\\left\\lbrace f\\left(x+\\frac{h}{2}\\Phi _t(x)\\right)-f\\left(x-\\frac{h}{2}\\Phi _t(x)\\right)\\right\\rbrace \\right\\Vert _{L_p(|x|\\le \\sigma (2t))}\\\\&& \\quad +\\inf _{c\\in \\mathbb {R}}\\left\\Vert w(x)(f-c)(x)\\right\\Vert _{L_p(|x|\\ge \\sigma (4t))}$ (see [2], [3]).", "Proposition 3.3 (cf.", "[3], [2]) Let $w\\in \\mathcal {F}(C^2+)$ .", "Let $0<p\\le \\infty $ .", "Then for $f: \\mathbb {R}\\rightarrow \\mathbb {R}$ which $fw\\in L_p(\\mathbb {R})$ (and for $p=\\infty $ , we require $f$ to be continuous, and $fw$ to vanish at $\\pm \\infty $ ), we have for $n\\ge C_3$ , $E_{p,n}\\left(f,w\\right)\\le C_1\\omega _{p}\\left(f,w,C_2\\frac{a_n}{n}\\right),$ where $C_j$ , $j=1,2,3$ , do not depend on $f$ and $n$ .", "Damelin and Lubinsky [3] or Damelin [2] have treated a certain class $\\mathcal {E}_1$ of weights containing the conditions (a)-(d) in Definition REF and $\\frac{yQ^{\\prime }(y)}{xQ^{\\prime }(x)}\\le C_1 \\left(\\frac{Q(y)}{Q(x)}\\right)^{C_2},\\quad y\\ge x\\ge C_3,$ where $C_i$ , $i=1,2,3>0$ are some constants, and they obtain this Proposition for $w\\in \\mathcal {E}_1$ .", "Therefore, we may show $\\mathcal {F}(C^2+)\\subset \\mathcal {E}_1$ .", "In fact, from Definition REF (d) and (e), we have for $y\\ge x>0$ , $\\frac{Q^{\\prime }(y)}{Q^{\\prime }(x)}=\\exp \\left(\\int _x^y\\frac{Q^{\\prime \\prime }(t)}{Q^{\\prime }(t)}dt\\right)\\le \\exp \\left(C_3\\int _x^y\\frac{Q^{\\prime }(t)}{Q(t)}dt\\right)=\\left(\\frac{Q(y)}{Q(x)}\\right)^{C_3},$ and $\\frac{y}{x}=\\exp \\left(\\int _x^y\\frac{1}{t}dt\\right)\\le \\exp \\left(\\frac{1}{\\Lambda }\\int _x^y\\frac{Q^{\\prime }(t)}{Q(t)}dt\\right)=\\left(\\frac{Q(y)}{Q(x)}\\right)^{\\frac{1}{\\Lambda }}.$ Therefore, we obtain (REF ) with $C_2=C_3+\\frac{1}{\\Lambda }$ , that is, we see $\\mathcal {F}(C^2+)\\subset \\mathcal {E}_1$ .", "Theorem 3.4 Let $w\\in \\mathcal {F}(C^2+)$ .", "(1) If $f$ is a function having bounded variation on any compact interval and if $\\int _{-\\infty }^{\\infty } w(x)|df(x)|<\\infty ,$ then there exists a constant $C>0$ such that for every $t>0$ , $\\omega _1(f,w,t)\\le C t\\int _{-\\infty }^{\\infty } w(x)|df(x)|,$ and so $E_{1,n}(f,w)\\le C\\frac{a_n}{n}\\int _{-\\infty }^{\\infty } w(x)|df(x)|.$    (2) Let us suppose that $f$ is continuous and $\\lim _{|x|\\rightarrow \\infty }|(\\sqrt{T}wf)(x)|=0$ , then we have $\\lim _{t\\rightarrow 0}\\omega _{\\infty }(f,w,t)=0.$ To prove this theorem we need the following lemma.", "Lemma 3.5 ([9]) (1) For $t>0$ there exists $a_u$ such that $t=\\frac{a_u}{u} \\quad \\textrm {and} \\quad \\sigma (t)=a_u.$    (2) If $t=a_u/u$ , $u>0$ large enough and $|x-y|\\le t\\Phi _t(x),$ then there exist $C_1,C_2>0$ such that $C_1w(x)\\le w(y)\\le C_2 w(x).$ (1) Let $g(x):=f(x)-f(0)$ .", "For $t>0$ small enough let $0<h\\le t$ and $|x|\\le \\sigma (2t)<\\sigma (t)$ .", "Hence we may consider $\\Phi _t(x)\\le 2$ .", "Then by Lemma REF , $&&\\int _{-\\infty }^{\\infty } w(x)\\left|g\\left(x+\\frac{h}{2}\\Phi _t(x)\\right)-g\\left(x-\\frac{h}{2}\\Phi _t(x)\\right)\\right|dx\\\\&&=\\int _{-\\infty }^{\\infty } w(x)\\left|\\int _{x-\\frac{h}{2}\\Phi _t(x)}^{x+\\frac{h}{2}\\Phi _t(x)}df(v)\\right| dx\\le C\\int _{-\\infty }^{\\infty } \\left|\\int _{x-\\frac{h}{2}\\Phi _t(x)}^{x+\\frac{h}{2}\\Phi _t(x)}w(v)df(v)\\right| dx\\\\&&\\le \\int _{-\\infty }^{\\infty } \\int _{x-h}^{x+h} w(v)|df(v)|dx\\le \\int _{-\\infty }^{\\infty } w(v)\\int _{v-h\\le x\\le v+h} dx|df(v)|\\\\&&\\le 2h\\int _{-\\infty }^{\\infty } w(v)|df(v)|.$ Hence we have $\\int _{-\\infty }^{\\infty } w(x)\\left|g\\left(x+\\frac{h}{2}\\Phi _t(x)\\right)-g\\left(x-\\frac{h}{2}\\Phi _t(x)\\right)\\right| dx\\le 2t \\int _{-\\infty }^{\\infty } w(x)|df(x)|.$ Moreover, we see $\\inf _{c\\in \\mathbb {R}}\\left\\Vert w(x)(f-c)(x)\\right\\Vert _{L_1(|x|\\ge \\sigma (4t))}\\le \\frac{1}{Q^{\\prime }(\\sigma (4t))}\\left\\Vert Q^{\\prime }(x)w(x)g(x)\\right\\Vert _{L_1(|x|\\ge \\sigma (4t))}.$ Here we see $\\frac{\\sqrt{T(\\sigma (t))}}{Q^{\\prime }(\\sigma (t))}\\sim t.$ In fact, from Lemma REF (2), for $t=\\frac{a_u}{u}$ $Q^{\\prime }(\\sigma (t))= Q^{\\prime }(a_u)\\sim \\frac{u\\sqrt{T(a_u)}}{a_u}\\sim \\frac{\\sqrt{T(\\sigma (t))}}{t}.$ On the other hand, we have $\\int _{0}^{\\infty } Q^{\\prime }(x)w(x)|g(x)|dx&=& \\int _{0}^{\\infty } Q^{\\prime }(x)w(x)\\left|\\int _0^xdg(u)\\right|dx\\\\&\\le & \\int _{0}^{\\infty } Q^{\\prime }(x)w(x)\\int _0^x|df(u)|dx\\\\&=& -w(x)\\int _0^x|df(u)|\\bigg |_0^{\\infty }+\\int _0^{\\infty } w(x)|df(x)|\\\\&=&\\int _0^{\\infty } w(x)|df(x)|$ because $\\lim _{x\\rightarrow \\infty } w(x) =0$ from (c) of Definition REF .", "Similarly we have $\\int _{-\\infty }^0 \\left| Q^{\\prime }(x)w(x)g(x)\\right|dx\\le \\int _{-\\infty }^0 w(x)|df(x)|.$ Hence we have $\\Vert Q^{\\prime }wg\\Vert _{L_1(\\mathbb {R})}\\le \\int _{-\\infty }^{\\infty } w(u)|df(u)|.$ Therefore, using (REF ), (REF ) and (REF ), we have $\\inf _{c\\in \\mathbb {R}}\\left\\Vert w(x)(f-c)(x)\\right\\Vert _{L_1(|x|\\ge \\sigma (4t))}= O(t) \\int _{-\\infty }^{\\infty } w(x)|df(x)|.$ Consequently, by (REF ) and (REF ) we have $\\omega _1(f,w,t)\\le C t\\int _{-\\infty }^{\\infty } w(x)|df(x)|.$ Hence, setting $t=C_2\\frac{a_n}{n}$ , if we use Proposition REF , then $E_{1,n}(f,w)\\le C\\frac{a_n}{n}\\int _{-\\infty }^{\\infty } w(x)|df(x)|.$ (2) Given $\\varepsilon >0$ , and let us take $L=L(\\varepsilon )>0$ large enough as $\\sup _{|x|\\ge L}|w(x)f(x)| \\le \\frac{1}{\\sqrt{T(L)}}\\sup _{|x|\\ge L}|\\sqrt{T(x)}w(x)f(x)|<\\varepsilon \\quad (\\textrm {by our assumption}).$ Then we have $\\inf _{c\\in \\mathbb {R}}\\sup _{|x|\\ge L} \\left|w(x)(f-c)(x)\\right|\\le \\frac{1}{\\sqrt{T(L)}}\\sup _{|x|\\ge L}\\left|\\sqrt{T(x)}w(x)f(x)\\right|<\\varepsilon .$ Now, there exists $\\varepsilon >0$ small enough such that $\\frac{h}{2}\\Phi _t(x)\\le \\varepsilon \\frac{1}{T(x)}, \\quad |x|\\le \\sigma (2t),$ because if we put $t=a_u/u$ , then we see $\\sigma (t)=a_u$ and $|x|\\le \\sigma (2t)<a_u$ .", "Hence, noting [8], that is, for some $\\varepsilon >0$ , and for large enough $t$ , $T(a_t) \\le C t^{2-\\varepsilon },$ and if $w$ is the Erdös-type weight, then from Lemma REF (4), we have $t\\Phi _t(x)\\le \\frac{a_u}{u}\\le \\varepsilon \\frac{1}{T(a_u)}\\le \\varepsilon \\frac{1}{T(x)}.$ If $w\\in \\mathcal {F}^*$ , we also have (REF ), because for some $\\delta >0$ and $u>0$ large enough, $t\\Phi _t(x)\\le \\frac{a_u}{u}\\le u^{-\\delta } \\le \\varepsilon \\frac{1}{T(x)}.$ Therefore, using Lemma REF (3), Lemma REF and the assumption $\\lim _{|x|\\rightarrow \\infty }\\sqrt{T\\left(x\\right)}w\\left(x\\right)f\\left(x\\right)=0,$ for $2L\\le |x|\\le \\sigma (2t)$ , $h>0$ , $&&\\left|w(x)\\left\\lbrace f\\left(x+\\frac{h}{2}\\Phi _t(x)\\right)-f\\left(x-\\frac{h}{2}\\Phi _t(x)\\right)\\right\\rbrace \\right|\\\\&\\le & C\\Bigg [\\frac{1}{\\sqrt{T(x)}}\\left|\\sqrt{T\\left(x+\\frac{h}{2}\\Phi _t(x)\\right)}w\\left(x+\\frac{h}{2}\\Phi _t(x)\\right)f\\left(x+\\frac{h}{2}\\Phi _t(x)\\right)\\right|\\\\&& \\qquad +\\frac{1}{\\sqrt{T(x)}}\\left|\\sqrt{T\\left(x-\\frac{h}{2}\\Phi _t(x)\\right)}w\\left(x-\\frac{h}{2}\\Phi _t(x)\\right)f\\left(x-\\frac{h}{2}\\Phi _t(x)\\right)\\right|\\Bigg ]\\\\&\\le & 2\\varepsilon .$ On the other hand, $\\lim _{t\\rightarrow 0}\\sup _{0<h\\le t}\\left\\Vert w(x) \\left\\lbrace f\\left(x+\\frac{h}{2}\\Phi _t(x)\\right)-f\\left(x-\\frac{h}{2}\\Phi _t(x)\\right)\\right\\rbrace \\right\\Vert _{L_\\infty (|x|\\le 2L)}=0.$ Therefore, we have the result.", "Lemma 3.6 (cf.", "[4]) Let $g$ be a real valued function on $\\mathbb {R}$ satisfying $\\Vert gw\\Vert _{L_\\infty (\\mathbb {R})}<\\infty $ and $\\int _{-\\infty }^{\\infty } gPw^2dt=0 \\quad P\\in \\mathcal {P}_n.$ Then we have $\\left\\Vert w(x)\\int _0^x g(t)dt \\right\\Vert _{L_\\infty (\\mathbb {R})}\\le C \\frac{a_n}{n}\\Vert gw\\Vert _{L_\\infty (\\mathbb {R})}.$ Especially, if $w\\in \\mathcal {F}_\\lambda (C^3+), 0<\\lambda <3/2$ , then we have $\\left\\Vert w(x)\\int _0^x \\left(f^{\\prime }(t)-v_n(f^{\\prime })(t)\\right)dt \\right\\Vert _{L_\\infty (\\mathbb {R})}\\le C \\frac{a_n}{n}E_{n} \\left(w_{1/4},f^{\\prime }\\right).$ When $w\\in \\mathcal {F}^*$ , we also have (REF ) replacing $w_{1/4}$ with $w$ .", "We let $\\phi _x(t)=\\left\\lbrace \\begin{array}{lr}w^{-2}(t),& 0\\le t\\le x; \\\\0,& otherwise,\\end{array}\\right.$ then we have for arbitrary $P_n\\in \\mathcal {P}_n$ , $\\left|\\int _0^x g(t)dt\\right|=\\left|\\int _{-\\infty }^{\\infty } g(t)\\phi _x(t)w^2(t)dt\\right|=\\left|\\int _{-\\infty }^{\\infty } g(t)(\\phi _x(t)-P_n(t))w^2(t)dt\\right|.$ Therefore, we have $\\left|\\int _0^x g(t)dt\\right|&\\le & \\Vert gw\\Vert _{L_\\infty (\\mathbb {R})}\\inf _{P_n\\in \\mathcal {P}_n}\\int _{-\\infty }^{\\infty } \\left|\\phi _x(t)-P_n(t)\\right|w(t)dt \\\\&=&\\Vert gw\\Vert _{L_\\infty (\\mathbb {R})}E_{1,n}(w:\\phi _x).$ Here, from Theorem REF we see that $E_{1,n}(w:\\phi _x)&\\le & C\\frac{a_n}{n}\\int _{-\\infty }^{\\infty } w(t)|d\\phi _x(t)|\\le C\\frac{a_n}{n}\\int _0^x w(t)|Q^{\\prime }(t)|w^{-2}(t)dt\\\\&=& C\\frac{a_n}{n}\\int _0^x Q^{\\prime }(t)w^{-1}(t)dt\\le C\\frac{a_n}{n}w^{-1}(x).$ So, we have $\\left|w(x)\\int _0^x g(t)dt\\right|\\le \\Vert gw\\Vert _{L_\\infty (\\mathbb {R})}w(x)E_{1,n}\\left(w:\\phi _x\\right)\\le C\\frac{a_n}{n}\\Vert gw\\Vert _{L_\\infty (\\mathbb {R})}.$ Therefore, we have (REF ).", "Next we show (REF ).", "Since $v_n(f^{\\prime })(t)=\\frac{1}{n}\\sum _{j=n+1}^{2n}s_j(f^{\\prime },t),$ and for any $P\\in \\mathcal {P}_n$ , $j\\ge n+1$ , $\\int _{-\\infty }^{\\infty } \\left(f^{\\prime }(t)-s_j(f^{\\prime };t)\\right)P(t)w^2(t)dt=0,$ we have $\\int _{-\\infty }^{\\infty } \\left(f^{\\prime }(t)-v_n(f^{\\prime })(t)\\right)P(t)w^2(t)dt=0.$ Using (REF ) and (REF ), we have (REF ).", "Lemma 3.7 Let $w=\\exp (-Q)\\in \\mathcal {F}_\\lambda (C^3+)$ , $0<\\lambda <3/2$ .", "Let $\\left\\Vert w_{1/4}f^{\\prime }\\right\\Vert _{L_\\infty (\\mathbb {R})}<\\infty $ , and let $q_{n-1}\\in \\mathcal {P}_n$ be the best approximation of $f^{\\prime }$ with respect to the weight $w$ , that is, $\\left\\Vert (f^{\\prime }-q_{n-1})w \\right\\Vert _{L_\\infty (\\mathbb {R})}=E_{n-1}(w,f^{\\prime }).$ Now we set $F(x):=f(x)-\\int _0^xq_{n-1}(t)dt,$ then there exists $S_{2n}\\in \\mathcal {P}_{2n}$ such that $\\left\\Vert w \\left(F-S_{2n}\\right) \\right\\Vert _{L_\\infty (\\mathbb {R})}\\le C \\frac{a_n}{n}E_n \\left(w_{1/4},f^{\\prime }\\right),$ and $\\left\\Vert wS_{2n}^{\\prime } \\right\\Vert _{L_\\infty (\\mathbb {R})}\\le C E_{n-1}\\left(w_{1/4},f^{\\prime }\\right).$ When $w\\in \\mathcal {F}^*$ , we also have same results replacing $w_{1/4}$ with $w$ .", "Let $S_{2n}(x)=f(0)+\\int _0^x v_n \\left(f^{\\prime }-q_{n-1}\\right)(t)dt,$ then by Lemma REF (REF ), $&&\\left\\Vert w\\left(F-S_{2n}\\right)\\right\\Vert _{L_\\infty (\\mathbb {R})} \\\\&=&\\left\\Vert w\\left(f-\\int _0^xq_{n-1}(t)dt -f(0)-\\int _0^x v_n\\left(f^{\\prime }-q_{n-1}\\right)(t)dt\\right)\\right\\Vert _{L_\\infty (\\mathbb {R})}\\\\&=&\\left\\Vert w\\left(\\int _0^x \\left[f^{\\prime }(t)-v_n(f^{\\prime })(t)\\right]dt\\right)\\right\\Vert _{L_\\infty (\\mathbb {R})}\\le C\\frac{a_n}{n}E_n\\left(w_{1/4},f^{\\prime }\\right).$ Now by Proposition REF (REF ), $\\left\\Vert wS_{2n}^{\\prime }\\right\\Vert _{L_\\infty (\\mathbb {R})}&=&\\left\\Vert w\\left(v_n(f^{\\prime }-q_{n-1})\\right)\\right\\Vert _{L_\\infty (\\mathbb {R})}\\\\&\\le &\\left\\Vert \\left(f^{\\prime }-v_n\\left(f^{\\prime }\\right)\\right)w\\right\\Vert _{L_\\infty (\\mathbb {R})}+\\left\\Vert \\left(f^{\\prime }-q_{n-1}\\right)w\\right\\Vert _{L_\\infty (\\mathbb {R})}\\\\&\\le & E_{n}\\left(w_{1/4},f^{\\prime }\\right)+E_{n-1}\\left(w,f^{\\prime }\\right)\\le E_{n-1}\\left(w_{1/4},f^{\\prime }\\right).$ To prove Theorem REF we need the following theorems.", "Theorem 3.8 ([9]) Let $w\\in \\mathcal {F}(C^2+)$ , and let $r\\ge 0$ be an integer.", "Let $1\\le p\\le \\infty $ , and let $wf^{(r)}\\in L_p(\\mathbb {R})$ .", "Then we have $E_{p,n}(f,w)\\le C \\left(\\frac{a_n}{n}\\right)^k \\left\\Vert f^{(k)}w\\right\\Vert _{L_p(\\mathbb {R})}, \\quad k=1,2,...,r,$ and equivalently, $E_{p,n}(f,w)\\le C \\left(\\frac{a_n}{n}\\right)^kE_{p,n-k}\\left(f^{(k)},w\\right).$ Theorem 3.9 ([10]) Let $r\\ge 1$ be an integer and $w\\in \\mathcal {F}_\\lambda (C^{r+2}+)$ , $0< \\lambda <(r+2)/(r+1)$ , and let $1\\le p\\le \\infty $ .", "Then there exists a constant $C>0$ such that for any $1\\le k\\le r$ , any integer $n\\ge 1$ and any polynomial $P\\in \\mathcal {P}_n$ , $\\left\\Vert P^{(k)}w\\right\\Vert _{L_p(\\mathbb {R})}\\le C\\left(\\frac{n}{a_n}\\right)^k\\left\\Vert T^{k/2}Pw\\right\\Vert _{L_p(\\mathbb {R})}.$ We prove the theorem only in case of unbounded $T(x)$ , in the case of Freud case $\\mathcal {F}^*$ we can prove it similarly.", "We show that for $k=0,1,...,r$ , $\\left|\\left(f^{(k)}(x)-P_{n;f,w}^{(k)}\\right)w(x)\\right|\\le CT^{k/2}(x)E_{n-k}\\left(w_{1/4},f^{(k)}\\right).$ If $r=0$ , then (REF ) is trivial.", "For some $r\\ge 0$ we suppose that (REF ) holds, and let $f\\in C^{(r+1)}(\\mathbb {R})$ .", "Then $f^{\\prime }\\in C^{(r)}(\\mathbb {R})$ .", "Let $q_{n-1}\\in \\mathcal {P}_{n-1}$ be the polynomial of best approximation of $f^{\\prime }$ with respect to the weight $w$ .", "Then, from our assumption we have for $0\\le k\\le r$ , $\\left|\\left(f^{(k+1)}(x)-q_{n-1}^{(k)}(x)\\right)w(x)\\right|\\le C T^{k/2}(x)E_{n-k}\\left(w_{1/4},f^{(k+1)}\\right),$ that is, for $1\\le k\\le r+1$ $\\left|\\left(f^{(k)}(x)-q_{n-1}^{(k-1)}(x)\\right)w(x)\\right|\\le C T^{\\frac{k-1}{2}}(x)E_{n-k+1}\\left(w_{1/4},f^{(k)}\\right).$ Let $F(x):=f(x)-\\int _0^x q_{n-1}(t)dt=f(x)-Q_n(x),$ then $|F^{\\prime }(x)w(x)|\\le C E_{n-1}\\left(w,f^{\\prime }\\right).$ As (REF ) we set $S_{2n}=\\int _0^x(v_n(f^{\\prime })(t)-q_{n-1}(t))dt+f(0)$ , then from Lemma REF $\\left\\Vert \\left(F-S_{2n}\\right)w\\right\\Vert _{L_\\infty (\\mathbb {R})}\\le C \\frac{a_n}{n}E_n\\left(w_{1/4},f^{\\prime }\\right),$ and $\\left\\Vert S_{2n}^{\\prime }w\\right\\Vert _{L_\\infty (\\mathbb {R})}\\le C E_{n-1}\\left(w_{1/4},f^{\\prime }\\right).$ Here we apply Theorem REF with the weight $w_{-(k-1)/2}$ .", "In fact, by Theorem REF we have $w_{-(k-1)/2}\\in \\mathcal {F}_\\lambda (C^{r+2}+)$ .", "Then, noting $a_{2n}\\sim a_n$ from Lemma REF (1), we see $|S_{2n}^{(k)}(x) w_{-(k-1)/2}(x))|&\\le & C \\left(\\frac{n}{a_n}\\right)^{k-1}\\Vert S_{2n}^{\\prime }w\\Vert _{L_\\infty (\\mathbb {R})}\\\\&\\le & C \\left(\\frac{n}{a_n}\\right)^{k-1}E_{n-1}\\left(w_{1/4},f^{\\prime }\\right),$ that is, $\\left|S_{2n}^{(k)}(x) w(x)\\right|\\le C \\left(\\frac{n\\sqrt{T(x)}}{a_n}\\right)^{k-1}E_{n-1}\\left(w_{1/4},f^{\\prime }\\right),\\quad 1\\le k\\le r+1.$ Let $R_{n}\\in \\mathcal {P}_{n}$ denote the polynomial of best approximation of $F$ with $w$ .", "By Theorem REF with $w_{-\\frac{k}{2}}$ again, for $0\\le k\\le r+1$ we have $ \\nonumber \\left|(R_{n}^{(k)}-S_{2n}^{(k)}(x)) w_{-\\frac{k}{2}}(x)\\right|&\\le & C \\left(\\frac{n}{a_n}\\right)^k\\Vert (R_{n}-S_{2n})w_{-\\frac{k}{2}}(x)T^{k/2}(x)\\Vert _{L_\\infty (\\mathbb {R})}\\\\&\\le & C \\left(\\frac{n}{a_n}\\right)^k\\Vert (R_{n}-S_{2n})w\\Vert _{L_\\infty (\\mathbb {R})}$ and by (REF ) $ \\nonumber \\Vert (R_{n}-S_{2n})w\\Vert _{L_\\infty (\\mathbb {R})}&\\le & C \\left[\\Vert (F-R_{n})w\\Vert _{L_\\infty (\\mathbb {R})}+\\Vert (F-S_{2n})w\\Vert _{L_\\infty (\\mathbb {R})}\\right]\\\\\\nonumber &\\le & C \\left[E_{n}(w,F)+\\frac{a_n}{n}E_{n}\\left(w_{1/4},f^{\\prime }\\right) \\right] \\\\ \\nonumber &\\le & C \\left[\\frac{a_n}{n}E_{n-1}(w,f^{\\prime })+\\frac{a_n}{n}E_{n-1}(w_{1/4},f^{\\prime })\\right]\\\\&\\le & C \\frac{a_n}{n}E_{n-1}\\left(w_{1/4},f^{\\prime }\\right).$ Hence, from (REF ) and (REF ) we have for $0\\le k\\le r+1$ $\\nonumber |(R_{n}^{(k)}-S_{2n}^{(k)}(x)) w(x)|&\\le & C\\left|T^{k/2}(x)\\right|\\left|(R_{n}^{(k)}-S_{2n}^{(k)}(x)) w_{-\\frac{k}{2}}(x)\\right|\\\\&\\le & C \\left(\\frac{n\\sqrt{T(x)}}{a_n}\\right)^k\\frac{a_n}{n}E_{n-1}\\left(w_{1/4},f^{\\prime }\\right).$ Therefore by (REF ), (REF ) and Theorem REF , $ \\nonumber |R_{n}^{(k)}(x) w(x))|&\\le & C T^{k/2}(x)\\left(\\frac{n}{a_n}\\right)^{k-1}E_{n-1}\\left(w_{1/4},f^{\\prime }\\right) \\\\&\\le & C T^{k/2}(x)E_{n-k}\\left(w_{1/4},f^{(k)}\\right).$ Since $E_{n}(F,w)=E_{n}(w,f)$ and $E_{n}\\left(F,w\\right)=\\left\\Vert w\\left(F-R_{n}\\right)\\right\\Vert _{L_\\infty (\\mathbb {R})}=\\left\\Vert w\\left(f-Q_n-R_{n}\\right)\\right\\Vert _{L_\\infty (\\mathbb {R})}$ (see (REF )), we know that $P_{n;f,w}:=Q_n+R_n$ is the polynomial of best approximation of $f$ with $w$ .", "Now, from (REF ), (REF ) and (REF ) we have for $1\\le k\\le r+1$ , $\\left|\\left(f^{(k)}(x)-P_{n;f.w}^{(k)}(x)\\right)w(x)\\right|&=&\\left|\\left(f^{(k)}(x)-Q_n^{(k)}(x)-R_{n}^{(k)}(x)\\right)w(x)\\right|\\\\&\\le & \\left|(f^{(k)}(x)-q_{n-1}^{(k-1)}(x))w(x)\\right|+\\left|R_{n}^{(k)}(x)w(x)\\right|\\\\&\\le & C T^{k/2}(x)E_{n-k}\\left(w_{1/4},f^{(k)}\\right).$ For $k=0$ it is trivial.", "Consequently, we have (REF ) for all $r\\ge 0$ .", "Moreover, using Theorem REF , we conclude Theorem REF .", "It follows from Theroem REF .", "Applying Theorem REF with $w_{k/2}$ , we have for $0\\le j\\le r$ $\\left\\Vert (f^{(j)}-P_{n;f,w_{k/2}}^{(j)})w\\right\\Vert _{L_\\infty (\\mathbb {R})}\\le C E_{n-k}\\left(w_{(2k+1)/4},f^{(j)}\\right).$ Especially, when $j=k$ , we obtain $\\left\\Vert \\left(f^{(k)}-P_{n;f,w_{k/2}}^{(k)}\\right)w\\right\\Vert _{L_\\infty (\\mathbb {R})}\\le CE_{p,n-k}\\left(w_{(2k+1)/4},f^{(k)}\\right).$ Theorems in $L_p(\\mathbb {R})$ $(1 \\le p \\le \\infty )$ In this section we will give an analogy of Theorem REF in $L_p(\\mathbb {R})$ -space ($1 \\le p \\le \\infty $ ) and we will prove it using the same method as the proof of Theorem REF .", "Let $1 \\le p \\le \\infty $ .", "Let $w=\\exp (-Q)\\in \\mathcal {F}_\\lambda (C^3+)$ , $0<\\lambda <3/2$ , and let $\\beta >1$ be fixed.", "Then we set $w^{\\sharp }$ and $w^{\\flat }$ as follows; $&& \\frac{w(x)}{\\left\\lbrace (1+|Q^{\\prime }(x)|)(1+|x|)^{\\beta }\\right\\rbrace ^{1/p}}\\sim w^{\\sharp }(x)\\in \\mathcal {F}(C^2+);\\\\&&w(x)\\left\\lbrace (1+|Q^{\\prime }(x)|)(1+|x|)^{\\beta }\\right\\rbrace ^{1/p} \\sim w^{\\flat }(x)\\in \\mathcal {F}(C^2+)$ (see Theorem REF ).", "Theorem 4.1 Let $r \\ge 0$ be an integer.", "Let $w=\\exp (-Q)\\in \\mathcal {F}_\\lambda (C^{r+2}+)$ , $0< \\lambda <(r+2)/(r+1)$ , and let $\\beta >1$ be fixed.", "Suppose that $T^{1/4}f^{(r)}w\\in L_p(\\mathbb {R})$ .", "Let $P_{p,n;f,w}\\in \\mathcal {P}_n$ be the best approximation of $f$ with respect to the weight $w$ in $L_p(\\mathbb {R})$ -space, that is, $E_{p,n}\\left(w,f\\right):=\\inf _{P\\in \\mathcal {P}_n}\\left\\Vert \\left(f-P\\right)w\\right\\Vert _{L_p(\\mathbb {R})}=\\left\\Vert \\left(f-P_{p,n;f,w}\\right)w\\right\\Vert _{L_p(\\mathbb {R})}.$ Then there exists an absolute constant $C_r>0$ which depends only on $r$ such that for $0\\le k\\le r$ and $x\\in \\mathbb {R}$ , $\\left\\Vert \\left(f^{(k)}-P_{p,n;f,w}^{(k)}\\right)w^{\\sharp }_{-k/2}\\right\\Vert _{L_p(\\mathbb {R})}&\\le & C_r E_{p,n-k}\\left(w_{1/4},f^{(k)}\\right)\\\\&\\le & C_r \\left(\\frac{a_n}{n}\\right)^{r-k}E_{p,n-r}\\left(w_{1/4},f^{(r)}\\right).$ When $w\\in \\mathcal {F}^*$ , we can replace $w_{1/4}$ and $w^{\\sharp }_{-k/2}$ with $w$ and $w^{\\sharp }$ , respectively in the above.", "If we apply Theorem REF with $w_{-1/4}$ , then we have the following.", "Corollary 4.2 Let $r \\ge 0$ be an integer.", "Let $w=\\exp (-Q)\\in \\mathcal {F}_\\lambda (C^{r+3}+)$ , $0< \\lambda <(r+3)/(r+2)$ , and let $\\beta >0$ be fixed.", "Suppose that $wf^{(r)}\\in L_p(\\mathbb {R})$ .", "Then for $0\\le k\\le r$ we have $\\left\\Vert \\left(f^{(k)}-P_{p,n;f,w_{-1/4}}^{(k)}\\right)w^{\\sharp }_{-(2k+1)/4}\\right\\Vert _{L_p(\\mathbb {R})}&\\le & C_r E_{p,n-k}\\left(w,f^{(k)}\\right)\\\\&\\le & C_r \\left(\\frac{a_n}{n}\\right)^{r-k}E_{p,n-r}\\left(w,f^{(r)}\\right).$ When $w\\in \\mathcal {F}^*$ , we can omit $T^{-(2k+1)/4}$ in the above.", "Corollary 4.3 Let $r \\ge 0$ be an integer.", "Let $w=\\exp (-Q)\\in \\mathcal {F}_\\lambda (C^{r+3}+)$ , $0< \\lambda <(r+3)/(r+2)$ , and let $\\beta >0$ be fixed.", "(1) Let $w_{(2r+1)/4}f^{(r)}\\in L_p(\\mathbb {R})$ .", "Then, for each $k$ $(0\\le k\\le r)$ and the best approximation polynomial $P_{p,n;f,w_{k/2}}$ , we have $\\left\\Vert \\left(f^{(k)}-P_{p,n;f,w_{k/2}}^{(k)}\\right)w^{\\sharp }\\right\\Vert _{L_p(\\mathbb {R})}&\\le & C_r E_{p,n-k}\\left(w_{(2k+1)/4},f^{(k)}\\right)\\\\&\\le & C_r \\left(\\frac{a_n}{n}\\right)^{r-k}E_{p,n-r}\\left(w_{(2k+1)/4},f^{(r)}\\right).$    (2) Let $w^{\\flat }_{(2r+1)/4}f^{(r)}\\in L_p(\\mathbb {R})$ .", "Then, for each $k$ $(0\\le k\\le r)$ and the best approximation polynomial $P_{p,n;f,w^{\\flat }_{k/2}}$ , we have $\\left\\Vert \\left(f^{(k)}-P_{p,n;f,w^{\\flat }_{k/2}}^{(k)}\\right)w\\right\\Vert _{L_p(\\mathbb {R})}&\\le & C_r E_{p,n-k}\\left(w^{\\flat }_{(2k+1)/4},f^{(k)}\\right)\\\\&\\le & C_r \\left(\\frac{a_n}{n}\\right)^{r-k}E_{p,n-r}\\left(w^{\\flat }_{(2k+1)/4},f^{(r)}\\right).$ When $w\\in \\mathcal {F}^*$ , we can replace $w_{(2k+1)/4}$ and $w^{\\flat }_{(2k+1)/4}$ with $w$ and $w^{\\flat }$ , respectively in the above.", "Especially when $p=\\infty $ , we can refer to $w^{\\sharp }$ or $w^{\\flat }$ as $w$ .", "In this case, we can note that Corollary REF and Corollary REF imply Corollary REF , and Corollary REF , respectively.", "To prove Theorem REF we need to prepare some notations and lemmas.", "Lemma 4.4 ([6]) Let $w\\in \\mathcal {F}(C^2+)$ and let $1\\le p\\le \\infty $ .", "If $g:\\mathbb {R}\\rightarrow \\mathbb {R}$ is absolutely continuous, $g(0)=0$ , and $wg^{\\prime }\\in L_p(\\mathbb {R})$ , then $\\left\\Vert Q^{\\prime }wg\\right\\Vert _{L_p(\\mathbb {R})}\\le C\\left\\Vert wg^{\\prime }\\right\\Vert _{L_p(\\mathbb {R})}.$ Lemma 4.5 (cf.", "[9]) Let $w\\in \\mathcal {F}(C^2+)$ and let $1\\le p\\le \\infty $ .", "If $wf^{\\prime }\\in L_p(\\mathbb {R})$ , then $E_{p,n}(w,f)\\le C\\omega _p\\left(f,w,\\frac{a_n}{n}\\right)\\le C \\frac{a_n}{n}\\left\\Vert wf^{\\prime }\\right\\Vert _{L_p(\\mathbb {R})}.$ The first inequality follows from Proposition REF .", "We show the second inequality.", "By [9] we have $&&\\left\\Vert w(x)\\left\\lbrace f\\left(x+\\frac{h}{2}\\Phi _t(x)\\right)-f\\left(x-\\frac{h}{2}\\Phi _t(x)\\right)\\right\\rbrace \\right\\Vert ^p_{L_p(|x|\\le \\sigma (2t))} \\\\&=&h^p\\int _{\\mathbb {R}}|w(x)\\Phi _t(x)f^{\\prime }(x_t)|^pdx\\le Ch^p\\int _{\\mathbb {R}}|w(x)f^{\\prime }(x)|^pdx.$ Hence we see $\\sup _{0<h\\le t}\\left\\Vert w(x)\\left\\lbrace f\\left(x+\\frac{h}{2}\\Phi _t(x)\\right)-f\\left(x-\\frac{h}{2}\\Phi _t(x)\\right)\\right\\rbrace \\right\\Vert _{L_p(|x|\\le \\sigma (2t))}\\le Ct\\left\\Vert wf^{\\prime }\\right\\Vert _{L_p(\\mathbb {R})}.$ Now, we estimate $\\left\\Vert w(x)(f-c)(x)\\right\\Vert _{L_p(|x|\\ge \\sigma (4t))}.$ Let $g(x):=f(x)-f(0)$ .", "$\\inf _{c\\in \\mathbb {R}}\\left\\Vert w(x)\\left(f-c\\right)(x)\\right\\Vert _{L_p(|x|\\ge \\sigma (4t))}\\le \\frac{1}{Q^{\\prime }(\\sigma (4t))} \\left\\Vert Q^{\\prime }wg\\right\\Vert _{L_p(\\mathbb {R})}.$ Then we have from (REF ), $\\inf _{c\\in \\mathbb {R}}\\left\\Vert w(x)(f-c)(x)\\right\\Vert _{L_p(|x|\\ge \\sigma (4t))}\\le Ct\\Vert Q^{\\prime }wg\\Vert _{L_p(\\mathbb {R})}.$ Here, from Lemma REF we have $\\Vert Q^{\\prime }w(f-f(0))\\Vert _{L_p(\\mathbb {R})}\\le C\\Vert wf^{\\prime }\\Vert _{L_p(\\mathbb {R})}.$ From (REF ) and (REF ) we have $\\inf _{c\\in \\mathbb {R}}\\left\\Vert w(x)(f-c)(x)\\right\\Vert _{L_p(|x|\\ge \\sigma (4t))}\\le Ct \\Vert wf^{\\prime }\\Vert _{L_p(\\mathbb {R})}.$ From (REF ) and (REF ) we have the result.", "Lemma 4.6 (cf.", "[4]) Let $1 \\le p \\le \\infty $ and $\\beta >1$ , and let us define $w^{\\sharp }$ with $p$ , $\\beta $ .", "Let $g$ be a real valued function on $\\mathbb {R}$ satisfying $\\Vert gw\\Vert _{L_p(\\mathbb {R})}<\\infty $ and (REF ), then we have $\\left\\Vert w^{\\sharp }(x)\\int _0^x g(t)dt\\right\\Vert _{L_p(\\mathbb {R})}\\le C \\frac{a_n}{n}\\Vert gw\\Vert _{L_p(\\mathbb {R})}.$ Especially, if $w\\in \\mathcal {F}_\\lambda (C^3+), 0<\\lambda <3/2$ , then we have $\\left\\Vert w^{\\sharp }(x)\\int _0^x \\left(f^{\\prime }(t)-v_n(f^{\\prime })(t)\\right)dt\\right\\Vert _{L_p(\\mathbb {R})}\\le C \\frac{a_n}{n}E_{p,n}\\left(w_{1/4},f^{\\prime }\\right).$ When $w\\in \\mathcal {F}^*$ , we also have (REF ) replacing $w_{1/4}$ with $w$ .", "For arbitrary $P_n\\in \\mathcal {P}_n$ , we have by (REF ) and Hölder inequality $\\left|\\int _0^x g(t)dt\\right|\\le \\left\\Vert g(t)w(t)\\right\\Vert _{L_p(\\mathbb {R})}E_{q,n}\\left(w,\\phi _x\\right), \\quad 1\\le p \\le \\infty , 1/p+1/q=1,$ where $\\phi $ is defined in (REF ).", "Then, we obtain by Lemma REF , $\\left|\\int _0^x g(t)dt\\right|&\\le & \\left\\Vert g(t)w(t)\\right\\Vert _{L_p(\\mathbb {R})}\\frac{a_n}{n}\\left(\\int _{\\mathbb {R}}|w(t)\\phi ^{\\prime }_x(t)|^qdt\\right)^{1/q} \\\\&\\le & \\frac{a_n}{n}\\left\\Vert g(t)w(t)\\right\\Vert _{L_p(\\mathbb {R})}|Q^{\\prime }(x)|^{1-1/q}\\left(\\int _0^x Q^{\\prime }(t)w^{-q}(t)dt\\right)^{1/q}\\\\&\\le & C\\frac{a_n}{n}\\left\\Vert g(t)w(t)\\right\\Vert _{L_p(\\mathbb {R})}|Q^{\\prime }(x)|^{1-1/q}w^{-1}(x).$ Here, for $p=1$ we may consider $\\lim _{q\\rightarrow \\infty } \\left( \\int _0^xQ^{\\prime }(t)w^{-q}(t)dt\\right)^{1/q}=\\lim _{q\\rightarrow \\infty } \\left( w^{-q}(t)\\right)^{1/q}=w^{-1}(x).$ Hence, we have $\\left\\Vert \\frac{w(x)}{\\left\\lbrace (1+|Q^{\\prime }(x)|)(1+|x|)^{\\beta }\\right\\rbrace ^{1/p}}\\int _0^x g(t)dt\\right\\Vert _{L_p(\\mathbb {R})}&\\le & C\\left(\\frac{a_n}{n}\\right)\\left\\Vert \\left(1+|x|\\right)^{-\\beta /p}\\right\\Vert _{L_p(\\mathbb {R})}\\left\\Vert gw\\right\\Vert _{L_p(\\mathbb {R})}\\\\&\\le & C\\left(\\frac{a_n}{n}\\right)\\left\\Vert gw\\right\\Vert _{L_p(\\mathbb {R})}.$ Therefore, we have (REF ).", "From (REF ), (REF ) and Proposition REF , we have (REF ).", "Lemma 4.7 Let $w=\\exp (-Q)\\in \\mathcal {F}_\\lambda (C^3+)$ , $0<\\lambda <3/2$ .", "Let $1 \\le p \\le \\infty $ , $\\Vert w_{1/4}f^{\\prime }\\Vert _{L_p(\\mathbb {R})}<\\infty $ , and let $q_{n-1}\\in \\mathcal {P}_n$ be the best approximation of $f^{\\prime }$ with respect to the weight $w$ on $L_p(\\mathbb {R})$ space, that is, $\\left\\Vert (f^{\\prime }-q_{n-1})w\\right\\Vert _{L_p(\\mathbb {R})}=E_{p,n-1}(w,f^{\\prime }).$ Using $q_{n-1}$ , define $F(x)$ and $S_{2n}$ as (REF ) and (REF ).", "Then we have $\\left\\Vert w^{\\sharp }(F-S_{2n})\\right\\Vert _{L_p(\\mathbb {R})}\\le C \\frac{a_n}{n}E_{p,n}\\left(w_{1/4},f^{\\prime }\\right),$ and $\\left\\Vert wS_{2n}^{\\prime }\\right\\Vert _{L_p(\\mathbb {R})}\\le C E_{p,n-1}\\left(w_{1/4},f^{\\prime }\\right).$ When $w\\in \\mathcal {F}^*$ , we also have same results replacing $w_{1/4}$ with $w$ .", "By Lemma REF (REF ), we have the result using the same method as the proof of Lemma REF .", "We will prove it similarly to the proof of Theorem REF .", "First, let $q_{n-1}\\in \\mathcal {P}_{n-1}$ be the polynomial of best approximation of $f^{\\prime }$ with respect to the weight $w$ on $L_p(\\mathbb {R})$ space.", "Then using $q_{n-1}$ , we define $F(x)$ and $S_{2n}$ in the same method as (REF ) and (REF ).", "Then we have using Lemma REF $\\left\\Vert F^{\\prime }w\\right\\Vert _{L_p(\\mathbb {R})}= E_{p,n-1}\\left(w,f^{\\prime }\\right),$ $\\Vert (F-S_{2n})w^{\\sharp }\\Vert _{L_p(\\mathbb {R})}\\le C \\frac{a_n}{n}E_{p,n}\\left(w_{1/4},f^{\\prime }\\right)$ and $\\left\\Vert S_{2n}^{\\prime }w \\right\\Vert _{L_p(\\mathbb {R})}\\le C E_{p,n-1}\\left(w_{1/4},f^{\\prime }\\right).$ Then we see from Theorem REF and (REF ), $\\left\\Vert S_{2n}^{(k)}w_{-(k-1)/2}\\right\\Vert _{L_p(\\mathbb {R})}\\le C \\left(\\frac{n}{a_n}\\right)^{k-1}E_{p,n-1}\\left(w_{1/4},f^{\\prime }\\right)$ and using $w^{\\sharp } \\le w$ and Theorem REF $\\left\\Vert \\left(R_{n}^{(k)}-S_{2n}^{(k)}\\right) w^{\\sharp }_{-k/2}\\right\\Vert _{L_p(\\mathbb {R})}\\le C \\left(\\frac{n}{a_n}\\right)^{k-1}E_{p,n-1}\\left(w_{1/4},f^{\\prime }\\right),$ where $R_{n}\\in \\mathcal {P}_{n}$ denotes the polynomial of best approximation of $F$ with $w$ on $L_p(\\mathbb {R})$ space(by the similar calculation as (REF ) and (REF )).", "Then, we see $w^{\\sharp }_{-k/2}(x) \\le w_{-(k-1)/2}(x)$ .", "By (REF ) and (REF ) and Theorem REF , we have $\\left\\Vert R_{n}^{(k)}w^{\\sharp }_{-k/2}\\right\\Vert _{L_p(\\mathbb {R})}\\le C \\left(\\frac{n}{a_n}\\right)^{k-1}E_{p,n-1}\\left(w_{1/4},f^{\\prime }\\right)\\le C E_{p,n-k}\\left(w_{1/4},f^{(k)}\\right).$ By the same reason to (REF ), we know that $P_{p,n;f,w}:=Q_n+R_n$ is the polynomial of best approximation of $f$ with $w$ on $L_p(\\mathbb {R})$ space.", "Therefore, using $P_{p,n;f,w}$ , (REF ) and the method of mathematical induction, we have for $1\\le k\\le r+1$ , $\\left\\Vert \\left(f^{(k)}-P_{p,n;f,w}^{(k)}\\right)w^{\\sharp }_{-k/2}\\right\\Vert _{L_p(\\mathbb {R})}\\le C E_{p,n-k}\\left(w_{1/4},f^{(k)}\\right).$ It follows from Theorem REF .", "If we apply Theorem REF with $w_{k/2}$ and $w^{\\flat }_{k/2}$ , then we can obtain the results.", "Monotone Approximation Let $r>0$ be an integer.", "Let $k$ and $\\ell $ be integers with $0\\le k\\le \\ell \\le r$ .", "In this section, we consider a real function $f$ on $\\mathbb {R}$ such that $f^{(r)}(x)$ is continuous in $\\mathbb {R}$ and we let $a_j(x)$ , $j=k,k+1,...,\\ell $ be bounded on $\\mathbb {R}$ .", "Now, we define the linear differential operator (cf.", "[1]) $L:=L_{k,\\ell }:=\\sum _{j=k}^\\ell a_j(x)[d^j/dx^j].$ G. A. Anastassiou and O. Shisha [1] consider the operator (REF ) with $a_j(x)$ under some condition on $[-1,1]$ .", "They showed that if $L(f) \\ge 0$ for $f\\in C^{(r)}[-1,1]$ , there exist $Q_n \\in \\mathcal {P}_n$ such that $L(Q_n)\\ge 0$ and for some constant $C>0$ , $\\left\\Vert f-Q_n\\right\\Vert _{L_{\\infty }([-1,1])}\\le C n^{\\ell -r}\\omega \\left(f^{(p)};\\frac{1}{n}\\right),$ where $\\omega \\left(f^{(p)};t\\right)$ is the modulus of continuity.", "In this section, we will obtain a similar result with exponential-type weighted $L_{\\infty }$ -norm as the above result.", "Our main theorem is as follows.", "Theorem 5.1 Let $k$ and $\\ell $ be integers with $0\\le k\\le \\ell \\le r$ .", "Let $w=\\exp (-Q)\\in \\mathcal {F}_\\lambda (C^{r+3}+)$ , and let $T(x)$ be continuous on $\\mathbb {R}$ .", "Suppose that $w(x)f^{(r)}(x)\\rightarrow 0$ as $|x|\\rightarrow \\infty $ .", "Let $P_{n;f,w_{-1/4}}\\in \\mathcal {P}_n$ be the best approximation for $f$ with the weight $w_{-1/4}$ on $\\mathbb {R}$ .", "Suppose that for a certain $\\delta >0$ , $L(f;x)\\ge \\delta , \\quad x\\in \\mathbb {R}.$ Then, for every integer $n\\ge 1$ and $j=0,1,...,\\ell $ , $\\left\\Vert \\left(f^{(j)}-P_{n;f,w_{-1/4}}^{(j)}\\right)wT^{-(2j+1)/4}(x)\\right\\Vert _{L_\\infty (\\mathbb {R})}\\le C_j\\left(\\frac{a_n}{n}\\right)^{r-j}E_{n-r}\\left(w,f^{(r)}\\right),$ where $C_j>0$ , $0\\le j\\le \\ell $ , are independent of $n$ or $f$ , and for any fixed number $M>0$ there exists a constant $N(M,\\ell ,\\delta )>0$ such that $L(P_{n;f,w_{-1/4}};x)\\ge \\frac{\\delta }{2}, \\quad |x|\\le M, \\quad n\\ge N(M,\\ell ,\\delta ).$ From Corollary REF , we have (REF ).", "Hence, we also have $&&\\left|\\left(L(f;x)-L\\left(P_{n;f,w_{-1/4}};x\\right)\\right)w(x)T^{-(2\\ell +1)/4}(x)\\right|\\\\&=&\\left|\\sum _{j=k}^\\ell a_j(x)\\left\\lbrace f^{(j)}(x)-P_{n;f,w_{-1/4}}^{(j)}(x)\\right\\rbrace w(x)T^{-(2\\ell +1)/4}(x)\\right|\\\\&\\le & E_{n-r}\\left(w,f^{(r)}\\right)\\sum _{j=k}^\\ell |a_j(x)|C_j\\left(\\frac{a_n}{n}\\right)^{r-j}\\\\&\\le & C_{k,\\ell }\\left(\\frac{a_n}{n}\\right)^{r-\\ell }E_{n-r}\\left(w,f^{(r)}\\right),$ where we set $C_{k,\\ell }:=\\sum _{j=k}^\\ell \\Vert a_j\\Vert _{L_\\infty (\\mathbb {R})}C_j$ .", "Then we have for $|x|\\le M$ $&&\\left|L(f;x)-L\\left(P_{n;f,w_{-1/4}};x\\right)\\right|\\\\&\\le & C_{k,\\ell }\\left\\Vert w^{-1}(x)T^{(2\\ell +1)/4}(x)\\right\\Vert _{L_\\infty (|x|\\le M)}\\left(\\frac{a_n}{n}\\right)^{r-\\ell }E_{n-r}\\left(w,f^{(r)}\\right).$ Here, for $\\delta >0$ there exists $N(M,\\ell ,\\delta )>0$ such that for $n\\ge N(M,\\ell ,\\delta )$ $C_{k,\\ell }\\left\\Vert w^{-1}(x)T^{(2\\ell +1)/4}(x)\\right\\Vert _{L_\\infty (|x|\\le M)}\\left(\\frac{a_n}{n}\\right)^{r-\\ell }E_{n-r}\\left(w,f^{(r)}\\right)\\le \\frac{\\delta }{2}.$ This follows from $w(x)f^{(r)}(x)\\rightarrow 0$ as $|x|\\rightarrow \\infty $ .", "Therefore we see $\\frac{\\delta }{2}\\le L(f;x)-\\frac{\\delta }{2}\\le L(P_{n;f,w_{-1/4}};x).$ Consequently, we have (REF )." ], [ "Theorems in $L_p(\\mathbb {R})$ {{formula:1ab433fd-f200-42ce-b691-f824d8ab1fa4}}", "In this section we will give an analogy of Theorem REF in $L_p(\\mathbb {R})$ -space ($1 \\le p \\le \\infty $ ) and we will prove it using the same method as the proof of Theorem REF .", "Let $1 \\le p \\le \\infty $ .", "Let $w=\\exp (-Q)\\in \\mathcal {F}_\\lambda (C^3+)$ , $0<\\lambda <3/2$ , and let $\\beta >1$ be fixed.", "Then we set $w^{\\sharp }$ and $w^{\\flat }$ as follows; $&& \\frac{w(x)}{\\left\\lbrace (1+|Q^{\\prime }(x)|)(1+|x|)^{\\beta }\\right\\rbrace ^{1/p}}\\sim w^{\\sharp }(x)\\in \\mathcal {F}(C^2+);\\\\&&w(x)\\left\\lbrace (1+|Q^{\\prime }(x)|)(1+|x|)^{\\beta }\\right\\rbrace ^{1/p} \\sim w^{\\flat }(x)\\in \\mathcal {F}(C^2+)$ (see Theorem REF ).", "Theorem 4.1 Let $r \\ge 0$ be an integer.", "Let $w=\\exp (-Q)\\in \\mathcal {F}_\\lambda (C^{r+2}+)$ , $0< \\lambda <(r+2)/(r+1)$ , and let $\\beta >1$ be fixed.", "Suppose that $T^{1/4}f^{(r)}w\\in L_p(\\mathbb {R})$ .", "Let $P_{p,n;f,w}\\in \\mathcal {P}_n$ be the best approximation of $f$ with respect to the weight $w$ in $L_p(\\mathbb {R})$ -space, that is, $E_{p,n}\\left(w,f\\right):=\\inf _{P\\in \\mathcal {P}_n}\\left\\Vert \\left(f-P\\right)w\\right\\Vert _{L_p(\\mathbb {R})}=\\left\\Vert \\left(f-P_{p,n;f,w}\\right)w\\right\\Vert _{L_p(\\mathbb {R})}.$ Then there exists an absolute constant $C_r>0$ which depends only on $r$ such that for $0\\le k\\le r$ and $x\\in \\mathbb {R}$ , $\\left\\Vert \\left(f^{(k)}-P_{p,n;f,w}^{(k)}\\right)w^{\\sharp }_{-k/2}\\right\\Vert _{L_p(\\mathbb {R})}&\\le & C_r E_{p,n-k}\\left(w_{1/4},f^{(k)}\\right)\\\\&\\le & C_r \\left(\\frac{a_n}{n}\\right)^{r-k}E_{p,n-r}\\left(w_{1/4},f^{(r)}\\right).$ When $w\\in \\mathcal {F}^*$ , we can replace $w_{1/4}$ and $w^{\\sharp }_{-k/2}$ with $w$ and $w^{\\sharp }$ , respectively in the above.", "If we apply Theorem REF with $w_{-1/4}$ , then we have the following.", "Corollary 4.2 Let $r \\ge 0$ be an integer.", "Let $w=\\exp (-Q)\\in \\mathcal {F}_\\lambda (C^{r+3}+)$ , $0< \\lambda <(r+3)/(r+2)$ , and let $\\beta >0$ be fixed.", "Suppose that $wf^{(r)}\\in L_p(\\mathbb {R})$ .", "Then for $0\\le k\\le r$ we have $\\left\\Vert \\left(f^{(k)}-P_{p,n;f,w_{-1/4}}^{(k)}\\right)w^{\\sharp }_{-(2k+1)/4}\\right\\Vert _{L_p(\\mathbb {R})}&\\le & C_r E_{p,n-k}\\left(w,f^{(k)}\\right)\\\\&\\le & C_r \\left(\\frac{a_n}{n}\\right)^{r-k}E_{p,n-r}\\left(w,f^{(r)}\\right).$ When $w\\in \\mathcal {F}^*$ , we can omit $T^{-(2k+1)/4}$ in the above.", "Corollary 4.3 Let $r \\ge 0$ be an integer.", "Let $w=\\exp (-Q)\\in \\mathcal {F}_\\lambda (C^{r+3}+)$ , $0< \\lambda <(r+3)/(r+2)$ , and let $\\beta >0$ be fixed.", "(1) Let $w_{(2r+1)/4}f^{(r)}\\in L_p(\\mathbb {R})$ .", "Then, for each $k$ $(0\\le k\\le r)$ and the best approximation polynomial $P_{p,n;f,w_{k/2}}$ , we have $\\left\\Vert \\left(f^{(k)}-P_{p,n;f,w_{k/2}}^{(k)}\\right)w^{\\sharp }\\right\\Vert _{L_p(\\mathbb {R})}&\\le & C_r E_{p,n-k}\\left(w_{(2k+1)/4},f^{(k)}\\right)\\\\&\\le & C_r \\left(\\frac{a_n}{n}\\right)^{r-k}E_{p,n-r}\\left(w_{(2k+1)/4},f^{(r)}\\right).$    (2) Let $w^{\\flat }_{(2r+1)/4}f^{(r)}\\in L_p(\\mathbb {R})$ .", "Then, for each $k$ $(0\\le k\\le r)$ and the best approximation polynomial $P_{p,n;f,w^{\\flat }_{k/2}}$ , we have $\\left\\Vert \\left(f^{(k)}-P_{p,n;f,w^{\\flat }_{k/2}}^{(k)}\\right)w\\right\\Vert _{L_p(\\mathbb {R})}&\\le & C_r E_{p,n-k}\\left(w^{\\flat }_{(2k+1)/4},f^{(k)}\\right)\\\\&\\le & C_r \\left(\\frac{a_n}{n}\\right)^{r-k}E_{p,n-r}\\left(w^{\\flat }_{(2k+1)/4},f^{(r)}\\right).$ When $w\\in \\mathcal {F}^*$ , we can replace $w_{(2k+1)/4}$ and $w^{\\flat }_{(2k+1)/4}$ with $w$ and $w^{\\flat }$ , respectively in the above.", "Especially when $p=\\infty $ , we can refer to $w^{\\sharp }$ or $w^{\\flat }$ as $w$ .", "In this case, we can note that Corollary REF and Corollary REF imply Corollary REF , and Corollary REF , respectively.", "To prove Theorem REF we need to prepare some notations and lemmas.", "Lemma 4.4 ([6]) Let $w\\in \\mathcal {F}(C^2+)$ and let $1\\le p\\le \\infty $ .", "If $g:\\mathbb {R}\\rightarrow \\mathbb {R}$ is absolutely continuous, $g(0)=0$ , and $wg^{\\prime }\\in L_p(\\mathbb {R})$ , then $\\left\\Vert Q^{\\prime }wg\\right\\Vert _{L_p(\\mathbb {R})}\\le C\\left\\Vert wg^{\\prime }\\right\\Vert _{L_p(\\mathbb {R})}.$ Lemma 4.5 (cf.", "[9]) Let $w\\in \\mathcal {F}(C^2+)$ and let $1\\le p\\le \\infty $ .", "If $wf^{\\prime }\\in L_p(\\mathbb {R})$ , then $E_{p,n}(w,f)\\le C\\omega _p\\left(f,w,\\frac{a_n}{n}\\right)\\le C \\frac{a_n}{n}\\left\\Vert wf^{\\prime }\\right\\Vert _{L_p(\\mathbb {R})}.$ The first inequality follows from Proposition REF .", "We show the second inequality.", "By [9] we have $&&\\left\\Vert w(x)\\left\\lbrace f\\left(x+\\frac{h}{2}\\Phi _t(x)\\right)-f\\left(x-\\frac{h}{2}\\Phi _t(x)\\right)\\right\\rbrace \\right\\Vert ^p_{L_p(|x|\\le \\sigma (2t))} \\\\&=&h^p\\int _{\\mathbb {R}}|w(x)\\Phi _t(x)f^{\\prime }(x_t)|^pdx\\le Ch^p\\int _{\\mathbb {R}}|w(x)f^{\\prime }(x)|^pdx.$ Hence we see $\\sup _{0<h\\le t}\\left\\Vert w(x)\\left\\lbrace f\\left(x+\\frac{h}{2}\\Phi _t(x)\\right)-f\\left(x-\\frac{h}{2}\\Phi _t(x)\\right)\\right\\rbrace \\right\\Vert _{L_p(|x|\\le \\sigma (2t))}\\le Ct\\left\\Vert wf^{\\prime }\\right\\Vert _{L_p(\\mathbb {R})}.$ Now, we estimate $\\left\\Vert w(x)(f-c)(x)\\right\\Vert _{L_p(|x|\\ge \\sigma (4t))}.$ Let $g(x):=f(x)-f(0)$ .", "$\\inf _{c\\in \\mathbb {R}}\\left\\Vert w(x)\\left(f-c\\right)(x)\\right\\Vert _{L_p(|x|\\ge \\sigma (4t))}\\le \\frac{1}{Q^{\\prime }(\\sigma (4t))} \\left\\Vert Q^{\\prime }wg\\right\\Vert _{L_p(\\mathbb {R})}.$ Then we have from (REF ), $\\inf _{c\\in \\mathbb {R}}\\left\\Vert w(x)(f-c)(x)\\right\\Vert _{L_p(|x|\\ge \\sigma (4t))}\\le Ct\\Vert Q^{\\prime }wg\\Vert _{L_p(\\mathbb {R})}.$ Here, from Lemma REF we have $\\Vert Q^{\\prime }w(f-f(0))\\Vert _{L_p(\\mathbb {R})}\\le C\\Vert wf^{\\prime }\\Vert _{L_p(\\mathbb {R})}.$ From (REF ) and (REF ) we have $\\inf _{c\\in \\mathbb {R}}\\left\\Vert w(x)(f-c)(x)\\right\\Vert _{L_p(|x|\\ge \\sigma (4t))}\\le Ct \\Vert wf^{\\prime }\\Vert _{L_p(\\mathbb {R})}.$ From (REF ) and (REF ) we have the result.", "Lemma 4.6 (cf.", "[4]) Let $1 \\le p \\le \\infty $ and $\\beta >1$ , and let us define $w^{\\sharp }$ with $p$ , $\\beta $ .", "Let $g$ be a real valued function on $\\mathbb {R}$ satisfying $\\Vert gw\\Vert _{L_p(\\mathbb {R})}<\\infty $ and (REF ), then we have $\\left\\Vert w^{\\sharp }(x)\\int _0^x g(t)dt\\right\\Vert _{L_p(\\mathbb {R})}\\le C \\frac{a_n}{n}\\Vert gw\\Vert _{L_p(\\mathbb {R})}.$ Especially, if $w\\in \\mathcal {F}_\\lambda (C^3+), 0<\\lambda <3/2$ , then we have $\\left\\Vert w^{\\sharp }(x)\\int _0^x \\left(f^{\\prime }(t)-v_n(f^{\\prime })(t)\\right)dt\\right\\Vert _{L_p(\\mathbb {R})}\\le C \\frac{a_n}{n}E_{p,n}\\left(w_{1/4},f^{\\prime }\\right).$ When $w\\in \\mathcal {F}^*$ , we also have (REF ) replacing $w_{1/4}$ with $w$ .", "For arbitrary $P_n\\in \\mathcal {P}_n$ , we have by (REF ) and Hölder inequality $\\left|\\int _0^x g(t)dt\\right|\\le \\left\\Vert g(t)w(t)\\right\\Vert _{L_p(\\mathbb {R})}E_{q,n}\\left(w,\\phi _x\\right), \\quad 1\\le p \\le \\infty , 1/p+1/q=1,$ where $\\phi $ is defined in (REF ).", "Then, we obtain by Lemma REF , $\\left|\\int _0^x g(t)dt\\right|&\\le & \\left\\Vert g(t)w(t)\\right\\Vert _{L_p(\\mathbb {R})}\\frac{a_n}{n}\\left(\\int _{\\mathbb {R}}|w(t)\\phi ^{\\prime }_x(t)|^qdt\\right)^{1/q} \\\\&\\le & \\frac{a_n}{n}\\left\\Vert g(t)w(t)\\right\\Vert _{L_p(\\mathbb {R})}|Q^{\\prime }(x)|^{1-1/q}\\left(\\int _0^x Q^{\\prime }(t)w^{-q}(t)dt\\right)^{1/q}\\\\&\\le & C\\frac{a_n}{n}\\left\\Vert g(t)w(t)\\right\\Vert _{L_p(\\mathbb {R})}|Q^{\\prime }(x)|^{1-1/q}w^{-1}(x).$ Here, for $p=1$ we may consider $\\lim _{q\\rightarrow \\infty } \\left( \\int _0^xQ^{\\prime }(t)w^{-q}(t)dt\\right)^{1/q}=\\lim _{q\\rightarrow \\infty } \\left( w^{-q}(t)\\right)^{1/q}=w^{-1}(x).$ Hence, we have $\\left\\Vert \\frac{w(x)}{\\left\\lbrace (1+|Q^{\\prime }(x)|)(1+|x|)^{\\beta }\\right\\rbrace ^{1/p}}\\int _0^x g(t)dt\\right\\Vert _{L_p(\\mathbb {R})}&\\le & C\\left(\\frac{a_n}{n}\\right)\\left\\Vert \\left(1+|x|\\right)^{-\\beta /p}\\right\\Vert _{L_p(\\mathbb {R})}\\left\\Vert gw\\right\\Vert _{L_p(\\mathbb {R})}\\\\&\\le & C\\left(\\frac{a_n}{n}\\right)\\left\\Vert gw\\right\\Vert _{L_p(\\mathbb {R})}.$ Therefore, we have (REF ).", "From (REF ), (REF ) and Proposition REF , we have (REF ).", "Lemma 4.7 Let $w=\\exp (-Q)\\in \\mathcal {F}_\\lambda (C^3+)$ , $0<\\lambda <3/2$ .", "Let $1 \\le p \\le \\infty $ , $\\Vert w_{1/4}f^{\\prime }\\Vert _{L_p(\\mathbb {R})}<\\infty $ , and let $q_{n-1}\\in \\mathcal {P}_n$ be the best approximation of $f^{\\prime }$ with respect to the weight $w$ on $L_p(\\mathbb {R})$ space, that is, $\\left\\Vert (f^{\\prime }-q_{n-1})w\\right\\Vert _{L_p(\\mathbb {R})}=E_{p,n-1}(w,f^{\\prime }).$ Using $q_{n-1}$ , define $F(x)$ and $S_{2n}$ as (REF ) and (REF ).", "Then we have $\\left\\Vert w^{\\sharp }(F-S_{2n})\\right\\Vert _{L_p(\\mathbb {R})}\\le C \\frac{a_n}{n}E_{p,n}\\left(w_{1/4},f^{\\prime }\\right),$ and $\\left\\Vert wS_{2n}^{\\prime }\\right\\Vert _{L_p(\\mathbb {R})}\\le C E_{p,n-1}\\left(w_{1/4},f^{\\prime }\\right).$ When $w\\in \\mathcal {F}^*$ , we also have same results replacing $w_{1/4}$ with $w$ .", "By Lemma REF (REF ), we have the result using the same method as the proof of Lemma REF .", "We will prove it similarly to the proof of Theorem REF .", "First, let $q_{n-1}\\in \\mathcal {P}_{n-1}$ be the polynomial of best approximation of $f^{\\prime }$ with respect to the weight $w$ on $L_p(\\mathbb {R})$ space.", "Then using $q_{n-1}$ , we define $F(x)$ and $S_{2n}$ in the same method as (REF ) and (REF ).", "Then we have using Lemma REF $\\left\\Vert F^{\\prime }w\\right\\Vert _{L_p(\\mathbb {R})}= E_{p,n-1}\\left(w,f^{\\prime }\\right),$ $\\Vert (F-S_{2n})w^{\\sharp }\\Vert _{L_p(\\mathbb {R})}\\le C \\frac{a_n}{n}E_{p,n}\\left(w_{1/4},f^{\\prime }\\right)$ and $\\left\\Vert S_{2n}^{\\prime }w \\right\\Vert _{L_p(\\mathbb {R})}\\le C E_{p,n-1}\\left(w_{1/4},f^{\\prime }\\right).$ Then we see from Theorem REF and (REF ), $\\left\\Vert S_{2n}^{(k)}w_{-(k-1)/2}\\right\\Vert _{L_p(\\mathbb {R})}\\le C \\left(\\frac{n}{a_n}\\right)^{k-1}E_{p,n-1}\\left(w_{1/4},f^{\\prime }\\right)$ and using $w^{\\sharp } \\le w$ and Theorem REF $\\left\\Vert \\left(R_{n}^{(k)}-S_{2n}^{(k)}\\right) w^{\\sharp }_{-k/2}\\right\\Vert _{L_p(\\mathbb {R})}\\le C \\left(\\frac{n}{a_n}\\right)^{k-1}E_{p,n-1}\\left(w_{1/4},f^{\\prime }\\right),$ where $R_{n}\\in \\mathcal {P}_{n}$ denotes the polynomial of best approximation of $F$ with $w$ on $L_p(\\mathbb {R})$ space(by the similar calculation as (REF ) and (REF )).", "Then, we see $w^{\\sharp }_{-k/2}(x) \\le w_{-(k-1)/2}(x)$ .", "By (REF ) and (REF ) and Theorem REF , we have $\\left\\Vert R_{n}^{(k)}w^{\\sharp }_{-k/2}\\right\\Vert _{L_p(\\mathbb {R})}\\le C \\left(\\frac{n}{a_n}\\right)^{k-1}E_{p,n-1}\\left(w_{1/4},f^{\\prime }\\right)\\le C E_{p,n-k}\\left(w_{1/4},f^{(k)}\\right).$ By the same reason to (REF ), we know that $P_{p,n;f,w}:=Q_n+R_n$ is the polynomial of best approximation of $f$ with $w$ on $L_p(\\mathbb {R})$ space.", "Therefore, using $P_{p,n;f,w}$ , (REF ) and the method of mathematical induction, we have for $1\\le k\\le r+1$ , $\\left\\Vert \\left(f^{(k)}-P_{p,n;f,w}^{(k)}\\right)w^{\\sharp }_{-k/2}\\right\\Vert _{L_p(\\mathbb {R})}\\le C E_{p,n-k}\\left(w_{1/4},f^{(k)}\\right).$ It follows from Theorem REF .", "If we apply Theorem REF with $w_{k/2}$ and $w^{\\flat }_{k/2}$ , then we can obtain the results.", "Monotone Approximation Let $r>0$ be an integer.", "Let $k$ and $\\ell $ be integers with $0\\le k\\le \\ell \\le r$ .", "In this section, we consider a real function $f$ on $\\mathbb {R}$ such that $f^{(r)}(x)$ is continuous in $\\mathbb {R}$ and we let $a_j(x)$ , $j=k,k+1,...,\\ell $ be bounded on $\\mathbb {R}$ .", "Now, we define the linear differential operator (cf.", "[1]) $L:=L_{k,\\ell }:=\\sum _{j=k}^\\ell a_j(x)[d^j/dx^j].$ G. A. Anastassiou and O. Shisha [1] consider the operator (REF ) with $a_j(x)$ under some condition on $[-1,1]$ .", "They showed that if $L(f) \\ge 0$ for $f\\in C^{(r)}[-1,1]$ , there exist $Q_n \\in \\mathcal {P}_n$ such that $L(Q_n)\\ge 0$ and for some constant $C>0$ , $\\left\\Vert f-Q_n\\right\\Vert _{L_{\\infty }([-1,1])}\\le C n^{\\ell -r}\\omega \\left(f^{(p)};\\frac{1}{n}\\right),$ where $\\omega \\left(f^{(p)};t\\right)$ is the modulus of continuity.", "In this section, we will obtain a similar result with exponential-type weighted $L_{\\infty }$ -norm as the above result.", "Our main theorem is as follows.", "Theorem 5.1 Let $k$ and $\\ell $ be integers with $0\\le k\\le \\ell \\le r$ .", "Let $w=\\exp (-Q)\\in \\mathcal {F}_\\lambda (C^{r+3}+)$ , and let $T(x)$ be continuous on $\\mathbb {R}$ .", "Suppose that $w(x)f^{(r)}(x)\\rightarrow 0$ as $|x|\\rightarrow \\infty $ .", "Let $P_{n;f,w_{-1/4}}\\in \\mathcal {P}_n$ be the best approximation for $f$ with the weight $w_{-1/4}$ on $\\mathbb {R}$ .", "Suppose that for a certain $\\delta >0$ , $L(f;x)\\ge \\delta , \\quad x\\in \\mathbb {R}.$ Then, for every integer $n\\ge 1$ and $j=0,1,...,\\ell $ , $\\left\\Vert \\left(f^{(j)}-P_{n;f,w_{-1/4}}^{(j)}\\right)wT^{-(2j+1)/4}(x)\\right\\Vert _{L_\\infty (\\mathbb {R})}\\le C_j\\left(\\frac{a_n}{n}\\right)^{r-j}E_{n-r}\\left(w,f^{(r)}\\right),$ where $C_j>0$ , $0\\le j\\le \\ell $ , are independent of $n$ or $f$ , and for any fixed number $M>0$ there exists a constant $N(M,\\ell ,\\delta )>0$ such that $L(P_{n;f,w_{-1/4}};x)\\ge \\frac{\\delta }{2}, \\quad |x|\\le M, \\quad n\\ge N(M,\\ell ,\\delta ).$ From Corollary REF , we have (REF ).", "Hence, we also have $&&\\left|\\left(L(f;x)-L\\left(P_{n;f,w_{-1/4}};x\\right)\\right)w(x)T^{-(2\\ell +1)/4}(x)\\right|\\\\&=&\\left|\\sum _{j=k}^\\ell a_j(x)\\left\\lbrace f^{(j)}(x)-P_{n;f,w_{-1/4}}^{(j)}(x)\\right\\rbrace w(x)T^{-(2\\ell +1)/4}(x)\\right|\\\\&\\le & E_{n-r}\\left(w,f^{(r)}\\right)\\sum _{j=k}^\\ell |a_j(x)|C_j\\left(\\frac{a_n}{n}\\right)^{r-j}\\\\&\\le & C_{k,\\ell }\\left(\\frac{a_n}{n}\\right)^{r-\\ell }E_{n-r}\\left(w,f^{(r)}\\right),$ where we set $C_{k,\\ell }:=\\sum _{j=k}^\\ell \\Vert a_j\\Vert _{L_\\infty (\\mathbb {R})}C_j$ .", "Then we have for $|x|\\le M$ $&&\\left|L(f;x)-L\\left(P_{n;f,w_{-1/4}};x\\right)\\right|\\\\&\\le & C_{k,\\ell }\\left\\Vert w^{-1}(x)T^{(2\\ell +1)/4}(x)\\right\\Vert _{L_\\infty (|x|\\le M)}\\left(\\frac{a_n}{n}\\right)^{r-\\ell }E_{n-r}\\left(w,f^{(r)}\\right).$ Here, for $\\delta >0$ there exists $N(M,\\ell ,\\delta )>0$ such that for $n\\ge N(M,\\ell ,\\delta )$ $C_{k,\\ell }\\left\\Vert w^{-1}(x)T^{(2\\ell +1)/4}(x)\\right\\Vert _{L_\\infty (|x|\\le M)}\\left(\\frac{a_n}{n}\\right)^{r-\\ell }E_{n-r}\\left(w,f^{(r)}\\right)\\le \\frac{\\delta }{2}.$ This follows from $w(x)f^{(r)}(x)\\rightarrow 0$ as $|x|\\rightarrow \\infty $ .", "Therefore we see $\\frac{\\delta }{2}\\le L(f;x)-\\frac{\\delta }{2}\\le L(P_{n;f,w_{-1/4}};x).$ Consequently, we have (REF )." ], [ "Monotone Approximation", "Let $r>0$ be an integer.", "Let $k$ and $\\ell $ be integers with $0\\le k\\le \\ell \\le r$ .", "In this section, we consider a real function $f$ on $\\mathbb {R}$ such that $f^{(r)}(x)$ is continuous in $\\mathbb {R}$ and we let $a_j(x)$ , $j=k,k+1,...,\\ell $ be bounded on $\\mathbb {R}$ .", "Now, we define the linear differential operator (cf.", "[1]) $L:=L_{k,\\ell }:=\\sum _{j=k}^\\ell a_j(x)[d^j/dx^j].$ G. A. Anastassiou and O. Shisha [1] consider the operator (REF ) with $a_j(x)$ under some condition on $[-1,1]$ .", "They showed that if $L(f) \\ge 0$ for $f\\in C^{(r)}[-1,1]$ , there exist $Q_n \\in \\mathcal {P}_n$ such that $L(Q_n)\\ge 0$ and for some constant $C>0$ , $\\left\\Vert f-Q_n\\right\\Vert _{L_{\\infty }([-1,1])}\\le C n^{\\ell -r}\\omega \\left(f^{(p)};\\frac{1}{n}\\right),$ where $\\omega \\left(f^{(p)};t\\right)$ is the modulus of continuity.", "In this section, we will obtain a similar result with exponential-type weighted $L_{\\infty }$ -norm as the above result.", "Our main theorem is as follows.", "Theorem 5.1 Let $k$ and $\\ell $ be integers with $0\\le k\\le \\ell \\le r$ .", "Let $w=\\exp (-Q)\\in \\mathcal {F}_\\lambda (C^{r+3}+)$ , and let $T(x)$ be continuous on $\\mathbb {R}$ .", "Suppose that $w(x)f^{(r)}(x)\\rightarrow 0$ as $|x|\\rightarrow \\infty $ .", "Let $P_{n;f,w_{-1/4}}\\in \\mathcal {P}_n$ be the best approximation for $f$ with the weight $w_{-1/4}$ on $\\mathbb {R}$ .", "Suppose that for a certain $\\delta >0$ , $L(f;x)\\ge \\delta , \\quad x\\in \\mathbb {R}.$ Then, for every integer $n\\ge 1$ and $j=0,1,...,\\ell $ , $\\left\\Vert \\left(f^{(j)}-P_{n;f,w_{-1/4}}^{(j)}\\right)wT^{-(2j+1)/4}(x)\\right\\Vert _{L_\\infty (\\mathbb {R})}\\le C_j\\left(\\frac{a_n}{n}\\right)^{r-j}E_{n-r}\\left(w,f^{(r)}\\right),$ where $C_j>0$ , $0\\le j\\le \\ell $ , are independent of $n$ or $f$ , and for any fixed number $M>0$ there exists a constant $N(M,\\ell ,\\delta )>0$ such that $L(P_{n;f,w_{-1/4}};x)\\ge \\frac{\\delta }{2}, \\quad |x|\\le M, \\quad n\\ge N(M,\\ell ,\\delta ).$ From Corollary REF , we have (REF ).", "Hence, we also have $&&\\left|\\left(L(f;x)-L\\left(P_{n;f,w_{-1/4}};x\\right)\\right)w(x)T^{-(2\\ell +1)/4}(x)\\right|\\\\&=&\\left|\\sum _{j=k}^\\ell a_j(x)\\left\\lbrace f^{(j)}(x)-P_{n;f,w_{-1/4}}^{(j)}(x)\\right\\rbrace w(x)T^{-(2\\ell +1)/4}(x)\\right|\\\\&\\le & E_{n-r}\\left(w,f^{(r)}\\right)\\sum _{j=k}^\\ell |a_j(x)|C_j\\left(\\frac{a_n}{n}\\right)^{r-j}\\\\&\\le & C_{k,\\ell }\\left(\\frac{a_n}{n}\\right)^{r-\\ell }E_{n-r}\\left(w,f^{(r)}\\right),$ where we set $C_{k,\\ell }:=\\sum _{j=k}^\\ell \\Vert a_j\\Vert _{L_\\infty (\\mathbb {R})}C_j$ .", "Then we have for $|x|\\le M$ $&&\\left|L(f;x)-L\\left(P_{n;f,w_{-1/4}};x\\right)\\right|\\\\&\\le & C_{k,\\ell }\\left\\Vert w^{-1}(x)T^{(2\\ell +1)/4}(x)\\right\\Vert _{L_\\infty (|x|\\le M)}\\left(\\frac{a_n}{n}\\right)^{r-\\ell }E_{n-r}\\left(w,f^{(r)}\\right).$ Here, for $\\delta >0$ there exists $N(M,\\ell ,\\delta )>0$ such that for $n\\ge N(M,\\ell ,\\delta )$ $C_{k,\\ell }\\left\\Vert w^{-1}(x)T^{(2\\ell +1)/4}(x)\\right\\Vert _{L_\\infty (|x|\\le M)}\\left(\\frac{a_n}{n}\\right)^{r-\\ell }E_{n-r}\\left(w,f^{(r)}\\right)\\le \\frac{\\delta }{2}.$ This follows from $w(x)f^{(r)}(x)\\rightarrow 0$ as $|x|\\rightarrow \\infty $ .", "Therefore we see $\\frac{\\delta }{2}\\le L(f;x)-\\frac{\\delta }{2}\\le L(P_{n;f,w_{-1/4}};x).$ Consequently, we have (REF )." ] ]
1403.0477
[ [ "Formation of Magnetized Prestellar Cores with Ambipolar Diffusion and\n Turbulence" ], [ "Abstract We investigate the roles of magnetic fields and ambipolar diffusion during prestellar core formation in turbulent giant molecular clouds (GMCs), using three-dimensional numerical simulations.", "Our simulations focus on the shocked layer produced by a converging flow within a GMC, and survey varying ionization and angle between the upstream flow and magnetic field.", "We also include ideal magnetohydrodynamic (MHD) and hydrodynamic models.", "From our simulations, we identify hundreds of self-gravitating cores that form within 1 Myr, with masses M ~ 0.04 - 2.5 solar-mass and sizes L ~ 0.015 - 0.07 pc, consistent with observations of the peak of the core mass function (CMF).", "Median values are M = 0.47 solar-mass and L = 0.03 pc.", "Core masses and sizes do not depend on either the ionization or upstream magnetic field direction.", "In contrast, the mass-to-magnetic flux ratio does increase with lower ionization, from twice to four times the critical value.", "The higher mass-to-flux ratio for low ionization is the result of enhanced transient ambipolar diffusion when the shocked layer first forms.", "However, ambipolar diffusion is not necessary to form low-mass supercritical cores.", "For ideal MHD, we find similar masses to other cases.", "These masses are 1 - 2 orders of magnitude lower than the value that defines a magnetically supercritical sphere under post-shock ambient conditions.", "This discrepancy is the result of anisotropic contraction along field lines, which is clearly evident in both ideal MHD and diffusive simulations.", "We interpret our numerical findings using a simple scaling argument which suggests that gravitationally critical core masses will depend on the sound speed and mean turbulent pressure in a cloud, regardless of magnetic effects." ], [ "Introduction", "The formation of stars begins with dense molecular cores , .", "These cores form through the concentration of overdense regions within turbulent, filamentary GMCs; subsequent core collapse leads to protostellar (or protobinary)/disk systems.", "Magnetic fields are important at all scales during this process , : the cloud-scale magnetic field can limit compression in interstellar shocks that create dense clumps and filaments in which cores form, while the local magnetic field within individual cores can prevent collapse if it is large enough , , , and can help to remove angular momentum during the disk formation process if cores are successful in collapsing , , , .", "The significance of magnetic fields in self-gravitating cores can be quantified by the ratio of mass to magnetic flux; only if the mass-to-flux ratio exceeds a critical value is gravitational collapse possible.", "How the mass-to-flux ratio increases from the strongly-magnetized interstellar medium to weakly-magnetized stars is a fundamental problem of star formation , .", "Here, as suggested in , we consider core formation in GMCs with highly supersonic turbulence and non-ideal MHD.", "Magnetic fields are coupled only to charged particles, while the gas in GMCs and their substructures is mostly neutral.", "The ability of magnetic fields to affect core and star formation thus depends on the collisional coupling between neutrals and ions.", "Ambipolar diffusion is the non-ideal MHD process that allows charged particles to drift relative to the neutrals, with a drag force proportional to the collision rate .", "Ambipolar drift modifies the dynamical effect of magnetic fields on the gas, and may play a key role in the star formation.", "In classical theory, quasi-static ambipolar diffusion is the main mechanism for prestellar cores to lose magnetic support and reach supercritical mass-to-flux ratios.", "Through ambipolar drift, the mass within dense cores can be redistributed, with the neutrals diffusing inward while the magnetic field threading the outer region is left behind .", "However, the quasi-static evolution model , gives a prestellar core lifetime considerably longer (up to a factor of 10) than the gravitational free-fall timescale, $t_\\mathrm {ff}$ , while several observational studies have shown that cores only live for $(2-5)$ $t_\\mathrm {ff}$ , .", "The failure of the traditional picture to predict core lifetimes indicates that supercritical cores may not have formed quasi-statically through ambipolar diffusion.", "Indeed, it is now generally recognized that, due to pervasive supersonic flows in GMCs, core formation is not likely to be quasi-static.", "Realistic star formation models should take both ambipolar diffusion and large-scale supersonic turbulence into consideration.", "This turbulence may accelerate the ambipolar diffusion process , , with an analytic estimate of the enhanced diffusion rate by a factor of 2$-$ 3 for typical conditions in GMCs .", "In our previous work (CO12), we investigated the physical mechanism driving enhanced ambipolar diffusion in one-dimensional C-type shocks.", "These shocks pervade GMCs, and are responsible for the initial compression of gas above ambient densities.", "We obtained a formula for the C-shock thickness as a function of density, magnetic field, shock velocity, and ionization fraction, and explored the dependence of shock-enhanced ambipolar diffusion on environment through a parameter study.", "Most importantly, we identified and characterized a transient stage of rapid ambipolar diffusion at the onset of shock compression, for one-dimensional converging flows.", "For an interval comparable to the neutral-ion collision time and before the neutral-ion drift reaches equilibrium, the neutrals do not experience drag forces from the ions.", "As a consequence, the initial shock in the neutrals is essentially unmagnetized, and the neutrals can be very strongly compressed.", "This transient stage, with timescale $t_\\mathrm {transient}\\sim 1$  Myr (but depending on ionization), can create dense structures with much higher $\\rho /B$ than upstream gas.", "CO12 suggested this could help enable supercritical core formation.", "CO12 also found that (1) the perpendicular component of the magnetic field is the main determinant of the shock compression, and (2) the perpendicular component of the magnetic field $B_\\perp $ must be weak ($\\lesssim 5~\\mu $ G) for transient ambipolar diffusion in shocks to significantly enhance $\\rho /B_\\perp $ .", "Observations of nearby clouds provide direct constraints on the role of magnetic fields, as well as other properties of prestellar cores.", "The typical mean mass-to-flux ratio of dark cloud cores is $\\Gamma \\sim 2$ (in units of critical value; see Equation (REF )) from Zeeman studies , .", "Due to the instrumental limitations, magnetic field observations in solar-mass and smaller scale regions are relatively lacking compared with observations of larger scales , however.", "Surveys in nearby clouds have found that prestellar cores have masses between $\\sim 0.1-10$  M$_\\odot $ and sizes $\\sim 0.01-1$  pc , , , .", "In addition, a mass-size relation has been proposed as a power law $M\\propto R^k$ , with $k=1.2-2.4$ dependent on various molecule tracers , , , .", "The magnetic field strength within prestellar cores is important for late evolution during core collapse, since disk formation may be suppressed by magnetic braking (for recent simulations see , , , ; or see review in ).", "However, many circumstellar disks and planetary systems have been detected , , suggesting that the magnetic braking “catastrophe\" seen in many simulations does not occur in nature.", "The proposed solutions include the misalignment between the magnetic and rotation axes , , , , turbulent reconnection and other turbulent processes during the rotating collapse , , , and non-ideal MHD effects including ambipolar diffusion, Hall effect, and Ohmic dissipation , , , , .", "If prestellar cores have sufficiently weak magnetic fields, however, braking would not be a problem during disk formation , , .", "Therefore, the magnetic field (and mass-to-flux ratio) within a prestellar core is important not just for the ability of the core to collapse, but also of a disk to form.", "Fragmentation of sheetlike magnetized clouds induced by small-amplitude perturbation and regulated by ambipolar diffusion has been widely studied , , , , .", "Analogous fully three-dimensional simulations have also been conducted .", "Supercritical cores formed in the flattened layer have masses $\\sim 0.1-10$  M$_\\odot $ , , at timescales $\\sim 1-10$  Myr dependent on the initial mass-to-flux ratio of the cloud , , .", "The above cited simulations start from relatively high densities ($\\sim 10^4$  cm$^{-3}$ ; e.g. )", "and included only the low-amplitude perturbations.", "Alternatively, and took the formation of these overdense regions into consideration by including a direct treatment of the large-scale supersonic turbulence.", "They demonstrated that ambipolar diffusion can be sped up locally by the supersonic turbulence, forming cores with masses $\\sim 0.5$  M$_\\odot $ and sizes $\\sim 0.1$  pc within $\\sim 2$  Myr, while the strong magnetic field keeps the star formation efficiency low ($1-10\\%$ ).", "Similarly, found that turbulence-accelerated, magnetically-regulated core formation timescales are $\\sim 1$  Myr in two-dimensional simulations of magnetized sheet-like clouds, with corresponding three-dimensional simulations showing comparable results , .", "In addition, measured the core properties in their three-dimensional simulations to find $L_\\mathrm {core}\\sim 0.04-0.14$  pc, $\\Gamma _\\mathrm {core}\\sim 0.3-1.5$ , and $M_\\mathrm {core}\\sim 0.15-12.5$  M$_\\odot $ , while found a broader core mass distribution $M_\\mathrm {core}\\sim 0.04-25$  M$_\\odot $ in their parameter study using thin-sheet approximation.", "Supersonic turbulence within GMCs extends over a wide range of spatial scales , .", "Although turbulence contains sheared, diverging, and converging regions in all combinations, regions in which there is a large-scale convergence in the velocity field will strongly compress gas, creating favorable conditions for the birth of prestellar cores.", "investigated core formation in an idealized model containing both a large-scale converging flow and multi-scale turbulence.", "These simulations showed that the time until the first core collapses depends on inflow Mach number ${\\cal M}$ as $t_\\mathrm {collapse}\\propto {\\cal M}^{-1/2}$ .", "With a parameter range ${\\cal M} = 1.1$ to 9, cores formed in the GO11 simulations had masses $0.05-50$  M$_\\odot $ .", "Following similar velocity power spectrum but including ideal MHD effects, performed simulations with sink particle, radiative transfer, and protostellar outflows to follow the protostar formation in turbulent massive clump.", "They demonstrated that the median stellar mass in the simulated star cluster can be doubled by the magnetic field, from $0.05$  M$_\\odot $ (unmagnetized case) to $0.12$  M$_\\odot $ (star cluster with initial mass-to-flux ratio $\\Gamma =2$ ).", "This is qualitatively consistent with the conclusion in , that the mass of the cores formed in the post-shock regions created by cloud-cloud collision is positively related to (and dominated by) the strong magnetic field in the shocked layer.", "Note that, though the main focus of is the cloud's ability to form massive cores ($\\sim 20-200$  M$_\\odot $ in their simulations), the idea of cloud-cloud collision is very similar to the converging flows setup adopted in GO11 and this study.", "In this paper, we combine the methods of CO12 for modeling ambipolar diffusion with the methods of GO11 for studying self-gravitating structure formation in turbulent converging flows.", "Our numerical parameter study focuses on the level of ambipolar diffusion (controlled by the ionization fraction of the cloud) and the obliquity of the shock (controlled by the angle between the magnetic field and the upstream flow).", "We show that filamentary structures similar to those seen in observations develop within shocked gas layers, and that cores form within these filaments.", "We measure core properties to test their dependence on these parameters.", "As we shall show, our models demonstrate that low-mass supercritical cores can form for all magnetic obliquities and all levels of ionization, including ideal MHD.", "However, our models also show that ambipolar diffusion affects the magnetization of dynamically-formed cores.", "The outline of this paper is as follows.", "We provide a theoretical analysis of oblique MHD shocks in Section , pointing out that a quasi-hydrodynamic compression ratio (which is $\\sim 5$ times stronger than in fast MHD shocks for the parameters we study) can exist when the converging flow is nearly parallel to the magnetic field.", "We also show that shock compression cannot increase the mass-to-flux ratio except in the nearly-parallel case or with ambipolar diffusion.", "Section  describes methods used in our numerical simulations and data analysis, including our model parameter set and method for measuring magnetic flux within cores.", "The evolution of gas structure (including development of filaments) and magnetic fields for varying parameters is compared in Section .", "In Section  we provide quantitative results for masses, sizes, magnetizations, and other physical properties of the bound cores identified from our simulations.", "Implications of these results for core formation is discussed in Section , where we argue that the similarity of core masses and sizes among models with different magnetizations and ionizations can be explained by anisotropic condensation preferentially along the magnetic field.", "Section  summarizes our conclusions.", "CO12 describe a one-dimensional simplified MHD shock system with velocity and magnetic field perpendicular to each other, including a short discussion of oblique shocks.", "Here we review the oblique shock equations and write them in a more general form to give detailed jump conditions.", "We shall consider a plane-parallel shock with uniform pre-shock neutral density $\\rho _0$ and ionization-recombination equilibrium everywhere.", "The shock front is in the $x$ -$y$ plane, the upstream flow is along the $z$ -direction ($\\mathbf {v}_{0} = v_{0}\\hat{\\mathbf {z}}$ ), and the upstream magnetic field is in the $x$ -$z$ plane, at an angle $\\theta $ to the inflow ($\\mathbf {B}_0 = B_0\\sin \\theta \\hat{\\mathbf {x}} + B_0\\cos \\theta \\hat{\\mathbf {z}}$ ) such that $B_x=B_0\\sin \\theta $ is the upstream component perpendicular to the flow.", "The parameters ${\\cal M}$ and $\\beta $ (upstream value of the Mach number and plasma parameter) defined in CO12 therefore become ${\\cal M} \\equiv {\\cal M}_z = \\frac{v_{0}}{c_{s}},\\ \\ \\ \\frac{1}{\\beta _0} \\equiv \\frac{B_{0}^2}{8\\pi \\rho _{0}c_{s}^2} = \\frac{1}{\\beta _x} \\frac{1}{\\sin ^2\\theta }.$ The jump conditions of MHD shocks are described by compression ratios of density and magnetic field: $r_f \\equiv \\frac{\\rho _{n,\\ \\mathrm {downstream}}}{\\rho _{n,\\ \\mathrm {upstream}}}, \\ \\ \\ r_{B_\\perp } \\equiv \\frac{B_{\\perp ,\\ \\mathrm {downstream}}}{B_{\\perp ,\\ \\mathrm {upstream}}}.$ From Equations (A10) and (A14) in CO12, we have $\\frac{\\sin ^2\\theta {r_f}^2}{\\beta _0}\\left(1-\\frac{2\\cos ^2\\theta }{\\beta _0{\\cal M}^2}\\right)^2 = \\left({\\cal M}^2 + 1 + \\frac{\\sin ^2\\theta }{\\beta _0} - \\frac{{\\cal M}^2}{r_f} - r_f\\right)\\left(1-\\frac{2 r_f \\cos ^2\\theta }{\\beta _0{\\cal M}^2}\\right)^2,$ which can be solved numerically to obtain explicit solution(s) $r_{f, \\mathrm {exp}}(\\theta )$ .", "The compression ratio for the magnetic field perpendicular to the inflow is $r_{B_\\perp }(\\theta ) = r_f(\\theta )\\frac{1 - \\frac{2\\cos ^2\\theta }{\\beta _0{\\cal M}^2}}{1 - \\frac{2 r_f(\\theta ) \\cos ^2\\theta }{\\beta _0{\\cal M}^2}}.$ Equation (A17) of CO12 gives an analytical approximation to $r_f(\\theta )$ : $r_{f, \\mathrm {app}}(\\theta ) = \\frac{\\sqrt{\\beta _0}{\\cal M}}{\\sin \\theta }\\left[\\frac{2\\sin \\theta }{\\sqrt{\\beta _0}{\\cal M}\\tan ^2\\theta } + \\frac{\\sqrt{\\beta _0}}{2{\\cal M}\\sin \\theta } + 1\\right]^{-1}.", "$ Since Equation (REF ) is a quartic function of $\\theta $ , there are four possible roots of $r_f$ for each angle, and $r_f(\\theta )=const.=1$ (no-shock solution) is always a solution.", "When $\\theta $ is large, Equation (REF ) has one simple root ($r_f=1$ ) and a multiple root with multiplicity $=3$ .", "When $\\theta $ drops below a critical value, $\\theta _\\mathrm {crit}$ , Equation (REF ) has four simple roots, which give us four different values of $r_{B_\\perp }$ .", "Figure REF shows the three explicit solutions for $r_f$ and $r_{B_\\perp }$ ($r_{f,\\mathrm {exp}}(\\theta )$ and $r_{B,\\mathrm {exp}}(\\theta )$ ) as well as the approximations ($r_{f,\\mathrm {app}}(\\theta )$ and $r_{B,\\mathrm {app}}(\\theta )$ ) that employ Equation (REF ).", "The fact that there are multiple solutions for post-shock properties is the consequence of the non-unique Riemann problem in ideal MHD , , , and whether all solutions are physically real is still controversial.", "The first set of solutions $r_{f, \\mathrm {exp}}(\\theta ), 1$ and $r_{B, \\mathrm {exp}}(\\theta ), 1$ shown in Figure REF gives positive $r_f$ and $r_{B_\\perp }$ , classified as fast MHD shocks , , and is the principal oblique shock solution referred to in this contributionWe use Equation (REF ) as analytical approximation for $r_f(\\theta )$ , if necessary..", "The other two solutions for post-shock magnetic field, $r_{B, \\mathrm {exp}}(\\theta ), 2$ and $r_{B, \\mathrm {exp}}(\\theta ), 3$ , both become negative when $\\theta < \\theta _\\mathrm {crit}$ , indicating that the tangential component of the magnetic field to the shock plane is reversed in the post-shock region.", "These two solutions are commonly specified as intermediate shocks , , .", "Among these two field-reversal solutions, we notice that $r_{f, \\mathrm {exp}}(\\theta ), 2$ approaches the hydrodynamic jump condition ($r_{f,\\mathrm {hydro}} = {\\cal M}^2$ ) when $\\theta \\rightarrow 0$ , and $r_{B, \\mathrm {exp}}(\\theta ), 2$ is smaller in magnitude than other solutions when $\\theta < \\theta _\\mathrm {crit}$ .", "Thus, we classify this set of solutions $r_{f, \\mathrm {exp}}(\\theta ), 2$ and $r_{B, \\mathrm {exp}}(\\theta ), 2$ as the quasi-hydrodynamic shock.", "This quasi-hydrodynamic solution can create gas compression much stronger than the regularly-applied fast shock condition, and may be the reason that when $\\theta < \\theta _\\mathrm {crit}$ , even ideal MHD simulations can generate shocked layers with relatively high mass-to-flux ratio (see Sections  and for more details).", "The definition of $\\theta _\\mathrm {crit}$ can be derived from Equation (REF ), which turns negative when $1 - \\frac{2\\cos ^2\\theta }{\\beta _0{\\cal M}^2} > 0$ and $1 - \\frac{2 r_f(\\theta ) \\cos ^2\\theta }{\\beta _0{\\cal M}^2} <0$ : $\\cos ^2\\theta > \\cos ^2\\theta _\\mathrm {crit} = \\frac{\\beta _0{\\cal M}^2}{2 r_f(\\theta _\\mathrm {crit})}.$ Using Equation (REF ) and considering only the terms $\\sim {\\cal M}$ , this becomes $\\frac{\\cos ^2\\theta _\\mathrm {crit}}{\\sin \\theta _\\mathrm {crit}} \\approx \\frac{\\sqrt{\\beta _0} {\\cal M}}{2},$ or $\\sin ^2\\theta _\\mathrm {crit} + \\frac{\\sqrt{\\beta _0} {\\cal M}}{2} \\sin \\theta _\\mathrm {crit} -1 = 0.$ Assuming $\\theta _\\mathrm {crit}\\ll 1$ , this gives $\\theta _\\mathrm {crit} \\sim \\frac{2}{\\sqrt{\\beta _0} {\\cal M}} = \\sqrt{2}\\frac{v_\\mathrm {A,0}}{v_0},$ where $v_\\mathrm {A,0} \\equiv B_0/\\sqrt{4\\pi \\rho _0}$ is the Alfvén speed in the cloud.", "Therefore, the criterion to have multiple solutions, $\\theta < \\theta _\\mathrm {crit}$ , is approximately equivalent to $v_\\perp = v_0\\sin \\theta \\lesssim v_0 \\cdot \\sqrt{2}\\frac{v_\\mathrm {A,0}}{v_0} \\sim v_\\mathrm {A,0}$ where $v_\\perp $ is the component of the inflow perpendicular to the magnetic field.", "Though Equation (REF ) only provides a qualitative approximationFor parameters used in Figure REF , Equation (REF ) gives $\\theta _\\mathrm {crit} = 18^\\circ $ , approximately 2 times larger than the exact solution.", "for $\\theta _\\mathrm {crit}$ , Equation (REF ) suggests that when $v_\\perp /v_\\mathrm {A,0}$ is sufficiently small, high-compression quasi-hydrodynamic shocks are possible." ], [ "Gravitational Critical Scales in Spherical Symmetry", "For a core to collapse gravitationally, its self-gravity must overcome both the thermal and magnetic energy.", "For a given ambient density $\\rho \\equiv \\mu _n n$ and assuming spherical symmetry, the mass necessary for gravity to exceed the thermal pressure support (with edge pressure $\\rho {c_s}^2$ ) is the mass of the critical Bonnor-Ebert sphere : $M_\\mathrm {th,sph} = 4.18\\frac{{c_s}^3}{\\sqrt{4\\pi G^3 \\rho }} = 4.4~\\mathrm {M}_\\odot \\left(\\frac{T}{10~\\mathrm {K}}\\right)^{3/2}\\left(\\frac{n}{1000~\\mathrm {cm}^{-3}}\\right)^{-1/2}$ (see Section REF for discussion about the value of $\\mu _n$ ).", "The corresponding length scale at the original ambient density is $R_\\mathrm {th,sph} \\equiv \\left(\\frac{3 M_\\mathrm {th,sph}}{4\\pi \\rho }\\right)^{1/3} = 2.3 \\frac{c_s}{\\sqrt{4\\pi G \\rho }} = 0.26~\\mathrm {pc} \\left(\\frac{T}{10~\\mathrm {K}}\\right)^{1/2}\\left(\\frac{n}{1000~\\mathrm {cm}^{-3}}\\right)^{-1/2},$ although the radius of a Bonnor-Ebert sphere with mass given by Equation (REF ) would be smaller than Equation (REF ) by $25\\%$ , due to internal stratification.", "In a magnetized medium with magnetic field $B$ , the ratio of mass to magnetic flux for a region to be magnetically supercriticalSee Section REF for more detailed discussion about the critical value of $M/\\Phi _B$ .", "can be written as $\\frac{M}{\\Phi _B}\\bigg |_\\mathrm {mag,crit} \\equiv \\frac{1}{2\\pi \\sqrt{G}}.$ With $M=4\\pi R^3 \\rho /3$ and $\\Phi _B = \\pi R^2 B$ for a spherical volume at ambient density $\\rho $ , this gives $M_\\mathrm {mag,sph} = \\frac{9}{128\\pi ^2G^{3/2}}\\frac{B^3}{\\rho ^2} = 14~\\mathrm {M}_\\odot \\left(\\frac{B}{10~\\mu \\mathrm {G}}\\right)^3\\left(\\frac{n}{1000~\\mathrm {cm}^{-3}}\\right)^{-2}.$ and $R_\\mathrm {mag,sph} = \\frac{3}{8\\pi \\sqrt{G}}\\frac{B}{\\rho } = 0.4~\\mathrm {pc} \\left(\\frac{B}{10~\\mu \\mathrm {G}}\\right) \\left(\\frac{n}{1000~\\mathrm {cm}^{-3}}\\right)^{-1},$ A spherical region must have $M>M_\\mathrm {th,sph}$ as well as $M>M_\\mathrm {mag, sph}$ to be able to collapse.", "In the cloud environment (the pre-shock region), $B\\sim 10~\\mu $ G and $n\\sim 1000$  cm$^{-3}$ are typical.", "Comparing Equation (REF ) and (REF ), the magnetic condition is more strict than the thermal condition; if cores formed from a spherical volume, the mass would have to exceed $\\sim 10$  M$_\\odot $ in order to collapse.", "This value is much larger than the typical core mass ($\\sim 1$  M$_\\odot $ ) identified in observations.", "This discrepancy is the reason why traditionally ambipolar diffusion is invoked to explain how low-mass cores become supercritical.", "We can examine the ability for magnetically supercritical cores to form isotropically in a post-shock layer.", "The normalized mass-to-flux ratio $\\Gamma \\equiv \\frac{M}{\\Phi _B}\\cdot 2\\pi \\sqrt{G}$ of a spherical volume with density $\\rho $ , magnetic field $B$ , and mass $M$ is $\\Gamma _\\mathrm {sph} &=\\frac{8\\pi \\sqrt{G}}{3}\\left(\\frac{3}{4\\pi }\\right)^{1/3} M^{1/3}\\rho ^{2/3}B^{-1}\\\\&= 0.4 \\left(\\frac{M}{\\mathrm {M}_\\odot }\\right)^{1/3}\\left(\\frac{n}{1000~\\mathrm {cm}^{-3}}\\right)^{2/3}\\left(\\frac{B}{10~\\mu \\mathrm {G}}\\right)^{-1}.$ Or, with $\\Sigma = 4R\\rho /3\\equiv \\mu _n N_n$ for a sphere, we have $\\Gamma _\\mathrm {sph} =2\\pi \\sqrt{G}\\cdot \\frac{\\Sigma }{B} = 0.6\\left(\\frac{N_n}{10^{21}~\\mathrm {cm}^{-2}}\\right)\\left(\\frac{B}{10~\\mu \\mathrm {G}}\\right)^{-1}.$ Considering the cloud parameters from Figure REF (${\\cal M} = 10$ , $B_0 = 10$  $\\mu $ G, $n_0 = 1000$  cm$^{-3}$ ), the post-shock density and magnetic field are approximately $n_\\mathrm {ps}\\sim 10^4$  cm$^{-3}$ and $B_\\mathrm {ps}\\sim 50$  $\\mu $ G when $\\theta >\\theta _\\mathrm {crit}$ .", "A solar-mass spherical region in this shocked layer will have $\\Gamma _\\mathrm {ps,sph}\\approx 0.37$ ; spherical contraction induced by gravity would be suppressed by magnetic fields.", "Thus, typical post-shock conditions are unfavorable for forming low-mass cores by spherical contraction in ideal MHD.", "Furthermore, using $r_f$ and $r_{B_\\perp }$ defined in Section REF , we can compare $\\Gamma _\\mathrm {ps,sph}$ and the pre-shock value $\\Gamma _\\mathrm {pre,sph}$ for spherical post-shock and pre-shock regions: $\\frac{\\Gamma _\\mathrm {ps,sph}}{\\Gamma _\\mathrm {pre,sph}} = \\left(\\frac{M_\\mathrm {ps}}{M_\\mathrm {pre}}\\right)^{1/3}\\left(\\frac{\\rho _\\mathrm {ps}}{\\rho _\\mathrm {pre}}\\right)^{2/3}\\left(\\frac{B_\\mathrm {ps}}{B_\\mathrm {pre}}\\right)^{-1} \\approx \\left(\\frac{M_\\mathrm {ps}}{M_\\mathrm {pre}}\\right)^{1/3} {r_f}^{2/3} {r_{B_\\perp }}^{-1}.$ Considering volumes containing similar mass, $M_\\mathrm {ps}\\sim M_\\mathrm {pre}$ , the ratio between the post-shock and pre-shock $\\Gamma _\\mathrm {sph}$ is smaller than unity when $\\theta > \\theta _\\mathrm {crit}$ , because Equation (REF ) shows that $r_{B_\\perp }$ is larger than $r_f$ .", "Thus, provided $\\theta > \\theta _\\mathrm {crit}$ , the post-shock layer will actually have stronger magnetic support than the pre-shock region for a given spherical mass.", "Based on the above considerations, formation of low-mass supercritical cores appears difficult in ideal MHD.", "Adapting classical ideas, one might imagine that low-mass subcritical cores form quasi-statically within the post-shock layer, then gradually lose magnetic support via ambipolar diffusion to become magnetically supercritical in a timescale $\\sim 1-10~$ Myr.", "A process of this kind would, however, give prestellar core lifetimes longer than observed, and most cores would have $\\Gamma < 1$ (inconsistent with observations).", "Two alternative scenarios could lead to supercritical core formation in a turbulent magnetized medium.", "First, the dynamic effects during a turbulence-induced shock (including rapid, transient ambipolar diffusion and the quasi-hydrodynamic compression when $\\theta <\\theta _\\mathrm {crit}$ ) may increase the compression ratio of neutrals, creating $r_f \\gg r_{B_\\perp }$ and $\\Gamma _\\mathrm {ps,sph} > 1$ , enabling low-mass supercritical cores to form.", "Second, even if the post-shock region is strongly magnetized, mass can accumulate through anisotropic condensation along the magnetic field until both the thermal and magnetic criteria are simultaneously satisfied.", "In this study, we carefully investigate these two scenarios, showing that both effects contribute to the formation of low-mass supercritical cores within timescale $\\lesssim 0.6$  Myr, regardless of ionization or magnetic obliquity." ], [ "Simulation Setup and Equations", "To examine core formation in shocked layers of partially-ionized gas, we employ a three-dimensional convergent flow model with ambipolar diffusion, self-gravity, and a perturbed turbulent velocity field.", "We conducted our numerical simulations using the Athena MHD code with Roe's Riemann solver.", "To avoid negative densities if the second-order solution fails, we instead use first-order fluxes for bad zones.", "The self-gravity of the domain, with an open boundary in one direction and periodic boundaries in the other two, is calculated using the fast Fourier transformation (FFT) method developed by .", "Ambipolar diffusion is treated in the strong coupling approximation, as described in , with super time-stepping to accelerate the evolution.", "The equations we solve are: $\\frac{\\partial \\rho _n}{\\partial t} &+ \\mathbf {\\nabla }\\cdot \\left(\\rho _n\\mathbf {v}\\right) = 0,\\\\\\frac{\\partial \\rho _n\\mathbf {v}}{\\partial t} &+ \\mathbf {\\nabla }\\cdot \\left(\\rho _n\\mathbf {v}\\mathbf {v} - \\frac{\\mathbf {B}\\mathbf {B}}{4\\pi }\\right) + \\mathbf {\\nabla }P^* = 0,\\\\\\frac{\\partial \\mathbf {B}}{\\partial t} &+ \\mathbf {\\nabla }\\times \\left(\\mathbf {B}\\times \\mathbf {v}\\right) = \\mathbf {\\nabla }\\times \\left[\\frac{\\left(\\left(\\mathbf {\\nabla }\\times \\mathbf {B}\\right)\\times \\mathbf {B}\\right)\\times \\mathbf {B}}{4\\pi \\rho _i\\rho _n\\alpha }\\right], $ where $P^* = P + B^2/(8\\pi )$ .", "For simplicity, we adopt an isothermal equation of state $P=\\rho {c_s}^2$ .", "The numerical setup for inflow and turbulence is similar to that adopted by GO11.", "For both the whole simulation box initially and the inflowing gas subsequently, we apply perturbations following a Gaussian random distribution with a Fourier power spectrum as described in GO11.", "The scaling law for supersonic turbulence in GMCs obeys the relation $\\frac{\\delta v_\\mathrm {1D} (\\ell )}{\\sigma _{v,\\mathrm {cloud}}} = \\left(\\frac{\\ell }{2R_\\mathrm {cloud}}\\right)^{1/2},$ where $\\delta v_\\mathrm {1D} (\\ell )$ represents the one-dimensional velocity dispersion at scale $\\ell $ , and $\\sigma _{v,\\mathrm {cloud}}$ is the cloud-scale one-dimensional velocity dispersion.", "In terms of the virial parameter $\\alpha _\\mathrm {vir} \\equiv 5{\\sigma _v}^2 R_\\mathrm {cloud}/ (GM_\\mathrm {cloud})$ with $M_\\mathrm {cloud} \\equiv 4\\pi \\rho _0 {R_\\mathrm {cloud}}^3 / 3$ , and for the inflow Mach number ${\\cal M}$ comparable to $\\sigma _v/c_s$ of the whole cloud, the three-dimensional velocity dispersion $\\delta v=\\sqrt{3}\\cdot \\delta v_\\mathrm {1D}$ at the scale of the simulation box would be $\\delta v (L_\\mathrm {box}) = \\sqrt{3} \\left(\\frac{\\pi G \\alpha _\\mathrm {vir}}{15}\\right)^{1/4} {\\cal M}^{1/2} {c_s}^{1/2} {\\rho _0}^{1/4} {L_\\mathrm {box}}^{1/2}.$ To emphasize the influence of the cloud magnetization instead of the perturbation field, our simulations are conducted with $10\\%$ of the value $\\delta v (L_\\mathrm {box})$ , or $\\delta v = 0.14$  km/s with $\\alpha _\\mathrm {vir} = 2$ .", "With larger $\\delta v (L_\\mathrm {box})$ , simulations can still form cores, but because non-self-gravitating clumps can easily be destroyed by strong velocity perturbations and no core can form before the turbulent energy dissipates, it takes much longer, with corresponding higher computational expense." ], [ "Model Parameters", "A schematic showing our model set-up is shown in Figure REF .", "Our simulation box is 1 pc on each side and represents a region within a GMC where a large-scale supersonic converging flow with velocity $\\mathbf {v}_0$ and $-\\mathbf {v}_0$ (i.e.", "in the center-of-momentum frame) collides.", "The $z$ -direction is the large-scale inflow direction, and we adopt periodic boundary conditions in the $x$ - and $y$ -directions.", "We initialize the background magnetic field in the cloud, $\\mathbf {B}_0$ , in the $x$ -$z$ plane, with an angle $\\theta $ with respect to the convergent flow.", "For simplicity, we treat the gas as isothermal at temperature $T=10$  K, such that the sound speed is $c_s=0.2$  km/s.", "The neutral density within the cloud, $\\rho _0$ , is set to be uniform in the initial conditions and in the upstream converging flow.", "It has been shown that ionization-recombination equilibrium generally provides a good approximation to the ionization fraction within GMCs for the regime under investigation (CO12).", "Thus, the number density of ions in our model can be written as $n_i = \\frac{\\rho _i}{\\mu _i}= 10^{-6} \\chi _{i0} \\left(\\frac{\\rho _n}{\\mu _n}\\right)^{1/2},$ with $\\chi _{i0} \\equiv 10^6 \\times \\sqrt{\\frac{\\zeta _\\mathrm {CR}}{\\alpha _\\mathrm {gas}}}$ determined by the cosmic-ray ionization rate ($\\zeta _\\mathrm {CR}$ ) and the gas-phase recombination rate ($\\alpha _\\mathrm {gas}$ ).", "The ionization coefficient, $\\chi _{i0}$ , has values $\\sim 1-20$ , and is the model parameter that controls ambipolar diffusion effects in our simulations, following CO12.", "We use typical values of the mean neutral and ion molecular weight $\\mu _n$ and $\\mu _i$ of $2.3 m_\\mathrm {H}$ and $30 m_\\mathrm {H}$ , respectively, which give the collision coefficient (see Equation (REF )) between neutrals and ions $\\alpha = 3.7\\times 10^{13}$  cm$^3$ s$^{-1}$ g$^{-1}$ .", "The physical parameters defining each model are $\\rho _0$ , $v_0 = |\\mathbf {v}_0|$ , $B_0 = |\\mathbf {B}_0|$ , $\\theta $ , and $\\chi _{i0}$ .", "We set the upstream neutral number density to be $ n_0 = \\rho _0 / \\mu _n = 1000$  cm$^{-3}$ in all simulations, consistent with typical mean molecular densities within GMCsNote that the upstream neutral number density we adopted here is $n_0 = n_{\\mathrm {neutral},0} \\equiv n_\\mathrm {H_2} + n_\\mathrm {He} = 0.6 n_\\mathrm {H} = 1.2 n_\\mathrm {H_2}$ , with GMC observations giving $n_\\mathrm {H_2} \\sim 10^2-10^3$  cm$^{-3}$ .", "Also note that $\\mu _n \\equiv \\rho _n/n_n = (\\rho _\\mathrm {H_2} + \\rho _\\mathrm {He}) / (n_\\mathrm {H_2} + n_\\mathrm {He}) = (0.5n_\\mathrm {H} \\times 2m_\\mathrm {H} + 0.1n_\\mathrm {H} \\times 4m_\\mathrm {H})/(0.5 n_\\mathrm {H} + 0.1 n_\\mathrm {H}) = 2.3 m_\\mathrm {H}$ .", ", , , .", "We choose the upstream $B_0 = 10$  $\\mu $ G as typical of GMC values , , , for all our simulations.", "To keep the total number of simulations practical, we set the large-scale inflow Mach number to ${\\cal M}=10$ for all models.", "Exploration of the dependence on Mach number of ambipolar diffusion and of core formation has been studied in previous simulations (CO12 and GO11, respectively).", "For our parameter survey, we choose $\\theta =5$ , 20, and 45 degrees to represent small ($\\theta <\\theta _\\mathrm {crit}$ ), intermediate ($\\theta >\\theta _\\mathrm {crit}$ ), and large ($\\theta \\gg \\theta _\\mathrm {crit}$ ) angles between the inflow velocity and cloud magnetic field.", "For each $\\theta $ , we conduct simulations with $\\chi _{i0} = 3$ , 10, and ideal MHD to cover situations with strong, weak, and no ambipolar diffusion.", "We also run corresponding hydrodynamic simulations with same $\\rho _0$ and $v_0$ for comparison.", "A full list of models is contained in Table REF .", "Table REF also lists the steady-state post-shock properties, as described in Section REF .", "Solutions for all three types of shocks are listed for the $\\theta =5^\\circ $ (A5) case.", "For the $\\theta =20^\\circ $ and $\\theta =45^\\circ $ cases, there is only one shock solution.", "Also included in Table REF are the nominal values of critical mass and radius for spherically symmetric volumes to be self-gravitating under these steady-state post-shock condition, as discussed in Section REF (see Equations (REF ), (REF ), (REF ) and (REF )).", "Both “thermal\" and “magnetic\" critical masses are listed.", "In most models, $M_\\mathrm {mag,sph} > M_\\mathrm {th,sph}$ and $M_\\mathrm {mag,sph} \\gg \\mathrm {M}_\\odot $ , indicating the post-shock regions are dominated by magnetic support, and either ambipolar diffusion or anisotropic condensation would be needed to form low-mass supercritical cores, as discussed in Section REF .", "On the other hand, the quasi-hydrodynamic shock solution for models with $\\theta <\\theta _\\mathrm {crit}$ (i.e.", "A5 cases) has $M_\\mathrm {mag,sph}<M_\\mathrm {th,sph}<\\mathrm {M}_\\odot $ downstream.", "If this shock solution could be sustained, then in principle low-mass supercritical cores could form by spherical condensation of post-shock gas.", "Table: Summary" ] ]
1403.0582
[ [ "Intrinsic Prices Of Risk" ], [ "Abstract We review the nature of some well-known phenomena such as volatility smiles, convexity adjustments and parallel derivative markets.", "We propose that the market is incomplete and postulate the existence of intrinsic risks in every contingent claim as a basis for understanding these phenomena.", "In a continuous time framework, we bring together the notion of intrinsic risk and the theory of change of measures to derive a probability measure, namely risk-subjective measure, for evaluating contingent claims.", "This paper is a modest attempt to prove that measure of intrinsic risk is a crucial ingredient for explaining these phenomena, and in consequence proposes a new approach to pricing and hedging financial derivatives.", "By adapting theoretical knowledge to practical applications, we show that our approach is consistent and robust, compared with the standard risk-neutral approach." ], [ "Introduction", "This section has two purposes.", "Firstly, we review some well-known phenomena in order to motivate subsequent developments.", "After that, we provide a background of the phenomena with some notation, terminology and notions.", "Volatility smiles.", "In a nutshell, vanilla options with different maturities and strikes have different volatilities implied by the well-known formula of [6].", "Implied volatility is quoted as the market expectation about the average future volatility of the underlying asset over the remaining life of the option.", "Thus compared to historical volatility it is the forward looking approach.", "For many years, practitioners and academics have tried to analyse the volatility smile phenomenon and understand its implications for derivatives pricing and risk management.", "In [10], their link between the real-world and risk-neutral processes of the underlying would be complete by non-traded sources of risk.", "[37] found that the dynamics of the risk premium, when volatility is stochastic, is not a traded security.", "A number of models and extensions of, or alternatives to, the Black-Scholes model, have been proposed in the literature: the local volatility models of [14], [13]; a jump-diffusion model of [33]; stochastic volatility models of [23], [22] and others; mixed stochastic jump-diffusion models of [2] and others; universal volatility models of [15], [29], [7], [3] and others; regime switching models, etc.", "From a hedging perspective, traders who use the Black-Scholes model must continuously change the volatility assumption in order to match market prices.", "Their hedge ratios change accordingly in an uncontrolled way: the models listed above bring some order into this chaos.", "In the course of time, the general consensus, as advocated by practitioners and academics, is to choose a model that produces hedging strategies for both vanilla and exotic options resulting in profit and loss distributions that are sharply peaked at zero.", "We argue that such a model, if recovered (or implied) from option prices, by no means nearly explains this phenomenon, but is a means only to describe the implied volatility surface.", "Convexity adjustments.", "One of many well-known adjustments is the convexity adjustment - the implied yield of a futures and the equivalent forward rate agreement contracts are different.", "This phenomenon implies that market participants need to be paid more (or less) premium.", "The common approach, as used by most practitioners and academics, is to adjust futures quotes such that they can be used as forward rates.", "Naturally, this approach depends on an model that is used for this purpose.", "For the extended Vasicek known as [24] and [11] model, explicit formulae can be derived.", "The situation is different for models whose continuous description gives the short rate a log-normal distribution such as the [4] and [5] models: for these, in their analytical form of continuous evolution, futures prices can be shown to be positively infinite [21] and [35].", "In subsequent developments, we shall offer a different approach to this phenomenon.", "Parallel derivative markets.", "In an economic system, a financial market consists of a risk-free money account, primary and parallel markets.", "Examples of primary markets are stocks and bonds, and examples of parallel markets are derivatives such as forward, futures, vanilla options, credits which are derived from the same primary asset.", "Market makers can trade and make prices for derivatives in a parallel market without references to another." ], [ "Background", "The framework is as follows: a complete probability space $(\\Omega ,\\mathcal {F},\\mathbb {P})$ with a filtration $\\mathcal {F}= \\mathcal {F}(t)$ satisfying the usual conditions of right-continuity and completeness.", "$T\\in \\mathbb {R}$ denotes a fixed and finite time horizon; furthermore, we assume that $\\mathcal {F}(0)$ is trivial and that $\\mathcal {F}(T) = \\mathcal {F}$ .", "Let $X = X(t)$ be a continuous semimartingale representing the price process of a risky asset.", "The absence of arbitrage opportunities implies the existence of an probability measure $\\mathbb {Q}$ equivalent to the probability measure $\\mathbb {P}$ (the real world probability), such that $X$ is a $\\mathbb {Q}$ -martingale.", "Denote by $\\mathcal {Q}$ the set of coexistent equivalent measures $\\mathbb {Q}$ .", "A financial market is considered such that $\\mathcal {Q} \\ne \\emptyset $ .", "Uniqueness of the equivalent probability measure $\\mathbb {Q}$ implies the market is complete.", "The fundamental theorem of asset pricing establishes the relationship between the absence of arbitrage opportunities and the existence of an equivalent martingale measure and in a basic framework was proved by [18], [19], [20].", "The modern version of this theorem, established by [12], states that the absence of arbitrage opportunities is “essentially\" equivalent to the existence of an equivalent martingale measure under which the discounted (primary asset) price process is a martingale.", "For simplicity, we consider only one horizon of uncertainty $[0,T]$ .", "A contingent claim, or a derivative, $H = H(\\omega )$ is a payoff at time $T$ , contingent on the scenario $\\omega \\in \\Omega $ .", "The derivative has the special form $H = h(X(T))$ for some function $h$ .", "Here, $X$ is referred to as the primary (or the `underlying').", "More generally, $H$ depends on the whole evolution of $X$ up to time $T$ and is a random variable $H \\in \\mathcal {L}^2(\\Omega ,\\mathcal {F},\\mathbb {P}).$ In financial terms, every contingent claim can be replicated by means of a trading strategy (or interchangeably known as hedging strategy or a replication portfolio) which is a portfolio consisting of the primary asset $X$ and a risk-free money account $D = D(t)$ .", "Let $\\alpha = \\alpha (t)$ and $\\beta =\\beta (t)$ be a predictable process and an adapted process, respectively.", "$\\alpha (t)$ and $\\beta (t)$ are the amounts of asset and money account, respectively, held at time $t$ .", "In this section, for ease of exposition, we assume that $D(t) = 1$ for all $0\\le t\\le T$ .", "The value of the portfolio at time $t$ is given by $V(t) = \\alpha (t)X(t) + \\beta (t)D(t)$ for $0\\le t\\le T$ .", "It can be shown that the trading strategy $(\\alpha ,\\beta )$ is admissible such that the value process $V = V(t)$ is square-integrable and have right-continuous paths and is defined by $V(t) := V_0 + \\int _0^t\\alpha (s)dX(s)$ for $0\\le t\\le T$ .", "For $\\mathbb {Q}$ -almost surely, every contingent claim $H$ is attainable and admits the following representation $V(T) = H = V_0 + \\int _0^T\\alpha (s)dX(s),$ where $V_0 = E_{\\mathbb {Q}}[H]$ .", "Moreover, the strategy is self-financing, that is the cost of the portfolio (also known as derivative price) is a constant $V_0$ $V(t) - \\int _0^t\\alpha (s)dX(s) = V_0.$ The constant value $V_0$ represents a perfect replication or a perfect hedge.", "Thus far, we have presented the well-known mathematical construction of a hedging strategy in a complete market where every contingent claim is attainable.", "In a complete market, derivative prices are unique - no arbitrage opportunities exist.", "Derivatives cannot be valuated in a parallel market at any price other than $V_0$ .", "From financial and economic point of view, the phenomena imply that the market is incomplete, arbitrage opportunities exist and may not be at all eliminated.", "A derivative can be valued at different prices and hedged by mutually exclusively trading in risky assets (or derivatives) in parallel markets where market makers engage in market activities: investments, speculative trading, hedging, arbitrage and risk management.", "In addition, market makers expose themselves to market conditions such as liquidity, see for instance [1].", "We argue that exposure to the variability of market activities, market conditions and generally to uncertain future events constitutes a basis of arbitrage opportunity which we shall call intrinsic risk.", "In general, market incompleteness is a principle under which every contingent claim bears intrinsic risks.", "Let us postulate an assumption as a basis for subsequent reasonings and discussions.", "Assumption.", "The market is incomplete and there exist intrinsic risks embedded in every contingent claim.", "While the assumption is theoretical, it is rather realistically a proposition with the phenomena as proof.", "In a mathematical context, let $\\Pi $ be the set of all intrinsic risks, that is the set of all real valued functions on $\\Omega $ .", "Denote by $G(\\pi )$ the measure to the intrinsic risk $\\pi = \\pi (\\omega )$ on the scenario $\\omega \\in \\Omega $ .", "As a measure of intrinsic risk, $G$ is a mapping from $\\Pi $ into $\\mathbb {R}$ .", "As a basic object of our study, $G$ shall therefore be the random variable on the set of states of nature at a future date $T$ .", "Generally, $G$ depends on the evolution of the primary asset up to time $T$ and may also depend on the contingent claim: $G^H \\in \\mathcal {L}^2(\\Omega ,\\mathcal {F},\\mathbb {P}).$ The superscript indicates the dependence of a particular contingent claim $H$ .", "This leads to a new representation of $H$ $H = V_0 + \\int _0^T\\alpha (s)dX(s) + G^H.$ We now introduce the Kunita-Watanabe decomposition $G^H = G_0 + \\int _0^T\\alpha ^H(s)dX(s) + N(T)$ where $N = N(t)$ is a square-integrable martingale orthogonal to $X$ .", "Thus, we have $H = V_0^* + \\int _0^T\\alpha ^*(s)dX(s) + N(T),$ where $V_0^* = V_0 + G_0$ and $\\alpha ^* = \\alpha + \\alpha ^H$ .", "This representation of $H$ have been extensively dealt with, see for example [17].", "By incompleteness, the derivative value $V^*_0$ represents a perfect hedge, which manifests an initial intrinsic value of risk $G_0$ .", "In relation to the hedging strategy (REF ), the measure of intrinsic risk shall be considered as the value of all possible future capital which, required to control the risk incurred by the market maker (such as hedger) and invested in the primary asset, makes not only the contingent claim acceptable, but its valuation fair.", "From a mathematical point of view, market incompleteness implies that there exists in the set $\\mathcal {Q}$ an equivalent measure, not necessarily a martingale and/or unique measure, that is assigned to a parallel market.", "Thus, intrinsic risk may depend on the derivative and is not necessarily unique, as such its measure takes many forms some of which we shall consider for applications.", "In the remaining of this paper, we shall not discuss further on the abstract representations (REF ) and (REF ), but present them in a more descriptive (down to earth) framework - the continuous time framework.", "In this section we propose a continuous time financial market consisting of a primary price process $X$ and a risk-free money account $D$ .", "We shall define a measure of intrinsic risk and show that perfect hedging strategies can be constructed.", "We also show that the existence of intrinsic risk provides an internal consistency in pricing and hedging a contingent claim.", "Let $B = B(t)$ be a Brownian motion on the complete probability space $(\\Omega ,\\mathcal {F},\\mathbb {P})$ .", "The underlying price process of $X$ satisfies the SDE $dX(t) = \\mu (t)X(t)dt + \\sigma (t)X(t)dB(t),$ where $\\mu = \\mu (t)$ and $\\sigma = \\sigma (t)$ are Lipschitz continuous functions so that a solution exists.", "$\\mu $ and $\\sigma $ can be functions of $X$ .", "The price process of $D$ is given by $dD(t) = \\nu (t)D(t)dt,$ where $\\nu = \\nu (t)$ is a Lipschitz continuous function.", "We expand the portfolio value process (REF ) as follows: $dV(t) &=& \\alpha (t)dX(t) + \\nu (t)\\beta (t)D(t)dt \\\\&=& \\alpha (t)\\mu (t)X(t)dt + \\alpha (t)\\sigma (t)X(t)dB(t) + \\nu (t)\\left(V(t) - \\alpha (t)X(t)\\right)dt \\nonumber \\\\&=& \\nu (t)V(t)dt + \\alpha (t)(\\mu (t) - \\nu (t))X(t)dt + \\alpha (t)\\sigma (t)X(t)dB(t)\\nonumber \\\\&=& \\nu (t)V(t)dt + \\alpha (t)\\sigma (t)X(t) \\left[ \\frac{\\mu (t) - \\nu (t)}{\\sigma (t)}dt + dB(t)\\right]\\nonumber \\\\&=& \\nu (t)V(t)dt + \\alpha (t)\\sigma (t)X(t) dW(t),\\nonumber $ where $W = W(t)$ is a $\\mathbb {Q}$ -Brownian motion and is defined by $dW(t) = \\lambda (t)dt + dB(t)$ and $\\lambda (t) = \\frac{\\mu (t) - \\nu (t)}{\\sigma (t)}.$ Here, $\\mathbb {Q}$ is some martingale measure.", "Indeed, the theory of the Girsanov change of measure, see for example [26], shows that there exists such a martingale measure $\\mathbb {Q}$ equivalent to $\\mathbb {P}$ and which excludes arbitrage opportunities.", "More precisely, there exists a probability measure $\\mathbb {Q} \\ll \\mathbb {P}$ such that $\\frac{d\\mathbb {Q}}{d\\mathbb {P}} \\in \\mathcal {L}^2(\\Omega ,\\mathcal {F},\\mathbb {P})$ and $X$ is a $\\mathbb {Q}$ -martingale.", "Such a martingale measure $\\mathbb {Q}$ is determined by the right-continuous square-integrable martingale $\\Lambda (t) = E_{\\mathbb {P}}\\left[ \\left.", "\\frac{d\\mathbb {Q}}{d\\mathbb {P}} \\right| \\mathcal {F}(t)\\right]$ for $0\\le t\\le T$ .", "And explicitly $\\Lambda (T) = \\exp \\left(-\\int _0^T\\lambda (t)dB(t) - \\frac{1}{2}\\int _0^T\\lambda ^2(t)dt\\right)$ and $\\lambda $ satisfies Novikov's condition $E_{\\mathbb {P}}\\left[\\exp \\left( \\frac{1}{2}\\int _0^T\\lambda ^2(t)dt \\right)\\right] < \\infty .$ It is not hard to see that the price process $X$ under $\\mathbb {Q}$ is given by $dX(t) = \\nu (t)X(t)dt + \\sigma (t)X(t)dW(t).$ Note that the martingale measure $\\mathbb {Q}$ and $\\lambda $ are, if unique, theoretically and practically well-known as the risk-neutral measure and the market price of risk, respectively.", "The risk-neutral valuation formula is given by $V(t) = D(t)E_{\\mathbb {Q}}\\left[\\left.", "\\frac{1}{D(T)}H \\right| \\mathcal {F}(t) \\right] = D(t)E_{\\mathbb {Q}}\\left[\\left.", "\\frac{1}{D(T)}h(X(T)) \\right| \\mathcal {F}(t) \\right].$ The expectation is taken under the measure $\\mathbb {Q}$ .", "It is important to note that in the risk-neutral world the essential theoretical assumptions are: (1) the true price process (REF ) is correctly specified and (2) prices of derivatives $H$ are drawn from this price process, that is derivative prices are uniquely determined by formula (REF ).", "These assumptions, if not violated, lead to a complete market and the trading strategy (REF ) and the measure $\\mathbb {Q}$ are unique.", "However, in practice as we argued earlier, these assumptions are strongly violated; as a result market completeness and uniqueness of derivative prices are no longer valid.", "That is $\\mathbb {Q}$ is no longer risk-neutral, but only an equivalent measure in the set $\\mathcal {Q}$ .", "We now consider the representation (REF ) in a continuous time framework, the measure of intrinsic risk can be defined, without loss of generality, in terms of changes in values in a future time interval $[t, t+dt]$ as follows.", "Definition.", "The measure of intrinsic risk in a time interval $dt$ is defined by $dG(t,T) = \\zeta (t,T)X(t)dt$ , where $\\zeta = \\zeta (t,T)$ is a continuous adapted process representing a rate of intrinsic risk.", "As was represented earlier in (REF ), the evolution of a trading strategy shall be adaptable to adjust for the measure of intrinsic risk which can be considered an additional/less capital required in a time interval $dt$ , that is $dV(t) &=& \\alpha (t)(dX(t) + dG(t)) + \\nu (t)\\beta (t)D(t)dt \\\\&=& \\alpha (t)(\\mu (t) + \\zeta (t,T))X(t)dt + \\alpha (t)\\sigma (t)X(t)dB(t) + \\nu (t)\\left(V(t) - \\alpha (t)X(t)\\right)dt \\nonumber \\\\&=& \\nu (t)V(t)dt + \\alpha (t)(\\mu (t) + \\zeta (t,T) - \\nu (t))X(t)dt + \\alpha (t)\\sigma (t)X(t)dB(t)\\nonumber \\\\&=& \\nu (t)V(t)dt + \\alpha (t)\\sigma (t)X(t) \\left[ \\frac{\\mu (t) + \\zeta (t,T) - \\nu (t)}{\\sigma (t)}dt + dB(t)\\right]\\nonumber \\\\&=& \\nu (t)V(t)dt + \\alpha (t)\\sigma (t)X(t)dZ(t),\\nonumber $ where $Z = Z(t)$ is a $\\mathbb {S}$ -Brownian motion and is given by $dZ(t) = \\frac{\\mu (t) + \\zeta (t,T) - \\nu (t)}{\\sigma (t)}dt + dB(t) = \\frac{\\zeta (t,T)}{\\sigma (t)}dt + dW(t)$ and $\\mathbb {S}$ is a measure equivalent to $\\mathbb {P}$ .", "Thus, $\\mathbb {S} \\in \\mathcal {Q}$ .", "Analogously, $\\zeta /\\sigma $ is defined as an intrinsic price of risk.", "Under $\\mathbb {S}$ measure, the price process of $X$ under $\\mathbb {S}$ is given by $dX(t) = (\\nu (t) - \\zeta (t,T)) X(t)dt + \\sigma (t)X(t)dZ(t).$ Consequently the fair value of a contingent claim is given by the formula $V(t) = D(t)E_{\\mathbb {S}}\\left[\\left.", "\\frac{1}{D(T)}H \\right| \\mathcal {F}(t) \\right] = D(t)E_{\\mathbb {S}}\\left[\\left.", "\\frac{1}{D(T)}h(X(T)) \\right| \\mathcal {F}(t) \\right].$ From a pragmatic standpoint, what is needed in determining prices of derivatives and managing their risks is to allow sources of uncertainty that are epistemic (or subjective) rather than aleatory in nature.", "In theory, the value of a derivative can be perfectly replicated by a combination of other derivatives provided that these derivatives are uniquely determined by the formula (REF ).", "In practice, prices of derivatives (such as futures, vanilla options) on the same primary asset are not determined by (REF ) from statistically or econometrically observed model (REF ), but made by individual market makers who, with little, if not at all, knowledge of the true price process, have used their personal perception of the future.", "We argue further on this point as follows.", "If we let $Y = Y(t)$ be the price process of a derivative in a derivative market (such as futures in particular, $Y(t,T) = D(t)E_{\\mathbb {S}}\\left[\\left.X(T)/D(T) \\right| \\mathcal {F}(t) \\right]$ , since its contract is not necessarily connected with a physical primary asset), $Y$ must have an abstract dynamics and is assumed to satisfy a SDE $dY(t,T) = \\nu (t)Y(t,T)dt + \\bar{\\sigma }(t,T)Y(t,T)dZ(t),$ where $T$ denotes a fixed time horizon larger than or equal to the maturity of any contingent claim, $\\bar{\\sigma }$ is a Lipschitz continuous function so that a solution exists.", "We now show that $Z$ is a $\\mathbb {S}$ -Brownian motion - the source of randomness that drives the derivative price process $Y$ .", "We introduce a change of time, see for example [28].", "Let $U(t)$ be a positive function such that $U(t) = \\int _0^t \\frac{\\bar{\\sigma }^2(s,T)}{\\sigma ^2(s)}ds$ which is finite for finite time $t \\le T$ and increases almost surely.", "Define $\\tau (t) = U^{-1}(t)$ , let $Y$ be a replacement of $X$ , i.e.", "$X(t) = Y(\\tau (t),\\tau (T))$ whose solution is given by $dX(t) = \\frac{\\nu (t)\\sigma ^2(t)}{\\bar{\\sigma }^2(t,T)}X(t)dt + \\sigma (t)X(t)dZ(t)$ with $X(0) = Y(0)$ .", "Rearranging the drift term leads to $dX(t) = (\\nu (t) - \\zeta (t,T))X(t)dt + \\sigma (t)X(t)dZ(t),$ where $\\zeta (t,T) = \\frac{\\nu (t)}{\\bar{\\sigma }^2(t,T)} \\left(\\bar{\\sigma }^2(t,T) - \\sigma ^2(t)\\right).$ Here, we see the concurrence of the SDEs (REF ) and (REF ), the source of randomness $Z$ is the very $\\mathbb {S}$ -Brownian motion (REF ).", "We have just shown that the measure $\\mathbb {S}$ is subjective in the sense that the valuation of a contingent claim is not only subjected to the dynamics of the primary asset price, but also subject to an exogenous measure of risk $\\zeta $ .", "We shall call the measure $\\mathbb {S}$ the risk-subjective measure.", "The connection between the risk-subjective measure and the risk-neutral measure described by (REF ) is far more precise than that found in [25].", "An important note here is that the trading strategy (REF ) is equivalent to the risk-free money account, that is the growth of portfolio value (REF ) is at the risk-free rate $\\nu $ .", "In terms of pricing and hedging, the presence of intrinsic risk imposes an internal consistency and implies that possible arbitrage exists in the market (the primary market and its associated derivative markets)." ], [ "Applications - pricing and hedging", "In this section, we shall first discuss some problems related to asset models in parallel markets so as to provide some background for subsequent applications.", "In the light of intrinsic risk, the SDE (REF ) in reality may represent a risky asset price process in parallel markets such as: (1) futures price process, or (2) an implied price process recovered from option prices where $\\bar{\\sigma }$ is the implied volatility.", "Attempts of recovering the implied price process were pioneered, for examples, by [36], [9], [30], [8] and references therein.", "Market makers indeed have dispensed with the correct specification (REF ) and directly use an implied price process as a tool to prescribe the dynamics of the implied volatility surface.", "A practice of recovering an implied price process from observed derivative prices (such as vanilla option prices) and use it to price derivatives is known as instrumental approach, described in [34].", "A practical point that is more pertinent to the instrumental approach is that the prices of exotic derivatives are given by the price dynamics that can take into account or recover the volatility smile.", "With reference to intrinsic risk, an implied price process is a mis-specification for the primary asset, this was discussed in [16] and was shown that successful hedging depends entirely on the relationship between the mis-specified volatility $\\bar{\\sigma }$ and the true local volatility $\\sigma $ , and the total hedging error is given by, assuming zero risk-free rate, $H - h(X(T)) = \\frac{1}{2}\\int _0^T X^2(t)\\frac{\\partial ^2V}{\\partial x^2} \\left(\\bar{\\sigma }^2(t,T) - \\sigma ^2(t)\\right)dt.$ Note that this hedge error resembles the term (REF ).", "Clearly, the hedging error is an intrinsic price of risk presented as traded asset in the hedging strategy (REF ), but not in (REF ).", "Before we illustrate a number of applications for pricing and hedging with specific form of the measure of intrinsic risk, let us state a general result for derivative valuation." ], [ "Risk-subjective valuation", "We have established the risk-subjective valuation formula (REF ) where the risk-subjective price process is given by (REF ).", "The risk-subjective value $V$ of a contingent claim $H = h(X(T))$ given by $V = V(X(t),t) = D(t)E_{\\mathbb {S}}\\left[\\left.", "\\frac{1}{D(T)}h(X(T)) \\right| \\mathcal {F}(t) \\right]$ is a unique solution to $\\frac{\\partial V}{\\partial t}(x,t) + \\frac{1}{2}\\sigma ^2(t)x^2\\frac{\\partial ^2V}{\\partial x^2} + (\\nu (t) - \\zeta (t,T))x\\frac{\\partial V}{\\partial x}(x,t) = \\nu (t) V(x,t),$ with $X(t) = x$ and $V(x,T) = h(x)$ .", "The result is obtained by directly applying the Feynman-Kac formula.", "We have shown that the trading strategy (REF ) yields the risk-free rate of return on the value of a derivative, and also the intrinsic risk is perfectly hedged by delta-hedging representation (REF )." ], [ "Modelling measure of intrinsic risk", "As unpredictable as a market, prices in a parallel market (such as futures and corresponding vanilla options) may not be driven by the same source of randomness that drives the primary asset (such as stock and bond).", "Motivated by results (REF ) and (REF ), in the present framework it makes sense to formulate $\\zeta $ by an abstract form $\\zeta (t,T) = \\gamma (t,T)\\left(\\bar{\\sigma }^2(t,T) - \\sigma ^2(t)\\right),$ where $\\sigma $ is the volatility of the underlying asset, $\\bar{\\sigma }$ the volatility of a risky asset in a parallel market.", "We propose that $\\zeta $ takes a general form of an exponential family $\\zeta = e^{\\xi (x) + \\eta (\\theta )\\phi (x) - \\psi (\\theta )},$ the parameter $\\theta = \\lbrace \\sigma , \\bar{\\sigma }\\rbrace $ and $X(t) = x$ .", "As a result, (REF ) is a special case.", "Remark.", "While the diffusion term $\\sigma $ accounts for the distributional property of the primary asset price, the exogenous term $\\zeta $ accounts for a phenomenon such as volatility smile.", "The existence of intrinsic risk appears to undermine the true probability distribution of the underlying, however it emphasises its important role in determining the values of derivatives.", "It ensures maximal consistency in pricing and hedging contingent claims that are path-dependent/independent and particularly derivatives on volatility (such as variance swap, volatility swap).", "It insists on a realistic dynamics for the underlying asset as far as delta-hedge is concerned." ], [ "Valuation of forward and futures contracts", "In practice, forward contracts are necessarily associated with the primary asset (such as stock and bond) and therefore their prices are determined by (REF ) and hedged by (REF ).", "As was illustrated in the previous section, can be determined by (REF ) which includes a measure of intrinsic risk, $\\zeta $ , as a convexity adjustment." ], [ "Contingent claims on dividend paying assets with default risk", "Hedgers holding the primary asset in their hedging portfolio would receive dividends which are assumed to be a continuous stream of payments, whereas hedgers holding other hedge instruments (such as futures, vanilla options) do not receive dividends.", "In this case, $\\zeta $ can be considered as dividend yield and $\\zeta X dt$ is the amount of dividend received in a time interval $dt$ .", "$\\zeta $ may also be a non-negative function representing the hazard rate of default in a time interval $dt$ , this well-known approach was proposed in [31] and references therein." ], [ "Foreign market derivatives", "Suppose that $r_f$ is the risk-free rate of return of a foreign money account and $\\zeta _v$ the measure of risk that accounts for volatility smile, (REF ) is then a direct application to foreign market derivatives where $\\zeta = r_f + \\zeta _v$ .", "This is indeed the simplest application of risk-subjective valuation." ], [ "Interest rate derivatives", "As an exogenous variable to the risk-subjective price process (REF ), $\\zeta $ of a particular form would become a mean of reversion.", "This is a desirable feature in a number of well-known interest rate models such as extended model of [24], [5] model.", "With reference to the liquidity preference theory or the preferred habitat theory of [27], a term premium for a bond can be represented as a measure of intrinsic risk." ], [ "Concluding remarks", "It is well-known among both academic and practitioners that the standard complete market framework often failed, see for example [32].", "Incomplete market framework becomes crucial in understanding and explaining well-known market anomalies.", "In this article we have introduced the notion of intrinsic risk and derived the risk-subjective measure $\\mathbb {S}$ equivalent to the real-world measure $\\mathbb {P}$ , where $\\mathbb {S}\\in \\mathcal {Q}$ .", "At a conceptual level, the theory of Girsanov change of measure allows us to recognise that the crucial role of $\\mathbb {S}$ , rather than the expectation $E_{\\mathbb {S}}[H]$ , is assigned to the price of a derivative (such as futures, vanilla option).", "In addition, the intrinsic risk as a structure is what needed to be imposed on the mutual movements of the primary and derivative markets so that, at least, the pricing and hedging derivatives (such as swaps and caplets) can be undertaken on a consistent basis.", "Apart from such conceptual aspect, the measure $\\mathbb {S}$ does not undermine the role of the measure $\\mathbb {P}$ in that a lot of knowledge about the primary market is known at any given time $t$ .", "More precisely, the market's expectation (predictions) in terms of a measure $\\mathbb {S}$ at time $t$ is given by the conditional probability distribution $\\mathbb {S}\\left[ \\left.", "\\cdot \\right| \\mathcal {F}(t) \\right] \\mbox{ on } \\mathcal {F}_{\\mathbb {S}}(t)$ where $\\mathcal {F}(t)$ is the information available given by the primary market at time $t$ , and $\\mathcal {F}_{\\mathbb {S}}(t)$ is the information generated by derivatives (such as vanilla options) with maturities $T > t$ .", "A final remark: In view of the last financial crises, the market has evolved and there is an apparent need, both among practitioners and in academia, to comprehend the problems caused by an excessive dependence on a specific asset modeling approach, by ambiguous specification of risks and/or by confusions between risks and uncertainties (volatilities).", "As a result, we presented a continuous time framework that, we believe, brings unity, simplicity and consistency to two important aspects: pricing with correctly specified model for a primary risky asset, and hedging risks that can be correctly understood and specified.", "In addition, the framework proposed in this article is rigorous in the sense that the true meanings of properties and relationships of intrinsic risk and volatility are self-consistent such that their values are not arbitrarily assigned nor should their properties be misused by ignorance." ] ]
1403.0333
[ [ "Tropical Images of Intersection Points" ], [ "Abstract A key issue in tropical geometry is the lifting of intersection points to a non-Archimedean field.", "Here, we ask: Where can classical intersection points of planar curves tropicalize to?", "An answer should have two parts: first, identifying constraints on the images of classical intersections, and, second, showing that all tropical configurations satisfying these constraints can be achieved.", "This paper provides the first part: images of intersection points must be linearly equivalent to the stable tropical intersection by a suitable rational function.", "Several examples provide evidence for the conjecture that our constraints may suffice for part two." ], [ "Introduction", "Let $K$ be an algebraically closed non-Archimedean field with a nontrivial valuation $\\text{val}:K^*\\rightarrow \\mathbb {R}$ .", "The examples throughout this paper will use $K=\\!\\lbrace t\\rbrace \\!\\rbrace $ , the field of Puiseux series over the complex numbers with indeterminate $t$ .", "This is the algebraic closure of the field of Laurent series over $, and can be defined as$$\\!\\lbrace t\\rbrace \\!\\rbrace =\\left\\lbrace \\sum _{i=k}^\\infty a_i t^{i/n}\\,:\\, a_i\\in n,k\\in \\mathbb {Z}, n>0 \\right\\rbrace , $$with $ val(i=kai ti/n)=k/n$ if $ ak0$.", "In particular, $ val(t)=1$.$ The tropicalization map $\\text{trop}:(K^*)^n\\rightarrow \\mathbb {R}^n$ sends points in the $n$ -dimensional torus $(K^*)^n$ into Euclidean space under coordinate-wise valuation: $\\text{trop}:(a_1,\\ldots ,a_n)\\rightarrow (\\text{val}(a_1),\\ldots ,\\text{val}(a_n)).$ In tropical geometry, we consider the tropicalization map on a variety $X\\subset (K^*)^n$ .", "Since the value group is dense in $\\mathbb {R}$ , we take the Euclidean closure of $\\text{trop}(X)$ in $\\mathbb {R}^n$ , and call this the tropicalization of $X$, denoted $\\text{Trop}(X)$ .", "The tropicalization of a variety is a piece-wise linear subset of $\\mathbb {R}^n$ , and has the structure of a balanced weighted polyhedral complex.", "In the case where $X$ is a hypersurface, the combinatorics of the tropicalization can be found from a subdivision of the Newton polytope of $X$ .", "For more background on tropical geometry, see [5] and [8].", "Consider two curves $X,Y\\subset (K^*)^2$ intersecting in a finite number of points.", "We are interested in the image of the intersection points under tropicalization; that is, in $\\text{Trop}(X\\cap Y)$ inside of $\\text{Trop}(X)\\cap \\text{Trop}(Y)\\subset \\mathbb {R}^2$ .", "It was shown in [10] that if $\\text{Trop}(X)\\cap \\text{Trop}(Y)$ is zero dimensional in a neighborhood of a point in the intersection, then that point is in $\\text{Trop}(X\\cap Y)$ .", "More generally, they showed this for varieties $X$ and $Y$ under the assumption that $\\text{Trop}(X)\\cap \\text{Trop}(Y)$ has codimension $\\text{codim }X+\\text{codim }Y$ in a neighborhood of the point.", "It follows that if $\\text{Trop}(X)\\cap \\text{Trop}(Y)$ is a finite set, then $\\text{Trop}(X\\cap Y)=\\text{Trop}(X)\\cap \\text{Trop}(Y)$ .", "It is possible for $\\text{Trop}(X)\\cap \\text{Trop}(Y)$ to have higher dimensional components, namely finite unions of line segments and rays.", "It was shown in [11] that if $\\text{Trop}(X)\\cap \\text{Trop}(Y)$ is bounded, then each connected component of $\\text{Trop}(X)\\cap \\text{Trop}(Y)$ has the “right” number of images of points in $X\\cap Y$ , counted with multiplicity.", "In this context, the “right” number is the number of points in the stable tropical intersection of that connected component; the stable tropical intersection is $\\lim _{\\varepsilon \\rightarrow 0}(\\text{Trop}(X)+\\varepsilon \\cdot v)\\cap \\text{Trop}(Y)$ , where $v$ is a generic vector and $\\varepsilon $ is a real number [11].", "They further showed that the theorem holds for components of $\\text{Trop}(X)\\cap \\text{Trop}(Y)$ that are unbounded, after a suitable compactification.", "We offer the following example to illustrate this higher dimensional component phenomenon.", "This will motivate the following question: as we vary $X$ and $Y$ over curves with the same tropicalizations, what are the possibilities for the varying set $\\text{Trop}(X\\cap Y)$ inside of the fixed set $\\text{Trop}(X)\\cap \\text{Trop}(Y)$ ?", "Example 1.1 Let $K=\\!\\lbrace t\\rbrace \\!\\rbrace $ and let $f,g\\in K[x,y]$ be $f(x,y)=c_1+c_2x+c_3y $ and $g(x,y)=c_4x+c_5xy+tc_6y$ , where $c_i\\in K$ and $\\text{val}(c_i)=0$ for all $i$ .", "Let $X,Y\\subset (K^*)^2$ be the curves defined by $f$ and $g$ , respectively.", "Figure: Trop(X)\\text{Trop}(X) and Trop(Y)\\text{Trop}(Y), before and after a small shift of Trop(Y)\\text{Trop}(Y).Regardless of our choice of $c_i$ , $\\text{Trop}(X)$ and $\\text{Trop}(Y)$ will be as pictured in Figure REF , with $\\text{Trop}(X)$ and $\\text{Trop}(Y)$ intersecting in the line segment $L$ from $(0,0)$ to $(1,0)$ .", "However, $X$ and $Y$ only intersect in two points (or one point with multiplicity 2).", "The natural question is: as we vary the coefficients while keeping valuations (and thus tropicalizations) fixed, what are the possible images of the two intersection points within $L$ ?", "A reasonable guess is that the intersection points map to the stable tropical intersection $\\lbrace (0,0),(1,0)\\rbrace $ , and indeed this does happen for a generic choice of coefficients.", "However, as shown in Example REF , one can choose coefficients such that the intersection points map to any pair of points in $L$ of the form $(r,0)$ and $(1-r,0)$ , where $0\\le r\\le \\frac{1}{2}$ .", "These possible configurations are illustrated in Figure REF .", "Figure: Possible images of X∩YX\\cap Y in Trop(X)∩Trop(Y)\\text{Trop}(X)\\cap \\text{Trop}(Y).The main result of this paper is that the points $\\text{Trop}(X\\cap Y)$ inside of $\\text{Trop}(X)\\cap \\text{Trop}(Y)$ must be linearly equivalent to the stable tropical intersection via particular tropical rational functions, defined in Section .", "To distinguish tropical rational functions from classical rational functions, they will be written as $f^{\\text{trop}}$ , $g^{\\text{trop}}$ , or $h^{\\text{trop}}$ instead of $f$ , $g$ , or $h$ .", "See [1] and [4] for more background.", "In all the examples discussed in Section , essentially every such configuration is achievable.", "Conjecture REF expresses our hope that this always holds.", "Theorem 1.2 Let $X,Y\\subset (K^*)^2$ where $X\\cap Y$ is equal to the multiset $\\lbrace p_1,\\ldots ,p_n\\rbrace $ and where $\\text{Trop}(X)$ is smooth.", "Let $E$ be the stable intersection divisor of $\\text{Trop}(X)$ and $\\text{Trop}(Y)$ , and let $D$ be $D=\\sum _i\\text{trop}(p_i).$ Then there exists a tropical rational function $h^{\\text{trop}}$ on $\\text{Trop}(X)$ such that $(h^{\\text{trop}})=D-E$ and $\\text{supp}(h^{\\text{trop}})\\subset \\text{Trop}(X)\\cap \\text{Trop}(Y)$ .", "We will present two proofs of this theorem.", "In Sections and we approach the question from the perspective of Berkovich theory, which in the smooth case allows us to tropicalize rational functions on classical curves.", "In Section we present an alternate argument using tropical modifications, which allows us to drop the smoothness assumption.", "Example 1.3 Let $X$ and $Y$ be as in Example REF .", "We will consider tropical rational functions on $\\text{Trop}(X)\\cap \\text{Trop}(Y)$ such that (i) the stable intersection points are the poles (possibly canceling with zeros), and (ii) the tropical rational function takes on the same value at every boundary point of $\\text{Trop}(X)\\cap \\text{Trop}(Y)$ .", "If we insist that the “same value” in condition (ii) is 0, we may extend these tropical rational functions to all of $\\text{Trop}(X)$ by setting them equal to 0 on $\\text{Trop}(X)\\setminus \\text{Trop}(Y)$ .", "This yields tropical rational functions on $\\text{Trop}(X)$ with $\\text{supp}(h^{\\text{trop}})\\subset \\text{Trop}(X)\\cap \\text{Trop}(Y)$ , as in Theorem REF .", "Instances of the types of such tropical rational functions on $L=\\text{Trop}(X)\\cap \\text{Trop}(Y)$ from our example are illustrated in Figure REF .", "Figure: Graphs of tropical rational functions on Trop(X)∩Trop(Y)\\text{Trop}(X)\\cap \\text{Trop}(Y).As asserted by Theorem REF , all possible image intersection sets in $\\text{Trop}(X)\\cap \\text{Trop}(Y)$ arise as the zero set of such a tropical rational function.", "Equivalently, the stable intersection divisor and the image of intersection divisor are linearly equivalent via one of these functions.", "Remark 1.4 It is not quite the case that the zero set of every such tropical rational function (from Example REF ) is attainable as the image of the intersections of $X$ and $Y$ (with changed coefficients).", "For instance, such a tropical rational function could have zeros at $(\\frac{\\sqrt{2}}{2},0)$ and $(1-\\frac{\\sqrt{2}}{2},0)$ , which cannot be the images of any points on $X$ and $Y$ since they have irrational coordinates.", "However, if we insist that the tropical rational functions have zeros at points with rational coefficients (since $\\mathbb {Q}=\\text{val}(K^*)$ ), all zero sets can be achieved as the images of intersections.", "This is the content of Conjecture REF ." ], [ "Acknowledgements", "The author would like to thank Matt Baker and Bernd Sturmfels for introducing him to these questions in tropical geometry.", "The author would also like to thank Sarah Brodsky, Melody Chan, Nikita Kalinin, Kristin Shaw, and Josephine Yu for helpful conversations and insights.", "The author was supported by the NSF through grant DMS-0968882 and a graduate research fellowship, and by the Max Planck Institute for Mathematics in Bonn." ], [ "Tropicalizations of Rational Functions", "In this section we present background information on tropical rational function theory, and use some Berkovich theory to define the tropicalization of a rational function.", "For the theory of tropical rational functions, we consider abstract tropical curves $\\Gamma $ , which are weighted metric graphs with finitely many edges and vertices, where the edges have possibly infinite lengths.", "See [2] for background on Berkovich spaces, and [9] for more background on tropical rational functions.", "Tropical rational functions on tropical curves are analogous to classical rational functions on classical curves.", "A divisor on a tropical curve $\\Gamma $ is a finite formal sum of points in $\\Gamma $ with coefficients in $\\mathbb {Z}$ .", "If $D=\\sum _ia_iP_i$ , the degree of $D$ is $\\deg D:=\\sum _ia_i$ .", "The support of $D$ is the set of all points $P_i$ with $a_i\\ne 0$ , and $D$ is called effective if all $a_i$ 's are nonnegative.", "Definition 2.1 A rational function on a tropical curve $\\Gamma $ is a continuous function $f^{\\text{trop}}:\\Gamma \\rightarrow \\mathbb {R}\\cup \\lbrace \\pm \\infty \\rbrace $ such that the restriction of $f^{\\text{trop}}$ to any edge of $\\Gamma $ is a piecewise linear function with integer slopes and only finitely many pieces.", "This means that $f^{\\text{trop}}$ can only take on the values of $\\pm \\infty $ at the unbounded ends of $\\Gamma $ .", "The associated divisor of $f^{\\text{trop}}$ is $(f^{\\text{trop}})=\\sum _{P\\in \\Gamma }\\text{ord}_P(f^{\\text{trop}})\\cdot P$ , where $\\text{ord}_P(f^{\\text{trop}})$ is minus the sum of the outgoing slopes of $f$ at a point $P$ .", "If $D$ and $E$ are divisors such that $D-E=(f^{\\text{trop}})$ for some tropical rational function $f$ , we say that $D$ and $E$ are linearly equivalent.", "Figure: The graph of a rational function f trop f^{\\text{trop}} on an abstract tropical curve Γ\\Gamma .As an example, consider Figure REF .", "Here $\\Gamma $ consists of four vertices and three edges arranged in a Y-shape, and the image of $\\Gamma $ under a rational function $f$ is illustrated lying above it.", "The leftmost vertex is a zero of order 2, since there is an outgoing slope of $-2$ and no other outgoing slopes.", "The next kink in the graph is a pole of order 1, since the outgoing slopes are 2 and $-1$ and $2+(-1)=1$ .", "Moving along in this direction we have a pole of order 4, a zero of order 4, at one endpoint a pole of order 1, and at the other endpoint no zeros or poles.", "Note that, counting multiplicity, there are six zeros and six poles.", "The numbers agree, as in the classical case.", "Since we can tropicalize a curve to obtain a tropical curve, we would like to tropicalize a rational function on a curve and obtain a tropical rational function on a tropical curve.", "A naïve definition of “tropicalizing a rational function” would be as follows.", "Naïve Definition 2.2 Let $h$ be a rational function on a curve $X$ .", "Define the tropicalization of $h$, denoted $\\text{trop}(h)$ , as follows.", "For every point $w$ in the image of $X\\setminus \\lbrace \\text{zeros and poles of $h$}\\rbrace $ under tropicalization, lift that point to $p\\in X$ , and define $\\text{trop}(h)(w)=\\text{val}(h( p)).$ Extend this function to all of $\\text{Trop}(X)$ by continuity.", "Unfortunately this is not quite well-defined, because $\\text{val}(h( p))$ depends on which lift $p$ of $w$ we choose.", "However, as suggested to the author by Matt Baker, this definition can be made rigorous if at least one of the tropicalizations is suitably faithful in a Berkovich sense.", "Let $h$ be a rational function on $X$ , and assume that there is a canonical section $s$ to the map $X^{an}\\rightarrow \\text{Trop}(X)$ , where $X^{an}$ is the analytification of $X$ .", "For $w\\in \\text{Trop}(X)$ , define $\\text{trop}(h)(w)=\\log |h|_{s(w)},$ where $|\\cdot |_{s(w)}$ is the seminorm corresponding to the point $s(w)$ in $X^{an}$ .", "This rational function has the desired properties.", "Remark 2.3 In [2] one can find conditions to guarantee that there exists a canonical section $s$ to the map $X^{an}\\rightarrow \\text{Trop}(X)$ .", "For instance, if $\\text{Trop}(X)$ is smooth in the sense that it comes from a unimodular triangulation of its Newton polygon, such a section will exist." ], [ "Main Result and a Conjecture", "We are ready to prove Theorem REF .", "Let $f$ and $g$ be the defining equations of $X$ and $Y$ , respectively.", "Let $g^{\\prime }\\in K[x,y]$ have the same tropical polynomial as $g$ , and let $Y^{\\prime }$ be the curve defined by $g^{\\prime }$ .", "We have that $\\text{Trop}(Y)=\\text{Trop}(Y^{\\prime })$ , and for generic $g^{\\prime }$ we have that $\\text{Trop}(X\\cap Y^{\\prime })$ is the stable tropical intersection of $\\text{Trop}(X)$ and $\\text{Trop}(Y)$ .", "Recall that $p_1,\\ldots ,p_n$ denote the intersection points of $X$ and $Y$ , possibly with repeats.", "Let $p^{\\prime }_1,\\ldots ,p^{\\prime }_m$ denote the intersection points of $X$ and $Y^{\\prime }$ , with duplicates in the case of multiplicity.", "Note that $m$ and $n$ will be equal unless $X$ and $Y$ have intersection points outside of $(K^*)^2$ ; this is discussed in Remark REF .", "Consider the rational function $h=\\frac{g}{g^{\\prime }}$ on $X$ , which has zeros at the intersection points of $X$ and $Y$ and poles at the intersection points of $X$ and $Y^{\\prime }$ .", "Since $\\text{Trop}(X)$ is smooth, by Remark REF we may tropicalize $h$ .", "This gives a tropical rational function $\\text{trop}(h)$ on $\\text{Trop}(X)$ with divisor $(\\text{trop}(h))=\\text{trop}(p_1)+\\ldots +\\text{trop}(p_n)-\\text{trop}(p^{\\prime }_1)-\\ldots -\\text{trop}(p^{\\prime }_m)=D-E.$ We claim that $\\text{trop}(h)$ is the desired $h^{trop}$ from the statement of the theorem.", "All that remains to show is that $\\text{supp}(\\text{trop}(h))\\subset \\text{Trop}(X)\\cap \\text{Trop}(Y)$ .", "If $w\\in \\text{Trop}(X)\\setminus \\text{Trop}(Y)$ , then $|g|_{s(w)}=|g^{\\prime }|_{s(w)}$ because $g$ and $g^{\\prime }$ both have bend locus $\\text{Trop}(Y)$ , and $w$ is away from $\\text{Trop}(Y)$ .", "This means that $\\text{trop}(h)(w)=|h|_{s(w)}=|g|_{s(w)}-|g^{\\prime }|_{s(w)}=0$ on $\\text{Trop}(X)\\setminus \\text{Trop}(Y)$ .", "This completes the proof.", "Remark 3.1 The argument and result will hold even if $\\text{Trop}(X)$ is not smooth as long as there exists a section $s$ to $X^{an}\\rightarrow \\text{Trop}(X)$ .", "Remark 3.2 Since we have our result in terms of linear equivalence, we get as a corollary that the configurations of points differ by a sequence of chip firing moves by [6].", "Remark 3.3 If $\\text{Trop}(X)\\cap \\text{Trop}(Y)$ is unbounded (for instance, if $\\text{Trop}(X)=\\text{Trop}(Y)$ ), then it is possible to have zeros of the rational function “at infinity.” This is OK, and can be made sense of using a compactifying fan as in [11].", "See Example REF for an instance of this phenomenon.", "Our theorem has placed a constraint on the configurations of intersection points mapping into tropicalizations.", "The following conjecture posits that essentially all these configurations are attainable.", "Conjecture 3.4 Assume we are given $\\text{Trop}(X)$ and $\\text{Trop}(Y)$ and a tropical rational function $h^{\\text{trop}}$ on $\\text{Trop}(X)$ with simple poles precisely at the stable tropical intersection points and zeros in some configuration (possibly canceling some of the poles) with coordinates in the value group ($\\mathbb {Q}$ for $\\lbrace t\\rbrace \\rbrace $ ), such that $\\text{supp}(h^{\\text{trop}})\\subset \\text{Trop}(X)\\cap \\text{Trop}(Y)$ .", "Then it is possible to find $X$ and $Y$ with the given tropicalizations such that $\\text{trop}(p_1),\\ldots ,\\text{trop}(p_n)$ are the zeros of $h^{\\text{trop}}$ .", "We will consider the space of all configurations of zeros of rational functions on $\\text{Trop}(X)\\cap \\text{Trop}(Y)$ satisfying the given properties.", "This will form a polyhedral complex.", "First, we will prove that we can achieve the configurations corresponding to the vertices of this complex.", "Next, let $E$ be an edge connecting $V$ and $V^{\\prime }$ , where the configuration given by $V$ is achieved by $X$ and $Y$ and the configuration given by $V^{\\prime }$ is achieved by $X^{\\prime }$ and $Y^{\\prime }$ .", "We will prove that we can achieve any configuration along the edge by somehow deforming $(X,Y)$ to $(X^{\\prime },Y^{\\prime })$ .", "This will show that all points on edges of the complex correspond to achievable configurations.", "We will continue this process (vertices give edges, edges give faces, etc.)", "to show that all points in the complex correspond to achievable configurations.", "For an illustration of this process, see Example REF and Figure REF ." ], [ "Tropical Modifactions", "In this section we outline an alternate proof to Theorem REF using tropical modifications.", "See [3] for background on this subject.", "Let $X$ , $Y$ , $f$ , $g$ , $g^{\\prime }$ , $D$ , and $E$ be as in the proof from Section .", "Let $g_{\\text{trop}}$ and $g^{\\prime }_{\\text{trop}}$ be the tropical polynomials defined by $g$ and $g^{\\prime }$ , respectively.", "Let $g(X)\\subset (K^*)^2\\times K$ be the curve that is the closure of $\\lbrace (p,g(p )\\,|\\,p\\in X\\rbrace $ .", "Its tropicalization $\\text{Trop}(g(X))$ is contained in the tropical hypersurface in $\\mathbb {R}^3$ determined by the polynomial $z=g_{\\text{trop}}$ , and projects onto $\\text{Trop}(X)$ .", "Call this projection $\\pi $ .", "Note that outside of $\\text{Trop}(Y)$ , $\\pi $ is one-to-one, and $\\text{Trop}(g(X))$ agrees with $\\text{Trop}(g^{\\prime }(X))$ .", "By [3], the infinite vertical rays in $\\pi ^{-1}(\\text{Trop}(X)\\cap \\text{Trop}(Y))$ correspond to the intersection points of $X$ and $Y$ , and so lie above the support of the divisor $D$ on $\\text{Trop}(X)$ .", "Delete the vertical rays from $\\pi ^{-1}(\\text{Trop}(X)\\cap \\text{Trop}(Y))$ , and decompose the remaining line segments into one or more layer, where each layer gives the graph of a piecewise linear function on $\\text{Trop}(X)\\cap \\text{Trop}(Y)$ .", "(If deleting the vertical rays makes $\\pi $ a bijection, there will be only one layer.)", "Call these piecewise linear functions $\\ell _1,\\ldots ,\\ell _k$ .", "The tropical rational function $h^{\\text{trop}}=\\sum _{i=1}^k(\\ell _i-g^{\\prime }_{\\text{trop}}) $ has value 0 outside of $\\text{Trop}(X)\\cap \\text{Trop}(Y)$ because of the agreement of $\\text{Trop}(g(X))$ and $\\text{Trop}(g^{\\prime }(X))$ , and has divisor $D-E$ .", "This argument gives us a slightly stronger version of Theorem REF , in that it does not require the assumption of smoothness on $X$ ." ], [ "Evidence for Conjecture ", "In these examples we consider curves $X$ and $Y$ over the field of Puiseux series $\\mathbb {C}\\lbrace \\lbrace t\\rbrace \\rbrace $ .", "Example 5.1 Let $f$ and $g$ be as in Example REF .", "Treating them as elements of $(K[x])[y]$ , their resultant is $-c_2c_5x^2 +(c_3c_4 -c_1c_5- tc_2c_6)x - tc_1c_6$ The two roots of this quadratic polynomial in $x$ , which are the $x$ -coordinates of the two points in $X\\cap Y$ , have valuations equal to the slopes of the Newton polygon.", "Generically the valuations of the coefficients are 0, 0, and 1, giving slopes 0 and 1.", "For any rational number $r>0$ we may choose $c_1=1-t^r-t$ and all other $c_i=1$ , giving $\\text{val}(c_3c_4 -c_1c_5- tc_2c_6)=\\text{val}(t^r)=r$ .", "If $r\\le \\frac{1}{2}$ this gives slopes of $r$ and $1-r$ , and if $r\\ge \\frac{1}{2}$ this gives two slopes of $\\frac{1}{2}$ .", "These cases are illustrated in Figure REF and correspond to rational functions illustrated in Figure REF .", "This means all possible images of intersections allowed by Theorem REF with rational coordinates are achievable, so Conjecture REF holds for this example.", "Example 5.2 Consider conic curves $X$ and $Y$ given by the polynomials $f(x,y)=c_1x+c_2y+c_3xy=0\\rbrace $ and $g(x,y)=c_4x+c_5y+c_6xy+t(c_7x^2+c_8y^2+c_9)=0$ , where $\\text{val}(c_i)=0$ for all $i$ .", "The tropicalizations of $X$ and $Y$ are shown in Figure REF , and intersect in three line segments joined at a point.", "Figure: Trop(X)\\text{Trop}(X) and Trop(Y)\\text{Trop}(Y), before and after a small shift of Trop(Y)\\text{Trop}(Y).The stable tropical intersection consists of four points: $(-1,0)$ , $(0,-1)$ , $(1,1)$ , and $(0,0)$ .", "The possible images of $\\text{Trop}(X\\cap Y)$ must be linearly equivalent to these via a rational function equal to 0 on the three exterior points.", "This gives us intersection configurations of three possible types: (i) $\\lbrace (-(p-r),0),(0,-p), (p,p), (-r,0)\\rbrace $ where $0\\le r\\le p/2$ ; (ii) $\\lbrace (-p,0),(0,-(p-r)), (p,p), (0,-r)\\rbrace $ where $0\\le r\\le p/2$ ; and (iii) $\\lbrace (-p,0),(0,-p), (p-r,p-r), (r,r)\\rbrace $ where $0\\le r\\le p/2$ .", "To achieve a type (i) configuration, set $f(x,y)=x+y+xy$ and $g(x,y)=(1+2t^{1-p+r})x+(1+t^{1-p})y+xy+t(x^2+y^2+1)$ ; if $r>0$ , the 2 can be omitted from the coefficient of $x$ in $g$ .", "The Newton polygons of two polynomials, namely the resultants of $f$ and $g$ with respect to $x$ and with respect to $y$ , show that $\\text{Trop}(X\\cap Y)=\\lbrace (-(p-r),0),(0,-p), (p,p), (-r,0)\\rbrace $ .", "Type (ii) and (iii) are achieved similarly, so Conjecture REF holds for this example.", "For instance, if $f(x,y)=x+y+xy$ and $g(x,y)=(1+t^{1/2})x+(1+t^{1/3})y+xy+t(x^2+y^2+1)$ , then $\\text{Trop}(X\\cap Y)=\\lbrace (2/3,2/3),(0,-2/3),(-1/2,0),(-1/6,0)\\rbrace $ .", "The formal sum of these points is linearly equivalent to the stable intersection divisor, as illustrated by the rational function in Figure REF .", "This is the tropicalization of the rational function $h(x,y)=\\frac{(1+t^{1/2})x+(1+t^{1/3})y+xy+t(x^2+y^2+1)}{2x+4y+xy+t(x^2+y^2+1)}$ , where $g^{\\prime }(x,y):=2x+4y+xy+t(x^2+y^2+1)$ was chosen so that $\\text{Trop}(X)\\cap \\text{Trop}(V(g^{\\prime }))$ is the stable tropical intersection of $\\text{Trop}(X)$ and $\\text{Trop}(Y)$ .", "Figure: The graph of trop(h)\\text{trop}(h) on Trop(X)∩Trop(Y)\\text{Trop}(X)\\cap \\text{Trop}(Y), with zeros at the dots and poles at the x's.Figure: The moduli space MM of intersection configurations, with six examples.We can also consider this example in view of the outlined method of proof for Conjecture REF .", "Considering each intersection configuration as a point in $\\mathbb {R}^8$ (natural for four points in $\\mathbb {R}^2$ ), we obtain a moduli space $M$ for the possible tropical images of $X\\cap Y$ .", "The structure of this space is related to the notion of tropical convexity, as discussed in [7].", "As illustrated in Figure REF , $M$ consists of three triangles glued along one edge.", "The hope is that if vertices like $A$ and $C$ can be achieved, then it is possible to slide along the edge and achieve points like $D$ .", "For instance, if we set $f_A(x,y)=f_C(x,y)=f_{{AC},r}=x+y+xy$ $g_A=(1+t^0)x+4y+xy+t(x^2+y^2+1)$ $g_C=(1+t^{1/2})x+4y+xy+t(x^2+y^2+1)$ $g_{{AC},r}=(1+t^{r})x+4y+xy+t(x^2+y^2+1),$ then $f_A$ and $g_A$ give configuration $A$ , $f_C$ and $g_C$ give configuration $C$ , and $f_{AC,r}$ and $g_{AC,r}$ give all configurations along the edge $AC$ as $r$ varies from 0 to $\\frac{1}{2}$ .", "Example 5.3 Let $X$ and $Y$ be distinct lines defined by $f(x,y)=c_1+c_2x+c_3y$ and $g(x,y)=c_6+c_4x+c_5y$ with $\\text{val}(c_i)=0$ for all $i$ .", "These lines tropicalize to the same tropical line centered at the origin, with stable tropical intersection equal to the single point $(0,0)$ .", "Any point on $\\text{Trop}(X)=\\text{Trop}(X)\\cap \\text{Trop}(Y)$ is linearly equivalent to $(0,0)$ via a tropical rational function on $X$ , so Theorem REF puts no restrictions on the image of $p=X\\cap Y$ under tropicalization.", "In keeping with Conjecture REF , all possibilities can be achieved: (i) For $\\text{trop}( p)=(r,0)$ , let $f(x,y)=1+x+y$ , $g(x,y)=(1+t^r)+x+y$ .", "(ii) For $\\text{trop}( p)=(0,r)$ , let $f(x,y)=1+x+y$ , $g(x,y)=1+(1+t^r)x+y$ .", "(iii) For $\\text{trop}(p )=(-r,-r)$ , let $f(x,y)=1+x+y$ , $g(x,y)=1+x+(1+t^r)y$ .", "The point $(0,0)$ is also linearly equivalent to points at infinity, as witnessed by rational functions with constant slope 1 on an entire infinite ray.", "Mapping $p$ “to infinity” means that $X$ and $Y$ cannot intersect in $(K^*)^2$ , so we can choose equations for $X$ and $Y$ that give $p$ a coordinate equal to 0, such as $x+y+1=0$ and $x+2y+1=0$ .", "Example 5.4 Let $X$ and $Y$ be the curves defined by $f(x,y)=xy+t(c_1x+c_2y^2+c_3x^2y)$ $g(x,y)=xy+t(d_1x+d_2y^2+d_3x^2y) $ respectively, where $\\text{val}(c_i)=\\text{val}(d_i)=0$ for all $i$ .", "This means $\\text{Trop}(X)$ and $\\text{Trop}(Y)$ are the same, and are as pictured in Figure REF .", "Figure: Trop(X)=Trop(Y)=Trop(X∩Y)\\text{Trop}(X)=\\text{Trop}(Y)=\\text{Trop}(X\\cap Y), with the vertices of the triangle at (-1,1)(-1,1), (2,1)(2,1), and (-1,-2)(-1,-2).The resultant of $f$ and $g$ with respect to the variable $y$ is $&t^4( c_2^2d_1^2 - 2c_1c_2d_1d_2 + c_1^2d_2^2)x^2+ t^2(c_1c_2 - c_2d_1 - c_1d_2 + d_1d_2)x^3\\\\&+ t^3(-c_2c_3d_1 - c_1c_3d_2 + 2c_3d_1d_2 + 2c_1c_2d_3 -c_2d_1d_3 - c_1d_2d_3)x^4\\\\&+t^4 (c_3^2d_1d_2 - c_2c_3d_1d_3 - c_1c_3d_2d_3 + c_1c_2d_3^2)x^5,$ and the resultant of $f$ and $g$ with respect to the variable $x$ is $&t^4(c_2c_3d_1^2 - c_1c_3d_1d_2 - c_1c_2d_1d_3 + c_1^2d_2d_3)y^3\\\\&+t^3(2c_2c_3d_1 - c_1c_3d_2 - c_3d_1d_2 - c_1c_2d_3 -c_2d_1d_3 + 2c_1d_2d_3)y^4\\\\&+t^2( c_2c_3 - c_3d_2 - c_2d_3 + d_2d_3)y^5+t^4( c_3^2d_2^2 - 2c_2c_3d_2d_3 + c_2^2d_3^2)y^6.$ The stable tropical intersection consists of the three vertices of the triangle.", "Let us consider possible configurations of the three intersection points that have all three intersection points lying on the triangle, rather than on the unbounded rays.", "These are the configurations of zeros of rational functions with poles precisely at the three vertices; let $h^{\\text{trop}}$ be such a function.", "Label the vertices clockwise starting with $(-1,1)$ as $v_1$ , $v_2$ , $v_3$ .", "Starting from $v_1$ and going clockwise, label the poles of $h^{\\text{trop}}$ as $w_1$ , $w_2$ , $w_3$ .", "Let $\\delta _i$ denote the signed lattice distance between $v_i$ and $w_i$ , with counterclockwise distance negative.", "Then a necessary condition for the $w_i$ 's to be the poles of $h^{\\text{trop}}$ is $\\delta _1+\\delta _2+\\delta _3=0$ ; and in fact this condition is sufficient to guarantee the existence of such an $h^{\\text{trop}}$ .", "It follows that the $w_i$ 's cannot be in all different or all the same line segment of triangle, as all different would have $\\delta _1+\\delta _2+\\delta _3>0$ and all the same would have $\\delta _1+\\delta _2+\\delta _3\\ne 0$ .", "Hence we need only show that each configuration with exactly two $w_i$ 's on the same edge satisfying $\\delta _1+\\delta _2+\\delta _3=0$ is achievable.", "There are six cases to handle, since there are three choices for the edge with a pair of points and then two choices for the edge with the remaining point point.", "We will focus on the case where $w_1$ and $w_2$ are on the edge connecting $v_1$ and $v_2$ , and $w_3$ is on the edge connecting $v_2$ and $v_3$ , as shown in Figure REF .", "Let $\\delta _1=r$ and $\\delta _2=-s$ , where $r,s>0$ , and $2-s\\ge -1+r$ .", "It follows that $\\delta _3=-(r-s)$ , and that $r>s$ by the position of $w_3$ .", "Figure: The desired configuration of intersection points, where δ 1 =r>0\\delta _1=r>0, δ 2 =-s<0\\delta _2=-s<0, and δ 3 =-(r-s)<0\\delta _3=-(r-s)<0.To achieve the configuration specified by $r$ and $s$ , set $c_1=3+t^r,c_2=3,c_3=1,d_1=3, d_2=d+2t^{r-s},d_3=2.", "$ The valuations of the coefficients of the resultant polynomial with $x$ terms are $4+2(r-s)$ for $x^2$ , $2+2r-3$ for $x^3$ , $3+r-s$ for $x^4$ , and 4 for $x^5$ .", "It follows that the valuations of the $x$ -coordinates are $2-s$ , $-1+r$ , and $-1-s+r$ .", "When coupled with rational function restrictions, this implies that the intersection points of $X$ and $Y$ tropicalize to $(-1+r,1)$ , $(2-s,1)$ , and $(-1-s+r, -2-s+r)$ , which are indeed the points $w_1$ , $w_2$ , and $w_3$ we desired.", "The five other cases with all three intersection points in the triangle are handled similarly, and the cases with one or more intersection point on an infinite ray are even simpler.", "These examples provide not only a helpful check of Theorem REF , but also evidence that all possible intersection configurations can in fact be achieved.", "Future work towards proving this might be of a Berkovich flavor, as in Sections and , or may have more to do with tropical modifications, as presented in Section .", "Regardless of the approach, future investigations should not only look towards proving Conjecture REF , but also towards algorithmically lifting tropical intersection configurations to curves yielding them." ] ]
1403.0548
[ [ "Electromagnetically induced transparency like transmission in a\n metamaterial composed of cut-wire pairs with indirect coupling" ], [ "Abstract We theoretically and numerically investigate metamaterials composed of coupled resonators with indirect coupling.", "First, we theoretically analyze a mechanical model of coupled resonators with indirect coupling.", "The theoretical analysis shows that an electromagnetically induced transparency (EIT)-like phenomenon with a transparency bandwidth narrower than the resonance linewidths of the constitutive resonators can occur in the metamaterial with strong indirect coupling.", "We then numerically examine the characteristics of the metamaterial composed of coupled cut-wire pairs using a finite-difference time-domain (FDTD) method.", "The FDTD simulation confirms that an EIT-like transparency phenomenon occurs in the metamaterial owing to indirect coupling.", "Finally, we compare the results of the theoretical and numerical analyses.", "The behavior of the EIT-like metamaterial is found to be well described by the mechanical model of the coupled resonators." ], [ "Introduction", "Metamaterials are arrays of artificial structures that are much smaller than the wavelength of electromagnetic waves.", "The macroscopic characteristics of metamaterials are determined by their constitutive elements and, therefore, electromagnetic media with desired properties can be created by designing the shape, material, and density of the constitutive elements.", "We can fabricate various media that do not occur in nature and can use such designed and fabricated metamaterials to control electromagnetic waves at will.", "Resonant structures such as split-ring resonators[1] and electric-field-coupled inductor–capacitor resonators[2] are typically used as constitutive elements of metamaterials.", "These structures are often called meta-atoms.", "The effective relative permittivity and permeability of metamaterials can be varied from unity and can even be made negative by using the strong response of these meta-atoms around the resonant frequencies.", "These resonant meta-atoms enable the realization of exotic phenomena such as negative refraction,[3], [4] simultaneous negative phase and group velocities,[5] subwavelength imaging,[6] and cloaking.", "[7] In addition, metamaterials composed of coupled resonators, which are often called metamolecules, have been investigated to realize useful characteristics such as giant optical activity[8] and enhancement of second-harmonic generation.", "[9], [10], [11] Dispersion control by introducing coupling between neighboring unit cells has also been studied.", "[12], [13] Although only electric and magnetic direct couplings (near-field couplings) have been introduced in the above mentioned metamaterials, indirect coupling mediated by radiative modes has been introduced in other coupled-resonator systems.", "For example, indirect coupling has been introduced to achieve electromagnetically induced transparency (EIT)-like transmission in a two-dimensional photonic crystal waveguide coupled with two resonators,[14] EIT-like scattering and superscattering in a double-slit structure in a metal film,[15] and control of light at the nanoscale using plasmonic antennas.", "[16] If indirect coupling could be introduced into metamaterials as well as these isolated coupled-resonator systems, further development of methods for controlling electromagnetic waves could be expected.", "However, it is not obvious whether indirect coupling can be induced only between meta-atoms in each unit cell of metamaterials composed of periodically arranged coupled resonators (metamolecules).", "In this paper, we analyze the characteristics of metamaterials composed of coupled resonators with indirect coupling.", "First, we use a mechanical model of coupled resonators to show that an EIT-like transparency phenomenon occurs when indirect coupling is introduced.", "The characteristics of the EIT-like metamaterial with indirect coupling are compared to those of previously investigated EIT-like metamaterials with direct coupling.", "[17], [18], [19], [20], [21], [22], [23] Then, we show through a finite-difference time-domain (FDTD) simulation[24] that the EIT-like transparency phenomenon caused by the indirect coupling is observed in the metamaterial composed of coupled cut-wire pairs.", "Finally, we discuss whether indirect coupling can be induced only between meta-atoms in each unit cell of metamaterials by comparing the theory based on the mechanical model and the results of the numerical analysis." ], [ "Models of coupled resonators", "We analyze the electromagnetic response of a metamaterial composed of metamolecules that are modeled by the mechanical model shown in Fig.", "REF (a).", "When the masses of the two particles are the same ($m_1 = m_2 = m$ ) and the elastic constant of the central spring is much smaller than that of the other springs ($K \\ll k_{1,2}$ ), the equation of motion is given by $&(-\\omega ^2 + \\omega _1^2 - {\\mathrm {i}}\\gamma _1 \\omega ) x_1 - \\kappa ^2 x_2 = \\alpha _1E_0, \\\\&(-\\omega ^2 + \\omega _2^2 - {\\mathrm {i}}\\gamma _2 \\omega ) x_2 - \\kappa ^2 x_1 = \\alpha _2E_0, $ where $\\omega _{1,2} = \\sqrt{k_{1,2} / m}$ , $\\kappa = \\sqrt{K/m}$ , $\\gamma _{1,2}$ are damping constants, and $F_{1,2} = m \\alpha _{1,2} E_0$ are external forces, which are assumed to be proportional to the electric field $E_0$ of the electromagnetic wave with proportionality constants of $\\alpha _{1,2}$ .", "Solving Eqs.", "(REF ) and () for $x_1$ and $x_2$ , the following equation is obtained: $\\begin{bmatrix}x_1\\\\x_2\\end{bmatrix}&=\\frac{-1}{(\\omega ^2 - \\omega _1^2 + {\\mathrm {i}}\\gamma _1 \\omega )(\\omega ^2 -\\omega _2^2 + {\\mathrm {i}}\\gamma _2 \\omega ) - \\kappa ^4} \\nonumber \\\\& \\hspace{14.22636pt} \\times \\begin{bmatrix}(\\omega ^2 - \\omega _2^2 + {\\mathrm {i}}\\gamma _2 \\omega )\\alpha _1 - \\kappa ^2 \\alpha _2 \\\\(\\omega ^2 - \\omega _1^2 + {\\mathrm {i}}\\gamma _1 \\omega )\\alpha _2 - \\kappa ^2 \\alpha _1\\end{bmatrix}E_0 .", "$ We assume that the electric susceptibility of the metamaterial is proportional to $x_1 + x_2$ .", "We further assume that the characteristics of the two resonators are similar to each other and that particles 1 and 2 are coupled only via indirect coupling, that is, $\\kappa ^2$ is a purely imaginary quantity, $\\kappa ^2 = {\\mathrm {i}}\\mathop {\\mathrm {Im}}{(\\kappa ^2)}$ .", "When $\\gamma _{1,2}$ are derived only from the radiations of the meta-atoms, both $\\gamma _{1,2}$ and $\\alpha _{1,2}$ represent the coupling between the meta-atoms and freespace.", "Thus, $\\alpha _1/\\alpha _2 = \\gamma _1 / \\gamma _2$ can be safely assumed.", "[16] Using the above assumptions, we obtain $x_1 + x_2$ at around $\\omega \\simeq \\omega _0$ as follows: $x_1 + x_2&\\approx -2\\alpha E_0 ( \\omega ^2 - \\omega _0^2 + {\\mathrm {i}}\\gamma _{\\scriptsize \\mbox{L}} \\omega ) \\nonumber \\\\& \\hspace{14.22636pt}\\times \\lbrace (\\omega ^2 - \\omega _0^2 + {\\mathrm {i}}\\gamma _1 \\omega )(\\omega ^2 - \\omega _0^2+ {\\mathrm {i}}\\gamma _2 \\omega ) \\nonumber \\\\& \\hspace{14.22636pt}- (\\omega _0^2 - \\omega _1^2)(\\omega _2^2 -\\omega _0^2) + [\\mathop {\\mathrm {Im}}{(\\kappa ^2)}]^2 \\rbrace ^{-1}, $ where $\\omega _0^2=(\\alpha _2 \\omega _1^2 + \\alpha _1\\omega _2^2)/ (\\alpha _1 + \\alpha _2)$ , $ {\\mathrm {i}}2 \\gamma _{\\scriptsize \\mbox{L}} \\omega \\alpha = {\\mathrm {i}}\\gamma _2 \\omega \\alpha _1 + {\\mathrm {i}}\\gamma _1 \\omega \\alpha _2 -(\\alpha _1 + \\alpha _2) \\kappa ^2$ , and $\\alpha = (\\alpha _1 + \\alpha _2)/2 $ .", "The right-hand side of Eq.", "(REF ) resembles the susceptibility of the classical model of EIT with direct coupling.", "[25] The correspondences of main parameters in the susceptibility of classical EIT with direct coupling to parameters in Eq.", "(REF ) are as follows: Both the resonant angular frequencies of the bright and dark modes are equal to $\\omega _0$ , the loss of the dark mode is nearly the same as $\\gamma _{\\scriptsize \\mbox{L}}$ , and the coupling factor between the bright and dark modes corresponds to $(\\omega _0^2 - \\omega _1^2)(\\omega _2^2 -\\omega _0^2) - [\\mathop {\\mathrm {Im}}{(\\kappa ^2)}]^2$ .", "This implies that an EIT-like transparency phenomenon occurs at the angular frequency $\\omega =\\omega _0 \\approx (\\omega _1 + \\omega _2)/2$ .", "Note that $\\gamma _{\\scriptsize \\mbox{L}}$ becomes less than $\\gamma _{1,2}$ if $\\mathop {\\mathrm {Im}}{(\\kappa ^2 )} >0$ .", "From the passive condition for the metamaterial, the imaginary part of $x_1 + x_2$ must be non-negative.", "To satisfy the passive condition at $\\omega = \\omega _0$ irrespective of $\\omega _1$ and $\\omega _2$ , $\\mathop {\\mathrm {Im}}{(\\kappa ^2)} \\le \\omega _0\\sqrt{\\gamma _1 \\gamma _2}$ is required.", "Next, we consider the group index at $\\omega = \\omega _0$ .", "When the absolute value of $= \\omega _1 - \\omega _2$ is larger than a certain value $_{\\scriptsize \\mbox{max}}$ , the transmission bandwidth decreases; that is, the group index increases with decreasing $| |$ .", "When $| |$ is smaller than $_{\\scriptsize \\mbox{max}}$ , the transmission window gradually disappears and the group index decreases with decreasing $| |$ due to the loss.", "The value of $_{\\scriptsize \\mbox{max}}$ can be regarded as the minimum transmission bandwidth.", "We calculate $_{\\scriptsize \\mbox{max}}$ that gives the condition for the largest group index below.", "We may assume that the value of $| |$ that maximizes $\\mathop {\\mathrm {Re}}{[{\\mathrm {d}}(x_1 + x_2)/{\\mathrm {d}}\\omega |_{\\omega =\\omega _0}]}$ is almost equal to $_{\\scriptsize \\mbox{max}}$ in the case of strongly dispersive media as in the present case.", "For simplicity of the analysis, we further assume that $\\gamma _1 = \\gamma _2 = \\gamma _0$ , from which $\\mathop {\\mathrm {Im}}{(\\kappa ^2)} = \\omega _0 (\\gamma _0 - \\gamma _{\\scriptsize \\mbox{L}})$ is obtained.", "This implies that $\\gamma _{\\scriptsize \\mbox{L}}$ that appears in Eq.", "(REF ) represents the leak of the indirect coupling.", "From Eq.", "(REF ), we obtain $\\mathop {\\mathrm {Re}}{\\left[ \\left.", "\\frac{{\\mathrm {d}}(x_1 + x_2)}{{\\mathrm {d}}\\omega } \\right|_{\\omega =\\omega _0} \\right] }\\approx \\frac{4 \\alpha E_0(^2 - \\gamma _{\\scriptsize \\mbox{L}}^2)}{\\omega _0(^2 + 2\\gamma _0 \\gamma _{\\scriptsize \\mbox{L}} - \\gamma _{\\scriptsize \\mbox{L}}^2)^2}.", "$ The right-hand side takes a maximum value for $|| =\\sqrt{2\\gamma _0\\gamma _{\\scriptsize \\mbox{L}}+\\gamma _{\\scriptsize \\mbox{L}}^2}$ , which we define as $_{\\scriptsize \\mbox{max}}$ .", "In the case of $\\gamma _{\\scriptsize \\mbox{L}} = 0$ , the transmission bandwidth can become infinitesimal and the group index can become infinite.", "Note that the transmission bandwidth can be smaller than the resonance linewidth $\\gamma _0$ of the meta-atoms when the leak $\\gamma _{\\scriptsize \\mbox{L}}$ of the indirect coupling is sufficiently small.", "It is also found from Eq.", "(REF ) that the group index becomes negative; that is, the transmission window disappears in the range $|| < \\gamma _{\\scriptsize \\mbox{L}}$ .", "Here we discuss the derivation of the leak $\\gamma _{\\scriptsize \\mbox{L}}$ of the indirect coupling, which is an important parameter that determines the minimum bandwidth of the transparency window.", "Since the indirect coupling is mediated by radiative modes, the indirect coupling takes a maximum value, that is, $\\gamma _{\\scriptsize \\mbox{L}}$ vanishes when all the energy dissipated from one meta-atom in each unit cell is absorbed by the other meta-atom.", "The dissipated energy is derived from Ohmic loss, dielectric loss, and radiative loss in most meta-atoms.", "The dissipated energy derived from Ohmic and dielectric losses cannot excite the other meta-atom, and thus, Ohmic and dielectric losses contribute to $\\gamma _{\\scriptsize \\mbox{L}}$ .", "In addition, the difference between the radiation modes of the two kinds of meta-atoms causes a reduction of the radiative coupling, i.e., an increase in $\\gamma _{\\scriptsize \\mbox{L}}$ .", "Thus, $\\gamma _{\\scriptsize \\mbox{L}}$ is derived from Ohmic loss, dielectric loss, and the difference between the radiation modes of the two kinds of meta-atoms.", "We also discuss the physical meaning of the narrower transmission bandwidth than the resonance linewidths of the meta-atoms in the transparency phenomenon.", "For $\\omega = \\omega _0$ , Eq.", "(REF ) is reduced to $x_1 \\approx -x_2$ .", "In this case, the dissipation terms derived from $\\gamma _{1,2}$ in Eqs.", "(REF ) and () are (partially) canceled out by the terms derived from the indirect coupling $\\kappa ^2 = {\\mathrm {i}}\\mathop {\\mathrm {Im}}{(\\kappa ^2)}$ .", "This implies that the energy radiated from one meta-atom in each unit cell is absorbed by the other meta-atom.", "That is, the radiated energy moves backward and forward between the two kinds of meta-atoms in each unit cell.", "Therefore, the effective radiation loss in the metamolecule is reduced and the narrowband transparency phenomenon can be achieved.", "We assumed $\\mathop {\\mathrm {Re}}{(\\kappa ^2)}=0$ in the above discussion.", "In the case of $\\mathop {\\mathrm {Re}}{(\\kappa ^2)} \\ne 0$ , it is found from Eq.", "(REF ) that the EIT-like transparency phenomenon occurs for $\\omega ^2 = \\omega _0^2 +\\mathop {\\mathrm {Re}}{(\\kappa ^2)}$ .", "It is useful for understanding the electromagnetic response of the metamaterial to analyze an electrical circuit model of the metamolecule.", "We consider an electrical circuit that consists of two coupled inductor–capacitor resonant circuits, shown in Fig.", "REF (b).", "Three kinds of couplings exist in the electrical circuit.", "Applying Kirchhoff's voltage law for the electrical circuit yields the following equations: $& \\left( - \\omega ^2 + \\frac{1}{L C_1} - {\\mathrm {i}}\\frac{R_1}{L} \\omega \\right) q_1 \\nonumber \\\\& \\hspace{14.22636pt} - \\frac{{\\mathrm {i}}\\omega }{L}\\left[ R_{\\scriptsize \\mbox{M}} -{\\mathrm {i}}\\left( \\omega M - \\frac{1}{\\omega C_{\\scriptsize \\mbox{M}}}\\right) \\right]q_2 = \\frac{V_1}{L}, \\\\& \\left( - \\omega ^2 + \\frac{1}{L C_2} - {\\mathrm {i}}\\frac{R_2}{L} \\omega \\right) q_2 \\nonumber \\\\& \\hspace{14.22636pt} - \\frac{{\\mathrm {i}}\\omega }{L}\\left[ R_{\\scriptsize \\mbox{M}} -{\\mathrm {i}}\\left( \\omega M - \\frac{1}{\\omega C_{\\scriptsize \\mbox{M}}}\\right) \\right]q_1 = \\frac{V_2}{L}, $ where $L_1 = L_2 = L$ is assumed.", "It is found by comparing Eqs.", "(REF ) and () with Eqs.", "(REF ) and () that the imaginary and real parts of $\\kappa ^2$ correspond to the real and imaginary parts, respectively, of the mutual impedance $Z_{\\scriptsize \\mbox{M}} = R_{\\scriptsize \\mbox{M}} - {\\mathrm {i}}[ \\omega M - (\\omega C_{\\scriptsize \\mbox{M}})^{-1}]$ .", "This relation shows the influences of the electric coupling $C_{\\scriptsize \\mbox{M}}$ , magnetic coupling $M$ , and energy coupling $R_{\\scriptsize \\mbox{M}}$ on $\\kappa ^2$ .", "We now discuss the difference between the EIT-like metamaterial with indirect coupling and that with direct coupling.", "[17], [18], [19], [20], [21], [22], [23] The unit cell of the latter metamaterial consists of two directly coupled meta-atoms: a low quality-factor meta-atom, which interacts with the incident electromagnetic wave, and a high quality-factor meta-atom, which does not interact with the incident wave.", "The loss in the high quality-factor meta-atom should be reduced to achieve a large group index.", "The resonant frequencies of the two kinds of meta-atoms should be identical to ensure that the transmission spectrum is symmetric to reduce higher-order dispersion.", "On the other hand, as described above, the unit cell of the former metamaterial consists of two similar meta-atoms.", "The indirect coupling should be strong to realize a large group index.", "The two meta-atoms should be coupled so that $\\mathop {\\mathrm {Re}}{(\\kappa ^2)}=0$ is satisfied to ensure that the transmission spectrum is symmetric.", "Both kinds of EIT-like metamaterials require efforts to achieve a large group index and symmetric transmission spectrum.", "However, the structure of the meta-atoms can be simple for the former metamaterial, because the unit cell consists of two similar meta-atoms and the radiation losses of the meta-atoms may be large.", "Therefore, the former metamaterial can be superior in terms of ease of design and fabrication to the latter metamaterial if $\\gamma _{\\scriptsize \\mbox{L}} \\simeq 0$ and $\\mathop {\\mathrm {Re}}{(\\kappa ^2)}=0$ are simultaneously satisfied." ], [ "FDTD analysis of EIT-like metamaterial with indirect coupling", "In order to confirm the validity of the above theory and to investigate whether $\\gamma _{\\scriptsize \\mbox{L}} \\simeq 0$ and $\\mathop {\\mathrm {Re}}{(\\kappa ^2)}=0$ are simultaneously satisfied, we design the EIT-like metamaterial with indirect coupling and analyze the characteristics of the metamaterial using an FDTD method.", "[24] Figure REF shows the unit cell of the EIT-like metamaterial whose electromagnetic response is modeled by the mechanical model shown in Fig.", "REF (a).", "Two kinds of cut-wire resonators (meta-atoms) with different resonant frequencies are placed with a gap of $g_2$ .", "The structures of the two kinds of cut-wire resonators are designed to be similar to each other in order to make their characteristics including the radiation modes similar.", "The configuration of the two kinds of cut-wire resonators is determined so that the radiation from one cut-wire resonator can excite the other cut-wire resonator; that is, indirect coupling is induced in the cut-wire pair.", "The unit structure can be regarded as a kind of asymmetric split-ring resonator.", "[26], [27] The resonant frequencies, $\\omega _1$ and $\\omega _2$ , of the two kinds of resonators are determined by the inductance derived from the metal wire with length $l$ and the capacitance derived from the gap $g_1$ .", "The difference $| |$ of the resonant angular frequencies between the two kinds of resonators is determined by the difference between $w_1$ and $w_2$ .", "The resonance linewidths, $\\gamma _1$ and $\\gamma _2$ , of the two kinds of resonators depend on the radiation loss, dielectric loss in the substrate, and Ohmic loss in the metal.", "The coupling factor $\\kappa $ between the two kinds of resonators can be controlled by varying $g_2$ .", "The FDTD simulation is performed in the microwave region where metals can be regarded as perfect electric conductors; that is, Ohmic loss in the metal is negligible.", "This enables us to investigate the influence of the leak $\\gamma _{\\scriptsize \\mbox{L}}$ of the indirect coupling on the transmission characteristics by only varying the dielectric loss of the substrate and to easily compare the theory based on the mechanical model with the results of the FDTD analysis.", "Figure: Snapshots of (a) the current and (b) electric fielddistributions at the transparency frequency.", "The incident electricfield is 1μΩV/m1\\,{\\mu \\Omega \\mathrm {V/m}}.We first calculated the transmission spectrum of the EIT-like metamaterial.", "The geometrical parameters of the metamaterial were set to $l = 28\\,{\\mu \\Omega \\mathrm {mm}}$ , $w_1 = 3\\,{\\mu \\Omega \\mathrm {mm}}$ , $w_2 = 4\\,{\\mu \\Omega \\mathrm {mm}}$ , $g_1 = 2\\,{\\mu \\Omega \\mathrm {mm}}$ , $g_2 = 2\\,{\\mu \\Omega \\mathrm {mm}}$ , $d =1 \\,{\\mu \\Omega \\mathrm {mm}}$ , $p_x = 29\\,{\\mu \\Omega \\mathrm {mm}}$ , and $p_y = 39\\,{\\mu \\Omega \\mathrm {mm}}$ .", "The relative permittivity of the substrate was set to 3.3, which is the real part of the relative permittivity of polyphenylene ether.", "The FDTD simulation space was discretized into uniform cubes with dimensions of $1\\,{\\mu \\Omega \\mathrm {mm}} \\times 1\\,{\\mu \\Omega \\mathrm {mm}} \\times 1\\,{\\mu \\Omega \\mathrm {mm}}$ .", "(Although the thickness of the substrate was modeled using only a single FDTD cell in the simulation, no significant errors were caused to the simulation results.)", "Periodic boundary conditions were applied to the $x$ and $y$ directions to realize periodically arranged metamolecules.", "Figure REF shows the transmission spectra of three kinds of metamaterials.", "The red solid curve represents the transmission spectrum of the metamaterial composed of coupled cut-wire pairs (metamolecules) described above.", "The green dashed curve (blue dashed-dotted curve) represents the transmission spectrum of the metamaterial composed of one kind of cut wires, i.e., meta-atoms, shown in the right-hand side (left-hand side) of Fig.", "REF .", "While simple absorption spectra are observed for the metamaterials composed of one kind of cut wires, a transmission window is observed in the absorption spectrum for the metamaterial composed of coupled cut-wire pairs.", "In addition, the transmission bandwidth is much smaller than the resonance linewidths (absorption bandwidths) of the metamaterials composed of one kind of cut wires.", "This implies that the EIT-like transparency phenomenon occurs for the metamaterial composed of coupled cut-wire pairs owing to the indirect coupling.", "Since the transmittance at the transparency frequency is nearly unity, $\\gamma _{\\scriptsize \\mbox{L}}$ seems to be negligibly small for the present geometrical parameters.", "Although the present metamaterial, which is composed of periodically arranged coupled resonators, is different from isolated coupled resonators such as the mechanical model shown in Fig.", "REF (a) and the coupled resonators in previous studies,[14], [15], [16] indirect coupling can be induced between the meta-atoms in each unit cell and $\\gamma _{\\scriptsize \\mbox{L}}$ seems to be negligibly small also in the metamaterial.", "(Note that it is still unclear whether indirect coupling is induced between different cells.)", "We next calculated the field distributions at the transparency frequency to understand the physical meaning of the EIT-like transparency phenomenon.", "Figure REF (a) shows the current distribution at the transparency frequency.", "An antisymmetric current flows in the coupled cut-wire pair and thus the total electric dipole moment vanishes.", "That is, the scattering is suppressed due to destructive interference between the radiations from the two kinds of cut wires.", "This observation is another aspect of the cancellation between the radiation loss and the indirect coupling described in Sec. II.", "Figure REF (b) shows the electric field distribution at the transparency frequency.", "A large quadrupole electric field is induced at the gap of the two kinds of cut wire.", "The electric field at the gap is about 200 times as large as the incident electric field, in which the narrow band effect is reflected.", "Figure: Dependence of the transmission spectrum on g 2 g_2.We next analyzed the dependence of the transmission spectrum on $g_2$ to investigate whether $\\mathop {\\mathrm {Re}}{(\\kappa ^2)}=0$ can be satisfied by varying $g_2$ .", "Figure REF shows the transmission spectra for $g_2 =1\\,{\\mu \\Omega \\mathrm {mm}}$ , 2 mm, 4 mm, and 8 mm.", "The other parameters are the same as those in the case of Fig.", "REF .", "With increasing $g_2$ , the transmission peak shifts to lower frequency; that is, $\\mathop {\\mathrm {Re}}{(\\kappa ^2)}$ decreases.", "The transmission spectrum for $g_2=1\\,{\\mu \\Omega \\mathrm {mm}}$ shows the opposite asymmetry to that for $g_2=2\\,{\\mu \\Omega \\mathrm {mm}}$ and, therefore, the condition that satisfies $\\mathop {\\mathrm {Re}}{(\\kappa ^2)}=0$ exists in the range $1\\,{\\mu \\Omega \\mathrm {mm}} < g_2 <2\\,{\\mu \\Omega \\mathrm {mm}}$ .", "The imaginary part of $\\kappa ^2$ seems to be a constant value in the range $1\\,{\\mu \\Omega \\mathrm {mm}} < g_2 < 8\\,{\\mu \\Omega \\mathrm {mm}}$ because the transmittance at the transparency frequency is nearly unity for these four calculated conditions.", "The dependence of $\\mathop {\\mathrm {Re}}{(\\kappa ^2)}$ on $g_2$ can be understood using the electrical circuit model of the coupled resonators shown in Fig.", "REF (b).", "As $g_2$ increases, both the mutual inductance $M$ and mutual capacitance $C_{\\scriptsize \\mbox{M}}$ decrease and thus the imaginary part of the mutual impedance $Z_{\\scriptsize \\mbox{M}}$ decreases.", "Therefore, $\\mathop {\\mathrm {Re}}{(\\kappa ^2)}$ decreases and the transmission peak shifts to lower frequency with increasing $g_2$ .", "Note that this discussion can be applied only to the case of $g_2 \\ll \\lambda $ , where $\\lambda $ is the wavelength of the electromagnetic waves.", "When $g_2$ is comparable to or larger than $\\lambda $ , the phase retardation of the coupling has to be taken into account; therefore, we cannot discuss the indirect and direct couplings separately.", "However, $g_2 \\ll \\lambda $ is safely satisfied in metamaterials.", "Figure: Transmission spectra for tanδ=0.005\\tan \\delta = 0.005.", "Thered solid curve represents the transmission spectrum of themetamaterial composed of coupled cut-wire pairs.The green dashed curve and blue dashed-dotted curverepresent the transmission spectra of the metamaterials composed of onekind of cut wires.Figure: Dependence of the group delay at the transparency frequencyon ||| | when tanδ\\tan \\delta is equal to(a) 0, (b) 0.005, (c) 0.05, and (d)0.1.", "The values of max _{\\scriptsize \\mbox{max}} and γ d \\gamma _{\\scriptsize \\mbox{d}} arerepresented by the vertical dashed lines in each case.", "Note that thescales of the vertical axes are different from each other.We next analyzed the influence of the dielectric loss of the substrate on the transmission characteristics.", "Figure REF shows the transmission spectra of the above mentioned three kinds of metamaterials when the loss tangent $\\tan \\delta $ of the dielectric substrate is 0.005, which is the loss tangent of polyphenylene ether at 3 GHz.", "The other parameters are the same as those in the case of Fig.", "REF .", "While the transmission spectra of the metamaterials composed of one kind of cut wires are almost the same as those in Fig.", "REF , the transmittance of the metamaterial composed of coupled cut-wire pairs at the transparency frequency is smaller than that in the case of Fig.", "REF due to the dielectric loss.", "This implies that the leak of the indirect coupling without dielectric loss is much smaller than the dielectric loss $\\gamma _{\\scriptsize \\mbox{d}}$ of the substrate.", "Thus, we may assume $\\gamma _{\\scriptsize \\mbox{L}}\\simeq \\gamma _{\\scriptsize \\mbox{d}}$ when $\\tan \\delta $ is larger than 0.005 at most.", "Finally, we calculated the dependence of the group delay at the transparency frequency on $| |$ to investigate the influence of $\\tan \\delta $ on $_{\\scriptsize \\mbox{max}}$ .", "The absolute value of $$ was varied by varying $w_2$ from $4\\,{\\mu \\Omega \\mathrm {mm}}$ to $10\\,{\\mu \\Omega \\mathrm {mm}}$ with steps of $1\\,{\\mu \\Omega \\mathrm {mm}}$ .", "The other parameters except $\\tan \\delta $ are the same as those in the case of Fig.", "REF .", "Figure REF shows the group delay at the transparency frequency as a function of $| |$ for $\\tan \\delta = 0$ , 0.005, 0.05, and 0.1.", "For the calculated conditions, the group delay monotonically increases with decreasing $| |$ for $\\tan \\delta = 0$ and 0.005, while the group delay first increases and then decreases with decreasing $| |$ for $\\tan \\delta = 0.05$ and 0.1.", "We now discuss the above results using the mechanical model of the coupled resonator shown in Fig.", "REF (a).", "We have to evaluate in advance the radiation loss $\\gamma _0$ of the cut-wire resonators and the dielectric loss $\\gamma _{\\scriptsize \\mbox{d}}$ of the substrate in order to use the mechanical model.", "For a Lorentz medium with a simple absorption line, $\\gamma _0$ is almost equal to the bandwidth of the negative group delay.", "Thus, $\\gamma _0$ can be estimated by calculating the group delay of the metamaterial composed of one kind of cut wires (not shown).", "The value of $\\gamma _{\\scriptsize \\mbox{d}}$ can be estimated as follows.", "In a series inductor–capacitor resonant circuit, the quality factor of the circuit can be approximated as the inverse of the loss tangent of the dielectric in the capacitor when the loss in the circuit is caused only by the dielectric loss in the capacitor.", "Since the thin metallic film is on the dielectric substrate in the metamaterial shown in Fig.", "REF , we assume that half of the capacitor in the metamaterial is filled with the dielectric and the other half is filled with vacuum.", "That is, the capacitance of the gap is assumed to be $(1+\\varepsilon _{\\scriptsize \\mbox{r}})/2$ times as large as that without the substrate, where $\\varepsilon _{\\scriptsize \\mbox{r}}$ is the relative permittivity of the substrate.", "From this assumption, the effective loss tangent $\\tan \\delta ^{\\prime }$ , which is the loss tangent when the capacitor is assumed to be filled with a uniform medium, is found to be $\\lbrace \\mathop {\\mathrm {Re}}{(\\varepsilon _{\\scriptsize \\mbox{r}})} / [ 1 + \\mathop {\\mathrm {Re}}{(\\varepsilon _{\\scriptsize \\mbox{r}})}] \\rbrace \\tan \\delta $ .", "Therefore, the dielectric loss $\\gamma _{\\scriptsize \\mbox{d}}$ in the metamaterial is estimated to be $\\omega _0 \\tan \\delta ^{\\prime }$ .", "We now show $_{\\scriptsize \\mbox{max}}$ calculated using the above estimated $\\gamma _0$ and $\\gamma _{\\scriptsize \\mbox{d}}$ in Fig.", "REF .", "The value of $\\gamma _{\\scriptsize \\mbox{d}}$ is also shown in the figure.", "It is found that the group delay increases or decreases with decreasing $| |$ when $| |$ is larger or smaller, respectively, than $_{\\scriptsize \\mbox{max}}$ and that the group delay seems to vanish at around $|| = \\gamma _{\\scriptsize \\mbox{d}}$ .", "This implies that the behavior of the coupled cut-wire pair metamaterial can be well described by the mechanical model.", "That is, indirect coupling can be induced only between meta-atoms in each unit cell of metamaterials composed of periodically arranged coupled resonators." ], [ "Conclusion", "We analyzed the EIT-like transparency phenomenon in metamaterials composed of coupled resonators with indirect coupling.", "The theoretical analysis based on the mechanical model showed that the transparency bandwidth can be narrower than the resonance linewidths of the constitutive resonators when strong indirect coupling is introduced.", "The FDTD simulation demonstrated that the EIT-like transparency phenomenon with $\\gamma _{\\scriptsize \\mbox{L}} \\simeq 0$ and $\\mathop {\\mathrm {Re}}{(\\kappa ^2)}=0$ can occur in metamaterials composed of coupled cut-wire pairs.", "The characteristics of the metamaterial was confirmed to be well described by the mechanical model, and indirect coupling was found to be induced only between meta-atoms in each unit cell of metamaterials composed of periodically arranged coupled resonators.", "Structures of meta-atoms with narrow resonance linewidth usually have complicated shapes.", "However, the narrow band transparency window can be obtained using simple structures such as cut wires when indirect coupling is introduced.", "If dielectric loss is prevented completely, an extremely narrow transparency window can be achieved below infrared frequencies where metals can be regarded as perfect electric conductors.", "For future studies, the minimum transparency bandwidth of the EIT-like metamaterial without dielectric loss needs to be investigated experimentally.", "In the optical region, metals exhibit relatively large Ohmic losses, and thus the metamaterial should be designed with low-loss dielectrics.", "Indirect coupling would be useful not only for realizing EIT-like phenomena but also for other techniques for controlling electromagnetic waves.", "This research was supported by a Grant-in-Aid for Scientific Research on Innovative Areas (No.", "22109004) from the Ministry of Education, Culture, Sports, Science, and Technology, Japan, and by a Grant-in-Aid for Research Activity Start-up (No.", "25889028) from the Japan Society for the Promotion of Science." ] ]
1403.0400
[ [ "Diffusive transport in two-dimensional nematics" ], [ "Abstract We discuss a dynamical theory for nematic liquid crystals describing the stage of evolution in which the hydrodynamic fluid motion has already equilibrated and the subsequent evolution proceeds via diffusive motion of the orientational degrees of freedom.", "This diffusion induces a slow motion of singularities of the order parameter field.", "Using asymptotic methods for gradient flows, we establish a relation between the Doi-Smoluchowski kinetic equation and vortex dynamics in two-dimensional systems.", "We also discuss moment closures for the kinetic equation and Landau-de Gennes-type free energy dissipation." ], [ "Introduction", "Dynamics of liquid crystalline systems is traditionally described in a framework of theories combining fluid dynamics equations, constitutive relations between the hydrodynamic stress tensor and liquid crystalline order parameters, and evolution equations for the latter [3], [6], [9], [12].", "In the absence of hydrodynamic motion, the relaxation of the orientational degrees of freedom is induced by the free energy dissipation.", "This relaxation is generally slow and can be characterized by the evolution of topological defects in the order parameter fields.", "Our goal in this work is to derive equations of motion for these defects starting from a kinetic Doi-Smoluchowski-type equation [6].", "To accomplish this task we use asymptotic methods for gradient flows similarly to the way it is used in the Ginzburg-Landau theory [4], [7], [13], [14], [16].", "We omit most of the technical details concentrating rather on the methodology and final results.", "Additionally, we discuss the possibility of describing the dissipative dynamics in terms of the order parameter fields, similar, to, e.g., Landau-de Gennes theory [5].", "We also discuss the extent to which the formal moment closures provide the correct evolution equations.", "We choose the Doi-Smoluchowski (D-S) model [6] as the starting point for our analysis, because, in some respect, it is a microscopic theory, in comparison to, e.g., Ericksen-Leslie [9], [12], or Beris-Edwards models [3].", "In the D-S theory, the state of a liquid crystalline system is described by means of a probability density of rods orientations; and the D-S equations are kinetic equations for this density.", "The other aforementioned models are based on description via its various moments, and should, in principle, be derived from a D-S-type model.", "Mathematically, the D-S equations describe gradient flow dynamics for the Onsager-Maier-Saupe free energy [10] in Wasserstein metric [17].", "This makes analysis of the system amenable to methods of the theory of gradient flows [2].", "In this paper we are interested in the two-dimensional model.", "One of the characteristic features of two-dimensional systems is that the energy of topological singularities, or vortices, diverges logarithmically in meaningful asymptotic limits.", "The Doi dynamics of orientation density reduces directly to the vortex dynamics.", "Due to this, it is impossible to find a nontrivial regime in which a Landau-de Gennes-type gradient flow evolution for the second moment can be derived from the D-S model.", "This is different from the three-dimensional theory, where one can reduce the D-S dynamics to equations for the second moment [8], [18]." ], [ "The paper is organized as follows:", "We start by reviewing the two-dimensional spatially extended Onsager-Maier-Saupe free energy, introduced in our earlier works [10], [11].", "Understanding the landscape of this free energy provides us with characterization of the states which posses “moderate” amounts of free energy, as compared to the ground, uniform nematic state; we call such states tempered.", "These states are uniquely characterized by the location of vortices, and some auxiliary function, so that the evolution of tempered states may be completely described by the evolution of these quantities.", "We then set up the D-S dynamics as a gradient flow dynamics for our free energy, and carry out an asymptotic reduction.", "In order to explain the methods and ideas of this reduction we consider a finite-dimensional example on a rather rigorous level.", "After that, we implement an analogous procedure for the infinite-dimensional system, deriving equations governing the vortex motion.", "Finally, we derive an infinite hierarchy of equations for moment of the orientation density, and discuss possible closures." ], [ "Review of the spatially-extended Onsager-Maier-Saupe model", "The main goal of the next two sections is to familiarize the reader with spatially extended Onsager-Maier-Saupe model [10], [11], and to state the principal results of this paper in its context.", "As mentioned in the Introduction, we specialize to two-dimensional systems.", "In the framework of this model, the state of a liquid crystalline system is characterized by the space-dependent orientation probability density of nematic molecules, $(\\varphi ,z)$ integrating to unity over $\\varphi $ for each $z\\in $ .", "Here $\\varphi \\in [0,2\\pi )$ is the orientation parameter of liquid crystalline molecules, and $z\\in \\subset is a spatial variable.", "Note that we employ complex notation $ z=x+y$ for the spatial coordinates, as this simplifies many calculations.", "Refer to appendix for additional details.$" ], [ "The free energy", "of the liquid crystalline system, $()$ , is a functional of orientation probability density $(\\varphi ,z)$ and is represented as an integral over the spatial domain $$ of the sum of two contributions: $()\\,=\\,\\int _\\Big [\\,_()+_()\\,\\Big ]\\upsilon (z).$ Hereafter we use $\\upsilon (z)=xy$ to denote the volume element in $$ .", "The orientational free energy density  $_$ is an Onsager-type functional, $_()=\\int _0^{2\\pi }(\\varphi ,z)\\,\\ln \\!\\big [2\\pi (\\varphi ,z)\\big ]\\,{\\varphi }\\,-\\frac{\\gamma }{2}\\iint _0^{2\\pi }\\cos 2(\\varphi -\\varphi ^\\prime )\\,(\\varphi ,z)(\\varphi ^\\prime ,z)\\,{\\varphi }{\\varphi ^\\prime }+\\,C_{\\gamma },$ where the constant $C_{\\gamma }$ is chosen to have $_\\ge 0$ .", "The positive parameter $\\gamma $ is referred to as concentration.", "The elastic free energy density is a quadratic functional of the order-parameter field equivalent to that of the Landau-de Gennes theory: $_()\\,=\\,\\frac{\\epsilon ^2}{2}\\,|\\!", "{}(z)|^2.$ Here ${}({z})$ is the order parameter field related to the orientation probability density function, $$ , via ${}({z})\\,=\\,\\int _0^{2\\pi }\\,^{\\,2\\varphi }(\\varphi ,z)\\varphi .$ The positive parameter $\\epsilon ^2$ in equation (REF ) is called the elastic modulus." ], [ "A useful observation", "is that the total free energy (REF ) may be decomposed in the following way: $()\\,&=\\,\\int _\\Big [\\,\\frac{\\epsilon ^2}{2}\\,\\big |\\!", "{}\\big |^2\\,+\\,W^{\\gamma }()\\Big ]z\\,+\\,\\int _(|\\hat{}{[]})\\upsilon (z).$ Here for a given probability density $$ , the order parameter field, $$ , is defined in (REF ).", "The potential $W^{\\gamma }$ is given by $W^{\\gamma }()\\,=\\,-\\,\\frac{\\gamma ^2}{2}\\,+\\,\\big [\\operatorname{A}()\\,-\\,\\ln \\operatorname{I}_0(\\operatorname{A}())\\big ]\\,+\\,C_{\\gamma },$ where we use notation $=| {} |$ .", "Some of the properties of $W^{\\gamma }()$ and related special functions are presented in Appendix REF .", "In particular, we must pick $\\gamma >2$ to assure the existence of nematic states.", "The locally-equilibrated probability density $\\hat{}{[]}$ in equation (REF ) is related to $$ via $\\hat{}{[]}(\\varphi ,z)=\\frac{\\exp \\big \\lbrace \\!\\big ((z)\\big )\\,\\cos \\big (2\\varphi -\\arg (z)\\big )\\big \\rbrace }{2\\pi _0\\!\\big (\\!", "()\\big )};$ and $(|\\hat{}{[]})$ is the relative entropy of $$ with respect to $\\hat{}{[]}$ , , $(|\\hat{}{[]})\\,=\\,\\int _0^{2\\pi }\\ln \\frac{(\\varphi )}{\\hat{}{[]}(\\varphi )}\\;(\\varphi )\\varphi .$ The field $\\hat{}{[]}$ has a straightforward interpretation: whenever $=\\hat{}{[]}$ , $(|\\hat{}{[]})=0$ , and thus $\\hat{}{[]}$ minimizes the total free energy in the class of all fields with prescribed order-parameter $(z)$ .", "We define the reduced free energy  $({})$ as a functional of the order parameter $$ alone, $({})\\,=\\,\\int _\\Big [\\,\\frac{\\epsilon ^2}{2}\\,\\big |\\!", "{}\\big |^2\\,+\\,W^{\\gamma }()\\Big ]\\upsilon (z)$ and notice its similarity to the Ginzburg-Landau energy.", "Thus, from (REF ) we see that the total liquid crystalline energy, $()$ , is decomposed in the sum of a Ginzburg-Landau-type energy, $({})$ , and the relative entropy $(|\\hat{}{[]})$ .", "This allows us to obtain an asymptotic limit in which the orientation density, $$ , becomes enslaved to its second moment, $$ via equation (REF )." ], [ "Multi-vortex patterns", "are configurations of the order parameter field, $$ , which appear in the limits as $\\gamma \\rightarrow \\infty $ or $\\epsilon \\rightarrow 0$ .", "In this work we are interested in the dynamics of liquid crystalline systems in the limit as $\\epsilon \\rightarrow 0$ .", "Even though this particular limit was not considered in [11], the results below can be easily obtained by combining analysis in [11] and results presented in [15] for the Ginzburg-Landau energy.", "Consider a family of order parameter fields, $^\\epsilon (z)$ , which satisfy the boundary condition, $^\\epsilon (z)\\,=\\,_\\exp \\lbrace \\Psi (z)\\rbrace ,\\quad z\\in ,$ where $_{}$ is the minimizer of $W^\\gamma ()$ .", "Assume that in the limit, as $\\epsilon \\rightarrow 0$ , the energy of these order parameter fields satisfies the bound $\\frac{1}{\\epsilon ^2}(^\\epsilon ) \\,\\le \\, \\pi _{}^2\\,|d|\\ln {\\epsilon }+C,$ where $d=\\deg ^\\epsilon |_{}$ ; $C$ is a positive constant independent of $\\epsilon $ .", "(Such configurations are called tempered.)", "Then, as $\\epsilon \\rightarrow 0$ , $^\\epsilon (z)$ , converge in appropriate sense (up to a subsequence) to $\\quad _*(z)\\,=\\,_{\\mathrm {eq}}\\exp \\big \\lbrace \\Phi (z)\\,+\\,\\,d\\sum _{k=1}^d\\arg (z-{z}_{k})\\big \\rbrace ,$ where $\\Phi (z)$ is a function with finite Dirichlet energy.", "Such a field $_*(z)$ is a particular example of the so-called multi-vortex field, which in general may be represented as $_*[{z}_{1},\\ldots ,{z}_{N};d_1,\\ldots ,d_N](z)\\,=\\,_{\\mathrm {eq}}\\exp \\bigg \\lbrace \\Phi (z)\\,+\\,\\,\\sum _{k=1}^Nd_k\\arg (z-{z}_{k})\\bigg \\rbrace .$ In this paper we allow the vortices to have different degrees $d_k=\\pm 1$ .", "It is possible to show, that the results valid for tempered states $^\\epsilon (z)$ hold in the setting when vortices have degrees $d_k=\\pm 1$ , provided $^\\epsilon (z)$ stays close to a multi-vortex configuration (REF ).", "In this case we still refer to such states as tempered.", "In particular, the limiting equations for the vortex dynamics remain valid until vortices of different signs approach each other (or the boundary) and undergo collision-annihilation process that we do not discuss here.", "Until that, the structure of all tempered states is completely characterized by the degrees and locations of vortices, $d_k$ , ${z}_{k}$ , $k=1,\\ldots ,N$ , and the function $\\Phi (z)$ .", "Our derivation of the limiting equations relies on the lower bound on the energy $(^\\epsilon )$ .", "This bound follows from the results obtained in [11] for Onsager-Maier-Saupe energy and in [1] for the Ginzburg-Landau energy: $\\frac{(^\\epsilon )}{\\epsilon ^2_^2}\\,\\,\\ge \\,\\pi N\\ln \\epsilon \\,+\\,N_0\\,+\\,\\tilde{}({z}_{1},\\ldots ,{z}_{N};\\Phi )\\,+\\,\\frac{1}{\\epsilon ^2_^2}\\,\\int _\\big (^\\epsilon \\big \\vert \\hat{}[_*]\\big )\\upsilon (z).$ Here $E_0$ is a fixed constant related to the optimal profile problem, the renormalized multi-vortex energy is given by $ \\tilde{}({z}_{1},\\ldots ,{z}_{N};\\Phi )\\,=\\,\\frac{1}{2}\\int _|\\Phi (z)|^2\\upsilon (z)\\,+\\,U({z}_{1},\\ldots ,{z}_{N}),$ where the first term is the Dirichlet energy of the field $\\Phi (z)$ , which satisfies the boundary condition, $\\Phi (z)\\,=\\,\\Psi (z)-\\,\\,\\sum _{k=1}^Nd_k\\arg (z-{z}_{k}),\\quad z\\in ;$ and the second term is the multi-vortex potential, $U({z}_{1},\\ldots ,{z}_{N})\\,=\\,&-\\,\\pi \\sum _{j,\\,k=1\\atop j\\ne k}^{N}d_kd_j\\,\\ln |{z}_{k}-{z}_{j}|\\\\\\nonumber &+\\,\\sum _{k=1}^{N}d_k\\oint _{\\partial } \\ln |z-{z}_{k}|\\Psi (z) \\,-\\,\\frac{1}{2}\\sum _{j,\\,k=1}^{N}d_kd_j\\oint _{\\partial }\\ln |z-{z}_{k}|\\,\\arg (z-{z}_{j}).$ In this work we impose the Dirichlet boundary condition on the order parameter fields, $(z)$ , on $$ , i.e., we prescribe the function $\\Psi (z)$ , cf.", "(REF ).", "Therefore the multi-vortex potential $U$ may be expressed explicitly as a function of vortex locations.", "This contrasts with a situation when the Neumann boundary condition is used.", "In that case, the multi-vortex potential also depends on the function $\\Phi (z)$ .", "This may be seen if we rewrite the second term on the right-hand side of (REF ) in terms of $\\Phi (z)$ using the identity (REF ) (as the function $\\Psi $ is not given).", "Therefore the multi-vortex potential, $U$ , also depends on the boundary value of $\\Phi (z)$ .", "The energy decomposition (REF ) allow us to undertake an asymptotic reduction in the limit of small $\\epsilon $ .", "We see, that, as $\\epsilon \\rightarrow 0$ , the relative entropy term forces $(\\varphi ,z)$ to remain close to $\\hat{}[_*](\\varphi ,z)$ at all times.", "Consequently, any gradient flow dynamics preserving the temperedness condition reduces to the motion of vortices and evolution of the field $\\Phi (z)$ ." ], [ "Dissipative Doi-Smoluchowski dynamics", "The generalized Doi-Smoluchowski kinetic equations [6] describe evolution of the density of liquid crystalline molecules $c(s, r; t)$ at a position $r\\in ^3$ and orientation $s \\in ^2$ .", "In general, the D-S dynamics includes the hydrodynamic interactions and diffusive transport of the spatial and orientational degrees of freedom.", "In this paper we consider the stage of evolution at which the hydrodynamic and diffusive transports of the spatial degrees of freedom have already equilibrated, and the evolution proceeds via diffusion of the orientational degrees of freedom.", "In this regime the concentration of liquid crystalline molecules is constant, $c(r;t)\\int _{^2} c(s, r; t) s\\,=\\,;$ and the D-S equations may be rewritten in terms of the space-dependent probability density of molecules orientations, $(s, r; t)c(s, r; t)/c(r;t)$ : $_t(\\varphi ,z;t)\\,=\\,_{s}\\cdot \\bigg [\\,_{s}\\!", "{\\bf }{}\\bigg ],$ where $_{s}$ and $_{s}\\cdot $ denote the gradient and divergence operators on the sphere $^2$ ; $\\delta /\\delta $ is the usual Euler-Lagrange variational derivative; and $()$ is the free energy of the system.", "In the two-dimensional model that we consider in this work, this equation becomes $_t(\\varphi ,z;t)\\,=\\,_{\\varphi }\\bigg [\\,_{\\varphi }\\!", "{}{}\\bigg ],$ where $\\varphi \\in [0,2\\pi )$ , and $()$ is the spatially-extended Onsager-Maier-Saupe free energy (REF ).", "Explicitly, equation (REF ) may be written as $_t(\\varphi ,z;t)\\,=\\,^2_{\\varphi \\varphi }\\,-\\,2\\big [_\\varphi (^{-2\\varphi })\\big ],\\qquad \\,=\\,\\epsilon ^2\\Delta +\\gamma .$ Prescribing the boundary conditions on $$ for $$ directly is physically meaningless (there is no physical mechanism which would allow us to manipulate the density of orientations directly), and mathematically ill-posed.", "In this work we impose the Dirichlet boundary condition on the order parameter field: $(z)\\,=\\,_\\exp \\big \\lbrace \\Psi (z)\\big \\rbrace ,\\qquad z\\in ,$ where $\\Psi $ is defined on the boundary, possibly with a $2\\pi k$ -jump discontinuity somewhere on $$ , as only values of $\\Psi (z)\\!\\mod {2}\\pi $ are relevant.", "Physically, this corresponds to the strong anchoring regime.", "Note, however, that Neumann, Robin, or mixed boundary conditions may be treated similarly, and result in different expressions for the renormalized multi-vortex energy.", "We would like to study the dynamics prescribed by (REF ) in the limit when $\\epsilon \\ll 1$ .", "This scaling corresponds to a regime when the defect cores shrink to a point and is motivated by our goal to understand the global evolution of patterns arising in the system, rather than the particular details of dynamics in vicinity of defect cores.", "Observe, that because of the energy decomposition (REF ) in order to obtain a nontrivial dynamics, when the system does not immediately relax to the equilibrium state, we must consider (REF ) on a slower timescale $t^{\\prime }=\\epsilon ^2 t$ (dropping primes in what follows).", "The dynamics on this timescale is given by $_t(\\varphi ,z;t)\\,=\\,\\frac{1}{\\epsilon ^2} _{\\varphi }\\bigg [\\;_{\\varphi }\\!", "{}{}\\bigg ].$" ], [ "Summary of the results.", "In this work we show that, as $\\epsilon \\rightarrow 0$ , the states of the system are close to multi-vortex configurations prescribed by (REF ), and the dynamics (REF ) may be reformulated in terms of the dynamics of vortices ${z}_{k}$ and the field $\\Phi (z)$ .", "In particular, on the $(1)$ timescale, the vortices are stationary, while the field $\\Phi (z)$ evolves according to heat equation, ${_t}\\Phi (z;t)\\,=\\,\\frac{4}{||\\tau _\\gamma }\\Delta \\Phi ,\\quad z\\in ,$ with the boundary condition $\\Phi (z;t)\\,=\\,\\Psi (z)\\,-\\,\\sum _{k=1}^N d_k\\,\\arg (z - {z}_{k}),\\quad z\\in .$ Here $\\tau _\\gamma $ is related to parameters of the system via formula (REF ).", "In order to obtain the motion of vortices, we must rescale the time yet again, introducing $t^\\prime \\,=\\,-\\frac{8t}{\\pi \\tau _\\gamma \\ln \\epsilon }.$ On this time scale, the function $\\Phi (z;t^\\prime )$ is a harmonic function satisfying the same boundary condition (REF ), while the vortices move according to the gradient flow equations generated by the renormalized multi-vortex energy given by (REF ): ${\\dot{z}}_{k}(t^\\prime )\\,&=\\,-_{\\bar{{z}}_{k}}U\\big ({z}_{1},\\ldots ,{z}_{N}\\big ).$ We want to remark that in the natural regime of the Doi-Smoluchowski dynamics, which we consider in this paper, it is impossible to obtain a closed evolution equation for the order parameter field $(z)$ , without immediately reducing it to the dynamics of vortices.", "This is a consequence of the fact that the reduction of the dynamics of $(\\varphi ,z)$ to configurations defined by $\\hat{}{[]}$ (which allows one to express all the relevant quantities in terms of $$ ) happens in the same limit as reduction of $(z)$ to multi-vortex configurations prescribed by $_*$ .", "This can be seen from equation (REF ), where the boundedness of the energy $()/\\epsilon ^2$ simultaneously imposes constraints on the relative entropy $(|\\hat{}{[]})$ and the potential $W^\\gamma ()$ , making the reduced free energy $()$ singular.", "Reduction to a theory involving exclusively the order parameter $$ would be possible if an additional large parameter appeared in front of the relative entropy term in (REF ), without affecting the potential $W^\\gamma $ .", "However this is impossible due to the fact that both these terms appear as parts of the same entropic term in the Onsager energy [11].", "The situation is somewhat different in three dimensions, because in three-dimensional systems, the topological singularities do not possess infinite energy and thus the analogue of the reduced free energy $()$ remains nonsingular in such a limit.", "See [8] and [18], where a reduction of Doi-Smoluchowski-type kinetic equations to Ericksen-Leslie equations was carried out.", "In the following sections we will derive equations (REF ) and (REF ) from the D-S evolution (REF ) using ideas from the theory of gradient flows.", "To familiarize the reader with these ideas, we first discuss a finite dimensional example, in which we explain the methodology and derive equations for gradient flow dynamics constrained to a submanifold by a large drift generated by the diverging part of the energy.", "Then we proceed to the analogous derivation for the D-S dynamics.", "In the final section of this paper, we rewrite equation (REF ) in terms of an infinite hierarchy of evolution equations for moments of the orientation density and discuss possibilities for various closures of this hierarchy." ], [ "A finite-dimensional example", "Suppose we are solving a (gradient flow) differential equation for $x(t)\\in ^n$ , $\\boxed{\\raisebox {-0.5em}{\\rule {0pt}{1.5em}}\\quad \\dot{x}(t)\\,=\\,-D(x)\\,{_{x}}E(x),\\quad }$ where $E:^n\\rightarrow $ is the energy function; $D(x)$ is a symmetric positive semi-definite $n\\times n$ matrix.", "Assume that all the quantities that we employ are sufficiently regular to guarantee well-posedness of our formal derivations.", "Our first goal is to show that the vector problem (REF ) is equivalent to a single scalar inequality, which allows us to interpret solutions of (REF ) as curves of maximal slope for the energy function, $E$ .", "Next, we will show, that if the energy depends on a small parameter, $\\epsilon $ , in such a way that, as $\\epsilon \\rightarrow 0$ , solutions become constrained to a submanifold of $^n$ , we can describe the limiting curves as curves of maximal slope in some native parameterization of this manifold." ], [ "Formulation via curves of maximal slope.", "Consider an arbitrary curve $x(t)$ , $t\\in [0,T]$ , which is such that $\\dot{x}\\in \\operatorname{rng}D(x)$ for all $t$ .", "For variation of $E(x)$ along this curve, we have $E(t)-E(0)\\,=\\,\\int _0^t{_{x}}E\\big (x(s)\\big )\\cdot {\\dot{x}}(s)s.$ Here we allow for a slight abuse of notation, employing $E(t)$ instead of $E\\big (x(t)\\big )$ .", "Let $G(x)$ be the generalized inverse of $D(x)$ , in the sense that $G$ is a symmetric matrix with the same range and kernel as $D$ , inverting $D$ in its range (for each $x$ ).", "Such generalized inverse is defined uniquelly.", "Observe that both $D$ and $G$ are positive semi-definite, they commute, and posses unique symmetric square roots.", "Observe also that $DG$ acts as identity on $\\dot{x}$ .", "Thus we have, $E(0)-E(t)\\,=\\,-\\int _0^t{_{x}}E\\cdot \\Big [\\sqrt{DG}\\,\\Big ]\\,{\\dot{x}}(s)s\\,=\\,\\int _0^t\\Big [-\\sqrt{D}\\,{_{x}}E\\Big ]\\cdot \\Big [\\sqrt{G}\\,{\\dot{x}}(s)\\Big ]s.$ Using elementary inequalities, we obtain (omitting $s$ -dependence), $E(0)-E(t)\\stackrel{^{(\\mathrm {a})}}{\\le }\\,\\int _0^t\\big |\\sqrt{D}\\,{_{x}}E\\big |\\,\\big |\\sqrt{G}\\,{\\dot{x}}\\big |s\\stackrel{^{(\\mathrm {b})}}{\\le }\\,\\frac{1}{2}\\int _0^t\\left(\\big |\\sqrt{D}\\,{_{x}}E\\big |^2\\,+\\,\\big |\\sqrt{G}\\,{\\dot{x}}\\big |^2\\right)s.$ Equality in (a) holds, if and only if $-\\sqrt{D}\\,{_{x}}E$ and $\\sqrt{G}\\,{\\dot{x}}$ are collinear.", "Equality in (b) holds, if and only if these quantities are equal by absolute values.", "Therefore, equalities in (REF ) are attained, if and only if $\\sqrt{G}\\,{\\dot{x}}\\,=\\,-\\sqrt{D}\\,{_{x}}E(x).$ Multiplying both sides by $\\sqrt{D}$ , we recover (REF ).", "Thus, by reversing the inequalities in (REF ), we obtain an inequality which is equivalent to the differential equation (REF ): $\\boxed{\\raisebox {-0.5em}{\\rule {0pt}{1.5em}}\\quad E(0)-E(t)\\,\\ge \\,\\frac{1}{2}\\int _0^t\\left(\\big \\Vert {_{x}}E\\big (x(s)\\big )\\big \\Vert _D^2\\,+\\,\\big \\Vert {\\dot{x}(s)}\\big \\Vert _G^2\\right)s.\\quad }$ Here we denoted (explicitly writing out the derivatives and employing Einstein summation rules) $\\big \\Vert {_{x}}E(x)\\big \\Vert _D^2\\,\\,D^{ij}(x)\\,{_j}E(x)\\,{_i}E(x);\\qquad \\big \\Vert {\\dot{x}}\\big \\Vert _G^2\\,\\,G_{ij}(x)\\,\\dot{x}^i\\dot{x}^j.$ We say that a curve $x(t)$ is a curve of maximal slope for the energy function $E(x)$ in metric prescribed by $G(x)$ , if $\\dot{x}\\in \\operatorname{rng}G(x)$ , and the inequality (REF ) holds for (almost) all $t$ .", "In this derivation we started from a differential equation and obtained a scalar inequality, i.e., we showed that solutions of (REF ) are curves of maximal slope for $E(x)$ , and vice versa.", "One could, however, start with (REF ) in a rather general metric space setting, and prove that the curves of maximal slope exist, posses certain regularity properties, and satisfy some differential equations, whenever the energy and the metric are sufficiently regular themselves.", "Such developments may be found in the book by Ambrosio, Gigli, and Savaré [2].", "Whenever we discuss gradient flow equations in this work, we understand them in terms of the curves of maximal slope formulation." ], [ "Change of variables.", "Suppose we want to study a family of curves of maximal slope which lie in an $m$ -dimensional submanifold, $\\mathcal {M}$ , of $^n$ in a parameterization native to $\\mathcal {M}$ .", "In other words, we assume that $x(t)=\\chi (y(t))$ for $y\\in ^m$ , $m\\le n$ , and some map ${\\chi }:^m\\rightarrow ^n$ ; and we want to obtain description of our curves using the $y$ -variables.", "We will always work within the same chart of $\\mathcal {M}$ , and will not worry about chart transitions here.", "Using the chain rule, we immediately get, (employing Greek indices for the $y$ -variables) $\\big \\Vert {\\dot{x}}\\big \\Vert _G^2\\,=\\,G_{ij}(x)\\,\\dot{x}^i\\dot{x}^j\\,=\\,G_{ij}\\big (\\chi (y)\\big )\\,\\Big [{_\\alpha }\\chi ^i(y)\\dot{y}^\\alpha \\Big ]\\Big [{_\\beta }\\chi ^j(y)\\dot{y}^\\beta \\Big ]\\,\\tilde{G}_{\\alpha \\beta }(y)\\,\\dot{y}^\\alpha \\dot{y}^\\beta \\,=\\,\\big \\Vert {\\dot{y}}\\big \\Vert _{\\tilde{G}}^2,$ where the $m\\times m$ matrix $\\tilde{G}(y)$ is defined as ${\\tilde{G}}_{\\alpha \\beta }(y)\\,=\\,{_\\alpha }\\chi ^i(y)\\,G_{ij}\\big (\\chi (y)\\big )\\,{_{\\beta }}\\chi ^j({y}).$ The matrix $\\tilde{G}(y)$ has a simple geometric interpretation: it is the metric induced by $G$ on $\\mathcal {M}$ , expressed in the $y$ -parameterization.", "Define $\\tilde{D}(y)$ as the generalized inverse of $\\tilde{G}(y)$ ; and set $\\tilde{E}(y)\\,\\,E\\big (\\chi (y)\\big ).$ Let us make a few additional assumptions, which are not required, but simplify some of the following arguments.", "Suppose there exists a neighborhood of $\\mathcal {M}$ , in which there exists a non-degenerate map $\\eta :^n\\rightarrow ^m$ , such that $y\\,=\\,\\eta \\big (\\chi (y)\\big ).$ Assume also that the exists another non-degenerate map $\\zeta :^n\\rightarrow ^{n-m}$ , such that $\\mathcal {M}$ is the 0-level set of $\\zeta $ , and ${_{x}}\\zeta $ is orthogonal to ${_{x}}\\eta $ , i.e., ${_i}\\zeta ^\\nu (x)\\, D^{ij}(x)\\, {_j}\\eta ^\\alpha (x)\\,=\\,0;\\qquad \\alpha =1\\ldots m;\\quad \\nu =1\\ldots n-m.$ Decomposing the $x$ -gradient of $E$ into the sum of gradients with respect to $\\eta $ and $\\zeta $ , we get $\\big \\Vert {_{x}}E\\big \\Vert ^2_D\\,&=\\,\\big \\Vert {_{\\eta }}E\\,{_{x}}\\eta \\,+\\,{_{\\zeta }}E\\,{_{x}}\\zeta \\big \\Vert ^2_D\\,=\\,\\big \\Vert {_{\\eta }}E\\,{_{x}}\\eta \\big \\Vert ^2_D\\,+\\,{_\\nu }E\\Big [{_i}\\zeta ^\\nu \\, D^{ij}\\, {_j}\\eta ^\\alpha \\Big ]{_\\alpha }E\\,+\\,\\big \\Vert {_{\\zeta }}E\\,{_{x}}\\zeta \\big \\Vert ^2_D\\nonumber \\\\&=\\,\\big \\Vert {_{\\eta }}E\\,{_{x}}\\eta \\big \\Vert ^2_D\\,+\\,\\big \\Vert {_{\\zeta }}E\\,{_{x}}\\zeta \\big \\Vert ^2_D\\,\\ge \\,\\big \\Vert {_{\\eta }}E\\,{_{x}}\\eta \\big \\Vert ^2_D.$ When $x=\\eta (y)\\in \\mathcal {M}$ , the last term in (REF ) is exactly $\\big \\Vert {_{y}}\\tilde{E}\\big \\Vert ^2_{\\tilde{D}}$ .", "Thus we get $\\big \\Vert {_{x}}E\\big (\\chi (y)\\big )\\big \\Vert _D^2\\,\\ge \\,\\big \\Vert {_{y}}\\tilde{E}(y)\\big \\Vert _{\\tilde{D}}^2.$ This inequality expresses the fact that by extending $\\tilde{E}$ as $E$ from $\\mathcal {M}$ into $^n$ , we can only increase the norm of its gradient.", "Combining equations (REF ), (REF ), and (REF ), we get $\\tilde{E}(0)-\\tilde{E}(t)\\,\\ge \\,\\frac{1}{2}\\int _0^t\\left(\\big \\Vert {_{y}}\\tilde{E}\\big (y(s)\\big )\\big \\Vert _{\\tilde{D}}^2\\,+\\,\\big \\Vert {\\dot{y}(s)}\\big \\Vert _{\\tilde{G}}^2\\right)s.$ Thus we demonstrated that curves of maximal slope in $x$ -parameterization of $^n$ are also curves of maximal slope in $y$ -parameterization of $^m$ , when the latter is equipped with metric inherited through its embedding as the submanifold of $^n$ ." ], [ "Asymptotic reduction.", "Suppose now, our energy depends on a small parameter, $\\epsilon $ , and the dynamics is such that, as $\\epsilon \\rightarrow 0$ , all the trajectories $x^\\epsilon (t)$ become constrained onto $\\mathcal {M}$ .", "Let us derive equations which describe this asymptotic dynamics in terms of the $y$ -variables.", "Consider the energy function of the following form: $E^\\epsilon (x)\\,=\\,U(x)\\,+\\,\\frac{1}{\\epsilon }V(\\zeta (x)),$ where $\\zeta (x)$ is as above.", "Assume that $U:^n\\rightarrow $ is bounded below; $V:^{n-m}\\rightarrow ^+$ has minimum at the origin and has no other critical points; without loss of generality, set $V(0)=0$ .", "This construction is designed so, that the “fast” flow generated by $V$ will quickly carry solutions to the vicinity of $\\mathcal {M}$ , while it will not affect the dynamics on $\\mathcal {M}$ itself.", "Pick a sequence of initial conditions, such that $x^\\epsilon (0)&\\rightarrow x^0(0)\\in \\mathcal {M};\\\\E^\\epsilon \\big (x^\\epsilon (0)\\big )\\,&\\rightarrow \\,U\\big (x^0(0)\\big ).$ The second condition assures that there is no excess energy in the system.", "Generally, one can show (by other methods) that this condition is not required, as it will be automatically satisfied after some initial time of $\\scriptstyle {\\mathcal {O}}\\displaystyle (1)$ .", "Assume that, as $\\epsilon \\rightarrow 0$ , $x^\\epsilon (t)\\rightarrow x^0(t),\\quad \\text{pointwise for}~t\\in [0,T].\\\\$ This may be proven for sufficiently regular $U$ and $V$ .", "The energy is non-increasing along the curves of maximal slope, therefore $V\\big (x^\\epsilon (t)\\big )$ must remain of $(\\epsilon )$ for all $t>0$ .", "This implies that $\\smash{x^0(t)\\in \\mathcal {M}}$ for all $t\\ge 0$ .", "We will now show that $x^0(t)$ is a curve of maximal slope for $U$ on $\\mathcal {M}$ equipped with metric inherited from $^n$ .", "First of all, observe, that due to positivity of $V$ , $E^\\epsilon \\big (x^\\epsilon (0)\\big )-E^\\epsilon \\big (x^\\epsilon (t)\\big )\\,\\le \\,E^\\epsilon \\big (x^\\epsilon (0)\\big )-U\\big (x^\\epsilon (t)\\big ).$ Passing to the limit as $\\epsilon \\rightarrow 0$ , using (REF ) and the continuity of $U$ , we get $\\lim _{\\epsilon \\rightarrow 0}\\Big [E^\\epsilon \\big (x^\\epsilon (0)\\big )-E^\\epsilon \\big (x^\\epsilon (t)\\big )\\Big ]\\,\\le \\,U\\big (x^0(0)\\big )-U\\big (x^0(t)\\big )\\,=\\,\\tilde{U}\\big (y(0)\\big )-\\tilde{U}\\big (y(t)\\big ).$ The pointwise convergence of $x^\\epsilon (t)$ to $x^0(t)$ implies $\\liminf _{\\epsilon \\rightarrow 0}\\int _0^t\\big \\Vert \\dot{x}^\\epsilon (s)\\big \\Vert _{G(x^\\epsilon )}^2s\\ge \\int _0^t\\big \\Vert \\dot{x}^0(s)\\big \\Vert _{G(x^0)}^2s\\,=\\,\\int _0^t\\big \\Vert \\dot{y}(s)\\big \\Vert _{\\tilde{G}(y)}^2s.$ From (REF ), using that $V$ only depends on $\\zeta $ , we get that $\\big \\Vert {_{x}}E^\\epsilon (x^\\epsilon )\\big \\Vert ^2_{D(x^\\epsilon )}\\,\\ge \\,\\big \\Vert {_{\\eta }}E^\\epsilon (x^\\epsilon )\\,{_{x}}\\eta (x^\\epsilon )\\big \\Vert ^2_{D(x^\\epsilon )}\\,=\\,\\big \\Vert {_{\\eta }}U(x^\\epsilon )\\,{_{x}}\\eta (x^\\epsilon )\\big \\Vert ^2_{D(x^\\epsilon )}.$ Therefore $\\liminf _{\\epsilon \\rightarrow 0}\\int _0^t\\big \\Vert {_{x}}E^\\epsilon \\big (x^\\epsilon \\big )\\big \\Vert ^2_{D(x^\\epsilon )}s\\,\\ge \\,\\int _0^t\\big \\Vert {_{\\eta }}U\\big (x^0\\big )\\,{_{x}}\\eta \\big (x^0\\big )\\big \\Vert ^2_{D(x^0)}s\\,=\\,\\int _0^t\\big \\Vert {_{y}}\\tilde{U}(y)\\big \\Vert ^2_{\\tilde{D}(y)}s.$ Using inequalities (REF ), (REF ), and (REF ) in (REF ), we get the desired result: $\\tilde{U}(0)-\\tilde{U}(t)\\,\\ge \\,\\frac{1}{2}\\int _0^t\\left(\\big \\Vert {_{y}}\\tilde{U}\\big (y(s)\\big )\\big \\Vert _{\\tilde{D}}^2\\,+\\,\\big \\Vert {\\dot{y}(s)}\\big \\Vert _{\\tilde{G}}^2\\right)s.$ Thus we see that the limiting trajectories are curves of maximal slope for the “slow” part of the energy, $U$ , constrained to $M$ .", "Using the equivalence of this formulation to formulation via gradient differential equations, we can also state this result in the following manner: the limiting trajectories may be obtained as $x^0(t)\\,=\\,\\chi (y(t))$ , where $y(0)=\\eta (x(0))$ and $y(t)$ satisfies $\\dot{y}(t)\\,=\\,-\\tilde{D}(y)\\,{_{y}}\\tilde{U}(y).$ Note that the matrix $\\tilde{D}(y)$ must be computed by inverting the matrix of the metric tensor $\\tilde{G}(y)$ given by (REF ).", "Calculation of these matrices becomes the only ingredient required for obtaining the limiting dynamics." ], [ "Derivation for the Doi-Smoluchowski dynamics", "The kinetic equation (REF ) formally resembles our finite-dimensional ODE example.", "The density of orientations $(\\varphi ,z)$ plays the role of $x$ -variables, while the vortex locations $z_k$ and the function $\\Phi (z)$ correspond to the reduced $y$ -variables.", "Thus we will proceed along the same lines in this derivation." ], [ "Mobility and metric.", "As in the finite-dimensional example, we can obtain that the dynamics (REF ) is equivalent to the following inequality: $(0) - (t) \\,\\ge \\,\\int _0^t\\, \\left( \\left\\Vert {}{}\\right\\Vert ^2_{\\hat{D}} \\,+ \\, \\big \\Vert _t \\big \\Vert ^2_{\\hat{G}} \\right) \\, t ,$ where the operator $\\hat{D}[]f(\\varphi )-{_\\varphi }\\big (\\,{_\\varphi }f(\\varphi )\\big )$ corresponds to the matrix $D(x)$ in (REF ), and $\\smash{\\hat{G} = \\hat{D}^{-1}}$ is the generalized inverse of $\\hat{D}$ .", "In order to determine $\\hat{G}$ , we must solve the differential equation, $-{_\\varphi }\\big ((\\varphi )\\,{_\\varphi }f(\\varphi )\\big )\\,=\\,u(\\varphi ).$ Let us assume that the support of $(\\varphi )$ is the entire interval, $[0,2\\pi )$ , and treat $\\smash{\\hat{D}}$ and $\\smash{\\hat{G}}$ as symmetric operators defined on smooth $2\\pi $ -periodic functions.", "Integrating (REF ) once, we get $v(\\varphi )\\,\\,{_\\varphi } f(\\varphi )\\,=\\,-\\frac{1}{(\\varphi )}\\int _0^{\\varphi } u(\\varphi ^\\prime )\\varphi ^\\prime \\,+\\,\\frac{C}{(\\varphi )}.$ The function $v(\\varphi )$ is a derivative, and thus its total integral must vanish; this gives us the condition, $C\\,=\\,\\bigg [\\int _0^{2\\pi }\\frac{\\varphi }{(\\varphi )}\\bigg ]^{-1}\\int _0^{2\\pi }\\frac{1}{(\\varphi )}\\int _0^{\\varphi } u(\\varphi ^\\prime )\\varphi ^\\prime \\varphi .$ We do not need to integrate equation (REF ) second time, as it is convenient to define $\\hat{G}$ using $v(\\varphi )$ .", "We only need $\\hat{G}$ as a bilinear form; for its action we have, whenever $u,\\tilde{u},f,\\tilde{f}\\in \\operatorname{rng}\\hat{G}$ , $(u,\\hat{G}\\tilde{u})\\,=\\,(\\hat{D} f,\\hat{G}\\hat{D}\\tilde{f})\\,=\\,(\\hat{D} f, \\tilde{f})\\,=\\,(-_\\varphi _\\varphi f,\\tilde{f})\\,=\\,(\\,{_\\varphi }f,{_\\varphi }\\tilde{f}).$ Writing this down explicitly in terms of $v(\\varphi )$ and $\\tilde{v}(\\varphi )$ , $(u,\\hat{G}\\tilde{u})\\,=\\,\\big (u,\\big [-_\\varphi _\\varphi \\big ]^{-1}\\tilde{u}\\big )\\,\\,\\int _\\int _0^{2\\pi }v(\\varphi ,z)\\,\\tilde{v}(\\varphi ,z)\\;(\\varphi ,z)\\varphi \\upsilon (z).$" ], [ "Structure of the slow manifold.", "The manifold $\\mathcal {M}$ corresponds to the set of optimal orientation densities produced by multi-vortex maps.", "Let us use the hat symbol, “$\\hat{~}$ ” to denote such configurations; we have, as in (REF ), $(\\varphi ,z)\\,=\\,\\hat{}{[\\hat{}]}(\\varphi ,z)\\,=\\,\\Big [2\\pi _0\\!\\big (\\!", "(\\hat{})\\big )\\Big ]^{-1}{\\exp \\Big [(\\hat{})\\cos (2\\varphi -\\arg \\hat{})\\Big ]},$ where the order parameter field is parameterized by vortex locations, ${z}_{k}$ , $k=1,\\ldots ,N$ , and the phase function $\\Phi (z)$ : $\\quad \\hat{}[{z}_{1},\\ldots ,{z}_{k};\\Phi ](z)\\,=\\,\\hat{}[{z}_{1},\\dots ,{z}_{N}](z)\\;\\exp \\Big [\\Phi (z)\\,+\\,\\sum _k \\;d_k\\arg (z-{z}_{k})\\Big ].$ Even though we do not know the exact shape of the function $\\hat{}[{z}_{1},\\dots ,{z}_{N}](z)$ , the finiteness of the energy $()$ implies that $\\hat{}(z)$ turns to zero at the location of vortices and approaches $_$ at distances larger than $(\\epsilon )$ .", "However, the specifics of this behavior are not important for our purposes.", "It is sufficient to utilize the following property, which is well-known in the context of the Ginzburg-Landau theory [15].", "Given a tempered family of order parameter fields converging to a multi-vortex configuration, there exists a covering of the vortices by disks ${{z}_{k}}{R_\\epsilon }$ with radii $R_\\epsilon =(\\epsilon )$ , such that $|^\\epsilon (z)-_|=\\operatorname{{\\scriptstyle \\mathcal {O}}}(1)$ in the exterior of these disks, while $\\int _{{{z}_{k}}{R_\\epsilon }} |\\nabla ^\\epsilon (z)|^2\\upsilon (z)\\,\\le \\,C.$ This bound will be used below to estimate the matrix elements of $\\hat{G}$ restricted to multi-vortex configurations." ], [ "Change of variables.", "In our liquid crystalline system, the role of the map $\\chi $ is played by $\\hat{}$ ; variables parameterizing the slow manifold, $\\mathcal {M}$ , are the vortex locations, ${z}_{k}$ , and the function $\\Phi (z)$ .", "Let us define $Y({z}_{1}, \\ldots , {z}_{N}, \\Phi )$ .", "Let the index $\\alpha $ run through these parameters.", "Using expression (REF ) for the decomposition of energy $()$ , and proceeding formally in the same way as in the finite dimensional example, we obtain, $\\tilde{}(0)\\, -\\, \\tilde{}(t) \\,\\ge \\, \\int _0^t \\left( \\left\\Vert {\\tilde{}}{Y}\\right\\Vert ^2_{D} + \\big \\Vert {_t}Y \\big \\Vert ^2_{G} \\right) \\, t.$ This inequality is equivalent to the following differential equations for the reduced dynamics: ${_t}Y = - D\\,{\\tilde{}}{Y}.$ Thus we need to compute the matrix $G$ and its generalized inverse $D = G^{-1}$ .", "We start by computing the analogues of the derivatives $_{\\alpha }\\chi ^{i}(y)$ .", "The chain rule gives us, ${_\\alpha }\\hat{}\\,=\\,\\Big [{_{}}\\hat{}\\Big ]\\,{_\\alpha }\\hat{}\\,+\\,\\Big [{_{\\arg }}\\hat{}\\Big ]\\,{_\\alpha }\\arg \\hat{}.$ Observing that ${_{\\arg }\\hat{}}\\,=\\,-_\\varphi \\hat{}/2$ , we can write ${_{\\Phi }}\\hat{}\\,&=\\,\\boxed{\\raisebox {0em}{\\rule {0pt}{0em}}\\quad -\\frac{1}{2}_\\varphi \\hat{};\\quad }^{~\\text{(a)}}\\\\{_{{z}_{k}}}\\hat{}\\,&=\\,\\boxed{\\raisebox {0em}{\\rule {0pt}{0em}}\\quad -\\frac{d_k}{4(z-{z}_{k})}_\\varphi \\hat{}\\quad }^{~\\text{(b)}}\\!+~\\boxed{\\raisebox {-0.95em}{\\rule {0pt}{2.4em}}\\quad {_}\\hat{}\\,{_{{z}_{k}}}\\hat{}.\\quad }^{~\\text{(c)}}$ In order to compute the matrix $G_{\\alpha \\beta }\\,\\,({_\\alpha }\\hat{},\\hat{G}\\,{_\\beta }\\hat{})\\,=\\,\\int _\\int _0^{2\\pi }v_\\alpha (\\varphi ,z)\\,v_\\beta (\\varphi ,z)\\;(\\varphi ,z)\\varphi \\upsilon (z),$ we must find the fields $v_\\alpha (\\varphi ,z)$ by solving the differential equations, $-_\\varphi (\\hat{}\\,v_\\alpha )\\,=\\,_\\alpha \\hat{},$ as described in formulas (REF ) and (REF ), and after that, evaluate the integrals in (REF ).", "Let us implement this plan for $G_{{z}_{k}\\bar{{z}}_{k}}$ .", "As equation (REF ) is linear, its solution may be represented as a sum of solutions with right-hand sides corresponding to terms labelled as (b) and (c) in formula (REF ).", "For the solution corresponding to term (b), omitting the $z$ -dependence, we get $v(\\varphi )\\,=\\,\\frac{2\\pi }{\\hat{}(\\varphi )}\\bigg [\\int _0^{2\\pi }\\frac{\\varphi ^\\prime }{\\hat{}(\\varphi ^\\prime )}\\bigg ]^{-1}-\\,1\\,=\\,\\frac{1}{2\\pi \\operatorname{I}_0^2(\\operatorname{A}\\!\\big (\\hat{})\\big )\\hat{}(\\varphi )}\\,-\\,1.$ For term (c), we first compute ${\\hat{}}{}\\,=\\,\\operatorname{A}^\\prime ()\\big [\\cos (2\\varphi -\\arg )\\,-\\,\\big ]\\hat{}(\\varphi );$ now we can see that the solution to (REF ) with right-hand side given by (REF ) may be represented as $\\operatorname{A}^\\prime ()F(\\varphi ,\\arg )$ , where $F$ is some bounded function.", "Thus we have $v_{{z}_{k}}(\\varphi )\\,=\\,-\\frac{d_k}{4(z-{z}_{k})}\\bigg [\\frac{1}{2\\pi \\operatorname{I}_0^2(\\operatorname{A}\\!\\big (\\hat{})\\big )\\hat{}(\\varphi )}\\,-\\,1\\bigg ]\\,+\\,\\operatorname{A}^\\prime ()F(\\varphi ,\\arg )\\,{_{{z}_{k}}}\\hat{}.$ For $v_{\\bar{{z}}_{k}}$ we get the expression, complex-conjugate to (REF ).", "Now we compute the integrals in (REF ).", "As, the only non-integrable singularity which appears in the calculation is $\\smash{1/|z|^2}$ , all the terms, except the product of the first term in (REF ) and its complex conjugate, contribute in $(1)$ as $\\epsilon \\rightarrow 0$ .", "Integrating over $\\varphi $ , we obtain $G_{{z}_{k}\\bar{{z}}_{k}}\\,=\\,\\frac{1}{16}\\int _\\bigg [1\\,-\\,\\frac{1}{\\operatorname{I}^2_0\\!\\big (\\!\\operatorname{A}(\\hat{}(z))\\big )}\\bigg ]\\frac{\\upsilon (z)}{|z-{z}_{k}|^2}\\,+\\,(1).$ In order to estimate this integral, we first split the domain $$ in two parts: ${R_\\epsilon }{z_k}$ and $\\setminus {R_\\epsilon }{z_k}$ , where the radius $R_\\epsilon =(\\epsilon )$ is chosen so that the inequality (REF ) holds.", "Using the properties of the function $\\operatorname{A}()$ (see formula (REF ) in the appendix), and taking into account (REF ), we obtain $G_{{z}_{k}\\bar{{z}}_{k}} \\le C_1\\int _{{R_\\epsilon }{z_k}} |\\nabla |^2\\upsilon (z)\\le C_2.$ For the integral over the exterior of the disk ${R_\\epsilon }{z_k}$ we can use Lemma REF from the appendix, obtaining in the end, $G_{{z}_{k}\\bar{{z}}_{k}}\\,=\\,-\\frac{\\pi }{8}\\left[1\\,-\\,\\frac{1}{\\operatorname{I}^2_0\\!\\big (\\!\\operatorname{A}(_)\\big )}\\right]\\ln \\epsilon \\,+\\,(1).$ The calculation of $G_{\\Phi \\Phi }$ is equivalent to this one.", "We do not provide detailed calculations for other matrix elements of $G$ here; it is only important to note that they all are of $(1)$ as $\\epsilon \\rightarrow 0$ .", "Summarizing all these calculations we have, $G_{{z}_{k}\\bar{{z}}_{k}}\\,&=\\,-\\frac{\\pi \\tau _\\gamma }{8}\\ln \\epsilon \\,+\\,(1),\\quad k=1,\\ldots ,N;\\\\G_{\\Phi \\Phi }\\,&=\\,\\frac{||\\tau _\\gamma }{4}\\,+\\,(\\epsilon );\\\\G_{\\alpha \\beta }\\,&=\\,(1),\\quad \\text{for all other combinations of~}\\alpha \\text{~and~}\\beta .\\rule {0em}{1.5em}$ Here we introduced a scaling factor, $\\tau _\\gamma \\,\\,1\\,-\\,\\frac{1}{\\operatorname{I}^2_0\\!\\big (\\!\\operatorname{A}(_)\\big )}\\,=\\,1\\,-\\,\\frac{1}{\\operatorname{I}^2_0(\\gamma _)};$ for the second equality we used that $_$ satisfies $\\gamma _\\,=\\,\\operatorname{A}(_)$ .", "The final step in our calculation is computation of $D$ , the inverse of $G$ .", "Using Lemma REF from the appendix (setting $\\delta =-1/\\ln \\epsilon $ ), we obtain $D_{{z}_{k}\\bar{{z}}_{k}}\\,&=\\,-\\frac{8}{\\pi \\tau _\\gamma \\ln \\epsilon }\\,+\\,(1/\\ln ^2\\epsilon ),\\quad D_{{z}_{k}{z}_{k}}\\,=\\,(1/\\ln ^2\\epsilon ),\\quad k=1,\\ldots ,N;\\\\D_{\\Phi \\Phi }\\,&=\\,\\frac{4}{||\\tau _\\gamma }\\,+\\,(1/\\ln \\epsilon );\\\\D_{\\alpha \\beta }\\,&=\\,(1/\\ln \\epsilon ),\\quad \\text{for all other combinations of~}\\alpha \\text{~and~}\\beta .\\rule {0em}{1.25em}$" ], [ "Evolution equations for ${z}_{k}$ , {{formula:a6ac628a-f1ef-4c7a-987b-654f40ef54f8}} , and timescales.", "Now, once we have computed the matrix $D$ , we are finally able to write down equations governing the evolution of vortices, ${z}_{k}$ , and the function $\\Phi (z)$ in the limit as $\\epsilon \\rightarrow 0$ : ${\\dot{z}}_{k}\\,&=\\,\\frac{8}{\\pi \\tau _\\gamma \\ln \\epsilon }_{\\bar{{z}}_{k}}\\tilde{E}({z}_{1},\\ldots ,{z}_{N};\\Phi )\\,+\\,(1/\\ln \\epsilon ){\\tilde{E}}{\\Phi }\\,+\\,(1/\\ln ^2\\epsilon );\\\\{_t}\\Phi \\,&=\\,-\\frac{4}{||\\tau _\\gamma }{\\tilde{E}}{\\Phi }\\,+\\,(1/\\ln \\epsilon )\\,=\\,\\frac{4}{||\\tau _\\gamma }\\,\\Delta \\Phi \\,+\\,(1/\\ln \\epsilon ).$ From these equations we can see that, as $\\epsilon \\rightarrow 0$ , the vortices are stationary, and the only evolution which occurs in our system is the relaxation of the field $\\Phi (z)$ governed by the heat equation, ${_t}\\Phi (z;t)\\,=\\,\\frac{4}{||\\tau _\\gamma }\\Delta \\Phi ,\\quad z\\in .$ The boundary condition is inherited from our Dirichlet boundary condition on $$ : $\\Phi (z;t)\\,=\\,\\Psi (z)\\,-\\,\\sum _{k=1}^N d_k\\,\\arg (z - {z}_{k}),\\quad z\\in .$ In order to capture the vortex dynamics, we introduce a slow, rescaled time, $t^\\prime \\,=\\,-\\frac{8t}{\\pi \\tau _\\gamma \\ln \\epsilon }.$ On this time scale, in the leading order, $\\Phi $ is a harmonic function satisfying the same boundary condition (REF ).", "We get that on the $t^\\prime $ -timescale, the evolution of vortices satisfies ${\\dot{z}}_{k}(t^\\prime )\\,&=\\,-_{\\bar{{z}}_{k}}U\\big ({z}_{1},\\ldots ,{z}_{N}\\big ).$ Note, that the second term on the right-hand side of equation (REF ) vanishes because on the $t^\\prime $ -timescale, $\\delta \\tilde{E}/\\delta \\Phi =0$ ." ], [ "Equations for the moments and closures", "For the sake of completeness, here we derive equations for the moments of the orientation density (its Fourier coefficients) and formally perform a closure at the level of the first moment.", "This closure, even though sensible from the physical standpoint, does not have a valid mathematical justification, and does not occur in some well-defined asymptotic limit." ], [ "Equations for the moments.", "Let us define $k$ -th moment (Fourier coefficient) of the orientations density $(\\varphi ,z;t)$ as $^{(k)}(z;t)\\,\\,\\int _0^{2\\pi }^{\\,2k\\varphi }(\\varphi ,z;t)\\varphi .$ The factor of 2 in $^{2k\\varphi }$ appears because, physically, in nematic systems, the orientations density is invariant with respect to inversion of liquid crystalline molecules, and thus $(\\varphi )=(\\varphi +\\pi )$ , and all the odd Fourier coefficients of the orientation density vanish.", "The first moment $\\smash{^{(1)}}$ is exactly the order parameter field, $$ , employed in our work.", "In order to obtain dynamic equations for $k$ -th moment we can differentiate equation (REF ) with respect to time $t$ and use evolution equation (REF ) for $$ , obtaining $_t{{}}^{(k)}(z;t)\\,=\\,-\\,4k^2\\,{}^{(k)}\\,+\\,2k\\,\\left[{}^{(k-1)}\\,-\\,{}^{(k+1)}\\bar{}\\right].$ It is possible to rewrite these equations in a gradient form using the energy decomposition (REF ): $_t{}^{(k)}(z;t)\\,=\\,4k\\,\\left[^{(k+1)}\\,{}{}\\,-\\,^{(k-1)}\\,{}{\\bar{}}\\right]\\,-\\,2k\\int _0^{2\\pi }^{\\,2k\\varphi }\\,\\left[{(|\\hat{}{[]})}{}\\right]_\\varphi \\,(z,\\varphi ;t)\\varphi $ Equations (REF ) or (REF ) form an infinite hierarchy, as equation for each $^{(k)}$ involves $^{(k+1)}$ , etc.", "In order to obtain a closed system of equations for some first few moments, one must find a way to decouple this hierarchy by expressing the higher-order moments via the lower-order ones.", "This requires some additional assumptions on the orientation density $$ ." ], [ "Maximal entropy closure.", "A natural physical assumption is that the orientation density relaxes to its optimal configuration, $\\hat{}{[]}$ , given by (REF ), which minimizes the relative entropy term in the energy (REF ).", "(Physical entropy is defined with a sign opposite to the one used here, and thus the name, “maximal entropy.”) This allows us to calculate the higher-order moments in terms of $$ explicitly: $^{(k)}(z)\\,=\\,\\int _0^{2\\pi }^{2k\\varphi }(\\varphi ,z)\\varphi \\,=\\,\\frac{\\operatorname{I}_k(\\operatorname{A}())}{\\operatorname{I}_0(\\operatorname{A}())}^{\\,k\\arg (z)}.$ In particular, we find that $^{(2)}(z)\\,= \\, \\frac{\\operatorname{I}_2(\\operatorname{A}())}{\\operatorname{I}_0(\\operatorname{A}())} ^{\\,2\\arg (z)} \\, =\\, \\frac{{}^2}{^2} \\left[ 1- \\-\\frac{2}{\\operatorname{A}()} \\right].$ Now it is possible to close the hierarchy (REF ) at the level of the first moment: $_t{{}}(z;t)\\,&=\\,-\\,4{}\\,+\\,2\\,\\left[\\,-\\,\\frac{{}^2}{^2}\\left(1-\\frac{2}{\\operatorname{A}()}\\right)\\bar{}\\right].$ Similarly, the same closure in the gradient form may be obtained from (REF ): $_t{}(z;t)\\,=\\,\\frac{{4}^2}{^2} \\left[ 1- \\-\\frac{2}{\\operatorname{A}()}\\right] \\,{}{}\\,-\\,4{}{\\bar{}}.$ This equation is quite similar to the canonical Landau-de Gennes equation (or Ginzburg-Landau equation) for the free energy dissipation in $L_2$ metric, $_t{}(z;t)\\,=\\,-{}{\\bar{}}.$ Curiously, (REF ) becomes exactly (REF ) (up to a time-scale change), when $(z)=_$ ; $\\gamma =2$ and $\\operatorname{A}(_)=2_$ , which corresponds to the isotropic-nematic phase transition.", "This is exactly when the Landau expansion of the free energy is valid.", "We would like to stress though, that in general, this maximal entropy closure is only mathematically justifiable when the relative entropy term in (REF ) is penalized in some appropriate asymptotic limit.", "In such a limit, however, the dynamics prescribed by equations (REF ) or (REF ) itself becomes singular and reduces to vortex dynamics, as explained in this work." ], [ "Notation and some useful facts", "We use bold face font to denote complex-valued functions and variables; regular font for their absolute values, e.g., $z=|z|$ .", "Given $z=x+y$ , the operators $_{z}$ and $_{\\bar{z}}$ are defined as $_{z}\\,=\\,\\frac{1}{2}\\big (_x\\,-\\,_y\\big ),\\qquad _{\\bar{z}}\\,=\\,\\frac{1}{2}\\big (_x\\,+\\,_y\\big ).$ The complex form of Stokes' theorem may be written as $\\oint _{} f(z,\\bar{z}){z}\\,=\\,2\\int __{\\bar{z}}f(z,\\bar{z})\\upsilon (z),$ where integral on the left is counter-clockwise contour integral, and integral on the right is the usual area integral, i.e., $\\upsilon (z)=xy$ ." ], [ "Useful identities", "involving $\\operatorname{Ln}z\\,=\\,\\ln z+\\arg z$ : $_{z}\\operatorname{Ln}z\\,&=\\,\\frac{1}{z}\\,=\\,2_{z}\\ln z\\,=\\,2_{z}\\arg z;\\qquad _{\\bar{z}}\\operatorname{Ln}z\\,=\\,0;\\\\\\,\\arg z\\,&=\\,_{z}\\arg zz\\,+\\,_{\\bar{z}}\\arg z\\bar{z}\\,=\\,\\frac{x}{z}y\\,-\\,\\frac{y}{z}x.$ Note that integration with $\\,\\arg z$ is well-defined in $\\lbrace 0\\rbrace $ , even though $\\arg z$ is a multivalued function with jump discontinuities on closed contours encircling the origin.", "Let us state a lemma which is used in estimation of some integrals; its proof is straightforward.", "Let $\\subset and $ BR (z0)$ be a disk of radius $ R=()$ centered at $ z0$.", "Assume that $ () 0$ as $ 0$ and a sequence $ f(z)$ satisfies $ (1- ())|f(z)| 1$ for all $ zBR (z0)$.", "Then\\begin{equation}\\int _{\\setminus B_{R_\\epsilon } ({z}_{0})} f^\\epsilon (z) \\frac{v(z)}{| z - {z}_{0}|^2} = - 2 \\pi \\ln \\epsilon + (1).\\end{equation}$" ], [ "Special functions.", "In our work we use several special functions, such as the modified Bessel functions of the first kind, $\\operatorname{I}_{\\nu }(\\lambda )$ , and the function $\\operatorname{A}()$ .", "Here is a brief summary of their properties.", "The function $\\operatorname{A}()$ is the inverse of $\\operatorname{I}_1/\\operatorname{I}_0$ , i.e., $\\frac{\\operatorname{I}_1\\big (\\operatorname{A}()\\big )}{\\operatorname{I}_0\\big (\\operatorname{A}()\\big )}\\,=\\,.$ Using the properties of modified Bessel functions, it is straightforward to show that $\\operatorname{A}()$ is a monotone increasing function defined on $(-1,1)$ with vertical asymptotes at $=\\pm 1$ ; it is odd and convex when $>0$ .", "The graph of $\\operatorname{A}(n)$ is shown in Figure REF .", "By direct computation we can verify that $0 < 1\\,-\\,\\frac{1}{\\operatorname{I}^2_0(\\operatorname{A}())}\\,\\le \\,\\min (1,C ^2),$ for some $C >0$ independent of $$ .", "This inequality is used in estimation of some of the integrals occurring in this paper.", "The potential $W^\\gamma ()$ is given by $W^{\\gamma }()\\,=\\,-\\,\\frac{\\gamma ^2}{2}\\,+\\,\\big [\\operatorname{A}()\\,-\\,\\ln \\operatorname{I}_0(\\operatorname{A}())\\big ]\\,+\\,C_{\\gamma },$ where $C_\\gamma $ is chosen so that $W^\\gamma () \\ge 0$ with equality achieved at $=^\\gamma _{}$ .", "The value of $^\\gamma _{}$ satisfies $\\smash{\\gamma ^\\gamma _= \\operatorname{A}\\big (^\\gamma _\\big )}$ .", "This equation has a nonzero solution for $\\gamma >2$ , which corresponds to the isotropic-nematic phase transition.", "The graph of $W^\\gamma ()$ is shown in Figure REF .", "Here is a lemma which we use to invert the matrix $G$ in Section : Let $A$ be a symmetric block-matrix , representable, when $\\delta \\rightarrow 0$ , as $A\\,=\\,\\frac{1}{\\delta }\\left[\\begin{array}{cc}A^{}_{11}&0\\\\0~&0\\end{array}\\right]\\,+\\,\\left[\\begin{array}{cc} B^{}_{11}&B^{}_{12}\\\\B^{}_{21}&B^{}_{22}\\end{array}\\right]\\,+\\,\\delta \\left[\\begin{array}{cc}C^{}_{11}&C^{}_{12}\\\\C^{}_{21}&C^{}_{22}\\end{array}\\right]\\,+\\,\\big (\\delta ^2\\big ),$ where $A_{ii}$ , $B_{ii}$ , $C_{ii}$ symmetric matrices; $A_{11}$ and $B_{22}$ are invertible; $B^{}_{12}=B^\\dag _{21}$ , $C^{}_{12}=C^\\dag _{21}$ .", "Then its inverse, $\\smash{A^{-1}}$ , exists for all sufficiently small $\\delta $ , and is given by $A^{-1}\\,=\\,\\left[\\begin{array}{cc} 0&0\\\\0&B^{-1}_{22}\\end{array}\\right]\\,+\\,\\delta \\left[\\begin{array}{cc}A^{-1}_{11}&D^{}_{12}\\\\D^{}_{21}&D^{}_{22}\\end{array}\\right]\\,+\\,\\big (\\delta ^2\\big ),$ where $D^{}_{12}=-A^{-1}_{11}B^{}_{12}B^{-1}_{22}=D^\\dag _{21}$ , $D^{}_{22}=B^{-1}_{22}\\big (C^{}_{22}-B^{}_{21}A^{-1}_{11}B^{}_{12}\\big )B^{-1}_{22}$ .", "Computing the determinant of $A$ via expansion with respect to rows corresponding to $A_{11}$ , we obtain an asymptotic formula, $\\smash{\\det A=\\delta ^{-d}\\det A_{11}\\det B_{22}+\\big (\\delta ^{1-d}\\big )}$ , where $d$ is the dimension of $A_{11}$ .", "Because $A_{11}$ and $B_{22}$ are invertible, $\\smash{\\det A^{-1}}\\ne 0$ for all sufficiently small $\\delta $ , and thus, within this range of $\\delta $ , $A^{-1}$ exists.", "Let us verify equation (REF ).", "Consider $B\\,=\\,\\left[\\begin{array}{cc} 0&0\\\\0&B^{-1}_{22}\\end{array}\\right]\\,+\\,\\delta \\left[\\begin{array}{cc}A^{-1}_{11}&D^{}_{12}\\\\D^{}_{21}&0\\end{array}\\right].$ By direct computation we obtain, $AB\\,=\\,I\\,+\\,\\delta C\\,+\\,(\\delta ^2);\\qquad C\\,=\\,\\left[\\begin{array}{cc}A^{-1}_{11}&B^{}_{11}D^{}_{12}+C^{}_{12}B^{-1}_{22}\\\\0&B^{}_{21}D^{}_{12}+C^{}_{22}B^{-1}_{22}\\end{array}\\right].$ Multiplying (on the left) both sides by $A^{-1}$ , we get $B\\,=\\,A^{-1}\\left\\lbrace I\\,+\\,\\delta C\\,+\\,(\\delta ^2)\\right\\rbrace .$ By Gershgorin circle theorem, all eigenvalues of the matrix in curly brackets in equation (REF ) lie within $(\\delta )$ distance of 1, thus it is invertible, and its inverse is given up to $\\big (\\delta ^2\\big )$ by $I-\\delta C$ .", "Therefore, $\\smash{A^{-1}\\,=B\\big (I-\\delta C+(\\delta ^2)\\big )\\,}$ , verifying the claim." ] ]
1403.0243
[ [ "Radial and vertical flows induced by galactic spiral arms: likely\n contributors to our \"wobbly Galaxy''" ], [ "Abstract In an equilibrium axisymmetric galactic disc, the mean galactocentric radial and vertical velocities are expected to be zero everywhere.", "In recent years, various large spectroscopic surveys have however shown that stars of the Milky Way disc exhibit non-zero mean velocities outside of the Galactic plane in both the Galactocentric radial and vertical velocity components.", "While radial velocity structures are commonly assumed to be associated with non-axisymmetric components of the potential such as spiral arms or bars, non-zero vertical velocity structures are usually attributed to excitations by external sources such as a passing satellite galaxy or a small dark matter substructure crossing the Galactic disc.", "Here, we use a three-dimensional test-particle simulation to show that the global stellar response to a spiral perturbation induces both a radial velocity flow and non-zero vertical motions.", "The resulting structure of the mean velocity field is qualitatively similar to what is observed across the Milky Way disc.", "We show that such a pattern also naturally emerges from an analytic toy model based on linearized Euler equations.", "We conclude that an external perturbation of the disc might not be a requirement to explain all of the observed structures in the vertical velocity of stars across the Galactic disc.", "Non-axisymmetric internal perturbations can also be the source of the observed mean velocity patterns." ], [ "Introduction", "The Milky Way has long been known to possess spiral structure, but studying the nature and the dynamical effects of this structure has proven to be elusive for decades.", "Even though its fundamental nature is still under debate today, it has nevertheless started to be recently considered as a key player in galactic dynamics and evolution (e.g., Antoja et al.", "2009; Quillen et al.", "2011; Lépine et al.", "2011; Minchev et al.", "2012; Roskar et al.", "2012 for recent works, or Sellwood 2013 for a review).", "However, zeroth order dynamical models of the Galaxy still mostly rely on the assumptions of a smooth time-independent and axisymmetric gravitational potential.", "For instance, recent determinations of the circular velocity at the Sun's position and of the peculiar motion of the Sun itself all rely on the assumption of axisymmetry and on minimizing the non-axisymmetric residuals in the velocity field (Reid et al.", "2009; McMillan & Binney 2010; Bovy et al.", "2012; Schönrich 2012).", "Such zeroth order assumptions are handy since they allow us to develop dynamical models based on a phase-space distribution function depending only on three isolating integrals of motion, such as the action integrals (e.g., Binney 2013; Bovy & Rix 2013).", "Actually, an action-based approach does not necessarily have to rely on the axisymmetric assumption, as it is also possible to take into account the main non-axisymmetric component (e.g., the bar, see Kaasalainen & Binney 1994) by modelling the system in its rotating frame (e.g., Kaasalainen 1995).", "However the other non-axisymmetric components such as spiral arms rotating with a different pattern speed should then nevertheless be treated through perturbations (e.g., Kaasalainen 1994; McMillan 2013).", "The main problem with such current determinations of Galactic parameters, through zeroth order axisymmetric models, is that it is not clear that assuming axisymmetry and dynamical equilibrium to fit a benchmark model does not bias the results, by e.g.", "forcing this benchmark model to fit non-axisymmetric features in the observations that are not present in the axisymmetric model itself.", "This means that the residuals from the fitted model are not necessarily representative of the true amplitude of non-axisymmetric motions.", "In this respect, it is thus extremely useful to explore the full range of possible effects of non-axisymmetric features such as spiral arms in both fully controlled test-particle simulations as well as self-consistent simulations, and to compare these with observations.", "With the advent of spectroscopic and astrometric surveys, observational phase-space information for stars in an increasingly large volume around the Sun have allowed us to see more and more of these dynamical effect of non-axisymmetric components emerge in the data.", "Until recently, the most striking features were found in the solar neighbourhood in the form of moving groups, i.e.", "local velocity-space substructures shown to be made of stars of very different ages and chemical compositions (e.g., Chereul et al.", "1998, 1999; Dehnen 1998; Famaey et al.", "2005, 2007, 2008; Pompéia et al. 2011).", "Various non-axisymmetric models have been argued to be able to represent these velocity structures equally well, using transient (e.g., De Simone et al.", "2004) or quasi-static spirals (e.g., Quillen & Minchev 2005; Antoja et al.", "2011), with or without the help of the outer Lindblad resonance from the central bar (e.g., Dehnen 2000; Antoja et al.", "2009; Minchev et al.", "2010; McMillan 2013; Monari et al.", "2013).", "The effects of non-axisymmetric components have also been analyzed a bit less locally by Taylor expanding to first order the planar velocity field in the cartesian frame of the Local Standard of Rest, i.e.", "measuring the Oort constants $A$ , $B$ , $C$ and $K$ (Kuijken & Tremaine 1994; Olling & Dehnen 2003), a procedure valid up to distances of less than 2 kpc.", "While old data were compatible with the axisymmetric values $C=K=0$ (Kuijken & Tremaine 1994), a more recent analysis of ACT/Tycho2 proper motions of red giants yielded $C = -10 \\, {\\rm km}\\,{\\rm s}^{-1}\\,{\\rm kpc}^{-1}$ (Olling & Dehnen 2003).", "Using line-of-sight velocities of 213713 stars from the RAVE survey (Steinmetz et al.", "2006; Zwitter et al.", "2008; Siebert et al.", "2011a; Kordopatis et al.", "2013), with distances $d<2 \\,$ kpc in the longitude interval $-140^\\circ < l < 10^\\circ $ , Siebert et al.", "(2011b) confirmed this value of $C$ , and estimated a value of $K= +6\\,{\\rm km}\\,{\\rm s}^{-1}\\,{\\rm kpc}^{-1}$ , implying a Galactocentric radial velocityIn this paper, 'radial velocity' refers to the Galactocentric radial velocity, not to be confused with the line-of-sight (l.o.s.)", "velocity.", "gradient of $C+K = \\partial V_R/ \\partial R \\simeq - 4\\,{\\rm km}\\,{\\rm s}^{-1}\\,{\\rm kpc}^{-1}$ in the solar suburb (extended solar neighbourhood, see also Williams et al.", "2013).", "The projection onto the plane of the mean line-of-sight velocity as a function of distance towards the Galactic centre ($|l|<5^\\circ $ ) was also examined by Siebert et al.", "(2011b) both for the full RAVE sample and for red clump candidates (with an independent method of distance estimation), and clearly confirmed that the RAVE data are not compatible with a purely axisymmetric rotating disc.", "This result is not owing to systematic distance errors as considered in Binney et al.", "(2013), because the geometry of the radial velocity flow cannot be reproduced by systematic distance errors alone (Siebert et al.", "2011b; Binney et al. 2013).", "Assuming, to first order, that the observed radial velocity map in the solar suburb is representative of what would happen in a razor-thin disc, and that the spiral arms are long-lived, Siebert et al.", "(2012) applied the classical density wave description of spiral arms (Lin & Shu 1964; Binney & Tremaine 2008) to constrain their parameters in the Milky Way.", "They found that the best-fit was obtained for a two-armed perturbation with an amplitude corresponding to $\\sim 15$ % of the background density and a pattern speed $\\Omega _P \\simeq 19 \\,{\\rm Gyr}^{-1}$ , with the Sun close to the 4:1 inner ultra-harmonic resonance (IUHR).", "This result is in agreement with studies based on the location of moving groups in local velocity space (Quillen & Minchev 2005; Antoja et al.", "2011; Pompéia et al. 2011).", "This study was advocated to be a useful first order benchmark model to then study the effect of spirals in three dimensions.", "In three dimensions, observations of the solar suburb from recent spectroscopic surveys actually look even more complicated.", "Using the same red clump giants from RAVE, it was shown that the mean vertical velocity was also non-zero and showed clear structure suggestive of a wave-like behaviour (Williams et al. 2013).", "Measurements of line-of-sight velocities for 11000 stars with SEGUE also revealed that the mean vertical motion of stars reaches up to 10 km/s at heights of 1.5 kpc (Widrow et al.", "2012), echoing previous similar results by Smith et al.", "(2012).", "This is accompanied by a significant wave-like North-South asymmetry in SDSS (Widrow et al.", "2012; Yanny & Gardner 2013).", "Observations from LAMOST in the outer Galactic disc (within 2 kpc outside the Solar radius and 2 kpc above and below the Galactic plane) also recently revealed (Carlin et al.", "2013) that stars above the plane exhibit a net outward motion with downward mean vertical velocities, whilst stars below the plane exhibit the opposite behaviour in terms of vertical velocities (moving upwards, i.e.", "towards the plane too), but not so much in terms of radial velocities, although slight differences are also noted.", "There is thus a growing body of evidence that Milky Way disc stars exhibit velocity structures across the Galactic plane in both the Galactocentric radial and vertical components.", "While a global radial velocity gradient such as that found in Siebert et al.", "(2011b) can naturally be explained with non-axisymmetric components of the potential such as spiral arms, such an explanation is a priori less self-evident for vertical velocity structures.", "For instance, it was recently shown that the central bar cannot produce such vertical features in the solar suburb (Monari et al.", "2014).", "For this reason, such non-zero vertical motions are generally attributed to vertical excitations of the disc by external means such as a passing satellite galaxy (Widrow et al. 2012).", "The Sagittarius dwarf has been pinpointed as a likely culprit for creating these vertical density waves as it plunged through the Galactic disc (Gomez et al.", "2013), while other authors have argued that these could be due to interaction of the disc with small starless dark matter subhalos (Feldmann & Spolyar 2013).", "Here, we rather investigate whether such vertical velocity structures can be expected as the response to disc non-axisymmetries, especially spiral arms, in the absence of external perturbations.", "As a first step in this direction, we propose to qualitatively investigate the response of a typical old thin disc stellar population to a spiral perturbation in controlled test particle orbit integrations.", "Such test-particle simulations have revealed useful in 2D to understand the effects of non-axisymmetries and their resonances on the disc stellar velocity field, including moving groups (e.g., Antoja et al.", "2009, 2011; Pompéia et al.", "2011), Oort constants (e.g., Minchev et al.", "2007), radial migrations (e.g., Minchev & Famaey 2010), or the dip of stellar density around corotation (e.g., Barros et al. 2013).", "Recent test-particle simulations in 3D have rather concentrated on the effects of the central bar (Monari et al.", "2013, 2014), while we concentrate here on the effect of spiral arms, with special attention to mean vertical motions.", "In Sect.", "2, we give details on the model potential, the initial conditions and the simulation technique, while results are presented in Sect.", "3, and discussed in comparison with solutions of linearized Euler equations.", "Conclusions are drawn in Sect.", "4." ], [ "Model", "To pursue our goal, we use a standard test-particle method where orbits of massless particles are integrated in a time-varying potential.", "We start with an axisymmetric background potential representative of the Milky Way (Sect.", "2.1), and we adiabatically grow a spiral perturbation on it within $\\sim 3.5$  Gyr.", "Once settled, the spiral perturbation is kept at its full amplitude.", "This is not supposed to be representative of the actual complexity of spiral structure in real galaxies, where self-consistent simulations indicate that it is often coupled to a central bar and/or a transient nature with a lifetime of the order of only a few rotations.", "Nevertheless, it allows us to investigate the stable response to an old enough spiral perturbation ($\\sim 600 \\,$ Myr to $1 \\,$ Gyr in the self-consistent simulations of Minchev et al.", "2012).", "The adiabatic growth of this spiral structure is not meant to be realistic, as we are only interested in the orbital structure of the old thin disk test population once the perturbation is stable.", "We generate initial conditions for our test stellar population from a discrete realization of a realistic phase-space distribution function for the thin disc defined in integral-space (Sect 2.2), and integrate these initial conditions forward in time within a given time-evolving background+spiral potential (Sect. 2.3).", "We then analyze the mean velocity patterns seen in configuration space, both radially and vertically, and check whether such patterns are stable within the rotating frame of the spiral." ], [ "Axisymmetric background potential", "The axisymmetric part of the Galactic potential is taken to be Model I of Binney & Tremaine (2008).", "Its main parameters are summarized in Table 1 for convenience.", "The central bulge has a truncated power-law density of the form $\\rho _b(R,z) = \\rho _{b0} \\times \\left( \\frac{\\sqrt{R^2 + (z/q_b)^2}}{a_b} \\right)^{-\\alpha _b} {\\rm exp}\\left( -\\frac{R^2+(z/q_b)^2}{r_b^2} \\right)$ where $R$ is the Galactocentric radius within the midplane, $z$ the height above the plane, $\\rho _{b0}$ the central density, $r_b$ the truncation radius, and $q_b$ the flattening.", "The total mass of the bulge is $M_b = 5.18\\times 10^9 {\\rm M}_\\odot $ .", "The stellar disc is a sum of two exponential profiles (for the thin and thick discs): $\\rho _d(R,z) = \\Sigma _{d0} \\times \\left( \\sum _{i=1}^{i=2} \\frac{\\alpha _{d,i}}{2z_{d,i}} {\\rm exp}\\left(-\\frac{|z|}{z_{d,i}}\\right) \\right) {\\rm exp}\\left(-\\frac{R}{R_d}\\right)$ where $\\Sigma _{d0}$ is the central surface density, $\\alpha _{d,1}$ and $\\alpha _{d,2}$ the relative contributions of the thin and thick discs, $z_{d,1}$ and $z_{d,2}$ their respective scale-heights, and $R_d$ the scale-length.", "The total mass of the disc is $M_d = 5.13 \\times 10^{10}{\\rm M}_\\odot $ .The disc potential also includes a contribution from the interstellar medium of the form $\\rho _g(R,z) = \\frac{\\Sigma _g}{2z_g} \\times {\\rm exp}\\left(-\\frac{R}{R_g} -\\frac{R_m}{R} - \\frac{|z|}{z_g}\\right)$ where $R_m$ is the radius within which there is a hole close to the bulge region, $R_g$ is the scale-length, $z_g$ the scale-height, and $\\Sigma _g$ is such that it contributes to 25% of the disc surface density at the galactocentric radius of the Sun.", "Finally, the dark halo is represented by an oblate two-power-law model with flattening $q_h$ , of the form $\\rho _{h}(R,z)&=&\\rho _{h0} \\times \\left( \\frac{\\sqrt{R^2 + (z/q_h)^2}}{a_h} \\right)^{-\\alpha _h}\\times \\nonumber \\\\&&\\left( 1 + \\frac{\\sqrt{R^2 + (z/q_h)^2}}{a_h} \\right)^{\\alpha _h - \\beta _h}.$ Table: Parameters of the axisymmetric background model potential (Binney & Tremaine 2008)The potential is calculcated using the GalPot routine (Dehnen & Binney 1998).", "The rotation curve corresponding to this background axisymmetric potential is displayed in Fig.", "REF .", "For radii smaller than 11 kpc, the total rotation curve (black line) is mostely infuenced by the disc (blue dashed line) and above by the halo (red dotted line).", "Figure: Rotation curve corresponding to the background axisymmetric potential" ], [ "Initial conditions", "The initial conditions for the test stellar population are set from a discrete realization of a phase-space distribution function (Shu 1969, Bienaymé & Séchaud 1997) which can be written in integral space as: $f(E_R,L_z,E_z)=\\frac{\\Omega \\, \\rho _d}{\\sqrt{2} \\kappa \\pi ^{\\frac{3}{2}} \\sigma ^2_R \\sigma _z} \\exp \\left( \\frac{-(E_R-E_c)}{\\sigma ^2_R}-\\frac{E_z}{\\sigma ^2_z} \\right)$ in which the angular velocity $\\Omega $ , the radial epicyclic frequency $\\kappa $ and the disc density in the plane $\\rho _d$ are all functions of $L_z$ , being taken at the radius $R_c(L_z)$ of a circular orbit of angular momentum $L_z$ .", "The scale-length of the disc is taken to be 2 kpc as for the background potential.", "The energy $E_c(L_z)$ is the energy of the circular orbit of angular momentum $L_z$ at the radius $R_c$ .", "Finally, the radial and vertical dispersions $\\sigma ^2_R$ and $\\sigma ^2_z$ are also function of $L_z$ and are expressed as: $\\sigma ^2_R=\\sigma ^2_{R_\\odot }\\exp \\left( \\frac{2R_{\\odot }-2R_c}{R_{\\sigma _R}}\\right),$ $\\sigma ^2_z=\\sigma ^2_{z_\\odot }\\exp \\left( \\frac{2R_{\\odot }-2R_c}{R_{\\sigma _z}}\\right)$ where $R_{\\sigma _R}/R_d = R_{\\sigma _z}/R_d = 5$ .", "The initial velocity dispersions thus decline exponentially with radius but at each radius, it is isothermal as a function of height.", "These initial values are set in such a way as to be representative of the old thin disc of the Milky Way after the response to the spiral perturbation.", "Indeed, the old thin disc is the test population we want to investigate the response of.", "From this distribution function, $4\\times 10^7$ test particle initial conditions are generated in a 3D polar grid between $R=4$  kpc and $R=15$  kpc (see Fig.", "REF ).", "This allows a good resolution in the solar suburb.", "Before adding the spiral perturbation, the simulation is run in the axisymmetric potential for two rotations ($\\sim 500$  Myr), and is indeed stable.", "Figure: Initial conditions.", "Left panel: Number of stars per kpc 2 {\\rm kpc}^2 (surface density) within the Galactic plane as a function of RR.", "Right panel: Stellar density as a function of zz at R=8R=8 kpc.Figure: Positions of the main radial resonances of the spiralpotential.", "Ω(R)=v c (R)/r\\Omega (R ) = v_c(R)/r is the local circular frequency, andv c (R)v_c(R ) is the circular velocity.", "The 2:12:1 ILR occurs along the curveΩ(R)-κ/2\\Omega (R ) - \\kappa /2, where κ\\kappa is the local radial epicyclicfrequency.", "The inner 4:14:1 IUHR occurs along the curve Ω(R)-κ/4\\Omega (R ) -\\kappa /4.Figure: Positions of the 4:1, 6:1 and 8:1 vertical resonances.", "When Ω-Ω P =ν/n\\Omega - \\Omega _P = \\nu /n, where ν\\nu is the vertical epicyclic frequency, thestar makes precisely nn vertical oscillations along one rotation withinthe rotating frame of the spiral." ], [ "Spiral perturbation and orbit integration", "In 3D, we consider a spiral arm perturbation of the Lin-Shu type (Lin & Shu 1964; see also Siebert et al.", "2012) with a sech$^2$ vertical profile (a pattern that can be supported by three-dimensional periodic orbits, see e.g.", "Patsis & Grosbøl 1996) and a small ($\\sim 100$  pc) scale-height: $\\Phi _{s}(R,\\theta ,z)=-A \\cos \\left[m\\left( \\Omega _P t -\\theta +\\frac{\\ln (R)}{\\tan p}\\right) \\right] {\\rm sech}^2 \\left(\\frac{z}{z_0} \\right)$ in which $A$ is the amplitude of the perturbation, $m$ is the spiral pattern mode ($m=2$ for a 2-armed spiral), $\\Omega _P$ is the pattern speed, $p$ the pitch angle, and $z_0$ is the spiral scale-height.", "The edge-on shapes of orbits of these thick spirals are determined by the vertical resonances existing in the potential.", "The parameters of the spiral potential used in our simulation are inspired by the analytic solution found in Siebert et al.", "(2012) using the classical 2D Lin-Shu formalism to fit the radial velocity gradient observed with RAVE (Siebert et al.", "2011b).", "The parameters used here are summarized in Table 2.", "The amplitude $A$ which we use corresponds to 1% of the background axisymmetric potential at the Solar radius (3% of the disc potential).", "The positions of the main radial resonances, i.e.", "the 2:1 inner Lindblad resonance (ILR) and 4:1 IUHR, are illustrated in Fig.", "REF .", "The presence of the 4:1 IUHR close to the Sun is responsible for the presence of the Hyades and Sirius moving groups in the local velocity space at the Solar radius (see Pompéia et al.", "2011), associated to square-shaped resonant orbital families in the rotating spiral frame.", "Vertical resonances are also displayed in Fig.", "REF .", "Such a spiral perturbation can grow naturally in self-consistent simulations of isolated discs without the help of any external perturber (e.g.", "Minchev et al.", "2012).", "As we are interested hereafter in the global response of the thin disc stellar population to a quasi-static spiral perturbation, we make sure to grow the perturbation adiabatically by multiplying the above potential perturbation by a growth factor starting at $t\\approx 0.5$  Gyr and finishing at $t\\approx 3.5$  Gyr: $\\epsilon (t)=\\frac{1}{2}(\\tanh (1.7\\times t-3.4)+1)$ .", "The integration of orbits is performed using a fourth order Runge-Kutta algorithm run on Graphics Processing Units (GPUs).", "The growth of this spiral is not meant to be realistic, as we are only interested in the orbital structure of the old thin disk once the perturbation is stable.", "Table: Parameters of the spiral potential and location of the main resonancesFigure: Left panel: Histogram of galactocentric radial velocities as afunction of time.", "Right panel: time evolution of the σ R (R)\\sigma _R(R) profileaveraged over all azimuths.", "The colour-scale indicates the time-steps inGyr.Figure: Top-left panel: Mean galactocentric radial velocity 〈v R 〉\\langle v_R\\rangle as a function of position in the Galactic plane soon after theadiabatic growth of the spiral (t=4t=4 \\,Gyr).", "Isocontours of the spiralpotential are overplotted, corresponding to 80% of the minimum of theperturber potential, and thus delimiting the region where the spiralpotential is between 80% and 100% of its minimum (or maximum in absolutevalue).", "Top right: Same at t=5t=5 \\,Gyr.", "Everything is plotted here withinthe rotating frame of the spiral, so that the spiral does not move fromone snapshot to the other.", "Bottom left: Same at t=6t=6 \\,Gyr.", "Bottom right:Same at the final time-step t=6.5t=6.5 \\,Gyr.Figure: Galactocentric radial velocities in the solar suburb, centered at(R,θ)=(8 kpc ,26 ∘ )(R,\\theta )=(8\\, {\\rm kpc}, 26^\\circ ) at t=4t=4 \\,Gyr.", "On this plot, theSun is centered on (x,y)=(0,0)(x,y)=(0,0), positive xx indicates the direction ofthe Galactic centre, and positive yy the direction of galactic rotation(as well as the sense of rotation of the spiral pattern).", "The spiralpotential contours overplotted (same as on Fig.", ", delimiting the region where the spiralpotential is between 80% and 100% of its absolute maximum) wouldcorrespond to the location of the Perseus spiral arm in the outerGalaxy.", "This Figure can be qualitatively compared to Fig.", "4 of Siebert etal.", "(2011b) and Fig.", "3 of Siebert et al.", "(2012)." ], [ "Radial velocity flow", "The histogram of individual galactocentric radial velocities, $v_R$ , as well as the time-evolution of the radial velocity dispersion profile starting from $t=3.5 \\,$ Gyr (once the steady spiral pattern is settled) are plotted on Fig.", "REF .", "It can be seen that these are reasonably stable, and that the mean radial motion of stars is very close to zero (albeit slightly positive).", "Our test population is thus almost in perfect equilibrium.", "However, due to the presence of spiral arms, the mean galactocentric radial velocity $\\langle v_R \\rangle $ of our test population is non-zero at given positions within the frame of the spiral arms.", "The map of $\\langle v_R\\rangle $ as a function of position in the plane is plotted on Fig.", "REF , for different time-steps (4 Gyr, 5 Gyr, 6 Gyr and 6.5 Gyr).", "Within the rotating frame of the spiral pattern, the locations of these non-zero mean radial velocities are stable over time: this means that the response to the spiral perturbation is stable, even though the amplitude of the non-zero velocities might slightly decrease with time.", "Within corotation, the mean $\\langle v_R \\rangle $ is negative within the arms (mean radial motion towards the Galactic centre) and positive (radial motion towards the anticentre) between the arms.", "Outside corotation, the pattern is reversed.", "This is exactly what is expected from the Lin-Shu density wave theory (see, e.g., Eq.", "3 in Siebert et al.", "2012).", "If we place the Sun at $(R,\\theta )=(8\\, {\\rm kpc}, 26^\\circ )$ in the frame of the spiral, we can plot the expected radial velocity field in the Solar suburb (Fig.", "REF ).", "We see that the galactocentric radial velocity is positive in the inner Galaxy, as observed by Siebert et al.", "(2011b), because the inner Galaxy in the local suburb corresponds to an inter-arm region located within the corotation of the spiral.", "Observations towards the outer arm (which should correspond to the Perseus arm in the Milky Way) should reveal negative galactocentric radial velocities.", "An important aspect of the present study is the behaviour of the response to a spiral perturbation away from the Galactic plane.", "The spiral perturbation of the potential is very thin in our model ($z_0 = 100 \\,$ pc) but as we can see on Figs.", "REF and REF , the radial velocity flow is not varying much as a function of $z$ up to five times the scale-height of the spiral perturber.", "This justifies the assumption made in Siebert et al.", "(2012) that the flow observed at $\\sim 500 \\,$ pc above the plane was representative of what was happening in the plane.", "Nevertheless, above these heights, the trend seems to be reversed, probably due to the higher eccentrities of stars, corresponding to different guiding radii.", "This could potentially provide a useful observational constraint on the scale height of the spiral potential, a test that could be conducted with the forthcoming surveys.", "Figure: Top left panel: Mean galactocentric radial velocity at t=4t=4 \\,Gyrin the meridional (R,z)(R,z)-plane for 21 ∘ <θ<31 ∘ 21^\\circ <\\theta <31^\\circ (within theframe of the spiral).", "The white line indicates the location of spiralarms, in terms of overdensities and underdensities generating the spiralpotential (normalized -Φ s -\\Phi _s, i.e.", "spiral arms are located at the peaks).", "Top right: Same at t=5t=5\\,Gyr.", "Bottom left: Same at t=6t=6 \\,Gyr.", "Bottom right: Same at the finaltime-step t=6.5t=6.5 \\,Gyr.Figure: Same as Fig.", ", but for six different azimuths at afixed time (t=6t=6 \\,Gyr)." ], [ "Non-zero mean vertical motions", "If we now turn our attention to the vertical motion of stars, we see on Fig.", "REF that the total mean vertical motion of stars remains zero at all times, but that there is still a slight, but reasonable, vertical heating going on in the inner Galaxy.", "What is most interesting is to concentrate on the mean vertical motion $\\langle v_z \\rangle $ as a function of position above or below the Galactic disc.", "As can be seen on Fig.", "REF and Fig.", "REF , while the vertical velocities are generally close to zero right within the plane, they are non-zero outside of it.", "At a given azimuth within the frame of the spiral, these non-zero vertical velocity patterns are extremely stable over time (Fig.", "REF ).", "Within corotation the mean vertical motion is directed away from the plane at the outer edge of the arm and towards the plane at the inner edge of the arm.", "The pattern of $\\langle v_z \\rangle $ above and below the plane are thus mirror-images, and the direction of the mean motion changes roughly in the middle of the interarm region.", "This produces diagonal features in terms of isocontours of a given $\\langle v_z \\rangle $ , corresponding precisely to the observation using RAVE by Williams et al.", "(2013, see especially their Fig.", "13), where the change of sign of $\\langle v_z \\rangle $ precisely occurs in between the Perseus and Scutum main arms.", "Our simulation predicts that the $\\langle v_z \\rangle $ pattern is reversed outside of corotation (beyond 12 kpc), where stars move towards the plane on the outer edge of the arm (rather than moving away from the plane): this can indeed be seen, e.g., on the right panel of the second row of our Fig.", "REF .", "If we now combine the information on $\\langle v_R \\rangle $ and $\\langle v_z \\rangle $ , we can plot the global meridional velocity flow $\\vec{\\langle v \\rangle } = \\langle v_R \\rangle \\vec{1}_R + \\langle v_z \\rangle \\vec{1}_z$ on Fig.", "REF and Fig.", "REF .", "The picture that emerges is the following: in the interarm regions located within corotation, stars move on average from the inner arm to the outer arm by going outside of the plane, and then coming back towards the plane at mid-distance between the two arms, to finally arrive back on the inner edge of the outer arm.", "For each azimuth, there are thus “source” points, preferentially on the outer edge of the arms (inside corotation, whilst on the inner edge outside corotation), out of which the mean velocity vector flows, while there are “sink” points, preferentially on the inner edge of the arms (inside corotation), towards which the mean velocity flows.", "This supports the interpretation of the observed RAVE velocity field of Williams et al.", "(2013) as “compression/rarefaction” waves." ], [ "Interpretation from linearized Euler equations", "In order to understand these features found in the meridional velocity flow of our test-particle simulation, we now turn to the fluid approximation based on linearized Euler equations, developed, e.g., in Binney & Tremaine (2008, Sect. 6.2).", "A rigorous analytical treatment of a quasi-static spiral perturbation in a three-dimensional stellar disk should rely on the linearized Boltzmann equations, which we plan to do in full in a forthcoming paper, but the fluid approximation can already give important insights on the shape of the velocity flow expected in the meridional plane.", "In the full Boltzmann-based treatment, the velocity flow will be tempered by reduction factors both in the radial (see, e.g., Binney & Tremaine 2008, Appendix K) and vertical directions.", "Let us rewrite our perturber potential of Eq.", "REF as $\\Phi _s= \\mathbf {Re} \\lbrace \\Phi _a(R,z) \\exp [i \\, m ( \\Omega _P t - \\theta )]\\rbrace $ with $\\Phi _a = - A \\, {\\mathrm {s}ech}^2 \\left(\\frac{z}{z_0} \\right) \\exp \\Big (i \\frac{m \\ln (R)}{\\tan p} \\Big ).$ Then if we write solutions to the linearized Euler equations for the response of a cold fluid as $\\left\\lbrace \\begin{array}{l}v_{Rs} = \\mathbf {Re} \\lbrace v_{Ra}(R,z) \\exp [i \\, m ( \\Omega _P t - \\theta )]\\rbrace \\\\\\\\v_{zs} = \\mathbf {Re} \\lbrace v_{za}(R,z)) \\exp [i \\, m ( \\Omega _P t - \\theta )]\\rbrace \\\\\\end{array}\\right.$ we find, following the same steps as in Binney & Tremaine (2008, Sect.", "6.2) $\\left\\lbrace \\begin{array}{l l}v_{Ra}=& - \\frac{ m (\\Omega - \\Omega _P)}{\\Delta } k \\Phi _a \\\\ &+ i \\frac{2 \\Phi _a}{\\Delta } \\left( \\frac{2 \\Omega {\\rm tanh}(z/z_0)}{m (\\Omega - \\Omega _P) z_0} + \\frac{m \\Omega }{R} \\right)\\\\\\\\v_{za} =& - \\frac{ 2 i}{m (\\Omega -\\Omega _P) z_0} {\\rm tanh}\\Big ( \\frac{z}{z_0}\\Big ) \\Phi _a \\\\\\end{array}\\right.$ where $k=m/(R \\, {\\rm tan \\,}p)$ is the radial wavenumber and $\\Delta = \\kappa ^2 - m^2(\\Omega -\\Omega _P)^2$ .", "If we plot these solutions for $v_{Rs}$ and $v_{zs}$ at a given angle (for instance $\\theta =30^\\circ $ ) we get the same pattern as in the simulation (Fig.", "REF ).", "Of course, the velocity flow plotted on Fig.", "REF would in fact be damped by a reduction factor depending on both radial and vertical velocity dispersions when treating the full linearized Boltzmann equation, which will be the topic of a forthcoming paper.", "Nevertheless, this qualitative consistency between analytical results and our simulations is an indication that the velocity pattern observed by Williams et al.", "(2013) is likely linked to the potential perturbation by spiral arms.", "Interestingly, this analytical model also predicts that the radial velocity gradient should become noticeably North/South asymmetric close to corotation." ], [ "Discussion and conclusions", "In recent years, various large spectroscopic surveys have shown that stars of the Milky Way disc exhibit non-zero mean velocities outside of the Galactic plane in both the Galactocentric radial component and vertical component of the mean velocity field (e.g., Siebert et al.", "2011b; Williams et al.", "2013; Carlin et al.", "2013).", "While it is clear that such a behaviour could be due to a large combination of factors, we investigated here whether spiral arms are able to play a role in these observed patterns.", "For this purpose, we investigated the orbital response of a test population of stars representative of the old thin disc to a stable spiral perturbation.", "This is done using a test-particle simulation with a background potential representative of the Milky Way.", "We found non-zero velocities both in the Galactocentric radial and vertical velocity components.", "Within the rotating frame of the spiral pattern, the location of these non-zero mean velocities in both components are stable over time, meaning that the response to the spiral perturbation is stable.", "Within corotation, the mean $\\langle v_R \\rangle $ is negative within the arms (mean radial motion towards the Galactic centre) and positive (radial motion towards the anticentre) between the arms.", "Outside corotation, the pattern is reversed, as expected from the Lin-Shu density wave theory (Lin & Shu 1964).", "On the other hand, even though the spiral perturbation of the potential is very thin, the radial velocity flow is still strongly affected above the Galactic plane.", "Up to five times the scale-height of the spiral potential, there are no strong asymmetries in terms of radial velocity, but above these heights, the trend in the radial velocity flow is reversed.", "This means that asymmetries could be observed in surveys covering different volumes above and below the Galactic plane.", "Also, forthcoming surveys like Gaia, 4MOST, WEAVE will be able to map this region of the disc of the Milky Way and measure the height at which the reversal occurs.", "Provided this measurement is successful, it would give a measurement of the scale height of the spiral potential.", "In terms of vertical velocities, within corotation, the mean vertical motion is directed away from the plane at the outer edge of the arms and towards the plane at the inner edge of the arms.", "The patterns of $\\langle v_z\\rangle $ above and below the plane are thus mirror-images (see e.g.", "Carlin et al.", "2013).", "The direction of the mean vertical motion changes roughly in the middle of the interam region.", "This produces diagonal features in terms of isocontours of a given $\\langle v_z \\rangle $ , as observed by Williams et al.", "(2013).", "The picture that emerges from our simulation is one of “source” points of the velocity flow in the meridional plane, preferentially on the outer edge of the arms (inside corotation, whilst on the inner edge outside corotation), and of “sink” points, preferentially on the inner edge of the arms (inside corotation), towards which the mean velocity flows.", "We have then shown that this qualitative structure of the mean velocity field is also the behaviour of the analytic solution to linearized Euler equations for a toy model of a cold fluid in response to a spiral perturbation.", "In a more realistic analytic model, this fluid velocity would in fact be damped by a reduction factor depending on both radial and vertical velocity dispersions when treating the full linearized Boltzmann equation.", "In a next step, the features found in the present test-particle simulations will also be checked for in fully self-consistent simulations with transient spiral arms, to check whether non-zero mean vertical motions as found here are indeed generic.", "The response of the gravitational potential itself to these non-zero motions should also have an influence on the long-term evolution of the velocity patterns found here, in the form of e.g.", "bending and corrugation waves.", "The effects of multiple spiral patterns (e.g., Quillen et al.", "2011) and of the bar (e.g., Monari et al.", "2013, 2014) should also have an influence on the global velocity fiel and on its amplitude.", "Once all these different dynamical effects and their combination will be fully understood, a full quantitative comparison with present and future datasets in 3D will be the next step.", "The present work on the orbital response of the thin disc to a small spiral perturbation by no means implies that no external perturbation of the Milky Way disc happened in the recent past, by e.g.", "the Sagittarius dwarf (e.g., Gomez et al.", "2013).", "Such a perturbation could of course be responsible for parts of the velocity structures observed in various recent large spectrosocpic surveys.", "For instance, concerning the important north-south asymmetry spotted in stellar densities at relatively large heights above the disc, spiral arms are less likely to play an important role.", "Nevertheless, any external perturbation will also excite a spiral wave, so that understanding the dynamics of spirals is also fundamental to understanding the effects of an external perturber.", "The qualitative similarity between our simulation (e.g., Fig.", "REF ), as well as our analytical estimates for the fluid approximation (Fig.", "REF ), and the velocity pattern observed by Williams et al.", "(2013, their Fig.", "13) indicates that spiral arms are likely to play a non-negligible role in the observed velocity pattern of our “wobbly Galaxy”." ] ]
1403.0587
[ [ "L\\'evy flights in inhomogeneous environments and 1/f noise" ], [ "Abstract Complex dynamical systems which are governed by anomalous diffusion often can be described by Langevin equations driven by L\\'evy stable noise.", "In this article we generalize nonlinear stochastic differential equations driven by Gaussian noise and generating signals with 1/f power spectral density by replacing the Gaussian noise with a more general L\\'evy stable noise.", "The equations with the Gaussian noise arise as a special case when the index of stability alpha=2.", "We expect that this generalization may be useful for describing 1/f fluctuations in the systems subjected to L\\'evy stable noise." ], [ "Introduction", "The Lévy $\\alpha $ -stable distributions, characterized by the index of stability $0<\\alpha \\leqslant 2$ , constitute the most general class of stable processes.", "The Gaussian distribution is their special case, corresponding to $\\alpha =2$ .", "If $\\alpha <2$ , the Lévy stable distributions have power-law tails $\\sim 1/x^{1+\\alpha }$ .", "There are many systems exhibiting Lévy $\\alpha $ -stable distributions: distribution function of turbulent magnetized plasma emitters [1] and step-size distribution of photons in hot vapors of atoms [2] have Lévy tails; theoretical models suggest that velocity distribution of particles in fractal turbulence is Lévy stable distribution [3] or at least has Lévy tails [4].", "If system behavior depends only on large noise fluctuations, such noise intensity distributions can by approximated by Lévy stable distribution, leading to Lévy flights.", "Lévy flight is a generalization of the Brownian motion which describes the motion of small macroscopic particles in a liquid or a gas experiencing unbalanced bombardments due to surrounding atoms.", "The Brownian motion mimics the influence of the “bath” of surrounding molecules in terms of time-dependent stochastic force which is commonly assumed to be white Gaussian noise.", "That postulate is compatible with the assumption of a short correlation time of fluctuations, much shorter than the time scale of the macroscopic motion, and the assumption of weak interactions with the bath.", "In contrast, the Lévy motions describe results of strong collisions between the particle and the surrounding environment.", "Lévy flights can be found in many physical systems: as an example we can point out anomalous diffusion of Na adatoms on solid Cu surface [5], anomalous diffusion of a gold nanocrystal, adsorbed on the basal plane of graphite [6] and anomalous diffusion in optical lattices [7].", "Lévy flights can be modeled by fractional Fokker-Planck equations [8] or Langevin equations with additive Lévy stable noise.", "Nonlinear stochastic differential equations (SDEs) with additive Lévy stable noise have been explored quite extensive for past 15 years [9], [10], [11], [12].", "Such stochastic differential equations lead to fractional Fokker-Planck equations with constant diffusion coefficient.", "Models with multiplicative Lévy stable noise have been used for modeling inhomogeneous media [13], ecological population density with fluctuating volume of resources [14].", "The relation between Langevin equation with multiplicative Lévy stable noise and fractional Fokker-Planck equation has been introduced in Ref.", "[15], where Langevin equation is interpreted in Itô sense [16].", "The relation between these two equation are not known in Stratonovich interpretation.", "Fractional Fokker-Planck equation models have been applied to model enzyme diffusion on polymer chain [17] and some cases of anomalous diffusion [18].", "However, application of Lévy stable noise driven SDEs can be problematic.", "We can always write Fokker-Planck equation corresponding to Langevin equation driven by Gaussian noise and vice versa, but such statement is not always true for Langevin equation with Lévy stable noise.", "For example, particle (enzyme) dispersion on rapidly folding random heteropolymer can be described by space fractional Fokker-Planck equation [19], but for such equation counterpart Langevin equation has not been found [20] and might not even exits [21].", "One of the characteristics of the signal is the power spectral density (PSD).", "Signals having the PSD at low frequencies $f$ of the form $S(f)\\sim 1/f^{\\beta }$ with $\\beta $ close to 1 are commonly referred to as “$1/f$ noise”, “$1/f$ fluctuations”, or “flicker noise”.", "Power-law distributions of spectra of signals with $0.5<\\beta <1.5$ , as well as scaling behavior are ubiquitous in physics and in many other fields [22], , , , , , .", "Despite the numerous models and theories proposed since its discovery more than 80 years ago [29], , the subject of $1/f$ noise remains still open for new discoveries.", "Most models and theories of $1/f$ noise are not universal because of the assumptions specific to the problem under consideration.", "A short categorization of the theories and models of $1/f$ noise is presented in the introduction of the paper [31], see also recent review by Balandin [32].", "Mostly $1/f$ noise is considered as Gaussian process [33], [34], but sometimes the signal exhibiting $1/f$ fluctuations are non-Gaussian [35], [36].", "Often $1/f$ noise is modeled as the superposition of Lorentzian spectra with a wide range distribution of relaxation times [37].", "An influential class of the models of $1/f$ noise involves self-organized criticality (SOC) [38], , .", "One more way of obtaining $1/f$ noise from a signal consisting of pulses has been presented in [41], , , : it has been shown that the intrinsic origin of $1/f$ noise may be a Brownian motion of the interevent time of the signal pulses.", "The nonlinear SDEs generating signals with $1/f$ noise were obtained in Refs.", "[45], [46] (see also papers [31], [47], ), starting from the point process model of $1/f$ noise.", "A special case of this SDE has been obtained using Kirman's agent model [49].", "Such nonlinear SDEs were used to describe signals in socio-economical systems [50], [51].", "The purpose of this paper is to generalize nonlinear SDEs driven by Gaussian noise and generating signals with $1/f$ PSD by replacing the Gaussian noise with a more general Lévy stable noise.", "The previously proposed SDEs then arise as a special case when $\\alpha =2$ .", "We can expect that this generalization may be useful for describing $1/f$ fluctuations in the systems subjected to Lévy stable noise.", "The paper is organized as follows: In Section  we search for the nonlinear SDE with Lévy stable noise yielding power law steady state probability density function (PDF) of the generated signal.", "In Section  we estimate when the signal generated by such an SDE has $1/f$ PSD in a wide region of frequencies.", "In Section  we numerically solve obtained equations and compare the PDF and PSD of the signal with analytical estimations.", "Section  summarizes our findings." ], [ "Stochastic differential equation with Lévy stable noise generating\nsignals with power law distribution", "In this Section we search for nonlinear SDEs with Lévy stable noise yielding power law steady state PDF of the generated signal.", "We consider the Langevin equation of the form [8], [52], [53] $\\frac{dx}{dt}=a(x)+b(x)\\xi (t)\\,,$ where $a(x)$ and $b(x)$ are given functions describing the deterministic drift term and the amplitude of the noise, respectively.", "The stochastic force $\\xi (t)$ is uncorrelated, $\\langle \\xi (t)\\xi (t^{\\prime })\\rangle =\\delta (t-t^{\\prime })$ and is characterized by Lévy $\\alpha $ -stable distribution.", "In this paper we will restrict our investigation only to symmetric stable distributions, thus the characteristic function of $\\xi (t)$ is $\\langle \\exp (ik\\xi )\\rangle =\\exp (-\\sigma ^{\\alpha }|k|^{\\alpha })\\,.$ Here $\\alpha $ is the index of stability and $\\sigma $ is the scale parameter.", "We interpret Eq.", "(REF ) in Itô sense.", "In mathematically more formal way Eq.", "(REF ) can be written in the form $dx=a(x)+b(x)dL_{t}^{\\alpha }\\,,$ where $dL_{t}^{\\alpha }$ stands for the increments of Lévy $\\alpha $ -stable motion $L_{t}^{\\alpha }$ [54], [55].", "For calculating of the steady state PDF of the signal $x$ we will use the fractional Fokker-Planck equation instead of stochastic differential equation (REF ).", "The fractional Fokker-Planck equation corresponding to Itô solution of Eq.", "(REF ) is [56], [15] $\\frac{\\partial }{\\partial t}P(x,t)=-\\frac{\\partial }{\\partial x}a(x)P(x,t)+\\sigma ^{\\alpha }\\frac{\\partial ^{\\alpha }}{\\partial |x|^{\\alpha }}b(x)^{\\alpha }P(x,t)\\,.$ Here $\\partial ^{\\alpha }/\\partial |x|^{\\alpha }$ is the Riesz-Weyl fractional derivative.", "The Riesz-Weyl fractional derivative of the function $f(x)$ is defined by its Fourier transform [57], $\\mathcal {F}\\left[\\frac{\\partial ^{\\alpha }}{\\partial |x|^{\\alpha }}f(x)\\right]=-|k|^{\\alpha }\\tilde{f}(k)\\,.$ One can get the following expression for the Riesz-Weyl derivative : $\\frac{\\partial ^{\\alpha }}{\\partial |x|^{\\alpha }}f(x)=-\\frac{1}{2\\cos \\left(\\frac{\\pi \\alpha }{2}\\right)}\\lbrace D_{+}^{-\\alpha }f(x)+D_{-}^{-\\alpha }f(x)\\rbrace \\,,$ where $D_{+}^{-\\alpha }$ and $D_{-}^{-\\alpha }$ are the left and right Riemann-Liouville derivatives [57]: $D_{\\pm }^{-\\alpha }=(\\pm 1)^{m}\\frac{d^{m}}{dx^{m}}D_{\\pm }^{m-\\alpha }\\,,\\qquad m-1<\\alpha <m\\,.$ Here $m$ is an integer and $D_{+}^{\\alpha }f(x) & = & \\frac{1}{\\Gamma (\\alpha )}\\int _{-\\infty }^{x}(x-z)^{\\alpha -1}f(z)\\, dz\\,,\\\\D_{-}^{\\alpha }f(x) & = & \\frac{1}{\\Gamma (\\alpha )}\\int _{x}^{+\\infty }(z-x)^{\\alpha -1}f(z)\\, dz\\,.$ When $\\alpha =1$ then the definition of the Riesz-Weyl derivative is $\\frac{d}{d|x|}f(x)=-\\frac{d}{dx}\\frac{1}{\\pi }\\int _{-\\infty }^{+\\infty }\\frac{f(z)}{x-z}\\, dz\\,.$ Eq.", "(REF ) leads to the following equation for the steady state PDF: $\\sigma ^{\\alpha }\\frac{\\partial ^{\\alpha }}{\\partial |x|^{\\alpha }}b(x)^{\\alpha }P_{0}(x)-\\frac{\\partial }{\\partial x}a(x)P_{0}(x)=0\\,.$ Equation (REF ) can be written as $-dJ(x)/dx=0$ , where $J(x)$ is the probability current.", "Reflective boundaries lead to the boundary condition $J(x)=0$ ." ], [ "Equation with only positive values of $x$", "We will search for the stochastic differential equation (REF ) generating signals with power law steady state PDF, $P_{0}(x)\\sim x^{-\\lambda }\\,.$ Since power law PDF cannot be normalized when $x$ can vary from zero to infinity, we will assume that the power law holds only in some wide region of $x$ , $x_{\\mathrm {min}}\\ll x\\ll x_{\\mathrm {max}}$ .", "One can expect that power law PDF can be obtained when the coefficients $a(x)$ and $b(x)$ in Eq.", "(REF ) themselves are of the power law form.", "Thus we will consider $b(x)=x^{\\eta }$ and $a(x)=\\sigma ^{\\alpha }\\gamma x^{\\mu }$ .", "Here $\\eta $ is the exponent of the multiplicative noise, $\\mu $ and $\\gamma $ are to be determined.", "With such a choice of $b(x)$ and power law form of $P_{0}(x)$ from Eq.", "(REF ) it follows that we need to calculate fractional derivative of the the power law function.", "Let us consider the function $f(x)={\\left\\lbrace \\begin{array}{ll}x^{\\rho }\\,, & x_{\\mathrm {min}}<x<x_{\\mathrm {max}}\\,,\\\\0 & \\mbox{otherwise .", "}\\end{array}\\right.", "}$ Using Eq.", "(REF ) we obtain the following approximate expressions for the fractional derivative of the function (REF ) when $x_{\\mathrm {min}}\\ll x\\ll x_{\\mathrm {max}}$ : $\\frac{d^{\\alpha }}{d|x|^{\\alpha }}f(x)\\approx {\\left\\lbrace \\begin{array}{ll}\\frac{\\sin \\left(\\pi \\left(\\frac{\\alpha }{2}-\\rho \\right)\\right)}{\\sin (\\pi (\\rho -\\alpha ))}\\frac{\\Gamma (1+\\rho )}{\\Gamma (1+\\rho -\\alpha )}x^{\\rho -\\alpha }\\,, & -1<\\rho <\\alpha \\,,\\\\\\frac{x_{\\mathrm {min}}^{1+\\rho }}{2\\cos \\left(\\frac{\\pi }{2}\\alpha \\right)(1+\\rho )\\Gamma (-\\alpha )}x^{-1-\\alpha }\\,, & \\rho <-1\\\\\\frac{x_{\\mathrm {max}}^{\\rho -\\alpha }}{2\\cos \\left(\\frac{\\pi }{2}\\alpha \\right)(\\alpha -\\rho )\\Gamma (-\\alpha )}\\,, & \\rho >\\alpha \\end{array}\\right.", "}\\,,\\qquad 0<\\alpha <2;\\,\\alpha \\ne 1$ and $\\frac{d}{d|x|}f(x)\\approx {\\left\\lbrace \\begin{array}{ll}-\\lambda \\cot (\\pi \\rho )x^{\\rho -1}\\,, & -1<\\rho <1\\\\-\\frac{x_{\\mathrm {min}}^{1+\\rho }}{\\pi (1+\\rho )}x^{-2}\\,, & \\rho <-1\\\\\\frac{x_{\\mathrm {max}}^{\\rho -1}}{\\pi (1-\\rho )}\\,, & \\rho >1\\end{array}\\right.", "}$ for $\\alpha =1$ .", "We see that the approximate expression for the fractional derivative does not depend on the limiting values $x_{\\mathrm {min}}$ and $x_{\\mathrm {max}}$ when $-1<\\rho <\\alpha $ .", "Using the power-law forms of the coefficients $a(x)$ and $b(x)$ , assuming that $-1<\\alpha \\eta -\\lambda <\\alpha $ and using Eq.", "(REF ) for the fractional derivative, from Eq.", "(REF ) we get $\\frac{\\sin \\left[\\pi \\left(\\frac{\\alpha }{2}-\\alpha \\eta +\\lambda \\right)\\right]}{\\sin [\\pi (\\alpha (\\eta -1)-\\lambda )]}\\frac{\\Gamma (1+\\alpha \\eta -\\lambda )}{\\Gamma (1+\\alpha (\\eta -1)-\\lambda )}x^{\\alpha (\\eta -1)-\\lambda }-\\gamma (\\mu -\\lambda )x^{\\mu -\\lambda -1}=0\\,.$ This equation should be valid for all values of $x$ .", "This can be only when $\\mu =\\alpha (\\eta -1)+1$ and $\\gamma =\\frac{\\sin \\left[\\pi \\left(\\frac{\\alpha }{2}-\\alpha \\eta +\\lambda \\right)\\right]}{\\sin [\\pi (\\alpha (\\eta -1)-\\lambda )]}\\frac{\\Gamma (\\alpha \\eta -\\lambda +1)}{\\Gamma (\\alpha (\\eta -1)-\\lambda +2)}\\,.$ Thus we will investigate the nonlinear SDE with Lévy stable noise of the form $dx=\\sigma ^{\\alpha }\\frac{\\sin \\left[\\pi \\left(\\frac{\\alpha }{2}-\\alpha \\eta +\\lambda \\right)\\right]}{\\sin [\\pi (\\alpha (\\eta -1)-\\lambda )]}\\frac{\\Gamma (\\alpha \\eta -\\lambda +1)}{\\Gamma (\\alpha (\\eta -1)-\\lambda +2)}x^{\\alpha (\\eta -1)+1}dt+x^{\\eta }dL_{t}^{\\alpha }\\,.$ This equation is a generalization of the nonlinear SDE with Gaussian noise proposed in Refs.", "[45], [46].", "Because of the divergence of the power law distribution and the requirement of the stationarity of the process, the SDE (REF ) should be analyzed together with the appropriate restrictions of the diffusion in some finite interval.", "The simplest choice of restriction is the reflective boundaries at $x=x_{\\mathrm {min}}$ and $x=x_{\\mathrm {max}}$ .", "However, other forms of restrictions are possible by introducing additional terms in the drift term of Eq.", "(REF ).", "From Eq.", "(REF ) it follows that the equation for the fractional derivative is valid when $-1<\\alpha \\eta -\\lambda <\\alpha $ .", "However, the condition $J(x)=0$ for the probability current leads to a stronger restriction than Eq.", "(REF ) which ensures only $dJ(x)/dx=0$ .", "Using Eq.", "(REF ) and the function (REF ) we see that the upper limiting value $x_{\\mathrm {max}}$ can be neglected in the probability current when $\\rho <\\alpha -1$ .", "Thus the power law exponent $\\lambda $ of the steady state PDF should be from the interval $\\alpha (\\eta -1)+1<\\lambda <\\alpha \\eta +1\\,.$ As a particular case when $\\alpha =2$ from Eq.", "(REF ) we get previously proposed SDE with the Gaussian noise [45], [46] $dx=\\sigma ^{2}(2\\eta -\\lambda )x^{2\\eta -1}dt+x^{\\eta }dL_{t}^{2}\\,.$ Note, that according to the definition (REF ), the scale parameter $\\sigma $ differs from the standard deviation of the Gaussian noise.", "Eq.", "(REF ) has a simple form when $\\alpha =1$ : $dx=\\sigma \\cot [\\pi (\\lambda -\\eta )]x^{\\eta }dt+x^{\\eta }dL_{t}^{1}\\,.$" ], [ "Equations allowing both positive and negative values of $x$", "In Eq.", "(REF ) the stochastic variable $x$ can acquire only positive values.", "Similarly as in Ref.", "[48] we can get the equations allowing $x$ to be negative.", "We will search for the stochastic differential equation (REF ) generating signals with power law steady state PDF $P_{0}(x)\\sim |x|^{-\\lambda }\\,.$ To have a normalizable PDF we will assume that the power law holds only in some wide region of $x$ , $x_{\\mathrm {min}}\\ll |x|\\ll x_{\\mathrm {max}}$ .", "In order to obtain such an equation we will consider Eq.", "(REF ) with the coefficients having the power law form $a(x)=\\sigma ^{\\alpha }\\gamma |x|^{\\mu -1}x$ and $b(x)=|x|^{\\eta }$ when $|x|\\gg x_{\\mathrm {min}}$ .", "Similarly as in the case of the positive $x$ we investigate the fractional derivative of the function $f(x)={\\left\\lbrace \\begin{array}{ll}|x|^{\\rho }\\,, & x_{\\mathrm {min}}<|x|<x_{\\mathrm {max}}\\,,\\\\x_{\\mathrm {min}}^{\\rho }\\,, & -x_{\\mathrm {min}}<x<x_{\\mathrm {min}}\\,,\\\\0 & \\mbox{otherwise.}\\end{array}\\right.", "}$ Using Eq.", "(REF ) we obtain the following approximate expressions for the fractional derivative of the function (REF ) when $x_{\\mathrm {min}}\\ll x\\ll x_{\\mathrm {max}}$ : $\\frac{d^{\\alpha }}{d|x|^{\\alpha }}f(x)\\approx \\frac{\\sin \\left(\\frac{\\pi }{2}\\rho \\right)}{\\sin \\left(\\frac{\\pi }{2}(\\alpha -\\rho )\\right)}\\frac{\\Gamma (1+\\rho )}{\\Gamma (1+\\rho -\\alpha )}x^{\\rho -\\alpha }\\,,\\qquad -1<\\rho <\\alpha \\,.$ Using Eq.", "(REF ) for the fractional derivative in Eq.", "(REF ), we obtain $\\mu =\\alpha (\\eta -1)+1$ and $\\gamma =\\frac{\\sin \\left[\\frac{\\pi }{2}(\\alpha \\eta -\\lambda )\\right]}{\\sin \\left[\\frac{\\pi }{2}(\\lambda -\\alpha (\\eta -1))\\right]}\\frac{\\Gamma (\\alpha \\eta -\\lambda +1)}{\\Gamma (\\alpha (\\eta -1)-\\lambda +2)}\\,.$ In addition, from Eq.", "(REF ) it follows that the power law exponent $\\lambda $ of the steady state PDF should be from the interval $\\alpha (\\eta -1)<\\lambda <\\alpha \\eta +1\\,.$ When $\\alpha =2$ , Eq.", "(REF ) simplifies to $\\gamma =2\\eta -\\lambda \\,.$ This expression is the same as the one for the SDE with only positive values of $x$ and $\\alpha =2$ .", "However, when $\\alpha <2$ , the coefficient $\\gamma $ given by Eq.", "(REF ) is different from $\\gamma $ given by Eq.", "(REF ), in contrast to the Gaussian case ($\\alpha =2$ ).", "This can be understood by noticing that the Lévy stable noise for $\\alpha <2$ has large jumps.", "Jumps from the regions with negative values of the stochastic variable $x$ to the regions with positive values influence the PDF $P_{0}(x)$ for the positive values of $x$ .", "The same situation is with the jumps from positive to negative regions.", "Eq.", "(REF ) also has a simple form $\\gamma =\\tan \\left[\\frac{\\pi }{2}(\\eta -\\lambda )\\right]$ for $\\alpha =1$ .", "The required form of the coefficients $\\alpha (x)$ and $b(x)$ has the equation $dx=\\sigma ^{\\alpha }\\gamma (x_{0}^{2}+x^{2})^{\\frac{\\alpha }{2}(\\eta -1)}xdt+(x_{0}^{2}+x^{2})^{\\frac{\\eta }{2}}dL_{t}^{\\alpha }$ and equation $dx=\\sigma ^{\\alpha }\\gamma (x_{0}^{\\alpha }+|x|^{\\alpha })^{\\eta -1}xdt+(x_{0}^{\\alpha }+|x|^{\\alpha })^{\\frac{\\eta }{\\alpha }}dL_{t}^{\\alpha }\\,.$ Here parameter $x_{0}$ plays the role of $x_{\\mathrm {min}}$ .", "The restriction the diffusion at the large absolute values of $x$ can be achieved by reflective boundaries at $\\pm x_{\\mathrm {max}}$ or by additional terms in the equations.", "Eq.", "(REF ) is a generalization of SDE with Gaussian noise from Ref. [48].", "The addition of the parameter $x_{0}$ restricts the divergence of the power law distribution of $x$ at $x\\rightarrow 0$ .", "Eqs.", "(REF ), (REF ) for $|x|\\ll x_{0}$ represents SDEs with additive Lévy stable noise and linear relaxation." ], [ "Power spectral density of the generated signals ", "In this Section we estimate the PSD of the signals generated by the SDE with Lévy stable noise $dx=\\sigma ^{\\alpha }\\gamma x^{\\alpha (\\eta -1)+1}dt+x^{\\eta }dL_{t}^{\\alpha }\\,,$ proposed in the previous Section.", "Here $\\gamma $ is given by Eq.", "(REF ).", "For this estimation we use the (approximate) scaling properties of the signals, as it is done in the Appendix A of Ref.", "[58] and in Ref. [59].", "Using Wiener-Khintchine theorem the PSD can be related to the autocorrelation function $C(t)$ , which can be calculated using the steady state PDF $P_{0}(x)$ and the transition probability $P(x^{\\prime },t|x,0)$ (the conditional probability that at time $t$ the signal has value $x^{\\prime }$ with the condition that at time $t=0$ the signal had the value $x$ ) [60]: $C(t)=\\int dx\\int dx^{\\prime }\\, xx^{\\prime }P_{0}(x)P(x^{\\prime },t|x,0)\\,.$ The transition probability can be obtained from the solution of the fractional Fokker-Planck equation (REF ) with the initial condition $P(x^{\\prime },t=0|x,0)=\\delta (x^{\\prime }-x)$ .", "The the increments of Lévy $\\alpha $ -stable motion $dL_{t}^{\\alpha }$ have the scaling property $dL_{at}^{\\alpha }=a^{1/\\alpha }dL_{t}^{\\alpha }$ [54].", "Changing the variable $x$ in Eq.", "(REF ) to the scaled variable $x_{s}=ax$ or introducing the scaled time $t_{s}=a^{\\alpha (\\eta -1)}t$ one gets the same resulting equation.", "Thus change of the scale of the variable $x$ and change of time scale are equivalent, leading to the following scaling property of the transition probability: $aP(ax^{\\prime },t|ax,0)=P(x^{\\prime },a^{\\mu }t|x,0)\\,,$ with the exponent $\\mu $ being $\\mu =\\alpha (\\eta -1)\\,.$ As has been shown in Ref.", "[59], the power law steady state PDF $P_{0}(x)\\sim x^{-\\lambda }$ and the scaling property of the transition probability (REF ) lead to the power law form PSD $S(f)\\sim f^{-\\beta }$ in a wide range of frequencies.", "From the equation $\\beta =1+(\\lambda -3)/\\mu \\,,$ obtained in Ref.", "[59], it follows that the power-law exponent in the PSD of the signal generated by SDE with Lévy stable noise (REF ) is $\\beta =1+\\frac{\\lambda -3}{\\alpha (\\eta -1)}\\,.$ This expression is the generalization of the expression for the power-law exponent in the PSD with $\\alpha =2$ , obtained in Ref. [46].", "As Eq.", "(REF ) shows, we get $1/f$ PSD when $\\lambda =3$ .", "The presence of the restrictions at $x=x_{\\mathrm {min}}$ and $x=x_{\\mathrm {max}}$ makes the scaling (REF ) not exact and this limits the power law part of the PSD to a finite range of frequencies $f_{\\mathrm {min}}\\ll f\\ll f_{\\mathrm {max}}$ .", "Similarly as in Ref.", "[59] we can estimate the limiting frequencies.", "Taking into account $x_{\\mathrm {min}}$ and $x_{\\mathrm {max}}$ the autocorrelation function has the scaling property [59] $C(t;ax_{\\mathrm {min}},ax_{\\mathrm {max}})=a^{2}C(a^{\\mu }t,x_{\\mathrm {min}},x_{\\mathrm {max}})\\,.$ This equation means that time $t$ in the autocorrelation function should enter only in combinations with the limiting values, $x_{\\mathrm {min}}t^{1/\\mu }$ and $x_{\\mathrm {max}}t^{1/\\mu }$ .", "We can expect that the influence of the limiting values can be neglected when the first combination is small and the second large, that is when time $t$ is in the interval $\\sigma ^{-\\alpha }x_{\\mathrm {max}}^{\\alpha (1-\\eta )}\\ll t\\ll \\sigma ^{-\\alpha }x_{\\mathrm {min}}^{\\alpha (1-\\eta )}$ .", "Then the frequency range where the PSD has $1/f^{\\beta }$ behavior can be estimated as $\\sigma ^{\\alpha }x_{\\mathrm {min}}^{\\alpha (\\eta -1)}\\ll 2\\pi f\\ll \\sigma ^{\\alpha }x_{\\mathrm {max}}^{\\alpha (\\eta -1)}\\,.$ This equation shows that the frequency range grows with increasing of the exponent $\\eta $ , the frequency range becomes zero when $\\eta =1$ .", "By increasing the ratio $x_{\\mathrm {max}}/x_{\\mathrm {min}}$ one can get arbitrarily wide range of the frequencies where the PSD has $1/f^{\\beta }$ behavior.", "Note, that pure $1/f^{\\beta }$ PSD is physically impossible because the total power would be infinite.", "Therefore, we consider signals with PSD having $1/f^{\\beta }$ behavior only in some wide intermediate region of frequencies, $f_{\\mathrm {min}}\\ll f\\ll f_{\\mathrm {max}}$ , whereas for small frequencies $f\\ll f_{\\mathrm {min}}$ PSD is bounded.", "The power spectral density of the form $1/f^{\\beta }$ is determined mainly by power law behavior of the coefficients of SDE (REF ) at large values of $x\\gg x_{\\mathrm {min}}$ .", "Changing the coefficients at small $x$ , the spectrum preserves the power law behavior.", "The modifications of the SDE (REF ), (REF ) and the introduction of negative values of the stochastic variable $x$ should not destroy the frequency region with $1/f^{\\beta }$ behavior of the power spectral density.", "This is confirmed by numerical solution of the equations." ], [ "Numerical examples", "When $\\lambda =3$ , we get that $\\beta =1$ and SDEs (REF ), (REF ), (REF ) should give a signal exhibiting $1/f$ noise.", "We will solve numerically two cases, corresponding to Eqs.", "(REF ) and (REF ), with the index of stability of Lévy stable noise $\\alpha =1$ and the power law exponent of the steady state PDF $\\lambda =3$ .", "Note, that for this value of $\\alpha $ the Lévy $\\alpha $ -stable distribution is the same as the Cauchy distribution.", "For simplicity we choose the exponent in the noise amplitude $\\eta $ such that the coefficient $\\gamma $ , given by Eqs.", "(REF ) or (REF ), becomes equal to $-1$ .", "For the numerical solution we use Euler's approximation, transforming differential equations to difference equations.", "Eq.", "(REF ) leads to the following difference equation $x_{k+1}=x_{k}+\\sigma ^{\\alpha }\\gamma x_{k}^{\\alpha (\\eta -1)+1}h_{k}+x_{k}^{\\eta }h_{k}^{1/\\alpha }\\xi _{k}^{\\alpha }\\,,$ where $h_{k}=t_{k+1}-t_{k}$ is the time step and $\\xi _{k}^{\\alpha }$ is a random variable having $\\alpha $ -stable Lévy distribution with the characteristic function (REF ).", "We can solve Eq.", "(REF ) numerically with the constant step $h_{k}=\\mathrm {const}$ .", "However, more effective method of solution of Eq.", "(REF ) is when the change of the variable $x_{k}$ in one step is proportional to the value of the variable, as has been done solving SDE with Gaussian noise in Ref. [45].", "Variable step of integration $h_{k}=\\frac{\\kappa ^{\\alpha }}{\\sigma ^{\\alpha }}x_{k}^{-\\alpha (\\eta -1)}$ results in the equation $x_{k+1}=x_{k}+\\kappa ^{\\alpha }\\gamma x_{k}+\\frac{\\kappa }{\\sigma }x_{k}\\xi _{k}^{\\alpha }\\,.$ Here $\\kappa \\ll 1$ is a small parameter.", "We include the reflective boundaries at $x=x_{\\mathrm {min}}$ and $x=x_{\\mathrm {max}}$ using the projection method [61], [62].", "According to the projection method, if the variable $x_{k+1}$ acquires the value outside of the interval $[x_{\\mathrm {min}},x_{\\mathrm {max}}]$ then the value of the nearest reflective boundary is assigned to $x_{k+1}$ .", "Figure: (Color online) (a) Signal generated by SDE with Lévy stable noise() with reflective boundaries at x=x min x=x_{\\mathrm {min}}and x=x max x=x_{\\mathrm {max}}.", "(b) Steady state PDF P 0 (x)P_{0}(x) of thesignal.", "The dashed green line shows the slope x -3 x^{-3}.", "(c) Powerspectral density S(f)S(f) of the signal.", "The dashed green line showsthe slope 1/f1/f.", "Parameters used are x min =1x_{\\mathrm {min}}=1, x max =10 4 x_{\\mathrm {max}}=10^{4},σ=1\\sigma =1.When $\\alpha =1$ , $\\lambda =3$ and $\\eta =9/4$ , the SDE (REF ) is $dx=-\\sigma x^{9/4}dt+\\sigma x^{9/4}dL_{t}^{1}\\,.$ The results obtained numerically solving this equation with reflective boundaries at $x=x_{\\mathrm {min}}$ and $x=x_{\\mathrm {max}}$ are shown in Fig.", "REF .", "A sample of the generated signal is shown in Fig.", "REF a.", "The signal exhibits peaks or bursts, corresponding to the large deviations of the variable $x$ .", "Comparison of the steady state PDF $P_{0}(x)$ and the PSD $S(f)$ with the analytical estimations is presented in Fig.", "REF b and Fig.", "REF c. There is quite good agreement of the numerical results with the analytical expressions.", "In Fig.", "REF b we see that near the reflecting boundaries the steady state PDF deviates from the power law prediction.", "This increase of the steady state PDF near boundaries is typical for equations with Lévy stable noise having $\\alpha <2$ [12].", "The behavior of the steady state PDF near the reflecting boundaries is similar to the behavior of the analytical expression obtained in Ref.", "[12] for the simplest stochastic differential equation Lévy stable noise having constant noise amplitude and zero drift.", "A numerical solution of the equations confirms the presence of the frequency region for which the PSD has $1/f$ dependence.", "The width of this region can be increased by increasing the ration between the minimum and the maximum values of the stochastic variable $x$ .", "In addition, the region in the PSD with the power law behavior depends on $\\alpha $ and the exponent $\\eta $ : the width increases with increasing the difference $\\eta -1$ and increasing $\\alpha $ ; when $\\eta =1$ then this width is zero.", "Such behavior is correctly predicted by Eq.", "(REF ).", "Figure: (Color online) (a) Signal generated by SDE with Lévy stable noise().", "(b) Steady state PDF P 0 (x)P_{0}(x) of the signal.The dashed green line shows the dependence on xx proportional to|x| -3 |x|^{-3}.", "(c) Power spectral density S(f)S(f) of the signal.", "Thedashed green line shows the slope 1/f1/f.", "Parameters used are x 0 =1x_{0}=1,x max =10 4 x_{\\mathrm {max}}=10^{4}, σ=1\\sigma =1.Similar schemes of numerical solution we use also for SDEs (REF ) and (REF ).", "Euler's approximation with variable step of integration $h_{k}=\\frac{\\kappa ^{\\alpha }}{\\sigma ^{\\alpha }}(x_{0}^{2}+x_{k}^{2})^{-\\frac{\\alpha }{2}(\\eta -1)}$ transforms SDE (REF ) to the difference equation $x_{k+1}=x_{k}+\\kappa ^{\\alpha }\\gamma x_{k}+\\frac{\\kappa }{\\sigma }\\sqrt{x_{0}^{2}+x_{k}^{2}}\\xi _{k}^{\\alpha }\\,.$ For SDE (REF ) we use the variable step of integration $h_{k}=\\frac{\\kappa ^{\\alpha }}{\\sigma ^{\\alpha }}(x_{0}^{\\alpha }+|x_{k}|^{\\alpha })^{-(\\eta -1)}$ resulting in the difference equation $x_{k+1}=x_{k}+\\kappa ^{\\alpha }\\gamma x_{k}+\\frac{\\kappa }{\\sigma }(x_{0}^{\\alpha }+|x_{k}|^{\\alpha })^{\\frac{1}{\\alpha }}\\xi _{k}^{\\alpha }\\,.$ Here $\\kappa \\ll 1$ is a small parameter.", "Reflective boundaries at $x=\\pm x_{\\mathrm {max}}$ we include using the projection method.", "When $\\alpha =1$ , $\\lambda =3$ and $\\eta =5/2$ , the SDE (REF ) with the coefficient $\\gamma $ given by Eq.", "(REF ) is $dx=-\\sigma (x_{0}^{2}+x^{2})^{3/4}xdt+(x_{0}^{2}+x^{2})^{5/4}dL_{t}^{1}\\,.$ The results obtained numerically solving this equation with reflective boundaries at $x=\\pm x_{\\mathrm {max}}$ are shown in Fig.", "REF .", "A sample of the generated signal is shown in Fig.", "REF a.", "Comparison of the steady state PDF $P_{0}(x)$ and the PSD $S(f)$ with the analytical estimations is presented in Fig.", "REF b and Fig.", "REF c. There is quite good agreement of the numerical results with the analytical expressions.", "As in the case with only positive values of $x$ , we see in Fig.", "REF b we see the increase of the steady state PDF near the reflecting boundaries $x=\\pm x_{\\mathrm {max}}$ in comparison to the power law prediction.", "Numerical solution of Eq.", "(REF ) confirms the presence of the frequency region where the PSD has $1/f^{\\beta }$ dependence." ], [ "Discussion", "Lévy flights have been modeled using Langevin equation with various subharmonic potentials and additive Lévy stable noise [9], [20], [10], [21].", "Proposed SDE (REF ) contains multiplicative Lévy stable noise and is a generalization of previous attempts to model Lévy flights.", "This SDE can be used to investigate Lévy flights in non-equilibrium and non-homogeneous environments, like porous media and some cases of polymer chains [19], [17].", "If specific conditions given by Eq.", "(REF ) are satisfied, our model generates Lévy flights exhibiting $1/f$ noise.", "The drift term $a(x)$ in Eq.", "(REF ) represents a subharmonic external force effecting the particle.", "Lévy flights in subharmonic potentials lead to various interesting phenomena such as stochastic resonance in singe well potential [63].", "The power law dependence of the diffusion coefficient $b^{2}(x)$ on the stochastic variable $x$ can be traced to the existence of the energy flux due to temperature gradient in a bath.", "Long jumps leading to Lévy stable noise can arise from a complex scale free structure of the bath as is in the case of enzyme diffusion on a polymer [19].", "There are suggestions that the non-homogeneity of the bath can be described by the dependence of the diffusion coefficient on the particle coordinate $x$ [13] and Lévy stable noise arises from the bath not being in an equilibrium.", "In the case of Gaussian noise ($\\alpha =2$ ) nonlinear SDE (REF ) that generates signal with $1/f$ spectrum can be obtained from various models.", "One of those models is a signal consisting form a sequence of pulses with a Brownian motion of the inter-pulse durations [45], [46].", "This suggests that our more general form of the SDE could be obtained from some kind of Lévy motion of the inter-pulse durations.", "However, we were unable to show this due to the complexity of Itô formula in case of equations driven by Lévy process [64].", "The special case of Eq.", "(REF ) for free particle ($a(x)=0$ ) with Lévy stable noise having $\\alpha <2$ has been derived from coupled continuous time random walk (CTRW) models [18], when jumping rate $\\nu $ of CTRW process depends on signal intensity as $\\nu (x)=x^{\\alpha \\eta }$ , $x>0$ .", "However, such derivation is quite complex and does little to help the understanding what kind of physical phenomena can be approximated by multiplicative Lévy stable noise.", "Thus instead of searching for underlying models in this article we have chosen an simpler approach: we have derived nonlinear SDEs using a simple reasoning about scaling properties of the steady state PDF.", "Taking into account of the scaling properties of the signal is one of the advantages of our model.", "In many theoretical models, such as diffusion of the particle in a fractal turbulence [3], ecological population density with fluctuating volume of resources [14], dynamics of two competing species [65] and tumor growth [66], an existence of Lévy stable noise instead of Gaussian noise is simply assumed.", "Such assumption might be incorrect, because the change of statistical properties of the noise change the scaling properties of the signal.", "In order to preserve original scaling properties of the signal the drift $a(x)$ or diffusion $b^{2}(x)$ coefficients must be changed as well.", "The required drift coefficient $a(x)$ can be found similarly as in Section .", "The scaling properties can be extracted from time series using fluctuation analysis methods [55].", "In summary, we have proposed nonlinear SDEs with Lévy stable noise and generating signals exhibiting $1/f$ noise in any desirably wide range of frequency.", "Proposed SDEs (REF ), (REF ) and (REF ) are a generalization of nonlinear SDEs driven by Gaussian noise and generating signals with $1/f$ PSD.", "The generalized equations can be obtained by replacing the Gaussian noise with the Lévy stable noise and changing the drift term to preserve statistical properties of the generated signal.", "We have investigated two cases: in the first case the stochastic variable can acquire only positive values (SDE (REF )), in the second case the stochastic variable can also be negative (SDEs (REF ) and (REF )).", "In contrast to the SDEs with the Gaussian noise, the constant in the drift term, given by Eqs.", "(REF ) and (REF ), is different in those two cases and becomes the same only for $\\alpha =2$ ." ] ]
1403.0409
[ [ "Individual dynamics induces symmetry in network controllability" ], [ "Abstract Controlling complex networked systems to a desired state is a key research goal in contemporary science.", "Despite recent advances in studying the impact of network topology on controllability, a comprehensive understanding of the synergistic effect of network topology and individual dynamics on controllability is still lacking.", "Here we offer a theoretical study with particular interest in the diversity of dynamic units characterized by different types of individual dynamics.", "Interestingly, we find a global symmetry accounting for the invariance of controllability with respect to exchanging the densities of any two different types of dynamic units, irrespective of the network topology.", "The highest controllability arises at the global symmetry point, at which different types of dynamic units are of the same density.", "The lowest controllability occurs when all self-loops are either completely absent or present with identical weights.", "These findings further improve our understanding of network controllability and have implications for devising the optimal control of complex networked systems in a wide range of fields." ], [ "1.1 Individual dynamics induces symmetry in network controllability Chen Zhao School of Systems Science, Beijing Normal University, Beijing, 10085, P. R. China Wen-Xu Wang School of Systems Science, Beijing Normal University, Beijing, 10085, P. R. China Yang-Yu [email protected] Channing Division of Network Medicine, Brigham and Women's Hospital, Harvard Medical School, Boston, Massachusetts 02115, USA Center for Complex Network Research and Department of Physics,Northeastern University, Boston, Massachusetts 02115, USA Jean-Jacques [email protected] Nonlinear Systems Laboratory, Massachusetts Institute of Technology, Cambridge, Massachusetts, 02139, USA Department of Mechanical Engineering and Department of Brain and Cognitive Sciences, Massachusetts Institute of Technology, Cambridge, Massachusetts, 02139, USA Controlling complex networked systems to a desired state is a key research goal in contemporary science.", "Despite recent advances in studying the impact of network topology on controllability, a comprehensive understanding of the synergistic effect of network topology and individual dynamics on controllability is still lacking.", "Here we offer a theoretical study with particular interest in the diversity of dynamic units characterized by different types of individual dynamics.", "Interestingly, we find a global symmetry accounting for the invariance of controllability with respect to exchanging the densities of any two different types of dynamic units, irrespective of the network topology.", "The highest controllability arises at the global symmetry point, at which different types of dynamic units are of the same density.", "The lowest controllability occurs when all self-loops are either completely absent or present with identical weights.", "These findings further improve our understanding of network controllability and have implications for devising the optimal control of complex networked systems in a wide range of fields.", "As a key notion in control theory, controllability denotes our ability to drive a dynamic system from any initial state to any desired final state in finite time [1], [2].", "For the canonical linear time-invariant (LTI) system $\\dot{\\bf x}= A{\\bf x} + B {\\bf u}$ with state vector ${\\bf x} \\in \\mathbb {R}^N$ , state matrix $A \\in \\mathbb {R}^{N\\times N}$ and control matrix $B \\in \\mathbb {R}^{N\\times M}$ , Kalman's rank condition $\\text{rank}[B,AB,\\cdots , A^{N-1}B] = N$ is sufficient and necessary to assure controllability.", "Yet, in many cases system parameters are not exactly known, rendering classical controllability tests impossible.", "By assuming that system parameters are either fixed zeros or freely independent, structural control theory (SCT) helps us overcome this difficulty for linear time-invariant systems [3], [4], [5], [6], [7].", "Quite recently, many research activities have been devoted to study the structural controllability of systems with complex network structure, where system parameters (e.g., the elements in $A$ , representing link weights or interaction strengths between nodes) are typically not precisely known, only the zero-nonzero pattern of $A$ is known [8], [9], [10], [11], [12], [13], [14], [15].", "Network controllability problem can be typically posed as a combinatorial optimization problem, i.e., identify a minimum set of driver nodes, with size denoted by $N_\\mathrm {D}$ , whose control is sufficient to fully control the system’s dynamics [8].", "Other controllability related issues, e.g., energy cost, have also been extensively studied for complex networked systems [16], [17], [18], [19].", "While the intrinsic individual dynamics can be incorporated in the network model, it would be more natural and fruitful to consider their effect separately.", "Hence, most of the previous studies focused on the impact of network topology, rather than the individual dynamics of nodes, on network controllability [8], [11].", "If one explores the impact of individual dynamics on network controllability in the SCT framework, a specious result would be obtained — a single control input can make an arbitrarily large linear system controllable.", "Although this result as a special case of the minimum inputs theorem can be proved [8] and its implication was further emphasized in [20], this result is inconsistent with empirical situations, implying that the SCT is inapplicable in studying network controllability, if individual dynamics of nodes are imperative to be incorporated to capture the collective dynamic behavior of a networked system.", "To overcome this difficulty, and more importantly, to understand the impact of individual dynamics on network controllability, we revisit the key assumption of SCT, i.e., the independency of system parameters.", "We anticipate that major new insights can be obtained by relaxing this assumption, e.g., considering the natural diversity and similarity of individual dynamics.", "This also offers a more realistic characterization of many real-world networked systems where not all the system parameters are completely independent.", "To solve the network controllability problem with dependent system parameters, we rely on the recently developed exact controllability theory (ECT) [21].", "ECT enables us to systematically explore the role of individual dynamics in controlling complex systems with arbitrary network topology.", "In particular, we consider prototypical linear forms of individual dynamics (from first-order to high-orders) that can be incorporated within the network representation of the whole system in a unified matrix form.", "This paradigm leads to the discovery of a striking symmetry in network controllability: if we exchange the fractions of any two types of dynamic units, the system's controllability (quantified by $N_\\mathrm {D}$ ) remains the same.", "This exchange-invariant property gives rise to a global symmetry point, at which the highest controllability (i.e., lowest number of driver nodes) emerges.", "This symmetry-induced optimal controllability holds for any network topology and various categories of individual dynamics.", "We substantiate these findings numerically in a variety of network models.", "Figure: Integration of network topology and (a)1st-order, (b) 2nd-order and (c) 3rd-order intrinsic individual dynamics.", "For a ddth-order individual dynamicsx (d) =a 0 x (0) +a 1 x (1) +⋯+a d-1 x (d-1) x^{(d)}=a_0x^{(0)}+a_1x^{(1)}+\\cdots + a_{d-1}x^{(d-1)},we denote each order by a colored square and the couplings among ordersare characterized by links or self-loops.", "This graphical representationallows individual dynamics to be integrated with their coupling network topology,giving rise to a unified matrix that reflects the dynamics ofthe whole system.", "In particular, each dynamic unit in the unified matrixcorresponds to a diagonal block and the nonzero elements (denoted by **)apart from the blocks stand for the couplings amongdifferent dynamic units.", "Therefore, the original network consisting of NN nodes withorder dd is represented in a dN×dNdN \\times dN matrix.Exact controllability theory (ECT) [21] claims that for arbitrary network topology and link weights characterized by the state matrix $A$ in the LTI system $\\dot{\\mathbf {x}}= A\\mathbf {x}+B\\mathbf {u} $ , the minimum number of driver nodes $N_\\text{D}$ required to be controlled by imposing independent signals to fully control the system is given by the maximum geometric multiplicity $\\max _i\\lbrace \\mu (\\lambda _i)\\rbrace $ of $A$ 's eigenvalues $\\lbrace \\lambda _i\\rbrace $  [22], [23], [24], [25], [26].", "Here $\\mu (\\lambda _i) \\equiv N-\\text{rank}(\\lambda _i I_N-A)$ is the geometric multiplicity of the eigenvalue $\\lambda _i$ and $I_N$ is the identity matrix.", "Calculating all the eigenvalues of $A$ and subsequently counting their geometric multiplicities are generally applicable but computationally prohibitive for large networks.", "If $A$ is symmetric, e.g., in undirected networks, $N_\\text{D}$ is simply given by the maximum algebraic multiplicity $\\max _i\\lbrace \\delta (\\lambda _i)\\rbrace $ , where $\\delta (\\lambda _i)$ denotes the degeneracy of eigenvalue $\\lambda _i$ .", "Calculating $N_\\text{D}$ in the case of symmetric $A$ is more computationally affordable than in the asymmetric case.", "Note that for structured systems where the elements in $A$ are either fixed zeros or free independent parameters, ECT offers the same results as that of the SCT [21].", "We first study the simplest case of first-order individual dynamics $\\dot{x}_i=a_0x_i$ .", "The dynamical equations of a linear time-invariant control system associated with first-order individual dynamics [27] can be written as $ \\dot{\\mathbf {x}}=\\Lambda \\mathbf {x}+ A\\mathbf {x}+B\\mathbf {u} = \\Phi \\mathbf {x}+B\\mathbf {u},$ where the vector $\\mathbf {x} = (x_1,\\cdots ,x_N)^\\text{T}$ captures the states of $N$ nodes, $\\Lambda \\in \\mathbb {R}^{N\\times N}$ is a diagonal matrix representing intrinsic individual dynamics of each node, $A\\in \\mathbb {R}^{N\\times N}$ denotes the coupling matrix or the weighted wiring diagram of the networked system, in which $a_{ij}$ represents the weight of a directed link from node $j$ to $i$ (for undirected networks, $a_{ij}=a_{ji}$ ).", "$\\mathbf {u}=(u_1,u_2,\\cdots ,u_M)^\\text{T}$ is the input vector of $M$ independent signals, $B \\in \\mathbb {R}^{N\\times M}$ is the control matrix, and $\\Phi \\equiv \\Lambda + A$ is the state matrix.", "Without loss of generality, we assume $\\Lambda $ is a “constant\" matrix over the field $\\mathbb {Q}$ (rational numbers), and $A$ is a structured matrix over the field $\\mathbb {R}$ (real numbers).", "In other words, we assume all the entries in $\\Phi $ have been rescaled by the individual dynamics parameters.", "The resulting state matrix $\\Phi $ is usually called a mixed matrix with respect to $(\\mathbb {Q}, \\mathbb {R})$  [28].", "The first-order individual dynamics in $\\Phi $ is captured by self-loops in the network representation of $\\Phi $ (see Fig.", "REF a).", "$N_\\text{D}$ can then be determined by calculating the maximum geometric multiplicity $\\max _i\\lbrace \\mu (\\lambda _i)\\rbrace $ of $\\Phi $ 's eigenvalues.", "We study two canonical network models (Erdös-Rényi and Scale-free) with random edge weights and a $\\rho _s$ fraction of nodes associated with identical individual dynamics (i.e., self-loops of identical weights).", "As shown in Fig.", "REF a,b, the fraction of driver nodes $n_\\text{D}\\equiv N_\\text{D}/N$ is symmetric about $\\rho _\\text{s}=0.5$ , regardless of the network topology.", "Note that the symmetry cannot be predicted by SCT in the sense that in case of completely independent self-loop weights $n_\\text{D}$ will monotonically decrease to $1/N$ as $\\rho _\\text{s}$ increases to 1, implying that a single driver node can fully control the whole network [20].", "The symmetry can be theoretically predicted (see SM Sec.2.2).", "An immediate but counterintuitive result from the symmetry is that $n_\\text{D}$ in the absence of self-loops is exactly the same as the case that each node has a self-loop with identical weight.", "This is a direct consequence of Kalman's rank condition for controllability [1]: $\\text{rank}[B,AB,\\cdots , A^{N-1}B] =\\text{rank}[B,(A+w_\\text{s}I_N)B,\\cdots , (A+w_\\text{s}I_N)^{N-1}B]$ where the left and the right hand sides are the rank of controllability matrix in the absence and full of identical self-loops, respectively (see SM Sec.1 for proof).", "Figure: (a)-(b) controllability measure n D n_\\text{D}in the presence of a single type of nonzero self-loops with fraction ρ s \\rho _\\text{s} for random (ER)networks (a) and scale-free (SF) networks (b) with different average degree 〈k〉\\langle k\\rangle .", "(c)-(d) n D n_\\text{D} of ER (c) and SF networks (d) with three types of self-loopss 1 s_1, s 2 s_2 and s 3 s_3 with density ρ s (1) \\rho _\\text{s}^{(1)}, ρ s (2) \\rho _\\text{s}^{(2)} and ρ s (3) \\rho _\\text{s}^{(3)},respectively.", "ECT denotes the results obtained from the exactcontrollability theory, ET denotes the results obtained from the efficient tool and GA denotesthe results obtained from the graphical approach (see SM Sec.3).The color bar denotes the value of n D n_\\text{D} and the coordinates in the triangle standsfor ρ s (1) \\rho _\\text{s}^{(1)} ρ s (2) \\rho _\\text{s}^{(2)} and ρ s (3) \\rho _\\text{s}^{(3)}.", "The networks are described by structured matrix AA and theirsizes in (a)-(d) are 2000.", "The results from ECT and ET are averaged over 30 different realizations, and those from GA are over 200 realizations.The presence of two types of nonzero self-loops $s_2$ and $s_3$ leads to even richer behavior of controllability.", "If the three types of self-loops (including self-loops of zero weights) are randomly distributed at nodes, the impact of their fractions on $n_\\text{D}$ can be visualized by mapping the three fractions into a 2D triangle (or 2-simplex), as shown in Fig.", "REF c,d.", "We see that $n_\\text{D}$ exhibits symmetry in the triangle and the minimum $n_\\text{D}$ occurs at the center that represents identical fractions of the three different self-loop types.", "The symmetry-induced highest controllability can be generalized to arbitrary number of self-loops.", "Assume there exist $n$ types of self-loops $s_1,\\cdots ,s_n$ with weights $w_\\text{s}^{(1)},\\cdots ,w_\\text{s}^{(n)}$ , respectively, we have $N_\\text{D}= N- \\min _i \\bigg \\lbrace \\text{rank}\\big (\\Phi -w_\\text{s}^{(i)} I_N \\big ) \\bigg \\rbrace $ for sparse networks with random weights (see SM Sec.", "2 for detailed derivation and the formula of dense networks).", "An immediate prediction of Eq.", "(REF ) is that $N_\\text{D}$ is primarily determined by the self-loop with the highest density, simplifying Eq.", "(REF ) to be $N_\\text{D} = N- \\text{rank}(\\Phi -w_\\text{s}^\\text{max} I_N )$ , where $w_\\text{s}^\\text{max}$ is the weight of the prevailing self-loop (see SM Sec.", "2).", "Using Eq.", "(REF ) and the fact that $\\Phi $ is a mixed matrix, we can predict that $N_\\text{D}$ remains unchanged if we exchange the densities of any two types of self-loops (see SM Sec.", "2), accounting for the symmetry of $N_\\text{D}$ for arbitrary types of self-loops.", "Due to the dominance of $N_\\text{D}$ by the self-loop with the highest density and the exchange-invariance of $N_\\text{D}$ , the highest controllability with the lowest value of $N_\\text{D}$ emerges when distinct self-loops are of the same density.", "To validate the symmetry-induced highest controllability predicted by our theory, we quantify the density heterogeneity of self-loops as follows: $\\Delta \\equiv \\sum _{i=1}^{N_\\text{s}}\\left| \\rho _\\text{s}^{(i)}-\\frac{1}{N_\\text{s}} \\right|,$ where $N_\\text{s}$ is the number of different types of self-loops (or the diversity of self-loops).", "Note that $\\Delta =0$ if and only if all different types of self-loops have the same density, i.e., $\\rho _\\text{s}^{(1)}=\\rho _\\text{s}^{(2)}=\\cdots \\rho _\\text{s}^{(N_\\text{s})}=\\frac{1}{N_\\text{s}}$ , and the larger value of $\\Delta $ corresponds to more diverse case.", "Figure REF a,b shows that $n_\\text{D}$ monotonically increases with $\\Delta $ and the highest controllability (lowest $n_\\text{D}$ ) arises at $\\Delta =0$ , in exact agreement with our theoretical prediction.", "Figure REF c,d display $n_\\text{D}$ as a function of $N_\\text{s}$ .", "We see that $n_\\text{D}$ decreases as $N_\\text{s}$ increases, suggesting that the diversity of individual dynamics facilitates the control of a networked system.", "When $N_\\text{s}=N$ (i.e., all the self-loops are independent), $n_\\text{D}=1/N$ , which is also consistent with the prediction of SCT [8], [20].", "Figure: a-b, n D n_\\text{D}as a function of the density heterogeneity of self-loops (Δ\\Delta ) for ER (a) and SF (b)networks.", "c-d, n D n_\\text{D} as a function of the number of different types of self-loopsfor ER (c) and SF (d) networks.", "The dotted line in (g) isn D =1/N s n_\\text{D} = 1/ N_\\text{s}.", "The networks are described by structured matrix AA and theirsizes in (a)-(d) are 1000.", "The results from ECT and ET are averaged over 30 different realizations,and those from GA are over 200 realizations.", "The notations are the same as Fig.", ".In some real networked systems, dynamic units are captured by high-order individual dynamics, prompting us to check if the symmetry-induced highest controllability still holds for higher-order individual dynamics.", "The graph representation of dynamic units with 2nd-order dynamics is illustrated in Fig.", "REF b.", "In this case, the eigenvalues of the dynamic unit's state matrix $\\left(\\begin{array}{cc}0 & 1 \\\\a_0 & a_1 \\\\\\end{array}\\right)$ play a dominant role in determining $N_\\text{D}$ .", "For two different units as distinguished by distinct ($a_0$ $a_1$ ) one can show that their state matrices almost always have different eigenvalues, except for some pathological cases of zero measure that occur when the parameters satisfy certain accidental constraints.", "The eigenvalues of the state matrix of dynamic units take over the roles of self-loops in the 1st-order dynamics, accounting for the following formulas for sparse networks $N_\\text{D} = 2N- \\min _i\\bigg \\lbrace \\text{rank}(\\Phi - \\lambda ^{(i)} I_{2N}) \\bigg \\rbrace ,$ where $\\lambda ^{(i)}$ is either one of the two eigenvalues of type-$i$ dynamic unit's state matrix.", "The formula implies that $N_\\text{D}$ is exclusively determined by the prevailing dynamic unit, (see SM Sec.", "2).", "The symmetry of $N_\\text{D}$ , i.e., exchanging the densities of any types of dynamic units, does not alter $N_\\text{D}$ (see SM Sec.", "2), and the emergence of highest controllability at the global symmetry point can be similarly proved as we did in the case of 1st-order individual dynamics.", "The 3rd-order individual dynamics are graphically characterized by a dynamic unit composed of three nodes (Fig.", "REF c), leading to a $3N\\times 3N$ state matrix (Fig.", "REF c).", "We can generalize Eq.", "(REF ) to arbitrary order of individual dynamics: $N_\\text{D} = dN- \\min _i\\bigg \\lbrace \\text{rank}(\\Phi - \\lambda ^{(i)}_d I_{dN}) \\bigg \\rbrace ,$ where $d$ is the order of the dynamic unit, $\\lambda ^{(i)}_d$ is any one of the $d$ eigenvalues of type-$i$ dynamic units and $I_{dN}$ is the identity matrix of dimension $dN$ .", "In analogy with the simplified formula for the 1st-order dynamics, insofar as a type of individual dynamics prevails in the system, Eq (REF ) is reduced to $N_\\text{D} = dN- \\text{rank}(\\Phi - \\lambda ^\\text{max}_d I_{dN})$ , where $\\lambda ^\\text{max}_d$ is one of the eigenvalues of the prevailing dynamic unit's state matrix.", "Similar to the case of 1st-order individual dynamics, the global symmetry of controllability and the highest controllability occurs at the global symmetry point can be proved for individual dynamics of any order and arbitrary network topology (see SM Sec.2 and 3 for theoretical derivations and see SM Sec.", "4 for numerical and analytical results of high-order individual dynamics).", "In summary, we map individual dynamics into dynamic units that can be integrated into the matrix representation of the system, offering a general paradigm to explore the joint effect of individual dynamics and network topology on the system's controllability.", "The paradigm leads to a striking discovery: the universal symmetry of controllability as reflected by the invariance of controllability with respect to exchanging the fractions of any two different types of individual dynamics, and the emergence of highest controllability at the global symmetry point.", "These findings generally hold for arbitrary networks and individual dynamics of any order.", "The symmetry-induced highest controllability has immediate implications for devising and optimizing the control of complex systems by for example, perturbing individual dynamics to approach the symmetry point without the need to adjust network structure.", "The theoretical paradigm and tools developed here also allow us to address a number of questions, answers to which could offer further insights into the control of complex networked systems.", "For example, we may consider the impact of general parameter dependency (e.g., link weight similarity), instead of focusing on self-loops or individual dynamics.", "Our preliminary results show that introducing more identical link weights will not affect the network controllability too much, unless the network is very dense and almost all link weights are identical (see SM Sec.5).", "We still lack a comprehensive understanding of the impact of parameter dependency on structural controllability for arbitrary complex networks.", "Moreover, at the present we are incapable of tackling general nonlinear dynamical systems in the framework of ECT, which is extremely challenging for both physicists and control theorists.", "Nevertheless, we hope our approach could inspire further research interests towards achieving ultimate control of complex networked systems." ] ]
1403.0041
[ [ "Matching Image Sets via Adaptive Multi Convex Hull" ], [ "Abstract Traditional nearest points methods use all the samples in an image set to construct a single convex or affine hull model for classification.", "However, strong artificial features and noisy data may be generated from combinations of training samples when significant intra-class variations and/or noise occur in the image set.", "Existing multi-model approaches extract local models by clustering each image set individually only once, with fixed clusters used for matching with various image sets.", "This may not be optimal for discrimination, as undesirable environmental conditions (eg.", "illumination and pose variations) may result in the two closest clusters representing different characteristics of an object (eg.", "frontal face being compared to non-frontal face).", "To address the above problem, we propose a novel approach to enhance nearest points based methods by integrating affine/convex hull classification with an adapted multi-model approach.", "We first extract multiple local convex hulls from a query image set via maximum margin clustering to diminish the artificial variations and constrain the noise in local convex hulls.", "We then propose adaptive reference clustering (ARC) to constrain the clustering of each gallery image set by forcing the clusters to have resemblance to the clusters in the query image set.", "By applying ARC, noisy clusters in the query set can be discarded.", "Experiments on Honda, MoBo and ETH-80 datasets show that the proposed method outperforms single model approaches and other recent techniques, such as Sparse Approximated Nearest Points, Mutual Subspace Method and Manifold Discriminant Analysis." ], [ "Introduction", " Compared to single image matching techniques, image set matching approaches exploit set information for improving discrimination accuracy as well as robustness to image variations, such as pose, illumination and misalignment [1], [5], [11], [20].", "Image set classification techniques can be categorised into two general classes: parametric and non-parametric methods.", "The former represent image sets with parametric distributions [1], [4], [12].", "The distance between two sets can be measured by the similarity between the estimated parameters of the distributions.", "However, the estimated parameters might be dissimilar if the training and test data sets of the same subject have weak statistical correlations [11], [21].", "State-of-the-art non-parametric methods can be categorised into two groups: single-model and multi-model methods.", "Single-model methods attempt to represent sets as linear subspaces [11], [23], or affine/convex hulls [5], [10].", "For single linear subspace methods, principal angles are generally used to measure the difference between two subspaces [16], [23].", "As the similarity of data structures is used for comparing sets, the subspace approach can be robust to noise and relatively small number of samples [21], [23].", "However, single linear subspace methods consider the structure of all data samples without selecting optimal subsets for classification.", "Figure: Illustration of artificially generated images from two training samples I 1 I_1 and I 2 I_2 via I gen =(1-w)I 1 +wI 2 I_{gen} = (1-w) I_{1} + w I_{2}.For images in a row, the three images (denoted as I gen I_{gen}) in the middle are generated from convex combinations of the first (denoted as I 1 I_{1}) and the last (denoted as I 2 I_{2}) images.Case (a): as I 1 I_{1} and I 2 I_{2} are very different from each other, the generated images may contain unrealistic artificial features, such as the profile in the middle of the face in the third and fourth image.Case (b): as I 1 I_{1} and I 2 I_{2} are similar to each other, the three generated images contain only minor artificial features.Convex hull approaches use geometric distances (eg.", "Euclidean distance between closest points) to compare sets.", "Given two sets, the closest points between two convex hulls are calculated by least squares optimisation.", "As such, these methods adaptively choose optimal samples to obtain the distance between sets, allowing for a degree of intra-class variations [10].", "However, as the closest points between two convex hulls are artificially generated from linear combinations of certain samples, deterioration in discrimination performance can occur if the nearest points between two hulls are outliers or noisy.", "An example is shown in Fig.", "REF , where unrealistic artificial variations are generated from combinations of two distant samples.", "Recent research suggest that creating multiple local linear models by clustering can considerably improve recognition accuracy [9], [20], [21].", "In [8], [9], Locally Linear Embedding [15] and $k$ -means clustering are used to extract several representative exemplars.", "Manifold Discriminant Analysis [20] and Manifold-Manifold Distance [21] use the notion of maximal linear patches to extract local linear models.", "For two sets with $m$ and $n$ local models, the minimum distance between their local models determines the set-to-set distance, which is acquired by $m \\times n$ local model comparisons (ie.", "an exhaustive search).", "A limitation of current multi-model approaches is that each set is clustered individually only once, resulting in fixed clusters of each set being used for classification.", "These clusters may not be optimal for discrimination and may result in the two closest clusters representing two separate characteristics of an object.", "For example, let us assume we have two face image sets of the same person, representing two conditions.", "The clusters in the first set represent various poses, while the clusters in the second set represent varying illumination (where the illumination is different to the illumination present in the first set).", "As the two sets of clusters capture two specific variations, matching image sets based on fixed cluster matching may result in a non-frontal face (eg.", "rotated or tilted) being compared against a frontal face.", "Contributions.", "To address the above problems, we propose an adaptive multi convex hull classification approach to find a balance between single hull and nearest neighbour method.", "The proposed approach integrates affine/convex hull classification with an adapted multi-model approach.", "We show that Maximum Margin Clustering (MMC) can be applied to extract multiple local convex hulls that are distant from each other.", "The optimal number of clusters is determined by restricting the average minimal middle-point distance to control the region of unrealistic artificial variations.", "The adaptive reference clustering approach is proposed to enforce the clusters of an gallery image set to have resemblance to the reference clusters of the query image set.", "Consider two sets $S_a$ and $S_b$ to be compared.", "The proposed approach first uses MMC to extract local convex hulls from $S_a$ to diminish unrealistic artificial variations and to constrain the noise in local convex hulls.", "We prove that after local convex hulls extraction, the noisy set will be reduced.", "The local convex hulls extracted from $S_a$ are treated as reference clusters to constrain the clustering of $S_b$ .", "Adaptive reference clustering is proposed to force the clusters of $S_b$ to have resemblance to the reference clusters in $S_a$ , by adaptively selecting the closest subset of images.", "The distance of the closest cluster from $S_b$ to the reference cluster of $S_a$ is taken to indicate the distance between the two sets.", "Fig.", "REF shows a conceptual illustration of the proposed approach.", "Figure: Framework of the proposed approach, including techniques used in conjunction with the proposed approach.Comparisons on three benchmark datasets for face and object classification show that the proposed method consistently outperforms single hull approaches and several other recent techniques.", "To our knowledge, this is the first method that adaptively clusters an image set based on the reference clusters from another image set.", "We continue the paper as follows.", "In Section  and , we briefly overview affine and convex hull classification and maximum margin clustering techniques.", "We then describe the proposed approach in Section , followed by complexity analysis in Section  and empirical evaluations and comparisons with other methods in Section .", "The conclusion and future research directions are summarised in Section ." ], [ "Affine and Convex Hull Classification", " An image set can be represented with a convex model, either an affine hull or a convex hull, and then the similarity measure between two sets can be defined as the distance between two hulls [5].", "This can be considered as an enhancement of nearest neighbour classifier with an attempt to reduce sensitivity of within-class variations by artificially generating samples within the set.", "Given an image set $S = [I_1, I_2, ... , I_n]$ , where each $I_k, k\\in [1,...,n]$ is a feature vector extracted from an image, the smallest affine subspace containing all the samples can be constructed as an affine hull model: $H^{aff} = \\lbrace y \\rbrace , \\forall y = \\sum \\nolimits _{k=1}^{n}w_k I_k, \\sum \\nolimits _{k=1}^{n} w_k = 1.$ Any affine combinations of the samples are included in this affine hull.", "If the affine hull assumption is too lose to achieve good discrimination, a tighter approximation can be achieved by setting constraints on $w_k$ to construct a convex hull via [5]: $H^{con} \\mbox{~=~} \\lbrace y \\rbrace , \\forall y \\mbox{~=~} \\hspace{-4.25pt} \\sum \\nolimits _{k=1}^{n} \\hspace{-4.25pt} w_k I_k, ~ \\sum \\nolimits _{k=1}^{n} \\hspace{-4.25pt} w_k \\mbox{~=~} 1, ~ w_k \\in [0,1].$ The distance between two affine or convex hulls $H_{1}$ and $H_{2}$ are the smallest distance between any point in $H_1$ and any point in $H_2$ : $D_{h}(H_1,H_2) = \\min _{y_1\\in H_1, y_2 \\in H_2} ||y_1 - y_2||.$ In [10], sparsity constraints are embedded into the distance when matching two affine hulls to enforce that the nearest points can be sparsely approximated by the combination of a few sample images." ], [ "Maximum Margin Clustering", "Maximum Margin Clustering (MMC) extends Support Vector Machines (SVMs) from supervised learning to the more challenging task of unsupervised learning.", "By formulating convex relaxations of the training criterion, MMC simultaneously learns the maximum margin hyperplane and cluster labels [22] by running an SVM implicitly.", "Experiments show that MMC generally outperforms conventional clustering methods.", "Given a point set $S = [{x}_1, {x}_2, \\ldots , {x}_n]$, MMC attempts to find the optimal hyperplane and optimal labelling for two clusters simultaneously, as follows: $\\min _{y \\in [-1,+1]} \\min _{{w},b,\\xi _{i}} \\frac{1}{2} {w}^{T}{w} + \\frac{C}{n}\\sum \\nolimits _{i=1}^{n}\\xi _{i} \\hspace{34.0pt}\\\\s.t.", "\\hspace{8.53581pt} y_{i}({w}^{T}\\phi ({x}_{i})+b) \\ge 1 - \\xi _{i}, \\hspace{4.25pt} \\xi _{i} \\ge 0, ~~~ i = 1,\\ldots ,n. \\nonumber $ where $y_i$ is the label learned for point ${x}_{i}$ .", "Several approaches has been proposed to solve this challenging non-convex integer optimisation problem.", "In [19], [22], several relaxations and class balance constraints are made to convert the original MMC problem into a semi-definite programming problem to obtain the solution.", "Alternating optimisation techniques and cutting plane algorithms are applied on MMC individually in [24] and [25] to speed up the learning for large scale datasets.", "MMC can be further extended for multi-class clustering from multi-class SVM [26]: $\\min _{y} \\min _{{w}_{1},...,{w}_{k}, \\xi } \\frac{\\beta }{2}\\sum \\nolimits _{p=1}^{k}||{w}_{p}||^{2}+\\frac{1}{n}\\sum \\nolimits _{i=1}^{n}\\xi _{i}\\\\ \\nonumber s.t.", "\\hspace{8.53581pt} {w}_{y_{i}}^{T}{x}_{i} + \\delta _{y_{i},r}-{w}_{r}^{T}{x}_{i} \\ge 1 - \\xi _{i} \\\\ \\nonumber \\forall i = 1,\\ldots ,n ~~~~~~ r = 1,\\ldots ,k. \\nonumber $" ], [ "Proposed Adaptive Multi Convex Hull", " The proposed adaptive multi convex hull classification algorithm consists of two main steps: extraction of local convex hulls and comparison of local convex hulls, which are elucidated in the following text." ], [ "Local Convex Hulls Extraction (LCHE)", " Existing single convex hull based methods assume that any convex combinations of samples represent intra-class variations and thus should also be included in the hull.", "However, as the nearest points between two hulls are normally artificially generated from samples, they may be noisy or outliers that lead to poor discrimination performance.", "On the contrary, nearest neighbour method only compare samples in image sets disregarding their combinations, resulting in sensitivity to within class variations.", "There should be a balance between these two approaches.", "We observe that under the assumption of affine (or convex) hull model, when sample points are distant from each other, linear combinations of samples will generate non-realistic artificial variations, as shown in Figure REF .", "This will result in deterioration of classification.", "Given an image set with a noisy sample $S = [{I}_1, {I}_2, \\ldots , {I}_N, {I}_{ns}]$, where ${I}_i, i\\in 1, \\ldots , N$ are normal sample images and ${I}_{ns}$ is a noisy sample, a single convex hull $H$ can be constructed using all the samples.", "The clear set $H_{cl}$ of $H$ is defined as a set of points whose synthesis does not require the noisy data ${I}_{ns}$.", "That is, $H_{cl}=\\lbrace p\\rbrace , \\forall p = \\sum \\nolimits _{i} \\alpha _{i} {I}_i, ~ \\alpha _{i} \\in [0,1], ~ \\sum \\nolimits _{i} \\alpha _{i} = 1.$ Accordingly, the remaining set of points in $H$ not in $H_{cl}$ is the noisy set $H_{ns}$.", "The synthesis of points in noisy set must require the noisy data ${I}_{ns}$.", "That is, $H_{ns}=\\lbrace p_{ns} \\rbrace , \\hspace{4.25pt} \\forall p_{ns} = \\alpha _{ns} {I}_{ns} + \\sum \\nolimits _{i} \\alpha _{i} {I}_i, \\hspace{4.25pt} p_{ns} \\notin H_{cl}, \\\\\\alpha _{ns} \\in (0,1], \\hspace{4.25pt} \\alpha _{i} \\in [0,1], \\hspace{4.25pt} \\alpha _{ns} + \\sum \\nolimits _{i} \\alpha _{i} = 1.\\nonumber $ All of the normal samples ${I}_i$ that involve the synthesis of points in $H_{ns}$ are called the noisy sample neighbours, as they are generally in the neighbourhood of the noisy sample.", "The noisy set $H_{ns}$ defines the set of points that is affected by the noisy data ${I}_{ns}$ .", "If the nearest point $p$ between $H$ and other convex hulls lie in the noisy set, the convex hull distance is inevitably affected by ${I}_{ns}$ .", "By dividing the single hull $H$ into multiple local convex hulls, $H_1,H_2, \\ldots , H_n$, the noisy set is constrained to only one local convex hull.", "Assume the noisy sample is in one of the local convex hulls $H_{j}$ , then the new noisy set $\\widehat{H}_{ns}$ is defined as: $\\widehat{H}_{ns} \\mbox{~=~} \\lbrace p_{ns} \\rbrace , \\hspace{4.25pt} \\forall p_{ns}= \\alpha _{ns} {I}_{ns} + \\sum \\nolimits _{k} \\alpha _{k} {I}_k, \\hspace{4.25pt} p_{ns} \\notin H_{cl}, \\\\{I}_{k} \\in H_{j}, \\hspace{4.25pt} \\alpha _{ns} \\in (0,1], \\hspace{4.25pt} \\alpha _{k} \\in [0,1], \\hspace{4.25pt} \\alpha _{ns} + \\sum \\nolimits _{k} \\alpha _{k} \\mbox{~=~} 1.", "\\nonumber $ Comparing (REF ) and (REF ), we notice that $\\widehat{H}_{ns} \\subseteq H_{ns}.$ Unless all the noisy sample neighbours are in the same local convex hull as the noisy sample, the noisy set will be reduced.", "By controlling the number of clusters to divide the noisy sample neighbours, the noise level can be controlled.", "Figure REF illustrates the effect of local convex hulls extraction on noisy set reduction.", "It is therefore necessary to divide samples from a set into multiple subsets to extract multiple local convex hulls, such that samples in each subset are similar with minimal artificial features generated.", "Moreover, subsets should be far from each other.", "By dividing a single convex hull into multiple local convex hulls, unrealistic artificial variations can be diminished and noise can be constrained to local hulls.", "Figure: Conceptual illustration of local convex hulls extraction.", "To compare image sets A and B,(a) existing single convex hull based methods use all samples in an image set to constructa model and calculate the distance between two closest points (indicated as the black line connecting the synthesisedblue point and the real image with blue frame).The black image in set A is a noisy sample and the red area indicates the noisy set of A.As the blue point is synthesised from the combinations of the noisy sample and a real sample,it is also noisy.", "The yellow areas in A and B illustrate the region of unrealistic artificialvariations generated from the combination of samples far away from each other.", "(b) The proposed method divides image set A and B into 4 and 2 clusters separately anda local convex hull is constructed for each cluster individually.The black lines indicate the distance between image subsets A1 and B1, and A2 and B2.Notice that the closest points between subsets in (b) are completely different from those in (a).The blue points are artificially generated from real samples of subset B1 and B2 separately.As blue points are closely surrounded by the real sample points, they contain less artificial features.Moreover, after clustering, the noise set (red area) is significantly reduced and theunrealistic artificial variations (yellow area) are diminished.A direct solution is to apply $k$ -means clustering to extract local convex hulls [18].", "However, the local convex hulls extracted by $k$ -means clustering are generally very close to each other, without maximisation of distance between local convex hulls.", "We propose to use MMC clustering to solve this problem.", "For simplicity, we first investigate the problem for two local convex hulls.", "Given an image set $S = [{I}_1, {I}_2, \\ldots , {I}_n]$, images ${I}_{i}, i \\in 1,\\ldots ,n$ should be grouped into two clusters $C_{1}$ and $C_{2}$ .", "Two local convex hulls $H_{1}$ and $H_{2}$ can be constructed from images in the two clusters individually.", "The two clusters should be maximally separated that any point inside the local convex hull is far from any point in the other convex hull.", "It is equivalent to maximise the distance between convex hulls: $\\max _{C_1,C_2}D_{h}(H_{1},H_{2}).$ The solution of Eqn.", "(REF ) is equivalent to Eqn.", "(REF ).", "Because finding the nearest points between two convex hulls is a dual problem and has been proved to be equivalent to the SVM optimisation problem [2].", "Thus, $\\max _{C_1,C_2} D_{h}(H_{1},H_{2}) \\mbox{~=~} \\max _{C_1,C_2} \\max _{{w},b, \\gamma } \\gamma , \\hspace{4.25pt} s.t.", "\\hspace{4.25pt} {w}^{T}{I}_{k}+b \\ge \\gamma , \\\\{w}^{T}{I}_{m}+b \\le -\\gamma , ~ ||{w}||_2 = 1, ~ {I}_{k} \\in C_1, ~ {I}_{m} \\in C_2.", "\\nonumber $ If we combine all of the images to make a set ${I} = \\lbrace {I}_k, \\hspace{4.25pt} {I}_m\\rbrace $, then (REF ) is equivalent to: $\\max _{C_1,C_2} D_{h}(H_{1},H_{2}) & \\mbox{~=~} & \\max _{y} \\max _{{w},b, \\gamma } \\gamma , \\hspace{4.25pt} s.t.", "\\hspace{4.25pt} y_i({w}^{T} {I}_i + b) \\ge \\gamma , \\nonumber \\\\& & ||{w}||_2 = 1, y_i = \\left\\lbrace ^{+1, ~ {I}_i \\in C_1}_{-1, ~{I}_i \\in C_2} \\right..$ Maximisation of distance between clusters is the same as maximising the discrimination margin in Eqn.", "(REF ) and is proved to be equivalent to Eqn.", "(REF ) in [22].", "We thus employ the maximum margin clustering method to cluster the sample images in a set to extract two distant local convex hulls.", "Similarly, multi-class MMC can be used to extract multiple local convex hulls as in Eqn.", "(REF )." ], [ "Local Convex Hulls Comparison (LCHC)", " In this section, we describe two approaches to compare the local convex hulls: Complete Cluster Pairs Comparison and Adaptive Reference Clustering." ], [ "Complete Cluster Pairs (CCP) Comparison", " Similar to other multi-model approaches, local convex hulls extracted from two image sets can be used for matching by complete cluster pairs (CCP) comparisons.", "Assuming multiple convex hulls $H_{a}^{1}, H_{a}^{2},..., H_{a}^{m}$ are extracted from image set $S_{a}$, and local convex hulls $H_{b}^{1}, H_{b}^{2},..., H_{b}^{n}$ are extracted from image set $S_{b}$.", "The distance between two sets $S_{a}$ and $S_{b}$ is defined as the minimal distance between all possible local convex hull pairs: $D_{ccp}(S_{a},S_{b}) \\mbox{~=~} \\min _{k,j} D_{h}(H_{a}^{k},H_{b}^{j}), \\forall k \\in [1,m], \\forall j \\in [1,n].$ Although LCHE can suppress noises in local convex hulls, CCP will still inevitably match noisy data between image sets.", "Another drawback of this approach is that fixed clusters are extracted from each image set individually for classification.", "There is no guarantee that the clusters from different sets capture similar variations.", "Moreover, this complete comparison requires $m \\times n$ convex hull comparisons, which is computational expensive." ], [ "Adaptive Reference Clustering (ARC)", " Figure: Conceptual illustration of adaptive reference clustering.Set C is separately clustered according to the clusters of sets A and B.Set C is divided into 4 clusters during comparison with set A.The number in each cluster of set C indicates that samples in thisconvex hull are close to the corresponding convex hull (with the same number) in set A.The grey cluster pair indicates the two most similar clusters in sets A and C.Set C is divided into 2 clusters when matching with set B.Cluster 1 in set B is a noisy cluster as it contains a noisy sample.", "All samples inset C are far from the noisy cluster in set B, therefore no samples are assigned to cluster 1.The light grey cluster pair indicates the closest two convex hulls between sets B and C.In order to address the problems mentioned above, we propose the adaptive reference clustering (ARC) to adaptively cluster each gallery image set according to the reference clusters from the probe image set (shown in Fig.", "REF ).", "Assuming image set $S_{a}$ is clustered to extract local convex hulls $H_{a}^{1}, H_{a}^{2}, \\ldots , H_{a}^{m}$.", "For all images $I_{b}^{1}, I_{b}^{2}, \\ldots , I_{b}^{N}$ from image set $S_{b}$, we cluster these images according to their distance to the reference local convex hulls from set $S_{a}$.", "That is, each image $I_{b}^{i}$ is clustered to the closest reference local convex hull $k$ : $\\min _{k} D_{h}(I_{b}^{i},H_{a}^{k}) \\hspace{4.25pt} \\forall k \\in [1,m].$ After clustering all the images from set $S_{b}$ , maximally $m$ clusters are obtained and local convex hulls $\\widehat{H}_{b}^{1}, \\ldots , \\widehat{H}_{b}^{m}$ can then be constructed from these clusters.", "Since each cluster $\\widehat{H}_{b}^{j}$ is clustered according to the corresponding reference cluster $H_{a}^{j}$, we only need to compare the corresponding cluster pairs instead of the complete comparisons.", "That is $D_{arc}(S_{a},S_{b}) = \\min _{k} D_{h}(H_{a}^{k},\\widehat{H}_{b}^{k}), \\hspace{8.53581pt} \\forall k \\in [1,m].$ ARC is helpful to remove noisy data matching (as shown in Fig.", "REF ).", "If there is a noisy cluster $H_{a}^{n}$ in $S_{a}$ , when no such noise exists in $S_{b}$ , then all the images $I_{b}^{i}$ are likely to be far from $H_{a}^{n}$ .", "Therefore, no images in $S_{b}$ is assigned for matching with the noisy cluster." ], [ "Adjusting Number of Clusters", " One important problem for local convex hull extraction is to determine the number of clusters.", "This is because the convex combination region ( regions of points generated from convex combination of sample points) will be reduced as the number of clusters is increased (as shown in Figure REF ).", "The reduction may improve the system performance if the convex combination region contains many non-realistic artificial variations.", "However, the reduction will negatively impact the system performance when the regions are too small that some reasonable combinations of sample points may be discarded as well.", "An extreme case is that each sample point is considered as a local convex hull, such as nearest neighbour method.", "We thus devise an approach, denoted as average minimal middle-point distance (AMMD), to indirectly “measure” the region of non-realistic artificial variations included in the convex hull.", "Let $I_a$ be a point generated from convex combination of two sample points $I_i$ , $I_j$ in the set $S$ , $I_a = w_i I_i + w_jI_j, w_i+w_j = 1, w_i, w_j > 0; I_i,I_j \\in S$ .", "The minimum distance $\\delta _i$ between $I_a$ to all the sample points in the set $S$ indicates the probability of $I_a$ not being unrealistic artificial variations.", "One extreme condition is when $\\delta _i =0$ , then $I_a = I_k, I_k \\in S$ , which means $I_a$ is equivalent to a real sample.", "The further the distance, the higher the “artificialness” of point $I_a$ .", "In addition, the distance between $I_a$ and $I_i$ , $I_j$ must also be maximised in order to avoid measurement bias.", "Thus, a setting of $w_i= 0.5,w_j=0.5$ ( $I_a$ as the middle point) is applied.", "Having this fact at our disposal, we are now ready to describe the AMMD.", "For each sample point $I_i$ in the set $S$ , we find $I_j$ which is its furthest sample point in $S$ .", "The minimal middle-point distance of the sample point $I_i$ is defined via: $\\vspace{-4.25pt}\\delta _i = \\min _{k} ||I_a - I_k||_{2}, \\forall I_k \\in S.$ where $I_a$ is defined as $I_a = 0.5 I_i + 0.5 I_j$ .", "Finally, the AMMD of the set $S$ is computed via: $\\vspace{-2.125pt}\\Delta (S) = \\frac{1}{N}\\sum \\nolimits _{i = 1}^{N} \\delta _i.$ where $N$ is the number of sample points included in $S$ .", "By setting a threshold $\\Delta _{thd}$ to constrain AMMD, the region of non-realistic artificial variations can be controlled.", "An image set can be recursively divided into small clusters until the AMMD of all clusters is less than the threshold." ], [ "Complexity Analysis", " Given two convex hulls with $m$ and $n$ vertexes, the basic Gilbert-Johnson-Keerthi (GJK) Distance Algorithm finds the nearest points between them with complexity $O(mn\\log (mn))$ [3].", "The proposed approach needs a pre-processing step to cluster two image sets by MMC with a complexity of $O(sm)$ and $O(sn)$ [26], where $s$ is the sparsity of the data.", "Assuming each image set is clustered evenly with $M$ and $N$ clusters, to compare two local convex hulls, the complexity is $O(\\frac{m}{M}\\frac{n}{N}\\log (\\frac{m}{M}\\frac{n}{N}))$.", "For the complete cluster pairs comparison, the total run time would be $O(MN\\frac{m}{M}\\frac{n}{N}\\log (\\frac{m}{M}\\frac{n}{N})) + O(sm) + O(sn) = O(mn\\log (\\frac{mn}{MN})) + O(sm) + O(sn)$.", "The dominant term is on complete local convex hulls comparison, ie.", "$O(mn\\log (\\frac{mn}{MN}))$.", "By applying the adaptive reference clustering technique, one of the image set needs to be clustered according to the reference clusters from the other set with a complexity of $O(nM\\frac{m}{M}\\log (\\frac{m}{M}))$.", "Thus the total run time is $O(M\\frac{m}{M}\\frac{n}{N}\\log (\\frac{m}{M}\\frac{n}{N})) + O(sm) + O(nM\\frac{m}{M}\\log (\\frac{m}{M})) = O(mn\\log (\\frac{(mn)^{1/N}}{(MN)^{1/N}}))+ O(sm) + O(mn\\log (\\frac{m}{M}))$.", "The dominant term is on the adaptive reference clustering, ie.", "$O(mn\\log (\\frac{m}{M}))$." ], [ "Experiments", " We first compare the design choices offered by the proposed framework and then compare the best variant of the framework with other state-of-the-art methods.", "The framework has two components: Local Convex Hulls Extraction (LCHE) and Local Convex Hulls Comparison (LCHC).", "There are two sub-options for LCHE: (1) Maximum Margin Clustering (MMC) versus $k$ -means clustering (Section ); (2) Fixed versus Adjustable number of clusters (Section  REF ).", "There are also two sub-options for LCHC: (1) Adaptive Reference Clustering (ARC) versus Complete Cluster Pairs (CCP) (Section REF and REF ); (2) Affine Hull Distance (AHD) versus Convex Hull Distance (CHD) (Section ).", "We use two single hull methods as baseline comparisons: the method using Affine Hull Distance (AHD) [5] and the method using Convex Hull Distance (CHD) [5].", "Honda/UCSD [12] is used to compare different variants of the framework to choose the best one.", "We use an implementation of the algorithm proposed in [14] to solve MMC optimisation problem.", "To eliminate the bias on large number of images in one cluster, we only select the top $m$ closest images to the reference cluster for the ARC algorithm, wherein $m$ is the number of images in the reference cluster.", "The clusters extracted from the query image set are used as reference clusters to adaptively cluster each gallery image set individually.", "In this way, the reference clusters stay the same for each query, thus the distances between the query set and each gallery set are comparable.", "The best performing variant of the framework will then be contrasted against the state-of-the-art approaches such as Sparse Approximated Nearest Points (SANP) [10] (a nearest point based method), Mutual Subspace Method (MSM) [23] (a subspace based method), and the Manifold Discriminant Analysis (MDA) method [20] (a multi-model based method).", "We obtained the implementations of all methods from the original authors.", "The evaluation is done in three challenging image-set datasets: Honda/UCSD [12], CMU MoBO [7] and the ETH-80 [13] datasets.", "Table: Comparing maximum margin clustering (MMC) vs kk-meansclustering as well as fix number vs adjustable number of clusters on local convex hulls extraction.The results are shown for AHD and CHD methods combined with adaptive reference clustering (ARC) technique on the Honda/UCSD dataset using 50 imagesper set.", "For fixed number of clusters, numbers shown in brackets indicate the corresponding optimal number of clusters.For adjustable number of clusters, numbers shown in square brackets are the optimal thresholds set foraverage minimal middle-point distance (AMMD).The evaluation results of different variants for local convex hulls extraction are shown in Table REF .", "We only show the hyper-parameters which give the best performance for each variant ( the best number of cluster is shown in bracket for fixed cluster number and the best threshold value is shown in square bracket for adjustable cluster number).", "From this table, it is clear that all the proposed variants outperform the baseline single hull approaches, validating our argument that it is helpful to use multiple local convex hulls.", "MMC variants outperform the $k$ -means counterparts indicating that maximising the local convex hull distance for clustering leads to more discrimination ability for system.", "The adjustable cluster number variant achieves significant performance increase over fixed number of cluster (between 1.5% points to 5.2% points).", "We also note that the performance for adjustable number of clusters is not very sensitive to the threshold.", "For instance, the performance only drops by 2% when the threshold $\\Delta _{thd}$ is set to 5000 for normalised images.", "In summary, MMC and adjustable number of clusters combined with ARC achieve the best performance over all.", "Table: Comparing the proposed method with SANP, MSM, MDA and Nearest Neighbour (NN) on Honda data set with strong noise.80% of the samples in each image set are replaced by noise.Table: Average time cost (in seconds) to compare two image sets on Honda dataset.", "The Convex Hull Distance (CHD) method combined with the proposed Complete Cluster Pairs (CCP) and Adaptive Reference Clustering (ARC) techniques are evaluated.", "`noc' indicates the number of clusters.Results in Table REF indicate that when strong noises occur in image sets, the proposed ARC approach considerably outperforms other methods, supporting our argument that ARC is helpful to remove noisy data matching.", "It is worthy to note that with strong noises, nearest neighbour (NN) method performs better than single hull methods.", "Figure  is the summary of the best variant found previously contrasted with the state-of-the-art methods.", "Normalised images are used for Honda dataset and raw images are used for ETH-80 and CMU-MoBo datasets.", "A fixed threshold $\\Delta _{thd}=5000$ is set for all datasets for adjustable number of clustering.", "It is clear that the proposed system consistently outperforms all other methods in all datasets regardless whether AHD or CHD are used.", "In the last evaluation, we compare the time complexity between the variants.", "The average time cost to compare two image sets is shown in Table REF .", "For small number of images per set, extracting multiple local convex hulls is slower than using only single convex hull because of extra time for MMC and adaptive reference clustering.", "However, for large number (greater than 100) of images per set, the proposed method is about three times faster than the CHD method.", "That is because the number of images in each cluster is significantly reduced, leading to considerably lower time cost for local convex hulls comparisons." ], [ "Conclusions and Future Directions", " In this paper, we have proposed a novel approach to find a balance between single hull methods and nearest neighbour method.", "Maximum margin clustering (MMC) is employed to extract multiple local convex hulls for each query image set.", "The adjustable number of clusters is controlled by restraining the average minimal middle-point distance to constrain the region of unrealistic artificial variations.", "Adaptive reference clustering (ARC) is proposed to cluster the gallery image sets resembling the clusters of the query image set.", "Experiments on three datasets show considerable improvement over single hull methods as well as other state-of-the-art approaches.", "Moreover, the proposed approach is faster than single convex hull based method and is more suitable for large image set comparisons.", "Currently, the proposed approach is only investigated for MMC and $k$ -means clustering.", "Other clustering methods for local convex hulls extraction, such as spectrum clustering [17] and subspace clustering [6] and their effects on various data distributions need to be investigated as well.", "Acknowledgements.", "This research was funded by Sullivan Nicolaides Pathology, Australia and the Australian Research Council Linkage Projects Grant LP130100230.", "NICTA is funded by the Australian Government through the Department of Communications and the Australian Research Council through the ICT Centre of Excellence program." ] ]
1403.0320
[ [ "Landauer, Kubo, and microcanonical approaches to quantum transport and\n noise: A comparison and implications for cold-atom dynamics" ], [ "Abstract We compare the Landauer, Kubo, and microcanonical [J. Phys.", "Cond.", "Matter {\\bf 16}, 8025 (2004)] approaches to quantum transport for the average current, the entanglement entropy and the semiclassical full-counting statistics (FCS).", "Our focus is on the applicability of these approaches to isolated quantum systems such as ultra-cold atoms in engineered optical potentials.", "For two lattices connected by a junction, we find that the current and particle number fluctuations from the microcanonical approach compare well with the values predicted by the Landauer formalism and FCS assuming a binomial distribution.", "However, we demonstrate that well-defined reservoirs (i.e., particles in Fermi-Dirac distributions) are not present for a substantial duration of the quasi-steady state.", "Thus, the Landauer assumption of reservoirs and/or inelastic effects is not necessary for establishing a quasi-steady state.", "Maintaining such a state indefinitely requires an infinite system, and in this limit well-defined Fermi-Dirac distributions can occur.", "A Kubo approach -- in the spirit of the microcanonical picture -- bridges the gap between the two formalisms, giving explicit analytical expressions for the formation of the steady state.", "The microcanonical formalism is designed for closed, finite-size quantum systems and is thus more suitable for studying particle dynamics in ultra-cold atoms.", "Our results highlight both the connection and differences with more traditional approaches to calculating transport properties in condensed matter systems, and will help guide the way to their simulations in cold-atom systems." ], [ "Introduction", "Experimental investigations of transport phenomena in ultra-cold atoms confined in engineered optical potentials offer a test bed for transport theories at the nanoscale.", "Several phenomena, such as the sloshing motion of an atomic cloud in optical lattices [1], directed transport using a quantum ratchet [2], relaxation of noninteracting and interacting fermions in optical lattices [3], and others have been demonstrated.", "Their applications in atomtronics [4], which aims at simulating electronics by using controllable atomic systems, are promising [5], [6], [7], [8], [9].", "It is thus important to develop proper theoretical and computational methods to direct future progress in this field.", "Due to the quantum nature of atoms, finite particle numbers, and small sizes of these systems, the applicability of semi-classical approaches, such as the Boltzmann equation, become questionable.", "The Landauer formalism [10], [11], which has been widely implemented in mesoscopic physics, is naturally appealing for studying transport phenomena in ultra-cold atoms.", "Those approaches and their generalizations have been applied to study various problems in cold atoms [12], [13], [14], [15], [16], [17].", "In addition to steady-state properties, one may want to study fluctuation effects and correlations using full-counting statistics (FCS) [18].", "An examination of the underlying assumptions of those well-known formalisms, however, raise questions on their applicability to ultra-cold atoms.", "The Landauer formalism, which is designed for open systems, assumes the existence of two reservoirs that supply particles to be transmitted through a junction region.", "Since the particle number and energy (when no external time-dependent fields are present) in ultra-cold atomic experiments are (to a very good approximation) conserved, the concept of a reservoir does not necessarily hold.", "FCS generally assumes the transmitted particles behave like billiards with a well-defined tunneling probability distribution.", "Whether such an assumption holds true in finite, closed systems will determine whether the formalism can be applied to cold atom experiments as well.", "An alternative approach for studying transport in quantum systems is within the microcanonical formalism (MCF) [19], [20], [11].", "This formalism is based on using closed quantum systems driven out of equilibrium by a change of parameters (e.g., an external bias or a density imbalance) to calculate transport properties.", "The conservation of particle number and energy are naturally built into this formalism, and there is no need to introduce reservoirs and one can fully preserve the wave nature of the particles.", "This formalism has also been integrated with density-functional theory for investigating quantum transport through atomic or molecular junctions [21], [22].", "The microcanonical formalism is particularly suitable for ultra-cold atoms, which are accurately modeled as isolated quantum systems.", "In this respect, the formalism has already been developed to study transport phenomena in these systems [23], [24], [25], [26], [27], [28].", "Figure: Schematic of a one dimensional lattice and transport induced by (a) application of a bias at t=0t=0, where a step-functionbias is applied to the system and a current flows through the middlelink or (b) connecting a link between the two initially disconnected parts at t=0t=0.The goal of this paper is to compare the microcanonical approach to the Landauer formalism and determine which assumptions lead to the same observables, such as the average current and FCS.", "The MCF is generically applicable to closed quantum systems, and here we use transport of ultra-cold non-interacting fermions in one-dimensional (1D) optical lattices as a particular example.", "A possible setup is shown in Figure REF .", "Unlike electronic systems where the Coulomb interactions cannot be really switched off, and therefore for which this comparison would be more academic, cold atoms experiments allow for a relatively easy tuning of interactions among particles down to the non-interacting limit.", "The microcanonical formalism can, of course, be applied to systems with Coulomb interactions.", "However, here we focus only on its applications to noninteracting cold-atom systems.", "While electrons are naturally confined in solid-state systems, a background harmonic trapping potential is often implemented in addition to the optical lattice for confining atoms.", "However, recent advance in trapping atoms in ring-shape geometries [5], [29] or a uniform potential [30] makes it possible to consider homogeneous cold-atom systems.", "Moreover, a weak background harmonic potential does not change the qualitative conclusions from the MCF, as illustrated in Ref. [23].", "Therefore we focus here on the dynamics of cold atoms in optical lattices without a background harmonic trapping potential.", "We find that the steady-state current and particle number fluctuations from the microcanonical formalism approach the values of the average current and FCS predicted from the Landauer formalism already at moderate system sizes.", "However, we also find – for finite times – that one of the assumptions of the Landauer formalism is unnecessary: The particle distributions in the two lattices supplying/absorbing particles do not need to be populated according to Fermi-Dirac distributions.", "In fact, their occupation deviates from the equilibrium distribution during the whole duration of a quasi-steady state.", "Furthermore, the results from the microcanonical formalism agree with the predictions from the FCS semi-classical formula by assuming a binomial distribution of the transmitted particles, ruling out alternative semi-classical descriptions.", "To connect the different approaches, we also develop a Kubo formalism based on the microcanonical picture of transport, which we use to calculate explicit expressions for transport in closed systems.", "This gives us an analytical method to investigate dynamical transport phenomena in nanoscale and ultracold atomic systems.", "In addition to the average current and FCS, we also investigate the dynamical evolution of the entanglement entropy, which quantifies the correlations between two connected systems.", "The entanglement entropy is of broad interest in many fields, ranging from black hole physics [31] to quantum information science [32].", "This quantity can be easily evaluated using the microcanonical formalism.", "A semi-classical formula based on FCS of two noninteracting fermionic systems connected by a junction has been derived in Ref.", "[18] and generalized to many-body systems [33], [34].", "Ref.", "[18] predicts a linear growth of the entanglement entropy as time increases.", "Again, we find that the results from the microcanonical formalism match the prediction from the semi-classical formula by assuming a binomial distribution of the transmitted particles.", "Assuming an alternative distribution results in predictions that are readily distinguishable.", "This paper is organized as follows: Section  reviews the Landauer formalism and its assumptions.", "Section  introduces the microcanonical formalism and its applications.", "The spatially resolved current from the MCF is discussed in Section .", "Section  reviews the FCS.", "Section  shows the absence of memory effects in transport of noninteracting fermions.", "Section  compares the results from the MCF and Landauer formalism.", "Importantly, the deviation from the equilibrium Fermi-Dirac distribution is clearly demonstrated.", "Section  shows the light-cone structure of wave propagation monitored by the MCF.", "Section  reviews the Kubo formalism and how it helps connect the two approaches.", "Finally, Section  concludes our study with suggestions of future work." ], [ "Landauer formalism", "By assuming the existence of a steady-state current between two reservoirs bridged by a central link, the current can be estimated from the Landauer formula with the help of, e.g., Green's functions [10], [11].", "For a detailed description of the physical assumptions behind this formalism we refer the reader to Ref.", "[11].", "Here, we mention only the assumptions that will be relevant for our comparison with the microcanonical formalism: (1) A steady-state current is assumed to exist.", "Whether a steady-state current always emerges from a given nonequilibrium condition is not at all obvious [11].", "(2) Two macroscopic reservoirs – holding noninteracting fermions populated according to Fermi-Dirac distributions – are also assumed.", "The separation of the system into reservoirs and a region of interest is not always easy to determine for an actual physical structure.", "(3) The transport at the junction does not provide any feedback to the reservoirs.", "While one can construct configurations where a steady-state current does not exist [35], in the case where two 1-D chains are connected by a central junction (as considered in this paper), there is always a steady-state current, as will be verified in the microcanonical formalism (see Section ).", "Therefore, we do not focus here on assumption (1), but rather on (2) and (3).", "As will be shown in Sec.", ", the distributions on both sides deviates from the Fermi-Dirac distribution when the system maintains a steady state so assumption (2) is not necessary for observing a steady-state current.", "Moreover, Section  will show that density changes can propagate into regimes away from the junction so there can be feedback and assumption (3) is also not necessary.", "Figure: (Color online) The transmission coefficients T(E)T(E) at μ B =0\\mu _B=0 for (a) the weak-linkcase (t ¯ ' /t ¯=1,0.5,0.2\\bar{t}^{\\prime }/\\bar{t}=1,0.5,0.2 from top to bottom)and (b) the central-site case (E C /t ¯=0,2,8E_{C}/\\bar{t}=0,2,8 from top tobottom).On the other hand, in this section we calculate the current using the Landauer formalism for two configurations of a junction between two 1D lattices.", "One can insert a link with a tunable hopping coefficient $\\bar{t}^{\\prime }$ in the middle of a chain, which we call the weak-link case, or insert a central site with tunable on-site energy $E_{C}$ , which we call the central-site case.", "In cold-atom experiments it has been shown that one can suppress the transmission of atoms by introducing an optical barrier [5] or by introducing a constriction in the trapping potential [6].", "Therefore the tunneling coefficient and onsite energy may be tuned simultaneously.", "Here we separate the effects of tuning the two parameters and one will see that there is no observable difference if the transmission coefficient $T$ can be found and physical quantities are compared at the same $T$ .", "We consider a uniform bias $E_{L}=\\mu _{B}/2$ on the left half and, similarly, $E_{R}=-\\mu _{B}/2$ on the right half.", "By making the two lattices on both sides semi-infinite, they behave as the two reservoirs with different electrochemical potentials.", "The hopping coefficient is denoted by $\\bar{t}$ and the unit of time is $t_{0}=\\hbar /\\bar{t}$ .", "We set the electric charge $e\\equiv 1$ and $\\hbar \\equiv 1$ .", "The length is measured in units of the lattice constant.", "The Green's function of the left (right) semi-infinite chain can be derived using recursive relations, which lead to [36] $G_{L(R)}(E)=1/[E-E_{L(R)}-\\Sigma _{L(R)}(E)]$ , where $\\Sigma _{L(R)}=(1/2)\\left[E-E_{L(R)}-i\\sqrt{4\\bar{t}^{2}-(E-E_{L(R)})^{2}}\\right]$ .", "The retarded Green's function of the junction is $G(E)=1/[E-E_{C}-\\Sigma _{CL}-\\Sigma _{CR}]$ , where $\\Sigma _{CL(CR)}=V_{CL(CR)}^{2}G_{L(R)}(E)$ and $V_{CL(CR)}$ is the coupling to the left (right) chain [11], [37].", "The current (including both spins) is [11] $I=\\frac{1}{\\pi }\\int _{-\\infty }^{\\infty }dE(f_{L}-f_{R})T(E)=\\frac{1}{\\pi }\\int _{-\\frac{\\mu _{B}}{2}}^{\\frac{\\mu _{B}}{2}}dET(E),$ where the reservoirs are taken to be at zero temperature, as we will throughout this work.", "The transmission coefficient is $T(E)=\\Gamma _{L}\\Gamma _{R}|G(E)|^{2},$ where $f_{L(R)}$ denotes the density distribution of the left (right) chain, i.e., the Fermi-Dirac distribution function, and $\\Gamma _{L (R)}=-2\\mbox{Im}\\Sigma _{CL (CR)}$ .", "For a uniform chain with $\\bar{t}^{\\prime }=\\bar{t}$ , $V_{CL(CR)}=\\bar{t}$ .", "After some algebra, the current is given by $I=\\frac{1}{\\pi }\\int _{-\\frac{\\mu _{B}}{2}}^{\\frac{\\mu _{B}}{2}}\\frac{4g_{L}g_{R}dE}{\\mu _{B}^{2}+(g_{L}+g_{R})^{2}},$ where $g_{L(R)}=\\sqrt{4\\bar{t}^{2}-(E-E_{L(R)})^{2}}$ .", "To the leading order of $\\mu _{B}$ , Eq.", "(REF ) gives $I \\simeq \\mu _B \\bar{t}/\\pi $ .", "Moreover, it can be shown that $T(E=0)\\rightarrow 1$ as $\\mu _{B}\\rightarrow 0$ .", "For the weak-link case, if we take the last site of the left chain as the central site, $V_{CL}=\\bar{t}$ , $V_{CR}=\\bar{t}^{\\prime }$ , and $E_{C}=E_{L}$ .", "The current is $I=\\frac{1}{\\pi }\\int _{-\\frac{\\mu _{B}}{2}}^{\\frac{\\mu _{B}}{2}}\\frac{4g^{2}g_{L}g_{R}dE}{[(E-E_L)-g^{2} (E-E_R)]^2+(g_L+ g^{2}g_R)^2},$ where $g\\equiv (\\bar{t}^{\\prime }/\\bar{t})$ .", "When $g\\ll 1$ , to the leading order of $g$ and then to the leading order of $\\mu _B$ , one obtains $I \\simeq 4\\mu _{B}g^2\\bar{t}/\\pi $ .", "For the central-site case, $V_{CL}=V_{CR}=\\bar{t}$ and $E_{C}$ can be tuned.", "The current is $I=\\frac{1}{\\pi }\\int _{-\\frac{\\mu _{B}}{2}}^{\\frac{\\mu _{B}}{2}}\\frac{4g_{L}g_{R}dE}{(E_L+E_R-2E_C)^2+(g_L+ g_R)^2}.$ Figure REF shows $T(E)$ , which is symmetric about $E=0$ , for both cases with selected parameters." ], [ "Micro-canonical formalism", "In the micro-canonical approach to quantum transport [19], one considers a finite system (say two electrodes and a junction) and a finite number of particles with Hamiltonian $H$ .", "The system is prepared in an initial state $|\\Psi _0\\rangle $ which is an eigenstate of some Hamiltonian $H_{0} \\ne H$ .", "From a physical point of view this initial state may represent, e.g., a charge, particle, or energy imbalance between the two finite electrodes that sandwich the junction.", "The system is then left to evolve from this initial condition under the dynamics of $H$ , and the average current across some surface or any other observable is monitored in time.", "The dynamics considered here may be considered as quantum quenches [38], [39].", "Note that, even if we assume the two electrodes biased as in the Landauer formalism, in this closed-system approach it is not at all obvious that the average current establishes any (quasi-)steady state in the course of time [19], [11], [40]." ], [ "Implementation of the MCF", "We adopt the implementation of the micro-canonical formalism as discussed in Refs.", "[23], [24], which is an extension of the scheme proposed in Ref. [20].", "One advantage of this extended scheme is that the dynamics of particle density fluctuations, entanglement entropy, and density distributions can be easily monitored.", "We consider a one-dimensional Hamiltonian $H=H_{L}+H_{R}+H_{C}$ , where $H_{L/R}$ is a lattice of $N/2$ sites.", "The system is filled with $N/2$ two-component fermions (with equal number in each species).", "In the tight-binding approximation we choose $H_{L/R}=-\\bar{t}\\sum _{\\langle ij\\rangle ,L/R}c_{i}^{\\dagger }c_{j}+E_{L/R}\\sum _{i\\in L/R}c_{i}^{\\dagger }c_{i}.$ Here $\\langle ij\\rangle $ denotes nearest-neighbor pairs and we suppress the spin index, and explicitly state where a summation over the spin is performed in our results.", "Here we only consider quadratic Hamiltonians.", "In the presence of other interaction terms, one may need to consider approximate methods [26].", "We consider two possible ways to set the system out of equilibrium.", "In the first scenario the system is initially prepared in the ground state of the unbiased Hamiltonian $H_{0}$ with $E_{L}=E_{R}=0$ and then it evolves according to a biased Hamiltonian $H$ .", "All conclusions in this work remain unchanged if we instead prepare the system in the ground state of the biased $H$ and then let it evolve according to the unbiased $H_{0}$ .", "In other words, there is a correspondence between a particle imbalance and an energy imbalance for the systems we consider.", "In the second scenario two initially disconnected lattices are connected, where one can not swapped the roles of $H$ and $H_0$ .", "We remark that the first scenario is closely related to the studies of Ref.", "[41] where photons are introduced to adjust the onsite energy of atoms in certain parts of the lattice.", "The second scenario is relevant to the case where an optical barrier separating the lattice into two parts is lifted [24].", "For the weak-link case $H_{C}=-\\bar{t}^{\\prime }(c_{N/2}^{\\dagger }c_{N/2+1}+c_{N/2+1}^{\\dagger }c_{N/2})$ , where $0\\le \\bar{t}^{\\prime }\\le \\bar{t}$ , while for the central-site case $H_{C}=E_{C}c_{N/2+1}^{\\dagger }c_{N/2+1}-\\bar{t}(c_{N/2}^{\\dagger }c_{N/2+1}+c_{N/2+1}^{\\dagger }c_{N/2})-\\bar{t}(c_{N/2+1}^{\\dagger }c_{N/2+2}+c_{N/2+2}^{\\dagger }c_{N/2+1})$ .", "For time $t<0$ , $E_{L/R}=0$ and the system is in the ground state of $H_{0}$ .", "For $t>0$ we set $E_{L}=\\mu _{B}/2$ and $E_{R}=-\\mu _{B}/2$ and let the system evolve.", "Figure REF illustrates this process for the weak-link case.", "A uniform chain with $\\bar{t}^{\\prime }=\\bar{t}$ in the weak-link case has been shown to have a quasi steady-state current (QSSC) at a small bias [20] for a system as small as $N=60$ .", "The QSSC is defined as a plateau in the current as a function of $t$ and it usually spans the range $(N/4)t_{0}\\le t\\le (N/2)t_{0}$ .", "In the thermodynamic limit with finite filling, the QSSC becomes a steady current [24].", "In contrast, noninteracting bosons in their ground state do not support a QSSC [23], [24].", "The dependence of the magnitude of the QSSC on the initial filling was discussed in Refs.", "[23], [24] and here we consider the case with $N_p/N=1/2$ , where $N_p$ denotes the number of particles in the system, unless specified otherwise.", "To gain more insight into the dynamics of the system, we write down the correlation matrix $C(t)$ with elements $c_{ij}(t)=\\langle GS_{0}|c_{i}^{\\dagger }(t)c_{j}(t)|GS_{0}\\rangle $ , where $|GS_{0}\\rangle $ denotes the ground state of $H_{0}$ , and derive the current and entanglement entropy from it.", "One can use unitary transformations $c_{j}=\\sum _{k}(U_{0})_{jk}a_{k}$ and $c_{j}=\\sum _{k}(U_{e})_{jk}d_{k}$ to rewrite $H_{0}$ and $H$ as $H_{0}=\\sum _{k}\\epsilon _{k}^{0}a_{k}^{\\dagger }a_{k};\\mbox{ }H=\\sum _{p}\\epsilon _{p}^{e}d_{p}^{\\dagger }d_{p}.$ Here $\\epsilon _{k}^{0}$ and $\\epsilon _{p}^{e}$ are the energy spectra of $H_{0}$ and $H$ , respectively.", "The initial state is then $|GS_{0}\\rangle =(\\Pi _{k=1}^{N/2}a_{k}^{\\dagger })|0\\rangle $ , where $|0\\rangle $ is the vacuum.", "From the equation of motion $i(dc_{j}(t)/dt)=[c_{j}(t),H]$ it follows $c_{j}(t)=\\sum _{p}(U_{e})_{jp}d_{p}(0)\\exp (-i\\epsilon _{p}^{e}t)$ .", "The initial correlation functions are $\\langle GS_{0}|a_{k}^{\\dagger }(0)a_{k^{\\prime }}(0)|GS_{0}\\rangle =\\theta (N/2-k)\\delta _{k,k^{\\prime }}$ since fermions occupy all states below the Fermi energy, where $\\theta (N/2-k)$ is 1 if $k\\le N/2$ , and 0 otherwise.", "Then it follows $c_{ij}(t) & = & \\sum _{p,p^{\\prime }=1}^{N}(U_{e}^{\\dagger })_{pi}(U_{e})_{jp^{\\prime }}D_{pp^{\\prime }}(0)e^{i(\\epsilon _{p}^{e}-\\epsilon _{p^{\\prime }}^{e})t};\\\\D_{pp^{\\prime }}(0) & = & \\sum _{m,m^{\\prime }=1}^{N}\\sum _{k=1}^{N/2}(U_{e}^{\\dagger })_{p^{\\prime }m^{\\prime }}(U_{0})_{m^{\\prime }k}(U_{0}^{\\dagger })_{km}(U_{e})_{mp}.\\nonumber $ Here $D_{pp^{\\prime }}(0)\\equiv \\langle GS_{0}|d_{p}^{\\dagger }(0)d_{p^{\\prime }}(0)|GS_{0}\\rangle $ ." ], [ "Current, entanglement entropy, and particle fluctuations", "The current flowing from left to right for one species is $I=-\\langle d\\hat{N}_{L}(t)/dt\\rangle $ , where $\\hat{N}_{L}(t)=\\sum _{i=1}^{N/2}c_{i}^{\\dagger }(t)c_{i}(t)$ .", "It can be shown that for the Hamiltonian considered here, $I=4\\bar{t}^{\\prime }\\mbox{Im}\\lbrace c_{(N/2),(N/2+1)}(t)\\rbrace $ , where a factor of 2 for the two spin components is included.", "This is equivalent to the expectation value of the current operator $\\hat{I}=-i\\bar{t}^{\\prime }(c_{N/2}^{\\dagger }c_{N/2+1}-c_{N/2+1}^{\\dagger }c_{N/2})$ .", "The MCF can be generalized to include finite-temperature effects in the initial state [23], but here we focus on the ground state.", "Figure: (Color online) The current from the Landauer formula for the weak-link case, Eq.", "(), (blackline) and the currents in the quasi-steady states of the micro-canonical simulations for the weak-link case (red circles)and the central-site case (green squares) as a function of the transmissioncoefficient T=T(E=0)T=T(E=0).", "Insets: Currents as a function of time fromthe micro-canonical simulations (solid lines; the dashed lines represent the Landauer value).The upper (lower) one corresponds to the weak-link (central-site)case.", "From top to bottom for the upper inset: t ¯ ' /t ¯=1.0,0.5,0.1\\bar{t}^{\\prime }/\\bar{t}=1.0,0.5,0.1and for the lower inset: E C /t ¯=0,2,8E_{C}/\\bar{t}=0,2,8.", "Here μ B =0.2t ¯\\mu _{B}=0.2\\bar{t}and N=512N=512.Figure REF compares the current predicted by the Landauer formula for the weak-link case, Eq.", "(REF ) to the simulations using the MCF for the weak-link as well as the central-site cases with $\\mu _{B}=0.2\\bar{t}$ .", "In the limit where $\\mu _{B}\\rightarrow 0$ , the Landauer formulas for the central-site case, Eq.", "(REF ), produces results that fully agree with the results from the weak-link case, Eq.", "(REF ).", "When $\\mu _{B}$ is finite, the two cases differ by a negligible amount due to the slightly different $T(E)$ .", "One can see that the currents from the MCF agree well with that from the Landauer formula.", "The entanglement entropy between the left and right halves, $s$ , for one species at time $t$ can be evaluated as follows [18].", "We define a $(N/2)\\times (N/2)$ matrix $M=P_{L}C(t)P_{L}$ with elements $M_{ij}$ , where the projection operator $P_{L}=diag({\\bf 1}_{N/2},{\\bf 0}_{N/2})$ .", "Then the entanglement entropy can be obtained from the expression $s=-\\mbox{Tr}[M\\log M+(1-M)\\log (1-M)].$ The Hermitian matrix $M_{ij}$ has eigenvalues $v_{i}$ , $i=1\\cdots N/2$ .", "Then $s(t)=\\sum _{i=1}^{N/2}[-v_{i}\\log (v_{i})-(1-v_{i})\\log (1-v_{i})].$ We use $\\log $ base 2, as is convention.", "This expression may be further simplified by using approximations from the semi-classical FCS [18] and will be discussed later on.", "The noninteracting fermions studied here may be regarded as a limiting case of a XXZ spin chain, whose entanglement entropy (due to the dynamics of magnetization) has been studied in Ref. [42].", "Now we derive the full quantum-mechanical expressions for the equal-time number fluctuations of the left half lattice.", "Let $\\hat{n}_{i}=c_{i}^{\\dagger }c_{i}$ and $\\hat{N}_{L}=\\sum _{i=1}^{N/2}\\hat{n}_{i}$ .", "Then the number of particles in the left part is $N_{L}=\\langle \\hat{N}_{L}\\rangle =\\sum _{i=1}^{N/2}c_{ii}$ .", "We define the equal-time number fluctuations of the left half as $\\Delta N_{L}^{2}=\\langle (\\hat{N}_{L}-N_{L})^{2}\\rangle =\\langle \\hat{N}_{L}^{2}\\rangle -N_{L}^{2}.$ The moments of $\\hat{N}_{L}$ can be obtained from $\\langle \\hat{N}_{L}^{2}\\rangle & = & \\sum _{i=1}^{N/2}\\langle \\hat{n}_{i}^{2}\\rangle +2\\sum _{i<j}^{N/2}\\langle \\hat{n}_{i}\\hat{n}_{j}\\rangle $ From Wick's theorem or exact calculations, $\\langle \\hat{n}_{i}^{\\alpha }\\rangle =\\langle \\hat{n}_{i}\\rangle =n_{i}$ for all positive integer $\\alpha $ , where $n_{i}=c_{ii}$ .", "The other correlation functions can be obtained from Wick's theorem so that $\\langle \\hat{n}_{i}\\hat{n}_{j}\\rangle & = & n_{i}n_{j}-|\\langle c_{i}^{\\dagger }c_{j}\\rangle |^{2}.$" ], [ "Spatial resolution of the current in MCF", "We stress an important feature of the MCF formalism.", "One can see from Eq.", "(REF ) and its context that MCF monitors the dynamics in both energy basis and real space.", "In contrast, the Landauer formalism as shown in Eq.", "(REF ) only reveals information in the energy basis.", "The ability of the MCF to trace the dynamics in real space allows us to address a crucial question: How do particles from different sites contribute to the current?", "To clearly demonstrate the importance of the information from the dynamics in real space, we consider a simplified initial condition where $N$ lattice sites are divided into the left $N/2$ sites and the right $N/2$ sites with each left site occupied by one fermion and each right site empty.", "We consider a uniform lattice here with a tunneling coefficient $\\bar{t}$ .", "The corresponding correlation matrix is $c_{ij}(t=0)=\\delta _{ij}$ if $1\\le i,j\\le (N/2)$ and zero otherwise.", "Eq.", "(REF ) becomes $c_{ij}(t)=\\sum _{m=1}^{N/2}\\sum _{p,p^{\\prime }=1}^{N}(U_{e}^{\\dagger })_{pi}(U_{e})_{jp\\prime }(U_{e}^{\\dagger })_{mp}(U_{e})_{p^{\\prime }m}e^{i(\\epsilon _{p}^{e}-\\epsilon _{p^{\\prime }}^{e})t}$ One important insight from this expression is that the index $m$ traces the contribution from the initially filled $m$ -th site on the left.", "Therefore in the current $I=-2\\bar{t}\\mbox{Im}(c_{N/2,N/2+1})$ it is meaningful to discuss where does the current come from as time evolves.", "This simplified case, despite its compactness and clarity, is relevant to several situations realizable in experiments.", "Two potential examples are: (1) initially a large step-function bias is applied to a nanowire with a small energy bandwidth so that all mobile particles are driven to the left half and then the bias is removed to allow a current to flow, and (2) ultra-cold atoms are loaded in an optical lattice so that there is one atom per lattice site.", "Then a focus laser beam excites the atoms on the right half lattice so that they leave the lattice and create a vacuum region.", "The atoms on the filled left part will then flow to the right and build a current.", "Thus the physics of this simplified case is relevant to both our deeper understanding of transport phenomena and advances in experiments.", "Figure: (Color online) Spatial decomposition of the contribution to the current.The black line labeled I tot I_{tot} showsthe current from an N=512N=512 lattice with the left half initiallyfilled with one fermion per site.", "We plot the contributions from sectionsof 32 sites each to the left of the middle of the whole latticeand the corresponding currents show up in bursts.", "The bursts, fromleft to right on the plot, correspond to the current from the first,second, ..., sixth sections of 32 sites to the left away from themiddle (the 256-th site).Figure REF shows the total current of this case with $N=512$ and clearly there is a quasi steady-state current.", "When we determine the contributions from each section of 32 lattices sites to the left of the middle (256-th site), each contribution comes in a burst following the previous burst from the section to its right.", "Thus the burst from the section of the 225-th site to the 256-th site crosses the middle first, followed by the burst from the section of the 193-th site to the 224-th sites, and so on, with each burst having a decaying tail.", "This succession of bursts gives a physical justification of the reason the semi-classical distribution assumed in FCS is binomial, and why other distributions can be excluded (see below).", "Each burst peak plus all the tails from previous bursts add up to maintain the observed quasi steady-state current.", "The MCF formalism thus provides more insights into how a quasi steady-state current forms and this is certainly beyond the scope of the Landauer's formalism.", "Since some spin chain problems can be mapped to fermions in 1D, our study is relevant to the dynamics of magnetization in these cases as well [43]." ], [ "Semi-classical FCS formalism", "For two 1D non-interacting fermionic systems connected by a central barrier, it has been proposed [18] that an expression for the entanglement entropy can be derived from FCS assuming a binomial distribution of the transmitted particle number.", "In linear response, it has the form $\\frac{\\Delta s}{\\Delta t}=-2\\frac{\\mu _{B}}{h}[T\\log T+(1-T)\\log (1-T)].$ Here, $T$ is the transmission coefficient at the Fermi energy.", "The second moment of transmitted particle numbers, $\\mathcal {C}_{2}$ , is important because it may be inferred from shot-noise measurements.", "Moreover, the spectrum of current fluctuations through the barrier, $P_{sn}$ , is related to $\\mathcal {C}_{2}$ by $P_{sn}=\\mathcal {C}_{2}/t$ .", "Refs.", "[18] gives the prediction for $P_{sn}$ : $P_{sn}=\\frac{\\mathcal {C}_{2}}{t}=\\frac{2\\mu _{B}}{h}T(1-T).$ We will briefly review the derivations for these expressions.", "In a semi-classical description, the second moment of transmitted particle numbers, $\\mathcal {C}_{2}$ , is equivalent to the number fluctuations of the left half of the system if the number of particles are conserved.", "This can be understood as follows.", "Let us assume that at time $t$ there are $N_{L0}$ particles on the left.", "At time $t+\\Delta t$ , if there are $N_{T}$ particles passing through the barrier, the total number of particles on the left becomes $N_{L}=N_{L0}-N_{T}$ .", "When $N_{L0}$ is treated as a number, one has $\\mathcal {C}_{2}=\\langle N_{T}^{2}\\rangle -\\langle N_{T}\\rangle ^{2}=\\langle N_{L}^{2}\\rangle -\\langle N_{L}\\rangle ^{2}=\\Delta N_{L}^{2}$ .", "In a fully quantum-mechanical description, however, $N_{L0}$ is an operator and the cross-correlation $\\langle \\hat{N}_{L0}\\hat{N}_{T}\\rangle \\ne \\langle \\hat{N}_{L0}\\rangle \\langle \\hat{N}_{T}\\rangle $ may introduce corrections to the expression.", "In the micro-canonical formalism, the fully quantum-mechanical equal-time number fluctuations, $\\Delta N_{L}^{2}$ , can be monitored.", "We will compare this with the prediction of $\\mathcal {C}_{2}$ from the semi-classical formula Eq.", "(REF ) and see how important the quantum corrections are.", "We summarize how the moments and entanglement entropy can be evaluated from semi-classical FSC [18].", "The characteristic function (CF) of transmission of fermions of one species is $\\chi (\\lambda )=\\sum _{n=-\\infty }^{\\infty }P_{n}e^{i\\lambda n}$ , where $P_{n}$ is the probability of $n$ fermions being transmitted.", "In terms of cumulants of FCS, $\\log \\chi (\\lambda )=\\sum _{m=1}^{\\infty }\\frac{(i\\lambda )^{m}}{m!", "}\\mathcal {C}_{m}.$ Importantly, the generating function is shown to be [18] $\\chi (\\lambda )=\\det \\left((1-M+Me^{i\\lambda })e^{-i\\lambda X}\\right),$ where $X=\\exp (iHt)C(0)P_{L}\\exp (-iHt)$ and $P_{L}$ is the projection operator into the left-half lattice.", "Using $\\det (AB)=\\det (A)\\det (B)$ and $\\log \\det (A)=Tr\\log (A)$ one obtains $\\log \\chi (\\lambda )=-i\\lambda x+\\log [\\det (1-M+Me^{i\\lambda })],$ where $x=Tr(X)$ and $Tr$ denotes the trace.", "The matrix $M$ can be diagonalized as $M=SD_{M}S^{\\dagger }$ , where $D_{M}=diag(v_{1},\\cdots ,v_{N/2})$ and $S$ is a unitary matrix.", "Then we get the final expression $\\log \\chi (\\lambda )=-i\\lambda x+\\log \\prod _{j=1}^{N/2}(1-v_{j}+v_{j}e^{i\\lambda }).$ The second cumulant can be obtained from $\\mathcal {C}_{2} & = & \\frac{\\partial ^{2}\\log \\chi (\\lambda )}{\\partial (i\\lambda )^{2}}\\Big |_{\\lambda \\rightarrow 0}=\\sum _{j=1}^{N/2}(v_{j}-v_{j}^{2}).$ The entanglement entropy defined in Eq.", "(REF ) can be calculated as $s=-\\int _{0}^{1}dz\\mu (z)[z\\log z+(1-z)\\log (1-z)].$ Here $z=1/(1-e^{i\\lambda })$ and the spectral weight $\\mu (z)$ is given by $\\mu (z)=\\frac{1}{\\pi }\\mbox{Im}\\partial _{z}\\log \\chi (z-i0^{+}).$ The CF of a binomial distribution with a transmitted probability $T$ is $\\chi (\\lambda )=(1-T+Te^{i\\lambda })^{\\mathcal {N}}=(1-T/z)^{\\mathcal {N}}$ , where $\\mathcal {N}=2\\mu _{B}\\Delta t/h$ is the flux of incoming particles.", "The spectral weight is then $\\mu (z) & = & \\frac{1}{\\pi }\\mbox{Im}\\partial _{z}\\log (1-\\frac{T}{z-i0^{+}})\\nonumber \\\\& = & \\mathcal {N}\\frac{T}{z}\\delta (z-T)\\nonumber \\\\& = & \\mathcal {N}\\delta (z-T).$ In this derivation we have used $1/(x-i0^{+})=P(1/x)+i\\pi \\delta (x)$ , $\\delta (z(z-T))=(1/z)\\delta (z-T)$ , and $(T/z)\\delta (z-T)=\\delta (z-T)$ , where $P$ denotes the Cauchy principal value.", "Then the entanglement entropy of Eq.", "(REF ) leads to the expression of Eq.", "(REF ).", "A similar calculation using Eq.", "(REF ) gives the expression of Eq.", "(REF )." ], [ "Absence of memory effects for non-interacting systems", "Before presenting a comparison of the MCF results with those of the Landauer formalism, we first investigate how sensitive the MCF results are to the time-dependence of the switch-on of the bias.", "This is important because in the Landauer formalism a steady state is assumed from the outset, while in the MCF a quasi-steady state develops in time and therefore its magnitude can be dependent on initial conditions and transient behavior of the bias.", "So far we only considered a sudden quench so that $\\mu _{B}$ is abruptly switched to its full value.", "The MCF can be applied to other scenarios beyond a sudden quench.", "Here we consider situations where $\\mu _{B}$ is switched on at a finite rate and reaches its full value at time $t_{m}$ .", "Here, we focus on the weak-link case and one has to monitor the dynamics of the correlation matrix by solving the equations of motion $i\\frac{\\partial \\langle c_{i}^{\\dagger }c_{j}\\rangle }{\\partial t} & = & X-\\frac{\\mu _{B}}{2}\\langle c_{i}^{\\dagger }c_{j}\\rangle _{i\\in L}+\\frac{\\mu _{B}}{2}\\langle c_{i}^{\\dagger }c_{j}\\rangle _{j\\in L}+\\nonumber \\\\& & \\frac{\\mu _{B}}{2}\\langle c_{i}^{\\dagger }c_{j}\\rangle _{i\\in R}-\\frac{\\mu _{B}}{2}\\langle c_{i}^{\\dagger }c_{j}\\rangle _{j\\in R}.$ Here, $X\\equiv [\\bar{t}^{\\prime }\\delta _{i,N/2}+\\bar{t}(1-\\delta _{i,N/2})]\\langle c_{i+1}^{\\dagger }c_{j}\\rangle +[\\bar{t}^{\\prime }\\delta _{i,N/2+1}+\\bar{t}(1-\\delta _{i,N/2+1})]\\langle c_{i-1}^{\\dagger }c_{j}\\rangle -[\\bar{t}^{\\prime }\\delta _{j,N/2}+\\bar{t}(1-\\delta _{j,N/2})]\\langle c_{i}^{\\dagger }c_{j+1}\\rangle -[\\bar{t}^{\\prime }\\delta _{j,N/2+1}+\\bar{t}(1-\\delta _{j,N/2+1})]\\langle c_{i}^{\\dagger }c_{j-1}\\rangle $ .", "The equations of motion are derived from $i(\\partial \\langle c_{i}^{\\dagger }c_{j}\\rangle /\\partial t)=\\langle [c_{i}^{\\dagger },H]c_{j}\\rangle +\\langle c_{i}^{\\dagger }[c_{j},H]\\rangle $ , where $[\\cdot ,\\cdot ]$ denotes the commutator of the corresponding operators.", "We assume that the dynamics of the two spins are identical and the initial condition is the same as that in the sudden-quench case.", "Figure REF shows the current and entanglement entropy from different cases with $\\mu _{B}(t)=(t/t_{m})^{\\alpha }\\bar{t}$ for $t<t_{m}$ and $\\mu _{B}=\\bar{t}$ for $t\\ge t_{m}$ .", "One can see that despite different transient behaviors, the currents reach the same magnitude when QSSCs emerge.", "Moreover, the slopes of the entanglement entropy are also the same in the regime where QSSCs emerge.", "We find the same conclusion when $t_{m}$ is varied.", "Importantly, one may over-excite the system by tuning the bias above its final constant value, and yet this spike does not affect the height of the QSSC or the slope of the entanglement entropy as shown by the dot-dash lines in Fig.", "REF .", "Figure: (Color online) (a) Current and (b) entanglement entropy for differenttime dependence of the bias μ B =(t/t m ) α t ¯\\mu _{B}=(t/t_{m})^{\\alpha }\\bar{t} fort<t m t<t_{m}.", "Here t m =10t 0 t_{m}=10t_{0}, t ¯ ' /t ¯=0.5\\bar{t}^{\\prime }/\\bar{t}=0.5,N=256N=256 and N p =128N_{p}=128.", "We show the results for α=0.1,1,10\\alpha =0.1,1,10labeled next to each curve along with the results from a sudden quench(dashed lines) and from a multi-step switching-on (dot-dash line).Our observations then suggest that there is no observable memory effect in the QSSC and entanglement entropy of non-interacting fermions driven by a step-function bias because those observables are not sensitive to the details of how the bias is turned on.", "However, the robustness of the QSSC against different time dependencies of the switch-on of the bias may not hold in the presence of interactions, and we leave this study for future work.", "Figure: (Color online) (a) Averaged current (Eq.", "()) and (b)slope of the entanglement entropy for different time dependence ofthe bias μ B (t/t m ) α t ¯\\mu _{B}(t/t_{m})^{\\alpha }\\bar{t} for t<t m t<t_{m}.", "Heret m =10t 0 t_{m}=10t_{0}, t ¯ ' /t ¯=0.5\\bar{t}^{\\prime }/\\bar{t}=0.5 (black) and t ¯ ' /t ¯=1\\bar{t}^{\\prime }/\\bar{t}=1(red), N=256N=256 and N p =128N_{p}=128.", "We show the results for α=0.01,0.1,1,10,100\\alpha =0.01,0.1,1,10,100along with the results from a sudden quench (dashed lines).Figure REF shows the averaged current $\\langle I\\rangle =\\frac{1}{50t_{0}}\\int _{50t_{0}}^{100t_{0}}dtI(t)$ and the slope of $s$ in the region $50t_{0}\\le t\\le 100t_{0}$ for $\\alpha =0.01,0.1,1,10,100$ along with the results from a sudden quench.", "The results from those cases where the bias is turned on in a finite time $t_{m}$ exhibit no observable deviation from the results from the case of a sudden quench.", "We choose $t_{m}=10t_{0}$ and $N=256$ with $N_{p}=128$ , but the conclusion holds for other parameters.", "Thus in the following we focus on the sudden-quench case when we compare the MCF and analytical formulas." ], [ "Comparisons", "Figure REF shows the current, entanglement entropy, and number fluctuations of the weak-link case for selected values of $\\bar{t}^{\\prime }/\\bar{t}$ .", "The currents clearly exhibit a quasi-steady state after a short transient time.", "We emphasize again that the steady-state current results from the quantum dynamics of the system and is not assumed a priori.", "The corresponding steady-state currents calculated from the Landauer formula, Eq.", "(REF ), are plotted on the same figure.", "The results from our simulations agree well with the predictions from Landauer formula.", "This agreement is in line with the observation that as the system approaches thermodynamic limit ($N\\rightarrow \\infty $ with finite filling), the microcanonical setup becomes indistinguishable from that in the Landauer formalism.", "For the case $\\bar{t}^{\\prime }=\\bar{t}$ , we recover the quantized conductance, $G_{0}=I/\\mu _{B}=2e^{2}/h$ (spin included).", "As expected, in the presence of a barrier ($\\bar{t}^{\\prime }<\\bar{t}$ ), the conductance is smaller than the quantized conductance.", "The suppression of the current by a weak central link was also shown in Ref. [44].", "We also test finite-size effects by comparing the currents from $N=256,512,1024$ with the prediction from Landauer formula in the inset of Fig.", "REF .", "One can see that while the oscillation amplitude decreases with increasing system size, the average currents of the three different sizes all agree well with the analytical result.", "For the central-site case we found similar results.", "When the link strength $\\bar{t}^{\\prime }$ or the central-site energy $E_{C}$ is tuned, the transmission coefficient $T(E)$ changes accordingly.", "Figure REF compares the quasi steady-state currents from Landauer formula (black line) and from micro-canonical simulations of the weak-link case (red circles) and the central-site case (green squares) as a function of the transmission coefficient $T=T(E=0)$ .", "The three results agree well and this supports the expectation that the Landauer formalism provides reasonable predictions.", "However, we will see that the agreement does not hold when we study the distributions on the two sides of the junction.", "The entanglement entropy is expected to be linear in time and our results support this claim.", "We found that the slope of $\\Delta s=s-s(\\mu _{B}=0)$ is proportional to $\\mu _{B}$ as predicted in Eq.", "(REF ).", "For different values of $\\bar{t}^{\\prime }/\\bar{t}$ , we test the predictions from the two formulas.", "From Fig.", "REF we find that in the range $-\\mu _{B}/2\\le E\\le \\mu _{B}/2$ the variation of $T(E)$ is within $3\\%$ for all cases we studied so we take $T(E=0)$ as the transmission coefficient in our evaluation of Eq.", "(REF ).", "The slope of the entanglement entropy from micro-canonical formalism and the predictions from Eq.", "(REF ) are shown in Figure REF for $\\mu _{B}/\\bar{t}=0.1$ and $0.2$ .", "One can see that our results agree well with Eq.", "(REF ) for all values of $T$ and this implies that the distribution of tunneling particles may be approximated by a binomial form as assumed in Ref. [18].", "In the derivation of Eq.", "(REF ) and in our simulation, fermions of different spins tunnel independently and do not generate spin-entangled states.", "The entanglement entropy comes from the correlation of partially tunneled and partially reflected wavefunctions of particles.", "Figure: (Color online) Comparison of the slope of s(t)s(t) from Eq.", "() (red) and simulations (symbols).", "Here the results for μ B /t ¯=0.1\\mu _B/\\bar{t}=0.1 are represented by the dashed line, circles (weak-link), and diamonds (central-site) while those for μ B /t ¯=0.2\\mu _B/\\bar{t}=0.2 are represented by the solid line, triangles (weak-link), and squares (central-site).", "We choose N=512N=512 and T=T(E=0)T=T(E=0).", "The thin red solid and dashed lines shows the results for a Gaussian distribution, Eq.", "(), for μ B /t ¯=0.2\\mu _B/\\bar{t}=0.2 and 0.10.1.", "Inset: The slopes (in units of t 0 -1 t_0^{-1}) from different system sizes N=256,512,1024,2048N=256,512,1024,2048 showing the convergence to the semi-classical value for t ¯ ' /t ¯=0.5\\bar{t}^{\\prime }/\\bar{t}=0.5 (solid line).We notice that the transient time, which is defined as the initial time interval during which the system has not reached a quasi-steady state, seems to differ in $I(t)$ and $s(t)$ (as illustrated in Fig.", "REF ).", "During the transient time, the currents fluctuate violently while the entropy exhibit a downward bending.", "From our simulations we found that the transient time for $s(t)$ is three times larger than that for $I(t)$ and this relation seems to be insensitive to the system size.", "To investigate how FCS depends on the underlying probability distribution, we study the behavior of Eq.", "(REF ) when a Gaussian (continuous) distribution is implemented.", "To make connections with the original binomial distribution, we choose the mean $m=\\mathcal {N}T$ and the variance $\\sigma ^{2}=\\mathcal {N}T(1-T)$ to match those of the binomial distribution.", "The CF is $\\chi (\\lambda )=\\exp (im\\lambda -\\frac{1}{2}\\sigma ^{2}\\lambda ^{2})$ .", "A change of variable $z=1/(e^{i\\lambda }-1)$ gives $\\chi (z)=\\exp (\\mu \\log (1-\\frac{1}{z})+\\frac{1}{2}\\sigma ^{2}[\\log (1-\\frac{1}{z})]^{2})$ .", "One then finds $\\mbox{Im}\\partial _{z}\\log \\chi (z)=\\mbox{Im}[\\frac{m}{z^{2}-z}+\\frac{\\sigma ^{2}}{z^{2}-z}\\log (1-\\frac{1}{z})]$ .", "When one changes $z$ to $z-i0^{+}$ and uses the formula $1/(x-i0^{+})=P(1/x)+i\\pi \\delta (x)$ , the imaginary part of $1/(z^{2}-z)$ does not contribute to the integral of $s$ because the delta function $\\delta (z^{2}-z)$ can only be satisfied at $z=0,1$ but those points do not have finite $s$ .", "The only contribution in the spectral weight is thus $\\mu (z)=\\frac{1}{\\pi }\\frac{\\sigma ^{2}}{z^{2}-z}\\mbox{Im}[\\log (1-\\frac{1}{z-i0^{+}})]$ .", "One can show that $\\mbox{Im}[\\log (1-\\frac{1}{z-i0^{+}})]=\\arg ((z-1)/z)=-\\pi $ (the choice of the sign will be clear in a moment) for $0<z<1$ .", "Thus the weight is $\\mu (z)=-\\frac{\\sigma ^{2}}{z^{2}-z}$ which is positive for $0<z<1$ .", "From $s=-\\int _{0}^{1}dz\\mu (z)[z\\log z+(1-z)\\log (1-z)]$ [18] one gets $s=\\alpha \\sigma ^{2}=\\alpha (2\\mu _{B}/h)T(1-T)\\Delta t$ .", "Thus $ \\frac{\\Delta s}{\\Delta t}= \\alpha \\left(\\frac{2\\mu _B}{h}\\right)T(1-T),$ where $\\alpha \\approx 3.3$ is a numerical factor.", "In Figure REF we show (in thin red lines) its values.", "It is clear that the data from the micro-canonical simulations can distinguish these distributions.", "As the system size increases, the small oscillation on top of the linear increase of the entanglement entropy decreases.", "We found that this reduces the difference between the slope from fitting the results from the MCF and the slope predicted by the semi-classical FCS formalism.", "In the inset of Fig.", "REF we show the slope from the MCF for $N=256,512,1024,2048$ with half-filling.", "One can see that as $N$ increases the agreement improves.", "However, optical lattices in real experiments are of limited sizes so one may expect observable finite-size effects in experimental results.", "Figure: Comparison of the slopes (in units of t 0 -1 t_0^{-1}) of ΔN L 2 \\Delta N_{L}^{2} (symbols) and 𝒞 2 \\mathcal {C}_{2} from Eq.", "() (curves).", "The circles (weak-link), diamonds (central-site), and dashed line correspond to μ B /t ¯=0.1\\mu _B/\\bar{t}=0.1 while the triangles (weak-link), squares (central-site), and solid line correspond to μ B /t ¯=0.2\\mu _B/\\bar{t}=0.2.", "Here N=512N=512 with half filling.", "Inset: The slope (in units of t 0 -1 t_0^{-1}) of ΔN L 2 \\Delta N_{L}^{2} for N=256,512,1024,2048N=256,512,1024,2048 at half filling with t ¯ ' /t ¯=0.5\\bar{t}^{\\prime }/\\bar{t}=0.5.", "The dashed line shows the result from Eq.", "().Next we extract the slopes of $\\Delta N_{L}^{2}$ and compare the results with the slopes predicted from the semi-classical formula of the second cumulant, Eq.", "(REF ), in Figure REF .", "The slopes agree reasonably well, which implies that quantum corrections to the semi-classical formula are insignificant.", "Moreover, we have compared the third and fourth moments with the quantum-mechanical fluctuations of the corresponding order.", "The results from micro-canonical simulations show observable deviations from those from semi-classical FCS in the fourth order but not in the third order.", "The slight difference between our results and the results from semiclassical FCS in Fig.", "REF is due to finite-size effects.", "We have checked our results for larger system sizes and the result converges to the FCS prediction, as shown in the inset of Fig.", "REF .", "So far the MCF agree reasonably with Landauer formalism and FCS.", "Now we will show several interesting phenomena associated with the finite size and conservation laws of isolated systems such as cold atoms.", "We first study the particle distribution functions on the left and the right sides.", "This can be done by first projecting the correlation matrix to the left (right) half uniform lattice and obtaining $M_{L}=P_{L}CP_{L}$ and $M_{R}=P_{R}CP_{R}$ .", "Next we find the eigenvalues and the corresponding unitary transformations of $H_{L}$ and $H_{R}$ (with the biases on) so that $H_{L/R}=U_{L/R}D_{L/R}U_{L/R}^{\\dagger }$ , where $D_{L,R}=diag(\\epsilon _{L/R,1},\\cdots ,\\epsilon _{L/R,N/2})$ .", "Then we construct the correlation matrix in energy space and get $\\tilde{D}_{qq^{\\prime }}^{L/R}=\\sum _{i,j\\in L/R}(U_{L/R}^{\\dagger })_{qi}(U_{L/R})_{jq^{\\prime }}(M_{L/R})_{ij}$ .", "For each eigenvalue $\\epsilon _{L/R,q}$ , the occupation number is given by $n_{L/R}(\\epsilon _{L/R,q})=\\tilde{D}_{qq}^{L/R}$ .", "In Fig.", "REF we show the particle distributions for the weak-link case with $\\bar{t}^{\\prime }=0.5\\bar{t}$ and $\\bar{t}^{\\prime }=\\bar{t}$ at $t=t_{0}$ , $100t_{0}$ , and $200t_{0}$ for $N=512$ with the system initially half-filled and $\\mu _{B}=0.2\\bar{t}$ .", "The particle distributions for the central-site case with similar parameters are shown in Fig.", "REF .", "Figure: (Coor online) The distribution function of the weak-link case withμ B =0.2t ¯\\mu _{B}=0.2\\bar{t} and t ¯ ' =0.5t ¯\\bar{t}^{\\prime }=0.5\\bar{t} (left column)and t ¯\\bar{t} (right column).", "From top to bottom, t=t 0 t=t_{0}, 100t 0 100t_{0},and 200t 0 200t_{0}.", "Here, N=512N=512 (with the lattice initially half-filled) and the quasi-steady state currentpersists to 240t 0 240t_{0}.Figure: (Color online) The distribution function of the central-site casewith μ B =0.2t ¯\\mu _{B}=0.2\\bar{t} and E c =2t ¯E_{c}=2\\bar{t} (left column) and 0(right column).", "From top to bottom, t=t 0 t=t_{0}, 100t 0 100t_{0}, and 200t 0 200t_{0}.Here, N=512N=512 and the quasi-steady state current persists to t=240t 0 t=240t_{0}.Clearly, the particle distributions on both sides vary dynamically but they evolve in a coordinated fashion so that the current across the junction remains constant for a long period of time.", "This is different from the picture behind the Landauer formula.", "In Landauer formula the distributions on the left (right) half lattice are fixed at $f_{L}$ ($f_{R}$ ) and a constant tunneling constant $T$ determines the rate at which particles move across the junction.", "On the other hand, for a finite system, the particle distributions must evolve with time.", "If Eq.", "(REF ) is naively used in this case, one may expect that the current decays with time because the difference between the distributions, $f_L(E)-f_R(E)$ , should be a decreasing function when particles are flowing from the left to the right.", "In contrast, a plateau in the current emerges in the full quantum dynamics.", "Even more surprisingly, there exists a time interval when a QSSC still flows from left to right, yet the right lattice has more particles, as shown in the bottom right panels of both Figs.", "REF and REF (for $t=200t_{0}$ ) [45].", "This highlights that this a highly correlated state that allows the QSSC to persist.", "We will see in the next section this is due to causality and the finite speed of propagation of information, and thus is analogous to the light cone in special relativity.", "There are recent proposals for designing batteries for atomtronic devices [8].", "However, an important message from our study is that an isolated quantum system can maintain a quasi steady-state current in many cases.", "The quasi-steady state, as we demonstrated, is maintained by internal dynamics so a battery may not be the only way for generating a steady current in atomtronic devices, one could instead engineer an appropriate initial state that will induce a QSSC." ], [ "Light-cone of wave propagation", "In the last section, we saw that a QSSC can continue to flow even when the particle imbalance would indicate otherwise.", "This effect is due to the finite speed of information.", "Recent experimental studies [46], [3] have shown that the density profile exhibits a “light cone” as an atomic cloud expands, and there are ongoing theoretical studies to support this fact [47].", "We can see this effect within the MCF (one of the many advantages of this formalism).", "We monitor the real-time dynamics of the density and current profiles for noninteracting fermions in a uniform lattice driven out of equilibrium by (1) a step-function potential as shown in Fig.", "REF , and (2) a sudden removal of atoms on the right-half lattice as discussed in Ref. [23].", "The time evolution of the first case is shown in Figure REF and that of the second case is shown in Fig.", "REF .", "In both cases one can see clearly a “light cone” within which the motion of atoms are confined.", "The propagation speed is limited by the Fermi velocity, which for filling $f$ is $v_F=2\\sin (f\\pi )/t_0$ .", "For $N=512$ at half filling ($N_p=256$ ), it takes about $128t_0$ for the wave front to reach the boundary and reflect back.", "Around $256t_0$ the two wave fronts propagating in the opposite directions meet again in the middle.", "That is when the current stops showing the quasi steady-state behavior.", "This applies to both cases, as shown in Fig.", "REF (c) and (d) and Fig.", "REF (c) and (d).", "This explains the paradoxical behavior of the QSSC flowing counter to the particle imbalance.", "This happens because the information regarding the population imbalance still has not been carried to the junction region where the current is being monitored.", "For the quarter filling ($N_p=128$ ), if the wave front propagates at the speed of the corresponding Fermi velocity $\\sqrt{2}/t_0$ , it takes about $181t_0$ for the wave front to reach the boundary and the two wave fronts meet again at around $362t_0$ .", "Although the main body of the wave propagates at this speed, there are \"leaks\" of the wave which propagate at speed higher than $\\sqrt{2}/t_0$ but they are limited by the maximal Fermi velocity $2/t_0$ , as shown in Figs.", "REF and REF .", "This “leaking” behavior is more prominent for the case of a sudden removal of half of the particles at higher filling.", "As shown in Fig.", "REF (e) and (f), for initial $(3/4)$ filling there is significant fraction of the wave propagation at $2/t_0$ .", "For the step-function bias case (Fig.", "REF (e) and (f)), the main wave propagates at $\\sqrt{2}/t_0$ and again the leak propagates at higher speed (limited by $2/t_0$ ).", "We also found that adding a weak central link or a central site with different onsite energy only decreases the magnitude of the current, but the speed of wave-front propagation remains the same for the same initial filling." ], [ "Kubo Formalism", "In order to connect the microcanonical and Landauer approaches, we apply leading order perturbation theory on finite systems by way of the Kubo formula [48], [49], [11] $\\langle A\\left(t\\right)\\rangle =\\langle A\\rangle _{0}-\\imath \\int _{0}^{t}dt^{\\prime }\\langle \\left[\\hat{A}\\left(t\\right),\\hat{H}^{\\prime }\\left(t^{\\prime }\\right)\\right]\\rangle $ for the observable $A$ .", "Here, $\\hat{O}=e^{\\imath H_{0}t}Oe^{-\\imath H_{0}t}$ indicates an operator in the interaction picture, $H^{\\prime }$ is the perturbing Hamiltonian, and $\\langle O\\rangle _{0}$ indicates an average with respect to the initial state.", "For all practical purposes, here we use the wavefunctions of finite-size systems without taking the thermodynamic limit commonly employed in solid-sate systems.", "We will consider a one-dimensional lattice set out of equilibrium by connecting two initially disconnected halves with a weak link or by the application of a bias to an initially connected system, as shown in Fig.", "REF ." ], [ "Connecting the $L$ and {{formula:94646524-6b43-463c-8c83-f93a90b1c77b}} lattices", "The initial Hamiltonian is $H_{0}=H_{L}+H_{R},$ where $H_{L}=-\\sum _{\\langle i,j\\rangle }\\bar{t}c_{i}^{\\dagger }c_{j}+\\mu _{L}\\sum _{i}c_{i}^{\\dagger }c_{i}$ and $H_{R}=-\\sum _{\\langle i,j\\rangle }\\bar{t}d_{i}^{\\dagger }d_{j}-\\mu _{R}\\sum _{i}d_{i}^{\\dagger }d_{i}.$ The left and the right lattices are both finite lattices of length $N$ with non-periodic (“open”) boundary conditions.", "We consider the ground state of $H_{L}$ and $H_{R}$ fixed at half filling – the bias can be thought of as added simultaneously with the connection of the two lattices.", "The diagonalization of the left lattice is performed by $c_{j}=\\sum _{k}U_{jk}a_{k}$ with $U_{jk}=\\sqrt{2/\\left(N+1\\right)}\\sin \\left(jk\\pi /\\left(N+1\\right)\\right)$ and $k=1,\\,\\ldots ,\\, N$ , yielding $H_{L}=\\sum _{k}\\epsilon _{k}^{L}a_{k}^{\\dagger }a_{k}$ and $\\epsilon _{k}^{L}=-2\\bar{t}\\cos \\left(k\\pi /\\left(N+1\\right)\\right)+\\mu _{L}$ .", "Similarly for the right lattice, using $d_{j}=\\sum _{k}U_{jk}b_{k}$ gives $H_{R}=\\sum _{k}\\epsilon _{k}^{R}b_{k}^{\\dagger }b_{k}$ and $\\epsilon _{k}^{R}=-2\\bar{t}\\cos \\left(k\\pi /\\left(N+1\\right)\\right)+\\mu _{R}$ .", "At $t=0$ , the lattices are connected by the perturbing Hamiltonian $H^{\\prime }=g\\bar{t}\\left(c_{1}^{\\dagger }d_{1}+d_{1}^{\\dagger }c_{1}\\right),$ where $g=\\bar{t}^{\\prime }/\\bar{t}$ .", "Note that the numbering of the sites in both lattices starts from the interface sites.", "The current is the quantity of interest, hence we will take $A=g\\bar{t}c_{1}^{\\dagger }d_{1},$ where $A$ gives the hopping between the two halves of the lattice, i.e., $c_{1}$ acts on the interface site on the left lattice and $d_{1}$ on the interface site of the right lattice.", "This will give the current through $I\\left(t\\right)=-2\\mbox{Im}\\langle A\\left(t\\right)\\rangle $ .", "The interaction picture operators are $\\hat{A}\\left(t\\right)=g\\bar{t}^2\\sum _{k,k^{\\prime }}U_{k1}^{\\dagger }U_{1k^{\\prime }}e^{\\imath \\left(\\epsilon _{k}^{L}-\\epsilon _{k^{\\prime }}^{R}\\right)t}a_{k}^{\\dagger }b_{k^{\\prime }}$ and $\\hat{H}^{\\prime }\\left(t^{\\prime }\\right)=\\left(\\hat{A}\\left(t^{\\prime }\\right)+\\hat{A}^{\\dagger }\\left(t^{\\prime }\\right)\\right).$ Putting these into Eq.", "(REF ) and using that $\\langle A\\rangle _{0}=0$ for two initially disconnected lattices, we obtain $\\langle A\\left(t\\right)\\rangle =g^2\\bar{t}\\sum _{k,k^{\\prime }}\\left|U_{k1}\\right|^{2}\\left|U_{1k^{\\prime }}\\right|^{2}\\frac{n_{k}-n_{k^{\\prime }}}{\\epsilon _{k}^{L}-\\epsilon _{k^{\\prime }}^{R}}\\left(1-e^{\\imath \\left(\\epsilon _{k}^{L}-\\epsilon _{k^{\\prime }}^{R}\\right)t}\\right).$ The current is then $I\\left(t\\right) & =-2\\mbox{Im}\\langle A\\left(t\\right)\\rangle \\nonumber \\\\& =2g^{2}\\bar{t}^{2}\\sum _{k,k^{\\prime }}\\left|U_{k1}\\right|^{2}\\left|U_{1k^{\\prime }}\\right|^{2}\\frac{n_{k}-n_{k^{\\prime }}}{\\epsilon _{k}^{L}-\\epsilon _{k^{\\prime }}^{R}}\\sin \\left[\\left(\\epsilon _{k}^{L}-\\epsilon _{k^{\\prime }}^{R}\\right)t\\right].$ When the $L$ and $R$ lattices are half filled, this double sum will be nonzero when either $k\\le N/2,\\, k^{\\prime }>N/2$ or $k>N/2,\\, k^{\\prime }\\le N/2$ .", "To simplify the expressions and show the correspondence with Landauer, we take the semi-infinite limit for the left and right lattices obtaining $I\\left(t\\right) & =\\frac{8g^{2}\\bar{t}^{2}}{\\pi ^{2}}\\int _{0}^{\\pi }dk\\int _{0}^{\\pi }dk^{\\prime }\\sin ^{2}k\\sin ^{2}k^{\\prime }\\\\& \\times \\frac{n_{k}-n_{k^{\\prime }}}{\\epsilon _{k}-\\epsilon _{k^{\\prime }}+\\mu _B}\\sin \\left[t\\left(\\epsilon _{k}-\\epsilon _{k^{\\prime }}+\\mu _B\\right)\\right],$ where $\\epsilon _{k}=-2\\bar{t}\\cos k$ and $\\mu _B=\\mu _{L}-\\mu _{R}$ .", "At this point, we have made no assumption about the strength of the bias, the filling, or the temperature.", "We will now restrict ourselves to the case of half filling and zero temperature.", "The two contributions to this expression give the forward and backward currents, integrating over energy instead of wave vector, $I_{\\rightarrow }\\left(t\\right)= & \\frac{2g^{2}\\bar{t}}{\\pi ^{2}}\\int _{-2}^{0}d\\epsilon \\int _{0}^{2}d\\epsilon ^{\\prime }\\left(1-\\frac{\\epsilon ^{2}}{4}\\right)^{1/2}\\left(1-\\frac{\\epsilon ^{\\prime 2}}{4}\\right)^{1/2}\\\\& \\times \\frac{\\sin \\left[t\\left(\\epsilon -\\epsilon ^{\\prime }+\\mu _b\\right)/t_0\\right]}{\\epsilon -\\epsilon ^{\\prime }+\\mu _b}$ and $I_{\\leftarrow }\\left(t\\right)= & \\frac{2g^{2}\\bar{t}}{\\pi ^{2}}\\int _{0}^{2}d\\epsilon \\int _{-2}^{0}d\\epsilon ^{\\prime }\\left(1-\\frac{\\epsilon ^{2}}{4}\\right)^{1/2}\\left(1-\\frac{\\epsilon ^{\\prime 2}}{4}\\right)^{1/2}\\\\& \\times \\frac{\\sin \\left[t\\left(\\epsilon -\\epsilon ^{\\prime }+\\mu _b\\right)/t_0\\right]}{\\epsilon -\\epsilon ^{\\prime }+\\mu _b}.$ Here, $\\mu _b=\\mu _B/\\bar{t}$ .", "As $t\\rightarrow \\infty $ , the fast oscillating function $\\sin \\left(tx\\right)/\\pi x$ enforces $\\epsilon ^{\\prime }=\\epsilon +\\mu _b$ .", "This latter equality can not be satisfied in the backward current as $\\epsilon $ is positive and $\\mu _b$ is also positive, but $\\epsilon ^{\\prime }$ is negative.", "Thus, only the forward current remains, giving $I= \\frac{g^{2}\\bar{t}^{2}}{\\pi }\\int _{-\\mu _b/2}^{\\mu _b/2}d\\epsilon \\bar{g}_L \\bar{g}_R$ in the steady state and including the factor of two for spin.", "Here, $\\bar{g}_{L(R)}=\\sqrt{4-\\left(\\epsilon \\mp \\mu _b/2\\right)^{2}}$ and the bias is applied symmetrically.", "The result is insensitive, though, to how the bias is applied – the left lattice can be shifted by $\\mu _B$ and right lattice by 0, or the left by $\\mu _B/2$ and the right by $-\\mu _B/2$ .", "This expression is valid for arbitrary bias and agrees with the Landauer expression, Eq.", "(REF ), to leading order in $g$ .", "For small bias, one obtains $I\\simeq \\frac{4g^{2}\\bar{t}}{\\pi }\\mu _{B}.$ Figure REF shows the agreement of this expression with the exact microcanonical expression for finite-size systems.", "Figure: (Color online) Current versus time for connection-induced transport.", "The Kubo result(blue crosses) compares very well with the exact microcanonical method(red, dashed line), with both approaching the steady-state current(green, dashed line) for long times.", "Here, the latticeis of length 1600 sites, the bias is μ B /t ¯=1/10\\mu _B/\\bar{t}=1/10, and the strengthof the weak link is g=1/100g=1/100.This Kubo approach is firmly rooted in the microcanonical picture – we have a finite, closed system (the semi-infinite limit is taken only for convenience) set out of equilibrium.", "The resulting expressions separate out the short time behavior – due to forward and backward fluctuations at all energy scales – and the long time behavior – the QSSC – that emerges from just the forward current." ], [ "Applied bias across the $L$ and {{formula:c9dbd2bd-b34a-4b10-8f01-b481915c9783}} lattices", "Let us now consider an initial Hamiltonian for a connected, homogeneous lattice of length $2N$ $H_{0}=-\\sum _{\\langle i,j\\rangle }\\bar{t}c_{i}^{\\dagger }c_{j}.$ We consider the ground state of $H_{0}$ fixed at half filling.", "The diagonalization is the same as above except with a lattice of length $2N$ , $c_{j}=\\sum _{k}U_{jk}a_{k}$ with $U_{jk}=\\sqrt{2/\\left(2N+1\\right)}\\sin \\left(jk\\pi /\\left(2N+1\\right)\\right)$ and $k=1,\\,\\ldots ,\\,2N$ , yielding $H_{0}=\\sum _{k}\\epsilon _{k}a_{k}^{\\dagger }a_{k}$ and $\\epsilon _{k}=-2\\bar{t}\\cos \\left(k\\pi /\\left(2N+1\\right)\\right)$ .", "At $t=0$ , the lattices are perturbed by the Hamiltonian $H^{\\prime }=\\frac{\\mu _B}{2}\\sum _{i\\in L}c_{i}^{\\dagger }c_{i}-\\frac{\\mu _B}{2}\\sum _{i\\in R}c_{i}^{\\dagger }c_{i},$ which applies the step potential bias as shown in Fig.", "REF .", "The strength of the perturbation is the bias $\\mu _B$ .", "The current between the two halves is of interest, and therefore we choose $A=\\bar{t}c_{N}^{\\dagger }c_{N+1}.$ The interaction picture operator is $\\hat{A}\\left(t\\right)=\\sum _{k,k^{\\prime }}U_{kN}^{\\dagger }U_{N+1k^{\\prime }}e^{\\imath \\left(\\epsilon _{k}-\\epsilon _{k^{\\prime }}\\right)t}\\bar{t}a_{k}^{\\dagger }a_{k^{\\prime }}.$ After some work, one finds that the current is $I\\left(t\\right) & =-2\\mbox{Im}\\langle A\\left(t\\right)\\rangle \\\\& =\\frac{2\\mu _B\\bar{t}}{\\left(2N+1\\right)^{2}}\\sum _{\\begin{array}{c} k\\,\\mathrm {Even}, \\\\ \\, k^{\\prime }\\,\\mathrm {Odd}\\end{array}}F_{kk^{\\prime }}\\frac{n_{k}-n_{k^{\\prime }}}{\\epsilon _{k}-\\epsilon _{k^{\\prime }}}\\sin \\left[t\\left(\\epsilon _{k}-\\epsilon _{k^{\\prime }}\\right)\\right],$ where $F_{kk^{\\prime }}=2\\bar{t}+\\frac{4\\bar{t}^2-\\epsilon _{k}\\epsilon _{k^{\\prime }}}{\\epsilon _{k}-\\epsilon _{k^{\\prime }}}.$ As $t\\rightarrow \\infty $ , one can compute the steady state current, $I \\simeq \\mu _B \\bar{t}/\\pi $ , which includes a factor of two for spins.", "This agrees with the Landauer expression, Eq.", "(REF ), expanded to the leading order of $\\mu _B$ .", "Figure REF shows the agreement of this expression with the exact microcanonical expression for finite-size systems.", "One may build connections between the Landauer formalism and the MCF via the use of non-equilibrium Green's functions [50].", "The Kubo approach here, however, gives explicit expressions for the effect of the reservoirs for discrete systems and thus explicitly connects closed, finite systems and their thermodynamic limit.", "Figure: (Color online) Current versus time for bias-induced transport.", "The Kubo result (bluecrosses) compares very well with the exact microcanonical method (red,dashed line), with both approaching the steady state current (green,dashed line) for long times.", "Here, the lattice is of length 1600 sites,the bias is μ B /t ¯=1/100\\mu _B/\\bar{t}=1/100." ], [ "Conclusion", "In summary, we have discussed different theoretical viewpoints for quantum transport phenomena that may be studied in ultra-cold atoms.", "In particular, we have compared the current, entanglement entropy, and number fluctuations from the Landauer approach, semi-classical FCS, and the micro-canonical formalism.", "In our study of two finite 1D lattices bridged by a junction, we found a quasi steady-state current from the quantum dynamics in the micro-canonical simulations.", "The magnitude of this quasi steady-state current agrees quantitatively with the value predicted by the Landauer approach.", "The underlying mechanisms, nevertheless, have been shown to be very different when the distributions of the two sides are analyzed.", "The distributions evolve in time and steadily deviate from Fermi-Dirac distributions even while the quasi steady-state current is maintained.", "Our work points out several key issues when applying different formalisms to closed quantum systems such as ultra-cold atoms in optical lattices.", "Of particular importance is the confirmation, using the micro-canonical approach, that a quasi-steady state fermionic current can be established in a 1-D closed system for a finite period of time without the need of inelastic effects or interaction effects beyond mean field [20].", "The magnitude of this non-interacting quasi-steady state current is independent of the way the bias is switched on.", "This also hints at the fact that the Landauer formalism may not be the best suited for the study of transport properties of these finite closed systems, even though the average current that it predicts is correct for long times.", "This is because, in the case of elastic scattering, the current is dominated by local properties at the junction.", "On the other hand, other quantities of interest, such as the occupation of particles – whether at a quasi-steady state or not – are very sensitive to the full spatial extent of the wavefunctions.", "The entanglement entropy from our simulations of the full quantum dynamics agrees with the formula derived from semi-classical FCS with a binomial distribution, which raises the question of how the wave nature of the transmitted particles can be well approximated by such a distribution.", "We also found that quantum corrections are not significant in the equal-time number fluctuations of these noninteracting systems.", "On the one hand, this supports the use of a semi-classical approach in studying certain transport phenomena.", "On the other hand, finding transport coefficients that are sensitive to quantum corrections is an interesting future direction.", "Extending those comparisons to higher dimensions or in the presence of interactions beyond mean field approximations could be very challenging, but they could lead to a deeper understanding of transport phenomena in closed quantum systems.", "We emphasize that these issues, such as the dynamics and feedback of reservoirs, quantum correlations, and matter-wave propagation, should be carefully investigated in more complex situations as we did here for non-interacting systems.", "C.C.C.", "acknowledges the support of the U.S. Department of Energy through the LANL/LDRD Program.", "MD acknowledges support from the DOE grant DE-FG02-05ER46204." ] ]
1403.0511
[ [ "Linear Stable Sampling Rate: Optimality of 2D Wavelet Reconstructions\n from Fourier Measurements" ], [ "Abstract In this paper we analyze two-dimensional wavelet reconstructions from Fourier samples within the framework of generalized sampling.", "For this, we consider both separable compactly-supported wavelets and boundary wavelets.", "We prove that the number of samples that must be acquired to ensure a stable and accurate reconstruction scales linearly with the number of reconstructing wavelet functions.", "We also provide numerical experiments that corroborate our theoretical results." ], [ "Introduction", "A problem that appears in multiple disciplines is the reconstruction of an object from linear measurements.", "One special situation of particular importance which we will focus on in this paper are Fourier measurements.", "This particular reconstruction problem occurs in numerous applications such as Fourier optics, radar imaging, magnetic resonance imaging (MRI) and X-ray CT (the latter after application of the Fourier slice theorem).", "One of the main issues is that we are only able to acquire finitely many samples, since we cannot process an infinite amount of information in practice.", "The reconstruction of an object from a finite collection of Fourier samples can be obtained by a truncated Fourier series.", "However, this typically leads to undesirable effects such as the Gibbs phenomenon, which are wild oscillation near points of discontinuity.", "Beside the Gibbs phenomenon, the convergence of the Fourier series (in the Euclidean norm) is slow.", "Conversely, wavelets bases are well-known to achieve much better results (see [22]).", "Indeed, wavelets have much better localization properties than the standard Fourier transform, leading to a better detection of image features.", "For this reason, wavelets have found widespread used in compression and denoising.", "For example the algorithm of JPEG 2000 is based on wavelets.", "Moreover, they come equipped with fast algorithms which are of great importance in today's age of technology.", "Wavelets also play a pivotal role in biomedical imaging, with an example being the technique of wavelet encoding in MRI (see [19], [23], [29], [30]).", "The issue, however, is that physical samplers such as an MRI scanner naturally yield Fourier measurements, not wavelet coefficients.", "Thus in order to exploit the power of wavelets, we need a reconstruction algorithm capable of producing wavelet coefficients given a fixed set of Fourier measurements." ], [ "Generalized sampling", "Generalized sampling is a framework for this problem developed by two of the authors in a series of papers [2], [3], [4], [5] and based on past work of Unser and Aldroubi [27], [28], Eldar [15] and Hrycak and Gröchenig [18].", "The theory allows for stable and accurate reconstructions in an arbitrary reconstruction system of choice given fixed measurements with respect to another system.", "The problem of reconstructing wavelet coefficients from Fourier samples is an important example of this abstract framework.", "Mathematically, the reconstruction problem can be modeled in a separable Hilbert space $\\mathcal {H}$ , resulting in an infinite-dimensional linear algebra problem.", "Generalized sampling provides a faithful discretization of such a problem.", "The stable sampling rate (see Section for details) is fundamental characteristic within generalized sampling that determines how many samples are needed in order to obtain stable and accurate reconstructions with a given number of reconstruction elements.", "It is therefore vital that this rate be determined for important instances of generalized sampling." ], [ "Our contribution", "In [7] the respective authors proved the linearity of the stable sampling rate for one-dimensional compactly supported wavelets based on finitely many Fourier samples.", "This means, up to a constant, one needs the same number of samples as reconstruction elements.", "Our results extend the previous one to dimension two, although higher dimensional results can be obtained in a straightforward manner.", "This is an important extension, since most of the above applications involve two- or three-dimensional images.", "The crucial part that makes our result non-trivial is the allowance of non-diagonal scaling matrices neglecting straightforward arguments for separable two dimensional wavelets from 1D to 2D.", "Moreover, we will not only prove the linearity for standard two- dimensional separable wavelets, but also for two-dimensional boundary wavelets which are of particular interest for smooth images.", "This case was not considered in [7] but was addressed recently in [1] for the case on 1D nonuniform Fourier samples.", "Here, for simplicity, we consider only uniform samples but in the 2D setting.", "At this stage we note that other higher dimensional concepts, such as curvelets and shearlets, can provide better approximations rates for cartoon-like-images; a specific class of functions ([10], [20]).", "However, this paper serves as an extension of known 1D results [7].", "It thus provides a necessary first step in the study of reconstructions from Fourier samples within the context of generalized sampling in higher-dimensional settings.", "We shall discuss shearlets in an upcoming paper.", "Let us now make one further remark.", "The reader may at this stage wonder why, given a vector $y$ of Fourier samples of a 2D image, one cannot simply form the vector $x = U^{-1}_{\\mathrm {dft}}y$ , and then form $z = V_{\\mathrm {dwt}}x$ and hope that $z$ would represent wavelet coefficients of the function $f$ to be reconstructed (here $U_{\\mathrm {dft}}$ and $V_{\\mathrm {dwt}}$ denote the discrete Fourier and wavelet transforms respectively).", "Unfortunately, $x$ represents a discretization of the truncated Fourier series of $f$ .", "Thus, ignoring the wavelet crime [25] for a moment, we find that $z$ represents the wavelet coefficients of the truncated Fourier series and not the wavelet coefficients of $f$ itself (taking the wavelet crime into account, $z$ would actually be an approximation to the wavelet coefficients of the truncated Fourier series).", "Thus, $z$ will typically have lost all the decay properties of the original wavelet coefficients.", "Moreover, if we map $z$ back to the image domain we get $x = V_{\\mathrm {dwt}}^{-1}z$ and thus we do not gain anything as $x$ is the discretized truncated Fourier series.", "This paper is about getting the actual wavelet coefficients of $f$ from the Fourier samples, thus preserving all the decay properties of the original coefficients.", "This is done using generalized sampling.", "We show that the number of samples that must be acquired to ensure a stable and accurate reconstruction scales linearly with the number of reconstructing wavelet functions.", "This means that one can reconstruct a function from its Fourier coefficients yet get error bounds on the reconstruction (up to a constant) in terms of the decay properties of the wavelet coefficients.", "Put in short: seeing Fourier coefficients of a function is asymptotically as good as seeing the wavelet coefficients directly.", "As we will see in the numerical experiments, the generalized sampling reconstruction provides a substantial gain over the classical Fourier reconstruction.", "The outline of the remainder of this paper is as follows.", "In Section 2 we give a more elaborate introduction into generalized sampling and present the main results of this method.", "Furthermore, we introduce the stable sampling rate to which our main focus is devoted.", "Having presented the framework we are dealing with, we introduce the wavelet reconstruction systems and the Fourier sampling systems in Section 3.", "The main results are then presented in Section 4.", "Finally we demonstrate our theoretical results in application by presenting some numerical experiments in Section 5.", "Proofs of the main results are given in Section 6." ], [ "Generalized Sampling", "In this section, we recall the main definitions and results of the methodology of generalized sampling from [2], [3], [4], [5].", "For this, we start by introducing a general model situation for reconstruction from samples with associated quality measures." ], [ "General Setting", "Let ${\\cal H}$ be a (separable) Hilbert space $\\mathcal {H}$ with an inner product $\\langle \\cdot , \\cdot \\rangle $ , which will be our ambient space throughout this section.", "For modeling the acquisition of samples, let $ \\mathcal {S}\\subset \\mathcal {H}$ be a closed subspace and $\\lbrace s_k\\rbrace _{k \\in \\mathbb {N}} \\subset \\mathcal {H}$ be an orthonormal basis for $\\mathcal {S}$ .", "We will refer to $\\lbrace s_k\\rbrace _{k \\in \\mathbb {N}}$ as the sampling system and $\\mathcal {S}$ as the sampling space.", "For a signal $ f\\in \\mathcal {H}$ , we then assume that the associated samples (also called measurements) are given by $m(f)_k := \\langle f, s_k \\rangle , \\quad k \\in \\mathbb {N}.$ Based on the measurements $m(f)=(\\langle f, s_k\\rangle )_{k \\in \\mathbb {N}}$ , we aim to reconstruct the original signal $f$ .", "To be able to utilize some prior knowledge concerning the initial signal $f$ , we also require a carefully chosen reconstruction system $\\lbrace r_i\\rbrace _{i \\in \\mathbb {N}} \\subset \\mathcal {H}$ .", "The space $\\mathcal {R}= \\overline{\\operatorname{span}}\\lbrace r_i \\, : \\, i \\in \\mathbb {N}\\rbrace $ , in which $f$ is assumed to lie or be well approximated in, is then referred to as the corresponding reconstruction space.", "Since in an algorithmic realization, only finitely many samples – and likewise a finite linear combination of reconstruction elements – is possible, we also introduce the finite-dimensional spaces $\\mathcal {S}_M = \\operatorname{span}\\lbrace s_1, \\ldots , s_M\\rbrace , \\quad M \\in \\mathbb {N}$ and $\\mathcal {R}_N = \\operatorname{span}\\lbrace r_1, \\ldots , r_N\\rbrace , \\quad N \\in \\mathbb {N}.$ Thus, the reconstruction problem can now be phrased as follows: Given samples $m(f)_1, \\ldots , m(f)_M$ of an initial signal $f \\in {\\cal H}$ , compute a good approximation to $f$ in the reconstruction space $\\mathcal {R}_N$ .", "Aiming to compare different methodologies for solving this problem, we next formally introduce the notion of reconstruction method.", "Definition 2.1 Let a sampling system $\\lbrace s_k\\rbrace _{k \\in \\mathbb {N}}$ and reconstruction spaces $\\mathcal {R}_N$ , $N \\in \\mathbb {N}$ be defined as before.", "Further, let $f \\in {\\cal H}$ and let $M, N \\in \\mathbb {N}$ .", "Then some mapping $F_{N,M} : \\mathcal {H}\\longrightarrow \\mathcal {R}_N$ is called reconstruction method, if, for every $f \\in {\\cal H}$ , the signal $F_{N,M}(f)$ depends only on $m(f)_1, \\ldots , m(f)_M$ , where the samples $m(f)_j$ are defined in (REF ).", "We can now strengthen the reconstruction problem in the following way: Given the dimension of the reconstruction space $N$ , how many samples $M$ are required to obtain a stable and optimally accurate reconstruction method?", "This intuitive phrasing will be made precise in the next subsections." ], [ "Quality Measures for Reconstruction Methods", "We start by introducing two quality measures for reconstruction methods that analyze the degree of approximation within the reconstruction space and robustness for reconstruction.", "For this, throughout this subsection, let $\\lbrace s_k\\rbrace _{k \\in \\mathbb {N}}$ be a sampling system, and let $\\mathcal {R}_N$ , $N \\in \\mathbb {N}$ be reconstruction spaces.", "The first measure quantifies the closeness of the reconstruction to the `best' reconstruction in the sense of the orthogonal projection onto the reconstruction space.", "In the sequel, for the orthogonal projection onto a closed subspace $U$ , we will always utilize the notation $P_{U} : \\mathcal {H}\\longrightarrow U$ .", "Definition 2.2 ([6]) Let $F_{N,M}: \\mathcal {H}\\longrightarrow \\mathcal {R}_N$ be a reconstruction method.", "Moreover, let $\\mu = \\mu (F_{N,M})>0$ be the least number such that $\\Vert f - F_{N,M}(f) \\Vert \\le \\mu \\Vert f - P_{\\mathcal {R}_N} (f) \\Vert \\quad \\text{for all } f \\in \\mathcal {H}.$ Then we call $\\mu $ the quasi-optimality constant of $F_{N,M}$ .", "If no such constant exists, then we write $\\mu = \\infty $ .", "If $\\mu $ is small, we say $F_{N,M}$ is quasi-optimal.", "The second measure quantifies stability in the sense of robustness against perturbations.", "Definition 2.3 ([6]) Let $F_{N,M}: \\mathcal {H}\\longrightarrow \\mathcal {R}_N$ be a reconstruction method.", "The (absolute) condition number $\\kappa = \\kappa (F_{N,M})>0$ is then given by $\\kappa = \\sup \\limits _{f \\in \\mathcal {H}} \\lim \\limits _{\\varepsilon \\searrow 0 } \\sup \\limits _{\\begin{array}{c} g \\in \\mathcal {H}, \\\\0 < \\Vert m(g) \\Vert _{\\ell ^2} \\le \\varepsilon \\end{array}} \\left( \\frac{\\Vert F_{N,M}(f+g) - F_{N,M}(f) \\Vert }{\\Vert m(g)\\Vert _{\\ell ^2}} \\right),$ where $m(g)= (m(g)_1, \\ldots , m(g)_M, 0, \\ldots )$ .", "If $\\kappa $ is small, then we say $F_{N,M}$ is well conditioned, otherwise $F_{N,M}$ is called ill-conditioned.", "We now merge both definitions in order to have only one single measure for a reconstruction method.", "Definition 2.4 ([6], [7]) Let $F_{N,M}: \\mathcal {H}\\longrightarrow \\mathcal {R}_N$ be a reconstruction method.", "The reconstruction constant of $F_{N,M}$ is defined as $C(F_{N,M}) = \\max \\lbrace \\mu (F_{N,M}), \\kappa (F_{N,M}) \\rbrace ,$ where $\\mu (F_{N,M})$ is the quasi-optimality constant and $\\kappa (F_{N,M})$ is the (absolute) condition number." ], [ "The Infimum Cosine Angle", "Intuitively, the angle between the sampling and reconstruction space should play a role in determining the reconstruction constant for a given reconstruction method.", "As we will see, this will be in particular the case for the reconstruction method of generalized sampling.", "To prepare those results, in this subsection, we first introduce a particularly useful notion of angle.", "The concept of principal angles between two Euclidean subspaces is well known in the literature.", "However, for arbitrary closed subspaces of a Hilbert space many different notions of an angle exist.", "We exemplarily mention the Friedrichs angle and the Dixmier angle (cf.", "[12], [14]).", "For our analysis, the notion of infimum cosine angle – utilized, for instance, in [8], [28] – will be the most appropriate.", "It is defined as follows.", "Definition 2.5 Let $U, V$ be closed subspaces of ${\\cal H}$ .", "Then the infimum cosine angle $\\cos (\\omega (U,V))$ between $U$ and $V$ is defined by $\\cos (\\omega (U,V)) = \\inf \\limits _{\\begin{array}{c}u \\in U, \\\\ \\Vert u \\Vert =1\\end{array}} \\Vert P_V u \\Vert , \\quad \\omega (U,V) \\in [0, \\pi /2].$ We remark that the infimum cosine angle is not symmetric in general.", "For example, if $U$ and $V$ are two non-trivial closed subspaces of $\\mathcal {H}$ and $U \\ne V$ with $U \\subset V$ , then $\\cos (\\omega (U,V)) =1$ , whereas $\\cos (\\omega (V,U))=0$ .", "The following general characterization of pairs of closed subspaces for which the infimum cosine angle is not symmetric was proven in [9].", "Lemma 2.6 ([9]) For two non-trivial closed subspaces $U,V \\subset \\mathcal {H}$ , we have $\\cos (\\omega (U,V)) \\ne \\cos (\\omega (V,U))$ if and only if one of these quantities is zero and the other is positive.", "A positive infimum cosine angle between the sampling and reconstruction space will be crucial for enabling reconstruction at all.", "The next theorem provides a characterization of subspaces which admit a positive infimum cosine angle.", "Theorem 2.7 ([26]) Let $U, V$ be closed subspaces of $\\mathcal {H}$ .", "Then, we have that $\\cos (\\omega (U,V)) >0$ if and only if $U \\cap V^\\perp = \\lbrace 0\\rbrace $ and $ U + V^\\perp $ is closed in $\\mathcal {H}$ .", "This characterization gives rise to the following definition.", "Definition 2.8 ([6], [8]) If $\\cos (\\omega (U,V^\\perp )) >0$ for two closed subspaces $U, V$ of ${\\cal H}$ , then we say $U$ and $V$ obey the subspace condition.", "In this case, we define the associated (oblique) projection $P_{U,V}: U \\oplus V\\longrightarrow \\mathcal {H}$ with range of $P$ equal to $U$ and kernel of $P$ equal to $V$ .", "We wish to mention that oblique projections are customarily used in sampling theory, such as, for instance, in [8], [15], [16], [28]." ], [ "Reconstruction Method of Generalized Sampling", "We are now ready to introduce the method of generalized sampling.", "To this end, we will always assume that the reconstruction space $\\mathcal {R}$ and the sampling space $\\mathcal {S}^\\perp $ fulfill the subspace condition.", "In other words, we have $\\cos (\\omega (\\mathcal {R},\\mathcal {S})) >0$ .", "For any $M \\in \\mathbb {N}$ , let $P_{\\mathcal {S}_M}$ be the orthogonal projection given by $P_{\\mathcal {S}_M} : \\mathcal {H}\\longrightarrow \\mathcal {S}_M, \\quad h \\mapsto \\sum \\limits _{k=1}^M \\langle h, s_k \\rangle s_k.$ This enables us to formally define the reconstruction method of generalized sampling.", "Definition 2.9 For $f \\in \\mathcal {H}$ and $N , M\\in \\mathbb {N}$ , we define the reconstruction method of generalized sampling $G_{N,M} : {\\cal H}\\rightarrow \\mathcal {R}_N$ by $\\langle P_{\\mathcal {S}_M} G_{N,M}(f), r_j \\rangle = \\langle P_{\\mathcal {S}_M} f, r_j \\rangle , \\quad j = 1, \\ldots , N.$ We also refer to $G_{N,M}(f)$ as the generalized sampling reconstruction of $f$ .", "We emphasize that this is indeed a reconstruction method in the sense of Definition REF , since the right-hand side of (REF ) only depends on the given samples $(\\left\\langle f,s_k\\right\\rangle )_{k = 1}^M$ and not on $f$ itself.", "Moreover, generalized sampling is a linear reconstruction method.", "Algorithmically, to determine $G_{N,M}(f)$ , i.e., solving (REF ), can be phrased as the numerical linear algebra problem of computing the coefficients $\\alpha ^{[N]} = (\\alpha _1, \\ldots , \\alpha _N )\\in N$ as the least-squares solution of $U^{[M,N]} \\alpha ^{[N]} = m(f)^{[M]}, \\qquad \\mbox{where } \\;U^{[M,N]}= \\begin{pmatrix}u_{11} & \\ldots & u_{1N} \\\\\\vdots & \\ddots & \\vdots \\\\u_{M1} & \\ldots & u_{MN}\\end{pmatrix}, \\; u_{ij} = \\langle r_j, s_i \\rangle .$ Note that, if $U^{[M,N]}$ is well-conditioned this requires $\\mathcal {O}(M N)$ operations in general.", "However, in the case of Fourier samples and wavelet reconstruction one can make use of the Fast Fourier Transform and the discrete wavelet transform and thus reduce this figure down to only $\\mathcal {O}(M \\log M)$ operations.", "The philosophy of generalized sampling is to allow the number of samples $M$ to grow independently of the fixed number of reconstruction elements $N$ .", "This flexibility of $M$ and $N$ is crucial for stable reconstruction.", "The next theorem guarantees the existence of the reconstruction for any $f \\in {\\cal H}$ provided that the number of samples $M$ is large enough.", "Theorem 2.10 ([6]) Let $N \\in \\mathbb {N}$ .", "Then, there exists an $M_0 \\in \\mathbb {N}$ , such that, for every $f \\in {\\cal H}$ , (REF ) has a unique solution $G_{N,M}(f)$ for all $M \\ge M_0$ .", "In particular, we then have $G_{N,M} = P_{\\mathcal {R}_N ,(P_{\\mathcal {S}_M}(\\mathcal {R}_N))^\\perp }.$ Moreover, the smallest $M_0$ is the least number such that $\\cos ( \\omega (\\mathcal {R}_N, \\mathcal {S}_{M_0} ))>0.$ It is a priori not clear how to find $M \\in \\mathbb {N}$ large enough such that $\\cos (\\omega (\\mathcal {R}_N, \\mathcal {S}_M)) >0$ , or even determine the smallest such value $M_0 \\in \\mathbb {N}$ .", "In the next subsection, it will turn out that this is intimately related to the reconstruction constant defined in Definition REF , and will lead to the notion of a stable sampling rate." ], [ "Stable Sampling Rate", "In Subsection REF , we introduced the reconstruction constant as the main quality measure for stable and accurate reconstructions.", "Intriguingly, we can now relate this notion to the infimum cosine angle in the case of generalized sampling as reconstruction method.", "Theorem 2.11 ([6]) Retaining the definitions and notations from Subsection REF , for all $f \\in {\\cal H}$ , we have $\\Vert G_{N,M}(f) \\Vert \\le \\frac{1}{\\cos ( \\omega (\\mathcal {R}_N, \\mathcal {S}_M))} \\Vert f \\Vert ,$ and $\\Vert f - P_{\\mathcal {R}_N} f \\Vert \\le \\Vert f- G_{N,M}(f) \\Vert \\le \\frac{1}{\\cos ( \\omega (\\mathcal {R}_N, \\mathcal {S}_M))} \\Vert f - P_{\\mathcal {R}_N} f \\Vert .$ In particular, these bounds are sharp.", "Moreover, the reconstruction constant of generalized sampling $C(G_{N,M})$ obeys $C(G_{N,M}) = \\mu (G_{N,M}) = \\kappa (G_{N,M}) = \\frac{1}{\\cos ( \\omega (\\mathcal {R}_N,\\mathcal {S}_M))}.$ Hence, in order to obtain stable and accurate reconstructions, it is both necessary and sufficient to control the angle between $\\mathcal {R}_N$ and $\\mathcal {S}_M$ .", "This leads us to the definition of the stable sampling rate.", "Definition 2.12 For $N \\in \\mathbb {N}$ and $\\theta >1$ , the stable sampling rate is defined as $\\Theta (N,\\theta ) = \\min \\left\\lbrace M \\in \\mathbb {N}\\, : \\, C(G_{N,M}) < \\theta \\right\\rbrace .$ Note that the stable sampling rate is of interest, since it determines the number of samples required for guaranteed, quasi-optimal and numerically stable reconstructions.", "We next aim to compare generalized sampling to other reconstruction methods.", "For this, we first introduce a class of reconstruction methods, which recover signals from the reconstruction space exactly.", "Definition 2.13 A reconstruction method $F_{N,M}: \\mathcal {H}\\longrightarrow \\mathcal {R}_N$ is called perfect, if $F_{N,M} (f) = f$ whenever $f \\in \\mathcal {R}_N$ .", "Note that any reconstruction method with finite quasi-optimality constant is automatically perfect.", "The next result proves that generalized sampling is in the sense superior to any other perfect reconstruction method that its condition number as well as even its reconstruction constant is smaller.", "Theorem 2.14 ([6]) Let $M \\ge N$ , and let $F_{N,M}: \\mathcal {H}\\longrightarrow \\mathcal {R}_N$ be a linear or non linear perfect reconstruction method.", "Then $\\kappa (G_{N,M}) \\le \\kappa (F_{N,M}),$ where $G_{N,M}$ is the reconstruction method of generalized sampling.", "In particular, we have $C(G_{N,M}) \\le C(F_{N,M}).$ Next, we aim to compare the quasi-optimality constant of generalized sampling to other reconstruction methods.", "For this, assume that the stable sampling rate of generalized sampling is linear in $N$ , i.e., $\\Theta (N, \\theta ) = \\mathcal {O}(N)$ as $N \\rightarrow \\infty $ , and assume that there exist constants $C_f,D_f, \\gamma _f >0$ depending on the initial signal $f \\in {\\cal H}$ such that $C_f N^{-\\gamma _f} \\le \\Vert f - P_{\\mathcal {R}_N}f \\Vert \\le D_f N^{-\\gamma _f}, \\quad \\mbox{for all } N \\in \\mathbb {N}.$ The following result then shows that the error of any reconstruction method $F_M$ can be only up to a constant better than the error of generalized sampling reconstruction.", "Theorem 2.15 ([6]) Suppose that the stable sampling rate $\\Theta (N, \\theta )$ is linear in $N$ , i.e.", "$\\Theta (N, \\theta ) = \\mathcal {O}(N)$ as $N \\rightarrow \\infty $ .", "Let $f \\in \\mathcal {H}$ be fixed and let $F_M: (m(f)_1, \\ldots , m(f)_M) \\mapsto F_M(f) \\in \\mathcal {R}_{\\psi _f(M)},$ be a reconstruction method, where $\\psi _f: \\mathbb {N}\\longrightarrow \\mathbb {N}$ with $\\psi _f(M) \\le \\lambda M$ for some $\\lambda >0$ .", "Assume that (REF ) holds.", "Then, for any $\\theta >1$ , there exist constants $d(\\theta ) \\in (0,1)$ and $c(\\theta , C_f, D_f)>0$ such that $\\Vert f - G_{d(\\theta )M,M}(f) \\Vert \\le c(\\theta , C_f, D_f) \\Vert f - F_M(f) ||, \\quad \\mbox{for all } M \\in \\mathbb {N},$ where $G_{N,M}$ denotes the generalized sampling reconstruction method." ], [ "Linear Sampling Rate for Compactly Supported 2D Wavelets", "As already elaborated upon in the introduction, the situation of taking the Fourier basis as sampling system is of particular importance to applications.", "A very common choice for a reconstruction system in imaging sciences are 2D wavelets, predominantly of compact support due to their high spatial localization.", "In this section, we will state our main result for the situation of Fourier samples and reconstruction within a wavelet basis using 2D compactly supported wavelets.", "More precisely, we will show linearity of the stable sampling rate for the associated generalized sampling scheme, which shows by Theorem REF the near-optimality of this reconstruction method.", "For this, we start by defining first the reconstruction and second the sampling space, followed by the statement of our main result.", "Due to its technical nature, we present its proof in a separate section, namely Section ." ], [ "Compactly Supported 2D Wavelets", "We start by recalling the notion of scaling matrices.", "Definition 3.1 Let $A$ be a $2 \\times 2$ matrix with non-negative integer entries and eigenvalues greater than one in modulus.", "Then we call $A$ a scaling matrix.", "For the sake of brevity, in the sequel, we will use the notation $A = \\begin{pmatrix} \\lambda _1 & \\lambda _2 \\\\ \\lambda _3 & \\lambda _4 \\end{pmatrix}.$ Moreover, for the entries of $A^j$ we write $A^j = \\begin{pmatrix} \\lambda _1^{(j)} & \\lambda _2^{(j)} \\\\ \\lambda _3^{(j)} & \\lambda _4^{(j)} \\end{pmatrix}, \\quad j \\in \\mathbb {N}.$ Notice that $\\lambda _i^{(j)} \\ne (\\lambda _i)^j$ in general.", "For the sake of completeness, we next give the definition of a 2D multiresolution analysis (MRA).", "For more details, we refer to the existing literature, e.g [13], [21].", "Definition 3.2 Let $A$ be a scaling matrix.", "Then a sequence of closed subspaces $(V_j)_{j \\in \\mathbb {Z}}$ of $L^2(\\mathbb {R}^2)$ is called a multiresolution analysis, if the following properties are satisfied.", "[i)] $ \\lbrace 0\\rbrace \\subset \\ldots \\subset V_j \\subset V_{j+1} \\subset \\ldots \\subset L^2(\\mathbb {R}^2)$ , $\\bigcap \\limits _{j \\in \\mathbb {Z}} V_j = \\lbrace 0 \\rbrace $ , $\\overline{\\bigcup \\limits _{j \\in \\mathbb {Z}} V_j} = L^2(\\mathbb {R}^2)$ , $f \\in V_j \\Leftrightarrow f(A \\cdot ) \\in V_{j+1}$ , there exists a function $\\phi \\in L^2(\\mathbb {R}^2)$ (called scaling function), such that $\\lbrace \\phi _{0,m} := \\phi ( \\cdot - m ) \\, : \\, m \\in \\mathbb {Z}^2 \\rbrace $ constitutes an orthonormal basis for $V_0$ .", "The associated wavelet spaces $(W_j)_{j \\in \\mathbb {Z}}$ are then defined by $V_{j+1} = V_j \\oplus W_j.$ It is well known that there exist $| \\det A | -1$ corresponding compactly supported wavelets $\\psi ^1, \\ldots , \\psi ^{|\\det A| -1}$ such that $\\lbrace \\psi _{j,m}^p := | \\det A |^{j/2} \\psi ^p(A^j \\cdot -m) : m= (m_1,m_2) \\in \\mathbb {Z}^2, p=1, \\ldots , |\\det A| -1\\rbrace $ forms an orthonormal basis for $W_j$ for each $j$ , see, e.g., .", "We now consider the decomposition $L^2(\\mathbb {R}^2) = V_0 \\oplus \\bigoplus _{j \\in \\mathbb {N}} W_j,$ where $V_0 :=\\overline{\\operatorname{span}}\\lbrace \\phi _{0,m} \\, : \\, m= (m_1,m_2) \\in \\mathbb {Z}^2\\rbrace $ and $W_j := \\overline{\\operatorname{span}}\\lbrace \\psi ^p_{j,m} \\, : \\, m= (m_1,m_2) \\in \\mathbb {Z}^2, p = 1, \\ldots , | \\det A | -1 \\rbrace , \\quad j \\in \\mathbb {N}.$ Reconstruction Space Our aim is to reconstruct functions that are supported on $[0,a]^2$ .", "To this end, suppose that the scaling function and wavelet functions are supported in $[0,a]^2$ .", "To mimic the fact that practical applications can only handle finite systems, we restrict to those functions whose support intersects $[0,a]^2$ , i.e., to the systems $\\Omega _1=\\lbrace \\phi _{0,m} \\, : \\, m= (m_1,m_2) \\in \\mathbb {Z}^2, -a< m_1, m_2 < a \\rbrace $ and $\\Omega _2 = \\lbrace \\psi ^p_{j,m} \\, : \\, j \\in \\mathbb {N}\\cup \\lbrace 0 \\rbrace , \\ &m= (m_1,m_2) \\in \\mathbb {Z}^2, -a < m_1 < a (\\lambda _1^{(j)}+ \\lambda _2^{(j)}), \\\\ & -a< m_2 < a (\\lambda _3^{(j)} + \\lambda _4^{(j)}), p = 1, \\ldots , | \\det A | -1 \\rbrace .$ The reconstruction space $\\mathcal {R}$ is then defined as the closed linear span of these functions, which is $\\mathcal {R}= \\overline{\\operatorname{span}} \\lbrace \\varphi \\, : \\, \\varphi \\in \\Omega _1 \\cup \\Omega _2\\rbrace .$ To define the finite-dimensional subspaces $\\mathcal {R}_N$ , we require an ordering for this system.", "The most natural way to order $\\Omega _1 \\cup \\Omega _2$ is starting from wavelets at coarsest scale and then continue to higher scales.", "Within one scale, one might order the translation $(m_1,m_2)$ in a lexicographical manner starting from the smallest number up the largest.", "More precisely, we fix $m_1$ and let $m_2$ run, increase $m_1$ by one and repeat.", "This then leads to the following ordering of $\\Omega _1 \\cup \\Omega _2$ : $\\nonumber {\\lbrace \\varphi _i\\rbrace _{i \\in \\mathbb {N}}}\\\\& = & \\lbrace \\phi _{0, (-a+1,-a+1)}, \\ldots \\phi _{0, (-a+1,a-1)}, \\phi _{0, (-a + 2,-a+1)} \\ldots , \\phi _{0, (-a+2,a-1)}, \\ldots ,\\phi _{0, (a-1,a-1)}, \\nonumber \\\\&& \\psi ^1_{0, (-a +1 , -a +1)}, \\ldots , \\psi ^1_{0, (-a +1 , m_2^{(0)} -1)}, \\ldots , \\psi ^1_{0, (m_1^{(0)} -1 , -a +1)},\\ldots , \\psi ^1_{0, (m_1^{(0)} -1 , m_2^{(0)} -1)}, \\ldots , \\nonumber \\\\&& \\psi ^{| \\det A | -1}_{0, (-a +1 , - a +1)}, \\ldots , \\psi ^{| \\det A | -1}_{0, (-a +1 , m_2^{(0)} -1)}, \\ldots ,\\psi ^{| \\det A | -1}_{0, (m_1^{(0)} -1 , - m_2^{(0)} +1)}, \\ldots , \\psi ^{| \\det A | -1}_{0, (m_1^{(0)} -1 , m_2^{(0)} -1)}, \\nonumber \\\\&& \\psi ^1_{1, (-a +1 , - m_2^{(1)} +1)}, \\ldots , \\psi ^1_{1, (-a +1 , m_2^{(1)} -1)}, \\psi ^1_{1, (-a +2 , - m_2^{(1)} +1)},\\ldots , \\psi ^1_{1, (-a +2 , m_2^{(1)} -1)}, \\ldots \\rbrace $ where $m_1^{(j)} = a (\\lambda _1^{(j)} + \\lambda _2^{(j)})$ and $m_2^{(j)} = a (\\lambda _3^{(j)} + \\lambda _4^{(j)}) $ .", "We emphasize that the presented results do not depend on the specific ordering within the scale.", "Finally, based on the chosen ordering, we define the reconstruction space $\\mathcal {R}_N$ by $\\mathcal {R}_N = \\operatorname{span}\\lbrace \\varphi _i \\, : \\, i = 1, \\ldots , N\\rbrace , \\quad N \\in \\mathbb {N}.$ In practise, it is much more common to use as approximation spaces those being generated up to a specific scale, say, $J$ .", "To mimic this approach, we coarsen the choices of $N \\in \\mathbb {N}$ for which we consider $\\mathcal {R}_N$ suitably in the following way.", "First, we observe that the number of functions being in $\\mathcal {R}$ up to a fixed scale $J-1 \\in \\mathbb {N}$ is $N_J = (2a - 1 )^2 + (|\\det A|-1) \\sum _{j=0}^{J-1} ( a(\\lambda _1^{(j)} + \\lambda _2^{(j)} +1) -1)( a(\\lambda _3^{(j)} + \\lambda _4^{(j)} +1) -1).$ Lemma 3.3 Retaining the definitions and notations above, then $N_J \\le C_a (\\lambda _1^{(J)} + \\lambda _2^{(J)})(\\lambda _3^{(J)} + \\lambda _4^{(J)})$ for some constant $C_a$ depending on $a$ .", "The total number of elements $N_J$ in the reconstruction space $\\mathcal {R}_{N_J}$ up to a scale $J-1$ is prescribed by the matrix $A$ and the support of the scaling function $\\phi $ and the wavelet $\\psi $ , respectively.", "However, since the support of both $\\phi $ and $\\psi $ is a square of the form $[0,a]^2$ , the total number of elements in the reconstruction space $N_J$ is the same as if we would have generated the wavelet system with the transpose of $A$ .", "Hence, it is sufficient to prove $N_J \\lesssim (\\lambda _1^{(J)} + \\lambda _2^{(J)})(\\lambda _3^{(J)} + \\lambda _4^{(J)}),$ where the constant depends on $a$ .", "We prove the result by induction.", "For $J = 1$ it is clearly true.", "Hence, $(2a - 1 )^2 + (|\\det A|-1) \\sum _{j=0}^{J-1}& ( a(\\lambda _1^{(j)} + \\lambda _2^{(j)} +1) -1)( a(\\lambda _3^{(j)} + \\lambda _4^{(j)} +1) -1) \\\\& \\lesssim (\\lambda _1^{(J-1)} + \\lambda _2^{(J-1)})(\\lambda _3^{(J-1)} + \\lambda _4^{(J-1)}) + (|\\det A|-1)( a(\\lambda _1^{(J-1)} \\\\& \\quad + \\lambda _2^{(J-1)} +1) -1)( a(\\lambda _3^{(J-1)} + \\lambda _4^{(J-1)} +1) -1)$ Now, $A^J &= \\begin{pmatrix} \\lambda _1^{(J)} & \\lambda _2^{(J)} \\\\ \\lambda _3^{(J)} & \\lambda _4^{(J)} \\end{pmatrix}= \\begin{pmatrix} \\lambda _1^{(J-1)} & \\lambda _2^{(J-1)} \\\\ \\lambda _3^{(J-1)} & \\lambda _4^{(J-1)} \\end{pmatrix}\\begin{pmatrix} \\lambda _1 & \\lambda _2 \\\\ \\lambda _3 & \\lambda _4 \\end{pmatrix}= \\begin{pmatrix} \\lambda _1^{(J-1)}\\lambda _1 + \\lambda _2^{(J-1)}\\lambda _3 & \\lambda _1^{(J-1)}\\lambda _2 + \\lambda _2^{(J-1)}\\lambda _4 \\\\ \\lambda _3^{(J-1)}\\lambda _1 + \\lambda _4^{(J-1)}\\lambda _3 & \\lambda _3^{(J-1)}\\lambda _2 + \\lambda _4^{(J-1)}\\lambda _4 \\end{pmatrix}$ which implies $\\lambda _1^{(J)} + \\lambda _2^{(J)} &= \\lambda _1^{(J-1)}( \\lambda _1 + \\lambda _2) + \\lambda _2^{(J-1)}(\\lambda _3 + \\lambda _4),\\\\\\lambda _3^{(J)} + \\lambda _4^{(J)} &= \\lambda _3^{(J-1)}( \\lambda _1 + \\lambda _2) + \\lambda _4^{(J-1)}(\\lambda _3 + \\lambda _4).$ Similar to the construction in [7], we now define the truncated scaling space by $V^{(a)}_0 := \\operatorname{span}\\lbrace \\phi _{0,m} \\, : \\, m= (m_1,m_2) \\in \\mathbb {Z}^2, -a< m_1, m_2 < a \\rbrace $ and the truncated wavelet spaces by $W^{(a)}_j := \\operatorname{span}\\lbrace \\psi ^p_{j,m} \\, : \\, \\ &m= (m_1,m_2) \\in \\mathbb {Z}^2, -a < m_1 < a(\\lambda _1^{(j)}+ \\lambda _{2}^{(j)}), \\\\&-a < m_2 < a(\\lambda _3^{(j)}+ \\lambda _{4}^{(j)}), p = 1, \\ldots , |\\det A| -1 \\rbrace .$ The reconstruction space of interest to us is then defined by $\\mathcal {R}_{N_J} = V^{(a)}_0 \\oplus W^{(a)}_0 \\oplus \\ldots \\oplus W^{(a)}_{J-1}.$ Sampling Space To define the sampling space consisting of elements of the Fourier basis, we first choose $T_1,T_2>0$ sufficiently large such that $\\mathcal {R}\\subset L^2([-T_1,T_2]^2).$ Thus, only functions supported on $[-T_1,T_2]^2$ are relevant to us.", "Indeed, choosing $T_1 \\ge a-1$ and $T_2 \\ge 2a-1$ is enough.", "Note that $\\lambda _1^{(0)}=\\lambda _4^{(0)} =1$ and $\\lambda _2^{(0)}=\\lambda _3^{(0)} =0$ .", "To allow an arbitrarily dense sampling, for each $\\varepsilon \\le \\frac{1}{T_1 + T_2}$ , we define the sampling vectors by $s_l^{(\\varepsilon )} = \\varepsilon e^{2 \\pi i \\varepsilon \\langle l, \\cdot \\rangle } \\cdot \\chi _{[-T_1,T_2]^2}, \\quad l \\in \\mathbb {Z}^2.$ Thus, we sample in each direction with the same sampling rate $\\varepsilon $ .", "Based on these sampling vectors, we now define the sampling space $\\mathcal {S}^{(\\varepsilon )}$ by $\\mathcal {S}^{(\\varepsilon )} = \\overline{\\operatorname{span}} \\left\\lbrace s_l^{(\\varepsilon )} \\, : \\, l \\in \\mathbb {Z}^2 \\right\\rbrace .$ The finite-dimensional subspaces $\\mathcal {S}^{(\\varepsilon )}_{M}$ , $M = (M_1,M_2 )\\in \\mathbb {N}\\times \\mathbb {N}$ , are then given by $\\mathcal {S}^{(\\varepsilon )}_{M} = \\operatorname{span}\\left\\lbrace s_l^{(\\varepsilon )} \\, : \\, l = (l_1,l_2) \\in \\mathbb {Z}^2, - M_i \\le l_i \\le M_i , i =1,2 \\right\\rbrace .$ Main Result Our main results concerns the stable sampling rate of the generalized sampling scheme for the sampling spaces $\\mathcal {S}^{(\\varepsilon )}_M$ and the reconstruction spaces $\\mathcal {R}_{N_J}$ .", "By Theorem REF , for this, we have to control the infimum cosine angle between the $\\mathcal {R}_{N_J}$ and $\\mathcal {S}^{(\\varepsilon )}_M$ .", "In particular, we wish to determine $M=(M_1,M_2) \\in \\mathbb {N}\\times \\mathbb {N}$ such that, for given $\\theta >1$ and $J-1 \\in \\mathbb {N}$ the term $\\cos (\\omega (\\mathcal {R}_{N_J},\\mathcal {S}_M^{(\\varepsilon )}))$ can be bounded from below by $\\theta ^{-1}$ , where $N_J$ denotes the total number of reconstruction elements up to scale $J-1$ .", "For this to work, we will assume that the scaling matrix does not distort the grid too much.", "To make this precise, let $I_M = \\lbrace Ê(l_1,l_2) \\in \\mathbb {Z}^2 \\, : \\, -M_i \\le l_i \\le M_i, i =1,2 \\rbrace , L_M = [-M_1,M_1]\\times [-M_2,M_2]$ for $(M_1,M_2) \\in \\mathbb {N}^2$ .", "Then, we assume that the so-called mesh norm $\\delta $ obeys $\\delta := \\max \\limits _{x \\in \\varepsilon A^{-J}(L_M)} \\min \\limits _{k \\in \\varepsilon A^{-J}(\\mathbb {Z}^2)} \\Vert x - y + k\\Vert _\\infty < \\frac{\\log \\left(\\frac{1}{\\sqrt{\\mu (L_M)}}+1\\right)}{ 4 \\pi \\max \\lbrace |L_1|,|L_2|,|L_3|,|L_4|\\rbrace }, $ for some $\\varepsilon $ independent of $J$ , where $\\mu $ denotes the 2D lebesgue measure and $L_i, i =1,2,3,4$ are bounds that are obtained by Lemma REF .", "We will also discuss this assumption in Example REF for better understanding.", "The following result shows that the stable sampling rate is indeed linear in the considered situation, showing that this scheme is superior to any other reconstruction method in the sense of Theorem REF .", "Theorem 3.4 Let $N_J$ be the number of reconstruction elements up to a fixed scale $J-1$ , and let $\\mathcal {R}_{N_J}$ and $\\mathcal {S}_M^{(\\varepsilon )}$ be the reconstruction space and sampling space, respectively.", "Furthermore, assume (REF ) is fulfilled.", "If $\\theta >1$ , then there exists a constant $S^{(\\theta )}$ independent of $J$ and $\\varepsilon $ such that, if $M_1 \\ge \\frac{ \\lambda _1^{(J)} + \\lambda _3^{(J)}}{\\varepsilon } S^{(\\theta )},\\qquad M_2 \\ge \\frac{ \\lambda _2^{(J)} + \\lambda _4^{(J)}}{\\varepsilon } S^{(\\theta )},$ then, for $M = (M_1,M_2)$ , $\\cos (\\omega (\\mathcal {R}_{N_J},\\mathcal {S}_M^{(\\varepsilon )})) \\ge \\frac{1}{\\theta }.$ In particular, the stable sampling rate obeys $\\Theta (N_J,\\theta ) = \\mathcal {O}(N_J)$ as $N_J \\rightarrow \\infty $ for every fixed $\\theta >1$ .", "Summarizing, this result shows that the required number of Fourier samples is optimally small up to a constant when utilizing 2D compactly supported wavelets for the generalized sampling reconstruction.", "This extends the result of [7] to the two-dimensional case.", "We next consider a special, yet widely used choice for scaling matrices, namely diagonal matrices.", "This includes two dimensional dyadic wavelets, which are those typically used in applications.", "This situation is also considered in the numerical experiments presented in Section .", "Example 3.5 In this example we want to demonstrate our results and, in particular, give some better understanding of the construction and assumption (REF ).", "Furthermore, this example shall show that there are large classes of 2D wavelets that fulfill (REF ).", "For this purpose, let $A = \\begin{pmatrix} 2 & 0 \\\\ 0 & 2 \\end{pmatrix}$ be the scaling matrix, which – as mentioned before – gives rise to three wavelet generators.", "Let $\\phi $ and $\\psi ^p, p = 1,2,3$ be two dimensional scaling and wavelet functions with compact support in $[0,a]^2, a\\in \\mathbb {N}$ , which might be obtained by tensor products of one dimensional scaling and wavelet functions, see [13], [22].", "As in Subsection REF , we define $\\Omega _1:=\\lbrace \\phi _{0,m} \\, : \\, m= (m_1,m_2) \\in \\mathbb {Z}^2, |m_i | < a, i =1,2 \\rbrace $ and $\\Omega _2:= \\lbrace \\psi ^p_{j,m} \\, : \\, j \\in \\mathbb {N}\\cup \\lbrace 0\\rbrace , m= (m_1,m_2) \\in \\mathbb {Z}^2, -a < m_i < 2^j a, i =1,2 , p = 1,2,3\\rbrace ,$ since these are the only functions whose support intersects $[0,a]^2$ .", "Again, in line with our previous approach, we then define the reconstruction space $\\mathcal {R}$ by $\\mathcal {R}= \\overline{\\operatorname{span}} \\lbrace \\varphi \\, : \\, \\varphi \\in \\Omega _1 \\cup \\Omega _2\\rbrace ,$ order the elements $\\Omega _1 \\cup \\Omega _2$ analogously to (REF ), and set $\\mathcal {R}_N = \\operatorname{span}\\lbrace \\varphi _i \\, : \\, i = 1, \\ldots , N\\rbrace , \\quad N \\in \\mathbb {N}.$ Now, each function $\\varphi $ in $\\mathcal {R}_N$ can be represented as a linear combination of scaling functions at highest level $J$ , in particular, there exist positive integers $L_1,L_2,L_3,$ and $L_4$ such that $\\varphi = \\sum _{l_1=L_3}^{L_1} \\sum _{l_2=L_4}^{L_2} \\alpha _{l_1,l_2} \\phi _{J,(l_1,l_2)}, \\quad \\alpha _{l_1,l_2} \\in $ This statement will be proven in Lemma REF .", "With a view to (REF ), the explicit expression of the bounds $L_i, i = 1,2,3,4$ are highly important.", "In fact, we should not choose them too large, otherwise (REF ) might not hold.", "Since $\\varphi = \\sum _{l_1=L_3}^{L_1} \\sum _{l_2=L_4}^{L_2} \\langle \\varphi , \\phi _{J,(l_1,l_2)} \\rangle \\phi _{J,(l_1,l_2)},$ and is compactly supported, shifting by $(l_1,l_2)$ far enough leads to the fact that the coefficients become zero.", "For the scaling matrix $A= \\operatorname{diag}(2,2)$ we have (see proof of Lemma REF ) $L_1=L_3= 2^{J}(3a-1), \\quad L_2 =L_4= -a+2^{J}(-a+1).$ Moreover, the mesh norm $\\delta $ obeys $\\delta \\le \\frac{\\varepsilon }{2^J}.$ Hence, for $\\varepsilon = \\frac{1}{4\\pi (3a-1)}$ $\\delta \\le \\frac{1}{4\\pi (3a-1)2^J} \\le \\frac{\\log \\left(\\left(\\frac{2^J}{\\varepsilon }\\right)^2+1\\right)}{2\\pi (3a-1)2^J}, $ so assumption (REF ) is fulfilled.", "Thus, we can apply Theorem REF to obtain a constant $S^\\theta $ independent of $J$ , such that for $M_1 , M_2 = \\left\\lceil \\frac{2^J S^\\theta }{\\varepsilon }\\right\\rceil $ we have $\\cos (\\omega (\\mathcal {R}_{N_J},\\mathcal {S}_M^{(\\varepsilon )}))> \\frac{1}{\\theta }.$ Counting the numbers of elements in the space $\\mathcal {R}_{N_J}$ gives $N_J =(2a - 1)^2 + 3\\sum \\limits _{j=0}^{J-1} (2^ja + a-1)^2 = (2^{2J}-1)a^2 + 6a(a-1)(2^J-1) + 3J(a-1)^2 + (2a-1)^2 = \\mathcal {O}(2^{2J}), $ which means the number of samples scales linearly with the number of reconstruction elements.", "Linear Sampling Rate for 2D Boundary Wavelets In applications, typically signals on a bounded domain are considered such as on the interval $[0,1]$ .", "Aiming to avoid artifacts at the boundaries $x=0, 1$ , in [11] (see also [22]) boundary wavelets were introduced.", "The considered wavelet system then consists of interior wavelets, which do not touch the boundary, and the just mentioned boundary wavelets, leading to a system with an associated multiresolution analysis and prescribed vanishing moments even at the boundary.", "Similar to the classical wavelet construction, this system can be lifted to 2D by tensor products.", "In this section, we will show that in fact also this (boundary) wavelet system allows a linear stable sampling rate, though with a proof differing significantly from the one of the `classical wavelet case' due to the different structure at the highest scale.", "For this, we start with formally introducing 2D boundary wavelets – which we use as an expression for the whole wavelet system – followed by our choice of reconstruction and sampling space.", "Construction of Boundary Wavelets We start with the 1D construction of wavelets on the interval as introduced in [11].", "For this, let $\\phi $ be a compactly supported Daubechies scaling function with $p$ vanishing moments.", "It is well known that $\\phi $ must then have a support of size $2p-1$ .", "By a shifting argument, we can assume that $\\operatorname{supp \\,}(\\phi ) = [-p+1,p]$ .", "In order to properly define interior wavelets and boundary wavelets, the scale need to be large enough – i.e., the support of the wavelets need to be small enough – to be able to distinguish between wavelets whose support fully lie in $[0,1]$ and wavelets whose support intersect the boundary $x=0$ and, likewise, $x =1$ .", "Therefore, we now let $j \\in \\mathbb {N}$ such that $2p \\le 2^j$ .", "Then there exist $2^j - 2p$ interior scaling functions, i.e., scaling functions which have support inside $[0,1]$ , defined by $\\phi ^{\\operatorname{b}}_{j,n} = \\phi _{j,n} = 2^{j/2} \\phi (2^j\\cdot - n), \\quad \\text{for } p\\le n < 2^j-p.$ Depending on boundary scaling functions $ \\lbrace \\phi ^{\\operatorname{left}}_n\\rbrace _{n=0, \\ldots , p-1}$ and $\\lbrace \\phi ^{\\operatorname{right}}_n\\rbrace _{n=0, \\ldots , p-1}$ , which we will introduce below, the $p$ left boundary scaling functions are defined by $\\phi ^{\\operatorname{b}}_{j,n} = 2^{j/2} \\phi ^{\\operatorname{left}}_n(2^j\\cdot ), \\quad \\text{for } 0 \\le n < p,$ and the $p$ right boundary scaling functions are $\\phi ^{\\operatorname{b}}_{j,n} = 2^{j/2} \\phi ^{\\operatorname{right}}_{2^j-1-n}(2^j(\\cdot -1)), \\quad \\text{for } 2^j-p \\le n < 2^j.$ We remark that this leads to $2^j$ scaling functions in total, which is the number of original scaling functions $(\\phi _{j,n})_n$ that intersect $[0,1]$ .", "We next sketch the idea of the construction of boundary scaling functions $ \\lbrace \\phi ^{\\operatorname{left}}_n\\rbrace _{n=0, \\ldots , p-1}$ as well as $\\lbrace \\phi ^{\\operatorname{right}}_n\\rbrace _{n=0, \\ldots , p-1}$ following [11], to the extent to which we require it in our proofs.", "One starts by defining edge functions $\\widetilde{\\phi }^k$ on the positive axis $[0,\\infty )$ by $\\widetilde{\\phi }^k(x) = \\sum _{n=0}^{2p-2} \\binom{n}{k} \\phi (x+n-p+1), \\quad k = 0, \\ldots , p-1,$ such that these edge functions are orthogonal to the interior scaling functions and such that they together generate all polynomials up to degree $p-1$ .", "After performing a Gram-Schmidt procedure one obtains the left boundary functions $\\phi ^{\\operatorname{left}}_k, k =0, \\ldots , p-1$ .", "The right boundary functions are then – after some minor adjustments – obtained by reflecting the left boundary functions.", "This construction from [11] allows one to obtain a multiresolution analysis.", "Theorem 4.1 ([11]) If $2^j \\ge 2p$ , then $\\lbrace \\phi ^{\\operatorname{b}}_{j,n}\\rbrace _{n=0, \\ldots , 2^j-1}$ is an orthonormal basis for a space $V^{\\operatorname{b}}_j$ that is nested, i.e.", "$V_j^{\\operatorname{b}} \\subset V_{j+1}^{\\operatorname{b}},$ and complete, i.e.", "$\\overline{\\bigcup _{j \\ge \\log _2{2p}} V^{\\operatorname{b}}_j} = L^2[0,1].$ Next, we define an orthonormal basis for the wavelet space $W^{\\operatorname{b}}_j$ , which is as usual defined as the orthogonal complement of $V_j^{\\operatorname{b}}$ in $V_{j+1}^{\\operatorname{b}}$ .", "For this, let $\\psi $ be the corresponding wavelet function to $\\phi $ with $p$ vanishing moments and $\\operatorname{supp \\,}\\psi = [-p+1,p]$ .", "Similar to the construction of the scaling functions, we will obtain interior wavelets and boundary wavelets, which then constitutes the set of wavelets in the interval.", "Again based on a careful choice of boundary wavelets $(\\psi ^{\\operatorname{left}}_n)_n$ and $(\\psi ^{\\operatorname{right}}_k)_k$ , for which we refer to [11], we define $2^j-2p$ interior wavelets by $\\psi ^{\\operatorname{b}}_{j,n} = \\psi _{j,n} = 2^{j/2} \\psi (2^j\\cdot - n), \\quad \\text{for } p\\le n < 2^j-p,$ $p$ left boundary wavelets $\\psi ^{\\operatorname{b}}_{j,n} = 2^{j/2} \\psi ^{\\operatorname{left}}_n(2^j\\cdot ), \\quad \\text{for } 0 \\le n < p,$ and $p$ right boundary wavelets $\\psi ^{\\operatorname{b}}_{j,n} = 2^{j/2} \\psi ^{\\operatorname{right}}_{2^j-1-n}(2^j(\\cdot -1)), \\quad \\text{for } 2^j-p \\le n < 1^j.$ Summarizing, the following result hold for these wavelet functions.", "Theorem 4.2 ([11]) Let $2^J \\ge 2p$ .", "Then the following properties hold: [i)] $\\lbrace \\psi ^{\\operatorname{b}}_{J,n} \\rbrace _{n = 0, \\ldots , 2^J-1}$ is an orthonormal basis for $W_J^{\\operatorname{b}}$ .", "$L^2[0,1]$ can be decomposed as $L^2[0,1] = V_J^{\\operatorname{b}} \\oplus W_{J}^{\\operatorname{b}} \\oplus W_{J+1}^{\\operatorname{b}} \\oplus W_{J+2}^{\\operatorname{b}} \\oplus \\ldots =V_J^{\\operatorname{b}} \\bigoplus \\limits _{j=J}^{\\infty } W_{j}^{\\operatorname{b}}.$ $\\left\\lbrace \\lbrace \\phi ^{\\operatorname{b}}_{J,m} \\rbrace _{m = 0, \\ldots , 2^J-1}, \\lbrace \\psi ^{\\operatorname{b}}_{j,n} \\rbrace _{j \\ge J,n = 0, \\ldots , 2^j-1}\\right\\rbrace $ is an orthonormal basis for $L^2[0,1]$ .", "If $\\phi , \\psi \\in C^r[0,1]$ , then $\\left\\lbrace \\lbrace \\phi ^{\\operatorname{b}}_{j,m} \\rbrace _{m = 0, \\ldots , 2^J-1}, \\lbrace \\psi ^{\\operatorname{b}}_{j,n} \\rbrace _{j \\ge J,n = 0,\\ldots , 2^j-1}\\right\\rbrace $ is an unconditional basis for $C^s[0,1]$ for all $s <r$ .", "As mentioned before, this system gives rise to a 2D system by tensor products, i.e., by the standard 2D separable wavelet construction.", "In particular, this 2D system again constitutes an MRA, see [22].", "Sampling and Reconstruction Space Reconstruction Space For defining the reconstruction space, we will now assume that the region of interest is $[0,1]^2$ instead of $[0,a]^2$ with $a \\in \\mathbb {N}$ , as previously chosen.", "Our starting point is a 1D compactly supported Daubechies scaling function with $p$ vanishing moments (cf.", "Subsection REF ).", "Recall that Daubechies wavelets have at least the following frequency decay, $|\\widehat{\\phi }(\\xi ) | \\lesssim \\frac{1}{1 + |\\xi |}, \\quad \\text{ as } |\\xi | \\rightarrow \\infty .", "$ Let $\\psi $ be a corresponding wavelet with $p$ vanishing moments, and let $\\phi ^{\\operatorname{b}}$ and $\\psi ^{\\operatorname{b}}$ be the wavelets on the interval as introduced in the previous subsection.", "Let $J_0$ be the smallest number such that $2^{J_0} \\ge 2p$ .", "Then the associated 2D scaling functions are of the form $\\phi ^{\\operatorname{b}}_{J_0,(n_1,n_2)} := \\phi ^{\\operatorname{b}}_{J_0,n_1} \\otimes \\phi ^{\\operatorname{b}}_{J_0,n_2}, \\quad 0\\le n_1,n_2 \\le 2^{J_0}-1$ and, for $0 \\le n_1,n_2 \\le 2^j -1$ with $j \\ge J_0$ , the 2D wavelet functions are defined by $\\psi ^{\\operatorname{b},k}_{j,(n_1,n_2)} := {\\left\\lbrace \\begin{array}{ll}\\phi ^{\\operatorname{b}}_{j,n_1} \\otimes \\psi ^{\\operatorname{b}}_{j,n_2}, & j \\ge J_0, k=1,\\\\\\psi ^{\\operatorname{b}}_{j,n_1} \\otimes \\phi ^{\\operatorname{b}}_{j,n_2}, & j \\ge J_0, k=2, \\\\\\psi ^{\\operatorname{b}}_{j,n_1} \\otimes \\psi ^{\\operatorname{b}}_{j,n_2}, & j \\ge J_0, k=3.\\end{array}\\right.", "}$ Next, let $\\Omega _1 = \\lbrace \\phi ^{\\operatorname{b}}_{J_0,(n_1,n_2)} \\, : \\, 0 \\le n_1,n_2 \\le 2^{J_0}-1 \\rbrace $ and $\\Omega _2 = \\lbrace \\psi ^{\\operatorname{b},k}_{j,(n_1,n_2)}\\, : \\, j =J_0,1, \\ldots , J-1, 0 \\le n_1,n_2\\le 2^j-1, k =1,2,3 \\rbrace .$ Then, for a fixed scale $J$ , and for $N_J=2^{2J}$ , the reconstruction space $\\mathcal {R}_{N_J}$ is given by $\\mathcal {R}_{N_J} = \\operatorname{span}\\lbrace \\varphi _i \\, : \\, \\varphi _i \\in \\Omega _1 \\cup \\Omega _2, i = 1, \\ldots , N_J\\rbrace .$ An ordering can be obtained analogously as in Subsection REF .", "Sampling Space The sampling space can be chosen similar to Subsection REF , i.e., for $\\varepsilon \\le 1$ , we set $s_l^{(\\varepsilon )} = \\varepsilon e^{2 \\pi i \\varepsilon \\langle l, \\cdot \\rangle } \\cdot \\chi _{[0,1]^2}, \\quad l \\in \\mathbb {Z}^2,$ and for $M = (M_1,M_2 )\\in \\mathbb {N}\\times \\mathbb {N}$ $\\mathcal {S}^{(\\varepsilon )}_{M} = \\operatorname{span}\\left\\lbrace s_l^{(\\varepsilon )} \\, : \\, l = (l_1,l_2) \\in \\mathbb {Z}^2, - M_i \\le l_i \\le M_i , i =1,2 \\right\\rbrace .$ Main Result Our main result of this section states the linearity of the stable sampling rate for boundary wavelets, whose proof is presented in Section .", "Theorem 4.3 Let $J \\in \\mathbb {N}$ , $\\varepsilon \\le 1$ , and $\\theta >1$ .", "Further, let $\\mathcal {S}_M^{(\\varepsilon )}$ and $\\mathcal {R}_{N_J}$ be defined as REF and (REF ), respectively.", "Then $\\inf \\limits _{\\begin{array}{c} \\varphi \\in \\mathcal {R}_{N_J} \\\\ \\Vert \\varphi \\Vert =1 \\end{array}} \\Vert P_{S_M^{(\\varepsilon )}} \\varphi \\Vert \\ge \\frac{1}{\\theta }$ for $M = (\\lceil S^\\theta /\\varepsilon \\rceil 2^J,\\lceil S^\\theta /\\varepsilon \\rceil 2^J) \\in \\mathbb {N}\\times \\mathbb {N}$ , where $S^\\theta $ is a constant independent of $J$ .", "In particular, the stable sampling rate obeys $\\Theta (N_J,\\theta ) = \\mathcal {O}(N_J)$ as $N_J \\rightarrow \\infty $ for every fixed $\\theta >1$ Remark 4.4 This result will be proved independently of Theorem REF , since boundary wavelets do constitute an MRA but at highest scaling level, the space $V_J$ contains more than one generating function $\\phi $ .", "It uses the reflected functions as well.", "Figure: Stable sampling rate Θ(N,θ)\\Theta (N,\\theta ) for two dimensional dyadic Haar wavelets and two dimensional dyadic Daubechies-4 wavelets.Table: Number of reconstruction elements and samples for Haar and Daubechies-4.", "Note that these numbers predict the jumps in Figure .", "Numerical Experiments In this section we numerically demonstrate the linearity of the stable sampling rate as stated in Theorem REF .", "We will also demonstrate how this combines with generalized sampling in practice.", "In particular, given this linearity, reconstructing from Fourier samples in smooth boundary wavelets will give an error decaying according to the smoothness and the number of vanishing moments.", "In this section we consider dyadic scaling matrices $A = \\begin{pmatrix} 2 & 0 \\\\ 0 & 2 \\end{pmatrix}.", "$ Furthermore, our focus are separable wavelets, i.e.", "wavelets that are obtained by tensor products of one dimensional scaling functions and one dimensional wavelet functions, respectively.", "Scaling matrices of the form (REF ) preserve the separability.", "Linearity examples with Haar and Daubechies-4 wavelets We use the description of Section 3.1 and Example REF in order to perform numerical experiments for some known wavelets.", "This gives the reconstruction space $\\mathcal {R}= \\overline{\\operatorname{span}} \\lbrace \\varphi \\, : \\, \\varphi \\in \\Omega _1 \\cup \\Omega _2\\rbrace $ where $\\Omega _1$ and $\\Omega _2$ are defined in (REF ) and (REF ) respectively.", "We order the reconstruction space $\\mathcal {R}$ in the same manner as presented at the end of Section 3.", "In (REF ) we counted the number of reconstruction elements up to level $J-1$ , which leads to $N_J = (2^{2J}-1)a^2 + 6a(a-1)(2^J-1) + 3J(a-1)^2 + (2a-1)^2 $ many elements, which is asymptotically of order $2^{2J}$ .", "We test Haar wavelets and 2D Daubechies-4 wavelets.", "Figure REF shows the linear behaviour of the stable sampling rate for these two types of wavelet generators.", "By a small abuse of notation, we also write $M$ for the total number of samples.", "In our analysis in Section 3 we estimated the angle $\\cos (\\omega (\\mathcal {R}_N, \\mathcal {S}_M^{(\\varepsilon )}))$ (recall that $\\epsilon $ is the sampling rate) with respect to some fixed $\\theta >1$ .", "In fact we computed $M$ such that $\\inf \\limits _{\\begin{array}{c} \\varphi \\in \\mathcal {R}_N \\\\ \\Vert \\varphi \\Vert =1\\end{array}} \\Vert P_{\\mathcal {S}_M^{(\\varepsilon )}} \\varphi \\Vert \\ge \\frac{1}{\\theta }$ holds.", "We proved that $M$ is up to a constant of the same size as $N_J$ .", "Figure REF shows the stable sampling rate (in blue) $\\Theta (N,\\theta ) = \\min \\left\\lbrace M \\in \\mathbb {N}, \\cos (\\omega (\\mathcal {R}_N,\\mathcal {S}_M)) \\ge \\frac{1}{\\theta }\\right\\rbrace $ and the linear function $f$ (in red) given by $f(N) = \\frac{M_{\\text{max}}}{N_{\\text{max}}}N,$ where $N_{\\text{max}}$ is the maximum value of $N$ used in the experiment and $M_{\\text{max}} = \\Theta (N_{\\text{max}},\\theta )$ .", "We computed the stable sampling rate up to level $J = 4$ .", "Note the (significant) jumps of the stable sampling rate occur whenever $N\\in \\mathbb {N}$ crosses the scaling level $N_J, J = 0, \\ldots , 4$ .", "In the Haar case these are $N_0 = 4, \\qquad N_1 = 16, \\qquad N_2 = 64, \\qquad N_3 = 256, \\quad N_4 = 1024,$ see (REF ).", "Note that $a=1$ in the Haar case.", "In particular, the jumps are linear, suggesting a linear stable sampling rate.", "Figure 1 (b), (c), and (d) are interpreted similarly.", "However, the theoretical results are asymptotic results.", "Therefore, it should not be surprising that the stable sampling rate is below the linear line in some cases.", "It aligns asymptotically.", "Fourier samples and boundary wavelet reconstruction Figure: Reconstruction of the function f 1 (x,y)=cos 2 (x)exp(-y)f_1(x,y)=\\cos ^2(x)\\exp (-y).The second row shows an 8 times zoomed-in version of the upper left corner.Left: original function.", "Middle: truncated Fourier serieswith 256 2 256^2 Fourier coefficients.", "Right: generalized sampling withDaubechies-3 wavelets computed from the same Fourier coefficients.Figure: Reconstruction of the function f 2 (x,y)=(1+x 2 )(2y-1 2 )f_2(x,y) = (1+x^2)(2y-1^2).", "Upper left:truncated Fourier series with 512 2 512^2 Fourier coefficients.", "Middleleft: 8 times zoomed-in version of the upper figure.", "Lower left:error committed by the truncated Fourier series.", "Upper right:generalized sampling with Daubechies-3 wavelets computed from the same 512 2 512^2 Fouriercoefficients.", "Middle right: 8 times zoomed-in version of the upperfigure.", "Lower right: error committed by generalized sampling.In this example we will demonstrate the efficiency of generalized sampling given the established linearity of the stable sampling rate.", "In particular, suppose that $f$ is a function we want to recover from its Fourier information.", "It is smooth, however, not periodic – a problem that occurs for example in electron microscopy and also in MRI.", "This causes the classical Fourier reconstruction to converge slowly, yet a smooth boundary wavelet basis will give much faster convergence (see [22]).", "As discussed, the issue is that we are given Fourier samples, not wavelet coefficients.", "However, this is not a problem in view of the linearity of the stable sampling rate.", "In particular, if $f \\in W^s(0,1)$ , where $\\mathrm {W}^s(0,1)$ denotes the usual Sobolev space, and $P_{\\mathcal {R}_N}$ denotes the projection onto the space $\\mathcal {R}_{N}$ of the first $N$ boundary wavelets (see (REF )), then $\\Vert f - P_{\\mathcal {R}_N} f \\Vert = \\mathcal {O}(N^{-s}),\\quad N \\rightarrow \\infty ,$ provided that the wavelet has sufficiently many vanishing moments.", "Now, if $G_{N,M}(f) \\in \\mathcal {R}_{N}$ is the generalized sampling solution from Definition REF given $M$ Fourier coefficients, and $M$ is chosen according to the stable sampling rate then $\\Vert f - G_{N,M}(f)\\Vert = \\mathcal {O}(N^{-s}) = \\mathcal {O}(M^{-s}),\\quad N \\rightarrow \\infty .$ Hence, we obtain the same convergence rate up to a constant, by simply postprocessing the given samples.", "To illustrate this advantage we will consider the following two functions: $f_1(x,y)=\\cos ^2(x)\\exp (-y), \\qquad f_2(x,y) = (1+x^2)(2y-1^2).$ In Figure REF we have shown the results for $f_1$ and compared the classical Fourier reconstruction with the generalized sampling reconstruction.", "Both examples use exactly the same samples, however, note the pleasant absence of the Gibb's ringing in the generalized sampling case.", "The same experiment is carried out for $f_2$ in Figure REF , however, here we have displayed the reconstructions in $3D$ in order to visualize the error.", "Proof of Theorem REF The proof of Theorem REF is somewhat technical, wherefore we divide the proof into several steps.", "First, in Subsection REF , the overall structure of the proof is presented, and the respective details can then be found in Subsection REF .", "Structure of the Proof Let $\\varepsilon \\in (0,1/(T_1+T_2)]$ and $\\theta >1$ .", "Then we have to prove that $\\inf \\limits _{\\begin{array}{c} \\varphi \\in \\mathcal {R}_{N_J} \\\\ \\Vert \\varphi \\Vert =1\\end{array}} \\Vert P_{\\mathcal {S}_M^{(\\varepsilon )}} \\varphi \\Vert \\ge \\frac{1}{\\theta },$ for $M=(M_1,M_2) = \\left(\\frac{ \\lambda _1^{(J)} + \\lambda _3^{(J)}}{\\varepsilon } S^{(\\theta )}, \\frac{ \\lambda _2^{(J)} + \\lambda _4^{(J)}}{\\varepsilon } S^{(\\theta )}\\right),$ and some $S^{(\\theta )}$ independent of $J$ .", "To this end, let $\\varphi \\in \\mathcal {R}_N$ be such that $\\Vert \\varphi \\Vert =1$ .", "Since the sampling functions form an orthonormal basis of $\\mathcal {S}_M^{(\\varepsilon )}$ , we obtain $\\Vert P_{\\mathcal {S}_M^{(\\varepsilon )}} \\varphi \\Vert ^2= \\sum \\limits _{l_1 = -M_1}^{M_1} \\sum \\limits _{l_2 = -M_2}^{M_2}|\\langle \\varphi , s_l^{(\\varepsilon )} \\rangle |^2, \\quad l = (l_1,l_2).$ By Lemma REF , which relies mainly on the underlying MRA structure, we can write $\\varphi $ in terms of scaling functions at highest scale, i.e., $\\varphi = \\sum \\limits _{l_1 = L_3}^{L_1} \\sum \\limits _{l_2 = L_4}^{L_2} \\alpha _{l_1,l_2}\\phi _{J,(l_1,l_2)},$ for some $L_1,L_2,L_3,L_4 \\in \\mathbb {Z}$ .", "Lemma REF , which is proven by direct computations, then shows that $\\langle \\varphi , s_l^{(\\varepsilon )} \\rangle = \\varepsilon |\\det A|^{-J/2} \\Phi (\\varepsilon (A^{-J})^Tl) \\widehat{\\phi }(\\varepsilon (A^{-J})^Tl),$ where $\\Phi $ is a trigonometric polynomial of the form $\\Phi (z) = \\sum \\limits _{m_1 = L_3}^{L_1} \\sum \\limits _{m_2 = L_4}^{L_2} \\alpha _{m_1,m_2} e^{-2\\pi i \\langle z, m \\rangle }, \\quad z \\in \\mathbb {R}^2, m = (m_1,m_2).$ Using (REF ), we conclude that $\\Vert P_{\\mathcal {S}_M^{(\\varepsilon )}} \\varphi \\Vert ^2= \\sum \\limits _{l_1 = -M_1}^{M_1} \\sum \\limits _{l_2 = -M_2}^{M_2} \\varepsilon ^2 |\\det A|^{-J}|\\Phi (\\varepsilon (A^{-J})^Tl)|^2 |\\widehat{\\phi }(\\varepsilon (A^{-J})^Tl)|^2, \\quad l = (l_1, l_2).$ By Theorem REF , there exist some $\\varepsilon _0 > 0$ and $S^{(\\theta )} = \\left(S^{(\\theta )}_1,S^{(\\theta )}_2\\right)\\in \\mathbb {N}\\times \\mathbb {N}$ that does not depend on $J$ such that $\\sum \\limits _{l_1 = -M_1}^{M_1} \\sum \\limits _{l_2 = -M_2}^{M_2} \\varepsilon _0 ^2 | \\det A^{-J}| | \\Phi (\\varepsilon _0 (A^{-J})^T l) |^2| \\widehat{\\phi }(\\varepsilon _0(A^{-J})^Tl)|^2 \\ge \\frac{\\Vert \\Phi \\Vert ^2}{\\theta ^2}$ for $M= \\frac{1}{\\varepsilon _0} (A^J)^T \\widetilde{S}^{(\\theta )}=\\begin{pmatrix} \\frac{ \\lambda _1^{(J)} + \\lambda _3^{(J)}}{\\varepsilon _0} S^{(\\theta )}_1 \\\\\\frac{ \\lambda _2^{(J)} + \\lambda _4^{(J)}}{\\varepsilon _0} S^{(\\theta )}_2 \\end{pmatrix}.$ Since $\\Vert \\Phi \\Vert ^2 =\\sum \\limits _{m_1 = L_3}^{L_1} \\sum \\limits _{m_2 = -L_4}^{L_2} |\\beta _{m_1,m_2}|^2 =\\Vert \\varphi \\Vert ^2 = 1,$ we obtain $\\Vert P_{\\mathcal {S}_M^{(\\varepsilon _0)}} \\varphi \\Vert ^2 \\ge \\frac{1}{\\theta ^2}.$ Finally, Lemma REF implies that a change of $\\varepsilon $ only changes the constant, showing that (REF ) is true for any $\\varepsilon \\in (0,1/(T_1+T_2)]$ , thereby proving Theorem REF .", "Auxiliary results In this section we will prove the results mentioned in Subsection REF , which are required for completing the proof of Theorem REF .", "The following result is an extension of a result from [6] to the two dimensional setting.", "Lemma 6.1 Let $J \\in \\mathbb {N}$ and $\\varphi \\in V_0^{(a)} \\oplus W_0^{(a)} \\oplus \\ldots \\oplus W_{J-1}^{(a)}$ .", "Then there exist $L_1,L_2,L_3,L_4 \\in \\mathbb {Z}$ dependent of $J$ such that $\\varphi = \\sum \\limits _{l_1 = L_3}^{L_1} \\sum \\limits _{l_2 = L_4}^{L_2} \\alpha _{l_1,l_2}\\phi _{J,(l_1,l_2)}.$ Let $\\varphi \\in V_0^{(a)} \\oplus W_0^{(a)} \\oplus \\ldots \\oplus W_{J-1}^{(a)}$ .", "Then $\\varphi $ has an expansion of the following form $\\varphi = \\sum _{m_1 = - a +1}^{m_1 = a-1} \\sum _{m_2 = - a +1}^{m_2 = a-1} \\alpha _{m_1,m_2} \\phi _{0,(m_1,m_2)} +\\sum _{p=1}^{|\\det A| -1} \\sum _{j=0}^{J-1} \\sum _{m_1 = -a +1}^{a(\\lambda _1^{j}+ \\lambda _{2}^{j})-1}\\sum \\limits _{m_2= -a +1}^{a(\\lambda _3^{j}+ \\lambda _{4}^{j})-1} \\beta _{j,(m_1,m_2)}^p \\psi _{j,(m_1,m_2)}^p.$ Since $V_0^{(a)} \\subset V_0$ and $W_j^{(a)} \\subset W_{j}$ for $j \\in \\mathbb {N}$ and the sequence $(V_j)_{j\\in \\mathbb {Z}}$ forms an MRA, it follows that $V_0^{(a)} \\oplus W_0^{(a)} \\oplus \\ldots \\oplus W_{J-1}^{(a)} \\subset V_J.$ Since an orthonormal basis for $V_J$ is given by the functions $\\lbrace \\phi _{J,m} \\, : \\, m \\in \\mathbb {Z}^2\\rbrace $ , for each $|m_i| < a, i =1,2$ , we have $\\phi _{0,(m_1,m_2)} = \\sum _{l \\in \\mathbb {Z}^2} \\langle \\phi _{0,(m_1,m_2)}, \\phi _{J,l} \\rangle \\phi _{J,l}$ Moreover, since $\\phi _{0,(m_1,m_2)}$ and $\\phi _{J,l}$ are compactly supported, we obtain $\\langle \\phi _{0,(m_1,m_2)}, \\phi _{J,l} \\rangle \\ne 0,$ if $-a+ (-a+1)( \\lambda _1^{(J)}+ \\lambda _2^{(J)} ) \\le l_1 \\le (2a-1)(\\lambda _1^{(J)} + \\lambda _2^{(J)})$ and $-a+ (-a+1)( \\lambda _3^{(J)}+ \\lambda _4^{(J)} ) \\le l_2 \\le (2a-1)(\\lambda _3^{(J)} + \\lambda _4^{(J)}).$ This follows by a straightforward computation from the support conditions of $\\phi _{0,(m_1,m_2)}$ and $\\phi _{J,l}$ together with $|m_i| < a, i =1,2$ .", "Similarly, we have $\\langle \\psi _{j,(m_1,m_2)}^p, \\phi _{J,(l_1,l_2)} \\rangle \\ne 0$ if Case I: $\\det A^j \\ge 0$ $-a& - \\lambda _1^{(J)}\\frac{(-a+1)\\lambda _4^{(j)}+\\lambda _2^{(j)}(a(\\lambda _3^{(j)}+\\lambda _4^{(j)})-1)}{\\det A^j} + \\lambda _2^{(J)} \\frac{(-a+1)\\lambda _1^{(j)}+(a(\\lambda _1^{(j)}+\\lambda _2^{(j)})-1)\\lambda _3^{(j)}}{\\det A^j} \\\\& < l_1 < \\lambda _1^{(J)} \\frac{a(\\lambda _4^{(j)}-\\lambda _2^{(j)})+(a(\\lambda _1^{(j)}+\\lambda _2^{(j)})-1)\\lambda _4^{(j)}+(a-1)\\lambda _2^{(j)}}{\\det A^j}+\\\\&\\qquad +\\lambda _2^{(J)} \\frac{a(\\lambda _1^{(j)}-\\lambda _3^{(j)})+(a(\\lambda _3^{(j)}+\\lambda _4^{(j)})-1)\\lambda _1^{(j)}+(a-1)\\lambda _3^{(j)}}{\\det A^j}$ and $-a& - \\lambda _3^{(J)}\\frac{(-a+1)\\lambda _4^{(j)}+\\lambda _2^{(j)}(a(\\lambda _3^{(j)}+\\lambda _4^{(j)})-1)}{\\det A^j} + \\lambda _4^{(J)} \\frac{(-a+1)\\lambda _1^{(j)}+(a(\\lambda _1^{(j)}+\\lambda _2^{(j)})-1)\\lambda _3^{(j)}}{\\det A^j} \\\\& < l_2 < \\lambda _3^{(J)} \\frac{a(\\lambda _4^{(j)}-\\lambda _2^{(j)})+(a(\\lambda _1^{(j)}+\\lambda _2^{(j)})-1)\\lambda _4^{(j)}+(a-1)\\lambda _2^{(j)}}{\\det A^j}+\\\\&\\qquad +\\lambda _4^{(J)} \\frac{a(\\lambda _1^{(j)}-\\lambda _3^{(j)})+(a(\\lambda _3^{(j)}+\\lambda _4^{(j)})-1)\\lambda _1^{(j)}+(a-1)\\lambda _3^{(j)}}{\\det A^j}.$ Case II: $\\det A^j < 0$ $-a& - \\lambda _1^{(J)}\\frac{(-a+1)\\lambda _4^{(j)}+\\lambda _2^{(j)}(a(\\lambda _3^{(j)}+\\lambda _4^{(j)})-1)}{\\det A^j} + \\lambda _2^{(J)} \\frac{(-a+1)\\lambda _1^{(j)}+(a(\\lambda _1^{(j)}+\\lambda _2^{(j)})-1)\\lambda _3^{(j)}}{\\det A^j} \\\\& > l_1 > \\lambda _1^{(J)} \\frac{a(\\lambda _4^{(j)}-\\lambda _2^{(j)})+(a(\\lambda _1^{(j)}+\\lambda _2^{(j)})-1)\\lambda _4^{(j)}+(a-1)\\lambda _2^{(j)}}{\\det A^j}+\\\\&\\qquad +\\lambda _2^{(J)} \\frac{a(\\lambda _1^{(j)}-\\lambda _3^{(j)})+(a(\\lambda _3^{(j)}+\\lambda _4^{(j)})-1)\\lambda _1^{(j)}+(a-1)\\lambda _3^{(j)}}{\\det A^j}$ and $-a& - \\lambda _3^{(J)}\\frac{(-a+1)\\lambda _4^{(j)}+\\lambda _2^{(j)}(a(\\lambda _3^{(j)}+\\lambda _4^{(j)})-1)}{\\det A^j} + \\lambda _4^{(J)} \\frac{(-a+1)\\lambda _1^{(j)}+(a(\\lambda _1^{(j)}+\\lambda _2^{(j)})-1)\\lambda _3^{(j)}}{\\det A^j} \\\\& > l_2 > \\lambda _3^{(J)} \\frac{a(\\lambda _4^{(j)}-\\lambda _2^{(j)})+(a(\\lambda _1^{(j)}+\\lambda _2^{(j)})-1)\\lambda _4^{(j)}+(a-1)\\lambda _2^{(j)}}{\\det A^j}+\\\\&\\qquad +\\lambda _4^{(J)} \\frac{a(\\lambda _1^{(j)}-\\lambda _3^{(j)})+(a(\\lambda _3^{(j)}+\\lambda _4^{(j)})-1)\\lambda _1^{(j)}+(a-1)\\lambda _3^{(j)}}{\\det A^j}.$ Minimizing the lower bounds with respect to $j \\in \\lbrace 0, \\ldots , J-1\\rbrace $ and maximizing the upper bounds with respect to $j$ , respectively, yields the claim.", "The following lemma is well known (see [22]).", "Lemma 6.2 Let $f \\in L^2(\\mathbb {R}^2)$ .", "Then $\\lbrace f( \\cdot - m) \\, : \\, m \\in \\mathbb {Z}^2\\rbrace $ is an orthonormal system if and only if $\\sum \\limits _{m \\in \\mathbb {Z}^2} | \\widehat{f}(\\xi + m )|^2 = 1 \\quad \\text{for almost every } \\xi \\in \\mathbb {R}^2.$ Finally, one more technical lemma is needed.", "Lemma 6.3 Let $A$ be a scaling matrix, $J \\in \\mathbb {Z}$ , and $m=(m_1,m_2) \\in \\mathbb {Z}^2$ .", "Further, let $\\varphi \\in \\overline{\\operatorname{span}} \\lbrace \\phi _{J,m} \\, : \\, m \\in \\mathbb {Z}^2\\rbrace $ be compactly supported in $[-T_1,T_2]^2$ , and let $L_1,L_2,L_3,L_4 \\in \\mathbb {Z}$ be such that $\\varphi = \\sum \\limits _{m_1 = L_3}^{L_1} \\sum \\limits _{m_2 = L_4}^{L_2} \\alpha _{m_1,m_2} \\phi _{J,(m_1,m_2)}, \\quad \\alpha _m \\in $ Then, for all $l \\in \\mathbb {Z}^2$ , $\\langle \\varphi , s_l^{(\\varepsilon )} \\rangle = \\varepsilon |\\det A|^{-J/2} \\Phi (\\varepsilon (A^{-J})^Tl) \\widehat{\\phi }(\\varepsilon (A^{-J})^Tl),$ where $s_l^{(\\varepsilon )}$ is defined in (REF ) and $\\Phi $ is the trigonometric polynomial given by $\\Phi (z) = \\sum \\limits _{\\begin{array}{c} L_3 \\le m_1 \\le L_1, \\\\ L_4 \\le m_2 \\le L_2,\\end{array}} \\alpha _m e^{-2\\pi i \\langle z, m \\rangle }, \\quad z \\in \\mathbb {R}^2, m = (m_1,m_2).$ Since $\\varphi $ is supported in $[-T_1,T_2]^2$ , we obtain $\\langle \\varphi , s_l^{(\\varepsilon )} \\rangle &= \\varepsilon \\int \\limits _{\\mathbb {R}^2} \\varphi (x) e^{-2 \\pi i \\varepsilon \\langle l, x \\rangle }\\cdot \\chi _{[-T_1,T_2]^2} \\, dx \\\\&= \\varepsilon \\widehat{\\varphi }( \\varepsilon l) \\\\&= \\varepsilon \\left(\\sum \\limits _{m_1 = L_1}^{L_2} \\sum \\limits _{m_2 = L_3}^{L_4} \\alpha _{m_1,m_2} \\phi _{J,(m_1,m_2)}\\right)^{\\wedge }(\\varepsilon l) \\\\&= \\varepsilon \\sum \\limits _{m_1 = L_1}^{L_2} \\sum \\limits _{m_2 = L_3}^{L_4} \\alpha _{m_1,m_2} \\left( \\phi _{J,(m_1,m_2)}\\right)^{\\wedge }(\\varepsilon l)\\\\&= \\varepsilon |\\det A|^{-J/2} \\Phi (\\varepsilon (A^{-J})^T l) \\widehat{\\phi }(\\varepsilon (A^{-J})^Tl).$ This proves the claim.", "Theorem REF The proof of Theorem REF requires a particular estimate (Proposition REF ) for the norm of trigonometric polynomials depending on their evaluations on a particular grid whose mesh norm and associated Voronoi regions come also into play.", "Mesh Norm We start with the definition of a mesh norm for the situation we are faced with.", "A mesh norm can be interpreted as the largest distance between neighboring nodes.", "Definition 6.4 Let $\\Lambda \\subset \\mathbb {Z}^2$ be an integer grid of the form $\\Lambda := \\left\\lbrace (l_1,l_2) \\in \\mathbb {Z}^2 \\, : \\, -M_i \\le l_i \\le M_i, i=1,2 \\right\\rbrace , \\quad M_1,M_2 \\in \\mathbb {N},$ and let $A$ be a scaling matrix.", "Set $\\Omega := \\overline{\\Lambda }^A := A ([-M_1,M_1] \\times [-M_2,M_2]) \\subset \\mathbb {R}^2,$ and define a metric $\\rho $ on $\\Omega $ by $\\rho : \\Omega \\times \\Omega \\longrightarrow \\mathbb {R}^+, \\quad (x,y) \\mapsto \\min \\limits _{k \\in A(\\mathbb {Z}^2)} \\Vert x - y + k\\Vert _\\infty .$ The mesh norm of $\\lbrace x_l \\in \\Omega \\, : \\, l \\in \\Lambda \\rbrace $ is then defined as $\\delta := \\max \\limits _{x \\in \\Omega } \\min \\limits _{l \\in \\Lambda } \\rho (x_l,x),$ where $x_l := A \\cdot l, l \\in \\Lambda $ denote the nodes in $\\Omega $ .", "Before we can continue, we require some notions and results on Voronoi regions and trigonometric polynomials.", "Our first result shows that if the distance between the nodes $\\lbrace x_l\\rbrace _l$ converges to zero, the mesh norm of the entire grid $\\Omega $ converges to zero.", "Lemma 6.5 Let $\\varepsilon >0$ and $\\Lambda \\subset \\mathbb {Z}^2$ be defined by $\\Lambda := \\left\\lbrace (l_1,l_2) \\in \\mathbb {Z}^2 \\, : \\, -M_i \\le l_i \\le M_i, i=1,2 \\right\\rbrace ,$ where $M_1,M_2 \\in \\mathbb {N}$ .", "Furthermore, suppose $A$ is a scaling matrix, and let $\\Omega ^{(\\varepsilon )} := \\overline{\\Lambda }^{\\varepsilon A}$ .", "If $\\varepsilon \\longrightarrow 0$ , then $\\delta ^{(\\varepsilon )} \\longrightarrow 0$ , where $\\delta ^{(\\varepsilon )}$ denotes the mesh norm of $\\lbrace \\varepsilon A l \\in \\Omega ^{(\\varepsilon )} \\, : \\, l \\in \\Lambda \\rbrace $ .", "If $\\varepsilon \\rightarrow 0$ , then $\\varepsilon A (\\Lambda ) = \\lbrace \\varepsilon A l \\, : \\, l \\in \\Lambda \\rbrace \\longrightarrow \\lbrace (0,0) \\rbrace $ with respect to the standard Euclidean distance.", "Furthermore, for $x_l := \\varepsilon A l$ , we have $\\lim \\limits _{\\varepsilon \\rightarrow 0} \\delta ^{(\\varepsilon )}= \\lim \\limits _{\\varepsilon \\rightarrow 0} \\max \\limits _{x \\in \\Omega ^{(\\varepsilon )}} \\min \\limits _{l \\in \\Lambda } \\min \\limits _{k \\in \\varepsilon A(\\mathbb {Z}^2)} \\Vert x - x_l + k\\Vert _\\infty \\le \\lim \\limits _{\\varepsilon \\rightarrow 0} \\max \\limits _{x \\in \\Omega ^{(\\varepsilon )}} \\min \\limits _{l \\in \\Lambda } \\Vert x - x_l \\Vert _\\infty .$ Since $x, x_l \\in \\Omega ^{(\\varepsilon )}$ for all $ l \\in \\Lambda $ , we obtain $\\lim \\limits _{\\varepsilon \\rightarrow 0} 2 \\max \\limits _{x \\in \\Omega ^{(\\varepsilon )}} \\min \\limits _{l \\in \\Lambda } \\Vert x - x_l \\Vert _\\infty \\le \\lim \\limits _{\\varepsilon \\rightarrow 0} \\max \\limits _{x,y \\in \\Omega ^{(\\varepsilon )}} \\Vert x - y \\Vert _\\infty .$ Inserting this estimate into (REF ) yields $\\lim \\limits _{\\varepsilon \\rightarrow 0} \\delta ^{(\\varepsilon )}\\le \\lim \\limits _{\\varepsilon \\rightarrow 0} \\max \\limits _{x,y \\in \\Omega ^{(\\varepsilon )}} \\Vert x - y \\Vert _\\infty \\le \\lim \\limits _{\\varepsilon \\rightarrow 0} \\operatorname{diam}\\Omega ^{(\\varepsilon )}= \\lim \\limits _{\\varepsilon \\rightarrow 0} \\operatorname{diam}\\overline{\\Lambda }^{\\varepsilon A}= \\lim \\limits _{\\varepsilon \\rightarrow 0} \\operatorname{diam}\\varepsilon \\overline{\\Lambda }^A= \\lim \\limits _{\\varepsilon \\rightarrow 0} \\varepsilon \\underbrace{ \\operatorname{diam}\\overline{\\Lambda }^A}_{<\\infty } =0,$ where $\\operatorname{diam}(F)$ denotes the diameter of a set $F\\subset \\mathbb {R}^d$ , i.e., $\\operatorname{diam}(F) = \\sup \\limits _{x,y \\in F} d_2(x,y),$ and $d_2$ denotes the Euclidean metric on $\\mathbb {R}^d$ .", "Since the mesh norm is always non-negative, the lemma is proven.", "Voronoi Regions The next result studies the volume of the Voronoi regions associated to the previously considered grid $\\Lambda $ with respect to the metric $\\rho $ defined in Definition REF .", "We start by formally defining the notion of Voronoi region in our setting.", "Definition 6.6 Let $\\Omega \\subset \\mathbb {R}^2$ , and let $(x_l)_{l \\in \\Lambda }$ be a sequence in $\\mathbb {R}^2$ .", "Then we refer to the sets $(V_l)_{l \\in \\Lambda }$ defined by $V_l := \\lbrace x \\in \\Omega \\, : \\, \\rho (x,x_l) \\le \\rho (x,x_k) \\; \\text{ for all } k \\ne l \\rbrace $ as Voronoi regions with respect to $\\Lambda $ , $\\Omega $ , $\\rho $ , and $x_l$.", "We can now state the previously announced result.", "Lemma 6.7 Let $M_1,M_2 \\in \\mathbb {N}$ and $\\Lambda := \\left\\lbrace (l_1,l_2) \\in \\mathbb {Z}^2 \\, : \\, -M_i \\le l_i \\le M_i, i=1,2 \\right\\rbrace .$ Moreover, let $\\Omega =\\overline{\\Lambda }^{\\operatorname{Id}}$ , where $\\operatorname{Id}$ denotes the $2\\times 2$ -identity matrix, and let $(V_l)_{l \\in \\Lambda }$ be the Voronoi regions with respect to $\\Lambda $ , $\\Omega $ , $\\rho $ , and $l$ .", "Then, for all $l \\in \\Lambda $ , $\\mu (V_l) \\le 1,$ where $\\mu $ denotes the 2D Lebesgue measure.", "Notice that the Voronoi regions $(V_l)_{l \\in \\Lambda }$ are in fact rectangles, since the grid is an integer grid with a constant step-size.", "Hence, for each $l \\in \\Lambda $ , $\\mu (V_l) = a_{l_1,l_2} \\cdot b_{l_1,l_2}, \\quad a_{l_1,l_2},b_{l_1,l_2} \\in \\mathbb {R},$ where $a_{l_1,l_2}$ denotes the width and $b_{l_1,l_2}$ the height of the rectangle $V_l$ .", "Towards a contradiction, assume that $V_l$ does contain two different nodes $x_k$ and $x_l$ with $k \\ne l$ .", "This implies $0\\ne \\rho (x_k,x_l) \\le \\rho (x_l,x_l) =0,$ which is a contradiction.", "Thus, we can conclude that $a_{l_1,l_2} \\le \\rho (x_{l_1+1,l_2},x_{l_1,l_2})\\quad \\mbox{and} \\quad b_l \\le \\rho (x_{l_1,l_2},x_{l_1,l_2+1}),$ which, by (REF ), proves the claim.", "We next obtain a slight generalization of the previous result.", "Lemma 6.8 Let $A$ be a (linear) bijective transformation acting on $\\mathbb {R}^2$ with matrix representation $A= \\begin{pmatrix} \\lambda _1& \\lambda _2 \\\\ \\lambda _3 & \\lambda _4 \\end{pmatrix},$ and let $\\Lambda $ and $(V_l)_{l \\in \\Lambda }$ be defined as in Lemma REF .", "Then $\\mu (A(V_l)) \\le |\\det A |.$ The result follows from Lemma REF by an integration by substitution.", "Trigonometric Polynomials The next theorem is an adapted version of a result presented in [24] which again is a reformulation of a result proven by Gröchenig in [17].", "Proposition 6.9 Let $J, L_1,L_2, L_3, L_4 \\in \\mathbb {Z}$ such that $L_1 \\ge L_3$ and $L_2 \\ge L_4$ , and let $\\Phi $ a trigonometric polynomial of the form $\\Phi (z) = \\sum \\limits _{m_1 = L_3}^{L_1} \\sum \\limits _{m_2 = L_4}^{L_2} \\alpha _{m_1,m_2} e^{-2\\pi i \\langle z, m \\rangle }, \\quad z \\in \\mathbb {R}^2, m = (m_1,m_2).$ Further, let the grid $\\Lambda $ be defined as in Lemma REF , let $A$ be a scaling matrix, and let $\\Omega ^{(\\varepsilon )} := \\overline{\\Lambda }^{\\varepsilon A^{-J}}$ for $\\varepsilon >0$ .", "Set $x_l :=\\varepsilon (A^{-J})^Tl$ , $l \\in \\Lambda $ .", "If the mesh norm $\\delta $ of $\\lbrace x_l \\in \\Omega ^{(\\varepsilon )} \\, : \\, l \\in \\Lambda \\rbrace $ obeys $\\delta < \\frac{\\log \\left(\\frac{1}{\\sqrt{\\mu ( \\Omega ^{(\\varepsilon )})}}+1\\right)}{2 \\pi \\max \\lbrace |L_1|,|L_2|,|L_3|,|L_4|\\rbrace },$ where $\\mu $ is the 2D Lebesgue measure, then there exists a positive constant $C(\\delta ,L_1,L_2,L_3,L_4)$ such that $C(\\delta ,L_1,L_2,L_3,L_4) \\Vert \\Phi \\Vert \\le \\left( \\sum \\limits _{l \\in \\Lambda } \\varepsilon ^2 |\\det A^{-J} | | \\Phi (\\varepsilon (A^{-J})^Tl|^2 \\right)^{\\frac{1}{2}},$ where $C(\\delta , L_1,L_2,L_3,L_4) = \\left( 1 - \\left(e^{2\\pi \\delta \\max \\lbrace |L_1|,|L_2|,|L_3|,|L_4|\\rbrace } -1\\right) \\sqrt{ \\mu (\\Omega ^{(\\varepsilon )})}\\right)$ and $\\Vert f \\Vert := \\left(\\int _{\\Omega ^{(\\varepsilon )}} |f(x)|^2 \\, dx\\right)^{1/2}, \\quad f\\in L^2(\\Omega ^{(\\varepsilon )}).$ We first observe that by the hypotheses, the constant $C(\\delta , L_1,L_2,L_3,L_4)$ is indeed positive.", "Second, let $(V_l)_{l \\in \\Lambda }$ be the Voronoi regions with respect to $\\Lambda $ , $\\Omega ^{(\\varepsilon )}$ , and $\\rho $ .", "For $l \\in \\Lambda $ , we define the weights $\\omega _l := \\mu (V_l)$ .", "As in Lemma REF , integration by substitution yields $\\omega _l = \\mu (V_l) \\le \\varepsilon ^2 |\\det A^{-J} |.$ Hence it suffices to prove the existence of a constant $C(\\delta , L_1,L_2,L_3,L_4) >0$ such that $C(\\delta ,L_1,L_2,L_3,L_4) \\Vert \\Phi \\Vert \\le \\left( \\sum \\limits _{l \\in \\Lambda } \\omega _l | \\Phi (\\varepsilon (A^{-J})^Tl|^2 \\right)^{\\frac{1}{2}}, $ For this, we first observe that $\\left( \\sum \\limits _{l \\in \\Lambda } \\omega _l | \\Phi (\\varepsilon (A^{-J})^Tl|^2 \\right)^{\\frac{1}{2}}= \\left( \\int \\limits _{\\Omega ^{(\\varepsilon )}} \\sum \\limits _{l \\in \\Lambda } | \\Phi (\\varepsilon (A^{-J})^Tl|^2 \\right)^{\\frac{1}{2}}= \\left\\Vert \\sum \\limits _{l \\in \\Lambda } \\Phi (x_l)\\chi _{V_l} \\right\\Vert .$ By the (inverse) triangle inequality, $\\left\\Vert \\sum \\limits _{l \\in \\Lambda } \\Phi (x_l)\\chi _{V_l} \\right\\Vert \\ge \\Vert \\Phi \\Vert - \\left\\Vert \\Phi - \\sum \\limits _{l \\in \\Lambda }\\Phi (x_l)\\chi _{V_l} \\right\\Vert .", "$ Hence, we require an upper bound for $\\Vert \\Phi - \\sum _{l \\in \\Lambda } \\Phi (x_l)\\chi _{V_l} \\Vert $ .", "By Taylor expansion and Bernstein's inequality, $\\left\\Vert \\Phi - \\sum \\limits _{l \\in \\Lambda } \\Phi (x_l) \\chi _{V_l} \\right\\Vert ^2& = \\int \\limits _{\\Omega ^{(\\varepsilon )}} \\left| \\Phi (x) - \\sum \\limits _{l \\in \\Lambda } \\Phi (x_l)\\chi _{V_l}(x) \\right|^2 \\, dx \\\\& = \\int \\limits _{\\Omega ^{(\\varepsilon )}} \\left| \\sum \\limits _{l \\in \\Lambda } \\Phi (x) \\chi _{V_l}(x) - \\Phi (x_l)\\chi _{V_l}(x) \\right|^2 \\, dx \\\\& \\le \\sum \\limits _{l \\in \\Lambda } \\int \\limits _{V_l} \\left| \\Phi (x) - \\Phi (x_l) \\right|^2 \\, dx \\\\& \\le \\sum \\limits _{l \\in \\Lambda } \\int \\limits _{V_l} \\left| \\sum _{\\alpha \\in \\mathbb {N}^2\\setminus \\lbrace (0,0)\\rbrace }\\frac{1}{\\alpha !}", "\\Vert x - x_l\\Vert ^\\alpha | D^\\alpha \\Phi (x)| \\right|^2 \\, dx \\\\& \\le \\sum \\limits _{l \\in \\Lambda } \\int \\limits _{V_l} \\left| \\sum _{\\alpha \\in \\mathbb {N}^2\\setminus \\lbrace (0,0)\\rbrace }\\frac{1}{\\alpha !}", "\\delta ^{|\\alpha |} ( \\max \\lbrace |L_1|,|L_2|,|L_3|,|L_4|\\rbrace \\pi )^{|\\alpha |} \\Vert \\Phi \\Vert \\right|^2 \\, dx \\\\& \\le \\left| \\sum _{\\alpha \\in \\mathbb {N}^2\\setminus \\lbrace (0,0)\\rbrace }\\frac{1}{\\alpha !}", "\\delta ^{|\\alpha |} ( \\max \\lbrace |L_1|,|L_2|,|L_3|,|L_4|\\rbrace \\pi )^{|\\alpha |} \\Vert \\Phi \\Vert \\right|^2\\cdot \\sum \\limits _{l \\in \\Lambda } \\int \\limits _{V_l} 1\\, dx$ Since the Voronoi regions build a partition of $\\Omega ^{(\\varepsilon )}$ , $\\sum \\limits _{l \\in \\Lambda } \\mu (V_l) = \\mu (\\Omega ^{(\\varepsilon )}),$ and we can continue by $\\left\\Vert \\Phi - \\sum \\limits _{l \\in \\Lambda } \\Phi (x_l) \\chi _{V_l} \\right\\Vert ^2 \\le \\left| \\left(e^{2\\pi \\delta \\max \\lbrace |L_1|,|L_2|,|L_3|,|L_4|\\rbrace } - 1\\right) \\right|^2\\Vert \\Phi \\Vert ^2 \\mu (\\Omega ^{(\\varepsilon )}).", "$ Using (REF ) and (REF ), we obtain $\\left\\Vert \\sum \\limits _{l \\in \\Lambda } \\Phi (x_l)\\chi _{V_l} \\right\\Vert &\\ge \\Vert \\Phi \\Vert - \\left\\Vert \\Phi - \\sum \\limits _{l \\in \\Lambda }\\Phi (x_l)\\chi _{V_l} \\right\\Vert \\\\&\\ge \\Vert \\Phi \\Vert - \\left| \\Vert \\Phi \\Vert \\left(e^{2\\pi \\delta \\max \\lbrace |L_1|,|L_2|,|L_3|,|L_4|\\rbrace } - 1\\right) \\sqrt{ \\mu (\\Omega ^{(\\varepsilon )})}\\right|\\\\&= \\Vert \\Phi \\Vert \\left( 1 - \\left(e^{2\\pi \\delta \\max \\lbrace |L_1|,|L_2|,|L_3|,|L_4|\\rbrace } -1\\right) \\sqrt{ \\mu (\\Omega ^{(\\varepsilon )})}\\right).$ Combining this estimate with (REF ) proves (REF ).", "Finally, we can state and prove Theorem REF , which is one main ingredient for the proof of Theorem REF in Subsection REF .", "Theorem 6.10 Let $L_1,L_2, L_3, L_4 \\in \\mathbb {Z}$ such that $L_1 \\ge L_3$ and $L_2 \\ge L_4$ and $\\alpha _{m_1,m_2} \\in , and let $$ be the trigonometric polynomial$$\\Phi ( \\cdot , \\cdot ) = \\sum _{m_1= L_3}^{L_1} \\sum _{m_2= L_4}^{L_2} \\alpha _{m_1,m_2} e^{-2 \\pi i (\\cdot )m_1}e^{-2 \\pi i (\\cdot )m_2}.$$Further, let $ A= 1 2 3 4 $ be a scaling matrix, $ J N$ a maximalscale, and $ Z2$ the grid defined by$$\\Lambda := \\left\\lbrace (l_1,l_2) \\in \\mathbb {Z}^2 \\, : \\, -M_i \\le l_i \\le M_i, i=1,2 \\right\\rbrace ,$$where $ M1,M2 N$.", "If there exists $ 1/(T1 + T2)$ independent of $ J$ such that $$ fulfills{\\begin{@align*}{1}{-1}\\delta < \\frac{\\log \\left(\\frac{1}{\\sqrt{\\mu ( \\Omega ^{(\\varepsilon )})}}+1\\right)}{ 4 \\pi \\max \\lbrace |L_1|,|L_2|,|L_3|,|L_4|\\rbrace },\\end{@align*}}then there exists $ S() =(S()1,S()2) NN$,independent of $ J$, such that, for all $ >1$, we have$$\\sum \\limits _{l \\in \\Lambda } \\varepsilon ^2 | \\det A^{-J}| | \\Phi (\\varepsilon (A^{-J})^T l) |^2| \\widehat{\\phi }(\\varepsilon (A^{-J})^Tl)|^2 \\ge \\frac{\\Vert \\Phi \\Vert ^2}{2\\theta ^2},$$for $ M= (M1,M2) = 1 (AJ)T S()$ and scaling function $$.$ Since $0<\\varepsilon <1$ , there exists an $m\\in \\mathbb {N}$ , such that $\\frac{1}{m+1} \\le \\varepsilon < \\frac{1}{m}.$ Set $\\varepsilon = \\frac{1}{m+1}$ , and note that (REF ) still holds, since the logarithm is monotonically increasing.", "Next, for some $S \\in \\mathbb {N}$ to be determined later, let $\\begin{pmatrix} M_1 \\\\ M_2 \\end{pmatrix} := \\frac{1}{\\varepsilon } (A^J)^T\\begin{pmatrix} S \\\\S \\end{pmatrix} = \\frac{1}{\\varepsilon }\\begin{pmatrix} \\lambda ^{(J)}_1 + \\lambda ^{(J)}_3 \\\\ \\lambda ^{(J)}_2 + \\lambda ^{(J)}_4 \\end{pmatrix} S,\\quad \\mbox{where as before }A^J=\\begin{pmatrix} \\lambda ^{(J)}_1 & \\lambda ^{(J)}_2 \\\\ \\lambda ^{(J)}_3 & \\lambda ^{(J)}_4 \\end{pmatrix}.$ For $l = (l_1,l_2) \\in \\mathbb {Z}^2$ , we then obtain $ \\nonumber {\\sum \\limits _{l \\in \\Lambda } \\varepsilon ^2 | \\det A^{-J}| | \\Phi (\\varepsilon (A^{-J})^T l) |^2 | \\widehat{\\phi }(\\varepsilon (A^{-J})^Tl)|^2}\\\\ \\nonumber &=&\\sum \\limits _{l_1 = - M_1}^{M_1 -1} \\sum \\limits _{ l_2 = M_2 }^{M_2-1} \\varepsilon ^2 | \\det A^{-J}| | \\Phi (\\varepsilon (A^{-J})^T l) |^2 |\\widehat{\\phi }(\\varepsilon (A^{-J})^Tl)|^2\\\\ \\nonumber &=& \\sum \\limits _{l_1=0 }^{\\frac{\\lambda ^{(J)}_1 + \\lambda ^{(J)}_3}{\\varepsilon } -1} \\sum \\limits _{l_2=0}^{\\frac{\\lambda ^{(J)}_2 + \\lambda ^{(J)}_4}{\\varepsilon } -1}\\sum \\limits _{s=-S} ^{S-1}\\sum \\limits _{t=-S} ^{S-1} \\bigg ( \\varepsilon ^2 | \\det A^{-J}| \\cdot \\left| \\Phi \\left(\\varepsilon (A^{-J})^T \\left(l+\\frac{1}{\\varepsilon } (A^J)^T\\begin{pmatrix} s \\\\ t \\end{pmatrix}\\right) \\right) \\right|^2 \\\\& & \\cdot \\left| \\widehat{\\phi }\\left(\\varepsilon (A^{-J})^T \\left(l+\\frac{1}{\\varepsilon } (A^J)^T\\begin{pmatrix} s \\\\ t \\end{pmatrix}\\right) \\right) \\right|^2\\bigg ), $ Integer periodicity of the trigonometric polynomial implies that, for all $s \\in \\mathbb {Z}$ , $\\Phi \\left(\\varepsilon (A^{-J})^T \\left(l+\\frac{1}{\\varepsilon } (A^J)^T \\begin{pmatrix} s \\\\ t \\end{pmatrix} \\right) \\right)= \\Phi \\left(\\varepsilon (A^{-J})^T l + \\begin{pmatrix} s \\\\ t \\end{pmatrix}\\right)= \\Phi \\left(\\varepsilon (A^{-J})^T l \\right).$ Therefore, by (REF ) $\\sum \\limits _{l \\in \\Lambda } \\; & \\varepsilon ^2 | \\det A^{-J}| | \\Phi (\\varepsilon (A^{-J})^T l) |^2 | \\widehat{\\phi }(\\varepsilon (A^{-J})^Tl)|^2 \\nonumber \\\\&= \\sum \\limits _{l_1=0 }^{\\frac{\\lambda ^{(J)}_1 + \\lambda ^{(J)}_3}{\\varepsilon } -1}\\sum \\limits _{l_2=0}^{\\frac{\\lambda ^{(J)}_2 + \\lambda ^{(J)}_4}{\\varepsilon } -1} \\bigg (\\varepsilon ^2 | \\det A^{-J}| | \\Phi (\\varepsilon (A^{-J})^T l) |^2 \\sum \\limits _{s=-S} ^{S-1} \\sum \\limits _{t=-S} ^{S-1} \\left| \\widehat{\\phi }\\left((A^{-J})^Tl+\\begin{pmatrix} s \\\\ t \\end{pmatrix}\\right)\\right|^2\\bigg ).", "$ Let $\\theta >1$ .", "Then, by Lemma REF , there exists $S^{(\\theta )} \\in \\mathbb {N}$ such that $\\sum \\limits _{s=-S^{(\\theta )}} ^{S^{(\\theta )}-1} \\sum \\limits _{t=-S} ^{S-1} \\left| \\widehat{\\phi }\\left((A^{-J})^Tl+\\begin{pmatrix} s \\\\ t \\end{pmatrix}\\right)\\right|^2 \\ge \\frac{1}{\\theta } \\ .", "$ We now choose $S := S^{(\\theta )}$ .", "Combining (REF ) and (REF ) yields $\\sum \\limits _{l \\in \\Lambda } \\ & \\varepsilon ^2 | \\det A^{-J}| | \\Phi (\\varepsilon (A^{-J})^T l) |^2 | \\widehat{\\phi }(\\varepsilon (A^{-J})^Tl)|^2 \\nonumber \\\\&\\ge \\sum \\limits _{l_1=0 }^{\\frac{\\lambda ^{(J)}_1 + \\lambda ^{(J)}_3}{\\varepsilon } -1}\\sum \\limits _{l_2=0}^{\\frac{\\lambda ^{(J)}_2 + \\lambda ^{(J)}_4}{\\varepsilon } -1} \\varepsilon ^2 | \\det A^{-J}| | \\Phi (\\varepsilon (A^{-J})^T l) |^2\\cdot \\frac{1}{\\theta } \\ .", "$ Next, we apply Proposition REF to (REF ) to obtain $\\sum \\limits _{l \\in \\Lambda } \\varepsilon ^2 | \\det A^{-J}|& | \\Phi (\\varepsilon (A^{-J})^T l) |^2 | \\widehat{\\phi }(\\varepsilon (A^{-J})^Tl)|^2 \\nonumber \\\\&\\ge \\left(\\Vert \\Phi \\Vert \\left( 1 - \\left(e^{2\\pi \\delta \\max \\lbrace |L_1|,|L_2|,|L_3|,|L_4|\\rbrace } -1\\right) \\sqrt{ \\mu (\\Omega ^{(\\varepsilon )})}\\right) \\right)^2\\cdot \\frac{1}{\\theta }$ Since $1+ &\\left( 1- e^{2\\pi \\delta \\max \\lbrace |L_1|,|L_2|,|L_3|,|L_4|\\rbrace }\\right) \\sqrt{\\mu (\\Omega ^{(\\varepsilon )})} \\\\&\\ge 1+ \\left( 1- e^{2\\pi \\max \\lbrace |L_1|,|L_2|,|L_3|,|L_4|\\rbrace \\frac{\\log \\left(\\frac{1}{\\sqrt{\\mu (\\Omega ^{(\\varepsilon )})}}+1\\right)}{4\\pi \\max \\lbrace |L_1|,|L_2|,|L_3|,|L_4|\\rbrace } }\\right) \\sqrt{\\mu (\\Omega ^{(\\varepsilon )})} \\\\&=1+ \\left( 1- \\left(\\sqrt{\\mu (\\Omega ^{(\\varepsilon )})}+1\\right)^{1/2}\\right)\\sqrt{\\mu (\\Omega ^{(\\varepsilon )})} \\\\&=1+ \\left( \\sqrt{\\mu (\\Omega ^{(\\varepsilon )})}- \\left(\\mu (\\Omega ^{(\\varepsilon )})^{3/2}+\\mu (\\Omega ^{(\\varepsilon )})\\right)^{1/2}\\right) \\\\&\\ge 1 -\\mu (\\Omega ^{(\\varepsilon )})^{3/4}\\\\&\\ge 1- \\varepsilon ^{3/4} \\left( \\det A^{-J}\\right)^{3/4}\\\\&\\ge \\frac{1}{2}$ we can conclude $\\sum \\limits _{l \\in \\Lambda } \\varepsilon ^2 | \\det A^{-J}|& | \\Phi (\\varepsilon (A^{-J})^T l) |^2 | \\widehat{\\phi }(\\varepsilon (A^{-J})^Tl)|^2 \\ge \\frac{\\Vert \\Phi \\Vert ^2}{2\\theta }.$ Hence the theorem is proven for $M_1 = \\frac{ \\lambda _1^{(J)} + \\lambda _3^{(J)}}{\\varepsilon } S^{(\\theta )}\\quad \\mbox{and} \\quad M_2 = \\frac{ \\lambda _2^{(J)} + \\lambda _4^{(J)}}{\\varepsilon } S^{(\\theta )}.$ Lemma REF We mention that this result is proved in the same manner as the one dimensional result from [6].", "Lemma 6.11 For $\\gamma >1$ and $\\varepsilon _1, \\varepsilon _2 \\in (0,1/(T_1+T_2)]$ , let $\\theta (\\gamma ),C(\\gamma )>1$ be such that $\\sqrt{\\frac{1}{\\theta (\\gamma )^{2}} - \\frac{16}{\\pi ^4(C(\\gamma ) -1)^2}} - \\sqrt{1-\\frac{1}{\\theta (\\gamma )}} > \\frac{1}{\\gamma }.", "$ If there exists $M=(M_1,M_2) \\in \\mathbb {N}\\times \\mathbb {N}$ such that $\\inf \\limits _{ \\begin{array}{c}\\varphi \\in \\mathcal {R}_N\\\\ \\Vert \\varphi \\Vert =1\\end{array}} \\Vert P_{\\mathcal {S}_M^{(\\varepsilon _1)}} \\varphi \\Vert \\ge \\frac{1}{\\theta (\\gamma )},$ then $\\inf \\limits _{ \\begin{array}{c}\\varphi \\in \\mathcal {R}_N\\\\ \\Vert \\varphi \\Vert =1\\end{array}} \\Vert P_{\\mathcal {S}_K^{(\\varepsilon _2)}} \\varphi \\Vert \\ge \\frac{1}{\\gamma },$ whenever $K = (K_1,K_2) =\\left( \\left\\lceil \\frac{C(\\gamma )M_1\\varepsilon _1}{\\varepsilon _2}\\right\\rceil ,\\left\\lceil \\frac{C(\\gamma )M_2\\varepsilon _1}{\\varepsilon _2}\\right\\rceil \\right)$ .", "First, notice that for any $\\gamma \\in (0,1)$ , there exist $\\theta (\\gamma )$ and $C(\\gamma )$ such that (REF ) is fulfilled.", "Now, let $\\gamma >1$ and $\\varepsilon _2 >0$ .", "Then, by (REF ), $\\nonumber \\inf \\limits _{ \\begin{array}{c}\\varphi \\in \\mathcal {R}_N\\\\ \\Vert \\varphi \\Vert =1\\end{array}} \\Vert P_{\\mathcal {S}_K^{(\\varepsilon _2)}} \\varphi \\Vert & \\ge \\inf \\limits _{ \\begin{array}{c}\\varphi \\in \\mathcal {R}_N\\\\ \\Vert \\varphi \\Vert =1\\end{array}} \\left( \\Vert P_{\\mathcal {S}_K^{(\\varepsilon _2)}} P_{\\mathcal {S}_M^{(\\varepsilon _1)}} \\varphi \\Vert - \\Vert P_{\\mathcal {S}_K^{(\\varepsilon _2)}} P_{\\mathcal {S}_M^{(\\varepsilon _1)}}^{\\perp } \\varphi \\Vert \\right) \\\\ & \\ge \\inf \\limits _{ \\begin{array}{c}\\varphi \\in \\mathcal {R}_N\\\\ \\Vert \\varphi \\Vert =1\\end{array}} \\left( \\Vert P_{\\mathcal {S}_K^{(\\varepsilon _2)}} P_{\\mathcal {S}_M^{(\\varepsilon _1)}} \\varphi \\Vert - \\sqrt{1 - \\frac{1}{\\theta (\\gamma )}} \\right).$ In order to estimate $\\Vert P_{\\mathcal {S}_K^{(\\varepsilon _2)}} P_{\\mathcal {S}_M^{(\\varepsilon _1)}} \\varphi \\Vert $ , we decompose this term by $\\Vert P_{\\mathcal {S}_K^{(\\varepsilon _2)}} P_{\\mathcal {S}_M^{(\\varepsilon _1)}} \\varphi \\Vert ^2= \\Vert P_{\\mathcal {S}_M^{(\\varepsilon _1)}} \\varphi \\Vert ^2 - \\Vert P_{\\mathcal {S}_K^{(\\varepsilon _2)}}^{\\perp } P_{\\mathcal {S}_M^{(\\varepsilon _1)}} \\varphi \\Vert ^2.$ Thus, we require a suitable upper bound for $\\Vert P_{\\mathcal {S}_K^{(\\varepsilon _2)}}^{\\perp } P_{\\mathcal {S}_M^{(\\varepsilon _1)}} \\varphi \\Vert ^2$ .", "For this, let $I_M &= \\lbrace l =(l_1, l_2) \\in \\mathbb {Z}^2 \\, : \\, -M_i/2 \\le l_i \\le M_i/2-1, i =1,2 \\rbrace , \\\\I_K &= \\lbrace j = (j_1, j_2) \\in \\mathbb {Z}^2 \\, : \\, -K_i/2 \\le j_i \\le K_i/2 -1, i =1,2 \\rbrace ,$ and for the complementary sets in $\\mathbb {Z}^2$ we write $I_M^c &= \\lbrace l=(l_1, l_2) \\in \\mathbb {Z}^2 \\, : l \\notin I_M \\rbrace , \\\\I_K^c &= \\lbrace j = (j_1, j_2) \\in \\mathbb {Z}^2 \\, : \\, j \\notin I_K \\rbrace .$ Then, we have $\\nonumber \\Vert P_{\\mathcal {S}_K^{(\\varepsilon _2)}}^{\\perp } P_{\\mathcal {S}_M^{(\\varepsilon _1)}} \\varphi \\Vert ^2&= \\left\\Vert \\sum _{j \\in I_K^c} \\langle P_{\\mathcal {S}_M^{(\\varepsilon _1)}} \\varphi , s_j^{(\\varepsilon _2)} \\rangle s_j^{(\\varepsilon _2)} \\right\\Vert ^2 \\\\ \\nonumber &= \\sum _{j \\in I_K^c} \\left| \\sum _{l \\in I_M} \\langle \\varphi , s_l^{(\\varepsilon _1)} \\rangle \\langle s_l^{(\\varepsilon _1)}, s_j^{(\\varepsilon _2)} \\rangle \\right|^2 \\\\ \\nonumber &\\le \\sum _{j\\in I_K^c} \\left( \\sum _{l \\in I_M} | \\langle \\varphi , s_l^{(\\varepsilon _1)} \\rangle |^2\\sum _{l \\in I_M} |\\langle s_l^{(\\varepsilon _1)}, s_j^{(\\varepsilon _2)} \\rangle |^2 \\right)\\\\ &\\le \\sum _{j \\in I_K^c} \\sum _{l \\in I_M} |\\langle s_l^{(\\varepsilon _1)}, s_j^{(\\varepsilon _2)} \\rangle |^2 .$ Now, for $\\varepsilon = \\max \\lbrace \\varepsilon _1, \\varepsilon _2 \\rbrace $ , we obtain $| \\langle s_l^{(\\varepsilon _1)}, s_j^{(\\varepsilon _2)} \\rangle |&= \\left| \\varepsilon _1 \\cdot \\varepsilon _2 \\cdot \\int \\limits _{\\left[-\\frac{1}{2\\varepsilon }, \\frac{1}{2\\varepsilon } \\right]^2}e^{2 \\pi i \\varepsilon _1 \\langle l, x \\rangle } e^{2 \\pi i \\varepsilon _2 \\langle j, x \\rangle } \\, dx \\right| \\\\&\\le \\left| \\frac{\\varepsilon _1 \\varepsilon _2}{\\pi ^2 (\\varepsilon _1 l_1 - \\varepsilon _2 j_1)( \\varepsilon _1 l_2 - \\varepsilon _2 j_2)} \\right|.$ Therefore, by using (REF ), $\\Vert P_{\\mathcal {S}_K^{(\\varepsilon _2)}}^{\\perp } P_{\\mathcal {S}_M^{(\\varepsilon _1)}} \\varphi \\Vert ^2 \\le \\sum _{j \\in I_K^c} \\sum _{l \\in I_M} \\left| \\frac{\\varepsilon _1 \\varepsilon _2}{\\pi ^2 (\\varepsilon _1 l_1 - \\varepsilon _2 j_1)( \\varepsilon _1 l_2 - \\varepsilon _2 j_2)} \\right|^2 .$ Assuming $K_i = \\frac{C(\\gamma ) M_i \\varepsilon _1}{\\varepsilon _2}, i = 1,2$ , we can continue by $\\Vert P_{\\mathcal {S}_K^{(\\varepsilon _2)}}^{\\perp } P_{\\mathcal {S}_M^{(\\varepsilon _1)}} \\varphi \\Vert ^2 & \\le \\left( \\frac{\\varepsilon _1 \\varepsilon _2}{\\pi ^2} \\right)^2 M_1 M_2 \\sum _{(j_1,j_2) \\notin I_K} \\frac{4}{ |( \\varepsilon _1 \\frac{M_1}{2} - \\varepsilon _2 j_1 )( \\varepsilon _1 \\frac{M_2}{2} - \\varepsilon _2 j_2)|^2} \\\\& \\le \\left( \\frac{\\varepsilon _1 \\varepsilon _2}{\\pi ^2} \\right)^2 M_1 M_2 \\frac{16}{\\varepsilon _2^2 |( \\varepsilon _1 M_1 - \\varepsilon _2 K_1 )( \\varepsilon _1 M_2 - \\varepsilon _2 K_2)|} \\\\& \\le \\left( \\frac{\\varepsilon _1 }{\\pi ^2} \\right)^2 M_1 M_2 \\frac{16}{ |( \\varepsilon _1 M_1 - \\varepsilon _2 K_1 )( \\varepsilon _1 M_2 - \\varepsilon _2 K_2)|} \\\\& \\le \\left( \\frac{\\varepsilon _1 }{\\pi ^2} \\right)^2 M_1 M_2 \\frac{16}{ |( \\varepsilon _1 M_1 - C(\\gamma ) M_1 \\varepsilon _1 )( \\varepsilon _1 M_2 - C(\\gamma ) M_2 \\varepsilon _1)|} \\\\&= \\frac{16}{\\left( \\pi ^2 (C(\\gamma )-1) \\right)^2}.$ Thus, by (REF ), $\\Vert P_{\\mathcal {S}_K^{(\\varepsilon _2)}} P_{\\mathcal {S}_M^{(\\varepsilon _1)}} \\varphi \\Vert ^2\\ge \\frac{1}{\\theta (\\gamma )^2} - \\frac{16}{\\left( \\pi ^2 (C(\\gamma )-1) \\right)^2}$ which, using (REF ), yields $\\inf \\limits _{ \\begin{array}{c}\\varphi \\in \\mathcal {R}_N\\\\ \\Vert \\varphi \\Vert =1\\end{array}} \\Vert P_{\\mathcal {S}_K^{(\\varepsilon _2)}} \\varphi \\Vert \\ge \\sqrt{\\frac{1}{\\theta (\\gamma )^2} - \\frac{16}{\\left( \\pi ^2 (C(\\gamma )-1) \\right)^2}} - \\sqrt{1 - \\frac{1}{\\theta }}.$ The lemma is proved.", "Proof of Theorem REF The following lemma will be used in the upcoming proof Theorem REF .", "A one dimensional analogue can be found in [6].", "The proof extends straightforwardly and we omit it here.", "Lemma 7.1 Let $A_1, A_2, A_3$ , and $A_4 \\in \\mathbb {Z}, A_1 \\le A_2, A_3 \\le A_4$ .", "Moreover, let $L_1, L_2 \\in \\mathbb {N}$ such that $2 L_1 \\ge A_2 - A_1 +1$ and $2L_2 \\ge A_4 - A_3 +1$ .", "Then the trigonometric polynomial $\\Phi (z_1,z_2) = \\sum _{k = A_1}^{A_2} \\sum _{l=A_3}^{A_4} \\alpha _{k,l} e^{2\\pi i k z_1} e^{2\\pi i l z_2}$ satisfies $\\sum \\limits _{m=0}^{2L_1 - 1} \\sum \\limits _{n=0}^{2L_2 - 1} \\frac{1}{4L_1 L_2} \\left| \\Phi \\left(\\frac{m}{2L_1}, \\frac{n}{2L_2}\\right) \\right|^2 = \\sum _{k = A_1}^{A_2} \\sum _{l=A_3}^{A_4} |\\alpha _{k,l} |^2 .$ [Proof of Theorem REF ] Let $\\varphi \\in \\mathcal {R}_N$ such that $\\Vert \\varphi \\Vert =1$ .", "Then $\\varphi $ can be expanded as $\\varphi = \\sum _{n_1,n_2=0}^{2^{J_0}-1} \\alpha _{J_0,(n_1,n_2)} \\phi _{J_0,(n_1,n_2)} + \\sum _{k=1}^3 \\sum _{j = J_0}^{J-1} \\sum _{n_1,n_2 =0}^{2^j-1} \\beta ^k_{j,(n_1,n_2)} \\psi _{j,(n_1,n_2)}^{\\operatorname{b},k} $ We will now use the nestedness of the two dimensional MRA that is generated by the one dimensional wavelet system $\\left\\lbrace \\lbrace \\phi ^{\\operatorname{b}}_{J_0,m} \\rbrace _{m = 0, \\ldots , 2^J-1}, \\lbrace \\psi ^{\\operatorname{b}}_{j,n} \\rbrace _{j \\ge J_0,n = 0, \\ldots , 2^j-1}\\right\\rbrace $ .", "In particular, we have $V_j^{\\operatorname{b},2} \\subset V_{j+1}^{\\operatorname{b},2} , \\quad j\\ge J_0,$ and $V_{j+1}^{\\operatorname{b},2} = V_{j}^{\\operatorname{b},2} \\oplus W_{j}^{\\operatorname{b},2}, \\quad j\\ge J_0,$ with $V_j^{\\operatorname{b},2} = V_j^{\\operatorname{b}} \\otimes V_j^{\\operatorname{b}}$ and $W_j^{\\operatorname{b},2} = (V^{\\operatorname{b}}_j \\otimes W^{\\operatorname{b}}_j) \\oplus (W^{\\operatorname{b}}_j \\otimes V^{\\operatorname{b}}_j) \\oplus (W^{\\operatorname{b}}_j \\otimes W^{\\operatorname{b}}_j)$ .", "Loosely speaking, due to the MRA embedding properties we can expand functions from the reconstructions space into scaling functions $(\\phi ^{\\operatorname{b}}_{J,(n_1,n_2)})_{n_1,n_2}$ at highest scale.", "Since the left boundary functions can be constructed by translates of the initial scaling function $\\phi $ and the right scaling function can be obtained by reflecting the left boundary functions.", "The reflected function will be denoted by $\\phi ^{\\#}$ .", "This gives us in (REF ) $\\varphi = \\sum _{n_1,n_2=0}^{2^J-p-1} \\alpha _{n_1,n_2} \\phi (2^{J} \\cdot -n) + \\sum _{n_1,n_2 =2^J-p}^{2^J} \\beta _{n_1,n_2} \\phi ^{\\#}(2^{J} \\cdot -n),$ where only finitely many $\\alpha _{n_1,n_2}$ and $\\beta _{n_1,n_2}$ are non-zero.", "Now, for any $l= (l_1,l_2) \\in \\mathbb {Z}^2$ we obtain by basic properties of the Fourier transform $\\langle \\varphi &, s_l^{(\\varepsilon )} \\rangle = \\sum _{n_1,n_2=0}^{2^J-p-1} \\alpha _{n_1,n_2} \\frac{\\varepsilon }{ 2^{J}} e^{-2\\pi i \\varepsilon \\langle n, 2^{-J}l\\rangle }\\widehat{\\phi }\\left(\\frac{\\varepsilon }{ 2^{J}}l\\right) + \\sum _{n_1,n_2 =2^J-p}^{2^J} \\beta _{n_1,n_2} \\frac{\\varepsilon }{ 2^{J}} e^{-2\\pi i \\varepsilon \\langle n, 2^{-J}l\\rangle } \\widehat{\\phi ^{\\#}}\\left(\\frac{\\varepsilon }{ 2^{J}}l\\right).", "$ For the sake of brevity, we shall write in the following $\\Phi _1(z) &= \\sum _{n_1,n_2=0}^{2^J-p-1} \\alpha _{n_1,n_2} \\frac{\\varepsilon }{ 2^{J}} e^{-2\\pi i \\varepsilon \\langle n, z\\rangle },\\nonumber \\\\\\Phi _2(z) &= \\sum _{n_1,n_2 =2^J-p}^{2^J} \\beta _{n_1,n_2} \\frac{\\varepsilon }{ 2^{J}} e^{-2\\pi i \\varepsilon \\langle n, z\\rangle }.", "$ By our assumptions on the scaling function $|\\widehat{\\phi }(\\xi _1, \\xi _2)| \\lesssim \\frac{1}{ (1+|\\xi _1|)(1+ |\\xi _2|)}, $ and by the same argument $|\\widehat{\\phi ^{\\#}}(\\xi _1, \\xi _2)| \\lesssim \\frac{1}{(1+|\\xi _1|)(1+ |\\xi _2|)}.", "$ Using (REF ), (REF ), and (REF ) in (REF ) yields $\\sum _{l \\notin I_M} |\\langle \\varphi , s_l^{(\\varepsilon )} \\rangle |^2 \\nonumber &\\le \\sum _{l \\notin I_M} \\left| \\frac{\\varepsilon }{ 2^{J}}\\Phi _1\\left(\\frac{\\varepsilon }{2^{J}}l\\right) \\widehat{\\phi }\\left(\\frac{\\varepsilon }{ 2^{J}}l\\right) \\right|^2 + \\sum _{l \\notin I_M}\\left|\\frac{\\varepsilon }{ 2^{J}} \\Phi _2\\left(\\frac{\\varepsilon }{2^{J}}l\\right) \\widehat{\\phi ^{\\#}}\\left(\\frac{\\varepsilon }{ 2^{J}}l\\right) \\right|^2 + \\nonumber \\\\&\\makebox{[}0.5cm][c]{}+ 2 \\left( \\sum _{l \\notin I_M} \\left|\\frac{\\varepsilon }{ 2^{J}} \\Phi _1\\left(\\frac{\\varepsilon }{2^{J}}l\\right) \\widehat{\\phi }\\left(\\frac{\\varepsilon }{ 2^{J}}l\\right) \\right|^2 \\right)^{1/2} \\left( \\sum _{l \\notin I_M} \\left|\\frac{\\varepsilon }{ 2^{J}} \\Phi _2\\left(\\frac{\\varepsilon }{2^{J}}l\\right) \\widehat{\\phi ^{\\#}}\\left(\\frac{\\varepsilon }{ 2^{J}}l\\right) \\right|^2\\right)^{1/2}.", "$ We assume $2^J/\\varepsilon \\in \\mathbb {N}$ and the number of samples $M=(M_1,M_2) \\in \\mathbb {N}\\times \\mathbb {N}$ is $M_i = S \\cdot \\frac{2^J}{\\varepsilon }, \\quad i = 1,2$ where $S$ is some positive constant.", "Now, $(l_1,l_2) \\notin I_M$ if Case I: $|l_1| > M_1$ and $|l_2|<M_2$ , Case II: $|l_1| < M_1$ and $|l_2|>M_2$ , or Case III: $|l_1| > M_1$ and $|l_2|>M_2$ .", "It is sufficient to consider the sum for Case I.", "Case II can be obtained by symmetry and Case III yields a smaller sum.", "For $K = 2^J/\\varepsilon $ , we have $\\sum _{|l_2| <M_2}& \\sum _{|l_1| >M_1} \\left| \\frac{\\varepsilon }{ 2^{J}} \\Phi _1\\left(\\frac{\\varepsilon }{2^{J}}l\\right) \\widehat{\\phi }\\left(\\frac{\\varepsilon }{ 2^{J}}l\\right) \\right|^2 \\nonumber \\\\&= \\sum _{j_2}^{K-1} \\sum _{j_1}^{K-1} \\frac{1}{K}\\frac{1}{K}\\left| \\Phi _1\\left(\\frac{j_1}{K}, \\frac{j_2}{K}\\right) \\right|^2 \\sum _{|s_2|<S} \\sum _{|s_1|>S} \\left|\\widehat{\\phi }\\left(\\frac{j_1}{K} + s_1,\\frac{j_2}{K} + s_2\\right) \\right|^2 \\nonumber \\\\&\\lesssim \\sum _{j_2}^{K-1} \\sum _{j_1}^{K-1} \\frac{1}{K}\\frac{1}{K}\\left| \\Phi _1\\left(\\frac{j_1}{K}, \\frac{j_2}{K}\\right) \\right|^2 \\sum _{|s_2|<S} \\sum _{|s_1|>S} \\frac{1}{(1 + |j_1/K + s_1|)^2}\\frac{1}{(1 + |j_2/K + s_2|)^2} \\nonumber \\\\&\\le C_1 \\sum _{j_2}^{K-1} \\sum _{j_1}^{K-1} \\frac{1}{K}\\frac{1}{K}\\left| \\Phi _1\\left(\\frac{j_1}{K}, \\frac{j_2}{K}\\right) \\right|^2 \\frac{1}{S}.", "$ By Lemma REF we obtain in (REF ) $\\sum _{|l_2| >M_2}& \\sum _{|l_1| >M_1} \\left| \\frac{\\varepsilon }{ 2^{J}} \\Phi _1\\left(\\frac{\\varepsilon }{2^{J}}l\\right) \\widehat{\\phi }\\left(\\frac{\\varepsilon }{ 2^{J}}l\\right) \\right|^2 \\le C_1 \\sum _{n_1,n_2=0}^{2^J-1} |\\alpha _{n_1,n_2}|^2 \\frac{1}{S}.", "$ Since the functions form an orthonormal basis and $\\Vert \\varphi \\Vert =1$ we have $\\sum _{n_1,n_2\\in \\mathbb {N}} |\\alpha _{n_1,n_2}|^2 + \\sum _{n_1,n_2\\in \\mathbb {N}} |\\beta _{n_1,n_2}|^2 = 1$ and hence $\\sum _{n_1,n_2\\in \\mathbb {N}} |\\alpha _{n_1,n_2}|^2 \\le 1$ Similarly, one shows $\\sum _{|l_2| >M_2}& \\sum _{|l_1| >M_1} \\left| \\frac{\\varepsilon }{ 2^{J}} \\Phi _2\\left(\\frac{\\varepsilon }{2^{J}}l\\right) \\widehat{\\phi ^{\\#}} \\left(\\frac{\\varepsilon }{ 2^{J}}l\\right) \\right|^2 \\le C_2 \\sum _{n_1,n_2 \\in \\mathbb {N}}|\\beta _{n_1,n_2}|^2 \\frac{1}{S}.", "$ Therefore, in (REF ) we have that $\\sum _{|l_2| <M_2} \\sum _{|l_1| >M_1} |\\langle \\varphi , s_l^{(\\varepsilon )} \\rangle |^2 & \\le C_1 \\sum _{n_1,n_2\\in \\mathbb {N}} |\\alpha _{n_1,n_2}|^2 \\frac{1}{S} + C_2 \\sum _{n_1,n_2 \\in \\mathbb {N}} |\\beta _{n_1,n_2}|^2 \\frac{1}{S} +\\\\&\\qquad + 2 \\left(C_1 \\sum _{n_1,n_2\\in \\mathbb {N}} |\\alpha _{n_1,n_2}|^2 \\frac{1}{S}\\right)^{1/2} \\left(C_2\\sum _{n_1,n_2\\in \\mathbb {N}}|\\beta _{n_1,n_2}|^2 \\frac{1}{S}\\right)^{1/2} \\nonumber \\\\&\\le \\left( \\frac{\\sqrt{2C_1}+\\sqrt{2C_2}}{\\sqrt{S}}\\right)^2$ Now, for $\\theta >1$ choosing $S$ large enough, such that $\\left( \\frac{\\sqrt{2C_1}+\\sqrt{2C_2}}{\\sqrt{S}}\\right)^2\\le \\frac{\\theta ^2 - 1}{3\\theta ^2}$ gives the claim.", "Acknowledgements The authors would like to thank Bogdan Roman for providing Figure 2 and Figure 3.", "BA acknowledges support from the NSF DMS grant 1318894.", "ACH acknowledges support from a Royal Society University Research Fellowship as well as the UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/L003457/1.", "GK was supported in part by the Einstein Foundation Berlin, by Deutsche Forschungsgemeinschaft (DFG) Grant KU 1446/14, by the DFG Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics”, and by the DFG Research Center Matheon “Mathematics for key technologies” in Berlin.", "JM acknowledges support from the Berlin Mathematical School." ], [ "Linear Sampling Rate for 2D Boundary Wavelets", "In applications, typically signals on a bounded domain are considered such as on the interval $[0,1]$ .", "Aiming to avoid artifacts at the boundaries $x=0, 1$ , in [11] (see also [22]) boundary wavelets were introduced.", "The considered wavelet system then consists of interior wavelets, which do not touch the boundary, and the just mentioned boundary wavelets, leading to a system with an associated multiresolution analysis and prescribed vanishing moments even at the boundary.", "Similar to the classical wavelet construction, this system can be lifted to 2D by tensor products.", "In this section, we will show that in fact also this (boundary) wavelet system allows a linear stable sampling rate, though with a proof differing significantly from the one of the `classical wavelet case' due to the different structure at the highest scale.", "For this, we start with formally introducing 2D boundary wavelets – which we use as an expression for the whole wavelet system – followed by our choice of reconstruction and sampling space." ], [ "Construction of Boundary Wavelets", "We start with the 1D construction of wavelets on the interval as introduced in [11].", "For this, let $\\phi $ be a compactly supported Daubechies scaling function with $p$ vanishing moments.", "It is well known that $\\phi $ must then have a support of size $2p-1$ .", "By a shifting argument, we can assume that $\\operatorname{supp \\,}(\\phi ) = [-p+1,p]$ .", "In order to properly define interior wavelets and boundary wavelets, the scale need to be large enough – i.e., the support of the wavelets need to be small enough – to be able to distinguish between wavelets whose support fully lie in $[0,1]$ and wavelets whose support intersect the boundary $x=0$ and, likewise, $x =1$ .", "Therefore, we now let $j \\in \\mathbb {N}$ such that $2p \\le 2^j$ .", "Then there exist $2^j - 2p$ interior scaling functions, i.e., scaling functions which have support inside $[0,1]$ , defined by $\\phi ^{\\operatorname{b}}_{j,n} = \\phi _{j,n} = 2^{j/2} \\phi (2^j\\cdot - n), \\quad \\text{for } p\\le n < 2^j-p.$ Depending on boundary scaling functions $ \\lbrace \\phi ^{\\operatorname{left}}_n\\rbrace _{n=0, \\ldots , p-1}$ and $\\lbrace \\phi ^{\\operatorname{right}}_n\\rbrace _{n=0, \\ldots , p-1}$ , which we will introduce below, the $p$ left boundary scaling functions are defined by $\\phi ^{\\operatorname{b}}_{j,n} = 2^{j/2} \\phi ^{\\operatorname{left}}_n(2^j\\cdot ), \\quad \\text{for } 0 \\le n < p,$ and the $p$ right boundary scaling functions are $\\phi ^{\\operatorname{b}}_{j,n} = 2^{j/2} \\phi ^{\\operatorname{right}}_{2^j-1-n}(2^j(\\cdot -1)), \\quad \\text{for } 2^j-p \\le n < 2^j.$ We remark that this leads to $2^j$ scaling functions in total, which is the number of original scaling functions $(\\phi _{j,n})_n$ that intersect $[0,1]$ .", "We next sketch the idea of the construction of boundary scaling functions $ \\lbrace \\phi ^{\\operatorname{left}}_n\\rbrace _{n=0, \\ldots , p-1}$ as well as $\\lbrace \\phi ^{\\operatorname{right}}_n\\rbrace _{n=0, \\ldots , p-1}$ following [11], to the extent to which we require it in our proofs.", "One starts by defining edge functions $\\widetilde{\\phi }^k$ on the positive axis $[0,\\infty )$ by $\\widetilde{\\phi }^k(x) = \\sum _{n=0}^{2p-2} \\binom{n}{k} \\phi (x+n-p+1), \\quad k = 0, \\ldots , p-1,$ such that these edge functions are orthogonal to the interior scaling functions and such that they together generate all polynomials up to degree $p-1$ .", "After performing a Gram-Schmidt procedure one obtains the left boundary functions $\\phi ^{\\operatorname{left}}_k, k =0, \\ldots , p-1$ .", "The right boundary functions are then – after some minor adjustments – obtained by reflecting the left boundary functions.", "This construction from [11] allows one to obtain a multiresolution analysis.", "Theorem 4.1 ([11]) If $2^j \\ge 2p$ , then $\\lbrace \\phi ^{\\operatorname{b}}_{j,n}\\rbrace _{n=0, \\ldots , 2^j-1}$ is an orthonormal basis for a space $V^{\\operatorname{b}}_j$ that is nested, i.e.", "$V_j^{\\operatorname{b}} \\subset V_{j+1}^{\\operatorname{b}},$ and complete, i.e.", "$\\overline{\\bigcup _{j \\ge \\log _2{2p}} V^{\\operatorname{b}}_j} = L^2[0,1].$ Next, we define an orthonormal basis for the wavelet space $W^{\\operatorname{b}}_j$ , which is as usual defined as the orthogonal complement of $V_j^{\\operatorname{b}}$ in $V_{j+1}^{\\operatorname{b}}$ .", "For this, let $\\psi $ be the corresponding wavelet function to $\\phi $ with $p$ vanishing moments and $\\operatorname{supp \\,}\\psi = [-p+1,p]$ .", "Similar to the construction of the scaling functions, we will obtain interior wavelets and boundary wavelets, which then constitutes the set of wavelets in the interval.", "Again based on a careful choice of boundary wavelets $(\\psi ^{\\operatorname{left}}_n)_n$ and $(\\psi ^{\\operatorname{right}}_k)_k$ , for which we refer to [11], we define $2^j-2p$ interior wavelets by $\\psi ^{\\operatorname{b}}_{j,n} = \\psi _{j,n} = 2^{j/2} \\psi (2^j\\cdot - n), \\quad \\text{for } p\\le n < 2^j-p,$ $p$ left boundary wavelets $\\psi ^{\\operatorname{b}}_{j,n} = 2^{j/2} \\psi ^{\\operatorname{left}}_n(2^j\\cdot ), \\quad \\text{for } 0 \\le n < p,$ and $p$ right boundary wavelets $\\psi ^{\\operatorname{b}}_{j,n} = 2^{j/2} \\psi ^{\\operatorname{right}}_{2^j-1-n}(2^j(\\cdot -1)), \\quad \\text{for } 2^j-p \\le n < 1^j.$ Summarizing, the following result hold for these wavelet functions.", "Theorem 4.2 ([11]) Let $2^J \\ge 2p$ .", "Then the following properties hold: [i)] $\\lbrace \\psi ^{\\operatorname{b}}_{J,n} \\rbrace _{n = 0, \\ldots , 2^J-1}$ is an orthonormal basis for $W_J^{\\operatorname{b}}$ .", "$L^2[0,1]$ can be decomposed as $L^2[0,1] = V_J^{\\operatorname{b}} \\oplus W_{J}^{\\operatorname{b}} \\oplus W_{J+1}^{\\operatorname{b}} \\oplus W_{J+2}^{\\operatorname{b}} \\oplus \\ldots =V_J^{\\operatorname{b}} \\bigoplus \\limits _{j=J}^{\\infty } W_{j}^{\\operatorname{b}}.$ $\\left\\lbrace \\lbrace \\phi ^{\\operatorname{b}}_{J,m} \\rbrace _{m = 0, \\ldots , 2^J-1}, \\lbrace \\psi ^{\\operatorname{b}}_{j,n} \\rbrace _{j \\ge J,n = 0, \\ldots , 2^j-1}\\right\\rbrace $ is an orthonormal basis for $L^2[0,1]$ .", "If $\\phi , \\psi \\in C^r[0,1]$ , then $\\left\\lbrace \\lbrace \\phi ^{\\operatorname{b}}_{j,m} \\rbrace _{m = 0, \\ldots , 2^J-1}, \\lbrace \\psi ^{\\operatorname{b}}_{j,n} \\rbrace _{j \\ge J,n = 0,\\ldots , 2^j-1}\\right\\rbrace $ is an unconditional basis for $C^s[0,1]$ for all $s <r$ .", "As mentioned before, this system gives rise to a 2D system by tensor products, i.e., by the standard 2D separable wavelet construction.", "In particular, this 2D system again constitutes an MRA, see [22].", "Sampling and Reconstruction Space Reconstruction Space For defining the reconstruction space, we will now assume that the region of interest is $[0,1]^2$ instead of $[0,a]^2$ with $a \\in \\mathbb {N}$ , as previously chosen.", "Our starting point is a 1D compactly supported Daubechies scaling function with $p$ vanishing moments (cf.", "Subsection REF ).", "Recall that Daubechies wavelets have at least the following frequency decay, $|\\widehat{\\phi }(\\xi ) | \\lesssim \\frac{1}{1 + |\\xi |}, \\quad \\text{ as } |\\xi | \\rightarrow \\infty .", "$ Let $\\psi $ be a corresponding wavelet with $p$ vanishing moments, and let $\\phi ^{\\operatorname{b}}$ and $\\psi ^{\\operatorname{b}}$ be the wavelets on the interval as introduced in the previous subsection.", "Let $J_0$ be the smallest number such that $2^{J_0} \\ge 2p$ .", "Then the associated 2D scaling functions are of the form $\\phi ^{\\operatorname{b}}_{J_0,(n_1,n_2)} := \\phi ^{\\operatorname{b}}_{J_0,n_1} \\otimes \\phi ^{\\operatorname{b}}_{J_0,n_2}, \\quad 0\\le n_1,n_2 \\le 2^{J_0}-1$ and, for $0 \\le n_1,n_2 \\le 2^j -1$ with $j \\ge J_0$ , the 2D wavelet functions are defined by $\\psi ^{\\operatorname{b},k}_{j,(n_1,n_2)} := {\\left\\lbrace \\begin{array}{ll}\\phi ^{\\operatorname{b}}_{j,n_1} \\otimes \\psi ^{\\operatorname{b}}_{j,n_2}, & j \\ge J_0, k=1,\\\\\\psi ^{\\operatorname{b}}_{j,n_1} \\otimes \\phi ^{\\operatorname{b}}_{j,n_2}, & j \\ge J_0, k=2, \\\\\\psi ^{\\operatorname{b}}_{j,n_1} \\otimes \\psi ^{\\operatorname{b}}_{j,n_2}, & j \\ge J_0, k=3.\\end{array}\\right.", "}$ Next, let $\\Omega _1 = \\lbrace \\phi ^{\\operatorname{b}}_{J_0,(n_1,n_2)} \\, : \\, 0 \\le n_1,n_2 \\le 2^{J_0}-1 \\rbrace $ and $\\Omega _2 = \\lbrace \\psi ^{\\operatorname{b},k}_{j,(n_1,n_2)}\\, : \\, j =J_0,1, \\ldots , J-1, 0 \\le n_1,n_2\\le 2^j-1, k =1,2,3 \\rbrace .$ Then, for a fixed scale $J$ , and for $N_J=2^{2J}$ , the reconstruction space $\\mathcal {R}_{N_J}$ is given by $\\mathcal {R}_{N_J} = \\operatorname{span}\\lbrace \\varphi _i \\, : \\, \\varphi _i \\in \\Omega _1 \\cup \\Omega _2, i = 1, \\ldots , N_J\\rbrace .$ An ordering can be obtained analogously as in Subsection REF .", "Sampling Space The sampling space can be chosen similar to Subsection REF , i.e., for $\\varepsilon \\le 1$ , we set $s_l^{(\\varepsilon )} = \\varepsilon e^{2 \\pi i \\varepsilon \\langle l, \\cdot \\rangle } \\cdot \\chi _{[0,1]^2}, \\quad l \\in \\mathbb {Z}^2,$ and for $M = (M_1,M_2 )\\in \\mathbb {N}\\times \\mathbb {N}$ $\\mathcal {S}^{(\\varepsilon )}_{M} = \\operatorname{span}\\left\\lbrace s_l^{(\\varepsilon )} \\, : \\, l = (l_1,l_2) \\in \\mathbb {Z}^2, - M_i \\le l_i \\le M_i , i =1,2 \\right\\rbrace .$ Main Result Our main result of this section states the linearity of the stable sampling rate for boundary wavelets, whose proof is presented in Section .", "Theorem 4.3 Let $J \\in \\mathbb {N}$ , $\\varepsilon \\le 1$ , and $\\theta >1$ .", "Further, let $\\mathcal {S}_M^{(\\varepsilon )}$ and $\\mathcal {R}_{N_J}$ be defined as REF and (REF ), respectively.", "Then $\\inf \\limits _{\\begin{array}{c} \\varphi \\in \\mathcal {R}_{N_J} \\\\ \\Vert \\varphi \\Vert =1 \\end{array}} \\Vert P_{S_M^{(\\varepsilon )}} \\varphi \\Vert \\ge \\frac{1}{\\theta }$ for $M = (\\lceil S^\\theta /\\varepsilon \\rceil 2^J,\\lceil S^\\theta /\\varepsilon \\rceil 2^J) \\in \\mathbb {N}\\times \\mathbb {N}$ , where $S^\\theta $ is a constant independent of $J$ .", "In particular, the stable sampling rate obeys $\\Theta (N_J,\\theta ) = \\mathcal {O}(N_J)$ as $N_J \\rightarrow \\infty $ for every fixed $\\theta >1$ Remark 4.4 This result will be proved independently of Theorem REF , since boundary wavelets do constitute an MRA but at highest scaling level, the space $V_J$ contains more than one generating function $\\phi $ .", "It uses the reflected functions as well.", "Figure: Stable sampling rate Θ(N,θ)\\Theta (N,\\theta ) for two dimensional dyadic Haar wavelets and two dimensional dyadic Daubechies-4 wavelets.Table: Number of reconstruction elements and samples for Haar and Daubechies-4.", "Note that these numbers predict the jumps in Figure .", "Numerical Experiments In this section we numerically demonstrate the linearity of the stable sampling rate as stated in Theorem REF .", "We will also demonstrate how this combines with generalized sampling in practice.", "In particular, given this linearity, reconstructing from Fourier samples in smooth boundary wavelets will give an error decaying according to the smoothness and the number of vanishing moments.", "In this section we consider dyadic scaling matrices $A = \\begin{pmatrix} 2 & 0 \\\\ 0 & 2 \\end{pmatrix}.", "$ Furthermore, our focus are separable wavelets, i.e.", "wavelets that are obtained by tensor products of one dimensional scaling functions and one dimensional wavelet functions, respectively.", "Scaling matrices of the form (REF ) preserve the separability.", "Linearity examples with Haar and Daubechies-4 wavelets We use the description of Section 3.1 and Example REF in order to perform numerical experiments for some known wavelets.", "This gives the reconstruction space $\\mathcal {R}= \\overline{\\operatorname{span}} \\lbrace \\varphi \\, : \\, \\varphi \\in \\Omega _1 \\cup \\Omega _2\\rbrace $ where $\\Omega _1$ and $\\Omega _2$ are defined in (REF ) and (REF ) respectively.", "We order the reconstruction space $\\mathcal {R}$ in the same manner as presented at the end of Section 3.", "In (REF ) we counted the number of reconstruction elements up to level $J-1$ , which leads to $N_J = (2^{2J}-1)a^2 + 6a(a-1)(2^J-1) + 3J(a-1)^2 + (2a-1)^2 $ many elements, which is asymptotically of order $2^{2J}$ .", "We test Haar wavelets and 2D Daubechies-4 wavelets.", "Figure REF shows the linear behaviour of the stable sampling rate for these two types of wavelet generators.", "By a small abuse of notation, we also write $M$ for the total number of samples.", "In our analysis in Section 3 we estimated the angle $\\cos (\\omega (\\mathcal {R}_N, \\mathcal {S}_M^{(\\varepsilon )}))$ (recall that $\\epsilon $ is the sampling rate) with respect to some fixed $\\theta >1$ .", "In fact we computed $M$ such that $\\inf \\limits _{\\begin{array}{c} \\varphi \\in \\mathcal {R}_N \\\\ \\Vert \\varphi \\Vert =1\\end{array}} \\Vert P_{\\mathcal {S}_M^{(\\varepsilon )}} \\varphi \\Vert \\ge \\frac{1}{\\theta }$ holds.", "We proved that $M$ is up to a constant of the same size as $N_J$ .", "Figure REF shows the stable sampling rate (in blue) $\\Theta (N,\\theta ) = \\min \\left\\lbrace M \\in \\mathbb {N}, \\cos (\\omega (\\mathcal {R}_N,\\mathcal {S}_M)) \\ge \\frac{1}{\\theta }\\right\\rbrace $ and the linear function $f$ (in red) given by $f(N) = \\frac{M_{\\text{max}}}{N_{\\text{max}}}N,$ where $N_{\\text{max}}$ is the maximum value of $N$ used in the experiment and $M_{\\text{max}} = \\Theta (N_{\\text{max}},\\theta )$ .", "We computed the stable sampling rate up to level $J = 4$ .", "Note the (significant) jumps of the stable sampling rate occur whenever $N\\in \\mathbb {N}$ crosses the scaling level $N_J, J = 0, \\ldots , 4$ .", "In the Haar case these are $N_0 = 4, \\qquad N_1 = 16, \\qquad N_2 = 64, \\qquad N_3 = 256, \\quad N_4 = 1024,$ see (REF ).", "Note that $a=1$ in the Haar case.", "In particular, the jumps are linear, suggesting a linear stable sampling rate.", "Figure 1 (b), (c), and (d) are interpreted similarly.", "However, the theoretical results are asymptotic results.", "Therefore, it should not be surprising that the stable sampling rate is below the linear line in some cases.", "It aligns asymptotically.", "Fourier samples and boundary wavelet reconstruction Figure: Reconstruction of the function f 1 (x,y)=cos 2 (x)exp(-y)f_1(x,y)=\\cos ^2(x)\\exp (-y).The second row shows an 8 times zoomed-in version of the upper left corner.Left: original function.", "Middle: truncated Fourier serieswith 256 2 256^2 Fourier coefficients.", "Right: generalized sampling withDaubechies-3 wavelets computed from the same Fourier coefficients.Figure: Reconstruction of the function f 2 (x,y)=(1+x 2 )(2y-1 2 )f_2(x,y) = (1+x^2)(2y-1^2).", "Upper left:truncated Fourier series with 512 2 512^2 Fourier coefficients.", "Middleleft: 8 times zoomed-in version of the upper figure.", "Lower left:error committed by the truncated Fourier series.", "Upper right:generalized sampling with Daubechies-3 wavelets computed from the same 512 2 512^2 Fouriercoefficients.", "Middle right: 8 times zoomed-in version of the upperfigure.", "Lower right: error committed by generalized sampling.In this example we will demonstrate the efficiency of generalized sampling given the established linearity of the stable sampling rate.", "In particular, suppose that $f$ is a function we want to recover from its Fourier information.", "It is smooth, however, not periodic – a problem that occurs for example in electron microscopy and also in MRI.", "This causes the classical Fourier reconstruction to converge slowly, yet a smooth boundary wavelet basis will give much faster convergence (see [22]).", "As discussed, the issue is that we are given Fourier samples, not wavelet coefficients.", "However, this is not a problem in view of the linearity of the stable sampling rate.", "In particular, if $f \\in W^s(0,1)$ , where $\\mathrm {W}^s(0,1)$ denotes the usual Sobolev space, and $P_{\\mathcal {R}_N}$ denotes the projection onto the space $\\mathcal {R}_{N}$ of the first $N$ boundary wavelets (see (REF )), then $\\Vert f - P_{\\mathcal {R}_N} f \\Vert = \\mathcal {O}(N^{-s}),\\quad N \\rightarrow \\infty ,$ provided that the wavelet has sufficiently many vanishing moments.", "Now, if $G_{N,M}(f) \\in \\mathcal {R}_{N}$ is the generalized sampling solution from Definition REF given $M$ Fourier coefficients, and $M$ is chosen according to the stable sampling rate then $\\Vert f - G_{N,M}(f)\\Vert = \\mathcal {O}(N^{-s}) = \\mathcal {O}(M^{-s}),\\quad N \\rightarrow \\infty .$ Hence, we obtain the same convergence rate up to a constant, by simply postprocessing the given samples.", "To illustrate this advantage we will consider the following two functions: $f_1(x,y)=\\cos ^2(x)\\exp (-y), \\qquad f_2(x,y) = (1+x^2)(2y-1^2).$ In Figure REF we have shown the results for $f_1$ and compared the classical Fourier reconstruction with the generalized sampling reconstruction.", "Both examples use exactly the same samples, however, note the pleasant absence of the Gibb's ringing in the generalized sampling case.", "The same experiment is carried out for $f_2$ in Figure REF , however, here we have displayed the reconstructions in $3D$ in order to visualize the error.", "Proof of Theorem REF The proof of Theorem REF is somewhat technical, wherefore we divide the proof into several steps.", "First, in Subsection REF , the overall structure of the proof is presented, and the respective details can then be found in Subsection REF .", "Structure of the Proof Let $\\varepsilon \\in (0,1/(T_1+T_2)]$ and $\\theta >1$ .", "Then we have to prove that $\\inf \\limits _{\\begin{array}{c} \\varphi \\in \\mathcal {R}_{N_J} \\\\ \\Vert \\varphi \\Vert =1\\end{array}} \\Vert P_{\\mathcal {S}_M^{(\\varepsilon )}} \\varphi \\Vert \\ge \\frac{1}{\\theta },$ for $M=(M_1,M_2) = \\left(\\frac{ \\lambda _1^{(J)} + \\lambda _3^{(J)}}{\\varepsilon } S^{(\\theta )}, \\frac{ \\lambda _2^{(J)} + \\lambda _4^{(J)}}{\\varepsilon } S^{(\\theta )}\\right),$ and some $S^{(\\theta )}$ independent of $J$ .", "To this end, let $\\varphi \\in \\mathcal {R}_N$ be such that $\\Vert \\varphi \\Vert =1$ .", "Since the sampling functions form an orthonormal basis of $\\mathcal {S}_M^{(\\varepsilon )}$ , we obtain $\\Vert P_{\\mathcal {S}_M^{(\\varepsilon )}} \\varphi \\Vert ^2= \\sum \\limits _{l_1 = -M_1}^{M_1} \\sum \\limits _{l_2 = -M_2}^{M_2}|\\langle \\varphi , s_l^{(\\varepsilon )} \\rangle |^2, \\quad l = (l_1,l_2).$ By Lemma REF , which relies mainly on the underlying MRA structure, we can write $\\varphi $ in terms of scaling functions at highest scale, i.e., $\\varphi = \\sum \\limits _{l_1 = L_3}^{L_1} \\sum \\limits _{l_2 = L_4}^{L_2} \\alpha _{l_1,l_2}\\phi _{J,(l_1,l_2)},$ for some $L_1,L_2,L_3,L_4 \\in \\mathbb {Z}$ .", "Lemma REF , which is proven by direct computations, then shows that $\\langle \\varphi , s_l^{(\\varepsilon )} \\rangle = \\varepsilon |\\det A|^{-J/2} \\Phi (\\varepsilon (A^{-J})^Tl) \\widehat{\\phi }(\\varepsilon (A^{-J})^Tl),$ where $\\Phi $ is a trigonometric polynomial of the form $\\Phi (z) = \\sum \\limits _{m_1 = L_3}^{L_1} \\sum \\limits _{m_2 = L_4}^{L_2} \\alpha _{m_1,m_2} e^{-2\\pi i \\langle z, m \\rangle }, \\quad z \\in \\mathbb {R}^2, m = (m_1,m_2).$ Using (REF ), we conclude that $\\Vert P_{\\mathcal {S}_M^{(\\varepsilon )}} \\varphi \\Vert ^2= \\sum \\limits _{l_1 = -M_1}^{M_1} \\sum \\limits _{l_2 = -M_2}^{M_2} \\varepsilon ^2 |\\det A|^{-J}|\\Phi (\\varepsilon (A^{-J})^Tl)|^2 |\\widehat{\\phi }(\\varepsilon (A^{-J})^Tl)|^2, \\quad l = (l_1, l_2).$ By Theorem REF , there exist some $\\varepsilon _0 > 0$ and $S^{(\\theta )} = \\left(S^{(\\theta )}_1,S^{(\\theta )}_2\\right)\\in \\mathbb {N}\\times \\mathbb {N}$ that does not depend on $J$ such that $\\sum \\limits _{l_1 = -M_1}^{M_1} \\sum \\limits _{l_2 = -M_2}^{M_2} \\varepsilon _0 ^2 | \\det A^{-J}| | \\Phi (\\varepsilon _0 (A^{-J})^T l) |^2| \\widehat{\\phi }(\\varepsilon _0(A^{-J})^Tl)|^2 \\ge \\frac{\\Vert \\Phi \\Vert ^2}{\\theta ^2}$ for $M= \\frac{1}{\\varepsilon _0} (A^J)^T \\widetilde{S}^{(\\theta )}=\\begin{pmatrix} \\frac{ \\lambda _1^{(J)} + \\lambda _3^{(J)}}{\\varepsilon _0} S^{(\\theta )}_1 \\\\\\frac{ \\lambda _2^{(J)} + \\lambda _4^{(J)}}{\\varepsilon _0} S^{(\\theta )}_2 \\end{pmatrix}.$ Since $\\Vert \\Phi \\Vert ^2 =\\sum \\limits _{m_1 = L_3}^{L_1} \\sum \\limits _{m_2 = -L_4}^{L_2} |\\beta _{m_1,m_2}|^2 =\\Vert \\varphi \\Vert ^2 = 1,$ we obtain $\\Vert P_{\\mathcal {S}_M^{(\\varepsilon _0)}} \\varphi \\Vert ^2 \\ge \\frac{1}{\\theta ^2}.$ Finally, Lemma REF implies that a change of $\\varepsilon $ only changes the constant, showing that (REF ) is true for any $\\varepsilon \\in (0,1/(T_1+T_2)]$ , thereby proving Theorem REF .", "Auxiliary results In this section we will prove the results mentioned in Subsection REF , which are required for completing the proof of Theorem REF .", "The following result is an extension of a result from [6] to the two dimensional setting.", "Lemma 6.1 Let $J \\in \\mathbb {N}$ and $\\varphi \\in V_0^{(a)} \\oplus W_0^{(a)} \\oplus \\ldots \\oplus W_{J-1}^{(a)}$ .", "Then there exist $L_1,L_2,L_3,L_4 \\in \\mathbb {Z}$ dependent of $J$ such that $\\varphi = \\sum \\limits _{l_1 = L_3}^{L_1} \\sum \\limits _{l_2 = L_4}^{L_2} \\alpha _{l_1,l_2}\\phi _{J,(l_1,l_2)}.$ Let $\\varphi \\in V_0^{(a)} \\oplus W_0^{(a)} \\oplus \\ldots \\oplus W_{J-1}^{(a)}$ .", "Then $\\varphi $ has an expansion of the following form $\\varphi = \\sum _{m_1 = - a +1}^{m_1 = a-1} \\sum _{m_2 = - a +1}^{m_2 = a-1} \\alpha _{m_1,m_2} \\phi _{0,(m_1,m_2)} +\\sum _{p=1}^{|\\det A| -1} \\sum _{j=0}^{J-1} \\sum _{m_1 = -a +1}^{a(\\lambda _1^{j}+ \\lambda _{2}^{j})-1}\\sum \\limits _{m_2= -a +1}^{a(\\lambda _3^{j}+ \\lambda _{4}^{j})-1} \\beta _{j,(m_1,m_2)}^p \\psi _{j,(m_1,m_2)}^p.$ Since $V_0^{(a)} \\subset V_0$ and $W_j^{(a)} \\subset W_{j}$ for $j \\in \\mathbb {N}$ and the sequence $(V_j)_{j\\in \\mathbb {Z}}$ forms an MRA, it follows that $V_0^{(a)} \\oplus W_0^{(a)} \\oplus \\ldots \\oplus W_{J-1}^{(a)} \\subset V_J.$ Since an orthonormal basis for $V_J$ is given by the functions $\\lbrace \\phi _{J,m} \\, : \\, m \\in \\mathbb {Z}^2\\rbrace $ , for each $|m_i| < a, i =1,2$ , we have $\\phi _{0,(m_1,m_2)} = \\sum _{l \\in \\mathbb {Z}^2} \\langle \\phi _{0,(m_1,m_2)}, \\phi _{J,l} \\rangle \\phi _{J,l}$ Moreover, since $\\phi _{0,(m_1,m_2)}$ and $\\phi _{J,l}$ are compactly supported, we obtain $\\langle \\phi _{0,(m_1,m_2)}, \\phi _{J,l} \\rangle \\ne 0,$ if $-a+ (-a+1)( \\lambda _1^{(J)}+ \\lambda _2^{(J)} ) \\le l_1 \\le (2a-1)(\\lambda _1^{(J)} + \\lambda _2^{(J)})$ and $-a+ (-a+1)( \\lambda _3^{(J)}+ \\lambda _4^{(J)} ) \\le l_2 \\le (2a-1)(\\lambda _3^{(J)} + \\lambda _4^{(J)}).$ This follows by a straightforward computation from the support conditions of $\\phi _{0,(m_1,m_2)}$ and $\\phi _{J,l}$ together with $|m_i| < a, i =1,2$ .", "Similarly, we have $\\langle \\psi _{j,(m_1,m_2)}^p, \\phi _{J,(l_1,l_2)} \\rangle \\ne 0$ if Case I: $\\det A^j \\ge 0$ $-a& - \\lambda _1^{(J)}\\frac{(-a+1)\\lambda _4^{(j)}+\\lambda _2^{(j)}(a(\\lambda _3^{(j)}+\\lambda _4^{(j)})-1)}{\\det A^j} + \\lambda _2^{(J)} \\frac{(-a+1)\\lambda _1^{(j)}+(a(\\lambda _1^{(j)}+\\lambda _2^{(j)})-1)\\lambda _3^{(j)}}{\\det A^j} \\\\& < l_1 < \\lambda _1^{(J)} \\frac{a(\\lambda _4^{(j)}-\\lambda _2^{(j)})+(a(\\lambda _1^{(j)}+\\lambda _2^{(j)})-1)\\lambda _4^{(j)}+(a-1)\\lambda _2^{(j)}}{\\det A^j}+\\\\&\\qquad +\\lambda _2^{(J)} \\frac{a(\\lambda _1^{(j)}-\\lambda _3^{(j)})+(a(\\lambda _3^{(j)}+\\lambda _4^{(j)})-1)\\lambda _1^{(j)}+(a-1)\\lambda _3^{(j)}}{\\det A^j}$ and $-a& - \\lambda _3^{(J)}\\frac{(-a+1)\\lambda _4^{(j)}+\\lambda _2^{(j)}(a(\\lambda _3^{(j)}+\\lambda _4^{(j)})-1)}{\\det A^j} + \\lambda _4^{(J)} \\frac{(-a+1)\\lambda _1^{(j)}+(a(\\lambda _1^{(j)}+\\lambda _2^{(j)})-1)\\lambda _3^{(j)}}{\\det A^j} \\\\& < l_2 < \\lambda _3^{(J)} \\frac{a(\\lambda _4^{(j)}-\\lambda _2^{(j)})+(a(\\lambda _1^{(j)}+\\lambda _2^{(j)})-1)\\lambda _4^{(j)}+(a-1)\\lambda _2^{(j)}}{\\det A^j}+\\\\&\\qquad +\\lambda _4^{(J)} \\frac{a(\\lambda _1^{(j)}-\\lambda _3^{(j)})+(a(\\lambda _3^{(j)}+\\lambda _4^{(j)})-1)\\lambda _1^{(j)}+(a-1)\\lambda _3^{(j)}}{\\det A^j}.$ Case II: $\\det A^j < 0$ $-a& - \\lambda _1^{(J)}\\frac{(-a+1)\\lambda _4^{(j)}+\\lambda _2^{(j)}(a(\\lambda _3^{(j)}+\\lambda _4^{(j)})-1)}{\\det A^j} + \\lambda _2^{(J)} \\frac{(-a+1)\\lambda _1^{(j)}+(a(\\lambda _1^{(j)}+\\lambda _2^{(j)})-1)\\lambda _3^{(j)}}{\\det A^j} \\\\& > l_1 > \\lambda _1^{(J)} \\frac{a(\\lambda _4^{(j)}-\\lambda _2^{(j)})+(a(\\lambda _1^{(j)}+\\lambda _2^{(j)})-1)\\lambda _4^{(j)}+(a-1)\\lambda _2^{(j)}}{\\det A^j}+\\\\&\\qquad +\\lambda _2^{(J)} \\frac{a(\\lambda _1^{(j)}-\\lambda _3^{(j)})+(a(\\lambda _3^{(j)}+\\lambda _4^{(j)})-1)\\lambda _1^{(j)}+(a-1)\\lambda _3^{(j)}}{\\det A^j}$ and $-a& - \\lambda _3^{(J)}\\frac{(-a+1)\\lambda _4^{(j)}+\\lambda _2^{(j)}(a(\\lambda _3^{(j)}+\\lambda _4^{(j)})-1)}{\\det A^j} + \\lambda _4^{(J)} \\frac{(-a+1)\\lambda _1^{(j)}+(a(\\lambda _1^{(j)}+\\lambda _2^{(j)})-1)\\lambda _3^{(j)}}{\\det A^j} \\\\& > l_2 > \\lambda _3^{(J)} \\frac{a(\\lambda _4^{(j)}-\\lambda _2^{(j)})+(a(\\lambda _1^{(j)}+\\lambda _2^{(j)})-1)\\lambda _4^{(j)}+(a-1)\\lambda _2^{(j)}}{\\det A^j}+\\\\&\\qquad +\\lambda _4^{(J)} \\frac{a(\\lambda _1^{(j)}-\\lambda _3^{(j)})+(a(\\lambda _3^{(j)}+\\lambda _4^{(j)})-1)\\lambda _1^{(j)}+(a-1)\\lambda _3^{(j)}}{\\det A^j}.$ Minimizing the lower bounds with respect to $j \\in \\lbrace 0, \\ldots , J-1\\rbrace $ and maximizing the upper bounds with respect to $j$ , respectively, yields the claim.", "The following lemma is well known (see [22]).", "Lemma 6.2 Let $f \\in L^2(\\mathbb {R}^2)$ .", "Then $\\lbrace f( \\cdot - m) \\, : \\, m \\in \\mathbb {Z}^2\\rbrace $ is an orthonormal system if and only if $\\sum \\limits _{m \\in \\mathbb {Z}^2} | \\widehat{f}(\\xi + m )|^2 = 1 \\quad \\text{for almost every } \\xi \\in \\mathbb {R}^2.$ Finally, one more technical lemma is needed.", "Lemma 6.3 Let $A$ be a scaling matrix, $J \\in \\mathbb {Z}$ , and $m=(m_1,m_2) \\in \\mathbb {Z}^2$ .", "Further, let $\\varphi \\in \\overline{\\operatorname{span}} \\lbrace \\phi _{J,m} \\, : \\, m \\in \\mathbb {Z}^2\\rbrace $ be compactly supported in $[-T_1,T_2]^2$ , and let $L_1,L_2,L_3,L_4 \\in \\mathbb {Z}$ be such that $\\varphi = \\sum \\limits _{m_1 = L_3}^{L_1} \\sum \\limits _{m_2 = L_4}^{L_2} \\alpha _{m_1,m_2} \\phi _{J,(m_1,m_2)}, \\quad \\alpha _m \\in $ Then, for all $l \\in \\mathbb {Z}^2$ , $\\langle \\varphi , s_l^{(\\varepsilon )} \\rangle = \\varepsilon |\\det A|^{-J/2} \\Phi (\\varepsilon (A^{-J})^Tl) \\widehat{\\phi }(\\varepsilon (A^{-J})^Tl),$ where $s_l^{(\\varepsilon )}$ is defined in (REF ) and $\\Phi $ is the trigonometric polynomial given by $\\Phi (z) = \\sum \\limits _{\\begin{array}{c} L_3 \\le m_1 \\le L_1, \\\\ L_4 \\le m_2 \\le L_2,\\end{array}} \\alpha _m e^{-2\\pi i \\langle z, m \\rangle }, \\quad z \\in \\mathbb {R}^2, m = (m_1,m_2).$ Since $\\varphi $ is supported in $[-T_1,T_2]^2$ , we obtain $\\langle \\varphi , s_l^{(\\varepsilon )} \\rangle &= \\varepsilon \\int \\limits _{\\mathbb {R}^2} \\varphi (x) e^{-2 \\pi i \\varepsilon \\langle l, x \\rangle }\\cdot \\chi _{[-T_1,T_2]^2} \\, dx \\\\&= \\varepsilon \\widehat{\\varphi }( \\varepsilon l) \\\\&= \\varepsilon \\left(\\sum \\limits _{m_1 = L_1}^{L_2} \\sum \\limits _{m_2 = L_3}^{L_4} \\alpha _{m_1,m_2} \\phi _{J,(m_1,m_2)}\\right)^{\\wedge }(\\varepsilon l) \\\\&= \\varepsilon \\sum \\limits _{m_1 = L_1}^{L_2} \\sum \\limits _{m_2 = L_3}^{L_4} \\alpha _{m_1,m_2} \\left( \\phi _{J,(m_1,m_2)}\\right)^{\\wedge }(\\varepsilon l)\\\\&= \\varepsilon |\\det A|^{-J/2} \\Phi (\\varepsilon (A^{-J})^T l) \\widehat{\\phi }(\\varepsilon (A^{-J})^Tl).$ This proves the claim.", "Theorem REF The proof of Theorem REF requires a particular estimate (Proposition REF ) for the norm of trigonometric polynomials depending on their evaluations on a particular grid whose mesh norm and associated Voronoi regions come also into play.", "Mesh Norm We start with the definition of a mesh norm for the situation we are faced with.", "A mesh norm can be interpreted as the largest distance between neighboring nodes.", "Definition 6.4 Let $\\Lambda \\subset \\mathbb {Z}^2$ be an integer grid of the form $\\Lambda := \\left\\lbrace (l_1,l_2) \\in \\mathbb {Z}^2 \\, : \\, -M_i \\le l_i \\le M_i, i=1,2 \\right\\rbrace , \\quad M_1,M_2 \\in \\mathbb {N},$ and let $A$ be a scaling matrix.", "Set $\\Omega := \\overline{\\Lambda }^A := A ([-M_1,M_1] \\times [-M_2,M_2]) \\subset \\mathbb {R}^2,$ and define a metric $\\rho $ on $\\Omega $ by $\\rho : \\Omega \\times \\Omega \\longrightarrow \\mathbb {R}^+, \\quad (x,y) \\mapsto \\min \\limits _{k \\in A(\\mathbb {Z}^2)} \\Vert x - y + k\\Vert _\\infty .$ The mesh norm of $\\lbrace x_l \\in \\Omega \\, : \\, l \\in \\Lambda \\rbrace $ is then defined as $\\delta := \\max \\limits _{x \\in \\Omega } \\min \\limits _{l \\in \\Lambda } \\rho (x_l,x),$ where $x_l := A \\cdot l, l \\in \\Lambda $ denote the nodes in $\\Omega $ .", "Before we can continue, we require some notions and results on Voronoi regions and trigonometric polynomials.", "Our first result shows that if the distance between the nodes $\\lbrace x_l\\rbrace _l$ converges to zero, the mesh norm of the entire grid $\\Omega $ converges to zero.", "Lemma 6.5 Let $\\varepsilon >0$ and $\\Lambda \\subset \\mathbb {Z}^2$ be defined by $\\Lambda := \\left\\lbrace (l_1,l_2) \\in \\mathbb {Z}^2 \\, : \\, -M_i \\le l_i \\le M_i, i=1,2 \\right\\rbrace ,$ where $M_1,M_2 \\in \\mathbb {N}$ .", "Furthermore, suppose $A$ is a scaling matrix, and let $\\Omega ^{(\\varepsilon )} := \\overline{\\Lambda }^{\\varepsilon A}$ .", "If $\\varepsilon \\longrightarrow 0$ , then $\\delta ^{(\\varepsilon )} \\longrightarrow 0$ , where $\\delta ^{(\\varepsilon )}$ denotes the mesh norm of $\\lbrace \\varepsilon A l \\in \\Omega ^{(\\varepsilon )} \\, : \\, l \\in \\Lambda \\rbrace $ .", "If $\\varepsilon \\rightarrow 0$ , then $\\varepsilon A (\\Lambda ) = \\lbrace \\varepsilon A l \\, : \\, l \\in \\Lambda \\rbrace \\longrightarrow \\lbrace (0,0) \\rbrace $ with respect to the standard Euclidean distance.", "Furthermore, for $x_l := \\varepsilon A l$ , we have $\\lim \\limits _{\\varepsilon \\rightarrow 0} \\delta ^{(\\varepsilon )}= \\lim \\limits _{\\varepsilon \\rightarrow 0} \\max \\limits _{x \\in \\Omega ^{(\\varepsilon )}} \\min \\limits _{l \\in \\Lambda } \\min \\limits _{k \\in \\varepsilon A(\\mathbb {Z}^2)} \\Vert x - x_l + k\\Vert _\\infty \\le \\lim \\limits _{\\varepsilon \\rightarrow 0} \\max \\limits _{x \\in \\Omega ^{(\\varepsilon )}} \\min \\limits _{l \\in \\Lambda } \\Vert x - x_l \\Vert _\\infty .$ Since $x, x_l \\in \\Omega ^{(\\varepsilon )}$ for all $ l \\in \\Lambda $ , we obtain $\\lim \\limits _{\\varepsilon \\rightarrow 0} 2 \\max \\limits _{x \\in \\Omega ^{(\\varepsilon )}} \\min \\limits _{l \\in \\Lambda } \\Vert x - x_l \\Vert _\\infty \\le \\lim \\limits _{\\varepsilon \\rightarrow 0} \\max \\limits _{x,y \\in \\Omega ^{(\\varepsilon )}} \\Vert x - y \\Vert _\\infty .$ Inserting this estimate into (REF ) yields $\\lim \\limits _{\\varepsilon \\rightarrow 0} \\delta ^{(\\varepsilon )}\\le \\lim \\limits _{\\varepsilon \\rightarrow 0} \\max \\limits _{x,y \\in \\Omega ^{(\\varepsilon )}} \\Vert x - y \\Vert _\\infty \\le \\lim \\limits _{\\varepsilon \\rightarrow 0} \\operatorname{diam}\\Omega ^{(\\varepsilon )}= \\lim \\limits _{\\varepsilon \\rightarrow 0} \\operatorname{diam}\\overline{\\Lambda }^{\\varepsilon A}= \\lim \\limits _{\\varepsilon \\rightarrow 0} \\operatorname{diam}\\varepsilon \\overline{\\Lambda }^A= \\lim \\limits _{\\varepsilon \\rightarrow 0} \\varepsilon \\underbrace{ \\operatorname{diam}\\overline{\\Lambda }^A}_{<\\infty } =0,$ where $\\operatorname{diam}(F)$ denotes the diameter of a set $F\\subset \\mathbb {R}^d$ , i.e., $\\operatorname{diam}(F) = \\sup \\limits _{x,y \\in F} d_2(x,y),$ and $d_2$ denotes the Euclidean metric on $\\mathbb {R}^d$ .", "Since the mesh norm is always non-negative, the lemma is proven.", "Voronoi Regions The next result studies the volume of the Voronoi regions associated to the previously considered grid $\\Lambda $ with respect to the metric $\\rho $ defined in Definition REF .", "We start by formally defining the notion of Voronoi region in our setting.", "Definition 6.6 Let $\\Omega \\subset \\mathbb {R}^2$ , and let $(x_l)_{l \\in \\Lambda }$ be a sequence in $\\mathbb {R}^2$ .", "Then we refer to the sets $(V_l)_{l \\in \\Lambda }$ defined by $V_l := \\lbrace x \\in \\Omega \\, : \\, \\rho (x,x_l) \\le \\rho (x,x_k) \\; \\text{ for all } k \\ne l \\rbrace $ as Voronoi regions with respect to $\\Lambda $ , $\\Omega $ , $\\rho $ , and $x_l$.", "We can now state the previously announced result.", "Lemma 6.7 Let $M_1,M_2 \\in \\mathbb {N}$ and $\\Lambda := \\left\\lbrace (l_1,l_2) \\in \\mathbb {Z}^2 \\, : \\, -M_i \\le l_i \\le M_i, i=1,2 \\right\\rbrace .$ Moreover, let $\\Omega =\\overline{\\Lambda }^{\\operatorname{Id}}$ , where $\\operatorname{Id}$ denotes the $2\\times 2$ -identity matrix, and let $(V_l)_{l \\in \\Lambda }$ be the Voronoi regions with respect to $\\Lambda $ , $\\Omega $ , $\\rho $ , and $l$ .", "Then, for all $l \\in \\Lambda $ , $\\mu (V_l) \\le 1,$ where $\\mu $ denotes the 2D Lebesgue measure.", "Notice that the Voronoi regions $(V_l)_{l \\in \\Lambda }$ are in fact rectangles, since the grid is an integer grid with a constant step-size.", "Hence, for each $l \\in \\Lambda $ , $\\mu (V_l) = a_{l_1,l_2} \\cdot b_{l_1,l_2}, \\quad a_{l_1,l_2},b_{l_1,l_2} \\in \\mathbb {R},$ where $a_{l_1,l_2}$ denotes the width and $b_{l_1,l_2}$ the height of the rectangle $V_l$ .", "Towards a contradiction, assume that $V_l$ does contain two different nodes $x_k$ and $x_l$ with $k \\ne l$ .", "This implies $0\\ne \\rho (x_k,x_l) \\le \\rho (x_l,x_l) =0,$ which is a contradiction.", "Thus, we can conclude that $a_{l_1,l_2} \\le \\rho (x_{l_1+1,l_2},x_{l_1,l_2})\\quad \\mbox{and} \\quad b_l \\le \\rho (x_{l_1,l_2},x_{l_1,l_2+1}),$ which, by (REF ), proves the claim.", "We next obtain a slight generalization of the previous result.", "Lemma 6.8 Let $A$ be a (linear) bijective transformation acting on $\\mathbb {R}^2$ with matrix representation $A= \\begin{pmatrix} \\lambda _1& \\lambda _2 \\\\ \\lambda _3 & \\lambda _4 \\end{pmatrix},$ and let $\\Lambda $ and $(V_l)_{l \\in \\Lambda }$ be defined as in Lemma REF .", "Then $\\mu (A(V_l)) \\le |\\det A |.$ The result follows from Lemma REF by an integration by substitution.", "Trigonometric Polynomials The next theorem is an adapted version of a result presented in [24] which again is a reformulation of a result proven by Gröchenig in [17].", "Proposition 6.9 Let $J, L_1,L_2, L_3, L_4 \\in \\mathbb {Z}$ such that $L_1 \\ge L_3$ and $L_2 \\ge L_4$ , and let $\\Phi $ a trigonometric polynomial of the form $\\Phi (z) = \\sum \\limits _{m_1 = L_3}^{L_1} \\sum \\limits _{m_2 = L_4}^{L_2} \\alpha _{m_1,m_2} e^{-2\\pi i \\langle z, m \\rangle }, \\quad z \\in \\mathbb {R}^2, m = (m_1,m_2).$ Further, let the grid $\\Lambda $ be defined as in Lemma REF , let $A$ be a scaling matrix, and let $\\Omega ^{(\\varepsilon )} := \\overline{\\Lambda }^{\\varepsilon A^{-J}}$ for $\\varepsilon >0$ .", "Set $x_l :=\\varepsilon (A^{-J})^Tl$ , $l \\in \\Lambda $ .", "If the mesh norm $\\delta $ of $\\lbrace x_l \\in \\Omega ^{(\\varepsilon )} \\, : \\, l \\in \\Lambda \\rbrace $ obeys $\\delta < \\frac{\\log \\left(\\frac{1}{\\sqrt{\\mu ( \\Omega ^{(\\varepsilon )})}}+1\\right)}{2 \\pi \\max \\lbrace |L_1|,|L_2|,|L_3|,|L_4|\\rbrace },$ where $\\mu $ is the 2D Lebesgue measure, then there exists a positive constant $C(\\delta ,L_1,L_2,L_3,L_4)$ such that $C(\\delta ,L_1,L_2,L_3,L_4) \\Vert \\Phi \\Vert \\le \\left( \\sum \\limits _{l \\in \\Lambda } \\varepsilon ^2 |\\det A^{-J} | | \\Phi (\\varepsilon (A^{-J})^Tl|^2 \\right)^{\\frac{1}{2}},$ where $C(\\delta , L_1,L_2,L_3,L_4) = \\left( 1 - \\left(e^{2\\pi \\delta \\max \\lbrace |L_1|,|L_2|,|L_3|,|L_4|\\rbrace } -1\\right) \\sqrt{ \\mu (\\Omega ^{(\\varepsilon )})}\\right)$ and $\\Vert f \\Vert := \\left(\\int _{\\Omega ^{(\\varepsilon )}} |f(x)|^2 \\, dx\\right)^{1/2}, \\quad f\\in L^2(\\Omega ^{(\\varepsilon )}).$ We first observe that by the hypotheses, the constant $C(\\delta , L_1,L_2,L_3,L_4)$ is indeed positive.", "Second, let $(V_l)_{l \\in \\Lambda }$ be the Voronoi regions with respect to $\\Lambda $ , $\\Omega ^{(\\varepsilon )}$ , and $\\rho $ .", "For $l \\in \\Lambda $ , we define the weights $\\omega _l := \\mu (V_l)$ .", "As in Lemma REF , integration by substitution yields $\\omega _l = \\mu (V_l) \\le \\varepsilon ^2 |\\det A^{-J} |.$ Hence it suffices to prove the existence of a constant $C(\\delta , L_1,L_2,L_3,L_4) >0$ such that $C(\\delta ,L_1,L_2,L_3,L_4) \\Vert \\Phi \\Vert \\le \\left( \\sum \\limits _{l \\in \\Lambda } \\omega _l | \\Phi (\\varepsilon (A^{-J})^Tl|^2 \\right)^{\\frac{1}{2}}, $ For this, we first observe that $\\left( \\sum \\limits _{l \\in \\Lambda } \\omega _l | \\Phi (\\varepsilon (A^{-J})^Tl|^2 \\right)^{\\frac{1}{2}}= \\left( \\int \\limits _{\\Omega ^{(\\varepsilon )}} \\sum \\limits _{l \\in \\Lambda } | \\Phi (\\varepsilon (A^{-J})^Tl|^2 \\right)^{\\frac{1}{2}}= \\left\\Vert \\sum \\limits _{l \\in \\Lambda } \\Phi (x_l)\\chi _{V_l} \\right\\Vert .$ By the (inverse) triangle inequality, $\\left\\Vert \\sum \\limits _{l \\in \\Lambda } \\Phi (x_l)\\chi _{V_l} \\right\\Vert \\ge \\Vert \\Phi \\Vert - \\left\\Vert \\Phi - \\sum \\limits _{l \\in \\Lambda }\\Phi (x_l)\\chi _{V_l} \\right\\Vert .", "$ Hence, we require an upper bound for $\\Vert \\Phi - \\sum _{l \\in \\Lambda } \\Phi (x_l)\\chi _{V_l} \\Vert $ .", "By Taylor expansion and Bernstein's inequality, $\\left\\Vert \\Phi - \\sum \\limits _{l \\in \\Lambda } \\Phi (x_l) \\chi _{V_l} \\right\\Vert ^2& = \\int \\limits _{\\Omega ^{(\\varepsilon )}} \\left| \\Phi (x) - \\sum \\limits _{l \\in \\Lambda } \\Phi (x_l)\\chi _{V_l}(x) \\right|^2 \\, dx \\\\& = \\int \\limits _{\\Omega ^{(\\varepsilon )}} \\left| \\sum \\limits _{l \\in \\Lambda } \\Phi (x) \\chi _{V_l}(x) - \\Phi (x_l)\\chi _{V_l}(x) \\right|^2 \\, dx \\\\& \\le \\sum \\limits _{l \\in \\Lambda } \\int \\limits _{V_l} \\left| \\Phi (x) - \\Phi (x_l) \\right|^2 \\, dx \\\\& \\le \\sum \\limits _{l \\in \\Lambda } \\int \\limits _{V_l} \\left| \\sum _{\\alpha \\in \\mathbb {N}^2\\setminus \\lbrace (0,0)\\rbrace }\\frac{1}{\\alpha !}", "\\Vert x - x_l\\Vert ^\\alpha | D^\\alpha \\Phi (x)| \\right|^2 \\, dx \\\\& \\le \\sum \\limits _{l \\in \\Lambda } \\int \\limits _{V_l} \\left| \\sum _{\\alpha \\in \\mathbb {N}^2\\setminus \\lbrace (0,0)\\rbrace }\\frac{1}{\\alpha !}", "\\delta ^{|\\alpha |} ( \\max \\lbrace |L_1|,|L_2|,|L_3|,|L_4|\\rbrace \\pi )^{|\\alpha |} \\Vert \\Phi \\Vert \\right|^2 \\, dx \\\\& \\le \\left| \\sum _{\\alpha \\in \\mathbb {N}^2\\setminus \\lbrace (0,0)\\rbrace }\\frac{1}{\\alpha !}", "\\delta ^{|\\alpha |} ( \\max \\lbrace |L_1|,|L_2|,|L_3|,|L_4|\\rbrace \\pi )^{|\\alpha |} \\Vert \\Phi \\Vert \\right|^2\\cdot \\sum \\limits _{l \\in \\Lambda } \\int \\limits _{V_l} 1\\, dx$ Since the Voronoi regions build a partition of $\\Omega ^{(\\varepsilon )}$ , $\\sum \\limits _{l \\in \\Lambda } \\mu (V_l) = \\mu (\\Omega ^{(\\varepsilon )}),$ and we can continue by $\\left\\Vert \\Phi - \\sum \\limits _{l \\in \\Lambda } \\Phi (x_l) \\chi _{V_l} \\right\\Vert ^2 \\le \\left| \\left(e^{2\\pi \\delta \\max \\lbrace |L_1|,|L_2|,|L_3|,|L_4|\\rbrace } - 1\\right) \\right|^2\\Vert \\Phi \\Vert ^2 \\mu (\\Omega ^{(\\varepsilon )}).", "$ Using (REF ) and (REF ), we obtain $\\left\\Vert \\sum \\limits _{l \\in \\Lambda } \\Phi (x_l)\\chi _{V_l} \\right\\Vert &\\ge \\Vert \\Phi \\Vert - \\left\\Vert \\Phi - \\sum \\limits _{l \\in \\Lambda }\\Phi (x_l)\\chi _{V_l} \\right\\Vert \\\\&\\ge \\Vert \\Phi \\Vert - \\left| \\Vert \\Phi \\Vert \\left(e^{2\\pi \\delta \\max \\lbrace |L_1|,|L_2|,|L_3|,|L_4|\\rbrace } - 1\\right) \\sqrt{ \\mu (\\Omega ^{(\\varepsilon )})}\\right|\\\\&= \\Vert \\Phi \\Vert \\left( 1 - \\left(e^{2\\pi \\delta \\max \\lbrace |L_1|,|L_2|,|L_3|,|L_4|\\rbrace } -1\\right) \\sqrt{ \\mu (\\Omega ^{(\\varepsilon )})}\\right).$ Combining this estimate with (REF ) proves (REF ).", "Finally, we can state and prove Theorem REF , which is one main ingredient for the proof of Theorem REF in Subsection REF .", "Theorem 6.10 Let $L_1,L_2, L_3, L_4 \\in \\mathbb {Z}$ such that $L_1 \\ge L_3$ and $L_2 \\ge L_4$ and $\\alpha _{m_1,m_2} \\in , and let $$ be the trigonometric polynomial$$\\Phi ( \\cdot , \\cdot ) = \\sum _{m_1= L_3}^{L_1} \\sum _{m_2= L_4}^{L_2} \\alpha _{m_1,m_2} e^{-2 \\pi i (\\cdot )m_1}e^{-2 \\pi i (\\cdot )m_2}.$$Further, let $ A= 1 2 3 4 $ be a scaling matrix, $ J N$ a maximalscale, and $ Z2$ the grid defined by$$\\Lambda := \\left\\lbrace (l_1,l_2) \\in \\mathbb {Z}^2 \\, : \\, -M_i \\le l_i \\le M_i, i=1,2 \\right\\rbrace ,$$where $ M1,M2 N$.", "If there exists $ 1/(T1 + T2)$ independent of $ J$ such that $$ fulfills{\\begin{@align*}{1}{-1}\\delta < \\frac{\\log \\left(\\frac{1}{\\sqrt{\\mu ( \\Omega ^{(\\varepsilon )})}}+1\\right)}{ 4 \\pi \\max \\lbrace |L_1|,|L_2|,|L_3|,|L_4|\\rbrace },\\end{@align*}}then there exists $ S() =(S()1,S()2) NN$,independent of $ J$, such that, for all $ >1$, we have$$\\sum \\limits _{l \\in \\Lambda } \\varepsilon ^2 | \\det A^{-J}| | \\Phi (\\varepsilon (A^{-J})^T l) |^2| \\widehat{\\phi }(\\varepsilon (A^{-J})^Tl)|^2 \\ge \\frac{\\Vert \\Phi \\Vert ^2}{2\\theta ^2},$$for $ M= (M1,M2) = 1 (AJ)T S()$ and scaling function $$.$ Since $0<\\varepsilon <1$ , there exists an $m\\in \\mathbb {N}$ , such that $\\frac{1}{m+1} \\le \\varepsilon < \\frac{1}{m}.$ Set $\\varepsilon = \\frac{1}{m+1}$ , and note that (REF ) still holds, since the logarithm is monotonically increasing.", "Next, for some $S \\in \\mathbb {N}$ to be determined later, let $\\begin{pmatrix} M_1 \\\\ M_2 \\end{pmatrix} := \\frac{1}{\\varepsilon } (A^J)^T\\begin{pmatrix} S \\\\S \\end{pmatrix} = \\frac{1}{\\varepsilon }\\begin{pmatrix} \\lambda ^{(J)}_1 + \\lambda ^{(J)}_3 \\\\ \\lambda ^{(J)}_2 + \\lambda ^{(J)}_4 \\end{pmatrix} S,\\quad \\mbox{where as before }A^J=\\begin{pmatrix} \\lambda ^{(J)}_1 & \\lambda ^{(J)}_2 \\\\ \\lambda ^{(J)}_3 & \\lambda ^{(J)}_4 \\end{pmatrix}.$ For $l = (l_1,l_2) \\in \\mathbb {Z}^2$ , we then obtain $ \\nonumber {\\sum \\limits _{l \\in \\Lambda } \\varepsilon ^2 | \\det A^{-J}| | \\Phi (\\varepsilon (A^{-J})^T l) |^2 | \\widehat{\\phi }(\\varepsilon (A^{-J})^Tl)|^2}\\\\ \\nonumber &=&\\sum \\limits _{l_1 = - M_1}^{M_1 -1} \\sum \\limits _{ l_2 = M_2 }^{M_2-1} \\varepsilon ^2 | \\det A^{-J}| | \\Phi (\\varepsilon (A^{-J})^T l) |^2 |\\widehat{\\phi }(\\varepsilon (A^{-J})^Tl)|^2\\\\ \\nonumber &=& \\sum \\limits _{l_1=0 }^{\\frac{\\lambda ^{(J)}_1 + \\lambda ^{(J)}_3}{\\varepsilon } -1} \\sum \\limits _{l_2=0}^{\\frac{\\lambda ^{(J)}_2 + \\lambda ^{(J)}_4}{\\varepsilon } -1}\\sum \\limits _{s=-S} ^{S-1}\\sum \\limits _{t=-S} ^{S-1} \\bigg ( \\varepsilon ^2 | \\det A^{-J}| \\cdot \\left| \\Phi \\left(\\varepsilon (A^{-J})^T \\left(l+\\frac{1}{\\varepsilon } (A^J)^T\\begin{pmatrix} s \\\\ t \\end{pmatrix}\\right) \\right) \\right|^2 \\\\& & \\cdot \\left| \\widehat{\\phi }\\left(\\varepsilon (A^{-J})^T \\left(l+\\frac{1}{\\varepsilon } (A^J)^T\\begin{pmatrix} s \\\\ t \\end{pmatrix}\\right) \\right) \\right|^2\\bigg ), $ Integer periodicity of the trigonometric polynomial implies that, for all $s \\in \\mathbb {Z}$ , $\\Phi \\left(\\varepsilon (A^{-J})^T \\left(l+\\frac{1}{\\varepsilon } (A^J)^T \\begin{pmatrix} s \\\\ t \\end{pmatrix} \\right) \\right)= \\Phi \\left(\\varepsilon (A^{-J})^T l + \\begin{pmatrix} s \\\\ t \\end{pmatrix}\\right)= \\Phi \\left(\\varepsilon (A^{-J})^T l \\right).$ Therefore, by (REF ) $\\sum \\limits _{l \\in \\Lambda } \\; & \\varepsilon ^2 | \\det A^{-J}| | \\Phi (\\varepsilon (A^{-J})^T l) |^2 | \\widehat{\\phi }(\\varepsilon (A^{-J})^Tl)|^2 \\nonumber \\\\&= \\sum \\limits _{l_1=0 }^{\\frac{\\lambda ^{(J)}_1 + \\lambda ^{(J)}_3}{\\varepsilon } -1}\\sum \\limits _{l_2=0}^{\\frac{\\lambda ^{(J)}_2 + \\lambda ^{(J)}_4}{\\varepsilon } -1} \\bigg (\\varepsilon ^2 | \\det A^{-J}| | \\Phi (\\varepsilon (A^{-J})^T l) |^2 \\sum \\limits _{s=-S} ^{S-1} \\sum \\limits _{t=-S} ^{S-1} \\left| \\widehat{\\phi }\\left((A^{-J})^Tl+\\begin{pmatrix} s \\\\ t \\end{pmatrix}\\right)\\right|^2\\bigg ).", "$ Let $\\theta >1$ .", "Then, by Lemma REF , there exists $S^{(\\theta )} \\in \\mathbb {N}$ such that $\\sum \\limits _{s=-S^{(\\theta )}} ^{S^{(\\theta )}-1} \\sum \\limits _{t=-S} ^{S-1} \\left| \\widehat{\\phi }\\left((A^{-J})^Tl+\\begin{pmatrix} s \\\\ t \\end{pmatrix}\\right)\\right|^2 \\ge \\frac{1}{\\theta } \\ .", "$ We now choose $S := S^{(\\theta )}$ .", "Combining (REF ) and (REF ) yields $\\sum \\limits _{l \\in \\Lambda } \\ & \\varepsilon ^2 | \\det A^{-J}| | \\Phi (\\varepsilon (A^{-J})^T l) |^2 | \\widehat{\\phi }(\\varepsilon (A^{-J})^Tl)|^2 \\nonumber \\\\&\\ge \\sum \\limits _{l_1=0 }^{\\frac{\\lambda ^{(J)}_1 + \\lambda ^{(J)}_3}{\\varepsilon } -1}\\sum \\limits _{l_2=0}^{\\frac{\\lambda ^{(J)}_2 + \\lambda ^{(J)}_4}{\\varepsilon } -1} \\varepsilon ^2 | \\det A^{-J}| | \\Phi (\\varepsilon (A^{-J})^T l) |^2\\cdot \\frac{1}{\\theta } \\ .", "$ Next, we apply Proposition REF to (REF ) to obtain $\\sum \\limits _{l \\in \\Lambda } \\varepsilon ^2 | \\det A^{-J}|& | \\Phi (\\varepsilon (A^{-J})^T l) |^2 | \\widehat{\\phi }(\\varepsilon (A^{-J})^Tl)|^2 \\nonumber \\\\&\\ge \\left(\\Vert \\Phi \\Vert \\left( 1 - \\left(e^{2\\pi \\delta \\max \\lbrace |L_1|,|L_2|,|L_3|,|L_4|\\rbrace } -1\\right) \\sqrt{ \\mu (\\Omega ^{(\\varepsilon )})}\\right) \\right)^2\\cdot \\frac{1}{\\theta }$ Since $1+ &\\left( 1- e^{2\\pi \\delta \\max \\lbrace |L_1|,|L_2|,|L_3|,|L_4|\\rbrace }\\right) \\sqrt{\\mu (\\Omega ^{(\\varepsilon )})} \\\\&\\ge 1+ \\left( 1- e^{2\\pi \\max \\lbrace |L_1|,|L_2|,|L_3|,|L_4|\\rbrace \\frac{\\log \\left(\\frac{1}{\\sqrt{\\mu (\\Omega ^{(\\varepsilon )})}}+1\\right)}{4\\pi \\max \\lbrace |L_1|,|L_2|,|L_3|,|L_4|\\rbrace } }\\right) \\sqrt{\\mu (\\Omega ^{(\\varepsilon )})} \\\\&=1+ \\left( 1- \\left(\\sqrt{\\mu (\\Omega ^{(\\varepsilon )})}+1\\right)^{1/2}\\right)\\sqrt{\\mu (\\Omega ^{(\\varepsilon )})} \\\\&=1+ \\left( \\sqrt{\\mu (\\Omega ^{(\\varepsilon )})}- \\left(\\mu (\\Omega ^{(\\varepsilon )})^{3/2}+\\mu (\\Omega ^{(\\varepsilon )})\\right)^{1/2}\\right) \\\\&\\ge 1 -\\mu (\\Omega ^{(\\varepsilon )})^{3/4}\\\\&\\ge 1- \\varepsilon ^{3/4} \\left( \\det A^{-J}\\right)^{3/4}\\\\&\\ge \\frac{1}{2}$ we can conclude $\\sum \\limits _{l \\in \\Lambda } \\varepsilon ^2 | \\det A^{-J}|& | \\Phi (\\varepsilon (A^{-J})^T l) |^2 | \\widehat{\\phi }(\\varepsilon (A^{-J})^Tl)|^2 \\ge \\frac{\\Vert \\Phi \\Vert ^2}{2\\theta }.$ Hence the theorem is proven for $M_1 = \\frac{ \\lambda _1^{(J)} + \\lambda _3^{(J)}}{\\varepsilon } S^{(\\theta )}\\quad \\mbox{and} \\quad M_2 = \\frac{ \\lambda _2^{(J)} + \\lambda _4^{(J)}}{\\varepsilon } S^{(\\theta )}.$ Lemma REF We mention that this result is proved in the same manner as the one dimensional result from [6].", "Lemma 6.11 For $\\gamma >1$ and $\\varepsilon _1, \\varepsilon _2 \\in (0,1/(T_1+T_2)]$ , let $\\theta (\\gamma ),C(\\gamma )>1$ be such that $\\sqrt{\\frac{1}{\\theta (\\gamma )^{2}} - \\frac{16}{\\pi ^4(C(\\gamma ) -1)^2}} - \\sqrt{1-\\frac{1}{\\theta (\\gamma )}} > \\frac{1}{\\gamma }.", "$ If there exists $M=(M_1,M_2) \\in \\mathbb {N}\\times \\mathbb {N}$ such that $\\inf \\limits _{ \\begin{array}{c}\\varphi \\in \\mathcal {R}_N\\\\ \\Vert \\varphi \\Vert =1\\end{array}} \\Vert P_{\\mathcal {S}_M^{(\\varepsilon _1)}} \\varphi \\Vert \\ge \\frac{1}{\\theta (\\gamma )},$ then $\\inf \\limits _{ \\begin{array}{c}\\varphi \\in \\mathcal {R}_N\\\\ \\Vert \\varphi \\Vert =1\\end{array}} \\Vert P_{\\mathcal {S}_K^{(\\varepsilon _2)}} \\varphi \\Vert \\ge \\frac{1}{\\gamma },$ whenever $K = (K_1,K_2) =\\left( \\left\\lceil \\frac{C(\\gamma )M_1\\varepsilon _1}{\\varepsilon _2}\\right\\rceil ,\\left\\lceil \\frac{C(\\gamma )M_2\\varepsilon _1}{\\varepsilon _2}\\right\\rceil \\right)$ .", "First, notice that for any $\\gamma \\in (0,1)$ , there exist $\\theta (\\gamma )$ and $C(\\gamma )$ such that (REF ) is fulfilled.", "Now, let $\\gamma >1$ and $\\varepsilon _2 >0$ .", "Then, by (REF ), $\\nonumber \\inf \\limits _{ \\begin{array}{c}\\varphi \\in \\mathcal {R}_N\\\\ \\Vert \\varphi \\Vert =1\\end{array}} \\Vert P_{\\mathcal {S}_K^{(\\varepsilon _2)}} \\varphi \\Vert & \\ge \\inf \\limits _{ \\begin{array}{c}\\varphi \\in \\mathcal {R}_N\\\\ \\Vert \\varphi \\Vert =1\\end{array}} \\left( \\Vert P_{\\mathcal {S}_K^{(\\varepsilon _2)}} P_{\\mathcal {S}_M^{(\\varepsilon _1)}} \\varphi \\Vert - \\Vert P_{\\mathcal {S}_K^{(\\varepsilon _2)}} P_{\\mathcal {S}_M^{(\\varepsilon _1)}}^{\\perp } \\varphi \\Vert \\right) \\\\ & \\ge \\inf \\limits _{ \\begin{array}{c}\\varphi \\in \\mathcal {R}_N\\\\ \\Vert \\varphi \\Vert =1\\end{array}} \\left( \\Vert P_{\\mathcal {S}_K^{(\\varepsilon _2)}} P_{\\mathcal {S}_M^{(\\varepsilon _1)}} \\varphi \\Vert - \\sqrt{1 - \\frac{1}{\\theta (\\gamma )}} \\right).$ In order to estimate $\\Vert P_{\\mathcal {S}_K^{(\\varepsilon _2)}} P_{\\mathcal {S}_M^{(\\varepsilon _1)}} \\varphi \\Vert $ , we decompose this term by $\\Vert P_{\\mathcal {S}_K^{(\\varepsilon _2)}} P_{\\mathcal {S}_M^{(\\varepsilon _1)}} \\varphi \\Vert ^2= \\Vert P_{\\mathcal {S}_M^{(\\varepsilon _1)}} \\varphi \\Vert ^2 - \\Vert P_{\\mathcal {S}_K^{(\\varepsilon _2)}}^{\\perp } P_{\\mathcal {S}_M^{(\\varepsilon _1)}} \\varphi \\Vert ^2.$ Thus, we require a suitable upper bound for $\\Vert P_{\\mathcal {S}_K^{(\\varepsilon _2)}}^{\\perp } P_{\\mathcal {S}_M^{(\\varepsilon _1)}} \\varphi \\Vert ^2$ .", "For this, let $I_M &= \\lbrace l =(l_1, l_2) \\in \\mathbb {Z}^2 \\, : \\, -M_i/2 \\le l_i \\le M_i/2-1, i =1,2 \\rbrace , \\\\I_K &= \\lbrace j = (j_1, j_2) \\in \\mathbb {Z}^2 \\, : \\, -K_i/2 \\le j_i \\le K_i/2 -1, i =1,2 \\rbrace ,$ and for the complementary sets in $\\mathbb {Z}^2$ we write $I_M^c &= \\lbrace l=(l_1, l_2) \\in \\mathbb {Z}^2 \\, : l \\notin I_M \\rbrace , \\\\I_K^c &= \\lbrace j = (j_1, j_2) \\in \\mathbb {Z}^2 \\, : \\, j \\notin I_K \\rbrace .$ Then, we have $\\nonumber \\Vert P_{\\mathcal {S}_K^{(\\varepsilon _2)}}^{\\perp } P_{\\mathcal {S}_M^{(\\varepsilon _1)}} \\varphi \\Vert ^2&= \\left\\Vert \\sum _{j \\in I_K^c} \\langle P_{\\mathcal {S}_M^{(\\varepsilon _1)}} \\varphi , s_j^{(\\varepsilon _2)} \\rangle s_j^{(\\varepsilon _2)} \\right\\Vert ^2 \\\\ \\nonumber &= \\sum _{j \\in I_K^c} \\left| \\sum _{l \\in I_M} \\langle \\varphi , s_l^{(\\varepsilon _1)} \\rangle \\langle s_l^{(\\varepsilon _1)}, s_j^{(\\varepsilon _2)} \\rangle \\right|^2 \\\\ \\nonumber &\\le \\sum _{j\\in I_K^c} \\left( \\sum _{l \\in I_M} | \\langle \\varphi , s_l^{(\\varepsilon _1)} \\rangle |^2\\sum _{l \\in I_M} |\\langle s_l^{(\\varepsilon _1)}, s_j^{(\\varepsilon _2)} \\rangle |^2 \\right)\\\\ &\\le \\sum _{j \\in I_K^c} \\sum _{l \\in I_M} |\\langle s_l^{(\\varepsilon _1)}, s_j^{(\\varepsilon _2)} \\rangle |^2 .$ Now, for $\\varepsilon = \\max \\lbrace \\varepsilon _1, \\varepsilon _2 \\rbrace $ , we obtain $| \\langle s_l^{(\\varepsilon _1)}, s_j^{(\\varepsilon _2)} \\rangle |&= \\left| \\varepsilon _1 \\cdot \\varepsilon _2 \\cdot \\int \\limits _{\\left[-\\frac{1}{2\\varepsilon }, \\frac{1}{2\\varepsilon } \\right]^2}e^{2 \\pi i \\varepsilon _1 \\langle l, x \\rangle } e^{2 \\pi i \\varepsilon _2 \\langle j, x \\rangle } \\, dx \\right| \\\\&\\le \\left| \\frac{\\varepsilon _1 \\varepsilon _2}{\\pi ^2 (\\varepsilon _1 l_1 - \\varepsilon _2 j_1)( \\varepsilon _1 l_2 - \\varepsilon _2 j_2)} \\right|.$ Therefore, by using (REF ), $\\Vert P_{\\mathcal {S}_K^{(\\varepsilon _2)}}^{\\perp } P_{\\mathcal {S}_M^{(\\varepsilon _1)}} \\varphi \\Vert ^2 \\le \\sum _{j \\in I_K^c} \\sum _{l \\in I_M} \\left| \\frac{\\varepsilon _1 \\varepsilon _2}{\\pi ^2 (\\varepsilon _1 l_1 - \\varepsilon _2 j_1)( \\varepsilon _1 l_2 - \\varepsilon _2 j_2)} \\right|^2 .$ Assuming $K_i = \\frac{C(\\gamma ) M_i \\varepsilon _1}{\\varepsilon _2}, i = 1,2$ , we can continue by $\\Vert P_{\\mathcal {S}_K^{(\\varepsilon _2)}}^{\\perp } P_{\\mathcal {S}_M^{(\\varepsilon _1)}} \\varphi \\Vert ^2 & \\le \\left( \\frac{\\varepsilon _1 \\varepsilon _2}{\\pi ^2} \\right)^2 M_1 M_2 \\sum _{(j_1,j_2) \\notin I_K} \\frac{4}{ |( \\varepsilon _1 \\frac{M_1}{2} - \\varepsilon _2 j_1 )( \\varepsilon _1 \\frac{M_2}{2} - \\varepsilon _2 j_2)|^2} \\\\& \\le \\left( \\frac{\\varepsilon _1 \\varepsilon _2}{\\pi ^2} \\right)^2 M_1 M_2 \\frac{16}{\\varepsilon _2^2 |( \\varepsilon _1 M_1 - \\varepsilon _2 K_1 )( \\varepsilon _1 M_2 - \\varepsilon _2 K_2)|} \\\\& \\le \\left( \\frac{\\varepsilon _1 }{\\pi ^2} \\right)^2 M_1 M_2 \\frac{16}{ |( \\varepsilon _1 M_1 - \\varepsilon _2 K_1 )( \\varepsilon _1 M_2 - \\varepsilon _2 K_2)|} \\\\& \\le \\left( \\frac{\\varepsilon _1 }{\\pi ^2} \\right)^2 M_1 M_2 \\frac{16}{ |( \\varepsilon _1 M_1 - C(\\gamma ) M_1 \\varepsilon _1 )( \\varepsilon _1 M_2 - C(\\gamma ) M_2 \\varepsilon _1)|} \\\\&= \\frac{16}{\\left( \\pi ^2 (C(\\gamma )-1) \\right)^2}.$ Thus, by (REF ), $\\Vert P_{\\mathcal {S}_K^{(\\varepsilon _2)}} P_{\\mathcal {S}_M^{(\\varepsilon _1)}} \\varphi \\Vert ^2\\ge \\frac{1}{\\theta (\\gamma )^2} - \\frac{16}{\\left( \\pi ^2 (C(\\gamma )-1) \\right)^2}$ which, using (REF ), yields $\\inf \\limits _{ \\begin{array}{c}\\varphi \\in \\mathcal {R}_N\\\\ \\Vert \\varphi \\Vert =1\\end{array}} \\Vert P_{\\mathcal {S}_K^{(\\varepsilon _2)}} \\varphi \\Vert \\ge \\sqrt{\\frac{1}{\\theta (\\gamma )^2} - \\frac{16}{\\left( \\pi ^2 (C(\\gamma )-1) \\right)^2}} - \\sqrt{1 - \\frac{1}{\\theta }}.$ The lemma is proved.", "Proof of Theorem REF The following lemma will be used in the upcoming proof Theorem REF .", "A one dimensional analogue can be found in [6].", "The proof extends straightforwardly and we omit it here.", "Lemma 7.1 Let $A_1, A_2, A_3$ , and $A_4 \\in \\mathbb {Z}, A_1 \\le A_2, A_3 \\le A_4$ .", "Moreover, let $L_1, L_2 \\in \\mathbb {N}$ such that $2 L_1 \\ge A_2 - A_1 +1$ and $2L_2 \\ge A_4 - A_3 +1$ .", "Then the trigonometric polynomial $\\Phi (z_1,z_2) = \\sum _{k = A_1}^{A_2} \\sum _{l=A_3}^{A_4} \\alpha _{k,l} e^{2\\pi i k z_1} e^{2\\pi i l z_2}$ satisfies $\\sum \\limits _{m=0}^{2L_1 - 1} \\sum \\limits _{n=0}^{2L_2 - 1} \\frac{1}{4L_1 L_2} \\left| \\Phi \\left(\\frac{m}{2L_1}, \\frac{n}{2L_2}\\right) \\right|^2 = \\sum _{k = A_1}^{A_2} \\sum _{l=A_3}^{A_4} |\\alpha _{k,l} |^2 .$ [Proof of Theorem REF ] Let $\\varphi \\in \\mathcal {R}_N$ such that $\\Vert \\varphi \\Vert =1$ .", "Then $\\varphi $ can be expanded as $\\varphi = \\sum _{n_1,n_2=0}^{2^{J_0}-1} \\alpha _{J_0,(n_1,n_2)} \\phi _{J_0,(n_1,n_2)} + \\sum _{k=1}^3 \\sum _{j = J_0}^{J-1} \\sum _{n_1,n_2 =0}^{2^j-1} \\beta ^k_{j,(n_1,n_2)} \\psi _{j,(n_1,n_2)}^{\\operatorname{b},k} $ We will now use the nestedness of the two dimensional MRA that is generated by the one dimensional wavelet system $\\left\\lbrace \\lbrace \\phi ^{\\operatorname{b}}_{J_0,m} \\rbrace _{m = 0, \\ldots , 2^J-1}, \\lbrace \\psi ^{\\operatorname{b}}_{j,n} \\rbrace _{j \\ge J_0,n = 0, \\ldots , 2^j-1}\\right\\rbrace $ .", "In particular, we have $V_j^{\\operatorname{b},2} \\subset V_{j+1}^{\\operatorname{b},2} , \\quad j\\ge J_0,$ and $V_{j+1}^{\\operatorname{b},2} = V_{j}^{\\operatorname{b},2} \\oplus W_{j}^{\\operatorname{b},2}, \\quad j\\ge J_0,$ with $V_j^{\\operatorname{b},2} = V_j^{\\operatorname{b}} \\otimes V_j^{\\operatorname{b}}$ and $W_j^{\\operatorname{b},2} = (V^{\\operatorname{b}}_j \\otimes W^{\\operatorname{b}}_j) \\oplus (W^{\\operatorname{b}}_j \\otimes V^{\\operatorname{b}}_j) \\oplus (W^{\\operatorname{b}}_j \\otimes W^{\\operatorname{b}}_j)$ .", "Loosely speaking, due to the MRA embedding properties we can expand functions from the reconstructions space into scaling functions $(\\phi ^{\\operatorname{b}}_{J,(n_1,n_2)})_{n_1,n_2}$ at highest scale.", "Since the left boundary functions can be constructed by translates of the initial scaling function $\\phi $ and the right scaling function can be obtained by reflecting the left boundary functions.", "The reflected function will be denoted by $\\phi ^{\\#}$ .", "This gives us in (REF ) $\\varphi = \\sum _{n_1,n_2=0}^{2^J-p-1} \\alpha _{n_1,n_2} \\phi (2^{J} \\cdot -n) + \\sum _{n_1,n_2 =2^J-p}^{2^J} \\beta _{n_1,n_2} \\phi ^{\\#}(2^{J} \\cdot -n),$ where only finitely many $\\alpha _{n_1,n_2}$ and $\\beta _{n_1,n_2}$ are non-zero.", "Now, for any $l= (l_1,l_2) \\in \\mathbb {Z}^2$ we obtain by basic properties of the Fourier transform $\\langle \\varphi &, s_l^{(\\varepsilon )} \\rangle = \\sum _{n_1,n_2=0}^{2^J-p-1} \\alpha _{n_1,n_2} \\frac{\\varepsilon }{ 2^{J}} e^{-2\\pi i \\varepsilon \\langle n, 2^{-J}l\\rangle }\\widehat{\\phi }\\left(\\frac{\\varepsilon }{ 2^{J}}l\\right) + \\sum _{n_1,n_2 =2^J-p}^{2^J} \\beta _{n_1,n_2} \\frac{\\varepsilon }{ 2^{J}} e^{-2\\pi i \\varepsilon \\langle n, 2^{-J}l\\rangle } \\widehat{\\phi ^{\\#}}\\left(\\frac{\\varepsilon }{ 2^{J}}l\\right).", "$ For the sake of brevity, we shall write in the following $\\Phi _1(z) &= \\sum _{n_1,n_2=0}^{2^J-p-1} \\alpha _{n_1,n_2} \\frac{\\varepsilon }{ 2^{J}} e^{-2\\pi i \\varepsilon \\langle n, z\\rangle },\\nonumber \\\\\\Phi _2(z) &= \\sum _{n_1,n_2 =2^J-p}^{2^J} \\beta _{n_1,n_2} \\frac{\\varepsilon }{ 2^{J}} e^{-2\\pi i \\varepsilon \\langle n, z\\rangle }.", "$ By our assumptions on the scaling function $|\\widehat{\\phi }(\\xi _1, \\xi _2)| \\lesssim \\frac{1}{ (1+|\\xi _1|)(1+ |\\xi _2|)}, $ and by the same argument $|\\widehat{\\phi ^{\\#}}(\\xi _1, \\xi _2)| \\lesssim \\frac{1}{(1+|\\xi _1|)(1+ |\\xi _2|)}.", "$ Using (REF ), (REF ), and (REF ) in (REF ) yields $\\sum _{l \\notin I_M} |\\langle \\varphi , s_l^{(\\varepsilon )} \\rangle |^2 \\nonumber &\\le \\sum _{l \\notin I_M} \\left| \\frac{\\varepsilon }{ 2^{J}}\\Phi _1\\left(\\frac{\\varepsilon }{2^{J}}l\\right) \\widehat{\\phi }\\left(\\frac{\\varepsilon }{ 2^{J}}l\\right) \\right|^2 + \\sum _{l \\notin I_M}\\left|\\frac{\\varepsilon }{ 2^{J}} \\Phi _2\\left(\\frac{\\varepsilon }{2^{J}}l\\right) \\widehat{\\phi ^{\\#}}\\left(\\frac{\\varepsilon }{ 2^{J}}l\\right) \\right|^2 + \\nonumber \\\\&\\makebox{[}0.5cm][c]{}+ 2 \\left( \\sum _{l \\notin I_M} \\left|\\frac{\\varepsilon }{ 2^{J}} \\Phi _1\\left(\\frac{\\varepsilon }{2^{J}}l\\right) \\widehat{\\phi }\\left(\\frac{\\varepsilon }{ 2^{J}}l\\right) \\right|^2 \\right)^{1/2} \\left( \\sum _{l \\notin I_M} \\left|\\frac{\\varepsilon }{ 2^{J}} \\Phi _2\\left(\\frac{\\varepsilon }{2^{J}}l\\right) \\widehat{\\phi ^{\\#}}\\left(\\frac{\\varepsilon }{ 2^{J}}l\\right) \\right|^2\\right)^{1/2}.", "$ We assume $2^J/\\varepsilon \\in \\mathbb {N}$ and the number of samples $M=(M_1,M_2) \\in \\mathbb {N}\\times \\mathbb {N}$ is $M_i = S \\cdot \\frac{2^J}{\\varepsilon }, \\quad i = 1,2$ where $S$ is some positive constant.", "Now, $(l_1,l_2) \\notin I_M$ if Case I: $|l_1| > M_1$ and $|l_2|<M_2$ , Case II: $|l_1| < M_1$ and $|l_2|>M_2$ , or Case III: $|l_1| > M_1$ and $|l_2|>M_2$ .", "It is sufficient to consider the sum for Case I.", "Case II can be obtained by symmetry and Case III yields a smaller sum.", "For $K = 2^J/\\varepsilon $ , we have $\\sum _{|l_2| <M_2}& \\sum _{|l_1| >M_1} \\left| \\frac{\\varepsilon }{ 2^{J}} \\Phi _1\\left(\\frac{\\varepsilon }{2^{J}}l\\right) \\widehat{\\phi }\\left(\\frac{\\varepsilon }{ 2^{J}}l\\right) \\right|^2 \\nonumber \\\\&= \\sum _{j_2}^{K-1} \\sum _{j_1}^{K-1} \\frac{1}{K}\\frac{1}{K}\\left| \\Phi _1\\left(\\frac{j_1}{K}, \\frac{j_2}{K}\\right) \\right|^2 \\sum _{|s_2|<S} \\sum _{|s_1|>S} \\left|\\widehat{\\phi }\\left(\\frac{j_1}{K} + s_1,\\frac{j_2}{K} + s_2\\right) \\right|^2 \\nonumber \\\\&\\lesssim \\sum _{j_2}^{K-1} \\sum _{j_1}^{K-1} \\frac{1}{K}\\frac{1}{K}\\left| \\Phi _1\\left(\\frac{j_1}{K}, \\frac{j_2}{K}\\right) \\right|^2 \\sum _{|s_2|<S} \\sum _{|s_1|>S} \\frac{1}{(1 + |j_1/K + s_1|)^2}\\frac{1}{(1 + |j_2/K + s_2|)^2} \\nonumber \\\\&\\le C_1 \\sum _{j_2}^{K-1} \\sum _{j_1}^{K-1} \\frac{1}{K}\\frac{1}{K}\\left| \\Phi _1\\left(\\frac{j_1}{K}, \\frac{j_2}{K}\\right) \\right|^2 \\frac{1}{S}.", "$ By Lemma REF we obtain in (REF ) $\\sum _{|l_2| >M_2}& \\sum _{|l_1| >M_1} \\left| \\frac{\\varepsilon }{ 2^{J}} \\Phi _1\\left(\\frac{\\varepsilon }{2^{J}}l\\right) \\widehat{\\phi }\\left(\\frac{\\varepsilon }{ 2^{J}}l\\right) \\right|^2 \\le C_1 \\sum _{n_1,n_2=0}^{2^J-1} |\\alpha _{n_1,n_2}|^2 \\frac{1}{S}.", "$ Since the functions form an orthonormal basis and $\\Vert \\varphi \\Vert =1$ we have $\\sum _{n_1,n_2\\in \\mathbb {N}} |\\alpha _{n_1,n_2}|^2 + \\sum _{n_1,n_2\\in \\mathbb {N}} |\\beta _{n_1,n_2}|^2 = 1$ and hence $\\sum _{n_1,n_2\\in \\mathbb {N}} |\\alpha _{n_1,n_2}|^2 \\le 1$ Similarly, one shows $\\sum _{|l_2| >M_2}& \\sum _{|l_1| >M_1} \\left| \\frac{\\varepsilon }{ 2^{J}} \\Phi _2\\left(\\frac{\\varepsilon }{2^{J}}l\\right) \\widehat{\\phi ^{\\#}} \\left(\\frac{\\varepsilon }{ 2^{J}}l\\right) \\right|^2 \\le C_2 \\sum _{n_1,n_2 \\in \\mathbb {N}}|\\beta _{n_1,n_2}|^2 \\frac{1}{S}.", "$ Therefore, in (REF ) we have that $\\sum _{|l_2| <M_2} \\sum _{|l_1| >M_1} |\\langle \\varphi , s_l^{(\\varepsilon )} \\rangle |^2 & \\le C_1 \\sum _{n_1,n_2\\in \\mathbb {N}} |\\alpha _{n_1,n_2}|^2 \\frac{1}{S} + C_2 \\sum _{n_1,n_2 \\in \\mathbb {N}} |\\beta _{n_1,n_2}|^2 \\frac{1}{S} +\\\\&\\qquad + 2 \\left(C_1 \\sum _{n_1,n_2\\in \\mathbb {N}} |\\alpha _{n_1,n_2}|^2 \\frac{1}{S}\\right)^{1/2} \\left(C_2\\sum _{n_1,n_2\\in \\mathbb {N}}|\\beta _{n_1,n_2}|^2 \\frac{1}{S}\\right)^{1/2} \\nonumber \\\\&\\le \\left( \\frac{\\sqrt{2C_1}+\\sqrt{2C_2}}{\\sqrt{S}}\\right)^2$ Now, for $\\theta >1$ choosing $S$ large enough, such that $\\left( \\frac{\\sqrt{2C_1}+\\sqrt{2C_2}}{\\sqrt{S}}\\right)^2\\le \\frac{\\theta ^2 - 1}{3\\theta ^2}$ gives the claim.", "Acknowledgements The authors would like to thank Bogdan Roman for providing Figure 2 and Figure 3.", "BA acknowledges support from the NSF DMS grant 1318894.", "ACH acknowledges support from a Royal Society University Research Fellowship as well as the UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/L003457/1.", "GK was supported in part by the Einstein Foundation Berlin, by Deutsche Forschungsgemeinschaft (DFG) Grant KU 1446/14, by the DFG Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics”, and by the DFG Research Center Matheon “Mathematics for key technologies” in Berlin.", "JM acknowledges support from the Berlin Mathematical School." ], [ "Numerical Experiments", "In this section we numerically demonstrate the linearity of the stable sampling rate as stated in Theorem REF .", "We will also demonstrate how this combines with generalized sampling in practice.", "In particular, given this linearity, reconstructing from Fourier samples in smooth boundary wavelets will give an error decaying according to the smoothness and the number of vanishing moments.", "In this section we consider dyadic scaling matrices $A = \\begin{pmatrix} 2 & 0 \\\\ 0 & 2 \\end{pmatrix}.", "$ Furthermore, our focus are separable wavelets, i.e.", "wavelets that are obtained by tensor products of one dimensional scaling functions and one dimensional wavelet functions, respectively.", "Scaling matrices of the form (REF ) preserve the separability." ], [ "Linearity examples with Haar and Daubechies-4 wavelets", "We use the description of Section 3.1 and Example REF in order to perform numerical experiments for some known wavelets.", "This gives the reconstruction space $\\mathcal {R}= \\overline{\\operatorname{span}} \\lbrace \\varphi \\, : \\, \\varphi \\in \\Omega _1 \\cup \\Omega _2\\rbrace $ where $\\Omega _1$ and $\\Omega _2$ are defined in (REF ) and (REF ) respectively.", "We order the reconstruction space $\\mathcal {R}$ in the same manner as presented at the end of Section 3.", "In (REF ) we counted the number of reconstruction elements up to level $J-1$ , which leads to $N_J = (2^{2J}-1)a^2 + 6a(a-1)(2^J-1) + 3J(a-1)^2 + (2a-1)^2 $ many elements, which is asymptotically of order $2^{2J}$ .", "We test Haar wavelets and 2D Daubechies-4 wavelets.", "Figure REF shows the linear behaviour of the stable sampling rate for these two types of wavelet generators.", "By a small abuse of notation, we also write $M$ for the total number of samples.", "In our analysis in Section 3 we estimated the angle $\\cos (\\omega (\\mathcal {R}_N, \\mathcal {S}_M^{(\\varepsilon )}))$ (recall that $\\epsilon $ is the sampling rate) with respect to some fixed $\\theta >1$ .", "In fact we computed $M$ such that $\\inf \\limits _{\\begin{array}{c} \\varphi \\in \\mathcal {R}_N \\\\ \\Vert \\varphi \\Vert =1\\end{array}} \\Vert P_{\\mathcal {S}_M^{(\\varepsilon )}} \\varphi \\Vert \\ge \\frac{1}{\\theta }$ holds.", "We proved that $M$ is up to a constant of the same size as $N_J$ .", "Figure REF shows the stable sampling rate (in blue) $\\Theta (N,\\theta ) = \\min \\left\\lbrace M \\in \\mathbb {N}, \\cos (\\omega (\\mathcal {R}_N,\\mathcal {S}_M)) \\ge \\frac{1}{\\theta }\\right\\rbrace $ and the linear function $f$ (in red) given by $f(N) = \\frac{M_{\\text{max}}}{N_{\\text{max}}}N,$ where $N_{\\text{max}}$ is the maximum value of $N$ used in the experiment and $M_{\\text{max}} = \\Theta (N_{\\text{max}},\\theta )$ .", "We computed the stable sampling rate up to level $J = 4$ .", "Note the (significant) jumps of the stable sampling rate occur whenever $N\\in \\mathbb {N}$ crosses the scaling level $N_J, J = 0, \\ldots , 4$ .", "In the Haar case these are $N_0 = 4, \\qquad N_1 = 16, \\qquad N_2 = 64, \\qquad N_3 = 256, \\quad N_4 = 1024,$ see (REF ).", "Note that $a=1$ in the Haar case.", "In particular, the jumps are linear, suggesting a linear stable sampling rate.", "Figure 1 (b), (c), and (d) are interpreted similarly.", "However, the theoretical results are asymptotic results.", "Therefore, it should not be surprising that the stable sampling rate is below the linear line in some cases.", "It aligns asymptotically." ], [ "Fourier samples and boundary wavelet reconstruction", "In this example we will demonstrate the efficiency of generalized sampling given the established linearity of the stable sampling rate.", "In particular, suppose that $f$ is a function we want to recover from its Fourier information.", "It is smooth, however, not periodic – a problem that occurs for example in electron microscopy and also in MRI.", "This causes the classical Fourier reconstruction to converge slowly, yet a smooth boundary wavelet basis will give much faster convergence (see [22]).", "As discussed, the issue is that we are given Fourier samples, not wavelet coefficients.", "However, this is not a problem in view of the linearity of the stable sampling rate.", "In particular, if $f \\in W^s(0,1)$ , where $\\mathrm {W}^s(0,1)$ denotes the usual Sobolev space, and $P_{\\mathcal {R}_N}$ denotes the projection onto the space $\\mathcal {R}_{N}$ of the first $N$ boundary wavelets (see (REF )), then $\\Vert f - P_{\\mathcal {R}_N} f \\Vert = \\mathcal {O}(N^{-s}),\\quad N \\rightarrow \\infty ,$ provided that the wavelet has sufficiently many vanishing moments.", "Now, if $G_{N,M}(f) \\in \\mathcal {R}_{N}$ is the generalized sampling solution from Definition REF given $M$ Fourier coefficients, and $M$ is chosen according to the stable sampling rate then $\\Vert f - G_{N,M}(f)\\Vert = \\mathcal {O}(N^{-s}) = \\mathcal {O}(M^{-s}),\\quad N \\rightarrow \\infty .$ Hence, we obtain the same convergence rate up to a constant, by simply postprocessing the given samples.", "To illustrate this advantage we will consider the following two functions: $f_1(x,y)=\\cos ^2(x)\\exp (-y), \\qquad f_2(x,y) = (1+x^2)(2y-1^2).$ In Figure REF we have shown the results for $f_1$ and compared the classical Fourier reconstruction with the generalized sampling reconstruction.", "Both examples use exactly the same samples, however, note the pleasant absence of the Gibb's ringing in the generalized sampling case.", "The same experiment is carried out for $f_2$ in Figure REF , however, here we have displayed the reconstructions in $3D$ in order to visualize the error." ], [ "Proof of Theorem ", "The proof of Theorem REF is somewhat technical, wherefore we divide the proof into several steps.", "First, in Subsection REF , the overall structure of the proof is presented, and the respective details can then be found in Subsection REF ." ], [ "Structure of the Proof", "Let $\\varepsilon \\in (0,1/(T_1+T_2)]$ and $\\theta >1$ .", "Then we have to prove that $\\inf \\limits _{\\begin{array}{c} \\varphi \\in \\mathcal {R}_{N_J} \\\\ \\Vert \\varphi \\Vert =1\\end{array}} \\Vert P_{\\mathcal {S}_M^{(\\varepsilon )}} \\varphi \\Vert \\ge \\frac{1}{\\theta },$ for $M=(M_1,M_2) = \\left(\\frac{ \\lambda _1^{(J)} + \\lambda _3^{(J)}}{\\varepsilon } S^{(\\theta )}, \\frac{ \\lambda _2^{(J)} + \\lambda _4^{(J)}}{\\varepsilon } S^{(\\theta )}\\right),$ and some $S^{(\\theta )}$ independent of $J$ .", "To this end, let $\\varphi \\in \\mathcal {R}_N$ be such that $\\Vert \\varphi \\Vert =1$ .", "Since the sampling functions form an orthonormal basis of $\\mathcal {S}_M^{(\\varepsilon )}$ , we obtain $\\Vert P_{\\mathcal {S}_M^{(\\varepsilon )}} \\varphi \\Vert ^2= \\sum \\limits _{l_1 = -M_1}^{M_1} \\sum \\limits _{l_2 = -M_2}^{M_2}|\\langle \\varphi , s_l^{(\\varepsilon )} \\rangle |^2, \\quad l = (l_1,l_2).$ By Lemma REF , which relies mainly on the underlying MRA structure, we can write $\\varphi $ in terms of scaling functions at highest scale, i.e., $\\varphi = \\sum \\limits _{l_1 = L_3}^{L_1} \\sum \\limits _{l_2 = L_4}^{L_2} \\alpha _{l_1,l_2}\\phi _{J,(l_1,l_2)},$ for some $L_1,L_2,L_3,L_4 \\in \\mathbb {Z}$ .", "Lemma REF , which is proven by direct computations, then shows that $\\langle \\varphi , s_l^{(\\varepsilon )} \\rangle = \\varepsilon |\\det A|^{-J/2} \\Phi (\\varepsilon (A^{-J})^Tl) \\widehat{\\phi }(\\varepsilon (A^{-J})^Tl),$ where $\\Phi $ is a trigonometric polynomial of the form $\\Phi (z) = \\sum \\limits _{m_1 = L_3}^{L_1} \\sum \\limits _{m_2 = L_4}^{L_2} \\alpha _{m_1,m_2} e^{-2\\pi i \\langle z, m \\rangle }, \\quad z \\in \\mathbb {R}^2, m = (m_1,m_2).$ Using (REF ), we conclude that $\\Vert P_{\\mathcal {S}_M^{(\\varepsilon )}} \\varphi \\Vert ^2= \\sum \\limits _{l_1 = -M_1}^{M_1} \\sum \\limits _{l_2 = -M_2}^{M_2} \\varepsilon ^2 |\\det A|^{-J}|\\Phi (\\varepsilon (A^{-J})^Tl)|^2 |\\widehat{\\phi }(\\varepsilon (A^{-J})^Tl)|^2, \\quad l = (l_1, l_2).$ By Theorem REF , there exist some $\\varepsilon _0 > 0$ and $S^{(\\theta )} = \\left(S^{(\\theta )}_1,S^{(\\theta )}_2\\right)\\in \\mathbb {N}\\times \\mathbb {N}$ that does not depend on $J$ such that $\\sum \\limits _{l_1 = -M_1}^{M_1} \\sum \\limits _{l_2 = -M_2}^{M_2} \\varepsilon _0 ^2 | \\det A^{-J}| | \\Phi (\\varepsilon _0 (A^{-J})^T l) |^2| \\widehat{\\phi }(\\varepsilon _0(A^{-J})^Tl)|^2 \\ge \\frac{\\Vert \\Phi \\Vert ^2}{\\theta ^2}$ for $M= \\frac{1}{\\varepsilon _0} (A^J)^T \\widetilde{S}^{(\\theta )}=\\begin{pmatrix} \\frac{ \\lambda _1^{(J)} + \\lambda _3^{(J)}}{\\varepsilon _0} S^{(\\theta )}_1 \\\\\\frac{ \\lambda _2^{(J)} + \\lambda _4^{(J)}}{\\varepsilon _0} S^{(\\theta )}_2 \\end{pmatrix}.$ Since $\\Vert \\Phi \\Vert ^2 =\\sum \\limits _{m_1 = L_3}^{L_1} \\sum \\limits _{m_2 = -L_4}^{L_2} |\\beta _{m_1,m_2}|^2 =\\Vert \\varphi \\Vert ^2 = 1,$ we obtain $\\Vert P_{\\mathcal {S}_M^{(\\varepsilon _0)}} \\varphi \\Vert ^2 \\ge \\frac{1}{\\theta ^2}.$ Finally, Lemma REF implies that a change of $\\varepsilon $ only changes the constant, showing that (REF ) is true for any $\\varepsilon \\in (0,1/(T_1+T_2)]$ , thereby proving Theorem REF ." ], [ "Auxiliary results", "In this section we will prove the results mentioned in Subsection REF , which are required for completing the proof of Theorem REF .", "The following result is an extension of a result from [6] to the two dimensional setting.", "Lemma 6.1 Let $J \\in \\mathbb {N}$ and $\\varphi \\in V_0^{(a)} \\oplus W_0^{(a)} \\oplus \\ldots \\oplus W_{J-1}^{(a)}$ .", "Then there exist $L_1,L_2,L_3,L_4 \\in \\mathbb {Z}$ dependent of $J$ such that $\\varphi = \\sum \\limits _{l_1 = L_3}^{L_1} \\sum \\limits _{l_2 = L_4}^{L_2} \\alpha _{l_1,l_2}\\phi _{J,(l_1,l_2)}.$ Let $\\varphi \\in V_0^{(a)} \\oplus W_0^{(a)} \\oplus \\ldots \\oplus W_{J-1}^{(a)}$ .", "Then $\\varphi $ has an expansion of the following form $\\varphi = \\sum _{m_1 = - a +1}^{m_1 = a-1} \\sum _{m_2 = - a +1}^{m_2 = a-1} \\alpha _{m_1,m_2} \\phi _{0,(m_1,m_2)} +\\sum _{p=1}^{|\\det A| -1} \\sum _{j=0}^{J-1} \\sum _{m_1 = -a +1}^{a(\\lambda _1^{j}+ \\lambda _{2}^{j})-1}\\sum \\limits _{m_2= -a +1}^{a(\\lambda _3^{j}+ \\lambda _{4}^{j})-1} \\beta _{j,(m_1,m_2)}^p \\psi _{j,(m_1,m_2)}^p.$ Since $V_0^{(a)} \\subset V_0$ and $W_j^{(a)} \\subset W_{j}$ for $j \\in \\mathbb {N}$ and the sequence $(V_j)_{j\\in \\mathbb {Z}}$ forms an MRA, it follows that $V_0^{(a)} \\oplus W_0^{(a)} \\oplus \\ldots \\oplus W_{J-1}^{(a)} \\subset V_J.$ Since an orthonormal basis for $V_J$ is given by the functions $\\lbrace \\phi _{J,m} \\, : \\, m \\in \\mathbb {Z}^2\\rbrace $ , for each $|m_i| < a, i =1,2$ , we have $\\phi _{0,(m_1,m_2)} = \\sum _{l \\in \\mathbb {Z}^2} \\langle \\phi _{0,(m_1,m_2)}, \\phi _{J,l} \\rangle \\phi _{J,l}$ Moreover, since $\\phi _{0,(m_1,m_2)}$ and $\\phi _{J,l}$ are compactly supported, we obtain $\\langle \\phi _{0,(m_1,m_2)}, \\phi _{J,l} \\rangle \\ne 0,$ if $-a+ (-a+1)( \\lambda _1^{(J)}+ \\lambda _2^{(J)} ) \\le l_1 \\le (2a-1)(\\lambda _1^{(J)} + \\lambda _2^{(J)})$ and $-a+ (-a+1)( \\lambda _3^{(J)}+ \\lambda _4^{(J)} ) \\le l_2 \\le (2a-1)(\\lambda _3^{(J)} + \\lambda _4^{(J)}).$ This follows by a straightforward computation from the support conditions of $\\phi _{0,(m_1,m_2)}$ and $\\phi _{J,l}$ together with $|m_i| < a, i =1,2$ .", "Similarly, we have $\\langle \\psi _{j,(m_1,m_2)}^p, \\phi _{J,(l_1,l_2)} \\rangle \\ne 0$ if Case I: $\\det A^j \\ge 0$ $-a& - \\lambda _1^{(J)}\\frac{(-a+1)\\lambda _4^{(j)}+\\lambda _2^{(j)}(a(\\lambda _3^{(j)}+\\lambda _4^{(j)})-1)}{\\det A^j} + \\lambda _2^{(J)} \\frac{(-a+1)\\lambda _1^{(j)}+(a(\\lambda _1^{(j)}+\\lambda _2^{(j)})-1)\\lambda _3^{(j)}}{\\det A^j} \\\\& < l_1 < \\lambda _1^{(J)} \\frac{a(\\lambda _4^{(j)}-\\lambda _2^{(j)})+(a(\\lambda _1^{(j)}+\\lambda _2^{(j)})-1)\\lambda _4^{(j)}+(a-1)\\lambda _2^{(j)}}{\\det A^j}+\\\\&\\qquad +\\lambda _2^{(J)} \\frac{a(\\lambda _1^{(j)}-\\lambda _3^{(j)})+(a(\\lambda _3^{(j)}+\\lambda _4^{(j)})-1)\\lambda _1^{(j)}+(a-1)\\lambda _3^{(j)}}{\\det A^j}$ and $-a& - \\lambda _3^{(J)}\\frac{(-a+1)\\lambda _4^{(j)}+\\lambda _2^{(j)}(a(\\lambda _3^{(j)}+\\lambda _4^{(j)})-1)}{\\det A^j} + \\lambda _4^{(J)} \\frac{(-a+1)\\lambda _1^{(j)}+(a(\\lambda _1^{(j)}+\\lambda _2^{(j)})-1)\\lambda _3^{(j)}}{\\det A^j} \\\\& < l_2 < \\lambda _3^{(J)} \\frac{a(\\lambda _4^{(j)}-\\lambda _2^{(j)})+(a(\\lambda _1^{(j)}+\\lambda _2^{(j)})-1)\\lambda _4^{(j)}+(a-1)\\lambda _2^{(j)}}{\\det A^j}+\\\\&\\qquad +\\lambda _4^{(J)} \\frac{a(\\lambda _1^{(j)}-\\lambda _3^{(j)})+(a(\\lambda _3^{(j)}+\\lambda _4^{(j)})-1)\\lambda _1^{(j)}+(a-1)\\lambda _3^{(j)}}{\\det A^j}.$ Case II: $\\det A^j < 0$ $-a& - \\lambda _1^{(J)}\\frac{(-a+1)\\lambda _4^{(j)}+\\lambda _2^{(j)}(a(\\lambda _3^{(j)}+\\lambda _4^{(j)})-1)}{\\det A^j} + \\lambda _2^{(J)} \\frac{(-a+1)\\lambda _1^{(j)}+(a(\\lambda _1^{(j)}+\\lambda _2^{(j)})-1)\\lambda _3^{(j)}}{\\det A^j} \\\\& > l_1 > \\lambda _1^{(J)} \\frac{a(\\lambda _4^{(j)}-\\lambda _2^{(j)})+(a(\\lambda _1^{(j)}+\\lambda _2^{(j)})-1)\\lambda _4^{(j)}+(a-1)\\lambda _2^{(j)}}{\\det A^j}+\\\\&\\qquad +\\lambda _2^{(J)} \\frac{a(\\lambda _1^{(j)}-\\lambda _3^{(j)})+(a(\\lambda _3^{(j)}+\\lambda _4^{(j)})-1)\\lambda _1^{(j)}+(a-1)\\lambda _3^{(j)}}{\\det A^j}$ and $-a& - \\lambda _3^{(J)}\\frac{(-a+1)\\lambda _4^{(j)}+\\lambda _2^{(j)}(a(\\lambda _3^{(j)}+\\lambda _4^{(j)})-1)}{\\det A^j} + \\lambda _4^{(J)} \\frac{(-a+1)\\lambda _1^{(j)}+(a(\\lambda _1^{(j)}+\\lambda _2^{(j)})-1)\\lambda _3^{(j)}}{\\det A^j} \\\\& > l_2 > \\lambda _3^{(J)} \\frac{a(\\lambda _4^{(j)}-\\lambda _2^{(j)})+(a(\\lambda _1^{(j)}+\\lambda _2^{(j)})-1)\\lambda _4^{(j)}+(a-1)\\lambda _2^{(j)}}{\\det A^j}+\\\\&\\qquad +\\lambda _4^{(J)} \\frac{a(\\lambda _1^{(j)}-\\lambda _3^{(j)})+(a(\\lambda _3^{(j)}+\\lambda _4^{(j)})-1)\\lambda _1^{(j)}+(a-1)\\lambda _3^{(j)}}{\\det A^j}.$ Minimizing the lower bounds with respect to $j \\in \\lbrace 0, \\ldots , J-1\\rbrace $ and maximizing the upper bounds with respect to $j$ , respectively, yields the claim.", "The following lemma is well known (see [22]).", "Lemma 6.2 Let $f \\in L^2(\\mathbb {R}^2)$ .", "Then $\\lbrace f( \\cdot - m) \\, : \\, m \\in \\mathbb {Z}^2\\rbrace $ is an orthonormal system if and only if $\\sum \\limits _{m \\in \\mathbb {Z}^2} | \\widehat{f}(\\xi + m )|^2 = 1 \\quad \\text{for almost every } \\xi \\in \\mathbb {R}^2.$ Finally, one more technical lemma is needed.", "Lemma 6.3 Let $A$ be a scaling matrix, $J \\in \\mathbb {Z}$ , and $m=(m_1,m_2) \\in \\mathbb {Z}^2$ .", "Further, let $\\varphi \\in \\overline{\\operatorname{span}} \\lbrace \\phi _{J,m} \\, : \\, m \\in \\mathbb {Z}^2\\rbrace $ be compactly supported in $[-T_1,T_2]^2$ , and let $L_1,L_2,L_3,L_4 \\in \\mathbb {Z}$ be such that $\\varphi = \\sum \\limits _{m_1 = L_3}^{L_1} \\sum \\limits _{m_2 = L_4}^{L_2} \\alpha _{m_1,m_2} \\phi _{J,(m_1,m_2)}, \\quad \\alpha _m \\in $ Then, for all $l \\in \\mathbb {Z}^2$ , $\\langle \\varphi , s_l^{(\\varepsilon )} \\rangle = \\varepsilon |\\det A|^{-J/2} \\Phi (\\varepsilon (A^{-J})^Tl) \\widehat{\\phi }(\\varepsilon (A^{-J})^Tl),$ where $s_l^{(\\varepsilon )}$ is defined in (REF ) and $\\Phi $ is the trigonometric polynomial given by $\\Phi (z) = \\sum \\limits _{\\begin{array}{c} L_3 \\le m_1 \\le L_1, \\\\ L_4 \\le m_2 \\le L_2,\\end{array}} \\alpha _m e^{-2\\pi i \\langle z, m \\rangle }, \\quad z \\in \\mathbb {R}^2, m = (m_1,m_2).$ Since $\\varphi $ is supported in $[-T_1,T_2]^2$ , we obtain $\\langle \\varphi , s_l^{(\\varepsilon )} \\rangle &= \\varepsilon \\int \\limits _{\\mathbb {R}^2} \\varphi (x) e^{-2 \\pi i \\varepsilon \\langle l, x \\rangle }\\cdot \\chi _{[-T_1,T_2]^2} \\, dx \\\\&= \\varepsilon \\widehat{\\varphi }( \\varepsilon l) \\\\&= \\varepsilon \\left(\\sum \\limits _{m_1 = L_1}^{L_2} \\sum \\limits _{m_2 = L_3}^{L_4} \\alpha _{m_1,m_2} \\phi _{J,(m_1,m_2)}\\right)^{\\wedge }(\\varepsilon l) \\\\&= \\varepsilon \\sum \\limits _{m_1 = L_1}^{L_2} \\sum \\limits _{m_2 = L_3}^{L_4} \\alpha _{m_1,m_2} \\left( \\phi _{J,(m_1,m_2)}\\right)^{\\wedge }(\\varepsilon l)\\\\&= \\varepsilon |\\det A|^{-J/2} \\Phi (\\varepsilon (A^{-J})^T l) \\widehat{\\phi }(\\varepsilon (A^{-J})^Tl).$ This proves the claim." ], [ "Theorem ", "The proof of Theorem REF requires a particular estimate (Proposition REF ) for the norm of trigonometric polynomials depending on their evaluations on a particular grid whose mesh norm and associated Voronoi regions come also into play." ], [ "Mesh Norm", "We start with the definition of a mesh norm for the situation we are faced with.", "A mesh norm can be interpreted as the largest distance between neighboring nodes.", "Definition 6.4 Let $\\Lambda \\subset \\mathbb {Z}^2$ be an integer grid of the form $\\Lambda := \\left\\lbrace (l_1,l_2) \\in \\mathbb {Z}^2 \\, : \\, -M_i \\le l_i \\le M_i, i=1,2 \\right\\rbrace , \\quad M_1,M_2 \\in \\mathbb {N},$ and let $A$ be a scaling matrix.", "Set $\\Omega := \\overline{\\Lambda }^A := A ([-M_1,M_1] \\times [-M_2,M_2]) \\subset \\mathbb {R}^2,$ and define a metric $\\rho $ on $\\Omega $ by $\\rho : \\Omega \\times \\Omega \\longrightarrow \\mathbb {R}^+, \\quad (x,y) \\mapsto \\min \\limits _{k \\in A(\\mathbb {Z}^2)} \\Vert x - y + k\\Vert _\\infty .$ The mesh norm of $\\lbrace x_l \\in \\Omega \\, : \\, l \\in \\Lambda \\rbrace $ is then defined as $\\delta := \\max \\limits _{x \\in \\Omega } \\min \\limits _{l \\in \\Lambda } \\rho (x_l,x),$ where $x_l := A \\cdot l, l \\in \\Lambda $ denote the nodes in $\\Omega $ .", "Before we can continue, we require some notions and results on Voronoi regions and trigonometric polynomials.", "Our first result shows that if the distance between the nodes $\\lbrace x_l\\rbrace _l$ converges to zero, the mesh norm of the entire grid $\\Omega $ converges to zero.", "Lemma 6.5 Let $\\varepsilon >0$ and $\\Lambda \\subset \\mathbb {Z}^2$ be defined by $\\Lambda := \\left\\lbrace (l_1,l_2) \\in \\mathbb {Z}^2 \\, : \\, -M_i \\le l_i \\le M_i, i=1,2 \\right\\rbrace ,$ where $M_1,M_2 \\in \\mathbb {N}$ .", "Furthermore, suppose $A$ is a scaling matrix, and let $\\Omega ^{(\\varepsilon )} := \\overline{\\Lambda }^{\\varepsilon A}$ .", "If $\\varepsilon \\longrightarrow 0$ , then $\\delta ^{(\\varepsilon )} \\longrightarrow 0$ , where $\\delta ^{(\\varepsilon )}$ denotes the mesh norm of $\\lbrace \\varepsilon A l \\in \\Omega ^{(\\varepsilon )} \\, : \\, l \\in \\Lambda \\rbrace $ .", "If $\\varepsilon \\rightarrow 0$ , then $\\varepsilon A (\\Lambda ) = \\lbrace \\varepsilon A l \\, : \\, l \\in \\Lambda \\rbrace \\longrightarrow \\lbrace (0,0) \\rbrace $ with respect to the standard Euclidean distance.", "Furthermore, for $x_l := \\varepsilon A l$ , we have $\\lim \\limits _{\\varepsilon \\rightarrow 0} \\delta ^{(\\varepsilon )}= \\lim \\limits _{\\varepsilon \\rightarrow 0} \\max \\limits _{x \\in \\Omega ^{(\\varepsilon )}} \\min \\limits _{l \\in \\Lambda } \\min \\limits _{k \\in \\varepsilon A(\\mathbb {Z}^2)} \\Vert x - x_l + k\\Vert _\\infty \\le \\lim \\limits _{\\varepsilon \\rightarrow 0} \\max \\limits _{x \\in \\Omega ^{(\\varepsilon )}} \\min \\limits _{l \\in \\Lambda } \\Vert x - x_l \\Vert _\\infty .$ Since $x, x_l \\in \\Omega ^{(\\varepsilon )}$ for all $ l \\in \\Lambda $ , we obtain $\\lim \\limits _{\\varepsilon \\rightarrow 0} 2 \\max \\limits _{x \\in \\Omega ^{(\\varepsilon )}} \\min \\limits _{l \\in \\Lambda } \\Vert x - x_l \\Vert _\\infty \\le \\lim \\limits _{\\varepsilon \\rightarrow 0} \\max \\limits _{x,y \\in \\Omega ^{(\\varepsilon )}} \\Vert x - y \\Vert _\\infty .$ Inserting this estimate into (REF ) yields $\\lim \\limits _{\\varepsilon \\rightarrow 0} \\delta ^{(\\varepsilon )}\\le \\lim \\limits _{\\varepsilon \\rightarrow 0} \\max \\limits _{x,y \\in \\Omega ^{(\\varepsilon )}} \\Vert x - y \\Vert _\\infty \\le \\lim \\limits _{\\varepsilon \\rightarrow 0} \\operatorname{diam}\\Omega ^{(\\varepsilon )}= \\lim \\limits _{\\varepsilon \\rightarrow 0} \\operatorname{diam}\\overline{\\Lambda }^{\\varepsilon A}= \\lim \\limits _{\\varepsilon \\rightarrow 0} \\operatorname{diam}\\varepsilon \\overline{\\Lambda }^A= \\lim \\limits _{\\varepsilon \\rightarrow 0} \\varepsilon \\underbrace{ \\operatorname{diam}\\overline{\\Lambda }^A}_{<\\infty } =0,$ where $\\operatorname{diam}(F)$ denotes the diameter of a set $F\\subset \\mathbb {R}^d$ , i.e., $\\operatorname{diam}(F) = \\sup \\limits _{x,y \\in F} d_2(x,y),$ and $d_2$ denotes the Euclidean metric on $\\mathbb {R}^d$ .", "Since the mesh norm is always non-negative, the lemma is proven." ], [ "Voronoi Regions", "The next result studies the volume of the Voronoi regions associated to the previously considered grid $\\Lambda $ with respect to the metric $\\rho $ defined in Definition REF .", "We start by formally defining the notion of Voronoi region in our setting.", "Definition 6.6 Let $\\Omega \\subset \\mathbb {R}^2$ , and let $(x_l)_{l \\in \\Lambda }$ be a sequence in $\\mathbb {R}^2$ .", "Then we refer to the sets $(V_l)_{l \\in \\Lambda }$ defined by $V_l := \\lbrace x \\in \\Omega \\, : \\, \\rho (x,x_l) \\le \\rho (x,x_k) \\; \\text{ for all } k \\ne l \\rbrace $ as Voronoi regions with respect to $\\Lambda $ , $\\Omega $ , $\\rho $ , and $x_l$.", "We can now state the previously announced result.", "Lemma 6.7 Let $M_1,M_2 \\in \\mathbb {N}$ and $\\Lambda := \\left\\lbrace (l_1,l_2) \\in \\mathbb {Z}^2 \\, : \\, -M_i \\le l_i \\le M_i, i=1,2 \\right\\rbrace .$ Moreover, let $\\Omega =\\overline{\\Lambda }^{\\operatorname{Id}}$ , where $\\operatorname{Id}$ denotes the $2\\times 2$ -identity matrix, and let $(V_l)_{l \\in \\Lambda }$ be the Voronoi regions with respect to $\\Lambda $ , $\\Omega $ , $\\rho $ , and $l$ .", "Then, for all $l \\in \\Lambda $ , $\\mu (V_l) \\le 1,$ where $\\mu $ denotes the 2D Lebesgue measure.", "Notice that the Voronoi regions $(V_l)_{l \\in \\Lambda }$ are in fact rectangles, since the grid is an integer grid with a constant step-size.", "Hence, for each $l \\in \\Lambda $ , $\\mu (V_l) = a_{l_1,l_2} \\cdot b_{l_1,l_2}, \\quad a_{l_1,l_2},b_{l_1,l_2} \\in \\mathbb {R},$ where $a_{l_1,l_2}$ denotes the width and $b_{l_1,l_2}$ the height of the rectangle $V_l$ .", "Towards a contradiction, assume that $V_l$ does contain two different nodes $x_k$ and $x_l$ with $k \\ne l$ .", "This implies $0\\ne \\rho (x_k,x_l) \\le \\rho (x_l,x_l) =0,$ which is a contradiction.", "Thus, we can conclude that $a_{l_1,l_2} \\le \\rho (x_{l_1+1,l_2},x_{l_1,l_2})\\quad \\mbox{and} \\quad b_l \\le \\rho (x_{l_1,l_2},x_{l_1,l_2+1}),$ which, by (REF ), proves the claim.", "We next obtain a slight generalization of the previous result.", "Lemma 6.8 Let $A$ be a (linear) bijective transformation acting on $\\mathbb {R}^2$ with matrix representation $A= \\begin{pmatrix} \\lambda _1& \\lambda _2 \\\\ \\lambda _3 & \\lambda _4 \\end{pmatrix},$ and let $\\Lambda $ and $(V_l)_{l \\in \\Lambda }$ be defined as in Lemma REF .", "Then $\\mu (A(V_l)) \\le |\\det A |.$ The result follows from Lemma REF by an integration by substitution." ], [ "Trigonometric Polynomials", "The next theorem is an adapted version of a result presented in [24] which again is a reformulation of a result proven by Gröchenig in [17].", "Proposition 6.9 Let $J, L_1,L_2, L_3, L_4 \\in \\mathbb {Z}$ such that $L_1 \\ge L_3$ and $L_2 \\ge L_4$ , and let $\\Phi $ a trigonometric polynomial of the form $\\Phi (z) = \\sum \\limits _{m_1 = L_3}^{L_1} \\sum \\limits _{m_2 = L_4}^{L_2} \\alpha _{m_1,m_2} e^{-2\\pi i \\langle z, m \\rangle }, \\quad z \\in \\mathbb {R}^2, m = (m_1,m_2).$ Further, let the grid $\\Lambda $ be defined as in Lemma REF , let $A$ be a scaling matrix, and let $\\Omega ^{(\\varepsilon )} := \\overline{\\Lambda }^{\\varepsilon A^{-J}}$ for $\\varepsilon >0$ .", "Set $x_l :=\\varepsilon (A^{-J})^Tl$ , $l \\in \\Lambda $ .", "If the mesh norm $\\delta $ of $\\lbrace x_l \\in \\Omega ^{(\\varepsilon )} \\, : \\, l \\in \\Lambda \\rbrace $ obeys $\\delta < \\frac{\\log \\left(\\frac{1}{\\sqrt{\\mu ( \\Omega ^{(\\varepsilon )})}}+1\\right)}{2 \\pi \\max \\lbrace |L_1|,|L_2|,|L_3|,|L_4|\\rbrace },$ where $\\mu $ is the 2D Lebesgue measure, then there exists a positive constant $C(\\delta ,L_1,L_2,L_3,L_4)$ such that $C(\\delta ,L_1,L_2,L_3,L_4) \\Vert \\Phi \\Vert \\le \\left( \\sum \\limits _{l \\in \\Lambda } \\varepsilon ^2 |\\det A^{-J} | | \\Phi (\\varepsilon (A^{-J})^Tl|^2 \\right)^{\\frac{1}{2}},$ where $C(\\delta , L_1,L_2,L_3,L_4) = \\left( 1 - \\left(e^{2\\pi \\delta \\max \\lbrace |L_1|,|L_2|,|L_3|,|L_4|\\rbrace } -1\\right) \\sqrt{ \\mu (\\Omega ^{(\\varepsilon )})}\\right)$ and $\\Vert f \\Vert := \\left(\\int _{\\Omega ^{(\\varepsilon )}} |f(x)|^2 \\, dx\\right)^{1/2}, \\quad f\\in L^2(\\Omega ^{(\\varepsilon )}).$ We first observe that by the hypotheses, the constant $C(\\delta , L_1,L_2,L_3,L_4)$ is indeed positive.", "Second, let $(V_l)_{l \\in \\Lambda }$ be the Voronoi regions with respect to $\\Lambda $ , $\\Omega ^{(\\varepsilon )}$ , and $\\rho $ .", "For $l \\in \\Lambda $ , we define the weights $\\omega _l := \\mu (V_l)$ .", "As in Lemma REF , integration by substitution yields $\\omega _l = \\mu (V_l) \\le \\varepsilon ^2 |\\det A^{-J} |.$ Hence it suffices to prove the existence of a constant $C(\\delta , L_1,L_2,L_3,L_4) >0$ such that $C(\\delta ,L_1,L_2,L_3,L_4) \\Vert \\Phi \\Vert \\le \\left( \\sum \\limits _{l \\in \\Lambda } \\omega _l | \\Phi (\\varepsilon (A^{-J})^Tl|^2 \\right)^{\\frac{1}{2}}, $ For this, we first observe that $\\left( \\sum \\limits _{l \\in \\Lambda } \\omega _l | \\Phi (\\varepsilon (A^{-J})^Tl|^2 \\right)^{\\frac{1}{2}}= \\left( \\int \\limits _{\\Omega ^{(\\varepsilon )}} \\sum \\limits _{l \\in \\Lambda } | \\Phi (\\varepsilon (A^{-J})^Tl|^2 \\right)^{\\frac{1}{2}}= \\left\\Vert \\sum \\limits _{l \\in \\Lambda } \\Phi (x_l)\\chi _{V_l} \\right\\Vert .$ By the (inverse) triangle inequality, $\\left\\Vert \\sum \\limits _{l \\in \\Lambda } \\Phi (x_l)\\chi _{V_l} \\right\\Vert \\ge \\Vert \\Phi \\Vert - \\left\\Vert \\Phi - \\sum \\limits _{l \\in \\Lambda }\\Phi (x_l)\\chi _{V_l} \\right\\Vert .", "$ Hence, we require an upper bound for $\\Vert \\Phi - \\sum _{l \\in \\Lambda } \\Phi (x_l)\\chi _{V_l} \\Vert $ .", "By Taylor expansion and Bernstein's inequality, $\\left\\Vert \\Phi - \\sum \\limits _{l \\in \\Lambda } \\Phi (x_l) \\chi _{V_l} \\right\\Vert ^2& = \\int \\limits _{\\Omega ^{(\\varepsilon )}} \\left| \\Phi (x) - \\sum \\limits _{l \\in \\Lambda } \\Phi (x_l)\\chi _{V_l}(x) \\right|^2 \\, dx \\\\& = \\int \\limits _{\\Omega ^{(\\varepsilon )}} \\left| \\sum \\limits _{l \\in \\Lambda } \\Phi (x) \\chi _{V_l}(x) - \\Phi (x_l)\\chi _{V_l}(x) \\right|^2 \\, dx \\\\& \\le \\sum \\limits _{l \\in \\Lambda } \\int \\limits _{V_l} \\left| \\Phi (x) - \\Phi (x_l) \\right|^2 \\, dx \\\\& \\le \\sum \\limits _{l \\in \\Lambda } \\int \\limits _{V_l} \\left| \\sum _{\\alpha \\in \\mathbb {N}^2\\setminus \\lbrace (0,0)\\rbrace }\\frac{1}{\\alpha !}", "\\Vert x - x_l\\Vert ^\\alpha | D^\\alpha \\Phi (x)| \\right|^2 \\, dx \\\\& \\le \\sum \\limits _{l \\in \\Lambda } \\int \\limits _{V_l} \\left| \\sum _{\\alpha \\in \\mathbb {N}^2\\setminus \\lbrace (0,0)\\rbrace }\\frac{1}{\\alpha !}", "\\delta ^{|\\alpha |} ( \\max \\lbrace |L_1|,|L_2|,|L_3|,|L_4|\\rbrace \\pi )^{|\\alpha |} \\Vert \\Phi \\Vert \\right|^2 \\, dx \\\\& \\le \\left| \\sum _{\\alpha \\in \\mathbb {N}^2\\setminus \\lbrace (0,0)\\rbrace }\\frac{1}{\\alpha !}", "\\delta ^{|\\alpha |} ( \\max \\lbrace |L_1|,|L_2|,|L_3|,|L_4|\\rbrace \\pi )^{|\\alpha |} \\Vert \\Phi \\Vert \\right|^2\\cdot \\sum \\limits _{l \\in \\Lambda } \\int \\limits _{V_l} 1\\, dx$ Since the Voronoi regions build a partition of $\\Omega ^{(\\varepsilon )}$ , $\\sum \\limits _{l \\in \\Lambda } \\mu (V_l) = \\mu (\\Omega ^{(\\varepsilon )}),$ and we can continue by $\\left\\Vert \\Phi - \\sum \\limits _{l \\in \\Lambda } \\Phi (x_l) \\chi _{V_l} \\right\\Vert ^2 \\le \\left| \\left(e^{2\\pi \\delta \\max \\lbrace |L_1|,|L_2|,|L_3|,|L_4|\\rbrace } - 1\\right) \\right|^2\\Vert \\Phi \\Vert ^2 \\mu (\\Omega ^{(\\varepsilon )}).", "$ Using (REF ) and (REF ), we obtain $\\left\\Vert \\sum \\limits _{l \\in \\Lambda } \\Phi (x_l)\\chi _{V_l} \\right\\Vert &\\ge \\Vert \\Phi \\Vert - \\left\\Vert \\Phi - \\sum \\limits _{l \\in \\Lambda }\\Phi (x_l)\\chi _{V_l} \\right\\Vert \\\\&\\ge \\Vert \\Phi \\Vert - \\left| \\Vert \\Phi \\Vert \\left(e^{2\\pi \\delta \\max \\lbrace |L_1|,|L_2|,|L_3|,|L_4|\\rbrace } - 1\\right) \\sqrt{ \\mu (\\Omega ^{(\\varepsilon )})}\\right|\\\\&= \\Vert \\Phi \\Vert \\left( 1 - \\left(e^{2\\pi \\delta \\max \\lbrace |L_1|,|L_2|,|L_3|,|L_4|\\rbrace } -1\\right) \\sqrt{ \\mu (\\Omega ^{(\\varepsilon )})}\\right).$ Combining this estimate with (REF ) proves (REF ).", "Finally, we can state and prove Theorem REF , which is one main ingredient for the proof of Theorem REF in Subsection REF .", "Theorem 6.10 Let $L_1,L_2, L_3, L_4 \\in \\mathbb {Z}$ such that $L_1 \\ge L_3$ and $L_2 \\ge L_4$ and $\\alpha _{m_1,m_2} \\in , and let $$ be the trigonometric polynomial$$\\Phi ( \\cdot , \\cdot ) = \\sum _{m_1= L_3}^{L_1} \\sum _{m_2= L_4}^{L_2} \\alpha _{m_1,m_2} e^{-2 \\pi i (\\cdot )m_1}e^{-2 \\pi i (\\cdot )m_2}.$$Further, let $ A= 1 2 3 4 $ be a scaling matrix, $ J N$ a maximalscale, and $ Z2$ the grid defined by$$\\Lambda := \\left\\lbrace (l_1,l_2) \\in \\mathbb {Z}^2 \\, : \\, -M_i \\le l_i \\le M_i, i=1,2 \\right\\rbrace ,$$where $ M1,M2 N$.", "If there exists $ 1/(T1 + T2)$ independent of $ J$ such that $$ fulfills{\\begin{@align*}{1}{-1}\\delta < \\frac{\\log \\left(\\frac{1}{\\sqrt{\\mu ( \\Omega ^{(\\varepsilon )})}}+1\\right)}{ 4 \\pi \\max \\lbrace |L_1|,|L_2|,|L_3|,|L_4|\\rbrace },\\end{@align*}}then there exists $ S() =(S()1,S()2) NN$,independent of $ J$, such that, for all $ >1$, we have$$\\sum \\limits _{l \\in \\Lambda } \\varepsilon ^2 | \\det A^{-J}| | \\Phi (\\varepsilon (A^{-J})^T l) |^2| \\widehat{\\phi }(\\varepsilon (A^{-J})^Tl)|^2 \\ge \\frac{\\Vert \\Phi \\Vert ^2}{2\\theta ^2},$$for $ M= (M1,M2) = 1 (AJ)T S()$ and scaling function $$.$ Since $0<\\varepsilon <1$ , there exists an $m\\in \\mathbb {N}$ , such that $\\frac{1}{m+1} \\le \\varepsilon < \\frac{1}{m}.$ Set $\\varepsilon = \\frac{1}{m+1}$ , and note that (REF ) still holds, since the logarithm is monotonically increasing.", "Next, for some $S \\in \\mathbb {N}$ to be determined later, let $\\begin{pmatrix} M_1 \\\\ M_2 \\end{pmatrix} := \\frac{1}{\\varepsilon } (A^J)^T\\begin{pmatrix} S \\\\S \\end{pmatrix} = \\frac{1}{\\varepsilon }\\begin{pmatrix} \\lambda ^{(J)}_1 + \\lambda ^{(J)}_3 \\\\ \\lambda ^{(J)}_2 + \\lambda ^{(J)}_4 \\end{pmatrix} S,\\quad \\mbox{where as before }A^J=\\begin{pmatrix} \\lambda ^{(J)}_1 & \\lambda ^{(J)}_2 \\\\ \\lambda ^{(J)}_3 & \\lambda ^{(J)}_4 \\end{pmatrix}.$ For $l = (l_1,l_2) \\in \\mathbb {Z}^2$ , we then obtain $ \\nonumber {\\sum \\limits _{l \\in \\Lambda } \\varepsilon ^2 | \\det A^{-J}| | \\Phi (\\varepsilon (A^{-J})^T l) |^2 | \\widehat{\\phi }(\\varepsilon (A^{-J})^Tl)|^2}\\\\ \\nonumber &=&\\sum \\limits _{l_1 = - M_1}^{M_1 -1} \\sum \\limits _{ l_2 = M_2 }^{M_2-1} \\varepsilon ^2 | \\det A^{-J}| | \\Phi (\\varepsilon (A^{-J})^T l) |^2 |\\widehat{\\phi }(\\varepsilon (A^{-J})^Tl)|^2\\\\ \\nonumber &=& \\sum \\limits _{l_1=0 }^{\\frac{\\lambda ^{(J)}_1 + \\lambda ^{(J)}_3}{\\varepsilon } -1} \\sum \\limits _{l_2=0}^{\\frac{\\lambda ^{(J)}_2 + \\lambda ^{(J)}_4}{\\varepsilon } -1}\\sum \\limits _{s=-S} ^{S-1}\\sum \\limits _{t=-S} ^{S-1} \\bigg ( \\varepsilon ^2 | \\det A^{-J}| \\cdot \\left| \\Phi \\left(\\varepsilon (A^{-J})^T \\left(l+\\frac{1}{\\varepsilon } (A^J)^T\\begin{pmatrix} s \\\\ t \\end{pmatrix}\\right) \\right) \\right|^2 \\\\& & \\cdot \\left| \\widehat{\\phi }\\left(\\varepsilon (A^{-J})^T \\left(l+\\frac{1}{\\varepsilon } (A^J)^T\\begin{pmatrix} s \\\\ t \\end{pmatrix}\\right) \\right) \\right|^2\\bigg ), $ Integer periodicity of the trigonometric polynomial implies that, for all $s \\in \\mathbb {Z}$ , $\\Phi \\left(\\varepsilon (A^{-J})^T \\left(l+\\frac{1}{\\varepsilon } (A^J)^T \\begin{pmatrix} s \\\\ t \\end{pmatrix} \\right) \\right)= \\Phi \\left(\\varepsilon (A^{-J})^T l + \\begin{pmatrix} s \\\\ t \\end{pmatrix}\\right)= \\Phi \\left(\\varepsilon (A^{-J})^T l \\right).$ Therefore, by (REF ) $\\sum \\limits _{l \\in \\Lambda } \\; & \\varepsilon ^2 | \\det A^{-J}| | \\Phi (\\varepsilon (A^{-J})^T l) |^2 | \\widehat{\\phi }(\\varepsilon (A^{-J})^Tl)|^2 \\nonumber \\\\&= \\sum \\limits _{l_1=0 }^{\\frac{\\lambda ^{(J)}_1 + \\lambda ^{(J)}_3}{\\varepsilon } -1}\\sum \\limits _{l_2=0}^{\\frac{\\lambda ^{(J)}_2 + \\lambda ^{(J)}_4}{\\varepsilon } -1} \\bigg (\\varepsilon ^2 | \\det A^{-J}| | \\Phi (\\varepsilon (A^{-J})^T l) |^2 \\sum \\limits _{s=-S} ^{S-1} \\sum \\limits _{t=-S} ^{S-1} \\left| \\widehat{\\phi }\\left((A^{-J})^Tl+\\begin{pmatrix} s \\\\ t \\end{pmatrix}\\right)\\right|^2\\bigg ).", "$ Let $\\theta >1$ .", "Then, by Lemma REF , there exists $S^{(\\theta )} \\in \\mathbb {N}$ such that $\\sum \\limits _{s=-S^{(\\theta )}} ^{S^{(\\theta )}-1} \\sum \\limits _{t=-S} ^{S-1} \\left| \\widehat{\\phi }\\left((A^{-J})^Tl+\\begin{pmatrix} s \\\\ t \\end{pmatrix}\\right)\\right|^2 \\ge \\frac{1}{\\theta } \\ .", "$ We now choose $S := S^{(\\theta )}$ .", "Combining (REF ) and (REF ) yields $\\sum \\limits _{l \\in \\Lambda } \\ & \\varepsilon ^2 | \\det A^{-J}| | \\Phi (\\varepsilon (A^{-J})^T l) |^2 | \\widehat{\\phi }(\\varepsilon (A^{-J})^Tl)|^2 \\nonumber \\\\&\\ge \\sum \\limits _{l_1=0 }^{\\frac{\\lambda ^{(J)}_1 + \\lambda ^{(J)}_3}{\\varepsilon } -1}\\sum \\limits _{l_2=0}^{\\frac{\\lambda ^{(J)}_2 + \\lambda ^{(J)}_4}{\\varepsilon } -1} \\varepsilon ^2 | \\det A^{-J}| | \\Phi (\\varepsilon (A^{-J})^T l) |^2\\cdot \\frac{1}{\\theta } \\ .", "$ Next, we apply Proposition REF to (REF ) to obtain $\\sum \\limits _{l \\in \\Lambda } \\varepsilon ^2 | \\det A^{-J}|& | \\Phi (\\varepsilon (A^{-J})^T l) |^2 | \\widehat{\\phi }(\\varepsilon (A^{-J})^Tl)|^2 \\nonumber \\\\&\\ge \\left(\\Vert \\Phi \\Vert \\left( 1 - \\left(e^{2\\pi \\delta \\max \\lbrace |L_1|,|L_2|,|L_3|,|L_4|\\rbrace } -1\\right) \\sqrt{ \\mu (\\Omega ^{(\\varepsilon )})}\\right) \\right)^2\\cdot \\frac{1}{\\theta }$ Since $1+ &\\left( 1- e^{2\\pi \\delta \\max \\lbrace |L_1|,|L_2|,|L_3|,|L_4|\\rbrace }\\right) \\sqrt{\\mu (\\Omega ^{(\\varepsilon )})} \\\\&\\ge 1+ \\left( 1- e^{2\\pi \\max \\lbrace |L_1|,|L_2|,|L_3|,|L_4|\\rbrace \\frac{\\log \\left(\\frac{1}{\\sqrt{\\mu (\\Omega ^{(\\varepsilon )})}}+1\\right)}{4\\pi \\max \\lbrace |L_1|,|L_2|,|L_3|,|L_4|\\rbrace } }\\right) \\sqrt{\\mu (\\Omega ^{(\\varepsilon )})} \\\\&=1+ \\left( 1- \\left(\\sqrt{\\mu (\\Omega ^{(\\varepsilon )})}+1\\right)^{1/2}\\right)\\sqrt{\\mu (\\Omega ^{(\\varepsilon )})} \\\\&=1+ \\left( \\sqrt{\\mu (\\Omega ^{(\\varepsilon )})}- \\left(\\mu (\\Omega ^{(\\varepsilon )})^{3/2}+\\mu (\\Omega ^{(\\varepsilon )})\\right)^{1/2}\\right) \\\\&\\ge 1 -\\mu (\\Omega ^{(\\varepsilon )})^{3/4}\\\\&\\ge 1- \\varepsilon ^{3/4} \\left( \\det A^{-J}\\right)^{3/4}\\\\&\\ge \\frac{1}{2}$ we can conclude $\\sum \\limits _{l \\in \\Lambda } \\varepsilon ^2 | \\det A^{-J}|& | \\Phi (\\varepsilon (A^{-J})^T l) |^2 | \\widehat{\\phi }(\\varepsilon (A^{-J})^Tl)|^2 \\ge \\frac{\\Vert \\Phi \\Vert ^2}{2\\theta }.$ Hence the theorem is proven for $M_1 = \\frac{ \\lambda _1^{(J)} + \\lambda _3^{(J)}}{\\varepsilon } S^{(\\theta )}\\quad \\mbox{and} \\quad M_2 = \\frac{ \\lambda _2^{(J)} + \\lambda _4^{(J)}}{\\varepsilon } S^{(\\theta )}.$" ], [ "Lemma ", "We mention that this result is proved in the same manner as the one dimensional result from [6].", "Lemma 6.11 For $\\gamma >1$ and $\\varepsilon _1, \\varepsilon _2 \\in (0,1/(T_1+T_2)]$ , let $\\theta (\\gamma ),C(\\gamma )>1$ be such that $\\sqrt{\\frac{1}{\\theta (\\gamma )^{2}} - \\frac{16}{\\pi ^4(C(\\gamma ) -1)^2}} - \\sqrt{1-\\frac{1}{\\theta (\\gamma )}} > \\frac{1}{\\gamma }.", "$ If there exists $M=(M_1,M_2) \\in \\mathbb {N}\\times \\mathbb {N}$ such that $\\inf \\limits _{ \\begin{array}{c}\\varphi \\in \\mathcal {R}_N\\\\ \\Vert \\varphi \\Vert =1\\end{array}} \\Vert P_{\\mathcal {S}_M^{(\\varepsilon _1)}} \\varphi \\Vert \\ge \\frac{1}{\\theta (\\gamma )},$ then $\\inf \\limits _{ \\begin{array}{c}\\varphi \\in \\mathcal {R}_N\\\\ \\Vert \\varphi \\Vert =1\\end{array}} \\Vert P_{\\mathcal {S}_K^{(\\varepsilon _2)}} \\varphi \\Vert \\ge \\frac{1}{\\gamma },$ whenever $K = (K_1,K_2) =\\left( \\left\\lceil \\frac{C(\\gamma )M_1\\varepsilon _1}{\\varepsilon _2}\\right\\rceil ,\\left\\lceil \\frac{C(\\gamma )M_2\\varepsilon _1}{\\varepsilon _2}\\right\\rceil \\right)$ .", "First, notice that for any $\\gamma \\in (0,1)$ , there exist $\\theta (\\gamma )$ and $C(\\gamma )$ such that (REF ) is fulfilled.", "Now, let $\\gamma >1$ and $\\varepsilon _2 >0$ .", "Then, by (REF ), $\\nonumber \\inf \\limits _{ \\begin{array}{c}\\varphi \\in \\mathcal {R}_N\\\\ \\Vert \\varphi \\Vert =1\\end{array}} \\Vert P_{\\mathcal {S}_K^{(\\varepsilon _2)}} \\varphi \\Vert & \\ge \\inf \\limits _{ \\begin{array}{c}\\varphi \\in \\mathcal {R}_N\\\\ \\Vert \\varphi \\Vert =1\\end{array}} \\left( \\Vert P_{\\mathcal {S}_K^{(\\varepsilon _2)}} P_{\\mathcal {S}_M^{(\\varepsilon _1)}} \\varphi \\Vert - \\Vert P_{\\mathcal {S}_K^{(\\varepsilon _2)}} P_{\\mathcal {S}_M^{(\\varepsilon _1)}}^{\\perp } \\varphi \\Vert \\right) \\\\ & \\ge \\inf \\limits _{ \\begin{array}{c}\\varphi \\in \\mathcal {R}_N\\\\ \\Vert \\varphi \\Vert =1\\end{array}} \\left( \\Vert P_{\\mathcal {S}_K^{(\\varepsilon _2)}} P_{\\mathcal {S}_M^{(\\varepsilon _1)}} \\varphi \\Vert - \\sqrt{1 - \\frac{1}{\\theta (\\gamma )}} \\right).$ In order to estimate $\\Vert P_{\\mathcal {S}_K^{(\\varepsilon _2)}} P_{\\mathcal {S}_M^{(\\varepsilon _1)}} \\varphi \\Vert $ , we decompose this term by $\\Vert P_{\\mathcal {S}_K^{(\\varepsilon _2)}} P_{\\mathcal {S}_M^{(\\varepsilon _1)}} \\varphi \\Vert ^2= \\Vert P_{\\mathcal {S}_M^{(\\varepsilon _1)}} \\varphi \\Vert ^2 - \\Vert P_{\\mathcal {S}_K^{(\\varepsilon _2)}}^{\\perp } P_{\\mathcal {S}_M^{(\\varepsilon _1)}} \\varphi \\Vert ^2.$ Thus, we require a suitable upper bound for $\\Vert P_{\\mathcal {S}_K^{(\\varepsilon _2)}}^{\\perp } P_{\\mathcal {S}_M^{(\\varepsilon _1)}} \\varphi \\Vert ^2$ .", "For this, let $I_M &= \\lbrace l =(l_1, l_2) \\in \\mathbb {Z}^2 \\, : \\, -M_i/2 \\le l_i \\le M_i/2-1, i =1,2 \\rbrace , \\\\I_K &= \\lbrace j = (j_1, j_2) \\in \\mathbb {Z}^2 \\, : \\, -K_i/2 \\le j_i \\le K_i/2 -1, i =1,2 \\rbrace ,$ and for the complementary sets in $\\mathbb {Z}^2$ we write $I_M^c &= \\lbrace l=(l_1, l_2) \\in \\mathbb {Z}^2 \\, : l \\notin I_M \\rbrace , \\\\I_K^c &= \\lbrace j = (j_1, j_2) \\in \\mathbb {Z}^2 \\, : \\, j \\notin I_K \\rbrace .$ Then, we have $\\nonumber \\Vert P_{\\mathcal {S}_K^{(\\varepsilon _2)}}^{\\perp } P_{\\mathcal {S}_M^{(\\varepsilon _1)}} \\varphi \\Vert ^2&= \\left\\Vert \\sum _{j \\in I_K^c} \\langle P_{\\mathcal {S}_M^{(\\varepsilon _1)}} \\varphi , s_j^{(\\varepsilon _2)} \\rangle s_j^{(\\varepsilon _2)} \\right\\Vert ^2 \\\\ \\nonumber &= \\sum _{j \\in I_K^c} \\left| \\sum _{l \\in I_M} \\langle \\varphi , s_l^{(\\varepsilon _1)} \\rangle \\langle s_l^{(\\varepsilon _1)}, s_j^{(\\varepsilon _2)} \\rangle \\right|^2 \\\\ \\nonumber &\\le \\sum _{j\\in I_K^c} \\left( \\sum _{l \\in I_M} | \\langle \\varphi , s_l^{(\\varepsilon _1)} \\rangle |^2\\sum _{l \\in I_M} |\\langle s_l^{(\\varepsilon _1)}, s_j^{(\\varepsilon _2)} \\rangle |^2 \\right)\\\\ &\\le \\sum _{j \\in I_K^c} \\sum _{l \\in I_M} |\\langle s_l^{(\\varepsilon _1)}, s_j^{(\\varepsilon _2)} \\rangle |^2 .$ Now, for $\\varepsilon = \\max \\lbrace \\varepsilon _1, \\varepsilon _2 \\rbrace $ , we obtain $| \\langle s_l^{(\\varepsilon _1)}, s_j^{(\\varepsilon _2)} \\rangle |&= \\left| \\varepsilon _1 \\cdot \\varepsilon _2 \\cdot \\int \\limits _{\\left[-\\frac{1}{2\\varepsilon }, \\frac{1}{2\\varepsilon } \\right]^2}e^{2 \\pi i \\varepsilon _1 \\langle l, x \\rangle } e^{2 \\pi i \\varepsilon _2 \\langle j, x \\rangle } \\, dx \\right| \\\\&\\le \\left| \\frac{\\varepsilon _1 \\varepsilon _2}{\\pi ^2 (\\varepsilon _1 l_1 - \\varepsilon _2 j_1)( \\varepsilon _1 l_2 - \\varepsilon _2 j_2)} \\right|.$ Therefore, by using (REF ), $\\Vert P_{\\mathcal {S}_K^{(\\varepsilon _2)}}^{\\perp } P_{\\mathcal {S}_M^{(\\varepsilon _1)}} \\varphi \\Vert ^2 \\le \\sum _{j \\in I_K^c} \\sum _{l \\in I_M} \\left| \\frac{\\varepsilon _1 \\varepsilon _2}{\\pi ^2 (\\varepsilon _1 l_1 - \\varepsilon _2 j_1)( \\varepsilon _1 l_2 - \\varepsilon _2 j_2)} \\right|^2 .$ Assuming $K_i = \\frac{C(\\gamma ) M_i \\varepsilon _1}{\\varepsilon _2}, i = 1,2$ , we can continue by $\\Vert P_{\\mathcal {S}_K^{(\\varepsilon _2)}}^{\\perp } P_{\\mathcal {S}_M^{(\\varepsilon _1)}} \\varphi \\Vert ^2 & \\le \\left( \\frac{\\varepsilon _1 \\varepsilon _2}{\\pi ^2} \\right)^2 M_1 M_2 \\sum _{(j_1,j_2) \\notin I_K} \\frac{4}{ |( \\varepsilon _1 \\frac{M_1}{2} - \\varepsilon _2 j_1 )( \\varepsilon _1 \\frac{M_2}{2} - \\varepsilon _2 j_2)|^2} \\\\& \\le \\left( \\frac{\\varepsilon _1 \\varepsilon _2}{\\pi ^2} \\right)^2 M_1 M_2 \\frac{16}{\\varepsilon _2^2 |( \\varepsilon _1 M_1 - \\varepsilon _2 K_1 )( \\varepsilon _1 M_2 - \\varepsilon _2 K_2)|} \\\\& \\le \\left( \\frac{\\varepsilon _1 }{\\pi ^2} \\right)^2 M_1 M_2 \\frac{16}{ |( \\varepsilon _1 M_1 - \\varepsilon _2 K_1 )( \\varepsilon _1 M_2 - \\varepsilon _2 K_2)|} \\\\& \\le \\left( \\frac{\\varepsilon _1 }{\\pi ^2} \\right)^2 M_1 M_2 \\frac{16}{ |( \\varepsilon _1 M_1 - C(\\gamma ) M_1 \\varepsilon _1 )( \\varepsilon _1 M_2 - C(\\gamma ) M_2 \\varepsilon _1)|} \\\\&= \\frac{16}{\\left( \\pi ^2 (C(\\gamma )-1) \\right)^2}.$ Thus, by (REF ), $\\Vert P_{\\mathcal {S}_K^{(\\varepsilon _2)}} P_{\\mathcal {S}_M^{(\\varepsilon _1)}} \\varphi \\Vert ^2\\ge \\frac{1}{\\theta (\\gamma )^2} - \\frac{16}{\\left( \\pi ^2 (C(\\gamma )-1) \\right)^2}$ which, using (REF ), yields $\\inf \\limits _{ \\begin{array}{c}\\varphi \\in \\mathcal {R}_N\\\\ \\Vert \\varphi \\Vert =1\\end{array}} \\Vert P_{\\mathcal {S}_K^{(\\varepsilon _2)}} \\varphi \\Vert \\ge \\sqrt{\\frac{1}{\\theta (\\gamma )^2} - \\frac{16}{\\left( \\pi ^2 (C(\\gamma )-1) \\right)^2}} - \\sqrt{1 - \\frac{1}{\\theta }}.$ The lemma is proved." ], [ "Proof of Theorem ", "The following lemma will be used in the upcoming proof Theorem REF .", "A one dimensional analogue can be found in [6].", "The proof extends straightforwardly and we omit it here.", "Lemma 7.1 Let $A_1, A_2, A_3$ , and $A_4 \\in \\mathbb {Z}, A_1 \\le A_2, A_3 \\le A_4$ .", "Moreover, let $L_1, L_2 \\in \\mathbb {N}$ such that $2 L_1 \\ge A_2 - A_1 +1$ and $2L_2 \\ge A_4 - A_3 +1$ .", "Then the trigonometric polynomial $\\Phi (z_1,z_2) = \\sum _{k = A_1}^{A_2} \\sum _{l=A_3}^{A_4} \\alpha _{k,l} e^{2\\pi i k z_1} e^{2\\pi i l z_2}$ satisfies $\\sum \\limits _{m=0}^{2L_1 - 1} \\sum \\limits _{n=0}^{2L_2 - 1} \\frac{1}{4L_1 L_2} \\left| \\Phi \\left(\\frac{m}{2L_1}, \\frac{n}{2L_2}\\right) \\right|^2 = \\sum _{k = A_1}^{A_2} \\sum _{l=A_3}^{A_4} |\\alpha _{k,l} |^2 .$ [Proof of Theorem REF ] Let $\\varphi \\in \\mathcal {R}_N$ such that $\\Vert \\varphi \\Vert =1$ .", "Then $\\varphi $ can be expanded as $\\varphi = \\sum _{n_1,n_2=0}^{2^{J_0}-1} \\alpha _{J_0,(n_1,n_2)} \\phi _{J_0,(n_1,n_2)} + \\sum _{k=1}^3 \\sum _{j = J_0}^{J-1} \\sum _{n_1,n_2 =0}^{2^j-1} \\beta ^k_{j,(n_1,n_2)} \\psi _{j,(n_1,n_2)}^{\\operatorname{b},k} $ We will now use the nestedness of the two dimensional MRA that is generated by the one dimensional wavelet system $\\left\\lbrace \\lbrace \\phi ^{\\operatorname{b}}_{J_0,m} \\rbrace _{m = 0, \\ldots , 2^J-1}, \\lbrace \\psi ^{\\operatorname{b}}_{j,n} \\rbrace _{j \\ge J_0,n = 0, \\ldots , 2^j-1}\\right\\rbrace $ .", "In particular, we have $V_j^{\\operatorname{b},2} \\subset V_{j+1}^{\\operatorname{b},2} , \\quad j\\ge J_0,$ and $V_{j+1}^{\\operatorname{b},2} = V_{j}^{\\operatorname{b},2} \\oplus W_{j}^{\\operatorname{b},2}, \\quad j\\ge J_0,$ with $V_j^{\\operatorname{b},2} = V_j^{\\operatorname{b}} \\otimes V_j^{\\operatorname{b}}$ and $W_j^{\\operatorname{b},2} = (V^{\\operatorname{b}}_j \\otimes W^{\\operatorname{b}}_j) \\oplus (W^{\\operatorname{b}}_j \\otimes V^{\\operatorname{b}}_j) \\oplus (W^{\\operatorname{b}}_j \\otimes W^{\\operatorname{b}}_j)$ .", "Loosely speaking, due to the MRA embedding properties we can expand functions from the reconstructions space into scaling functions $(\\phi ^{\\operatorname{b}}_{J,(n_1,n_2)})_{n_1,n_2}$ at highest scale.", "Since the left boundary functions can be constructed by translates of the initial scaling function $\\phi $ and the right scaling function can be obtained by reflecting the left boundary functions.", "The reflected function will be denoted by $\\phi ^{\\#}$ .", "This gives us in (REF ) $\\varphi = \\sum _{n_1,n_2=0}^{2^J-p-1} \\alpha _{n_1,n_2} \\phi (2^{J} \\cdot -n) + \\sum _{n_1,n_2 =2^J-p}^{2^J} \\beta _{n_1,n_2} \\phi ^{\\#}(2^{J} \\cdot -n),$ where only finitely many $\\alpha _{n_1,n_2}$ and $\\beta _{n_1,n_2}$ are non-zero.", "Now, for any $l= (l_1,l_2) \\in \\mathbb {Z}^2$ we obtain by basic properties of the Fourier transform $\\langle \\varphi &, s_l^{(\\varepsilon )} \\rangle = \\sum _{n_1,n_2=0}^{2^J-p-1} \\alpha _{n_1,n_2} \\frac{\\varepsilon }{ 2^{J}} e^{-2\\pi i \\varepsilon \\langle n, 2^{-J}l\\rangle }\\widehat{\\phi }\\left(\\frac{\\varepsilon }{ 2^{J}}l\\right) + \\sum _{n_1,n_2 =2^J-p}^{2^J} \\beta _{n_1,n_2} \\frac{\\varepsilon }{ 2^{J}} e^{-2\\pi i \\varepsilon \\langle n, 2^{-J}l\\rangle } \\widehat{\\phi ^{\\#}}\\left(\\frac{\\varepsilon }{ 2^{J}}l\\right).", "$ For the sake of brevity, we shall write in the following $\\Phi _1(z) &= \\sum _{n_1,n_2=0}^{2^J-p-1} \\alpha _{n_1,n_2} \\frac{\\varepsilon }{ 2^{J}} e^{-2\\pi i \\varepsilon \\langle n, z\\rangle },\\nonumber \\\\\\Phi _2(z) &= \\sum _{n_1,n_2 =2^J-p}^{2^J} \\beta _{n_1,n_2} \\frac{\\varepsilon }{ 2^{J}} e^{-2\\pi i \\varepsilon \\langle n, z\\rangle }.", "$ By our assumptions on the scaling function $|\\widehat{\\phi }(\\xi _1, \\xi _2)| \\lesssim \\frac{1}{ (1+|\\xi _1|)(1+ |\\xi _2|)}, $ and by the same argument $|\\widehat{\\phi ^{\\#}}(\\xi _1, \\xi _2)| \\lesssim \\frac{1}{(1+|\\xi _1|)(1+ |\\xi _2|)}.", "$ Using (REF ), (REF ), and (REF ) in (REF ) yields $\\sum _{l \\notin I_M} |\\langle \\varphi , s_l^{(\\varepsilon )} \\rangle |^2 \\nonumber &\\le \\sum _{l \\notin I_M} \\left| \\frac{\\varepsilon }{ 2^{J}}\\Phi _1\\left(\\frac{\\varepsilon }{2^{J}}l\\right) \\widehat{\\phi }\\left(\\frac{\\varepsilon }{ 2^{J}}l\\right) \\right|^2 + \\sum _{l \\notin I_M}\\left|\\frac{\\varepsilon }{ 2^{J}} \\Phi _2\\left(\\frac{\\varepsilon }{2^{J}}l\\right) \\widehat{\\phi ^{\\#}}\\left(\\frac{\\varepsilon }{ 2^{J}}l\\right) \\right|^2 + \\nonumber \\\\&\\makebox{[}0.5cm][c]{}+ 2 \\left( \\sum _{l \\notin I_M} \\left|\\frac{\\varepsilon }{ 2^{J}} \\Phi _1\\left(\\frac{\\varepsilon }{2^{J}}l\\right) \\widehat{\\phi }\\left(\\frac{\\varepsilon }{ 2^{J}}l\\right) \\right|^2 \\right)^{1/2} \\left( \\sum _{l \\notin I_M} \\left|\\frac{\\varepsilon }{ 2^{J}} \\Phi _2\\left(\\frac{\\varepsilon }{2^{J}}l\\right) \\widehat{\\phi ^{\\#}}\\left(\\frac{\\varepsilon }{ 2^{J}}l\\right) \\right|^2\\right)^{1/2}.", "$ We assume $2^J/\\varepsilon \\in \\mathbb {N}$ and the number of samples $M=(M_1,M_2) \\in \\mathbb {N}\\times \\mathbb {N}$ is $M_i = S \\cdot \\frac{2^J}{\\varepsilon }, \\quad i = 1,2$ where $S$ is some positive constant.", "Now, $(l_1,l_2) \\notin I_M$ if Case I: $|l_1| > M_1$ and $|l_2|<M_2$ , Case II: $|l_1| < M_1$ and $|l_2|>M_2$ , or Case III: $|l_1| > M_1$ and $|l_2|>M_2$ .", "It is sufficient to consider the sum for Case I.", "Case II can be obtained by symmetry and Case III yields a smaller sum.", "For $K = 2^J/\\varepsilon $ , we have $\\sum _{|l_2| <M_2}& \\sum _{|l_1| >M_1} \\left| \\frac{\\varepsilon }{ 2^{J}} \\Phi _1\\left(\\frac{\\varepsilon }{2^{J}}l\\right) \\widehat{\\phi }\\left(\\frac{\\varepsilon }{ 2^{J}}l\\right) \\right|^2 \\nonumber \\\\&= \\sum _{j_2}^{K-1} \\sum _{j_1}^{K-1} \\frac{1}{K}\\frac{1}{K}\\left| \\Phi _1\\left(\\frac{j_1}{K}, \\frac{j_2}{K}\\right) \\right|^2 \\sum _{|s_2|<S} \\sum _{|s_1|>S} \\left|\\widehat{\\phi }\\left(\\frac{j_1}{K} + s_1,\\frac{j_2}{K} + s_2\\right) \\right|^2 \\nonumber \\\\&\\lesssim \\sum _{j_2}^{K-1} \\sum _{j_1}^{K-1} \\frac{1}{K}\\frac{1}{K}\\left| \\Phi _1\\left(\\frac{j_1}{K}, \\frac{j_2}{K}\\right) \\right|^2 \\sum _{|s_2|<S} \\sum _{|s_1|>S} \\frac{1}{(1 + |j_1/K + s_1|)^2}\\frac{1}{(1 + |j_2/K + s_2|)^2} \\nonumber \\\\&\\le C_1 \\sum _{j_2}^{K-1} \\sum _{j_1}^{K-1} \\frac{1}{K}\\frac{1}{K}\\left| \\Phi _1\\left(\\frac{j_1}{K}, \\frac{j_2}{K}\\right) \\right|^2 \\frac{1}{S}.", "$ By Lemma REF we obtain in (REF ) $\\sum _{|l_2| >M_2}& \\sum _{|l_1| >M_1} \\left| \\frac{\\varepsilon }{ 2^{J}} \\Phi _1\\left(\\frac{\\varepsilon }{2^{J}}l\\right) \\widehat{\\phi }\\left(\\frac{\\varepsilon }{ 2^{J}}l\\right) \\right|^2 \\le C_1 \\sum _{n_1,n_2=0}^{2^J-1} |\\alpha _{n_1,n_2}|^2 \\frac{1}{S}.", "$ Since the functions form an orthonormal basis and $\\Vert \\varphi \\Vert =1$ we have $\\sum _{n_1,n_2\\in \\mathbb {N}} |\\alpha _{n_1,n_2}|^2 + \\sum _{n_1,n_2\\in \\mathbb {N}} |\\beta _{n_1,n_2}|^2 = 1$ and hence $\\sum _{n_1,n_2\\in \\mathbb {N}} |\\alpha _{n_1,n_2}|^2 \\le 1$ Similarly, one shows $\\sum _{|l_2| >M_2}& \\sum _{|l_1| >M_1} \\left| \\frac{\\varepsilon }{ 2^{J}} \\Phi _2\\left(\\frac{\\varepsilon }{2^{J}}l\\right) \\widehat{\\phi ^{\\#}} \\left(\\frac{\\varepsilon }{ 2^{J}}l\\right) \\right|^2 \\le C_2 \\sum _{n_1,n_2 \\in \\mathbb {N}}|\\beta _{n_1,n_2}|^2 \\frac{1}{S}.", "$ Therefore, in (REF ) we have that $\\sum _{|l_2| <M_2} \\sum _{|l_1| >M_1} |\\langle \\varphi , s_l^{(\\varepsilon )} \\rangle |^2 & \\le C_1 \\sum _{n_1,n_2\\in \\mathbb {N}} |\\alpha _{n_1,n_2}|^2 \\frac{1}{S} + C_2 \\sum _{n_1,n_2 \\in \\mathbb {N}} |\\beta _{n_1,n_2}|^2 \\frac{1}{S} +\\\\&\\qquad + 2 \\left(C_1 \\sum _{n_1,n_2\\in \\mathbb {N}} |\\alpha _{n_1,n_2}|^2 \\frac{1}{S}\\right)^{1/2} \\left(C_2\\sum _{n_1,n_2\\in \\mathbb {N}}|\\beta _{n_1,n_2}|^2 \\frac{1}{S}\\right)^{1/2} \\nonumber \\\\&\\le \\left( \\frac{\\sqrt{2C_1}+\\sqrt{2C_2}}{\\sqrt{S}}\\right)^2$ Now, for $\\theta >1$ choosing $S$ large enough, such that $\\left( \\frac{\\sqrt{2C_1}+\\sqrt{2C_2}}{\\sqrt{S}}\\right)^2\\le \\frac{\\theta ^2 - 1}{3\\theta ^2}$ gives the claim." ], [ "Acknowledgements", "The authors would like to thank Bogdan Roman for providing Figure 2 and Figure 3.", "BA acknowledges support from the NSF DMS grant 1318894.", "ACH acknowledges support from a Royal Society University Research Fellowship as well as the UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/L003457/1.", "GK was supported in part by the Einstein Foundation Berlin, by Deutsche Forschungsgemeinschaft (DFG) Grant KU 1446/14, by the DFG Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics”, and by the DFG Research Center Matheon “Mathematics for key technologies” in Berlin.", "JM acknowledges support from the Berlin Mathematical School." ], [ "Proof of Theorem ", "The following lemma will be used in the upcoming proof Theorem REF .", "A one dimensional analogue can be found in [6].", "The proof extends straightforwardly and we omit it here.", "Lemma 7.1 Let $A_1, A_2, A_3$ , and $A_4 \\in \\mathbb {Z}, A_1 \\le A_2, A_3 \\le A_4$ .", "Moreover, let $L_1, L_2 \\in \\mathbb {N}$ such that $2 L_1 \\ge A_2 - A_1 +1$ and $2L_2 \\ge A_4 - A_3 +1$ .", "Then the trigonometric polynomial $\\Phi (z_1,z_2) = \\sum _{k = A_1}^{A_2} \\sum _{l=A_3}^{A_4} \\alpha _{k,l} e^{2\\pi i k z_1} e^{2\\pi i l z_2}$ satisfies $\\sum \\limits _{m=0}^{2L_1 - 1} \\sum \\limits _{n=0}^{2L_2 - 1} \\frac{1}{4L_1 L_2} \\left| \\Phi \\left(\\frac{m}{2L_1}, \\frac{n}{2L_2}\\right) \\right|^2 = \\sum _{k = A_1}^{A_2} \\sum _{l=A_3}^{A_4} |\\alpha _{k,l} |^2 .$ [Proof of Theorem REF ] Let $\\varphi \\in \\mathcal {R}_N$ such that $\\Vert \\varphi \\Vert =1$ .", "Then $\\varphi $ can be expanded as $\\varphi = \\sum _{n_1,n_2=0}^{2^{J_0}-1} \\alpha _{J_0,(n_1,n_2)} \\phi _{J_0,(n_1,n_2)} + \\sum _{k=1}^3 \\sum _{j = J_0}^{J-1} \\sum _{n_1,n_2 =0}^{2^j-1} \\beta ^k_{j,(n_1,n_2)} \\psi _{j,(n_1,n_2)}^{\\operatorname{b},k} $ We will now use the nestedness of the two dimensional MRA that is generated by the one dimensional wavelet system $\\left\\lbrace \\lbrace \\phi ^{\\operatorname{b}}_{J_0,m} \\rbrace _{m = 0, \\ldots , 2^J-1}, \\lbrace \\psi ^{\\operatorname{b}}_{j,n} \\rbrace _{j \\ge J_0,n = 0, \\ldots , 2^j-1}\\right\\rbrace $ .", "In particular, we have $V_j^{\\operatorname{b},2} \\subset V_{j+1}^{\\operatorname{b},2} , \\quad j\\ge J_0,$ and $V_{j+1}^{\\operatorname{b},2} = V_{j}^{\\operatorname{b},2} \\oplus W_{j}^{\\operatorname{b},2}, \\quad j\\ge J_0,$ with $V_j^{\\operatorname{b},2} = V_j^{\\operatorname{b}} \\otimes V_j^{\\operatorname{b}}$ and $W_j^{\\operatorname{b},2} = (V^{\\operatorname{b}}_j \\otimes W^{\\operatorname{b}}_j) \\oplus (W^{\\operatorname{b}}_j \\otimes V^{\\operatorname{b}}_j) \\oplus (W^{\\operatorname{b}}_j \\otimes W^{\\operatorname{b}}_j)$ .", "Loosely speaking, due to the MRA embedding properties we can expand functions from the reconstructions space into scaling functions $(\\phi ^{\\operatorname{b}}_{J,(n_1,n_2)})_{n_1,n_2}$ at highest scale.", "Since the left boundary functions can be constructed by translates of the initial scaling function $\\phi $ and the right scaling function can be obtained by reflecting the left boundary functions.", "The reflected function will be denoted by $\\phi ^{\\#}$ .", "This gives us in (REF ) $\\varphi = \\sum _{n_1,n_2=0}^{2^J-p-1} \\alpha _{n_1,n_2} \\phi (2^{J} \\cdot -n) + \\sum _{n_1,n_2 =2^J-p}^{2^J} \\beta _{n_1,n_2} \\phi ^{\\#}(2^{J} \\cdot -n),$ where only finitely many $\\alpha _{n_1,n_2}$ and $\\beta _{n_1,n_2}$ are non-zero.", "Now, for any $l= (l_1,l_2) \\in \\mathbb {Z}^2$ we obtain by basic properties of the Fourier transform $\\langle \\varphi &, s_l^{(\\varepsilon )} \\rangle = \\sum _{n_1,n_2=0}^{2^J-p-1} \\alpha _{n_1,n_2} \\frac{\\varepsilon }{ 2^{J}} e^{-2\\pi i \\varepsilon \\langle n, 2^{-J}l\\rangle }\\widehat{\\phi }\\left(\\frac{\\varepsilon }{ 2^{J}}l\\right) + \\sum _{n_1,n_2 =2^J-p}^{2^J} \\beta _{n_1,n_2} \\frac{\\varepsilon }{ 2^{J}} e^{-2\\pi i \\varepsilon \\langle n, 2^{-J}l\\rangle } \\widehat{\\phi ^{\\#}}\\left(\\frac{\\varepsilon }{ 2^{J}}l\\right).", "$ For the sake of brevity, we shall write in the following $\\Phi _1(z) &= \\sum _{n_1,n_2=0}^{2^J-p-1} \\alpha _{n_1,n_2} \\frac{\\varepsilon }{ 2^{J}} e^{-2\\pi i \\varepsilon \\langle n, z\\rangle },\\nonumber \\\\\\Phi _2(z) &= \\sum _{n_1,n_2 =2^J-p}^{2^J} \\beta _{n_1,n_2} \\frac{\\varepsilon }{ 2^{J}} e^{-2\\pi i \\varepsilon \\langle n, z\\rangle }.", "$ By our assumptions on the scaling function $|\\widehat{\\phi }(\\xi _1, \\xi _2)| \\lesssim \\frac{1}{ (1+|\\xi _1|)(1+ |\\xi _2|)}, $ and by the same argument $|\\widehat{\\phi ^{\\#}}(\\xi _1, \\xi _2)| \\lesssim \\frac{1}{(1+|\\xi _1|)(1+ |\\xi _2|)}.", "$ Using (REF ), (REF ), and (REF ) in (REF ) yields $\\sum _{l \\notin I_M} |\\langle \\varphi , s_l^{(\\varepsilon )} \\rangle |^2 \\nonumber &\\le \\sum _{l \\notin I_M} \\left| \\frac{\\varepsilon }{ 2^{J}}\\Phi _1\\left(\\frac{\\varepsilon }{2^{J}}l\\right) \\widehat{\\phi }\\left(\\frac{\\varepsilon }{ 2^{J}}l\\right) \\right|^2 + \\sum _{l \\notin I_M}\\left|\\frac{\\varepsilon }{ 2^{J}} \\Phi _2\\left(\\frac{\\varepsilon }{2^{J}}l\\right) \\widehat{\\phi ^{\\#}}\\left(\\frac{\\varepsilon }{ 2^{J}}l\\right) \\right|^2 + \\nonumber \\\\&\\makebox{[}0.5cm][c]{}+ 2 \\left( \\sum _{l \\notin I_M} \\left|\\frac{\\varepsilon }{ 2^{J}} \\Phi _1\\left(\\frac{\\varepsilon }{2^{J}}l\\right) \\widehat{\\phi }\\left(\\frac{\\varepsilon }{ 2^{J}}l\\right) \\right|^2 \\right)^{1/2} \\left( \\sum _{l \\notin I_M} \\left|\\frac{\\varepsilon }{ 2^{J}} \\Phi _2\\left(\\frac{\\varepsilon }{2^{J}}l\\right) \\widehat{\\phi ^{\\#}}\\left(\\frac{\\varepsilon }{ 2^{J}}l\\right) \\right|^2\\right)^{1/2}.", "$ We assume $2^J/\\varepsilon \\in \\mathbb {N}$ and the number of samples $M=(M_1,M_2) \\in \\mathbb {N}\\times \\mathbb {N}$ is $M_i = S \\cdot \\frac{2^J}{\\varepsilon }, \\quad i = 1,2$ where $S$ is some positive constant.", "Now, $(l_1,l_2) \\notin I_M$ if Case I: $|l_1| > M_1$ and $|l_2|<M_2$ , Case II: $|l_1| < M_1$ and $|l_2|>M_2$ , or Case III: $|l_1| > M_1$ and $|l_2|>M_2$ .", "It is sufficient to consider the sum for Case I.", "Case II can be obtained by symmetry and Case III yields a smaller sum.", "For $K = 2^J/\\varepsilon $ , we have $\\sum _{|l_2| <M_2}& \\sum _{|l_1| >M_1} \\left| \\frac{\\varepsilon }{ 2^{J}} \\Phi _1\\left(\\frac{\\varepsilon }{2^{J}}l\\right) \\widehat{\\phi }\\left(\\frac{\\varepsilon }{ 2^{J}}l\\right) \\right|^2 \\nonumber \\\\&= \\sum _{j_2}^{K-1} \\sum _{j_1}^{K-1} \\frac{1}{K}\\frac{1}{K}\\left| \\Phi _1\\left(\\frac{j_1}{K}, \\frac{j_2}{K}\\right) \\right|^2 \\sum _{|s_2|<S} \\sum _{|s_1|>S} \\left|\\widehat{\\phi }\\left(\\frac{j_1}{K} + s_1,\\frac{j_2}{K} + s_2\\right) \\right|^2 \\nonumber \\\\&\\lesssim \\sum _{j_2}^{K-1} \\sum _{j_1}^{K-1} \\frac{1}{K}\\frac{1}{K}\\left| \\Phi _1\\left(\\frac{j_1}{K}, \\frac{j_2}{K}\\right) \\right|^2 \\sum _{|s_2|<S} \\sum _{|s_1|>S} \\frac{1}{(1 + |j_1/K + s_1|)^2}\\frac{1}{(1 + |j_2/K + s_2|)^2} \\nonumber \\\\&\\le C_1 \\sum _{j_2}^{K-1} \\sum _{j_1}^{K-1} \\frac{1}{K}\\frac{1}{K}\\left| \\Phi _1\\left(\\frac{j_1}{K}, \\frac{j_2}{K}\\right) \\right|^2 \\frac{1}{S}.", "$ By Lemma REF we obtain in (REF ) $\\sum _{|l_2| >M_2}& \\sum _{|l_1| >M_1} \\left| \\frac{\\varepsilon }{ 2^{J}} \\Phi _1\\left(\\frac{\\varepsilon }{2^{J}}l\\right) \\widehat{\\phi }\\left(\\frac{\\varepsilon }{ 2^{J}}l\\right) \\right|^2 \\le C_1 \\sum _{n_1,n_2=0}^{2^J-1} |\\alpha _{n_1,n_2}|^2 \\frac{1}{S}.", "$ Since the functions form an orthonormal basis and $\\Vert \\varphi \\Vert =1$ we have $\\sum _{n_1,n_2\\in \\mathbb {N}} |\\alpha _{n_1,n_2}|^2 + \\sum _{n_1,n_2\\in \\mathbb {N}} |\\beta _{n_1,n_2}|^2 = 1$ and hence $\\sum _{n_1,n_2\\in \\mathbb {N}} |\\alpha _{n_1,n_2}|^2 \\le 1$ Similarly, one shows $\\sum _{|l_2| >M_2}& \\sum _{|l_1| >M_1} \\left| \\frac{\\varepsilon }{ 2^{J}} \\Phi _2\\left(\\frac{\\varepsilon }{2^{J}}l\\right) \\widehat{\\phi ^{\\#}} \\left(\\frac{\\varepsilon }{ 2^{J}}l\\right) \\right|^2 \\le C_2 \\sum _{n_1,n_2 \\in \\mathbb {N}}|\\beta _{n_1,n_2}|^2 \\frac{1}{S}.", "$ Therefore, in (REF ) we have that $\\sum _{|l_2| <M_2} \\sum _{|l_1| >M_1} |\\langle \\varphi , s_l^{(\\varepsilon )} \\rangle |^2 & \\le C_1 \\sum _{n_1,n_2\\in \\mathbb {N}} |\\alpha _{n_1,n_2}|^2 \\frac{1}{S} + C_2 \\sum _{n_1,n_2 \\in \\mathbb {N}} |\\beta _{n_1,n_2}|^2 \\frac{1}{S} +\\\\&\\qquad + 2 \\left(C_1 \\sum _{n_1,n_2\\in \\mathbb {N}} |\\alpha _{n_1,n_2}|^2 \\frac{1}{S}\\right)^{1/2} \\left(C_2\\sum _{n_1,n_2\\in \\mathbb {N}}|\\beta _{n_1,n_2}|^2 \\frac{1}{S}\\right)^{1/2} \\nonumber \\\\&\\le \\left( \\frac{\\sqrt{2C_1}+\\sqrt{2C_2}}{\\sqrt{S}}\\right)^2$ Now, for $\\theta >1$ choosing $S$ large enough, such that $\\left( \\frac{\\sqrt{2C_1}+\\sqrt{2C_2}}{\\sqrt{S}}\\right)^2\\le \\frac{\\theta ^2 - 1}{3\\theta ^2}$ gives the claim." ], [ "Acknowledgements", "The authors would like to thank Bogdan Roman for providing Figure 2 and Figure 3.", "BA acknowledges support from the NSF DMS grant 1318894.", "ACH acknowledges support from a Royal Society University Research Fellowship as well as the UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/L003457/1.", "GK was supported in part by the Einstein Foundation Berlin, by Deutsche Forschungsgemeinschaft (DFG) Grant KU 1446/14, by the DFG Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics”, and by the DFG Research Center Matheon “Mathematics for key technologies” in Berlin.", "JM acknowledges support from the Berlin Mathematical School." ] ]
1403.0172
[ [ "A New Hypothesis On The Origin and Formation of The Solar And Extrasolar\n Planetary Systems" ], [ "Abstract A new theoretical hypothesis on the origin and formation of the solar and extrasolar planetary systems is summarized and briefly discussed in the light of recent detections of extrasolar planets, and studies of shock wave interaction with molecular clouds, as well as H. Alfven's work on Sun's magnetic field and its effect on the formation of the solar system (1962).", "We propose that all objects in a planetary system originate from a small group of dense fragments in a giant molecular cloud (GMC).", "The mechanism of one or more shock waves, which propagate through the protoplanetary disk during the star formation is necessary to trigger rapid cascade fragmentation of dense clumps which in turn collapse quickly, simultaneously, and individually to form multi-planet and multi-satellite systems.", "Magnetic spin resonance may be the cause of the rotational directions of newly formed planets to couple and align in the strong magnetic field of a younger star." ], [ "Summary of The New Hypothesis", "The lifetime of giant molecular clouds and massive stars may be short, but the impact of their birth and death on the surrounding intercloud (ICM) and interstellar mediums (ISM) is profound [34], [33].", "The strong stellar winds and supernova (SN) explosions from hundreds to thousands of the massive stars create a rapid expanding hot bubble.", "The kinetic energy in such supersonic expansion is thermalized by a stand-off shock.", "The high pressure downstream drives a strong shock into the ambient ISM, or even breaks through the cloud into the adjacent ICM.", "These strong and fast shock waves contribute significantly to the formation and rotation of the GMCs.", "On the other hand, thermal and dynamical instabilities produce strong cooling and turbulence in shocked gas on small scales, which play a dominant role in providing a non-linear density perturbation through the cloud [18].", "This mechanism is needed during a GMC formation to create supersonic turbulence and to allow rapid fragmentation and formation of dense clumps locally and simultaneously before the onset of global gravitational collapse of the cloud.", "As observed in recent studies, for example, the interaction between W44 supernova remnant (SNR) and the adjacent GMC [8], [12], [26], a secondary shock wave may produce further cascade fragmentations, hence formation of smaller and higher density clumps in the cloud (see Fig.", "REF and Fig.", "REF ).", "These small dense cores in turn quickly collapse to form stars or a star cluster [28], [29].", "When a cloud (or a clump) has an initial non-zero angular momentum it will collapse to an accretion disk surrounding its central mass.", "Observations show that the mass spectrum of the star clusters follows the same power law distribution as that of the GMCs in the ISM [30], [25], [20].", "Figure: Schematic view of W44 shock wave propagates through the GMC and dense clumps detected by HCO+ J = 1 →\\rightarrow 0 and CO J = 3 →\\rightarrow 2 observations.", "Figure is obtained from Sashida et al.", "(2013), see details in Figure 8 of their paper.Figure: Schematic of rotation of a small group of dense clumps in a GMC; clump fragmentation and rotation of protostellar and protoplanet clumps.Core accretion and disk instability are the two currently accepted theories explaining how solar planets form [17], [22], [19].", "Core accretion model requires a few to a hundred million years to form a gas giant planet through the accretion of planetesimals from the protoplanetary disk, depending on the mass of the planet and how far it is to the parent star.", "This is a major problem for core accretion theory, because the time it takes to form a super giant planet (greater than a few times of the Jupiter mass) is much longer than the observed disk dissipation timescale of younger stars.", "For example, how can extrasolar planet GJ 504b, which has four times the Jupiter mass and about 160 million years old, form through planetesimals' collision and then core accretion at an orbital distance of 43 AU to its parent star GJ 504 [21]?", "The core accretion model also presents a problem for various planet migration theories [3].", "On the other hand, disk instability can form a super giant planet in a few hundred years, but the theory assumes that the protoplanetary disk is massive enough and can cool quickly enough to allow fragmentation to occur [5].", "In addition, sources and mechanisms for rapid fragmentation on small scales is needed to overcome global gravitational instability in the disk [18].", "Neither core accretion nor disk instability can explain the spin directions of solar planets, the conservation of angular momentum of the protostars, the evaporation of the protodisk, the formation of gas giant planet, and incredibly stable near-circular planetary orbits for billions of years, as well as why some low mass stars have planets while others have nothing around them, not even dust rings.", "In this study, we propose that 4.6 billion years ago our Sun along with its planets and satellites were born from a small group of dense clumps, located in one of the star clusters with thousands of stars [7], [11], [32].", "Like many other group of dense clumps, this group of clumps was initially compressed by a strong shock wave.", "The densest and largest clump quickly collapsed and formed a protostar of the Sun.", "Other compressed clumps that orbit around the protostar will eventually collapse due to gravitational instability (i.e.", "disk instability theory).", "However, if there is a secondary shock wave, e.g.", "an expanding thin shell of cold gas (produced by a nearby star formation) propagates into the disk before the onset of global gravitational collapse of dense clumps in the protoplanetary disk, the dense clumps behind the shock will then be further compressed.", "These compressed clumps may undergo rapid cascade fragmentation which break into smaller ones, and in turn become the seeds of multiple planets and their satellites.", "The secondary shock wave (SW$_2$ ) may be weaker than the supersonic shock wave that creates the GMC and star clusters, and it should have arrived at the solar protoplanetary disk before the Sun is completely formed.", "This shock wave sweeps up most of the interclump gas in the protoplanetary disk into its shell before it collides with the ionized gas shock (SW$_i$ ) front near the radius of $R_i$ = 0.277 AU corresponding to a 80 km s$^{-1}$ of critical velocity of ionized gas particles around the forming Sun [2], [1].", "The denser fragments behind the shock waves quickly collapse and form protoplanets and protosatellites that orbit around them, as a result of strong thermal and dynamical instabilities that dominate on the small scales [18], [6].", "The shock waves are partially refracted and partially reflected.", "Hence, part of the cold gas in the shell of SW$_2$ merges with the warm and (partially) ionized gas of SW$_i$ leaving behind the reflected shock waves an expanding non-uniform refracted gas region (0.3 AU $\\rightarrow $ 10 AU), where the cooling effect converts more warm gas into the cold gas (see Fig.", "REF ).", "The reflected shock waves of SW$_2$ and SW$_i$ are rarefaction waves indicated by SW$_2$$^\\prime $ and SW$_i$$^\\prime $ in Fig.", "REF .", "The inward expansion of the refracted gas is stalled by the high pressure plasma gas front at R$_i$ , so is the reflected shock wave of SW$_i$ .", "The outward expansion that is confined by the reflected shock wave of SW$_2$ becomes the gas supply for building gas giant planets.", "The dense clumps in warm gas zone (WGZ, 0.3 AU - 2.7 AU) collapse instantly and form the rocky planets, i.e.", "Mercury, Venus, Earth, and Mars, while collapsing clumps in neutral cold gas zone (CGZ, 2.7 AU - 10 AU) accrete sufficient cold gas to form gas giant planets Jupiter, Saturn, and their satellites.", "During the shock collision of SW$_2$ and SW$_i$ very small clumps (smaller than the Earth moon) that are supposed to form the satellites of rocky planets may attain or lose their orbital velocities and escape.", "Some of these clumps may be thrown inward toward the star where they will be heated and ionized, while other clumps may be ripped into very tiny fragments and pushed outward into the discontinuous gas region (at $\\sim $ 2.7 AU) between WGZ and CGZ.", "Eventually these tiny fragments from WGZ along with the condensed tiny cold gas clumps form the ring of the Asteroid belt (2 AU and 4 AU).", "The reflected shock wave of SW$_2$ continues propagating outward and beyond Uranus, Neptune, and Pluto, then gradually slows down and eventually stalls.", "It soon becomes fragmented and dispersed to be part of the cold icy gas rings (i.e.", "Kuiper belt), when it reaches the outer region of the solar system (30 - 50 AU from the Sun).", "This explains why materials from the early formation of the solar system are found in the Kuiper belt, when examining stardust samples return from Comet Wild 2 [9].", "In addition, during the stellar core accretion, much of the angular momentum is transferred from the forming star to its surrounding ionized gas, and then from the ionized gas to the angular momentum of protoplanets and their satellites during the head-on shock wave collision.", "The orbital parameters (especially their semi-major axis) for all planets and their satellites have changed very little since the time of their formation.", "A majority of the planets and their satellites have the same spin direction as the their orbital direction and spin direction of the Sun.", "If these planets and their satellites, the asteroids, were formed from merely random core accretions (i.e.", "planetesimal accretion theory), we would see an even mixture of directions of orbital revolution and rotation of planets and their satellites.", "However, if there is a certain mechanism as proposed in this study, which allows a small group of dense clumps that are orbiting around the forming Sun to undergo rapid fragmentation followed by a quick collapsing both locally and simultaneously, we are expected to see the formation of multiple planets and satellites like our solar system.", "Hence, the newly formed planets may experience a strong spin-spin coupling and begin to align each other in the external magnetic field generated by the plasma gas surrounding the forming parent star (see Fig.", "REF ).", "The spin axes of the planets will precess around the magnetic field.", "Magnetic spin resonance is the origin of the observed rotational directions of solar planets which follows the Pascal's rule of a quintet configuration (1:4:6:4:1), as indicated by blue-dashed box in Fig.", "REF .", "This also implies that water may exist in all newly formed planets in our solar system.", "Magnetic field of planets may also play a dominate role in spin-spin coupling of large satellites at the time of their formation.", "The magnetic fields of the Sun and its planets may have evolved over the past 4.6 billion years, but the spin coupling among the planets remains strong and stable.", "On the other hand, orbital coupling or resonance in our solar system is governed by the gravitational influences between the parent star, the planets, and their satellites [14], [15].", "Figure: Schematic of expanding shocked gas regions produced by two reflected shock waves (SW i _i ' ^\\prime and SW 2 _2 ' ^\\prime ) around a forming star.", "Rocky planets form in warm gas zone (WGZ), while gas giant planets form in cold gas zone (CGZ).Figure: Schematic of spin-spin coupling of planets in an external strong magnetic field of a younger star.", "The blue-dashed box shows the spin-spin coupling of eight planets in solar system.The initial conditions that determine the formation and final configuration of a star/planet system may vary from one group of dense clumps to another.", "Building such a stabilized system will strongly depend not only on the physical and chemical properties of its parent GMC and distribution of groups of dense clumps, but also on the triggering mechanisms for rapid fragmentation, core accretion, planet/satellite formation in the protoplanetary disk around a forming star.", "Therefore, each star/planet system may have its own unique characteristics depending on the initial conditions of the cloud, the clumps, and the ambient environment at the time of their formation.", "For example, recent detection of single planet systems found that HD 106906b (11 M$_{Jupiter}$ ) are located at a very large distance 650 AU to its parent star HD 106906 (K-type star) with only 13 million years age [4], and PSO J318.5-22 is a free-floating gas giant (6.5 M$_{Jupiter}$ ) formed without a parent star 12 million years ago [23].", "These two exoplanets may have formed directly from clump collapsing due to the onset of global gravitational instability in a cloud before they become fragmented.", "On the other hand, the HD 10180 is a multi-planet system; all of its planets are located at very close distances (0.02 - 3.5 AU) to their parent star HD 10180 (G1V star, 7.3 Gyr), but none of its planets has mass like gas giants (Jupiter or Saturn) [31].", "The Gliese 876 system has four planets, two of them have Jupiter mass but are much closer to their parent star (0.1 - 0.2 AU) than those in our solar system [27].", "The Gliese 876 system also has a notable orbital arrangement between its four planets, i.e.", "Laplace resonance configuration, which is similar to that of Jupiter's closest Galilean moons.", "On the other hand, four detected planets in the Upsilon Andromedae system are gas giant planets (0.6 - 4 M$_{Jupiter}$ ), located within 5 AU distance to their parent star [16].", "The star of the Upsilon Andromedae system is about 3.5 times more luminous than the Sun, and the distance of its inner most planet to its star is about six times closer than that of the Mercury to the Sun.", "Another multi-planet system that is similar to the Upsilon Andromedae system is the HR 8799 W system, but the inner most planet locates at a distance of 14 AU to its parent star that is 35 times further than that of the Mercury to the Sun [24].", "These extrasolar systems that have multiple planets may form in a similar way like our solar system, but have different clump properties, distribution, and protoplanetary disk conditions.", "From what has been said here, a hypothesis is an idea.", "More theoretical computation and observational studies are needed to draw more definite conclusions.", "These studies may include the properties of small groups of dense clumps and the interclump conditions in the GMCs, the role of shock waves, the interplay among thermal, dynamic and gravitational instabilities on large and small scales, their effects on planetary system formation and evolution, as well as observational measurements of gas and dust properties in nearby protoplanetary disks around forming/younger stars, e.g.", "the HD 142527 system [13].", "I would like to thank my colleagues Dr. Claus Leitherer at Space Telescope Science Institute and Dr. Joe Ganem at Loyola University Maryland for their encouragement." ] ]
1403.0168
[ [ "Local quanta, unitary inequivalence, and vacuum entanglement" ], [ "Abstract In this work we develop a formalism for describing localised quanta for a real-valued Klein-Gordon field in a one-dimensional box $[0, R]$.", "We quantise the field using non-stationary local modes which, at some arbitrarily chosen initial time, are completely localised within the left or the right side of the box.", "In this concrete set-up we directly face the problems inherent to a notion of local field excitations, usually thought of as elementary particles.", "Specifically, by computing the Bogoliubov coefficients relating local and standard (global) quantizations, we show that the local quantisation yields a Fock space $\\mathfrak F^L$ which is unitarily inequivalent to the standard one $\\mathfrak F^G$.", "In spite of this, we find that the local creators and annihilators remain well defined in the global Fock space $\\mathfrak F^G$, and so do the local number operators associated to the left and right partitions of the box.", "We end up with a useful mathematical toolbox to analyse and characterise local features of quantum states in $\\mathfrak F^G$.", "Specifically, an analysis of the global vacuum state $|0_G\\rangle\\in\\mathfrak F^G$ in terms of local number operators shows, as expected, the existence of entanglement between the left and right regions of the box.", "The local vacuum $|0_L\\rangle\\in\\mathfrak F^L$, on the contrary, has a very different character.", "It is neither cyclic nor separating and displays no entanglement.", "Further analysis shows that the global vacuum also exhibits a distribution of local excitations reminiscent, in some respects, of a thermal bath.", "We discuss how the mathematical tools developed herein may open new ways for the analysis of fundamental problems in local quantum field theory." ], [ "Introduction", "Quantum Field Theory (QFT in short) has proven to be one of the most successful theories in Physics.", "Its potential to describe the properties of elementary particles has been richly demonstrated within the framework of the Standard Model of Particle Physics.", "The extraordinary agreement between theoretical and experimental values of the muon $g-2$ anomaly [1], or the recent experimental success vindicating the Higgs mechanism after decades of search [2], [3], are just two examples among many.", "Elementary particles in modern physics are commonly thought of as small localised entities moving around in space.", "A careful examination, however, reveals such an interpretation to be problematic: in QFT a free particle is represented by a superposition of positive-frequency complex-valued modes which satisfy some field equation (e.g.", "the Klein-Gordon equation).", "Yet, no superposition of positive-frequency modes can be localised within a region of space, even for an arbitrarily small period of time [4].", "This confusing issue is sometimes mistaken as superluminality, see [5] for a clarification.", "In fact, it can be shown that the time derivative $\\dot{\\psi }$ , for any wave-packet $\\psi $ composed exclusively out of positive frequency modes, is non-zero almost everywhere in space.One way of seeing this is by noting that positive frequency solutions also satisfy the square root of the Klein-Gordon equation, i.e.", "the Schrödinger equation $i\\dot{\\phi }(\\vec{x},t)=\\sqrt{-\\nabla ^2 +m^2} \\phi (\\vec{x},t)$ .", "From there, using the antilocality property of the operator $\\sqrt{-\\nabla ^2 +m^2}$ , it follows that the time derivative $\\dot{\\phi }$ is necessarily non-zero almost everywhere in space [6].", "For that reason, even if $\\psi $ propagates in a perfectly causal manner according to the Klein-Gordon equation, it can hardly represent a localised entity.", "It is problematic to think of the fundamental field excitations of QFT as `particles' in any common sense of the word.", "The problem of localisation can be analysed from other angles, for example in terms of localisation systems.", "These are defined in terms of a set of projectors $E_\\Delta $ on bounded spatial regions $\\Delta $ whose expectation values yield the probability of a position measurement to find the particle within $\\Delta $ .", "A theorem by Malament [7] shows that in a Minkowski spacetime, under reasonable assumptions for the projector algebra, no such non-trivial set of projectors exists.", "There is also a general result (valid for both, relativistic or non-relativistic cases) due to Hegerfeldt [4] proving that, assuming a Hamiltonian with spectrum bounded from below, the expectation value of those projectors is non-zero for almost all times.", "In particular this applies also to states naively thought to be localised.", "Also along this line, but in order to describe unsharp localisation systems, Busch [8] replaced the use of projectors by more general operators, \"effects\" (or Positive-Operator Value Measures – POVM), showing that it is impossible to localise with certainty a particle in any bounded region of space.", "Furthermore, completing the collection of no-go theorems, Clifton and Halvorson [9] have shown, under a set of natural requirements, that it is not possible to define local number operators associated to any finite region of space.", "At this point it is also worthwhile mention the well-known problems of other efforts, based on the use of putative observables such as the Newton-Wigner position operator [10], [11], [12].", "In addition, there is also a different notion of localisation called strict localisability [13], [14].", "The basic idea is that a state, localised within a region of space at some specific moment in time, should be such that the expectation value of any operator associated to a spacelike separated region should be the same as in the vacuum.", "In other words, average values of local operators will depend on the state only if the observation is made in the region where the state is localised.", "However, as shown by Knight, no finite superposition of $N$ -particle states can be strictly localised.", "Some researchers have adopted the view that the notion of strict localizability is therefore too strong, and suggested that it should be relaxed by allowing for asymptotic localization, implemented by exponential fall-offs out of the localisation region.", "This was called essential localization and proposed as a criterion for deciding whether a QFT could describe particles [15].", "Although the results and theorems discussed above are well-understood mathematically, they nevertheless remain puzzling from a physical point of view, as they indicate that the quanta of QFT are not, at the fundamental level, particles in any common sense of the word.", "The situation is further complicated when we consider quantum fields in curved spacetimes, or in the presence of an external field, where there is, in general, no well-defined notion of a particle.", "This is the well-known particle number ambiguity, which have led some researchers to claim that the notion of particle is ultimately not a useful concept.", "For example, in his book [16], Wald writes: “Indeed, I view the lack of an algorithm for defining a preferred notion of `particles' in QFT in curved spacetime to be closely analogous to the lack of an algorithm for defining a preferred system of coordinates in classical general relativity.", "(Readers familiar only with presentations of special relativity based on the use of global coordinates might well find this fact to be alarming.)", "In both cases, the lack of an algorithm does not, by itself, pose any difficulty for the formulation of the theory.” R. Wald We shall not be concerned in this paper with the usefulness of the particle concept in QFT.", "We will rather make practical use of this ambiguity to provide a non-standard quantisation procedure yielding a QFT which, by construction, contains strictly localised one-particle states.", "Our approach can be viewed as a modification and further elaboration on a previous work by Colosi and Rovelli [17].", "Instead of quantizing the field using the standard stationary modes, we employ non-stationary modes which are, together with their time-derivatives, completely localised within a region of space at some arbitrary chosen time.", "These modes then evolve freely and spread out to become completely de-localised.", "The associated creation and annihilation operators can then be used to construct a local Fock space $\\mathfrak {F}^L$ which is distinct from the Fock space $\\mathfrak {F}^G$ associated with the standard quantisation based on the global (i.e.", "non-localisable) stationary modes.", "The local quantisation brings along a notion of strictly localised particle states which means that one or more assumptions of the theorems and results discussed above do not hold in our construction.", "Intriguingly, the local Fock space $\\mathfrak {F}^L$ can be shown to be unitarily inequivalent to the global Fock space $\\mathfrak {F}^G$ .", "This could be taken as an indication that the local quantization, and the associated localised particle states, are problematic.", "However, they yield a self-consistent QFT with well-defined state evolution and quanta having a well-defined energy expectation value after the relevant local vacuum energy has been subtracted.", "This paper is organised as follows.", "Section serves to fix notation and conventions as well as to provide the basic background material.", "In particular, we make explicit the arbitrariness of the choice of a complete set of orthonormal modes for the quantisation procedure.", "In Section we briefly discuss the standard quantisation based on stationary modes yielding the standard Fock space $\\mathfrak {F}^G$ .", "We then discuss the relationship between quantum theories obtained by different choices of modes and provide a sufficient condition for unitary inequivalence.", "In Section a new set of local modes is introduced in order to construct the local Fock space $\\mathfrak {F}^L$ .", "Later, in Section , we prove that the local and the global representations, are unitarily inequivalent.", "In Section we show the local one-particle states are strictly localised and evolve causally.", "We also prove that the Hamiltonian can be regularised by subtraction of the local vacuum energy.", "By showing in Section that the local creators and annihilators are well-defined operators in the global Fock space $\\mathfrak {F}^G$ , we end up with a mathematical toolbox enabling us to analyse and characterise states in $\\mathfrak {F}^G$ .", "We later check the properties of the vacuum in terms of local number operators.", "We exhibit the expectation values of the local number operators and quantify their correlations between the two regions.", "We also introduce a set of quasi-local states on $\\mathfrak {F}^G$ .", "In the section we study the properties of these quasi-local states, including the positivity of energy and their failure to be strictly localised, while comparing them to local and global states.", "Then we discuss the possibility of quantum steering using the vacuum and how it relates to the Reeh-Schlieder theorem.", "We end up with an outline of future extensions of this work and a summary of the conclusions." ], [ "Background material, notation, and conventions", "In this section we shall review some background material while fixing notations and conventions used throughout this paper." ], [ "Classical scalar field", "Consider a free real scalar field $\\phi (x,t)$ in a one dimensional cavity of size $R$ .", "Varying the Klein-Gordon action $S=\\frac{1}{2}\\int \\mathrm {d}x \\left(\\eta ^{\\mu \\nu }\\partial _\\mu \\phi \\partial _\\nu \\phi -\\mu ^2\\phi ^2\\right),$ and imposing Dirichlet boundary conditions $\\phi (0,t)=\\phi (R,t)=0$ , we obtain the Klein-Gordon equation $\\partial _\\mu \\partial ^\\mu \\phi +\\mu ^2\\phi =(\\Box + \\mu ^2)\\phi (x,t)=0,$ where we have put $\\hbar =c=1$ and $\\eta _{\\mu \\nu }=diag(+1,-1)$ .", "The linearity of the equation implies that the space of solutions forms a vector space $\\mathfrak {S}$ ." ], [ "Klein-Gordon inner product", "The classical field is throughout this paper taken to be real valued $\\phi (x,t) : [0, R] \\times \\mathbb {R}\\rightarrow \\mathbb {R}$ .", "Nevertheless, at the QFT level, complex valued solutions $\\phi : [0, R] \\times \\mathbb {R}\\rightarrow \\mathbb {C}$ occur naturally and describe one-particle states.", "The vector space $\\mathfrak {S}^{\\mathbb {C}}$ of complex valued solutions of (REF ) is equipped with a sesqui-linear (pseudo) inner product called the Klein-Gordon inner product: $(\\phi _1|\\phi _2)=i\\int _0^R dx\\phi _1^*(x,t)\\overset{\\leftrightarrow }{\\partial }_t\\phi _2(x,t)=i\\int _0^R dx(\\phi _1^*(x,t)\\dot{\\phi }_2(x,t)-\\dot{\\phi }_1^*(x,t)\\phi _2(x,t)),$ with $\\dot{}\\equiv \\partial _t$ .", "The quantity $(\\phi _1|\\phi _2)$ is conserved in time only if $\\phi _1$ and $\\phi _2$ are both solutions and subject to the same boundary conditions, i.e.", "$\\phi _1,\\phi _2 \\in \\mathfrak {S}^{\\mathbb {C}}$ .", "We note that the Klein-Gordon inner product is not positively definite on $\\mathfrak {S}^{\\mathbb {C}}$ .", "Thus, although $\\mathfrak {S}^{\\mathbb {C}}$ is a vector space, it is not a Hilbert space.", "In fact, the Klein-Gordon product partitions the solutions space $\\mathfrak {S}^{\\mathbb {C}}$ into three subsets of solutions: $\\phi \\in \\mathfrak {S}_+^{\\mathbb {C}}\\quad \\Rightarrow \\quad (\\phi |\\phi )>0,\\nonumber \\\\\\phi \\in \\mathfrak {S}_-^{\\mathbb {C}}\\quad \\Rightarrow \\quad (\\phi |\\phi )<0,\\nonumber \\\\\\phi \\in \\mathfrak {S}_0^{\\mathbb {C}}\\quad \\Rightarrow \\quad (\\phi |\\phi )=0,\\nonumber \\\\$ corresponding to solutions with positive, negative, and zero Klein-Gordon norm.", "Real-valued solutions are members of $\\mathfrak {S}_0^{\\mathbb {C}}$ .", "Moreover, neither of the three subsets $\\mathfrak {S}_+^{\\mathbb {C}}$ , $\\mathfrak {S}_-^{\\mathbb {C}}$ , and $\\mathfrak {S}_0^{\\mathbb {C}}$ form vector spaces, let alone Hilbert spaces." ], [ "Mode bases and the one-particle Hilbert space", "We can isolate a one-particle Hilbert space by introducing a complete and orthonormal basis, $\\lbrace f_m(x,t),f_m^*(x,t)\\rbrace $ with $m\\in \\mathbb {N}^+$ , of the vector space $\\mathfrak {S}^{\\mathbb {C}}$ .The structure we need in order to isolate a one-particle Hilbert space $\\mathfrak {H}$ in the solution space $\\mathfrak {S}^{\\mathbb {C}}$ is a complex structure [16].", "In our notation it takes the form $\\mathfrak {J}=i\\left( \\sum _N|f_m)(f_m|+|f_m^*)(f_m^*|\\right)$ .", "We will require all $f_m(x,t)$ to have positive norm, which implies that the complex conjugate ones $f^*_m(x,t)$ have negative norm.", "The orthonormality conditions read $(f_m|f_n)=\\delta _{mn},\\quad (f_m^*|f_n^*)=-\\delta _{mn},\\quad (f_m^*|f_n)=0.$ A set of modes form a complete set if for any solution $\\phi (x,t)\\in \\mathfrak {S}^{\\mathbb {C}}$ we have the following identity $\\phi (x,t)=\\sum _m(f_m|\\phi )f_m(x,t)-(f_m^*|\\phi )f_m^*(x,t),$ up to a zero measure set of points $x\\in [0, R]$ .", "Writing out this identity using the definition of the Klein-Gordon inner product (REF ) yields $\\phi (x,t)&=i\\int dx^{\\prime }\\sum _m \\left(f_m^*(x^{\\prime },t)f_m(x,t)-f_m(x^{\\prime },t)f_m^*(x,t)\\right)\\dot{\\phi }(x^{\\prime },t)\\nonumber \\\\&\\qquad -\\left(\\dot{f}_m^*(x^{\\prime },t)f_m(x,t)-\\dot{f}_m(x^{\\prime },t)f_m^*(x,t)\\right)\\phi (x^{\\prime },t).$ Since the Klein-Gordon equation is a second-order partial differential equation, $\\phi (x^{\\prime },t)$ and $\\dot{\\phi }(x^{\\prime },t)$ are independently specifiable.", "Thus, for the identity to hold for any solution $\\phi $ , and at any time $t$ , we deduce the following completeness relations $0&=\\sum _mf_m^*(x^{\\prime },t)f_m(x,t)-f_m(x^{\\prime },t)f_m^*(x,t),\\nonumber \\\\\\delta (x-x^{\\prime })&=i\\sum _m\\dot{f}_m(x^{\\prime },t)f_m^*(x,t)-\\dot{f}_m^*(x^{\\prime },t)f_m(x,t).$ If we restrict ourselves to real fields, any such a field $\\phi (x,t)$ can be expanded as $\\phi (x,t)&=\\sum f_m(x,t)a_m+f_m^*(x,t)a_m^*,$ where $a_m=(f_m|\\phi )$ are complex numbers and $a_m^*=-(f_m^*|\\phi )$ , the complex conjugates of $a_m$ .", "The Hilbert space of one-particle states $\\mathfrak {H}$ is then defined to be the vector space spanned by the positive norm modes $f_m$ , i.e.", "$\\mathfrak {H}=span(f_m).$ The Klein-Gordon product, when restricted to the subspace $\\mathfrak {H}\\subset \\mathfrak {S}^{\\mathbb {C}}$ , is by construction a positive definite sesqui-linear product.", "Therefore, $\\mathfrak {H}$ is a Hilbert space.", "In general $\\mathfrak {H}$ will depend on the choice of basis $\\lbrace f_m,f_m^*\\rbrace $ as defined in (REF ), leading to the well-known particle number ambiguity in QFT [18]." ], [ "Dirac notation", "To keep notation tidy and transparent it will be useful to introduce a Dirac notation to denote the vectors of $\\mathfrak {S}^{\\mathbb {C}}$ .", "To that end we make the identification $\\phi (x,t)\\sim |\\phi )\\in \\mathfrak {S}^{\\mathbb {C}}$ .", "We will also consider the dual space $\\mathfrak {S}^{\\mathbb {C}*}$ ; the vector space of linear maps $\\mathfrak {m}:\\mathfrak {S}^{\\mathbb {C}}\\rightarrow \\mathbb {C}$ .", "The Klein-Gordon product $(\\cdot |\\cdot )\\rightarrow \\mathbb {C}$ associates any vector $|\\phi )\\in \\mathfrak {S}^{\\mathbb {C}}$ to a member of the dual vector space through $(\\phi |\\cdot )\\in \\mathfrak {S}^{\\mathbb {C}*}$ and so we will write $(\\phi |\\in \\mathfrak {S}^{\\mathbb {C}*}$ .", "In this notation the completeness relations (REF ) take the succinct form $&\\sum _m |f_m)(f_m|-|f_m^*)(f_m^*|=1,$ where 1 denotes the identity operator on the vector space of solutions $\\mathfrak {S}^{\\mathbb {C}}$ .", "Note that we use `round' brackets $|\\phi )$ for vectors in $\\mathfrak {S}^{\\mathbb {C}}$ .", "In contrast we will use the standard brackets $|\\psi \\rangle $ to denote states of the corresponding QFT to which we now turn." ], [ "Quantization", "In order to quantise the real-valued classical field $\\phi (x,t)$ we first perform the Legendre transformation, which yields the Hamiltonian and canonical momenta $ H=\\int dx \\frac{1}{2}\\left[\\pi ^2 + (\\nabla \\phi )^2 +\\mu ^2\\phi ^2\\right],\\qquad \\pi =\\dot{\\phi }.$ Standard Dirac quantisation now requires us to turn $\\phi $ and $\\pi $ into operators $\\hat{\\phi }$ and $\\hat{\\pi }$ , satisfying equal-time canonical commutation relations $[\\hat{\\phi }(x,t), \\hat{\\pi }(y,t)]=i\\delta (x-y),\\quad [\\hat{\\phi }(x,t),\\hat{\\phi }(y,t)]=0, \\quad [\\hat{\\pi }(x,t),\\hat{\\pi }(y,t)]=0.$ For notational convenience and since no confusion arises, we will refer to these operators from now on as $\\phi $ and $\\pi $ , with the hats ` $\\hat{}$ ' omitted.", "In order to provide a Fock space representation of the commutator algebra (REF ) we expand the field in some complete and orthonormal basis $\\lbrace f_m,f_m^*\\rbrace $ and write $\\phi (x,t)&=\\sum _m f_m(x,t)a_m+f_m^*(x,t)a_m^{\\dagger },\\nonumber \\\\\\pi (x,t)&=\\dot{\\phi }(x,t)=\\sum _m \\dot{f}_m(x,t)a_m+\\dot{f}_m^*(x,t)a_m^{\\dagger },$ where $\\dot{f}_m\\equiv \\partial _t f_m$ , and $a_m$ and $a_m^\\dagger $ have been promoted into operators.", "If the modes $\\lbrace f_m,f_m^*\\rbrace $ satisfy the (second of the) completeness relations (REF ) then the following standard commutator algebra of creation and annihilation operators $[a_m,a_n^\\dagger ]=\\delta _{mn},\\qquad [a_m^{\\dagger },a_n^{\\dagger }]=0, \\quad [a_m,a_n]=0,$ ensures that we satisfy (REF ).", "As usual, we will define the vacuum state $|0\\rangle $ to be the state annihilated by all operators $a_m$ , i.e.", "$a_m|0\\rangle =0\\ \\forall m\\in \\mathbb {N}^+.$ A complete and orthonormal set of basis vectors $|n_1,n_2,\\dots \\rangle $ of the corresponding Fock space $\\mathfrak {F}$ is obtained by repeated application of the creation operators on the vacuum state: $|n_1,n_2,\\dots \\rangle =\\prod _m\\frac{(a_m^\\dagger )^{n_m}}{\\sqrt{n_m!", "}}|0\\rangle ,$ where the total number of particles in each basis state is required to be finite, $\\sum _{k} n_k < \\infty $ , ensuring that $\\mathfrak {F}$ is a separable Hilbert space [19].", "$\\mathfrak {F}$ is, as spanned by this basis, nothing but the symmetrised Fock space associated with the bosonic one-particle Hilbert space $\\mathfrak {H}$ , i.e.", "$\\mathfrak {F}(\\mathfrak {H})=\\bigoplus _{n=0}^{\\infty }\\mathop {\\mmlmultiscripts{\\bigotimes {S}{\\mmlnone }}}\\limits ^n \\mathfrak {H} =\\mathbb {C}\\oplus \\mathfrak {H} \\oplus (\\mathfrak {H}\\otimes _S\\mathfrak {H})\\oplus \\ldots .$ Here we note that the one-particle subspace spanned by the states $|1_m\\rangle \\equiv a_m^\\dagger |0\\rangle $ is indeed the same as $\\mathfrak {H}$ , or explicitly $f_m(x,t)=\\langle 0|\\phi (x,t)|1_m\\rangle $ [20]." ], [ "Non-uniqueness of the quantisation procedure", "In the previous section we described how to quantise a classical real-valued Klein-Gordon field, and deliberately kept the choice of orthonormal modes $\\lbrace f_k,f_k^*\\rbrace $ unspecified.", "Although this choice does not affect the classical field theory the situation is different at the QFT level.", "In fact, different choices of modes may lead to unitarily inequivalent Fock space representations.", "A well-known example in this regard is of course the Fulling-Rindler quantisation [21].", "Examples of a different kind are given in [22].", "By the Stone-von-Neumann theorem [23] this is something that can happen only for systems with infinitely many degrees of freedom, which is precisely the case of QFT [24]." ], [ "Standard (global) quantization", "The standard set of complete and orthonormal modes for a quantum field in a cavity is given by the normal modes $U_N(x,t)=\\mathcal {U}_N(x)e^{-i\\Omega _Nt}=\\frac{1}{\\sqrt{R\\Omega _N}}\\sin \\frac{\\pi N x}{R}e^{-i\\Omega _Nt},\\quad U_N^*(x,t)=\\mathcal {U}_N(x)e^{+i\\Omega _Nt},$ with $\\Omega _N^2=\\frac{\\pi ^2 N^2}{R^2}+\\mu ^2$ .", "We note that $\\lbrace U_N,U_N^*\\rbrace $ are all stationary solutions with the time dependence confined to a complex phase $e^{\\pm i\\Omega _Nt}$ .", "By computing the Klein-Gordon inner products, e.g.", "$(U_N|U_M)$ , it is easily checked that these modes satisfy the orthogonality conditions (REF ).", "That they form a complete set of modes, and so satisfy the completeness relations (REF ), follows from the fact that they are stationary: the first of the completeness relations is identically satisfied, while the second one is satisfied because of the identity of Fourier analysis $\\sum _N\\frac{2}{R}\\sin \\frac{N\\pi x}{R}\\sin \\frac{N\\pi x^{\\prime }}{R}=\\delta (x-x^{\\prime }).$ Thus we have: $\\sum _N |U_N)(U_N|-|U_N^*)(U_N^*| = 1.$ With this choice of modes, the field operator $\\phi $ and its conjugate momentum $\\pi $ take the form $\\phi (x,t)&=\\sum _N U_N(x,t)A_N+U^*_N(x,t)A_N^{\\dagger },\\nonumber \\\\\\pi (x,t)&=\\sum _N-i\\Omega _N\\left(U_N(x,t)A_N-U^*_N(x,t)A_N^{\\dagger }\\right).$ Now, by making use of the commutation relations (REF ), a very simple expression of the (regularised) Hamiltonian operator can be obtained $H^G=H-\\langle 0_G|H|0_G\\rangle =\\sum _N \\Omega _NA_N^\\dagger A_N,$ where the infinite vacuum energy $\\langle 0_G|H|0_G\\rangle =\\sum _N\\frac{1}{2}\\Omega _N$ has been removed.", "The state $|0_G\\rangle $ annihilated by all $A_N$ , will be referred to hereafter as the global vacuum.", "The basis vectors of the corresponding global Fock space, denoted by $\\mathfrak {F}^G$ , are then $|n_1,n_2,\\dots \\rangle =\\prod _N\\frac{(A_N^\\dagger )^{n_N}}{\\sqrt{n_N!", "}}|0\\rangle ,$ and correspond to energy eigenstates of the Hamiltonian $H^G$ .", "Needless to say, the usefulness of the global modes (REF ) stems from the fact that they diagonalise the Hamiltonian operator.", "We call the basis (REF ) a global basis, since no state in the corresponding one-particle Hilbert space $\\mathfrak {H}^G=span(U_N)$ can be fully contained within a subregion of $[0, R]$ for any arbitrarily small time interval $\\Delta t$ .", "As follows from a theorem by Hegerfeldt [4], there is no state such that $\\phi (x,\\tau )=\\dot{\\phi }(x,\\tau )=0$ for all $r<x<R$ at any time instant $t=\\tau $ .", "Instead, all states in $\\mathfrak {H}^G$ have, at almost all time, support in the entire cavity, i.e.", "they are global.", "As a matter of fact, the non-localizability of one-particle states in Minkowski spacetime is well-known and it has been noted and widely studied in several works, e.g.", "[13], [14], [5]." ], [ "Positive norm vs positive frequency", "It is important to stress that in the standard global quantisation the positive (negative) norm modes coincide with positive (negative) frequency modes; two conceptually distinct notions, which should not be confused.", "In fact, what is important for the quantisation procedure and the construction of a Fock space is not the partitioning of modes into positive and negative frequencies, but rather the partitioning into positive and negative norm modes.", "The latter notion does not require the basic field equations to admit symmetry under time translations but generalizes straightforwardly to non-stationary equations such as a quantum field in a time-dependent spacetime, or in the presence of a varying external field.", "This is so since the Klein-Gordon inner product, which defines the partitioning into positive and negative norm solutions, remains well defined also in these situations.", "We shall exploit this fact in the next section." ], [ "Bogoliubov transformations", "Let us explore here the relationship between quantizations based on different choices of modes.", "To this end, let $\\lbrace f_m,f^*_m\\rbrace $ and $\\lbrace \\tilde{f}_m,\\tilde{f}^*_m\\rbrace $ be two complete sets of orthonormal modes.", "Then we can expand the quantum field in two distinct ways: $\\phi (x,t)=&\\sum _m f_m(x,t)a_m+f_m^*(x,t)a_m^\\dagger =\\sum _m \\tilde{f}_m(x,t)\\tilde{a}_m+\\tilde{f}_m^*(x,t)\\tilde{a}_m^\\dagger .$ Using the orthogonality relations (REF ) we can immediately read off the relations $\\tilde{a}_m&=\\sum _n(\\tilde{f}_m|f_n)a_n+(\\tilde{f}_m|f_n^*)a_n^\\dagger ,\\\\\\tilde{a}_m^\\dagger &=\\sum _n(f_n|\\tilde{f}_m)a_n^\\dagger +(f_n^*|\\tilde{f}_m)a_n.$ The complex coefficients $(\\tilde{f}_m|f_n)$ , $(\\tilde{f}_m|f_n^*)$ , $(f_n|\\tilde{f}_m)$ , and $(f_n^*|\\tilde{f}_m)$ are the Bogoliubov coefficientsMore formally speaking, a Bogoliubov transformation is a transformation that preserves the symplectic structure in the case of classical fields, or the canonical commutation relations in a QFT..", "In the literature they are commonly denoted by $\\alpha $ and $\\beta $ (and its complex conjugate), defined by $\\tilde{f}_m=\\sum _n \\alpha _{mn}f_n+\\beta _{mn}f_n^*$ so that $\\alpha _{mn}\\equiv (f_n|\\tilde{f}_m),\\qquad \\beta _{mn}\\equiv -(f_n^*|\\tilde{f}_m).$" ], [ "A sufficient condition for unitary inequivalence", "We say that two Fock space representations are unitarily equivalent if there exists a unitary map $\\mathfrak {B}:\\mathfrak {F}\\rightarrow \\tilde{\\mathfrak {F}}$ that relates the Fock spaces associated with the representations, $\\mathfrak {F}$ and $\\tilde{\\mathfrak {F}}$ .", "Necessary and sufficient conditions for two Fock space representations to be unitarily equivalent are given in [16].", "In this paper we shall demonstrate unitary inequivalence of two Fock space representations and will therefore only need the following condition.", "Sufficient condition for unitary inequivalence: Two Fock-space representations are unitarily inequivalent if the vacuum state of one representation has infinitely many particles in terms of the number operator of the other representation, i.e.", "$\\sum _m\\langle \\tilde{0}|N_m|\\tilde{0}\\rangle =\\sum _m\\langle 0|\\tilde{N}_m|0\\rangle =\\sum _{mn}|(\\tilde{f}_m|f^*_n)|^2=\\infty ,$ where $a_m|0\\rangle =\\tilde{a}_m|\\tilde{0}\\rangle =0\\ \\forall m\\in \\mathbb {N}^+$ , $N_m\\equiv a_m^\\dagger a_m$ , and $\\tilde{N}_m\\equiv \\tilde{a}_m^\\dagger \\tilde{a}_m$ .", "Well-known cases of unitarily inequivalent representations can be found in [25], [21], [22], [15].", "In this paper we shall provide a new example.", "The elementary excitations of the field, defined by the Fock quantisation described in Section REF , consist of global modes which are also stationary.", "As already mentioned before, using only positive frequencies it is not possible to construct wave packets that are completely localised within a subregion $\\mathfrak {R}\\subset [0, R]$ of the cavity in the sense that $\\phi (x,t)=\\dot{\\phi }(x,t)=0$ if $x\\notin \\mathfrak {R}$ .", "This feature is a consequence of Hegerfeldt's theorem [4].", "Forcing $\\phi =0$ outside the region $\\mathfrak {R}$ implies a non-zero $\\dot{\\phi }$ outside the subregion resulting in a wave-packet that at an infinitesimal time later would become non-zero almost everywhere outside the subregion of localisation.", "For such a case, the Hamiltonian density would be non-zero outside the subregion, and in this sense, states in $\\mathfrak {H}^G$ cannot be localised.", "The standard quantisation of a free field relies on global non-localised excitations.", "Given the freedom in the choice of modes when quantizing a field (see Section REF ) it is suggestive to try, alternatively, to quantise the scalar field using modes representing local excitations.", "Such an excitation would be, at some instant $t=\\tau $ , localised and hereafter free to evolve and causally spread out.", "These local modes can then be used to find a Fock space representation of the canonical commutation relations as outlined previously, and a `local' Fock space $\\mathfrak {F}^L$ which hopefully admits strictly localised one-particle states.", "Nevertheless, as will be demonstrated in Section , the local Fock space $\\mathfrak {F}^L$ will turn out to be unitarily inequivalent to $\\mathfrak {F}^G$ .", "Figure: a) Simple scheme for quantisation in a cavity.", "Global modes are used to define the one-particle Hilbert space.", "b) The local modes (defined by imagining an instant partitioning of the cavity), can be used to define a local one-excitation space.", "They form a complete set of modes, which can expand the global modes almost everywhere.", "In particular we see in the figure how a decomposition in local modes (up to a cutoff) would look for a global mode N=1N=1 at t=0t=0." ], [ "Defining a new set of local modes", "In order to motivate the form of the local modes we consider what happens if we place a perfect mirror at $x=r$ , imposing a Dirichlet boundary condition at that point, $\\phi (x=r,t)=0,\\ \\forall t\\in \\mathbb {R}$ .", "Mathematically speaking we now have two distinct cavities, each with a quantum field.", "The complete set of orthonormal modes, $\\lbrace v_l(x,t),v_l^*(x,t)\\rbrace $ and $\\lbrace \\bar{v}_l(x,t),\\bar{v}_l^*(x,t)\\rbrace $ for the left and right cavities respectively, are taken to be the usual stationary modes $v_l(x,t)&=\\frac{1}{\\sqrt{r\\omega _l}}\\sin \\frac{l\\pi x}{r}e^{-i\\omega _l t},\\qquad v_l^*(x,t)=\\frac{1}{\\sqrt{r\\omega _l}}\\sin \\frac{l\\pi x}{r}e^{+i\\omega _l t},\\nonumber \\\\\\bar{v}_l(x,t)&=\\frac{1}{\\sqrt{\\bar{r}\\bar{\\omega }_l}}\\sin \\frac{l\\pi (x-r)}{\\bar{r}}e^{-i\\omega _l t},\\qquad \\bar{v}_l^*(x,t)=\\frac{1}{\\sqrt{r\\bar{\\omega }_l}}\\sin \\frac{l\\pi (x-r)}{\\bar{r}}e^{+i\\bar{\\omega }_l t},$ where $\\omega _l^2=\\frac{\\pi ^2 l^2}{r^2}+\\mu ^2$ , and $\\bar{\\omega }_{l}^2=\\frac{\\pi ^2 l^2}{\\bar{r}^2}+\\mu ^2$ , with $\\bar{r}=R-r$ .", "We now quantise the two systems yielding two quantum fields in two distinct cavities.", "The Fock spaces of the quantum excitations for each cavity are, by construction, localised within $[0, r]$ and $[r, R]$ , respectively.", "We could now try to analyse the quantum field in the entire cavity $[0, R]$ using such local excitations.", "It is clear that the introduction of a mirror at $x=r$ necessarily changes the physical conditions and we therefore are no longer dealing with the same physical system, i.e.", "the original cavity in $[0, R]$ .", "At the mathematical level, the introduction of the Dirichlet boundary condition changes the solution space to something different than $\\mathfrak {S}^{\\mathbb {C}}$ .", "Specifically, the modes $\\lbrace v_l(x,t),v_l^*(x,t)\\rbrace $ and $\\lbrace \\bar{v}_l(x,t),\\bar{v}_l^*(x,t)\\rbrace $ no longer form a basis for $\\mathfrak {S}^{\\mathbb {C}}$ .", "For this reason, modes of this type are not appropriate for quantizing the field of the full cavity $[0, R]$ .", "The remedy, however, is simple: instead we will use the local modes (REF ) to define the Cauchy initial conditions.", "Although we could take modes well localised at different moments in time, we shall only consider here, for simplicity, modes $\\lbrace u_l,u_l^ *\\rbrace $ and $\\lbrace \\bar{u}_l,\\bar{u}_l^{ *}\\rbrace $ localised at time $t=0$ .", "These modes are then free to spread out over the entire box $[0, R]$ with no Dirichlet boundary condition imposed at $x=r$ .", "This guarantees that they are still members of the complex solution space, i.e.", "$u_l,u_l^{*},\\bar{u}_l,\\bar{u}_l^{ *}\\in \\mathfrak {S}^{\\mathbb {C}}$ .", "In order to mimic the local modes we simply read off the initial conditions from the modes (REF ) evaluated at $t=0$ .", "This yields, $u_l(x,t=0)&=\\frac{\\theta (r-x)}{\\sqrt{r\\omega _l}}\\sin \\frac{l \\pi x}{r}=\\chi _l(x),\\qquad \\dot{u}_l(x,t=0)=-i\\omega _l\\chi _l(x),\\nonumber \\\\\\bar{u}_l(x,t=0)&=\\frac{\\theta (x-r)}{\\sqrt{\\bar{r}\\bar{\\omega }_l}}\\sin \\frac{l \\pi (x-r)}{\\bar{r}}=\\bar{\\chi }_l(x),\\qquad \\dot{\\bar{u}}_l(x,t=0)=-i\\bar{\\omega }_l\\bar{\\chi }_l(x).$ Before we determine the form of the local modes $\\lbrace u_l(x,t),u_l^*(x,t)\\rbrace $ and $\\lbrace \\bar{u}_l(x,t),\\bar{u}_l^*(x,t)\\rbrace $ for an arbitrary time $t$ , i.e.", "solve the Cauchy problem, we should make sure that they do indeed provide a complete and orthonormal basis for the complex solutions space $\\mathfrak {S}^{\\mathbb {C}}$ .", "Indeed, by explicit calculation (conveniently done at the specific time $t=0$ ) we can verify that $(u_m|u_l)=\\delta _{ml},\\qquad (u_m^*|u_l^*)=-\\delta _{ml}, \\qquad (\\bar{u}_m^*|\\bar{u}_l^*)=-\\delta _{ml},\\qquad ( \\bar{u}_m|\\bar{u}_)=\\delta _{ml}.$ That the modes form a complete set of solutions for $\\mathfrak {S}^{\\mathbb {C}}$ can be seen as follows.", "First we note that at time $t=0$ the modes coincide with the Fourier basis on $[0, r]$ and $[r, R]$ .", "By Carleson's theorem of Fourier analysis [26] we have pointwise convergence for almost all points $x\\in [0, R]$ , i.e.", "we have convergence in $L^2([0, R],\\mathbb {C})$ norm.We note that if the field $\\phi $ is expanded using the local modes, its value in that mode basis at $x=r$ at time $t=0$ is identically zero.", "Thus, we cannot expect to have convergence at $x=r$ .", "Nevertheless, for almost all other points in $[0, R]$ we will have pointwise convergence.", "This means that we can generate any initial conditions at $t=0$ (up to equivalence in $L^2([0, R],\\mathbb {C})$ norm) and thus any solution of $\\mathfrak {S}^{\\mathbb {C}}$ (Check Figure REF for an illustration).", "By relating the local modes to the global ones through the Bogoliobov transformations and using the well-known completeness properties for the latter, one can also show that the local modes satisfy (REF ) for an arbitrary time $t$ .", "Hence, in Dirac notation we have $\\sum _l |u_l)(u_l|+|\\bar{u}_l)(\\bar{u}_l|-|u_l^*)( u_l^*|-|\\bar{u}_l^*)( \\bar{u}_l^*|=1.$" ], [ "Bogoliubov coefficients and evolution", "In order to obtain the modes $u_m(x,t)$ and $\\bar{u}_m(x,t)$ for any time $t$ we simply make use of the completeness property (REF ): $|u_m)&=\\left(\\sum _N |U_N)(U_N|-|U_N^*)(U_N^*|\\right)|u_m)=\\sum _N (U_N|u_m)|U_N)-(U_N^*|u_m)|U_N^*),\\nonumber \\\\|\\bar{u}_m)&=\\left(\\sum _N |U_N)(U_N|-|U_N^*)(U_N^*|\\right)|\\bar{u}_m)=\\sum _N (U_N|\\bar{u}_m)|U_N)-(U_N^*|\\bar{u}_m)|U_N^*),$ or equivalently $u_m(x,t)&=\\sum _N (U_N|u_m)U_N(x,t)-(U_N^*|u_m)U_N^*(x,t),\\nonumber \\\\\\bar{u}_m(x,t)&=\\sum _N (U_N|\\bar{u}_m)U_N(x,t)-(U_N^*|\\bar{u}_m)U_N^*(x,t).$ The Bogoliubov coefficients, $(u_m|U_N)$ , $(u_m|U^*_N)$ , etc., are independent of which time $t$ we calculate them.", "Indeed, they can be conveniently calculated by easily taking $t=0$ and using the relations (REF ).", "A straightforward calculation then yields $(u_m|U_N)&=(\\omega _m+\\Omega _N)\\mathcal {V}_{mN},\\nonumber \\\\(u_m|U_N^*)&=(\\omega _m-\\Omega _N)\\mathcal {V}_{mN}, \\nonumber \\\\(\\bar{u}_m|U_N)&=(\\bar{\\omega }_m+\\Omega _N)\\bar{ \\mathcal {V}}_{mN}, \\nonumber \\\\(\\bar{u}_m|U_N^*)&=(\\bar{\\omega }_m-\\Omega _N)\\bar{ \\mathcal {V}}_{mN},$ where $\\mathcal {V}_{mN}&=\\int _0^R dx \\mathcal {U}_N(x)\\chi _m(x)=\\frac{1}{\\sqrt{Rr\\Omega _N\\omega _m}}\\frac{\\frac{m\\pi }{r}(-1)^m}{\\Omega _N^2-\\omega _m^2}\\sin \\frac{N\\pi r}{R},\\\\\\bar{\\mathcal {V}}_{mN}&=\\int _0^R dx \\mathcal {U}_N(x)\\bar{\\chi }_m(x)=-\\frac{1}{\\sqrt{R\\bar{r}\\Omega _N\\bar{\\omega }_m}}\\frac{\\frac{m\\pi }{\\bar{r}}(-1)^{m+N}}{\\Omega _N^2-\\bar{\\omega }_m^2}\\sin \\frac{N\\pi r}{R}.$ Using (REF ) we can see that the local modes at any time $t$ are given by: $u_m(x,t)&=\\sum _N\\left((\\omega _m+\\Omega _N)e^{-i\\Omega _Nt}-(\\omega _m-\\Omega _N)e^{i\\Omega _Nt}\\right) \\mathcal {V}_{mN} \\mathcal {U}_N(x),\\nonumber \\\\\\bar{u}_m(x,t)&=\\sum _N\\left((\\bar{\\omega }_m+\\Omega _N)e^{-i\\Omega _Nt}-(\\bar{\\omega }_m -\\Omega _N)e^{i\\Omega _Nt}\\right) \\bar{\\mathcal {V}}_{mN} \\mathcal {U}_N(x).$ Although it is not manifest from the form of the mode expansions (REF ), at $t=0$ the local modes $u_m$ and $\\bar{u}_m$ and their time derivatives $\\dot{u}_m$ and $\\dot{\\bar{u}}_m$ are zero outside their respective region of localisation.", "Furthermore, the local modes $u_k(x,t)$ and $\\bar{u}_k(x,t)$ and their time-derivatives spread out causally from the initial region .", "This is illustrated for the first-excited mode $u_{m=1}$ in Figure REF .", "Figure: Evolution of the first-excited local mode u m=1 (x,t)u_{m=1}(x,t) for different times t=0,0.1R...0.5Rt=0,0.1R \\ldots 0.5R.", "The mode is localised at t=0 in ℜ=[0,0.21R]\\mathfrak {R} =[0, 0.21R] within a cavity of size RR .", "The blue dashed line represents the light-cone.", "a) Massless case.", "b) Same but with μ=1/r=1/(0.21R)\\mu =1/r=1/(0.21R).", "c) Same but with μ=5/r=5/(0.21R)\\mu =5/r=5/(0.21R).We can verify that, after the localisation event in ℜ\\mathfrak {R} at t=0t=0 the elementary excitation causally spreads out, and so does its time derivative u ˙ k=1 (x,t)\\dot{u}_{k=1}(x,t).", "The mixing of both positive and negative global frequencies has allowed us to build up a localised mode avoiding the non-causal infinite tails that Hegerfeldt's theorem would imply." ], [ "Local quantization", "We now turn to the quantisation using these local modes.", "First we expand the field operator $\\phi (x,t)$ using the local modes $\\phi (x,t)=\\sum _m u_m(x,t)a_m+\\bar{u}_m(x,t)\\bar{a}_m+u_m^*(x,t)a_m^\\dagger +\\bar{u}^*_m(x,t)\\bar{a}_m^\\dagger .$ The expressions relating the local and global annihilators are given by $a_m&=\\sum _N (u_m|U_N)A_N+(u_m|U_N^*)A_N^\\dagger ,\\qquad a_m^\\dagger =\\sum _N (U_N|u_m)A_N^\\dagger +(U_N^*|u_m)A_N,\\nonumber \\\\\\bar{a}_m&=\\sum _N (\\bar{u}_m|U_N)A_N+(\\bar{u}_m|U_N^*)A_N^\\dagger ,\\qquad \\bar{a}_m^\\dagger =\\sum _N (U_N|\\bar{u}_m)A_N^\\dagger +(U_N^*|\\bar{u}_m)A_N.$ The commutation relations $[a_m,a_n]=0,\\quad [a_m, a_n^\\dagger ]=\\delta _{mn},\\quad [\\bar{a}_m,\\bar{a}_n]=0,\\quad [\\bar{a}_m, \\bar{a}_n^\\dagger ]=\\delta _{mn}, \\quad [a_m,\\bar{a}_n]=0,\\quad [a_m, \\bar{a}_n^\\dagger ]=0,\\nonumber $ and their Hermitian conjugates ensure that the canonical commutation relations (REF ) are satisfied.", "Besides, the local vacuum state $|0_L\\rangle $ is defined as the state annihilated by both $a_m$ and $\\bar{a}_m$ $a_m|0_L\\rangle =\\bar{a}_m|0_L\\rangle =0\\ \\forall m\\in \\mathbb {N}^+.$ The orthonormal basis vectors are given, as usual, by the repeated application of the creation operators $|n_1,n_2,\\dots \\rangle =\\prod _m\\frac{(a_m^\\dagger )^{n_m}}{\\sqrt{n_m!", "}}|0\\rangle _L, \\qquad |\\bar{n}_1,\\bar{n}_2,\\dots \\rangle =\\prod _{m}\\frac{(\\bar{a}_{m}^\\dagger )^{\\bar{n}_{m}}}{\\sqrt{\\bar{n}_{m!", "}}}|0\\rangle _L,$ We note that the creator and annihilation operators corresponding to different subregions necessarily commute.", "From this we see that the Fock space built from local modes has a tensor product structure $\\mathfrak {F}^L= \\mathfrak {f}\\otimes \\bar{\\mathfrak {f}},$ where $\\mathfrak {f}$ and $\\bar{\\mathfrak {f}}$ are Fock spaces associated with the two regions $[0, r]$ and $[r, R]$ .", "These Fock spaces are defined in the usual fashion by first defining vacuum states $|0\\rangle \\in \\mathfrak {f}$ and $|\\bar{0}\\rangle \\in \\bar{\\mathfrak {f}}$ and then the basis states by repeated application of the creators $a_m^\\dagger $ and $\\bar{a}_m^\\dagger $ .", "For example, the local vacuum for the whole cavity is then the tensor product $|0_L\\rangle =|0\\rangle \\otimes |\\bar{0}\\rangle $ and product states can be written as $|\\psi ,\\phi \\rangle =|\\psi \\rangle \\otimes |\\phi \\rangle $ .", "Notice that $|0_L\\rangle $ is not a standard vacuum [19].", "Indeed, $|0_L\\rangle $ is neither separating nor cyclic.", "It is not separating since $a_l|0_L\\rangle =0$ does not imply $a_l=0$ .", "It is not cyclic since it is a product state." ], [ "Unitary inequivalence", "So far we have shown that a quantisation based on a different choice of modes, i.e.", "the local modes, yields to a different Fock space $\\mathfrak {F}^L$ .", "However, as we shall now see, this Fock space is not unitarily related to the standard $\\mathfrak {F}^G$ ." ], [ "The unitary inequivalence of $\\mathfrak {F}^G$ and {{formula:81ac0f61-fb20-472a-955f-70de51932b9e}}", "By the sufficient condition for unitary inequivalence stated in Section REF , all we have to do is to demonstrate that the sum $\\sum _m\\langle 0_G|n_m+\\bar{n}_m|0_G\\rangle =\\sum _N\\langle 0_L|N_N|0_L\\rangle =\\sum _{m,N}\\left|(U^*_N|u_m)\\right|^2+\\left|(U^*_N|\\bar{u}_m)\\right|^2,$ diverges.", "To that end it is enough to establish that (REF ) diverges for each value of $N\\in \\mathbb {N}^+$ .", "Explicitly evaluating the sum yields $\\sum _{m}\\left|(U^*_N|u_m)\\right|^2+\\left|(U^*_N|\\bar{u}_m)\\right|^2=\\sum _{m} \\left|\\frac{\\sin \\frac{N\\pi r}{R}}{\\sqrt{Rr\\Omega _N\\omega _m}}\\frac{\\frac{m\\pi }{r}}{\\Omega _N+\\omega _m}\\right|^2+\\left|\\frac{\\sin \\frac{N\\pi r}{R}}{\\sqrt{R\\bar{r}\\Omega _N\\bar{\\omega }_m}}\\frac{\\frac{m\\pi }{\\bar{r}}}{\\Omega _N+\\bar{\\omega }_m}\\right|^2.$ We now proceed by making use of the integral test for convergence: the sum diverges iff the corresponding integral diverges.", "The integral is obtained by simply replacing the index $m$ with a continuous variable $x$ , i.e.", "$\\int _1^\\infty \\!\\!\\!\\!dx\\!\\left(\\!", "\\frac{\\sin ^2\\frac{N\\pi r}{R}}{Rr\\Omega _N\\sqrt{\\frac{\\pi ^2x^2}{r^2}+\\mu ^2}}\\frac{\\frac{x^2\\pi ^2}{r^2}}{\\left(\\Omega _N+\\sqrt{\\frac{\\pi ^2x^2}{r^2}+\\mu ^2}\\right)^2}+\\frac{\\sin ^2\\frac{N\\pi r}{R}}{Rr\\Omega _N\\sqrt{\\frac{\\pi ^2x^2}{\\bar{r}^2}+\\mu ^2}}\\frac{\\frac{x^2\\pi ^2}{\\bar{r}^2}}{\\left(\\Omega _N+\\sqrt{\\frac{\\pi ^2x^2}{\\bar{r}^2}+\\mu ^2}\\right)^2}\\!\\right)\\!.$ This integrand has the asymptotic behaviour $\\sim 1/x$ and therefore (REF ) diverges, which implies that $\\sum _N\\langle 0_L|N_N|0_L\\rangle = \\langle 0_L|N|0_L\\rangle =\\infty .$" ], [ "Analysis of the divergences", "In order to proceed, it is important to understand why the sum (REF ) diverges.", "As shown in the previous section this behaviour comes from summing over $m$ and not $N$ .", "Specifically, it is easy to show that although summing over $m$ yields an infinite result $\\langle 0_L|N_N|0_L\\rangle =\\sum _{m}\\left|(U^*_N|u_m)\\right|^2+\\left|(U^*_N|\\bar{u}_m)\\right|^2=\\infty .$ The same is not true when summing only over $N$ , i.e.", "we have $\\langle 0_G|n_m+\\bar{n}_m|0_G\\rangle =\\sum _{N}\\left|(U^*_N|u_m)\\right|^2+\\left|(U^*_N|\\bar{u}_m)\\right|^2<\\infty .$ Thus, the global number operators $N_N$ are ill defined in the local Fock space $\\mathfrak {F}^L$ , which also implies that $A_N$ and $A_N^\\dagger $ are not well-defined operators in $\\mathfrak {F}^L$ .", "Nevertheless, as we shall see in Section REF , it will turn out that the local number operators $n_m$ and $\\bar{n}_m$ are perfectly well defined in the global Fock space $\\mathfrak {F}^G$ .", "This mathematical asymmetry could be taken as a sign that the global Fock space $\\mathfrak {F}^G$ is in this respect preferred.", "However, as we shall see below in Section REF , the canonical Hamiltonian (REF ) can be regularised by subtracting the relevant infinite (local) vacuum energy thus rendering the energy expectation values of all basis states in $\\mathfrak {F}^L$ finite and well-defined.", "Furthermore, we will see in Section REF that states in $\\mathfrak {F}^L$ can be consistently evolved.", "In this sense, it seems that the unitarily inequivalent global and local quantum field theories are both possible quantizations of the real Klein-Gordon field in the one-dimensional box." ], [ "Strictly localised one-particle states in $\\mathfrak {F}^L$ and their causal evolution", "In this section we shall see that the local quantisation leads to a mathematically meaningful notion of local particles.", "We show that these states are strictly localised and that the evolution is causal." ], [ "Local quanta and their average energy", "The canonical Hamiltonian $H$ defined by equation (REF ) contains an infinite vacuum energy, which is regularised by subtraction, i.e.", "$H^G=H-\\langle 0_G|H|0_G\\rangle .$ This regularised Hamiltonian $H^G$ defines a notion of energy of states in the global Fock space $\\mathfrak {F}^G$ .", "Although the global Hamiltonian $H^G$ is an operator in $\\mathfrak {F}^G$ , some states in $\\mathfrak {F}^G$ may lie outside its domain and thus have an infinite/ill-defined average energy, being for this reason unphysical.", "We now turn to the question of whether we can define a meaningful notion of energy in the local Fock space $\\mathfrak {F}^L$ .", "A good guess is that the regularised Hamiltonian $H^L=H-\\langle 0_L|H|0_L\\rangle ,$ obtained by subtracting the infinite energy of the local vacuum, is well defined in the local Fock space $\\mathfrak {F}^L$ .", "Let us see how this works out.", "We first define $\\mathcal {E}$ as the expectation value of $H^G$ on the local vacuum, i.e.", "$\\mathcal {E}\\equiv \\langle 0_L|H^G|0_L\\rangle .$ Next we compute the energy expectation value of a local $n$ -particle state $\\langle m_l,\\bar{0}|H^G|m_l,\\bar{0}\\rangle $ .", "Substituting the Bogoliobov relations $A_N=\\sum _l (U_N|u_l)a_l+(U_N|u_l^*)a_l^\\dagger +(U_N|\\bar{u}_l)\\bar{a}_l+(U_N|\\bar{u}_l^*)\\bar{a}_l^\\dagger \\nonumber ,\\\\A_N^\\dagger =\\sum _l (u_l|U_N)a_l^\\dagger +(u_l^*|U_N)a_l+(\\bar{u}_l|U_N)\\bar{a}_l^\\dagger +(\\bar{u}_l^*|U_N)\\bar{a}_l,$ into the definition of $H^G$ we obtain $\\langle m_l,\\bar{0}|H^G|m_l,\\bar{0}\\rangle =m_l\\sum _N\\Omega _N\\left(|(u_l|U_N)|^2+|(u_l^*|U_N)|^2\\right)+\\mathcal {E}\\nonumber .$ For $m_l=0$ we would have $\\langle 0_L|H^G|0_L\\rangle =\\mathcal {E}$ and therefore we can write $\\langle m_l,\\bar{0}|H^L|m_l,\\bar{0}\\rangle =m_l\\sum _N\\Omega _N\\left(|(u_l|U_N)|^2+|(u_l^*|U_N)|^2\\right).$ From here we see that the local $n$ -particle state $|m_l,\\bar{0}\\rangle $ contains $m_l$ units of quanta with the manifestly positive average energy $\\epsilon _l=\\sum _N\\Omega _N\\left(|(u_l|U_N)|^2+|(u_l^*|U_N)|^2\\right).$ A simple integral test of convergence reveals that $\\epsilon $ is indeed convergent (the corresponding integrand has the asymptotic behaviour $\\sim \\frac{\\sin ^2x}{x^2}$ ).", "Thus, the regularised Hamiltonian $H^L$ yields finite expectation values for all $n$ -particle particle states $|m_l,\\bar{0}\\rangle $ .", "Repeating the above calculations we can also see that the $n$ -particle states $|0,\\bar{n}_l\\rangle $ have finite energy and so do all basis states $|n_l,\\bar{n}_m\\rangle $ .", "Thus, all basis states of $\\mathfrak {F}^L$ and finite superpositions of them will have finite average energy." ], [ "Strict localisation on the local vacuum", "We now proceed to construct strictly localised one-particle states in $\\mathfrak {F}^L$ .", "As briefly mentioned in the introduction, a state $|\\psi \\rangle $ is said to be strictly localised [13] within a region of space $\\mathfrak {R}$ if the expectation value of any local operator $\\mathcal {O}(x)$ outside that region (i.e.", "$x\\notin \\mathfrak {R}$ ) is identical to that of the vacuum, i.e.", "$\\langle \\psi |\\mathcal {O}(x)|\\psi \\rangle =\\langle 0|\\mathcal {O}(x)|0\\rangle \\,\\, \\mbox{if} \\,x\\notin \\mathfrak {R}.\\nonumber $ Since we have based our local quantisation on modes $u_m$ and $\\bar{u}_m$ which are localised within the regions $[0, r]$ and $[r, R]$ it is reasonable to expect that the one-particle excitation $|1_m,\\bar{0}\\rangle \\equiv a_m^\\dagger |0_L\\rangle =a_m^\\dagger |0,\\bar{0}\\rangle \\nonumber ,$ is strictly localised within $[0, r]$ .", "Indeed this is the case.", "The only operators we can build outside the region $[0, r]$ , i.e.", "in $[r, R]$ , are expansions in the annihilators and creators $\\bar{a}_m$ and $\\bar{a}_m^\\dagger $ , and these all commute with $a_m^\\dagger $ .", "Hence, we have $\\langle \\psi |\\mathcal {O}(\\bar{a}_m,\\bar{a}_m^\\dagger )|\\psi \\rangle &=\\langle 0_L|a_m\\mathcal {O}(\\bar{a}_m,\\bar{a}_m^\\dagger )a_m^\\dagger |0_L\\rangle =\\langle 0_L|\\mathcal {O}(\\bar{a}_m,\\bar{a}_m^\\dagger )a_m a_m^\\dagger |0_L\\rangle =\\langle 0_L|\\mathcal {O}(\\bar{a}_m,\\bar{a}_m^\\dagger )|0_L\\rangle ,\\nonumber $ verifying that the state $|1_m,\\bar{0}\\rangle $ is a strictly localised one-particle state.", "Clearly, the quantisation based on local non-stationary modes provides us with a natural notion of a local particle within the local QFT.", "Notice however that the notion of strict localisation introduced by Knight in [13] made use of the Minkowski vacuum based on stationary solutions of the Klein-Gordon equation.", "The analogous vacuum state would not be the local vacuum $|0_L\\rangle $ , but rather the global vacuum $|0_G\\rangle $ , which is also constructed using stationary modes.", "As a matter of fact, the possibility of strictly localised states in $\\mathfrak {F}^L$ has to do with the separability of $|0_L\\rangle =|0\\rangle \\otimes |\\bar{0}\\rangle $ , a property not shared by $|0_G\\rangle $ .", "Furthermore, local one-particle states do not belong to the global Fock space $\\mathfrak {F}^G$ , which is, as we have shown above, unitarily inequivalent to $\\mathfrak {F}^L$ .", "We see here that the possibility of local particle states is in our construction intimately related to the existence of unitarily inequivalent representations within QFT.", "This construction result should not be considered a mathematical counter-example to the no-go theorems presented in [13], [7], [12].", "Indeed, our system does not exhibit translational covariance since we are dealing with a finite box with Dirichlet boundary conditions imposed at the endpoints.", "It seems nonetheless plausible to us that additional assumptions might be violated in the limit of an infinite unbounded box admitting translation invariance.", "This possibility should be investigated further." ], [ "Causal propagation of local states", "The evolution of states in $\\mathfrak {F}^L$ is defined by the unitary operator $U^L(t)=\\exp (-iH^Lt)$ which trivially commutes with $H^L$ , implying that the total energy is conserved.", "We also note that none of the local $n$ -particle states are eigenstates of $H^L$ , in particular not the local vacuum $|0_L\\rangle $ .", "For this reason it will be interesting to study the evolution of these strictly localised states and verify whether they propagate causally, or not.", "To do this we shall have to introduce a third region $[\\tilde{r},R]$ with $\\tilde{r}>r$ and the local modes associated with it.", "We define these modes to be completely localised within $[\\tilde{r}, R]$ at a later moment in time $t=\\tau >0$ : $\\tilde{u}_l(x,t=\\tau )&=\\frac{\\theta (x-\\tilde{r})}{\\sqrt{\\tilde{r}\\tilde{\\omega }_l}}\\sin \\frac{l \\pi (x-\\tilde{r})}{R-\\tilde{r}}=\\bar{\\chi }_l(x),\\quad \\dot{\\tilde{u}}_l(x,t=0)=-i\\tilde{\\omega }_l\\tilde{\\chi }_l(x)\\quad \\tilde{\\omega }_l^2=\\frac{\\pi ^2 l^2}{(R-\\tilde{r})^2}+\\mu ^2\\nonumber $ This defines a new set of creators and annihilators $\\tilde{a}_l$ and $\\tilde{a}_l^\\dagger $ related to the global ones as $\\tilde{a}_l&=\\sum _N (\\tilde{u}_l|U_N)A_N+(\\tilde{u}_l|U_N^*)A_N^\\dagger \\nonumber ,\\\\\\tilde{a}_l^\\dagger &=\\sum _N (U_N|\\tilde{u}_l)A_N^\\dagger +(U_N^*|\\tilde{u}_l|)A_N.$ The local operators $\\tilde{\\mathcal {O}}(\\tau )$ associated with the region $[\\tilde{r}, R]$ at time $t=\\tau $ will be generated by series expansions in $\\tilde{a}_l$ and $\\tilde{a}_l^\\dagger $ .", "We can now calculate the commutator $[a_m,\\tilde{a}_n^\\dagger ]$ obtaining $[\\tilde{a}_n,a_m^\\dagger ]&=\\sum _{M,N}\\left[(\\tilde{u}_n|U_N)A_N+(\\tilde{u}_n|U_N^*)A_N^\\dagger ,(U_M|u_m)A_M^\\dagger +(U_M^*|u_m)A_M\\right]\\nonumber \\\\&=\\sum _{M,N}\\left[(\\tilde{u}_n|U_N)(U_M|u_m)[A_N,A_M^\\dagger ]+(\\tilde{u}_n|U_N^*)(U_M^*|u_m)[A_N^\\dagger ,A_M]\\right]\\nonumber \\\\&=(\\tilde{u}_n|\\left(\\sum _{N}|U_N)(U_N-|U_N^*)(U_N^*|\\right)|u_m)=(\\tilde{u}_n|u_m)\\nonumber .$ An identical calculation yields $[\\tilde{a}_n,a_m]=-(\\tilde{u}_n|u_m^*)$ .", "The fact that the local modes propagate causally (see Section REF ) means that $(\\tilde{u}_n|u_m)$ and $(\\tilde{u}_n|u_m^*)$ are zero whenever $\\tau <|r-\\tilde{r}|$ , which in turn implies that $a_m$ and $a_m^\\dagger $ commute with $\\tilde{a}_n$ and $\\tilde{a}_n^\\dagger $ .", "Thus, any local observable $\\tilde{\\mathcal {O}}(\\tau )$ will commute with $a_m^\\dagger $ and $a_m$ whenever $\\tau <|r-\\tilde{r}|$ , that is, whenever the spacetime regions associated with the operators $\\tilde{\\mathcal {O}}(\\tau )$ and the pair $\\lbrace a_m,a_n^\\dagger \\rbrace $ are spacelike.", "This way, micro-causality is built into the construction.", "Besides, we have clearly that $\\langle 1_m,\\bar{0}|\\tilde{\\mathcal {O}}(\\tau )|1_m,\\bar{0}\\rangle =\\langle 0_L|a_m\\tilde{\\mathcal {O}}(\\tau )a_m^\\dagger |0_L\\rangle =\\langle 0_L|\\tilde{\\mathcal {O}}(\\tau )a_ma_m^\\dagger |0_L\\rangle =\\langle 0_L|\\tilde{\\mathcal {O}}(\\tau )|0_L\\rangle ,$ for $\\tau <|r-\\tilde{r}|$ , which implies that the local one-particle state $|1_m,0\\rangle $ propagates causally as it should.", "This situation should be contrasted to Knight's strict localisation [13] which would state $\\langle 0_G|a_m\\tilde{\\mathcal {O}}(\\tau )a_m^\\dagger |0_G\\rangle =\\langle 0_G|\\tilde{\\mathcal {O}}(\\tau )|0_G\\rangle ,$ which in fact does not hold since, as will become clear below, $a_ma_m^\\dagger |0_G\\rangle \\ne |0_G\\rangle $ ." ], [ "Local analysis of the global vacuum", "In Section REF , we pointed to a mathematical asymmetry between the local and global quantum theories.", "We saw that, while the global number operators are ill defined in $\\mathfrak {F}^L$ , the case is different for the local number operators as defined in $\\mathfrak {F}^G$ .", "In this section we shall demonstrate that the local creators and annihilators are indeed well-defined operators in $\\mathfrak {F}^G$ , which will allows us to analyse global states using number operators associated with the local quantisation.", "In particular, we will examine the spectrum of local particles and numerically quantify existent space-like correlations of the global vacuum $|0_G\\rangle $ ." ], [ "Local operators in $\\mathfrak {F}^G$", "Let us now show that the local creator and annihilators $a_l$ , $a_l^\\dagger $ , $\\bar{a}_l$ , and $\\bar{a}_l^\\dagger $ are well-defined operators in $\\mathfrak {F}^G$ .", "Here we will prove this for $a_l$ .", "The proof is identical for $a_l^\\dagger $ , $\\bar{a}_l$ , and $\\bar{a}_l^\\dagger $ .", "It suffices to show that $\\langle \\psi |a_l^\\dagger Na_l|\\psi \\rangle <\\infty $ for any basis state $|\\psi \\rangle =|n_1,n_2,\\dots \\rangle $ of $\\mathfrak {F}^G$ .", "We first expand the local annihilator $a_l=\\sum _N (u_l|U_N)A_N+(u_l|U_N^*)A_N^\\dagger .$ We have that $a_l|n_1,\\dots ,n_N,\\dots \\rangle &=\\left(\\sum _N (u_l|U_N)A_N+(u_l|U_N^*)A_N^\\dagger \\right)|n_1,\\dots ,n_N,\\dots \\rangle \\nonumber \\\\&\\!\\!\\!\\!=\\sum _N (u_l|U_N)\\sqrt{n_N}|n_1,\\dots ,n_N-1,\\dots \\rangle +(u_l|U_N^*)\\sqrt{n_N+1}|n_1,\\dots ,n_N+1,\\dots \\rangle \\nonumber .$ Multiplying by the number operator $N=\\sum _NN_N$ , we obtain $Na_l|n_1,\\dots ,n_N,\\dots \\rangle &=\\sum _N (u_l|U_N)\\sqrt{n_N}(n-1)|n_1,\\dots ,n_N-1,\\dots \\rangle \\nonumber \\\\&\\qquad +(u_l|U_N^*)\\sqrt{n_N+1}(n+1)|n_1,\\dots ,n_N+1,\\dots \\rangle ,$ where $n$ is the number of particles of the basis state, i.e.", "$N|n_1,n_2,\\dots \\rangle \\equiv n|n_1,n_2,\\dots \\rangle $ .", "Sandwiching with $\\langle n_1,\\dots ,n_N,\\dots |a_l^\\dagger $ now gives $\\langle n_1,\\dots ,n_N,\\dots |a_l^\\dagger N a_l|n_1,\\dots ,n_N,\\dots \\rangle &=\\sum _N |(u_l|U_N)|^2n_N(n-1)+|(u_l|U_N^*)|^2(n_N+1)(n+1)\\nonumber .$ Since $\\sum _N |(u_l|U_N)|^2<\\infty ,\\qquad \\sum _N |(u_l|U_N^*)|^2<\\infty ,$ and given that $|n_1,n_2,\\dots \\rangle $ is a basis state of $\\mathfrak {F}^G$ (therefore satisfying $n=\\sum _Nn_N<\\infty $ ), we see that the action of $a_l$ on any basis state is not pathological .", "An analogous demonstration with minor changes shows that finite expectation values for the global Hamiltonian are also obtained for these vectors.", "Thus, since both demonstrations also go through for $a_l^\\dagger $ , $\\bar{a}_l$ , and $\\bar{a}_l^\\dagger $ , we have shown that the local creators and annihilators are well-defined linear operators in $\\mathfrak {F}^G$ and gi.", "Nevertheless, as we have stressed above in Section REF , the situation is not symmetric since $A_N$ and $A_N^\\dagger $ are not well defined in $\\mathfrak {F}^L$ ." ], [ "Local particle spectrum of the global vacuum", "The global vacuum is defined to have zero global particles, i.e $\\langle 0_G|N_N|0_G\\rangle =0$ .", "On the other hand, the local quantisation developed above yields a natural notion of local particle number $n_l=a_l^\\dagger a_l$ , $\\bar{n}_l=\\bar{a}_l^\\dagger \\bar{a}_l$ , corresponding to the number of local excitations we have in the left and right regions of the box, $[0, r]$ and $[r, R]$ , respectively.", "Figure: Number of local quanta of energy ω l \\omega _l expected value for the global vacuum for different masses μ=μ ˜ R\\mu =\\frac{\\tilde{\\mu }}{R} with μ ˜∈(10,50)\\tilde{\\mu }\\in (10,50).", "The region of localisation is taken to be r ˜ -1 =R/r=π\\tilde{r}^{-1}=R/r=\\pi .", "The inset shows discrete values in the same interval for the masses.", "Higher plots correspond to smaller values.", "The distribution of local particles in the global vacuum resembles a Planckian spectrum, i.e.", "a thermal bath of particles.Let us now ask what the distribution of local particles is for the global vacuum.", "To see this we compute the expectation values $\\langle 0_G|n_l|0_G\\rangle =\\langle 0_G|a^\\dagger _l a_l|0_G\\rangle =\\sum _{N}\\frac{l^2\\pi ^2}{Rr^3\\Omega _N\\omega _l}\\frac{1}{(\\Omega _N+\\omega _l)^2}\\sin ^2\\frac{\\pi N r}{R}.$ These depend on three distinct quantities: the size of the cavity $R$ , the size of the region of localisation $r<R$ , and the mass $\\mu $ .", "We could plot the expectation values for different values of these three magnitudes.", "However, it is more adequate to vary dimensionless quantities, e.g.", "$r/R$ , $r\\mu $ , and $R\\mu $ .", "We might as well fix $R=1$ , ending up with two independent dimensionless quantities $\\tilde{r}=r/R$ and $\\tilde{\\mu }=R\\mu $ .", "Figures REF and REF show the dependence of the expectation values (REF ) on these two variables.", "In Figure REF we see that when we increase the mass $\\mu $ we have that $\\langle 0_G|n_l|0_G\\rangle =\\sum _N|(u_l|U_N^*)|^2\\rightarrow 0.\\nonumber $ In fact, in the large mass limit the coefficients $(u_l|U_N^*)$ have the asymptotic behaviour $\\sim \\mu ^{-2}$ while $(u_l|U_N)$ converge to a non-zero value.", "Indeed, it is well known that the Compton wavelength $\\lambda _C=\\mu ^{-1}$ determines how well localised a wave-packet, made out of positive frequency modes, can be [10], [27], [28].", "Thus, in the limit $\\lambda _C\\rightarrow 0$ , or equivalently $\\mu \\rightarrow \\infty $ , the $\\beta $ -coefficients $(u_l|U_N^*)$ should approach zero.", "Another interesting limit is when $r\\rightarrow R$ , case in which local and global modes converge.", "Intuitively we would expect the local description to approach the global one so that the expectation value of local particles goes to zero (since the global vacuum is defined to have zero global particles).", "This is illustrated in Figure REF .", "This intuition can be made mathematically precise by studying the convergence of the operators $a_m\\rightarrow A_m$ as $r\\rightarrow R$ .", "We note that for any notion of convergence to make mathematical sense, the operators must act in the same vector space.", "For example, it is meaningless to claim that $a_k$ converges to $A_N$ as operators defined in the local Fock space $\\mathfrak {F}^L$ .", "Indeed, the operators $A_N$ are not even well defined in $\\mathfrak {F}^L$ .", "Nonetheless, it is meaningful to study the convergence $a_k\\rightarrow A_k$ as operators defined in $\\mathfrak {F}^G$ .", "The relationship between the operators is given by $a_l=\\sum _N(u_l|U_N)A_N+(u_l|U_N^*)A_N^\\dagger ,$ where $(u_l|U_N)&=\\frac{1}{\\sqrt{Rr\\Omega _N\\omega _l}}\\frac{\\frac{l\\pi }{r}(-1)^l}{\\Omega _N-\\omega _l}\\sin \\frac{N\\pi r}{R},\\nonumber \\\\(u_l|U_N^*)&=-\\frac{1}{\\sqrt{Rr\\Omega _N\\omega _l}}\\frac{\\frac{l\\pi }{r}(-1)^l}{\\Omega _N+\\omega _l}\\sin \\frac{N\\pi r}{R}.$ From here it is easy to show that $(u_l|U_N^*)\\rightarrow 0$ and $(u_l|U_N)\\rightarrow \\delta _{lN}$ in the limit $r\\rightarrow R$ .", "It is now clear that we have convergence of $a_l$ and $A_l$ in the strong operator topology.", "It is important to note that because of unitary inequivalence, the total number of local particles is necessarily infinite, i.e.", "$\\sum _m\\langle 0_G|n_m+\\bar{n}_m|0_G\\rangle =\\infty $ .", "In fact, even though Bogoliobov coefficients converge to finite values when $r\\rightarrow R$ , the sum diverges for any $r$ arbitrarily close to $R$ .", "This is due to the fact that the sum over $m$ and the limit $r \\rightarrow R$ do not commute, i.e.", "$\\lim _{r\\rightarrow R} \\sum _m\\langle 0_G|n_m+\\bar{n}_m|0_G\\rangle \\ne \\sum _m \\lim _{r\\rightarrow R}\\langle 0_G|n_m+\\bar{n}_m|0_G\\rangle =0.$ Figure: Number of particles for the global vacuum for different sizes of the localisation region r∈(0.25R,R)r \\in (0.25R,R) with fixed R=1R=1 and λ c =R 10\\lambda _c=\\frac{R}{10}.", "As expected, when r=Rr=R the expectation value of the vacuum is zero for all modes, since local and global modes are the same.Another interesting case is the limit $r\\rightarrow 0$ .", "Inspecting the coefficients reveal that both $(u_l|U_N)$ and $(u_l|U_N^*)$ have the asymptotic behaviour $\\sim r$ and thus vanish in the limit.", "However, the sum $\\sum _m\\langle 0_G|n_m|0_G\\rangle $ approaches a finite non-zero value when $r\\rightarrow 0$ ." ], [ "Vacuum entanglement", "As a second application we will look at vacuum entanglement.", "We shall study the entanglement between the two regions $[0, r]$ and $[r, R]$ by computing the correlations between local particle numbers as given by $\\text{cov}(n_m,\\bar{n}_l)$ defined by $\\text{cov}(n_m,\\bar{n}_l)&\\equiv \\langle \\psi |n_m\\bar{n}_n|\\psi \\rangle -\\langle \\psi |n_m|\\psi \\rangle \\langle \\psi |\\bar{n}_n|\\psi \\rangle .$ We note that if we choose $|\\psi \\rangle =|0_L\\rangle $ then $\\text{cov}(n_n,\\bar{n}_m)$ is identically zero.", "However, this is not so for the global vacuum $|\\psi \\rangle =|0_G\\rangle $ .", "The correlations of the global vacuum are more conveniently characterised by the dimensionless values $\\text{corr}(n_m,\\bar{n}_n)&=\\frac{\\langle 0_G|n_m\\bar{n}_n|0_G\\rangle -\\langle 0_G|n_m|0_G\\rangle \\langle 0_G|\\bar{n}_n|0_G\\rangle }{\\sqrt{\\langle 0_G|n_m^2|0_G\\rangle -\\langle 0_G|n_m|0_G\\rangle ^2}\\sqrt{\\langle 0|\\bar{n}_n^2|0_G\\rangle -\\langle 0_G|\\bar{n}_n|0_G\\rangle ^2}}\\nonumber \\\\&=\\frac{\\text{cov}(n_m, \\bar{n}_n)}{\\sqrt{\\text{cov}(n_m, n_m) \\text{cov}(\\bar{n}_n, \\bar{n}_n)}},$ which are known as the correlation coefficients.", "Figure: Values for the dimensionless correlation coefficients 𝒞(n m ,n ¯ n )\\mathcal {C}(n_m, \\bar{n}_n) for the extreme cases of a massless field (a), and highly massive field (b), with μ=1000/R\\mu = 1000/R.", "For both cases the localisation region ℜ\\mathfrak {R} has a size r=R/πr= R/\\pi .", "As we can see, modes with the same frequency are the most correlated ones.", "Notice that instead of plotting with respect to the mode indexes, we are using the mode frequencies.Figure: (a) A 2D sketch of the features observed in figure a.", "(b) Exact 2D plot of the correlation coefficients as shown in figure a.From equation (REF ) we have $\\langle 0_G|n_m\\bar{n}_n|0_G\\rangle =\\langle 0_G|a_m^\\dagger a_m\\bar{a}_n^\\dagger \\bar{a}_n|0_G\\rangle =\\\\=\\sum _{M,N}(U_M^*|u_m)(u_m|U_N)(U_N|\\bar{u}_n)(\\bar{u}_n|U_M^*) +(U_M^*|u_m)(u_m|U_N)(U_M|\\bar{u}_n)(\\bar{u}_n|U_N^*)\\\\+(U_M^*|u_m)(u_m|U_M^*)(U_N^*|\\bar{u}_n)(\\bar{u}_n|U_N^*)$ On the other hand we have that $\\langle 0_G|n_m|0_G\\rangle \\langle 0_G|\\bar{n}_n|0_G\\rangle =\\sum _{M,P}(U_M^*|u_m)(u_m|U_M^*)(U_P^*|\\bar{u}_n)(\\bar{u}_n|U_P^*)$ and thus $\\langle 0_G|n_m\\bar{n}_n|0_G\\rangle -\\langle 0_G|n_m|0_G\\rangle \\langle 0_G|\\bar{n}_n|0_G\\rangle =\\\\=\\sum _{M,P}(U_M^*|u_m)(u_m|U_P)(U_P|\\bar{u}_n)(\\bar{u}_n|U_M^*)+ (U_M^*|u_m)(u_m|U_P)(U_M|\\bar{u}_n)(\\bar{u}_n|U_P^*)$ and using the computed inner products (REF ) and equation (REF ) we obtain $\\text{corr}(n_m,\\bar{n}_n)&=\\frac{\\frac{2\\pi ^4m^2n^2}{R^2r^3\\bar{r}^3\\omega _m\\bar{\\omega }_n}\\sum _{N,P}\\left[\\frac{(-1)^{N+P}\\sin ^2\\frac{N\\pi r}{R}}{\\Omega _N\\Omega _P(\\Omega _N+\\omega _m)}\\frac{\\sin ^2\\frac{P\\pi r}{R}(\\Omega _N\\Omega _P-\\bar{\\omega }_n^2)}{(\\Omega _P-\\omega _m)(\\Omega _N^2-\\bar{\\omega }_n^2)(\\Omega _P^2-\\bar{\\omega }_n^2)}\\right]}{\\sqrt{\\sum _{l,N}\\frac{l^2\\pi ^2}{Rr^3\\Omega _N\\omega _l}\\frac{1}{(\\Omega _N+\\omega _l)^2}\\sin ^2\\frac{\\pi N r}{R}}\\sqrt{\\sum _{l,N}\\frac{l^2\\pi ^2}{R\\bar{r}^3\\Omega _N\\bar{\\omega }_l}\\frac{1}{(\\Omega _N+\\bar{\\omega }_l)^2}\\sin ^2\\frac{\\pi N \\bar{r}}{R}}}$ an expression that can be numerically evaluated, see Figure REF .", "Even just a quick look to the figures REF a and REF b reveals the existence of certain patterns: the extension of correlations along the axis of the small region's local modes (vertical), or the alternance of those extensions (vertical bars) from relevant values to almost zero along the axis of the big region's local modes (horizontal).", "Although it is out of the scope of this paper to discuss those patterns in detail, we can give a simple explanation of why they would exist, just by thinking in terms of the Fourier decomposition of global modes in terms of small and big local modes (check Figure REF ).", "In order to expand the same global mode, for example $U_{N=1}$ , the number of local modes with a relevant contribution will be much higher for the small side than for the big side, the reason for that being, that a smaller section of a global mode requires more frequencies to be expanded.", "So as a matter of fact, those lines also exist along the big region's axis, but they are just much shorter, and so they pass unnoticed.", "Regarding the alternating pattern we can just mention that it has to do mainly with the existence of noticeable differences in the values of the Bogoliubov coefficients for consecutive modes $U_N, U_{N+1}$ ." ], [ "Properties of quasi-local states on $\\mathfrak {F}^G$", "As we have seen in Section REF , the local quantisation based on non-stationary modes yields a natural notion of local one-particle states in $\\mathfrak {F}^L$ defined by $a_m^\\dagger |0_L\\rangle $ .", "On the other hand, since the local creators are well-defined in $\\mathfrak {F}^G$ , this suggests a natural class of one-particle states $a_m^\\dagger |0_G\\rangle $ that we will call quasi-local states defined in $\\mathfrak {F}^G$ .", "In this section we shall examine the properties of these states.", "In particular, their failure to be strictly localised states is directly related to the Reeh-Schlieder theorem and vacuum entanglement." ], [ "Positivity of energy", "For historical reasons – coming from the early attempts of interpreting the solutions of second order Klein-Gordon equation as one-particle wave-functions – it is commonplace to associate the negative frequency states $U_N^*$ with negative energies, and for this reason to regard them as unphysical states.", "From that point of view it might seem alarming that we have constructed our local modes using both positive and negative frequency energy-eigenstates, i.e.", "both $U_M$ and $U_N^*$ .", "Nonetheless, the problem with negative frequencies is a problem in that interpretation and not in relativistic QFT.", "Indeed, when we adopt the perspective that relativistic QFT arises from the quantisation of a relativistic field, no problems associated with negative frequencies appear.", "Instead the frequencies are related to energy changes associated with the creation or annihilation of individual quanta.", "The classical canonical Hamiltonian $H=\\int dx \\frac{1}{2}(\\pi ^2+(\\partial _x\\phi )^2+\\mu ^2\\phi ^2)\\ge 0,$ being a sum of squares, is manifestly positive definite and is thus bounded from below by zero.", "As a quantum operator in the corresponding QFT, it is of course ill-defined due to the infinite vacuum energy.", "Notwithstanding, the regularised Hamiltonian is a sum of the positive operators $N_N$ , i.e.", "$H^G\\equiv H-\\langle 0_G|H|0_G\\rangle =\\sum _N\\Omega _NN_N.$ It is thus clear that any state in $\\mathfrak {F}^G$ has manifestly positive energy and the problem with negative energies is thus avoided by viewing the system, to be quantised, as a classical field rather than a classical relativistic particle [29].", "One may be worried that acting with the local creators and annihilators (which were constructed using both positive and negative frequencies) on the global vacuum $|0_G\\rangle $ , one would obtain unphysical states, perhaps with negative energy.", "However, as we have demonstrated, the action of the local creators and annihilators on any state $|\\psi \\rangle \\in \\mathfrak {F}^G$ is well defined.", "Since all states in $\\mathfrak {F}^G$ have manifestly positive energy expectation value it is clear that no problems with negative energy arise.", "Nevertheless it is instructive to elaborate on this a bit further.", "To that end let us investigate whether the state $|\\psi _l\\rangle =a_l^\\dagger |0_G\\rangle $ has negative energy.", "Calculating explicitly the average energy of a state $|\\psi _l\\rangle =a_l^\\dagger |0_G\\rangle $ , we get $\\langle \\psi _l|H^G|\\psi _l\\rangle &=\\sum _N\\Omega _N\\langle 0_G|a_l N_Na_l^\\dagger |0_G\\rangle =\\sum _{M,N,P}\\Omega _N(u_l|U_M)(U_P|u_l)\\langle 0_G|A_M A_N^\\dagger A_NA_P^\\dagger |0_G\\rangle \\nonumber \\\\&=\\sum _N\\Omega _N(u_l|U_N)(U_N|u_l)=\\sum _N\\Omega _N|(U_N|u_l)|^2>0,$ verifying that the energy is manifestly positive.", "To demonstrate that the energy is finite we first note that $a_l|0_G\\rangle $ is not yet normalised: $\\langle \\psi _l|\\psi _l\\rangle =\\langle 0_G|a_la_l^\\dagger |0_G\\rangle =1+\\langle 0_G|n_l|0_G\\rangle \\ne 1.$ The normalised state is therefore given by $|\\psi _l\\rangle =\\frac{a_l^\\dagger |0_G\\rangle }{\\sqrt{1+\\langle 0_G|n_l|0_G\\rangle }}.$ By inspecting the Bogoliubov coefficients (REF ) and making use of the integral test of convergence we see that $\\langle \\psi _l|H^G|\\psi _l\\rangle <\\infty $ .", "Hence, we see that the application of the local creation operator $a_k^\\dagger $ on the global vacuum $|0_G\\rangle $ keeps the state in the global Fock space $\\mathfrak {F}^G$ , i.e.", "$|\\psi \\rangle \\in \\mathfrak {F}^G$ .", "We can also consider the state $|\\phi _l\\rangle =\\frac{a_l|0_G\\rangle }{\\sqrt{\\langle 0_G|n_l|0_G\\rangle }},$ which is not zero since $a_l$ contains both $A_N$ and $A_N^\\dagger $ , nor does it have less energy than the global vacuum state.", "A calculation similar to the one above shows that the energy is manifestly positive $\\langle \\phi |H|\\phi \\rangle >0$ .", "Again by the integral test of convergence we could check that the state has, in fact, a finite energy expectation value." ], [ "Quantum steering and the Reeh-Schlieder theorem", "We are now in a position to address the question of whether the normalised state $|\\psi _m\\rangle =\\frac{a_m^\\dagger |0_G\\rangle }{\\sqrt{1+\\langle 0_G|n_m|0_G\\rangle }},$ can be viewed as a strictly localised one-particle state.", "The associated wave-packet defined by $\\psi _m(x,t)\\equiv \\langle 0_G|\\phi (x,t)|\\psi _m\\rangle $ is in fact the positive frequency part of $u_m$ , defined in (REF ).", "One might naively suspect that these states should be localised states since they are created by a local operation on the vacuum state, i.e.", "$|0_G\\rangle \\rightarrow a_m^\\dagger |0_G\\rangle $ .", "The components of this state in the global basis (REF ) are given by $\\frac{a_m^\\dagger |0_G\\rangle }{\\sqrt{1+\\langle 0_G|n_m|0_G\\rangle }}=\\frac{\\sum _N (U_N|u_m)A_N^\\dagger +(U_N^*|u_m)A_N|0_G\\rangle }{\\sqrt{1+\\langle 0_G|n_m|0_G\\rangle }}=\\frac{\\sum _N (U_N|u_m)|1_N\\rangle }{\\sqrt{1+\\langle 0_G|n_m|0_G\\rangle }},$ which we recognise as a superposition of global one-particle excitations.", "From an analysis by Knight [13] showing that no finite superposition of $N$ -particle states can be strictly localised, we already know that $|\\psi _m\\rangle $ is not strictly localised.", "We could stop here, but it is interesting to gain more understanding why this happens.", "To investigate this fact, let us see whether the expectation value $\\langle \\psi _m|\\bar{n}_l|\\psi _m\\rangle $ is different from $\\langle 0_G|\\bar{n}_l|0_G\\rangle $ .", "Computing this difference yields $\\langle \\psi _m|\\bar{n}_l|\\psi _m\\rangle -\\langle 0_G|\\bar{n}_l|0_G\\rangle &=\\frac{\\langle 0_G|a_m\\bar{n}_l a_m^\\dagger |0_G\\rangle }{1+\\langle 0_G|n_m|0_G\\rangle }-\\langle 0_G|\\bar{n}_l|0_G\\rangle ,\\nonumber \\\\&=\\frac{\\langle 0_G|n_m\\bar{n}_l|0_G\\rangle -\\langle 0_G|\\bar{n}_l|0_G\\rangle \\langle 0_G|n_m|0_G\\rangle }{1+\\langle 0_G|n_m|0_G\\rangle }\\propto \\text{corr}(n_m,\\bar{n}_l),$ which not only shows that the one-particle state $|\\psi _m\\rangle $ is not strictly localised, but also tells us that the reason for it is vacuum entanglement.", "Indeed, making the replacement $|0_G\\rangle \\rightarrow |0_L\\rangle $ and $|\\psi _m\\rangle \\rightarrow |1_m,\\bar{0}\\rangle $ we have $\\text{corr}(n_m,\\bar{n}_l)=0$ and the above difference disappears.", "It may seem puzzling that we can change the expectation values in the region $[r, R]$ by performing a local operation in $[0, r]$ .", "Does this not imply the possibility of superluminal signaling?", "The answer is no, the reason being that the operation $|0_G\\rangle \\rightarrow |\\psi _m\\rangle $ is not a unitary operation on the vacuum state since $a_m a_m^\\dagger \\ne 1$ .", "This local operation does not correspond to something which can be achieved physically by local manipulations solely in $[0, r]$ .", "However, with suitable post-selection, the operation $|0_G\\rangle \\rightarrow |\\psi _m\\rangle $ could perhaps be implemented, but only by informing the observer in the region $[r, R]$ which states to post-select.", "This of course would require classical communication, limited by the speed of light [30].", "We can view this in the context of the Reeh-Schlieder theorem [31].", "This theorem states that by a local non-unitary operation in a finite region in space we can obtain, to arbitrary precision, any state at a spatially separated region.", "The theorem does not go through if we restrict ourselves to local unitary operations.", "The situation is different when we replace the global vacuum $|0_G\\rangle $ with the local vacuum $|0_L\\rangle $ .", "As seen in Section REF the key difference is that the local vacuum $|0_L\\rangle $ neither cyclic nor separating, or more simply, it is a product state $|0_L\\rangle =|0\\rangle \\otimes |\\bar{0}\\rangle $ which is therefore not entangled.", "Thus, no steering whatsoever could take place in this case." ], [ "Further properties", "In the section REF we analysed the positivity of energy of the pseudo-local states $|\\psi _l\\rangle =\\frac{1}{\\sqrt{1+\\langle 0_G|n_l|0_G\\rangle }}a_l^\\dagger |0_G\\rangle \\qquad |\\phi _l\\rangle =\\frac{1}{\\sqrt{\\langle 0_G|n_l|0_G\\rangle }} a_l|0_G\\rangle ,$ which are in fact superpositions of global one-particle states $|1_N\\rangle =A_N^\\dagger |0_G\\rangle $ , i.e.", "$|\\psi _l\\rangle =\\frac{1}{\\sqrt{1+\\langle 0_G|n_l|0_G\\rangle }}\\sum _N(u_l|U_N)|1_N\\rangle \\qquad |\\phi _l\\rangle =\\frac{1}{\\sqrt{\\langle 0_G|n_l|0_G\\rangle }} \\sum _N(u_l|U_N^*)|1_N\\rangle $ Figure: Quasi-local modes as compared with local modes.", "The picture shows the particular case of r=0.21Rr=0.21R, μ=1/r\\mu = 1/r with mode number m=1m=1.", "It portraits local modes (zero valued out of the light cone) and quasi-local modes, showing exponential decaying fall-offs around the light cone.", "The inset shows the difference of both modes at the same scale.Let's define $|\\psi ^{(r)}_l\\rangle = a^{(r)\\dagger }_l|0_G\\rangle $ , where the $(r)$ superindex refers to the operator corresponding to a localisation region of size $r$ .", "We would expect that state to resemble a one-particle local state, in the sense that the corresponding mode would just be the positive frequency part of the one-particle local mode.", "That is indeed the case.", "Figure REF illustrates this case for a particular case of those shown in figure REF .", "We would therefore call these modes, which lie in the global Fock space $\\mathfrak {F}^G$ , quasi-local modes.", "For all practical purposes this kind of states could be used as localised and causal to a very good approximation.", "Besides that, it is interesting to study how much $ | \\psi ^{(r)}_l\\rangle $ states resemble to the one-particle global states, and therefore we will calculate the expectation value : $\\langle \\psi ^{(r)}_l | A^{\\dagger }_N A_N | \\psi ^{(r)}_l\\rangle $ which happens to be identically equal to $|\\langle 1_N | \\psi ^{(r)}_l\\rangle |^2 = |\\langle 0_G| A_N | \\psi ^{(r)}_l\\rangle |^2 = \\frac{|\\langle 0_G| A_N a^{(r)\\dagger }_l | 0_G\\rangle |^2}{1+\\langle 0_G|n_l|0_G\\rangle } = \\frac{|(U_N|u_l)|^2}{1+\\langle 0_G|n_l|0_G\\rangle }$ Figure REF a shows the expansion of $| \\psi ^{(r)}_l\\rangle $ in terms of global particle states $| 1_N\\rangle $ for the massless case.", "We can see that the decomposition is a rather peaked one, and in particular, we can estimate a bandwidth $\\Delta \\Omega $ for the expansion in global modes.", "We can define it as the smallest $\\Delta \\Omega $ for which: $\\sum _{\\Omega _N \\in (\\omega _l- \\Delta \\Omega /2,\\omega _l+\\Delta \\Omega /2)} |\\langle 1_{N} | \\psi ^{(r)}_l\\rangle |^2 > 0.95$ In the general case, $\\Delta \\Omega $ depends on the frequency of the mode $\\omega _l$ , but tends to an asymptotic value in the limit of big $l$ 's, as we can see in the inset of Figure REF b, where the dependence with the Klein Gordon mass $\\mu $ is also plotted.", "The asymptotic value is independent on the mass, and only dependent on the $r/R$ value.", "The relationship between these two can be seen in Figure REF b.", "In the limit of small values of $r/R$ , which would correspond to strongly “localised particles”, the bandwidth tends to infinity, i.e.", "we need an infinite amount of global modes to describe the quasi-local particle.", "For high values of $r/R$ the bandwidth approaches a minimum and we can approximately identify the quasi-local particle states with global states.", "Figure: Quasi-local state analysis.", "a) Decomposition of |ψ l 〉|\\psi _l\\rangle states in terms of |1 N 〉|1_N\\rangle global states for a massless case with r=R/9.", "In the inset, the particular case for l=20,ω l =πl/r≃571/Rl=20, \\omega _l = \\pi l /r \\simeq 571/R.", "b) The estimated bandwidth ΔΩ\\Delta \\Omega for quasi-local states is independent of the mode ll for big ll, but shows a strong dependence with r/Rr/R.", "The inset shows the dependence of ΔΩ\\Delta \\Omega with ll for different masses μ\\mu for the case r=R/9r=R/9." ], [ "Conclusions and outlook", "In the extant literature there are several theorems and results that indicate the impossibility of having local particle states, e.g.", "[7], [12], [4], [13].", "We believe that the main obstruction comes from postulating that the one-particle Hilbert space is spanned by positive frequency modes.", "In particular, no wave-packet built from these modes can be localized within a finite spatial region, even for an arbitrarily small time interval.", "However, as pointed out in Wald's exposition of the quantization procedure [16], there is nothing preventing us from making use of a different set of modes.", "The basic idea of this paper was that, basing the quantization procedure on localized modes, we might account for localized one-particle states.", "Indeed, this turns out to be the case.", "These local modes are defined by their initial data.", "Both the value and time-derivative of the modes are taken to be completely localized within either the right or left partition of the box.", "This data defines a well-posed Cauchy problem.", "By Hegerfeldt's theorem, these solutions of the Cauchy problem must contain both positive and negative frequency modes.", "This marks, at the classical level, a point of departure from the standard quantization procedure.", "The creation and annihilation operators associated with these local modes can then be used to build a Fock space $\\mathfrak {F}^L$ , whose basis states describe local elementary excitations of the quantum field.", "A set of these basis states, e.g.", "$|n_k,\\bar{0}\\rangle $ , does in fact represent strictly localized states with respect to the local vacuum $|0_L\\rangle \\in \\mathfrak {F}^L$ .", "This vacuum state, however, does not share the typical properties of a quantum field vacuum.", "In particular, it is neither cyclic not separating, as it is free from correlations between left and right partitions.", "Intriguingly, the local and standard (global) quantum field theories turn out to be unitarily inequivalent.", "Specifically, by computing the Bogoliobov coefficients relating the global and local quantum theories we have found that $Tr\\ \\beta ^\\dagger \\beta \\equiv \\sum _{k,N}|(u_k|U_N)^*|^2+|(\\bar{u}_k|U_N^*)|^2,$ diverges, which is a sufficient condition for establishing unitary inequivalence.", "Nevertheless, it is important to note that both standard and local quantizations produce self-consistent quantum theories of the field.", "As a matter of fact, as we have demonstrated, we can evolve states and we also have a well-defined notion of energy after the local vacuum energy has been subtracted from the canonical Hamiltonian.", "The existence of unitarily inequivalent representations would seem to confront us with a problem of which Fock space representation to choose [32].", "The problem of unitary inequivalence disappears, however, when some form of regularisation is introduced [33].", "Imposing of a wave-number $k=\\pi m/r$ cutoff, for example, could solve the issue.", "Such a cut-off would come naturally, for example, from a quantum theory of gravity requiring a discretisation of space(time).", "A restatement of the theory, which considers the use of measurement apparatuses for a finite time, would also imply the introduction of a frequency cut-off, circumventing the divergences present in (REF ).", "Within our approach we nevertheless find that there is a mathematical asymmetry between the two Fock space representations.", "In fact, the divergence of the sum (REF ) originates from the summation over the local-mode numbers $m$ .", "On the other hand, the sum over global-mode numbers $N$ is finite for each specific value of $m$ .", "A consequence of this fact is that the local creators and annihilators are well-defined operators in the global Fock space, and so are the local number operators.", "However, the global creators and annihilators turn out to be ill-defined on $\\mathfrak {F}^L$ .", "This asymmetry could perhaps be taken as an indication that the global Fock space representation is preferred.", "In any case, the fact that both local creators and annihilators are well-defined in $\\mathfrak {F}^G$ provides us with a useful set of mathematical tools to analyse the properties of the states in $\\mathfrak {F}^G$ .", "In particular, by computing the expectation values of the local number operators, we have shown that the global vacuum $|0_G\\rangle $ is characterised by a bath of local particles.", "We also showed, by calculating the correlation coefficients of local number operators, that the local particles associated with the left and right regions are highly entangled in the global vacuum, a feature not shared by the local vacuum $|0_L\\rangle $ .", "Again, the well-defined character of local creators and annihilators in $\\mathfrak {F}^G$ also allows us to introduce a new set of quasi-local states defined by applying the local creation operator on the global vacuum, i.e.", "$|\\psi _m\\rangle \\sim a_k^\\dagger |0_G\\rangle $ .", "These are natural candidates for essentially localized states [15].", "We have also shown how these states fail to be strictly localized, a fact related to vacuum entanglement and the Reeh-Schieder theorem.", "Unitary inequivalence seems to be the key problem in the construction of particle localised states, and that could connect with the abstract no-go results by Malament [7] and Clifton et.", "al [12].", "However, a proper analysis of this matter would require an adaptation of our setup to incorporate translation covariance, which is an essential assumption in the theorems mentioned.", "Clearly there are several topics that deserve further exploration.", "Here we mention a few of them.", "For example, it would be nice to express the global vacuum state using the eigenstates $|n_k,\\bar{n}_l\\rangle $ of the local number operators.Although the local number operators $n_m$ and $\\bar{n}_l$ are well-defined Hermitian operators in $\\mathfrak {F}^G$ we note that their eigenstates $|n_m,\\bar{n}_l\\rangle $ belong to $\\mathfrak {F}^L$ and not to $\\mathfrak {F}^G$ .", "The situation is similar for a non-relativistic quantum particle in a box where the eigenstates of the self-adjoint momentum operator $p=-i\\partial _x$ do not belong to the Hilbert space because they do not satisfy the Dirichlet conditions at the boundary.", "Such an expression would allow us to construct the reduced density matrix for the regions $[0, r]$ and $[r, R]$ by partial tracing.", "From there it would be interesting to see whether the reduced state takes the form of a KMS state.", "Hopefully we could make contact with existing literature, which examines the entanglement and thermality connected to localised regions of space [34].", "In that respect it is perhaps interesting to note that our construction, in contrast to the Minkowski and Rindler quantizations, was not based on standard stationary states.", "Indeed, while the Minkowski and Rindler quantizations both rely on stationary modes with respect to time translation and boost operators respectively, our construction makes use of manifestly non-stationary states.", "Whether this provides some advantage remains to be seen.", "In any case, it would be of interest to analyse in detail the differences and similarities between the Rindler quantisation and the one presented in this paper.", "Acknowledgements: We have benefited from discussions with Luis Garay, Guillermo A. Mena Marugán, Juan Manuel Pérez-Pardo, Hans Halvorson and Jakob Yngvason.", "H. Westman is grateful for initial discussions with Fay Dowker on the possibility of having local qubits in QFT.", "This work is supported by Spanish MICINN Projects FIS2011-29287 and CAM research consortium QUITEMAD S2009-ESP-1594.", "M. del Rey was supported by a CSIC JAE-PREDOC grant.", "H. Westman was supported by the CSIC JAE-DOC 2011 program." ] ]
1403.0073
[ [ "Quantum Mechanics as Classical Physics" ], [ "Abstract Here I explore a novel no-collapse interpretation of quantum mechanics which combines aspects of two familiar and well-developed alternatives, Bohmian mechanics and the many-worlds interpretation.", "Despite reproducing the empirical predictions of quantum mechanics, the theory looks surprisingly classical.", "All there is at the fundamental level are particles interacting via Newtonian forces.", "There is no wave function.", "However, there are many worlds." ], [ "Introduction", "On the face of it, quantum physics is nothing like classical physics.", "Despite its oddity, work in the foundations of quantum theory has provided some palatable ways of understanding this strange quantum realm.", "Most of our best theories take that story to include the existence of a very non-classical entity: the wave function.", "Here I offer an alternative which combines elements of Bohmian mechanics and the many-worlds interpretation to form a theory in which there is no wave function.", "According to this theory, all there is at the fundamental level are particles interacting via Newtonian forces.", "In this sense, the theory is classical.", "However, it is still undeniably strange as it posits the existence of a large but finite collection of worlds, each completely and utterly real.", "When an experiment is conducted, every result with appreciable Born Rule probability does actually occur in one of these worlds.", "Unlike the many worlds of the many-worlds interpretation, these worlds are fundamental, not emergent; they are interacting, not causally isolated; and they never branch.", "In each of these worlds, particles follow well-defined trajectories and move as if they were being guided by a wave function in the familiar Bohmian way.", "In this paper I will not attempt to argue that this theory is unequivocally superior to its competitors.", "Instead, I would like to establish it as a surprisingly successful alternative which deserves attention and development, hopefully one day meriting inclusion among the list of promising realist responses to the measurement problem.", "In §, I briefly review why quantum mechanics is in need of a more precise formulation and discuss two no-collapse theories: the many-worlds interpretation and Bohmian mechanics.", "I then go on to offer a rather unlikable variant of Bohmian mechanics which adds to the standard story a multitude of worlds all guided by the same wave function.", "This theory is useful as a stepping stone on the way to Newtonian QM.", "Newtonian QM is then introduced.", "As soon as Newtonian QM is on the table, § & present one of the most significant costs associated with the theory: the space of states must be restricted if the theory is to recover the experimental predictions of quantum mechanics.", "In §, , & , I discuss the advantages of this new theory over Everettian and Bohmian quantum mechanics in explaining the connection between the squared amplitude of the wave function and probability.", "In §, I consider the possibility of modifying the theory so that it describes a continuous infinity of worlds instead of a finite collection, concluding that such a modification would be inadvisable.", "In §, I propose two options for the fundamental ontology of Newtonian QM.", "In §, I use Newtonian QM to explain the way the wave function transforms under time reversal and Galilean boosts.", "Spin is then discussed in §.", "Some limitations of the theory presented here are worth stating up front.", "First, just as hydrodynamics relies on approximating a discrete collection of particles as a continuum, in its current form this theory must treat the discrete collection of worlds as a continuum.", "As this is merely an approximation, empirical equivalence with standard quantum mechanics is likely only approximate (§).", "Second, one must impose a significant restriction on the space of states if the predictions of QM are to be reproduced (the Quantization Condition, §).", "Third, I will not discuss extending the theory to handle multiple particles with spin or relativistic quantum physics.", "bostrom2012Böstrom's madelung1927Madelung's Newtonian QM is a realist version of quantum mechanics based on the theory's hydrodynamic formulation (originally due to [23]).", "For recent and relevant discussions of quantum hydrodynamics, see [36]; [20].", "An approach much like Newtonian QM was independently arrived at by [19].", "Newtonian QM is somewhat similar to bostrom2012 [8] metaworld theoryThe key difference with Newtonian QM being that Böstrom's theory does not as thoroughly excise the wave function (the dynamics being given by (REF ) not (REF )).", "and the proposal in [29].", "Related ideas about how to remove the wave function are explored in [25], [27], including a suggestion of many worlds.", "To avoid confusion, throughout the paper I'll use “universe” to denote the entirety of reality, what philosophers call “the actual world” and what in these contexts is sometimes called the “multiverse,” reserving “world” for the many worlds of quantum mechanics." ], [ "The Measurement Problem", "If the state of the universe is given by a wave function and that wave function always evolves in accordance with the Schrödinger equation, then quantum measurements will typically not have single definite outcomes.", "Actual measurements of quantum systems performed in physics laboratories do seem to yield just one result.", "This, in brief, is the measurement problem.", "There are various ways of responding.", "According to Everettian quantum mechanics, a.k.a.", "the many-worlds interpretation, the wave function $\\Psi $ is all there is.", "The evolution of the wave function is always given by the Schrödinger equation, $i\\hbar \\frac{\\partial }{\\partial t}\\Psi ({x}_1,{x}_2,...,t)=\\left(\\sum _k{\\frac{-\\hbar ^2}{2 m_k}\\nabla _k^2}+V({x}_1,{x}_2,...,t)\\right)\\Psi ({x}_1,{x}_2,...,t)\\ ,$ where $\\Psi $ is a function of particle configuration $({x}_1,{x}_2,...)$ and time $t$ , $m_k$ is the mass of particle $k$ , $\\nabla _k^2$ is the Laplacian with respect to ${x}_k$ , and $V$ is the classical potential energy of particle configuration $({x}_1,{x}_2,...)$ at $t$ .", "When an observer performs a quantum measurement, the universal wave function enters a superposition of the observer seeing each possible outcome.", "This is not to be understood as one observer seeing many outcomes, but as many observers each seeing a single outcome.", "Thus, the theory is not obviously inconsistent with our experience of measurements appearing to have unique outcomes.", "According to Everettian quantum mechanics, there is nothing more than the wave function and therefore things like humans, measuring devices, and cats must be understood as being somehow composed of or arising out of wave function.", "([33], [34] takes these things to be patterns or structures in the universal wave function.)", "To summarize, here is what the Everettian QM says that there is (the ontology) and how it evolves in time (the dynamical laws).", "Ontology: (I) universal wave function $\\Psi ({x}_1,{x}_2,...,t)$          Law: (I) Schrödinger equation (REF ) A second option in responding to the measurement problem is to expand the ontology so that the universe contains both a wave function evolving according to (REF ) and particles with definite locations.", "The time-dependent position of particle $k$ can be written as ${x}_k(t)$ and its velocity as ${v}_k(t)$ .", "The wave function pushes particles around by a specified law, ${v}_k(t)=\\frac{\\hbar }{m_k}\\mbox{Im}\\left[\\frac{{\\nabla }_k\\Psi ({x}_1,{x}_2,...,t)}{\\Psi ({x}_1,{x}_2,...,t)}\\right]\\ .$ Experiments are guaranteed to have unique outcomes because humans and their scientific instruments are made of particles (not wave function).", "These particles follow well-defined trajectories and are never in two places at once.", "This theory is Bohmian mechanics, a.k.a.", "de Broglie-Bohm pilot wave theory.", "Ontology: (I) universal wave function $\\Psi ({x}_1,{x}_2,...,t)$          (II) particles with positions ${x}_k(t)$ and velocities ${v}_k(t)$          Laws: (I) Schrödinger equation (REF )          (II) guidance equation (REF ) From (REF ) and (REF ), one can derive an expression for the acceleration of each particle, $m_j {a}_j(t)=-{\\nabla }_j\\Big [Q({x}_1,{x}_2,...,t)+V({x}_1,{x}_2,...,t)\\Big ]\\ ,$ where $Q({x}_1,{x}_2,...,t)$ is the quantum potential, defined by $Q({x}_1,{x}_2,...,t)=\\sum _k \\frac{-\\hbar ^2}{2 m_k }\\left(\\frac{\\nabla _k^2 |\\Psi ({x}_1,{x}_2,...,t)|}{|\\Psi ({x}_1,{x}_2,...,t)|}\\right)\\ .$ Since the focus of this paper is not on Everettian or Bohmian quantum mechanics, I've sought to present each as simply as possible.", "The best way to formulate each theory—ontology and laws—is a matter of current debate." ], [ "Prodigal QM", "As a precursor to the theory I'll propose, consider the following interpretation of quantum mechanics which has both a many-worlds and a Bohmian flavor.", "The wave function always obeys the Schrödinger equation.", "There are many different worlds, although a finite number, each represented by a point in configuration spaceThe location of a single particle is given by a point in space, $({x})$ .", "The locations of all particles are given by a point in configuration space, $({x}_1,{x}_2,...)$ , where ${x}_i$ is the location of particle $i$ ..", "There are more worlds where $|\\Psi |^2$ is large and less where it is small.", "Each world is guided by the single universal wave function in accordance with the Bohmian guidance equation and thus each world follows a Bohmian trajectory through configuration space.", "Let's call this ontologically extravagant theory Prodigal QM.With a continuous infinity of worlds, Prodigal QM is mentioned in [31] and in [6] (in Barrett's terminology, it is a Bohmian many-threads theory in which all of the threads are taken to be completely real); a closely related proposal is discussed in [14].", "Why include a multitude of worlds when we only ever observe one, our own?", "We could simplify the theory by removing all of the worlds but one, arriving at Bohmian mechanics [31].", "But, less obviously, it turns out that there is another route to simplification: keep the multitude of worlds but remove the wave function.", "This option will be explored in the next section.", "According to Prodigal QM, the universe contains a wave function $\\Psi ({x}_1,{x}_2,...,t)$ on configuration space and a large number of worlds which can be represented as points moving around in configuration space.", "The arrangement of the worlds in configuration space is described by a number density, $\\rho ({x}_1,{x}_2,...,t)$ , normalized so that integrating $\\rho $ over all of configuration space gives one, $\\int \\!", "d^3 x_1 d^3 x_2...\\: \\rho =1$ .", "Integrating $\\rho ({x}_1,{x}_2,...,t)$ over a not-too-small volume of configuration space gives the proportion of all of the worlds that happen to be in that volume at $t$ .", "By hypothesis, worlds are initially distributed so that $\\rho ({x}_1,{x}_2,...,t)=|\\Psi ({x}_1,{x}_2,...,t)|^2\\ .$ The velocities of the particles are described by a collection of velocity fields indexed by particle number, $k$ , ${v}_k({x}_1,{x}_2,...,t)=\\frac{\\hbar }{m_k}\\mbox{Im}\\left[\\frac{{\\nabla }_k\\Psi ({x}_1,{x}_2,...,t)}{\\Psi ({x}_1,{x}_2,...,t)}\\right]\\ ,$ In Prodigal QM, if there is a world at $({x}_1,{x}_2,...)$ at $t$ the velocity of the $k$ th particle in that world is ${v}_k({x}_1,{x}_2,...,t)$ .This is not true for Newtonian QM (see §).", "With these velocity fields, the equivariance property of the Bohmian guidance equation (REF ) ensures that $\\rho $ is always equal to $|\\Psi |^2$ if it ever is [16].", "Ontology: (I) universal wave function $\\Psi ({x}_1,{x}_2,...,t)$          (II) particles in many worlds described by a world density $\\rho ({x}_1,{x}_2,...,t)$ and velocity          fields ${v}_k({x}_1,{x}_2,...,t)$          Laws: (I) Schrödinger equation (REF )          (II) guidance equation (REF )Actually, the second dynamical law is more specific than (REF ) since it requires not just that the velocity fields obey (REF ) but that each world follows an exact Bohmian trajectory (see §).", "The connection between $\\rho $ and $\\Psi $ in (REF ), though not a dynamical law, might best be thought of as a third law of Prodigal QM.", "The use of densities and velocity fields is familiar from fluid dynamics.", "A quick review will be helpful.", "Consider a fluid composed of $N$ point particles which each have mass $m$ .", "The number density of these particles is $n({x},t)$ , normalized so that $\\int \\!", "d^3 x_1 d^3 x_2...\\: n =N$ .", "The mass density is $m\\!\\times \\!", "n({x},t)$ .", "Integrating $n({x},t)$ over a not-too-small volume gives the number of particles in that volume at $t$ .", "Whereas $n({x},t)$ gives the density of particles in three-dimensional space, $\\rho $ gives the density of worlds in configuration space.", "The velocity field for the fluid is ${u}({x},t)$ , defined as the mean velocity of particles near ${x}$ at $t$ .More precisely, the number density and velocity field provide a good description of the particle trajectories if to a good approximation: $n({x},t)$ gives the average number of particles in a small-but-not-too-small region $\\mathcal {R}$ centered about ${x}$ over a short-but-not-too-short period of time $\\mathcal {T}$ around $t$ divided by the volume of $\\mathcal {R}$ , and ${u}({x},t)$ gives the average velocities of the particles in $\\mathcal {R}$ over $\\mathcal {T}$ .", "For more detail, see [10].", "The connection between $\\rho $ and the ${v}_k$ s and the trajectories of individual worlds could be spelled out along similar lines, but full rigor in the context of Newtonian QM would require a better understanding of the dynamics (see § and [19]).", "For an inviscid compressible fluid with zero vorticity, the time evolution of $n$ and ${u}$ are determined by a continuity equation $\\frac{\\partial n({x},t)}{\\partial t}=-{\\nabla } \\cdot \\Big (n\\left({x},t\\right){u}\\left({x},t\\right)\\Big )\\ ,$ and a Newtonian force law $m {a}({x},t)=-{\\nabla }\\left[\\frac{p({x},t)}{n({x},t)}+V\\left({x},t\\right)\\right]\\ ,$ where $V$ is the external potential, $p$ is the pressure, and ${a}({x},t)=\\frac{D{u}({x},t)}{Dt}=\\left({u}({x},t)\\cdot {\\nabla }\\right){u}({x},t)+\\frac{\\partial {u}({x},t)}{\\partial t}\\ .$ The acceleration is given by the material derivative of ${u}$ not the partial derivative because a particle's position in the fluid is time dependent.", "The three quantum theories on the table thus far are applied to the double-slit experiment in figure REF .", "In the bottom-right diagram is Everettian QM where the universe is just a wave function.", "The particle's wave function is initially peaked at the two slits and then spreads out and interferes as time progresses.", "When the particle hits the detector, a multitude of worlds will separate via decoherence and in each the particle will be observed hitting at a particular point on the screen.", "In Bohmian mechanics, one adds to the wave function an actual particle which follows a definite trajectory in accordance with the guidance equation.", "In Prodigal QM, there is a wave function and a collection of worlds, each of which contains a particle following a Bohmian trajectory.", "In Newtonian QM, which will be introduced at the end of §, one retains the multitude of worlds but removes the wave function.", "Figure: Diagrams of the evolution of a single particle in the double-slit experiment according to four different no-collapse theories.", "The vertical axis gives the position of the single particle and the horizontal axis time.", "|Ψ| 2 |\\Psi |^2 is shown as a contour plot and particle trajectories as lines." ], [ "Removing the Wave Function", "One can derive an equation for the dynamics of particles in Prodigal QM that makes no reference to the wave function.", "Once this is done, we can formulate an alternate theory where the superfluous wave function has been removed.", "This new theory, Newtonian QM, will be the focus of the remainder of the article.", "The mathematical manipulations presented in this section are familiar from discussions of Bohmian mechanics, but take on a different meaning as derivations of particle dynamics in Prodigal QM.", "Those who wish to skip the derivation should simply note that (REF ) is derivable from (REF ), (REF ), and (REF ).", "As $\\rho =|\\Psi |^2$ (REF ), the wave function can be written in terms of the world-density and a phase factor as $\\Psi ({x}_1,{x}_2,...,t)=\\sqrt{\\rho ({x}_1,{x}_2,...,t)}e^{i\\theta ({x}_1,{x}_2,...,t)}\\ .$ Plugging (REF ) into the guidance equation (REF ) generates ${v}_k({x}_1,{x}_2,...,t)=\\frac{\\hbar }{m_k}{\\nabla }_k\\theta ({x}_1,{x}_2,...,t)\\ ,$ relating ${v}_k$ and $\\theta $ .", "(At this point, I will stop repeating the arguments of $\\Psi $ , $\\rho $ , $\\theta $ , and ${v}_k$ ; they all depend on the configuration of particles and the time.)", "The evolution of the wave function $\\Psi $ is given by the Schrödinger equation (REF ).", "Dividing both sides of (REF ) by $\\Psi $ and using (REF ), one can derive that $\\frac{i\\hbar }{2\\rho }\\frac{\\partial \\rho }{\\partial t}-\\hbar \\frac{\\partial \\theta }{\\partial t} = \\sum _k{\\frac{-\\hbar ^2}{2 m_k}\\left[\\frac{\\nabla _k^2 \\sqrt{\\rho }}{\\sqrt{\\rho }}+\\frac{2 i}{\\sqrt{\\rho }} \\left({\\nabla }_k \\sqrt{\\rho }\\right)\\cdot \\left({\\nabla }_k \\theta \\right) + i \\: \\nabla _k^2 \\theta - \\left|{\\nabla }_k \\theta \\right|^2\\right]}+V\\ .$ Equating the imaginary parts, using (REF ), yields $\\frac{\\partial \\rho }{\\partial t}=-\\sum _k{{\\nabla }_k \\cdot \\left(\\rho {v}_k\\right)}\\ ,$ a continuity equation similar to (REF ).", "Equating the real parts of (REF ), using (REF ), yields $\\frac{\\partial \\theta }{\\partial t}=\\sum _k{\\left\\lbrace \\frac{\\hbar }{2 m_k}\\frac{\\nabla _k^2 \\sqrt{\\rho }}{\\sqrt{\\rho }}-\\frac{m_k}{2 \\hbar }|{v}_k|^2\\right\\rbrace }-\\frac{V}{\\hbar }\\ .$ Acting with $\\frac{\\hbar }{m_j}{\\nabla }_j$ on both sides of (REF ) and rearranging, making use of (REF ) and the fact that ${a}_j=\\sum _k ({v}_k\\cdot {\\nabla }_k){v}_j+\\frac{\\partial {v}_j}{\\partial t}$ gives $m_j {a}_j=-{\\nabla }_j\\left[\\sum _k \\frac{-\\hbar ^2}{2 m_k }\\left(\\frac{\\nabla _k^2 \\sqrt{\\rho }}{\\sqrt{\\rho }}\\right)+V\\right]\\ .$ We have derived an equation of motion of the form $F=ma$ , similar to both (REF ) and (REF ).This is the multi-particle version of [36]; [20].", "The last term in the brackets gives the classical potential energy of the configuration of particles and makes no reference to the other worlds.", "The other term looks like an interaction between the worlds.", "This term is the quantum potential $Q$ familiar from Bohmian mechanics (REF ), with $|\\Psi |$ replaced by $\\sqrt{\\rho }$ .", "Within Prodigal QM, we've seen that one can derive an equation which determines the dynamics for all of the particles in all of the worlds without ever referencing the wave function.", "(REF ) gives a way of calculating the acceleration of a particle that doesn't mention $\\Psi $ , as (REF ) does, but only depends on the density of worlds $\\rho $ and the potential $V$ .", "In Prodigal QM, this equation is derived, not part of the statement of the theory in the previous section.", "But, what if we took it to be the primary equation of motion for the particles?", "One can remove the wave function from Prodigal QM leaving only the corresponding $\\rho $ and ${v}_k$ s. So long as one enforces (REF ), the dynamics for particles will be essentially as they were in Prodigal QM.", "Now we can formulate a new theory: Newtonian QM.", "Reality consists of a large but finite number of worlds whose distribution in configuration space is described by $\\rho ({x}_1,{x}_2,...,t)$ .", "The velocities of the particles in the worlds are described by the velocity fields ${v}_k({x}_1,{x}_2,...,t)$ .", "The dynamical law for the velocity fields is (REF ), a Newtonian force law.", "As the particles move, the resultant shift in the distribution $\\rho $ is determined by (REF ).", "According to Newtonian QM, quantum mechanics is nothing but the Newtonian mechanics of particles in many different worlds.", "Ontology: (I) particles in many worlds described by a world density $\\rho ({x}_1,{x}_2,...,t)$          and velocity fields ${v}_k({x}_1,{x}_2,...,t)$          Law: (I) Newtonian force law (REF )The continuity equation (REF ), although used alongside (REF ) to calculate the dynamics, is not here considered a dynamical law since it merely encodes the fact that worlds are neither created nor destroyed.", "As is mentioned in §, the Quantization Condition might be considered a non-dynamical law.", "Comparing this statement of Newtonian QM to the formulation of Bohmian mechanics in §, Newtonian QM is arguably the simpler theory.", "The theory has a single dynamical law and the fundamental ontology consists only of particles.", "However, this quick verdict could certainly be contested, especially in light of the discussion below: (REF ) is not a fundamental law (§); an unnatural restriction must be put on the space of states (§); there are multiple ways to precisify ontology of the theory (§)." ], [ "The Continuum Approximation", "Since the number of worlds is taken to be finite, the actual distribution of worlds will be highly discontinuous; some locations in configuration space will contain worlds and others will not.", "Still, we can use a smooth density function $\\rho $ to describe the distribution of worlds well enough at a coarse-grained level (see footnote REF ).", "The velocity field ${v}_k({x}_1,{x}_2,...)$ gives the mean velocity of the $k$ -th particle in worlds near $({x}_1,{x}_2,...)$ , but the $k$ -th particle in a world at $({x}_1,{x}_2,...)$ may have a somewhat different velocity from ${v}_k({x}_1,{x}_2,...)$ .", "So, in Newtonian QM worlds will typically only approximately follow Bohmian trajectories through configuration space just as fluid particles do not exactly follow pathlines.A pathline gives the trajectory of a particle always traveling at the mean velocity ${u}$ .", "In fluid dynamics, the use of a description of the fluid in terms of $n$ and ${u}$ is justified by the fact that we can calculate the dynamics of these coarse-grained properties (and others) without needing to know exactly what all the particles are doing.", "Also, it is the coarse-grained properties that we measure ([7]; [10]).", "What justifies the use of $\\rho $ and the ${v}_k$ s to describe the collection of worlds?", "As it turns out, we can calculate the dynamics of these properties without worrying about the exact locations of worlds via (REF ) and (REF ).", "Once the evolution of $\\rho $ and the ${v}_k$ s are known, we can use $\\rho (t)$ to get probabilities (§) and the ${v}_k(t)$ s to determine pathlines (showing that particles follow Bohmian trajectories).", "The equation of motion for the theory (REF ) treats the collection of worlds as a continuum.", "It fails to be a fundamental law since it does not describe the precise evolution of each world and is not valid if there are too few worlds to be well-described as a continuum.", "Slight deviations from standard quantum mechanical behavior should be expected due to the fact that there are only a finite number of worlds; worse deviations the fewer worlds there are.", "Future experiments may observe such deviations and support Newtonian QM.", "As textbook quantum mechanics works well, we have reason to believe there are a very large number of worlds.", "(The situation here is similar to that of spontaneous collapse theories, which are in principle empirically testable.)", "Ultimately, the quantum contribution to the force in (REF ) should be derivable from a more fundamental inter-world interaction.", "One should be able to calculate the forces when there are only a handful of worlds.", "Hopefully future research will explain how the continuum approximation arises from a “micro-dynamics” of worlds just as fluid dynamics arises from the micro-dynamics of molecules.", "For some progress in this direction, see [19]." ], [ "Reintroducing the Wave Function", "In § we saw that for any wave function $\\Psi (t)$ obeying the Schrödinger equation, there exists a world-density $\\rho (t)$ and a collection of velocity fields ${v}_k(t)$ obeying (REF ) such that the relations between $\\Psi $ , $\\rho $ , and the ${v}_k$ s expressed in (REF ) and (REF ) are satisfied at all times.", "The converse does not hold.", "There are some combinations of $\\rho $ and the ${v}_k$ s, that is, some ways the universe might be according to Newtonian QM, that do not correspond to any wave function.", "In general, we'll restrict our attention to combinations of $\\rho $ and the ${v}_k$ s that can be derived from a wave function via (REF ) and (REF ) as it is these states which reproduce the predictions of quantum physics.", "For such states, it may be useful to introduce a wave function, $\\Psi $ , even though it is not a fundamental entity and does not appear in the equation of motion of the theory (REF ).", "The wave function serves as a convenient way of summarizing information about the positions and velocities of particles in the various worlds; the magnitude encodes the density of worlds (REF ) and the phase encodes the velocities of particles (REF ).", "The wave function need not be mentioned in stating the theory or (in principle) for deriving empirical predictions, but introducing a wave function is useful for making contact with standard treatments of quantum mechanics.", "As was just mentioned, there are some states of the universe in Newtonian QM that do not correspond to quantum wave functions.This point was made concisely by [35] in the context of quantum hydrodynamics; it was noted earlier by [28]; see also [20].", "That is, there are some combinations of $\\rho $ and the ${v}_k$ s for which one cannot find a wave function $\\Psi $ that satisfies (REF ) and (REF ).", "The amplitude of $\\Psi $ follows straightforwardly from $\\rho $ , but not every set of velocity fields ${v}_k$ can be expressed as $\\frac{\\hbar }{m_k}$ times the gradient of a phase (REF ).", "For this to be the case, we must impose a constraint on the velocity fields.This is loosely analogous to the constraint on the fluid velocity field ${u}$ that it be irrotational (everywhere zero vorticity) which is required to introduce a velocity potential (and for the validity of (REF )).", "Quantization Condition Integrating the momenta of the particles along any closed loop in configuration space gives a multiple of Planck's constant, $h=2\\pi \\hbar $ .", "$\\oint {\\left\\lbrace \\sum _k{\\left[m_k{v}_k \\cdot d{\\ell }_k\\right]}\\right\\rbrace }=nh\\ .$ If the Quantization Condition is satisfied initially, (REF ) ensures that it will be satisfied at all times.", "To see one sort of constraint this requirement imposes, think about the following case: a single electron orbiting a hydrogen nucleus in the $n=2$ , $l=1$ , $m=1$ energy eigenstate.", "For simplicity, take the nucleus to provide an external potential and the universe to contain many worlds with a single electron in each.", "The electron's wave function is $\\Psi _{2,1,1}(r,\\theta ,\\phi )=\\frac{-1}{8 \\sqrt{a^5 \\pi }}e^{\\frac{-r}{2a}}e^{i\\phi }r\\sin \\theta \\ ,$ where $a$ is the Bohr radius.", "The guidance equation tells us that the particle in each world executes a circle around the $z$ -axis with velocity $v_{\\phi }=\\frac{\\hbar }{m r sin{\\theta }}$ , entirely in the $\\widehat{\\phi }$ direction (here $\\phi $ is the azimuthal angle).", "(REF ) is trivially satisfied since $\\frac{m \\hbar }{m r sin{\\theta }} \\times 2 \\pi r sin{\\theta }=h$ .", "But, if the electrons were circling the $z$ -axis a bit faster or a bit slower the integral wouldn't turn out right and (REF ) wouldn't be satisfied; they could orbit twice as fast but not $1.5$ times as fast.", "Without the Quantization Condition, Newtonian QM has too large a space of states.", "There are ways the universe might be that are quantum mechanical and others that are not.", "It is easy to specify what universes should be excluded, those that violate (REF ), but hard to give a principled reason why those states should be counted as un-physical, improbable, or otherwise ignorable.", "For now, I think it is best to understand the Quantization Condition as an empirically discovered feature of the current state of the universe, or equivalently, of the initial conditions.", "However, one might prefer to think of it as a non-dynamical law.", "A better explanation of the Quantization Condition's satisfaction would help strengthen Newtonian QM as it might seem that the best possible explanation of the condition's satisfaction is the existence of a wave function (backtracking to Prodigal QM).", "In the remainder of the article I will assume that the Quantization Condition is satisfied.", "Suppose the world density and the velocity fields at a time are given.", "Provided the Quantization Condition is satisfied, there exists a wave function satisfying (REF ) and (REF ).", "But, is it unique?Here the question is considered at the level of the continuum description.", "Because there are multiple ways of coarse-graining, there will be multiple not-too-different $\\rho $ s and ${v}_k$ s that well-describe any finite collection of worlds and thus many wave functions.", "It may be that some ways of coarse-graining avoid the problems raised below by ensuring that the velocity fields are always well-defined.", "If they do, the derivability of $\\Psi $ from $\\rho $ and the ${v}_k$ s comes at the cost of limiting the wave functions one can recover, losing those in (REF ), (REF ), and (REF ).", "That is, can (REF ) and (REF ) be used to define $\\Psi $ in terms of $\\rho $ and the ${v}_k$ s?See also the discussion in [20].", "First consider the case where $\\rho $ is everywhere nonzero.", "The magnitude of $\\Psi $ can be derived from (REF ), and (REF ) gives the phase up to a global constant.", "The wave function can be determined up to a global phase.", "This would be insufficient if the overall phase mattered, but as the global phase is arbitrary this gives exactly what we need.", "Actually, it's even better this way.", "The fact that the dynamics don't care about the overall phase is explained in Newtonian QM by the fact that changes in the global phase of the wave function don't change the state of the universe; that is, they don't change $\\rho $ or the ${v}_k$ s. If the region in which $\\rho \\ne 0$ is not connected, the wave function is not uniquely determined by $\\rho $ and the ${v}_k$ s—one can introduce arbitrary phase differences between the separate regions.", "As an example of the breakdown of uniqueness, consider the second energy eigenstate of a single particle in a one-dimensional infinite square well of length $L$ .", "In this case the wave function is $\\psi _a(x)=\\sqrt{\\frac{2}{L}}\\sin \\left(\\frac{2\\pi x}{L}\\right)\\ .$ This describes a universe with $\\rho $ and ${v}$ given by $\\rho (x)&=\\frac{2}{L}\\sin ^2\\left(\\frac{2\\pi x}{L}\\right)\\nonumber \\\\{v}(x)&=\\left\\lbrace \\begin{array}{ll} 0 & \\mbox{ if }x\\ne \\frac{L}{2} \\\\ undefined & \\mbox{ if }x= \\frac{L}{2} \\end{array}\\right.\\ .$ The velocity field ${v}$ is undefined where there are no worlds.", "These expressions for $\\rho $ and ${v}$ are also compatible withThe wave function $\\psi _b$ has the disreputable property of not being smooth.", "It should be noted that there exist pairs of distinct smooth non-analytic wave functions which agree on $\\rho $ and ${v}$ at a time.", "(Thanks to Gordon Belot for suggesting an example like this.)", "For example, $\\psi _{\\alpha }(x)=&\\left\\lbrace \\begin{array}{ll}C e^{\\frac{-1}{1-(x+2)^2}} & \\mbox{ if }-3<x<-1 \\\\ -C e^{\\frac{-1}{1-(x-2)^2}} & \\mbox{ if }1<x<3 \\\\ 0 & \\mbox{ else}\\end{array}\\right.\\nonumber \\\\\\psi _{\\beta }(x)=& \\ |\\psi _{\\alpha }(x)| \\ .\\nonumber $ $\\psi _b(x)=\\sqrt{\\frac{2}{L}}\\left|\\sin \\left(\\frac{2\\pi x}{L}\\right)\\right| \\ .$ This exposes an inconvenient indeterminism: The time evolution of $\\psi _a$ is trivial as it is an energy eigenstate.", "Since $\\psi _b$ is not differentiable at $L/2$ , its time evolution cannot be calculated straightforwardly using the Schrödinger equation (REF ).", "As (REF ) and (REF ) do not determine which wave function is to be used to describe the state in (REF ), it is not clear how the state will evolve.", "The future evolution of the universe is not uniquely determined by the instantaneous state (REF ), the continuity equation (REF ), and the equation of motion (REF ).", "This indeterminacy arises because $\\rho $ is zero and the velocity field is undefined at $L/2$ , so $\\frac{\\partial \\rho }{\\partial t}$ and ${a}$ are undefined at $L/2$ .", "There is reason to think this indeterminism is an artifact of the continuum approximation where (REF ) and (REF ) need the velocity fields to be well-defined at every point in configuration space—even where there are no worlds—to yield a unique time evolution.", "The fundamental dynamics should take as input a specification of the position of each world in configuration space and the velocities of the particles in those worlds, all of which will be well-defined (§).", "Consider a slightly different problem from that just considered: Suppose one would would like to find a wave function $\\Psi (t)$ which describes a history of $\\rho (t)$ and the ${v}_k(t)$ s, satisfying (REF ) and (REF ) over some time interval.", "There will be a collection of wave functions which satisfy (REF ) and (REF ) at each time.", "For any such wave function, one can multiply it by a spatially homogeneous time-dependent phase factor, $e^{if(t)}$ , to get another wave function which always satisfies (REF ) and (REF ).", "(The global phase at each time is arbitrary and (REF ) and (REF ) do nothing to stop you from picking whatever global phase you'd like at each time.)", "In general, some of these wave functions will satisfy the Schr equation (REF ) and others will not.", "To constrain the time-dependence of the phase when using a wave function to describe histories, (REF ) can be imposed as a third link between the wave function and the particles (in addition to (REF ) and (REF )).", "Because (REF ), (REF ), (REF ), and (REF ) hold, the wave function must obey the Schrödinger equation.", "This section began with the observation that there are states in Newtonian QM that cannot be described by a wave function.", "However, these can be excised by imposing the Quantization Condition.", "Given a state that can be described by a wave function, one might hope that this wave function would be unique.", "Sometimes it is not.", "A wave function aptly describes a state in Newtonian QM at a time if (REF ) and (REF ) are satisfied.", "But, if these are the only constraints, a history in Newtonian QM can always be described by many wave functions.", "So, there is freedom to add a third connection between the wave function and the particles.", "Imposing (REF ) proves a convenient choice as it guarantees that the wave function obeys the Schrödinger equation—a desirable feature since the point of introducing a wave function was to clarify the connection between Newtonian QM and standard treatments of quantum mechanics.", "Because a wave function can be introduced to describe the world density and the velocity fields, one is free to use well-known techniques to calculate the time evolution of the wave function and use that to determine how the world density and velocity fields evolve.", "However, there is evidence that it is sometimes easier to use the trajectories of worlds to calculate the time evolution [36], [19]." ], [ "Probability: Versus Everettian Quantum Mechanics", "The Born Rule is easier to justify in Newtonian QM than in the many-worlds interpretation.", "In Everettian QM, there is dispute over how one can even make sense of assigning probabilities to measurement outcomes when the way the universe will branch is deterministic and known (the incoherence problem).", "There is also the quantitative problem of why the Born Rule probabilities are the right ones to assign.", "Recent derivations tend to appeal to complex decision-theoretic arguments, which, although they may ultimately be successful, are not uncontroversially accepted [26].", "Things look worrisome because there are some prima facie plausible ways of counting agents which yield the result that the vast majority of agents see relative frequencies of experimental outcomes which deviate significantly from those predicted by the Born Rule (although the total amplitude-squared weight of the branches in which agents see anomalous statistics is small).", "Newtonian QM does not run into similar problems since the number of worlds in a particular region of configuration space is always proportional to $|\\Psi |^2$ .", "At any time, most agents are in high amplitude regions.", "So, in typical measurement scenarios, most agents will see long-run frequencies which agree with the predictions of the Born Rule.", "Were a proponent of Prodigal QM to claim similar advantages over Everettian QM, one could reasonably object that the Born Rule is recovered only because it was put in by hand.", "In Prodigal QM, (REF ) is an additional postulate.", "In Newtonian QM, it is not.", "The density of worlds is given by $|\\Psi |^2$ because $\\Psi $ is definitionally related to the density of worlds by (REF ) (see §).", "The wave function is, after all, not fundamental but a mere description of $\\rho $ and the ${v}_k$ s." ], [ "Probability: Versus Bohmian Mechanics", "Although it is widely agreed that the Born Rule can be justified in Bohmian mechanics, there is disagreement about how exactly the story should go.", "In this section I will briefly discuss three ways of justifying the Born Rule in Bohmian mechanics and then argue that Newtonian QM can give a cleaner story.", "First, though, note an important similarity between the two theories.", "According to Newtonian QM each world follows an approximately Bohmian path through configuration space.", "So if you think that worlds in which particles follow Bohmian trajectories are able to reproduce the results of familiar quantum experiments, you should think worlds in Newtonian QM can too.", "In Bohmian mechanics, not all initial conditions reproduce the statistical predictions of quantum mechanics.", "That is, not all specifications of the initial wave function $\\Psi (0)$ and particle configuration $({x}_1(0), {x}_2(0), ...)$ yield a universe in which experimenters would see long-run statistics of measurements on subsystems which agree with the predictions of the Born Rule.", "Why should we expect to be in one of the universes with Born Rule statistics?", "One way to respond to this problem is to add a postulate to the theory which ensures that ensembles of particles in the universe will (or almost certainly will) display Born Rule statistics upon measurement (e.g., [21]).", "A second option is to argue that typical universes are such that Born Rule frequencies will be observed when measurements are made [16].", "To say that such results are “typically” observed is to say that: for any initial wave function $\\Psi (0)$ , the vast majority of initial particle configurations reproduce Born Rule statistics.", "Speaking of the “vast majority” of initial configurations only makes sense relative to a way of measuring the size of regions of configuration space; here the measure used is given by $|\\Psi |^2$ .", "A third option: one could argue that many initial states will start to display Born Rule statistics sufficiently rapidly that, since we are not at the beginning of the universe, we should expect to see Born Rule frequencies now even if such frequencies were not displayed in the distant past [32].", "Each of these proposals faces challenges.", "The additional postulates which might be added to the theory look ad hoc.", "The measure used to determine typicality must be satisfactorily justified.For a statement of the objection, see [13].", "For a variety of reasons to regard the measure as natural, see [18].", "The desirable evolution of states described in the third option has only been demonstrated in relatively simple cases.", "Also, there will certainly exist initial conditions that do not come to display Born Rule statistics sufficiently rapidly and these must somehow be excluded.", "To the extent that one finds these objections to Bohmian strategies worrisome, it is an advantage of the new theory that it avoids them.", "Although Newtonian QM, like Bohmian mechanics, permits a particular world to have a history of measurement results where the frequencies of outcomes do not match what one would expect from the Born Rule, it is impossible for the density of worlds to deviate from $|\\Psi |^2$ .", "So, in light of the results in [16], it will always be the case that Born Rule statistics are observed in the vast majority of worlds in any universe of Newtonian QM.", "Since we're not sure which world we are in, we should expect to be in one in which Born Rule statistics are observed." ], [ "Probability: Newtonian QM", "If the universe's evolution is deterministic and the initial state is known, what is there left for an agent to assign probabilities to?", "There is no incoherence problem in Newtonian QM since, given the state of the universe, one is generally uncertain which of the many distinct worlds one is in.", "There will always be many possibilities consistent with one's immediate experiences.", "The uncertainty present here is self-locating uncertainty [22].", "Of course, there will generally also be uncertainty about the state of the universe.", "On to the quantitative problem:See also the discussion in [8].", "Given a particular distribution of worlds $\\rho $ and set of velocity fields ${v}_k$ , that is, given a specification of the state of the universe, one ought to assign equal credence to being in any of the worlds consistent with one's evidence.This follows from a more general epistemic principle defended in [17].", "Because there are only a finite number worlds, this advice is unambiguous.", "As it turns out, this basic indifference principle suffices to derive the correct quantum probabilities.", "Consider an idealized case in which the agent knows the world density and the velocity fields, and knows that there is an agent in each of these worlds having experiences indistinguishable from her own.", "In this case, the above indifference principle tells her to assign probabilities to being in different regions of configuration space in accordance with $\\rho $ .", "Since $\\rho =|\\Psi |^2$ , she must assign credences in accordance with $|\\Psi |^2$ and thus in agreement with the Born Rule.", "Next, suppose this agent learns the outcome of an experiment.", "Then she ought to assign zero credence to the worlds inconsistent with her evidence and reapportion that credence among those which remain (keeping the probability of each non-eliminated world equal).", "This updating is analogous to learning which branch you are on after a measurement in Everettian QM.", "In general, the probability agent $S$ ought to assign to her own world having property $A$ , conditional on a particular state of the universe at a certain time, is $\\mbox{Pr}\\big (A\\big |\\rho ,{v}_1, {v}_2, ...)&=\\frac{\\mbox{\\# of worlds with property }A\\mbox{ and a copy of }S}{\\mbox{\\# of worlds with a copy of }S}\\nonumber \\\\&=\\frac{\\!\\!", "d V_{AS} \\ \\rho ({x}_1,{x}_2,...)}{\\!\\!", "d V_S \\ \\rho ({x}_1,{x}_2,...)}=\\frac{\\!\\!", "d V_{AS} \\ |\\Psi ({x}_1,{x}_2,...)|^2}{\\!\\!", "d V_S \\ |\\Psi ({x}_1,{x}_2,...)|^2}\\ .$ Here $A$ could be something like, “the pointer indicates 7” or “the particle just fired will hit in the third band of the interference pattern.” The volume $V_{S}$ delimits the set of worlds, specified by a region of configuration space, compatible with $S$ 's data.", "Worlds in this region are such that previous experiments had the outcomes $S$ remembers them having, macroscopic arrangements of particles match what $S$ currently observes, and some person is having the same conscious experiences as $S$ .For simplicity, I have neglected the possibility that $S$ 's memories or current observations are deceptive.", "The volume $V_{AS}$ gives the set of worlds compatible with $S$ 's data in which $A$ holds.Note that the boundaries of $V_{S}$ and $V_{AS}$ will often depend on $\\rho $ and the ${v}_k$ s. These conditional probabilities can be used to test hypotheses about $\\rho $ and the ${v}_k$ s and thus to learn about the state of the universe (not just one's own world) from experience." ], [ "Continuous Infinity or Mere Multitude of Worlds?", "So far, we have taken $\\rho $ to describe the distribution of a large but finite number of worlds.", "But, one might be tempted to defend a variant of Newtonian QM in which there are a continuous infinity of worlds, one at every point at which $\\rho $ is non-zero.", "This causes trouble.", "The meaning of $\\rho $ becomes unclear if we move to a continuous infinity of worlds since we can no longer understand $\\rho $ as yielding the proportion of all worlds in a given volume of configuration space upon integration over that volume.", "There would be infinite worlds in any finite volume (where $\\rho \\ne 0$ ) and infinite total worlds.", "If $\\rho $ doesn't give the proportion of worlds in a region, it is unclear why epistemic agents should apportion credences as recommended in the previous section.", "So, the continuous variant, if sense can be made of it, faces the quantitative probability problem head on.", "As discussed in §, the dynamical law proposed for Newtonian QM (REF ) is not fundamental.", "If it somehow turns out that we cannot view the force caused by the quantum potential as arising from an interaction between individual worlds, this would provide a reason to accept a continuous infinity of quantum worlds over a mere multitude.", "It might appear to be a strength of the continuous variant that its laws can already be precisely stated, but I expect that this advantage will evaporate when possible fundamental interactions are formulated for the discrete variant.", "The continuous variant does have a serious advantage: the continuum approximation (§) is no approximation.", "Particles will unerringly follow Bohmian trajectories." ], [ "Ontology", "According to Newtonian QM, what the universe contains is a finite collection of worlds.", "There are at least two ways to precisify this idea.", "First, one might take configuration space to be the fundamental space, inhabited by point-particles (worlds).", "Second, one might take the fundamental space to be ordinary three-dimensional space, inhabited by particles in different worlds.", "According to the first picture, on the fundamental level, the universe is 3$N$ -dimensional and contains a large number of point particles, each of which has dynamics so complex that it merits the name of “world” or “world-particle.” Forces between these world-particles are Newtonian and the dynamics are local.", "Here Newtonian QM is a theory of the Newtonian dynamics of a fluid of world-particles in 3$N$ -dimensional space.", "[2] has argued that the one world of Bohmian mechanics can be understood as a world-particle which moves around in configuration space guided by the wave function.", "He provides a way of explaining how the appearance of a three-dimensional world arises from the motion of this world-particle which applies mutatis mutandis to Newtonian QM in which there are more world-particles executing the same old Bohmian dances.", "On the second picture there are particles interacting in three-dimensional space, nothing more.This second option resembles the novel ontology for the many-worlds interpretation proposed by [4].", "Space is very densely packed with particles, but not all particles are members of the same world.", "Some particles are members of world #1, some of world #2, etc.", "What world a particle belongs to might be a primitive property, like its mass or charge.", "The equation of motion for a particle in world #827, (REF ), says that the force from the potential $V$ depends only on the positions of the other particles in world #827.", "However, the quantum potential introduces an inter-world force whereby particles that are not members of world #827 can still impact the trajectory of a particle in this world.", "So, particles which happen to be members of the same world interact in one way, whereas particles which are members of different worlds interact another way.", "In the many-worlds interpretation, one must tell a somewhat complicated story about how people and quantum worlds arise as emergent entities in the time-evolving quantum state [33].", "This story may not be successful.", "It might be the case that wave functions evolving in accordance with the Schrödinger equation are incapable of supporting life or at least lives that feel like ours [24].", "If that's right, Newtonian QM has a potential advantage.", "On the second ontological picture, people are built from particles in the usual way.", "On the first ontological picture, there is a story about emergence that must be told but the details of the story are very different from the Everettian one and it succeeds or fails independently.", "If, on the other hand, the Everettian story about emergence is successful, then Bohmian mechanics (as formulated here) faces the Everett-in-denial objection [11], [9], [31].", "Both Everettian QM and Bohmian mechanics contain in their fundamental ontology a wave function which always obeys the Schrödinger equation.", "If such a wave function is sufficient for there to be creatures experiencing what appears upon not-too-close inspection to be a classical world, then Bohmian mechanics, like Everettian QM, includes agents who see every possible outcome of a quantum measurement.", "If the Everettian story about emergence works and the Everett-in-denial objection against Bohmian mechanics is successful, then Newtonian QM has a serious advantage over Bohmian mechanics.", "Newtonian QM cannot be accused of being a many worlds theory in disguise since the theory embraces its many worlds ontology." ], [ "Symmetries: Time Reversal and Galilean Boosts", "Newtonian QM can help us understand symmetry transformations in quantum mechanics.", "First, consider time reversal.", "[3] proposes an intuitive and general account of time reversal symmetry in physical theories which judges QM, in all of its familiar precisifications, to fail to be time-reversal invariant.", "A deterministic physical theory specifies which sequences of instantaneous states are allowed and which are forbidden through dynamical laws.", "If the laws allow the time-reversed history of instantaneous states for any allowed history of instantaneous states, then the theory is deemed time-reversal invariant.", "In theories like Bohmian mechanics or Everettian QM, the instantaneous state includes the wave function at a time $\\Psi ({x}_1,{x}_2,...,t)$ and a complete history includes the wave function at all times.", "The time reverse of the history is $\\Psi ({x}_1,{x}_2,...,-t)$ .", "$\\Psi ({x}_1,{x}_2,...,-t)$ will not necessarily satisfy the Schrödinger equation whenever $\\Psi ({x}_1,{x}_2,...,t)$ does—so quantum mechanics is judged not to be time-reversal invariant.", "However, $\\Psi ^*({x}_1,{x}_2,...,-t)$ will always satisfy the Schrödinger equation whenever $\\Psi ({x}_1,{x}_2,...,t)$ does (standard textbook accounts take this to be the time reversed history and thus judge the theory to be time-reversal invariant).", "In Newtonian QM, it is straightforward to show that time reversing the history of particle trajectories amounts to changing the history of the wave function from $\\Psi ({x}_1,{x}_2,...,t)$ to $\\Psi ^*({x}_1,{x}_2,...,-t)$ .", "The instantaneous state of the world is specified by giving the locations (but not the velocities) of all of the particles in all of the worlds.", "The time reversal operation thus takes the history $\\rho ({x}_1,{x}_2,...,t)$ and ${v}_k({x}_1,{x}_2,...,t)$ to $\\rho ({x}_1,{x}_2,...,-t)$ and $-{v}_k({x}_1,{x}_2,...,-t)$ .", "By (REF ), flipping the phase generates a wave function which describes the flipped velocities of particles in the time-reversed history.", "The complex conjugation in the textbook time reversal operation for quantum mechanics can be explained as deriving from a reversal in the velocities of the particles.", "Newtonian QM is time-reversal invariant according to Albert's account.", "Even if one doesn't agree with Albert's account of time-reversal invariance, it is a virtue of this theory over others that it can give a simple explanation of why the wave function transforms in the textbook way under time-reversal.", "Next, consider Galilean boosts.", "In a similar spirit to Albert's criticism of the standard account of time-reversal, one could argue that quanrtum mechanics is not invariant under Galilean boosts since the equations of motion are not generally obeyed when we take $\\Psi ({x}_1,{x}_2,...,t)$ to $\\Psi ({x}_1-{w}t,{x}_2-{w}t,...,t)$ .A point made by Albert in presentations.", "See also [30].", "The invariance of quantum mechanics under Galilean boosts is sometimes demonstrated by showing that, for certain potentials, there exists a transformation of the state which appropriately shifts the probability density and guarantees satisfaction of the Schrödinger equation (e.g., [5]).", "Under a boost by ${w}$ , the wave function is supposed to transform as $\\Psi _0(\\vec{x}_1,{x}_2,...,t)\\stackrel{{w}}{\\longrightarrow }\\Psi (\\vec{x}_1,{x}_2,...,t)= e^{\\frac{i}{\\hbar }\\sum _k{\\left\\lbrace m_k {w}\\cdot {x}_k-\\frac{1}{2}m_k |{w}|^2 t\\right\\rbrace }}\\Psi _0({x}_1-{w}t,{x}_2-{w}t,...,t)\\ .$ It's interesting that there exists a transformation which moves probability densities in the right way and guarantees that the Schrödinger equation is invariant under boosts, but it is unclear why this particular transformation is the one that really represents Galilean boosts.", "In Newtonian QM this transformation of the wave function results from boosting the velocities of all of the particles in all of the worlds.", "Adding ${w}$ to the velocity of each particle transforms the original density $\\rho _0(t)$ and the original velocity fields ${v}_{0k}(t)$ to $\\rho ({x}_1,{x}_2,...,t) &= \\rho _0({x}_1-{w}t,{x}_2-{w}t,...,t)\\nonumber \\\\{v}_{k}({x}_1,{x}_2,...,t) &= {v}_{0k}({x}_1-{w}t,{x}_2-{w}t,...,t)+{w}\\ .$ Suppose $\\Psi _0(t)$ , $\\rho _0(t)$ , and the ${v}_{0k}(t)$ s satisfy (REF ), (REF ), and (REF ); that is, $\\Psi _0(t)$ describes this density and these velocity fields.", "Then, the new wave function $\\Psi (t)$ generated by the transformation in (REF ) will satisfy (REF ), (REF ), and (REF ) for the $\\rho (t)$ and ${v}_k(t)$ s in (REF ), provided that the potential $V$ is translation invariant (as the reader can verify).", "Thus, (REF ) gives a general recipe for finding a wave function which correctly describes the boosted particles." ], [ "Spin-1/2 Particles", "There appears to be serious trouble on the horizon for this new theory.", "In Bohmian mechanics spin is often treated as a property of the wave function, not the particles pushed along by it.e.g., [15] and [1] So, if we remove the wave function, it looks like we'll lose all of the information about the spin of the system!", "Actually, there is a very natural way to extend Newtonian QM to a single particle with spin.", "If we endow the particle with a definite spin in every world, we can recover the standard dynamics.", "Here I'll apply to Newtonian QM a strategy which has been used in quantum hydrodynamics and (a version of) Bohmian mechanics (see [21] and references therein).", "Consider the dynamics of a single spin-$1/2$ particle.", "To our basic ontology, consisting of a distribution of worlds $\\rho ({x},t)$ where the particle has velocity ${v}({x},t)$ in each world, let us add a property to the particle in each world: spin magnetic moment.", "The spin magnetic moment ${u}({x},t)$ of a particle can be specified by a polar angle $\\alpha ({x},t)$ , an azimuthal angle $\\beta ({x},t)$ , and a constant $\\mu $ (for an electron, $\\mu \\approx \\frac{-e \\hbar }{2 m}$ , where $e$ is the magnitude of the electron's charge).", "${\\mu }=\\mu \\left(\\begin{array}{c}\\sin \\alpha \\cos \\beta \\\\ \\sin \\alpha \\sin \\beta \\\\ \\cos \\alpha \\end{array} \\right)$ Alternatively, we can speak of the particle's internal angular momentum ${S}$ , which is related to ${\\mu }$ by ${S}=\\frac{\\hbar }{2 \\mu } {\\mu }\\ .$ With the magnetic moment in hand, we can partially defineThis definition is only partial as $\\theta $ is left unspecified.", "the spinor wave function $\\chi $ from $\\rho $ and ${\\mu }$ by $\\chi =\\left(\\begin{array}{c}\\chi _+ \\\\ \\chi _- \\end{array}\\right)=\\left(\\begin{array}{c} \\sqrt{\\rho }\\cos \\frac{\\alpha }{2} \\: e^{i\\theta } \\\\ \\sqrt{\\rho }\\sin \\frac{\\alpha }{2} \\: e^{i\\theta +i\\beta } \\end{array}\\right)\\ ,$ similar to (REF ).", "Here the $z$ -spin basis is used to represent the spinor.", "The Bohmian guidance equation for a spin-$1/2$ particle is ${v}=\\frac{\\hbar }{m}\\mbox{Im}\\left[\\frac{\\chi ^\\dagger {\\nabla } \\chi }{\\chi ^\\dagger \\chi }\\right]\\ .$ Inserting the expression for $\\chi $ in (REF ) yields ${v}=\\frac{\\hbar }{m}{\\nabla }\\theta +\\frac{\\hbar }{m}\\sin ^2\\frac{\\alpha }{2}\\: {\\nabla }\\beta \\ ,$ similar to (REF ).", "The Pauli equation for a spin-$1/2$ particle in the presence of an external magnetic field is $i\\hbar \\frac{\\partial }{\\partial t} \\chi = \\left\\lbrace \\frac{-\\hbar ^2}{2 m}\\nabla ^2+V-\\mu {B} \\cdot \\sigma \\right\\rbrace \\chi \\ ,$ where $\\sigma $ are the Pauli spin matrices.", "To focus on spin, the contributions to the Hamiltonian arising because the particle has a charge (not just a magnetic moment) have been omitted.", "From (REF ), (REF ), and (REF ) one can derive the time dependence of ${\\mu }$ and ${v}$ .", "The magnetic moment vector evolves as $\\frac{\\hbar }{2 \\mu }\\frac{d{\\mu }}{dt}=&\\frac{\\hbar ^2}{4 m \\mu ^2 \\rho } {\\mu } \\times \\left[ \\partial _a \\left(\\rho \\: \\partial _a {\\mu }\\right) \\right] + {\\mu } \\times {B}\\nonumber \\\\\\frac{d {S}}{dt}=&{\\mu } \\times {B}_{\\text{Tot}}\\ ,$ using the Einstein summation convention over spatial index $a$ .Result as in [21].", "Note that different conventions are adopted for the sign of $\\mu $ .", "(REF ) is in agreement with Holland's eq.", "9.3.19, although written in a more suggestive form.", "The right hand side gives the net torque on the particle, which arises from a quantum and a classical contribution.", "These torques can be combined by defining ${B}_{\\text{Tot}}\\equiv {B}+\\frac{\\hbar ^2 \\left[ \\partial _a \\left(\\rho \\: \\partial _a {\\mu }\\right) \\right]}{4 m \\mu ^2 \\rho }\\ .$ The net magnetic field ${B}_{\\text{Tot}}$ is the sum of a classical and a quantum contribution.", "(REF ) gives the classical dynamics for the angular momentum of a magnetic dipole in the presence of the magnetic field ${B}_{\\text{Tot}}$ .", "From (REF ), (REF ), and (REF ), it follows that the acceleration can be expressed as $m {a}=-{\\nabla }\\left[Q+Q_{P}+V\\right]+\\mu _a {\\nabla } {B_{\\text{Tot}}}_a\\ .$ This is simply the equation of motion for a particle without spin (REF ) with two new terms: the classical force on a particle with magnetic moment ${\\mu }$ from a magnetic field ${B}_{\\text{Tot}}$ and a spin-dependent contribution to the quantum potential, $Q_P=\\frac{\\hbar ^2}{8 m \\mu ^2}{\\mu }\\cdot \\left(\\nabla ^2 {\\mu }\\right)=\\frac{1}{2 m}{S}\\cdot \\left(\\nabla ^2 {S}\\right)\\ .$ As with the quantum potential $Q$ discussed in §, this new term represents an interaction between worlds (as does the quantum contribution to the net magnetic field ${B}_{\\text{Tot}}$ ).", "Together, the above equations of motion for ${\\mu }$ and ${v}$ , (REF ) and (REF ), serve to define Newtonian QM for a single spin-$1/2$ particle.", "We can omit any mention of the spinor wave function $\\chi $ or the phase $\\theta $ in the fundamental laws.", "The equations of motion for ${\\mu }$ and ${v}$ , which govern the evolution of $\\rho $ via (REF ), will guarantee that $\\rho $ , ${\\mu }$ , and ${v}$ will evolve as if they were governed by a spinor wave function satisfying the Pauli equation, provided that the velocity field obeys a constraint like the one imposed for spin-0 particles in §, $\\oint {\\left(m{v}-\\hbar \\sin ^2\\frac{\\alpha }{2} \\: {\\nabla }\\beta \\right)\\cdot d{\\ell }}=nh\\ .$ In Newtonian QM, particles have well-defined spin magnetic moments at all times.", "How can the theory recover the results of standard experiments involving spin if particles are never in superpositions of different spin states?", "Consider, for example, a $z$ -spin “measuring” Stern-Gerlach apparatus.", "Suppose the wave function is in a superposition $z$ -spin up and $z$ -spin down: $\\frac{1}{\\sqrt{2}}\\left|\\uparrow _z\\right\\rangle +\\frac{1}{\\sqrt{2}}\\left|\\downarrow _z\\right\\rangle $ .", "When passed through the inhomogeneous magnetic field, the wave function will split in two.", "On the standard account, the particle will be found in either the upper region (corresponding to $z$ -spin up) or the lower region (corresponding to $z$ -spin down) upon measurement with equal probability.", "In Newtonian QM, there is initially an ensemble of worlds, in each of which the particle has some initial position in the wave packet and in all of which the particle's spin magnetic moment points squarely in the $x$ -direction.", "A particle in the top half of the initial wave packet has its spin rotated to point in the $z$ -direction as it passes through the Stern-Gerlach apparatus (in accordance with (REF )); a particle in the lower portion will end up with spin pointing in the negative $z$ -direction.", "In this theory, the Stern-Gerlach apparatus does not measure $z$ -spin, but instead forces particles to align their magnetic moments along the $z$ -axis.", "This is also how Stern-Gerlach measurements are interpreted in versions of Bohmian mechanics where particles have definite spins [12], [21]." ], [ "Conclusion", "An optimistic synopsis: Once we realize that Newtonian QM is a viable way of understanding non-relativistic quantum mechanics, we see that we never needed to overthrow Newtonian mechanics with a quantum revolution.", "One can formulate quantum mechanics in terms of point particles interacting via Newtonian forces.", "The mysterious wave function is merely a way of summarizing the properties of particles, not a piece of fundamental reality.", "There are a variety of reasons not to like this theory.", "First, there is arguably a cost associated with the abundance of other worlds which, although detectable via their interactions with our own world, are admittedly odd.", "Second, the space of states for the theory is larger than one might like in two distinct ways: There are possible combinations of $\\rho $ and the ${v}_k$ s that do not correspond to any wave function because the velocity fields cannot be expressed as the gradient of a phase (§).", "There are also states of the universe where the number of worlds is not sufficiently large for the continuum description to be valid (§).", "Even if there are a great many worlds, slight divergence from the predictions of standard quantum mechanics is to be expected.", "Third, it is a shortcoming of the current formulation of Newtonian QM that we must approximate the actual distribution of worlds as continuous and cannot yet formulate the fundamental equation of motion precisely for a discrete collection of worlds (§).", "Finally, the theory is limited in that it is not here extended to systems of multiple particles with spin or to relativistic quantum physics.", "In addition to its seductive conservatism, I view the following comparative strengths as most compelling.", "Against the many-worlds interpretation, Newtonian QM has two main advantages.", "First, there is no incoherence problem or quantitative probability problem—the Born Rule can be justified quickly from self-locating uncertainty (§).", "Second, the theory avoids the need to explain how worlds emerge from the wave function—worlds are taken to be fundamental (§).", "Compared to Bohmian mechanics, the theory is arguably simpler—it replaces an ontology of wave functions and particles with one just containing particles (§).", "Newtonian QM's explanation of why we should expect our world to reproduce Born Rule statistics is potentially more compelling than the Bohmian stories (§).", "Also, Newtonian QM is forthright about its many worlds character, sidestepping the Everett-in-denial objection (§).", "Acknowledgements Thanks to David Baker, Gordon Belot, Cian Dorr, Detlef Dürr, J. Dmitri Gallow, Sheldon Goldstein, Michael Hall, Daniel Peterson, Laura Ruetsche, Ward Struyve, Nicola Vona, and two anonymous referees for very useful feedback on drafts of this article.", "Thank you to Adam Becker, Sean Carroll, Dirk-André Deckert, Neil Dewar, Benjamin Feintzeig, Sophie Monahan, Cat Saint-Croix, Jonathan Shaheen, and Howard Wiseman for helpful discussions.", "This material is based upon work supported by the National Science Foundation Graduate Research Fellowship under Grant No.", "DGE 0718128." ] ]
1403.0014
[ [ "Domain wall network as QCD vacuum and the chromomagnetic trap formation\n under extreme conditions" ], [ "Abstract The ensemble of Euclidean gluon field configurations represented by the domain wall network is considered.", "A single domain wall is given by the sine-Gordon kink for the angle between chromomagnetic and chromoelectric components of the gauge field.", "The domain wall separates the regions with self-dual and anti-self-dual fields.", "The network of the domain wall defects is introduced as a combination of multiplicative and additive superpositions of kinks.", "The character of the spectrum and eigenmodes of color-charged fluctuations in the presence of the domain wall network is discussed.", "The concept of the confinement-deconfinement transition in terms of the ensemble of domain wall networks is outlined.", "Conditions for the formation of thick domain wall junction during heavy ion collisions are discussed, and the spectrum of color charged quasiparticles inside the trap is evaluated.", "An important observation is the existence of the critical size $L_c$ of the trap stable against gluon tachyonic modes, which means that deconfinement can occur only in a finite region of space-time in principle.", "The size $L_c$ is related to the value of gluon condensate $\\langle g^2F^2\\rangle$." ], [ "Introduction", "In general, diffusion of the relativized versions of ideas born in condensed matter and solid state physics to the quantum field theory has been proven to be extremely fruitful.", "It was realised long time ago that a complex of problems associated with investigation of the QCD vacuum structure appeared as particularly suitable object in this respect.", "This paper is focused on the further development of approach to QCD vacuum as a medium describable in terms of statistical ensemble of domain wall networks.", "This concept plays important role in description of condensed matter systems with rival order and disorder but has been insufficiently explored in application to QCD vacuum.", "The identification of the properties of nonperturbative gauge field configurations relevant to a coherent resolution of confinement, chiral symmetry breaking, $U_{\\rm A}(1)$ and strong CP problems is an overall task pursued by most approaches to the QCD vacuum structure.", "As a rule, analytical as well as Lattice QCD studies of QCD vacuum structure are focused on localized topological configurations (instantons, monopoles and dyons, vortices) which via condensation could be seen as appropriate gauge field configurations responsible for confinement of static color charges and other nonperturbative features of strong interactions.", "In recent years, three-dimensional configurations akin to domain walls became popular as well [1], [2], [3], [4], [5], [6].", "First of all, these are the $Z(3)$ domain walls related to the center symmetry of the pure Yang-Mills theory [4] and double-layer domain wall structures in topological charge density [6].", "Lattice QCD serves as a main source of motivation and verification tool for these studies in pair with the theoretically appealing scenario of static quark confinement in the spirit of the dual Meissner effect equipped with the Wilson and Polyakov loop criteria.", "The localized configurations are characterized by the vanishing ratio of the action to the 4-volume in the infinite volume limit.", "In this sense, instantons, monopoles, vortices and double-layer domain walls are localized configurations.", "A complementary treatment of the above mentioned overall task is based on the investigation of the properties of quantum effective action of QCD.", "As in other quantum systems with infinitely many degrees of freedom, the global minima of the effective action define the phase structure of QCD.", "The identification of global minima in different regimes (high energy density, high baryon density, strong external electromagnetic fields) has highest priority for understanding the phase transformations in hadronic matter.", "In general, a nontrivial global minimum corresponds to a gauge field with the strength not vanishing at space-time infinity and, hence, extensive action proportional to the four dimensional space-time volume of the system, unlike the localized configurations.", "A variety of essentially equivalent statements of the problem in the context of QCD can be found, for instance, in [7], [8], [9], [10], [11], [12].", "Global minima related by discrete symmetry transformations like CP, Weyl symmetry in the root space of $su(N_{\\rm c})$ , center symmetry in particular, is a reason to look for field configurations interpolating between them.", "First of all, these are domain wall configurations, but also lower dimensional topological defects.", "There is among others one difference between this treatment and approaches based on localized objects: the last one intends to merge the initially isolated objects (e.g., instanton gas or liquid) while the former collects defects in an initially homogeneous background.", "At first sight, both ways seem to lead to a similar outcome - a class of nonperturbative gluon field configurations with a self-consistent balance of order and disorder which can be characterized, in particular, by nonzero gluon condensate and topological charge density.", "However, essential disparity can arise since a superposition of localized objects inherits the properties of isolated objects while the superposition of defects in the initially homogeneous ordered background brings some disorder and merely refines the overall properties of the background.", "For instance, the superposition of infinitely many instantons and anti-instantons is not a configuration with a finite classical action but it maintains the property to have integer-valued topological charge.", "On the contrary, the configuration obtained by implanting infinitely many domain wall defects into the Abelian covariantly constant (anti-)self-dual field can have any real value of the mean topological charge density as well as any real value of topological charge fraction per domain [13].", "Both configurations can be seen as lumps of the topological charge density distributed in the Euclidean space-time like in Fig.REF or in the lattice configurations [2], [1].", "However, in the instanton picture each lump carries an integer charge while in the treatment of global minima the charge is any real, irrational, for instance, number.", "This can have dramatic consequences for the fate of $\\theta $ parameter in QCD and the natural resolution of the strong CP-problem [13].", "In the Euclidean formulation, the statement of the problem starts with the very basic symbol of the functional integral $&&Z=N\\int \\limits _{{\\cal F}} DA \\exp \\lbrace -S[A]\\rbrace ,$ where the functional space ${\\cal F}$ is subject to the condition ${\\cal F}=\\lbrace A: \\lim _{V\\rightarrow \\infty } \\frac{1}{V}\\int \\limits _V d^4xg^2F^a_{\\mu \\nu } (x)F^a_{\\mu \\nu }(x) =B_{\\mathrm {v}ac}^2\\rbrace .$ The constant $B_{\\mathrm {v}ac}$ is not equal to zero in the general case, which is equivalent to nonzero gluon condensate $\\langle g^2F^2 \\rangle $ .", "Condition (REF ) singles out fields $B_\\mu ^a$ with the strength which is constant almost everywhere in $R^4$ .", "It is a necessary requirement to allow gluon condensate to be nonzero.", "It does not forbid also fields with a finite action since the case $B_{\\mathrm {v}ac}=0$ has to be also studied.", "The dynamics chooses the value of $B_{\\mathrm {v}ac}$ .", "However, the phenomenology of strong interactions has already required nonzero gluon and quark condensates.", "Hence, they must be allowed in the QCD functional integral from the very beginning.", "Separation of the long range modes $B_\\mu ^a$ responsible for gluon condensate and the local fluctuations $Q_\\mu ^a$ in the background $B_\\mu ^a$ , must be supplemented by the gauge fixing condition.", "The background gauge condition for fluctuations $D(B)Q=0$ is the most natural choice.", "Further steps include integration over the fluctuation fields resulting in the effective action for the long-range fields and identification of the minima of this effective action (for more details see [11], [13], [14]) which dominate over the integral in the limit $V\\rightarrow \\infty $ and define the phase structure of the system.", "As soon as minima are identified, this setup defines a principal scheme for self-consistent identification of the class of gauge fields which almost everywhere coincide with the global minima of the quantum effective action.", "A treatment of these “vacuum fields” in the functional integral $Z &=&N^{\\prime }\\int \\limits _{{\\cal B}}DB \\int \\limits _{{\\cal Q}} DQ \\det [D(B)D(B+Q)]\\\\&&\\times \\delta [D(B)Q] \\exp \\lbrace -S[B+Q]+S[B]\\rbrace .$ must be nonperturbative.", "The fields $B_\\mu ^a\\in {\\cal B}$ are subject to condition (REF ) with the fixed vacuum value of the condensate $B_{\\mathrm {v}ac}^2$ .", "The condensate plays the role of the scale parameter of QCD to be identified from the hadron phenomenology.", "The fluctuations $Q$ in the background of the vacuum fields can be seen as perturbations.", "The homogeneous fields with the domain wall defects are the most natural and simplest example of gluon configurations which are homogeneous almost everywhere in $R^4$ and satisfy the basic condition Eq.", "(REF ).", "Basic argumentation in favour of the Abelian (anti-)self-dual homogeneous fields as global minima of the effective action originates from papers [7], [10], [8], [16], [17], [15].", "Within the Ginzburg-Landau approach to the effective action the domain wall is described simply by the sine-Gordon kink for the angle between chromomagnetic and chromoelectric components of the gluon field [15].", "This kink configuration can be seen as either Bloch or Néel domain wall separating the regions with self-dual and anti-self-dual Abelian gauge fields.", "On the domain wall the gluon field is Abelian with orthogonal to each other chromomagnetic and chromoelectric fields.", "We shall not repeat here arguments leading to this conclusion but just refer to papers [15], [14] where a more detailed discussion can be found.", "Group theoretical analysis of the Weyl symmetry and subgroup embeddings behind the domain wall formation in the effective gauge theories is given in a recent paper [18].", "It should be also mentioned that the model of confinement, chiral symmetry breaking and hadronization based on the dominance of the gluon fields which are (anti-)self-dual Abelian almost everywhere demonstrated high phenomenological performance [19], [11], [13], [20].", "The purpose of the present paper is to evolve the approach outlined in article [15] in two respects: explicit analytical construction of the domain wall network in $R^4$ through a combination of additive and multiplicative superpositions of kinks, and refining the spectrum and eigenmodes of the color charged scalar, spinor and vector fields in the background of a domain wall.", "In particular the spectrum of quasiparticles inside the thick domain wall junction is evaluated.", "It is shown that the standard methods of the sine-Gordon model [21] allow one to generate various domain wall networks.", "The eigenvalues and eigenfunctions are found for the Laplace operator in the background of a single infinitely thin domain wall.", "In this case, the eigenvalue problem has to be solved separately in the bulk of $R^4$ and on the 3-dimensional hyperplane of the wall.", "The continuity of the charge current through the wall is required together with the square integrability of the bulk eigenfunctions.", "For the infinitely thin wall the bulk eigenfunction possesses the purely discrete spectrum which coincides with the spectrum for the case of a homogeneous (anti-)self-dual field without a kink defect.", "The eigenfunctions differ in a certain way but have the same harmonic oscillator type as the ones in the absence of the kink defect.", "These modes describe confined color charged fields.", "The eigenfunctions localized on the wall have continuous spectrum with the dispersion relation of charged quasi-particles.", "This confirms qualitative conjectures of [15], [14].", "It is argued that thick domain wall junction may be formed during heavy ion collisions and play the role of a trap for charged quasi-particles.", "Confinement is lost inside the trap of a finite size.", "There exists a critical size of the stable trap, beyond which the emerging tachyonic gluon modes destroy it.", "The paper is organized as follows.", "Section II is devoted to the domain wall network construction.", "The spectrum of scalar color charged field in the background of infinitely thin domain wall is discussed in section III.", "In the fourth section we discuss the chromomagnetic trap formation and evaluate the spectrum and eigenmodes of the color charged scalar, vector and spinor quasiparticles inside the trap." ], [ "Nonzero gluon condensate $\\langle g^2F^2\\rangle $ and domain wall network in QCD vacuum", "The calculation of the effective quantum action for the Abelian (anti-)self-dual homogeneous gluon field within the functional renormalization group approach [17] has indicated that this configuration is a serious candidate for the role of global minimum of QCD effective action and has enhanced the older one-loop results [7], [10], [8].", "The functional RG result also supported conclusions of [11], [15] based on the Ginzburg-Landau type effective Lagrangian of the form $\\mathcal {L}_{\\mathrm {eff}} &=& - \\frac{1}{4\\Lambda ^2}\\left(D^{ab}_\\nu F^b_{\\rho \\mu } D^{ac}_\\nu F^c_{\\rho \\mu } + D^{ab}_\\mu F^b_{\\mu \\nu } D^{ac}_\\rho F^c_{\\rho \\nu }\\right)\\nonumber \\\\&-&U_{\\mathrm {eff}}\\nonumber \\\\U_{\\mathrm {eff}}&=&\\frac{\\Lambda ^4}{12} {\\rm Tr}\\left(C_1\\breve{ f}^2 + \\frac{4}{3}C_2\\breve{ f}^4 - \\frac{16}{9}C_3\\breve{ f}^6\\right),$ where $\\Lambda $ is a scale of QCD related to gluon condensate, $\\breve{f}=\\breve{F}/\\Lambda ^2$ , and $&& D^{ab}_\\mu = \\delta ^{ab} \\partial _\\mu - i\\breve{ A}^{ab}_\\mu = \\partial _\\mu - iA^c_\\mu {(T^c)^{ab}},\\\\&& F^a_{\\mu \\nu } = \\partial _\\mu A^a_\\nu - \\partial _\\nu A^a_\\mu - if^{abc} A^b_\\mu A^c_\\nu ,\\\\&& \\breve{ F}_{\\mu \\nu } = F^a_{\\mu \\nu } T^a,\\ \\ \\ T^a_{bc} = -if^{abc}\\\\&& {\\rm Tr}\\left(\\breve{ F}^2\\right) = \\breve{ F}^{ab}_{\\mu \\nu }\\breve{ F}^{ba}_{\\nu \\mu } = -3 F^a_{\\mu \\nu }F^a_{\\mu \\nu } \\le 0,\\\\ && C_1>0, \\ C_2>0, \\ C_3 > 0.$ Detailed discussion of this expression can be found in [15].", "Here it should be noted that all symmetries of QCD are respected and the signs of the constants are chosen so that the action is bounded from below and its minimum corresponds to the fields with nonzero strength, i.e.", "$F^2\\ne 0$ at the minimum.", "Thus, an important input is the existence of the nonzero gluon condensate.", "By inspection, one gets as an output twelve (for $SU(3)$ ) global degenerate discrete minima.", "The minima are achieved for covariantly constant Abelian (anti-)self-dual fields $\\breve{ A}_{\\mu } = -\\frac{1}{2}\\breve{ n}_k F_{\\mu \\nu }x_\\nu , \\, \\tilde{F}_{\\mu \\nu }=\\pm F_{\\mu \\nu }$ where the matrix $\\breve{n}_k$ belongs to the Cartan subalgebra of $su(3)$ $\\breve{n}_k &=& T^3\\ \\cos \\left(\\xi _k\\right) + T^8\\ \\sin \\left(\\xi _k\\right),\\nonumber \\\\\\xi _k&=&\\frac{2k+1}{6}\\pi , \\, k=0,1,\\dots ,5.$ The values $\\xi _k$ correspond to the boundaries of the Weyl chambers in the root space of $su(3)$ .", "The minima are connected by the discrete parity and Weyl transformations, which indicates that the system is prone to existence of solitons (in real space-time) and kink configurations (in Euclidean space).", "Below we shall concentrate on the simplest configuration – kink interpolating between self-dual and anti-self-dual Abelian vacua.", "If the angle $\\omega $ between chromoelectric and chromomagnetic fields is allowed to deviate from the constant vacuum value and all other parameters are fixed to the vacuum values, then the Lagrangian takes the form $\\mathcal {L}_{\\textrm {eff}} &=& -\\frac{1 }{2}\\Lambda ^2 b_{\\textrm {vac}}^2 \\partial _\\mu \\omega \\partial _\\mu \\omega \\\\&-& b_{\\textrm {vac}}^4 \\Lambda ^4 \\left(C_2+3C_3b_{\\textrm {vac}}^2 \\right){\\sin ^2\\omega },$ with the corresponding sine-Gordon equation $\\partial ^2\\omega = m_\\omega ^2 \\sin 2\\omega ,\\ \\ m_\\omega ^2 = b_{\\textrm {vac}}^2 \\Lambda ^2\\left(C_2+3C_3b_{\\textrm {vac}}^2 \\right),$ and the standard kink solution $\\omega (x_\\mu ) = 2\\ {\\rm arctg} \\left(\\exp (\\mu x_\\mu )\\right)$ interpolating between 0 and $\\pi $ .", "Here $x_\\mu $ stays for one of the four Euclidean coordinates.", "The kink describes a planar domain wall between the regions with almost homogeneous Abelian self-dual and anti-self-dual gluon fields.", "Chromomagnetic and chromoelectric fields are orthogonal to each other on the wall, see Fig.REF .", "Far from the wall, the topological charge density is constant, its absolute value is equal to the value of the gluon condensate.", "The topological charge density vanishes on the wall.", "The upper plot shows the profiles of the components of the chromomagnetic and chromoelectric fields corresponding to the Bloch domain wall – the chromomagnetic field flips in the direction parallel to the wall plane.", "Figure: Kink profile in terms of the components of the chromomagnetic and chromoelectric field strengths (upper plot), and a two-dimensional slice for the topological charge density in the presence of a single kink measured in units of g 2 F αβ b F αβ b g^2F^b_{\\alpha \\beta }F^b_{\\alpha \\beta } (lower plot).Here ω\\omega is the angle between the chromomagnetic and chromoelectric fields, cosω=F μν a F ˜ μν a /F αβ b F αβ b \\cos \\omega =F^a_{\\mu \\nu }\\tilde{F}^a_{\\mu \\nu }/F^b_{\\alpha \\beta }F^b_{\\alpha \\beta }.", "The three-dimensional planar domain wall separates the four-dimensional regions filled with the self-dual (blue color) and anti-self-dual (red color) Abelian covariantly constant gluon fields.", "The chromomagnetic and chromoelectric fields are orthogonal to each other inside the wall (green color).Figure: Two-dimensional slice of a multiplicative superposition of two kinks.Figure: Two-dimensional slice of the layered topological charge distribution in R 4 R^4 according to Eq.().", "The action density is equal to the same nonzero constant value for all three configurations.", "The LHS plot represents a configuration with infinitely thin planar Bloch domain wall defects, which is the Abelian homogeneous (anti-)self-dual field almost everywhere in R 4 R^4, characterized by the nonzero absolute value of the topological charge density almost everywhere proportional to the value of the action density.The most RHS plot shows the opposite case of very thick kink network.", "Green color corresponds to the gauge field with infinitesimally small topological charge density.", "Most LHS configuration is confining (only colorless hadrons can be excited) while most RHS one supports the color charged quasiparticles as elementary excitations.Figure: A two-dimensional slice of the four-dimensional lump of anti-self-dual field in the background of the self-dual configuration.", "The domain wall surrounding the lump in the four-dimensional space is given by the multiplicative superposition of eight kinks as it is defined by Eq.", "().The domain wall network can be now constructed by the standard methods [21].", "Let us denote the general kink configuration as $\\zeta (\\mu _i,\\eta _\\nu ^{i}x_\\nu -q^{i})=\\frac{2}{\\pi }\\arctan \\exp (\\mu _i(\\eta _\\nu ^{i}x_\\nu -q^{i})),$ where $\\mu _i$ is the inverse width of the kink, $\\eta _\\nu ^{i}$ is a normal vector to the plane of the wall, $q^{i}=\\eta _\\nu ^{i}x^{i}_\\nu $ with $x^{i}_\\nu $ - coordinates of the wall.", "The topological charge density for the multiplicative superposition of two kinks with the normal vectors anti-parallel to each other $\\omega (x_1)=\\pi \\zeta (\\mu _1,x_1-a_1)\\zeta (\\mu _2,-x_1-a_2)$ is shown in Fig.REF .", "The additive superposition of infinitely many pairs $\\omega (x_1)=\\pi \\sum \\limits _{j=1}^{\\infty }\\zeta (\\mu _j,x_1-a_j)\\zeta (\\mu _{j+1},-x_1-a_{j+1})$ gives a layered topological charge structure in $R^4$ , Fig.REF .", "Formally, one may try to go further and consider the product $\\omega (x)=\\pi \\prod _{i=1}^k \\zeta (\\mu _i,\\eta _\\nu ^{i}x_\\nu -q^{i}).$ For an appropriate choice of normal vectors $\\eta ^i$ this superposition represents a lump of anti-self-dual field in the background of the self-dual one, in two, three and four dimensions for $k=4,6,8$ , respectively.", "The case $k=8$ is illustrated in Fig.REF .", "The general kink network is then given by the additive superposition of lumps (REF ) $\\omega =\\pi \\sum _{j=1}^{\\infty }\\prod _{i=1}^k \\zeta (\\mu _{ij},\\eta _\\nu ^{ij}x_\\nu -q^{ij}).$ The correponding topological charge density is shown in Fig.", "REF .", "This figure as well as the LHS of Fig.", "REF represents the configuration with infinitely thin domain wall defects, that is the Abelian homogeneous (anti-)self-dual field almost everywhere in $R^4$ characterized by the nonzero absolute value of the topological charge density which is constant and proportional to the value of the action density almost everywhere.", "The most RHS plots in Figs.", "REF and REF show the opposite case of the network composed of very thick kinks.", "Green color corresponds to the gauge field with an infinitesimally small topological charge density.", "Study of the spectrum of colorless and color charged fluctuations indicates that the LHS configuration is expected to be confining (only colorless hadrons can be excited as particles) while the RHS one (crossed orthogonal field) supports the color charged quasiparticles as the elementary excitations.", "It is expected that the RHS configuration can be triggered by external electromagnetic fields  [14], [22], [23].", "Strong electromagnetic fields emerge in relativistic heavy ion collisions [26], [25], [24].", "Even after switching off the external electromagnetic field the nearly pure chromomagnetic vacuum configuration (RHS Fig.REF ) can support strong anisotropies [27] and, in particular, influence the chiral symmetry realization in the collision region [28].", "More detailed consideration of the spectrum of elementary color charged excitations at the domain wall junctions (the green regions) is given in the section .", "A comment on representation of the domain wall network in terms of the vector potential is in order.", "The domain wall network constructed in this section relies on the separation of the Abelian part from the general gauge field.", "The vector potential representation can be easily realized for the planar Bloch domain wall and their layered superposition, Fig.", "REF .", "The same is true also for the interior of a thick domain wall junction, where field is almost homogeneous.", "The description of the domain walls in the general network Fig.", "REF in terms of the vector potential requires application of the gauge field parametrization suggested in a series of papers by Y.M.", "Cho [29], [30], S. Shabanov [31], [32], L.D.", "Faddeev and A. J. Niemi [33] and, recently, by K.-I.", "Kondo [34].", "In this parameterization the Abelian part ${\\hat{V}}_\\mu (x)$ of the gauge field ${\\hat{A}}_\\mu (x)$ is separated manifestly, ${\\hat{A}}_\\mu (x) &=& {\\hat{V}}_\\mu (x) + {\\hat{X}}_\\mu (x), \\,{\\hat{V}}_\\mu (x) = {\\hat{B}}_\\mu (x) + {\\hat{C}}_\\mu (x), \\\\{\\hat{B}}_\\mu (x) &=& [n^aA^a_\\mu (x)]\\hat{n} (x)=B_\\mu (x)\\hat{n}(x), \\nonumber \\\\{\\hat{C}}_\\mu (x) &=& g^{-1}\\partial _\\mu \\hat{n}(x)\\times \\hat{n}(x), \\nonumber \\\\{\\hat{X}}_\\mu (x) &=& g^{-1}{\\hat{n}}(x) \\times \\left( \\partial _\\mu {\\hat{n}}(x) + g {\\hat{A}}_\\mu (x) \\times {\\hat{n}}(x) \\right), \\nonumber $ where ${\\hat{A}}_\\mu (x) = A^a_\\mu (x) t^a$ , ${\\hat{n}} (x) = n_a (x) t^a$ , $n^a n^a = 1$ , and ${\\partial _\\mu \\hat{n}}\\times {\\hat{n}} = i f^{abc}\\partial _\\mu n^a n^b t^c,\\,\\, [t^a,t^b]=if^{abc}t^c.$ The field ${\\hat{V}}_\\mu $ is seen as the Abelian field in the sense that $[{\\hat{V}}_\\mu (x),{\\hat{V}}_\\nu (x)]=0$ .", "The color vector field $n^a(x)$ may be used for detailed description of the thin domain wall junctions in general case.", "This issue is beyond the scope of the present paper and will be considered elsewhere.", "Figure: Three-dimensional slices of the kink network - additive superposition of numerous four-dimensional lumps as it is given byEq. ().", "The correspondence of colors to the character of the configuration is the same as in Fig.", "." ], [ "Boundary condition", "In this section we study the spectrum of color charged field fluctuations in the background of a single planar domain wall of the Bloch type.", "The best thing to do would be to solve the eigenvalue problem for the kink of the finite width.", "However, the problem turns out to be not that simple.", "Let us consider the problem for the scalar field in the adjoint representation, that is just the Faddeev-Popov ghost field in the background gauge.", "The quadratic part of the action for the scalar field in the background field of a planar kink with the finite width placed at $x_1=0$ looks like $S[\\Phi ]&=&-\\int d^4x (D_\\mu \\Phi )^\\dagger (x)D_\\mu \\Phi (x)\\\\ \\nonumber &=&\\int d^4x \\Phi ^\\dagger (x)D^2\\Phi (x),\\\\\\nonumber D_\\mu &=&\\partial _\\mu +i\\breve{B}_\\mu , \\, \\breve{B}_\\mu =-\\breve{n} B_\\mu (x).$ Here $\\breve{n}$ is the constant color matrix, $B_\\mu $ is the vector potential for the planar Bloch domain wall.", "For our purposes the most convenient gauge for $B_\\mu $ is $&& B_1=H_2(x_1)x_3+H_3(x_1)x_2,\\\\ \\nonumber && B_2=B_3=0,\\quad B_4=-Bx_3,\\\\ \\nonumber &&H_2=B\\sin \\omega (x_1), \\,H_3=-B\\cos \\omega (x_1),\\\\ \\nonumber &&\\omega (x_1)=2\\ {\\rm arctg} \\exp \\mu x_1.$ A kink with the finite width is a regular everywhere in $R^4$ function, the scalar field is assumed to be a continuous square integrable function.", "Integration by parts in Eq.", "(REF ) does not generate surface terms either at infinity or at the location of the kink.", "However, there is a peculiarity related to the chosen gauge of the background field.", "According to Eq.", "(REF ), $D^2&=& \\tilde{D}^2+i\\partial _\\mu \\breve{B}_\\mu ,\\\\\\tilde{D}^2&=& \\partial ^2+2i\\breve{B}_\\mu \\partial _\\mu -i\\breve{B}_\\mu \\breve{B}_\\mu \\nonumber \\\\&=& (\\partial _1-i\\breve{n} H_2(x_1)x_3-i\\breve{n}H_3(x_1)x_2)^2\\nonumber \\\\&& + \\partial _2^2 +\\partial _3^2 + (\\partial _4+i\\breve{n} Bx_3)^2 -i\\partial _1B_1\\nonumber \\\\\\partial _\\mu \\breve{B}_\\mu &=&-\\breve{n} H^\\prime _2(x_1)x_3-\\breve{n} H^\\prime _3(x_1)x_2.$ The action can be written as $S[\\Phi ]&=&\\int d^4x \\Phi ^\\dagger (x)\\tilde{D}^2\\Phi (x)\\\\&-&i\\int d^4x \\Phi ^\\dagger (x)\\breve{n}\\Phi (x) \\left[H^\\prime _2(x_1)x_3+H^\\prime _3(x_1)x_2\\right] .\\nonumber $ It should be noted that the integral in the second line is equal to zero if $\\Phi ^\\dagger (x)\\breve{n}\\Phi (x)$ is an even function of $x_2$ and $x_3$ .", "The structure of $D^2$ in Eq.", "(REF ) is quite complicated.", "In the eigenvalue problem the variables can hardly be separated in the case of the finite width of the kink.", "The problem becomes much simpler and tractable in the limit of the infinitely thin domain wall $\\mu \\rightarrow \\infty $ .", "This limit brings discontinuity into the background field and thus creates a sharp boundary – the hyperplane of the domain wall.", "In such a situation one has to solve the problem in the bulk and on the wall and match the solutions according to some appropriate conditions.", "For our choice of the kink location there are three regions to be studied: $x_1<0$ with the self-dual field $B_\\mu $ , $x_1>0$ with the anti-self-dual field, and $x_1=0$ with the chromomagnetic and chromoelectric fields orthogonal to each other.", "Conditions imposed onto the eigenmodes of color charged fields on the sharp wall can be obtained from the requirement of preservation of the properties of eigenmodes for finite $\\mu $ as far as they can be identified.", "The continuity of the normal to the wall component of the total (through the whole hypersurface of the wall) charged current offers a reliable guiding principle for identification of the matching conditions.", "Continuity of the total current means that the surface terms do not appear under integration by parts in the action, $&&\\lim _{\\varepsilon \\rightarrow 0}\\left[J_1(\\varepsilon )-J_1(-\\varepsilon )\\right]=0,\\\\\\nonumber &&J_\\mu (x_1)=\\int d^3x \\Phi ^\\dagger (x)D_\\mu \\Phi (x),\\\\\\nonumber &&d^3x=dx_2dx_3dx_4.$ Moreover, this requirement restricts the form of the eigenfunctions in such a way that the surface terms associated with the gauge dependent delta-function singularuties in $\\partial _\\mu \\breve{B}_\\mu $ , Eq.", "() vanish as well.", "Figure: Derivatives of the components of the chromomagnetic field are plotted for two values of the width parameter μ/B=3,10\\mu /\\sqrt{B}=3,10.The coordinate x 1 x_1 is given in units of 1/B1/\\sqrt{B}.", "In the limit of the infinitely thin domain wall (μ/B→∞\\mu /\\sqrt{B}\\rightarrow \\infty ) the derivatives develop the delta-function singularities at the location of the wall." ], [ "Confined fluctuations in the bulk", "Let us consider the eigenvalue problem $&&-\\tilde{D}^2\\Phi =\\lambda \\Phi .$ for the functions square integrable in $R^4$ and satisfying the condition (REF ).", "For all $x_1\\ne 0$ the operator $\\tilde{D}^2$ takes the form $\\tilde{D}^2&=& (\\partial _1 \\pm i\\breve{n} B x_2)^2\\nonumber \\\\&& + \\partial _2^2 +\\partial _3^2 + (\\partial _4+i\\breve{n} Bx_3)^2\\nonumber $ where plus corresponds to the anti-self-dual configuration and minus is for the self-dual one.", "By inspection one can see that the eigenfunctions satisfy the relation $\\Phi ^{(+)}(x_1,x_\\perp )=\\Phi ^{(-)}(-x_1,x_\\perp ),$ where $(\\pm )$ denotes the duality of the background field for a given $x_1$ .", "Respectively, the square integrable solutions are $\\Phi ^{(\\pm )}_{kl}(x)&=&\\phi ^{(\\pm )}_k(x_1,x_2)\\chi _l(x_3,x_4)\\\\\\phi ^{(\\pm )}_k(x_1,x_2)&=&\\int dp_1 f(p_1)e^{\\pm ip_1x_1-\\frac{1}{2}|\\breve{n}|B(x_2+p_1/|\\breve{n}|B)^2}\\nonumber \\\\&\\times & H_k\\left(\\sqrt{|\\breve{n}|B}\\left[x_2+\\frac{p_1}{|\\breve{n}|B}\\right]\\right)\\nonumber \\\\\\chi _k(x_3,x_4)&=&\\int dp_4 g(p_4)e^{ip_4x_4-\\frac{1}{2}|\\breve{n}|B(x_3+p_4/|\\breve{n}|B)^2 }\\nonumber \\\\&\\times & H_l\\left(\\sqrt{|\\breve{n}|B}\\left[x_3+\\frac{p_4}{|\\breve{n}|B}\\right]\\right),\\nonumber $ where $H_m$ are the Hermite polynomials.", "The eigenvalues are $\\lambda _{kl}=2|\\breve{n}|B(k+l+1), \\, \\, k,l=0,1,\\dots .$ The amplitudes $f(p_1)$ and $g(p_4)$ have to provide square integrability of the eigenfunctions in $x_1$ and $x_4$ .", "In order to satisfy condition (REF ) one has to restrict the amplitude $f(p_1)$ additionally.", "The integral current through the domain wall is continuous if both $f$ and $H_k$ are odd or even functions simultaneously under the combined change $p_1\\rightarrow -p_1$ and $x_2\\rightarrow -x_2$ $f(-p_1)H_k(-z)=f(p_1)H_k(z).$ This property also guarantees the absence of the gauge specific contribution to the action related to the derivative of $H_3$ in Eqs.", "(,REF ).", "A combination of (REF ) and (REF ) obviously leads to the relation $\\phi _{k}^{(\\pm )}(x_1,x_2)=\\phi _{k}^{(\\pm )}(-x_1,-x_2),$ where $(\\pm )$ denotes the duality of the background field for a given $x_1$ .", "This identity allows one to show that the eigenfunctions $\\Phi _{kl}(x) = \\left\\lbrace \\begin{array}{l}\\Phi ^{(+)}_{kl}(x), \\, \\, x_1\\in L_+\\\\\\Phi ^{(-)}_{kl}(x), \\, \\, x_1\\in L_-\\end{array}\\right., \\, k,l=0,1\\dots $ form a complete orthogonal set in the space of square integrable functions which are even with respect to simultaneous reflection $x_1\\rightarrow -x_1$ and $x_2\\rightarrow -x_2$ .", "The eigenfunctions are of the bound state type with the purely discrete spectrum.", "Field fluctuations of this type can be seen as confined.", "It should be noted that the eigenvalues coincide with those for the purely homogeneous (anti-)self-dual Abelian field.", "In this sense, the domain wall defect does not destroy dynamical confinement of color charged fields.", "The eigenfunctions are restricted by the correlated evenness condition (REF ), while in the case of the homogeneous field the properties of the amplitude $f(p_1)$ and the polynomial $H_k$ are mutually independent." ], [ "Color charged quasiparticles on the wall", "Let us now consider the eigenvalue problem on the domain wall, i.e.", "for the region $x_1=0$ .", "On the wall the chromomagnetic and chromoelectric fields are orthogonal to each other (see Fig.REF ).", "In conformity with (REF ) the absence of the charged current off the infinitely thin domain wall requires $\\partial _1\\Phi |_{x_1=0}=0,$ and the eigenvalue problem on the wall takes the form $\\left[- \\partial _2^2 -\\partial _3^2 +\\breve{n}^2 B^2 x_3^2 + (i\\partial _4-\\breve{n} Bx_3)^2\\right]\\Phi =\\lambda \\Phi \\nonumber $ with the solution $\\Phi _k(x_2,x_3,x_4)&=&e^{ip_2x_2+ip_4x_4}e^{-\\frac{|\\breve{n}|B}{\\sqrt{2}}\\left(x_3-\\frac{p_4}{2|\\breve{n}|B}\\right)^2}\\nonumber \\\\&\\times &H_k\\left[\\sqrt{\\sqrt{2}|\\breve{n}|B}\\left(x_3-\\frac{p_4}{2|\\breve{n}|B}\\right)\\right],\\nonumber \\\\\\lambda _k(p^2_2,p_4^2)&=&\\sqrt{2}|\\breve{n}|B(2k+1)+\\frac{p_4^2}{2}+\\frac{p_2^2}{2},\\nonumber \\\\&&k=0,1,2,\\dots \\nonumber $ The spectrum of the eigenmodes on the wall is continuous, it depends on the momentum $p_2$ longitudinal to the chromomagnetic field and Euclidean energy $p_4$ , the corresponding eigenfunctions are oscillating in $x_2$ and $x_4$ .", "In the direction $x_3$ transverse to the chromomagnetic field the eigenfunctions are bounded and the eigenvalues display the Landau level structure.", "The continuation $p_4^2=-p_0^2$ leads to the dispersion relation $p_0^2=p_2^2+\\mu ^2_k,\\quad \\mu ^2_k=2\\sqrt{2}(2k+1)|\\breve{n}|B,\\, \\, k=0,1,2,\\dots .$ This can be treated as the lack of confinement - the color charged quasiparticles with masses $\\mu _k$ and momentum $\\mathbf {p}$ parallel to the chromomagnetic field $\\mathbf {H}$ can be excited on the wall.", "The case of the planar domain wall configuration (two infinite parts of the space-time separated by a three-dimentional hypersurface like in Fig.REF ) is rather artificial.", "Its weight in the whole ensemble of the gluon field configurations with the constant scalar condensate $\\langle g^2F_{\\mu \\nu }F_{\\mu \\nu }\\rangle $ and the lumpy structured distribution of the topological charge density $\\langle g^2\\tilde{F}_{\\mu \\nu }F_{\\mu \\nu }\\rangle $ is negligible.", "The entropy-energy balance implies that the typical configuration should be highly disordered (see Fig.REF ).", "Moreover, in the case of the planar domain wall the eigenvalue problem for the square integrable vector gauge fields $\\left[-D^2\\delta _{\\mu \\nu }+2i\\breve{F}_{\\mu \\nu }\\right]Q_\\nu =\\lambda Q_\\mu $ leads to the negative eigenvalues and corresponding tachyonic modes on the wall where $\\tilde{F}_{\\mu \\nu }F_{\\mu \\nu }=0$ .", "This is a well-known instability of the Nielsen-Olesen type [35].", "The presence of the tachyonic mode is due to the three infinite dimensions of the planar domain wall hypersurface.", "One can expect that finite size of boundaries between lumps in the typical kink network configuration, Fig.REF , removes the tachyonic modes.", "This is manifestly exemplified in the next section where the color charged field eigenvalues and modes are studied for thick cylindrical domain wall junction.", "The relatively stable defect of this type can occur in the ensemble of confining gluon fields due to the influence of the strong electromagnetic fields on the QCD vacuum structure." ], [ "Heavy ion collisions: the strong electromagnetic field as a trigger for deconfinement", "It has been observed that the strong electromagnetic fields generated in relativistic heavy ion collisions can play the role of a trigger for deconfinement [14].", "The mechanism discussed in [14] is as follows.", "The electric $\\mathbf {E}_{\\rm el}$ and magnetic $\\mathbf {H}_{\\rm el}$ fields are practically orthogonal to each other [25], [26]: $\\mathbf {E}_{\\rm el}\\mathbf {H}_{\\rm el}\\approx 0 $ .", "For this configuration of the external electromagnetic field the one-loop quark contribution to the QCD effective potential for the homogeneous Abelian gluon fields is minimal for the chromoelectric and chromomagnetic fields directed along the electric and magnetic fields respectively.", "The orthogonal chromo-fields are not confining: color charged quasiparticles can move along the chromomagnetic field.", "It has been noted also that this mechanism assumes the strong azimuthal anisotropy in momentum distribution of color charged quasiparticles.", "Deconfined quarks as well as gluons will move preferably along the direction of the magnetic field but this will happen due to the gluon field configuration even after switching the electromagnetic field off.", "A detailed and systematic analytical one-loop calculation of the QCD effective potential for the pure chromomagnetic field was performed recently in [36] and confirmed the result that the chromomagnetic field prefers to be parallel (or anti-parallel) to the external magnetic field.", "Another important source of verification of the basic observations of paper [14] is due to the recent Lattice QCD studies of the response of the QCD vacuum to external electromagnetic fields [22], [23], [37], [38].", "In particular, in qualitative agreement with [14] Lattice QCD study [23] has demonstrated that in the presence of external magnetic field the gluonic action develops an anisotropy: the chromomagnetic field parallel to the external field is enhanced, while the chromo-electric field in this direction is suppressed.", "The results of [37] indicated that the magnetic field can affect the azimuthal structure of the expansion of the system during heavy ion collisions.", "Figure: Examples of two-dimensional slice of the cylindrical thick domain wall junctions.", "The correspondence of colors is the same as in Fig.. Blue and red regions represent self-dual and anti-self-dual lumps.", "Confinement is lost in the green region whereg 2 F ˜ μν (x)F μν (x)=0g^2\\tilde{F}_{\\mu \\nu }(x)F_{\\mu \\nu }(x)=0.", "The scalar condensate density g 2 F μν (x)F μν (x)g^2F_{\\mu \\nu }(x)F_{\\mu \\nu }(x) is nonzero and homogeneous everywhere.Within the context of the confining domain wall network these observations mean that a flash of the strong electromagnetic field during heavy ion collisions produces a kind of defect in the form of the thick domain wall junction in the confining gluon background exactly in the region where collision occurs (see Fig.REF ).", "The electromagnetic flash can act as one of the preconditions for conversion of the high energy density and baryon density to the thermodynamics of color charged degrees of freedom.", "Since topological charge density is zero in the interior of the trap ($g^2\\tilde{F}_{\\mu \\nu }(x)F_{\\mu \\nu }(x)=0$ ) there exists a specific reference frame where one can use the pure chromomagnetic field for description of the gluon background inside the trap.", "For simplicity we take cylindrical geometry of the trap and study the properties of scalar and vector (gluon) color charged field eigenmodes.", "Extension of the present consideration to more realistic form of the trap is straightforward.", "Consider the eigenvalue problem for the massless scalar field $\\Phi ^a$ $-\\left(\\partial _\\mu -i\\breve{B}_\\mu \\right)^2\\Phi (x)=\\lambda ^2\\Phi (x)$ in the cylindrical region $x\\in \\mathcal {T}=\\left\\lbrace x_1^2+x_2^2<R^2, \\ (x_3,x_4)\\in \\mathrm {R^2} \\right\\rbrace $ with the homogeneous Dirichlet condition at the boundary $&&\\Phi (x)=0, \\ x\\in \\partial \\mathcal {T}\\\\&&\\partial \\mathcal {T} =\\left\\lbrace x_1^2+x_2^2=R^2, \\ (x_3,x_4)\\in \\mathrm {R^2} \\right\\rbrace .\\nonumber $ Here $\\breve{B}_\\mu $ stays for adjoiunt representaion of the homogeneous chromomagnetic field $H^a_i=\\delta _{i3}n^a H$ with the vector potential taken in the symmetric gauge $\\breve{B}_\\mu =-\\frac{1}{2} \\breve{n} B_{\\mu \\nu }x_\\nu ,\\\\\\breve{B}_4=\\breve{B}_3=0, \\ B_{12}=-B_{21}=H,\\nonumber \\\\\\breve{n}=T_3\\cos (\\xi )+T_8\\sin (\\xi ).\\nonumber $ The eigenvalues of the matrix $\\breve{n}$ are $\\breve{v}=\\mathrm {diag}\\left[\\cos \\left(\\xi \\right),-\\cos \\left(\\xi \\right),0,\\cos \\left(\\xi -\\frac{\\pi }{3}\\right),-\\cos \\left(\\xi -\\frac{\\pi }{3}\\right),\\cos \\left(\\xi +\\frac{\\pi }{3}\\right),-\\cos \\left(\\xi +\\frac{\\pi }{3}\\right),0\\right].$ For any value of the angle $\\xi $ there are two zero eigenvalues $\\breve{v}_3=\\breve{v}_8=0$ .", "Two additional zero elements occur in $\\breve{v}$ if the angle takes values $\\xi _k$ (see Eq.", "(REF )) minimizing the effective potential (REF ) and corresponding to the boundaries of the Weyl chambers.", "By inspection one can check that nonzero eigenvalues $\\breve{v}$ take values $\\pm v$ with $v=\\sqrt{3}/2$ .", "Below we use notation $\\nonumber \\breve{v}^a=v\\kappa ^a.$ For example, if $\\xi =\\xi _0=\\pi /6$ then the nonzero values of $v^a$ correspond to $a=1,2,4,5$ and $\\nonumber \\kappa _1=1, \\kappa _2=-1, \\kappa _4=1, \\ \\kappa _5=-1.$ It has to be noted that the effective Lagrangian (REF ) leads to the kink configuration (for details see [15]) $\\xi _{k}(x_i) =\\frac{1}{3} \\arctan \\left[\\sinh (m_\\xi x_i)\\right]+\\frac{\\pi k}{3}, \\ k=0,\\dots ,5,$ interpolating between boundaries $\\xi _k$ and $\\xi _{k+1}$ of the $k$ -th Weyl chamber.", "Superposition of these \"color\" domain walls can be arranged in a complete analogy with the \"duality\" domain walls.", "The only new feature of the \"color\" domains is that there are six different types interrelated by the Weyl reflections instead of two types as in the case of duality domains.", "Solution of the problem (REF ) is straightforward.", "We give it below just for completeness.", "It is convenient to introduce dimensionless variables using the strength of the chromomagnetic field as a basic scale.", "Below all quantities are assumed to be measured in terms of this scale, for instance $\\sqrt{H}x_\\mu \\equiv x_\\mu ,\\quad \\frac{\\lambda }{\\sqrt{H}}\\equiv \\lambda .$ After diagonalization with respect to color indices and transformation to the cylindrical coordinates Eq.", "(REF ) takes the form $-\\left[\\partial _4^2+\\partial _3^2+\\frac{\\partial ^2}{\\partial r^2}+\\frac{1}{r}\\frac{\\partial }{\\partial r}+\\frac{1}{r^2}\\frac{\\partial ^2}{\\partial \\vartheta ^2}-i\\kappa ^a v \\frac{\\partial }{\\partial \\vartheta }\\right.\\nonumber \\\\\\left.-\\frac{1}{4} v^2r^2\\right]\\Phi ^a=\\lambda ^2\\Phi ^a,\\ \\ \\ $ where it has been used that $&&x_1=r\\cos \\vartheta ,\\ x_2=r\\sin \\vartheta ,\\\\\\\\&&\\frac{\\partial }{\\partial x_1}=\\cos \\vartheta \\frac{\\partial }{\\partial r}-\\frac{\\sin \\vartheta }{r}\\frac{\\partial }{\\partial \\vartheta },\\\\&&\\frac{\\partial }{\\partial x_2}=\\sin \\vartheta \\frac{\\partial }{\\partial r}+\\frac{\\cos \\vartheta }{r}\\frac{\\partial }{\\partial \\vartheta },\\\\&&\\partial _1^2+\\partial _2^2=\\frac{\\partial ^2}{\\partial r^2}+\\frac{1}{r}\\frac{\\partial }{\\partial r}+\\frac{1}{r^2}\\frac{\\partial ^2}{\\partial \\vartheta ^2}.$ The variables in Eq.", "(REF ) are separated by substitution $\\Phi ^a=\\phi ^a(r)e^{il\\vartheta }\\exp \\left(ip_3x_3+ip_4x_4\\right).$ Periodicity of the solution in angle $\\vartheta \\in [0,2\\pi ]$ requires integer values of parameter $l$ .", "The radial part $\\phi (r)$ should satisfy equation $-\\left[\\frac{\\partial ^2}{\\partial r^2}+\\frac{1}{r}\\frac{\\partial }{\\partial r}-\\frac{1}{r^2}\\left(\\frac{1}{2}\\breve{v} r^2-l\\right)^2\\right]\\phi =\\mu ^2\\phi ,$ where $\\mu $ is related to the original eigenvalue $\\lambda $ , $\\lambda ^2=p_4^2+p_3^2+\\mu ^2.$ By means of the substitution $\\phi =r^le^{-\\frac{1}{4}\\breve{v}r^2}\\chi ,$ one arrives at the Kummer equation ($z=\\breve{v}r^2/2$ ) $\\left[z\\frac{d^2}{dz^2}+(l+1-z)\\frac{d}{dz}-\\frac{\\breve{v}-\\mu ^2}{2\\breve{v}}\\right]\\chi =0.$ The complete solution can be chosen in the form $\\chi (z)=C_1M\\left(\\frac{\\breve{v}-\\mu ^2}{2\\breve{v}},1+l,z\\right)+C_2z^{-l}M\\left(\\frac{\\breve{v}-\\mu ^2}{2\\breve{v}}-l,1-l,z\\right),$ where $M(a,b,z)$ is Kummer function.", "General solution of equation (REF ) takes the form $\\phi _{l}(r)=e^{-\\frac{1}{4}\\breve{v}r^2}\\left[C_1r^lM\\left(\\frac{\\breve{v}-\\mu ^2}{2\\breve{v}},1+l,\\frac{1}{2}\\breve{v}r^2\\right)+C_2r^{-l}M\\left(\\frac{\\breve{v}-\\mu ^2}{2\\breve{v}}-l,1-l,\\frac{1}{2}\\breve{v}r^2\\right)\\right]$ The first term is regular at $r=0$ provided $l\\geqslant 0$ while the second one is well-defined for $l\\leqslant 0$ .", "Therefore, the solution regular inside the cylinder is $\\phi _{al}&=&e^{-\\frac{1}{4}\\breve{v}_ar^2}r^lM\\left(\\frac{\\breve{v}_a-\\mu ^2}{2\\breve{v}_a},1+l,\\frac{1}{2}\\breve{v}_ar^2\\right), \\ \\ l\\geqslant 0,\\\\\\phi _{al}&=&e^{-\\frac{1}{4}\\breve{v}_ar^2}r^{-l}M\\left(\\frac{\\breve{v}_a-\\mu ^2}{2\\breve{v}_a}-l,1-l,\\frac{1}{2}\\breve{v}_ar^2\\right), \\ \\ l< 0,$ where the color index $a$ has been explicitly indicated.", "The color matrix elements $\\breve{v}_a$ can be negative.", "In this case one has to apply Kummer transformation [39] $M(a,b,z)=e^zM(b-a,b,-z).$ Dirichlet boundary condition (REF ) defines the infinite discrete set of eigenvalues as the solutions $\\mu ^2_{alk}$ ($k=0,1\\dots \\infty $ ) of the equations $M\\left(\\frac{\\hat{v}_a-\\mu ^2}{2\\hat{v}_a},1+l,\\frac{1}{2}\\hat{v}_aR^2\\right)=0,\\quad l\\geqslant 0,\\\\M\\left(\\frac{\\hat{v}_a-\\mu ^2}{2\\hat{v}_a}-l,1-l,\\frac{1}{2}\\hat{v}_aR^2\\right)=0,\\quad l< 0.$ If $\\mu ^2_{alk}$ satisfies equation (REF ), than $\\tilde{\\mu }^2_{alk}=\\mu ^2_{alk}-2\\hat{v}_al$ is a solution of ().", "Finally the complete orthogonal set of eigenfunctions for the problem (REF ) and (REF ) reads $&&\\Phi _{alk}(p_3, p_4|r,\\vartheta ,x_3,x_4)=e^{ip_3x_3+ip_4x_4}e^{il\\vartheta }\\phi _{alk}(r),\\nonumber \\\\&&\\lambda _{alk}^2=p_4^2+p_3^2+\\mu _{akl}^2,\\\\&& k=0,1,\\dots ,\\infty , \\ \\ l=-\\infty \\dots \\infty ,\\nonumber $ where functions $\\phi _{alk}$ are defined by (REF ) with $\\mu ^2=\\mu _{akl}^2$ solving the boundary condition (REF ).", "Unlike Landau levels in the infinite space the eigenvalues $\\mu _{akl}^2$ are not equidistant in $k$ and non-degenerate in $l$ as it is illustrated in Fig.REF ).", "The dependence of several low-lying eigenvalues $\\mu _{akl}^2$ on the dimensionless size parameter $\\sqrt{H}R$ is shown in Fig.REF ." ], [ "Vector field eigenmodes", "For pure chromomagnetic field (REF ) the adjoint representation vector field Eq.", "(REF ) takes the form $\\left[-\\breve{D}^2\\delta _{\\mu \\nu }+2i\\breve{n} B_{\\mu \\nu }\\right]Q_\\nu =\\lambda Q_\\mu ,$ and the boundary conditions are $&&\\breve{n} Q_{\\mu }(x)=0, \\ x\\in \\partial \\mathcal {T}\\nonumber \\\\&&\\partial \\mathcal {T} =\\left\\lbrace x_1^2+x_2^2=R^2, \\ (x_3,x_4)\\in \\mathrm {R^2} \\right\\rbrace $ In terms of the eigenvectors $\\breve{Q}_\\mu ^a$ of matrices $B_{\\mu \\nu }$ and $\\breve{n}$ Eqs.", "(REF ) and (REF ) take the form $\\left[-\\breve{D}^2+2s_{\\mu }\\breve{v} H\\right]^a\\breve{Q}^{a}_\\mu =\\lambda _{a\\mu }\\breve{Q}^{a}_\\mu ,\\\\\\nonumber \\breve{v} \\breve{Q}_{\\mu }(x)=0, \\ x\\in \\partial \\mathcal {T}.$ Omitting obvious well-known details we just note that equation (REF ) describes sixteen charged with respect to $\\breve{n}$ spin-color polarizations of the gluon fluctuations with $(s_1=1, s_2=-1, s_3=s_4=0)$ and $\\breve{v}^a\\ne 0$ as well as sixteen \"color neutral\" with respect to $\\breve{n}$ modes $&&-\\partial ^2\\breve{Q}_\\mu ^{(0)}=p^2 \\breve{Q}_\\mu ^{(0)}.$ Neutral mode $\\breve{Q}_\\mu ^{(0)}$ is a zero mode of $\\breve{n}$ , and it is insensitive to the boundary condition (REF ).", "We shall briefly discuss the possible role of the neutral modes in the last section.", "Equations for the color charged modes have the same form as the scalar field equation in the previous subsection.", "The only essential difference is that the eigenvalues $\\lambda _{alk\\nu }$ for nonzero $v^a$ have an addition $\\pm 2vH$ to the eigenvalues $\\mu ^2_{akl}$ of the scalar case: $&&\\lambda ^2_{alk\\nu }=p_4^2+p_3^2+\\mu _{alk}^2+ 2s_\\nu \\kappa _a v,\\\\&& k=0,1,\\dots ,\\infty , \\ \\ l\\in Z,\\nonumber \\\\&& s_1=1, \\ s_2=-1, \\ s_3=s_4=0, \\ \\kappa _a=\\pm 1.\\nonumber $ where $\\mu ^2_{akl}$ are the same as in the scalar case.", "If we were considering the square integrable solutions in $R^4$ then the lowest mode $\\lambda ^2_{a00\\nu }$ with $s_\\nu \\kappa _a=-1$ would be tachyonic.", "In the finite trap the lowest eigenvalue is $\\lambda ^2_{a00\\nu }=p_4^2+p_3^2+\\mu _{a00}^2-2v, \\ s_\\nu \\kappa _a=-1.$ The dependence of $\\mu _{a00}^2$ on dimensionless size parameter $\\sqrt{H}R$ is strongly nonlinear.", "Few lowest eigenvalues $\\mu _{akl}^2$ as functions of $\\sqrt{H}R$ are shown in Fig.", "REF .", "One concludes that if the dimensionless size $\\sqrt{H}R$ of the trap is sufficiently small $\\sqrt{H}R<\\sqrt{H}R_{\\rm c}\\approx 1.91,$ then there are no unstable tachyonic modes in the spectrum of color charged vector fields.", "To estimate the critical size one may use the mean phenomenological value of the gluon condensate (gauge coupling constant $g$ is included into the field strength tensor) $\\langle F^a_{\\mu \\nu }F^{a\\mu \\nu }\\rangle = 2H^2\\approx 0.5{\\rm GeV}^4.$ Equation (REF ) leads to the critical radius $R_{\\rm c}\\approx 0.51 \\ {\\rm fm} \\ (2R_{\\rm c}\\approx 1 \\ {\\rm fm}).$ Thus the tachyonic mode is absent if the diameter of the cylindrical trap is less or equal to $1 \\ {\\rm fm}$ .", "Figure: Eigenvalues μ alk 2 \\mu ^2_{alk} for the scalar field problem, l=-2,-1,0,1,2l=-2,-1,0,1,2 and k=0,1,2k=0,1,2, for HR=1.6\\sqrt{H}R=1.6.Eigenvalues are denoted by asterisks in the case of positive v a v_a andby circles in the case of negative v a v_a.Figure: The lowest eigenvalues corresponding to positive color orientation κ a =1\\kappa ^a=1 as functions of HR\\sqrt{H}R. The critical radius R c R_{\\rm c} corresponds to μ a00 2 =2v=3\\mu _{a00}^2=2v=\\sqrt{3}.", "For large HR\\sqrt{H}R eigenvalues approach correct Landau levels, the degeneracy in ll is restored." ], [ "Quark field eigen modes", "In this subsection we address the eigenvalue problem for Dirac operator in the cylindrical region in the presence of chromomagnetic background field (REF ) $&&\\!\\lnot \\!\\!D\\psi (x)=\\lambda \\psi (x),\\\\&&D_\\mu =\\partial _\\mu +\\frac{i}{2}\\hat{n} B_{\\mu \\nu }x_\\nu ,\\nonumber \\\\&&\\hat{n}= t_3\\cos \\xi +t_8\\sin \\xi \\\\&&=\\frac{1}{2}{\\rm diag}\\left(\\cos \\xi +\\frac{\\sin \\xi }{\\sqrt{3}}, -\\cos \\xi +\\frac{\\sin \\xi }{\\sqrt{3}}, -\\frac{2\\sin \\xi }{\\sqrt{3}}\\right).\\nonumber $ Euclidean Dirac matrices are taken in the anti-hermitian representation.", "The angle $\\xi $ is assumed to take one of the vacuum values $\\xi _k$ , and according to Eq.", "(REF ) the following forms of the matrix $\\hat{n}$ can occur $&&\\hat{n}=\\left\\lbrace \\pm \\frac{1}{\\sqrt{3}}{\\rm diag}\\left(1, -\\frac{1}{2}, -\\frac{1}{2}\\right),\\right.\\\\&&\\left.\\pm \\frac{1}{\\sqrt{3}}{\\rm diag}\\left(\\frac{1}{2}, \\frac{1}{2},-1\\right), \\pm \\frac{1}{\\sqrt{3}}{\\rm diag}\\left(-\\frac{1}{2},1, -\\frac{1}{2}\\right) \\right\\rbrace .\\nonumber $ Below we use notation $\\hat{n}_{ij}=\\delta _{ij}\\hat{u}_j.\\nonumber \\\\$ The boundary conditions are $i\\!\\lnot \\!\\eta (x)e^{i\\theta \\gamma _5}\\hat{n}\\psi (x) = \\hat{n}\\psi (x), \\ x\\in \\partial \\mathcal {T},\\nonumber \\\\\\bar{\\psi }(x)e^{i\\theta \\gamma _5} \\hat{n} i \\!", "\\!\\lnot \\!\\eta (x) = -\\bar{\\psi }(x) \\hat{n},\\ x\\in \\partial \\mathcal {T},$ where $\\eta _\\mu $ is a unit vector normal to the cylinder surface $\\partial \\mathcal {T}$ , see Eq.", "(REF ).", "These are simply the bag boundary conditions.", "This choice appears to be rather natural.", "Indeed, inside the thick domain wall junction one expects an existence of the color charged quasiparticles (quarks) being the carriers of the color current, but outside the junction gluon configurations are confining (see Fig.", "REF )) and the current has to vanish at the boundary.", "Unlike the adjoint representation of color matrix () the matrix $\\hat{n}$ in fundamental representation () has no zero eigenvalues for any value of the angle $\\xi $ corresponding to the boundaries of Weyl chambers, see Eq.", "(REF ).", "Boundary condition (REF ) restricts all three color components of the quark field.", "Substitution $\\psi =\\left(\\lnot {\\hspace*{-3.00003pt}D}+\\lambda \\right)\\varphi $ leads to the equation $-\\left(D^2+\\hat{u}H\\Sigma _3\\right)\\varphi =\\lambda ^2\\varphi ,$ where it has been used that in the pure chromomagnetic field (REF ) $\\frac{1}{2}\\sigma _{\\mu \\nu }\\hat{B}_{\\mu \\nu }=\\Sigma _3H\\hat{u},\\ \\Sigma _i=\\frac{1}{2}\\varepsilon _{ijk}\\sigma _{jk}.$ Equation (REF ) is essentially the same as (REF ).", "Its solution in cylindrical coordinates ($2\\pi $ -periodic in $\\vartheta $ and regular at $r=0$ ) is given by four independent components $\\varphi _l^\\alpha $ ($\\alpha =1,\\dots ,4$ , $l\\in Z$ ): $\\varphi _l^\\alpha =e^{-ip_3x_3-ip_4x_4}e^{il\\vartheta }\\phi _l^\\alpha (r)$ with $\\phi _{l}^\\alpha =e^{-\\frac{1}{4}\\hat{u}r^2}r^{l}M\\left(\\frac{1+s_\\alpha }{2}-\\frac{\\mu ^2}{2\\hat{u}},1+l,\\frac{\\hat{u}r^2}{2}\\right)$ for the case $l\\geqslant 0$ and $\\phi _{l}^{\\alpha }=e^{-\\frac{1}{4}\\hat{u}r^2}r^{-l}M\\left(\\frac{1+s_{\\alpha }}{2}-\\frac{\\mu ^2}{2\\hat{u}}-l,1-l,\\frac{\\hat{u}r^2}{2}\\right)$ for $l<0$ .", "Here $s_\\alpha =(-1)^\\alpha , \\ \\alpha =1,\\dots ,4$ denotes the sign of the quark spin projection on the direction of chromomagnetic field, and therefore $\\phi _l^3=\\phi _l^1=\\Phi _l^{\\uparrow \\uparrow }(r), \\ \\phi _l^4=\\phi _l^2=\\Phi _l^{\\uparrow \\downarrow }(r).$ The variable $\\mu $ is related to the Dirac eigenvalues as $\\mu ^2=\\lambda ^2-p_3^2-p_4^2.$ Finally the Dirac operator eigenfunction $\\psi $ can be obtained by means of relation (REF ) with $\\lnot {\\hspace*{-3.00003pt}D}+\\lambda =\\left(\\begin{array}{cccc}\\lambda &0&i\\partial _4+\\partial _3&D_1-iD_2\\\\0&\\lambda &D_1+iD_2&i\\partial _4-\\partial _3\\\\i\\partial _4-\\partial _3&-D_1+iD_2&\\lambda &0\\\\-D_1-iD_2&i\\partial _4+\\partial _3&0&\\lambda \\end{array}\\right),\\nonumber $ where $D_1+iD_2=e^{i\\vartheta }\\left(\\frac{\\partial }{\\partial r}+\\frac{i}{r}\\frac{\\partial }{\\partial \\vartheta }+\\frac{1}{2}\\hat{u}r\\right),[0]\\\\D_1-iD_2=e^{-i\\vartheta }\\left(\\frac{\\partial }{\\partial r}-\\frac{i}{r}\\frac{\\partial }{\\partial \\vartheta }-\\frac{1}{2}\\hat{u}r\\right).$ Four solutions are for $l\\geqslant 0$ $\\psi _l^{(1)}=e^{-ip_3x_3-ip_4x_4}\\left(\\begin{array}{c}\\lambda \\Phi _l^{\\uparrow \\uparrow }(r)e^{il\\vartheta }\\\\0\\\\(p_4+ip_3)\\Phi _l^{\\uparrow \\uparrow }(r)e^{il\\vartheta }\\\\\\frac{\\mu ^2}{2(l+1)}\\Phi _{l+1}^{\\uparrow \\downarrow }(r)e^{i(l+1)\\vartheta }\\end{array}\\right)$ $\\psi _l^{(2)}=e^{-ip_3x_3-ip_4x_4}\\left(\\begin{array}{c}0\\\\\\lambda \\Phi _{l+1}^{\\uparrow \\downarrow }(r)e^{i(l+1)\\vartheta }\\\\-2(l+1)\\Phi _l^{\\uparrow \\uparrow }(r)e^{il\\vartheta }\\\\(p_4-ip_3)\\Phi _{l+1}^{\\uparrow \\downarrow }(r)e^{i(l+1)\\vartheta }\\end{array}\\right)$ $\\psi ^{(3)}_l=e^{-ip_3x_3-ip_4x_4}\\left(\\begin{array}{c}(p_4-ip_3)\\Phi _{l}^{\\uparrow \\uparrow }(r)e^{il\\vartheta }\\\\-\\frac{\\mu ^2}{2(1+l)}\\Phi _{l+1}^{\\uparrow \\downarrow }(r)e^{i(l+1)\\vartheta }\\\\\\lambda \\Phi _l^{\\uparrow \\uparrow }(r)e^{il\\vartheta }\\\\0\\end{array}\\right)$ $\\psi ^{(4)}_l=e^{-ip_3x_3-ip_4x_4}\\left(\\begin{array}{c}2(l+1)\\Phi _l^{\\uparrow \\uparrow }(r)e^{il\\vartheta }\\\\(p_4+ip_3)\\Phi _{l+1}^{\\uparrow \\downarrow }(r)e^{i(l+1)\\vartheta }\\\\0\\\\\\lambda \\Phi _{l+1}^{\\uparrow \\downarrow }(r)e^{i(l+1)\\vartheta }\\\\\\end{array}\\right),$ and for $l<0$ $\\psi _l^{(1)}=e^{-ip_3x_3-ip_4x_4}\\left(\\begin{array}{c}\\lambda \\Phi _l^{\\uparrow \\uparrow }(r)e^{il\\vartheta }\\\\0\\\\(p_4+ip_3)\\Phi _l^{\\uparrow \\uparrow }(r)e^{il\\vartheta }\\\\2l\\Phi _{l+1}^{\\uparrow \\downarrow }(r)e^{i(l+1)\\vartheta }\\end{array}\\right)$ $\\psi _l^{(2)}=e^{-ip_3x_3-ip_4x_4}\\left(\\begin{array}{c}0\\\\\\lambda \\Phi _{l+1}^{\\uparrow \\downarrow }(r)e^{i(l+1)\\vartheta }\\\\-\\frac{\\mu ^2}{2l}\\Phi _l^{\\uparrow \\uparrow }(r)e^{il\\vartheta }\\\\(p_4-ip_3)\\Phi _{l+1}^{\\uparrow \\downarrow }(r)e^{i(l+1)\\vartheta }\\end{array}\\right)$ $\\psi ^{(3)}_l=e^{-ip_3x_3-ip_4x_4}\\left(\\begin{array}{c}(p_4-ip_3)\\Phi _{l}^{\\uparrow \\uparrow }(r)e^{il\\vartheta }\\\\-2l\\Phi _{l+1}^{\\uparrow \\downarrow }(r)e^{i(l+1)\\vartheta }\\\\\\lambda \\Phi _l^{\\uparrow \\uparrow }(r)e^{il\\vartheta }\\\\0\\end{array}\\right)$ $\\psi ^{(4)}_l=e^{-ip_3x_3-ip_4x_4}\\left(\\begin{array}{c}\\frac{\\mu ^2}{2l}\\Phi _l^{\\uparrow \\uparrow }(r)e^{il\\vartheta }\\\\(p_4+ip_3)\\Phi _{l+1}^{\\uparrow \\downarrow }(r)e^{i(l+1)\\vartheta }\\\\0\\\\\\lambda \\Phi _{l+1}^{\\uparrow \\downarrow }(r)e^{i(l+1)\\vartheta }\\\\\\end{array}\\right).$ All four spinors are eigenfunctions, $J_3\\psi _l^{(m)}=\\left(l+\\frac{1}{2}\\right)\\psi _l^{(m)},$ of the total momentum projection operator onto $x_3$ .", "$J_i=L_i+S_i,\\quad L_i=-i\\varepsilon _{ijk}x_j\\partial _k,\\quad S_i=\\frac{1}{2}\\Sigma _i,\\\\J_3=-i\\frac{\\partial }{\\partial \\vartheta }+\\frac{1}{2}\\left(\\begin{array}{cc}\\sigma _3&0\\\\0&\\sigma _3\\end{array}\\right).$ Only two of these solutions at given $l$ are linearly independent.", "We select $\\psi _l=A\\psi ^{(1)}_l+B\\psi ^{(4)}_l$ as a general solution to equation (REF ) for the reason that $\\psi ^{(1)}_l$ and $\\psi ^{(4)}_l$ remain linearly independent in the limit $\\lambda \\rightarrow 0$ .", "The limit will be used in the next section for the solving the Dirac equation in Minkowski space-time.", "Boundary condition (REF ) with $\\theta =\\pi /2$ leads to the equation defining the values of the parameter $\\mu $ as well as the ratio of $A$ and $B$ .", "For $l\\geqslant 0$ one gets $&& A\\left(\\frac{\\mu ^2}{2(1+l)}\\Phi ^{\\uparrow \\downarrow }_{l+1}(R)+\\lambda \\Phi ^{\\uparrow \\uparrow }_{l}(R)\\right)+B\\left(\\lambda \\Phi ^{\\uparrow \\downarrow }_{l+1}(R)+2(l+1)\\Phi ^{\\uparrow \\uparrow }_{l}(R)\\right)=0,\\nonumber \\\\&&A\\Phi ^{\\uparrow \\uparrow }_l(R)+B \\Phi ^{\\uparrow \\downarrow }_{l+1}(R)=0.\\nonumber $ This system has a nontrivial solution for $A$ and $B$ if the determinant of the matrix composed of the coefficients in front of them is equal to zero $\\left[\\Phi ^{\\uparrow \\uparrow }_l(R)\\right]^2=\\left[\\frac{\\mu }{2(1+l)}\\Phi ^{\\uparrow \\downarrow }_{l+1}(R)\\right]^2.$ This equation defines the spectrum of $\\mu ^2$ .", "States with definite spin orientation with respect to the chromomagnetic field are mixed in the boundary condition, and the spin projection onto the direction of the field is not a good quantum number unlike the projection of the total momentum $j_3$ as it is taken into account in Fig.REF .", "As is illustrated in Fig.REF there is a discrete set of solutions $\\mu _{ilk}>0$ which depend also on the color orientation $\\hat{u}_i$ ($j_3=(2l+1)/2$ with $l\\in Z$ , $k\\in N$ , $j=1,2,3$ ).", "As a rule one can omit the color index $j$ assuming that $\\mu _{lk}$ is a diagonal color matrix for any $l,k$ .", "The values $\\mu ^2_{lk}$ has to be used to find the relation between $A$ and $B$ $\\frac{B_{lk}}{A_{lk}}=-\\left.\\frac{\\Phi ^{\\uparrow \\uparrow }_l(R)}{\\Phi ^{\\uparrow \\downarrow }_{l+1}(R)}\\right|_{\\mu ^2=\\mu ^2_{lk}}= (-1)^{k+1}\\frac{\\mu _{lk}}{2(l+1)},\\\\\\lambda _{lk}=\\pm \\sqrt{\\mu _{lk}^2+p_3^2+p_4^2}=\\pm |\\lambda _{lk}|.\\nonumber $ Here $\\mu _{lk}$ is taken to be positive, and $\\lambda _{lk}$ takes both positive and negative values.", "Equation (REF ) has been used in combination with observation (by inspection) that the sign of the ratio $B_{lk}$ and $A_{lk}$ depends on $k\\in N$ as it is indicated in (REF ) irrespectively to $l$ and color orientation.", "Analogous consideration for the case $l<0$ leads to the equation for $\\mu $ $\\left[\\Phi ^{\\uparrow \\downarrow }_{l+1}(R)\\right]^2=\\left[\\frac{\\mu }{2l}\\Phi ^{\\uparrow \\uparrow }_{l}(R)\\right]^2,$ and for the ratio of coefficients $\\frac{A_{lk}}{B_{lk}}=-\\left.\\frac{\\Phi ^{\\uparrow \\uparrow }_l(R)}{\\Phi ^{\\uparrow \\downarrow }_{l+1}(R)}\\right|_{\\mu ^2=\\mu ^2_{lk}}= (-1)^{k}\\frac{\\mu _{lk}}{2l},\\\\\\lambda _{lk}=\\pm \\sqrt{\\mu _{lk}^2+p_3^2+p_4^2}=\\pm |\\lambda _{lk}|.\\nonumber $ The orthogonal normalized set of solutions has the form for $l\\geqslant 0$ $\\psi _{lk}^{(\\pm )}=\\frac{A_{lk}}{(2\\pi )^\\frac{3}{2}\\sqrt{2|\\lambda _{lk}|}}\\left(\\begin{array}{c}\\frac{\\pm |\\lambda _{lk}|+(-1)^{k+1}\\mu _{lk}}{\\sqrt{p_4+ip_3}}\\Phi _l^{\\uparrow \\uparrow }(r)e^{il\\vartheta }\\\\(-1)^{k+1}\\frac{\\mu _{lk}\\sqrt{p_4+ip_3}}{2(l+1)}\\Phi _{l+1}^{\\uparrow \\downarrow }(r)e^{i(l+1)\\vartheta }\\\\\\sqrt{p_4+ip_3}\\Phi _l^{\\uparrow \\uparrow }(r)e^{il\\vartheta }\\\\\\frac{\\mu _{lk}(\\mu _{lk}\\pm (-1)^{k+1}|\\lambda _{lk}|)}{2(l+1)\\sqrt{p_4+ip_3}}\\Phi _{l+1}^{\\uparrow \\downarrow }(r)e^{i(l+1)\\vartheta }\\end{array}\\right)e^{-ip_3x_3-ip_4x_4},$ and for $l<0$ $\\psi _{lk}^{(\\pm )}=\\frac{B_{lk}}{(2\\pi )^\\frac{3}{2}\\sqrt{2|\\lambda _{lk}|}}\\left(\\begin{array}{c}\\frac{\\mu _{lk}(\\mu _{lk}\\pm (-1)^k|\\lambda _{lk}|)}{2l\\sqrt{p_4+ip_3}}\\Phi _l^{\\uparrow \\uparrow }(r)e^{il\\vartheta }\\\\\\sqrt{p_4+ip_3}\\Phi _{l+1}^{\\uparrow \\downarrow }(r)e^{i(l+1)\\vartheta }\\\\(-1)^k\\frac{\\mu _{lk}\\sqrt{p_4+ip_3}}{2l}\\Phi _l^{\\uparrow \\uparrow }(r)e^{il\\vartheta }\\\\\\frac{\\pm |\\lambda _{lk}|+(-1)^k\\mu _{lk}}{\\sqrt{p_4+ip_3}}\\Phi _{l+1}^{\\uparrow \\downarrow }(r)e^{i(l+1)\\vartheta }\\end{array}\\right)e^{-ip_3x_3-ip_4x_4}.\\nonumber $ The spinors $\\psi _{lk}^{(+)}$ and $\\psi _{lk}^{(-)}$ correspond to the positive and negative eigenvalues $\\lambda _{lk}$ in (REF ) respectively, they are eigenfunctions of $J_3$ with $j_3=l+1/2$ .", "Normalization constants are $A^{-2}_{jlk}(R)&=&\\int _0^Rdrr\\left[\\left(\\frac{\\mu _{jlk}}{2(l+1)}\\Phi _{l+1}^{\\uparrow \\downarrow }(r)\\right)^2+\\left(\\Phi _{l+1}^{\\uparrow \\uparrow }(r)\\right)^2\\right]\\nonumber \\\\B^{-2}_{jlk}(R)&=&\\int _0^Rdrr\\left[\\left(\\frac{\\mu _{jlk}}{2l}\\Phi _{l+1}^{\\uparrow \\uparrow }(r)\\right)^2+\\left(\\Phi _{l+1}^{\\uparrow \\downarrow }(r)\\right)^2\\right]$ The same procedure applied to the equation $\\bar{\\psi }(x)\\stackrel{\\leftarrow }{\\!\\lnot \\!\\!D}=\\lambda \\bar{\\psi }(x)$ leads to the solutions for $l\\geqslant 0$ $\\bar{\\psi }_{lk}^{(\\pm )}=\\frac{A_{lk}}{(2\\pi )^\\frac{3}{2}\\sqrt{2|\\lambda _{lk}|}}\\left(\\begin{array}{c}\\pm \\sqrt{p_4+ip_3}\\Phi ^{\\uparrow \\uparrow }_l(r)e^{-il\\vartheta }\\\\\\frac{\\mu _{lk}(\\mp \\mu _{lk}+(-1)^{k+1}|\\lambda _{lk}|)}{2(1+l)\\sqrt{p_4+ip_3}}\\Phi ^{\\uparrow \\downarrow }_{l+1}(r)e^{-i(l+1)\\vartheta }\\\\\\frac{\\pm (-1)^{k}\\mu _{lk}+|\\lambda _{lk}|}{\\sqrt{p_4+ip_3}}\\Phi ^{\\uparrow \\uparrow }_l(r)e^{-il\\vartheta }\\\\\\mp (-1)^{k}\\frac{\\mu _{lk}\\sqrt{p_4+ip_3}}{2(1+l)}\\Phi ^{\\uparrow \\downarrow }_{l+1}(r)e^{-i(l+1)\\vartheta }\\end{array}\\right)^\\textrm {T}e^{ip_3x_3+ip_4x_4},$ and for $l<0$ $\\bar{\\psi }_{lk}^{(\\pm )}=\\frac{B_{lk}}{(2\\pi )^\\frac{3}{2}\\sqrt{2|\\lambda _{lk}|}}\\left(\\begin{array}{c}\\pm (-1)^{k}\\frac{\\mu _{lk}\\sqrt{p_4+ip_3}}{2l}\\Phi ^{\\uparrow \\uparrow }_l(r)e^{-il\\vartheta }\\\\\\frac{|\\lambda _{lk}|\\mp (-1)^k\\mu _{lk}}{\\sqrt{p_4+ip_3}}\\Phi ^{\\uparrow \\downarrow }_{l+1}(r)e^{-i(l+1)\\vartheta }\\\\\\frac{\\mu _{lk}(\\mp \\mu _{lk}+(-1)^k|\\lambda _{lk}|)}{2l\\sqrt{p_4+ip_3}}\\Phi ^{\\uparrow \\uparrow }_l(r)e^{-il\\vartheta }\\\\\\pm \\sqrt{p_4+ip_3}\\Phi ^{\\uparrow \\downarrow }_{l+1}(r)e^{-i(l+1)\\vartheta }\\\\\\end{array}\\right)^\\textrm {T}e^{ip_3x_3+ip_4x_4}.\\nonumber $ Figure: The lowest values of μ\\mu solving Eq.", "() for HR=1.6\\sqrt{H}R=1.6.", "Here j 3 =l+1/2j_3=l+1/2 is the projection of the total momentum on the direction of the chromomagnetic field.Eigenvalues are denoted by asterisks in the case of positive u j u_j andby circles in the case of negative u j u_j.To get insight into the physical treatment of above-considered Euclidean eigenmodes one has to solve the Minkowski space Klein-Gordon and Dirac equations in the presence of chromomagnetic field inside the cylinder with the bag-like boundary conditions.", "Solutions describe the elementary quasiparticle excitations inside the thick cylindrical domain wall junction.", "Quite detailed analysis of the notion of quasiparticles in relativistic quantum field theory can be found in [40].", "Unlike the fundamental elementary and composite particles the characteristic properties of quasiparticles (for instance the specific form of the dispersion relation) need not be necessarily Lorentz invariant or even gauge invariant.", "The overall statement of the problem under consideration necessarily assumes that space direction along the chromomagnetic field is singled out by underlining experimental setup as it coincides with the direction of the strong magnetic field generated for short time in heavy ion collision.", "In generic relativistic frame both chromoelectric and chromomagnetic fields are present inside the domain wall junction.", "However since the topological charge density vanishes in the region (see Fig.REF ) there exists specific frame where chromoelectric field is absent.", "This frame is the most convenient for our purposes." ], [ "Adjoint representation: color charged bosons", "In Minkowski space-time the problem (REF ) and (REF ) turns to the wave equation $-\\left(\\partial _\\mu -i\\breve{B}_\\mu \\right)^2\\phi (x)=0$ for color charged adjoint spin zero field inside a cylindrical wave guide.", "As it follows from (REF ) and futher discussion the charged components of the adjoint field of the color matrix $\\breve{n}$ comes in complex conjugate pairs.", "For instance if $\\xi =\\pi /6$ then there are two pairs $\\phi _1\\pm i\\phi _2$ and $\\phi _4\\pm i\\phi _5$ .", "Thus $\\phi ^a$ is a complex scalar field, the corresponding solution of (REF ) satisfying boundary condition (REF ) takes the form $&&\\phi ^a(x)=\\sum _{lk}\\int \\limits _{-\\infty }^{+\\infty }\\frac{dp_3}{2\\pi }\\frac{1}{\\sqrt{2\\omega _{alk}}} \\left[a^{+}_{akl}(p_3)e^{ix_0\\omega _{akl}-ip_3x_3}+b_{akl}(p_3)e^{-ix_0\\omega _{akl}+ip_3x_3}\\right]e^{il\\vartheta }\\phi _{alk}(r),\\\\&&\\phi ^{a\\dagger }(x)=\\sum _{lk}\\int \\limits _{-\\infty }^{+\\infty }\\frac{dp_3}{2\\pi }\\frac{1}{\\sqrt{2\\omega _{alk}}} \\left[b^{+}_{akl}(p_3)e^{-ix_0\\omega _{akl}+ip_3x_3}+a_{akl}(p_3)e^{ix_0\\omega _{akl}-ip_3x_3}\\right]e^{-il\\vartheta }\\phi _{alk}(r),\\nonumber \\\\&&p_0^2=p_3^2+\\mu _{akl}^2,\\nonumber \\\\&&p_0=\\pm \\omega _{akl}(p_3), \\ \\omega _{akl}=\\sqrt{p_3^2+\\mu _{akl}^2},\\\\&& k=0,1,\\dots ,\\infty , \\ \\ l\\in Z,\\nonumber $ with $\\phi _{alk}(r)$ defined in (REF ) but here it is assumed to be normalized $\\int \\limits _0^\\infty dr r\\int \\limits _0^{2\\pi } d\\vartheta e^{i(l-l^{\\prime })\\vartheta } \\phi _{alk}(r) \\phi _{al^{\\prime }k^{\\prime }}(r)=\\delta _{ll^{\\prime }}\\delta _{kk^{\\prime }}.$ Equation () can be treated as the dispersion relation between energy $p_0$ and momentum $p_3$ for the quasiparticles with masses $\\mu _{akl}$ .", "These quasiparticles are extended in $x_1$ and $x_2$ directions and are classified by the quantum numbers $l,k$ .", "The orthogonality, normalization and completeness of the set of functions $e^{il\\vartheta }\\phi _{alk}(r)$ guarantees the standard canonical commutation relations for the field $\\phi ^a$ and its canonically conjugated momentum if $a^\\dagger _{akl}(p_3)$ , $a_{akl}(p_3)$ , $b^\\dagger _{akl}(p_3)$ and $b_{akl}(p_3)$ are assumed to satisfy the standard commutation relations for creation and annihilation operators.", "The Fock space of states for the quasiparticles with masses $\\mu _{akl}$ can be constructed by means of the standard QFT methods.", "This treatment provides one with a suitable terminology and formalism for discussion of the confining properties of various gluon field configurations in the context of QFT: unlike the chromomagnetic field the (anti-)self-dual fields characteristic for the bulk of domain network configuration (see the LHS plot in Fig.", "REF ) lead to purely discrete spectrum of eigenmodes in Euclidean space and do not possess any quasiparticle treatment in terms of dispersion relation between energy and momentum for elementary color charged excitations.", "If there is a reason for long-lived defect in the form of thick domain wall junction then its boundary defines a shape and a size for the space region which can be populated by color charged quasiparticles.", "The vector adjoint field can be elaborated in the similar to the scalar case way.", "A modification relates just to the inclusion of polarization vectors.", "As it has already been mentioned the most important feature is the absence of tachyonic mode of the vector color charged field if $R<R_{\\mathrm {c}}$ .", "Disappearance of the tachyonic mode for subcritical size of the trap is one of the most important observations of this paper." ], [ "Fundamental representation: color charged fermions", "Neither the background field nor the boundary condition involve the time coordinate.", "The solution of the Dirac equation $&&i\\!\\lnot \\!\\!D\\psi (x)=0,$ satisfying condition (REF ) can be obtained from Euclidean solutions (REF ) (unnormalized solutions have to be used) by the analytical continuation $p_4\\rightarrow ip_0$ , $x_4\\rightarrow ix_0$ and the requirement $\\lambda _{lk}=0$ , which leads to the energy-momentum relation for the solutions with definite $j_3$ , $k$ and color $j$ $p_0^2=p_3^2+\\mu _{jlk}^2 ,\\ \\ p_0=\\pm \\omega _{jlk}(p_3),\\\\ \\omega _{jlk}=\\sqrt{p_3^2+\\mu _{jlk}^2}.$ Finally the solution of the Dirac equation takes the form $\\psi ^{j}(x)=\\sum _{lk}\\int \\limits _{-\\infty }^{+\\infty }\\frac{dp_3}{2\\pi }\\frac{1}{\\sqrt{2\\omega _{jlk}}}\\left[a^{\\dagger }_{jlk}(p_3)\\chi _{jlk}(p_3|r,\\vartheta )e^{ix_0\\omega _{jlk}-ix_3p_3}+b_{jlk}(p_3)\\upsilon _{jlk}(p_3|r,\\vartheta )e^{-ix_0\\omega _{jlk}+ix_3p_3}\\right],\\\\\\bar{\\psi }^{j}(x)=\\sum _{lk}\\int \\limits _{-\\infty }^{+\\infty }\\frac{dp_3}{2\\pi }\\frac{1}{\\sqrt{2\\omega _{jlk}}}\\left[b^{\\dagger }_{jlk}(p_3)\\bar{\\chi }_{jlk}(p_3|r,\\vartheta )e^{-ix_0\\omega _{jlk}+ix_3p_3}+a_{jlk}(p_3){\\bar{\\upsilon }}_{jlk}(p_3|r,\\vartheta )e^{ix_0\\omega _{jlk}-ix_3p_3}\\right].$ Here the pair of spinors for positive $\\chi _{lk}$ and negative $\\upsilon _{lk}$ energy solutions are $\\chi _{lk}=A_{lk}\\left(\\begin{array}{c}(-1)^{k+1}\\frac{\\mu _{lk}}{\\sqrt{\\omega _{lk}+p_3}} \\Phi _l^{\\uparrow \\uparrow }(r)e^{il\\vartheta }\\\\i(-1)^{k+1}\\frac{\\mu _{lk}\\sqrt{\\omega _{lk}+p_3}}{2(l+1)}\\Phi _{l+1}^{\\uparrow \\downarrow }(r)e^{i(l+1)\\vartheta }\\\\i\\sqrt{\\omega _{lk}+p_3}\\Phi _l^{\\uparrow \\uparrow }(r)e^{il\\vartheta }\\\\\\frac{\\mu ^2_{lk}}{2(l+1)\\sqrt{\\omega _{lk}+p_3}}\\Phi _{l+1}^{\\uparrow \\downarrow }(r)e^{i(l+1)\\vartheta }\\end{array}\\right),\\ \\ \\ \\ \\upsilon _{lk}=A_{lk}\\left(\\begin{array}{c}(-1)^{k+1}\\frac{\\mu _{lk}}{\\sqrt{\\omega _{lk}+p_3}} \\Phi _l^{\\uparrow \\uparrow }(r)e^{il\\vartheta }\\\\i(-1)^{k}\\frac{\\mu _{lk}\\sqrt{\\omega _{lk}+p_3}}{2(l+1)}\\Phi _{l+1}^{\\uparrow \\downarrow }(r)e^{i(l+1)\\vartheta }\\\\-i\\sqrt{\\omega _{lk}+p_3}\\Phi _l^{\\uparrow \\uparrow }(r)e^{il\\vartheta }\\\\\\frac{\\mu ^2_{lk}}{2(l+1)\\sqrt{\\omega _{lk}+p_3}}\\Phi _{l+1}^{\\uparrow \\downarrow }(r)e^{i(l+1)\\vartheta }\\end{array}\\right),$ for $l\\geqslant 0$ and $\\chi _{lk}=B_{lk}\\left(\\begin{array}{c}\\frac{\\mu ^2_{lk}}{2l\\sqrt{\\omega _{lk}+p_3}} \\Phi _l^{\\uparrow \\uparrow }(r)e^{il\\vartheta }\\\\i\\sqrt{\\omega _{lk}+p_3}\\Phi _{l+1}^{\\uparrow \\downarrow }(r)e^{i(l+1)\\vartheta }\\\\i(-1)^k\\frac{\\mu _{lk}\\sqrt{\\omega _{lk}+p_3}}{2l}\\Phi _l^{\\uparrow \\uparrow }(r)e^{il\\vartheta }\\\\(-1)^k\\frac{\\mu _{lk}}{\\sqrt{\\omega _{lk}+p_3}}\\Phi _{l+1}^{\\uparrow \\downarrow }(r)e^{i(l+1)\\vartheta }\\end{array}\\right),\\ \\ \\ \\ \\upsilon _{lk}=B_{lk}\\left(\\begin{array}{c}\\frac{\\mu ^2_{lk}}{2l\\sqrt{\\omega _{lk}+p_3}} \\Phi _l^{\\uparrow \\uparrow }(r)e^{il\\vartheta }\\\\-i\\sqrt{\\omega _{lk}+p_3}\\Phi _{l+1}^{\\uparrow \\downarrow }(r)e^{i(l+1)\\vartheta }\\\\i(-1)^{k+1}\\frac{\\mu _{lk}\\sqrt{\\omega _{lk}+p_3}}{2l}\\Phi _l^{\\uparrow \\uparrow }(r)e^{il\\vartheta }\\\\(-1)^k\\frac{\\mu _{lk}}{\\sqrt{\\omega _{lk}+p_3}}\\Phi _{l+1}^{\\uparrow \\downarrow }(r)e^{i(l+1)\\vartheta }\\end{array}\\right)$ for $l<0$ .", "The spinors are normalized as $\\int \\limits _{0}^{2\\pi }d\\vartheta \\int \\limits _0^R dr r\\chi ^\\dagger _{jlk}(p_3|r,\\vartheta )\\chi _{jlk}(p_3|r,\\vartheta )=\\int \\limits _{0}^{2\\pi }d\\vartheta \\int \\limits _0^R dr r\\upsilon ^\\dagger _{jlk}(p_3|r,\\vartheta )\\upsilon _{jlk}(p_3|r,\\vartheta )= 2\\omega _{jlk}$ The Dirac conjugated spinors are $\\bar{\\psi }^{j}(x)=\\psi ^{j\\dagger }(x)\\gamma _0$ as usual.", "The Fock space can be constructed by means of the creation and annihilation operators $\\left\\lbrace a^{\\dagger }_{jlk}(p_3), a_{jlk}(p_3), b^{\\dagger }_{jlk}(p_3), b_{jlk}(p_3)\\right\\rbrace $ satisfying the standard anticommutation relations.", "The one-particle state is characterized by a color orientation $j$ , momentum $p_3$ , projection $j_3=(l+1/2)$ of the total angular momentum and the energy $\\omega _{jlk}=\\sqrt{p_3^2+\\mu ^2_{jlk}}$ .", "Since the boundary condition mixes the states with spin parallel and anti-parallel to the chromomagnetic field the spin projection is not a good quantum number unlike the half-integer valued projection of the total angular momentum $j_3$ ." ], [ "Discussion", "An ensemble of confining gluon configurations has been constructed explicitly as a domain wall networks representing the almost everywhere homogeneous Abelian (anti-)self-dual gluon fields.", "Confinement is understood here as the absence of the color charged wave-like elementary excitations.", "The dynamical quark confinement occurs in the (four-dimensional) bulk of the domain wall network.", "Inside the (three-dimensional) domain walls topological charge density vanishes and the color charged quasiparticles can be excited.", "Under extreme conditions, in particular under the influence of the strong electromagnetic field specific for relativistic heavy ion collisions, a relatively stable defect in the confining ensemble, a thick domain wall junction, can be formed.", "Though the scalar gluon condensate is nonzero everywhere $\\langle g^2F^2\\rangle \\ne 0$ , the region of defect is characterized by the vanishing topological charge density $\\langle |g^2\\tilde{F}F|\\rangle $ =0 unlike the rest of the space, which indicates the lack of confinement in the junction.", "The quark field excitations inside the junction are represented by the color charged quasiparticles.", "The spectrum of gluon excitations besides the trapped color charged modes contains also the color neutral with respect to the background field modes.", "Almost obvious but important observation is that there exists a critical size $L_{\\rm c}$ of the junction beyond which the tachyonic gluon modes emerge in the excitation spectrum and destabilize the defect.", "The critical size can be related to the value of the gluon condensate $\\langle g^2F^2\\rangle $ and in the case of the considered in the paper cylindrical trap $L_{\\rm c}\\approx 1$ fm for the standard value of the condensate, see (REF ).", "The specific value of the critical size depends on the geometry of the trap but its very existence and its commensurability with a distance of order of 1fm is a generic feature.", "This observation underlines the generic necessity of accounting for the essentially finite size of the space-time region in which deconfinement may occur.", "The reason is that thermodynamic limit does not exist as the system under consideration disappears as soon as the typical size of the space volume exceeds the critical value.", "Excess of the internal pressure of the trap filled by many charged quasiparticles leads to its expansion and breakdown of stability followed by its disintegration to many smaller traps (or bags), which is reminiscent of the heterophase fluctuations studied in [41] as well as the dynamics and statistical mechnaics of bags with a surface tension [42].", "The dynamics of the color charged quasiparticles as it is described above is strictly one-dimentional in space.", "This feature can be a source of the azimuthal asymmetries in heavy ion collisions, similarly to the approach of paper [27] upto a substitution of the magnetic field by the Abelian chromomagnetic field (see also [37]).", "However it should be noted that the one-dimensional dynamics is a property of the zero-th order approximation based on the quadratic part of the action.", "Taking into account interactions between the quasiparticles according to the interaction terms in the action should certainly dither the direction of the quasiparticle momenta, leaving just some degree of azimuthal asymmetry." ], [ "ACKNOWLEDGMENTS", "We acknowledge fruitful discussions with V.Toneev, S. Molodtsov, J. Pawlowski, M.Ilgenfritz, A.Dorokhov, K.Bugaev S.Vinitsky, G.Efimov, V.Yukalov, A.Efremov, A.Titov.", "tocsectionBibliography1" ] ]
1403.0415
[ [ "Improving contact prediction along three dimensions" ], [ "Abstract Correlation patterns in multiple sequence alignments of homologous proteins can be exploited to infer information on the three-dimensional structure of their members.", "The typical pipeline to address this task, which we in this paper refer to as the three dimensions of contact prediction, is to: (i) filter and align the raw sequence data representing the evolutionarily related proteins; (ii) choose a predictive model to describe a sequence alignment; (iii) infer the model parameters and interpret them in terms of structural properties, such as an accurate contact map.", "We show here that all three dimensions are important for overall prediction success.", "In particular, we show that it is possible to improve significantly along the second dimension by going beyond the pair-wise Potts models from statistical physics, which have hitherto been the focus of the field.", "These (simple) extensions are motivated by multiple sequence alignments often containing long stretches of gaps which, as a data feature, would be rather untypical for independent samples drawn from a Potts model.", "Using a large test set of proteins we show that the combined improvements along the three dimensions are as large as any reported to date." ], [ "Results using the $8.5$ Å heavy atom criterion", "In this work we have used the C$\\beta $ criterion, that a pair of amino acids are in contact in a crystal structure if their C$\\beta $ atoms (C$\\alpha $ in case of Glycines) are not more than $\\le 8$ Å apart.", "This kind of contact evaluation criterion has been used for some time in the biannual Critical Assessment of protein Structure Prediction (CASP) competition, and is considered standard in the field of protein structure prediction [5], [6].", "In the context of DCA it was used, with a threshold of 8Å, in [3] and [8].", "No fixed criterion of this kind will be perfect.", "The threshold to use depends on the desired trade-off between false negatives and false positives, and that depends on the intended application.", "For a reasonable threshold, such as around 8Å, there will be some pairs of amino acids which satisfy the C$\\beta $ criterion, but nevertheless probably do not make any contact, at least in one given crystal structure, and there will be some pairs of amino acids which do not satisfy the C$\\beta $ criterion, but where the side chains in fact do make contact.", "In both cases the residue types will matter, an aspect which is not taken into account by the C$\\beta $ criterion.", "Several earlier publications on DCA have, following [7], used an alternative criterion which we call the heavy atom criterion, where a a pair of amino acids are taken to be in contact if the distance between the two closest heavy (i.e.", "non-hydrogen) atoms of the two amino acids in question is less than some threshold around 8Å.", "To faciliate comparision we show in this supplementary material results using the heavy atom criterion with a threshold of $8.5$ Å, as used by one of us in [1].", "Figures and tables in this supplementary material are numbered identically as in the main paper, and are based on the same data, the only difference being that we use throughout the $8.5$ Å heavy atom criterion.", "It is evident that the 8Å C$\\beta $ criterion is more stringent than the $8.5$ Å (or 8Å) heavy atom criterion.", "A main difference will therefore be that nominal PPVs will be higher using the heavy atom criterion, other differences will be pointed out in figure captions.", "We note that a heavy atom criterion with substantially smaller thresholds have also been proposed in the literature, such as 6Å in [3], 5Å in [4] and $4.5$ Å in a paper from before the DCA era [2].", "We do not here make any comparisons to the heavy atom criterion with these choices of thresholds.", "References cited in this supplementary material are listed in a separate bibliography.", "Figure: Examples of qualitative contact prediction improvement.", "Left panel: contact predictionmaps built by plmDCA and gplmCDA using protein sequences homologous to 1JFU as explained in Methods.plmDCA here predicts a number of strong couplings at both the N-terminus and the C-terminus whicharise from the high sequence variability at both ends of proteins homologous to 1JFU,and the many gaps in the multiple sequence alignments at these positions.", "In gplmDCA these gapslead to adjustment of gap parameters and not to contact predictions.", "Right panel: analogousresults using protein sequences homologous to 1ATZ where gplmDCA removes strong spurious couplings at theC-terminus.Remarks pertaining to the 8.5Å8.5 \\hbox{Å{}} heavy atom criterion:The PPVs are substantially higher, in the range 0.89-0.950.89-0.95, andthe relative improvement is less pronounced than using the Cβ\\beta criterion.Figure: Absolute PPV, average over all proteins in the main test data set.", "The curves show for PSICOV, plmDCA and gplmDCAthe average of the number of correct predictions in the nn highest scoring pairs divided by nn;Left panel: PPV for absolute contact index.", "The horizontal axis shows nn.", "According to the8.5Å8.5 \\hbox{Å{}} heavy atom criterion plmDCA performs on par with gplmDCA at nn greater than about 100.Right panel: PPV for relative contact index (fraction of protein length).", "The horizontal axis shows (n/N)·100(n/N) \\cdot 100.Remarks pertaining to the 8.5Å8.5 \\hbox{Å{}} heavy atom criterion:Left panel: plmDCA performs on par with gplmDCA at values of nngreater than about 100.", "Right panel: in difference to the data shown in right panelof Figure 2 in main paper,plmDCA here performs on par with gplmDCA over the full range of nn.Figure: Contact prediction accuracy (mean absolute PPV)for proteins in the main test set by plmDCA (abscissa) and gplmDCA (ordinate).Data points can be fitted by astraight line with slope 0.996±0.0040.996 \\pm 0.004 (R 2 =0.986R^2=0.986).Remarks pertaining to the 8.5Å8.5 \\hbox{Å{}} heavy atom criterion:By the 8.5 Å heavy atom criterion there is no differencebetween plmDCA and gplmDCA.Table: Numbers and fraction of proteins where gplmDCA performs better than plmDCA.In each row all proteins in the data set are included for which the PPV from both plmDCAand gplmDCA is larger than the cutoff value given in the first column.The full data set (last row) consists of 801 proteins for 369 (46%) of which gplmDCAperforms better than plmDCA.", "In the most stringent selection (first row) there are108 proteins where both plmDCA and gplmDCA have a PPV at least 0.8.", "In this setgplmDCA performs better on 89 (82%) of the instances.Figure: Contact prediction accuracy for proteins in the test set by gplmDCA and plmDCA vs number of sequences in the alignment, when considering top 10%, 25% (top row), 50% and 100% (bottom row) contacts,100% being the same number of contacts as the number of amino acids in the protein.The advantage of gplmDCA is particularly interesting in ranges highlighted by verticaldotted lines.", "For the top 10% and top 25% (top row) these rangesare approximately 350-3000 and 500-7000 sequences(279 and 359 out of 801 proteins), while for the top50% and top 100% (bottom row) they correspondingly in the ranges of 1000-7000 sequences (291 proteins) and 2000 sequences in the alignmentand upwards (454 out of 801 proteins).PSICOV outperforms both plmDCA and gplmDCA when there are less than about100 sequences in the alignment.", "The peak around500-sequence point is due to concentration of β\\beta -sheet rich proteins (mostlyhydrolases), that seem to be particularly amiable to contact prediction.Figure: Prediction by absolute PPV and 8.5Å8.5 Å{} heavy atom criterion for gplmDCA and plmDCA run on Pfam and HHblits alignmentsin the reduced test data set.", "The reduced test data set comprises the proteins in the main test data set wherea comparison can be made to Pfam alignments, as described in Methods.Remarks pertaining to the 8.5Å8.5 \\hbox{Å{}} heavy atom criterion:As in the data shown in Figure 5 in the main paper, contact predictionis more effective using HHblits alignments.", "In contrast to Figure 5 in themain paper, gplmDCA here does not show an advantage over plmDCA.Figure: Scatter plots of prediction by absolute PPV and 8.58.5Å heavy atom criterion for individual proteins in thereduced test data set.", "Top row shows, analogously to Figure (for the main data set),gplmDCA vs plmDCA for Pfam alignments (left panel) and for HHblits alignments (right panel).Bottom row shows prediction for HHblits alignments vs Pfam alignmentsusing plmDCA (left panel) and gplmDCA (right panel).Remarks pertaining to the 8.5Å8.5 \\hbox{Å{}} heavy atom criterion:As in Figure  there ishere no advantage of gplmDCA over plmDCA." ] ]
1403.0379
[ [ "Unavoidable collections of balls for processes with isotropic unimodal\n Green function" ], [ "Abstract Let us suppose that we have a right continuous Markov semigroup on $R^d$, $d\\ge 1$, such that its potential kernel is given by convolution with a function $G_0=g(|\\cdot|)$, where $g$ is decreasing, has a mild lower decay property at zero, and a very weak decay property at infinity.", "This captures not only the Brownian semigroup (classical potential theory) and isotropic $\\alpha$-stable semigroups (Riesz potentials), but also more general isotropic L\\'evy processes, where the characteristic function has a certain lower scaling property, and various geometric stable processes.", "There always exists a corresponding Hunt process.", "A subset $A$ of $R^d$ is called unavoidable, if the process hits $A$ with probability $1$, wherever it starts.", "It is known that, for any locally finite union of pairwise disjoint balls $B(z,r_z)$, $z\\in Z$, which is unavoidable, $\\sum_{z\\in Z} g(|z|)/g(r_z)=\\infty$.", "The converse is proven assuming, in addition, that, for some $\\varepsilon>0$, $|z-z'|\\ge \\varepsilon |z| (g(|z|)/g(r_z))^{1/d}$, whenever $z,z'\\in Z$, $z\\ne z'$.", "It also holds, if the balls are regularly located, that is, if their centers keep some minimal mutual distance, each ball of a certain size intersects $Z$, and $r_z=g(\\phi(|z|))$, where $\\phi$ is a decreasing function.", "The results generalize and, exploiting a zero-one law, simplify recent work by A. Mimica and Z. Vondracek." ], [ "Introduction and main results", "Let $\\mathbb {P}=(P_t)_{t>0} $ be a right continuous Markov semigroup on ${\\mathbb {R}^d}$ , $d\\ge 1$ , such that its potential kernel $V_0:=\\int _0^\\infty P_t\\,dt$ is given by convolution with a $\\mathbb {P}$ -excessive function $G_0=g(|\\cdot |),$ where $g$ is a decreasing function on $\\left[0,\\infty \\right)$ such that $0<g<\\infty $ on $(0,\\infty )$ , $\\lim _{r\\rightarrow 0} g(r)=g(0)=\\infty $ and the following holds: (LD) Lower decay property: There are $R_0\\ge 0$ and $C_G\\ge 1$ such that $d \\int _0^r s^{d-1}g(s) \\,ds \\le C_G \\, r^d g(r), \\qquad \\mbox{ for all~$r>R_0$}.$ (UD) Upper decay property at infinity: There are $R_1\\ge 0$ , $\\eta \\in (0,1) $ , and $K>1$ such that $g(Kr)\\le \\eta g(r), \\qquad \\mbox{ for all $r> R_1$}.$ REMARK 1.1 The inequality (REF ) has a very intuitive meaning: If $B$ is a ball with radius $r$ and center 0, and $\\lambda _B$ denotes normed Lebesgue measure on $B$ , then the potential $G\\lambda _B:=G_0\\ast \\lambda _B$ of $\\lambda _B$ satisfies $G\\lambda _B(0) \\le C_G g(r)$ (and hence $G\\lambda _B \\le C_G g(r)$ on ${\\mathbb {R}^d}$ ), where $g(r)$ is the value of $G_0$ at the boundary of $B$ .", "In Section we shall see the following.", "1.", "For the Brownian semigroup (classical potential theory) and isotropic $\\alpha $ -stable semigroups (Riesz potentials) we have $g(r)=r^{\\alpha -d}$ , $\\alpha \\in (0,2]$ , $\\alpha <d$ , and our assumptions are satisfied with $R_0=R_1=0$ .", "This holds as well for the more general isotropic unimodal Lévy semigroups considered in [11].", "2.", "If (LD) is satisfied for some $R_0>0$ , then, for every $R>0$ , there exists $C_G\\ge 1$ such that (REF ) holds for all $r>R$ (and hence the restriction $r_z>R_0$ , for all $z\\in Z$ , imposed below, reduces to the requirement that $\\inf _{z\\in Z} r_z>0$ ).", "Analogously for (UD).", "3.", "If $\\int _0^1 s^{d-1} g(s)\\, ds<\\infty $ and $g(r)\\approx r^{\\beta -d}$ , $0<\\beta <d$ , as $r\\rightarrow \\infty $ , then (LD) and (UD) hold with arbitrary $R_0, R_1\\in (0,\\infty )$ .", "Subordinate Brownian semigroups with subordinators having Laplace exponents of the form $\\phi (\\lambda )=\\ln ^\\delta (1+\\lambda ^{\\alpha /2}), \\qquad \\mbox{ $ 0<\\delta \\le 1$, $\\alpha \\in (0,2]$, $\\alpha <d$},$ provide examples (symmetric geometric stable processes, if $\\delta =1$ ), where (LD) does not hold with $R_0=0$ .", "Here $g$ satisfies $g(r)\\approx r^{-d}\\ln ^{-(1+\\delta )} (1/r) \\mbox{ as } r\\rightarrow 0 \\quad \\mbox{ and }\\quad g(r)\\approx r^{\\delta \\alpha -d} \\mbox{ as } r\\rightarrow \\infty .$ Let $\\mathfrak {X}=(\\Omega ,\\mathfrak {M}, \\mathfrak {M}_t, X_t, \\theta _t, P^x)$ be an associated Hunt process on ${\\mathbb {R}^d}$ (for its existence see Remark REF ,1).", "A Borel measurable set $A$ in ${\\mathbb {R}^d}$ is called unavoidable, if $P^x[T_A<\\infty ]=1 \\qquad \\mbox{ for every } x\\in {\\mathbb {R}^d},$ where $T_A(\\omega ):=\\inf \\lbrace t\\ge 0\\colon X_t(\\omega )\\in A\\rbrace $ .", "Otherwise, it is called avoidable, that is, $A$  is avoidable, if there exists $x\\in {\\mathbb {R}^d}$ such that $P^x[T_A<\\infty ]<1$ .", "For all $x\\in {\\mathbb {R}^d}$ and $r>0$ , let $B(x,r)$ denote the open ball with center $x$ and radius $r$ , and let $\\overline{B}(x,r)$ be its closure.", "Let us introduce two properties for families of balls which, in the the classical case, have already been considered in [3], [4] (and where it does not make a real difference, if we look at open or closed balls, since a union of open balls is unavoidable if and only if the union of the corresponding closed balls is unavoidable; see Remark REF ,2).", "Let $Z$ be a countable set in ${\\mathbb {R}^d}\\setminus \\lbrace 0\\rbrace $ and $r_z>R_0$ , $z\\in Z$ , such that the balls $B(z,r_z)$ are pairwise disjoint.", "We say that the balls $B(z,r_z)$ , $z\\in Z$ , satisfy the separation condition, if $Z$ is locally finite and $\\inf \\nolimits _{z,z^{\\prime }\\in Z,\\, z\\ne z^{\\prime }} \\frac{|z-z^{\\prime }|^d}{|z|^d} \\, \\frac{g(r_z)}{g(|z|)} >0.$ We say that they are regularly located, if the following holds: (a) There exists $\\varepsilon >0$ such that $|z-z^{\\prime }|\\ge \\varepsilon $ , for all $z,z^{\\prime }\\in Z$ , $z\\ne z^{\\prime }$ .", "(b) There exists $R>0$ such that $B(x,R)\\cap Z\\ne \\emptyset $ , for every $x\\in {\\mathbb {R}^d}$ .", "(c) There exists a decreasing function $\\phi \\colon [0,\\infty )\\rightarrow (0,\\infty )$ such that $r_z=\\phi (|z|)$ .", "Our main results are the following (where we might bear in our mind that $1/g(r)$ is approximately the capacity of balls having radius $r$ , that is, the total mass of their equilibrium measure; see Proposition REF ).", "THEOREM 1.2 If the balls $B(z,r_z)$ , $z\\in Z$ , satisfy the separation condition, then their union $A$ is unavoidable provided $\\sum \\nolimits _{z\\in Z} \\frac{g(|z|)}{g(r_z)}=\\infty .$ COROLLARY 1.3 Suppose that the balls $B(z,r_z)$ , $z\\in Z$ , are regularly located.", "Then their union $A$ is unavoidable if and only if $\\int _1^\\infty \\frac{r^{d-1} g(r)}{g(\\phi (r))}\\,dr =\\infty .$ The converse in Theorem REF is already known without any restriction on the balls and assuming only $\\lim _{r\\rightarrow \\infty } g(r)=0$ instead of (UD) (see [8]; the inequality $R_1^{\\overline{B}(z,r_z)}\\le g(|z|)/g(r_z)$ , which is used in its proof, holds trivially, since $g$ is decreasing).", "PROPOSITION 1.4 Let $A$ be an unavoidable union of balls $B(z,r_z)$ , $z\\in Z$ .", "Then $\\sum _{z\\in Z} g(|z|)/g(r_z)=\\infty $ and $\\sum _{z\\in Z} 1/g(r_z)=\\infty $ .", "REMARK 1.5 1.", "In the classical case, Theorem REF is [4] (for unavoidableness under a  weaker separation property see [3]) and Corollary REF is [3].", "2.", "In the more general case of isotropic unimodal Lévy processes, where the characteristic function satisfies a lower scaling condition (and (LD), (UD) hold with $R_0=R_1=0$ ), both Theorem REF , its converse, and Corollary REF are proven in [11].", "We shall use the same method of considering finitely many countable unions of concentric shells, but have to overcome additional difficulties caused by having only a rather weak estimate for the exit distribution of balls (compare [11], going back to [5], and Proposition REF ).", "Nevertheless our proof for Theorem REF can be simpler, since starting with an avoidable union $A$ and an arbitrary $\\delta >0$ , we may assume without loss of generality that $P^0[T_A<\\infty ]< \\delta $ (using Proposition REF and translation invariance).", "3.", "If the balls $B(z,r_z)$ , $z\\in Z$ , are regularly located, then $\\sum \\nolimits _{z\\in Z} \\frac{g(|z|)}{g(r_z)}=\\infty \\quad \\mbox{ if and only if } \\quad \\int _1^\\infty \\frac{r^{d-1} g(r)}{g(\\phi (r))}\\,dr =\\infty .$ This is fairly obvious (see [11]) and allows us to reduce Corollary REF to a consequence of Theorem REF by first treating a simple case (see Proposition REF ).", "In view of the second statement in Proposition REF let us mention the following part of [8] (where only $\\lim _{r\\rightarrow \\infty } g(r)=0$ instead of (UD) is needed).", "See also [7] for the result in classical potential theory.", "THEOREM 1.6 Suppose that (LD) holds with $R_0=0$ .", "Let $h\\colon (0,1)\\rightarrow (0,1)$ with $\\lim _{t\\rightarrow 0} h(t)=0$ , let $\\varphi \\in {\\mathbb {R}^d})$ , $\\varphi >0$ , and $\\delta >0$ .", "Then there exist a locally finite set $Z$ in ${\\mathbb {R}^d}$ and $0<r_z<\\varphi (z)$ , $z\\in Z$ , such that the balls $\\overline{B}(z,r_z)$ are pairwise disjoint, the union of all $\\overline{B}(z,r_z)$ is unavoidable, and $\\sum \\nolimits _{z\\in Z} h(r_z)/g(r_z)<\\delta .$ In Section 2, we shall first take a closer look at the properties (LD) and (UD) and then show that our assumptions cover the isotropic unimodal processes considered in [11] and geometric stable processes.", "In Section 3, we shall discuss some general potential theory of the semigroup $\\mathbb {P}$ , where, as in [8], at the beginning (LD) and (UD) are replaced by the weaker properties $\\int _0^1 r^{d-1}g(r)\\, dr<\\infty $ and $\\lim _{r\\rightarrow \\infty }g(r)=0$ .", "In Section 4, we prove Theorem REF , and the proof of Corollary REF is given in Section 5." ], [ "Examples", "Let us first consider an arbitrary positive decreasing function $g$ on $(0,\\infty )$ and write down a few elementary facts justifying, in particular, our statements in Remark REF .", "Given $R_0\\ge 0 $ , we say that (LD) holds on $(R_0,\\infty )$ , if there exists $C\\ge 1$ such that $d \\int _0^r s^{d-1} g(s)\\,ds \\le C r^d g(r), \\qquad \\mbox{ for every } r>R_0.$ Similarly, given $0\\le R_1<\\infty $ , we say that (UD) holds on $(R_1,\\infty )$ , if there exist $K>1$ and $\\eta \\in (0,1)$ such that $g(Kr)\\le \\eta g(r), \\qquad \\mbox{ for every }r>R_1.$ LEMMA 2.1 1.", "If there is a function $\\varphi >0$ on $(0,1)$ with $\\int _0^1 \\gamma ^{d-1}\\varphi (\\gamma ) \\, d\\gamma <\\infty $ and $g(\\gamma r)\\le \\varphi (\\gamma ) g(r), \\qquad \\mbox{ for all }\\gamma \\in (0,1) \\mbox{ and } r>0,$ then (LD) holds on $(0,\\infty ) $ .", "2.", "Let $f(r):=r^d g(r)$ , $R_0\\ge 0$ , $\\kappa , C\\in (0,\\infty )$ .", "If $\\int _0^1 s^{-1}f(s)\\,ds<\\infty $ , $f\\ge \\kappa $ on $(R_0,\\infty )$ , and $\\int _{R_0}^r s^{-1}f(s)\\,ds \\le C f(r),\\qquad \\mbox{ for every $r>R_0$},$ then (LD) holds on $(R_0,\\infty )$ .", "3.", "If $0<R<R_0$ and (LD) holds on $(R_0,\\infty )$ , then (LD) holds on $( R,\\infty )$ .", "1.", "For every $r>0$ , $\\int _0^r s^{d-1} g(s)\\,ds = r^d\\int _0^1 \\gamma ^{d-1} g(\\gamma r)\\,d\\gamma \\le r^d g(r) \\int _0^1 \\gamma ^{d-1} \\varphi (\\gamma )\\, d\\gamma .$ 2.", "Clearly, $c:= \\int _0^{R_0} s^{d-1} g(s)\\,ds<\\infty $ .", "For every $r>R_0$ , $\\int _0^r s^{d-1} g(s)\\,ds= c + \\int _{R_0}^r s^{-1}f(s)\\,ds \\le (c\\kappa ^{-1}+C) f(r)= (c\\kappa ^{-1}+C) r^d g(r).$ 3.", "Let $0<R< R_0<R_1$ and assume that (REF ) holds.", "Defining $\\tilde{C}:=C( R_1/R)^d$ we obtain that, for every $r\\in [R,R_0]$ , $d \\int _0^r s^{d-1} g(s) \\,ds\\le C R_1^d g( R_1) =\\tilde{C} R^d g( R_1)\\le \\tilde{C} r^d g(r).$ LEMMA 2.2 Let $0\\le R_1<\\infty $ .", "1.", "If there is a function $\\varphi >0$ on $(R,\\infty )$ , $R>0$ , with $\\lim _{\\lambda \\rightarrow \\infty } \\varphi (\\lambda )=0$ and $g(\\lambda r)\\le \\varphi (\\lambda ) g(r)$ , for all $\\lambda \\ge R$ and $r>R_1$ , then (UD) holds on $(R_1,\\infty )$ .", "2.", "If $0<R<R_1$ and (UD) holds on $(R_1,\\infty )$ , then (UD) holds on $( R,\\infty )$ .", "3.", "If (UD) holds on $(R_1,\\infty )$ , then, for every $\\delta >0$ , there exists $K>1$ such that $g(Kr)\\le \\delta g(r)$ for every $r>R_1$ .", "1.", "We take $K\\ge R$ such that $\\varphi (K) <\\eta $ .", "2.", "Let $c:=R_1/R$ , $r> R$ .", "Then $cr>R_1$ and $g(Kcr)\\le \\eta g(cr)\\le \\eta g(r)$ .", "3.", "We choose $m\\in \\mathbb {N}$ such that $\\eta ^m<\\delta $ and replace $K$ by $K^m$ .", "If $0<\\alpha <d$ and $g(r)=r^{\\alpha -d}$ , then, by Lemmas REF and REF , (LD) and (UD) hold on $(0,\\infty )$ .", "So our assumptions are satisfied by Brownian motion and isotropic $\\alpha $ -stable processes with $0<\\alpha \\le 2$ , $\\alpha <d$ .", "Let us observe next that, more generally, our assumptions are satisfied by the isotropic unimodal Lévy processes $\\mathfrak {X}=(X_t, P^x)$ studied in [11], where the characteristic function $\\psi $ for  $\\mathfrak {X}$ (characterized by $ e^{-t\\psi (|x|)}= E^0 [e^{i\\langle x,X_t\\rangle }]$ , $t>0$ ) is supposed to satisfy the following weak lower scaling condition: There exist $\\alpha >0$ and $C_L>0$ such that $\\psi (\\lambda r)\\ge C_L\\lambda ^\\alpha \\psi (r),\\qquad \\mbox{ for all }\\lambda \\ge 1\\mbox{ and }r>0$ (see [11] and the subsequent list of examples in [11]).", "Then, by [11] (see also [5]), there exists a constant $C\\ge 1$ such that, for all $r>0$ and $0<\\gamma \\le 1$ , $\\frac{C^{-1}}{r^{d} \\psi (1/r)} \\le &g(r)&\\le \\frac{C}{r^{d} \\psi (1/r)} ,\\\\[1.5mm]C^{-1}\\gamma ^{2-d} g(r)\\le &g(\\gamma r)&\\le C\\gamma ^{\\alpha -d} g(r).", "$ By Lemma REF ,1 and the second inequality of (), (UD) holds on $(0,\\infty )$ .", "Replacing, in the first inequality of (), $r$ by $\\lambda r$ and $\\gamma $ by $1/\\lambda $ , we see that $g(\\lambda r)\\le C \\lambda ^{2-d} g(r)$ , for all $r>0$ and $\\lambda \\ge 1$ .", "Hence (UD) holds on $(0,\\infty )$ , by Lemma REF ,1, provided $d\\ge 3$ .", "For the case $d\\le 2$ , see [11].", "Further, since the transition kernels $P_t$ are given by convolution with positive functions $p_t$ (see, for example, [9]) satisfying $p_s\\ast p_t=p_{s+t}$ , $s,t>0$ , we have $G_0=\\int _0^\\infty p_t\\, dt \\in E_{\\mathbb {P}}.$ Moreover, the separation condition (1.6) in [11] is our separation condition (REF ).", "Now let us look at a subordinate Brownian semigroup, where $\\alpha \\in (0,2]$ , $\\alpha <d$ , $0<\\delta \\le 1$ , and the Laplace exponent of the subordinator is $\\phi (\\lambda )=\\ln ^\\delta (1+\\lambda ^{\\alpha /2}).$ If $\\delta =1$ , then, by [13]), $g(r)\\approx r^{-d}\\ln ^{-2} (1/r) \\mbox{ as } r\\rightarrow 0 \\quad \\mbox{ and }\\quad g(r)\\approx r^{\\alpha -d} \\mbox{ as } r\\rightarrow \\infty ,$ and (LD) certainly does not hold with $R_0=0$ , since, for $r>0$ , $\\int _0^r s^{-1}\\ln ^{-2}(1/s)\\,ds=\\ln ^{-1}(1/r).$ In the general caseThe author is indebted to T. Grzywny for informations in this case., we have $\\phi ^{\\prime }(\\lambda )/\\phi ^2(\\lambda )\\approx \\lambda ^{-1}\\ln ^{-(1+\\delta )}(\\lambda )$ .", "By [10], we obtain that $g(r) \\approx r^{-d-2} \\phi ^{\\prime }(r^{-2})/\\phi ^2(r^{-2}) \\approx r^{-d} \\ln ^{-(1+\\delta )} (1/r) \\quad \\mbox{ as }r\\rightarrow 0.$ Further, by [12], $g(r)\\approx r^{\\delta \\alpha -d}$ as $r\\rightarrow \\infty $ .", "Thus, by Lemmas REF and REF , our assumptions in Section 1 are satisfied taking any $R_0,R_1\\in (0,\\infty )$ ." ], [ "Potential theory of $\\mathbb {P}$", "For the moment, let us assume that the right continuous semigroup $\\mathbb {P}$ is only sub-Markov and that, instead of (LD) and (UD), $\\int _0^1 r^{d-1}g(r)\\,dr<\\infty \\quad \\mbox{ and }\\quad \\lim \\nolimits _{r\\rightarrow \\infty } g(r)=0.$ Let $\\mathcal {B}({\\mathbb {R}^d})$ (${\\mathbb {R}^d})$ , respectively) denote the set of all Borel measurable numerical functions (continuous real functions, respectively) on ${\\mathbb {R}^d}$ .", "We recall that the potential kernel $V_0=\\int _0^\\infty P_t\\,dt$ is given by $V_0 f(x):=G_0\\ast f (x)=\\int G_0(x-y) f(y)\\,dy,\\qquad f\\in \\mathcal {B}^+({\\mathbb {R}^d}), \\, x\\in {\\mathbb {R}^d}.$ Let $E_{\\mathbb {P}}$ denote the set of all $\\mathbb {P}$ -excessive functions, that is, $E_{\\mathbb {P}}$ is the set of all $v\\in \\mathcal {B}^+(X)$ such that $\\sup _{t>0} P_t v=v$ .", "We note that $V_0(\\mathcal {B}^+({\\mathbb {R}^d}))\\subset E_{\\mathbb {P}}$ .", "If $f\\in \\mathcal {B}^+({\\mathbb {R}^d})$ is bounded and has compact support, then $V_0f\\in {\\mathbb {R}^d})$ and $V_0f$ vanishes at infinity, by (REF ).", "This leads to the following results in [8] (for the definition of balayage spaces and their connection with sub-Markov semigroups see [2], [6], or [8]).", "THEOREM 3.1 $({\\mathbb {R}^d},E_{\\mathbb {P}})$ is a balayage space such that every point in ${\\mathbb {R}^d}$ is polar and Borel measurable finely open sets $U\\ne \\emptyset $ have strictly positive Lebesgue measure.", "REMARK 3.2 1.", "There exists a  Hunt process $\\mathfrak {X}=(\\Omega ,\\mathfrak {M}, \\mathfrak {M}_t, X_t, \\theta _t, P^x)$ on ${\\mathbb {R}^d}$ with transition semigroup $\\mathbb {P}$ (see [2]).", "2.", "Every open ball $B(x,r)$ , $x\\in {\\mathbb {R}^d}$ , $r>0$ , is finely dense in the closed ball $\\overline{B}(x,r)$ (see [8]; the fine topology is the coarsest topology such that every function in $E_{\\mathbb {P}}$ is continuous).", "For every subset $A$ of ${\\mathbb {R}^d}$ , we have a reduced function $R_1^A$ : $R_1^A:=\\inf \\lbrace v\\in E_{\\mathbb {P}}\\colon v\\ge 1\\mbox{ on } A\\rbrace .$ Obviously, $R_1^A\\le 1$ , since $1\\in E_{\\mathbb {P}}$ .", "Hence $R_1^A=1$ on $A$ .", "If $A$ is open, then $R_1^A\\in E_{\\mathbb {P}}$ .", "For a general subset $A$ , the greatest lower semicontinuous minorant $\\hat{R}_1^A$ of $R_1^A$ (which is also the greatest finely lower semicontinuous minorant of $R_1^A$ ) is contained in $E_{\\mathbb {P}}$ .", "It is known that $\\hat{R}_1^A=R_1^A$ on $A^c$ .", "If $A$  is Borel measurable, then, for every $x\\in {\\mathbb {R}^d}$ , ${ R_1^A(x)= P^x[T_A<\\infty ]}$ (see [2]).", "The zero-one law (REF ) will be the key to our proofs of both Theorem REF and Corollary REF .", "PROPOSITION 3.3 Suppose that $\\mathbb {P}$ is a Markov semigroup.", "Then the constant function 1 is harmonic and, for each set $A$ in ${\\mathbb {R}^d}$ , $R_1^A=1 \\quad \\mbox{ or }\\quad \\inf \\nolimits _{x\\in {\\mathbb {R}^d}} R_1^A(x)=0.$ Having $P_t1=1$ , for every $t>0$ , we know, by [8]), that 1 is harmonic.", "Moreover, by [8]), (REF ) holds.", "To illustrate that (REF ) is almost trivial, let us suppose that $R_1^A\\in E_{\\mathbb {P}}$ (which is true in our applications) and let $\\gamma :=\\inf _{x\\in {\\mathbb {R}^d}} R_1^A(x)$ .", "Since $E_{\\mathbb {P}}$  is a cone, we trivially have $R_{1-\\gamma }^A=(1-\\gamma )R_1^A\\in E_{\\mathbb {P}}$ .", "So $v:=\\gamma +R_{1-\\gamma }^A\\in E_{\\mathbb {P}}$ and $v= 1$ on $A$ , hence $v\\ge R_1^A$ .", "Moreover, $w:=R_1^A-\\gamma \\in E_{\\mathbb {P}}$ and $w= 1-\\gamma $ on $A$ , hence $w\\ge R_{1-\\gamma }^A$ .", "Therefore $R_1^A=v=\\gamma +R_{1-\\gamma }^A=\\gamma +(1-\\gamma ) R_1^A$ .", "Thus $\\gamma R_1^A=\\gamma $ , that is, $\\gamma =0$ or $R_1^A=1$ .", "For all $x,y\\in {\\mathbb {R}^d}$ , let $G_y(x):= G(x,y):= G_0(x-y),$ and let us recall that, by definition, a potential is a positive superharmonic function with greatest harmonic minorant 0.", "The next result is essentially [8].", "THEOREM 3.4 1.", "The function $G$ is symmetric and continuous.", "2.", "For every $y\\in {\\mathbb {R}^d}$ , $G_y$ is a potential with superharmonic support $\\lbrace y\\rbrace $ .", "3.", "If $\\mu $ is a measure on ${\\mathbb {R}^d}$ with compact support, then $G\\mu :=\\int G_y\\,d\\mu (y)$ is a potential, and the support of $\\mu $ is the superharmonic support of $G\\mu $ .", "4.", "For every potential $p$ on ${\\mathbb {R}^d}$ , there exists a (unique) measure $\\mu $ on ${\\mathbb {R}^d}$ such that $p=G\\mu $ .", "For every ball $B$ let $|B|$ denote the Lebesgue measure of $B$ and let $\\lambda _B$ denote normalized Lebesgue measure on $B$ (the measure on $B$ having density $1/|B|$ with respect to Lebesgue measure).", "Let us now fix $R_0\\ge 0$ and assume that (REF ) holds, that is, for $r>R_0$ , $G\\lambda _{B(0,r)}(0) = \\frac{1}{ |B(0,r)|} \\int _{B(0,r)} G_y(0)\\, dy \\le C _G \\,g(r).$ Then, in fact (see [8]), $G\\lambda _{B(0,r)} \\le C_G g(r), \\quad \\mbox{ for every }r>R_0.$ Moreover, since $g(r/2)\\le g$ on $(0,r/2)$ and $d\\int _0^{r/2} s^{d-1}\\,ds=(r/2)^d$ , we see that there exists $1\\le C_D\\le 2^d C_G$ such that, for all $r>R_0$ , $g(r/2) \\le C_D g(r) \\qquad \\mbox{ (doubling property)}.$ To simplify our estimates, let us define, once and for all, ${ c:=\\max \\lbrace C_D,C_G\\rbrace .", "}$ If $B$ is an open ball, then $R_1^B=R_1^{\\overline{B}}$ is a continuous potential and, by Theorem REF , there exists a unique measure $\\mu $ on $\\overline{B}$ , the equilibrium measure for $B$ , such that $R_1^B=G\\mu $ .", "For measures $\\nu $ on ${\\mathbb {R}^d}$ , let $\\Vert \\nu \\Vert $ denote their total mass.", "The following holds (cf.", "[8] in the case $r_0=\\infty $ ; assuming $r>R_0$ its simple proof carries over word by word).", "PROPOSITION 3.5 Let $r>R_0$ , $B:=B(0,r)$ , and let $\\mu $ be the equilibrium measure for $B$ .", "Then $\\frac{g(|\\cdot |)}{g(r)}\\ge R_1^B\\ge c^{-1}\\, \\frac{g(|\\cdot |+r)}{g(r)}\\quad \\mbox{ and }\\quad c^{-1}\\, \\frac{1}{g(r)}\\le \\Vert \\mu \\Vert \\le c\\,\\frac{1}{g(r)}.$ The following well known fact will be used in the proofs of Proposition REF and Lemma REF .", "By (REF ), it is an immediate consequence of the strong Markov property.", "For the convenience of the reader we write down its short proof (a corresponding argument based on iterated balayage can be given using [2]).", "LEMMA 3.6 Let $A$ be a Borel measurable set in an open set $U\\subset {\\mathbb {R}^d}$ and $\\gamma >0$ such that $R_1^A\\le \\gamma $ on $U^c$ .", "Then $P^x[T_A<T_{U^c}]\\ge R_1^A(x)- \\gamma $ , for every $x\\in U$ .", "Let $\\tau :=T_{U^c}$ and $x\\in U$ .", "We obviously have the identity $[T_A<\\infty ]\\setminus [T_A<\\tau ] =[\\tau \\le T_A<\\infty ] = [\\tau \\le T_A]\\cap \\theta _\\tau ^{-1}( [ T_A<\\infty ]).$ Since $X_\\tau \\in {U^c}$ on $[\\tau <\\infty ]$ , the strong Markov property yields that $P^x([\\tau <T_A]\\cap \\theta _\\tau ^{-1}[ T_A<\\infty ])=\\int _{[\\tau <T_A]} P^{X_\\tau } [T_A<\\infty ]\\,dP^x\\le \\gamma ,$ and hence $P^x[T_A<\\infty ]-P^x[T_A<\\tau ]\\le \\gamma $ .", "For every $r>0$ , we introduce the (closed) shell $S(r):=\\overline{B}(0,3r)\\setminus B(0,r).$ The following estimate of the probability for hitting a shell $S(r)$ before leaving a ball $B(0,Mr)$ , $M$ large, will be sufficient for us (see [11], going back to [5], for a much stronger estimate which is used [11]).", "PROPOSITION 3.7 Let $r>R_0$ , $\\eta :=c^{-3}/2$ , $M > 3$ , and $g((M-2) r)\\le \\eta g(r)$ .", "Then $P^0[T_{S(r)} < T_{B(0,Mr)^c}] \\ge \\eta .$ We choose $z\\in \\partial B(0,2r)$ and take $B:=B(z,r)$ .", "Then $B$ is contained in $S(r)$ .", "By Proposition REF , $R_1^B(0)\\ge c^{-1}\\,\\frac{g(|z|+r)}{g(r)}= c^{-1}\\, \\frac{g(3r)}{g(r)}\\ge c^{-3}=2\\eta ,$ whereas, for every $y\\in B(0,Mr)^c$ , $R_1^B(y)\\le \\frac{g(|y-z|)}{g(r)}\\le \\frac{g((M-2)r)}{g(r)}\\le \\eta .$ The proof is finished by Lemma REF .", "The next simple result on comparison of potentials will be sufficient for us (see the proof of [8] for a much more delicate version; cf.", "also the proof of [1]).", "LEMMA 3.8 Let $Z\\subset {\\mathbb {R}^d}$ be finite and $r_z>R_0$ , $z\\in Z$ , such that, for $z\\ne z^{\\prime }$ , $B(z,r_z)\\cap B(z^{\\prime },3r_{z^{\\prime }})=\\emptyset $ .", "Let $w\\in E_{\\mathbb {P}}$ and, for every $z\\in Z$ , let $\\mu _z,\\nu _z$ be measures on $\\overline{B}(z,r_z)$ such that $G\\mu _z\\in {\\mathbb {R}^d})$ , $G\\mu _z\\le w$ , and $\\Vert \\mu _z\\Vert \\le \\Vert \\nu _z\\Vert $ .", "Then $\\mu :=\\sum _{z\\in Z} \\mu _z$ and $\\nu :=\\sum _{z\\in Z}\\nu _z$ satisfy $G\\mu \\le w+ c G\\nu .$ Let $z,z^{\\prime }\\in Z$ , $z^{\\prime }\\ne z$ , and $x\\in \\overline{B}(z,r_z)$ .", "For all $y,y^{\\prime }\\in \\overline{B}(z^{\\prime },r_{z^{\\prime }})$ , $|y-y^{\\prime }|\\le 2r_{z^{\\prime }}\\le |x-y^{\\prime }|$ , hence $R_0<r_{z^{\\prime }}<|x-y|\\le 2|x-y^{\\prime }|$ and $g(|x-y^{\\prime }|)\\le c g(|x-y|)$ .", "By integration, $G\\mu _{z^{\\prime }} (x)\\le c G\\nu _{z^{\\prime }}(x)$ .", "Therefore $G\\mu (x)= G\\mu _z(x)+\\sum \\nolimits _{z^{\\prime }\\in Z, z^{\\prime }\\ne z} G\\mu _{z^{\\prime }}(x)\\le w(x)+c G\\nu (x).$ Thus $G\\mu \\le w+c G\\nu $ on the union of the balls $\\overline{B}(z,r_z)$ , $z\\in Z$ .", "By the minimum principle [2], the proof is finished.", "REMARK 3.9 If each $G\\mu _z$ is only bounded by some potential in ${\\mathbb {R}^d})$ , but there exists $\\gamma >1$ such that $B(z,\\gamma r_z)\\cap B(z^{\\prime }, 3 r_{z^{\\prime }})=\\emptyset $ , whenever $z\\ne z^{\\prime }$ , then (REF ) holds for all $x\\in B(z,\\gamma r_z)$ , $z\\in Z$ , and (REF ) follows as well." ], [ "Proof of Theorem ", "From now on let us suppose that the assumptions introduced at the beginning of Section 1 are satisfied.", "We recall that in many cases (LD) and (UD) hold with $R_0=0$ and $R_1=0$ .", "If not, we may assume without loss of generality that $R_0$ and $R_1$ , respectively, while being strictly positive, are as small as we want.", "We prepare the proof of Theorem REF by a first application of Lemma REF .", "LEMMA 4.1 Let $\\rho >\\max \\lbrace R_0,R_1\\rbrace $ , $0<\\varepsilon \\le 1/4 $ .", "Let $Z$ be a finite subset of $S(\\rho )$ and $R_0<r_z\\le |z|/4$ , $z\\in Z$ , such that the balls $B(z,4r_z)$ are pairwise disjoint and $|z-z^{\\prime }| \\ge 4\\varepsilon |z| \\bigl ({g(|z|)}/{g(r_z)}\\bigr )^{1/d},\\qquad \\mbox{ whenever } z\\ne z^{\\prime }.$ Let $ C:= 1+ (4/\\varepsilon )^dc^3$ , $\\delta :=(2C c^4)^{-1}$ , $M>4$ , and suppose $g((M-3)\\rho )\\le \\delta g(\\rho )$ .", "Then the union $A$ of the balls $B(z,r_z)$ , $z\\in Z$ , satisfies $P^x[T_A<T_{B(0,M\\rho )^c}] \\ge \\delta \\sum \\nolimits _{z\\in Z} g(|z|)/g(r_z), \\qquad \\mbox{for every $x\\in B(0,3\\rho )$}.$ Let $B:=B(0,4\\rho )$ .", "By (REF ), $G\\lambda _B\\le c g(4 \\rho ).$ For $z\\in Z$ , let $\\tilde{r}_z:=\\max \\bigl \\lbrace r_z, \\varepsilon |z| \\bigl ({g(|z|)} /{g(r_z)}\\bigr )^{1/d}\\bigr \\rbrace $ so that $ B(z,\\tilde{r}_z)\\cap B(z^{\\prime },3\\tilde{r}_{z^{\\prime }})=\\emptyset $ , whenever $ z\\ne z^{\\prime }$ .", "For the moment, fix $z\\in Z$ .", "Since $\\max \\lbrace r_z,\\varepsilon |z|\\rbrace \\le |z|/4<\\rho $ and $g(|z|)/g(r_z)\\le 1$ , we know that $B(z,\\tilde{r}_z)\\subset B$ .", "Moreover, $\\tilde{r}_z^{-d}\\le \\varepsilon ^{-d} \\,\\frac{g(r_z)}{|z|^dg(|z|)}\\le \\varepsilon ^{-d} \\,\\frac{g(r_z)}{\\rho ^d g( 4\\rho )}.$ Let $\\mu _z$ be the equilibrium measure for $B(z,r_z)$ , that is, $G\\mu _z=R_1^{B(z,r_z)}$ .", "Then $\\Vert \\mu _z\\Vert \\le c g(r_z)^{-1}$ , by Proposition REF .", "We define $\\nu _z:= \\Vert \\mu _z\\Vert \\lambda _{B(z,\\tilde{r}_z)}=\\beta _z 1_{B(z,\\tilde{r}_z)} \\lambda _B,$ where, by (REF ), $\\beta _z= \\Vert \\mu _z\\Vert \\, \\frac{|B|}{|B(z,\\tilde{r}_z)|} \\le cg(r_z)^{-1}(4\\rho /\\tilde{r}_z)^d \\le (4/\\varepsilon )^{d} c g(4\\rho )^{-1}=:\\beta .$ Let $\\nu :=\\sum _{z\\in Z} \\nu _z$ .", "Since the balls $B(z,\\tilde{r}_z)$ , $z\\in Z$ , are pairwise disjoint subsets of $B$ , we conclude, by (REF ), (REF ), and (REF ), that $G\\nu \\le \\beta G\\lambda _B\\le (4/\\varepsilon )^d c^2.$ Next let $\\mu :=\\sum _{z\\in Z} \\mu _z$ so that $p:= \\sum \\nolimits _{z\\in Z} R_1^{B(z,r_z)} = G\\mu .$ By Lemma REF , $ G\\mu \\le 1+c G\\nu $ , and hence $p\\le C$ , by (REF ) and our definition of $C$ .", "Therefore, by the minimum principle [2], we obtain that $ C^{-1}p\\le R_1^{\\overline{A}}= R_1^{A}$ .", "Trivially, $R_1^{A}\\le p$ .", "Thus $C^{-1}p\\le R_1^{A}\\le p.$ Let $U:=B(0,M\\rho )$ and $z\\in Z$ .", "By Proposition REF , for $y\\in U^c$ , $g(r_z) R_1^{B(z,r_z)} (y) \\le g(|y-z|)\\le g((M-3)\\rho )\\le \\delta g(\\rho ),$ whereas, for every $x\\in B(0,3\\rho )$ , $g(r_z) R_1^{B(z,r_z)} (x) \\ge c^{-1}g(|x-z|+\\rho )\\ge c^{-1}g(7\\rho ) \\ge c^{-4} g(\\rho ).$ Defining $\\gamma :=\\sum \\nolimits _{z\\in Z} g(\\rho )/g(r_z)$ we hence see, by (REF ), that $R_1^{A} \\le \\delta \\gamma \\quad \\text{ on }U^c, \\qquad \\quad R_1^{A} \\ge 2\\delta \\gamma \\quad \\text{ on }B(0,3\\rho ).$ By Lemma REF , for every $x\\in B(0,3\\rho )$ , $P^x[T_{A}<T_{B(0,M\\rho )^c}] \\ge \\delta \\gamma .$ Observing that $g(\\rho )\\ge g$ on $S(\\rho )$ the proof is finished.", "Now let us fix a locally finite subset $Z$ of ${\\mathbb {R}^d}\\setminus \\lbrace 0\\rbrace $ and $r_z>4R_0$ , $z\\in Z$ , such that the balls $ B(z,r_z)$ are pairwise disjoint and satisfy the separation condition (REF ).", "Let $A$ denote the union of these balls.", "For a proof of Theorem REF we show the following.", "PROPOSITION 4.2 If $A$ is avoidable, then $\\sum _{z\\in Z} {g(|z|)} /{g(r_z)}<\\infty $ .", "So let us suppose that $A$ is avoidable.", "To prove that $\\sum _{z\\in Z} {g(|z|)} /{g(r_z)}<\\infty $ we may assume that $|z|>8R_0$ , for every $z\\in Z$ (we simply omit finitely many points from $Z$ ).", "Further, we may assume that the balls $B(z, 4r_z)$ are pairwise disjoint.", "Indeed, since $g(r)\\le g(r/4) \\le c^2 g(r)$ , $r>R_0$ , a replacement of $r_z$ by $r_z/4$ does neither affect (REF ) nor the convergence of $\\sum _{z\\in Z} {g(|z|)} /{g(r_z)}$ , and the new, smaller union is, of course, avoidable.", "Moreover, similarly as at the beginning of the proof of [11]), we may assume without loss of generality that $r_z\\le |z|/8, \\qquad \\mbox{ for every }z\\in Z.$ Indeed, replacing $r_z$ by $r_z^{\\prime }:=\\min \\lbrace r_z,|z|/8\\rbrace $ our assumptions are preserved as well.", "Suppose we have shown that $\\sum _{z\\in Z} {g(|z|)} /{g(r_z^{\\prime })}<\\infty $ .", "Since $g(|z|)/g(|z|/8)\\ge c^{-3}$ , we see that the set $Z^{\\prime }$ of all $z\\in Z$ such that $r_z^{\\prime }=|z|/8$ is finite, and hence certainly $\\sum _{z\\in Z^{\\prime }} g(|z|)/g(r_z)<\\infty $ .", "So we may assume without loss of generality that $r_z^{\\prime }=r_z$ , for all $z\\in Z$ , that is, (REF ) holds.", "By (REF ), we may choose $0<\\varepsilon < 1/4$ such that, for $z,z^{\\prime }\\in Z$ , $ z\\ne z^{\\prime }$ , $|z-z^{\\prime }| \\ge 8c^{1/d}\\varepsilon |z| \\bigl ({g(|z|)}/{g(r_z)}\\bigr )^{1/d}.$ As in Lemma REF , we define $C:= 1+ (4/\\varepsilon )^dc^3, \\qquad \\delta :=(2C c^4)^{-1}.$ By Lemma REF , there exists $M:=3^m$ , $m\\in \\mathbb {N}$ , such that $g((M-3)\\rho )\\le \\delta g(\\rho ),\\qquad \\mbox{ for every }\\rho > R_1.$ Moreover, let us define $R:=1+\\max \\lbrace R_0,R_1\\rbrace .$ By Proposition REF , there is a point $x_0$ in ${\\mathbb {R}^d}$ such that $P^{x_0}[T_A<\\infty ] = R_1^A(x_0)< \\delta /2.$ Deleting finitely many points from $Z$ , we obtain $Z\\cap B(0,2|x_0|+R)=\\emptyset $ .", "Then, for every $z\\in Z$ , $|z|/2 \\le |z-x_0|\\le 2|z|, \\qquad c^{-1}g(|z|)\\le g(|z-x_0|)\\le c g(|z|).$ Hence, by (REF ) and (REF ), $r_z<|z-x_0|/4$ and, for $ z,z^{\\prime }\\in Z$ , $ z\\ne z^{\\prime }$ , $|z-z^{\\prime }|\\ge 4\\varepsilon |z-x_0| \\bigl ( {g(|z-x_0|)}/{g(r_z)}\\bigr )^d.$ By translation invariance, we may therefore assume without loss of generality that $x_0=0$ , $Z\\cap B(0,R)=\\emptyset $ , and (REF ) holds instead of (REF ).", "For every $0\\le j<m$ , let $Z_j:=\\bigcup \\nolimits _{n=0}^\\infty Z\\cap S(3^{nm+j}R).$ Then $Z$ is the union of $Z_0,Z_1,\\dots , Z_{m-1}$ .", "Therefore it suffices to show that $\\sum \\nolimits _{z\\in Z_j} g(|z|)/g(r_z) <\\infty , \\qquad \\mbox{ for every } 0\\le j<m.$ So let us fix $0\\le j< m$ .", "For the moment, we also fix $n\\in \\lbrace 0,1,2,\\dots \\rbrace $ and define $ \\rho :=3^{nm+j}R$ , $S:=T_{S(\\rho )} , \\quad \\tau :=T_{B(0,\\rho )^c}, \\quad \\tau ^{\\prime }:=T_{B(0,M\\rho )^c}, \\quad T:=\\min \\lbrace T_A,\\tau ^{\\prime }\\rbrace .$ By Lemma REF , $P^y[T_A<\\tau ^{\\prime }]\\ge \\delta \\sum \\nolimits _{z\\in Z\\cap S(\\rho )} g(|z|)/g(r_z), \\quad \\mbox{ for every } y\\in S(\\rho ).$ By Proposition REF , $P^0[S<\\tau ^{\\prime }] \\ge \\delta $ , and hence, by (REF ), $P^0[S<T] \\ge P^0[S<\\tau ^{\\prime }]- P^0[T_A<\\infty ] >\\delta /2.$ Clearly, $S+T_A\\circ \\theta _{S}= T_A$ and $S+\\tau ^{\\prime }\\circ \\theta _{S}= \\tau ^{\\prime }$ on $[S<T] $ .", "Hence $[S<T_A<\\tau ^{\\prime }]=[S< T, T_A<\\tau ^{\\prime }]=[S< T]\\cap \\theta _{S}^{-1}([T_A<\\tau ^{\\prime }]).$ Since $X_{S}\\in S(\\rho )$ on $[S<\\infty ]$ , the strong Markov property yields that $P^0[S<T_A<\\tau ^{\\prime }]=\\int _{[S< T]} P^{X_{S}}[T_A<\\tau ^{\\prime }]\\,dP^0 \\ge (\\delta ^2/2) \\sum \\nolimits _{z\\in Z\\cap S(\\rho ) } \\frac{g(|z|)}{g(r_z)}.$ Of course, $\\tau \\le S$ .", "Hence the sets $[S<T_A<\\tau ^{\\prime }]$ , obtained for different $n$ , are pairwise disjoint subsets of $[T_A<\\infty ]$ (recall that $M=3^m$ ).", "Thus, by (REF ), $\\sum \\nolimits _{z\\in Z_j} g(|z|)/g(r_z)\\le (2/\\delta ^2) P^0[T_A<\\infty ] \\le 1/\\delta .$ Let us note that the preceding proof could also be presented in a purely analytic way using iterated balayage of measures." ], [ "Proof of Corollary ", "Again we suppose that the assumptions from the beginning of Section 1 are satisfied.", "Let $Z$ be a countable set in ${\\mathbb {R}^d}$ and $r_z>0$ , $z\\in Z$ , such that the balls $B(z,r_z)$ are pairwise disjoint and regularly located.", "So there exist $\\varepsilon ,R\\in (0,\\infty )$ such that the points in $Z$ have a mutual distance which is at least $\\varepsilon $ and every open ball of radius $R$ contains some point of $Z$ .", "Moreover, $r_z=\\phi (|z|)$ , where the function $\\phi $ is decreasing.", "If (LD) does not hold with $R_0=0$ , we assume that $\\kappa :=\\inf _{x\\in {\\mathbb {R}^d}} \\phi (x)>0$ .", "By Lemma REF , we then know that (LD) holds, if we define $R_0:=\\kappa /8$ .", "By Lemma REF , (UD) holds with, say, $R_1:=R_0+1$ .", "Of course, we may assume that $R\\ge 1+\\phi (1)$ .", "We already know (see Remark REF ,3 and Proposition REF ) that it suffices to show that the union $A$ of all $B(z,r_z)$ , $z\\in Z$ , is unavoidable provided $\\sum \\nolimits _{z\\in Z} g(|z|)/g(r_z)=\\infty .$ So let us suppose that (REF ) holds.", "Moreover, let us assume for the moment that $\\limsup \\nolimits _{\\rho \\rightarrow \\infty } \\rho ^dg(\\rho )/g(\\phi (\\rho ))<\\infty .$ Then $\\beta := \\inf \\nolimits _{z\\in Z} g(r_z)(|z|^d g(|z|))^{-1}>0$ .", "Since $|z-z^{\\prime }|\\ge \\varepsilon >0$ , whenever $z\\ne z^{\\prime }$ , this implies that $\\inf \\nolimits _{z,z^{\\prime }\\in Z, z\\ne z^{\\prime }} \\frac{|z-z^{\\prime }|^d}{|z|^d} \\, \\frac{g(r_z)}{g(|z|)} \\ge \\varepsilon ^d\\beta .$ Hence the balls $B(z,r_z)$ , $z\\in Z$ , satisfy the separation condition (REF ), and $A$  is unavoidable, by Theorem REF .", "Thus already the following lemma would finish the proof of Corollary REF .", "LEMMA 5.1 If $\\limsup _{\\rho \\rightarrow \\infty } \\rho ^dg(\\rho )/g(\\phi (\\rho ))=\\infty $ , the set $A$ is unavoidable.", "For Lévy processes considered in [11], this is [11].", "However, its proof (by contradiction) is almost as involved as the proof of [11].", "By Proposition REF , we only have to show that $\\inf _{x\\in {\\mathbb {R}^d}} R_1^A(x)>0$ .", "Hence a second application of Lemma REF , which is yet another variation of the arguments for quasi-additivity of capacities in [1], will allow us even to prove the following.", "PROPOSITION 5.2 Suppose that $\\limsup _{\\rho \\rightarrow \\infty } \\rho ^dg(\\rho )/g(\\phi (\\rho ))>\\eta >0$ .", "Then the union $A$ of all $B(z,r_z)$ , $z\\in Z$ , is unavoidable.", "We define $a:= (2c)^{-1}(18R)^{-d} , \\qquad b:= c^2 R^{-d},$ and fix $x\\in {\\mathbb {R}^d}$ .", "There exists $\\rho > 9R +2|x|+4 R_1$ such that $\\gamma :=\\rho ^d g(\\rho )/g(\\phi (\\rho ))> \\eta .$ Let $r:=\\phi (\\rho ),\\qquad B:=B(0,\\rho ) \\quad \\mbox{ and }\\quad S:=\\overline{B}(0,\\rho /2)\\setminus B(\\rho /4).$ There exist finitely many points $y_1,\\dots ,y_m\\in S$ such that $B(y_1,3R),\\dots , B(y_m,3R)$ are pairwise disjoint and $S$ is covered by the balls $B(y_1,9R), \\dots , B(y_m,9R)$ .", "Obviously, $m\\ge (1/2) (\\rho /18R)^d$ .", "There exist points $z_j\\in Z\\cap B(y_j,R)$ , $1\\le j\\le m$ .", "Then, for all $i,j\\in \\lbrace 1,\\dots , m\\rbrace $ with $i\\ne j$ , $|z_i-z_j|\\ge |y_i-y_j|-2R\\ge 4R$ , and hence $B(z_i,R)\\cap B(z_j,3R)=\\emptyset .$ Let $1\\le j\\le m$ .", "Clearly, $\\rho \\ge \\rho /2+R\\ge |z_j| \\ge \\rho /4-R\\ge R\\ge 1$ , and hence $r=\\phi (\\rho )\\le \\phi (|z_j|)=r_{z_j} \\le \\phi (1)\\le R.$ Moreover, $r+|x-z_j| \\le R+|x|+ \\rho /2+R\\le \\rho $ , and hence $g(|x-z_j|+r)\\ge g(\\rho )$ .", "So, by translation invariance and Proposition REF , $R_1^{B(z_j,r)}(x)\\ge c^{-1}g(|x-z_j|+r)/g(r) \\ge c^{-1}g(\\rho )/g(r).$ Let $A_x:=B(z_1,r)\\cup \\dots \\cup B(z_m,r) \\quad \\mbox{ and }\\quad p:= \\sum \\nolimits _{j=1}^m R_1^{B(z_j,r)}.$ Then $A_x\\subset A$ , by (REF ).", "So, by (REF ) and our definitions of $\\gamma $ , $a$ , and $r$ , $R_1^{A_x}\\le R_1^A \\quad \\mbox{ and }\\quad p(x) \\ge m c^{-1}g(\\rho )/g(r ) \\ge a\\gamma .$ Now let $\\mu _0$ denote the equilibrium measure for $B(0,r)$ , $G\\mu _0=R_1^{B(0,r)}$ .", "By Proposition REF , $\\Vert \\mu _0\\Vert \\le c g(r)^{-1}$ .", "We define $\\nu _j:=\\Vert \\mu _0\\Vert \\lambda _{B(z_j,R)}=\\Vert \\mu _0\\Vert (\\rho /R)^d 1_{B(z_j,R)} \\lambda _B, \\qquad 1\\le j\\le m,$ and $\\nu :=\\sum _{j=1}^m \\nu _j$ .", "Since $B(z_1,R), \\dots , B(z_j, R)$ are pairwise disjoint subsets of $B$ and $G\\lambda _B\\le c g(\\rho )$ , we see that $G\\nu \\le \\Vert \\mu _0\\Vert (\\rho /R)^d G\\lambda _B \\le c^2 R^{-d}g(r)^{-1}\\rho ^d g(\\rho ) = b\\gamma .$ For every $1\\le j\\le m$ , $R_1^{B(z_j,r)}=G\\mu _j$ , where $\\mu _j$ is obtained from $\\mu _0$ translating by $z_j$ .", "Let $\\mu :=\\sum _{j=1}^m \\mu _j$ .", "By (REF ), (REF ), and Lemma REF , $p= G\\mu \\le 1+c G\\nu \\le 1+c b\\gamma .$ Since $\\mu $ is supported by $\\overline{A}_x$ and $p$ is continuous, we get that $R_1^{A_x}=R_1^{\\overline{A}_x}\\ge (1+c b\\gamma )^{-1}p,$ by the minimum principle [2].", "In particular, $R_1^A(x)\\ge R_1^{A_x}(x)\\ge \\frac{a\\gamma }{1+cb\\gamma }=\\frac{a}{\\gamma ^{-1}+cb} > \\frac{a}{\\eta ^{-1}+c b}\\,$ by (REF ) and (REF ).", "Thus $A$ is unavoidable, by Proposition REF ." ] ]
1403.0076
[ [ "G-valued crystalline representations with minuscule p-adic Hodge type" ], [ "Abstract We study G-valued semi-stable Galois deformation rings where G is a reductive group.", "We develop a theory of Kisin modules with G-structure and use this to identify the connected components of crystalline deformation rings of minuscule p-adic Hodge type with the connected components of moduli of \"finite flat models with G-structure.\"", "The main ingredients are a construction in integral p-adic Hodge theory using Liu's theory of $(\\varphi, \\widehat{G})$-modules and the local models constructed by Pappas and Zhu." ], [ "Overview", "One of the principal challenges in the study of modularity lifting or more generally automorphy lifting via the techniques introduced in Taylor-Wiles [46] is understanding local deformation conditions at $\\ell = p$ .", "In [24], Kisin introduced a ground-breaking new technique for studying one such condition, flat deformations, which led to better modularity lifting theorems.", "[26] extends those techniques to construct potentially semistable deformation rings with specified Hodge-Tate weights.", "In this paper, we study Galois deformations valued in a reductive group $G$ and extend Kisin's techniques to this setting.", "In particular, we define and prove structural results about “flat” $G$ -valued deformations.", "Let $G$ be a reductive group over a $\\mathbb {Z}_p$ -finite flat local domain $\\Lambda $ with connected fibers.", "Let $\\mathbb {F}$ be the residue field of $\\Lambda $ and $F := \\Lambda [1/p]$ .", "Let $K/\\mathbb {Q}_p$ be a finite extension with absolute Galois group $\\Gamma _K$ and fix a representation $\\overline{\\eta }:\\Gamma _K \\rightarrow G(\\mathbb {F})$ .", "The (framed) $G$ -valued deformation functor is represented by a complete local Noetherian $\\Lambda $ -algebra $R^{\\square }_{G, \\overline{\\eta }}$ .", "For any geometric cocharacter $\\mu $ of $\\mathrm {Res}_{(K \\otimes _{\\mathbb {Q}_p} F)/F} G_F$ , there exists a quotient $R^{\\operatorname{st}, \\mu }_{\\overline{\\eta }}$ (resp.", "$R^{\\operatorname{cris}, \\mu }_{\\overline{\\eta }}$ ) of $R^{\\square }_{G, \\overline{\\eta }}$ whose points over finite extensions $F^{\\prime }/F$ are semi-stable (resp.", "crystalline) representations with $p$ -adic Hodge type $\\mu $ (see [1]).", "When $G = \\mathrm {GL}_n$ and $\\mu $ is minuscule, $R^{\\operatorname{cris}, \\mu }_{\\overline{\\eta }}$ is a quotient of a flat deformation ring.", "For modularity lifting, it is important to know the connected components of $\\mathrm {Spec}\\ R^{\\operatorname{cris}, \\mu }_{\\overline{\\eta }}[1/p]$ .", "Intuitively, Kisin's technique introduced in [24] is to resolve the flat deformation ring by “moduli of finite flat models” of deformations of $\\overline{\\eta }$ .", "When $K/\\mathbb {Q}_p$ is ramified, the resolution is not smooth, but its singularities are relatively mild, which allowed for the determination of the connected components in many instances when $G = \\mathrm {GL}_2$ [24].", "Kisin's technique extends beyond the flat setting (for $\\mu $ arbitrary) where one resolves deformation rings by moduli spaces of integral $p$ -adic Hodge theory data called $\\mathfrak {S}$ -modules of finite height also known as Kisin modules.", "In this paper, we define a notion of Kisin module with $G$ -structure or as we call them $G$ -Kisin modules (Definition REF ), and we construct a resolution $\\Theta :X^{\\operatorname{cris}, \\mu }_{\\bar{\\eta }} \\rightarrow \\mathrm {Spec}\\ R^{\\operatorname{cris}, \\mu }_{\\overline{\\eta }}$ where $\\Theta $ is a projective morphism and $\\Theta [1/p]$ is an isomorphism (see Propositions REF , REF ) .", "The same construction works for $R^{\\operatorname{st}, \\mu }_{\\overline{\\eta }}$ as well.", "The goal then is to understand the singularities of $X^{\\operatorname{cris}, \\mu }_{\\bar{\\eta }}$ .", "The natural generalization of the flat condition for $\\mathrm {GL}_n$ to an arbitrary group $G$ is minuscule $p$ -adic Hodge type $\\mu $ .", "A cocharacter $\\mu $ of a reductive group $H$ is minuscule if its weights when acting on $\\operatorname{Lie}H$ lie in $\\lbrace -1, 0, 1\\rbrace $ (see Definition REF and discussion afterward).", "Our main theorem is a generalization of the main result of [24] on the geometry of $X^{\\operatorname{cris}, \\mu }_{\\bar{\\eta }}$ for $G$ reductive and $\\mu $ minuscule: Theorem REF Assume $p \\nmid \\pi _1(G^{\\mathrm {der}})$ where $G^{\\mathrm {der}}$ is the derived subgroup of $G$ .", "Let $\\mu $ be a minuscule geometric cocharacter of $\\mathrm {Res}_{(K \\otimes _{\\mathbb {Q}_p} F)/F} G_F$ .", "Then $X^{\\operatorname{cris}, \\mu }_{\\overline{\\eta }}$ is normal and $X^{\\operatorname{cris}, \\mu }_{\\overline{\\eta }} \\otimes _{\\Lambda _{[\\mu ]}} \\mathbb {F}_{[\\mu ]}$ is reduced where $\\Lambda _{[\\mu ]}$ is the ring of integers of the reflex field of $\\mu $ .", "When $G = \\mathrm {GSp}_{2g}$ , this is a result of Broshi [7]; also, this is a stronger result than in [27] where we made a more restrictive hypothesis on $\\mu $ (see Remark REF ).", "The significance of Theorem REF is that it allows one to identify the connected components of $\\mathrm {Spec}\\ R^{\\operatorname{cris}, \\mu }_{\\overline{\\eta }}[1/p]$ with the connected components of the fiber in $X^{\\operatorname{cris}, \\mu }_{\\overline{\\eta }}$ over the closed point of $\\mathrm {Spec}\\ R^{\\operatorname{cris}, \\mu }_{\\overline{\\eta }}$ , a projective scheme over $\\mathbb {F}_{[\\mu ]}$ (see Corollary REF ).", "This identification led to the successful determination of the connected components of $\\mathrm {Spec}\\ R^{\\operatorname{cris}, \\mu }_{\\overline{\\eta }}[1/p]$ in the case when $G = \\mathrm {GL}_2$ ([24], [16], [21], [22], [20]).", "Outside of $\\mathrm {GL}_2$ , relatively little is known about the connected components of these deformations rings without restricting the ramification in $K$ .", "When $K/\\mathbb {Q}_p$ is unramified, we have a stronger result: Theorem REF Assume $K/\\mathbb {Q}_p$ is unramified, $p > 3$ , and $p \\nmid \\pi _1(G^{\\mathrm {ad}})$ .", "Then the universal crystalline deformation ring $R^{\\operatorname{cris}, \\mu }_{\\overline{\\eta }}$ is formally smooth over $\\Lambda _{[\\mu ]}$ .", "In particular, $\\mathrm {Spec}\\ R^{\\operatorname{cris}, \\mu }_{\\overline{\\eta }}[1/p]$ is connected.", "Remark 1.1.1 In [27], we made the assumption on the cocharacter $\\mu $ that there exists a representation $\\rho :G \\rightarrow \\mathrm {GL}(V)$ such that $\\rho \\circ \\mu $ is minuscule.", "This extra hypothesis on $\\mu $ excluded most adjoint groups like $\\mathrm {PGL}_n$ as well as exceptional types like $E_6$ and $E_7$ both of which have minuscule cocharacters.", "One can weaken the assumptions in Theorem REF if one assumes this stronger condition on $\\mu $ .", "Remark 1.1.2 The groups $\\pi _1(G^{\\mathrm {der}})$ and $\\pi _1(G^{\\mathrm {ad}})$ appearing in Theorems REF and REF are the fundamental groups in the sense of semisimple groups.", "Note that $\\pi _1(G^{\\mathrm {der}})$ is a subgroup of $\\pi _1(G^{\\mathrm {ad}})$ .", "The assumption that $p \\nmid \\pi _1(G^{\\mathrm {der}})$ insures that the local models we use have nice geometric property.", "The stronger assumption in Theorem REF that $p \\nmid \\pi _1(G^{\\mathrm {ad}})$ is probably not necessary and is a byproduct of the argument which involves reduction to the adjoint group.", "There are two main ingredients in the proof of Theorem REF and its applications, one coming from integral $p$ -adic Hodge theory and the other from local models of Shimura varieties.", "In Kisin's original construction, a key input was an advance in integral $p$ -adic Hodge theory, building on work of Breuil, which allows one to describe finite flat group schemes over $\\mathcal {O}_K$ in terms of certain linear algebra objects called Kisin modules of height in $[0,1]$ ([24], [25]).", "More precisely, then, $X^{\\operatorname{cris}, \\mu }_{\\overline{\\eta }}$ is a moduli space of $G$ -Kisin modules with “type” $\\mu $ .", "Intuitively, one can imagine $X^{\\operatorname{cris}, \\mu }_{\\overline{\\eta }}$ as a moduli of finite flat models with additional structure.", "The proof of Theorem REF uses a recent advance of Liu [30] in integral $p$ -adic Hodge theory to overcome a difficulty in identifying the local structure of $X^{\\operatorname{cris}, \\mu }_{\\overline{\\eta }}$ .", "Heuristically, the difficulty arises because for a general group $G$ one cannot work only in the setting of Kisin modules of height in $[0,1]$ where one has a nice equivalence of categories between finite flat group schemes and the category of Kisin modules with height in $[0,1]$ .", "Beyond the height in $[0,1]$ situation, the Kisin module only remembers the Galois action of the subgroup $\\Gamma _{\\infty } \\subset \\Gamma _K$ which fixes the field $K( \\pi ^{1/p}, \\pi ^{1/p^2}, \\ldots )$ for some compatible system of $p$ -power roots of uniformizer $\\pi $ of $K$ .", "Liu [30] introduced a more complicated linear algebra structure on a Kisin module, called a $(\\varphi , \\widehat{G})$ -module, which captures the action of the full Galois group $\\Gamma _K$ .", "We call them $(\\varphi , \\widehat{\\Gamma })$ -modules to avoid confusion with the group $G$ .", "Let $A$ be a finite local $\\Lambda $ -algebra which is either Artinian or flat.", "Our principal result (Theorem REF ) says roughly that if $\\rho :\\Gamma _{\\infty } \\rightarrow G(A)$ has “type” $\\mu $ , i.e., comes from a $G$ -Kisin module $(\\mathfrak {P}_A, \\phi _A)$ over $A$ of type $\\mu $ with $\\mu $ minuscule, then there exists a canonical extension $\\widetilde{\\rho }:\\Gamma _K \\rightarrow G(A)$ and furthermore if $A$ is flat over $\\mathbb {Z}_p$ then $\\widetilde{\\rho }[1/p]$ is crystalline.", "This is rough in the sense that what we actually prove is an isomorphism of certain deformation functors.", "As a consequence, we get that the local structure of $X^{\\operatorname{cris}, \\mu }_{\\overline{\\eta }}$ at a point $(\\mathfrak {P}_{\\mathbb {F}^{\\prime }}, \\phi _{\\mathbb {F}^{\\prime }}) \\in X^{\\operatorname{cris}, \\mu }_{\\overline{\\eta }}(\\mathbb {F}^{\\prime })$ is smoothly equivalent to the deformation groupoid $D^{\\mu }_{\\mathfrak {P}_{\\mathbb {F}^{\\prime }}}$ of $\\mathfrak {P}_{\\mathbb {F}^{\\prime }}$ with type $\\mu $ .", "To prove Theorem REF , one studies the geometry of $D^{\\mu }_{\\mathfrak {P}_{\\mathbb {F}^{\\prime }}}$ .", "Here, the key input comes from the theory of local models of Shimura varieties.", "A local model is a projective scheme $X$ over the ring of integers of a $p$ -adic field $F$ such that $X$ is supposed to étale-locally model the integral structure of a Shimura variety.", "Classically, local models were built out of moduli spaces of linear algebra structures.", "Rapoport and Zink [38] formalized the theory of local models for Shimura varieties of PEL-type.", "Subsequent refinements of these local models were studied mostly on a case by case basis by Faltings, Görtz, Haines, Pappas, and Rapoport, among others.", "Pappas and Zhu [37] define for any triple $(G, P, \\mu )$ , where $G$ is a reductive group over $F$ (which splits over a tame extension), $P$ is a parahoric subgroup, and $\\mu $ is any cocharacter of $G$ , a local model $M(\\mu )$ over the ring of integers of the reflex field of $\\mu $ .", "Their construction, unlike previous constructions, is purely group-theoretic, i.e.", "it does not rely on any particular representation of $G$ .", "They build their local models inside degenerations of affine Grassmanians extending constructions of Beilinson, Drinfeld, Gaitsgory, and Zhu to mixed characteristic.", "The geometric fact we will use is that $M(\\mu )$ is normal with special fiber reduced ([37]).", "The significance of local models in this paper is that the singularities of $X^{\\operatorname{cris}, \\mu }_{\\overline{\\eta }}$ are smoothly equivalent to those of a local model $M(\\mu )$ for the Weil-restricted group $\\mathrm {Res}_{(K \\otimes _{\\mathbb {Q}_p} F)/F} G_F$ .", "This equivalence comes from a diagram of formally smooth morphisms (REF ): $ {& \\widetilde{D}^{(\\infty ), \\mu }_{\\mathfrak {P}_{\\mathbb {F}}} [dl] [dr] & \\\\D^{\\mu }_{\\mathfrak {P}_{\\mathbb {F}}} & & \\overline{D}^{\\mu }_{Q_{\\mathbb {F}}}, \\\\}$ which generalizes constructions from [24] and [35].", "The deformation functor $\\overline{D}^{\\mu }_{Q_{\\mathbb {F}}}$ is represented by the completed local ring at an $\\mathbb {F}$ -point of $M(\\mu )$ .", "Intuitively, the above modification corresponds to adding a trivialization to the $G$ -Kisin module and then taking the “image of Frobenius.” We construct the diagram (REF ) in §3 with no assumptions on the cocharacter $\\mu $ (to be precise $D^{\\mu }_{\\mathfrak {P}_{\\mathbb {F}}}$ is deformations of type $\\le \\mu $ in general).", "It is intriguing to wonder whether $D^{\\mu }_{\\mathfrak {P}_{\\mathbb {F}}}$ and diagram (REF ) has any relevance to studying higher weight Galois deformation rings, i.e., when $\\mu $ is not minuscule.", "As a remark, we usually cannot apply [37] directly since the group $\\mathrm {Res}_{(K \\otimes _{\\mathbb {Q}_p} F)/F} G$ will generally not split over a tame extension.", "In [27], we develop a theory of local models following Pappas and Zhu's approach but adapted to these Weil-restricted groups (for maximal special parahoric level).", "These results are reviewed in §3.2 and are studied in more generality in [28].", "We now a give brief outline of the article.", "In §2, we define and develop the theory of $G$ -Kisin modules and construct resolutions of semi-stable and crystalline $G$ -valued deformation rings (REF , REF ).", "This closely follows the approach of [26].", "The proof that “semi-stable implies finite height” (Proposition REF ) requires an extra argument not present in the $\\mathrm {GL}_n$ -case (Lemma REF ).", "In §3, we study the relationship between deformations of $G$ -Kisin modules and local models.", "We construct the big diagram (Theorem REF ) and then impose the $\\mu $ -type condition to arrive at the diagram (REF ).", "We also give an initial description of the local structure of $X^{\\operatorname{cris}, \\mu }_{\\overline{\\eta }}$ in Corollary REF .", "§4.1 develops the theory of $(\\varphi , \\widehat{\\Gamma })$ -modules with $G$ -structure, and §4.2 is devoted to the proof of our key result (Theorem REF ) in integral $p$ -adic Hodge theory.", "In the last section §4.3, we prove Theorems REF and REF which follow relatively formally from the results of §3.3 and §4.2." ], [ "Acknowledgments", "The enormous influence of the work of Mark Kisin and Tong Liu in this paper will be evident to the reader.", "This paper is based on the author's Stanford Ph.D. thesis advised by Brian Conrad to whom the author owes a debt of gratitude for his generous guidance and his feedback on multiple drafts of the thesis.", "It is a pleasure to thank Tong Liu, Xinwen Zhu, George Pappas, Bhargav Bhatt, and Mark Kisin for many helpful discussions and exchanges related to this project.", "The author is additionally grateful for the support of the National Science Foundation and the Department of Defense in the form of NSF and NDSEG fellowships.", "Part of this work was completed while the author was a visitor at the Institute for Advanced Study supported by the National Science Foundation grant DMS-1128155.", "The author is grateful to the IAS for its support and hospitality.", "Finally, we would like to thank the referee for a number of valuable suggestions." ], [ "Notations and conventions", "We take $F$ to be our coefficient field, a finite extension of $\\mathbb {Q}_p$ .", "Let $\\Lambda $ be the ring of integers of $F$ with residue field $\\mathbb {F}$ .", "Let $G$ be reductive group scheme over $\\Lambda $ with connected fibers and $\\mathop {\\mmlmultiscripts{\\operatorname{Rep}{\\Lambda }{\\mmlnone }\\mmlprescripts {\\mmlnone }{f}}}\\limits (G)$ the category of representations of $G$ on finite free $\\Lambda $ -modules.", "We will use $V$ to denote a fixed faithful representation of $G$ , i.e., $V \\in \\mathop {\\mmlmultiscripts{\\operatorname{Rep}{\\Lambda }{\\mmlnone }\\mmlprescripts {\\mmlnone }{f}}}\\limits (G)$ such that $G \\rightarrow \\mathrm {GL}(V)$ is a closed immersion.", "The derived subgroup of $G$ will be denoted by $G^{\\mathrm {der}}$ and its adjoint quotient by $G^{\\mathrm {ad}}$ .", "All $G$ -bundles will be with respect to the fppf topology.", "If $X$ is a $\\Lambda $ -scheme, then $\\operatorname{GBun}(X)$ will denote the category of $G$ -bundles on $X$ .", "We will denote the trivial $G$ -bundle by $\\mathcal {E}^0$ .", "For any $G$ -bundle $P$ on a $\\Lambda $ -scheme $X$ and any $W \\in \\mathop {\\mmlmultiscripts{\\operatorname{Rep}{\\Lambda }{\\mmlnone }\\mmlprescripts {\\mmlnone }{f}}}\\limits (G)$ , $P(W)$ will denote the pushout of $P$ with respect to $W$ (see discussion before Theorem REF ).", "Let $\\overline{F}$ be an algebraic closure of $F$ .", "For a linear algebraic $F$ -group $H$ , $X_*(H)$ will denote the group of geometric cocharacters, i.e., $\\mathrm {Hom}(\\operatorname{\\mathbb {G}_m}, H_{\\overline{F}})$ .", "For $\\mu \\in X_*(H)$ , $[\\mu ]$ will denote its conjugacy class .", "The reflex field $F_{[\\mu ]}$ of $[\\mu ]$ is the smallest subfield of $\\overline{F}$ over which the conjugacy class $[\\mu ]$ is defined.", "If $\\Gamma $ is a pro-finite group and $B$ is a finite $\\Lambda $ -algebra, then $\\mathop {\\mmlmultiscripts{\\operatorname{Rep}{B}{\\mmlnone }\\mmlprescripts {\\mmlnone }{f}}}\\limits (\\Gamma )$ will be the category of continuous representations of $\\Gamma $ on finite projective $B$ -modules where $B$ is given the $p$ -adic topology.", "More generally, $\\operatorname{GRep}_{B}(\\Gamma )$ will denote the category of pairs $(P, \\eta )$ where $P$ is a $G$ -bundle over $\\mathrm {Spec}\\ B$ and $\\eta :\\Gamma \\rightarrow \\mathrm {Aut}_G(P)$ is a continuous homomorphism.", "Let $K$ be a $p$ -adic field with rings of integers $\\mathcal {O}_K$ and residue field $k$ .", "Denote its absolute Galois group by $\\Gamma _K$ .", "We furthermore take $W := W(k)$ and $K_0 := W[1/p]$ .", "We fix a uniformizer $\\pi $ of $K$ and let $E(u)$ the minimal polynomial of $\\pi $ over $K_0$ .", "Our convention will be to work with covariant $p$ -adic Hodge theory functors so we take the $p$ -adic cyclotomic character to have Hodge-Tate weight $-1$ .", "For any local ring $R$ , we let $m_R$ denote the maximal ideal.", "We will denote the completion of $B$ with respect to a specified topology by $\\widehat{B}$ ." ], [ "Kisin modules with $G$ -structure", "In this section, we construct resolutions of Galois deformation rings by moduli spaces of Kisin modules (i.e.", "$\\mathfrak {S}$ -modules) with $G$ -structure.", "For $\\mathrm {GL}_n$ , this technique was introduced in [24] to study flat deformation rings.", "In [26], the same technique is used to construct potentially semi-stable deformation rings for $\\mathrm {GL}_n$ .", "Here we develop a theory of $G$ -Kisin modules (Definition REF ).", "In particular, in §2.4, we show the existence of a universal $G$ -Kisin module over these deformation rings (Theorem REF ) and relate the filtration defined by a $G$ -Kisin module to $p$ -adic Hodge type.", "One can construct $G$ -valued semi-stable and crystalline deformation rings with fixed $p$ -adic Hodge type without $G$ -Kisin modules [1].", "However, the existence of a resolution by a moduli space of Kisin modules allows for finer analysis of the deformation rings as is carried out in §4." ], [ "Background on $G$ -bundles", "All bundles will be for the fppf topology.", "For any $G$ -bundle $P$ on a $\\Lambda $ -scheme $X$ and any $W \\in \\mathop {\\mmlmultiscripts{\\operatorname{Rep}{\\Lambda }{\\mmlnone }\\mmlprescripts {\\mmlnone }{f}}}\\limits (G)$ , define $P(W) := P \\times ^G W = (P \\times W)/\\sim $ to be the pushout of $P$ with respect to $W$ .", "This is a vector bundle on $X$ .", "This defines a functor from $\\mathop {\\mmlmultiscripts{\\operatorname{Rep}{\\Lambda }{\\mmlnone }\\mmlprescripts {\\mmlnone }{f}}}\\limits (G)$ to the category $\\operatorname{Vec}_X$ of vector bundles on $X$ .", "Theorem 2.1.1 Let $G$ be a flat affine group scheme of finite type over $\\mathrm {Spec}\\ \\Lambda $ with connected fibers.", "Let $X$ be an $\\Lambda $ -scheme.", "The functor $P \\mapsto \\lbrace P(W)\\rbrace $ from the category of $G$ -bundles on $X$ to the category of fiber functors $($ i.e., faithful exact tensor functors$)$ from $\\mathop {\\mmlmultiscripts{\\operatorname{Rep}{\\Lambda }{\\mmlnone }\\mmlprescripts {\\mmlnone }{f}}}\\limits (G)$ to $\\operatorname{Vec}_X$ is an equivalence of categories.", "When the base is a field, this is a well-known result ([12]) in Tannakian theory.", "When the base is a Dedekind domain, see [8] or [27].", "We will also need the following gluing lemma for $G$ -bundles: Lemma 2.1.2 Let $B$ be any $\\Lambda $ -algebra.", "Let $f \\in B$ be a non-zero divisor and $G$ be a flat affine group scheme of finite type over $\\Lambda $ .", "The category of triples $(P_f, \\widehat{P}, \\alpha )$ , where $P_f \\in \\operatorname{GBun}(\\mathrm {Spec}\\ B_f)$ , $\\widehat{P} \\in \\operatorname{GBun}(\\mathrm {Spec}\\ \\widehat{B})$ , and $\\alpha $ is an isomorphism between $P_f$ and $\\widehat{P}$ over $\\mathrm {Spec}\\ \\widehat{B}_f$ , is equivalent to the category of $G$ -bundles on $B$ .", "This is a generalization of the Beauville-Laszlo formal gluing lemma for vector bundles.", "See [37] or [27].", "Let $i:H \\subset G$ be a flat closed $\\Lambda $ -subgroup.", "We are interested in the “fibers” of the pushout map $i_*:\\operatorname{HBun}\\rightarrow \\operatorname{GBun}$ carrying an $H$ -bundle $Y$ to the $G$ -bundle $Y \\times ^H G$ .", "Let $Q$ be a $G$ -bundle on a $\\Lambda $ -scheme $S$ .", "For any $S$ -scheme $X$ , define $\\mathrm {Fib}_Q(X)$ to be the category of pairs $(P, \\alpha )$ , where $P \\in \\operatorname{HBun}(X)$ and $\\alpha :i_*(P) \\cong Q_X$ is an isomorphism in $\\operatorname{GBun}(X)$ .", "A morphism $(P, \\alpha ) \\rightarrow (P^{\\prime }, \\alpha ^{\\prime })$ is a map $f:P \\rightarrow P^{\\prime }$ of $H$ -bundles such that $\\alpha ^{\\prime } \\circ i_*(f) \\circ \\alpha ^{-1}$ is the identity.", "Proposition 2.1.3 The category of $\\mathrm {Fib}_Q(X)$ has no non-trivial automorphisms for any $S$ -schemes $X$ .", "Furthermore, the underlying functor $|\\mathrm {Fib}_Q|$ is represented by the pushout $Q \\times ^G (G / H)$ .", "In particular, if $G / H$ is affine $($ resp.", "quasi-affine$)$ over $S$ then $|\\mathrm {Fib}_Q|$ is affine $($ resp.", "quasi-affine$)$ over $X$ .", "See [44] or [27].", "Proposition 2.1.4 Let $G$ be a smooth affine group scheme of finite type over $\\mathrm {Spec}\\ \\Lambda $ with connected fibers.", "Let $R$ any $\\Lambda $ -algebra and $I$ a nilpotent ideal of $R$ .", "For any $G$ -bundle $P$ on $\\mathrm {Spec}\\ R$ , $P$ is trivial if and only if $P \\otimes _R R/I$ is trivial.", "Let $R$ be any complete local $\\Lambda $ -algebra with finite residue field.", "Any $G$ -bundle on $\\mathrm {Spec}\\ R$ is trivial.", "For (1), because $G$ is smooth, $P$ is also smooth.", "Thus, $P(R) \\rightarrow P(R/I)$ is surjective.", "A $G$ -bundle is trivial if and only if it admits a section.", "Part (2) reduces to the case of $R = \\mathbb {F}$ using part (1).", "Lang's Theorem says that $H^1_{\\text{ét}} (\\mathbb {F}, G)$ is trivial for any smooth connected algebraic group over $\\mathbb {F}$ (see Theorem 4.4.17 [45])" ], [ "Definitions and first properties", "Let $K$ be a $p$ -adic field with rings of integers $\\mathcal {O}_K$ and residue field $k$ .", "Set $W := W(k)$ and $K_0 := W[1/p]$ .", "Recall Breuil/Kisin's ring $\\mathfrak {S}:= W[\\![u]\\!", "]$ and let $E(u) \\in W[u]$ be the Eisenstein polynomial associated to a choice of uniformizer $\\pi $ of $K$ which generates $K$ over $K_0$ .", "Fix a compatible system $\\lbrace \\pi ^{1/p}, \\pi ^{1/p^2}, \\ldots \\rbrace $ of $p$ -power roots of $\\pi $ and let $K_{\\infty } = K(\\pi ^{1/p}, \\pi ^{1/p^2}, \\ldots )$ .", "Set $\\Gamma _{\\infty } := \\mathrm {Gal}(\\overline{K}/K_{\\infty })$ .", "Let $\\mathcal {O}_{\\mathcal {E}}$ denote the $p$ -adic completion of $\\mathfrak {S}[1/u]$ .", "We equip both $\\mathcal {O}_{\\mathcal {E}}$ and $\\mathfrak {S}$ with a Frobenius endomorphism $\\varphi $ defined by taking the ordinary Frobenius lift on $W$ and $u \\mapsto u^p$ .", "For any $\\mathbb {Z}_p$ -algebra $B$ , let $\\mathcal {O}_{\\mathcal {E}, B} := \\mathcal {O}_{\\mathcal {E}} \\otimes _{\\mathbb {Z}_p} B$ and $\\mathfrak {S}_B := \\mathfrak {S}\\otimes _{\\mathbb {Z}_p} B$ .", "We equip both of these rings with Frobenii having trivial action on $B$ .", "Note that all tensor products are over $\\mathbb {Z}_p$ even though the group $G$ may only be defined over the $\\Lambda $ .", "Definition 2.2.1 Let $B$ be any $\\Lambda $ -algebra.", "For any $G$ -bundle on $\\mathrm {Spec}\\ \\mathcal {O}_{\\mathcal {E}, B}$ , we take $\\varphi ^*(P) := P \\otimes _{\\mathcal {O}_{\\mathcal {E}, B}, \\varphi } \\mathcal {O}_{\\mathcal {E}, B}$ to be the pullback under Frobenius.", "An $(\\mathcal {O}_{\\mathcal {E}, B}, \\varphi )$ -module with $G$ -structure is a pair $(P, \\phi _{P})$ where $P$ is a $G$ -bundle on $\\mathrm {Spec}\\ \\mathcal {O}_{\\mathcal {E}, B}$ and $\\phi _{P}:\\varphi ^*(P) \\cong P$ is an isomorphism.", "Denote the category of such pairs by $\\operatorname{GMod}^{\\varphi }_{\\mathcal {O}_{\\mathcal {E}, B}}$ .", "Remark 2.2.2 When $G = \\mathrm {GL}_d$ , $\\operatorname{GMod}^{\\varphi }_{\\mathcal {O}_{\\mathcal {E}, B}}$ is equivalent to the category of rank $d$ étale $(\\mathcal {O}_{\\mathcal {E}, B}, \\varphi )$ -modules via the usual equivalence between $\\mathrm {GL}_d$ -bundles and rank $d$ vector bundles.", "When $B$ is $\\mathbb {Z}_p$ -finite and Artinian, the functor $T_B$ defined by $T_B(M, \\phi ) = (M \\otimes _{\\mathcal {O}_{\\mathcal {E}}} \\mathcal {O}_{\\widehat{\\mathcal {E}}^{\\operatorname{un}}})^{\\phi = 1}$ induces an equivalence of categories between étale $(\\mathcal {O}_{\\mathcal {E}, B}, \\varphi )$ -modules (which are $\\mathcal {O}_{\\mathcal {E}, B}$ -projective) and the category of representations of $\\Gamma _{\\infty }$ on finite projective $B$ -modules (see [24]).", "A quasi-inverse is given by $\\underline{M}_B(V) := (V \\otimes _{\\mathbb {Z}_p} \\mathcal {O}_{\\widehat{\\mathcal {E}}^{\\operatorname{un}}})^{\\Gamma _{\\infty }}.$ This equivalence extends to algebras which are finite flat over $\\mathbb {Z}_p$ .", "Definition 2.2.3 For any profinite group $\\Gamma $ and any $\\Lambda $ -algebra $B$ , define $\\operatorname{GRep}_B(\\Gamma )$ to be the category of pairs $(P, \\eta )$ where $P$ is a $G$ -bundle over $\\mathrm {Spec}\\ B$ and $\\eta :\\Gamma \\rightarrow \\mathrm {Aut}_G(P)$ is a continuous homomorphism (where $B$ is given the $p$ -adic topology).", "In the $G$ -setting, $\\operatorname{GRep}_B(\\Gamma )$ will play the role of representation of $\\Gamma $ on finite projective $B$ -modules.", "We have the following generalization of $T_B$ : Proposition 2.2.4 Let $B$ be any $\\Lambda $ -algebra which is $\\mathbb {Z}_p$ -finite and either Artinian or $\\mathbb {Z}_p$ -flat.", "There exists an equivalence of categories $T_{G, B}:\\operatorname{GMod}^{\\varphi }_{\\mathcal {O}_{\\mathcal {E}, B}} \\rightarrow \\operatorname{GRep}_B(\\Gamma _{\\infty })$ with a quasi-inverse $\\underline{M}_{G, B}$ .", "Furthermore, for any finite map $B \\rightarrow B^{\\prime }$ and any $(P, \\phi _P) \\in \\operatorname{GMod}^{\\varphi }_{\\mathcal {O}_{\\mathcal {E}, B}}$ , there is a natural isomorphism $T_{G, B^{\\prime }}(P \\otimes _B B^{\\prime }) \\cong T_{G, B}(P) \\otimes _B B^{\\prime }.$ Using Theorem REF , we can give Tannakian interpretations of both $\\operatorname{GMod}^{\\varphi }_{\\mathcal {O}_{\\mathcal {E}, B}}$ and $\\operatorname{GRep}_B(\\Gamma _{\\infty })$ .", "The former is equivalent to the category $[\\mathop {\\mmlmultiscripts{\\operatorname{Rep}{\\Lambda }{\\mmlnone }\\mmlprescripts {\\mmlnone }{f}}}\\limits (G),\\operatorname{Mod}^{\\varphi , \\text{ét}}_{\\mathcal {O}_{\\mathcal {E}, B}}]^{\\otimes }$ of faithful exact tensor functors.", "The latter is equivalent to the category of faithful exact tensor functors from $\\mathop {\\mmlmultiscripts{\\operatorname{Rep}{\\Lambda }{\\mmlnone }\\mmlprescripts {\\mmlnone }{f}}}\\limits (G)$ to $\\mathop {\\mmlmultiscripts{\\operatorname{Rep}{B}{\\mmlnone }\\mmlprescripts {\\mmlnone }{f}}}\\limits (\\Gamma _{\\infty })$ .", "We define $T_{G, B}(P, \\phi _P)$ to be the functor which assigns to any $W \\in \\mathop {\\mmlmultiscripts{\\operatorname{Rep}{\\Lambda }{\\mmlnone }\\mmlprescripts {\\mmlnone }{f}}}\\limits (G)$ the $\\Gamma _{\\infty }$ -representation $T_B(P(W), \\phi _{P(W)})$ .", "This is an object of $\\operatorname{GRep}_B(\\Gamma _{\\infty })$ because $T_B$ is a tensor exact functor (see [7] or [27]).", "Similarly, one can define $\\underline{M}_{G, B}$ which is quasi-inverse to $T_{G,B}$ .", "Compatibility with extending the coefficients follows from [24].", "Definition 2.2.5 Let $B$ be any $\\mathbb {Z}_p$ -algebra.", "A Kisin module with bounded height over $B$ is a finitely generated projective $\\mathfrak {S}_B$ -module $\\mathfrak {M}_B$ together with an isomorphism $\\phi _{\\mathfrak {M}_B}:\\varphi ^*(\\mathfrak {M}_B)[1/E(u)] \\cong \\mathfrak {M}_B[1/E(u)]$ .", "We say that $(\\mathfrak {M}_B, \\phi _{\\mathfrak {M}_B})$ has height in $[a, b]$ if $E(u)^{a} \\mathfrak {M}_B \\supset \\phi _{\\mathfrak {M}_B}(\\varphi ^*(\\mathfrak {M}_B)) \\supset E(u)^{b} \\mathfrak {M}_B$ as submodules of $\\mathfrak {M}_B[1/E(u)]$ .", "Let $\\operatorname{Mod}^{\\varphi , \\operatorname{bh}}_{\\mathfrak {S}_B}$ (resp.", "$\\operatorname{Mod}^{\\varphi , [a,b]}_{\\mathfrak {S}_B}$ ) denote the category of Kisin modules with bounded height (resp.", "height in $[a,b]$ ) with morphisms being $\\mathfrak {S}_B$ -module maps respecting Frobenii.", "$\\operatorname{Mod}^{\\varphi , [0,h]}_{\\mathfrak {S}_B}$ is the usual category of Kisin modules with height $\\le h$ as in [6], [24], [25].", "Example 2.2.6 Let $\\mathfrak {S}(1)$ be the Kisin module whose underlying module is $\\mathfrak {S}$ and whose Frobenius is given by $c_0^{-1} E(u) \\varphi _{\\mathfrak {S}}$ where $E(0) = c_0 p$ .", "For any $\\mathbb {Z}_p$ -algebra, we define $\\mathfrak {S}_B(1)$ by base change from $\\mathbb {Z}_p$ and define $\\mathcal {O}_{\\mathcal {E}, B}(1) := \\mathfrak {S}_B(1) \\otimes _{\\mathfrak {S}_B} \\mathcal {O}_{\\mathcal {E}, B}$ , an étale $(\\mathcal {O}_{\\mathcal {E}, B}, \\varphi )$ -module.", "In order to reduce to the effective case (height in $[0,h]$ ), it is often useful to “twist” by tensoring with $\\mathfrak {S}_B(1)$ .", "For any $\\mathfrak {M}_B \\in \\operatorname{Mod}^{\\varphi , \\mathrm {bh}}_{\\mathfrak {S}_B}$ and any $n \\in \\mathbb {Z}$ , define $\\mathfrak {M}_B(n)$ by $n$ -fold tensor product with $\\mathfrak {S}_B(1)$ (negative $n$ being tensoring with the dual).", "It is not hard to see that if $\\mathfrak {M}_B \\in \\operatorname{Mod}^{\\varphi , [a,b]}_{\\mathfrak {S}_B}$ then $\\mathfrak {M}_B(n) \\in \\operatorname{Mod}^{\\varphi , [a + n,b + n]}_{\\mathfrak {S}_B}$ .", "Definition 2.2.7 Let $B$ be any $\\Lambda $ -algebra.", "A $G$ -Kisin module over $B$ is a pair $(\\mathfrak {P}_B, \\phi _{\\mathfrak {P}_B})$ where $\\mathfrak {P}_B$ is a $G$ -bundle on $\\mathfrak {S}_{B}$ and $\\phi _{\\mathfrak {P}_B}: \\varphi ^*(\\mathfrak {P}_B)[1/E(u)] \\cong \\mathfrak {P}_B[1/E(u)]$ is an isomorphism of $G$ -bundles.", "Denote the category of such objects by $\\operatorname{GMod}^{\\varphi , \\mathrm {bh}}_{\\mathfrak {S}_B}$ .", "Remark 2.2.8 Unlike Kisin module for $\\mathrm {GL}_n$ , $G$ -bundles do not have endomorphisms.", "Additionally, there is no reasonable notion of effective $G$ -Kisin module.", "The Frobenius on a $G$ -Kisin module is only ever defined after inverting $E(u)$ .", "Later, we use auxiliary representations of $G$ to impose height conditions.", "The category $\\operatorname{Mod}^{\\varphi , \\mathrm {bh}}_{\\mathfrak {S}_B}$ is a tensor exact category where a sequence of Kisin modules $0 \\rightarrow \\mathfrak {M}^{\\prime }_B \\rightarrow \\mathfrak {M}_B \\rightarrow \\mathfrak {M}^{\\prime \\prime }_B \\rightarrow 0$ is exact if the underlying sequence of $\\mathfrak {S}_B$ -modules is exact.", "For any $W \\in \\mathop {\\mmlmultiscripts{\\operatorname{Rep}{\\Lambda }{\\mmlnone }\\mmlprescripts {\\mmlnone }{f}}}\\limits (G)$ , the pushout $(\\mathfrak {P}_B(W), \\phi _{\\mathfrak {P}_B}(W))$ is a Kisin module with bounded height.", "Using Theorem REF , one can interpret $\\operatorname{GMod}^{\\varphi , \\mathrm {bh}}_{\\mathfrak {S}_B}$ as the category of faithful exact tensor functors from $\\mathop {\\mmlmultiscripts{\\operatorname{Rep}{\\Lambda }{\\mmlnone }\\mmlprescripts {\\mmlnone }{f}}}\\limits (G)$ to $\\operatorname{Mod}^{\\varphi , \\operatorname{bh}}_{\\mathfrak {S}_B}$ .", "Since $E(u)$ is invertible in $\\mathcal {O}_{\\mathcal {E}}$ , there is natural map $\\mathfrak {S}_B[1/E(u)] \\rightarrow \\mathcal {O}_{\\mathcal {E}, B}$ for any $\\mathbb {Z}_p$ -algebra $B$ .", "This induces a functor $\\Upsilon _G:\\operatorname{GMod}^{\\varphi , \\mathrm {bh}}_{\\mathfrak {S}_B} \\rightarrow \\operatorname{GMod}^{\\varphi }_{\\mathcal {O}_{\\mathcal {E}, B}}$ for any $\\Lambda $ -algebra $B$ .", "Definition 2.2.9 Let $B$ be any $\\Lambda $ -algebra and let $P_B \\in \\operatorname{GMod}^{\\varphi }_{\\mathcal {O}_{\\mathcal {E}, B}}$ .", "A $G$ -Kisin lattice of $P_B$ is a pair $(\\mathfrak {P}_B, \\alpha )$ where $\\mathfrak {P}_B \\in \\operatorname{GMod}^{\\varphi , \\mathrm {bh}}_{\\mathfrak {S}_B}$ and $\\alpha :\\Upsilon _G(\\mathfrak {P}_B) \\cong P_B$ is an isomorphism.", "From the Tannakian perspective, a $G$ -Kisin lattice of $P$ is equivalent to Kisin lattices $\\mathfrak {M}_W$ in $P(W)$ for each $W \\in \\mathop {\\mmlmultiscripts{\\operatorname{Rep}{\\Lambda }{\\mmlnone }\\mmlprescripts {\\mmlnone }{f}}}\\limits (G)$ functorial in $W$ and compatible with tensor products.", "Furthermore, we have the following which says that the bounded height condition can be checked on a single faithful representation.", "Proposition 2.2.10 Let $P_B \\in \\operatorname{GMod}^{\\varphi }_{\\mathcal {O}_{\\mathcal {E}, B}}$ .", "A $G$ -Kisin lattice of $P_B$ is equivalent to an extension $\\mathfrak {P}_B$ of the bundle $P_B$ to $\\mathrm {Spec}\\ \\mathfrak {S}_B$ such that for a single faithful representation $V \\in \\mathop {\\mmlmultiscripts{\\operatorname{Rep}{\\Lambda }{\\mmlnone }\\mmlprescripts {\\mmlnone }{f}}}\\limits (G)$ , $\\mathfrak {P}_B(V) \\subset P_B(V)$ is a Kisin lattice of bounded height.", "The only claim which is does not follow from unwinding definitions is that if we have an extension $\\mathfrak {P}_{B}$ such that $\\mathfrak {P}_{B}(V) \\subset P_B(V)$ is a Kisin lattice for a single faithful representation $V$ , then $\\mathfrak {P}_{B}(W) \\subset P_B(W)$ is a Kisin lattice for all representations $W$ of $G$ .", "By [27], any $W \\in \\mathop {\\mmlmultiscripts{\\operatorname{Rep}{\\Lambda }{\\mmlnone }\\mmlprescripts {\\mmlnone }{f}}}\\limits (G)$ can be written as a subquotient of direct sums of tensor products of $V$ and the dual of $V$ .", "It suffices then to prove that bounded height is stable under duals, tensor products, quotients, and saturated subrepresentations.", "Duals and tensor products are easy to check.", "For subquotients, let $0 \\rightarrow M_B \\rightarrow N_B \\rightarrow L_B \\rightarrow 0$ be an exact sequence of étale $(\\mathcal {O}_{\\mathcal {E}, B}, \\varphi )$ -modules.", "Suppose that the sequence is induced by an exact sequence $0 \\rightarrow \\mathfrak {M}_B \\rightarrow \\mathfrak {N}_B \\rightarrow \\mathfrak {L}_B \\rightarrow 0$ of projective $\\mathfrak {S}_B$ -lattices.", "Assume $\\mathfrak {N}_B$ has bounded height with respect to $\\phi _{N_B}$ .", "By twisting, we can assume $\\mathfrak {N}_B$ has height in $[0,h]$ .", "Since $\\mathfrak {M}_B = M_B \\cap \\mathfrak {N}_B$ , $\\mathfrak {M}_B$ is $\\phi _{M_B}$ -stable.", "Similarly, $\\mathfrak {L}_B$ is $\\phi _{L_B}$ -stable.", "Consider the diagram ${0 [r] & \\varphi ^*(\\mathfrak {M}_B) [r] [d]^{\\phi _{M_B}} & \\varphi ^*(\\mathfrak {N}_B) [r] [d]^{\\phi _{N_B}} & \\varphi ^*(\\mathfrak {L}_B) [r] [d]^{\\phi _{L_B}} & 0 \\\\0 [r] & \\mathfrak {M}_B [r] & \\mathfrak {N}_B [r] & \\mathfrak {L}_B [r] & 0.", "\\\\}$ All the linearizations are injective because they are isomorphisms at the level of $\\mathcal {O}_{\\mathcal {E},B}$ -modules.", "By the snake lemma, the sequence of cokernels is exact.", "If $E(u)^h$ kills $\\mathrm {Coker}(\\phi _{N_B})$ , then it kills $\\mathrm {Coker}(\\phi _{M_B})$ and $\\mathrm {Coker}(\\phi _{P_B})$ as well.", "Thus, $\\mathfrak {M}_B$ and $\\mathfrak {P}_B$ both have height in $[0,h]$ whenever $\\mathfrak {N}_B$ does.", "Definition 2.2.11 For any $B$ as in Proposition REF , define $T_{G, \\mathfrak {S}_B}:\\operatorname{GMod}^{\\varphi , \\mathrm {bh}}_{\\mathfrak {S}_B} \\rightarrow \\operatorname{GRep}_B(\\Gamma _{\\infty })$ to be the composition $T_{G, \\mathfrak {S}_B} := T_{G, B} \\circ \\Upsilon _G$ .", "We end this section with an important full faithfulness result: Proposition 2.2.12 Assume $B$ is finite flat over $\\Lambda $ .", "Then the natural extension map $\\Upsilon _G:\\operatorname{GMod}^{\\varphi , \\mathrm {bh}}_{\\mathfrak {S}_B} \\rightarrow \\operatorname{GMod}^{\\varphi }_{\\mathcal {O}_{\\mathcal {E}, B}}$ is fully faithful.", "This follows from full faithfulness of $\\Upsilon _{\\mathrm {GL}_n}$ for all $n \\ge 1$ by considering a faithful representation of $G$ .", "When $B = \\mathbb {Z}_p$ , this is [6].", "One can reduce to this case by forgetting coefficients since any finitely generated projective $\\mathfrak {S}_B$ -module is finite free over $\\mathfrak {S}$ ." ], [ "Resolutions of $G$ -valued deformations rings", "Fix a faithful representation $V$ of $G$ over $\\Lambda $ and integers $a, b$ with $a \\le b$ .", "We will use $V$ and $a, b$ to impose finiteness conditions on our moduli space.", "Definition 2.3.1 Let $B$ be any $\\Lambda $ -algebra.", "We say a $G$ -Kisin lattice $\\mathfrak {P}_{B}$ in $(P_{B}, \\phi _{P_B}) \\in \\operatorname{GMod}^{\\varphi }_{\\mathcal {O}_{\\mathcal {E}, B}}$ has height in $[a,b]$ if $\\mathfrak {P}_{B}(V)$ in $P_B(V)$ has height in $[a,b]$ .", "For any finite local Artinian $\\Lambda $ -algebra $A$ and any $(P_A, \\phi _{P_A}) \\in \\operatorname{GMod}^{\\varphi }_{\\mathcal {O}_{\\mathcal {E}, A}}$ , consider the following moduli problem over $\\mathrm {Spec}\\ A$ : $X^{[a,b]}_{P_A}(B):= \\lbrace G \\text{-Kisin lattices in }P_A \\otimes _{\\mathcal {O}_{\\mathcal {E}, A}} \\mathcal {O}_{\\mathcal {E}, B} \\text{ with height in }[a,b] \\rbrace / \\cong $ for any $A$ -algebra $B$ .", "Theorem 2.3.2 Assume that $P_A$ is a trivial bundle over $\\mathrm {Spec}\\ \\mathcal {O}_{\\mathcal {E}, A}$ .", "The functor $X^{[a,b]}_{P_A}$ is represented by a closed finite type subscheme of the affine Grassmanian $\\operatorname{Gr}_{G^{\\prime }}$ over $\\mathrm {Spec}\\ A$ where $G^{\\prime }$ is the Weil restriction $\\mathrm {Res}_{(W \\otimes _{\\mathbb {Z}_p} \\Lambda )/\\Lambda } G$ .", "By Proposition REF , $X^{[a,b]}_{P_A}(B)$ is the set of bundles over $\\mathfrak {S}_B$ extending $P_B := P_A \\otimes _{\\mathcal {O}_{\\mathcal {E}, A}} \\mathcal {O}_{\\mathcal {E}, B}$ with height in $[a,b]$ with respect to $V$ .", "We want to identify this set with a subset of $\\operatorname{Gr}_{G^{\\prime }}(B)$ .", "Consider the following diagram ${\\mathfrak {S}\\otimes _{\\mathbb {Z}_p} B [r] [d] & (W \\otimes _{\\mathbb {Z}_p} B)[\\![u]\\!]", "[d] \\\\\\mathcal {O}_{\\mathcal {E}, B} [r] & (W \\otimes _{\\mathbb {Z}_p} B)((u)),}$ where the vertical arrows are localization at $u$ and the top horizontal arrow is $u$ -adic completion.", "The Beauville-Laszlo gluing lemma (REF ) says that the set of extensions of $P_B$ to $\\mathfrak {S}_B$ is in bijection with the set of extensions of $\\widehat{P}_B$ to $W_B[\\![u]\\!", "]$ , where $\\widehat{P}_B$ is the $u$ -adic completion.", "This second set is in bijection with the $B$ -points of the Weil restriction $\\mathrm {Res}_{(W \\otimes _{\\mathbb {Z}_p} \\Lambda )/\\Lambda } \\operatorname{Gr}_G$ which is isomorphic to $\\operatorname{Gr}_{G^{\\prime }}$ by [41] or [27].", "Set $M_A := P_A(V)$ .", "By [26], the functor $X_{M_A}^{[a,b]}$ of Kisin lattices in $M_A$ with height in $[a,b]$ is represented by a closed subscheme of $\\operatorname{Gr}_{\\mathrm {Res}_{(W \\otimes _{\\mathbb {Z}_p} \\Lambda )/\\Lambda } \\mathrm {GL}(V)}$ .", "Evaluation at $V$ induces a map of functors, $ X^{[a,b]}_{P_A} \\rightarrow X_{M_A}^{[a,b]}.$ The subset $X^{[a,b]}_{P_A}(B) \\subset \\operatorname{Gr}_{G^{\\prime }}(B)$ is exactly the preimage of $X^{[a,b]}_{M_A}(B)$ by Proposition REF .", "We now extend the construction beyond the Artinian setting by passing to the limit.", "Let $R$ be a complete local Noetherian $\\Lambda $ -algebra with residue field $\\mathbb {F}$ .", "Let $\\eta :\\Gamma _{\\infty } \\rightarrow G(R)$ be a continuous representation.", "Proposition 2.3.3 For any $n \\ge 1$ , let $\\eta _n:\\Gamma _{\\infty } \\rightarrow G(R/m_R^n)$ denote the reduction mod $m_R^n$ .", "From $\\lbrace \\eta _n\\rbrace $ , we construct a system $\\underline{M}_{G, R/m_R^n}(\\eta _n) =: (P_{\\eta _n}, \\phi _n) \\in \\operatorname{GMod}^{\\varphi }_{\\mathcal {O}_{\\mathcal {E}, R/m_R^n}}$ .", "Assume that $P_{\\eta _1}$ is a trivial $G$ -bundle.", "There exists a projective $R$ -scheme $\\Theta :X^{[a,b]}_{\\eta } \\rightarrow \\mathrm {Spec}\\ R,$ whose reduction modulo $m_R^n$ is $X^{[a,b]}_{P_{\\eta _n}}$ for any $n \\ge 1$ .", "By Proposition REF , there are natural isomorphisms $P_{\\eta _{n+1}} \\otimes _{\\mathcal {O}_{\\mathcal {E}, R/m_R^{n+1}}} \\mathcal {O}_{\\mathcal {E}, R/m_R^{n}} \\cong P_{\\eta _n}$ for all $n \\ge 1$ .", "Since $P_{\\eta _1}$ is a trivial $G$ -bundle, all $P_{\\eta _n}$ are trivial by Proposition REF (1) so we can apply Theorem REF .", "Consider then the system $\\lbrace X^{[a,b]}_{P_{\\eta _n}} \\rbrace $ of schemes over $\\lbrace R/m_R^n \\rbrace $ .", "Since $G^{\\prime }$ is reductive, the affine Grassmanian $\\operatorname{Gr}_{G^{\\prime }}$ is ind-projective ([27]).", "In particular, any ample line bundle on $\\operatorname{Gr}_{G^{\\prime }}$ will restrict to a compatible system of ample line bundles on $\\lbrace X^{[a,b]}_{P_{\\eta _n}} \\rbrace .$ By formal GAGA (EGA $\\mathrm {III}_1$ 5.4.5), there exists a projective $R$ -scheme $X^{[a,b]}_{\\eta }$ whose reductions modulo $m_R^n$ are $X^{[a,b]}_{P_{\\eta _n}}$ .", "Remark 2.3.4 Unlike for $\\mathrm {GL}_n$ , there are non-trivial $G$ -bundles over $\\mathrm {Spec}\\ \\mathbb {F}(\\!(u)\\!", ")$ which is why we need the assumption in Proposition REF .", "If $P_{\\eta _1}$ admits any $G$ -Kisin lattice $\\mathfrak {P}_{\\eta _1}$ , by Proposition REF (2), the $G$ -bundle $\\mathfrak {P}_{\\eta _1}$ is trivial since $\\mathfrak {S}_{\\mathbb {F}}$ is a semi-local ring with finite residue fields.", "Thus, the assumption in Proposition REF is natural if you are interested in studying $\\Gamma _{\\infty }$ -representations of finite height.", "By Steinberg's Theorem, one can always make $P_{\\eta _1}$ trivial by passing to finite extension $\\mathbb {F}^{\\prime }$ of $\\mathbb {F}$ .", "We record for reference the following compatibility with base change: Proposition 2.3.5 Let $f:R \\rightarrow S$ be a local map of complete local Noetherian $\\Lambda $ -algebras with finite residue fields of characteristic $p$ .", "Let $\\eta _S$ be the induced map $\\Gamma _{\\infty } \\rightarrow G(S)$ .", "Then, there is a natural map $f^{\\prime }:X^{[a,b]}_{\\eta _S} \\rightarrow X^{[a,b]}_{\\eta }$ which makes the following diagram Cartesian: ${X^{[a,b]}_{\\eta _S} [r]^{f^{\\prime }} [d] & X^{[a,b]}_{\\eta } [d] \\\\\\mathrm {Spec}\\ S [r]^{f} & \\mathrm {Spec}\\ R. \\\\}$ In particular, if $R \\rightarrow S$ is surjective, then $f^{\\prime }$ is a closed immersion.", "We will now study the projective $F$ -morphism $\\Theta [1/p]: X^{[a,b]}_{\\eta }[1/p] \\rightarrow \\mathrm {Spec}\\ R[1/p].$ We show it is a closed immersion (this is essentially a consequence of REF ) and that the closed points of the image are $G$ -valued representations with height in $[a,b]$ in a suitable sense (REF ).", "Next, we show that if $\\eta $ is the restriction of $\\eta ^{\\prime }:\\Gamma _K \\rightarrow G(R)$ , then the image of $\\Theta [1/p]$ contains all semi-stable representations with $\\eta ^{\\prime }(V)$ having Hodge-Tate weights in $[a,b]$ .", "These are generalizations of results from [26].", "The following lemma will be useful at several key points: Lemma 2.3.6 (Extension Lemma) Let $G$ be a smooth affine group scheme over $\\Lambda $ .", "Let $C$ be a finite flat $\\Lambda $ -algebra and let $U$ be the open complement of the finite set of closed points of $\\mathrm {Spec}\\ \\mathfrak {S}_C$ .", "There is an equivalence of categories between $G$ -bundles $Q$ on $U$ and the category of triples $(\\mathfrak {P}^*, P, \\gamma )$ where $\\mathfrak {P}^*$ is a $G$ -bundle on $\\mathrm {Spec}\\ \\mathfrak {S}_C[1/p]$ , $P$ is $G$ -bundle $\\mathrm {Spec}\\ \\mathcal {O}_{\\mathcal {E}, C}$ , and $\\gamma $ is an isomorphism of their restrictions to $\\mathrm {Spec}\\ \\mathcal {O}_{\\mathcal {E}, C}[1/p]$ .", "Assume $G$ is a reductive group scheme with connected fibers.", "Let $V$ be a faithful representation of $G$ over $\\Lambda $ .", "If $Q$ be a $G$ -bundle on $U$ such that the locally free coherent sheaf $Q(V)$ on $U$ extends to a projective $\\mathfrak {S}_C$ -module $\\mathfrak {M}_C$ , then there exists a unique $($ up to unique isomorphism$)$ $G$ -bundle $\\widetilde{Q}$ over $\\mathrm {Spec}\\ \\mathfrak {S}_C$ such that $\\widetilde{Q}|_U \\cong Q$ and $\\widetilde{Q}(V) = \\mathfrak {M}_C$ .", "First note that we can write $U$ as the union of $\\mathrm {Spec}\\ \\mathfrak {S}_C[1/u]$ and $\\mathrm {Spec}\\ \\mathfrak {S}_C[1/p]$ .", "Recall also that $\\mathcal {O}_{\\mathcal {E}, C}$ is the $p$ -adic completion of $\\mathfrak {S}_C[1/u]$ .", "Since $p$ is non-zero divisor in $\\mathfrak {S}_C[1/u]$ , we can apply the gluing lemma (REF ) to $P$ and $\\mathfrak {P}^*[1/u]$ to construct a $G$ -bundle $Q^{\\prime }$ on $\\mathrm {Spec}\\ \\mathfrak {S}_C[1/u]$ which by construction is isomorphic to $\\mathfrak {P}^*$ along $\\mathrm {Spec}\\ \\mathfrak {S}_C[1/u, 1/p]$ .", "The $G$ -bundles $\\mathfrak {P}^*$ and $Q^{\\prime }$ glue to give bundle $Q$ over $U$ .", "Each step in the construction is a categorical equivalence.", "For part (2), consider the functor $|\\text{Fib}_{\\mathfrak {M}_C}|$ which by Lemma REF and [27] is represented by an affine scheme $Y$ .", "$\\mathfrak {M}_C$ defines a $U$ -point of $\\text{Fib}_{\\mathfrak {M}_C}$ .", "Since $\\Gamma (U, \\mathcal {O}_U) = \\mathfrak {S}_C$ , we deduce that $\\mathrm {Hom}_{\\mathfrak {S}_C}(\\mathrm {Spec}\\ \\mathfrak {S}_C, \\text{Fib}_{\\mathfrak {M}_C}) = \\mathrm {Hom}_{\\mathfrak {S}_C}(U, \\text{Fib}_{\\mathfrak {M}_C}).$ A $\\mathfrak {S}_C$ -point of $\\text{Fib}_{\\mathfrak {M}_C}$ is exactly a bundle $\\widetilde{Q}$ extending $Q$ and mapping to $\\mathfrak {M}_C$ .", "A similar argument, using that the Isom-scheme between $G$ -bundles is representable by an affine scheme, shows that if an extension exists it is unique up to unique isomorphism (without any reductivity hypotheses).", "Let $B$ be any finite local $F$ -algebra with residue field $F^{\\prime }$ .", "Define $B^0$ to be the subring of elements which map to $\\mathcal {O}_{F^{\\prime }}$ modulo the maximal ideal of $B$ .", "Let $\\text{Int}_B$ denote the set of finitely generated $\\mathcal {O}_{F^{\\prime }}$ -subalgebras $C$ of $B^0$ such that $C[1/p] = B$ .", "Definition 2.3.7 A continuous homomorphism $\\eta :\\Gamma _{\\infty } \\rightarrow G(B)$ has bounded height if there exists a $C \\in \\text{Int}_B$ and $g \\in G(B)$ such that $\\eta ^{\\prime }_C := g \\eta g^{-1}$ factors through $G(C)$ ; $\\underline{M}_{G,C}(\\eta ^{\\prime }_C) \\in \\operatorname{GMod}^{\\varphi }_{\\mathcal {O}_{\\mathcal {E}, C}}$ admits a $G$ -Kisin lattice of bounded height.", "We define height in $[a,b]$ with respect to the chosen faithful representation $V$ by replacing bounded height in (2) with height in $[a,b]$ .", "Lemma 2.3.8 Let $B$ be a finite local $\\mathbb {Q}_p$ -algebra and choose $C \\in \\mathrm {Int}_B$ and $M_C \\in \\operatorname{Mod}^{\\varphi , \\text{ét}}_{\\mathcal {O}_{\\mathcal {E}, C}}$ .", "If $M_C$ considered as an $\\mathcal {O}_{\\mathcal {E}}$ -module has bounded height $($ resp.", "height in $[a,b])$ , then there exists some $C^{\\prime } \\supset C$ in $\\mathrm {Int}_B$ , such that $M_C \\otimes _C C^{\\prime }$ has bounded height $($ resp.", "height in $[a,b])$ .", "This is the main content in the proof of part (2) of Proposition 1.6.4 in [26].", "If $F^{\\prime }$ is the residue field of $B$ , then one first constructs a Kisin lattice $\\mathfrak {M}_{\\mathcal {O}_{F^{\\prime }}}$ in $M_C \\otimes _C \\mathcal {O}_{F^{\\prime }}$ .", "The Kisin lattice in $M_C \\otimes _C {C^{\\prime }}$ is constructed by lifting $\\mathfrak {M}_{\\mathcal {O}_{F^{\\prime }}}$ (the extension to $C^{\\prime }$ is required to insure that the lift is $\\phi $ -stable).", "Proposition 2.3.9 The morphism $\\Theta $ becomes a closed immersion after inverting $p$ .", "Furthermore, if $\\mathrm {Spec}\\ R_{\\eta }^{[a,b]} \\subset \\mathrm {Spec}\\ R$ is the scheme-theoretic image of $\\Theta $ , then for any finite $F$ -algebra $B$ , a $\\Lambda $ -algebra map $x:R \\rightarrow B$ factors through $R_{\\eta }^{[a,b]}$ if and only if $\\eta \\otimes _{R, x} B$ has height in $[a,b]$ .", "The map $\\Theta $ is injective on $C$ -points for any finite flat $\\Lambda $ -algebra $C$ by Proposition REF .", "The proof of the first assertion is then the same as in [26].", "For the second assertion, say $x:R \\rightarrow B$ factors through the $R_{\\eta }^{[a,b]}$ .", "Because $\\Theta [1/p]$ is a closed immersion, $x:R \\rightarrow B$ comes from a $B$ -point $y$ of $X^{[a,b]}_{\\eta }$ .", "Any such $x$ is induced by $x_C:R \\rightarrow C$ for some $C \\in \\mathrm {Int}_B$ .", "By properness of $\\Theta $ , there exists $y_C \\in X^{[a,b]}_{\\eta }(C)$ such that $\\Theta (y_C) = x_C$ .", "This implies that $\\eta \\otimes _{R, x_C} C$ has height in $[a,b]$ as a $G$ -valued representation and hence $\\eta \\otimes _{R, x} B$ also has height in $[a,b]$ (see Definition REF ).", "Now, let $x:R \\rightarrow B$ be a homomorphism such that $\\eta _B := \\eta \\otimes _{R, x} B$ has height in $[a,b]$ as a $G$ -valued representations.", "Any homomorphism $R \\rightarrow B$ factors through some $C \\in \\mathrm {Int}_B$ so that $\\eta _B$ has image in $G(C)$ ; call this map $\\eta _C$ .", "We claim that there exist some $C^{\\prime } \\supset C$ in $\\mathrm {Int}_B$ such that $\\eta _{C^{\\prime }} = \\eta _C \\otimes _C C^{\\prime }$ has height in $[a,b]$ and hence $x$ is in the image of $X^{[a,b]}_{\\eta }(B)$ .", "Essentially, we have to show that if one Galois stable “lattice” in $\\eta _B$ has finite height then all “lattices” do.", "For $\\mathrm {GL}_n$ , this is Lemma 2.1.15 in [25].", "We invoke the $\\mathrm {GL}_n$ result below.", "Since $\\eta _B$ has height in $[a,b]$ , there exists $C^{\\prime } \\in \\mathrm {Int}_B$ and $g \\in G(B)$ such that $\\eta ^{\\prime } = g \\eta _B g^{-1}$ factors through $G(C^{\\prime })$ and has height in $[a,b]$ .", "Enlarging $C$ if necessary, we assume both $\\eta _C$ and $\\eta ^{\\prime }$ are valued in $G(C)$ .", "Let $P_{\\eta } := \\underline{M}_{G, C} (\\eta )$ and $P_{\\eta ^{\\prime }} := \\underline{M}_{G, C}(\\eta ^{\\prime })$ .", "Then $g$ induces an isomorphism $P_{\\eta ^{\\prime }}[1/p] \\cong P_{\\eta _C}[1/p].$ Since $P_{\\eta ^{\\prime }}$ has a $G$ -Kisin lattice with height in $[a,b]$ , we get a bundle $\\mathfrak {Q}_C$ over $\\mathfrak {S}_{C}[1/p]$ extending $P_{\\eta _C}[1/p]$ .", "By Lemma REF (1), $P_{\\eta ^{\\prime }}$ and $\\mathfrak {Q}_C$ glue to give a bundle $Q_C$ over the complement of the closed points of $\\mathrm {Spec}\\ \\mathfrak {S}_{C}$ .", "We would like to apply Lemma REF (2).", "$P_{\\eta _C}(V)$ has height in $[a,b]$ as an $\\mathcal {O}_{\\mathcal {E}}$ -module by [25] since it corresponds to a lattice in $\\eta _C(V)[1/p] \\cong \\eta ^{\\prime }(V)[1/p]$ .", "By Lemma REF , there exists $\\widetilde{C} \\supset C$ in $\\mathrm {Int}_B$ such that $P_{\\eta _C}(V) \\otimes _C \\widetilde{C}$ has height in $[a,b]$ as a $\\mathcal {O}_{\\mathcal {E}, \\widetilde{C}}$ -module.", "Replace $C$ by $\\widetilde{C}$ .", "Then, if $\\mathfrak {M}_C$ is the unique Kisin lattice in $P_{\\eta _C}(V)$ , we have $\\mathfrak {M}^{\\prime }_{C}[1/p] \\cap P_{\\eta _C}(V) = \\mathfrak {M}_C$ where $\\mathfrak {M}^{\\prime }_C$ is the unique Kisin lattice in $P_{\\eta ^{\\prime }}(V)$ .", "This shows that $Q_C(V)$ extends across the closed points so we can apply Lemma REF (2) to construct a $G$ -Kisin lattice of $P_{\\eta _C}$ .", "Now, assume that $\\eta $ is the restriction to $\\Gamma _{\\infty }$ of a continuous representation of $\\Gamma _K$ which we continue to call $\\eta $ .", "Recall the definition of semi-stable for a $G$ -valued representation: Definition 2.3.10 If $B$ is a finite $F$ -algebra, a continuous representation $\\eta _B:\\Gamma _K \\rightarrow G_F(B)$ is semi-stable (respectively crystalline) if for all representations $W \\in \\operatorname{Rep}_F (G_F)$ the induced representation $\\eta _B(W)$ on $W \\otimes _{F} B$ is semi-stable (respectively crystalline).", "Note that because the semi-stable and crystalline conditions are stable under tensor products and subquotients, it suffices to check these conditions on a single faithful representation of $G_F$ .", "Remark 2.3.11 Since we are working with covariant functors, our convention will be that the cyclotomic character has Hodge-Tate weight $-1$ .", "This is unfortunately opposite the convention in [26].", "The following Theorem generalizes [26]: Theorem 2.3.12 Let $R$ be a complete local Noetherian $\\Lambda $ -algebra with finite residue field and $\\eta :\\Gamma _K \\rightarrow G(R)$ a continuous representation.", "Given any $a, b$ integers with $a < b$ , there exists a quotient $R^{[a,b], \\operatorname{st}}_{\\eta }$ $($ resp.", "$R^{[a,b], \\operatorname{cris}}_{\\eta })$ of $R^{[a,b]}_{\\eta }$ with the property that if $B$ is any finite $F$ -algebra and $x:R \\rightarrow B$ a map of $\\Lambda $ -algebras, then $x$ factors through $R^{[a,b],\\operatorname{st}}_{\\eta }$ $($ resp.", "$R^{[a,b], \\operatorname{cris}}_{\\eta })$ if and only if $\\eta _x:\\Gamma _K \\rightarrow G(B)$ is semi-stable $($ resp.", "crystalline$)$ and $\\eta _x(V)$ has Hodge-Tate weights in $[a,b]$ .", "Since the semi-stable and crystalline properties can be checked on a single faithful representation, the quotients $R^{[a,b], \\text{st}}_{\\eta (V)}$ and $R^{[a,b], \\operatorname{cris}}_{\\eta (V)}$ of $R$ constructed by applying [26] to $\\eta (V)$ satisfy the universal property in Theorem REF with respect to maps $x:R \\rightarrow B$ where $B$ is a finite $F$ -algebra.", "What remains is to show that $R^{[a,b], \\text{st}}_{\\eta } : = R^{[a,b], \\text{st}}_{\\eta (V)}$ is a quotient of $R^{[a,b]}_{\\eta }$ , i.e., that “semi-stable implies finite height.” Proposition 2.3.13 Let $R$ and $\\eta $ be as in $\\ref {stlocus}$ .", "For any map $x:R \\rightarrow B$ with $B$ a finite local $F$ -algebra, if the representation $\\eta _x$ is semi-stable and if $\\eta _x(V)$ has Hodge-Tate weights in $[a,b]$ then $x$ factors through $R_{\\eta }^{[a,b]}$ .", "By Lemma REF , there exists $C \\in \\mathrm {Int}_B$ such that $\\eta _x$ factors through $\\mathrm {GL}(V_{C})$ , hence $G(C)$ , and that $M_{C} := P_{\\eta _x}(V)$ admits a Kisin lattice $\\mathfrak {M}_C$ with height in $[a,b]$ .", "It suffices by REF to extend the bundle $P_{\\eta _x}$ to $\\mathrm {Spec}\\ \\mathfrak {S}_C$ such that $\\mathfrak {P}_{\\eta _x}(V) = \\mathfrak {M}_{C}$ .", "We will apply Lemma REF .", "Consider a candidate fiber functor $\\mathfrak {F}_{\\eta _x}$ for $\\mathfrak {P}_{\\eta _x}$ which assigns to any $W \\in \\mathop {\\mmlmultiscripts{\\operatorname{Rep}{\\Lambda }{\\mmlnone }\\mmlprescripts {\\mmlnone }{f}}}\\limits (G)$ the unique Kisin lattice of bounded height in $\\mathfrak {M}_W \\subset P_{\\eta _x}(W) = M_W$ (as an $\\mathcal {O}_{\\mathcal {E}}$ -module not as an $\\mathcal {O}_{\\mathcal {E}, C^{\\prime }}$ -module).", "Such a lattice exists since $\\eta _x(W)$ is semi-stable.", "The difficulties are that $\\mathfrak {M}_W$ may not be $\\mathcal {O}_{\\mathcal {E}, C^{\\prime }}$ -projective and that it is not obvious whether $\\mathfrak {F}_{\\eta _x}$ is exact.", "It can happen that a non-exact sequence of $\\mathfrak {S}$ -module can map under $T_{\\mathfrak {S}}$ to an exact sequence of $\\Gamma _{\\infty }$ -representations (see [32]).", "Let $B = C[1/p]$ .", "By [26], $\\mathfrak {M}_W[1/p]$ is finite projective over $\\mathfrak {S}_{C}[1/p] = \\mathfrak {S}_B$ for all $W$ .", "We claim furthermore that $\\mathfrak {F}_{\\eta _x} \\otimes _{\\mathfrak {S}_{C}} \\mathfrak {S}_B$ is exact.", "For any exact sequence $0 \\rightarrow W^{\\prime \\prime } \\rightarrow W \\rightarrow W^{\\prime } \\rightarrow 0$ in $\\mathop {\\mmlmultiscripts{\\operatorname{Rep}{\\Lambda }{\\mmlnone }\\mmlprescripts {\\mmlnone }{f}}}\\limits (G)$ , we have a left-exact sequence $0 \\rightarrow \\mathfrak {M}_{W^{\\prime \\prime }}[1/p] \\rightarrow \\mathfrak {M}_W[1/p] \\rightarrow \\mathfrak {M}_{W^{\\prime }}[1/p].$ Exactness on the right follows from [27] on the behavior of exactness for sequences of $\\mathfrak {S}$ -modules.", "Thus, $\\mathfrak {F}_{\\eta _x} \\otimes _{\\mathfrak {S}_{C}} \\mathfrak {S}_B$ defines a bundle $\\mathfrak {P}_B$ over $\\mathfrak {S}_B$ .", "Clearly, $\\mathfrak {P}_B \\otimes _{\\mathfrak {S}_B} \\mathcal {O}_{\\mathcal {E}, B} \\cong P_{\\eta _x} \\otimes _{\\mathcal {O}_{\\mathcal {E}, C}} \\mathcal {O}_{\\mathcal {E}, B}$ .", "By Lemma REF (1), we get a bundle $Q$ over $U$ such that $Q(W) = \\mathfrak {M}_W|_U$ .", "Since $\\mathfrak {M}_V$ is a projective $\\mathfrak {S}_{C}$ -module by our choice of $C$ , $Q$ extends to a bundle $\\widetilde{Q}$ over $\\mathfrak {S}_{C}$ by Lemma REF (2)." ], [ "Universal $G$ -Kisin module and filtrations", "For this section, we make a small change in notation.", "Let $R_0$ be a complete local Noetherian $\\Lambda $ -algebra with finite residue field and let $R = R_0[1/p]$ .", "Define $\\widehat{\\mathfrak {S}}_{R_0}$ to be the $m_{R_0}$ -adic completion of $\\mathfrak {S}\\otimes _{\\mathbb {Z}_p} R_0$ .", "The Frobenius on $\\mathfrak {S}\\otimes _{\\mathbb {Z}_p} R_0$ extends to a Frobenius on $\\widehat{\\mathfrak {S}}_{R_0}$ .", "Definition 2.4.1 A $(\\widehat{\\mathfrak {S}}_{R_0}[1/p], \\varphi )$ -module of bounded height is a finitely generated projective $\\widehat{\\mathfrak {S}}_{R_0}[1/p]$ -module $\\mathfrak {M}_R$ together with an isomorphism $\\phi _{R}:\\varphi ^*(\\mathfrak {M}_R)[1/E(u)] \\cong \\mathfrak {M}_R[1/E(u)].$ Let $\\eta :\\Gamma _{\\infty } \\rightarrow G(R_0)$ be continuous representation.", "If $\\widehat{\\mathcal {O}}_{\\mathcal {E}, R_0}$ is the $m_{R_0}$ -adic completion of $\\mathcal {O}_{\\mathcal {E}, R_0}$ , then the inverse limit of $\\varprojlim \\underline{M}_{G, R_0/m_{R_0}^n} (\\eta _n)$ defines a pair $(P_{\\eta }, \\phi _{\\eta })$ over $\\widehat{\\mathcal {O}}_{\\mathcal {E}, R_0}$ ([27]).", "Assume $R_0 = R_{0, \\eta }^{[a,b]}$ .", "For any finite $F$ -algebra $B$ and any homomorphism $x:R_0 \\rightarrow B$ , there is a unique $G$ -Kisin lattice in $P_{\\eta } \\otimes _{\\widehat{\\mathcal {O}}_{\\mathcal {E}, R_0}, x} \\mathcal {O}_{\\mathcal {E}, B}$ (REF ), call it $(\\mathfrak {P}_x, \\phi _x)$ .", "In the following theorem, we construct a universal $G$ -bundle over $\\widehat{\\mathfrak {S}}_{R_0}[1/p]$ with a Frobenius which specializes to $(\\mathfrak {P}_x, \\phi _x)$ at every $x$ .", "Theorem 2.4.2 Assume that $R_0 = R_{0, \\eta }^{[a,b]}$ .", "Let $B$ be a finite $F$ -algebra.", "The pair $(P_{\\eta }[1/p], \\phi _{\\eta }[1/p])$ extends to a $G$ -bundle $\\widetilde{\\mathfrak {P}}_{\\eta }$ over $\\widehat{\\mathfrak {S}}_{R_0}[1/p]$ together with a Frobenius $\\phi _{\\widetilde{\\mathfrak {P}}_{\\eta }}:\\varphi ^*(\\widetilde{\\mathfrak {P}}_{\\eta })[1/E(u)] \\cong \\widetilde{\\mathfrak {P}}_{\\eta }[1/E(u)]$ such that for any $x:R_0[1/p] \\rightarrow B$ , the base change $\\left(\\widetilde{\\mathfrak {P}}_{\\eta } \\otimes _{\\widehat{\\mathfrak {S}}_{R_0}[1/p]} \\mathfrak {S}_B, \\phi _{\\widetilde{\\mathfrak {P}}_{\\eta }} \\otimes _{\\widehat{\\mathfrak {S}}_{R_0}[1/p, 1/E(u)]} \\mathfrak {S}_B[1/E(u)]\\right)$ is $(\\mathfrak {P}_x, \\phi _x)$ .", "Let $X_n := X_{\\eta _n}^{[a,b]}$ be the projective $R_0/m_{R_0}^n$ -scheme as in §4.3.", "Take $Y_n := X_n \\times _{\\mathrm {Spec}\\ R_0/m_{R_0}^n} \\mathrm {Spec}\\ \\mathfrak {S}_{R_0/m_{R_0}^n}$ , a projective $\\mathfrak {S}_{R_0/m_{R_0}^n}$ -scheme.", "Let $X_{\\eta }^{[a,b]} \\rightarrow \\mathrm {Spec}\\ R_0$ be the algebraization of $\\varprojlim X_n$ as before.", "The base change $Y$ of $X_{\\eta }^{[a,b]}$ along the map $R_0 \\rightarrow \\widehat{\\mathfrak {S}}_{R_0}$ has the property that $Y \\mod {m}_{R_0}^n \\cong Y_n.$ Furthermore, $Y$ is a proper $\\widehat{\\mathfrak {S}}_{R_0}$ -scheme.", "Over each $Y_n$ , we have a universal $G$ -Kisin lattice $(\\mathfrak {P}_n, \\phi _n)$ with height in $[a,b]$ .", "By [27], there exists a $G$ -bundle $\\mathfrak {P}_{\\eta }$ on $Y$ such that $\\mathfrak {P}_{\\eta } \\mod {m}_{R_0}^n = \\mathfrak {P}_n$ .", "We would like to construct a Frobenius $\\phi $ over $Y[1/E(u)]$ which reduces to $\\phi _n$ modulo $m_R^n$ for each $n \\ge 1$ .", "A priori, the Frobenius is only defined over the $m_{R_0}$ -adic completion of $\\widehat{\\mathfrak {S}}_{R_0}[1/E(u)]$ which we denote by $\\widehat{S}$ .", "We have a projective morphism $Y_{\\widehat{S}} \\rightarrow \\mathrm {Spec}\\ \\widehat{S},$ where $Y_{\\widehat{S}}$ is the base change of $Y[1/E(u)]$ along $\\mathrm {Spec}\\ \\widehat{S} \\rightarrow \\mathrm {Spec}\\ \\widehat{\\mathfrak {S}}_{R_0}[1/E(u)]$ .", "$Y_{\\widehat{S}}$ is faithfully flat over $Y[1/E(u)]$ since $\\widehat{\\mathfrak {S}}_{R_0}[1/E(u)]$ is Noetherian.", "Let $\\mathrm {Isom}_G := \\mathrm {Isom}_G(\\varphi ^*(\\mathfrak {P}_{\\eta }), \\mathfrak {P}_{\\eta })$ be the affine finite type $Y$ -scheme of $G$ -bundle isomorphisms.", "The compatible system $\\lbrace \\phi _n \\rbrace $ lifts to an element $\\widehat{\\phi } \\in \\mathrm {Isom}_G(Y_{\\widehat{S}}).$ We would like to descend $\\widehat{\\phi }$ to a $Y[1/E(u)]$ -point of $\\mathrm {Isom}_G$ .", "Let $i:G \\hookrightarrow \\mathrm {GL}(V)$ be our chosen faithful representation.", "Consider the closed immersion $i_*:\\mathrm {Isom}_G \\hookrightarrow \\mathrm {Isom}_{\\mathrm {GL}(V)}(\\varphi ^*(\\mathfrak {P}_{\\eta })(V), \\mathfrak {P}_{\\eta }(V)).$ The image $i_*(\\widehat{\\phi })$ descends to a $Y[1/E(u)]$ -point of $\\mathrm {Isom}_{\\mathrm {GL}(V)}(\\varphi ^*(\\mathfrak {P}_{\\eta })(V), \\mathfrak {P}_{\\eta }(V))$ (twist to reduce to the effective case).", "Since $Y_{\\widehat{S}}$ is faithfully flat over $Y[1/E(u)]$ , for any closed immersion $Z \\subset Z^{\\prime }$ of $Y$ -schemes, we have $Z(Y[1/E(u)]) = Z(Y_{\\widehat{S}}) \\cap Z^{\\prime }(Y[1/E(u)]).$ Applying this with $Z^{\\prime } = \\mathrm {Isom}_G$ and $Z = \\mathrm {Isom}_{\\mathrm {GL}(V)}(\\varphi ^*(\\mathfrak {P}_{\\eta })(V), \\mathfrak {P}_{\\eta }(V))$ , we get a universal pair $(\\mathfrak {P}_{\\eta }, \\phi _{\\eta })$ over $Y$ respectively $Y[1/E(u)]$ .", "Since $R_0 = R_{0, \\eta }^{[a,b]}$ , $\\Theta [1/p]: X^{[a,b]}_{\\eta }[1/p] \\rightarrow R_0[1/p]$ is an isomorphism and the pair $\\widetilde{\\mathfrak {P}}_{\\eta } := \\mathfrak {P}_{\\eta }[1/p]$ and $\\phi _{\\mathfrak {P}_{\\eta }}[1/p]$ over $\\widehat{\\mathfrak {S}}_{R_0}[1/p]$ has the desired properties.", "We now discuss the notion of $p$ -adic Hodge type for $G$ -valued representation and relate this to a filtration associated to a $G$ -Kisin module.", "Let $B$ be any finite $F$ -algebra.", "For any representation of $\\Gamma _K$ on a finite free $B$ -module $V_B$ , set $D_{\\operatorname{dR}}(V_B) := (V_B \\otimes _{\\mathbb {Q}_p} B_{\\operatorname{dR}})^{\\Gamma _K},$ a filtered $(K \\otimes _{\\mathbb {Q}_p} B)$ -module whose associated graded is projective (see [1]).", "Furthermore, $D_{\\operatorname{dR}}$ defines a tensor exact functor from the category of de Rham representations on projective $B$ -modules to the category $\\operatorname{Fil}_{K \\otimes _{\\mathbb {Q}_p} B}$ of filtered $(K \\otimes _{\\mathbb {Q}_p} B)$ -modules (see [1]).", "For any field $\\kappa $ , $\\operatorname{Fil}_{\\kappa }$ will be the tensor category of $\\mathbb {Z}$ -filtered vectors spaces $(V, \\lbrace \\operatorname{Fil}^{i} V \\rbrace )$ where $\\operatorname{Fil}^i(V) \\supset \\operatorname{Fil}^{i+1} (V)$ .", "We recall a few facts from the Tannakian theory of filtrations: Definition 2.4.3 Let $H$ be any reductive group over a field $\\kappa $ .", "For any extension $\\kappa ^{\\prime } \\supset \\kappa $ , an $H$ -filtration over $\\kappa ^{\\prime }$ is a tensor exact functor from $\\operatorname{Rep}_{\\kappa }(H)$ to $\\operatorname{Fil}_{\\kappa ^{\\prime }}$ .", "Associated to any cocharacter $\\nu :\\operatorname{\\mathbb {G}_m}\\rightarrow H_{\\kappa ^{\\prime }}$ is a tensor exact functor from $\\operatorname{Rep}_\\kappa (H)$ to graded $\\kappa ^{\\prime }$ -vector spaces which assigns to each representation $W$ the vector space $W_{\\kappa ^{\\prime }}$ with its weight grading defined by the $\\operatorname{\\mathbb {G}_m}$ -action through $\\nu $ which we denote $\\omega _{\\nu }$ (see [12]).", "Definition 2.4.4 For any $H$ -filtration $\\mathcal {F}$ over $\\kappa ^{\\prime }$ , a splitting of $\\mathcal {F}$ is an isomorphism between the $\\operatorname{gr}(\\mathcal {F})$ and $\\omega _{\\nu }$ for some $\\nu :\\operatorname{\\mathbb {G}_m}\\rightarrow H_{\\kappa ^{\\prime }}$ .", "By [43], all $H$ -filtrations over $\\kappa ^{\\prime }$ are splittable.", "For any given $\\mathcal {F}$ , the cocharacters $\\nu $ for which there exists an isomorphism $\\operatorname{gr}(\\mathcal {F}) \\cong \\omega _{\\nu }$ lie in the common $H(\\kappa ^{\\prime })$ -conjugacy class.", "If $\\kappa ^{\\prime }$ is a finite extension of $\\kappa $ contained in $\\overline{\\kappa }$ , then the type $[\\nu _{\\mathcal {F}}]$ of the filtration $\\mathcal {F}$ is the geometric conjugacy class of $\\nu $ for any splitting $\\omega _{\\nu }$ over $\\kappa ^{\\prime }$ .", "For any conjugacy class $[\\nu ]$ of geometric cocharacters of $H$ , there is a smallest field of definition contained in a chosen separable closure of $\\kappa $ called the reflex field of $[\\nu ]$ .", "We denote this by $\\kappa _{[\\nu ]}$ .", "Let $G$ be as before so that $G_F$ is a (connected) reductive group over $F$ , and let $\\eta :\\Gamma _K \\rightarrow G(B)$ be a continuous representation which is de Rham.", "Then, $D_{\\operatorname{dR}}$ defines a tensor exact functor from $\\operatorname{Rep}_F(G_F)$ to $\\operatorname{Fil}_{K \\otimes _{\\mathbb {Q}_p} B}$ (see Proposition 3.2.2 in [1]) which we denote by $\\mathcal {F}^{\\operatorname{dR}}_{\\eta }$ .", "Fix a geometric cocharacter $\\mu \\in X_*((\\mathrm {Res}_{(K \\otimes _{\\mathbb {Q}_p} F)/F} G)_{\\overline{F}})$ and denote its conjugacy class by $[\\mu ]$ .", "The cocharacter $\\mu $ is equivalent to a set $(\\mu _{\\psi })_{\\psi :K \\rightarrow \\overline{F}}$ of cocharacters $\\mu _{\\psi }$ of $G_{\\overline{F}}$ indexed by $\\mathbb {Q}_p$ -embeddings of $K$ into $\\overline{F}.$ Definition 2.4.5 Let $F_{[\\mu ]}$ be the reflex field of $[\\mu ]$ .", "For any embedding $\\psi :K \\rightarrow \\overline{F}$ over $\\mathbb {Q}_p$ , let $\\operatorname{pr}_{\\psi }:K \\otimes _{\\mathbb {Q}_p} \\overline{F} \\rightarrow \\overline{F}$ denote the projection.", "If $F^{\\prime }$ is a finite extension of $F_{[\\mu ]}$ , a $G$ -filtration $\\mathcal {F}$ over $K \\otimes _{\\mathbb {Q}_p} F^{\\prime }$ has type $[\\mu ]$ if $\\operatorname{pr}_{\\psi }^*(\\mathcal {F}\\otimes _{F^{\\prime }, i} \\overline{F})$ has type $[\\mu _{\\psi }]$ for any $F_{[\\mu ]}$ -embedding $i:F^{\\prime } \\hookrightarrow \\overline{F}$ .", "A de Rham representation $\\eta :\\Gamma _K \\rightarrow G(F^{\\prime })$ has $p$ -adic Hodge type $\\mu $ if $\\mathcal {F}^{\\operatorname{dR}}_{\\eta }$ has type $[\\mu ]$ .", "Let $\\Lambda _{[\\mu ]}$ denote the ring of integers of $F_{[\\mu ]}$ .", "For any $\\mu $ in the conjugacy class $[\\mu ]$ , $\\operatorname{\\mathbb {G}_m}$ acts on $V \\otimes _{\\Lambda } \\overline{F}$ through $\\mu _{\\psi }$ for each $\\psi :K \\rightarrow \\overline{F}$ .", "We take $a$ and $b$ be the minimal and maximal weights taken over all $\\mu _{\\psi }$ .", "Theorem 2.4.6 Let $R_0$ be a complete local Noetherian $\\Lambda _{[\\mu ]}$ -algebra with finite residue field and $\\eta :\\Gamma _K \\rightarrow G(R_0)$ a continuous homomorphism.", "Let $R^{[a,b], \\operatorname{st}}_{0, \\eta }$ be as in $\\ref {stlocus}.$ There exists a quotient $R^{\\operatorname{st}, \\mu }_{0, \\eta }$ of $R^{[a,b], \\operatorname{st}}_{0, \\eta }$ such that for any finite extension $F^{\\prime }$ of $F_{[\\mu ]}$ , a homomorphism $\\zeta :R_0 \\rightarrow F^{\\prime }$ factors through $R^{\\operatorname{st}, \\mu }_{0, \\eta }$ if and only if the $G(F^{\\prime })$ -valued representation corresponding to $\\zeta $ is semi-stable with $p$ -adic Hodge type $[\\mu ]$ .", "See [1].", "Remark 2.4.7 One can deduce from the construction in [1] or by other arguments ([27]) that the $p$ -adic Hodge type on the generic fiber of the semi-stable deformation ring $R^{[a,b], \\operatorname{st}}_{0, \\eta }$ is locally constant so that $\\mathrm {Spec}\\ R^{\\operatorname{st}, \\mu }_{0, \\eta }[1/p]$ is a union of connected components of $\\mathrm {Spec}\\ R^{[a,b], \\operatorname{st}}_{0, \\eta }[1/p].$ Finally, we recall how the de Rham filtration is obtained from the Kisin module.", "Definition 2.4.8 Let $B$ be a finite $\\mathbb {Q}_p$ -algebra.", "Let $(\\mathfrak {M}_B, \\phi _B)$ be a Kisin module over $B$ with bounded height.", "Define $\\operatorname{Fil}^i(\\varphi ^*(\\mathfrak {M}_B)) := \\phi _B^{-1}(E(u)^i \\mathfrak {M}_B) \\cap \\varphi ^*(\\mathfrak {M}_B).$ Set $\\mathfrak {D}_B := \\varphi ^*(\\mathfrak {M}_B)/E(u) \\varphi ^*(\\mathfrak {M}_B)$ , a finite projective $(K \\otimes _{\\mathbb {Q}_p} B)$ -module.", "Define $\\operatorname{Fil}^i(\\mathfrak {D}_B)$ to be the image of $\\operatorname{Fil}^i(\\varphi ^*(\\mathfrak {M}_B))$ in $\\mathfrak {D}_B$ .", "Proposition 2.4.9 Let $B$ be a finite $\\mathbb {Q}_p$ -algebra and let $V_B$ be a finite-free $B$ -module with an action of $\\Gamma _K$ which is semi-stable with Hodge-Tate weights in $[a,b]$ .", "Any $\\mathbb {Z}_p$ -stable lattice in $V_B$ has finite height.", "If $\\mathfrak {M}_B$ is the $(\\mathfrak {S}_B, \\varphi )$ -module of bounded height attached to $V_B$ , then there is a natural isomorphism $\\mathfrak {D}_B \\cong D_{\\operatorname{dR}}(V_B)$ of filtered $(K \\otimes _{\\mathbb {Q}_p} B)$ -modules.", "The relevant results are in the proof of Corollary 2.6.2 and Theorem 2.5.5(2) in [26].", "Since [26] works with contravariant functors, one has to do a small translation.", "Under the conventions of [26], $\\mathfrak {M}_B$ would be associated to the $B$ -dual $V_B^*$ and it is shown there that $D_B \\cong D_{\\operatorname{dR}}^*(V_B^*)$ as filtered $K \\otimes _{\\mathbb {Q}_p} B$ -modules in the case where $[a,b] = [0,h]$ .", "By compatibility with duality ([1]), $D_{\\operatorname{dR}}^*(V_B^*) \\cong D_{\\operatorname{dR}}(V_B)$ .", "The general case follows by twisting." ], [ "Deformations of $G$ -Kisin modules", "In this section, we study the local structure of the “moduli space\" of $G$ -Kisin modules.", "This generalizes results of [24] and [35].", "$G$ -Kisin modules may have non-trivial automorphisms and so it is more natural as was done in [24] to work with groupoids.", "The goal of the section is to smoothly relate the deformation theory of a $G$ -Kisin module to the local structure of a local model for the group $\\mathrm {Res}_{(K \\otimes _{\\mathbb {Q}_p} F)/F} G_F$ .", "Intuitively, the smooth modication (chain of formally smooth morphisms) corresponds to adding a trivialization to the $G$ -Kisin module and then taking the “image of Frobenius” similar to Proposition 2.2.11 of [24].", "The target of the modification is a deformation functor for the moduli space $\\operatorname{Gr}_G^{E(u), W}$ discussed in §3.3 which is a version of the affine Grassmanian which appears in the work of [37] on local models.", "Finally, we show that the condition of having $p$ -adic Hodge type $\\mu $ is related to a (generalized) local model $M(\\mu ) \\subset \\operatorname{Gr}_G^{E(u), W}$ .", "In this section, there are no conditions on the cocharacter $\\mu $ .", "We will impose conditions on $\\mu $ only in the next section when we study the analogue of flat deformations." ], [ "Definitions and representability results", "Let $\\mathbb {F}$ be the residue field of $\\Lambda $ .", "Define the categories $\\mathcal {C}_{\\Lambda } = \\lbrace \\text{Artin local } \\Lambda \\text{-algebras with residue field } \\mathbb {F}\\rbrace $ and $\\widehat{\\mathcal {C}}_{\\Lambda } = \\lbrace \\text{complete local Noetherian } \\Lambda \\text{-algebras with residue field } \\mathbb {F}\\rbrace .$ Morphisms are local $\\Lambda $ -algebra maps.", "Recall that fiber products in the category $\\widehat{\\mathcal {C}}_{\\Lambda }$ exist and are represented by completed tensor products.", "A groupoid over $\\mathcal {C}_{\\Lambda }$ (or $\\widehat{\\mathcal {C}}_{\\Lambda }$ ) will be in the sense of Definition A.2.2 of [24]; this is also known as a category cofibered in groupoids over $\\mathcal {C}_{\\Lambda }$ (or $\\widehat{\\mathcal {C}}_{\\Lambda }$ ).", "Recall also the notion of a 2-fiber product of groupoids from (A.4) in [24].", "See Appendix §10 of [23] for more details related to groupoids.", "Choose a bounded height $G$ -Kisin module $(\\mathfrak {P}_{\\mathbb {F}}, \\phi _{\\mathbb {F}}) \\in \\operatorname{GMod}^{\\varphi , \\mathrm {bh}}_{\\mathfrak {S}_{\\mathbb {F}}}$ .", "Define $D_{\\mathfrak {P}_{\\mathbb {F}}} = \\cup _{a < b} D_{\\mathfrak {P}_{\\mathbb {F}}}^{[a,b]}$ to be the deformation groupoid of $\\mathfrak {P}_{\\mathbb {F}}$ as a $G$ -Kisin module of bounded height over $\\widehat{\\mathcal {C}}_{\\Lambda }$ .", "The morphisms $D_{\\mathfrak {P}_{\\mathbb {F}}}^{[a,b]} \\subset D_{\\mathfrak {P}_{\\mathbb {F}}}$ are relatively representable closed immersions so intuitively $D_{\\mathfrak {P}_{\\mathbb {F}}}$ is an ind-object built out of the finite height pieces.", "Let $\\mathcal {E}^0$ denote the trivial $G$ -bundle over $\\Lambda $ .", "Throughout we will be choosing various trivializations of the $G$ -bundle $\\mathfrak {P}_{\\mathbb {F}}$ and other related bundles.", "This is always possible because $\\mathfrak {S}_{\\mathbb {F}}$ is a complete semi-local ring with all residue fields finite (see Proposition REF (2)).", "Proposition 3.1.1 For any $\\mathfrak {P}_{\\mathbb {F}}$ with height in $[a,b]$ , the deformation groupoid $D_{\\mathfrak {P}_{\\mathbb {F}}}^{[a,b]}$ admits a formally smooth morphism $\\pi :\\mathrm {Spf}\\ R \\rightarrow D_{\\mathfrak {P}_{\\mathbb {F}}}^{[a,b]}$ for some $R \\in \\widehat{\\mathcal {C}}_{\\Lambda }$ $($ i.e., has a versal formal object in the sense of [42]$)$ .", "One can check the abstract Schlessinger's criterion in [42].", "However, it will be useful to have an explicit versal formal object.", "Fix a trivialization $\\beta _{\\mathbb {F}}$ of $\\mathfrak {P}_{\\mathbb {F}} \\mod {E}(u)^N$ for any $N \\ge 1$ , and define $\\widetilde{D}_{\\mathfrak {P}_{\\mathbb {F}}}^{[a,b], (N)}(A) := \\lbrace (\\mathfrak {P}_A, \\beta _A) \\mid \\mathfrak {P}_A \\in D^{[a,b]}_{\\mathfrak {P}_{\\mathbb {F}}}(A), \\beta _A:\\mathfrak {P}_A \\cong \\mathcal {E}^0_{\\mathfrak {S}_A} \\!", "\\!", "\\mod {E}(u)^N \\rbrace ,$ where $\\beta _A$ lifts $\\beta _{\\mathbb {F}}$ .", "Since $G$ is smooth, the forgetful morphism $\\pi ^{(N)}:\\widetilde{D}_{\\mathfrak {P}_{\\mathbb {F}}}^{[a,b], (N)} \\rightarrow D_{\\mathfrak {P}_{\\mathbb {F}}}^{[a,b]}$ is formally smooth for any $N$ .", "If $N > \\frac{b-a}{p-1}$ , then $\\widetilde{D}_{\\mathfrak {P}_{\\mathbb {F}}}^{[a,b], (N)}$ is pro-representable by a complete local Noetherian $\\Lambda $ -algebra.", "The proof uses Schlessinger's criterion.", "The two key points are that objects in $\\widetilde{D}_{\\mathfrak {P}_{\\mathbb {F}}}^{[a,b], (N)}(A)$ have no non-trivial automorphisms for which one inducts on the power of $p$ which kills $A$ (see [27]) and that the tangent space of the underlying functor is finite dimensional which uses a successive approximation argument (see [27]).", "It will also be useful to have an infinite version of $\\widetilde{D}_{\\mathfrak {P}_{\\mathbb {F}}}^{[a,b], (N)}$ .", "Fix a trivialization $\\beta _{\\mathbb {F}}:\\mathfrak {P}_{\\mathbb {F}} \\cong \\mathcal {E}^0_{\\mathfrak {S}_{\\mathbb {F}}}$ .", "Define a groupoid on $\\mathcal {C}_{\\Lambda }$ by $\\widetilde{D}^{[a,b], (\\infty )}_{\\mathfrak {P}_{\\mathbb {F}}}(A) := \\lbrace (\\mathfrak {P}_A, \\beta _A) \\mid \\mathfrak {P}_A \\in D^{[a,b]}_{\\mathfrak {P}_{\\mathbb {F}}}(A), \\beta _A:\\mathfrak {P}_A \\cong \\mathcal {E}^0_{\\mathfrak {S}_A} \\rbrace ,$ where $\\beta _A$ lifts $\\beta _{\\mathbb {F}}$ .", "Define $\\widetilde{D}^{(\\infty )}_{\\mathfrak {P}_{\\mathbb {F}}} := \\cup _{a < b} \\widetilde{D}^{[a,b], (\\infty )}_{\\mathfrak {P}_{\\mathbb {F}}}$ ." ], [ "Local models for Weil-restricted groups", "In this section, we associate to any geometric conjugacy class $[\\mu ]$ of cocharacters of $\\mathrm {Res}_{(K \\otimes _{\\mathbb {Q}_p} F)/F} G_F$ a local model $M(\\mu )$ (Definition REF ) over the ring of integers $\\Lambda _{[\\mu ]}$ of the reflex field $F_{[\\mu ]}$ of $[\\mu ]$ (the relevant parahoric here is $\\mathrm {Res}_{(\\mathcal {O}_K \\otimes _{\\mathbb {Z}_p} \\Lambda )/\\Lambda } G$ ).", "By construction, $M(\\mu )$ is a flat projective $\\Lambda _{[\\mu ]}$ -scheme.", "The principal result (Theorem REF ) says that $M(\\mu )$ is normal and its special fiber is reduced.", "The details of the proof of Theorem REF are in Chapter §10 of [27] where we follow the strategy introduced in [37].", "We cannot apply Pappas and Zhu's result directly because the group $\\mathrm {Res}_{(K \\otimes _{\\mathbb {Q}_p} F)/F} G_F$ usually does not split over a tame extension of $F$ .", "In [28], we generalize [27] and [37] to groups of the form $\\mathrm {Res}_{L/F} H$ where $H$ is reductive group over $L$ which splits over a tame extension of $L$ and allow arbitrary parahoric level structure.", "Here we recall the relevant definitions and results leaving the details to [27], [28].", "For any $\\Lambda $ -algebra $R$ , set $R_W := R \\otimes _{\\mathbb {Z}_p} W$ .", "Our local models are constructed inside the following moduli space: Definition 3.2.1 For any $\\Lambda $ -algebra $R$ , let $\\widehat{R_W[u]}_{(E(u))}$ denote the $E(u)$ -adic completion of $R_W[u]$ .", "Define $\\operatorname{Gr}^{E(u), W}_G(R) := \\lbrace \\text{isomorphism classes of pairs } (\\mathcal {E}, \\alpha ) \\rbrace ,$ where $\\mathcal {E}$ is a $G$ -bundle on $\\widehat{R_W[u]}_{(E(u))}$ and $\\alpha :\\mathcal {E}|_{\\widehat{R_W[u]}_{(E(u))}[E(u)^{-1}]} \\cong \\mathcal {E}^0_{\\widehat{R_W[u]}_{(E(u))}[E(u)^{-1}]}$ .", "Proposition 3.2.2 The functor $\\operatorname{Gr}^{E(u), W}_{G}$ is an ind-scheme which is ind-projective over $\\Lambda $ .", "Furthermore, the generic fiber $\\operatorname{Gr}^{E(u), W}_{G}[1/p]$ is naturally isomorphic to the affine Grassmanian of ${\\mathrm {Res}_{(K \\otimes _{\\mathbb {Q}_p} F)/F} G_F}$ over the field $F$ ; if $k_0$ is the residue field of $W$ , then the special fiber $\\operatorname{Gr}^{E(u), W}_{G} \\otimes _{\\Lambda } \\mathbb {F}$ is naturally isomorphic to the affine Grassmanian of $\\mathrm {Res}_{(k_0 \\otimes _{\\mathbb {F}_p} \\mathbb {F})/\\mathbb {F}} (G_{\\mathbb {F}})$ .", "See §10.1 in [27].", "Let $H$ be any reductive group over $F$ and $\\operatorname{Gr}_H$ be the affine Grassmanian of $H$ .", "Associated to any geometric conjugacy class $[\\mu ]$ of cocharacters there is an affine Schubert variety $S(\\mu )$ in $(\\operatorname{Gr}_H)_{F_{[\\mu ]}}$ where $F_{[\\mu ]}$ is the reflex field of $[\\mu ]$ .", "These are the closures of orbits for the positive loop group $L^+H$ .", "The geometric conjugacy classes of cocharacters of $H$ can be identified with the set of dominant cocharacters for a choice of maximal torus and Borel over $\\overline{F}$ .", "The dominant cocharacters have partial ordering defined by $\\mu \\ge \\lambda $ if and only if $\\mu - \\lambda $ is a non-negative sum of positive coroots.", "Then, $S(\\mu )_{\\overline{F}}$ is then the union of the locally closed affine Schubert cells for all $\\mu ^{\\prime } \\le \\mu $ ([40]).", "Definition 3.2.3 Let $F_{[\\mu ]}/F$ be the reflex field of $[\\mu ]$ with ring of integers $\\Lambda _{[\\mu ]}$ .", "If $S(\\mu ) \\subset \\operatorname{Gr}_{\\mathrm {Res}_{(K \\otimes _{\\mathbb {Q}_p} F)/F} G_F} \\otimes _F F_{[\\mu ]}$ is the closed affine Schubert variety associated to $\\mu $ , then the local model $M(\\mu )$ associated to $\\mu $ is the flat closure of $S(\\mu )$ in $\\operatorname{Gr}^{E(u), W}_{G} \\otimes _{\\Lambda } \\Lambda _{[\\mu ]}$ .", "It is a flat projective scheme over $\\mathrm {Spec}\\ \\Lambda _{[\\mu ]}$ .", "The main theorem on the geometry of local models is: Theorem 3.2.4 Suppose that $p \\nmid |\\pi _1(G^{\\mathrm {der}})|$ where $G^{\\mathrm {der}}$ is the derived subgroup of $G$ .", "Then $M(\\mu )$ is normal.", "The special fiber $M(\\mu ) \\otimes _{\\Lambda _{\\mu }} \\overline{\\mathbb {F}}$ is reduced, irreducible, normal, Cohen-Macaulay and Frobenius-split.", "For the next subsection, it will useful to recall a group which acts on $\\operatorname{Gr}^{E(u), W}_{G}$ and $M(\\mu )$ .", "Define $L^{+, E(u)} G (R) := G (\\widehat{R_W[u]}_{(E(u))}) = \\varprojlim _{i \\ge 1} G (R_W[u]/(E(u)^i))$ for all $\\Lambda $ -algebras $R$ .", "$L^{+, E(u)} G$ is represented by a group scheme which is the projective limit of the affine flat finite type group schemes $\\mathrm {Res}_{((\\Lambda \\otimes _{\\mathbb {Z}_p} W)[u]/E(u)^i)/\\Lambda } G$ .", "The group $L^{+, E(u)} G$ acts on $\\operatorname{Gr}^{E(u), W}_{G}$ by changing the trivialization.", "This action is nice in the sense of [15], i.e., $\\operatorname{Gr}^{E(u), W}_{G} \\cong \\varinjlim _i Z_i$ where $Z_i$ are $L^{+, E(u)} G$ -stable closed subschemes on which $L^{+, E(u)} G$ acts through the quotient $\\mathrm {Res}_{((\\Lambda \\otimes _{\\mathbb {Z}_p} W)[u]/E(u)^i)/\\Lambda } G$ .", "Corollary 3.2.5 For any $\\mu $ , the local model $M(\\mu )$ is stable under the action of $L^{+, E(u)} G$ .", "Since everything is flat, it suffices to show that $M(\\mu )[1/p]$ is stable under $L^{+, E(u)} G[1/p]$ .", "The functor $L^{+, E(u)} G [1/p]$ on $F$ -algebras is naturally isomorphic to the positive loop group $L^+ \\mathrm {Res}_{(K \\otimes _{\\mathbb {Q}_p} F)/F}(G)$ such that the isomorphism in Proposition REF (1) is equivariant.", "$M(\\mu )[1/p]$ is the closed affine Schubert variety $S(\\mu )$ which is stable under the action of this group." ], [ "Smooth modification", "We begin by defining the deformation functor which will be the target of our modification.", "Definition 3.3.1 Choose a $G$ -bundle $Q_{\\mathbb {F}}$ over $\\mathfrak {S}_{\\mathbb {F}}$ together with a trivialization $\\delta _0$ of $Q_{\\mathbb {F}}$ over $\\mathfrak {S}_{\\mathbb {F}}[1/E(u)]$ .", "Define a deformation functor on $\\mathcal {C}_{\\Lambda }$ by $\\overline{D}_{Q_{\\mathbb {F}}}(A) := \\lbrace \\text{isomorphism classes of triples } (\\mathcal {E}, \\delta , \\psi ) \\rbrace ,$ where $\\mathcal {E}$ is a $G$ -bundle on $\\mathfrak {S}_A$ , $\\delta :\\mathcal {E}|_{\\mathfrak {S}_A[E(u)^{-1}]} \\cong \\mathcal {E}^0_{\\mathfrak {S}_A[E(u)^{-1}]}$ , and $\\psi :\\mathcal {E}\\otimes _{\\mathfrak {S}_A} \\mathfrak {S}_{\\mathbb {F}} \\cong Q_{\\mathbb {F}}$ compatible with $\\delta $ and $\\delta _0$ .", "Example 3.3.2 Let $G = \\mathrm {GL}(V)$ .", "For any $(Q_{A}, \\delta _A) \\in \\overline{D}_{Q_{\\mathbb {F}}}(A)$ , $\\delta _A$ identifies $Q_{A}$ with a “lattice” in $(V \\otimes _{\\Lambda } \\mathfrak {S}_A)[1/E(u)]$ , i.e., a finitely generated projective $\\mathfrak {S}_A$ -module $L_{A}$ such that $L_{A}[1/E(u)] = (V \\otimes _{\\Lambda } \\mathfrak {S}_A)[1/E(u)]$ .", "The main result of this section is the following: Theorem 3.3.3 Let $\\Lambda $ be a $\\mathbb {Z}_p$ -finite flat local domain with residue field $\\mathbb {F}$ .", "Let $G$ be a connected reductive group over $\\Lambda $ and $\\mathfrak {P}_{\\mathbb {F}}$ a $G$ -Kisin module with coefficients in $\\mathbb {F}$ .", "Fix a trivialization $\\beta _{\\mathbb {F}}$ of $\\mathfrak {P}_{\\mathbb {F}}$ as a $G$ -bundle.", "There exists a diagram of groupoids over $\\mathcal {C}_{\\Lambda }$ , ${& \\widetilde{D}^{(\\infty )}_{\\mathfrak {P}_{\\mathbb {F}}} [dl]_{\\pi ^{(\\infty )}} [dr]^{\\Psi } & \\\\D_{\\mathfrak {P}_{\\mathbb {F}}} & & \\overline{D}_{Q_{\\mathbb {F}}}, \\\\}$ where $Q_{\\mathbb {F}} := (\\varphi ^*(\\mathfrak {P}_{\\mathbb {F}}), \\beta _{\\mathbb {F}}[1/E(u)] \\circ \\phi _{\\mathfrak {P}_{\\mathbb {F}}})$ .", "Both $\\pi ^{(\\infty )}$ and $\\Psi $ are formally smooth.", "Later in the section, we will refine this modification by imposing appropriate conditions on both sides.", "Intuitively, the above modification corresponds to adding a trivialization to the $G$ -Kisin module and then taking the “image of Frobenius.” The groupoid $\\widetilde{D}^{(\\infty )}_{\\mathfrak {P}_{\\mathbb {F}}}$ is defined at the end of §3.1 and $\\pi ^{(\\infty )}$ is formally smooth since $G$ is smooth.", "Next, we construct the morphism $\\Psi $ and show that it is formally smooth.", "To avoid excess notation, we sometimes omit the data of the residual isomorphisms modulo $m_A$ .", "One can check that the everything is compatible with such isomorphisms.", "Definition 3.3.4 For any $(\\mathfrak {P}_A, \\phi _{\\mathfrak {P}_A}, \\beta _A) \\in \\widetilde{D}^{(\\infty )}_{\\mathfrak {P}_{\\mathbb {F}}}(A)$ , we set $\\Psi ((\\mathfrak {P}_A, \\phi _{\\mathfrak {P}_A}, \\beta _A)) = (\\varphi ^*(\\mathfrak {P}_A), \\delta _A),$ where $\\delta _A$ is the composite $\\varphi ^*(\\mathfrak {P}_A)[1/E(u)] \\xrightarrow{} \\mathfrak {P}_A[1/E(u)] \\xrightarrow{} \\mathcal {E}^0_{\\mathfrak {S}_A} [1/E(u)].$ Proposition 3.3.5 The morphism $\\Psi $ of groupoids is formally smooth.", "Choose $A \\in \\mathcal {C}_{\\Lambda }$ and $I$ an ideal of $A$ .", "Consider a pair $(Q_A, \\delta _A) \\in \\overline{D}_{Q_{\\mathbb {F}}}(A)$ over a pair $(Q_{A/I}, \\delta _{A/I})$ .", "Let $(\\mathfrak {P}_{A/I}, \\phi _{A/I}, \\beta _{A/I})$ be an element in the fiber over $(Q_{A/I}, \\delta _{A/I})$ .", "The triple $(\\mathfrak {P}_{A/I}, \\phi _{A/I}, \\beta _{A/I})$ is isomorphic to a triple of the form $(\\mathcal {E}^0_{\\mathfrak {S}_{A/I}}, \\phi ^{\\prime }_{A/I}, \\text{Id}_{A/I})$ .", "Let $\\gamma _{A/I}$ be the isomorphism between $\\varphi ^*(\\mathcal {E}^0_{\\mathfrak {S}_{A/I}})$ and $Q_{A/I}$ .", "We want to construct a lift $(\\mathfrak {P}_A, \\phi _A, \\beta _A)$ such that $\\Psi (\\mathfrak {P}_A, \\phi _A, \\beta _A) = (Q_A, \\delta _A)$ .", "Take $\\mathfrak {P}_A = \\mathcal {E}^0_{\\mathfrak {S}_A}$ to be the trivial bundle and $\\beta _A$ to be the identity.", "Now, pick any lift $\\gamma _A:\\varphi ^*(\\mathcal {E}^0_{\\mathfrak {S}_{A}}) \\cong Q_{A}$ of $\\gamma _{A/I}$ which exists since $G$ is smooth.", "We can define the Frobenius by $\\phi _{A} = \\delta _A \\circ \\gamma _A[1/E(u)].$ It is easy to check that $\\Psi (\\mathfrak {P}_A, \\phi _A, \\beta _A) \\cong (Q_A, \\delta _A)$ .", "We would now like to relate $\\overline{D}_{Q_{\\mathbb {F}}}$ to $\\operatorname{Gr}^{E(u), W}_G$ from the previous section.", "Proposition 3.3.6 A pair $(Q_{\\mathbb {F}}, \\delta _0)$ as in Definition $\\ref {bardefn}$ defines a point $x_{\\mathbb {F}} \\in \\operatorname{Gr}_G^{E(u), W}(\\mathbb {F})$ .", "Furthermore, for any $A \\in \\mathcal {C}_{\\Lambda }$ , there is a natural functorial bijection between $\\overline{D}_{Q_{\\mathbb {F}}}(A)$ and the set of $x_A \\in \\operatorname{Gr}_G^{E(u), W}(A)$ such that $x_A \\mod {m}_A = x_{\\mathbb {F}}$ .", "Recall that $\\mathfrak {S}_A = (W \\otimes _{\\mathbb {Z}_p} A)[\\![u]\\!", "]$ because $A$ is finite over $\\mathbb {Z}_p$ .", "$\\operatorname{Gr}_G^{E(u), W}(A)$ is the set of isomorphism classes of bundles on the $E(u)$ -adic completion of $(W \\otimes _{\\mathbb {Z}_p} A)[u]$ together with a trivialization after inverting $E(u)$ .", "Since $p$ is nilpotent in $A$ , we can identify $(W \\otimes _{\\mathbb {Z}_p} A)[\\![u]\\!", "]$ and the $E(u)$ -adic completion $\\widehat{(W \\otimes _{\\mathbb {Z}_p} A)[u]}_{(E(u))}$ .", "This identifies $\\overline{D}_{Q_{\\mathbb {F}}}(A)$ with the desired subset of $\\operatorname{Gr}_G^{E(u), W}(A)$ .", "For any $\\mathbb {Z}_p$ -algebra $A$ , let $\\widehat{S}_{A}$ denote the $E(u)$ -adic completion of $(W \\otimes _{\\mathbb {Z}_p} A)[u]$ .", "Lemma 3.3.7 For any finite flat $\\mathbb {Z}_p$ -algebra $\\Lambda ^{\\prime }$ , there is a $(W \\otimes _{\\mathbb {Z}_p} \\Lambda ^{\\prime })[u]$ -algebra isomorphism $\\mathfrak {S}_{\\Lambda ^{\\prime }} \\rightarrow \\widehat{S}_{\\Lambda ^{\\prime }}.$ For any $n \\ge 1$ , we have an isomorphism $\\mathfrak {S}_{\\Lambda ^{\\prime }}/p^n \\cong \\widehat{S}_{\\Lambda ^{\\prime }}/p^n$ since $(E(u))$ and $u$ define the same adic topologies mod $p^n$ .", "Passing to the limit, we get an isomorphism of their $p$ -adic completions.", "Both $\\mathfrak {S}_{\\Lambda ^{\\prime }}$ and $\\widehat{S}_{\\Lambda ^{\\prime }}$ are already $p$ -adically complete and separated.", "Fix a geometric cocharacter $\\mu $ of $\\mathrm {Res}_{(K \\otimes _{\\mathbb {Q}_p} F)/F} G_F$ ; we can write $\\mu = (\\mu _{\\psi })_{\\psi :K \\rightarrow \\overline{F}}$ where the $\\mu _{\\psi }$ are cocharacters of $G_{\\overline{F}}$ .", "Assume that $F = F_{[\\mu ]}$ so that the generalized local model $M(\\mu )$ is then a closed subscheme of $\\operatorname{Gr}_G^{E(u), W}$ over $\\Lambda $ (REF ).", "Recall that $V$ is a fixed faithful representation of $G$ .", "For each $\\psi $ , $\\mu _{\\psi }$ induces an action of $\\operatorname{\\mathbb {G}_m}$ on $V_{\\overline{F}}$ .", "Define $a$ (resp.", "$b$ ) to the smallest (resp.", "largest) weight appearing in $V_{\\overline{F}}$ over all $\\mu _{\\psi }$ .", "Definition 3.3.8 Define a closed subfunctor $\\overline{D}^{\\mu }_{Q_{\\mathbb {F}}}$ of $\\overline{D}_{Q_{\\mathbb {F}}}$ by $\\overline{D}^{\\mu }_{Q_{\\mathbb {F}}}(A) := \\lbrace (Q_A, \\delta _A) \\in \\overline{D}_{Q_{\\mathbb {F}}}(A) \\mid (Q_A, \\delta _A) \\in M(\\mu )(A) \\rbrace $ under the identification in Proposition REF .", "Define $\\widetilde{D}^{(\\infty ), \\mu }_{\\mathfrak {P}_{\\mathbb {F}}}$ to be the base change of $\\overline{D}^{\\mu }_{\\mathfrak {P}_{\\mathbb {F}}}$ along $\\Psi $ .", "It is a closed subgroupoid of $\\widetilde{D}^{(\\infty )}_{\\mathfrak {P}_{\\mathbb {F}}}$ .", "The following proposition says that $\\widetilde{D}^{(\\infty ), \\mu }_{\\mathfrak {P}_{\\mathbb {F}}}$ descends to a closed subgroupoid $D^{\\mu }_{\\mathfrak {P}_{\\mathbb {F}}}$ of $D_{\\mathfrak {P}_{\\mathbb {F}}}$ : Proposition 3.3.9 Let $a$ and $b$ be as in the discussion before Definition REF .", "There is a closed subgroupoid $D^{\\mu }_{\\mathfrak {P}_{\\mathbb {F}}} \\subset D^{[a,b]}_{\\mathfrak {P}_{\\mathbb {F}}} \\subset D_{\\mathfrak {P}_{\\mathbb {F}}}$ such that $\\pi ^{(\\infty )}|_{\\widetilde{D}^{(\\infty ), \\mu }_{\\mathfrak {P}_{\\mathbb {F}}}}$ factors through $D^{\\mu }_{\\mathfrak {P}_{\\mathbb {F}}}$ and $\\widetilde{D}^{(\\infty ), \\mu }_{\\mathfrak {P}_{\\mathbb {F}}} \\rightarrow D^{\\mu }_{\\mathfrak {P}_{\\mathbb {F}}} \\times _{D_{\\mathfrak {P}_{\\mathbb {F}}}} \\widetilde{D}^{(\\infty )}_{\\mathfrak {P}_{\\mathbb {F}}}$ is an equivalence of closed subgroupoids.", "Furthermore, the map $\\pi ^{\\mu }:\\widetilde{D}^{(\\infty ), \\mu }_{\\mathfrak {P}_{\\mathbb {F}}} \\rightarrow D^{\\mu }_{\\mathfrak {P}_{\\mathbb {F}}}$ is formally smooth.", "For any $A \\in \\mathcal {C}_{\\Lambda }$ define $D^{\\mu }_{\\mathfrak {P}_{\\mathbb {F}}}(A)$ to be the full subcategory whose objects are $\\pi ^{(\\infty )}(\\widetilde{D}^{(\\infty ), \\mu }_{\\mathfrak {P}_{\\mathbb {F}}}(A))$ .", "Observe that for any $x \\in D^{\\mu }_{\\mathfrak {P}_{\\mathbb {F}}}(A)$ the group $G(\\mathfrak {S}_A)$ acts transitively on the fiber $(\\pi ^{(\\infty )})^{-1}(x) \\subset \\widetilde{D}^{(\\infty )}_{\\mathfrak {P}_{\\mathbb {F}}}(A)$ by changing the trivialization.", "The key point is that by Corollary REF , $\\widetilde{D}^{(\\infty ), \\mu }_{\\mathfrak {P}_{\\mathbb {F}}}(A)$ is stable under $G(\\mathfrak {S}_A)$ .", "Hence $ (\\pi ^{(\\infty )})^{-1}(x) \\subset \\widetilde{D}^{(\\infty ), \\mu }_{\\mathfrak {P}_{\\mathbb {F}}}(A).$ It is not hard to see then that the map to the fiber product is an isomorphism and that $\\pi ^{\\mu }$ is formally smooth.", "It remains to show that $D^{\\mu }_{\\mathfrak {P}_{\\mathbb {F}}} \\rightarrow D_{\\mathfrak {P}_{\\mathbb {F}}}$ is closed.", "Let $\\mathfrak {P}_A \\in D_{\\mathfrak {P}_{\\mathbb {F}}}(A)$ and choose a trivialization $\\beta _A$ of $\\mathfrak {P}_A$ , i.e., a lift to $\\widetilde{D}^{(\\infty )}_{\\mathfrak {P}_{\\mathbb {F}}}(A)$ .", "We want a quotient $A \\rightarrow A^{\\prime }$ such that for any $f:A \\rightarrow B$ , $\\mathfrak {P}_A \\otimes _{A,f} B \\in D^{\\mu }_{\\mathfrak {P}_{\\mathbb {F}}}(B)$ if and only if $f$ factors through $A^{\\prime }$ .", "Let $A \\rightarrow A^{\\prime }$ represent the closed condition $\\widetilde{D}^{(\\infty ), \\mu }_{\\mathfrak {P}_{\\mathbb {F}}} \\subset \\widetilde{D}^{(\\infty )}_{\\mathfrak {P}_{\\mathbb {F}}}$ .", "Clearly, $\\mathfrak {P}_A \\otimes _A A^{\\prime } \\in D^{\\mu }_{\\mathfrak {P}_{\\mathbb {F}}}(A^{\\prime })$ and so any further base change is as well.", "Now, let $f:A \\rightarrow B$ be such that $\\mathfrak {P}_A \\otimes _{A,f} B \\in D^{\\mu }_{\\mathfrak {P}_{\\mathbb {F}}}(B)$ .", "The trivialization $\\beta _A$ induces a trivialization $\\beta _B$ on $\\mathfrak {P}_B$ .", "The pair $(\\mathfrak {P}_B, \\beta _B)$ lies in $\\widetilde{D}^{(\\infty ), \\mu }_{\\mathfrak {P}_{\\mathbb {F}}}(B)$ by (REF ).", "We have constructed a diagram of formally smooth morphisms $ {& \\widetilde{D}^{(\\infty ),\\mu }_{\\mathfrak {P}_{\\mathbb {F}}} [dl]_{\\pi ^{\\mu }} [dr]^{\\Psi ^{\\mu }} & \\\\D^{\\mu }_{\\mathfrak {P}_{\\mathbb {F}}} & & \\overline{D}^{\\mu }_{Q_{\\mathbb {F}}}, \\\\}$ where $\\overline{D}^{\\mu }_{Q_{\\mathbb {F}}}$ is represented by the completed local ring at the $\\mathbb {F}$ -point of $M(\\mu )$ corresponding to $(Q_{\\mathbb {F}}, \\delta _{\\mathbb {F}})$ .", "Next, we would like to replace $\\widetilde{D}^{(\\infty ), \\mu }_{\\mathfrak {P}_{\\mathbb {F}}}$ by a “smaller” groupoid which is representable.", "Let $a, b$ be as in the discussion before Definition REF and choose $N > b-a$ .", "Recall the representable groupoid $\\widetilde{D}^{[a,b], (N)}_{\\mathfrak {P}_{\\mathbb {F}}}$ (Proposition REF ).", "Define a closed subgroupoid $\\widetilde{D}^{(N), \\mu }_{\\mathfrak {P}_{\\mathbb {F}}} := D^{\\mu }_{\\mathfrak {P}_{\\mathbb {F}}} \\times _{D_{\\mathfrak {P}_{\\mathbb {F}}}} \\widetilde{D}^{[a,b], (N)}_{\\mathfrak {P}_{\\mathbb {F}}}$ of $\\widetilde{D}^{[a,b], (N)}_{\\mathfrak {P}_{\\mathbb {F}}}$ .", "By Proposition REF , the morphism $\\widetilde{D}^{(\\infty ), \\mu }_{\\mathfrak {P}_{\\mathbb {F}}} \\rightarrow D^{(N), \\mu }_{\\mathfrak {P}_{\\mathbb {F}}}$ is formally smooth.", "Proposition 3.3.10 For any $N > b-a$ , the morphism $\\Psi ^{\\mu }:\\widetilde{D}^{(\\infty ), \\mu }_{\\mathfrak {P}_{\\mathbb {F}}} \\rightarrow \\overline{D}^{\\mu }_{\\mathfrak {P}_{\\mathbb {F}}}$ factors through $\\widetilde{D}^{(N), \\mu }_{\\mathfrak {P}_{\\mathbb {F}}}$ .", "Furthermore, $\\widetilde{D}^{(N), \\mu }_{\\mathfrak {P}_{\\mathbb {F}}}$ is formally smooth over $\\overline{D}^{\\mu }_{Q_{\\mathbb {F}}}$ .", "By our assumption on $N$ , $\\widetilde{D}^{(N), \\mu }_{\\mathfrak {P}_{\\mathbb {F}}}$ is representable so it suffices to define the factorization $\\Psi ^{\\mu }_N:\\widetilde{D}^{(N), \\mu }_{\\mathfrak {P}_{\\mathbb {F}}} \\rightarrow \\overline{D}^{\\mu }_{\\mathfrak {P}_{\\mathbb {F}}}$ on underlying functors.", "For any $x \\in \\widetilde{D}^{(N), \\mu }_{\\mathfrak {P}_{\\mathbb {F}}}(A)$ , set $\\Psi ^{(N), \\mu }(x) := \\Psi ^{\\mu }(\\widetilde{x})$ for any lift $\\widetilde{x}$ of $x$ to $\\widetilde{D}^{(\\infty ), \\mu }_{\\mathfrak {P}_{\\mathbb {F}}}(A)$ .", "The image is independent of the choice of lift by Corollary REF .", "The map $\\Psi ^{(N),\\mu }$ is formally smooth since $\\Psi ^{\\mu }$ is.", "In the remainder of this section, we discuss the relationship between $D^{\\mu }_{\\mathfrak {P}_{\\mathbb {F}}}$ and $p$ -adic Hodge type $\\mu $ .", "For this, it will useful to work in a larger category than $\\widehat{\\mathcal {C}}_{\\Lambda }$ .", "All of our deformation problems can be extended to the category of complete local Noetherian $\\Lambda $ -algebras $R$ with finite residue field.", "For any such $R$ , we define $D^{\\star }_{\\mathfrak {P}_{\\mathbb {F}}}(R)$ (resp.", "$\\widetilde{D}^{\\star }_{\\mathfrak {P}_{\\mathbb {F}}}(R)$ , $\\overline{D}^{\\star }_{Q_{\\mathbb {F}}}(R)$ ) to be the category of deformations to $R$ of $\\mathfrak {P}_{\\mathbb {F}} \\otimes _{\\mathbb {F}} R/m_R$ with condition $\\star $ , where $\\star $ is any of our various conditions.", "For any finite local $\\Lambda $ -algebra $\\Lambda ^{\\prime }$ , the category $\\widehat{\\mathcal {C}}_{\\Lambda ^{\\prime }}$ is a subcategory of the category of complete local Noetherian $\\Lambda $ -algebras with finite residue field.", "The functors $\\widetilde{D}^{[a,b], (N)}_{\\mathfrak {P}_{\\mathbb {F}}}, \\widetilde{D}^{(N), \\mu }_{\\mathfrak {P}_{\\mathbb {F}}}$ and $\\overline{D}^{\\mu }_{Q_{\\mathbb {F}}}$ are all representable on $\\widehat{\\mathcal {C}}_{\\Lambda }$ .", "It is easy to check using the criterion in [10] that these functors commute with change in coefficients, i.e., if $\\widetilde{R}^{[a,b], (N)}$ represents $\\widetilde{D}^{[a,b], (N)}_{\\mathfrak {P}_{\\mathbb {F}}}$ over $\\mathcal {C}_{\\Lambda }$ then $\\widetilde{R}^{[a,b], (N)} \\otimes _{\\Lambda } \\Lambda ^{\\prime }$ represents the extension of $\\widetilde{D}^{[a,b], (N)}_{\\mathfrak {P}_{\\mathbb {F}}}$ restricted to the category $\\widehat{\\mathcal {C}}_{\\Lambda ^{\\prime }}$ and similarly for $ \\widetilde{D}^{(N), \\mu }_{\\mathfrak {P}_{\\mathbb {F}}}$ and $\\overline{D}^{\\mu }_{Q_{\\mathbb {F}}}$ .", "An argument as in Theorem REF shows that an object of $D^{[a,b]}_{\\mathfrak {P}_{\\mathbb {F}}}(R)$ is the same as a $G$ -bundle $\\mathfrak {P}_R$ on $\\widehat{\\mathfrak {S}}_R$ together with a Frobenius $\\phi _{\\mathfrak {P}_R}: \\varphi ^*(\\mathfrak {P}_R)[1/E(u)] \\cong \\mathfrak {P}_R[1/E(u)]$ deforming $\\mathfrak {P}_{\\mathbb {F}} \\otimes _{\\mathbb {F}} R/m_R$ and having height in $[a,b]$ .", "The height in $[a,b]$ condition is essential in order to define the Frobenius over $R$ .", "We would like to give a criterion for when $(\\mathfrak {P}_R, \\phi _{\\mathfrak {P}_R})$ lies in $D^{\\mu }_{\\mathfrak {P}_{\\mathbb {F}}}(R)$ .", "Choose $(\\mathfrak {P}_R, \\phi _{\\mathfrak {P}_R}) \\in D^{[a,b]}_{\\mathfrak {P}_{\\mathbb {F}}}(R)$ .", "For any finite extension $F^{\\prime }$ of $F$ and any homomorphism $x:R \\rightarrow F^{\\prime }$ , denote the base change of $\\mathfrak {P}_R$ to $\\mathfrak {S}_{F^{\\prime }}$ by $(\\mathfrak {P}_x, \\phi _x)$ .", "Associated to $(\\mathfrak {P}_x, \\phi _x)$ is a functor $\\mathfrak {D}_x$ from $\\operatorname{Rep}_F(G_F)$ to filtered $(K \\otimes _{\\mathbb {Q}_p} F^{\\prime })$ -modules given by $\\mathfrak {D}_x(W) = \\varphi ^*(\\mathfrak {P}_x)(W)/E(u) \\varphi ^*(\\mathfrak {P}_x)(W)$ with the filtration defined as in Definition REF .", "Lemma 3.3.11 For any finite extension $F^{\\prime }$ of $F$ and any $x:R \\rightarrow F^{\\prime }$ , the functor $\\mathfrak {D}_x$ is a tensor exact functor.", "Any such $x$ factors through the ring of integers $\\Lambda ^{\\prime }$ of $F^{\\prime }$ so that $(\\mathfrak {P}_x, \\phi _x)$ comes from a pair $(\\mathfrak {P}_{x_0}, \\phi _{x_0})$ over $\\mathfrak {S}_{\\Lambda ^{\\prime }}$ .", "Let $\\widehat{S}_{\\Lambda ^{\\prime }}$ (resp.", "$\\widehat{S}_{F^{\\prime }}$ ) to be the $E(u)$ -adic completion of $(W \\otimes _{\\mathbb {Z}_p} \\Lambda ^{\\prime })[u]$ (resp.", "$(W \\otimes _{\\mathbb {Z}_p} F^{\\prime })[u]$ ).", "By Lemma REF , we can think of $(\\mathfrak {P}_{x_0}, \\phi _{x_0})$ equivalently as a pair over $\\widehat{S}_{\\Lambda ^{\\prime }}$ .", "Choose a trivialization $\\beta _0$ of $\\mathfrak {P}_{x_0}$ and set $Q_{x_0} := \\varphi ^*(\\mathfrak {P}_{x_0})$ with trivialization $\\delta _{x_0} := \\beta _0[1/E(u)] \\circ \\phi _{x_0}$ .", "Define $(Q_{x}, \\delta _x)$ to be $(Q_{x_0}, \\delta _{x_0}) \\otimes _{\\widehat{S}_{\\Lambda ^{\\prime }}} \\widehat{S}_{F^{\\prime }}$ and define a filtration on $\\mathfrak {D}_{Q_x} := Q_{x} \\mod {E}(u) $ by $\\operatorname{Fil}^i(\\mathfrak {D}_{Q_x}(W)) = (Q_{x}(W) \\cap E(u)^i (W \\otimes \\widehat{S}_{F^{\\prime }}))/ (E(u)Q_{x}(W) \\cap E(u)^i (W \\otimes \\widehat{S}_{F^{\\prime }}))$ for any $W \\in \\operatorname{Rep}_F(G_F)$ .", "Since $\\widehat{S}_{\\Lambda ^{\\prime }}[1/p] /(E(u)) = \\widehat{S}_{F^{\\prime }}/(E(u))$ , there is an isomorphism $\\mathfrak {D}_x \\cong \\mathfrak {D}_{Q_x}$ of tensor exact functors to $\\operatorname{Mod}_{K \\otimes _{\\mathbb {Q}_p} F^{\\prime }}$ identifying the filtrations.", "It suffices then to show that $\\mathfrak {D}_{Q_x}$ is a tensor exact functor to the category of filtered $(K \\otimes _{\\mathbb {Q}_p} F^{\\prime })$ -modules.", "Without loss of generality, we assume that $F^{\\prime }$ contains a Galois closure of $K$ .", "Then $\\widehat{S}_{F^{\\prime }} \\cong \\prod _{\\psi } F^{\\prime }[[u - \\psi (\\pi )]]$ over embeddings $\\psi :K \\rightarrow F^{\\prime }$ (first decompose $W \\otimes _{\\mathbb {Z}_p} F^{\\prime }$ and then decompose $E(u)$ in each factor).", "Thus, $(Q_x, \\delta _x)$ decomposes as a product $\\prod _{\\psi } (Q^{\\psi }_x, \\delta ^{\\psi }_x)$ where each pair defines a point $z_{\\psi }$ of the affine Grassmanian of $G_{F^{\\prime }}$ .", "The quotient $\\mathfrak {D}_{Q_x}$ decomposes compatibly as $\\prod _{\\psi } \\mathfrak {D}_{Q^{\\psi }_x}$ .", "We are reduced then to a computation for a point $z_{\\psi } \\in \\operatorname{Gr}_{G_{F^{\\prime }}}(F^{\\prime })$ .", "Without loss of generality, we can assume $G_{F^{\\prime }}$ is split.", "Up to translation by the positive loop group (which induces an isomorphism on filtrations), $z_{\\psi }$ is the image $[g]$ for some $g \\in T(F^{\\prime }((t)))$ where $T$ is maximal split torus of $G_{F^{\\prime }}$ .", "Using the weight space decomposition for $T$ on any representation $W$ , one can compute directly that $\\mathfrak {D}_{Q^{\\psi }_x}$ is a tensor exact functor.", "For more details, see [27].", "Definition 3.3.12 Let $F^{\\prime }$ be any finite extension of $F$ with ring of integers $\\Lambda ^{\\prime }$ .", "We say a $G$ -Kisin module $(\\mathfrak {P}_{\\Lambda ^{\\prime }}, \\phi _{\\Lambda ^{\\prime }})$ over $\\Lambda ^{\\prime }$ has $p$ -adic Hodge type $\\mu $ if the $G_F$ -filtration associated to $\\mathfrak {P}_{\\Lambda ^{\\prime }} [1/p]$ as above has type $\\mu $ .", "Theorem 3.3.13 Assume that $F = F_{[\\mu ]}$ .", "Let $R$ be any complete local Noetherian $\\Lambda $ -algebra with finite residue field which is $\\Lambda $ -flat and reduced.", "Then $\\mathfrak {P}_R \\in D^{[a,b]}_{\\mathfrak {P}_{\\mathbb {F}}}(R)$ lies in $D^{\\mu }_{\\mathfrak {P}_{\\mathbb {F}}}(R)$ if and only if for all finite extensions $F^{\\prime }/F$ and all homomorphisms $x:R \\rightarrow F^{\\prime }$ , the $G_{F}$ -filtration $\\mathfrak {D}_x$ has type less than or equal to $[\\mu ]$ .", "Choose a lift $\\widetilde{y}$ of $\\mathfrak {P}_R$ to $\\widetilde{D}^{[a,b], (N)}_{\\mathfrak {P}_{\\mathbb {F}}}(R)$ .", "Clearly, $\\mathfrak {P}_R \\in D^{\\mu }_{\\mathfrak {P}_{\\mathbb {F}}}(R)$ if and only if $\\tilde{y} \\in \\widetilde{D}^{(N), \\mu }_{\\mathfrak {P}_{\\mathbb {F}}}(R)$ which happens if and only $\\Psi (\\tilde{y}) \\in \\overline{D}^{\\mu }_{Q_{\\mathbb {F}}}(R)$ .", "Let $R^{\\mu }$ be the quotient of $R$ representing the fiber product $\\mathrm {Spf}\\ R \\times _{\\overline{D}_{Q_{\\mathbb {F}}}^{[a,b]}} \\overline{D}^{\\mu }_{Q_{\\mathbb {F}}}$ .", "To show that $R^{\\mu } = R$ , it suffices to show that $\\mathrm {Spec}\\ R^{\\mu }[1/p]$ contains all closed points of $\\mathrm {Spec}\\ R[1/p]$ since $R$ is flat and $R[1/p]$ is reduced and Jacobson.", "The groupoid $\\overline{D}^{\\mu }_{Q_{\\mathbb {F}}}$ is represented by a completed stalk on the local model $M(\\mu ) \\subset \\operatorname{Gr}_{G}^{E(u), W}$ so that for any $x:R \\rightarrow F^{\\prime }$ , $\\Psi (\\widetilde{y})[1/p]$ defines a $F^{\\prime }$ -point $(Q_x, \\delta _x)$ of $\\operatorname{Gr}_{G}^{E(u), W}$ .", "Since $M(\\mu )(F^{\\prime }) = S(\\mu )(F^{\\prime })$ , $(Q_x, \\delta _x) \\in S(\\mu )(F^{\\prime })$ if and only if the filtration $\\mathfrak {D}_{Q_x}$ has type $\\le [\\mu ]$ ([27]).", "The proof of Lemma REF shows that the two filtrations agree, i.e., $\\mathfrak {D}_x \\cong \\mathfrak {D}_{Q_x}.$ Thus, $x$ factors through $R^{\\mu }$ exactly when the type of the filtration $\\mathfrak {D}_x$ is less than or equal to $[\\mu ]$ .", "Fix a continuous representation $\\overline{\\eta }:\\Gamma _K \\rightarrow G(\\mathbb {F})$ .", "Let $R^{[a,b], \\operatorname{cris}}_{\\overline{\\eta }}$ be the universal framed $G$ -valued crystalline deformation ring with Hodge-Tate weights in $[a,b]$ , and let $\\Theta :X^{[a,b], \\operatorname{cris}}_{\\overline{\\eta }} \\rightarrow \\mathrm {Spec}\\ R^{[a,b], \\operatorname{cris}}_{\\overline{\\eta }}$ be as in REF .", "Definition 3.3.14 Assume $F = F_{[\\mu ]}$ .", "Define $R^{\\operatorname{cris}, \\le \\mu }_{\\overline{\\eta }}$ to be the flat closure of the connected components of $\\mathrm {Spec}\\ R^{[a,b], \\operatorname{cris}}_{\\overline{\\eta }}[1/p]$ with type $\\le \\mu $ (see REF ).", "Define $X^{\\operatorname{cris}, \\le \\mu }_{\\overline{\\eta }}$ to be the flat closure in $X^{[a,b], \\operatorname{cris}}_{\\overline{\\eta }}$ of the same connected components (since $\\Theta [1/p]$ is an isomorphism).", "Corollary 3.3.15 Let $X^{\\operatorname{cris}, \\le \\mu }_{\\overline{\\eta }}$ be as in Definition $\\ref {leqmu}$ .", "A point $\\overline{x} \\in X^{\\operatorname{cris}, \\le \\mu }_{\\overline{\\eta }}(\\mathbb {F}^{\\prime })$ corresponds to a $G$ -Kisin lattice $\\mathfrak {P}_{\\mathbb {F}^{\\prime }}$ over $\\mathfrak {S}_{\\mathbb {F}^{\\prime }}$ .", "The deformation problem $D^{\\operatorname{cris}, \\mu }_{\\overline{x}}$ which assigns to any $A \\in \\mathcal {C}_{\\Lambda \\otimes _{\\mathbb {Z}_p} W(\\mathbb {F}^{\\prime })}$ the set of isomorphisms classes of triples $\\lbrace (y:R^{\\operatorname{cris}, \\le \\mu }_{\\overline{\\eta }} \\rightarrow A, \\mathfrak {P}_A \\in D^{\\mu }_{\\mathfrak {P}_{\\mathbb {F}^{\\prime }}}(A), \\delta _A:T_{G, \\mathfrak {S}_A}(\\mathfrak {P}_A) \\cong \\eta _y|_{\\Gamma _{\\infty }})\\rbrace $ is representable.", "Furthermore, if $\\widehat{\\mathcal {O}}^{\\mu }_{\\overline{x}}$ is the completed local ring of $X^{\\operatorname{cris}, \\le \\mu }_{\\overline{\\eta }}$ at $\\overline{x}$ , then the natural map $\\mathrm {Spf}\\ \\widehat{\\mathcal {O}}^{\\mu }_{\\overline{x}} \\rightarrow D^{\\operatorname{cris}, \\mu }_{\\overline{x}}$ is a closed immersion which is an isomorphism modulo $p$ -power torsion.", "Without loss of generality, we can replace $\\Lambda $ by $\\Lambda \\otimes _{W(\\mathbb {F})} W(\\mathbb {F}^{\\prime })$ .", "By construction and Proposition REF , for any $A \\in \\mathcal {C}_{\\Lambda }$ , the deformation functor $D^{\\operatorname{cris}, \\mu , \\mathrm {bc}}_{\\overline{x}}(A) = \\lbrace y:R^{\\operatorname{cris}, \\le \\mu }_{\\overline{\\eta }} \\rightarrow A, \\mathfrak {P}_A \\in D^{[a,b]}_{\\mathfrak {P}_{\\mathbb {F}^{\\prime }}}(A), \\delta _A:T_{G, \\mathfrak {S}_A}(\\mathfrak {P}_A) \\cong \\eta _y|_{\\Gamma _{\\infty }}\\rbrace /\\cong $ is representable.", "That is, $D^{\\operatorname{cris}, \\mu , \\mathrm {bc}}_{\\overline{x}}$ represents the completed stalk at a point of the fiber product $X^{[a,b], \\operatorname{cris}}_{\\overline{\\eta }} \\times _{\\mathrm {Spec}\\ R^{[a,b], \\operatorname{cris}} _{\\overline{\\eta }}} \\mathrm {Spec}\\ R^{\\operatorname{cris}, \\le \\mu }_{\\overline{\\eta }}$ .", "Since $D^{\\mu }_{\\mathfrak {P}_{\\mathbb {F}^{\\prime }}} \\subset D^{[a,b]}_{\\mathfrak {P}_{\\mathbb {F}^{\\prime }}}$ is closed so is $D^{\\operatorname{cris}, \\mu }_{\\overline{x}} \\subset D^{\\operatorname{cris}, \\mu , \\mathrm {bc}}_{\\overline{x}}$ and hence $D^{\\operatorname{cris}, \\mu }_{\\overline{x}}$ is representable by $R^{\\operatorname{cris}, \\mu }_{\\overline{x}}$ .", "To see that the closed immersion $\\mathrm {Spf}\\ \\widehat{\\mathcal {O}}^{\\mu }_{\\overline{x}} \\rightarrow D^{\\operatorname{cris}, \\mu , \\mathrm {bc}}_{\\overline{x}}$ factors through $D^{\\operatorname{cris}, \\mu }_{\\overline{x}}$ it suffices to show that the “universal\" lattice $\\mathfrak {P}_{\\widehat{\\mathcal {O}}^{\\mu }_{\\overline{x}}} \\in D^{[a,b]}_{\\mathfrak {P}_{\\mathbb {F}^{\\prime }}}(\\widehat{\\mathcal {O}}^{\\mu }_{\\overline{x}})$ lies in $D^{\\mu }_{\\mathfrak {P}_{\\mathbb {F}^{\\prime }}}(\\widehat{\\mathcal {O}}^{\\mu }_{\\overline{x}})$ .", "By Theorem REF and REF , $\\Theta [1/p]$ is an isomorphism.", "Furthermore, by [1], $R^{[a,b], \\operatorname{cris}}_{\\overline{\\eta }}[1/p]$ and $R^{\\operatorname{cris}, \\le \\mu }_{\\overline{\\eta }}[1/p]$ are formally smooth over $F$ .", "Hence, $\\widehat{\\mathcal {O}}^{\\mu }_{\\overline{x}}$ satisfies the hypotheses of Theorem REF .", "By Theorem REF , we are reducing to showing that for any finite $F^{\\prime }/F$ and any homomorphism $x:\\widehat{\\mathcal {O}}^{\\mu }_{\\overline{x}} \\rightarrow F^{\\prime }$ the filtration $\\mathfrak {D}_x$ corresponding to the base change $\\mathfrak {P}_x := \\mathfrak {P}_{\\widehat{\\mathcal {O}}^{\\mu }_{\\overline{x}}} \\otimes _x F^{\\prime }$ has type less than or equal to $\\mu $ .", "The homomorphism $x$ corresponds to closed point of $\\mathrm {Spec}\\ R^{\\operatorname{cris}, \\le \\mu }_{\\overline{\\eta }}[1/p]$ , i.e., a crystalline representation $\\rho _x$ with $p$ -adic Hodge type $\\le \\mu $ .", "Furthermore, $\\mathfrak {P}_x$ is the unique $(\\mathfrak {S}_{F^{\\prime }}, \\varphi )$ -module of bounded height associated to $\\rho _x$ .", "By Proposition REF , the de Rham $\\mathcal {F}^{\\mathrm {dR}}_{\\rho _x}$ filtration associated to $\\rho _x$ is isomorphic to the filtration $\\mathfrak {D}_x$ associated to $(\\mathfrak {P}_x, \\phi _x)$ .", "Thus $\\mathfrak {D}_x$ has type $\\le \\mu $ for all points $x$ and so $\\mathfrak {P}_{\\widehat{\\mathcal {O}}^{\\mu }_{\\overline{x}}} \\in D^{\\mu }_{\\mathfrak {P}_{\\mathbb {F}^{\\prime }}}(\\widehat{\\mathcal {O}}^{\\mu }_{\\overline{x}})$ by Theorem REF .", "By the argument above, $\\mathrm {Spec}\\ \\widehat{\\mathcal {O}}^{\\mu }_{\\overline{x}}$ and $\\mathrm {Spec}\\ R^{\\operatorname{cris}, \\mu }_{\\overline{x}}$ have the same $F^{\\prime }$ -points for any finite extension of $F$ .", "Since $R^{\\operatorname{cris}, \\le \\mu }_{\\overline{\\eta }}[1/p]$ is formally smooth over $F$ , the kernel of $R^{\\operatorname{cris}, \\mu }_{\\overline{x}} \\rightarrow \\widehat{\\mathcal {O}}^{\\mu }_{\\overline{x}}$ is $p$ -power torsion.", "Remark 3.3.16 In fact, Corollary REF holds as well for semistable deformation rings with $p$ -adic Hodge type $\\le \\mu $ .", "To apply Theorem REF and make the final deduction, we needed that the generic fiber of the crystalline deformation ring was reduced (to argue at closed points).", "This is true for $G$ -valued semistable deformation rings by the main result of [4]." ], [ "Local analysis", "In this section, we analyze finer properties of crystalline $G$ -valued deformation rings with minuscule $p$ -adic Hodge type.", "The techniques in this section are inspired by [24] and [31].", "We develop a theory of $(\\varphi , \\widehat{\\Gamma })$ -modules with $G$ -structure and our main result, Theorem REF , is stated in these terms.", "However, the idea is the following: given a $G$ -Kisin module $(\\mathfrak {P}_A, \\phi _A)$ over some finite $\\Lambda $ -algebra $A$ , we get a representation of $\\Gamma _{\\infty }$ via the functor $T_{G, \\mathfrak {S}_A}$ .", "In general, this representation need not extend (and certainly not in a canonical way) to a representation of the full Galois group $\\Gamma _K$ .", "When $G = \\mathrm {GL}_n$ and $\\mathfrak {P}_A$ has height in $[0, 1]$ then via the equivalence between Kisin modules with height in $[0,1]$ and finite flat group schemes [25], one has a canonical extension to $\\Gamma _K$ which is flat.", "We show (at least when $A$ is a $\\Lambda $ -flat domain) that the same holds for $G$ -Kisin modules of minuscule type: there exists a canonical extension to $\\Gamma _K$ which is crystalline.", "This is stated precisely in Corollary REF .", "We end by applying this result to identify the connected components of $G$ -valued crystalline deformation rings with the connected components of a moduli space of $G$ -Kisin modules (Corollary REF )." ], [ "Minuscule cocharacters", "We begin with some preliminaries on minuscule cocharacters and adjoint representations which we use in our finer analysis with $(\\varphi , \\widehat{\\Gamma })$ -modules in the subsequent sections.", "Let $H$ be a reductive group over field $\\kappa $ .", "The conjugation action of $H$ on itself gives a representation $ \\operatorname{Ad}:H \\rightarrow \\mathrm {GL}(\\operatorname{Lie}(H)).$ This is an algebraic representation so for any $\\kappa $ -algebra $R$ , $H(R)$ acts on $\\operatorname{Lie}(H_R) = \\operatorname{Lie}H \\otimes _{\\kappa } R$ .", "We will use $\\operatorname{Ad}$ to denote these actions as well.", "We can define $\\operatorname{Ad}$ for $G$ over $\\mathrm {Spec}\\ \\Lambda $ in the same way.", "Definition 4.1.1 Any cocharacter $\\lambda :\\operatorname{\\mathbb {G}_m}\\rightarrow H$ gives a grading on $\\operatorname{Lie}H$ defined by $\\operatorname{Lie}H (i) := \\lbrace Y \\in \\operatorname{Lie}H \\mid \\operatorname{Ad}(\\lambda (a)) Y = a^i Y \\rbrace .$ A cocharacter $\\lambda $ is called minuscule if $\\operatorname{Lie}H(i) = 0$ for $i \\notin \\lbrace -1, 0, 1 \\rbrace $ .", "Minuscule cocharacters were studied by Deligne [11] in connection with the theory of Shimura varieties.", "A detailed exposition of their main properties can be found §1 of [18].", "Assume now that $H$ is split and fix a maximal split torus $T$ contained in Borel subgroup $B$ .", "This gives rise to a set of simple roots $\\Delta $ and a set of simple coroots $\\Delta ^{\\vee }$ .", "In each conjugacy class of cocharacters, there is a unique dominant cocharacter valued in $T$ .", "The set of dominant cocharacters is denoted by $X_*(T)^{+}$ .", "Recall the Bruhat (partial) ordering on $X_*(T)^{+}$ : given two dominant cocharacter $\\mu , \\mu ^{\\prime }:\\operatorname{\\mathbb {G}_m}\\rightarrow T$ , we say $\\mu ^{\\prime } \\le \\mu $ if $\\mu - \\mu ^{\\prime } = \\sum _{\\alpha \\in \\Delta ^{\\vee }} n_{\\alpha } \\alpha $ with $n_{\\alpha } \\ge 0$ .", "Proposition 4.1.2 Let $\\mu $ be a dominant minuscule cocharacter.", "Then there are no dominant $\\mu ^{\\prime }$ such that $\\mu ^{\\prime } < \\mu $ in the Bruhat order.", "See Exercise 24 from Chapter IV.1 of [5].", "Proposition 4.1.3 If $\\mu $ is a minuscule cocharacter, then the $($ open$)$ affine Schubert variety $S^0(\\mu )$ is equal to $S(\\mu )$ .", "Furthermore, $S(\\mu )$ is smooth and projective.", "In fact, $S(\\mu ) \\cong H/P(\\mu )$ where $P(\\mu )$ is a parabolic subgroup associated to the cocharacter $\\mu $ .", "Since the closure $S(\\mu ) = \\cup _{\\mu ^{\\prime } \\le \\mu } S^0(\\mu ^{\\prime })$ ([40]), the first part follows from Proposition REF .", "For the remaining facts, we refer to discussion after [36] and [27].", "For any $\\mu :\\operatorname{\\mathbb {G}_m}\\rightarrow T$ , we get an induced map $\\operatorname{\\mathbb {G}_m}(\\kappa (\\!(t)\\!))", "\\rightarrow T(\\kappa (\\!(t)\\!))", "\\subset H(\\kappa (\\!(t)\\!", "))$ on loop groups.", "We let $\\mu (t)$ denote the image of $t \\in \\kappa (\\!(t)\\!", ")^{\\times }$ .", "Proposition 4.1.4 For any $X \\in \\operatorname{Lie}H \\otimes _{\\kappa } \\kappa [\\![t]\\!", "]$ , we have $\\operatorname{Ad}(\\mu (t))(X) \\in \\frac{1}{t}(\\operatorname{Lie}H \\otimes _{\\kappa } \\kappa [\\![t]\\!", "]).$ As in Definition REF , we can decompose $\\operatorname{Lie}H = \\operatorname{Lie}H(-1) \\oplus \\operatorname{Lie}H \\oplus \\operatorname{Lie}H(1)$ .", "Then $\\operatorname{Ad}(\\mu (t))$ acts on $\\operatorname{Lie}H(i) \\otimes \\kappa (\\!(t)\\!", ")$ by multiplication by $t^{i}$ .", "The largest denominator is then $t^{-1}$ ." ], [ "$(\\varphi , \\widehat{\\Gamma })$ -modules with {{formula:e16dcfb4-67f6-42ee-b341-a8abe4217f42}} -structure", "We review Liu's theory of $(\\varphi , \\widehat{G})$ as in [30], [9].", "We will call them $(\\varphi , \\widehat{\\Gamma })$ -modules to avoid confusion with the algebraic group $G$ .", "The theory of $(\\varphi , \\widehat{\\Gamma })$ -modules is an adaptation of the theory of $(\\varphi , \\Gamma )$ -modules to the non-Galois extension $K_{\\infty } = K(\\pi ^{1/p}, \\pi ^{1/p^2}, \\ldots )$ .", "The $\\widehat{\\Gamma }$ refers to an additional structure added to a Kisin module which captures the full action of $\\Gamma _K$ as opposed to just the subgroup $\\Gamma _{\\infty } := \\mathrm {Gal}(\\overline{K}/K_{\\infty })$ .", "The main theorem in [30] is an equivalence of categories between (torsion-free) $(\\varphi , \\widehat{\\Gamma })$ -modules and $\\Gamma _K$ -stable lattices in semi-stable $\\mathbb {Q}_p$ -representations.", "Let $\\widetilde{\\mathbf {E}}^+$ denote the perfection of $\\mathcal {O}_{\\overline{K}}/(p)$ .", "There is a unique surjective map $\\Theta :W(\\widetilde{\\mathbf {E}}^+) \\rightarrow \\widehat{\\mathcal {O}}_{\\overline{K}}$ which lifts the projection $\\widetilde{\\mathbf {E}}^+ \\rightarrow \\mathcal {O}_{\\overline{K}}/(p)$ .", "The compatible system $(\\pi ^{1/p^n})_{n \\ge 0}$ of the $p^n$ th roots of $\\pi $ defines an element $\\underline{\\pi }$ of $\\widetilde{\\mathbf {E}}^+$ .", "Let $[\\underline{\\pi }]$ denote the Teichmüller representative in $W(\\widetilde{\\mathbf {E}}^+)$ .", "There is an embedding $\\mathfrak {S}\\hookrightarrow W(\\widetilde{\\mathbf {E}}^+)$ defined by $u \\mapsto [\\underline{\\pi }]$ which is compatible with the Frobenii.", "If $\\widetilde{\\mathbf {E}}$ is the fraction field of $\\widetilde{\\mathbf {E}}^+$ , then $W(\\widetilde{\\mathbf {E}}^+) \\subset W(\\widetilde{\\mathbf {E}})$ .", "The embedding $\\mathfrak {S}\\hookrightarrow W(\\widetilde{\\mathbf {E}}^+)$ extends to an embedding $\\mathcal {O}_{\\mathcal {E}} \\hookrightarrow W(\\widetilde{\\mathbf {E}}).$ As before, let $K_{\\infty } = \\bigcup K(\\pi ^{1/p^n})$ .", "Set $K_{p^{\\infty }} := \\bigcup K(\\zeta _{p^n})$ where $\\zeta _{p^n}$ is a primitive $p^n$ th root of unity.", "Denote the compositum of $K_{\\infty }$ and $K_{p^{\\infty }}$ by $K_{\\infty , p^{\\infty }}$ ; $K_{\\infty , p^{\\infty }}$ is Galois over $K$ .", "Definition 4.2.1 Define $\\widehat{\\Gamma } := \\mathrm {Gal}(K_{\\infty , p^{\\infty }}/K) \\text{ and } \\widehat{\\Gamma }_{\\infty } := \\mathrm {Gal}(K_{\\infty , p^{\\infty }}/K_{\\infty }).$ There is a subring $\\widehat{R} \\subset W(\\widetilde{\\mathbf {E}}^+)$ which plays a central role in the theory of $(\\varphi , \\widehat{\\Gamma })$ -modules.", "The definition can be found on page 5 of [30].", "The relevant properties of $\\widehat{R}$ are: $\\widehat{R}$ is stable by the Frobenius on $W(\\widetilde{\\mathbf {E}}^+)$ ; $\\widehat{R}$ contains $\\mathfrak {S}$ ; $\\widehat{R}$ is stable under the action of the Galois group $\\Gamma _K$ and $\\Gamma _K$ acts through the quotient $\\widehat{\\Gamma }$ .", "For any $\\mathbb {Z}_p$ -algebra $A$ , set $\\widehat{R}_A := \\widehat{R} \\otimes _{\\mathbb {Z}_p} A$ with a Frobenius induced by the Frobenius on $\\widehat{R}$ .", "Similarly, define $W(\\widetilde{\\mathbf {E}}^+)_A := W(\\widetilde{\\mathbf {E}}^+) \\otimes _{\\mathbb {Z}_p} A$ and $W(\\widetilde{\\mathbf {E}})_A := W(\\widetilde{\\mathbf {E}}) \\otimes _{\\mathbb {Z}_p} A$ .", "For any $\\mathfrak {S}_A$ -module $\\mathfrak {M}_A$ , define $\\widehat{\\mathfrak {M}}_A := \\widehat{R}_A \\otimes _{\\varphi , \\mathfrak {S}_A} \\mathfrak {M}_A = \\widehat{R}_A \\otimes _{\\mathfrak {S}_A} \\varphi ^*(\\mathfrak {M}_A)$ and $\\widetilde{\\mathfrak {M}}_A := W(\\widetilde{\\mathbf {E}}^+)_A \\otimes _{\\varphi , \\mathfrak {S}_A} \\mathfrak {M}_A = W(\\widetilde{\\mathbf {E}}^+)_A \\otimes _{\\widehat{R}_A} \\widehat{\\mathfrak {M}}_A.$ Recall that $\\varphi ^*(\\mathfrak {M}_A) := \\mathfrak {S}_A \\otimes _{\\varphi , \\mathfrak {S}_A} \\mathfrak {M}_A$ and that the linearized Frobenius is a map $\\phi _{\\mathfrak {M}_A}:\\varphi ^*(\\mathfrak {M}_A) \\rightarrow \\mathfrak {M}_A$ (when $\\mathfrak {M}_A$ has height in $[0, \\infty )$ ).", "If $\\mathfrak {M}_A$ is a projective $\\mathfrak {S}_A$ -module then, by Lemma 3.1.1 in [9], $\\varphi ^*(\\mathfrak {M}_A) \\subset \\widehat{\\mathfrak {M}}_A \\subset \\widetilde{\\mathfrak {M}}_A$ .", "Although the map $m \\mapsto 1 \\otimes m$ from $\\mathfrak {M}_A$ to $\\widehat{\\mathfrak {M}}_A$ is not $\\mathfrak {S}_A$ -linear, it is injective when $\\mathfrak {M}_A$ is $\\mathfrak {S}_A$ -projective.", "The image is a $\\varphi (\\mathfrak {S}_A)$ -submodule of $\\widehat{\\mathfrak {M}}_A$ .", "We will think of $\\mathfrak {M}_A$ inside of $\\widehat{\\mathfrak {M}}_A$ in this way.", "Finally, for any étale $(\\mathcal {O}_{\\mathcal {E}, A}, \\varphi )$ -module $\\mathcal {M}_A$ , we define $\\widetilde{\\mathcal {M}}_A := W(\\widetilde{\\mathbf {E}})_A \\otimes _{\\varphi , \\mathcal {O}_{\\mathcal {E}, A}} \\mathcal {M}_A = W(\\widetilde{\\mathbf {E}})_A \\otimes _{\\mathcal {O}_{\\mathcal {E}, A}} \\varphi ^*(\\mathcal {M}_A)$ with semi-linear Frobenius extending the Frobenius on $\\mathcal {M}_A$ .", "To summarize, for any Kisin module $(\\mathfrak {M}_A, \\phi _A)$ , we have the following diagram ${(\\mathfrak {M}_A, \\phi _A) @{~>}[d] @{~>}[r] & \\widehat{\\mathfrak {M}}_A @{~>}[r] & \\widetilde{\\mathfrak {M}}_A @{~>}[d] \\\\(\\mathcal {M}_A, \\phi _A) @{~>}[rr] & & (\\widetilde{\\mathcal {M}}_A, \\widetilde{\\phi }_A).", "\\\\}$ Now, let $\\gamma \\in \\widehat{\\Gamma }$ and let $\\widehat{\\mathfrak {M}}_A$ be an $\\widehat{R}_A$ -module.", "A map $g:\\widehat{\\mathfrak {M}}_A \\rightarrow \\widehat{\\mathfrak {M}}_A$ is a $\\gamma $ -semilinear if $g(a m) = \\gamma (a) g(m)$ for any $a \\in \\widehat{R}_A, m \\in \\widehat{\\mathfrak {M}}_A$ .", "A (semilinear) $\\widehat{\\Gamma }$ -action on $\\widehat{\\mathfrak {M}}_A$ is a $\\gamma $ -semilinear map $g_{\\gamma }$ for each $\\gamma \\in \\widehat{\\Gamma }$ such that $g_{\\gamma ^{\\prime }} \\circ g_{\\gamma } = g_{\\gamma ^{\\prime } \\gamma }$ as $(\\gamma ^{\\prime } \\gamma )$ -semilinear morphisms.", "A (semilinear) $\\widehat{\\Gamma }$ -action on $\\widehat{\\mathfrak {M}}_A$ extends in the natural way to a (semilinear) $\\Gamma _K$ -action on $\\widetilde{\\mathfrak {M}}_A$ and on $\\widetilde{\\mathcal {M}}_A$ .", "For any local Artinian $\\mathbb {Z}_p$ -algebra $A$ , choose a $\\mathbb {Z}_p$ -module isomorphism $A \\cong \\bigoplus \\mathbb {Z}/p^{n_i} \\mathbb {Z}$ so that as a $W(\\widetilde{\\mathbf {E}})$ -module, $W(\\widetilde{\\mathbf {E}})_A \\cong \\bigoplus W_{n_i} (\\widetilde{\\mathbf {E}})$ .", "We equip $W(\\widetilde{\\mathbf {E}})_A$ with the product topology where $W_{n_i}(\\widetilde{\\mathbf {E}})$ has a topology induced by the isomorphism $W_{n_i}(\\widetilde{\\mathbf {E}}) \\cong \\widetilde{\\mathbf {E}}^{n_i}$ given by Witt components (see §4.3 of [6] for more details on the topology of $\\widetilde{\\mathbf {E}}$ ).", "We can similarly define a topology on $W(\\widetilde{\\mathbf {E}}^+)_A$ using the topology on $\\widetilde{\\mathbf {E}}^+$ , and it is clear that this is the same as the subspace topology from the inclusion $W(\\widetilde{\\mathbf {E}}^+)_A \\subset W(\\widetilde{\\mathbf {E}})_A$ .", "Finally, we give $\\widehat{R}_A$ the subspace topology from the inclusion $\\widehat{R}_A \\subset W(\\widetilde{\\mathbf {E}}^+)_A$ .", "The same procedure works for $A$ finite flat over $\\mathbb {Z}_p$ .", "A $\\widehat{\\Gamma }$ -action on $\\widehat{\\mathfrak {M}}_A$ is continuous if for any basis (equivalently for all bases) of $\\widehat{\\mathfrak {M}}_A$ the induced map $\\widehat{\\Gamma } \\rightarrow \\mathrm {GL}_r(\\widehat{R}_A)$ is continuous where $r$ is the rank of $\\widehat{\\mathfrak {M}}_A$ (such a basis exists by [24]).", "Definition 4.2.2 Let $A$ be a finite $\\mathbb {Z}_p$ -algebra.", "A $(\\varphi , \\widehat{\\Gamma })$ -module with height in $[a,b]$ over $A$ is a triple $(\\mathfrak {M}_A, \\phi _{\\mathfrak {M}_A}, \\widehat{\\Gamma })$ , where $(\\mathfrak {M}_A, \\phi _{\\mathfrak {M}_A}) \\in \\operatorname{Mod}^{\\varphi , [a,b]}_{\\mathfrak {S}_A}$ ; $\\widehat{\\Gamma }$ is a continuous (semilinear) $\\widehat{\\Gamma }$ -action on $\\widehat{\\mathfrak {M}}_A$ ; The $\\Gamma _K$ -action on $\\widetilde{\\mathfrak {M}}_A$ commutes with $\\widetilde{\\phi }_{\\mathfrak {M}_A}$ (as endomorphisms of $\\widetilde{\\mathcal {M}}_A$ ); Regarding $\\mathfrak {M}_A$ as a $\\varphi (\\mathfrak {S}_A)$ -submodule of $\\widehat{\\mathfrak {M}}_A$ , we have $\\mathfrak {M}_A \\subset \\widehat{\\mathfrak {M}}^{\\widehat{\\Gamma }_{\\infty }}_A$ ; $\\widehat{\\Gamma }$ acts trivially on $\\widehat{\\mathfrak {M}}_A/I_+(\\widehat{\\mathfrak {M}}_A)$ (see §3.1 of [9] for the definition of $I_+(\\widehat{\\mathfrak {M}}_A)$ ).", "We often refer to the additional data of a $(\\varphi , \\widehat{\\Gamma })$ -module on a Kisin module as a $\\widehat{\\Gamma }$ -structure.", "Remark 4.2.3 Although we allow arbitrary height $[a,b]$ (in particular, negative height), the ring $\\widehat{R}$ is still sufficient for defining the $\\widehat{\\Gamma }$ -action.", "This follows from the fact that the $\\widehat{\\Gamma }$ -action on $\\mathfrak {S}(1)$ is given by $\\widehat{c}$ (see [30]) which is a unit in $\\widehat{R}$ .", "See also [27].", "Proposition 4.2.4 Choose $(\\mathfrak {M}_A, \\phi _{\\mathfrak {M}_A}) \\in \\operatorname{Mod}^{\\varphi , [a,b]}_{\\mathfrak {S}_A}$ of rank $r$ .", "Fix a basis $\\lbrace f_i \\rbrace $ for $\\mathfrak {M}_A$ .", "Let $C^{\\prime }$ be the matrix for $\\phi _{\\mathfrak {M}_A}$ with respect to $\\lbrace 1 \\otimes _{\\varphi } f_i\\rbrace $ .", "Then a $\\widehat{\\Gamma }$ -structure on $\\mathfrak {M}_A$ is the same as a continuous map $B_{\\bullet }:\\widehat{\\Gamma } \\rightarrow \\mathrm {GL}_r(\\widehat{R}_A)$ such that $C^{\\prime } \\cdot \\varphi (B_{\\gamma }) = B_{\\gamma } \\cdot \\gamma (C^{\\prime }) $ in $\\mathrm {Mat}(W(\\widetilde{\\mathbf {E}})_A)$ for all $\\gamma \\in \\widehat{\\Gamma }$ ; $B_{\\gamma } = \\mathrm {Id}$ for all $\\gamma \\in \\widehat{\\Gamma }_{\\infty }$ ; $B_{\\gamma } \\equiv \\mathrm {Id}\\mod {I}_+(\\widehat{R})_A$ for all $\\gamma \\in \\widehat{\\Gamma }$ ; $B_{\\gamma \\gamma ^{\\prime }} = B_{\\gamma } \\cdot \\gamma (B_{\\gamma ^{\\prime }})$ for all $\\gamma , \\gamma ^{\\prime } \\in \\widehat{\\Gamma }$ .", "Let $\\operatorname{Mod}^{\\varphi , [a,b], \\widehat{\\Gamma }}_{\\mathfrak {S}_A}$ denote the category of $(\\varphi , \\widehat{\\Gamma })$ -modules with height in $[a,b]$ over $A$ .", "A morphism between $(\\varphi , \\widehat{\\Gamma })$ -modules is a morphism in $\\operatorname{Mod}^{\\varphi , [a,b]}_{\\mathfrak {S}_A}$ that is $\\widehat{\\Gamma }$ -equivariant when extended to $\\widehat{R}_A$ .", "Let $\\operatorname{Mod}^{\\varphi , \\operatorname{bh}, \\widehat{\\Gamma }}_{\\mathfrak {S}_A} := \\bigcup _{h > 0} \\operatorname{Mod}^{\\varphi , [-h,h], \\widehat{\\Gamma }}_{\\mathfrak {S}_A}$ so $\\operatorname{Mod}^{\\varphi , \\operatorname{bh}, \\widehat{\\Gamma }}_{\\mathfrak {S}_A}$ has a natural tensor product operation which at the level of $\\operatorname{Mod}^{\\varphi , \\operatorname{bh}}_{\\mathfrak {S}_A}$ is tensor product of bounded height Kisin modules.", "The $\\widehat{\\Gamma }$ -structure on the tensor product is defined via $ \\widehat{R}_A \\otimes _{\\varphi , \\mathfrak {S}_A} (\\mathfrak {M}_A \\otimes _{\\mathfrak {S}_A} \\mathfrak {N}_A) \\cong (\\widehat{R}_A \\otimes _{\\varphi , \\mathfrak {S}_A} \\mathfrak {M}_A) \\otimes _{\\widehat{R}_A} (\\widehat{R}_A \\otimes _{\\varphi , \\mathfrak {S}_A} \\mathfrak {N}_A) = \\widehat{\\mathfrak {M}}_A \\otimes _{\\widehat{R}_A} \\widehat{\\mathfrak {N}}_A.$ One also defines a $\\widehat{\\Gamma }$ -structure on the dual $\\mathfrak {M}^{\\vee }_A := \\mathrm {Hom}_{\\mathfrak {S}_A}(\\mathfrak {M}_A, \\mathfrak {S}_A)$ in the natural way (see discussion after Proposition 9.1.5 [27]).", "Note that, unlike in other references (for example [33]), we do not include any Tate twist in our definition of duals.", "We will now relate these $(\\varphi , \\widehat{\\Gamma })$ -modules to $\\Gamma _K$ -representations.", "For this, we require that $A$ be $\\mathbb {Z}_p$ -finite and either $\\mathbb {Z}_p$ -flat or Artinian.", "Define a functor $\\widehat{T}_A$ from $\\operatorname{Mod}^{\\varphi , \\operatorname{bh}, \\widehat{\\Gamma }}_{\\mathfrak {S}_A}$ to Galois representations by $\\widehat{T}_A(\\widehat{\\mathfrak {M}}_A) := (W(\\widetilde{\\mathbf {E}}) \\otimes _{\\widehat{R}} \\widehat{\\mathfrak {M}}_A)^{\\widetilde{\\phi }_A = 1} = (\\widetilde{\\mathcal {M}}_A)^{\\widetilde{\\phi }_A = 1}.$ The semilinear $\\Gamma _K$ -action on $\\widetilde{\\mathcal {M}}_A$ commutes with $\\widetilde{\\phi }_A$ so $\\widehat{T}_A(\\widehat{\\mathfrak {M}}_A)$ is a $\\Gamma _K$ -stable $A$ -submodule of $W(\\widetilde{\\mathbf {E}}) \\otimes _{\\widehat{R}} \\widehat{\\mathfrak {M}}_A$ .", "We now recall the basic facts we will need about $\\widehat{T}_A$ : Proposition 4.2.5 Let $A$ be $\\mathbb {Z}_p$ -finite and either $\\mathbb {Z}_p$ -flat or Artinian.", "If $\\widehat{\\mathfrak {M}}_A \\in \\operatorname{Mod}^{\\varphi , \\operatorname{bh}, \\widehat{\\Gamma }}_{\\mathfrak {S}_A}$ , then there is a natural $A[\\Gamma _{\\infty }]$ -module isomorphism $\\theta _A:T_{\\mathfrak {S}_A}(\\mathfrak {M}_A) \\rightarrow \\widehat{T}_A(\\widehat{\\mathfrak {M}}_A).$ Furthermore, $\\theta _A$ is functorial with respect to morphisms in $\\operatorname{Mod}^{\\varphi , \\operatorname{bh}, \\widehat{\\Gamma }}_{\\mathfrak {S}_A}$ .", "$\\widehat{T}_A$ is an exact tensor functor from $\\operatorname{Mod}^{\\varphi , \\operatorname{bh}, \\widehat{\\Gamma }}_{\\mathfrak {S}_A}$ to $\\operatorname{Rep}_A(\\Gamma _K)$ which is compatible with duals.", "See Propositions 9.1.6 and 9.1.7 [27].", "We are now ready to add $G$ -structure to $(\\varphi , \\widehat{\\Gamma })$ -modules.", "Let $G$ be a connected reductive group over a $\\mathbb {Z}_p$ -finite and flat local domain $\\Lambda $ as in previous sections.", "Definition 4.2.6 Define $\\operatorname{GMod}^{\\varphi , \\widehat{\\Gamma }}_{\\mathfrak {S}_A}$ to be the category of faithful exact tensor functors $[\\mathop {\\mmlmultiscripts{\\operatorname{Rep}{\\Lambda }{\\mmlnone }\\mmlprescripts {\\mmlnone }{f}}}\\limits (G), \\operatorname{Mod}^{\\varphi , \\operatorname{bh}, \\widehat{\\Gamma }}_{\\mathfrak {S}_A}]^{\\otimes }$ .", "We will refer to these as $(\\varphi , \\widehat{\\Gamma })$ -modules with $G$ -structure.", "Recall the category $\\operatorname{GRep}_A(\\Gamma _K)$ from Definition REF .", "By Proposition REF (2), $\\widehat{T}_A$ induces a functor $\\widehat{T}_{G, A}:\\operatorname{GMod}^{\\varphi , \\widehat{\\Gamma }}_{\\mathfrak {S}_A} \\rightarrow \\operatorname{GRep}_A(\\Gamma _K).$ Furthermore, if $\\omega _{\\Gamma _{\\infty }}:\\operatorname{GRep}_A(\\Gamma _K) \\rightarrow \\operatorname{GRep}_A(\\Gamma _{\\infty })$ is the forgetful functor then there is an natural isomorphism $T_{G, \\mathfrak {S}_A} \\cong \\omega _{\\Gamma _{\\infty }} \\circ \\widehat{T}_{G, A}.$ The functor $\\widehat{T}_{G, A}$ behaves well with respect to base change along finite maps $A \\rightarrow A^{\\prime }$ by the same argument as in Proposition REF .", "We end this section by adding $G$ -structure to the main result of [30].", "For $A$ finite flat over $\\Lambda $ , an element $(P_A, \\rho _A)$ of $\\operatorname{GRep}_A(\\Gamma _{K})$ is semi-stable (resp.", "crystalline) if $\\rho _A[1/p]:\\Gamma _K \\rightarrow \\mathrm {Aut}_G(P_A) (A[1/p])$ is semi-stable (resp.", "crystalline).", "For $A$ a local domain, and $\\rho _A$ semi-stable, we say $\\rho _A$ has $p$ -adic Hodge type $\\mu $ if $\\rho _A[1/p]$ does for any trivialization of $P_A$ (see Definition REF ).", "Theorem 4.2.7 Let $F^{\\prime }$ be a finite extension of $F$ with ring of integers $\\Lambda ^{\\prime }$ .", "The functor $\\widehat{T}_{G, \\Lambda ^{\\prime }}$ induces an equivalence of categories between $\\operatorname{GMod}^{\\varphi , \\widehat{\\Gamma }}_{\\mathfrak {S}_{\\Lambda ^{\\prime }}}$ and the full subcategory of semi-stable representations of $\\operatorname{GRep}_{\\Lambda ^{\\prime }}(\\Gamma _K)$ .", "Using the Tannakian description of both categories, it suffices to show that $\\widehat{T}_{\\Lambda ^{\\prime }}$ defines a tensor equivalence between $\\operatorname{Mod}^{\\varphi , \\operatorname{bh}, \\widehat{\\Gamma }}_{\\mathfrak {S}_{\\Lambda ^{\\prime }}}$ and semi-stable representations of $\\Gamma _K$ on finite free $\\Lambda ^{\\prime }$ -modules.", "When $F = \\mathbb {Q}_p$ and the Hodge-Tate weights are negative (in our convention), this is Theorem 2.3.1 in [30].", "Note that [30] is using contravariant functors so that our $\\widehat{T}_{\\Lambda ^{\\prime }}$ is obtained by taking duals.", "The restriction on Hodge-Tate weights can be removed by twisting by $\\widehat{\\mathfrak {S}}(1)$ , the $(\\varphi , \\widehat{\\Gamma })$ -module corresponding to the inverse of the $p$ -adic cyclotomic character.", "To define a quasi-inverse to $\\widehat{T}_{\\Lambda ^{\\prime }}$ , let $L$ be a semi-stable $\\Gamma _K$ -representation on a finite free $\\Lambda ^{\\prime }$ -module.", "Forgetting the coefficients, [30] constructs a $\\widehat{\\Gamma }$ -structure $\\widehat{T}^{-1}(L)$ on the unique Kisin lattice in $\\underline{M}(L)$ .", "This $(\\varphi , \\widehat{\\Gamma })$ -module over $\\mathbb {Z}_p$ has an action of $\\Lambda ^{\\prime }$ by functoriality of the construction.", "By an argument as in [26], the resulting $\\mathfrak {S}_{\\Lambda ^{\\prime }}$ -module is projective and so this defines an object of $\\operatorname{Mod}^{\\varphi , \\operatorname{bh}, \\widehat{\\Gamma }}_{\\mathfrak {S}_{\\Lambda ^{\\prime }}}$ which we call $\\widehat{T}_{\\Lambda ^{\\prime }}^{-1}(L)$ .", "Finally, we appeal to Proposition I.4.4.2 in [43] to conclude that $\\widehat{T}_{\\Lambda ^{\\prime }}$ and $\\widehat{T}_{\\Lambda ^{\\prime }}^{-1}$ define a tensor equivalence of categories given that $\\widehat{T}_{\\Lambda ^{\\prime }}$ respects tensor products (Proposition REF )." ], [ "Faithfulness and existence result", "Fix an element $\\tau $ in $\\widehat{\\Gamma }$ such that $\\tau (\\underline{\\pi }) = \\underline{\\varepsilon } \\cdot \\underline{\\pi }$ where $\\underline{\\varepsilon }$ is a compatible system of primitive $p^n$ th roots of unity.", "If $p \\ne 2$ , then $\\tau $ is a topological generator for $\\widehat{\\Gamma }_{p^{\\infty }} := \\mathrm {Gal}(K_{\\infty , p^{\\infty }}/K_{p^{\\infty }})$ .", "If $p = 2$ , then some power of $\\tau $ will generate $\\widehat{\\Gamma }_{p^{\\infty }}$ .", "In both cases, $\\tau $ together with $\\widehat{\\Gamma }_{\\infty }$ topologically generate $\\widehat{\\Gamma }$ (see [30]).", "Given condition (4) in Definition REF the $\\widehat{\\Gamma }$ -action is determined by the action of $\\tau $ .", "Recall the element $\\mathfrak {t} \\in W(\\widetilde{\\mathbf {E}}^+)$ which is the period for $\\mathfrak {S}(1)$ in the sense that $\\varphi (\\mathfrak {t}) = c_0^{-1} E(u) \\mathfrak {t}$ .", "We will need a few structural results about $W(\\widetilde{\\mathbf {E}}^+)$ .", "Lemma 4.3.1 For any $\\widetilde{\\gamma } \\in \\Gamma _K$ , we have the following divisibilities in $W(\\widetilde{\\mathbf {E}}^+):$ $\\widetilde{\\gamma }(u) \\mid u, \\quad \\widetilde{\\gamma }(\\varphi (\\operatorname{\\mathfrak {t}})) \\mid \\varphi (\\operatorname{\\mathfrak {t}}), \\text{ and } \\quad \\widetilde{\\gamma }(E(u)) \\mid E(u).$ See [27].", "The $(\\varphi , \\widehat{\\Gamma })$ -modules which give rise to crystalline representations satisfy an extra divisibility condition on the action of $\\tau $ , i.e., [17] and [27].", "We call this the crystalline condition.", "Definition 4.3.2 An object $\\widehat{\\mathfrak {M}}_A \\in \\operatorname{Mod}^{\\varphi , [a,b], \\widehat{\\Gamma }}_{\\mathfrak {S}_A}$ is crystalline if for any $x \\in \\mathfrak {M}_A$ there exists $y \\in \\widetilde{\\mathfrak {M}}_A$ such that $\\tau (x) - x = \\varphi (\\mathfrak {t}) u^p y$ .", "Proposition 4.3.3 If $\\widehat{\\mathfrak {M}}_A$ is crystalline, then for all $x \\in \\mathfrak {M}_A$ and $\\gamma \\in \\widehat{\\Gamma }$ there exists $y \\in \\widetilde{\\mathfrak {M}}_A$ such that $\\gamma (x) - x = \\varphi (\\mathfrak {t}) u^p y$ .", "This is an easy calculation using that $\\widehat{\\Gamma }$ is topologically generated by $\\widehat{\\Gamma }_{\\infty }$ and $\\tau $ ([27]).", "Definition 4.3.4 We say an object $\\widehat{\\mathfrak {P}}_{A} \\in \\operatorname{GMod}^{\\varphi , [a,b], \\widehat{\\Gamma }}_{\\mathfrak {S}_A}$ is crystalline if $\\widehat{\\mathfrak {P}}_{A}(W)$ is crystalline for all $W \\in \\mathop {\\mmlmultiscripts{\\operatorname{Rep}{\\Lambda }{\\mmlnone }\\mmlprescripts {\\mmlnone }{f}}}\\limits (G)$ .", "For $\\widehat{\\mathfrak {P}}_{\\mathbb {F}} \\in \\operatorname{GMod}^{\\varphi , [a,b], \\widehat{\\Gamma }}_{\\mathfrak {S}_{\\mathbb {F}}}$ , define the crystalline $(\\varphi , \\widehat{\\Gamma })$ -module deformation groupoid over $\\mathcal {C}_{\\Lambda }$ by $D^{\\text{cris}, [a,b]}_{\\widehat{\\mathfrak {P}}_{\\mathbb {F}}}(A) = \\lbrace (\\widehat{\\mathfrak {P}}_A, \\psi _0) \\in D^{[a,b]}_{\\widehat{\\mathfrak {P}}_{\\mathbb {F}}} (A) \\mid \\widehat{\\mathfrak {P}}_A \\text{ is crystalline} \\rbrace $ for any $A \\in \\mathcal {C}_{\\Lambda }$ .", "Proposition 4.3.5 Let $F^{\\prime }$ be a finite extension of $F$ with ring of integers $\\Lambda ^{\\prime }$ .", "The equivalence from Theorem REF induces an equivalence between the full subcategory of crystalline objects in $\\operatorname{GMod}^{\\varphi , \\widehat{\\Gamma }}_{\\mathfrak {S}_{\\Lambda ^{\\prime }}}$ with the category of crystalline representations in $\\operatorname{GRep}_{\\Lambda ^{\\prime }}(\\Gamma _K)$ .", "It suffices to show that if $\\widehat{T}_A(\\widehat{\\mathfrak {P}}_{A}(W))$ is a lattice in a crystalline representation then $\\widehat{\\mathfrak {P}}_{A}(W)$ satisfies the crystalline condition.", "This only depends on the underlying $(\\varphi , \\widehat{\\Gamma })$ -module so we can take $A = \\mathbb {Z}_p$ .", "When $p > 2$ , this is proven in Corollary 4.10 in [17].", "The argument for $p = 2$ is essentially the same and was omitted only because in [17] they need further divisibilities on $(\\tau - 1)^n$ for which $p = 2$ becomes more complicated.", "Details can be found in [27].", "Choose a crystalline object $\\widehat{\\mathfrak {P}}_{\\mathbb {F}} \\in \\operatorname{GMod}^{\\varphi , [a,b], \\widehat{\\Gamma }}_{\\mathfrak {S}_{\\mathbb {F}}}$ .", "If $\\mathfrak {P}_{\\mathbb {F}}$ is the underlying $G$ -Kisin module of $\\widehat{\\mathfrak {P}}_{\\mathbb {F}}$ , then we would like to study the forgetful functor $\\widehat{\\Delta }:D^{\\text{cris}, [a,b]}_{\\widehat{\\mathfrak {P}}_{\\mathbb {F}}} \\rightarrow D^{[a,b]}_{\\mathfrak {P}_{\\mathbb {F}}}.$ More specifically, if $\\mu $ and $a, b$ are as in the discussion before Definition REF and $F = F_{[\\mu ]}$ , we consider $\\widehat{\\Delta }^{\\mu }: D^{\\text{cris}, \\mu }_{\\widehat{\\mathfrak {P}}_{\\mathbb {F}}} := D^{\\text{cris}, [a,b]}_{\\widehat{\\mathfrak {P}}_{\\mathbb {F}}} \\times _{ D^{[a,b]}_{\\mathfrak {P}_{\\mathbb {F}}}} D^{\\mu }_{\\mathfrak {P}_{\\mathbb {F}}} \\rightarrow D^{\\mu }_{\\mathfrak {P}_{\\mathbb {F}}}.$ We can now state our main theorem: Theorem 4.3.6 Assume that $p$ does not divide $\\pi _1(G^{\\mathrm {der}})$ where $G^{\\mathrm {der}}$ is the derived group of $G$ and that $F = F_{[\\mu ]}$ .", "If $\\mu $ is a minuscule geometric cocharacter of $\\mathrm {Res}_{(K \\otimes _{\\mathbb {Q}_p} F)/F} G_F$ then $\\widehat{\\Delta }^{\\mu }:D^{\\mathrm {cris}, \\mu }_{\\widehat{\\mathfrak {P}}_{\\mathbb {F}}} \\rightarrow D^{\\mu }_{\\mathfrak {P}_{\\mathbb {F}}}$ is an equivalence of groupoids over $\\mathcal {C}_{\\Lambda }$ .", "Remark 4.3.7 This generalizes Theorem 9.3.13 in [27] where we worked with $G$ -Kisin modules with height in $[0,1]$ .", "See Remark REF for more information.", "Corollary 4.3.8 Assume $F = F_{[\\mu ]}$ and that $\\mu $ is minuscule.", "Let $F^{\\prime }$ be finite extension of $F$ with ring of integers $\\Lambda ^{\\prime }$ .", "There is an equivalence of categories between $G$ -Kisin modules over $\\mathfrak {S}_{\\Lambda ^{\\prime }}$ with $p$ -adic Hodge type $\\mu $ and the subcategory of $\\operatorname{GRep}_{\\Lambda ^{\\prime }}(\\Gamma _K)$ consisting of crystalline representations with $p$ -adic Hodge type $\\mu $ .", "Corollary REF follows from the proof of Theorem REF .", "It generalizes the equivalence between Kisin modules of Barsotti-Tate type and lattices in crystalline representations with Hodge-Tate weights in $\\lbrace -1, 0\\rbrace $ ([25]).", "Note that we do not require $p \\nmid |\\pi _1(G^{\\mathrm {der}})|$ here.", "For the relevant definitions, see Definition REF and the discussion before Theorem REF .", "Before proving Theorem REF and Corollary REF , we begin with some preliminaries on crystalline $(\\varphi , \\widehat{\\Gamma })$ -modules with $G$ -structure.", "Definition 4.3.9 Define $G(u^{p^i})$ to be the kernel of the reduction map $G(W(\\widetilde{\\mathbf {E}}^+)_A) \\rightarrow G(W(\\widetilde{\\mathbf {E}}^+)_A /(\\varphi (\\operatorname{\\mathfrak {t}}) u^{p^i}))$ .", "Proposition 4.3.10 Choose $(\\mathfrak {P}_A, \\phi _{\\mathfrak {P}_A}) \\in \\operatorname{GMod}_{\\mathfrak {S}_A}^{\\varphi , \\operatorname{bh}}$ .", "Fix a trivialization $\\beta _A$ of $\\mathfrak {P}_A$ .", "Let $C^{\\prime } \\in G(\\mathfrak {S}_A[1/(\\varphi (E(u))])$ be $\\phi _{\\mathfrak {P}_A}$ with respect to the trivialization $1 \\otimes _{\\varphi } \\beta _A$ .", "Then a crystalline $\\widehat{\\Gamma }$ -structure on $\\mathfrak {P}_A$ is the same as a continuous map $B_{\\bullet }:\\widehat{\\Gamma } \\rightarrow G( \\widehat{R}_A)$ satisfying the following properties: $C^{\\prime } \\cdot \\varphi (B_{\\gamma }) = B_{\\gamma } \\cdot \\gamma (C^{\\prime }) $ in $G(W(\\widetilde{\\mathbf {E}})_A)$ for all $\\gamma \\in \\widehat{\\Gamma }$ ; $B_{\\gamma } = \\mathrm {Id}$ for all $\\gamma \\in \\widehat{\\Gamma }_{\\infty }$ ; $B_{\\gamma } \\in G(u^{p})$ for all $\\gamma \\in \\widehat{\\Gamma }$ ; $B_{\\gamma \\gamma ^{\\prime }} = B_{\\gamma } \\cdot \\gamma (B_{\\gamma ^{\\prime }})$ for all $\\gamma , \\gamma ^{\\prime } \\in \\widehat{\\Gamma }$ .", "Everything follows directly from Proposition REF .", "The only remark to make is that because $u \\in I_+(\\widehat{R})$ , $(u^p \\varphi (\\operatorname{\\mathfrak {t}})) \\subset I_+(\\widehat{R})_A$ .", "Hence, the crystalline condition which is equivalent to condition (c) implies condition (5) from Definition REF .", "Before we begin the proof of Theorem REF , we have two important lemmas.", "Lemma 4.3.11 Let $\\mathfrak {P}_A \\in D^{\\mu }_{\\mathfrak {P}_{\\mathbb {F}}}(A)$ and choose a trivialization $\\beta _A$ of the bundle $\\mathfrak {P}_A$ .", "If $C \\in G(\\mathfrak {S}_A[1/E(u)])$ is the Frobenius with respect to $\\beta _A$ , then for any $Y \\in G(u^{p^i})$ $\\varphi (C)\\varphi (Y) \\varphi (C)^{-1} \\in G(u^{p^{i+1}}),$ where $\\varphi (C) = C^{\\prime } \\in G(W(\\widetilde{\\mathbf {E}})_A)$ is the Frobenius with respect to $1 \\otimes _{\\varphi } \\beta _A$ .", "Let $\\mathcal {O}_{G}$ denote the coordinate ring of $G$ and let $I_e$ be the ideal defining the identity so that $\\mathcal {O}_G/I_e = \\Lambda $ and $I_e/I_e^2 \\cong (\\operatorname{Lie}(G))^{\\vee }$ .", "Then $G(u^{p^i})$ is identified with $\\lbrace Y \\in \\mathrm {Hom}_{\\Lambda } (\\mathcal {O}_G, W(\\widetilde{\\mathbf {E}}^+)_A) \\mid Y(I_e) \\subset (\\varphi (\\operatorname{\\mathfrak {t}})u^{p^i}) \\rbrace .$ Conjugation by $C$ induces an automorphism of $G_{\\mathfrak {S}_A[1/E(u)]}$ .", "Let $\\operatorname{Ad}_{\\mathcal {O}_G}(C)^*:\\mathcal {O}_G \\otimes _{\\Lambda } \\mathfrak {S}_A[1/E(u)] \\rightarrow \\mathcal {O}_G \\otimes _{\\Lambda } \\mathfrak {S}_A[1/E(u)]$ be the corresponding map on coordinate rings.", "The key observation is that $ \\operatorname{Ad}_{\\mathcal {O}_G}(C)^*(I_e \\otimes 1) \\subset \\sum _{j \\ge 1} I_e^j \\otimes _{\\Lambda } E(u)^{-j} \\mathfrak {S}_A.$ By successive approximation, one is reduced to studying the induced automorphism of $\\oplus _{j \\ge 0} (I_e^j/I_e^{j+1} \\otimes _{\\Lambda } \\mathfrak {S}_A[1/E(u)])$ .", "The $j$ th graded piece is $\\mathrm {Sym}^j(\\operatorname{Lie}(G)^{\\vee }) \\otimes _{\\Lambda } \\mathfrak {S}_A[1/E(u)]$ as a representation of $G(\\mathfrak {S}_A[1/E(u)])$ .", "Since $\\mu $ is minuscule, $\\operatorname{Lie}(G) \\otimes _{\\Lambda } \\mathfrak {S}_A$ has height in $[-1, 1]$ and so $\\mathrm {Sym}^j(\\operatorname{Lie}(G)^{\\vee } \\otimes _{\\Lambda } \\mathfrak {S}_A)$ has height in $[-j, j]$ .", "Thus, $\\operatorname{Ad}_{\\mathcal {O}_G}(C)^*(\\mathrm {Sym}^j(\\operatorname{Lie}(G)^{\\vee } \\otimes _{\\Lambda } \\mathfrak {S}_A) \\subset E(u)^{-j} (\\mathrm {Sym}^j(\\operatorname{Lie}(G)^{\\vee }) \\otimes _{\\Lambda } \\mathfrak {S}_A)$ from which one deduces (REF ).", "Let $Y \\in G(u^{p^i})$ .", "Then $\\varphi (Y)(I_e) \\subset \\varphi ( \\varphi (\\operatorname{\\mathfrak {t}})u^{p^i}) \\subset (\\varphi (E(u)) \\varphi (\\operatorname{\\mathfrak {t}}) u^{p^{i+1}})$ .", "For any $x \\in I_e$ , $(\\varphi (C)\\varphi (Y) \\varphi (C)^{-1})(x) = (\\varphi (Y) \\otimes 1)((1 \\otimes \\varphi )(\\operatorname{Ad}_{\\mathcal {O}_G}(C)^*(x)))$ which is a priori only in $W(\\widetilde{\\mathbf {E}})_A$ .", "But since for any $b \\in I_e^j$ , $\\varphi (Y)(b)$ is divisible by $\\varphi (E(u))^j \\varphi (\\operatorname{\\mathfrak {t}})^j u^{jp^{i+1}}$ , we have $\\operatorname{Ad}(\\varphi (C))(\\varphi (Y))(x) \\in (\\varphi (\\operatorname{\\mathfrak {t}}) u^{p^{i+1}})$ so $\\varphi (C)\\varphi (Y) \\varphi (C)^{-1}$ lies in $G(u^{p^{i+1}})$ .", "By [25], a $\\Gamma _{\\infty }$ -representation coming from a finite height torsion-free Kisin module $\\mathfrak {M}$ extends to a crystalline $\\Gamma _K$ -representation if and only if the canonical Frobenius equivariant connection on $\\mathfrak {M}\\otimes _{\\mathfrak {S}} {\\mathcal {O}[1/\\lambda ]}$ has at most logarithmic poles.", "[25] states furthermore that if $\\mathfrak {M}$ has height in $[0,1]$ then the condition of logarithmic poles is always satisfied.", "The following lemma is a version of [25] for $G$ -Kisin modules with minuscule type: Lemma 4.3.12 Let $F^{\\prime }/F$ be any finite extension containing $F_{[\\mu ]}$ and let $(\\mathfrak {P}_{F^{\\prime }}, \\phi _{F^{\\prime }})$ be any $G$ -Kisin module over $F^{\\prime }$ .", "Fix a trivialization of $\\mathfrak {P}_{F^{\\prime }}$ and let $C \\in G(\\mathfrak {S}_{F^{\\prime }}[1/E(u)])$ be the Frobenius with respect to this trivialization.", "If the $G$ -filtration $\\mathfrak {D}_{\\mathfrak {P}_{F^{\\prime }}}$ over $K \\otimes _{\\mathbb {Q}_p} F^{\\prime }$ defined before Lemma $\\ref {Dx}$ has type $\\mu $ , then the right logarithmic derivative $\\frac{dC}{du} \\cdot C^{-1} \\in (\\operatorname{Lie}G \\otimes \\mathfrak {S}_{F^{\\prime }}[1/E(u)])$ has at most logarithmic poles along $E(u)$ , i.e., lies in $E(u)^{-1}(\\operatorname{Lie}G \\otimes \\mathfrak {S}_{F^{\\prime }})$ .", "Choose an embedding $\\sigma :K_0 \\rightarrow F^{\\prime }$ .", "Without loss of generality, we assume that $\\sigma (E(u))$ splits in $F^{\\prime }$ and write $\\sigma (E(u)) = (u- \\psi _1(\\pi ))(u - \\psi _2(\\pi )) \\ldots (u- \\psi _e(\\pi ))$ over embeddings $\\psi _i:K \\rightarrow F^{\\prime }$ which extend $\\sigma $ .", "Let $C_{\\sigma }$ denote the $\\sigma $ -component of $C$ under the decomposition of $\\mathfrak {S}_{F^{\\prime }}[1/E(u)]$ as a $W \\otimes _{\\mathbb {Z}_p} F^{\\prime } \\cong \\prod _{K_0 \\rightarrow F^{\\prime }} F^{\\prime }$ -algebra.", "We can furthermore compute the “pole\" at $\\psi _i(\\pi )$ by working in the completion at $u - \\psi _i(\\pi )$ which is isomorphic to $F^{\\prime }[\\![t]\\!", "]$ with $t = u - \\psi _i(\\pi )$ .", "Let $\\mu _{\\psi _i} \\in X_*(G_F)$ be the $\\psi _i$ -component of $\\mu $ .", "Fix a maximal torus $T$ of $G_{F^{\\prime }}$ such that $\\mu _{\\psi _i}$ factors through $T$ .", "The Cartan decomposition for $G(F^{\\prime }(\\!(t)\\!", "))$ combined with the assumption that $\\mathfrak {D}_{\\mathfrak {P}_{F^{\\prime }}}$ has type $\\mu $ implies that $C_{\\sigma } = B_i \\mu _{\\psi _i}(t) D_i$ where $B_i$ and $D_i$ are in $G(F^{\\prime }[\\![t]\\!", "])$ (see discussion before Proposition REF for definition of $\\mu _{\\psi _i}(t)$ ).", "Finally, we compute that $\\frac{dC_{\\sigma }}{du} C_{\\sigma }^{-1}$ equals $\\frac{dB_i}{dt} B_i^{-1} + \\operatorname{Ad}(B_i)\\left(\\frac{d\\mu _{\\psi _i}(t)}{dt} \\mu _{\\psi _i}(t)^{-1}\\right) + \\operatorname{Ad}(B_i) \\left(\\operatorname{Ad}(\\mu _{\\psi _i}(t)) \\left(\\frac{dD_i}{dt} D_i^{-1}\\right) \\right).$ We have $\\frac{dB_i}{dt} B_i^{-1} \\in (\\operatorname{Lie}G \\otimes F^{\\prime }[\\![t]\\!", "])$ .", "Using a faithful representation on which $T$ acts diagonally, we have $\\frac{d\\mu _{\\psi _i}(t)}{dt} \\mu _{\\psi _i}(t)^{-1} \\in \\frac{1}{t} (\\operatorname{Lie}G \\otimes F^{\\prime }[\\![t]\\!", "])$ .", "Finally, since $\\mu _{\\psi _i}$ is minuscule, $\\operatorname{Ad}(\\mu _{\\psi _i}(t)) (X) \\in \\frac{1}{t} (\\operatorname{Lie}G \\otimes F^{\\prime }[\\![t]\\!", "])$ for any $X \\in \\operatorname{Lie}G$ so in particular for $\\frac{dD_i}{dt} D_i^{-1}$ by Proposition REF .", "The faithfulness of $\\widehat{\\Delta }^{\\mu }$ is clear.", "For fullness, let $\\widehat{\\mathfrak {P}}_A, \\widehat{\\mathfrak {P}}_A^{\\prime } \\in D^{\\text{cris}, \\mu }_{\\widehat{\\mathfrak {P}}_{\\mathbb {F}}}(A)$ and let $\\psi :\\mathfrak {P}_A \\cong \\mathfrak {P}^{\\prime }_A$ be an isomorphism of underlying $G$ -Kisin modules.", "To show $\\psi $ is equivariant for the $\\widehat{\\Gamma }$ -actions, we can identify $\\mathfrak {P}_A$ and $\\mathfrak {P}_A^{\\prime }$ using $\\psi $ and choose a trivialization of $\\mathfrak {P}_A$ .", "Then, it suffices to show that $(\\mathfrak {P}_A, \\phi _{\\mathfrak {P}_A})$ has at most one crystalline $\\widehat{\\Gamma }$ -structure.", "Let $B_{\\tau }$ and $B^{\\prime }_{\\tau }$ in $G(W(\\widetilde{\\mathbf {E}}^+)_A)$ define the action of $\\tau $ with respect to the chosen trivialization of $\\varphi ^*(\\mathfrak {P}_A)$ for the two $\\widehat{\\Gamma }$ -structures.", "By the crystalline property, $B_{\\tau } (B^{\\prime }_{\\tau })^{-1} \\in G(u^{p})$ .", "By Proposition REF if Frobenius is given by $C^{\\prime }$ with respect to the trivialization, $B_{\\tau } (B^{\\prime }_{\\tau })^{-1} = C^{\\prime } \\varphi (B_{\\tau } (B^{\\prime }_{\\tau })^{-1}) (C^{\\prime })^{-1}.$ But then by Lemma REF , $B_{\\tau } (B^{\\prime }_{\\tau })^{-1} = I$ since it is in $G(u^{p^i})$ for all $i \\ge 1$ .", "We next attempt to construct a crystalline $\\widehat{\\Gamma }$ -structure on any $\\mathfrak {P}_A \\in D^{\\mu }_{\\mathfrak {P}_{\\mathbb {F}}}(A)$ .", "Along the way, we will have to impose certain closed conditions on $D^{\\mu }_{\\mathfrak {P}_{\\mathbb {F}}}$ to make our construction work.", "In the end, we will reduce to $A$ flat over $\\mathbb {Z}_p$ to show that these conditions are always satisfied.", "Fix a trivialization $\\beta _A$ of $\\mathfrak {P}_A$ .", "We want elements $\\lbrace B_{\\gamma } \\rbrace \\in G(\\widehat{R}_A)$ for all $\\gamma \\in \\widehat{\\Gamma }$ satisfying the conditions from Proposition REF .", "Choose an element $\\gamma \\in \\widehat{\\Gamma }$ .", "Let $C$ denote the Frobenius with respect to $\\beta _A$ and let $C^{\\prime } = \\varphi (C)$ be the Frobenius with respect to $1 \\otimes _{\\varphi } \\beta _A$ .", "We use the topology on $G(W(\\widetilde{\\mathbf {E}})_A)$ induced from the topology on $W(\\widetilde{\\mathbf {E}})_A$ (see the discussion before Definition REF ).", "Take $B_0 = I$ .", "For all $i \\ge 1$ , define $ B_i := C^{\\prime } \\varphi (B_{i-1}) \\gamma (C^{\\prime })^{-1} \\in G(W(\\widetilde{\\mathbf {E}})_A).$ If $\\mathfrak {P}_A$ admits a $\\widehat{\\Gamma }$ -structure, then the $B_i$ converge to $B_{\\gamma }$ in $G(\\widehat{R}_A)$ or equivalently in $G(W(\\widetilde{\\mathbf {E}})_A)$ .", "Base case: $B_1 = C^{\\prime } \\gamma (C^{\\prime })^{-1} \\in G(u^{p}).$ Let $V$ be a faithful $n$ -dimensional representation of $G$ such that $\\mathfrak {P}_A(V)$ has height in $[a,b]$ .", "Set $r = b-a$ .", "Consider $C$ as an element of $\\mathrm {GL}_n(\\mathfrak {S}_A[1/E(u)])$ such that $C^{\\prime \\prime } := E(u)^{-a} C \\in \\mathrm {Mat}_n(\\mathfrak {S}_A) \\text{ and } D^{\\prime \\prime } := E(u)^b C^{-1} \\in \\mathrm {Mat}_n(\\mathfrak {S}_A)$ with $C^{\\prime \\prime } D^{\\prime \\prime } = E(u)^{r} I$ .", "Working in $\\mathrm {Mat}_n( W(\\widetilde{\\mathbf {E}})_A)$ , we compute that $C^{\\prime } \\gamma (C^{\\prime })^{-1} - I = \\varphi \\left( \\frac{1}{E(u)^{-a} \\gamma (E(u))^b} (C^{\\prime \\prime } \\gamma (D^{\\prime \\prime }) - E(u)^{-a} \\gamma (E(u))^b I) \\right).$ It would suffice then to show that $u \\varphi (\\operatorname{\\mathfrak {t}}) E(u)^{r - 1}$ divides $C^{\\prime \\prime } \\gamma (D^{\\prime \\prime }) - E(u)^{-a} \\gamma (E(u))^b I$ in $\\mathrm {Mat}_n( W(\\widetilde{\\mathbf {E}}^+)_A)$ as then $u \\operatorname{\\mathfrak {t}}$ divides $\\frac{1}{E(u)^{-a} \\gamma (E(u))^b} (C^{\\prime \\prime } \\gamma (D^{\\prime \\prime }) - E(u)^{-a} \\gamma (E(u))^b I)$ using Lemma REF .", "Consider $P(u_1, u_2) = C^{\\prime \\prime }(u_1) D^{\\prime \\prime }(u_2)$ where we replace $u$ by $u_1$ in $C^{\\prime \\prime } \\in \\mathrm {Mat}_n(\\mathfrak {S}_A)$ and $u$ in $u_2$ for $D^{\\prime \\prime }$ .", "Let $P_{ij}(u_1, u_2) = \\sum _{k \\ge 0} c_k^{ij}(u_1) u_2^k$ be the $(i,j)$ th entry where $c_k^{ij}(u_1)$ is a power series in $u_1$ with coefficients in $W \\otimes _{\\mathbb {Z}_p} A$ .", "We have that $P_{ij}(u,u) = \\delta _{ij} E(u)^r$ .", "The $(i, j)$ th entry of $C^{\\prime \\prime } \\gamma (D^{\\prime \\prime })$ is $P_{ij}(u, [\\underline{\\varepsilon }] u) = \\sum _{k \\ge 0} [\\underline{\\varepsilon }] ^k c_k^{ij}(u) u^k$ where $\\underline{\\varepsilon } = (\\zeta _{p^i})_{i \\ge 0}$ is the sequence of $p^n$ -th roots of unity such that $\\gamma (\\pi ^{1/p^n}) = \\zeta _{p^n} \\pi ^{1/p^n}$ .", "Note that $ \\varphi (\\operatorname{\\mathfrak {t}})$ divides $[\\underline{\\varepsilon }] - 1$ since $[\\underline{\\varepsilon }] - 1 \\in I^{[1]} W(\\widetilde{\\mathbf {E}}^+)$ (see [13]) and $\\varphi (\\operatorname{\\mathfrak {t}})$ is a generator for this ideal.", "Then, $P_{ij}(u, [\\underline{\\varepsilon }] u) = \\sum _{k \\ge 0} ([\\underline{\\varepsilon }]^k - 1) c_k^{ij}(u) u^k + \\delta _{ij} E(u)^r.$ Since $u ([\\underline{\\varepsilon }] - 1) E(u)^{r-1}$ divides $E(u)^r - E(u)^{-a} \\gamma (E(u))^b$ , it suffices to show that $u ([\\underline{\\varepsilon }] - 1) E(u)^{r-1}$ divides $\\sum _k ([\\underline{\\varepsilon }]^k - 1) c_k^{ij}(u) u^k$ .", "Using the Taylor expansion for $x^k - 1$ at $x = 1$ , we have $[\\underline{\\varepsilon }]^k - 1 = \\sum _{\\ell =1}^{k} \\binom{k}{\\ell } ([\\underline{\\varepsilon }] - 1)^{\\ell }$ from which we deduce that $\\sum _{k \\ge 0} ([\\underline{\\varepsilon }]^k - 1) c_k^{ij}(u) u^k = u ([\\underline{\\varepsilon }] - 1) \\left( \\sum _{\\ell \\ge 1} ([\\underline{\\varepsilon }] - 1)^{\\ell -1} u^{\\ell - 1} \\sum _{k \\ge 0} \\binom{k+\\ell }{\\ell } c_{k +\\ell }^{ij}(u) u^{k} \\right)$ Since $E(u)$ divides $[\\underline{\\varepsilon }] - 1$ , we are reducing to showing that $E(u)^{r-\\ell } \\mid u^{\\ell - 1} \\sum _{k \\ge 0} \\binom{k+\\ell }{\\ell } c_{k +\\ell }^{ij}(u) u^{k}$ for $1 \\le \\ell \\le r -1$ where the expression on the right is exactly $\\frac{u^{\\ell - 1}}{\\ell !}", "\\left( \\frac{d^{\\ell } P_{ij}(u_1, u_2)}{du_2^{\\ell }}|_{(u, u)} \\right)$ .", "Let $(\\star _1)$ be the condition that $E(u)^{r-\\ell }$ divides $\\frac{d^{\\ell } P_{ij}(u_1, u_2)}{du_2^{\\ell }}|_{(u, u)}$ for all $(i, j)$ and $1 \\le \\ell \\le r -1$ .", "This is a closed condition on $D^{\\mu }_{\\mathfrak {P}_{\\mathbb {F}}}$ .", "Induction step: Let $\\mathfrak {P}_A \\in D^{\\mu }_{\\mathfrak {P}_{\\mathbb {F}}}(A)$ satisfying $(\\star _1)$ with trivialization as above so that $B_1 = C^{\\prime } \\gamma (C^{\\prime })^{-1} \\in G(u^{p}).$ We have $B_{i+1} B_i^{-1} = C \\varphi (B_i B_{i-1}^{-1}) C^{-1}.$ As $C = \\varphi (C^{\\prime })$ , we can apply Lemma REF to conclude that $B_{i+1} B_i^{-1} \\in G(u^{p^{i+1}})$ , i.e., $B_{i+1} B_i^{-1} \\equiv I \\mod {\\varphi }(\\operatorname{\\mathfrak {t}}) u^{p^{i+1}} W(\\widetilde{\\mathbf {E}}^+)_A$ .", "Since $W(\\widetilde{\\mathbf {E}}^+)_A$ is separated and complete, $\\varinjlim B_i = B_{\\gamma } \\in G(W(\\widetilde{\\mathbf {E}}^+)_A)$ and $B_{\\gamma }$ satisfies $B_{\\gamma } \\gamma (C) = C \\varphi (B_{\\gamma })$ .", "It is easy to see that for any $\\gamma , \\gamma ^{\\prime }$ , $B_{\\gamma } \\gamma ^{\\prime }(B_{\\gamma }) = B_{\\gamma \\gamma ^{\\prime }}$ by continuity so we have a $\\widehat{\\Gamma }$ -action.", "If $\\gamma \\in \\widehat{\\Gamma }_{\\infty }$ , then $\\gamma $ acts trivially on $\\mathfrak {S}_A$ and so on $C$ as well so $B_{\\gamma } = I$ .", "Let $(\\star _2)$ denote the condition that $B_{\\gamma } \\in G(\\widehat{R}_A)$ for all $\\gamma \\in \\widehat{\\Gamma }$ .", "We claim this is also a closed condition on $D^{\\mu }_{\\mathfrak {P}_{\\mathbb {F}}}$ .", "Since $W(\\widetilde{\\mathbf {E}}^+)/\\widehat{R}$ is $\\mathbb {Z}_p$ -flat, the sequence $0 \\rightarrow \\widehat{R}_A \\rightarrow W(\\widetilde{\\mathbf {E}}^+)_A \\rightarrow (W(\\widetilde{\\mathbf {E}}^+)/\\widehat{R}) \\otimes _{\\mathbb {Z}_p} A \\rightarrow 0$ is exact for any $A$ .", "Any flat module over an Artinian ring is free so vanishing of an element $f \\in (W(\\widetilde{\\mathbf {E}}^+)/\\widehat{R}) \\otimes _{\\mathbb {Z}_p} A$ is a closed condition on $\\mathrm {Spec}\\ A$ .", "We have shown that any element $\\mathfrak {P}_A \\in D^{\\mu }_{\\mathfrak {P}_{\\mathbb {F}}}(A)$ which satisfies $(\\star _1)$ and $(\\star _2)$ admits a crystalline $\\widehat{\\Gamma }$ -structure and so lies in $D^{\\text{cris}, \\mu }_{\\widehat{\\mathfrak {P}}_{\\mathbb {F}}}(A)$ .", "It suffices then to show that the closed subgroupoid defined by the conditions $(\\star _1)$ and $(\\star _2)$ is all of $D^{\\mu }_{\\mathfrak {P}_{\\mathbb {F}}}$ .", "Recall that $D^{\\mu }_{\\mathfrak {P}_{\\mathbb {F}}}$ admits a formally smooth representable hull $D^{(N), \\mu }_{\\mathfrak {P}_{\\mathbb {F}}} = \\mathrm {Spf}\\ R^{(N), \\mu }_{\\mathfrak {P}_{\\mathbb {F}}}$ where $R^{(N), \\mu }_{\\mathfrak {P}_{\\mathbb {F}}}$ is flat and reduced by Theorem REF and Proposition REF .", "Since $R^{(N), \\mu }_{\\mathfrak {P}_{\\mathbb {F}}}$ is flat and $R^{(N), \\mu }_{\\mathfrak {P}_{\\mathbb {F}}}[1/p]$ is reduced and Jacobson, any closed subscheme of $\\mathrm {Spec}\\ R^{(N), \\mu }_{\\mathfrak {P}_{\\mathbb {F}}}$ which contains $\\mathrm {Hom}_{\\Lambda } (R^{(N), \\mu }_{\\mathfrak {P}_{\\mathbb {F}}}, F^{\\prime })$ for all $F^{\\prime }/F$ finite is the whole space.", "It suffices then to show that for any $F^{\\prime }/F$ finite and $\\Lambda ^{\\prime }$ the ring of integers of $F^{\\prime }$ every object of $D^{\\mu }_{\\mathfrak {P}_{\\mathbb {F}}}(\\Lambda ^{\\prime })$ satisfies $(\\star _1)$ and $(\\star _2)$ .", "For $(\\star _1)$ , choose $\\gamma \\in \\widehat{\\Gamma }$ .", "Then, set $Q_{\\ell }(u) := \\left( \\frac{d^{\\ell } P_{ij}(u_1, u_2)}{du_2^{\\ell }}|_{(u, u)} \\right) \\in \\mathrm {Mat}_n(\\mathfrak {S}_{\\Lambda ^{\\prime }})$ (we ignore $\\frac{u^{\\ell - 1}}{\\ell !", "}$ since we are in the torsion-free setting).", "We can check that $E(u)^{r-\\ell } \\mid Q_{\\ell }(u)$ working over $F^{\\prime } = \\Lambda ^{\\prime }[1/p]$ or any finite extension thereof.", "In particular, we can put ourselves in the situation of Lemma REF .", "We compute then that $\\begin{split}Q_{\\ell }(u) &= (E(u)^{-a} C) \\frac{d^{\\ell }}{du^{\\ell }} (E(u)^b C^{-1})\\\\&= (E(u)^{-a} C) \\sum ^{\\ell }_{m = 0} \\binom{\\ell }{m} \\frac{d^{m} E(u)^b}{du^{m}} \\frac{d^{\\ell - m} C^{-1}}{du^{\\ell - m}}\\\\&= \\sum ^{\\ell }_{m = 0} \\binom{\\ell }{m} \\left( E(u)^{-a} \\frac{d^{m} E(u)^b}{du^{m}}\\right) \\left(C \\frac{d^{\\ell - m} C^{-1}}{du^{\\ell - m}}\\right).\\end{split}$ Since $E(u)^{r - m}$ divides $E(u)^{-a} \\frac{d^{m} E(u)^b}{du^{m}}$ , it suffices to show that $Y_k := E(u)^k \\left(C \\frac{d^{k} C^{-1}}{du^{k}}\\right) \\in \\mathrm {Mat}_n(\\mathfrak {S}_{F^{\\prime }})$ for all $k \\ge 0$ (applied with $k = \\ell - m$ ).", "The case $k = 0$ is trivial.", "By Lemma REF , $X_C := E(u) \\frac{dC}{du} C^{-1} = - E(u) C \\frac{d (C^{-1})}{du}$ is an element of $\\operatorname{Lie}G \\otimes \\mathfrak {S}_{F^{\\prime }}$ considered as subset of $\\operatorname{Lie}(\\mathrm {GL}(V)) \\otimes \\mathfrak {S}_{F^{\\prime }}$ so in particular $Y_1 \\in \\mathrm {Mat}_n(\\mathfrak {S}_{F^{\\prime }})$ .", "The product rule applied to $\\frac{d}{du}(E(u)^k C \\frac{d^{k-1} C^{-1}}{du})$ implies that $Y_k = \\frac{d}{du} (E(u) Y_{k-1}) - k \\frac{d E(u)}{du} Y_{k-1} + Y_1 Y_{k -1}$ so by induction on $k$ , $Y_k \\in \\mathrm {Mat}_n(\\mathfrak {S}_{F^{\\prime }})$ for all $k \\ge 0$ .", "For $(\\star _2)$ , recall that $\\widehat{R} = R_{K_0} \\cap W(\\widetilde{\\mathbf {E}}^+)$ (see pg.", "5 of [30]) so it suffices to show that $B_{\\gamma } \\in G(R_{K_0} \\otimes _{\\mathbb {Z}_p} \\Lambda ^{\\prime })$ or equivalently $B_{\\gamma } \\in \\mathrm {GL}_n(R_{K_0} \\otimes _{\\mathbb {Z}_p} \\Lambda ^{\\prime })$ with respect to $V$ .", "Denote by $\\mathfrak {M}_V$ the Kisin module $\\mathfrak {P}_{\\Lambda ^{\\prime }}(V)$ of rank $n$ .", "Since $\\varphi (E(u))$ is invertible in $S_{K_0} $ , $C^{\\prime }$ lies in $\\mathrm {GL}_n(S_{K_0} \\otimes _{\\mathbb {Z}_p} \\Lambda ^{\\prime })$ and defines a Frobenius on the Breuil module $\\mathcal {M}_V := S_{K_0} \\otimes _{\\mathfrak {S}, \\varphi } \\mathfrak {M}_V $ .", "Using a similar argument to above, one can construct the monodromy operator $N_{\\mathcal {M}_V}$ on $\\mathcal {M}_V$ inductively taking $N_0 = 0$ and setting $ N_{i+1} := p C^{\\prime } \\varphi (N_i)( C^{\\prime })^{-1} + u \\frac{dC^{\\prime }}{du} (C^{\\prime })^{-1}.$ The sequence $\\lbrace N_i \\rbrace $ converges to an element of $\\mathrm {Mat}_n(u^p S_{K_0})$ .", "For each $N_i$ , let $\\widetilde{N}_i$ be the induced derivation on $\\mathcal {M}_V$ over $-u \\frac{d}{du}$ which on the chosen basis is given by $N_i$ .", "Equation (REF ) is equivalent to $ \\widetilde{N}_{i+1} \\phi _{\\mathcal {M}_V} = p \\phi _{\\mathcal {M}_V} \\widetilde{N}_{i}.$ Let $\\underline{\\varepsilon }(\\gamma ) := \\gamma ([\\underline{\\pi }] )/[\\underline{\\pi }]$ .", "Define a $\\gamma $ -semilinear map $\\widetilde{B}_i$ on $R_{K_0} \\otimes _{S_{K_0}} \\mathcal {M}_V$ by $\\widetilde{B}_i(x) = \\sum _{j \\ge 0} \\frac{(- \\log \\underline{\\varepsilon }(\\gamma ))^j}{j!", "}\\otimes (\\widetilde{N}_i)^j (x)$ for all $x \\in \\mathcal {M}_V$ .", "Equation (REF ) implies that $\\widetilde{B}_{i + 1} \\phi _{\\mathcal {M}_V} = \\phi _{\\mathcal {M}_V} \\widetilde{B}_i.$ By induction on $i$ , one deduces that $\\widetilde{B}_i$ is exactly the $\\gamma $ -semilinear morphism induced by the matrix $B_i$ defined in (REF ).", "If $N_{\\mathcal {M}_V}$ is the limit of the $\\widetilde{N}_i$ and $\\widetilde{B}_{\\gamma }$ is the $\\gamma $ -semilinear morphism induced by $B_{\\gamma }$ , then we have the following formula $\\widetilde{B}_{\\gamma }(x) := \\sum _{j \\ge 0} \\frac{(- \\log \\underline{\\varepsilon }(\\gamma ))^j}{j!", "}\\otimes N_{\\mathcal {M}}^j (x)$ for all $x \\in \\mathcal {M}_V$ .", "Working with respect to the chosen basis for $\\mathcal {M}_V$ , we deduce that $B_{\\gamma } \\in \\mathrm {GL}_n(R_{K_0} \\otimes _{\\mathbb {Z}_p} \\Lambda ^{\\prime })$ as desired." ], [ "Applications to $G$ -valued deformation rings", "Let $\\overline{\\eta }:\\Gamma _K \\rightarrow G(\\mathbb {F})$ be a continuous representation.", "As before, $\\mu $ is a minuscule geometric cocharacter of $\\mathrm {Res}_{(K \\otimes _{\\mathbb {Q}_p} F)/F} G_F$ .", "Let $R^{\\operatorname{cris}, \\mu }_{\\overline{\\eta }}$ be the univeral $G$ -valued framed crystalline deformation ring with $p$ -adic Hodge type $\\mu $ over $\\Lambda _{[\\mu ]}$ .", "Let $X^{\\operatorname{cris}, \\mu }_{\\overline{\\eta }}$ be the projective $R^{\\operatorname{cris}, \\mu }_{\\overline{\\eta }}$ -scheme as in Corollary REF .", "The following theorem on the geometry of $X^{\\operatorname{cris}, \\mu }_{\\overline{\\eta }}$ has a number of important corollaries.", "The proof uses the main results from §3.2 and §4.2.", "We can say more about the connected components when $K$ is unramified over $\\mathbb {Q}_p$ (see Theorem REF ).", "Theorem 4.4.1 Assume $p \\nmid \\pi _1(G^{\\mathrm {der}})$ .", "Let $\\mu $ be a minuscule geometric cocharacter of $\\mathrm {Res}_{(K \\otimes _{\\mathbb {Q}_p} F)/F} G_F$ .", "Then $X^{\\operatorname{cris}, \\mu }_{\\overline{\\eta }}$ is normal and $X^{\\operatorname{cris}, \\mu }_{\\overline{\\eta }} \\otimes _{\\Lambda _{[\\mu ]}} \\mathbb {F}_{[\\mu ]}$ is reduced.", "Corollary 4.4.2 Assume $p \\nmid \\pi _1(G^{\\mathrm {der}})$ .", "Let $X^{\\operatorname{cris}, \\mu }_{\\overline{\\eta }, 0}$ denote the fiber of $X^{\\operatorname{cris}, \\mu }_{\\overline{\\eta }}$ over the closed point of $\\mathrm {Spec}\\ R^{\\operatorname{cris}, \\mu }_{\\overline{\\eta }}$ .", "The connected components of $\\mathrm {Spec}\\ R^{\\operatorname{cris}, \\mu }_{\\overline{\\eta }}[1/p]$ are in bijection with the connected components of $X^{\\operatorname{cris}, \\mu }_{\\overline{\\eta }, 0}$ .", "By Theorem REF , $\\mathrm {Spec}\\ R^{\\operatorname{cris}, \\mu }_{\\overline{\\eta }}[1/p] = X^{\\operatorname{cris}, \\mu }_{\\overline{\\eta }}[1/p].$ Since $X^{\\operatorname{cris}, \\mu }_{\\overline{\\eta }} \\otimes _{\\Lambda } \\mathbb {F}$ is reduced (by Theorem REF ), the bijection between $\\pi _0(X^{\\operatorname{cris}, \\mu }_{\\overline{\\eta }}[1/p])$ and $\\pi _0(X^{\\operatorname{cris}, \\mu }_{\\overline{\\eta },0})$ follows from the “reduced fiber trick” [24].", "Remark 4.4.3 Both Theorem REF and Corollary REF hold for unframed $G$ -valued crystalline deformation functors when they are representable by exactly the same arguments.", "Before we begin the proof, we introduce a few auxiliary deformation groupoids.", "The relationship between various deformation spaces is described in the diagram below.", "Let $D^{\\square }_{\\overline{\\eta }}$ be the deformation functor of $\\overline{\\eta }$ , that is, $D^{\\square }_{\\overline{\\eta }}(A)$ is the set of homomorphism $\\eta :\\Gamma _K \\rightarrow G(A)$ lifting $\\overline{\\eta }$ .", "Let $\\mathfrak {P}_{\\mathbb {F}}$ be the $G$ -Kisin module associated to a $\\mathbb {F}$ -point $\\overline{x}$ of $X^{\\operatorname{cris}, \\mu }_{\\overline{\\eta }}$ .", "Definition 4.4.4 Define $D^{[a,b]}_{\\overline{x}}(A)$ to be the category of triples $\\lbrace \\eta _A \\in D^{\\square }_{\\overline{\\eta }}(A) , \\mathfrak {P}_A \\in D^{[a,b]}_{\\mathfrak {P}_{\\mathbb {F}}}(A), \\delta _A:T_{G,\\mathfrak {S}_A}(\\mathfrak {P}_A) \\cong \\eta _A|_{\\Gamma _{\\infty }} \\rbrace .$ Let $\\widehat{\\mathfrak {P}}_{\\mathbb {F}}$ be a crystalline $\\widehat{\\Gamma }$ -structure on $\\mathfrak {P}_{\\mathbb {F}}$ together with an isomorphism $\\widehat{T}_{G, \\mathbb {F}}(\\widehat{\\mathfrak {P}}_{\\mathbb {F}}) \\cong \\overline{\\eta }$ .", "Define $D^{\\text{cris}, \\mu , \\square }_{\\widehat{\\mathfrak {P}}_{\\mathbb {F}}}(A)$ to be the category of triples $\\lbrace \\eta _A \\in D^{\\square }_{\\overline{\\eta }}(A) , \\widehat{\\mathfrak {P}}_A \\in D^{\\operatorname{cris}, \\mu }_{\\widehat{\\mathfrak {P}}_{\\mathbb {F}}}(A), \\delta _A:\\widehat{T}_{G,A}(\\widehat{\\mathfrak {P}}_A) \\cong \\eta _A \\rbrace .$ Proposition 4.4.5 For any $\\widehat{\\mathfrak {P}}_{\\mathbb {F}}$ , the forgetful functor from $D^{\\operatorname{cris}, \\mu , \\square }_{\\widehat{\\mathfrak {P}}_{\\mathbb {F}}}$ to $D^{[a,b]}_{\\overline{x}}$ is fully faithful.", "One reduces immediately to the case of $\\mathrm {GL}_n$ and then we have the following more general fact: Choose any $\\widehat{\\mathfrak {M}}^{\\prime }_A, \\widehat{\\mathfrak {M}}_A \\in \\operatorname{Mod}^{\\varphi , \\operatorname{bh}, \\widehat{\\Gamma }}_{\\mathfrak {S}_A}$ .", "Let $f:\\mathfrak {M}^{\\prime }_A \\rightarrow \\mathfrak {M}_A$ be a map of underlying Kisin modules such that $T_{\\mathfrak {S}_A}(f)$ is $\\Gamma _K$ -equivariant $($ under the identification of $\\widehat{T}_{\\mathfrak {S}_A} \\cong T_{\\mathfrak {S}_A})$ .", "Then, $f$ is a map of $(\\varphi , \\widehat{\\Gamma })$ -modules.", "This is proven in [33] when height is in $[0,h]$ but can be easily extended to bounded height.", "The key input is a weak form of Liu's comparison isomorphism ([29] which can be found in [27].", "The diagram below illustrates some of the relationships between the different deformation problems.", "The diagonal maps on the left and the map labeled sm are formally smooth.", "Maps labeled with $c \\sim $ indicate that the complete stalk at a point of the target represents that deformation functor.", "The horizontal equivalences are consequences of the Theorem REF and the proof of Theorem REF respectively.", "$ {& \\widetilde{D}^{(\\infty ),\\mu }_{\\mathfrak {P}_{\\mathbb {F}}} [dl]_{\\pi ^{\\mu }} [dr]^{\\Psi ^{\\mu }} & & D^{\\operatorname{cris}, \\mu , \\square }_{\\widehat{\\mathfrak {P}}_{\\mathbb {F}}} [d]^{sm} @{^{(}->}[dr] [r]^{\\sim } & D^{\\operatorname{cris}, \\mu }_{\\overline{x}} [r]^{c \\sim } @{^{(}->}[d] & X^{\\operatorname{cris}, \\mu }_{\\overline{\\eta }} @{^{(}->}[d] \\\\\\overline{D}^{\\mu }_{Q_{\\mathbb {F}}}& & D^{\\mu }_{\\mathfrak {P}_{\\mathbb {F}}} & D^{\\operatorname{cris}, \\mu }_{\\widehat{\\mathfrak {P}}_{\\mathbb {F}}} [l]_{\\sim } & D_{\\overline{x}}^{[a,b]} [r]^{c \\sim } & X^{[a,b]}_{\\overline{\\eta }}}$ Let $\\overline{x}$ be a point of the special fiber of $X^{\\operatorname{cris}, \\mu }_{\\overline{\\eta }}$ defined over a finite field $\\mathbb {F}^{\\prime }$ .", "Since $X^{\\operatorname{cris}, \\mu }_{\\overline{\\eta }}[1/p] = \\mathrm {Spec}\\ R^{\\operatorname{cris}, \\mu }_{\\overline{\\eta }}[1/p]$ is formally smooth over $F$ ([1]), it suffices to show that the completed stalk $\\widehat{\\mathcal {O}}^{\\mu }_{\\overline{x}}$ at $\\overline{x}$ is normal and that $\\widehat{\\mathcal {O}}^{\\mu }_{\\overline{x}} \\otimes _{\\Lambda _{[\\mu ]}} \\mathbb {F}_{[\\mu ]}$ is reduced.", "To accomplish this, we compare $\\widehat{\\mathcal {O}}^{\\mu }_{\\overline{x}}$ with $\\overline{D}^{\\mu }_{Q_{\\mathbb {F}^{\\prime }}}$ from §3.3 and then use as input the corresponding results for the local model $M(\\mu )$ .", "These properties can be checked after an étale extension of $\\Lambda _{[\\mu ]}$ .", "$R^{\\operatorname{cris}, \\mu }_{\\overline{\\eta }}$ commutes with changing coefficients using the abstract criterion in [10] as does the formation of $X^{\\operatorname{cris}, \\mu }_{\\overline{\\eta }}$ by Proposition REF .", "We can assume then, without loss of generality, that $\\Lambda = \\Lambda _{[\\mu ]}$ and $\\mathbb {F}^{\\prime } = \\mathbb {F}$ .", "Let $\\mathfrak {P}_{\\mathbb {F}}$ be the $G$ -Kisin module defined by $\\overline{x}$ .", "Since $\\mu $ is minuscule, $X^{\\operatorname{cris}, \\mu }_{\\overline{\\eta }} = X^{\\operatorname{cris}, \\le \\mu }_{\\overline{\\eta }}$ (see Corollary REF ).", "Since $\\widehat{\\mathcal {O}}^{\\mu }_{\\overline{x}}$ is non-empty and $\\Lambda $ -flat (assuming that $R^{\\operatorname{cris}, \\mu }_{\\overline{\\eta }}$ is non-empty), it has an $F^{\\prime }$ -point for some finite extension $F^{\\prime }/F$ .", "Any such point gives rise to a crystalline lift $\\rho $ of $\\overline{x}$ to $\\mathcal {O}_{F^{\\prime }}$ such that the unique Kisin lattice in $\\underline{M}_{G, \\mathcal {O}_{F^{\\prime }}}(\\rho )$ reduces to $\\mathfrak {P}_{\\mathbb {F}} \\otimes _{\\mathbb {F}} \\mathbb {F}^{\\prime }$ .", "Replace $\\mathbb {F}^{\\prime }$ by $\\mathbb {F}$ .", "Then by Proposition REF , the corresponding $G(\\mathcal {O}_{F^{\\prime }})$ -valued representation is isomorphic to $\\widehat{T}_{G, \\mathcal {O}_{F^{\\prime }}}(\\widehat{\\mathfrak {P}}_{\\mathcal {O}_{F^{\\prime }}})$ for some crystalline $(\\varphi , \\widehat{\\Gamma })$ -module with $G$ -structure.", "Reducing modulo the maximal ideal, we obtain a crystalline $\\widehat{\\Gamma }$ -structure $\\widehat{\\mathfrak {P}}_{\\mathbb {F}}$ on $\\mathfrak {P}_{\\mathbb {F}}$ .", "By Proposition REF , this is the unique such structure.", "Recall the deformation problem $D^{\\operatorname{cris}, \\mu }_{\\overline{x}}$ from Corollary REF and $D^{[a,b]}_{\\overline{x}}$ from Definition REF .", "The natural map $D^{\\operatorname{cris}, \\mu }_{\\overline{x}} \\rightarrow D^{[a,b]}_{\\overline{x}}$ is a closed immersion (by Theorem REF ).", "By Corollary REF , $\\mathrm {Spf}\\ \\mathcal {O}^{\\mu }_{\\overline{x}}$ is closed in $D^{\\operatorname{cris}, \\mu }_{\\overline{x}}$ .", "Fix the isomorphism $\\beta _{\\mathbb {F}}:\\widehat{T}_{G, \\mathbb {F}}(\\widehat{\\mathfrak {P}}_{\\mathbb {F}}) \\cong \\overline{\\eta }$ .", "Consider the groupoid $D^{\\text{cris}, \\mu , \\square }_{\\widehat{\\mathfrak {P}}_{\\mathbb {F}}}$ from Definition REF .", "There is a natural morphism then from $D^{\\text{cris}, \\mu , \\square }_{\\widehat{\\mathfrak {P}}_{\\mathbb {F}}}$ to $D^{[a,b]}_{\\overline{x}}$ given by forgetting the $\\widehat{\\Gamma }$ -structure.", "By Proposition REF , this morphism is fully faithful, hence a closed immersion by considering tangent spaces.", "We claim that $D^{\\text{cris}, \\mu , \\square }_{\\widehat{\\mathfrak {P}}_{\\mathbb {F}}} = \\mathrm {Spf}\\ \\mathcal {O}^{\\mu }_{\\overline{x}}$ as closed subfunctors of $D^{[a,b]}_{\\overline{x}}$ .", "Since they are both representable, we look at their $F^{\\prime }$ -points for any finite extension $F^{\\prime }$ of $F$ .", "By Theorem REF and Corollary REF , $D^{\\text{cris}, \\mu , \\square }_{\\widehat{\\mathfrak {P}}_{\\mathbb {F}}}(F^{\\prime }) = D^{\\operatorname{cris}, \\mu }_{\\overline{x}}(F^{\\prime }) = \\mathrm {Spf}\\ \\mathcal {O}^{\\mu }_{\\overline{x}}(F^{\\prime }).$ Since $\\mathcal {O}^{\\mu }_{\\overline{x}}$ is $\\Lambda $ -flat and $\\mathcal {O}^{\\mu }_{\\overline{x}}[1/p]$ is formally smooth over $F$ , we deduce that $\\mathrm {Spf}\\ \\mathcal {O}^{\\mu }_{\\overline{x}} \\subset D^{\\text{cris}, \\mu , \\square }_{\\widehat{\\mathfrak {P}}_{\\mathbb {F}}}$ .", "Finally, $D^{\\text{cris}, \\mu , \\square }_{\\widehat{\\mathfrak {P}}_{\\mathbb {F}}}$ is formally smooth over $D^{\\mu }_{\\mathfrak {P}_{\\mathbb {F}}}$ by Theorem REF .", "By REF , there exists a diagram ${& \\mathrm {Spf}\\ S^{\\mu } [dl] [dr] & \\\\D^{\\text{cris}, \\mu , \\square }_{\\widehat{\\mathfrak {P}}_{\\mathbb {F}}}& & \\overline{D}^{\\mu }_{Q_{\\mathbb {F}}}, \\\\}$ where $S^{\\mu } \\in \\widehat{\\mathcal {C}}_{\\Lambda }$ and both morphisms are formally smooth ($Q_{\\mathbb {F}}$ is as in §3.2).", "The functor $ \\overline{D}^{\\mu }_{Q_{\\mathbb {F}}}$ is represented by a completed stalk $R^{\\mu }_{Q_{\\mathbb {F}}}$ on $M(\\mu )$ .", "In particular, $R^{\\mu }_{Q_{\\mathbb {F}}}$ is $\\Lambda $ -flat so the same is true of $D^{\\text{cris}, \\mu , \\square }_{\\widehat{\\mathfrak {P}}_{\\mathbb {F}}}$ .", "Thus, $D^{\\text{cris}, \\mu , \\square }_{\\widehat{\\mathfrak {P}_{\\mathbb {F}}}} = \\mathrm {Spf}\\ \\mathcal {O}^{\\mu }_{\\overline{x}}$ .", "By Theorem REF , $R^{\\mu }_{Q_{\\mathbb {F}}}$ is normal, Cohen-Macaulay and $R^{\\mu }_{Q_{\\mathbb {F}}} \\otimes _{\\Lambda } \\mathbb {F}$ is reduced so the same is true for $\\widehat{\\mathcal {O}}^{\\mu }_x$ .", "Theorem 4.4.6 Assume $K/\\mathbb {Q}_p$ is unramified, $p > 3$ , and $p \\nmid \\pi _1(G^{\\mathrm {ad}})$ .", "Then the universal crystalline deformation ring $R^{\\operatorname{cris}, \\mu }_{\\overline{\\eta }}$ is formally smooth over $\\Lambda _{[\\mu ]}$ .", "First, replace $\\Lambda $ by $\\Lambda _{[\\mu ]}$ .", "Without loss of generality, we can assume that $F$ contains all embeddings of $K$ since this can be arranged by a finite étale base change.", "When $K/\\mathbb {Q}_p$ is unramified, $\\operatorname{Gr}_G^{E(u), W}$ is a product of $[K:\\mathbb {Q}_p]$ copies of the affine Grassmanian $\\operatorname{Gr}_G$ (see [27]).", "If $\\mu = (\\mu _{\\psi })_{\\psi :K \\rightarrow F}$ , then $M(\\mu )_F = \\prod _{\\psi } S(\\mu _{\\psi })$ where $S(\\mu _{\\psi })$ are affine Schubert varieties of $\\operatorname{Gr}_{G_F}$ .", "Under the assumption that $p \\nmid \\pi _1(G^{\\mathrm {der}})$ , there is a flat closed $\\Lambda $ -subscheme of $\\operatorname{Gr}_G$ which abusing notation we denote by $S(\\mu _{\\psi })$ , whose fibers are the affine Schubert varieties for $\\mu _{\\psi }$ (see Theorem 8.4 in [34], especially the discussions in §8.e.3 and 8.e.4).", "Thus, $M(\\mu ) = \\prod _{\\psi :K \\rightarrow F} S(\\mu _{\\psi }).$ Since $\\mu _{\\psi }$ is minuscule, $S(\\mu _{\\psi })$ is isomorphic to a flag variety for $G$ hence $M(\\mu )$ is smooth (see Proposition REF ).", "The proof of Theorem REF shows that the local structure of $X^{\\operatorname{cris}, \\mu }_{\\overline{\\eta }}$ is smoothly equivalent to the local structure of $M(\\mu )$ .", "Thus, $X^{\\operatorname{cris}, \\mu }_{\\overline{\\eta }}$ is formally smooth over $\\Lambda $ .", "Finally, we have to show that $\\Theta :X^{\\operatorname{cris}, \\mu }_{\\overline{\\eta }} \\rightarrow \\mathrm {Spec}\\ R^{\\operatorname{cris}, \\mu }_{\\overline{\\eta }}$ is an isomorphism.", "Since $\\Theta [1/p]$ is an isomorphism and $R^{\\operatorname{cris}, \\mu }_{\\overline{\\eta }}$ is $\\Lambda $ -flat, it suffices to show that $\\Theta $ is a closed immersion.", "Let $m_R$ be the maximal ideal of $R^{\\operatorname{cris}, \\mu }_{\\overline{\\eta }}$ .", "Consider the reductions $\\Theta _n:X^{\\operatorname{cris}, \\mu }_{\\overline{\\eta }, n} \\rightarrow \\mathrm {Spec}\\ R^{\\operatorname{cris}, \\mu }_{\\overline{\\eta }}/m_R^n.$ We appeal to an analogue of Raynaud's uniqueness result for finite flat models ([39]).", "For any Artin local $\\mathbb {Z}_p$ -algebra $A$ and any finite $A$ -algebra $B$ , let $\\mathfrak {P}_B$ and $\\mathfrak {P}_B^{\\prime }$ be two distinct points in the fiber of $\\Theta _n$ over $x:R^{\\operatorname{cris}, \\mu }_{\\overline{\\eta }} \\rightarrow A$ , i.e., $G$ -Kisin lattices in $P_{x} \\otimes _A B$ .", "Let $V^{\\mathrm {ad}}$ denote the adjoint representation of $G$ .", "Under the assumption that $p >3$ , [29] (which generalizes Raynaud's result) implies that $\\mathfrak {P}_A(V^{\\mathrm {ad}}) = \\mathfrak {P}_A^{\\prime }(V^{\\mathrm {ad}})$ as Kisin lattices in $(P_{x} \\otimes _A B)(V^{\\mathrm {ad}})$ using that $\\mu $ is minuscule.", "Since $B$ is Artinian, without loss of generality we can assume it is local ring.", "Choose a trivialization of $\\mathfrak {P}_B$ .", "There exists $g \\in G(\\mathcal {O}_{\\mathcal {E}, B})$ such that $\\mathfrak {P}_{B}^{\\prime } = g.\\mathfrak {P}_B$ (working inside the affine Grassmanian as in Theorem REF ).", "The results above implies that $\\operatorname{Ad}(g) \\in G^{\\mathrm {ad}}(\\mathfrak {S}_A)$ .", "By assumption, $Z := \\ker (G \\rightarrow G^{\\mathrm {ad}})$ is étale so after possibly extending the residue field $\\mathbb {F}$ we can lift $\\operatorname{Ad}(g)$ to an element $\\widetilde{g} \\in G(\\mathfrak {S}_A)$ so that $g = \\widetilde{g} z$ where $z \\in Z(\\mathcal {O}_{\\mathcal {E}, A})$ .", "We want to show that $z \\in Z(\\mathfrak {S}_A)$ .", "We can write $Z$ as a product $Z_{\\operatorname{tors}} \\times (\\operatorname{\\mathbb {G}_m})^s$ for some $s \\ge 0$ .", "Since $Z_{\\operatorname{tors}}$ has order prime to $p$ by assumption, $Z_{\\operatorname{tors}}(\\mathcal {O}_{\\mathcal {E}, A}) = Z_{\\operatorname{tors}}(\\mathfrak {S}_A)$ so we can assume $z \\in (\\operatorname{\\mathbb {G}_m}(\\mathcal {O}_{\\mathcal {E}, A}))^s = ((A \\otimes _{\\mathbb {Z}_p} W)((u))^{\\times })^s.", "$ For any embedding $\\psi :W \\rightarrow \\mathcal {O}_F$ , we associate to $z$ the $s$ -tuple $\\lambda _{\\psi }$ of integers of the degrees of the leading terms of each component base changed by $\\psi $ .", "To show that $\\lambda _{\\psi } = 0$ we can work over $A/m_A = \\mathbb {F}$ .", "We think of $\\lambda _{\\psi }$ as a cocharacter of $Z$ .", "Consider the quotient of $G$ by its derived group $Z^{\\prime } := G/G^{\\mathrm {der}}$ .", "The map $X_*(Z) \\rightarrow X_*(Z^{\\prime })$ is injective.", "Any character $\\chi $ of $Z^{\\prime }$ defines a one-dimensional representation $L_{\\chi }$ of $G$ so in particular, we can consider $\\mathfrak {P}_B(L_{\\chi })$ and $\\mathfrak {P}_B^{\\prime }(L_{\\chi })$ as Kisin lattices in $P_x(L_{\\chi })$ .", "Writing $\\mathfrak {S}_{\\mathbb {F}} \\cong \\oplus _{\\psi :W \\rightarrow \\mathcal {O}_F} \\mathbb {F}[[u_{\\psi }]]$ , a Kisin lattice of $P_x(L_{\\chi })$ has type $(h_{\\psi })$ exactly when $\\phi _{P_x}(e) = (a_{\\psi } u^{h_{\\psi }})e$ for a basis element $e$ and $a_{\\psi } \\in \\mathbb {F}$ .", "Since both $\\mathfrak {P}_B$ and $\\mathfrak {P}_{B}^{\\prime }$ have type $\\mu $ , $\\mathfrak {P}_B(L_{\\chi })$ and $\\mathfrak {P}_B^{\\prime }(L_{\\chi })$ both have type $h_{\\psi } := \\langle \\chi , \\mu _{\\psi } \\rangle $ .", "However, a direct computation shows that $\\mathfrak {P}_B^{\\prime }(L_{\\chi })$ has type $h_{\\psi } + \\langle \\chi , p \\lambda _{\\psi ^{\\prime }} - \\lambda _{\\psi } \\rangle $ where $\\psi ^{\\prime } = \\varphi \\circ \\psi $ .", "Thus, $\\lambda _{\\psi } = p \\lambda _{\\psi ^{\\prime }}$ .", "We deduce that $p^{[K:\\mathbb {Q}_p]} \\lambda _{\\psi } = \\lambda _{\\psi }$ and so $\\lambda _{\\psi } = 0$ .", "We are reduced to the following general situation: $X \\rightarrow \\mathrm {Spec}\\ A$ is proper morphism which is injective on $B$ -points for all $A$ -finite algebras $B$ where $A$ is a local Artinian ring.", "By consideration of the one geometric fiber, $X \\rightarrow \\mathrm {Spec}\\ A$ is quasi-finite, hence finite.", "Thus, $X = \\mathrm {Spec}\\ A^{\\prime }$ .", "By Nakayama, it suffices to show $A/m_A \\rightarrow A^{\\prime }/(m_A) A^{\\prime }$ is surjective so we can assume $A = k$ is a field.", "Surjectivity follows from considering the two morphisms $A^{\\prime } \\rightrightarrows A^{\\prime } \\otimes _{k} A^{\\prime }$ which agree by injectivity of $X \\rightarrow \\mathrm {Spec}\\ A$ on $A$ -finite points." ] ]
1403.0553
[ [ "Poincar\\'e inverse problem and torus construction in phase space" ], [ "Abstract The phase space of an integrable Hamiltonian system is foliated by invariant tori.", "For an arbitrary Hamiltonian H such a foliation may not exist, but we can artificially construct one through a parameterised family of surfaces, with the intention of finding, in some sense, the closest integrable approximation to H. This is the Poincar\\'e inverse problem (PIP).", "In this paper, we review the available methods of solving the PIP and present a new iterative approach which works well for the often problematic thin orbits." ], [ "Introduction", "Integrability of a dynamical system, witnessed by a foliation of invariant toroidal manifolds in the phase space, is a highly useful property.", "For a Hamiltonian $H_0$ which is autonomous and integrable, we have a symplectic map (often explicit through the Hamilton-Jacobi approach) $(\\theta ,J)\\leftrightarrow (q,p),$ between the action-angle variables $(\\theta ,J)$ and the Cartesian phase-space coordinates $(q,p)$ .", "Not only does this eliminate the need for any numerical integration of the equations of motion, but also allows one to study the canon of near-integrable Hamiltonians $H$ in a form of power series; $H(\\theta ,J)=H_0(J)+\\varepsilon H_1(\\theta ,J)+\\varepsilon ^2 H_2(\\theta ,J)+\\ldots ,$ where the strength of the perturbation $\\varepsilon $ is small.", "To study Eq.", "(REF ) and the corresponding equations of motion, according to Poincaré [18], is “the general problem of dynamics”.", "We consider it as the direct problem, and ask the inverse: “given a near-integrable Hamiltonian in the form $H(q,p)$ , what is the best way of defining the maps (REF ) and $J\\mapsto H_0(J)$ ?”, i.e., we are looking for the closest integrable approximation to $H$ .", "Indeed, there are many cases where a system shows near-integrable behaviour numerically, but there is no prior information on $H_0$ , underlining the relevance of the question above.", "The Poincaré inverse problem (PIP) is obviously an ill-posed problem (in the sense of Hadamard); particularly, in regions of phase space, where invariant toroidal manifolds are absent, the closeness of $H$ and $H_0$ is an insufficient requirement for a unique solution.", "Simply minimising $\\Vert H-H_0\\Vert $ everywhere can lead to unacceptable results [7].", "Generally, in order to guarantee that the solution is a reasonable approximation to the actual motion, additional constraints (regularisation) are necessary.", "On the other hand, since $H$ is a known function, the data are error-free sampling points, and their number is only limited by the available computational resources.", "Therefore, “model noise” is the fundamental source of error, which is typical for many inverse problems.", "The PIP can also be a subproblem of a larger inverse one.", "For example, in dynamical or phase-space tomography [9] we want to solve for a potential field $\\Phi $ and phase-space distribution functions best corresponding to observed positions and velocities of matter.", "In this process, we repeatedly construct an $H_0$ corresponding to a trial $\\Phi $ in order to describe the phase-space distribution of observed matter along the tori of $H_0$ .", "Our approach towards the solution of the PIP is based on numerical construction of phase-space tori which should coincide with the existing Kolmogorov-Arnold-Moser (KAM) tori of $H$ (if any); i.e., on these, $\\Vert H-H_0\\Vert =0$ within the available accuracy.", "This ensures that the tori of $H=\\mathcal {H}(\\mathcal {J})$ , where $\\mathcal {H}$ is any specific integrable Hamiltonian and $\\mathcal {J}$ are its actions, are faithfully reconstructed.", "Also, for a perturbed $H=\\mathcal {H}+\\varepsilon \\mathcal {H}_1+\\varepsilon ^2\\mathcal {H}_2+\\ldots $ of the form (REF ), we have the natural limit $\\lim _{\\varepsilon \\rightarrow 0} H_0(J;\\varepsilon )=\\lim _{\\varepsilon \\rightarrow 0} H(q,p;\\varepsilon )=\\mathcal {H}(\\mathcal {J}).$ Note that it is not necessary to have $H_0=\\mathcal {H}$ , when $\\varepsilon >0$ .", "Even here there is some ambiguity; a torus can be analytically defined as a KAM one only by analysing it via the perturbation (REF ); whether we can directly deem an orbit of $H$ to correspond to a KAM torus is usually up to numerical resolution.", "As implied by the above principles, our approach is geometric rather than dynamical (which was the line of thought preferred by Poincaré as well).", "We construct the tori one by one, and shall refer to this process as torus construction, torus modelling, or torus embedding.", "The full solution, defining the maps (REF ) and $J\\mapsto H_0(J)$ everywhere, requires an interpolation scheme between the constructed tori.", "What interpolations produce a foliated global set of tori that really correspond to an $H_0$ is an open question.", "We shall not attempt to answer it in this paper; instead, we concentrate on the fundamental geometric problem of defining the local tori.", "This part of the PIP reduces to the problem of surface reconstruction, if only the KAM tori are modelled.", "However, this is generally not sufficient; when $H$ is far from integrable, its set of KAM tori may be too sparse to form an interpolation grid.", "One may even have a case where $H$ has no KAM tori at all [8].", "Hence, the ability of embedding approximating tori in the absence of KAM ones is also of practical importance, and it shall be a priority in our approach.", "This paper is organised as follows.", "In Sect.", "we review and classify some of the available methods of torus construction, and in Sect.", "we introduce a new one.", "In Sect.", "and we present numerical examples and discuss some practical choices during the implementation.", "In Sect.", "we sum up and discuss the performance of the method." ], [ "Overview of methods", "Two important questions that characterise any approach to torus construction are: a) how to define and parametrize a model torus, i.e., the map (REF ), and b) how to fit it in the phase space.", "In order to guarantee periodicity in the angle coordinates $\\theta $ , a typical, and natural, torus model involves Fourier series approximation.", "The approximated quantities can be simply the phase-space coordinates in the form $q(\\theta )$ and $p(\\theta )$ (direct model), or alternatively, the map (REF ) can be carried through using the action-angle variables $(\\vartheta ,\\mathcal {J})$ of some known integrable Hamiltonian $\\mathcal {H}$ as an intermediate step.", "In this case, a Fourier series is used to define a canonical transformation $(\\theta ,J)\\leftrightarrow (\\vartheta ,\\mathcal {J})$ by a generating function (GF model).", "Among the methods of fitting the model torus to the Hamiltonian flow of $H$ , a characterising feature is whether phase-space quantities are sampled along numerically integrated orbits (trajectory method), or are they iteratively adjusted on a grid of angles (iterative method).", "Worth noting are also the variables ($J$ , $(q,p)$ , or the frequencies $\\omega $ ) which label, i.e., define along with the Fourier coefficients, the torus to be modelled.", "A diverse selection of methods have been developed, each with their own strengths and weaknesses.", "Valluri & Merritt [21] give a more thorough review of the subject.", "For non-resonant regular orbits, direct trajectory methods, where the fundamental frequencies are determined from the Fourier spectra (e.g., [2], [13], [16]), are a robust choice.", "Having been refined by many authors, they show good accuracy which, however, decays near resonances, where it takes longer for a sample orbit to fill the torus.", "In addition, since chaotic orbits do not lie on tori at all, they are a questionable target for all trajectory methods.", "The hybrid GF-method of Warnock [22] is efficient, but also affected by the limitations above.", "For solving the PIP, it is important that tori can be embedded everywhere in the phase space.", "In this respect, the iterative method of McGill & Binney [14] with its further refinements [1], [10], [12] stands out, because it is unaffected by resonances, and can construct approximations to invariant tori also in phase-space regions where KAM tori do not exist [7], [8].", "As a GF-model it has the advantage that the constructed tori are inherently canonical.", "On the other hand, it relies on the existence of an integrable $\\mathcal {H}$ whose tori resemble those of $H$ .", "The wide class of Stäckel Hamiltonians is available for this purpose, but at the cost of increased analytical and computational effort [12].", "The use of simpler, but more restricted $\\mathcal {H}$ emphasises a general limitation of GF models; their inability to produce orbits which are arbitrarily thin; i.e., orbits for which $J_\\rho \\rightarrow 0$ , where $J_\\rho $ is the generalised transversal action, describing the thickness of the orbit in the configuration space [10], [11].", "This happens whenever orbits of $H$ and $\\mathcal {H}$ with $J_\\rho =0$ do not align in configuration space, which is usually the case.", "In two-dimensional systems in configuration space, this can sometimes be circumvented by using (rather cumbersome) point transformations [10].", "In three dimensions, this becomes much more difficult, and in any case there are tori in systems of all dimensions that cannot be modelled via such transformations.", "This is an important point when considering the general applicability of a method; indeed, we argue that only the type of method presented in this paper can be called generally applicable.", "The main problem with point transformations is that they are only based on the shape of the orbit in configuration space.", "Thus they separate, e.g., librating and circulation motion explicitly into two topologically different categories though in phase space they share the same topology.", "Furthermore, the point transformations are used to describe a coordinate system around the orbit $J_\\rho =0$ , in which one hopes to be able to describe the orbits $J_\\rho \\approx 0$ in a consistent way.", "In general, there is no guarantee that such a system exists; for example, some orbits in a rotating potential [8] are obviously not describable in this manner.", "In some cases, the image of the orbit in such a system becomes extremely complicated.", "Also, in two dimensions, the orbits at $J_\\rho =0$ are closed and thus easily integrated, whereas in three dimensions orbits for which $J_\\rho =0$ describe a surface in configuration space.", "Constructing this surface to define the desired coordinate system is more difficult than determining a curve.", "In this paper, acknowledging the limitations related to trajectory and GF-methods above, we propose a new approach which is direct and iterative.", "By doing this, we abandon the automatic canonicity of the model, which must be compensated by introducing additional optimisation constraints.", "A somewhat similar approach was taken by Ratcliff et al.", "[19], but the present algorithm is less restricted and more robust, since the model is fitted in the least-squares sense, and physically relevant optimisation functions are used.", "What is more, we introduce a principle for creating tori by letting model tori to adjust themselves by “floating”; i.e., their labels themselves are let to wander locally to some extent, which gives additional flexibility.", "We illustrate the new method by applying it to selected one- and two-dimensional systems commonly used in galaxy modelling." ], [ "The torus model", "We represent the model torus as a Fourier series of the Cartesian phase-space coordinates $q\\in \\mathbb {R}^n$ and velocities $p\\in \\mathbb {R}^n$ ; $p(\\theta )=\\sum _{k\\in A}\\alpha _ke^{i(k\\cdot \\theta )},\\qquad q(\\theta )=\\sum _{l\\in B}\\beta _le^{i(l\\cdot \\theta )},$ where $A,B\\subset \\mathbb {Z}^n$ are sets of multi-indices, $\\alpha _k,\\beta _l\\in \\mathbb {C}^n$ are Fourier coefficients, and $\\theta \\in [0,2\\pi )^n$ are canonical angles, conjugate to some actions $J_i\\ge 0$ , $i=1,\\ldots ,n$ which label the model torus.", "The real-valuedness of $p$ and $q$ dictates that $\\alpha _{-k}=\\overline{\\alpha }_k$ and $\\beta _{-l}=\\overline{\\beta }_l$ .", "The strict consistency of the Cartesian momenta, $p\\equiv \\dot{q}$ , would fix half of the coefficients: $A=B$ , $\\alpha _k=i(k\\cdot \\omega )\\beta _k$ .", "However, at this point, we wish to retain the maximal flexibility of the model (as a general $\\theta $ -parameterised surface in the $pq$ -space), and do not enforce this identically.", "In order to find the optimal $\\alpha _k$ and $\\beta _l$ for a given Hamiltonian $H:\\mathbb {R}^n\\times \\mathbb {R}^n\\rightarrow \\mathbb {R}$ , $(q,p)\\mapsto H(q,p)$ , we build an objective function $R$ which is a weighted sum $R(\\theta )=\\sum _i\\lambda _i\\Vert \\mathcal {E}_i(\\theta )\\Vert ^2$ of error functions $\\mathcal {E}_i$ that vanish on a KAM torus of $H$ .", "The norm is Euclidean, and $\\lambda _i>0$ .", "We will minimise $R$ on a grid of angles $\\theta _{(m)}$ , $m=1,\\ldots ,M$ with respect to the Fourier coefficients $\\alpha _k,\\beta _l$ .", "Obviously, success in torus construction requires that the error functions $\\mathcal {E}_i$ are chosen wisely.", "This is our goal in the following.", "Hamilton's equations of motion under $H$ are $\\dot{p}=-\\partial _qH,\\quad \\dot{q}=\\partial _pH,$ and supposing that $(\\theta ,J)$ are action-angle variables of $H$ , we also have $\\dot{J}=0,\\quad \\dot{\\theta }=\\partial _JH=\\omega .$ Applying the chain rule to the time derivatives of the model (REF ), inserting (REF ) and equating to (REF ) yields $\\frac{\\partial {p}}{\\partial {\\theta }}\\omega =-\\frac{\\partial {H}}{\\partial {q}},\\quad \\frac{\\partial {q}}{\\partial {\\theta }}\\omega =\\frac{\\partial {H}}{\\partial {p}}$ which identifies the vector field produced by the model with the Hamiltonian vector field of $H$ .", "Assuming that $H$ is differentiable, the partial derivatives in Eq.", "(REF ) can be evaluated for all $\\theta $ .", "We define two error functions, $\\mathcal {E}_1(\\theta )\\frac{\\partial {p}}{\\partial {\\theta }}\\omega +\\frac{\\partial {H}}{\\partial {q}},\\quad \\mathcal {E}_2(\\theta )\\frac{\\partial {q}}{\\partial {\\theta }}\\omega -\\frac{\\partial {H}}{\\partial {p}},$ which penalise for any deviation from the Hamiltonian flow of $H$ .", "Obviously, in order to use Eq.", "(REF ) we need the values of the frequencies $\\omega $ .", "A simple solution would be to use $\\omega $ for labelling the model torus, i.e., to use $\\omega $ as constants.", "However, sets of valid frequencies can be difficult to sample, and they may overlap for different orbit families, which makes $J$ the preferred label.", "Actually, we can avoid labelling the model torus (beforehand) altogether, if we dictate that $\\omega $ is a least-squares estimator, obtained by fitting the $2n$ equations (REF ) at all of the grid points.", "More specifically, we have an overdetermined linear system of the form $A\\omega =b$ , where $A$ is an $(2nM\\times n)$ -matrix obtained by concatenating the $(n\\times n)$ -matrices $\\partial {p}/\\partial {\\theta _{(m)}}$ and $\\partial {q}/\\partial {\\theta _{(m)}}$ for all $m=1,\\ldots ,M$ .", "The solution is given by the normal equations; $\\omega =(A^TA)^{-1}A^Tb.$ This complicates the dependence of $\\mathcal {E}_1$ and $\\mathcal {E}_2$ on the Fourier coefficients somewhat, but proved to be a numerically robust solution in our experiments.", "Convergence in minimisation towards a KAM torus of $H$ is possible with just the two error functions (REF ).", "This principle was employed by Ratcliff et al.", "[19], with the aim of having the two vanish at a given number of phase-space points.", "This, however, causes uncontrollable fluctuations on the rest of the constructed torus when it does not approximate a KAM one of $H$ .", "Our reconstruction is based on the geometric principle of least-squares fitting a number of error functions $\\mathcal {E}_i$ that are equivalent on KAM tori (subsets of them are necessary and sufficient), but different elsewhere.", "In other words, we construct the desired manifolds by regularisation and global smoothing.", "From Hamilton's equations of motion (REF ) we can easily derive $\\mathcal {E}_3(\\theta )\\frac{\\mathrm {d}{J}}{\\mathrm {d}{t}}=\\frac{\\partial {H(q(\\theta ),p(\\theta ))}}{\\partial {\\theta }}=\\frac{\\partial {H}}{\\partial {q}}\\frac{\\partial {q}}{\\partial {\\theta }}+\\frac{\\partial {H}}{\\partial {p}}\\frac{\\partial {p}}{\\partial {\\theta }}$ which gives additional support to, and falls into the same category as, $\\mathcal {E}_1$ and $\\mathcal {E}_2$ .", "In numerical exercises, where Eq.", "(REF ) was used, we found that the inclusion of $\\mathcal {E}_3$ improved our results noticeably.", "Further regularisation is probably necessary, at least when constructing tori in non-regular regions of phase space of $H$ .", "A useful error function is obtained by dictating that the value of the Hamiltonian $H$ should be constant on the minimisation grid; $\\mathcal {E}_4(\\theta )H(\\theta )-\\bar{H},$ where $\\bar{H}$ is the arithmetic mean of $H$ over the grid.", "This also yields the local solution of PIP: we define $H_0:=\\bar{H}$ on the constructed torus.", "The global interpolation scheme over the tori, must, of course, fulfill $\\omega (J)=\\partial H_0/\\partial J$ everywhere.", "Finally, we introduce an error function which ensures that the model converges towards a specific torus with given values of $J$ .", "For this purpose, we need to derive expressions for the model actions.", "By definition, $J_h=\\frac{1}{2\\pi }\\oint _{\\gamma _h}p\\cdot \\mathrm {d}q,\\quad h=1,\\ldots ,n,$ where $\\gamma _h$ is a closed path that cannot be continuously deformed into a point.", "For a model of the form $q(\\theta )$ we have $\\mathrm {d}q=\\sum _{j=1}^n\\frac{\\partial {q}}{\\partial {\\theta _j}}\\mathrm {d}\\theta _j,$ and if we select the path $\\gamma _h$ in such a way that $\\mathrm {d}\\theta _j=0$ for $j\\ne h$ , we can write $J_h=\\frac{1}{2\\pi }\\sum _{j=1}^n\\int _0^{2\\pi }p_j\\frac{\\partial {q_j}}{\\partial {\\theta _h}}\\mathrm {d}\\theta _h.$ By inserting the model equations (REF ) and integrating, we obtain $J_h=\\sum _{j=1}^n\\sum _{(k,l)\\in D_h}ik_h\\alpha _{j,k}\\beta _{j,l}e^{i\\left[(k+l)\\cdot \\theta \\right]},$ where $D_h=\\lbrace (k,l)\\in A\\times B:l_h=-k_h\\rbrace $ , and $\\alpha _{j,k}$ and $\\beta _{j,l}$ are the $j$ :th components of the corresponding Fourier coefficients.", "Note that $J_h$ does not depend on $\\theta _h$ .", "On a KAM torus, the dependence on $\\theta _j$ , $j\\ne h$ should vanish as well.", "In order to enforce this, and to punish for any deviation from a given set of actions $J=\\bar{J}$ , we introduce an error function $\\mathcal {E}_5(\\theta )J(\\theta )-\\bar{J}.$ As our notation suggests, $\\bar{J}$ could also be computed as an arithmetic mean over the grid, supplementing the label-free version of the torus construction algorithm." ], [ "Numerical example: isochrone potential", "Let us first check how the method reconstructs simple integrable Hamiltonians.", "The harmonic oscillator is trivial, since it is given directly in the Fourier form of Eq.", "(REF ) with only first-order terms.", "As a more interesting sanity check for the presented torus model and minimisation criteria, we apply them to the isochrone Hamiltonian [6] with a single degree of freedom; $H(q,p)=\\frac{1}{2}p^2-\\frac{c_1}{c_2+\\sqrt{c_2^2+q^2}},$ where $q,p\\in \\mathbb {R}$ , and the parameters have values $c_1=1$ and $c_2=0.15$ .", "By fixing the origin of $\\theta $ to $q=0$ , and acknowledging the symmetries of the system, we express the torus model (REF ) in trigonometric form; $p(\\theta )=\\sum _{k\\in X}a_k\\cos (k\\theta ),\\qquad q(\\theta )=\\sum _{l\\in Y}d_l\\sin (l\\theta ),$ where $X=Y=\\lbrace 1,3,5,7,\\ldots ,N-1\\rbrace $ .", "Since the system is one-dimensional and integrable, we can, in this case, safely label the model tori by their frequencies $\\omega $ , which simplifies the error functions (REF ).", "We have $\\mathcal {E}_1(\\theta )&=\\omega p^{\\prime }(\\theta )+\\frac{c_1q(\\theta )}{\\sqrt{c_2^2+q^2(\\theta )}(c_2+\\sqrt{c_2^2+q^2(\\theta )})},\\\\\\mathcal {E}_2(\\theta )&=\\omega q^{\\prime }(\\theta )-p(\\theta ),$ where $p^{\\prime }(\\theta )=-\\sum _{k\\in X}ka_k\\sin (k\\theta ),\\qquad q^{\\prime }(\\theta )=\\sum _{l\\in Y}ld_l\\cos (l\\theta ).$ The rest of the error functions $\\mathcal {E}_i$ , $i=3,4,5$ will not be used in this example.", "We minimise the objective function (REF ) (with $\\lambda _1=\\lambda _2=1$ ) using the Levenberg-Marquardt algorithm (LMA).", "This requires the calculation of the partial derivatives $\\partial {\\mathcal {E}_i}/\\partial {a_k}$ and $\\partial {\\mathcal {E}_i}/\\partial {d_l}$ , $i=1,2$ , which is straightforward using the equations above.", "With the purpose of studying the convergence radius and accuracy of the method, we ran the LMA several times, while varying the number of coefficients $N=16,32,48,\\ldots ,512$ and the frequency $\\omega =0.1,0.15,0.2,\\ldots ,2.0$ .", "The angle grid contained $M=1024$ equally spaced values (matching the Nyquist rate for $N=512$ ), but due to the symmetry of the model, only half of them were actually needed and used.", "As an initial guess for the model torus we set all the coefficients to zero except $a_1=d_1=1$ , which corresponds to a unit circle in the $qp$ -plane (i.e., a harmonic oscillator).", "For each pair $(N,\\omega )$ , after the LMA was well converged, we evaluated the standard deviation $\\sigma $ of the Hamiltonian $H(q(\\theta ),p(\\theta ))$ over the grid points.", "Figure REF shows a contour plot of the results.", "Figure: Standard deviation σ\\sigma of the one-dimensional isochrone Hamiltonian function values on the θ\\theta -grid after running the torus embedding LMA.For all $\\omega \\ge 0.4$ the LMA converged, and $H$ attained a constant value throughout the grid, within the accuracy seemingly bestowed by $N$ and also $\\omega $ .", "In the upper right portion of the figure, the values of $\\sigma $ are saturated due to the finite computational precision (64-bit floating point).", "For the isochrone, we know the relation $H=-(2c_1\\omega )^{2/3}/2$ , and hence, we could verify that Fig.", "REF also closely represents the deviation from the analytical values $H(\\omega )$ .", "For $\\omega <0.4$ there was no proper convergence.", "However, this could be remedied by compensating the increased amplitude of the motion in the initial guess ($\\omega \\rightarrow 0\\Rightarrow J\\rightarrow \\infty $ ).", "For example, with an initial guess $q(\\theta )=2\\sin \\theta $ , $p(\\theta )=2\\cos \\theta $ the LMA converged nicely for $\\omega =0.2,\\ldots ,0.4$ .", "We expect that a decent initial guess is increasingly important for more complex systems." ], [ "Numerical example: logarithmic potential and the PPS", "We raise the degrees of freedom, and formulate the torus modelling algorithm for planar systems of the form $H(q,p)=\\frac{1}{2}p\\cdot p+\\Phi (q),$ where $q,p\\in \\mathbb {R}^2$ and $\\Phi :\\mathbb {R}^2\\rightarrow \\mathbb {R}$ is a potential function.", "Depending on $\\Phi $ , there may be structural features in phase space, such as differentiation into orbit families, and stochastic regions surrounding resonant islands (for near-integrable potentials).", "In order to obtain a basic understanding how the torus algorithm copes with these features, we apply it to two selected potentials.", "First, we have the near-integrable logarithmic potential [20], [4] $\\Phi (q)=\\frac{1}{2}\\ln \\left(q_1^2+\\frac{q_2^2}{c_1^2}+c_2^2\\right)$ where $c_1=0.9$ and $c_2=1$ .", "The phase space of the logarithmic system is dominated by KAM tori, and with the selected parameter values, most of the orbits are either boxes (i.e.", "butterflies) or loops [15].", "For comparison, as an integrable example, we use a Stäckel potential in elliptic coordinates $u$ ; $\\Phi (u)=-\\frac{f(u_1)-f(u_2)}{u_1-u_2},$ where $f$ is a smooth function.", "The coordinate transformation $q\\mapsto u$ is given by $u_1+u_2=-c_1-c_2+q_1^2+q_2^2,\\qquad u_1u_2=c_1c_2-c_2q_1^2-c_1q_2^2,$ where $c_1<c_2$ are parameters.", "By selecting $u_2\\le u_1$ , we have $-c_2\\le u_2\\le -c_1\\le u_1$ .", "With the choice $f(u_1)&=-2\\pi (u_1+c_1)c_2c_3\\sqrt{\\frac{-c_1}{u_1+c_1}}\\arctan \\sqrt{\\frac{u_1+c_1}{-c_1}},\\\\f(u_2)&=-2\\pi (u_2+c_1)c_2c_3\\sqrt{\\frac{c_1}{u_2+c_1}}\\operatorname{artanh}\\sqrt{\\frac{u_2+c_1}{c_1}},$ the potential (REF ) becomes the perfect prolate spheroid (PPS), a special case of the perfect ellipsoid [5].", "We set the parameter values to $c_1=-1$ , $c_2=-0.25$ , and $c_3=1$ .", "In the $q_1q_2$ -plane, the PPS hosts the same major orbit families, the boxes and loops, as the logarithmic potential (REF ).", "In fact, the phase-space structures of the two systems are very similar, and in a side-by-side comparison of the corresponding torus models, integrability of the PPS supports the analysis on the logarithmic side as well.", "Since we expect to model only KAM tori, our main concern is how the orbit family, or the shape of an orbit within the family, affects the outcome of the torus modelling.", "Compared to the one-dimension case, a big difference is that we now use the actions $J$ to label the tori.", "There are two reasons for not using $\\omega $ .", "First, for the PPS (and thus possibly for the logarithmic system as well), the values of $\\omega $ overlap for boxes and loops (Fig.", "REF and REF ).", "Second, in some of our experiments while using $\\omega $ as a label, the LMA converged to a trivial 1-torus ($J_1=0$ or $J_2=0$ ).", "Figure: The loops (black circles) and boxes (gray crosses) in the action space of the PPS.Figure: The orbits of Fig.", "mapped into the PPS frequency space.A general trigonometric version of the torus model (REF ) is now $\\begin{aligned}p(\\theta )&=\\sum _{k\\in X}a_k\\cos (k\\cdot \\theta )+b_k\\sin (k\\cdot \\theta ),\\\\q(\\theta )&=\\sum _{l\\in Y}c_l\\cos (l\\cdot \\theta )+d_l\\sin (l\\cdot \\theta ),\\end{aligned}$ where $q,p\\in \\mathbb {R}^2$ , $a_k,b_k,c_l,d_l\\in \\mathbb {R}^2$ , $\\theta \\in [0,2\\pi )^2$ , and $X,Y\\subset \\mathbb {Z}^2$ are sets which do not contain opposite indices.", "Due to symmetries and other characteristics of a given orbit (family), some terms in Eq.", "(REF ) are always zero.", "Significant savings in computation time of the LMA are obtained, if the corresponding indices are identified and omitted from $X$ and $Y$ .", "For the integrable PPS there is a shortcut for doing this numerically.", "Namely, since we know the transformation $(\\theta ,J)\\mapsto (q,p)$ , we can compute sample points $q(\\theta )$ and $p(\\theta )$ on an even $\\theta $ -grid, and perform a discrete Fourier transformation (DFT) which yields a representation exactly of the form of our model.", "The computation is relatively fast (with optimised FFT-routines), and the DFT orbit can also be used to verify the result of the LMA.", "For the PPS box orbits, we found the nonzero coefficients to be $\\begin{aligned}&a_{1,k},\\text{ if }k=(\\text{odd, even}),\\\\&a_{2,k},\\text{ if }k=(\\text{even, odd}),\\end{aligned}\\qquad \\begin{aligned}&d_{1,l},\\text{ if }l=(\\text{odd, even}),\\\\&d_{2,l},\\text{ if }l=(\\text{even, odd}),\\end{aligned}$ and for the loops; $\\begin{aligned}&b_{1,k},\\text{ if }k=(\\text{even, odd}),\\\\&a_{2,k},\\text{ if }k=(\\text{even, odd}),\\end{aligned}\\qquad \\begin{aligned}&c_{1,l},\\text{ if }l=(\\text{even, odd}),\\\\&d_{2,l},\\text{ if }l=(\\text{even, odd}).\\end{aligned}$ We define two separate initial guesses for the model (REF ); one for box orbits, and the other for loops, containing only coefficients of the form (REF ) and (REF ), respectively.", "Because of the common orbit families, these initial guesses work for both the PPS and the logarithmic potential.", "Let $N\\in \\mathbb {N}$ limit the range of the indices such that $\\vert k_i\\vert \\le N$ , $\\vert l_i\\vert \\le N$ , $i=1,2$ .", "By setting $N=16$ , we end up with a total of 544 coefficients per orbit which are initially set to zero except the ones appearing below.", "As an initial box orbit, we use independent harmonic oscillators in both coordinate directions; $q(\\theta )=\\begin{pmatrix}\\displaystyle \\sin \\theta _1\\\\\\displaystyle \\sin \\theta _2\\end{pmatrix},\\qquad p(\\theta )=\\begin{pmatrix}\\displaystyle \\cos \\theta _1\\\\\\displaystyle \\cos \\theta _2\\end{pmatrix},$ which corresponds to $\\omega =(1,1)$ .", "For a loop orbit we must include more nonzero terms, in order to have the initial $J_1>0$ in Eq.", "(REF ).", "After some experiments, we found that $q(\\theta )=\\begin{pmatrix}\\displaystyle \\cos \\theta _2+\\frac{1}{20}\\cos (2\\theta _1+\\theta _2)-\\frac{1}{2}\\cos (-2\\theta _1+\\theta _2)\\\\\\displaystyle \\frac{3}{2}\\sin \\theta _2+\\frac{1}{10}\\sin (2\\theta _1+\\theta _2)-\\frac{1}{2}\\sin (-2\\theta _1+\\theta _2)\\end{pmatrix},$ and the $p(\\theta )$ that follows from selecting $\\omega =(\\frac{1}{2},\\frac{1}{2})$ are adequate.", "Using the least-squares frequencies (REF ), the error functions $\\mathcal {E}_1$ and $\\mathcal {E}_2$ in Eq.", "(REF ) for all grid points are given by $f(\\theta _{(m)},m=1,\\ldots ,M)=A\\omega -b$ .", "Differentiation with respect to the model coefficients gives $\\frac{\\partial {f}}{\\partial {x}}=A(A^TA)^{-1}\\frac{\\partial {A^T}}{\\partial {x}}(b-A\\omega )+(A(A^TA)^{-1}A^T-I)\\left(\\frac{\\partial {b}}{\\partial {x}}-\\frac{\\partial {A}}{\\partial {x}}\\omega \\right),$ where $x=a_k,b_k,c_l,d_l$ .", "Since the partial derivatives required for $\\partial {\\mathcal {E}_3}/\\partial {x}$ and $\\partial {\\mathcal {E}_4}/\\partial {x}$ already appear in Eq.", "(REF ), there is no trouble in adding $\\mathcal {E}_3$ and $\\mathcal {E}_4$ to the objective function (REF ).", "In order to use $\\mathcal {E}_5$ we must evaluate the actions (REF ) for the trigonometric version of the model.", "Taking care of the indexing, we have, for any $n$ , $\\begin{split}J_h=-\\frac{1}{2}\\sum _{j=1}^n\\sum _{(k,l)\\in Z_h}k_h\\Bigl \\lbrace (&a_{j,k}d_{j,l}+b_{j,k}c_{j,l})\\cos \\left[(k+l)\\cdot \\theta \\right]\\\\+(&b_{j,k}d_{j,l}-a_{j,k}c_{j,l})\\sin \\left[(k+l)\\cdot \\theta \\right]\\Bigr \\rbrace ,\\end{split}$ where $Z_h=\\lbrace (k,l)\\in X\\times Y\\cup X\\times -Y:l_h=-k_h\\rbrace $ .", "We set the $\\theta $ -grid to be a $32\\times 32$ square lattice.", "This fulfils the requirements of the Petersen-Middleton theorem [17] for the chosen number of model coefficients (i.e., there should be no aliasing).", "However, due to model symmetries with respect to both coordinate axes, we can omit everything but the values $\\theta \\in [0,\\pi )^2$ .", "This leaves us with only 256 grid points per orbit.", "Nevertheless, compared to the one-dimensional case, the computational effort to construct a torus is now far greater.", "Therefore, we do not try to optimise near to machine precision in the LMA.", "Instead, we set modest goals for the objective function value and concentrate on finding out whether the algorithm converges towards the correct solution or not." ], [ "Unlabelled torus construction", "In our first set of experiments, we use an objective function (REF ) which is unlabelled, consisting of the error functions $\\mathcal {E}_i$ , $i=1,\\ldots ,4$ with the weights $\\lambda _i$ set to unity.", "We run the LMA twice for each system (the PPS and logarithmic); first using the fixed initial guess for box orbits (REF ) and then using the one for loops (REF ).", "In all four cases, the LMA converges fast during the first few iteration steps, before reaching a plateau of slow convergence.", "During the phase of slow convergence, there may still be apparent changes in the model parameters.", "We interpret this behaviour being caused by the algorithm first finding a valley of KAM tori, and then advancing along its bottom.", "Since we have not specified a label, any KAM torus will suffice, and we tune the LMA to stop as soon as the bottom of the valley is reached.", "For comparing LMA results we again use $\\sigma $ ; the standard deviation of the corresponding Hamiltonian function $H$ over the $\\theta $ -grid.", "Table REF lists the final values of the actions, frequencies, and the values of $\\sigma $ .", "Table: Initially unlabelled constructed tori for the PPS and the logarithmic potential.It seems that the logarithmic tori are easier to construct than the PPS ones.", "For both systems, the constructed boxes are more accurate than the corresponding loops.", "It can be verified that the actions and frequencies in Table REF all correspond to invariant tori.", "For the PPS we can do this analytically (cf.", "Fig.", "REF ), and for both systems we can visually compare constructed orbits to numerically integrated ones, when the orbits share the same initial values.", "Fig.", "REF shows the Poincaré section $y=0$ , $\\dot{y}>0$ for the orbits in Table REF .", "The common initial point is $\\theta =(0,\\pi /2)$ on the constructed torus.", "The numerical integration of Hamilton's equations of motion in Cartesian coordinates is done using the Gragg-Bulirsch-Stoer (GBS) method with high accuracy settings (the standard deviation of $H$ along the integrated orbits is below $10^{-12}$ ).", "Figure: Poincaré sections y=0y=0, y ˙>0\\dot{y}>0 for the unlabelled torus construction, superimposed over numerically integrated reference orbits (gray): the PPS box (upper left) and loop (upper right), and logarithmic box (lower left) and loop (lower right).The plotted orbits coincide in all four cases.", "Hence, the tori are indeed close to invariant.", "For the PPS, increasing the number terms by setting $N=24$ (total of 1200 coefficients), and resizing the $\\theta $ -grid accordingly, resulted in less than an order of magnitude gain in the final $\\sigma $ .", "For our purposes, this was insignificant compared to the increased computation time, and we decided to stick with $N=16$ in the following." ], [ "Labels from an action grid", "Our next goal is to form an overall picture of how the torus construction performs when the actions $J$ are given as a label.", "An obvious problem in this approach is that we should know the orbit family in advance in order to supply the proper initial torus to the algorithm.", "Also, based on the isochrone experiment, the LMA may fail, if the size of the initial torus is far from correct.", "However, in the case of the integrable PPS, we have an opportunity to cheat, and to bypass these difficulties by using an initial torus given by the DFT.", "For a given action label, the set of non-zero Fourier coefficients from the DFT automatically determines the orbit family, and the truncated series is an excellent initial guess for the torus algorithm.", "What we obtain is a best-case scenario for the chosen size of the model $(N,M)$ .", "Figure REF shows the results for action labels matching the range in Fig.", "REF .", "Figure: The final standard deviation σ\\sigma of the PPS Hamiltonian HH on the θ\\theta -grid using initial guesses from the DFT.Clearly, the accuracy of the fit decreases as the tori get thicker and also near the separatrix of the box and loop orbits.", "The tori found by the unlabelled torus construction (Table REF ) are located in areas of low $\\sigma $ , which seems natural.", "Next, we shall take a brute force approach which does not rely on integrability, and run the LMA twice for each action label, first using the initial box orbit (REF ), and then the initial loop (REF ).", "From now on, the labels are taken from a rectangular grid of actions, consisting of the values $J_1=0,0.05,0.1,\\ldots ,1.3$ and $J_2=0,0.05,0.1,\\ldots ,1.3$ (note the inclusion of orbits with zero thickness).", "The action grid contains both boxes and loops for the PPS, and we shall see that this is also the case for the logarithmic potential.", "The same grid is used for both systems.", "The objective function (REF ) is now fully fledged, including all $\\mathcal {E}_i$ , $i=1,\\ldots ,5$ , and the corresponding weights $\\lambda _i$ set to unity.", "The results for each potential-orbit-family pair are gathered in Fig.", "REF .", "Figure: The final standard deviation σ\\sigma of the Hamiltonian HH on the θ\\theta -grid for the labelled torus construction using the fixed initial guesses: the PPS box (upper left) and loop (upper right), and logarithmic box (lower left) and loop (lower right).What we would like to see, is that there would be a clear threshold in $\\sigma $ , differentiating the regions where the initial orbit family is correct and where it is not.", "This would give us a tool for determining the orbit family by trial and error.", "Unfortunately, at least alone, $\\sigma $ is inadequate for such a purpose.", "However, most of the structures in Fig.", "REF can be identified in Fig.", "REF , and one can clearly visualise the separatrix of boxes and loops for the logarithmic potential.", "It seems that in regions where the initial orbit family is correct, the algorithm is not overly sensitive to the choice of the initial torus.", "There are a few isolated high values of $\\sigma $ for the PPS boxes, and a systematic change for the logarithmic boxes as $J_1$ increases (cf.", "Fig.", "REF ).", "These (local minima) can be avoided by improving the initial guess, as we shall see later.", "Although not emphasised in Fig.", "REF , the torus construction can run into various problems near the separatrices.", "First of all, there are some tori which attain low values of $\\sigma $ despite being stuck in a local minimum.", "Fortunately, these can be identified by having an inconsistent set of Fourier coefficients in such a way that the values $\\vert \\alpha _k-i(k\\cdot \\omega )\\beta _k\\vert $ , $k\\in A\\cap B$ in (REF ) are large.", "Then, even if the LMA is heading towards the correct solution, it may still suffer from slow convergence.", "Finally, if a good fit is obtained, the orbit family may still be incorrect, since $\\sigma $ is transparent in this respect (Fig.", "REF ).", "Figure: Poincaré sections y=0y=0, y ˙>0\\dot{y}>0 for selected constructed tori near the separatrix, superimposed over numerically integrated reference orbits (gray): the logarithmic loop J 1 =0.5,J 2 =0.65J_1=0.5,J_2=0.65 (left), and the PPS box J 1 =0.65,J 2 =0.75J_1=0.65,J_2=0.75 (right).On the other hand, the algorithm does not seem to be bothered by thin orbits; even the ones with zero thickness are modelled without a notable drop in accuracy (Fig.", "REF ; the box orbit on the right is shown for symmetry, though it is one of the rare cases when GF-methods work for thin orbits without point transformations).", "Figure: Constructed thin orbits in the xyxy-plane: the logarithmic loop J 1 =0,J 2 =1J_1=0,J_2=1 (left), and box J 1 =1,J 2 =0J_1=1,J_2=0 (right), superimposed over numerically integrated orbits (gray).Curiously, it seems that the model for loop orbits (REF ) is able to mimic a box orbit for small values of the actions, by setting a large amount of its coefficients to zero.", "Although $\\sigma $ for these orbits is relatively low, they are not close to the corresponding solutions (equal action labels) obtained by using the box model (REF ).", "We are not too concerned about this anomaly, though, since it does not play a role, if we, instead of blindly going through the whole action grid, use the following algorithm to construct tori only for selected grid points." ], [ "Action-grid probing", "Based on the the experiments above, we propose a scheme for probing the action grid, and isolating the regions where the torus construction best succeeds.", "The idea is to start from a point in the action grid which is closest to the actions obtained from the unlabelled torus construction.", "If the construction for the starting point is successful, we use it as the initial torus for all the adjacent grid points.", "When this procedure is iteratively repeated, we obtain an expanding region of successfully constructed tori in the action grid.", "Besides avoiding local minima, this technique enhances the rate of convergence in the LMA.", "A detailed description of the algorithm is given in Appendix .", "The algorithm must repeatedly evaluate whether a constructed torus is good or not.", "The choice is simply based on the final value of the objective function.", "If the value exceeds a certain threshold, the torus is rejected.", "In the following, we use a safe value of $10^{-6}$ , but this is subject to fine tuning.", "In order to rule out the specific local minima near the separatrices, we modify the objective function by adding the term $R_0=\\rho \\sum _{k\\in C}\\vert \\alpha _k-i(k\\cdot \\omega )\\beta _k\\vert ^2,$ where $C=A\\cap B$ , and $\\rho $ is a weighting parameter.", "We set $\\rho =0.01/\\#C$ .", "The results of action-grid probing, using the tori in Table REF as initial values, are displayed in Fig.", "REF -REF .", "For the PPS, the results in Fig.", "REF -REF are partly reconstructed, allowing a direct comparison.", "Note, however, that the thinnest orbits are only present in the constructed case.", "Figure: The PPS loops (black circles) and boxes (gray crosses) constructed by the action-grid probing algorithm.Figure: The constructed PPS frequencies, corresponding to the orbits in Fig.", ".Figure: The logarithmic loops (black circles) and boxes (gray crosses) constructed by the action-grid probing algorithm.Figure: The constructed logarithmic frequencies, corresponding to the orbits in Fig.", ".The points in the action-space map to frequency space as follows.", "In Fig.", "REF and REF the cusps pointing northeast correspond to the origin $J_1=J_2=0$ .", "The points $J_2=0$ form the northern facets.", "The eastern facets correspond to the box orbits with $J_1=0$ .", "The southern facet in Fig.", "REF (and partly in Fig.", "REF ) corresponds to the separatrix of the boxes and loops.", "For both the PPS and logarithmic potential, a few box orbits with zero thickness near the separatrix seem to have slightly incorrect values of $\\omega $ .", "Otherwise, the results seem consistent." ], [ "Discussion", "We have introduced the Poincaré inverse problem and proposed a new method towards its solution via torus construction.", "We have applied the method to the isochrone, PPS, and logarithmic systems.", "For the one-dimensional isochrone, the torus construction trivialises, but the results still prove the point that the LMA may not converge, if the initial torus is far away from the labelled KAM torus.", "Also, it is evident that the number of Fourier terms, needed to maintain a certain level of accuracy, increases as the tori get thicker ($\\Vert J\\Vert $ increases).", "These observations can be made for the PPS and logarithmic systems as well.", "In addition, in the two-dimensional cases, torus construction becomes more difficult near the separatrix between the orbit families.", "However, the thin orbits pose no problem, which may be a major selling point for our method.", "Especially for the PPS, the size of the torus model (set according to our computational budget) limits the accuracy of the results.", "Nevertheless, by using the action-grid probing algorithm, one is able to cherry-pick the most accurately constructed tori.", "For the specific goal of constructing KAM tori (boxes and loops) for the near-integrable logarithmic potential, the algorithm works quite well.", "This culminates in Fig.", "REF -REF where we provide a (partial) distribution of orbit families in the action and frequency spaces of a system which is inherently non-integrable.", "However, a question yet unanswered is how the modelling succeeds for the boxlet orbits; the bananas, fish, and pretzels (see, e.g., [15]), which appear when $c_2\\rightarrow 0$ in the logarithmic potential (REF ).", "Our results are similar to the action diagrams of Binney & Spergel [3] which were obtained with slightly different potential parameters.", "If increasing the size of the model is not an option, there are a couple of ideas that one could try to improve the quality of the constructed tori.", "First, the Fourier coefficients could be included according to $\\Vert k\\Vert \\le N$ , instead of $\\vert k_i\\vert \\le N$ , $i=1,\\ldots ,n$ .", "This, in combination with a hexagonal angle grid, would provide a more natural sampling scheme and could have a positive effect on the results.", "Second, the choice that the elementary phase-space variables $q$ and $p$ are Cartesian were made purely based on convenience; in the torus model (REF ) they can be defined to be any set of variables, as long as those uniquely define a point in the phase space.", "The Fourier series will certainly be shorter, if the geometry of the system matches the selected variables.", "However, one should note that angle-like variables (e.g., among polar, cylindrical, or spherical coordinates) are not suitable to be modelled using Fourier series because of their discontinuous nature.", "A trigonometric function of such a variable is obviously fine, but introduces ambiguity.", "Also, in curvilinear coordinates the computation of the partial derivatives needed for the LMA becomes more cumbersome.", "In our two-dimensional experiments, the Cartesian model was better suited for the logarithmic system.", "This might be explained by the fact that the PPS separates in elliptic, not in Cartesian, coordinates, although the orbit shapes are similar in the two systems.", "How does the new method with a direct torus model compare against iterative GF-methods?", "Its biggest advantage is undoubtedly flexibility; the method performs well for thin orbits and can also handle orbits near the separatrix.", "As a definite downside, it seems that increasing the size of the direct model is more expensive in computation time.", "This is understandable, since the LMA for a GF-method has to cope with only one Fourier series, and its objective function is simpler.", "Also, in the direct model, the built-in least squares solution (REF ) gets computationally heavy for large $\\theta $ -grids.", "Finding a suitable initial torus is important in both approaches; with the GF-model, we need to find a nearby integrable toy torus in order to initialise the construction process.", "This can be nontrivial.", "For the direct model, in order to avoid local minima, an equally nontrivial set of Fourier coefficients is required.", "This, however, can be obtained in various ways.", "The GF-model may have an edge when the full solution $J\\mapsto H_0(J)$ to the PIP is sought, since the Poincaré integral conditions are automatically satisfied on the interpolating tori; the direct model may require a denser grid of constructed tori.", "In the big picture, the algorithm presented in this paper complements the arsenal of torus construction methods.", "Within our classification, all four combinations of iterative/trajectory and direct/GF methods, have been tried.", "This, of course, does not mean that everything is invented; tori are geometrical objects in phase-space, and if viewed purely as such, they can be represented in many ways.", "One possibility could be to compute a best-fit tessellation and an angle-coordinate system for a bunch of numerically integrated orbit strips.", "As a simple example, consider Poincaré sections, cut along consecutive azimuthal angles, of a two-dimensional loop orbit with a constant value of $H$ , and visualise the torus model.", "As a benefit, this kind of approach would not be tied to Hamiltonian systems; any torus could be modelled.", "Also, Fourier series are not the only way of describing the points on a torus: one could also use other suitable cyclic tailored functions such as splines or wavelets." ], [ "Acknowledgements", "This work was supported by the Academy of Finland (the project “Inverse problems of regular and stochastic surfaces” and the centre of excellence in inverse problems)." ], [ "Action-grid probing algorithm", "Let $J_{(m)}$ , $m\\in \\mathbb {N}^n$ be a point in the action grid in such a way that $J_h=m_h\\Delta J_h$ , $h=1,\\ldots ,n$ , and $\\Delta J_h$ is the interval between grid points.", "We denote the whole action grid as a set $J_{(X)}$ , $X\\subset \\mathbb {N}^n$ .", "Suppose that we have a function nearestPoint which returns the closest grid point in $J_{(X)}$ to a given set of actions.", "We will also need adjacentPoints which returns the adjacent and diagonally adjacent points in $J_{(X)}$ to a given grid point.", "During the probing algorithm, we remove elements from $J_{(X)}$ and put them into new sets $J_{(S_0)},J_{(S_1)},J_{(S_2)},\\ldots $ which contain the action labels of successfully constructed tori.", "Let $T_{(m)}$ , $m\\in X$ be a torus (a set of Fourier coefficients) obtained by running the torus construction with $J_{(m)}$ given as a label.", "Suppose that this is implemented as a function constructTorus which also takes a second argument, an initial torus.", "A pseudocode for the probing algorithm is given in figure .", "As input, the algorithm requires a torus $T_*$ and a corresponding action label $J_*$ .", "These are given by the unlabelled torus construction, for each potential-orbit-family pair.", "[h] Inputinput Outputoutput constructTorusconstructTorus nearestPointnearestPoint adjacentPointsadjacentPoints $T_{(S_0)},T_{(S_1)},T_{(S_2)},\\ldots $ $J_{(k)}\\leftarrow $ $J_*$ $T_{(k)}\\leftarrow $ $J_{(k)},T_*$ $T_{(k)}$ is good$S_0\\leftarrow \\lbrace k\\rbrace $ $S_0\\leftarrow \\emptyset $ $X\\leftarrow X\\backslash \\lbrace k\\rbrace $ $i\\leftarrow 0$ $S_i\\ne \\emptyset $ $S_{i+1}\\leftarrow \\emptyset $ $m\\in S_i$ $J_{(A)}\\leftarrow $ $J_{(m)}$ $k\\in A$ $T_{(k)}\\leftarrow $ $J_{(k)},T_{(m)}$ $T_{(k)}$ is good$S_{i+1}\\leftarrow S_{i+1}\\cup \\lbrace k\\rbrace $ $X\\leftarrow X\\backslash \\lbrace k\\rbrace $ $i\\leftarrow i+1$ Action-grid probing algorithm.", "Received xxxx 20xx; revised xxxx 20xx." ] ]
1403.0395
[ [ "Anomalous nanoscale diffusion in Pt/Ti: superdiffusive intermixing" ], [ "Abstract Probing the anomalous nanoscale intermixing using molecular dynamics (MD) simulations in Pt/Ti bilayer we characterize the superdiffusive nature of interfacial atomic transport.", "In particular, the low-energy ($0.5$ keV) ion-sputtering induced transient enhanced intermixing has been studied by MD simulations.", "{\\em Ab initio} density functional calculations have been used to check and reparametrize the employed heteronuclear interatomic potential.", "We find a robust intermixing in Pt/Ti driven by nanoscale mass-anisotropy .", "The sum of the square of atomic displacements $\\langle R^2 \\rangle$ asymptotically scales nonlinearly ($\\langle R^2 \\rangle \\propto t^2$), where $t$ is the time of ion-sputtering, respectively which is the fingerprint of superdiffusive features.", "This anomalous behavior explains the high diffusity tail in the concentration profile obtained by Auger electron spectroscopy depth profiling (AES-DP) analysis in Pt/Ti bilayer (reported in ref.", ": P. S\\\"ule, et al., J. Appl.", "Phys., 101, 043502 (2007)).", "In Ti/Pt bilayer a linear time scaling of $\\langle R^2 \\rangle \\propto t$ has been found at the Ti/Pt interface indicating the suppression of superdiffusive features.", "These findings are inconsistent with the standard ion-mixing models.", "Instead a simple accelerative effect of the downward fluxes of energetic particles on the unidirectional fluxes of preferential intermixing of Pt atoms seems to explain the enhancement of interfacial broadening in Pt/Ti.", "Contrary to this in Ti/Pt the fluxes of recoils are in counterflow with intermixing Pt atoms and hence slows down the nanoscale mass-effect driven ballistic preferential mobility of Pt atoms." ], [ "Introduction", "The nanoscale production of nano-devices constitute a topic of high current interest due to numerous potential applications [1], [2].", "The construction of sharp interfaces in the nanoscale, however, faces many difficulties which requires the fundamental understanding of nanoscale interfacial diffusion [3].", "There are a growing number of evidences emerged in the last few years that anomalous nanoscale broadening of interfaces or high diffusity tail in the impurity concentration profile occur during ion-irradiation [4], [5], [6], [7], [8], [9], [10], sputter deposition of solids [11] or during thin film reactions and reactive front propagation [12], [13].", "The anomalous nanoscale bulk diffusional effects could be due to still not clearly established accelerative effects leading to anomalously fast and possibly athermal diffusion at interfaces or during impurity diffusion [4], [5], [6], [7], [8], [9], [10], [11], [12], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24].", "Anomalous atomic transport (AAT) in the bulk or on surfaces can be categorized into few groups of phenomena: (i) Post-irradiation induced enhancement of impurity [4], [5] or dopant diffusion [7], [18], [19], [20], [21], [22], [23], [24] and transient enhanced interfacial broadening [6], [8], [9], [10].", "Computer simulations reveal AAT during low-energy cluster and sigle-atomic deposition [31].", "(ii) The amplification of intermixing during thin film growth (sputter deposition) [11], [31] or forced alloying [14].", "(iii) Anomalous growth rates and front or interface propagation [12], [13].", "(iv) Superdiffusion, random walk, Levy flight [25], [26], [27], [28], [29] or quantum tunneling diffusion [3], [32] on solid surfaces.", "(v) Ultrafast atomic mobility with the coherent or collective movement of an ensemble of atoms (not cluster diffusion) [31], [33], [34], [35].", "The physics behind these processes is not clearly established yet hence theoretical works are needed to model AAT which could explain the relation between the diversity of such processes.", "This is because AAT could not simply be explained by the conventional theories of atom movements, such as vacancy, interstitial diffusion, simple atomic site exchanges or hopping mechanisms [3], [28].", "In the present paper we show that a simple explanation, such as atomic mass anisotropy induced amplification of ion-intermixing explains the occurrence of AAT in the prototypical mass-anisotropic system of Pt/Ti.", "Computer atomistic simulations also reveal the occurrence of the enhancement of atomic transport upon ion-bombardment of various solids beyond a level explained by radiation enhanced diffusion [9], [33], [36].", "The characterization and understanding of the nanoscale amplification or weakening of atomic transport in the bulk, at interfaces and on the surface could be an important ingredient of the efficient production of nanostructures [12], [37].", "In particular, it has been shown that nanoscale mass anisotropy influences seriously the sharpness of anisotropic interfaces [9], [36] as well as surface morphology [29], [30] and adatom yield [29].", "Under forced conditions (such as low-energy ion-sputtering or ion-beam deposition, ball milling) otherwise not easily observable anomalous atomic transport processes can be amplified and could be detected.", "Such conditions have widely been applied in the last decades to force e.g.", "intermixing or alloying between immiscible elements [14], [17], [38], [39], [40], [41].", "The athermal broadening of the interface in strongly intermixing ion-irradiated bilayers such as Ni/Al and Cr/Al have been found which are inconsistent with the standard ion beam mixing models, such as ballistic, thermal spike or radiation enhanced diffusion [6].", "In these materials, such as Ni/Al or Co/Al, the exotermic solid state reaction can even lead to extremely fast burn rates [12], [42].", "The anomalously strong asymmetric ion-beam mixing has been found recently in Cr/Si multilayer using focused ion beam [10].", "Moreover, transient enhanced diffusion (TED) and intermixing have also been reported in post-annealed dopant implanted semiconductors [18], [20], [21], [22], [23], [24].", "TED occurs when the depth distribution of the dopant exceeds the ion range [4].", "Using low-energy ion-sputtering [38], [45], [46] and Auger electron depth profiling analysis [40] it has been concluded recently that transient enhanced intermixing could occur in the film/substrate bilayer Pt/Ti while no such behavior is seen in the Ti/Pt bilayer, hence the magnitude of intermixing might depend on the succession of the film and the substrate [9].", "Anomalously long interdiffusion depths have been found in various diffusion couples [4], [6], [11], [18], [20], [21], [22], [23], [24].", "Transient enhanced intermixing has been reported in nonstochiometric AlAs/GaAs quantum well structures [43] or in AsSb/GaSb superlattices [44] and has been attributed to vacancy or self-interstitial supersaturation in annealed samples.", "It has also been reported that in few cases anomalous intermixing is neither driven by bulk diffusion parameters nor by thermodynamic forces (such as heats of alloying) [11] or nor by heats of mixing [36], [47].", "It has also been found that during low-energy ion-bombardment of bilayers the intermixing length (and the mixing efficiency) scales nonlinearly with the mass anisotropy (mass ratio) leading to the abrupt increase of the mixing efficiency in mass-anisotropic bilayers [36].", "In the present work, computer atomistic simulations have been carried out to explain the occurrence of the ion-sputtering induced high diffusity tail in the concentration profile of the film/substrate system of Pt/Ti.", "The enhanced intermixing in Pt/Ti reported in ref.", "[9] is attempted to interpret as a superdiffusive atomic transport process since it fulfills the most important condition of superdiffusion: the square of atomic displacements scales nonlinearly with the time of IM [25].", "Superdiffusion has only been reported until now on solid surfaces [26], [27], [28] and no reports have been found for bulk superdiffusion except for ultralight particles such as the migration of H in metals [3].", "We find the long range atomic transport of Pt in Ti is highly unusual and which could be explained as a mass anisotropy driven superdiffusive intermixing (SIM) process.", "Moreover, we conclude that SIM occurs due to the accelerative effect of unidirectional fluxes of energetic particles." ], [ "The setup of the atomistic simulations", "Classical molecular dynamics simulations have been used to simulate the ion-solid interaction (using the PARCAS code [53]).", "A variable timestep and the Berendsen temperature control is used to maintain the thermal equilibrium of the entire system.", "[54].", "The global coupling to the heat bath can be adjusted by the so called Berendsen temperature which we set to 70 K. Temperature controll has been applied at the cell borders of the simulation cell to maintain constant temperature conditions.", "The bottom layers are held fixed in order to avoid the rotation of the cell.", "Since the z direction is open, rotation could start around the z axis.", "The bottom layer fixation is also required to prevent the translation of the cell.", "Periodic boundary conditions are imposed laterarily and a free surface is left for the ion-impacts.", "The simulation uses the Gear's predictor-corrector algorithm to calculate atomic trajectories [54].", "The detailed description of other technical aspects of the MD simulations are given in refs.", "[53] and [54] and details specific to the current system in recent communications [9], [36], [47].", "Our primary purpose is to simulate the conditions occur during ion-sputtering [9], [45] and Auger electron spectroscopy depth profiling analysis (AES-DP) [9] using molecular dynamics simulations [54].", "Recently, MD simulation has been used to simulate ion-sputtering induced surface roughening [30], [48], [49].", "Following our previous work [9] we ion bombard the bilayers Pt/Ti and Ti/Pt with 0.5 keV Ar$^+$ ions repeatedly (consecutively) with a time interval of 10-20 ps between each of the ion-impacts at 300 K which we find sufficiently long time for most of the structural relaxations and the termination of atomic mixing, such as sputtering induced intermixing (IM) [47].", "Since we focus on the occurrence of transient intermixing atomic transport processes, the relaxation time of $10-20$ ps should be appropriate for getting adequate information on transient enhanced intermixing.", "Pair potentials have been used for the interaction of the Ar$^+$ ions with the metal atoms derived using ab initio density functional calculations.", "The initial velocity direction of the impacting ion was 10 degrees with respect to the surface of the crystal (grazing angle of incidence) to avoid channeling directions and to simulate the conditions applied during ion-sputtering [9].", "The impact positions have been randomly varied on the surface of the film/substrate system and the azimuth angle $\\phi $ (the direction of the ion-beam).", "In order to simulate ion-sputtering a large number of ion irradiation are employed using script governed simulations conducted subsequently together with analyzing the history files (movie files) in each irradiation steps.", "In this article results are shown up to 150 ion irradiation (in a similar way to that given in ref.", "[9]).", "The impact positions of the $100-150$ ions are randomly distributed over a $20 \\times 20$ Å$^2$ area on the surface.", "The volume of the cubic simulation cell is $110 \\times 110 \\times 90$ $\\hbox{Å}^3$ including $\\sim 57000$ atoms (with 9 monolayers (ML) film/substrate).", "The film and the substrate are $\\sim 20$ and $\\sim 68$ $\\hbox{Å}$ thick, respectively.", "The setup of the simulation cell, in particular the 20 $\\hbox{Å}$ film thickness is assumed to be appropriate for simulating broadening.", "Our experience shows that the variation of the film thickness does not affect the final result significantly, except if ultrathin film is used (e.g.", "if less than $\\sim 10$ $\\hbox{Å}$ thick film).", "At around 5 or less ML thick film surface roughening could affect mixing [30].", "The (111) interface of the fcc crystal is parallel to (0001) of the hcp crystal and the close packed directions are parallel.", "The interfacial system has been created as follows: the hcp Ti is put by hand on the (111) Pt bulk (and vice versa) and various structures with different lateral sizes have been probed and are put together with fixed orientation mentioned above.", "This must be done to avoid built in stress in the initial stucture.", "Therefore we put together slabs of the film and the substrate with different width while keeping (111) interfacial orientation.", "This is because the lattice mismatch is sensitive to the relative positions of the atoms at the interface.", "In order to minimize lattice misfit in the initial structure the relative lateral positions of the film and the substrate has been varied and the interfacial system with the smallest misfit strain has been selected.", "In fact this has been monitored via the starting average temperature of the system which indicates us how the generated strucure is relaxed.", "The closer this temperature to 0 K the more relaxed the system is.", "The remaining misfit is properly minimized below $\\sim 6 \\%$ during the relaxation process so that the Ti and Pt layers keep their original crystal structure and we get an atomically sharp interface.", "During the relaxation (equilibration) process the temperature is softly scaled down to zero and a sufficiently relaxed structure has been obtained.", "According to our practice we find that during the temperature scaling down the structure becomes sufficiently relaxed.", "Then the careful heating up of the system to 300 K has also been carried out.", "We believe that the lattice mismatch is minimized to the lowest possible level and we are convinced that no serious built-in stress remained in the simulation cell which could cause the explored anomalous atomic transport behaviors.", "Table: The parameters used in the tight binding potential given in Eqs.", "(1)-(2) We used a tight-binding many body potential on the basis of the second moment approximation (TB-SMA) to the density of states [58], to describe interatomic interactions.", "Within the TB-SMA, the band energy the attractive part of the potential reads, Figure: The crosspotential energy (eV) for the Ti-Pt dimer as a function ofthe interatomic distance (Å\\hbox{Å}) obtained bythe first-principles PBE/DFT method.", "For comparison the fitted interpolated TB-SMA potentialis also shown calculated for the Ti-Pt dimer.$E_b^i=-\\biggm [ \\sum _{j, r_{ij} < r_c} \\xi ^2 exp \\biggm [-2q \\biggm (\\frac{r_{ij}}{r_0}-1 \\biggm ) \\biggm ] \\biggm ]^{1/2},$ where $r_c$ is the cutoff radius of the interaction and $r_0$ is the first neighbor distance (atomic size parameter).", "The repulsive term is a Born-Mayer type phenomenological core-repulsion term: $E_r^i=A \\sum _{j, r_{ij}<r_c} exp \\biggm [-p \\biggm (\\frac{r_{ij}}{r_0}-1 \\biggm ) \\biggm ].$ The parameters ($\\xi , q, A, p, r_0$ ) are fitted to experimental values of the cohesive energy, the lattice parameter, the bulk modulus and the elastic constants $c_{11}$ , $c_{12}$ and $c_{44}$ [58] and which are given in Table 1.", "$r_{ij}$ is the internuclear separation between atoms $i$ and $j$ .", "The total cohesive energy of the system is $E_c=\\sum _i^{Nat} (E_r^i+E_b^i),$ where $Nat$ is the number of atoms in the system.", "The TB-SMA potential gives a good description of lattice vacancies, including atomic migration properties and a reasonable description of solid surfaces and melting [58].", "In ref.", "[9].", "it has been shown that the TB-SMA potential gives the reasonable description of IM in Pt/Ti and gives interfacial broadening comparable with AES-DP measurements.", "Cutoff is imposed out of the 2nd nearest neighbors when the interatomic interactions have been calculated which we find sufficient for simulating ion-mixing [36], [47].", "Simulations have also been conducted using larger cutoff distances (up to 4th neighbors), however, no serious change has been observed in the final results.", "An interpolation scheme has been employed for the crosspotential Ti-Pt [9], [36], [29], [57].", "The employed potentials and the interpolation scheme for heteronuclear interactions have successfully been used for MD simulations [36], [50], [55], [56].", "The Ti-Pt interatomic crosspotential of the TB-SMA potentials [58] type has been fitted recently to the experimental heat of mixing of the corresponding alloy system [47], [30].", "The scaling factor $r_0$ (the heteronuclear first neighbor distance) is given as the average of the elemental first neighbor distances.", "In this paper we use instead of our recent fit of the Ti-Pt potential [47] a more sophisticated potential.", "The crosspotential energy has been calculated for the Ti-Pt dimer using ab initio local spin density functional calculations [59] together with quadratic convergence self-consistent field (SCF) method.", "The G03 code is well suited for molecular calculations, hence it can be used for checking pair-potentials.", "The interatomic potential $V(dr)$ between two atoms is defined as the difference of total energy at an interatomic separation $dr$ and the total energy of the isolated atoms $V(dr)=E(dr)-E(\\infty ).$ The Kohn-Sham equations (based on density functional theory, DFT) [60] are solved in an atom centered Gaussian basis set and the core electrons are described by effective core potentials (using the LANL2DZ basis set) [61] and we used the Perwed-Burke-Ernzerhof (PBE) gradient corrected exchange-correlation potential [62].", "Fist principles calculations based on density functional theory (DFT) have been applied in various fields in the last few years [63].", "The obtained profile is plotted in Fig.", "REF together with our interpolated TB-SMA potential for the Ti-Pt dimer.", "We find that our interpolated TB-SMA potential nearly perfectly matches the ab initio one hence we are convinced that the TB-SMA model accurately describes the heteronuclear interaction in the Ti-Pt dimer.", "In fact we fitted only parameter $\\xi =4.2$ , which influences the deepness of the potential well.", "The rest of the parameters are obtained using the interpolation scheme outlined in the caption of Table 1.", "We assume that this dimer potential is transferable for those cases when the Pt atom is embedded in Ti.", "This can be done because, as we outlined above, the interpolated Ti-Pt potential properly reproduces the available experimental results for the Ti-Pt alloy [47].", "The crossectional computer animations of simulated ion-sputtering can be seen in our web page [64].", "Further details are given in refs.", "[9], [47], [36]." ], [ "Results", "The cartoons of the simulation cells (crossectional slabs as a 3D view) can be seen in Fig  REF which show the strong mixing at the interface in Fig.", "1a (Pt/Ti) and a much weaker mixing in Fig.", "1b (Ti/Pt).", "In Fig  REF the evolution of the sum of the square of atomic displacements (SD) $\\langle R^2 \\rangle = \\sum _i^{N_{atom}} [{\\bf r_i}(t)-{\\bf r_i}(t=0)]^2,$ of all intermixing atoms obtained by molecular dynamics simulations, where (${\\bf r_i}(t)$ is the position vector of atom 'i' at time $t$ , $N_{atom}$ is the total number of atoms included in the sum), can be followed as a function of the ion fluence.", "Figure: The cartoons of the simulation cells after 100 ions bombardmentas a crossectional view (cut in the middle of the cell).The incorporated Ar + ^+ ions are also shown as larger light spheres.The smaller light and dark spheres denote the Pt and Ti atoms, respectively.Fig 2a: Pt/Ti,Fig 2b: Ti/PtLateral components ($x,y$ ) are excluded from $\\langle R^2 \\rangle $ and only contributions from IM atomic displacements perpendicular to the layers are included ($z$ components).", "We follow during simulations the time evolution of $\\langle R^2 \\rangle $ which reflects the atomic migration through the interface (no other atomic transport processes are included).", "Note, that we do not calculate the mean square of atomic displacements (MSD) which is an averaged SD over the number of atoms included in the sum (MSD=$\\langle R^2 \\rangle /N_{atom}$ ).", "MSD does not reflect the real physics when localized events take place, e.g.", "when only few dozens of atoms are displaced and intermixed.", "In such cases the divison by $N_{atom}$ , when $N_{atom}$ is the total number of the atoms in the simulation cell leads to the meaningless $\\langle R^2 \\rangle /N_{atom} \\rightarrow 0$ result when $N_{atom} \\rightarrow \\infty $ , e.g.", "with the increasing number of atoms in the simulation cell.", "Also, it is hard to give the number of \"active\" particles which really take place in the transient atomic transport processes.", "Hence we prefer to use the more appropriate quantity SD.", "In Fig.", "REF we present $\\langle R^2 \\rangle $ as a function of the number of ion impacts $N_i$ (ion-number fluence).", "$\\langle R^2 \\rangle (N_i)$ corresponds to the final value of $\\langle R^2 \\rangle $ obtained during the $N_i$ th simulation.", "The final relaxed structure of the simulation of the $(N_i-1)$ th ion-bombardment is used as the input structure for the $N_i$ th ion-irradiation.", "The asymmetry of mixing can clearly be seen when $\\langle R^2 \\rangle (N_i)$ and the depth profiles given in ref.", "[9] are compared in Ti/Pt and in Pt/Ti.", "Figure: The simulated square of IM atomic displacements 〈R 2 〉\\langle R^2 \\rangle (Å 2 \\hbox{Å}^2) in Pt/Ti, Ti/Pt and in Cu/Co as a function of theion-fluence (number of ions) obtained during the ion-sputtering of these bilayers at 500 eV ion energy (results are shown up to 100 ions).The dotted lines (iso-Pt/Ti and iso-Ti/Pt) denote the results obtained for the artificial mass-isotropicPt/Ti and Ti/Pt bilayers, respectively.The computer animations of the simulations [64] together with the plotted broadening values at the interface in ref.", "[9] also reveal the stronger IM in Pt/Ti.", "Moreover we find the strong divergence of $\\langle R^2 \\rangle $ from linear scaling for Pt/Ti while linear scaling has been found for Ti/Pt.", "As it has already been shown in ref.", "[9] AES-DP found a relatively weak IM in Ti/Pt (the interface broadening $\\sigma \\approx 20$ $\\hbox{Å}$ ) whil e an unusually high IM occurs in the Pt/Ti bilayer ($\\sigma \\approx 70$ $\\hbox{Å}$ ).", "MD simulations provide $\\sim 20$ $\\hbox{Å})$ and $\\sim 40$ $\\hbox{Å})$ thick interface after 200 ion impacts, respectively." ], [ "The effect of mass anisotropy", "In order to clarify the mechanism of intermixing and to understand how much the interfacial anisotropy influences IM, simulations have been carried out with atomic mass ratio $\\delta =m_{Pt}/m_{Ti}$ , where $m_{Pt}$ and $m_{Ti}$ are the atomic masses, is artificially set to $\\delta \\approx 1$ (mass-isotropic).", "We find that $\\langle R^2 \\rangle $ is below the corresponding curve of Pt/Ti (see Fig  REF , iso-Pt/Ti, dotted line, see also the corresponding animation [64]).", "The $\\langle R^2 \\rangle $ scales nearly linearily as a function of the number of ions (and with $t$ ) for iso-Pt/Ti.", "Hence the asymptotics of $\\langle R^2 \\rangle $ is sensitive to the effect of $\\delta $ .", "We reach the conclusion that the mass-effect is robust and the magnitude of IM is weakened significantly.", "Actually the system undergoes the transition in the asymptotics of $\\langle R^2 \\rangle \\propto t^2 \\rightarrow \\langle R^2 \\rangle \\propto t$ .", "This finding together with our AES measurements (with the long-range tail shown in ref.", "[9]) confirms our recent results reported for various bilayers in which a strong correlation has been obtained between the experimental and simulated mixing efficiencies and mass anisotropy in various metallic bilayers [36].", "In that article we found that below a certain threshold mass ratio value ($\\delta \\le 0.33$ ) the rate of intermixing increases abruptly [36].", "On the basis of the results obtained in this paper this surprising interdiffusive behavior of bilayers could be explained by the anomalous nature of mixing which can be tuned by the mass-anisotropy (mass ratio) of the systems.", "The experimentally observed mixing asymmetry can also partly be explained by the mass effect.", "The interchange of atomic masses: To further test mass-effect on IM, we carried out simulations for the Pt/Ti system in which the atomic masses have been interchanged (Ti possesses the atomic mass of Pt and vice versa) setting in an artificial mass ratio (the inverse of the normal one).", "We find that this artificial setup of atomic masses results in the suppression of IM in Pt/Ti.", "Moerover, if we interchange the masses in Ti/Pt, we find strong IM and nonlinear scaling of $\\langle R^2 \\rangle $ , while we find a much weaker one with natural atomic masses.", "The effect of mass anisotropy seems to be decisive in the occurrence of transient enhanced IM in Pt/Ti and of the asymmetry of IM.", "In recent papers we explained single-ion impact induced intermixing governed by the mass-anisotropy parameter (mass ratio) in these bilayers [36], [29].", "It has already been shown in refs.", "[36], [47] that the backscattering of the light hyperthermal particles (BHP) at the mass-anisotropic interface leads to the increase in the energy density of the displacement cascade.", "We have found that the jumping rate of atoms through the interface is seriously affected by the mass-anisotropy of the interface when energetic atoms (hyperthermal particles) are present and which leads to the preferential IM of Pt to Ti [36], [47].", "Although we find in accordance with ref.", "[36] that thermal spike occurs in both systems with the lifetime of few ps, however, we rule out thermal spike effects on IM in Pt/Ti.", "In particular, the observed insensitivity of IM to the choice of the heat of mixing $\\Delta H$ in Ti/Pt [47] is in contrast with the thermal spike model [36].", "The appearance of $\\langle R^2 \\rangle \\propto t^2$ scaling requires the presence of hyperthermal particles which are present only during the collisional cascade.", "These particles are also present in Ti/Pt, however in Pt/Ti we find the further acceleration of the hot atoms due to unknown origin.", "We reach the conclusion that there must be an accelerative force field which speeds up few of the Pt particles to ballistic transport.", "The process must be active during the cascade.", "This is reflected by the divergence of $\\langle R^2 \\rangle $ from linear scaling in Fig  REF for Pt/Ti.", "Figure: The crossectional view of a typical collisional displacement cascade at the interfacewith atomic trajectories (two monolayers are shown at the interfaces as a crossectionalslab cut in the middle of the simulation cell)in Pt/Ti (upper panel) and in Ti/Pt (lower panel).The positions of the energetic particles are collected up to 500 fs duringa 500 eV single ion-impact event.The vertical axis corresponds to the depth position given in Å\\hbox{Å}.The position z=0z=0 is the depth position of the interface.xx is the horizonthal position (Å\\hbox{Å}).Interdiffusion takes place via ballistic jumps (ballistic mixing), when $\\langle R^2 \\rangle $ grows asymptotically as $N^2$ , where $N$ is the number fluence (the same asymptotics holds as a function of ion-dose or ion-fluence).", "This can clearly be seen in Fig.", "REF for Pt/Ti.", "The horizontal axis is proportional to the time of ion-sputtering, hence $\\langle R^2 \\rangle \\propto t^2$ which is the time scaling of ballistic atomic transport [25].", "In our particular case we follow the time evolution of the simulation cell after each of the ion impacts until $t \\sim 10-20$ ps which we find sufficient time for the evolution of $\\langle R^2 \\rangle $ .", "Anyhow, above this $t$ value the asymptotics of $\\langle R^2 \\rangle (t)$ is invariant to the choice of the elapsed time/ion-bombardment induced evolution of $\\langle R^2 \\rangle (t)$ .", "Hence the transformation between ion-fluence and time scale is allowed.", "$\\langle R^2 \\rangle \\propto t^2$ and $\\langle R^2 \\rangle \\propto t$ time scalings have been found even for the single-ion impacts averaged for few events (when $\\langle R^2 \\rangle (t)$ is plotted only for a single-ion impact) for Pt/Ti and Ti/Pt, respectively.", "The $\\langle R^2 \\rangle \\propto t^n$ , scaling, where $n \\ge 2$ , used to be considered as the signature of anomalous diffusion (superdiffusion) in the literature [25].", "Such kind of time scaling has been repoted until now during the random walk or flight of particles and clusters on solid surfaces [25], [26], [27], [28], [29].", "These processes are inherently athermal due to the vanisingly small activation energy of surface diffusion.", "We would like to show that it might also be the case that transient IM takes place in Pt/Ti which resembles in many respect the superdiffusive atomic transport processes known on solid surfaces [25].", "In ref.", "[9] the concentration profiles measured by Auger electron spectroscopy (AES) depth profiling analysis have been reported.", "The obtained results are in agreement with the findings presented in this article.", "However, in that paper it has not been realized that the fingerprint of superdiffusive feature of IM is detected by AES as a diffusity tail in the concentration profile of Pt at the Pt/Ti interface in the Pt/Ti bilayer.", "No such tail occurs in the concentration profile of Ti/Pt shown in ref.", "[9] where the profile can be characterized by \"normal\" $erf$ functions.", "Hence we find that the succession of the film and substrate could determine the magnitude of IM (the asymmetry of IM).", "No such ballistic behavior can be seen for Ti/Pt in Fig  REF .", "In Ti/Pt we find $\\langle R^2 \\rangle \\propto t$ time scaling.", "The mean free path of the energetic particles are much shorter in Ti/Pt.", "This can be seen qualitatively in Fig  REF if we compare the length of the atomic trajectories for Pt between upper and lower panels of Fig  REF .", "In the plot of Pt/Ti in Fig  REF we can see ballistic trajectories which result in the superdiffusive spread of Pt atoms.", "The trajectories of the reversed Ti recoils can be seen in upper Fig.", "REF and no intermixing Ti atomic positions can be found in the upper panel of Fig  REF (Ti/Pt).", "Although, Fig.", "REF has no any statistical meaning, however, the atomic trajectories are plotted from a typical cascade event hence some useful information can be obtained for the transport properties of energetic Pt atoms.", "In the lower panel of Fig  REF we can see the ballistic trajectories of intermixing hyperthermal Pt atoms (Pt/Ti).", "The reversed Ti particles at the interface and the weaker IM of Pt atoms to the Ti phase result in the weaker IM in Ti/Pt than in the Pt/Ti system.", "Hence Fig.", "REF depicts us at atomistic level what we see in the more statistical quantity $\\langle R^2 \\rangle $ .", "In Ti/Pt we see much shorter inter-layer atomic trajectories while in Pt/Ti ballistic trajectories of Pt atoms can be seen (moving through the interface)." ], [ "The ballistic model: deposited energy", "The ballistic model and the ballistic regime in the collisional cascade can also be ruled out as the source of the asymmetry of IM.", "We calculated the magnitude of the total deposited energy $F_D$ in Pt and in Ti, and get the values of $127.6$ and $148.5$ eV for Pt/Ti and for Ti/Pt, respectively at $0.5$ keV incident ion energy.", "Although the $F_D$ is larger in Ti/Pt, the intermixing is weaker in this material.", "Using TRIDYN calculations one can estimate the magnitude of the deposited energy at the interface [38], [51], [52].", "We calculate few eV/$\\hbox{Å}$ at the interface and again $F_D$ is smaller in the case of Pt/Ti.", "We conclude from this that not the larger $F_D$ and the smaller stopping power of Pt causes the anomalous IM in Pt/Ti.", "The larger the $E_d$ smaller the number of displaced atoms $n_d$ at the interface which leads to smaller $F_D$ following the Kinchin-Pease formula of $F_D=2 n_d E_d$ [39].", "Although we have no values for $n_d$ in various materials, however, the corresponding $E_d$ values for Pt (33 eV) and Ti (19 eV) can be found in text books [38].", "One can see that the $E_d$ of Pt is much larger than that of Ti, which again suggest that simply if we rely only on the ballistic model we expect larger IM in Ti/Pt.", "This is because such a big difference in $E_d$ values would strongly suggest that much larger IM should occur in Ti/Pt than in Pt/Ti.", "Also we can expect that the number of Frenkel pairs is much larger in Ti than in Pt under the same iradiation conditions.", "The Kinchin-Pease formula should give somewhat larger $F_D$ for Ti/Pt.", "However, we get comparable values as obtained by MD and SRIM simulations.", "Than we can conclude that the difference in the deposited energy at the interface can not be the reason of IM amplification in Pt/Ti.", "Also, at low-energy ion bombardment of 500 eV the deposited energy at the depth of the interface (few eV) is insufficient to create Frenkel pairs (few tens of eV).", "Hence this energy is dissipated into the lattice which leads to local thermalization.", "On the other hand displaced atoms close to the surface of the film if becomes not sputtered atoms, travel towards the interface as recoils.", "These recoils create second or higher order generation of energetic atoms.", "However, at this low incident ion energy regime the mean free path of these atoms do not exceed the distance of few times of the lattice constant.", "Hence their direct effect at the interface is negligible and these ballistic collisions of the recoils with the lattice atoms can not cause TED in Pt/Ti.", "The simulated nonlinear time evolution of $\\langle R^2 \\rangle $ indicates the acceleration of particles.", "During the cascade intermixing of particles with the mass ratio of $\\delta \\approx 1$ no such nonlinear scaling of $\\langle R^2 \\rangle $ can be found.", "This is because particles with nearly equal masses loose their kinetic energy during elastic collisions and the lifetime of the cascade and spike period remains short and which does not allow the evolution of $\\langle R^2 \\rangle \\propto t^2$ time scaling.", "Hence from the conventional picture of collisional cascades no intermixing acceleration can be expected.", "The projectile-to-target mass ratio could also play some role in the magnitude of $F_D$ at the interface [38].", "Due to the large mass difference in the case of $Ar^+$ $\\rightarrow $ Pt (Pt/Ti) elastic collisions the energy loss is larger than for $Ar^+$ $\\rightarrow $ Ti impacts (Ti/Pt).", "Contrary to this we get the stronger IM in Pt/Ti.", "It seems again that not simple binary collision effects the reason of the asymmetry of IM." ], [ "Thermal spike: heat of mixing", "In recent publications we have already shown that we find the lack of the effect of heat of mixing $\\Delta H$ of the corresponding alloy phases on ion-mixing in Ti/Pt [47].", "This is in constrast with the predictions of the thermal spike model which suggest that $\\Delta H$ governs intermixing during the ion bombardment of various bilayers [38], [39], [65].", "Repeating simulated ion-sputtering with varying $\\Delta H$ in Pt/Ti, we find again that the magnitude of ion-mixing is insensitive to the choice of the $\\Delta H$ which can be tuned by adjusting parameter $\\xi $ (the preexponential parameter in the attractive term in Eq.", "(1)) [47].", "Even if $\\Delta H \\approx 0$ ($\\xi =$ 0) strong IM occurs, although in the alloy phase decomposition takes place.", "Therefore the influence of thermodynamic driving forces can be ruled out and we conclude that the thermal spike model might not be consistent with the occurrence of TED in Pt/Ti." ], [ "Radiation enhanced diffusion: TED is athermal", "Since neither the ballistic nor the TS model are consistent with our findings in Pt/Ti we check whether other system parameters govern IM.", "First, we discuss, whether the thermally activated radiation enhanced diffusion, which ususally takes place after the TS period is responsible for the asymmetry of IM in Pt/Ti and in Ti/Pt.", "No temperature dependence has been found.", "The simulations provide nearly the same results in $\\sim 0$ K and at room temperature events.", "Large athermal experimental mixing rates ($k > 10^4$ $\\hbox{Å}^4$ ) have also been found for Ni/Al, Cu/Al and Al/Mo bilayers by other groups [6], [66], [67].", "These results are inconsistent with the operation of RED induced TED in these materials [6], [38].", "Because of the insensitivity of TED in Pt/Ti to temperature effects we can also rule out the influence of any conventional thermally activated vacancy and interstitial diffusion mechanisms [3]." ], [ "The proposed mechanism", "The radiation induced enhancement of IM in Ti/Pt has been studied in detail in ref.", "[47].", "The mass anisotropic interface stops the light energetic particles which leads to overheating in the Ti phase in Ti/Pt.", "This leads to the temporal decoupling of the IM of Pt and Ti with the preferential IM of Pt [47].", "The IM of Ti is delayed to the end of the TS period (retarded IM) [47].", "In Ti/Pt the IM of Pt atoms take place, however, upwards to the film against the downward ion and recoil fluxes which slows down IM.", "In Pt/Ti, however, the preferential IM of Pt goes downwards to the substrate accelerated by the unidirectional incoming ions and displaced energetic atoms.", "Hence the heavier particle Pt behaves like a ballistic first diffuser in Pt/Ti and as a slowed down particle by the counterflow of downward moving (from the film towards the substrate) energetic particles of ion irradiation in Ti/Pt.", "The ballistic preferential IM of Pt is governed by mass effect: the light energetic particles (Ti) are backscattered at the heavy interface leading to the retardation of IM of them [47].", "Figure: The schematic diagram of various ion-sputtering induced fluxes occur ina general mass-anisotropic film/substrate bilayer.Intermixing fluxes at the interface correspond toJ ⇑,S (δ)J_{\\Uparrow ,S}(\\delta ) (with upward arrow) and J ⇓,F (δ)J_{\\Downarrow ,F}(\\delta ) (downward arrow) whilethe downward arrow in the film to the recoil flux ofJ ⇓ rec J_{\\Downarrow }^{rec}.This configuration holds for m F ≫m S m_F \\gg m_S where m F m_F and m S m_S denote the atomic masses of the film andsubstrate constituents, respectively (FIG.", "5a corresponds to Pt/Ti).Note the unidirectional fluxes of J ⇓ rec J_{\\Downarrow }^{rec} and J ⇓,F (δ)J_{\\Downarrow ,F}(\\delta ).Moreover, fluxes J ⇑,S (δ)J_{\\Uparrow ,S}(\\delta ) and J ⇓,F (δ)J_{\\Downarrow ,F}(\\delta ) aredecoupled in time as it has been shown in ref.", ":the mass current of Pt atoms J ⇓,F (δ)J_{\\Downarrow ,F}(\\delta ) goes predominantlywhile J ⇑,S (δ)J_{\\Uparrow ,S}(\\delta ) is delayed by few ps.FIG.", "5b (Ti/Pt): In this case fluxes J ⇓ rec J_{\\Downarrow }^{rec} andJ ⇑,S (δ)J_{\\Uparrow ,S}(\\delta ) (IM Pt atoms) are in opposite direction hence weakens each others effectwhich leads to the suppression of IM.This situation corresponds to the case when m F ≪m S m_F \\ll m_S.Also, the reversed flux of the light particles increases the energy density in the Ti phase promoting the IM of the Pt atoms.", "Therefore we find the ion irradiation induced athermal preferential intermixing of Pt atoms in Pt/Ti.", "Two accelerating effects amplify each others effect: the mass effect induced preferential interfacial mixing of energetic Pt atoms and the downward fluxes of the incident ions and recoils contribute to ballistic IM in Pt/Ti and to the emergence of nonlinear time scaling of $\\langle R^2 \\rangle $ .", "The mass anisotropy induced enhancement of preferential Pt interdiffusion occurs in both bilayers, however, in Ti/Pt the fluxes of IM ballistic Pt atoms are somewhat slowed down by the counterflow of the downward movement of recoils.", "These simple reasonings explain the emergence of the asymmetry of IM with respect to the interchange of film and substrate constituents." ], [ "Phenomenology for $\\delta $ -driven TED", "The phenomenological description of the asymmetric TED might help in understanding and explaining the results obtained by experiment and MD simulations.", "Following the Martin's theory of irradiation induced ballistic diffusion [68] and the Cahn-Hilliard theory of thermal diffusion [69] the diffusion constant can be written formally as the sum of thermally activated and ballistic (athermal) terms [68]: $D_{irrad}=D_{th}(T)+D_{\\Downarrow }^{rec}.$ The interdiffusion drift due to cascade mixing with recoils (hyperthermal particles) is given via $D_{\\Downarrow }^{rec}$ .", "Within our picture of TED the mass effect induced amplification of IM over the thermal and cascade mixing rates can be written as $D_{TED}(T,\\delta )=D_{th}(T)+D_{\\Downarrow }^{rec}+D_{enhan}(\\delta ),$ where $D_{th}(T)$ is a normal thermally activated diffusion constant, where $T$ is the temperature in the thermal spike (irradiation induced molten phase) and $D_{enhan}(\\delta )$ is an enhancement term depending on $\\delta $ .", "This model explains TED as an amplification of atomic intermixing on top of radiation enhanced diffusion (thermally activated and collisional cascade ballistic interdiffusion $D_{RED}=D_{th}(T)+D_{\\Downarrow }^{rec}$ .", "Eq.", "REF should give $D=D_{th}+D_{\\Downarrow }^{rec}$ when $\\delta \\approx 1$ .", "It is not our intention in this paper to derive an explicit analytic expression which could reproduce the MD results (nonlinear time scaling of $\\langle R^2 \\rangle $ in Pt/Ti) as well as the experimental IM depth (long range diffusity tail) in the concentration profile.", "Simply we would like to explain in detail the mechanism of TED within a phenomenological picture which helps understanding the amplification of IM.", "Unidirectional and counterflow atomic flux: One possible way is to model mass effect by taking into account the counterflow ballistic intermixing mass flows $J_{\\Uparrow }$ and $J_{\\Downarrow }$ normal to the surface appear during ion-sputtering of bilayer systems.", "As mentioned above particle flow of heavy particles (Pt) takes place downwards in Pt/Ti and upwards in Ti/Pt.", "The total mass flow is the sum of these terms, $J(\\delta )=J_{\\Uparrow ,A}(\\delta )+J_{\\Downarrow ,B}(\\delta )+J_{\\Downarrow }^{rec},$ where $J_{\\Uparrow ,A}$ and $J_{\\Downarrow ,B}$ are the intermixing mass flow of constituents A and B through the interface (the interface currents in a A/B bilayer, respectively.", "The term $J_{\\Downarrow }^{rec}$ is the downward flux of energetic particles occurs upon external forced conditions (ion-sputtering) towards the interface.", "This term corresponds to the case when $\\delta \\approx 1$ .", "The $\\delta $ -induced downward flux amplification is $\\Delta J(\\delta )=J_{\\Downarrow ,B}(\\delta )-J_{\\Downarrow }^{rec}$ , which leads to superdiffusion.", "The schematic view of the fluxes is shown in Fig.", "REF for a situation, when the film component (F) is B and the substrate (S) is A.", "Heavy particles have the tendency to IM preferentially over the light components [47] which results in $J_{Pt} > J_{Ti}$ .", "In Pt/Ti, $J_{\\Downarrow ,B}(\\delta )=J_{Pt}$ , while in Ti/Pt $J_{\\Uparrow ,A}(\\delta )=J_{Pt}$ .", "It has also been shown recently that the IM of the light and heavy components is decoupled in time by few ps due to the predominant IM of the heavy atoms [47] which results in the robust amplification of the interface current of Pt $J_{Pt} \\gg J_{Ti}$ .", "Intermixing atomic fluxes $J_{\\Uparrow ,A}$ and $J_{\\Downarrow ,B}$ (see Fig.", "REF ) are created indirectly upon ion-sputtering, while flux $J_{\\Downarrow }^{rec}$ appears directly upon ion-bombardment.", "However, $J_{\\Uparrow ,A}(\\delta )$ and $J_{\\Downarrow ,B}(\\delta )$ are directly tuned by $\\delta $ , while flux $J_{\\Downarrow }^{rec}$ is nearly independent of $\\delta $ .", "This is because flux $J_{\\Downarrow }^{rec}$ appears in the film while the IM fluxes operate at the interface, where mass-anisotropy influences atomic transport directly.", "This rationalizes the separation of ion-induced atomic fluxes into $\\delta $ - dependent and independent terms.", "If $\\delta \\rightarrow 1$ , the sum of fluxes $J_{\\Uparrow ,A}+J_{\\Downarrow ,B} \\approx 0$ vanishes and $J=J_{\\Downarrow }^{rec}$ .", "The $\\delta $ -independent part of Eq.", "REF is simply the fluxes of cascade mixing term.", "Fluxes $J_{\\Downarrow }^{rec}$ and $J_{\\Downarrow ,B}(\\delta )$ appear nearly in the same time (ballistic or cascade period), although flux $J_{\\Downarrow ,B}(\\delta )$ is induced by $J_{\\Downarrow }^{rec}$ .", "This is because the mass-anisotropic system gives an ultrafast response to ion-irradiation and the $J_{\\Downarrow ,Pt}(\\delta )$ flux in Pt/Ti and $J_{\\Uparrow ,Pt}(\\delta )$ in Ti/Pt occur due to downwards recoils and flux $J_{\\Downarrow }^{rec}$ .", "$\\delta $ -driven particle acceleration: $J_{\\Downarrow }^{rec}$ and $J_{\\Downarrow ,B}(\\delta )$ are unidirectional in Pt/Ti, hence the ion-irradiation induced flux $J_{\\Downarrow }^{rec}$ accelerates Pt atoms in Pt/Ti and slows down $\\delta $ -driven IM in Ti/Pt because $J_{\\Downarrow }^{rec}$ and $J_{\\Uparrow ,A}$ are in contrary direction and nearly counteract or weakens each others effect.", "The impinging ions always generate energetic particles with downward momentum which is added to the momentum of $\\delta $ -driven heavy particles leading to the huge amplification of atomic mobility of few intermixing atoms.", "Hence we explain the huge interfacial broadening in Pt/Ti by the cumulative effect of the two types of downward atomic mobilities.", "The net intermixing mass flow can be described by the Fick's first law [3], $\\Delta J(\\delta ) = -\\frac{\\partial c}{\\partial z} (D_{\\Uparrow }(\\delta )-D_{\\Downarrow }(\\delta )),$ where the concentration gradient occurs due to the mass anisotropy [47].", "The thermal term does not lead to concentration difference and the broadening at the interface is symmetrical, the IM of the components is nearly the same on both side of the interface.", "The athermal interdiffusion drift is the balance of the upward and downward mass transport driven by $\\delta $ and characterized by the ballistic diffusion constants $D_{\\Uparrow }(\\delta )$ and $D_{\\Downarrow }(\\delta )$ .", "Finally the amplification term of TED for the diffusion constant can be given as $D_{enhan}(\\delta )=-(J_{\\Downarrow }(\\delta )-J_{\\Downarrow }^{rec}) \\biggm (\\frac{\\partial c}{\\partial z}\\biggm )^{-1}.$ Therefore, if $\\delta =1$ , purely thermally activated diffusion (atomic transport) takes place.", "If $\\delta \\gg 1$ , TED occurs (the amplification factor $D_{enhan}(\\delta ) \\gg D_{RED}$ ).", "When $\\delta < 1$ , TED features are suppressed, although we get a stronger IM than in the case of mass isotropy $\\delta \\approx 1$ ($D_{enhan}(\\delta ) \\approx D_{RED}$ )." ], [ "Superdiffusion", "Superdiffusive features, have never been reported before for intermixing, only for e.g.", "random walk of adatoms and clusters (Levy flight) on solid surfaces [25], [26], [27], [28].", "Transient mobility in the bulk has long been known only in collisional cascades [38], [39], [45] during quantum diffusion of light particles [3] or in shock loaded and stressed systems [3].", "However, these bulk phenomena are driven mostly by external stimulus (cascades, thermal spike, shock loaded rearrangements) and can be characterized as driven systems [14], [68].", "The only exception is the ultra low temperature ballistic diffusion of light particles in the lattice which is driven by (intrinsic) quantum effects [3].", "Brockmann et al.", "has been attempted to interpret reacting particle systems with front propagation driven by reaction-superdiffusion [13].", "The mass anisotropy driven TED in Pt/Ti can also be considered as a driven system.", "In $\\delta > 1$ systems the strong mass anisotropy driven acceleration of the particles leads to superdiffusion of the heavier atoms.", "However, the observed acceleration of the heavy particles is an intrinsic feature of $\\delta > 1$ systems.", "The situation is somewhat similar to that found in the anomalous impurity diffusion of N in stainless steel [4] and the observed large IM depths in various transition metal/Al diffusion couples [11] which could also be understood as super-interdiffusive processes.", "In Ni/Al bi-, multi-, and marker layers an unsually high mixing rate has been observed and which could not be understood within the picture of standard ion mixing models [6] as well as in our case in Pt/Ti.", "These results suggest that anomalous and superdiffusive mass transport could occur in various driven systems in which still unknown intrinsic system parameters govern TED.", "These parameters exist independently from the externally forced condition (ion bombardment).", "The external perturbation of these systems is necessary, however, to induce the transient atomic rearrangements.", "The superdiffusive features can be tuned by adjusting the mass ratio $\\delta $ in Pt/Ti.", "Setting in artificial mass isotropy, the nonlinear scaling of $\\langle R^2 \\rangle $ vanishes (see Fig.", "REF ).", "Hence mass anisotropy $\\delta $ operates as a system parameter ($\\delta = m_{film}/m_{substrate}$ , where the atomic mass of the film has been divided by the atomic mass of the substrate atoms), If $\\delta \\gg 1$ , super-interdiffusion appears, however, if $\\delta \\le 1$ , IM slows down because of the counterflow of incident particles with IM Pt atoms.", "In ref.", "[36] it has been shown, that the mixing efficiency $k/F_D$ , where $k$ is the mixing rate ($k=\\langle R^2 \\rangle /\\Phi $ ) and $F_D$ and $\\Phi $ are the deposited energy at the interface and the ion fluence, respectively [39], scales nonlinearily with $\\delta $ .", "At $\\delta < 0.33$ $k/F_D$ increases abruptly.", "In that article we studied the ion mixing of A/B bilayers with $\\delta \\le 1$ .", "However, we did not study the inverted systems (B/A, $\\delta > 1$ , film atoms are much heavier than the substrate one).", "In B/A systems in general, in which $\\delta \\gg 1$ , mass anisotropy driven superdiffusion might occur on the basis of the present results.", "This can be shown by computer experiments: if we simulate ion-mixing in mass isotropic systems, such as Co/Cu e.g., we get a very weak interfacial mixing.", "However, if we set in artificially $\\delta \\gg 1$ , strong IM takes place as in Pt/Ti.", "We reach the conclusion that the simple system parameter $\\delta $ governs the enhancement of IM in mass anisotropic bilayers.", "In the inverted case (A/B), when the lighter constituents are placed in the film, no transient enhanced intermixing occurs." ], [ "Conclusions", "The most important findings are the following: (i) Mass effect and asymmetric mixing: We find a robust mass effect on interfacial mixing in Pt/Ti which supports our finding published in ref.", "[36] in which a strong mass effect on ion-beam intermixing (IM) has been found for various mass-anisotropic bilayers.", "In order to increase the credibility of the employed computational approach, we fitted the crossinteraction atomic interaction potential to that of obtained from first principles calculations.", "(ii) Nonlinear time scaling: We find that the sum of the squares of atomic displacements through the anisotropic interface ($\\langle R^2 \\rangle $ ) scales nonlinearly in Pt/Ti ($\\langle R^2 \\rangle \\propto t^2$ ) as a function of the time (and the ion-number fluence) as shown in Fig.", "REF .", "The nonlinear time scaling of $\\langle R^2 \\rangle $ together with the long range (high diffusity) tail in the AES profile shown in ref.", "[9] might support the operation of a superdiffusive transport (athermal) process of Pt atoms in Pt/Ti.", "In Ti/Pt a nearly linear scaling ($\\langle R^2 \\rangle \\propto t$ ) is found.", "The lack of a tail in the AES concentration profile for Ti/Pt ([9]) is explained by the suppression of the preferentail IM of Pt into the Ti phase due to the counterflow fluxes of downward moving recoils and the upwards mobility Pt atoms.", "(iii) Preferential mixing of Pt and further acceleration: The atomistic mechanism of the asymmetry and TED in Pt/Ti is the following: The mass-anisotropy induced predominant intermixing of the heavier Pt atoms into the Ti phase has been found in both materials in accordance with previous findings [47].", "The new finding is that in Ti/Pt the Pt atoms slow down during IM because of the slowing down effect of the counterflow of the incident energetic particles (recoils) during the cascade period.", "In Pt/Ti we find the contrary situation: the IM Pt atoms are accelerated by the unidirectional current of the hyperthermal particles (downward Pt recoils), hence the originally already preferentially intermixing Pt atoms even further accelerated.", "Hence the superdiffusion of these particles is driven by double acceleration: first driven by the mass ratio induced preferential transport and additionally by the ballistic particles in the collisional cascade with downward mobility (their momentum directed from the film towards the substrate).", "Such kind of an acceleration of particles in the bulk has never been reported before at best of our knowledge, although, this situation can be rather general in $\\delta > 1$ systems.", "The phenomenon is governed by the intrinsic system parameter mass anisotropy and not or only weakly influenced by external parameters, such as the ion specie, projectile to target mass ratio, the ion energy above few hundred eV or the external temperature.", "This is because the process requires only a unidirectional current of energetic particles with different origin, which is present in the system nearly independently from the parameters mentioned above.", "In this sense this unique feature of the superdiffusive TED makes it different from other ballistic atomic transport processes occuring in e.g.", "collisional cascades which strongly influenced by external parameters.", "(iv) Unconventional mechanism: We conclude that the observed and simulated long range depth distribution of Pt atoms in the Ti phase of Pt/Ti cannot be understood by any established mechanisms of radiation-enhanced diffusion (RED).", "We find that the occurrence of the long range diffusity tail, whose penetration depth exceeds the ion range could be understood as a superdiffusive process in the bulk.", "Moreover, normally, RED could not lead to the asymmetry of IM.", "That is because intermixing in the collisional cascade is normally insensitive to the succession of the layers.", "Also, the thermal spike (which is rather short at this ion energy) in principle could not provide asymmetric IM.", "Instead we speculate on the possible operation of accelerative effects which could enhance atomic mixing and penetration.", "The divergence of $\\langle R^2 \\rangle $ clearly indicates that accelerative effects are present in the lattice leading to ballistic transport.", "$\\langle R^2 \\rangle \\propto t^2$ is the signature of ballistic atomic transport [25].", "The time evolution of nonlinear time scaling of $\\langle R^2 \\rangle $ requires the sufficiently large number of ballistic particles with large mean free path during most of the cascade events.", "During the \"normal\" cascade events of Ti/Pt $\\langle R^2 \\rangle $ does not exceed linear scaling.", "Hence a specific mechanism might come into play which speeds up Pt particles in the Ti bulk or at the interface during the collisional cascade period in Pt/Ti.", "(v) Unidirectional energetic atomic fluxes: The $\\delta $ -driven TED has also been explained using a phenomenological model based on a simple Fickian model.", "This helps in explaining the occurrence of the established accelerative effect on IM driven by unidirectional atomic fluxes.", "The fluxes of energetic particles can be separated into $\\delta $ -dependent and independent terms.", "The corresponding diffusion constant is constructed as a sum of thermally activated, cascade (recoil) mixing and $\\delta $ -dependent parts.", "The unidirectional fluxes in Pt/Ti leads to huge amplification of downwards fluxes while in Ti/Pt these fluxes are in counterflow direction leading to the weakening of the upwards intermixing fluxes of Pt atoms.", "(vi) The $\\delta $ -driven AAT might be a general mechanism: Finally we conclude that the established mechanism of TED might not be a specific one in nature.", "Nanoscale mass-anisotropy induced AAT could be a general feature of various multicomponent systems and could occur during the ion-beam processing (ion-sputtering, dopant implantation or ion-beam deposition) in various thin film multilayers, nanoinclusions, nanoislands, quantumdots embedded in light atomic host matrices (substrates) or in quantum well structures.", "Such kind of nanostructures and processing technologies are widely used in the production of heterostructure nanodevices [1] or magnetic nano-objects which are of high current interest due to numerous potential applications in various fields [2].", "In particular, TED has been found in nonstochiometric AlAs/GaAs quantum well structures [43] or in AsSb/GaSb superlattices [44] which can be considered as nanoscale mass-anisotropic systems with well-defined interfaces.", "The reported AAT in these systems could also be, at least partly, due to $\\delta $ -driven AAT mechanism.", "Also, possibly there are couple of other systems in which the $\\delta $ -driven AAT could take place.", "Just to mention few examples: sputter deposition of transition metals on Al [11], [31], low energy cluster deposition and pinning on various substrates [31].", "In these system it has already been shown by computer simulations that a $\\delta $ -driven AAT mechanism plays a significant role during thin film growth [31].", "The most important structural condition which must be fullfield is that the occurrence of $\\delta $ -driven TED requires the presence of a mass-anisotropic interface with $\\delta \\gg 1$ .", "Hence when the primary goal is the production of nanostructured surfaces sharp interfaces are required for the efficient operation of nanodevices.", "According to our results, however, nanoscale mass-anisotropy might deteriorate the sharpness of mass-anisotropic interfaces, especially when ion-sputtering have been used during the processing of the nanostructured surfaces and interfaces.", "One possible way of avoiding $\\delta $ -driven interface broadening is the construction of $\\delta \\le 1$ multilayer thin films which do not allow the amplification of intermixing and even lead to the suppression of interdiffusion as it has been found in the Ti/Pt bilayer.", "The better understanding of $\\delta $ -driven AAT could help in the more efficient production of nanothin films with sharp interfaces.", "This work is supported by the OTKA grant F037710 from the Hungarian Academy of Sciences.", "We wish to thank to K. Nordlund for helpful discussions and constant help.", "The work has been performed partly under the project HPC-EUROPA (RII3-CT-2003-506079) with the support of the European Community using the supercomputing facility at CINECA in Bologna.", "The help of the NKFP project of 3A/071/2004 is also acknowledged.", "The G03 code is available at NIIF center at Budapest." ] ]
1403.0237
[ [ "Asymptotic behavior of Type III mean curvature flow on noncompact\n hypersurfaces" ], [ "Abstract In this paper, we introduce a monotonicity formula for the mean curvature flow which is related to self-expanders.", "Then we use the monotonicity to study the asymptotic behavior of Type III mean curvature flow on noncompact hypersurfaces." ], [ "introduction", "Let $x_0:M^n\\rightarrow \\mathbb {R}^{n+1}$ be a complete immersed hypersurface.", "Consider the mean curvature flow $\\frac{\\partial x}{\\partial t}=\\vec{\\mathbf {H}},$ with the initial data $x_0$ , where $\\vec{\\mathbf {H}}=-H\\nu $ is the mean curvature vector and $\\nu $ is the outer unit normal vector.", "One of the main topics of interest in the study of mean curvature flow (REF ) is that of singularity formation.", "Since mean curvature flow always blows up at finite time on closed hypersurfaces, the singularity formation of the mean curvature flow (REF ) on closed hypersurfaces at the first singular time is described by Huisken [6] as follows.", "Let $x(\\cdot ,t)$ be the solution to the mean curvature flow (REF ).", "Let $A(\\cdot ,t)$ be the second fundamental form of $x(\\cdot ,t)$ .", "The solution to mean curvature flow (REF ) on closed hypersurfaces which blows up at finite time $T$ forms a (1) Type I singularity if $\\sup \\limits _{M\\times [0,T)}(T-t)|A|^2<\\infty $ , (2) Type II singularity if $\\sup \\limits _{M\\times [0,T)}(T-t)|A|^2=\\infty $ .", "For noncompact hypersurfaces, solution to the mean curvature flow may exist for all times.", "In [4], Ecker and Huisken showed that the mean curvature flow on locally Lipschitz continuous entire graph in $\\mathbb {R}^{n+1}$ exists for all time.", "In this paper, we study the asymptotic behavior of Type III mean curvature flow for which we have a long time existence by definition.", "In [6] Hamilton defined the Type III Ricci flow as the flow which has a long time existence and such that $\\sup _{t\\in (0,\\infty )} t \\Vert Rm(g_t)\\Vert < \\infty $ , where $\\Vert Rm(g_t)\\Vert $ is the norm of the Riemannian curvature of metric $g_t$ .", "Analogous to the Ricci flow we have the following definition for the mean curvature flow.", "Definition 1.1 We say the solution $x(\\cdot ,t)$ to the mean curvature flow (REF ) on noncompact hypersurfaces which exists for all time forms a Type III singularity if $\\sup \\limits _{M\\times [0,+\\infty )}t|A|^2<\\infty $ .", "Typical examples of Type III mean curvature flow are evolving entire graphs satisfying the linear growth condition, or equivalently the entire graphs having the bounded gradient, which in particular implies $V:=\\langle \\nu ,w\\rangle ^{-1}\\le c,$ where $\\nu $ is the unit normal vector of the graph and $w$ is a fixed unit vector such that $\\langle \\nu ,w\\rangle >0$ .", "Ecker and Huisken showed that the mean curvature flow on entire graphs satisfying the linear growth condition is Type III (see Corollary 4.4 in [3]).", "Bobe [1] has also related results for the cylindrical graphs.", "Huisken [6] introduced his entropy which becomes one of the most powerful tools in studying the mean curvature flow.", "Recall the Huisken's entropy is defined as the integral of backward heat kernel: $\\int _{M}(4\\pi (T-t)) ^{-\\frac{n}{2}}e^{-\\frac{|x|^2}{4(T-t)}}d\\mu _t.$ Huisken proved his entropy (REF ) is monotone non-increasing in $t$ under the mean curvature flow (REF ).", "By using this monotonicity formula, Huisken also showed that Type I singularities of mean curvature flow are smooth asymptotically like shrinking self-shinkers, characterized by the equation $\\vec{\\mathbf {H}}=-x^{\\perp },$ where $x^{\\perp }=\\langle x,\\nu \\rangle \\nu $ .", "By using the Hamilton's Harnack estimate for the mean curvature flow [9], Huisken and Sinestrari ([7], [8]) proved suitable rescaled sequence of the $n$ -dimensional compact Type II mean curvature flow with positive mean curvature converges to a translating soliton $\\mathbb {R}^{n-k}\\times \\Sigma ^k$ , where $\\Sigma ^k$ is strictly convex.", "In this paper, we study the singularity formation of the Type III mean curvature flow.", "For the entire graphs satisfying the linear growth condition (REF ) and in addition the estimate $\\langle x_0,\\nu \\rangle ^2 \\le c(1+|x_0|^2)^{1-\\delta }$ at time $t = 0$ , where $c<\\infty $ and $\\delta >0$ , Ecker and Huisken [3] proved the solution to the normalized mean curvature flow $\\frac{\\partial \\overline{x}}{\\partial s}=\\vec{\\overline{\\mathbf {H}}}-\\overline{x}$ with initial data $x_0$ converges as $s\\rightarrow \\infty $ to a self-expander.", "More precisely, in the case of entire graphs satisfying conditions (REF ) and (REF ), they showed the following strong estimate $\\sup \\limits _{\\overline{x}(M,s)}\\frac{|\\vec{\\overline{\\mathbf {H}}}+\\overline{x}_s^{\\perp }|^2 \\overline{V}^2}{(1+\\alpha |\\overline{x}_s|^2)^{1-\\epsilon }}\\le e^{-\\beta s} \\sup \\limits _{x_0(M)}\\frac{|\\vec{\\overline{\\mathbf {H}}}+ x_{0}^{\\perp }|^2 \\overline{V}^2}{(1+\\alpha | x_{0}|^2)^{1-\\epsilon }},$ for all $\\epsilon <\\delta $ by applying the maximum principle under the flow (REF ), where $\\overline{V}=\\langle \\overline{\\nu },w\\rangle ^{-1}$ , $\\overline{\\nu }$ is the unit normal vector of the graph and $w$ is a fixed unit vector such that $\\langle \\overline{\\nu },w\\rangle >0$ .", "In particular, this implies exponentially fast convergence on compact subsets.", "In order to study the singularity formation of Type III mean curvature flow, we introduce some monotonicity formulas which are related to self-expanders.", "We remark that there is a dual version of Huisken's entropy due to Ilmannen [13]: $\\frac{d}{dt}\\int _{M}\\rho d\\mu _t=-\\int _{M}|\\vec{\\mathbf {H}}-\\frac{x^{\\perp }}{2t+1}|^2\\rho d\\mu _t$ where $\\rho =(t+\\frac{1}{2})^{-\\frac{n}{2}}e^{\\frac{|x|^2}{4(t+\\frac{1}{2})}}$ and where the surfaces evolve by the mean curvature flow (REF ).", "Unfortunately, the monotonicity formula (REF ) only makes sense on closed hypersurfaces.", "Note that the density term $\\rho d\\mu _t$ is not pointwise monotone under the mean curvature flow (REF ).", "Actually, we can calculate that $\\frac{\\partial }{\\partial t} \\rho d\\mu _t=-|\\vec{\\mathbf {H}}-\\frac{x^{\\perp }}{2t+1}|^2\\rho d\\mu _t -\\left(\\frac{n}{2t+1}+\\frac{<x,\\vec{\\mathbf {H}}>}{2t+1}+\\frac{|x^T|^2}{(2t+1)^2}\\right)\\rho d\\mu _t.$ If we could integrate above formula (which is for example the case when we are on compact surfaces), we would have found the second term on the right hand side is zero by the divergence theorem.", "In this paper, we find that $\\rho d\\mu _t$ is monotone non-increasing under the following flow, which we call the drifting mean curvature flow, $\\frac{\\partial x}{\\partial t}=\\vec{\\mathbf {H}}+\\frac{x^T}{2t+1}.$ It turns out the drifting mean curvature flow (REF ) is equivalent to mean curvature flow (REF ) up to tangent diffeomorphisms.", "We have the following result.", "Theorem 1.2 Let $x(\\cdot ,t)$ be the solution to the drifting mean curvature flow (REF ) with the initial data $x_0:M\\rightarrow \\mathbb {R}^{n+1}$ being an immersed hypersurface.", "Set $\\rho =(t+\\frac{1}{2})^{-\\frac{n}{2}}e^{\\frac{|x|^2}{4(t+\\frac{1}{2})}}$ .", "We have $\\frac{\\partial }{\\partial t} \\rho d\\mu _t=-|\\vec{\\mathbf {H}}-\\frac{x^{\\perp }}{2t+1}|^2 \\rho d\\mu _t.$ Rescale the flow (REF ) as $\\widetilde{x}(\\cdot ,s)=\\frac{1}{\\sqrt{2t+1}}x(\\cdot ,t),$ where $s$ is given by $s=\\frac{1}{2}\\log (2t+1)$ .", "The normalized drifting mean curvature flow then becomes $\\frac{\\partial \\widetilde{x}}{\\partial s}=\\vec{\\widetilde{\\mathbf {H}}}-\\widetilde{x}^{\\perp },\\ \\ s\\ge 0.$ Moreover, $(t+\\frac{1}{2})^{-\\frac{n}{2}}e^{\\frac{|x|^2}{4(t+\\frac{1}{2})}}d\\mu _t$ becomes $e^{\\frac{1}{2}|\\widetilde{x}|^2}d\\widetilde{\\mu }_s$ under this rescaling.", "Note the stationary solutions to the normalized drifting mean curvature flow (REF ) are exactly self-expanders which are characterized by the equation $\\vec{\\mathbf {H}}=x^{\\perp }.$ That is why we consider the normalized drifting mean curvature flow (REF ).", "An immediate corollary of Theorem REF is the following monotonicity property for the normalized drifting mean curvature flow.", "Corollary 1.3 Let $\\widetilde{x}(\\cdot ,s)$ be the solution to the normalized drifting mean curvature flow (REF ) with the initial data $x_0:M\\rightarrow \\mathbb {R}^{n+1}$ being an immersed hypersurface.", "Set $\\widetilde{\\rho }=e^{\\frac{1}{2}|\\widetilde{x}|^2}$ .", "We have $\\frac{\\partial }{\\partial s} (\\widetilde{\\rho } d\\widetilde{\\mu }_s)=-|\\vec{\\widetilde{\\mathbf {H}}}-\\widetilde{x}^{\\perp }|^2 \\widetilde{\\rho } d\\widetilde{\\mu }_s.$ Next we introduce a global monotonicity formula for the normalized drifting mean curvature flow (REF ).", "The idea is the following: since $\\frac{\\partial }{\\partial s} (\\widetilde{\\rho } d\\widetilde{\\mu }_s)\\le 0$ pointwise, we can choose a time-independent positive function $f_0$ such that $\\int _{M}\\widetilde{\\rho } f_0 d\\widetilde{\\mu }_s$ is finite at $s=0$ , and therefore $\\int _{M}\\widetilde{\\rho } f_0 d\\widetilde{\\mu }_s$ is finite for all $s\\ge 0$ since $\\widetilde{\\rho }f_0 d\\widetilde{\\mu }_s$ is monotone nonincreasing.", "Theorem 1.4 Let $\\widetilde{x}(\\cdot ,s)$ be the solution to the normalized drifting mean curvature flow (REF ) with the initial data $x_{0}:M\\rightarrow \\mathbb {R}^{n+1}$ being an immersed hypersurface.", "Assume that $\\int _M e^{-\\frac{1}{2}|x_0|^2}d \\mu _0=C_0<\\infty $ .", "Then $\\int _M e^{\\frac{1}{2}|\\widetilde{x}|^2-|x_{0}|^2}d\\widetilde{\\mu }_s\\le C_0, \\qquad \\mbox{for all} \\qquad s\\ge 0,$ where the term $ e^{\\frac{1}{2}|\\widetilde{x}|^2-|x_{0}|^2}d\\widetilde{\\mu }_s$ means $ e^{\\frac{1}{2}|\\widetilde{x}|^2(p,s)-|x_{0}(p)|^2}d\\widetilde{\\mu }_s(p)$ for $p\\in M$ and $\\int ^{\\infty }_{0}\\int _M |\\vec{\\widetilde{\\mathbf {H}}}-\\widetilde{x}^{\\perp }|^2 e^{\\frac{1}{2}|\\widetilde{x}|^2-|x_{0}|^2} d\\widetilde{\\mu }_s\\le C_0.$ Moreover, we have the following monotonicity formula $\\frac{d}{d s}\\int _Me^{\\frac{1}{2}|\\widetilde{x}|^2-|x_{0}|^2}d\\widetilde{\\mu }_s=-\\int _M|\\vec{\\widetilde{\\mathbf {H}}}-\\widetilde{x}^{\\perp }|^2 e^{\\frac{1}{2}|\\widetilde{x}|^2-|x_{0}|^2} d\\widetilde{\\mu }_s.$ The theorem also holds when we replace the term $e^{-|x_{0}|^2}$ in (REF )-(REF ) by a time-independent positive function $f_0$ satisfying $\\int _M e^{\\frac{1}{2}|x_0|^2}f_0d \\mu _0<\\infty .$ We remark that the motivation for such monotonicity formulas originates from Perelman's work on the Ricci flow [15].", "Perelman [15] introduced the reduced volume $\\int _{M}\\tau ^{-\\frac{n}{2}}e^{-l(y,\\tau )}dvol(y)$ which is monotone non-increasing under the backward Ricci flow, where $l(y,\\tau )$ is the reduced length.", "However the density term $\\tau ^{-\\frac{n}{2}}e^{-l(y,\\tau )}dvol(y)$ is not pointwise monotone non-increasing quantity under the backward Ricci flow.", "Perelman showed that the density term is monotone non-increasing under the backward Ricci flow along the $\\mathcal {L}$ -geodesic.", "Feldman, Ilmanen, Ni [5] also observed that there is a dual version of Perelman's reduced entropy related to the Ricci flow expanders and defined by $\\int _{M} t^{-\\frac{n}{2}}e^{l_+(y,t)}dvol(y)$ , along the forward Ricci flow.", "It only makes sense on closed manifolds.", "The first author and Zhu [2] observed the density term $t^{-\\frac{n}{2}} e^{l_+(\\gamma _V(t),t)} \\mathcal {L_+}J_V(t)dV^n$ is pointwise monotone non-increasing along the $\\mathcal {L}_+$ -geodesics under the forward Ricci flow in the similar way as in [15].", "Hence one can add the weight and get that $\\int _{T_xM^n} t^{-\\frac{n}{2}}e^{l_+(\\gamma _V(t),t)}\\mathcal {L_+}J_V(t)e^{-2|V|^2_{g(0)}}dx_{g(0)}(V),$ is well-defined on noncompact manifolds and monotone non-increasing under the forward Ricci flow (see [2]).", "As an immediate application of Theorem REF , using (REF ), we have the following result.", "Corollary 1.5 Let $\\widetilde{x}(\\cdot ,s)$ be the normalized drifting mean curvature flow that exists for $s\\in [0,\\infty )$ , with initial data $ x_{0}:M\\rightarrow \\mathbb {R}^{n+1}$ being an immersed hypersurface and satisfying $\\int _M e^{-\\frac{1}{2}| x_{0}|^2}d \\mu _0=C_0<\\infty $ .", "Then the normalized drifting mean curvature flow (REF ) asymptotically looks like the self-expander as time approaches infinity in the sense $\\lim _{\\tau ,t\\rightarrow \\infty }\\int _{\\tau }^t \\int _M |\\vec{\\widetilde{\\mathbf {H}}}-\\widetilde{x}^{\\perp }|^2 e^{\\frac{1}{2}|\\widetilde{x}|^2-| x_{0}|^2} d\\widetilde{\\mu }_s\\, ds = 0.$ There exists a sequence of times $s_i\\rightarrow \\infty $ such that $\\lim _{i\\rightarrow \\infty } \\int _M |\\vec{\\widetilde{\\mathbf {H}}}-\\widetilde{x}^{\\perp }|^2 e^{\\frac{1}{2}|\\widetilde{x}|^2-| x_{0}|^2} d\\widetilde{\\mu }_{s_i} = 0,$ where $\\widetilde{x}$ stands for $\\widetilde{x}(\\cdot ,s_i)$ .", "Remark 1.6 If the mean curvature flow (REF ) exists for all times, then the corresponding normalized drifting mean curvature flow (REF ) also exists for all times.", "Since the normalized drifting mean curvature flow (REF ) is equivalent to the mean curvature flow (REF ) up to tangent diffeomorphisms and rescaling given by REF , we can view CorollaryREF giving us the asymptotic behavior at infinite time for the mean curvature flow in the distribution sense.", "Next we use the monotonicity formula (REF ) to study the asymptotic behavior of Type III mean curvature flow.", "We first have the following.", "Theorem 1.7 Let $x( \\cdot ,t):M\\rightarrow \\mathbb {R}^{n+1}$ be the Type III solution to the mean curvature flow (REF ) with initial data $x_{0}:M\\rightarrow \\mathbb {R}^{n+1}$ being a complete immersed hypersurface and satisfying $\\int _M e^{-\\frac{1}{2}|x_{0}|^2}d\\mu _{0}<\\infty $ .", "Assume that $\\widetilde{x}(\\cdot ,s)$ is the corresponding normalized drifting mean curvature flow (REF ) for $x( \\cdot ,t)$ .", "Denote by $ B(q,R)$ the ball in $\\mathbb {R}^{n+1}$ for some $q\\in \\mathbb {R}^{n+1}$ and $R>0$ .", "If there exists an $s_N>0$ such that $\\widetilde{x}(M,s)\\cap B(q,R)\\ne \\emptyset $ for $s>s_N$ and $|x_0(p)|\\le C_R$ for any $p$ such that $ \\widetilde{x}(p,s)\\in B(q,R)$ , where $C_R$ is a constant dependent on $R$ and independent of time $s$ , then $\\widetilde{x}(M,s)\\cap B(q,R)$ subconverges smoothly to the self-expander in $B(q,R)$ .", "Remark 1.8 Let $N_s(q,R)=\\widetilde{x}^{-1}(\\widetilde{x}(M,s)\\cap B(q,R))$ .", "Note that condition (REF ) is needed when using the monotonicity formula (REF ) since the weighted term $e^{-| x_{0}|^2}$ may go to zero in $N_s(q,R)$ (see the proof of Theorem REF ).", "In view of (REF ) and the proof of Theorem REF , we can see that Theorem REF still holds if the condition (REF ) is generalized by $f_0(p)\\ge c_0(R)>0$ for $p \\in N_s(q,R)$ , where $f_0$ satisfies (REF ) in Theorem REF and $c_0(R)$ is constant dependent on $R$ and independent of time.", "The condition (REF ) or (REF ) can not be removed due to an example by Huisken and Ecker ([3]) (see Remark REF ).", "We will show there exists an $R_0$ such that $\\widetilde{x}(\\cdot ,s) \\cap B(o,R_0) \\ne \\emptyset $ for $s$ sufficiently large.", "As an immediate corollary of this fact and Theorem REF , we have the following.", "Corollary 1.9 Let $x( \\cdot ,t)$ and $\\widetilde{x}(\\cdot ,s)$ be as in Theorem REF .", "Then if for any $R > 0$ we have $|x_0(p)|\\le C_R,$ for any $p$ such that $ \\widetilde{x}(p,s)\\in B(o,R)$ , where $C_R$ is a constant dependent of $R$ and independent of time $s$ , then $\\widetilde{x}(M,s)$ subconverges smoothly to the limiting self-expander.", "Next we give an application of Theorem REF in which given conditions depend only on the initial time.", "It turns out that they imply condition (REF ).", "Theorem 1.10 Let $x( \\cdot ,t):M\\rightarrow \\mathbb {R}^{n+1}$ be Type III solution to the mean curvature flow (REF ) with initial data $x_{0}:M\\rightarrow \\mathbb {R}^{n+1}$ being a complete immersed hypersurface satisfying $\\int _M e^{-\\frac{1}{2}|x_{0}|^2}d\\mu _{0}<\\infty $ and $\\mu H\\ge -\\langle x_0-q_0 ,\\nu \\rangle \\ge 0$ for some positive constant $\\mu $ and a fixed vector $q_0$ at the initial time.", "Then its corresponding normalized drifting mean curvature flow (REF ) subconverges smoothly to the limiting self-expander.", "Remark 1.11 The two-sheeted hyperboloid of revolution is an example satisfying Theorem REF .", "We write the two-sheeted hyperboloid of revolution as $x_0(u,v)=(a\\sqrt{u^2-1}\\cos {v},a\\sqrt{u^2-1}\\sin {v},cu)$ , $a>0$ , $c>0$ and $|u|\\ge 1$ .", "Then the unit normal vector field $\\nu =\\partial _vx_0\\times \\partial _ux_0=\\frac{(c\\sqrt{u^2-1}\\cos {v},c\\sqrt{u^2-1}\\sin {v},-au)}{\\sqrt{(a^2+c^2)u^2-c^2}},$ the mean curvature $H=\\frac{c[c^2(u^2-1)+a^2(u^2+1)]}{a[(a^2+c^2)u^2-c^2]^{\\frac{3}{2}}},$ and $<x_0,\\nu >=-\\frac{ac}{\\sqrt{(a^2+c^2)u^2-c^2}}.$ It easy see that the condition (REF ) is satisfied for $p=0$ and $\\mu $ large enough.", "Since each component of the two-sheeted hyperboloid of revolution $x_0(u,v)$ is the entire graph satisfying the linear growth condition (REF ), the mean curvature flow on $x_0(u,v)$ must be Type III by the result of Ecker and Huisken.", "Then by Theorem REF , the normalized drifting mean curvature flow (REF ) of the two-sheeted hyperboloid of revolution subconverges smoothly to the limiting self-expander.", "Note that $|\\langle x_0,\\nu \\rangle |\\le C$ , implying condition (REF ), and hence the convergence of $M_s$ towards the limiting self-expander was also confirmed by Ecker and Huisken in [3].", "If we assume that the Type III mean curvature flow only has positive mean curvature, one may not have the convergence towards a self-expander (see Example REF ).", "We also give another proof of Theorem REF .", "This proof is based on following observation: Let $x( \\cdot ,t):M\\rightarrow \\mathbb {R}^{n+1}$ be solution to the mean curvature flow (REF ) satisfying $\\mu H\\ge -\\langle x_0-q_0 ,\\nu \\rangle \\ge 0$ for some positive constant $\\mu $ and a fixed vector $q_0$ at the initial time.", "Rescale the flow $x_{\\mu }(\\cdot ,t)=\\mu ^{-\\frac{1}{2}}(x(\\cdot , \\mu t)-q_0 ).$ Let $\\widetilde{x}_{\\mu }(,s)$ be the corresponding normalized drifting mean curvature flow of $x_{\\mu }(\\cdot ,t)$ .", "Then $\\int _{M}e^{-\\frac{1}{2}|\\widetilde{x}_{\\mu }|^2}d\\widetilde{\\mu }_s$ is monotone nonincreasing.", "On the other hand, we also observe that if the solution to the mean curvature flow (REF ) has the initial data $x_{0}$ satisfying $-\\langle x_0-q_0 ,\\nu \\rangle \\ge \\mu H\\ge 0$ , then $\\int _{M}e^{-\\frac{1}{2}|\\widetilde{x}_{\\mu }|^2}d\\widetilde{\\mu }_s$ is monotone nondecreasing.", "Theorem 1.12 Let $x( \\cdot ,t):M\\rightarrow \\mathbb {R}^{n+1}$ be the Type III solution to the mean curvature flow (REF ) with initial data $x_{0}:M\\rightarrow \\mathbb {R}^{n+1}$ being a complete immersed hypersurface satisfying $\\int _M e^{-\\frac{1}{2}|x_{0}|^2}d\\mu _{0}<\\infty $ and $-\\langle x_0-q_0 ,\\nu \\rangle \\ge \\mu H\\ge 0$ for some positive constant $\\mu $ and a fixed vector $q_0$ at the initial time.", "Let $\\widetilde{x}(,s)$ be the corresponding normalized drifting mean curvature flow of $x_{\\mu }(\\cdot ,t)$ defined in (REF ).", "Then $\\int _{M}e^{-\\frac{1}{2}|\\widetilde{x}_{\\mu }|^2}d\\widetilde{\\mu }_s$ is monotone nondecreasing in $s$ .", "If $\\lim \\limits _{s\\rightarrow +\\infty }\\int _{M}e^{-\\frac{1}{2}|\\widetilde{x}_{\\mu }|^2}d\\widetilde{\\mu }_s<\\infty ,$ then its corresponding normalized drifting mean curvature flow (REF ) subconverges smoothly to the limiting self-expander.", "In particular, if $vol({\\widetilde{M}_s\\cap B(o,R)})\\le CR^m,$ for some $m>0$ , then $\\lim \\limits _{s\\rightarrow +\\infty }\\int _{M}e^{-\\frac{1}{2}|\\widetilde{x}_{\\mu }|^2}d\\widetilde{\\mu }_s\\le C$ and the result above holds.", "Recall that Ecker and Huisken ([3]) proved that the normalized mean curvature flow of entire graphs satisfying the linear growth condition (REF ) and the estimate $\\langle x_0,\\nu \\rangle ^2 \\le c(1+|x_0|^2)^{1-\\delta }$ at time $t = 0$ , where $c<\\infty $ and $\\delta >0$ , converges to the self-expander.", "There exists an example showing that the normalized mean curvature flow of entire graphs satisfying only the linear growth condition (REF ) and failing to satisfy (REF ) may not subconverge to a self-expander even if it has the positive mean curvature (see Example REF ).", "As the application to Theorem REF , we have Corollary 1.13 Let $x_0$ be the entire graph which has the nonnegative mean curvature and satisfies the linear growth condition (REF ).", "Moreover, assume there exists a fixed vector $q_0$ such that $\\langle x_0-q_0 ,\\nu \\rangle \\le C$ , where $C$ is a positive constant.", "Then normalized drifting mean curvature flow (REF ) with initial data $x_0$ subconverges to the limiting self-expander.", "The structure of this paper is as follows.", "In section 2 we give proofs of Theorem REF , Corollary REF , Theorem REF and Corollary REF .", "In section 3 we give the proofs of Theorem REF and Theorem REF .", "In section we give the proof of Theorem REF , Theorem REF and Corollary REF ." ], [ "Monotonicity formulas", "Recall that the drifting mean curvature flow (REF ) is equivalent to (REF ) up to tangent diffeomorphisms defined by $\\frac{x^T}{2t+1}$ .", "Indeed, let $x$ solve $\\frac{\\partial }{\\partial t} x = -H\\nu $ and let $\\phi _t = \\phi (\\cdot ,t)$ be a family of diffeomorphisms on $M$ satisfying $2D_q \\left(\\frac{x}{t+\\frac{1}{2}}(\\phi (p,t),t\\right)\\left(\\frac{\\partial \\phi }{\\partial t}(p,t)\\right) = \\left(\\frac{\\partial }{\\partial t}\\left(\\frac{x}{t+\\frac{1}{2}}\\right)(\\phi (p,t),t)\\right)^T,$ implying $D_q x(\\phi (p,t),t) \\left(\\frac{\\partial \\phi }{\\partial t}(p,t)\\right) = \\frac{x(\\phi (p,t),t)^T}{2t+1}.$ Define $y(p,t) = x(\\phi (p,t),t)$ .", "Then $y(p,t)$ solves the drifting mean curvature flow equation, $\\frac{\\partial }{\\partial t}y = \\frac{\\partial }{\\partial t} x + D_q x(\\phi (p,t),t) \\left(\\frac{\\partial }{\\partial t}\\phi (p,t)\\right) = -H\\nu + \\frac{y^T}{2t+1}$ Similarly, one can easily see that reparametrizing drifting mean curvature flow (REF ) by diffeomorphisms leads to the normalized mean curvature flow (REF ).", "Under the drifting mean curvature flow (REF ), we have $\\frac{\\partial }{\\partial t}g_{ij} &=2\\partial _i (\\vec{\\mathbf {H}}+\\frac{x^T}{2t+1}) \\partial _jx \\nonumber \\\\&=-2Hh_{ij}+\\frac{1}{t+\\frac{1}{2}}\\partial _i(x-x^{\\perp })\\partial _j x\\nonumber \\\\&=-2Hh_{ij}+\\frac{1}{t+\\frac{1}{2}}g_{ij}+\\frac{1}{t+\\frac{1}{2}}x^{\\perp }\\partial _i\\partial _j x \\nonumber \\\\&=-2Hh_{ij}+\\frac{1}{t+\\frac{1}{2}}g_{ij}-\\frac{1}{t+\\frac{1}{2}}\\langle x,\\nu \\rangle h_{ij},$ where we use $x^{\\perp }=<x,\\nu >\\nu $ and $h_{ij}=-\\nu \\cdot \\partial _i\\partial _j x$ .", "It follows that $\\frac{\\partial }{\\partial t}d\\mu _t &=(-|\\vec{\\mathbf {H}}|^2+\\frac{n}{2t+1}+\\frac{1}{2t+1}\\langle x^{\\perp },\\vec{\\mathbf {H}}\\rangle )d\\mu _t.$ Recall that $\\rho = (t + \\frac{1}{2})^{-n/2}\\, e^{\\frac{|x|^2}{4(t+\\frac{1}{2})}}$ .", "By (REF ) and (REF ), we get that $\\frac{\\partial }{\\partial t} \\rho d\\mu _t&=(-\\frac{n}{2t+1}-\\frac{|x|^2}{(2t+1)^2}+\\frac{\\langle x,\\frac{\\partial }{\\partial t}x\\rangle }{2t+1})\\rho d\\mu _t+ \\rho \\frac{\\partial }{\\partial t} d\\mu _t\\nonumber \\\\&=-|\\vec{\\mathbf {H}}-\\frac{x^{\\perp }}{2t+1}|^2 \\rho d\\mu _t\\nonumber .$ Using the scaling $\\widetilde{x}(\\cdot ,s) = \\frac{x(\\cdot ,t)}{\\sqrt{2t+1}}$ along with $s = \\frac{1}{2}\\log (2t+1)$ , and Theorem REF , we get $\\frac{\\partial }{\\partial s} \\widetilde{\\rho } d\\widetilde{\\mu }_s &= \\frac{\\partial }{\\partial t} \\left( e^{\\frac{|x|^2}{4(t+\\frac{1}{2})}} \\frac{d\\mu _t}{(2t+1)^{\\frac{n}{2}}}\\right) \\frac{d t}{ds} \\\\&= -(2t+1)\\,|\\vec{\\mathbf {H}} - \\frac{x^{\\perp }}{2t+1}|^2\\, \\rho \\, (d\\mu _t 2^{-\\frac{n}{2}}) \\\\&= -|\\vec{\\widetilde{\\mathbf {H}}} - \\widetilde{x}^{\\perp }|^2 \\widetilde{\\rho } d\\widetilde{\\mu }_s$ Next we give the proof of Theorem REF whose immediate consequence is Corollary REF .", "Since the weighted term $e^{-| x_{0}|^2}$ is independent of time, we have $\\frac{\\partial }{\\partial s} e^{\\frac{1}{2}|\\widetilde{x}|^2-| x_{0}|^2} d\\widetilde{\\mu }_s=-|\\vec{\\widetilde{\\mathbf {H}}}-\\widetilde{x}^{\\perp }|^2 e^{\\frac{1}{2}|\\widetilde{x}|^2-| x_{0}|^2} d\\widetilde{\\mu }_s.$ Integrate above over compact domain $\\Omega $ in $M$ , we get $\\frac{d}{d s}\\int _{\\Omega } e^{\\frac{1}{2}|\\widetilde{x}|^2-| x_{0}|^2} d\\widetilde{\\mu }_s=-\\int _{\\Omega }|\\vec{\\widetilde{\\mathbf {H}}}-\\widetilde{x}^{\\perp }|^2 e^{\\frac{1}{2}|\\widetilde{x}|^2-| x_{0}|^2} d\\widetilde{\\mu }_s\\le 0.$ Then $\\int _{\\Omega } e^{\\frac{1}{2}(|\\widetilde{x}|^2-2| x_{0}|^2)}d\\widetilde{\\mu }_s\\le \\int _{\\Omega } e^{-\\frac{1}{2}| x_{0}|^2}d \\mu _0.$ Taking $\\Omega \\rightarrow M$ , we conclude that (REF ) holds.", "Integrate (REF ) over time interval $[0,s]$ , we get $\\int _{\\Omega } e^{-\\frac{1}{2}| x_{0}|^2}d\\mu _0-\\int _{\\Omega } e^{\\frac{1}{2}|\\widetilde{x}|^2-| x_{0}|^2} d\\widetilde{\\mu }_s=\\int ^s_{0}\\int _{\\Omega }|\\vec{\\widetilde{\\mathbf {H}}}-\\widetilde{x}^{\\perp }|^2 e^{\\frac{1}{2}|\\widetilde{x}|^2-| x_{0}|^2} d\\widetilde{\\mu }_s.$ Then we have $\\int ^s_{0}\\int _{\\Omega }|\\vec{\\widetilde{\\mathbf {H}}}-\\widetilde{x}^{\\perp }|^2 e^{\\frac{1}{2}|\\widetilde{x}|^2-| x_{0}|^2} d\\widetilde{\\mu }_s\\le \\int _{\\Omega } e^{-\\frac{1}{2}| x_{0}|^2}d\\mu _0\\le C_0.$ Taking $\\Omega \\rightarrow M$ in (REF ) and (REF ), we conclude that (REF ) and (REF ) hold.", "Then Corollary REF follows from (REF ) and (REF ) directly." ], [ "Convergence to an expander", "In this section we present the proofs of Theorem REF and Corollary REF .", "They give us sufficient conditions under which we have that the rescaled Type III mean curvature flow converges to an expander.", "Let $N_s(q,R)=\\widetilde{x}^{-1}(\\widetilde{x}(M,s)\\cap B(q,R))$ .", "Using $|\\widetilde{x}_0|^2 \\le C_R^2$ on $N_s(q,R)$ and Corollary REF we have $\\mathcal {H}^n(\\widetilde{x}(M,s)\\cap B(q,R))&=\\int _{M}\\chi (N_s(q,R)) d\\widetilde{\\mu }_s\\\\&\\le \\int _{M}\\chi (N_s(q,R))e^{C_R^2+\\frac{1}{2}(|\\widetilde{x}|^2-2| x_{0}|^2)} d\\widetilde{\\mu }_s\\\\&\\le \\int _{M}\\chi (N_s(q,R))e^{C_R^2-\\frac{1}{2}| x_{0}|^2} d \\mu _0\\\\&\\le e^{C_R^2}C_0,$ for $s>s_N$ .", "Since the drifting mean curvature flow (REF ) only differs from (REF ) by the tangent diffeomorphisms, the drifting mean curvature flow (REF ) is also Type III.", "By rescaling (REF ) we have $|\\widetilde{A}(\\cdot ,s)|\\le C$ for $0< s<+\\infty $ , where $\\widetilde{A}(\\cdot ,s)$ is the second fundamental form of immersion $\\widetilde{x}(\\cdot ,s)$ .", "Moreover, we also have $|\\widetilde{\\nabla }^m \\widetilde{A}(\\cdot ,s)|\\le C(m)$ by Ecker and Huisken's derivative estimates for the mean curvature flow (see [4]).", "Moreover, $\\widetilde{x}(M,s)\\cap B(q,R)\\ne \\emptyset $ for $s>s_N$ by the assumption.", "Therefore, by the result of Langer ([14]) we conclude that $\\widetilde{x}(M,s_i)\\cap B(q,R)$ (under reparametrization), subconverges smoothly along sequences $s_i\\rightarrow \\infty $ to a limiting immersion $\\widetilde{x}_{\\infty }$ in $B(q,R)$ .", "We have $& \\int _M e^{\\frac{1}{2}(|\\widetilde{x}|^2 - |\\widetilde{x}_0 |^2)}\\, d\\widetilde{\\mu }_t - \\int _M e^{\\frac{1}{2}(|\\widetilde{x}|^2 - |\\widetilde{x}_0 |^2)}\\, d\\widetilde{\\mu }_s \\\\&= -\\int _s^t\\int _M e^{\\frac{1}{2}(|\\widetilde{x}|^2 - |\\widetilde{x}_0 |^2)}|\\vec{\\widetilde{\\mathbf {H}}}-\\widetilde{x}^{\\perp }|^2 d\\widetilde{\\mu }_{\\tau }\\, d\\tau .$ Since $\\int _M e^{\\frac{1}{2}(|\\widetilde{x}|^2 - |\\widetilde{x}_0 |^2)}\\, d\\widetilde{\\mu }_t$ is uniformly bounded and decreasing function in $t$ , there exists a finite $\\lim _{t\\rightarrow \\infty } \\int _M e^{\\frac{1}{2}(|\\widetilde{x}|^2 - |\\widetilde{x}_0 |^2)}\\, d\\widetilde{\\mu }_t$ implying that $\\lim _{s\\rightarrow \\infty } \\int _s^{\\infty } \\int _M e^{\\frac{1}{2}(|\\widetilde{x}|^2 - |\\widetilde{x}_0 |^2)}\\, |\\vec{\\widetilde{\\mathbf {H}}}-\\widetilde{x}^{\\perp }|^2\\, d\\widetilde{\\mu }_{\\tau }\\, d\\tau = 0.$ Using that $|\\widetilde{x}_0(p)| \\le C_R$ in $N_s(q,R)$ for all $s$ sufficiently big we get $&0 = \\lim _{s\\rightarrow \\infty } \\int _s^{\\infty }\\int _M e^{\\frac{1}{2}(|\\widetilde{x}|^2 - |\\widetilde{x}_0 |^2)}\\, |\\vec{\\widetilde{\\mathbf {H}}}-\\widetilde{x}^{\\perp }|^2\\, d\\widetilde{\\mu }_{\\tau }\\, d\\tau \\nonumber \\\\&\\ge e^{-C_R^2}\\int _s^{\\infty } \\int _{N_s(q,R)} e^{\\frac{1}{2}\\,|\\widetilde{x}|^2}\\, |\\vec{\\widetilde{\\mathbf {H}}} - \\widetilde{x}^{\\perp }|^2\\, d\\widetilde{\\mu }_{\\tau }\\, d\\tau .", "\\nonumber \\\\&= e^{-C_R^2}\\int _s^{\\infty } \\int _{\\widetilde{x}(M,s)\\cap B(q,R)} e^{\\frac{1}{2}\\,|\\widetilde{x}|^2}\\, |\\vec{\\widetilde{\\mathbf {H}}} - \\widetilde{x}^{\\perp }|^2\\, d\\widetilde{\\mu }_{\\tau }\\, d\\tau .$ Recall that for every sequence $s_i\\rightarrow \\infty $ , there exists a subsequence so that hypersurfaces $\\widetilde{x}(M,s)\\cap B(q,R)$ converge uniformly on compact sets to a limiting hypersurface in $B(q,R)$ which is defined by an immersion $\\widetilde{x}_{\\infty }$ .", "Estimate (REF ) implies $\\widetilde{x}_{\\infty }$ satisfies $\\vec{\\widetilde{\\mathbf {H}}}_{\\infty } = \\widetilde{x}_{\\infty }^{\\perp }$ in $B(q,R)$ .", "Let $x( \\cdot ,t):M\\rightarrow \\mathbb {R}^{n+1}$ be a Type III solution to the mean curvature flow (REF ) with $\\sup \\limits _{M\\times [0,\\infty )}t|A|^2=C<\\infty $ and let $\\widetilde{x}(\\cdot ,s)$ be its corresponding normalized mean curvature flow.", "By Theorem REF , we only need to prove there exists $R_0$ such that $\\widetilde{x}(M,s)\\cap B(o,R_0)\\ne \\emptyset $ for $s$ sufficiently large.", "Let $\\overline{x}(\\cdot ,s)$ be the solution to the normalized mean curvature flow $\\frac{\\partial \\overline{x}}{\\partial s}=\\vec{\\overline{\\mathbf {H}}}-\\overline{x},$ with the initial data $ x_{0}$ .", "Then we have $\\frac{\\partial }{\\partial s} |\\overline{x}|^2=2\\langle \\vec{\\overline{\\mathbf {H}}},\\overline{x}\\rangle -2|\\overline{x}|^2.$ Since the mean curvature flow is Type III and the normalized mean curvature flow (REF ) is obtained by $\\overline{x}(\\cdot ,s)=\\frac{1}{\\sqrt{2t+1}}x(\\cdot ,t),$ where $s$ is given by $s=\\frac{1}{2}\\log (2t+1)$ .", "Then $|\\vec{\\overline{\\mathbf {H}}}|\\le C(n)$ for $[0,+\\infty )$ .", "It follows from (REF ) that $|\\overline{x}|(p,s)\\le e^{-s}| x_{0}|(p)+C(n) (1 - e^{-s}).$ Hence $\\overline{x}(M,s)\\cap B(o,C(n)+1)\\ne \\emptyset $ for $s$ sufficiently large.", "Since $\\bar{x}(M,s)\\cap B(o,C(n)+1) \\ne \\emptyset $ and since the normalized drifting mean curvature flow (REF ) differs from the normalized mean curvature flow (REF ) only by the tangent diffeomorphisms, (implying $\\widetilde{x}(M,s)=\\overline{x}(M,s)$ ), we have that $\\widetilde{x}(M,s) \\cap B(o,C(n)+1) \\ne \\emptyset $ for $s$ sufficiently large.", "By Theorem REF , we conclude $\\widetilde{x}(M,s)\\cap B(o,R)$ subconverges to the limiting self-expander in $B(o,R)$ for all $R\\ge C(n)+1$ .", "Remark 3.1 In [3], Ecker and Huisken proved the following proposition showing that the normalized mean curvature flow (REF ) on entire graphs satisfying the linear growth condition (REF ) can not subconverge to a self-expander if the condition (REF ) fails.", "Proposition 3.2 ([3]) Let $\\overline{x}:M\\rightarrow \\mathbb {R}^{n+1}$ be the entire graph solution to the normalized mean curvature flow (REF ) whose initial data $x_{0}$ satisfies the linear growth condition (REF ) and $|\\nabla ^m A_0|\\le c(m)(1+|x|^2)^{-m-1}$ for $m=0,1$ , where $A_0$ is the second fundamental form of $x_{0}$ .", "Suppose there exists a sequence of points $p_k$ such that $|x_{0}(p_k)|\\rightarrow \\infty $ and $\\langle x_{0}(p_k),\\nu \\rangle ^2=\\gamma |x_{0}(p_k)|^2$ for some $\\gamma >0$ .", "Then there exists a sequence of times $s_k\\rightarrow \\infty $ for which $c_1\\le |x(p_k,s_k)|\\le c_2$ and $(H+\\langle x,\\nu \\rangle )(p_k,s_k)$ has a uniform positive lower bound.", "They also gave the following explicit example which satisfies the conditions of Proposition REF .", "Example 3.3 The graph $u_0(\\hat{x})=u_0(|\\hat{x}|)=\\left\\lbrace \\begin{array}{ll}|\\hat{x}|\\sin \\log |\\hat{x}|, & \\hbox{$|\\hat{x}|\\ge 1$;} \\\\smooth, & \\hbox{$|\\hat{x}|\\le 1$,}\\end{array}\\right.$ where $\\hat{x}$ is the coordinate on $\\mathbb {R}^2$ satisfies conditions of Proposition REF .", "It follows from Proposition REF that $\\overline{x}(M,s)\\cap B(o,c_2)$ can not converge to the self-expander in the case of Example REF , where $c_2$ is the same as in Proposition REF .", "Since the normalized drifting mean curvature flow (REF ) only differs from normalized mean curvature flow (REF ) by tangent diffeomorphisms, it follows that $\\widetilde{x}(M,s)\\cap B(o,c_2)$ can not converge to the self-expander in the case of Example REF , where $\\widetilde{x}(\\cdot ,s)$ is the corresponding solution to the normalized drifting flow (REF ).", "By Theorem REF and Remark REF , we know that the conditions (REF ) and (REF ) must fail in this case.", "Next we give an example which shows that one may not have that the asymptotic limit of Type III mean curvature is the self-expander even if the initial data $x_0$ is an entire graph satisfying the linear growth condition (REF ) and having the positive mean curvature.", "Example 3.4 Let $x_0(r,\\theta )=(r\\cos {\\theta },r\\sin {\\theta },f(r))$ be a surface of revolution.", "We calculate that the first fundamental form $g=(1+f^{\\prime }(r)^2)dr^2+r^2d\\theta ^2,$ the second fundamental form $h=\\frac{f^{\\prime \\prime }(r)}{\\sqrt{1+f^{\\prime }(r)^2}}dr^2+\\frac{rf^{\\prime }(r)}{\\sqrt{1+f^{\\prime }(r)^2}}d\\theta ^2,$ mean curvature $H=\\frac{f^{\\prime }(r)(1+f^{\\prime }(r)^2)+rf^{\\prime \\prime }(r)}{r(1+f^{\\prime }(r)^2)^{\\frac{3}{2}}}$ , $\\nu =\\frac{(f^{\\prime }(r)\\sin {\\theta },f^{\\prime }(r)\\cos {\\theta },-1)}{\\sqrt{1+f^{\\prime }(r)^2}}$ , and $<x_0,\\nu >=\\frac{f^{\\prime }(r)r-f(r)}{\\sqrt{1+f^{\\prime }(r)^2}}.$ Here we choose $f(r)=\\left\\lbrace \\begin{array}{ll}r\\sin \\log r+6r, & \\hbox{$r\\ge 1$;} \\\\\\text{smooth and $f^{\\prime }(r)\\ge 0$, $f^{\\prime \\prime }(r)\\ge 0$}, & \\hbox{$r \\le 1$,}\\end{array}\\right.$ Such $f(r)$ exists since $f^{\\prime }(1)=7$ and $f^{\\prime \\prime }(1)=1$ .", "It is easy to check that $f(r)$ satisfies $|f^{\\prime }(r)|\\le C$ , $|rf^{\\prime \\prime }(r)|\\le C$ , $|r^2f^{\\prime \\prime \\prime }(r)|\\le C$ and $|f^{\\prime }(r_k)r_k-f(r_k)|=\\gamma r_k$ for some sequence $r_k\\rightarrow +\\infty $ and a constant $\\gamma >0$ .", "Then it easily follows the surface $x_0$ satisfies the conditions of Proposition REF and that $H > 0$ ." ], [ "More on the convergence to an expander", "In this section we give the proofs of Theorem REF , Theorem REF and Corollary REF where we impose conditions on the initial data and then show that condition (REF ) is satisfied, so that the conclusion of Theorem (REF ) still holds.", "We will need to apply the maximum principle for complete noncompact one parameter family of hypersurfaces that has been proved for example in [4].", "We state this maximum principle result below for the convenience of a reader.", "Theorem 4.1 (Maximum principle for complete manifolds in [4]) Suppose that the manifold $M^n$ with Riemannian metrics $g(\\cdot ,t)$ satisfies a uniform volume growth restriction, namely $vol_t(B_{g(t)}(p,r)) \\le e^{k(1+r^2)},$ holds for some point $p\\in M^n$ and a uniform constant $k > 0$ for all $t \\in [0,T]$ , where $B_{g(t)}(p,r)$ is the intrinsic ball on $M^n$ .", "Let $f$ be a function on $M^n\\times [0,T]$ which is smooth on $M^n\\times (0,T]$ and continuous on $M^n\\times [0,T]$ .", "Assume that $f$ and $g(t)$ satisfy $\\frac{\\partial }{\\partial t} f \\le \\Delta _t f + {\\bf a} \\cdot \\nabla f + b f$ where the function $b$ satisfies $\\sup _{M^n\\times [0,T]} |b| \\le \\alpha _0$ for some $\\alpha _0 < \\infty $ and the vector $a$ satisfies $\\sup _{M^n\\times [0,T]} |{\\bf a}| \\le \\alpha _1$ for some $\\alpha _1 < \\infty $ , $f(p,0) \\le 0$ for all $p\\in M^n$ , $\\int _0^T \\int _M e^{-\\alpha _2^2 dist_t(p,y)^2}\\, |\\nabla f|^2(y,t)\\, d\\mu _t\\, dt < \\infty , \\qquad \\mbox{for some} \\qquad \\alpha _1 < \\infty $ , $\\sup _{M^n\\times [0,T]} \\left|\\frac{\\partial }{\\partial t} g_{ij}\\right| \\le \\alpha _3, \\qquad \\mbox{for some} \\qquad \\alpha _3< \\infty $ .", "Then we have $f \\le 0$ on $M^n\\times [0,T]$ .", "Remark 4.2 In the case the second fundamental form is at each time slice uniformly bounded in space, since $Ric_{M_t} \\ge -2|A|^2 \\ge -C$ , for $t\\in [0,T]$ (where $C$ may depend on $T$ ), the uniform volume growth condition (REF ) of Theorem REF holds for $t \\in [0,T]$ .", "Hence, we can apply the maximum principle for complete hypersurfaces moving by the mean curvature flow, such that the second fundamental form is bounded at each time slice and such that (i)-(iv) of Theorem REF hold.", "Before presenting the proof of Theorem REF we need the following lemma.", "Lemma 4.3 Let $\\widetilde{x}(\\cdot ,s)$ be the solution to the normalized drifting mean curvature flow (REF ) with the second fundamental form bounded at each time slice.", "Assume the initial data $x_{0}:M\\rightarrow \\mathbb {R}^{n+1}$ is a complete immersed hypersurface satisfying $H+\\langle x_0,\\nu \\rangle \\ge 0$ (resp.", "$H+\\langle x_0,\\nu \\rangle \\le 0$ ) at $s=0$ .", "Then $\\widetilde{H}+\\langle \\widetilde{x},\\widetilde{\\nu }\\rangle \\ge 0$ (resp.$\\widetilde{H}+\\langle \\widetilde{x},\\widetilde{\\nu }\\rangle \\le 0$ ) for all $s\\ge 0$ .", "Moreover, if $H\\ge 0$ and $\\langle x_0,\\nu \\rangle \\le 0$ at initial time, then $\\widetilde{H}\\ge 0$ and $\\langle \\widetilde{x},\\widetilde{\\nu }\\rangle \\le 0$ for all $s\\ge 0$ .", "Let $\\overline{x}(\\cdot ,s)$ be the solution to the normalized mean curvature flow (REF ) with the initial data $x_{0}$ .", "It follows from Lemma 5.5 in [3] that $(\\frac{\\partial }{\\partial s}-\\overline{\\Delta })\\overline{H}=|\\overline{A}|^2\\overline{H}+\\overline{H},$ $(\\frac{\\partial }{\\partial s}-\\overline{\\Delta })\\langle \\overline{x},\\overline{\\nu }\\rangle =|\\overline{A}|^2\\langle \\overline{x},\\overline{\\nu }\\rangle -2\\overline{H}-\\langle \\overline{x},\\overline{\\nu }\\rangle ,$ $(\\frac{\\partial }{\\partial s}-\\overline{\\Delta })(\\overline{H}+\\langle \\overline{x},\\overline{\\nu }\\rangle )=(|\\overline{A}|^2-1)(\\overline{H}+\\langle \\overline{x},\\overline{\\nu }\\rangle ),$ where $\\overline{A}(\\cdot ,s)$ is the seconded fundamental form of $\\overline{x}(\\cdot ,s)$ .", "By Remark REF and the maximum principle for noncompact manifolds (Theorem REF ), we have $\\overline{H}+\\langle \\overline{x},\\overline{\\nu }\\rangle \\ge 0$ (resp.$\\overline{H}+\\langle \\overline{x},\\overline{\\nu }\\rangle \\le 0$ ) for all $s\\ge 0$ if $H+\\langle x_0,\\nu \\rangle \\ge 0$ (resp.", "$H+\\langle x_0,\\nu \\rangle \\le 0$ ) at $s=0$ .", "Moreover, $\\overline{H}\\ge 0$ and $\\langle \\overline{x},\\overline{\\nu }\\rangle \\le 0$ for all $s\\ge 0$ if $H\\ge 0$ and $\\langle x_0,\\nu \\rangle \\le 0$ at initial time.", "Since the normalized drifting flow (REF ) only differs from (REF ) by the tangent diffeomorphisms, we conclude that Lemma REF holds.", "The rescaled solution to the mean curvature flow $x_{\\mu }(\\cdot ,t) := \\mu ^{-\\frac{1}{2}}(x(\\cdot , \\mu t)-q_0 )$ is also the Type III solution, with the initial data $\\mu ^{-\\frac{1}{2}}(x_0-q_0 )$ satisfying $\\langle x_{\\mu }(\\cdot ,0), \\nu \\rangle = \\langle \\mu ^{-\\frac{1}{2}}(x_0-q_0 ),\\nu \\rangle \\le 0$ , by the assumptions in Theorem REF .", "Moreover, $H_{\\mu }(\\cdot ,0) + \\langle x_{\\mu }(\\cdot ,0), \\nu \\rangle = \\mu ^{-\\frac{1}{2}}\\, \\big (H(\\cdot ,0) + \\langle x_0 - q_0 , \\nu \\rangle \\big ) \\ge 0.$ Let $\\widetilde{x}(\\cdot ,s)$ and $\\widetilde{x}_{\\mu }(\\cdot ,s)$ be the corresponding normalized drifting mean curvature flow for $x(\\cdot ,t)$ and $x_{\\mu }(\\cdot ,t)$ respectively.", "It is easy to see that $\\widetilde{x}_{\\mu }(\\cdot ,s)$ subconverges to the limiting self-expander if and only if $\\widetilde{x}(\\cdot ,s)$ subconverges to the limiting self-expander.", "Hence, as a matter of scaling with $\\mu = 1$ , without losing the generality we can assume that $q_0 = 0$ and that $\\langle x_0, \\nu \\rangle \\le 0, \\qquad \\mbox{and} \\qquad H(\\cdot ,0)+ \\langle x_0, \\nu \\rangle \\ge 0.$ By Lemma REF , $\\langle \\widetilde{x},\\widetilde{\\nu }\\rangle \\le 0$ and $\\widetilde{H}+\\langle \\widetilde{x},\\widetilde{\\nu }\\rangle \\ge 0$ for all $s\\ge 0$ .", "Moreover, we have $\\frac{\\partial }{\\partial s}|\\widetilde{x}|^2=2\\langle \\vec{\\widetilde{H}}-\\widetilde{x}^{\\perp }, \\widetilde{x}\\rangle =-2\\langle \\widetilde{x},\\widetilde{\\nu }\\rangle (\\widetilde{H}+\\langle \\widetilde{x},\\widetilde{\\nu }\\rangle )\\ge 0.$ This implies that for every $p$ such that $\\widetilde{x}(p,s) \\in B(o,R)$ , that is, $|\\widetilde{x}(p,s)| \\le R$ , we have $|\\widetilde{x}(p,0)| \\le |\\widetilde{x}(p,s)| \\le R.$ Then Theorem REF follows from Corollary REF immediately.", "Remark 4.4 We also give another proof of Theorem REF without applying Corollary REF .", "As shown in the proof presented above we can assume that $p = 0$ and that $\\langle x_0, \\nu \\rangle \\le 0, \\qquad \\mbox{and} \\qquad \\langle x_0, \\nu _0\\rangle + H(\\cdot ,0) \\ge 0,$ without losing the generality.", "We know that $\\langle \\widetilde{x},\\widetilde{\\nu }\\rangle \\le 0$ and $\\widetilde{H}+\\langle \\widetilde{x},\\widetilde{\\nu }\\rangle \\ge 0$ for all $s\\ge 0$ by Lemma REF .", "We compute that $\\frac{\\partial }{\\partial s}\\widetilde{g}_{ij} &=2\\partial _i (\\vec{\\widetilde{\\mathbf {H}}}-\\widetilde{x}^{\\perp }) \\partial _j\\widetilde{x} \\nonumber \\\\&=-2\\widetilde{H}\\widetilde{h}_{ij}+2\\widetilde{x}^{\\perp }\\partial _i\\partial _j \\widetilde{x} \\nonumber \\\\&=-2\\widetilde{H}\\widetilde{h}_{ij}-2\\langle \\widetilde{x},\\widetilde{\\nu }\\rangle \\widetilde{h}_{ij},$ where we use $\\widetilde{x}^{\\perp }=\\langle \\widetilde{x},\\widetilde{\\nu }\\rangle \\widetilde{\\nu }$ and $\\widetilde{h}_{ij}=-\\widetilde{\\nu }\\cdot \\partial _i\\partial _j \\widetilde{x}$ .", "It follows that $\\frac{\\partial }{\\partial s}d\\widetilde{\\mu }_s &=(-|\\vec{\\widetilde{\\mathbf {H}}}|^2+\\langle \\widetilde{x}^{\\perp },\\vec{\\widetilde{\\mathbf {H}}}\\rangle )d\\widetilde{\\mu }_s\\nonumber \\\\&=-\\widetilde{H}(\\widetilde{H}+\\langle \\widetilde{x},\\widetilde{\\nu }\\rangle )d\\widetilde{\\mu }_s.$ We have $\\frac{\\partial }{\\partial s}|\\widetilde{x}|^2=2\\langle \\vec{\\widetilde{H}}-\\widetilde{x}^{\\perp }, \\widetilde{x}\\rangle =-2\\langle \\widetilde{x},\\widetilde{\\nu }\\rangle (\\widetilde{H}+\\langle \\widetilde{x},\\widetilde{\\nu }\\rangle ).$ It follows that $\\frac{\\partial }{\\partial s}e^{-\\frac{1}{2}|\\widetilde{x}|^2}d\\widetilde{\\mu }_s=\\langle \\widetilde{x},\\widetilde{\\nu }\\rangle ^2-\\widetilde{H}^2=(\\langle \\widetilde{x},\\widetilde{\\nu }\\rangle -\\widetilde{H})(\\langle \\widetilde{x},\\widetilde{\\nu }\\rangle +\\widetilde{H})\\le 0.$ Then we have $\\frac{d}{d s}\\int _Me^{-\\frac{1}{2}|\\widetilde{x}|^2}d\\widetilde{\\mu }_s=\\int _M(\\langle \\widetilde{x},\\widetilde{\\nu }\\rangle -\\widetilde{H})(\\langle \\widetilde{x},\\widetilde{\\nu }\\rangle +\\widetilde{H})e^{-\\frac{1}{2}|\\widetilde{x}|^2}d\\widetilde{\\mu }_s\\le 0.$ It follows that $e^{-\\frac{1}{2}R^2}H^n(\\widetilde{x}(M,s)\\cap B(o,R))\\le \\int _{\\widetilde{M}_s\\cap B(o,R)}e^{-\\frac{1}{2}|\\widetilde{x}|^2}d\\widetilde{\\mu }_s\\le \\int _{M}e^{-\\frac{1}{2}|\\widetilde{x}_0|^2}d\\widetilde{\\mu }_0\\le C_0.$ Hence $H^n(\\widetilde{x}(M,s)\\cap B(o,R))\\le C_0e^{\\frac{1}{2}R^2}$ for any $R>0$ .", "Note that all derivatives of the second fundamental form of hypersurfaces $\\widetilde{x}(M,s)$ are uniformly bounded, which is a consequence of our Type III assumption, rescaling (REF ) and Ecker and Huisken's gradient estimates (see [4]).", "Moreover, by the Type III assumption, similarly as in the proof of Corollary REF we can show that $\\widetilde{x}(M,s)\\cap B(o,R_0))\\ne \\emptyset $ for some $R_0$ .", "As a result we conclude that $\\widetilde{x}(M,s)\\cap B(q,R)$ (under reparametrization) along sequences (as $s_i\\rightarrow \\infty $ ) subconverges smoothly to a limiting immersion $\\widetilde{x}_{\\infty }$ in $B(o,R)$ for any $R\\ge R_0$ by Langer's result [14].", "Moreover, we have $\\langle \\widetilde{x}_{\\infty },\\widetilde{\\nu }_{\\infty }\\rangle \\le 0$ and $\\widetilde{H}_{\\infty }+\\langle \\widetilde{x}_{\\infty },\\widetilde{\\nu }_{\\infty }\\rangle \\ge 0$ .", "By the (REF ), we have $(\\langle \\widetilde{x}_{\\infty },\\widetilde{\\nu }_{\\infty }\\rangle -\\widetilde{H}_{\\infty })(\\langle \\widetilde{x}_{\\infty },\\widetilde{\\nu }_{\\infty }\\rangle +\\widetilde{H}_{\\infty })=0$ .", "Hence $\\langle \\widetilde{x}_{\\infty },\\widetilde{\\nu }_{\\infty }\\rangle +\\widetilde{H}_{\\infty }=0$ .", "By rescaling the flow as in the proof of Theorem REF allows us to assume without losing the generality that $x_0$ satisfies $H+\\langle x_0,\\nu \\rangle \\le 0$ .", "By Lemma REF , $\\widetilde{H}\\ge 0$ , $\\langle \\widetilde{x},\\nu \\rangle \\le 0$ and $\\widetilde{H}+\\langle \\widetilde{x},\\widetilde{\\nu }\\rangle \\le 0$ for all $s\\ge 0$ .", "By (REF ), we have $\\frac{\\partial }{\\partial s}e^{-\\frac{1}{2}|\\widetilde{x}|^2}d\\widetilde{\\mu }_s=\\langle \\widetilde{x},\\widetilde{\\nu }\\rangle ^2-\\widetilde{H}^2=(\\langle \\widetilde{x},\\widetilde{\\nu }\\rangle -\\widetilde{H})(\\langle \\widetilde{x},\\widetilde{\\nu }\\rangle +\\widetilde{H})\\ge 0.$ Then $\\frac{d}{d s}\\int _Me^{-\\frac{1}{2}|\\widetilde{x}|^2}d\\widetilde{\\mu }_s=\\int _M(\\langle \\widetilde{x},\\widetilde{\\nu }\\rangle -\\widetilde{H})(\\langle \\widetilde{x},\\widetilde{\\nu }\\rangle +\\widetilde{H})e^{-\\frac{1}{2}|\\widetilde{x}|^2}d\\widetilde{\\mu }_s\\ge 0.$ and hence $\\int ^{\\infty }_0\\int _M(\\langle \\widetilde{x},\\widetilde{\\nu }\\rangle -\\widetilde{H})(\\langle \\widetilde{x},\\widetilde{\\nu }\\rangle +\\widetilde{H})e^{-\\frac{1}{2}|\\widetilde{x}|^2}d\\widetilde{\\mu }_s\\le \\lim \\limits _{s\\rightarrow \\infty }\\int _Me^{-\\frac{1}{2}|\\widetilde{x}|^2}d\\widetilde{\\mu }_s<\\infty .$ From this point on we can argue as in the proof of Theorem REF given in Remark REF to conclude that the normalized drifting mean curvature flow subconverges smoothly to the limiting self-expander.", "Moreover, if we have $H^n(\\widetilde{x}(M,s)\\cap B(o,R))\\le CR^m$ , then $&\\int _{M}e^{-\\frac{1}{2}|\\widetilde{x}|^2}d\\widetilde{\\mu }_s=\\sum \\limits _{j=1}^{\\infty }\\int _{\\widetilde{x}(M,s)\\cap B(o,R^j)\\setminus \\widetilde{x}(M,s)\\cap B(o,R^{j-1})}e^{-\\frac{1}{2}|\\widetilde{x}|^2}d\\widetilde{\\mu }_s\\\\\\le & \\sum \\limits _{j=0}^{\\infty }H^n(\\widetilde{x}(M,s)\\cap B(o,R^j)\\setminus \\widetilde{x}(M,s)\\cap B(o,R^{j-1}))e^{-\\frac{1}{2}R^{2j-2}}\\\\\\le & \\sum \\limits _{j=0}^{\\infty }CR^{mj}e^{-\\frac{1}{2}R^{2j-2}}<\\infty .$ The example REF illustrates that the asymptotic limit of Type III mean curvature flow which is an entire graph with positive mean curvature may not always be an expander.", "Corollary REF describes under which additional condition on an initial hypersurface we can guarantee that the asymptotic limit in the situation described as above is always an expander.", "By the result of Ecker and Huisken [3](see Proposition 4.4 in [3]), we know that the mean curvature flow of an entire graph satisfying the linear growth condition $\\langle \\nu ,w\\rangle ^{-1}\\le c,$ must be Type III, where $w$ is a fixed vector such that $\\langle \\nu ,w\\rangle >0$ .", "If $\\langle x_0-q_0 ,\\nu \\rangle \\le C$ for some fixed vector $q_0$ , then we have $\\langle x_0-q_0 -c_1w,\\nu \\rangle \\le C-c^{-1}c_1\\le 0$ for $c_1$ large enough.", "By Corollary 3.2 in [3], $\\langle \\nu ,w\\rangle ^{-1}\\le c$ remains valid under the mean curvature flow.", "Since $\\nu $ is scaling invariant, we conclude that $\\langle \\widetilde{\\nu },w\\rangle ^{-1}\\le c$ remains valid under the normalized drifting mean curvature flow and $H^n({\\widetilde{x}(M,s)\\cap B(o,R)})= \\int _{|x|\\le R}\\langle \\widetilde{\\nu },w\\rangle ^{-1} dx^n\\le cR^n$ .", "By Theorem REF , we know that its corresponding normalized drifting mean curvature flow (REF ) subconverges smoothly to the limiting self-expander." ] ]
1403.0235
[ [ "Nonparametric test for a constant beta between \\Ito semi-martingales\n based on high-frequency data" ], [ "Abstract We derive a nonparametric test for constant beta over a fixed time interval from high-frequency observations of a bivariate \\Ito semimartingale.", "Beta is defined as the ratio of the spot continuous covariation between an asset and a risk factor and the spot continuous variation of the latter.", "The test is based on the asymptotic behavior of the covariation between the risk factor and an estimate of the residual component of the asset, that is orthogonal (in martingale sense) to the risk factor, over blocks with asymptotically shrinking time span.", "Rate optimality of the test over smoothness classes is derived." ], [] ]
1403.0349
[ [ "Possible Implications of a Vortex Gas Model and Self-Similarity for\n Tornadogenesis and Maintenance" ], [ "Abstract We describe tornadogenesis and maintenance using the 3-dimensional vortex gas model presented in Chorin (1994) and developed further in Flandoli and Gubinelli (2002).", "We suggest that high-energy, super-critical vortices in the sense of Benjamin (1962), that have been studied by Fiedler and Rotunno (1986), have negative temperature in the sense of Onsager (1949) play an important role in the model.", "We speculate that the formation of high-temperature vortices is related to the helicity inherited as they form or tilt into the vertical and their interaction with the surface and boundary layer.", "We also exploit the notion of self-similarity to justify power laws derived from observations of weak and strong tornadoes presented in Cai (2005); Wurman and Gill (2000); Wurman and Alexander (2005).", "Analysis of a Bryan Cloud Model (CM1) simulation of a tornadic supercell reveals scaling consistent with the observational studies." ], [ "Introduction", "In a recent paper, [21] defined the pseudovorticity by $\\zeta _\\text{pv}=\\frac{\\Delta V}{L}$ , where $\\Delta V=|(V_r)_\\text{max}-(V_r)_\\text{min}|$ is the difference between the maximum and minimum radial velocity of the mesocyclone (rotating updraft) and $L$ is the distance between them.", "Mobile Doppler radar observations of past tornadic and nontornadic storms were filtered using a range of Cressman influence radii (lower-bounded by the native data resolution and upper-bounded by mesocyclone diameter) to obtain points $(\\log (\\varepsilon ),\\log (\\zeta _\\text{pv}))$ , where $\\varepsilon $ is the finest resolvable scale of the filtered radar data.", "Cai then calculated the regression line for each storm and found that steeper negative slopes are indicative of tornadic storms, and that the threshold slope for strong tornadoes in his sample was approximately $-1.6$ .", "The regression lines Cai calculated strongly fit the data over scales between that of the mesocyclone core and that of the “edge” of the mesocyclonic tangential flow, indicating a vorticity vs. scale power law is valid over those scales.", "This suggests a power law for the decay of the vertical component of vorticity outside the solid-body mesocyclone core of the form $\\zeta \\propto r^b$ for some $b<0$ , where $r$ is the radial distance from the axis of the vortex.", "Cai observed that it may be correct to interpret the exponent as a fractal dimension associated with the vortex.", "The vorticity power law, if valid, may extend to smaller (including tornadic) scales, but this could not be determined given the limited resolution of the radar observations used in Cai's study.", "In a paper devoted to analyzing mobile radar data obtained from a tornado that occurred in Dimmit, Texas (June 2, 1995) and was rated F2–F4, [107] found $v\\propto r^b$ , i.e., the tangential winds outside the tornado core roughly fit the modified Rankine vortex model.", "They calculated the exponent $b$ and found it to vary from $-0.5$ to $-0.7$ .", "From data obtained in the intercept of the Spencer, South Dakota F4 tornado (May 31, 1998), [106] calculated $b=-0.67$ .", "Cai noticed that the (threshold) vorticity power law exponent for strongly tornadic mesocyclones in his sample differs from the velocity power law exponent calculated for the strong tornado in [107] by 1, which is consistent with the vorticity being the curl of the velocity.", "The larger vorticity/velocity power law exponents found in strongly tornadic mesocyclones are consistent with the observation of [98] that “parcels that nearly conserve angular momentum penetrate closer to the central axis of the tornadic mesocyclones, resulting in large tangential velocities.” As noted by Cai, hurricanes exhibit a similar velocity power law and exponent outside their eyewall (see, e.g., [79]).", "This suggests that roughly the same vorticity power law may apply over a range of atmospheric vortex scales.", "It is curious that the power laws obtained by [107], [106] are consistent with the results obtained earlier by [72] in a vortex simulator.", "Power laws with a particular scaling exponent arise when a phenomenon “repeats itself on changing scales” (see, e.g., [74], [9], [7], [8]).", "This property is called self-similarity.", "We propose that to more completely understand strong atmospheric vortices requires further exploration of their self-similarity.", "Self-similarity, especially when exhibited across objects ranging in scale, can point to important properties of the underlying dynamics.", "We focus on the possible self-similarity of tornadoes in this paper.", "As will be shown, tornadoes appear to exhibit local self-similarity and local homogeneity suggesting fractal phenomena.", "Some tracks left by high-energy vortices within a tornado are as narrow as 30 cm.", "Some of these paths appear to originate outside the tornado and intensify as they move into the tornado.", "We identify these vortices as supercritical in the sense of [45].", "Their analysis the work of [6], [19], [10], suggests to us that the super-critical vortex below a vortex breakdown has its volume and its length decrease as the the energy of the super-critical vortex increases.", "This would suggests that the entropy (randomness of the vortex) is decreasing when the energy is increased.", "Hence the inverse temperature, which is the rate of change of the entropy with respect to the energy, of the vortex is negative.", "This temperature is not related to the molecular temperature of the atmosphere.", "These vortices would be barotropic, however their origin could very well be baroclinic.", "Recent results suggest vorticity produced baroclinicly in the rear-flank downdraft and that then decends to the surface and is tilted into the vertical is linked to tornadogenesis.", "Once these vortices come in contact with the surface, and the stretching and surface friction related swirl (boundary layer effects) are in the appropriate ratio, then by analogy with the work of [45] the vortex would have negative temperature and the vortex would now be barotropic.", "The paper is organized as follows.", "In Section we describe the tornadogenesis problem.", "In Section , motivated by the results of [72], [107], [106], [21], we discuss a numerical experiment producing a time series of slopes of vorticity lines and its connection with previous results.", "In Section we address self-similarity of tornadoes and mesocyclones and the way it might be manifested.", "We also give a heuristic argument supporting Cai's power law and its associated exponent for strong tornadoes.", "In Section we discuss the vortex gas theory of [87], [23], [46] in two and three dimensions and give arguments for its role in modeling tornadogenesis and tornado maintenance.", "We discuss the influence of the boundary layer on the possible formation of negative temperature vortices.", "We suggest that supercritical vortices have negative temperature.", "One important point is that while in classical statistical mechanics of a molecular gas a large number of particles is assumed, for a three dimensional vortex gas one can study a single vortex and its properties use using an ensemble much like the ensembles used to model weather forecasts.", "In Section we discuss suction vortices, give conclusions, and describe future work." ], [ "Tornadogenesis", "The search to understand tornadogenesis invariably involves the question: “Where does the vorticity in the tornado originate?” The most penetrating studies of this question have lead to the study of two types of vorticity: barotropic vorticity and baroclinic vorticity.", "Barotropic vorticity is vorticity that exists in the ambient environment and is frozen in the fluid and stretched and advected by the fluid.", "Baroclinic vorticity is vorticity that is generated by density currents in the fluid and is stretched and advected by the fluid.", "Definitive discussions of the role of barotropic and baroclinic vorticity in tornadogenesis and the mathematical decomposition of vorticity into barotropic and baroclinic parts and its consequences are given in [29], [30], [31], [32], [33], [34], [35].", "Both types of vorticity or combinations thereof have been suggested as the origins of the vorticity in tornadogenesis.", "Based on film footage of tornadoes showing sheets of precipitation spiraling into tornadoes, [49], [50] suggested a “barotropic” method called “Fujita's recycling hypothesis.” In this process the precipitation falling near the interface between the updraft and downdraft transported vorticity to the surface and into the tornado, and the precipitation-rich air was recycled into the thunderstorm updraft by the tornado.", "The numerical model and experiment of [35] shows that tornadogenesis can take place by a purely barotropic process.", "[77] have also found through numerical studies that baroclinic vorticity can be important, if not dominant, in tornadogenesis in a recycling-type process as well.", "Additional evidence supporting the recycling hypothesis is the observation that downdrafts associated with tornadic storms are generally warmer than downdrafts associated with storms that were nontornadic; this would make the updraft more buoyant in the tornado-producing storms and less buoyant in storms that do not produce tornadoes (see [76]).", "Recent studies of radar data by [78] and numerical simulations by [96] reveal arching vortex lines (or vortex tubes) in the rear flank of supercell storms.", "These vortex lines appear to be almost synonymous with supercell thunderstorms.", "If these vortex lines can be focused into a small region, as more and more vortex lines enter this region, viscous interactions between neighboring vortex lines are believed to lead to mergers.", "This can ultimately lead to the creation of a strong vortex.", "Several theories have been given for the production of the arching vortex lines.", "In one theory vorticity lines (or rings) baroclinically generated around the rear flank downdraft are advected toward the updraft as they descend (see [96], [78]).", "The downstream portions of the vorticity lines are subsequently lifted and stretched by the updraft while the upstream portions continue to descend, forming vortex arches.", "A second theory holds that horizontal shear across the rear flank gust front is the source of the vortex arches (see [59], [60], [61], [99]).", "This would create a vortex sheet that could be stretched and rolled up into a tornado vortex.", "It seems plausible that a combination of these two processes could be present, with a reconnection of vortex lines produced by the two different processes.", "However, there may be other vorticity sources as well.", "As discussed later, we hypothesize that the possible fractal dimension of $1.6$ found by Cai and implied in Wurman's observational studies comes from the interactions of vortices produced in these shear regions.", "Observational analysis of videos of the tornadogenesis phase in large-diameter tornadoes forming under low-cloud bases (see, e.g., [38]) suggests that vortices (vortex lines) which enter the developing tornado make a partial revolution about the ambient tornado vortex before folding up and dissipating.", "The vortex gas model described in Section follows [23] and suggests this folding up is necessary to conserve energy as the vortex stretches and/or interacts with other vortices.", "In this process some energy is transmitted to much smaller scales, the so-called inertial range, beginning the Kolmogorov cascade to the viscous range and then dissipating as heat.", "However, as the vortex stretches before kinking up, much energy is transmitted to the ambient vortex as kinetic energy of the flow, and this increases the vorticity of the tornado.", "In this theory, as more and more vortices successively enter the developing tornado, the process repeats itself many times gradually increasing the vorticity of the tornado vortex.", "Subsequently, the vorticity of the ambient vortex is increased (assuming the vortex lines are produced uniformly), which can be further enhanced by stretching, eventually achieving quasi-equilibrium with its environment.", "During this process, energy is transferred from the smaller scales to the larger scales in an inverse energy cascade.", "As more vortices enter the ambient tornado vortex, larger vortices tend to form; the stronger vortices at the core and slightly weaker vortices wrapping around them.", "Visually, the resulting flow could manifest itself as multiple vortices or as a large single vortex.", "The stretching of the vortices that enter the tornado eventually leads to the dissipation of their vorticity in a Kolmogorov cascade.", "The process just described contrasts with the transition to a multiple-vortex configuration that can occur when the critical swirl ratio (a measure of tornado-scale helicity) is exceeded (see, e.g., [27]).", "It has been shown by [81], [82] that twisting of subvortices about one another is measured by the helicity of the parent vortex.", "Helicity is the integral over physical space of the dot product of the velocity and the vorticity.", "It is thought that helicity of a flow inhibits the dissipation of energy and helps maintain the intensity of the flow (see [68], [62], [69], [70]).", "Helicity and its slightly modified form have been used as a parameter to study its effect on supercell storms by [68], [30], [70], [36], [42].", "Lilly thought of a vortex as a coiled spring that unwinds as it stretches.", "If we think of the helicity as measuring how much the vortex is wound up, the stretching unwinds the spring and releases energy to the surrounding flow.", "This unwinding could manifest itself as vortex breakdown and/or the fractalization and kinking up that is predicted in the vortex gas theory (described in Section ).", "[59], [60], [61] studied non-supercell tornadogenesis due to vertical shear in the boundary layer.", "They considered a weak cold pool (outflow boundary) advancing from the west into an ambient flow from the south to the north.", "This led to a south-to-north oriented vortex sheet forming at the interface of the two flows.", "They noted first-generation vortices rolling up into stronger second-generation vortices.", "It seems plausible that if one did finer-grid simulations of the situation considered in the papers, the first-generation vortices would have formed from roll-ups at smaller scales, resulting in a self-similar structure.", "The structure of the resulting vortex sheet resembled that of [5].", "The roll-up process has also been studied in [95], [92].", "Snow concludes that “the subsidiary vortices are integral parts of the overall flow pattern and should not be viewed as interacting independent vortices.” Once a vortex sheet roll-up has occurred, tornadogenesis can be induced by convection moving over and stretching one of the vortex sheet vortices.", "The first of three non-supercell tornadoes studied by [91] formed in this manner.", "It is shown in [91] that with sufficient stretching, non-supercell tornado vortices can have EF3 strength, but virtually all violent (EF4–EF5) tornadoes occur within supercells.", "While this suggests important differences between tornadogenesis mechanisms in supercells versus non-supercells, it does not preclude the possibility that vortex sheet roll-up can also play an important role in supercell tornadogenesis.", "Radar analyses in [39], [12], [14], [40], [41], [15], [13] and observational analyses in [17], [104], [101], [100] have revealed vortices along the rear flank gust front of both tornadic and nontornadic supercell storms.", "The role of these vortices in tornado formation is a subject of current research.", "As with non-supercells, supercell tornadogenesis has been observed by [100] to sometimes be triggered when one of these vortices is stretched by an updraft (but this process alone probably cannot produce violent tornadoes).", "Consistent with observations, numerical supercell simulations by [1] have shown vortices that appear to form along the edge of the rear flank gust front or a secondary gust front, and then roll up into a tornado vortex.", "The resulting vorticity distribution at one stage of tornadogenesis in [1] resembles the two-dimensional vortex sheet roll-up modeled by [25] and [57].", "The remarkable photo taken by Gene Moore (see Figure REF ) strongly implies vortex sheet roll-up.", "A sequence of vortices appears to be spiraling into a tornado as it crosses a lake.", "These feeder vortices have cross-sections too small to be resolved in all but the highest-resolution simulations currently achievable and appear to be very intense.", "Evidence of similar feeder vortices is also shown in Figure REF , in which tracks left in corn fields appear to show vortices spiraling into the tornadoes and then dissipating as they stretch.", "Subsequent to vortex sheet roll-up, vortex mergers may play a critical role.", "Observations of tornadogenesis near Bassett, NE by [12] suggested that a larger vortex ($\\sim 500$ m scale) and a smaller vortex ($\\sim 100$ –200 m scale), both of which were conjectured to arise from vortex sheet roll-up along the rear flank gust front, interacted so that the smaller vortex was absorbed by the larger vortex, possibly triggering tornadogenesis.", "The authors suggested that the origin of the vorticity was tilting of stream-wise vorticity along the rear flank gust front (see also [41]).", "Tilting of stream-wise vorticity would result in vortices that have large helicity and are more resistant to dissipation due to stretching.", "In such a scenario, it is plausible that stretching and subsequent intensification of a larger vortex could draw other vortices through their mutual interaction, resulting in vortex mergers that further intensify the dominant vortex, and so on to tornadogenesis.", "Further evidence that vortex sheet roll-up contributes to tornadogenesis in supercells is provided by [25], which showed that vortex sheets consisting of cyclonically rotating vortices roll up into a cyclonic vortex.", "The rolled up vortex sheet resembled the hook echo region of a supercell thunderstorm.", "We believe the roll-up in [25] is representative of the roll-up in [2].", "When vortices of opposite sign were placed in the two halves of the vortex sheet segment, the sheet rolled up into a cyclonic–anticyclonic couplet resembling that often observed to straddle the hook echo and recently associated with arching vortex lines.", "The process of tornadogenesis by roll-up of a vortex sheet undergoing stretching by the mesocyclone updraft occurs in the numerical simulation described in Section 3 and is illustrated in Figure REF .", "The time series for the maximum surface vorticity within the developing tornado can be found in Figure REF .", "Note how the vortex sheet vortices intensify as they stretch and approach the developing tornado.", "As the vortex sheet rolls up, the vortices transfer energy to the developing tornado vortex, increasing its maximum vorticity from $\\sim 0.1$ s$^{-1}$ to $\\sim 0.7$ s$^{-1}$ over the 30-minute period shown.", "In Figure REF , taken from [108], maximum gate-to-gate shear in an observed tornado exhibits marked oscillations superimposed on an upward trend.", "Given that the tornado did not display multiple-vortex behavior, we speculate that the shear oscillations and gradual increase in kinetic energy resulted from absorption of successive vortex sheet vortices, similar to that in the simulation.", "As such vortices are absorbed they would contribute not only energy but also helicity, which would decrease energy dissipation in the tornado (see [3], [69], [70], [110])." ], [ "Vorticity lines computed from numerical simulation", "A supercell thunderstorm simulation was investigated to help confirm the conclusions of [21] regarding the evolution of mesocyclone vorticity lines prior to and proceeding tornadogenesis.", "The supercell was simulated using the compressible mode of the non-hydrostatic Bryan Cloud Model 1 (CM1; [18]).", "The simulation proceeded on a $112.5$ km $\\times $ $112.5$ km $\\times $ $20.0$ km domain with horizontal grid spacing of 75 m and vertical grid spacing increasing from 50 m at the lowest layer to 750 m at the highest layer.", "The large and small time steps were $1/4$ s and $1/16$ s, respectively.", "Typical of idealized storm simulations, a horizontally uniform analytical base state was used (see Figure REF ), terrain, surface fluxes, radiative transfer, and Coriolis acceleration were omitted, and radiative (free slip) lateral (vertical) boundary conditions were imposed.", "Microphysical processes were parametrized using the double-moment [83] scheme.", "The subgrid turbulence scheme was similar to [37].", "The simulated supercell exhibits features commonly observed in real supercells, including a hook echo reflectivity signature with a cyclonic–anticyclonic vorticity couplet (see Figure REF ).", "In order to compute maximum vorticity at different length scales $\\varepsilon $ , the vorticity field valid on the 75-m simulation grid was filtered using the [28] interpolation method with the cutoff radius set to $2\\varepsilon $ (consistent with [21]).", "Vorticity lines were then computed near the low-level mesocyclone $\\sim 500$ m above ground level (AGL) every 5 minutes once a distinct low-level mesocyclone had formed (as discerned from visual inspect of the 75-m vorticity field).", "As in [21], vorticity lines were fit to 300 m $\\le \\varepsilon \\le 9600$ m. Tornadogenesis was considered to occur once the maximum axisymmetric tangential wind velocity, $V_T$ , around the intensifying surface vortex associated with the low-level mesocyclone exceeded 20 m s$^{-1}$ .", "The $V_T$ was retrieved using the vortex detection and characterization technique of [89].", "As in [21], the vorticity lines steepen prior to tornadogenesis (see Figures REF and REF ), consistent with the concentration of vorticity from larger to smaller scales.", "Also as in [21], a power law for vorticity appears to hold for scales exceeding that of the low-level mesocyclone core, but breaks down at smaller scales.", "[21] attributed this to smaller scales being more poorly resolved in his radar dataset.", "A similar effect occurs in our scenario: the effective model resolution of [47] artificially decreases the energy contained at scales approaching the grid spacing.", "In the absence of positive evidence that the vorticity power law indeed extends to tornadic and smaller scales, we can only offer this as a speculative explanation for the flattening of the vorticity lines at sub-mesocyclone scales.", "As discussed later, however, Cai's vorticity power law hypothesis (including its validity at tornadic scales) finds support in heuristic considerations of Kelvin–Helmholtz instability in vortex sheets." ], [ "Self-Similarity ", "In this section we give possible ways mesocyclones and tornadoes might acquire self-similarity and so give rise to the hypothesized vorticity and velocity power laws discussed above.", "Self-similarity can manifest itself in several ways in atmospheric flows.", "One such manifestation is scale-invariance of some characteristic of the flow, which may be demonstrated by the existence of a power law for the characteristic.", "Examples include the scenarios discussed above, where a power law for vorticity/pseudovorticity ([21]) or velocity ([107], [106]) are hypothesized.", "Another manifestation is geometric self-similarity: features having similar shape occur at different scales.", "For example, the left image in Figure REF , taken from [27], illustrates a hierarchy of known vortex scales in tornadic supercells.", "This figure is strikingly similar to images in [4] and the Smale–Williams attractor cross-section in [55], shown as the right image in Figure REF .", "Videos of recent large tornadoes show subvortices of subvortices within tornadoes (see [38], [16]).", "Though these sub-subvortices are transient and short lived, their existence suggests that near the surface the tornadic flow is approximating a vertically periodic flow similar to the Smale–Williams attractor.", "Recently, [58] produced a dynamical system with a Smale–Williams attractor.", "The system was forced by periodic pulses.", "We will revisit the idea of periodic pulses in the conclusions.", "Geometric self-similarity is occasionally seen in high-resolution numerical simulations of tornadic supercells ([85], [1]) and also in Doppler radar and reflectivity observations ([14], [85]); see Figure REF for an example.", "High-quality video recordings of some recent tornadoes depict mini-suction vortices (subvortices of suction vortices), confirming the smallest scale of the hierarchy in Figure REF .", "In a related work, [20] revisit the “swirling vortex” model of [94] and investigate solutions to the Navier–Stokes and Euler equation $v= r^b$ , where $b$ is not necessarily equal to $-1$ .", "The streamlines of the modeled vortices exhibit self-similarity, i.e., both the power law and the geometric manifestations of self-similarity are addressed in this work.", "[23] in his study of turbulent flows found quantities with fractal dimensions.", "In numerical experiments he found the fractal dimensions of the axes of vortices he studied to be related to the “temperature” of the vortex.", "“Hot” negative-temperature vortices had a smooth axis, while at temperatures of positive or negative infinity (Kolmogorov cascade) the vortex had a fractal axis (and cross-section).", "We hypothesize that high-energy vortices entering the tornado acquire fractal axes upon being stretched and kinked up (transition from negative-temperature vortices to infinite-temperature vortices).", "One would expect a mixture of fractal dimensions for these axes in the turbulent region surrounding the solid body tornado core.", "In their study of the effect of rotation and helicity on self-similarity, [90] comment that “when comparing numerical simulations, it was found that two runs at similar Rossby number and at similar times (albeit at different Reynolds number) display self-similar behavior or decreased intermittency depending on whether the flow had helicity or not.” That tornadoes form in helical environments may largely account for the degree of self-similarity that is often observed in them (e.g., the presence of suction vortices), and suggests self-similarity may extend to smaller scales than currently known.", "We propose that such self-similarity can arise within persistent vortex sheets along the rear flank downdraft gust fronts of tornadic supercells.", "In the proposed scenario, a sequence of vortex roll-ups occurs, with each new generation of vortices forming from previous-generation vortices wrapping around each other, ultimately resulting in vortices with roughly fractal cross-sections (geometric self-similarity).", "We now give a heuristic argument to support Cai's power law for the vertical vorticity of strong tornadoes, i.e., $\\zeta =\\mathcal {O}(\\epsilon ^{\\sim (-1.6)})$ .", "From Kelvin's circulation theorem, the product of vorticity $\\zeta $ and the cross-sectional area of a vortex tube is constant for Eulerian barotropic flows.", "Hence, $\\zeta =C/A$ , where $A$ is the cross-sectional area of the vortex.", "A numerical study of Kelvin–Helmholtz instability in [5] identifies a relationship between the thickness of the vortex sheet, $h$ , and the cross-sectional area of the vortices, $A$ .", "The result is that $A$ scales like $\\mathcal {O}(h^{1.55})$ as $h\\rightarrow 0$ .", "To the degree that a tornado is formed from vortex sheet vortices, and that the hypothesized vorticity power law (and self-similarity) extends to tornadic and smaller scales, $\\zeta $ would then scale as $\\mathcal {O}(h^{-1.55})$ .", "If the vortices stretch within the updraft, their cross-sections decrease and their vorticity increases, causing the slope of the vorticity line to decrease.", "Under this scenario, $-1.55$ would provide an upper bound for the vorticity line slope of a tornado forming by vortex sheet roll-up.", "That tornadoes may derive much of their energy from vortex sheet vortices is made plausible by the fact that energy can cascade from smaller to larger scales in two-dimensional flows ([23]).", "Vortex sheet vortices and tornadoes (at least near the surface) are approximately two-dimensional." ], [ "Vortex gases", "In this section we give an overview of a vortex gas theory for two dimensions and a vortex gas model for three dimensions.", "We introduce a notion of entropy and temperature that is different from the usual notions of entropy and temperature of gases of molecules.", "We use the theory and models described here to address the question of tornadogenesis and maintenance.", "The interaction of large numbers of vortices in two- and three-dimensional space has been studied by modeling the vortices as part of a vortex gas.", "This theory has its origins in the 19th century in the works of [54] and [97].", "The theory is the analogue of the classical statistical mechanics of gases, which attempts to explain the macroscopic behavior of gases by using the statistics of modeled microscopic behavior of molecules.", "In the vortex gas case the molecules are replaced by vortices.", "These could also be arching vortex lines (tubes).", "Just as in the case of gases, our development includes specialized entropy and temperature.", "It is important to note that one does not need an inordinately large number of vortices to use this theory.", "[87] first suggested the notion of temperature for vortex gases and formulated a two-dimensional theory.", "For a discussion of these ideas see [22], [23].", "We follow the development of [26], [23], [84], being guided by the work and energy balance analysis of [69] and finding some analogues in turbulence theory that one could apply to the development of rotation in tornadoes.", "In classical thermodynamics a body in contact with a heat bath will heat up until it achieves equilibrium with its environment.", "By analogy, a vortex will heat up if it interacts with “hotter” vortices.", "As this process repeats itself many times, the vortex eventually achieves quasi-equilibrium with its environment.", "The frequency and intensity of these the \"hotter\" vortices vortices will determine the intensity of the tornado.", "These “hot” vortices will have negative temperature (in the sense of vortex gas theory) and will increase the energy of a developing vortex.", "In the case where the vortex is a tornado or pre-tornado, this gives us a mechanism for understanding some aspects of tornadogenesis and maintenance.", "A critique of the notion of negative temperature in the two-dimensional vortex gas theory has been given by [48] and [80].", "One does not need a large number of vortices to use the approximations.", "The modeling of vortices in three dimensions has been carried out by [24] using the Ising model.", "It uses this simplified approach to the vortex gas to study the relationship of stretching and temperature of the vortex and other quantities associated to a vortex gas.", "In this approach the vortices appear as either horizontal or vertical segments joining adjacent points in a three-dimensional lattice.", "The lattice is formed from the points in $\\mathbb {Z}^3$ , the three-dimensional space with integer coordinates.", "As time advances, the vortex configuration is allowed to change without allowing the vortices to self-intersect.", "The future configurations of the vortices are then studied using a Monte Carlo Markov chain algorithm.", "The integer lattice could be replaced by a lattice with smaller grid spacing for finer resolution and a closer vortex approximation.", "Chorin uses ensembles to approximate the behavior of a vortex, similar to the ensembles of models used in weather modeling.", "Motivated by the ideas of Chorin, [46] developed a stochastic theory for vortex filaments in three dimensions which included fractal cross sections.", "This theory was not restricted to vortices on a lattice, but more general vortices in three dimensions.", "Based on the notion of capacity, cross sections of the vortices had to be fractal for the vortices to have finite energy.", "The capacity of the vortex is related to its energy and as the fractal dimension of the cross section of the vortex increases, the capacity increases as well.", "We do not claim tornadic vortices are exactly modeled by an Ising model.", "However the model can give insight into the qualitative behavior of vortices.", "While Chorin's Ising model may seem unrelated to the case of tornadic vortices, the ideas of Flandoli and Gubinelli support ideas in this paper and Chorin's experiments give qualitative support to the paper.", "In the last part of this section we take into account the cross-sections of vortices and combine the results of [107], [106], [21] with an argument of [23] and use it to obtain information about the possible source of increase in vorticity at tornado scales." ], [ "The two-dimensional vortex gas theory", "The Euler equation for incompressible fluid flow is $\\frac{D{\\bf V}}{Dt}=\\frac{\\partial {\\bf V}}{\\partial t}+ ({\\bf V}\\cdot \\nabla ){\\bf V}=-\\nabla p+{\\bf f},$ where $\\bf V$ is the velocity, $p$ is the pressure, and $\\bf f$ an external body force.", "To obtain the equation for vorticity, ${\\omega }=\\operatorname{curl}{\\bf V}=\\nabla \\times {\\bf V}$ , we take the curl of the above equation and obtain $\\frac{D\\omega }{Dt}=(\\omega \\cdot \\nabla ){\\bf V}.$ It is possible to extract the $z$ (vertical) component of vorticity, $\\zeta $ , from the above equation to obtain ([56]) $\\frac{\\partial \\zeta }{\\partial t}=-{{\\bf V}\\cdot \\nabla \\zeta }+\\zeta \\,\\frac{\\partial w}{\\partial z}+{\\omega }_H\\cdot {\\nabla _Hw},$ where $w$ is the vertical component of the velocity, $\\omega _H$ is the horizontal component of the vorticity, and $\\nabla _H$ is the horizontal gradient.", "The three terms on the right-hand side of the above equation represent the change in vertical vorticity due to the advection, stretching and tilting of vorticity, respectively.", "Vortices that form in strongly sheared environments have a two-dimensional structure before they are stretched.", "Recently, [86] has shown that a significant amount of the perturbation energy in tornadoes is due to stretching by the updraft near the surface.", "To model the flow behavior of an intense vortex at the surface, we assume the flow is essentially two-dimensional, although this does not fully capture three-dimensional behavior of the vortex.", "Under the influence of strong rotation, turbulent flow becomes anisotropic and the flow tends to become two-dimensional but never quite reaches that state ([90]).", "Comparing the tracks left by suction spots in tornadoes (Figure REF ) and plots of interacting two-dimensional vortices (Figure REF ) suggests that there is a connection between the two.", "The tracks suggest that the vortices behave like two-dimensional vortices, then dissipate due to stretching.", "This phenomenon can be observed in videos of intense tornadoes.", "The existence of such vortices and their behavior may be related to the vortices studied in papers by [43], [45], [44], [109], [63].", "[45] show that vortices in contact with ground that undergo stretching and an appropriate addition of swirl at the surface can become supercritical.", "The behavior of the vortices is related to the swirl ratio.", "The radius of the vortex is related to the thickness of the boundary layer and one thinks of the vortex erupting form the surface as an extension of the boundary layer.", "If the stretching is dominant then the vortex will stretch and eventually dissipate.", "[45] show if the swirl is dominant the supercritical vortex will narrow as the swirl increases and will become shorter.", "As this occurs the vortex also intensifies until a critical threshold is reached and the vortex undergoes a vortex breakdown.", "The intensification of the vortex, as swirl is added, is consistent with an increase in energy, while the decrease in length and radius is consistent with a decrease in entropy.", "This suggests that the vortices have negative inverse temperature in the vortex gas sense, $dS/dE<0$ , that is, when energy is added to the vortex, its entropy (randomness of the vortex) decreases.", "If stretching dominates, the supercritical vortex would decrease in temperature and lead to its dissipation.", "If the swirl is dominant, then the vortex reaches a maximum temperature consistent with ideas of Joyce and Montgomery ([23]).", "The vortex gas theory does not say what will happen with an increase in energy beyond this value, however based on the analogy above, vortex breakdown appears to be the outcome.", "We proceed to develop the two-dimensional theory to model the two-dimensional behavior.", "We then develop the three-dimensional model to fully understand dissipation.", "The two-dimensional vorticity equation for incompressible fluid flow is $\\frac{D\\omega }{Dt}=0.$ Writing the velocity in the component form, ${\\bf V}=(u,v)$ , the incompressibility condition, $\\operatorname{div}{\\bf V}=0$ , together with the assumption that the underlying domain is simply connected (has no holes) implies that there exists a stream function, $\\psi (x,y)$ , such that $u=\\frac{\\partial \\psi }{\\partial y},\\qquad v=-\\frac{\\partial \\psi }{\\partial x}$ and $-\\Delta \\psi =\\zeta .$ Assume that vorticity is concentrated at discrete points ${\\bf x}_i=(x_i,y_i)$ for $i=1,\\dots ,n$ , each with circulation $\\Gamma _i$ , so that $\\zeta ({\\bf x})=\\sum _{i=1}^n \\Gamma _i \\delta ({\\bf x-x}_i),$ where $\\delta $ denotes the Dirac delta function and ${\\bf x}=(x,y)$ .", "The solution to (REF ) is given by $\\psi ({\\bf x})=-\\sum _{i=1}^n\\frac{\\Gamma _i}{2\\pi } \\log {\\Vert {\\bf x-x}_i\\Vert },$ and, using (REF ), the velocity field induced by the $j$ th vortex is, $V_j({\\bf x})=\\frac{\\Gamma _j}{2\\pi r^2}\\left(y_j-y,-(x_j-x)\\right),$ where $r=\\Vert {\\bf x-x}_j\\Vert $ .", "If one assumes that each of the vortices moves under the influence of the combined velocity field of the remaining vortices, then $\\frac{d{\\bf x}_i}{dt}=\\sum _{\\begin{array}{c}j=1\\\\j\\ne i\\end{array}}^nV_j({\\bf x}_i)=\\sum _{\\begin{array}{c}j\\ne i\\end{array}}^nV_j({\\bf x}_i),$ or, in the component form, $\\frac{dx_i}{dt}&=\\frac{1}{2\\pi }\\sum _{j\\ne i}\\frac{\\Gamma _j({y_j-y_i})}{r_{ij}^2}\\\\\\frac{dy_i}{dt}&=-\\frac{1}{2\\pi }\\sum _{j\\ne i}\\frac{\\Gamma _j({x_j-x_i})}{r_{ij}^2},$ where $r_{ij}=\\Vert {\\bf x}_i-{\\bf x}_j\\Vert $ .", "These equations form a Hamiltonian system that has rigorous connections with the Euler equation ([26], [75]).", "The corresponding Hamiltonian is $H=-\\frac{1}{4\\pi }\\sum _{\\begin{array}{c}1\\le i,j\\le n\\\\i\\ne j\\end{array}}\\Gamma _i\\Gamma _j\\log {\\Vert {\\bf x}_i-{\\bf x}_j\\Vert }.$ It is easy to check that the Hamiltonian is conserved, i.e., $\\frac{dH}{dt}=0,$ which, in particular, implies that if all the circulations are of the same sign, then the vortices cannot merge in finite time.", "Other conserved quantities are the total vorticity, $\\Gamma $ , the center of vorticity, $\\bf M$ , and the moment of inertia, $I$ , given by $\\Gamma =\\sum \\Gamma _i,\\qquad {\\bf M}=\\frac{\\sum \\Gamma _i\\bf {x}_i}{\\Gamma },\\qquad I=\\sum \\Gamma _i\\Vert {\\bf x}_i-{\\bf M}\\Vert ^2,$ where all the sums are for $i=1,\\dots ,n$ .", "Using this representation, one can model the behavior of vortex configurations in the plane ([75], [84], [71], [25]).", "For example, a pair of vortices of equal circulations will move about the midpoint of the segment joining them.", "For a line of vortices of equal circulations, the vortices stay in a line.", "If one has a half-line of vortices located at integer points on the $x$ -axis, the vortex half-line rolls up into a spiral.", "We next give a brief exposition of a two-dimensional theory of vortex gases by proceeding in analogy with the development of the Boltzmann distribution in the theory of statistical mechanics in three dimensions.", "The particles are replaced by vortices and the assumptions on the distribution of vortices in a region in two-dimensional space is used to define a distribution in the corresponding phase space.", "The entropy of the distribution is defined.", "The discussion that follows is general and can be used in the context of both two- and three-dimensional flows.", "Consider a vortex system, with $k$ possible energy and moments of inertia levels $(E_j,I_j)$ , $j=1,\\ldots ,k$ .", "Let $0\\le p_j\\le 1$ represent the probabilities that the system is in the energy state $E_j$ with the moment of inertia $I_j$ .", "We assume that the average energy of the ensemble, $\\langle E\\rangle $ , and the average moment of inertia, $\\langle I\\rangle $ , are fixed.", "Then $\\sum _{j=1}^k p_j=1,$ $\\sum _{j=1}^k p_j E_j=\\langle E\\rangle ,$ $\\sum _{j=1}^k p_j I_j=\\langle I\\rangle .$ We define the entropy of the ensemble corresponding to the macrostate $(\\langle E\\rangle ,\\langle I\\rangle )$ , up to an additive constant, as $S=-\\sum _{j=1}^k p_j\\log {p_j}.$ To maximize the entropy, we consider the Lagrangian $\\begin{split}& L(p_1,\\ldots ,p_k)=-\\sum _{j=1}^k p_j\\log {p_j}-\\alpha \\left(\\sum _{j=1}^k p_j-1\\right)\\\\&\\quad -\\beta \\left(\\sum _{j=1}^k p_j E_j-\\langle E\\rangle \\right)-\\gamma \\left(\\sum _{j=1}^k p_j I_j-\\langle I\\rangle \\right),\\end{split}$ where $\\alpha $ , $\\beta $ and $\\gamma $ are Lagrange multipliers.", "Differentiating (REF ) with respect to each of the $p_j$ and setting these partial derivatives equal to zero, we obtain $-\\log {p_j}-1-\\alpha -\\beta E_j-\\gamma I_j=0,\\qquad j=1,\\dots ,k,$ while the derivatives with respect to the Lagrange multipliers return the constraints (REF )–(REF ).", "This results in $p_j=e^{-1-\\alpha -\\beta E_j-\\gamma I_j},\\qquad j=1,\\dots ,k.$ From (REF ) and (REF ) we now have $e^{1+\\alpha }=\\sum _{j=1}^k e^{-\\beta E_j-\\gamma I_j}\\equiv Z,$ where $Z$ is called a partition function.", "It follows that $p_j=\\frac{e^{-\\beta E_j-\\gamma I_j}}{Z},$ and from (REF ) and (REF ) we have $\\langle E\\rangle =\\sum _{j=1}^k E_j \\frac{e^{-\\beta E_j-\\gamma I_j}}{Z},\\quad \\langle I\\rangle =\\sum _{j=1}^k I_j \\frac{e^{-\\beta E_j-\\gamma I_j}}{Z}.$ Consequently, using the definition of the partition function (REF ), we obtain $\\begin{split}-\\frac{\\partial \\log {Z}}{\\partial \\beta }&=\\frac{\\sum _{j=1}^k E_j e^{-\\beta E_j-\\gamma I_j}}{Z}=\\langle E\\rangle ,\\\\-\\frac{\\partial \\log {Z}}{\\partial \\gamma }&=\\frac{\\sum _{j=1}^k I_j e^{-\\beta E_j-\\gamma I_j}}{Z}=\\langle I\\rangle .\\end{split}$ The expression (REF ) for the entropy can be written as $S=\\sum _{j=1}^k p_j(\\beta E_j+\\gamma I_j+\\log {Z})=\\beta \\langle E\\rangle +\\gamma \\langle I\\rangle +\\log {Z},$ and differentiating it with respect to $\\langle E\\rangle $ and using (REF ) gives $\\begin{split}\\frac{\\partial S}{\\partial \\langle E\\rangle }&=\\frac{\\partial \\beta }{\\partial \\langle E\\rangle }\\langle E\\rangle +\\beta +\\frac{\\partial \\gamma }{\\partial \\langle E\\rangle }\\langle I\\rangle \\\\&\\quad +\\frac{\\partial \\log {Z}}{\\partial \\beta }\\frac{\\partial \\beta }{\\partial \\langle E\\rangle }+\\frac{\\partial \\log {Z}}{\\partial \\gamma }\\frac{\\partial \\gamma }{\\partial \\langle E\\rangle }\\\\&=\\beta \\equiv \\frac{1}{T},\\end{split}$ where $T$ is called the temperature associated with the vortex configuration, and $\\beta $ is the corresponding inverse temperature, or “coldness” according to [52].", "From now on we will use the term “temperature” in this particular sense.", "By a process similar to (REF ), we can also obtain $\\dfrac{\\partial S}{\\partial \\langle I\\rangle }=\\gamma $ , etc.", "in case of more constraints." ], [ "Three-dimensional vortex gas model", "A model of three-dimensional vortex gases is much more difficult and has been developed only in special cases.", "Given a vorticity field ${\\omega }({\\bf x})$ in $\\mathbb {R}^3$ , the system ${\\omega }=\\nabla \\times {\\bf V},\\qquad \\nabla \\cdot {\\bf V}=0$ can be solved for the velocity field $\\bf V(x)$ under the assumption that ${\\omega }({\\bf x})$ decays sufficiently fast as $\\Vert {\\bf x}\\Vert \\rightarrow \\infty $ ([73]).", "For the velocity field we get ${\\bf V(x)}=-\\frac{1}{4\\pi }\\int \\frac{({\\bf x-x^{\\prime }})\\times {\\omega }({\\bf x^{\\prime }})}{\\Vert {\\bf x-x^{\\prime }}\\Vert ^3}\\,d\\bf x^{\\prime },$ and the kinetic energy can be written as $E=\\frac{1}{8\\pi }\\iint \\frac{{\\omega }({\\bf x})\\cdot {\\omega }({\\bf x^{\\prime }})}{\\Vert {\\bf x-x^{\\prime }}\\Vert }\\,d{\\bf x^{\\prime }}\\,d{\\bf x}.$" ], [ "Chorin's Monte Carlo Setup", "Vortex gases in three dimensions have been modeled in [24] and [23] assuming that vortices were made up of vertical and horizontal line segments connecting adjacent points on an integer lattice in the three-dimensional space, $\\mathbb {Z}^3$ .", "The vortices were supported on an oriented, self-avoiding random walk on the lattice.", "In the model, the expression (REF ) for the kinetic energy of one vortex becomes $E=\\frac{1}{8\\pi }\\sum _i\\sum _{j\\ne i}\\frac{{\\omega _i}\\cdot {\\omega _j}}{\\Vert i-j\\Vert }+\\frac{1}{8\\pi }\\sum _ iE_{ii},$ where $i$ and $j$ are the three-dimensional coordinates of the locations of the centers of the vortex segments making up the vortex, $E_{ii}$ is the “self-energy” term of the $i$ th segment that is a constant and is, therefore, neglected, $\\Vert i-j\\Vert $ is the distance between the $i$ th and $j$ th segment, and $\\omega _i$ is the vorticity of the $i$ th segment.", "The velocity field was carefully defined using a cut-off function to prevent singularities and keep the terms in the sum defined.", "Similar ideas can be also used in the two-dimensional case.", "To describe the state of such a vortex from a statistical viewpoint, [23] defines the probability of a vortex with energy $E$ and inverse temperature $\\beta $ via $P(E)=\\dfrac{e^{-{\\beta E}}}{Z}$ , where $Z$ is a partition function.", "This probability is then maximized with respect to permissible configurations of the vortex for various fixed values of $\\beta $ (positive, negative, and zero) using a Metropolis rejection algorithm.", "Note that from (REF ), as $\\beta $ decreases from $+\\infty $ to 0, and then to $-\\infty $ , the temperature $T=1/\\beta $ increases from 0 through positive values to $+\\infty =-\\infty $ (denoted by $\\infty $ in what follows), and then increases through negative values to $-0$ .", "To estimate the probability of various configurations Chorin considered ensembles of identical models and allowed them to evolve using the Metropolis algorithm.", "Then Chorin estimated probabilities similar to the way one forecasts the weather using ensembles of weather models.", "This does not take a large number of vortices.", "The main information gained from the model is about qualitative behavior of the vortex, we are not suggesting that vortices are on a lattice.", "However we think that this model provides useful qualitative information about the energy transfer from the outcome of the rather rapid dissipation that a vortex undergoes as it is stretched.", "Chorin studied vortices of various lengths, and concluded that a stretched negative-temperature ($\\beta <0$ ) vortex must fold assuming energy is conserved.", "In addition, as vortices with negative temperatures stretch, their temperature decreases to $T=\\infty $ ($\\beta =0$ ), and they become fractal.", "Note that as temperature crosses from negative to positive through $T=\\infty $ , the vortices tangle into three-dimensional configurations (see [24], [23]).", "It is also possible to model a fluid made up of several sparsely distributed vortices, but not without a significant increase in the computational complexity ([75]).", "The experimental probabilities of the various configurations of the vortices can be found by performing Markov Chain Monte Carlo simulations (see [24])." ], [ "Entropy and Temperature", "A physical system is said to have negative temperature if when energy is added to the system the entropy of the system decreases, that is the system becomes less random.", "We can give an example of such a system in fluid dynamics.", "The supercritical vortex below a breakdown bubble in a vortex undergoing vortex breakdown is an example of such a system.", "Two parameters have been used to study such vortices in a Ward chamber: the volumetric flow of the updraft and the swirl or angular momentum added to the flow.", "It has been shown that when the ratio of the swirl to the volumetric flow of the updraft is in a certain range the flow configuration takes on certain structure.", "This resembles a side-view of a champagne glass: the stem of the glass would correspond to the supercritical (what we will show to be the negative temperature vortex) vortex and the breakdown bubble corresponds to the part of the glass above.", "Theoretical studies by [6], [19], and [45] and experimental studies by [103], and later [27], resulted in the identification of relationships between the radius of the supercritical vortex and the amount of swirl added to the flow, as well as relationships between the angular momentum added to the flow and the azimuth velocity and the vertical component of the velocity in the supercritical part of the vortex.", "Experimentally the following has been found: Holding the volumetric flow upward in the vortex chamber constant, as angular momentum is increased, the radius of the supercritical vortex decreases, as does the length of the supercritical vortex, while the vertical component of the velocity and the azimuthal component of the velocity increase.", "This suggests that when the angular momentum is increased, the vortex is becoming less random as its volume decreases (entropy is decreasing) and the energy of the vortex is increasing as the velocity increases.", "Hence the ratio of the change in entropy to the change in energy is negative and the temperature of the vortex is negative.", "From the point of view of statistical mechanics such vortices transfer momentum to the larger scale flow when they dissipate.", "This can happen in a couple of different ways.", "One way is by stretching (until the vortex kinks up and dissipates) and the other is by adding swirl until vortex breakdown occurs.", "Video of tornados appears to show both possibilities occurring.", "Thus, negative temperature vortices form in vortex chambers when swirl and stretching are in a certain range.", "As in the paper by [45] who argue that tornados can undergo the above conditions have similar behavior to vortices within vortex chambers, we believe that under similar conditions supercritical vortices in nature can also have negative temperatures.", "The critical fact here is that the vortices must be in contact with the ground for the effect we are describing to occur i.e.", "for the negative temperature vortex to form.", "The effects of the boundary layer are critical in that as the swirl increases the thickness of the boundary layer decreases, and the radius of the supercritical vortex (thought of as an extension of the boundary layer) also decreases as well as the length of the vortex.", "On the other hand, as the swirl increases the vertical velocity in the core and azimuthal velocity increase suggesting that the energy of the vortex increases.", "We emphasize here that the swirl ratio must be in a certain range for the vortex to be supercritical, for a swirl ratio outside of this range the vortex behavior is different, and the temperature would not necessarily be negative.", "We can think of the entropy of a collection of vortices (vortex gas) as a sum of its configurational entropy and a structural entropy corresponding to stretching, kinking, and collapse of each vortex.", "The structural entropy is described in the preceding paragraph.", "Vortices moving into a region where there is a convergent updraft and an appropriate amount of low-level swirl will become more spatially organized (i.e., the configurational entropy decreases; see Figure REF ).", "We now argue that the vertical axis of vortices with negative temperature are straight and those with infinite temperature are fractal.", "Let us assume that $T<0$ , so that also $\\beta <0$ .", "Since the probability of a configuration of a vortex with energy $E$ is $P(E)=e^{-\\beta E}/Z$ , the most likely configurations are those with a large positive energy $E$ .", "For a vortex with a small cross-section and nearly constant cross-sectional vorticity, the energy (REF ) will be maximized if the dot products, ${\\omega }({\\bf x})\\cdot {\\omega }({\\bf x^{\\prime }})$ , are as large as possible, hence the vortex should be straight.", "These comments can be generalized to a configuration with more than one vortex in which case they would be nearly parallel.", "Therefore, the highest-temperature (hottest) vortices (negative temperatures near $-0$ ) are the straightest.", "On the other hand, as $\\beta =1/T$ approaches $-0$ (i.e., $T$ approaches $-\\infty $ ), the probability distribution becomes uniform.", "In that case, all the orientations are equally likely, and the vortex configuration can be fractal.", "Hence, as negative-temperature vortices are stretched, they cool down and kink up as they dissipate.", "Next we show, following [23], that if $T_1$ is the temperature of a vortex with mean energy $\\langle E_1\\rangle $ , and $T_2$ is the temperature of a vortex with mean energy $\\langle E_2\\rangle $ , and the vortex systems are combined, then, assuming conservation of energy, three cases are possible: If $T_2>T_1>0$ , then $\\dfrac{d\\langle E_2\\rangle }{dt}<0 $ and $\\dfrac{d\\langle E_1\\rangle }{dt}>0$ ; If $T_1>0>T_2$ , then $\\dfrac{d\\langle E_2\\rangle }{dt}<0 $ and $\\dfrac{d\\langle E_1\\rangle }{dt}>0$ ; If $0>T_2>T_1$ , then $\\dfrac{d\\langle E_2\\rangle }{dt}<0 $ and $\\dfrac{d\\langle E_1\\rangle }{dt}>0$ .", "Hence, for a vortex with a negative temperature, the closer the temperature is to 0, the “hotter” the vortex is.", "If two vortices interact as a combined system, in which the hotter vortex will lose energy to the lower-temperature vortex, the system moves to an equilibrium state.", "To show the three claims above, consider two vortices, one with energy $\\langle E_1\\rangle $ and a developing tornado with energy $\\langle E_2\\rangle $ .", "We may regard this as two disjoint vortex systems, each separately in equilibrium.", "Assume the vortex with energy $\\langle E_1\\rangle $ moves into a developing tornado with energy $\\langle E_2\\rangle $ and that their probability densities are independent.", "Then the energy, $\\langle E\\rangle $ , and the entropy, $\\langle S\\rangle $ , of the combined vortex system are $\\langle E \\rangle =\\langle E_1\\rangle +\\langle E_2\\rangle \\qquad \\text{and}\\qquad S=S_1+S_2.$ As the combined vortex system adjusts to equilibrium, the time rate of change of the total entropy satisfies $\\frac{dS}{dt}=\\frac{dS_1}{dt}+\\frac{dS_2}{dt}=\\frac{dS_1}{d\\langle E_1\\rangle }\\frac{d\\langle E_1\\rangle }{dt}+\\frac{dS_2}{d\\langle E_2\\rangle }\\frac{d\\langle E_2\\rangle }{dt}>0.$ Conservation of energy implies that $\\frac{d\\langle E\\rangle }{dt}=\\frac{d\\langle E_1\\rangle }{dt}+\\frac{d\\langle E_2\\rangle }{dt}=0.$ Hence, $\\frac{dS}{dt}=\\left[\\frac{dS_1}{d\\langle E_1\\rangle }-\\frac{dS_2}{d\\langle E_2\\rangle }\\right]\\frac{d\\langle E_1\\rangle }{dt}=\\left[\\frac{1}{T_1}-\\frac{1}{T_2}\\right]\\frac{d\\langle E_1\\rangle }{dt}>0,$ and the three cases i–iii above follow.", "Therefore, negative temperatures are “warmer” than positive temperatures, and negative temperatures that are closer to zero are warmer than negative temperatures that are farther from zero.", "Hence, as hot, negative-temperature vortices move into the developing tornado, they stretch and cool down, and the ambient vortex heats up.", "As this process repeats itself many times, the tornado vortex achieves a quasi-equilibrium with the environment.", "Generalizing Chorin's argument used in the two-dimensional model in [23], we consider a three-dimensional system in equilibrium, partition the system into boxes, and assume that the vortices are sparsely located, one per box, and nearly parallel and vertical.", "Let $E_{tot}=\\frac{1}{2}\\sum _im_iU_i^2+\\sum _i\\langle E_i\\rangle $ , where $m_i$ is the mass of the $i$ th box in the partition, and $U_i$ and $\\langle E_i\\rangle $ are the velocity and potential energy of the vortex in the $i$ th box, respectively.", "If $T_i<0$ , then $dS_i/d\\langle E_i\\rangle <0$ , and hence, as the entropy increases, the energy is transferred from $\\langle E_i\\rangle $ to $U_i$ .", "This explains the increase in the vorticity of the tornado as hot (negative-temperature) vortices enter the tornado.", "The vortices entering the tornado are stretched and begin to cool down and they kink up (In this situation the stretching dominates the swirl and the swirl ratio decreases.).", "As this happens, the entropy increases ($\\Delta S>0$ ) and energy of the vortex decreases ($\\Delta \\langle E\\rangle <0$ ) as it is transferred to the larger-scale flow, increasing the kinetic energy.", "This is the opposite of the situation considered above, where $\\Delta S<0$ and $\\Delta \\langle E\\rangle >0$ , but $\\beta =dS/d\\langle E\\rangle <0$ .", "We can illustrate a train of vortices in a vortex sheet entering the tornado and transferring the energy to the large scale by replacing the stretching term in the vertical vorticity evolution equation (REF ) by a train of delta functions in the form $\\sum _{i} \\Gamma _i \\delta _i = \\sum _{i} \\Gamma _i \\delta (t - i T)$ which entails $\\frac{\\partial \\zeta }{\\partial t}=-{{\\bf V}\\cdot \\nabla \\zeta }+\\sum _{i} \\Gamma _i \\delta (t - i T)+{\\omega }_H\\cdot {\\nabla _Hw},$ where $T$ is the time interval between successive vortices entering the tornado.", "The response of the vertical vorticity to this forcing is a step function with jumps of the size $\\Gamma _i$ .", "Hence the rate at which the vortices enter the ambient tornado vortex and their strength, and the rate at which the vorticity is removed from the ambient tornado vortex determine the eventual strength of the tornado." ], [ "Energy Spectrum and Power Laws of Cai and Wurman", "[23] gives two possible power laws of dissipation of energy with scale: $\\langle E\\rangle (k)\\sim k^{-5/3}$ and $\\langle E\\rangle (k)\\sim k^{-2}$ , where $k$ is the wave number.", "The former law is derived by a scaling argument due to Kolmogorov and is also supported by Chorin's filament model using results from a Monte Carlo simulation.", "The latter law is derived as an alternative and is based on a possible form of the energy cascade.", "Chorin also claims that the latter one is a “better candidate for the mean field result.” Chorin's filament model can be applied to analyze a vortex tube in a sparse, homogeneous suspension of tubes.", "Consider a narrow and straight enough vortex tube $\\mathcal {T}$ that can be uniquely described by its center line, $\\mathcal {C}$ , parametrized by $s$ , and cross-sections through $\\mathcal {C}(s)$ , denoted by $\\mathcal {S}(s)$ , orthogonal to the center line and such that $\\mathcal {S}(s_1)$ and $\\mathcal {S}(s_2)$ do not intersect for $s_1\\ne s_2$ .", "Given a point ${\\bf x}$ in the vortex tube and $r>0$ , we define the ball $B_r({\\bf x})=\\lbrace {\\bf x^{\\prime }}\\colon |{\\bf x^{\\prime }}-{\\bf x}|<r\\rbrace $ .", "We take $r$ small enough so that $B_r({\\bf x})$ contains no points that belong to other vortex tubes in the suspension.", "We denote by $\\Sigma (s)$ the part of the cross-section $\\mathcal {S}(s)$ inside $B_r({\\bf x})$ and by $\\mathcal {C}_r$ the part of the center line of the vortex tube for which $\\Sigma (s)$ is non-empty, i.e., $\\Sigma (s)=\\mathcal {S}(s)\\cap B_r({\\bf x})\\quad \\text{and}\\quad \\mathcal {C}_r=\\left\\lbrace \\mathcal {C}(s):\\ \\Sigma (s)\\ne \\emptyset \\right\\rbrace .$ In order to compute the energy spectrum, Chorin defines, for $r>0$ , the vorticity correlation integral $S_r=\\left\\langle \\int _{\\mathcal {T}\\cap B_r({\\bf x})}\\omega ({\\bf x})\\cdot \\omega ({\\bf x^{\\prime }})\\,d\\mathcal {H}_\\mathcal {T}\\right\\rangle ,$ where $d\\mathcal {H}_\\mathcal {T}$ denotes the appropriate Hausdorff measure, related to the set capacity on $\\mathcal {T}$ , and the average is taken over the ensemble of all possible configurations.", "Then, using disintegration of measure (see [93]), we have $S_r=\\left\\langle \\omega ({\\bf x })\\cdot \\int _{\\mathcal {C}_r}\\,ds\\int _{\\Sigma (s)}\\omega ({\\bf x^{\\prime }})\\,d\\mathcal {H}_\\Sigma \\right\\rangle .$ If vorticity is roughly uniform throughout the cross-section $\\Sigma (s)$ so that $\\omega ({\\bf x^{\\prime }})\\approx \\omega (s)$ , and if $|\\Sigma |(s)=\\mathcal {H}_\\Sigma (\\Sigma (s))$ denotes the Hausdorff measure of $\\Sigma (s)$ , then $S_r\\approx \\left\\langle \\omega ({\\bf x})\\cdot \\int _{\\mathcal {C}_r}|\\Sigma |(s)\\omega (s)\\,ds\\right\\rangle .$ Assuming that the Hausdorff dimension of $\\Sigma (s)$ remains constant throughout $\\mathcal {T}\\cap B_r({\\bf x})$ and denoting it by $D_{\\Sigma }$ , we obtain $S_r=\\mathcal {O}(r^{D_{\\Sigma }+1}).$ To obtain the vorticity spectrum, $Z(k)$ , we integrate the Fourier transform of $S_r$ over a sphere of radius $k=|{\\bf k}|$ ([24], [23]).", "This gives $Z(k)=\\mathcal {O}(k^{-D_{\\Sigma }+1})$ , and, consequently, the energy spectrum satisfies $\\langle E\\rangle (k)=Z(k)/k^2=\\mathcal {O}(k^{-D_{\\Sigma }-1}).$ Let $D_c$ be the dimension of the center line $\\mathcal {C}$ of the vortex, and $D$ the dimension of the support of the vorticity in the vortex filament $\\mathcal {T}$ .", "We argued above that for vortices with negative temperature the center line of the vortex has Hausdorff dimension one, i.e., $D_c=1$ , and therefore (see [88]) $D=D_c+D_\\Sigma .$ Supported by the results of [51], [45], [43], [65], [105], [109], [63], [64], we suggest that tornadoes have fractal cross-sections and subvortices moving into the larger tornadic flow have negative temperature.", "Graphical evidence is provided in Figures REF , REF , and REF .", "This suggestion is further supported by the existence of subvortices within subvortices (see Figure REF ) as shown in recent videos ([38], [16]).", "Assume now that the cross-section of the tornado is fractal for $T<0$ with cross-sectional dimension $D_\\Sigma $ .", "For $ 1<D_\\Sigma \\le 2 $ the energy (see (REF )) satisfies $\\langle E\\rangle (k)=\\mathcal {O}(k^{-\\gamma })$ with $2<\\gamma \\le 3$ .", "It follows that for large scales (small $k$ ), an increase in $\\gamma $ in the range from 2 to 3 corresponds to an increase of the energy $\\langle E\\rangle (k)$ .", "This is consistent with the idea that vortices from a vortex sheet feeding a larger tornado cause an increase in the dimension of the tornado's cross-sectional area and an increase in its energy.", "Thus, an increase in the dimension of the cross-sectional area may be associated with tornadogenesis or strengthening of an existing tornado.", "Note that this is analogous to Cai's power law, in which a decrease in the (negative) exponent in the power law (REF ) leads to tornadogenesis or a stronger tornado.", "To gain further insight into the processes that might contribute to tornadogenesis, we consider the effects of helicity.", "First we consider its effects in the simplified case of homogeneous isotropic turbulence (see [69], [70]).", "Writing the Navier–Stokes equation in energy form and taking the Fourier transform, [23] obtains $\\partial _t\\langle E\\rangle (k)+2k^2R^{-1}\\langle E\\rangle (k)=Q(k),$ where $Q(k)$ comes from the nonlinear term in the Navier–Stokes equation.", "This term represents the transfer of energy between wave numbers and has been studied extensively for the case of homogeneous turbulence (see, e.g., [102]).", "Certain terms have been singled out and studied in relation to inverse energy cascades.", "These interactions involve three wave numbers.", "It was found that the net effect of the so-called nonlocal interactions is to transfer energy from intermediate scales to larger scales.", "These interactions occur between modes with helicity of the same sign.", "[3] and [70] discuss flows without helicity and dissipation of energy as shown in Figure REF , and flows with helicity and low dissipation of energy as shown in Figure REF .", "The images represent the results of two numerical experiments and show that isotropic turbulence with helicity inhibits the dissipation of energy at large scales.", "Under the effects of strong rotation, the flow has the tendency to become anisotropic ([90]).", "Studies have shown that the presence of helicity and low-energy dissipation are unlinked unless the helicity is continuously supplied and/or generated at the energy-containing scales; this is associated with inhomogeneity in the mean field ([69], [70], [110]).", "Such an inhomogeneity would be supplied by surface friction and the rear flank and forward flank downdrafts and/or their gust fronts.", "The increase in the exponents for the power laws for the vorticity as tornadogenesis approaches is consistent with the helicity production of the flow at the energy-containing scales.", "Idealized cross-sections of vortices with high helicity exhibit self-similarity (compare Figure REF and the image in [4]).", "Thus the tornado develops at a “focus” scale at which much energy and helicity is transferred among other scales.", "This is consistent with the mean power law and helicity contribution to the flow ([90])." ], [ "Conclusions and Future Work", "The two-dimensional point vortex theory presented in Section can be applied to pairs of cyclonically rotating vortices in the half-plane.", "The paths of the pairs of interacting vortices (see Figure REF ) form the same type of pattern as the tracks of overlapping suction vortices moving through fields (see Figure REF ) as observed by [51] and others ([53]) from the air.", "As noted earlier (cf.", "Section ), under the influence of strong rotation, turbulent flow becomes anisotropic with the flow tending toward, but never fully becoming, a two-dimensional flow ([90]).", "This suggests that the two-dimensional flow may be an attractor in this situation.", "We suggest that the increase in the power in Wurman's and Cai's power laws is an indication of the anisotropy of the flow.", "The paths in Figure REF can be modeled using the two-dimensional vortex gas theory with translating and interacting point vortices (Figure REF ).", "In the first pair of paths in Figure REF , one can identify the two overlapping paths of cyclonically rotating vortices in the left half-plane and two overlapping paths of counter-rotating, anticyclonic, mirror vortices in the right half-plane as the other ends of the arching vortex lines.", "The image on the right in Figure REF shows the paths left by two translating interacting vortices.", "Both the first pair of paths and the last path in Figure REF look similar to paths in Figure REF .", "The tracks left by the suction vortices (Figure REF ) suggest the vortices either originate in the larger tornado vortex or move into the tornado vortex, after which they intensify due to stretching, make a partial revolution, and then dissipate.", "In either case they appear to occur as a result of the roll-up of a vortex sheet as observed in numerical simulations (see, e.g., [92] and Figure REF ).", "Some of these intense vortex paths are from one to two yards in diameter (Figure REF ), and some are as narrow as 30 cm in diameter ([51]).", "These vortices, which may be extremely intense, have been observed to pull cornstalks out of clay soil by their roots.", "We suggest that these vortices have negative temperature and transfer energy to the larger vortex as they dissipate.", "This could manifest itself as a vortex breakdown.", "Indeed, video footage of subvortices in tornadoes suggests that they behave as negative-temperature vortices would.", "For example, in some instances the vortices' appearance is associated with stretching and with strong convergence.", "This may indicate that the vortex intensification is related to a decrease in entropy and an increase in energy.", "Numerical simulations of intense vortices in [45], [43], [65], [109], [63], [64] show that the maximum wind speeds in intense narrow vortices undergoing vortex breakdown may exceed the speed of sound in the vertical direction.", "Mobile Doppler radar observations of the vortices apparently originating in the vortex core suggest that they make a partial revolution about the ambient tornado vortex and then dissipate.", "This might indicate a Hopf bifurcation of the horizontal component of the flow field, creating a two-cell flow structure: downdraft core and a surrounding updraft.", "[105] has found evidence supporting both the creation of vortices inside the tornado and outside the tornado resulting in flows potentially enhancing the tornado's strength.", "He has also noted that these secondary vortices have a different velocity and shear profile than the parent tornadoes.", "The tornadoes appear to have a two-cell structure and a modified Rankine combined profile, with mean azimuthal velocity, $v$ , depending linearly on radius inside the tornado core, $v=Cr$ , and a power-law outside the core, $v=Cr^{-b}$ , where $0.5\\le b\\le 0.6$ .", "In an extreme case, [105] found a power law $v=Cr^{-1}$ on one side of an intense tornado.", "Such a power law would be consistent with no vorticity outside the tornado core on that side.", "On the other side of the tornado, the power law was found to be $v=Cr^{-b}$ with $0.5\\le b\\le 0.6$ , which is consistent with vorticity being advected into the tornado, possibly along a vortex sheet spiraling into it.", "The secondary vortices appear to be single-celled with extreme values of shear and extreme transient updrafts.", "This is consistent with these vortices having negative temperatures in the vortex gas sense ([23]).", "The results of Section , the last part of Section , and Section suggest that the increase in strength of a developing tornado occurs as a result of the increase in vorticity due to an inverse energy cascade from smaller scales.", "This process appears to be related to a negative viscosity phenomenon described by [66], [67] and [11].", "Photographic evidence supporting this is given in the photo by Gene Moore in Figure REF .", "Numerical models have also supported this.", "A possible connection between tornadoes and nearly continuously (periodically) produced vortices (vortex lines (tubes)) is that the vortices stir or pump the tornado and increase the vorticity.", "The frequency with which vortex lines are produced, their strength, and the stretching of the vortices determine the eventual strength of the tornado.", "This can be seen from the point of view of the vortex gas theory (Section ).", "Supercell thunderstorms have a quasi-periodic nature, cycling between destructive and rebuilding phases.", "They typically have a longer life span than generic storms.", "Curiously, most classic supercells have common features which make them distinctive from other storms of the same scale.", "These features include, for example, a wall cloud, a tail cloud, and a flanking line.", "This commonality of features suggests that atmospheric flows that demonstrate themselves as supercells fluctuate near “attractor” flows that share some common structure.", "In the radar reflectivity image of the hook echo region of a supercell thunderstorm shown in Figure REF , the hooks on the boundary of the region represent successive vortices in a vortex sheet.", "Such vortices could provide periodic pulses of energy to the tornado.", "An analogy can thus be drawn with the work of [58], in which periodic pulses introduced into a dynamical system lead to a Smale–Williams attractor.", "As noted in Section , the hierarchy of vortices shown in Figure REF is reminiscent of the cross-sections of the Smale–Williams attractor.", "In addition, the boundary in Figure REF exhibits fractal structure, similar to that of the twindragon shown in Figure REF as well as other “dragon” fractals.", "Interestingly, the fractal dimensions of the boundaries of the dragon fractals are in the range of the exponents in the power laws discussed in this work (e.g., $\\sim 1.52$ for the twindragon or $\\sim 1.62$ for the “golden dragon”).", "We recommend further exploration of the relationships among helicity, temperature (in the vortex gas sense), self-similarity, and the power laws proposed by Cai and Wurman.", "The resulting benefits of our understanding of helical atmospheric vortices could improve operational tornadic prediction.", "To the degree that the vorticity power law extends from observable to tornadic scales, it may also be possible to improve tornado detection and perhaps even estimate maximum tangential winds in tornadoes, as discussed in [21].", "These hypotheses should be tested using real radar observations of tornadic and nontornadic supercells.", "The four authors, Bělík, Dokken, Scholz, and Shvartsman were supported by National Science Foundation grant DMS-0802959.", "Funding for Potvin was provided by the NOAA/Office of Oceanic and Atmospheric Research under NOAA–University of Oklahoma Cooperative Agreement #NA11OAR4320072, U.S. Department of Commerce.", "Funding for Dahl and McGovern was provided by NSF/IIS grant 0746816." ] ]
1403.0197
[ [ "On the non-existence of small positive loops of contactomorphisms on\n overtwisted contact manifolds" ], [ "Abstract We prove that on overtwisted contact manifolds there can be no positive loops of contactomorphisms that are generated by a $C^0$-small Hamiltonian function." ], [ "Introduction", "In 2000 Eliashberg and Polterovich [14] noticed that the natural notion of positive contact isotopies, i.e.", "contact isotopies that move every point in a direction positively trasverse to the contact distribution, induces for certain contact manifolds a partial order on the universal cover of the contactomorphism group.", "Such contact manifolds are called orderable.", "Since the work of Eliashberg and Polterovich orderability has become an important subject in the study of contact topology.", "In particular it has been discovered to be deeply related to the contact non–squeezing phenomenon [13], [16] and, more recently, to the non–degeneracy of a natural bi–invariant metric that is defined on the universal cover of the contactomorphism group [6].", "As Eliashberg and Polterovich explained, orderability of a contact manifold is equivalent to the non–existence of a positive contractible loop of contactomorphisms.", "By now many contact manifolds are known to be orderable and many are known not to be, but it is still not well–understood where the boundary between the orderable and non–orderable world lies.", "In particular it is not known whether there is a relation between overtwistedness and orderability, since not a single overtwisted contact manifold is known to be orderable or not to be.", "In this article we prove the following result.", "Theorem 1 Let $\\big (M,\\xi = \\ker \\alpha \\big )$ be a closed overtwisted contact 3–manifold.", "Then there exists a real positive constant $C(\\alpha )$ such that any positive loop $\\lbrace \\phi _{\\theta }\\rbrace $ of contactomorphisms which is generated by a contact Hamiltonian $H: M \\times \\mathbb {S}^1 \\longrightarrow \\mathbb {R}^+$ satisfies $\\Vert H\\Vert _{\\mathcal {C}^0} \\ge C(\\alpha )\\,.$ In other words, on closed overtwisted contact 3–manifolds there are no positive loops of contactomorphisms that are generated by a $\\mathcal {C}^0$ –small contact Hamiltonian.", "Note that there is no loss of generality in assuming the contact Hamiltonian to be 1-periodic, see Lemma 3.1.A in [14].", "It is important to notice that our result does not imply that overtwisted contact manifolds are orderable, because the contraction of a positive contractible loop of contactomorphisms is not necessarily performed via positive loops.", "For instance, it was even proved in [13] that for the standard tight contact sphere any contraction of a positive contractible loop must be sufficiently negative somewhere.", "Theorem REF states though that there exists a lower bound for a Hamiltonian function that generates a positive loop of contactomorphisms.", "Intuitively, in the presence of an overtwisted disc a positive isotopy returning to the identity requires a minimal amount of energy.", "The specificity of our result is that we deal with $\\mathcal {C}^0$ –small contact Hamiltonians.", "Indeed, let us prove that the non–existence of a positive loop of contactomorphisms that is generated by a $\\mathcal {C}^1$ –small Hamiltonian holds on any contact manifold.", "Consider first the $\\mathcal {C}^2$ –small case.", "If the Hamiltonian $H_{\\theta }: M \\longrightarrow \\mathbb {R}$ is $\\mathcal {C}^2$ –small then the generated loop $\\lbrace \\phi _{\\theta }\\rbrace $ is $\\mathcal {C}^1$ –small and so the contact graphsSee for example [26], [6] for the definition of contact graphs, contact products and more details on arguments similar to the one that follows.", "$\\text{gr}(\\phi _{\\theta })$ are Legendrian sections in a Weinstein neighborhood of the diagonal $\\Delta $ in the contact product $M \\times M \\times \\mathbb {R}$ .", "Since a Weinstein neighborhood is contactomorphic to a neighborhood of the zero section of $J^1(\\Delta )=J^1(M)$ , the graphs $\\text{gr}(\\phi _{\\theta })$ are of the form $\\lbrace j^1f_{\\theta }\\rbrace $ for a family of smooth functions $f_{\\theta }$ on $M$ .", "Because of the Hamilton–Jacobi equation (see [1]), positivity of the loop $\\lbrace \\phi _{\\theta }\\rbrace $ implies that the family $f_{\\theta }$ is strictly increasing, yielding a contradiction.", "If the Hamiltonian function is only $\\mathcal {C}^1$ –small, and thus the loop $\\lbrace \\phi _{\\theta }\\rbrace $ is $\\mathcal {C}^0$ –small, then the graphs $\\text{gr}(\\phi _{\\theta })$ are still contained in a Weinstein neighborhood of the diagonal in the contact product but they are not necessarily sections anymore, and so they cannot be written as 1-jet of functions.", "However it follows from Chekanov theorem [3], [2] that they have generating functions quadratic at infinity and so an argument similar to the one above (or the results in [5], [4]) allows to conclude also in this case.", "As far as we know, Theorem REF is the first result in the literature that shows the non–existence of a positive loop in the case when the Hamiltonian is $\\mathcal {C}^0$ –small.", "Our proof strongly uses overtwistedness in several points, and does not give an intuition of whether or not the result should also be true for tight contact manifolds.", "However it seems plausible to us that this might be the case.", "Although Theorem REF only applies to overtwisted contact 3–manifolds, a higher–dimensional analogue can also be stated.", "The careful reader can try to generalize the result to non–fillable contact manifolds containing a PS–structure [23], [25], a GPS–structure [24] or a blob [22] with the appropriate hypotheses on the Chern class of the contact distributions.", "The precise statement is not part of this article due to its technicality and to the fact that no new geometric ideas are required for the argument.", "As a consequence of Theorem REF , we can bound from below not only the supremum norm of the Hamiltonian of a positive loop but also its $L^1$ -norm, in the following sense.", "Corollary 2 Let $\\big (M,\\xi =\\ker \\alpha \\big )$ be a closed overtwisted contact 3–manifold.", "Then there exists a real positive constant $C(\\alpha )$ such that any positive loop of contactomorphisms $\\lbrace \\phi _\\theta \\rbrace $ which is generated by a contact Hamiltonian $H_{\\theta }$ , $\\theta \\in \\mathbb {S}^1$ , satisfies $\\int _0^1 \\Vert H_{\\theta }\\Vert _{\\mathcal {C}^0}\\,d\\theta \\ge C(\\alpha )\\,.$ Corollary REF can be deduced from Theorem REF as follows.", "Suppose that there is a positive loop $\\lbrace \\phi _\\theta \\rbrace $ which is generated by a contact Hamiltonian $H_{\\theta }$ that satisfies $\\int _0^1 \\Vert H_{\\theta }\\Vert _{\\mathcal {C}^0}\\,d\\theta \\le C(\\alpha )\\,.$ Define a reparametrization $\\beta : [0,1] \\rightarrow [0,1]$ of the time–coordinate by requiring $\\dot{\\beta }(\\theta ) = \\frac{\\Vert H_{\\theta }\\Vert _{\\mathcal {C}^0}}{\\int _0^1 \\Vert H_{\\theta }\\Vert _{\\mathcal {C}^0}\\,d\\theta }$ and write $\\phi _{\\theta } = \\psi _{\\beta (\\theta )}$ .", "Then $H_{\\theta } = \\dot{\\beta }(\\theta ) G_{\\beta (\\theta )}$ where $G_{\\beta (\\theta )}$ is the Hamiltonian of the reparametrized loop $\\psi _{\\beta (\\theta )}$ .", "For all $\\theta \\in \\mathbb {S}^1$ we then have $\\max _{x} G_{\\beta (\\theta )}(x) = \\frac{\\int _0^1 \\Vert H_{\\theta }\\Vert _{\\mathcal {C}^0}\\,d\\theta }{\\Vert H_{\\theta }\\Vert _{\\mathcal {C}^0}}\\Vert H_{\\theta }\\Vert _{\\mathcal {C}^0} = \\int _0^1 \\Vert H_{\\theta }\\Vert _{\\mathcal {C}^0}\\,d\\theta \\le C_{\\alpha }$ contradicting Theorem REF .", "The geometric core of the proof of Theorem REF can be shortly described in two parts.", "First, any overtwisted contact manifold $(M,\\xi )$ can be embedded with trivial symplectic normal bundle in an exact symplectically fillable contact 5–manifold $(X,\\xi _X)$ .", "Second, the existence of a small positive loop of contactomorphisms on $(M,\\xi )$ implies the existence of a PS–structure on $(X,\\xi _X)$ .", "This yields a contradiction, according to the main result of [23].", "The construction of a PS–structure on $X$ is based on techniques similar to those used by Niederkrüger and the second author [24] to study the size of tubular neighborhoods of contact submanifolds.", "The paper is organized as follows.", "Section recalls basic definitions and facts about overtwisted contact manifolds.", "In Section we explain how to construct a PS–structure in the total space of the contact fibration $M \\times \\mathbb {D}^2$ , where $M$ is an overtwisted contact 3–manifold, starting from a small positive loop of contactomorphisms of $M$ .", "Theorem REF is proved in Section assuming an embedding result that will be proved in Section .", "Acknowledgments.", "This work started five years ago as an attempt to prove that overtwisted contact manifolds are orderable.", "Many visits at that time of the third author to Madrid were supported by the CAMGSD of the IST Lisbon and by the CSIC-IST joint project 2007PT0014.", "More recently, we wish to thank the participants of the AIM Workshop Contact topology in higher dimensions in May 2012 for discussions that encouraged us to pursue and write down this partial result.", "The article was written during the stay of the third author at the UMI–CNRS of the CRM Montréal, and she would like to thank Laurent Habsieger, François Lalonde and Octav Cornea for their support and hospitality.", "The first author is grateful to A. Zamorzaev.", "The present work is part of the authors activities within CAST, a Research Network Program of the European Science Foundation.", "The third author is also supported by the ANR grant COSPIN." ], [ "Preliminaries on overtwisted contact manifolds", "We refer to the book of Geiges [15] for an introduction to Contact Topology, and recall here only the definitions and facts about overtwisted contact manifolds that will be needed in the rest of the article.", "A 3–dimensional contact manifold $(M,\\xi )$ is said to be overtwisted if it contains an overtwisted disc, i.e.", "an embedded 2–disk $\\Delta $ such that the characteristic foliation $T\\Delta \\cap \\xi $ contains a unique singular point in the interior of $\\Delta $ and $\\partial \\Delta $ is the only closed leaf of this foliation.", "A contact manifold is said to be tight if it is not overtwisted.", "As follows from the results of Lutz and Martinet [20], [21], there exists an overtwisted contact structure in any homotopy class of 2–plane fields.", "Moreover, by the classification of overtwisted contact structures achieved by Eliashberg [11], we also know that on a given homotopy class of 2–plane fields there exists exactly one overtwisted contact structure.", "More precisely we have the following result.", "Theorem 3 ([11]) Let $\\xi $ and $\\xi ^{\\prime }$ be overtwisted contact structures on a 3–dimensional manifold $M$ , and suppose that they are homotopic as 2–plane fields.", "Then $\\xi $ and $\\xi ^{\\prime }$ are isotopic contact structures.", "The notion of an overtwisted contact structure does not readily generalize to higher–dimensional contact manifolds.", "The following geometric model was proposed by Niederkrüger [23].", "Definition 4 Let $(M,\\xi )$ be a contact 5–manifold.", "A plastikstufe PS$(\\mathbb {S}^1)$ in $M$ with singular set $\\mathbb {S}^1$ is an embedding of a solid torus $\\iota :2\\times \\mathbb {S}^1\\longrightarrow M$ with the following properties: a.", "The boundary $\\partial 2\\times \\mathbb {S}^1$ is the unique closed leaf of the foliation $\\ker (\\iota ^*\\alpha )$ on $2\\times \\mathbb {S}^1$ .", "b.", "The interior of $2\\times \\mathbb {S}^1$ is foliated by an $\\mathbb {S}^1$ –family of stripes $(0,1)\\times \\mathbb {S}^1$ spanned between $\\mathbb {S}^1\\times \\lbrace 0\\rbrace $ and asymptotically approaching $\\partial 2\\times \\mathbb {S}^1$ on the other side.", "In particular, Property a. implies that the boundary of the solid torus is a Legendrian torus and the core $\\lbrace 0\\rbrace \\times \\mathbb {S}^1$ is transverse to the contact distribution $\\xi $ .", "Figure: An embedded PS–structure in a contact 5–fold.A plastikstufe is also referred to in the literature as a PS–structure.", "By results of Gromov and Eliashberg [17], [12], in dimension 3 the presence of an overtwisted disc obstructs the existence of symplectic fillings.", "The higher–dimensional analogue of this fact is the following theorem by Niederkrüger.", "Theorem 5 ([23]) Let $(M,\\xi )$ be a contact 5–manifold with a PS–structure.", "Then $M$ does not admit an exact symplectic filling.", "As we will explain, the argument used to prove Theorem REF is based on the insertion of a PS–structure in an exact symplectically fillable manifold, thus yielding a contradiction with Theorem REF .", "The techniques that provide such embedding are based on the study of certain contact structures on the manifold $M\\times 2$ .", "This will be explained in the next section." ], [ "PS–structures and contact fibrations", "In the first part of this section we will recall, following Lerman [19] and [25], the notion of a contact fibration and its relation to the group of contactomorphisms via the monodromy diffeomorphism.", "We will then show in Proposition REF how to apply these concepts to construct a PS–structure on the total space of the contact fibration $M \\times \\mathbb {D}^2$ , starting from a sufficiently small positive loop of contactomorphisms of $M$ .", "A smooth fiber bundle $\\pi :X\\longrightarrow B$ is said to be a contact fibration if there exists a hyperplane distribution $\\xi _X = \\ker \\alpha _X$ on $X$ such that its restriction $\\xi = \\ker (\\pi ) \\cap \\xi _X$ defines a contact structure in each fiber.", "In particular $\\big (\\xi ,d \\alpha _X|_{\\ker (\\pi )}\\big )$ is a symplectic subbundle of the –not necessarily symplectic– bundle $\\xi _X$ .", "This data leads to a natural choice of connection.", "Definition 6 Let $\\pi :\\big (X,\\xi _X = \\ker \\alpha _X\\big )\\longrightarrow B$ be a contact fibration.", "Then the distribution $\\xi ^{\\perp d\\alpha _X}\\subset \\xi _X$ is called the contact connection associated to the contact fibration.", "In other words, for a point $p$ of $B$ and a tangent vector $v \\in T_pB$ , the horizontal lift of $v$ at some $\\tilde{p} \\in \\pi ^{-1}(p)$ with respect to the contact connection is the unique vector $\\tilde{v} \\in T_{\\tilde{p}}X$ such that $\\pi _{\\ast } \\tilde{v} = v$ , $\\tilde{v} \\in \\ker (\\xi _X)$ and $\\iota _{\\tilde{v}}d\\alpha _X = 0$ on $\\xi $ .", "Note that the contact connection only depends on $\\xi _X$ , not on the choice of the 1–form $\\alpha _X$ with $\\xi _X = \\ker \\alpha _X$ .", "The parallel transport along a segment joining two points $q,p\\in B$ is defined as in the smooth case, but in the contact framework it is enhanced from a diffeomorphism to a contactomorphism between the fibers of $q$ and $p$ .", "Moreover, the definition of the contact connection implies that the trace by parallel transport of a submanifold that is tangent to the contact structure on the fibers is also tangent to the distribution on the total space.", "A precise statement of these properties is the content of the following proposition.", "Proposition 7 $($[19], [25]$)$ Let $\\pi :\\big (X,\\xi _X=\\ker \\alpha _X\\big )\\longrightarrow B$ be a contact fibration with closed fibers.", "Consider a point $p\\in B$ and an immersed path $\\gamma :[0,1]\\longrightarrow B$ with $\\gamma (0) = p$ .", "Then parallel transport along $\\gamma $ with respect to the contact connection defines a path of diffeomorphisms $\\widetilde{\\gamma }_t:\\pi ^{-1}(p)\\longrightarrow \\pi ^{-1}(\\gamma (t))$ with the following properties: a.", "The diffeomorphisms $\\widetilde{\\gamma }_t$ are contactomorphisms.", "b.", "Let L be an isotropic submanifold of $\\pi ^{-1}(p)$ and consider the map $\\mathfrak {t}:L\\times [0,1]\\longrightarrow X,\\quad (p,t)\\longmapsto \\widetilde{\\gamma }_t(p),$ then $im(\\mathfrak {t})$ is an immersed isotropic submanifold of $(X,\\xi _X)$ .", "It is an embedded isotropic submanifold if $\\gamma $ is an embedded path.", "Note that the closedness condition for the fibers is technical and only used to ensure that the vector fields implicitly appearing in the statement are complete.", "There are instances in which the contactomorphisms generated via parallel transport have a simple description.", "The following example will be used in the proof of our results.", "Let $\\big (M,\\xi =\\ker \\alpha \\big )$ be a contact manifold.", "A time–dependent function $H_{\\theta }$ on $M$ induces a path of contactomorphisms $\\lbrace \\phi _{\\theta }\\rbrace $ , which is defined to be the flow of the time–dependent vector field $X_{\\theta }$ satisfying $\\iota _{X_{\\theta }} \\alpha & = & H_{\\theta }, \\\\\\iota _{X_{\\theta }} d\\alpha &= & -dH_{\\theta } + dH_{\\theta }(R_{\\alpha })\\,\\alpha \\nonumber $ where $R_{\\alpha }$ is the Reeb vector field associated to $\\alpha $ .", "The function $H_{\\theta }$ is called the contact Hamiltonian with respect to the contact form $\\alpha $ of the contact isotopy $\\lbrace \\phi _{\\theta }\\rbrace $ .", "In contrast to the symplectic case, any contact isotopy can be written as the flow of a contact Hamiltonian, see [15].", "Consider the manifold $M \\times 2$ , where 2 denotes the 2–disc with polar coordinates $(r,\\theta )$ .", "Let $H:M\\times 2 \\longrightarrow \\mathbb {R}$ be a function such that $H\\in O(r^2)$ at the origin and $\\partial _r H>0$ .", "Then the 1–form $\\alpha _H=\\alpha + H(p,r,\\theta )d\\theta $ defines a contact structure $\\xi _H$ on the manifold $M\\times 2$ .", "In particular, suppose that $H:M\\times \\mathbb {S}^1\\longrightarrow \\mathbb {R}$ is a positive function.", "Then $\\alpha _H = \\alpha + H(p,\\theta )\\cdot r^2d\\theta $ is a contact form in $M\\times 2$ .", "Lemma 8 Let $\\big (M,\\xi =\\ker \\alpha \\big )$ be a contact manifold, and $H_\\theta :M\\longrightarrow \\mathbb {R}$ an $\\mathbb {S}^1$ –family of positive smooth functions.", "Consider the contact fibration $\\pi :\\big (M\\times 2,\\ker \\alpha _H\\big )\\longrightarrow 2.$ Then parallel transport along $\\gamma (\\theta )=(1,-\\theta )$ is the contact flow of the Hamiltonian $H_\\theta $ .", "The horizontal lift with respect to the contact connection of the vector field $\\partial _\\theta $ at a point $(1,\\theta )$ is of the form $\\widetilde{X}=\\partial _\\theta -X_\\theta ,$ where $X_\\theta $ satisfies the equations $\\iota _{X_{\\theta }} \\alpha = H_{\\theta }$ and $\\iota _{X_{\\theta }} d\\alpha = -dH_{\\theta } + dH_{\\theta }(R_{\\alpha })\\,\\alpha $ .", "Indeed, the lift is unique and $\\widetilde{X}$ satisfies both $\\alpha _H(\\widetilde{X})=0$ and $\\iota _{\\widetilde{X}}d\\alpha _H = 0$ on $\\xi $ .", "The statement then follows from equations (REF ).", "Let us explain how to use Lemma REF to construct a PS–structure in $\\big (M \\times 2(\\delta ),\\ker (\\alpha +r^2d\\theta )\\big )$ , where $2(\\delta )$ denotes the 2–disc of radius $\\delta $ , assuming that there is a sufficiently small positive loop of contactomorphisms in $M$ .", "Proposition 9 Assume that $\\lbrace \\phi _t\\rbrace $ is a positive loop of contactomorphisms of an overtwisted contact manifold $\\big (M,\\xi =\\ker \\alpha \\big )$ which is generated by a contact Hamiltonian $H_{\\theta }$ , $\\theta \\in S^1$ , with $H_{\\theta } < \\delta ^2$ for some $\\delta \\in \\mathbb {R}^+$ .", "Then there is a PS–structure on $\\big (M \\times 2(\\delta ),\\ker (\\alpha +r^2d\\theta )\\big )$ .", "Note first the following general fact.", "Suppose that that $\\pi :(X,\\xi _X) \\longrightarrow \\Sigma $ is a contact fibration over a smooth compact surface $\\Sigma $ , such that the fibers are closed overtwisted contact manifolds.", "Suppose also that there exists an embedded loop $\\gamma : \\mathbb {S}^1 \\longrightarrow \\Sigma $ whose time–1 parallel transport $\\widetilde{\\gamma }_1$ is the identity.", "Then there exists a PS–structure in the pre–image $\\pi ^{-1}(\\gamma (\\mathbb {S}^1))$ .", "Indeed, since the fiber $\\pi ^{-1}(\\gamma (0))$ is overtwisted we can consider an embedded overtwisted disk $\\Delta $ in it and define the map $\\rho : \\Delta \\times \\mathbb {S}^1 & \\longrightarrow & X \\\\(p, \\theta ) &\\longmapsto & \\rho (r,\\theta )=\\widetilde{\\gamma }_\\theta (p).$ Then property b. in Proposition REF implies that $im(\\rho )$ is a PS–structure.", "By combining this fact with Lemma REF we see that if $\\lbrace \\phi _{\\theta }\\rbrace $ is a positive loop of contactomorphisms on a contact manifold $\\big (M,\\xi = \\ker \\alpha \\big )$ then there is a PS–structure on $\\big (M\\times \\mathbb {D}^2(1),\\text{ker}(\\alpha +H_{\\theta }r^2d\\theta )\\big )$ where $H_{\\theta }$ is the Hamiltonian function of $\\lbrace \\phi _{\\theta }\\rbrace $ .", "The PS–structure is at the level $\\lbrace r=1\\rbrace $ .", "Note that if $H_\\theta < \\delta ^2$ for some $\\delta \\in \\mathbb {R}^+$ then there exists a strict contact embedding $\\big (M\\times \\mathbb {D}^2(1),\\text{ker}(\\alpha +H_{\\theta }r^2d\\theta )\\big ) \\longrightarrow \\big (M\\times \\mathbb {D}^2(\\delta ),\\text{ker}(\\alpha +r^2d\\theta )\\big )$ given by the map $(p, r, \\theta ) \\mapsto (p, \\sqrt{H_\\theta (p)}r, \\theta )$ .", "A PS–structure in $\\big (M\\times \\mathbb {D}^2(1),\\text{ker}(\\alpha +H_{\\theta }r^2d\\theta )\\big )$ at the level $\\lbrace r=1\\rbrace $ is sent to a PS–structure in $\\big (M\\times \\mathbb {D}^2(\\delta ),\\text{ker}(\\alpha +r^2d\\theta )\\big )$ at the level $\\lbrace r = \\sqrt{H_{\\theta }}\\rbrace $ .", "We have thus obtained the required PS–structure in $\\big (M \\times 2(\\delta ),\\ker (\\alpha +r^2d\\theta )\\big )$ ." ], [ "Proof of Theorem ", "In this section we prove Theorem REF in the case when $c_1(\\xi ) = 0$ .", "As we will explain, the general case also follows from the same argument modulo Proposition REF that will be proved in the last section.", "Let $(M,\\xi )$ be a 3–dimensional overtwisted contact manifold and assume that $\\lbrace \\phi _{\\theta }\\rbrace $ is a positive loop of contactomorphisms, generated by a contact Hamiltonian $H_{\\theta }$ , $\\theta \\in S^1$ .", "We want to show that if $H_{\\theta }$ is small in the $\\mathcal {C}^0$ –norm then the existence of $\\lbrace \\phi _{\\theta }\\rbrace $ gives a contradiction with Theorem REF .", "Recall from the previous section that if $\\lbrace \\phi _t\\rbrace $ is a positive loop of contactomorphisms of $M$ which is generated by a sufficiently small contact Hamiltonian $H_{\\theta }$ ($\\theta \\in S^1$ ) then there is a PS–structure on $\\big (M \\times 2(\\delta ),\\ker (\\alpha +r^2d\\theta )\\big )$ for some small $\\delta \\in \\mathbb {R}^+$ .", "Note that the manifold $\\big (M\\times \\mathbb {D}^2(\\delta ),\\text{ker}(\\alpha +r^2d\\theta )\\big )$ is the standard contact neighborhood of a codimension–2 contact submanifold with trivial symplectic normal bundle, see [15].", "The result of the previous section implies thus the following proposition.", "Proposition 10 Let $(X,\\xi _X)$ be a contact 5–manifold and $(M,\\xi )$ a codimension–2 overtwisted contact 3–manifold with trivial symplectic normal bundle.", "Suppose that $\\lbrace \\phi _\\theta \\rbrace $ is a positive loop of contactomorphisms which is generated by a sufficiently small contact Hamiltonian.", "Then there exists a PS–structure in a neighborhood of $M$ in $X$ .", "In the case when $c_1(\\xi ) = 0$ , Theorem REF follows from Proposition REF .", "Indeed any contact manifold $(M,\\xi )$ can be embedded as a contact submanifold into its unit cotangent bundle $\\mathbb {S}T^{\\ast }M$ , for example by the map $e_{\\alpha }: M \\longrightarrow \\mathbb {S}T^{\\ast }M$ defined by $e_{\\alpha }(p) = \\big (p,\\alpha (p)\\big )$ where $\\alpha $ is a contact form for $\\xi $ .", "Note that if $c_1(\\xi ) = 0$ then $e_{\\alpha }(M) \\subset \\mathbb {S}T^{\\ast }M$ has trivial symplectic normal bundle.", "Certainly, the symplectic normal bundle of $e_{\\alpha }(M)$ inside $\\mathbb {S}T^{\\ast }M$ is isomorphic to $\\xi $ , and thus it is trivial if its Euler class $c_1(\\xi )$ vanishes.", "If there was a small positive contact Hamiltonian $H_{\\theta }$ that generates a loop of contactomorphisms then Proposition REF would give a PS–structure inside $\\mathbb {S}T^{\\ast }M$ .", "But the existence of a PS–structure inside $\\mathbb {S}T^{\\ast }M$ is impossible by Theorem REF because $\\mathbb {S}T^{\\ast }M$ is an exact symplectially fillable manifold, a filling being given by $T̥^{\\ast }M$ .", "More precisely, a tubular neighborhood of $e_{\\alpha }(M)$ inside $\\mathbb {S}T^{\\ast }M$ is contactomorphic to $\\big (M\\times \\mathbb {D}^2(\\delta ), \\text{ker}(\\alpha + r^2d\\theta )\\big )$ for some $\\delta > 0$ .", "If the $\\mathcal {C}^0$ –norm of $H_{\\theta }$ is smaller than $\\delta ^2$ then we would obtain a PS–structure inside $\\mathbb {S}T^{\\ast }M$ .", "The square of the maximal size $\\delta $ of a tubular neighborhood $M\\times \\mathbb {D}^2(\\delta )$ of $M$ inside $\\mathbb {S}T^{\\ast }M$ gives in this case the constant $C(\\alpha )$ that appears in the statement of Theorem REF .", "In the general case, i.e.", "when $c_1(\\xi )$ does not necessarily vanish, the proof of Theorem REF follows from the same argument, combined with the following proposition.", "Proposition 11 Every 3–dimensional overtwisted contact manifold can be embedded as a contact submanifold with trivial symplectic normal bundle into an exact symplectically fillable contact 5–manifold.", "The proof of this result will be given in the next section.", "Assuming it, Theorem REF is proved as follows.", "Given an overtwisted contact 3–manifold $(M,\\xi )$ , by Proposition REF it can be embedded with trivial symplectic normal bundle into an exact symplectically fillable contact 5–manifold $(X,\\xi _X)$ .", "If there was a sufficiently small contact Hamiltonian $H_{\\theta }$ generating a positive loop of contactomorphisms then Proposition REF would give a PS–structure in $X$ , contradicting Theorem REF ." ], [ "Contact embeddings with trivial normal bundle", "In this section we will prove Proposition REF , i.e.", "that every overtwisted contact 3–manifold $(M,\\xi )$ can be embedded with trivial symplectic normal bundle into an exact symplectially fillable contact 5–manifold $(X,\\xi _X)$ .", "The idea of the proof is to start with a contact embedding of $(M,\\xi )$ into its unit cotangent bundle $\\mathbb {S}T^{\\ast }M$ and then perform contact surgeries in an appropriate way in order to make the symplectic normal bundle trivial while keeping the symplectic fillability of the resulting 5–manifold.", "As we will see the process will also modify the contact structure on the initial overtwisted 3–manifold $M$ .", "One of the crucial points of the proof will be to make sure that the modified contact structure on the 3–manifold will still be overtwisted and moreover in the same homotopy class as cooriented 2–plane fields as the initial one.", "Then it will be isotopic to it according to Theorem REF .", "We start by briefly recalling the notion of Lutz twist, its effect on the homotopy class of the contact structure and its relation to contact surgery and symplectic cobordism.", "See [15] for more details on these notions.", "Let $K$ be a positive transverse knot in $(M,\\xi )$ .", "A Lutz twist along $K$ is an operation that deforms in a certain way (see [15]) the contact structure in a neighborhood of $K$ .", "The resulting contact structure $\\xi ^K$ on $M$ is always overtwisted.", "The effect of a Lutz twist on the homotopy class of the contact structure can be described as follows.", "Given two 2–plane fields $\\xi _0$ and $\\xi _1$ there are two cohomology classes $d^2(\\xi _0,\\xi _1)\\in H^2(M,\\mathbb {Z})$ and $d^3(\\xi _0,\\xi _1)\\in H^3(M,\\mathbb {Z})$ that measure the obstruction for $\\xi _0$ and $\\xi _1$ to belong to the same homotopy class of plane fields.", "We refer to [15] for details and for a proof of the following results.", "Proposition 12 Let $K\\subset M$ be a positive transverse knot on $\\xi $ .", "Then $d^2(\\xi ,\\xi ^K) = -pd([K])$ .", "Proposition 13 Let $K\\subset M$ be a null–homologous positive transverse knot on $\\xi $ with self–linking number $sl(K)$ .", "Then $d^2(\\xi ,\\xi ^K)=0$ and $d^3(\\xi ,\\xi ^K)=sl(K)$ .", "Following Eliashberg [10] and Weinstein [27], fix a Legendrian knot $L$ on a contact manifold 3–manifold $M$ and fix the relative ($-1$ )–framing with respect to the canonical contact framing associated to the knot.", "If we perform on $M$ a handle attachment along $L$ , then the resulting cobordism has a natural symplectic structure.", "The bottom boundary of this cobordism, i.e.", "the initial contact manifold, is a concave boundary of the symplectic structure.", "The upper boundary is convex and therefore it has an induced contact structure, which is said to be obtained from the initial one by contact $(-1)$ –surgery.", "The inverse operation is called a contact $(+1)$ –surgery.", "As proved by Ding, Geiges and Stipsicz [7], the effect of a Lutz twist on a contact manifold can be described in terms of contact surgery as follows.", "Given a Legendrian knot $L\\subset (M,\\xi )$ , denote by $t(L)$ a positive transverse push–off of $L$ and by $\\sigma (L)$ a Legendrian push–off of $L$ with two added zig–zags.", "Then we have the following result.", "Proposition 14 ([7]) Let $(M,\\xi )$ be a contact 3–manifold and $L$ a Legendrian knot for $\\xi $ .", "The contact structure obtained by a Lutz twist along $t(L)$ is isotopic to the contact structure resulting from a contact $(+1)$ –surgery along $L$ and $\\sigma (L)$ .", "Being the inverse of a ($-1$ )–surgery, the contact ($+1$ )–surgery in a contact 3–fold corresponds to a symplectic 2–handle attachment to the concave boundary of a bounded part of the symplectization, i.e.", "we obtain a symplectic cobordism in which the new boundary is concave.", "Consider the transverse knot $K=t(L)\\subset (M,\\xi ^K)$ and the belt spheres $\\lambda _K,\\lambda ^\\sigma _K\\subset (M,\\xi ^K)$ corresponding to the contact $(+1)$ –surgeries along $L$ and $\\sigma (L)$ in $(M,\\xi )$ described in Proposition REF .", "Then $\\lambda _K$ and $\\lambda ^\\sigma _K$ are two Legendrian knots in $(M,\\xi ^K)$ .", "Since Proposition REF is a local result, both Legendrian knots can be assumed arbitrarily close to $K$ .", "The following observation will be used in our argument.", "Lemma 15 $[K]=[L]=[\\lambda _K]$ .", "By definition $[K]=[t(L)]=[L]$ .", "The equality $[L]=[\\lambda _K]$ follows from the fact that the surgery in Proposition REF is smoothly trivial.", "This implies the statement.", "See Proposition 6.4.5 in [15] for further details.", "A consequence of the description in Proposition REF is the existence of an exact symplectic cobordism realizing a Lutz twist.", "More precisely we have the following result.", "Corollary 16 Let $(M,\\xi )$ be a contact 3–manifold and $K$ a positive transverse knot.", "Then there exists an exact symplectic cobordism $(W,\\omega )$ from $(M,\\xi ^K)$ to $(M,\\xi )$ , which is realized by a 2–handle attachment along the Legendrian link $\\lambda _K\\cup \\lambda ^\\sigma _K$ .", "The convex end of $(W,\\omega )$ is the contact boundary $(M,\\xi )$ , the concave end is $(M,\\xi ^K)$ .", "A Lutz untwist is thus tantamount to an exact symplectic cobordism.", "It is central to note that the convex end of an exact symplectic cobordism is exact symplectically fillable if the concave end is.", "This fact will be crucial in our proof of Proposition REF , because it will ensure that the 5–manifold $X$ into which we will embed $(M,\\xi )$ will still be fillable.", "Indeed, as we will see, $X$ will be obtained by constructing an exact symplectic cobordism between contact 5–manifolds with an exact symplectically fillable concave end.", "This cobordism will restrict to a cobordism between contact 3–manifolds as the one described in Corollary REF .", "In our argument we will also use the following result.", "Lemma 17 Let $(M,\\xi )$ be an overtwisted contact 3–manifold and $\\Delta $ a fixed overtwisted disk.", "Consider a Legendrian link $L$ in $M$ disjoint from $\\Delta $ .", "Then there exists a Legendrian link $\\Lambda $ disjoint from $L\\cup \\Delta $ such that $\\xi $ is isotopic to $\\xi ^{t(L\\cup \\Lambda )}$ .", "Consider a Legendrian link $\\widetilde{L}$ disjoint from $L\\cup \\Delta $ and with homology class $[\\widetilde{L}]=-[L]$ .", "Then Proposition REF implies that $d^2(\\xi ,\\xi ^{t(L\\cup \\widetilde{L})})=0$ .", "Let $K$ be a null–homologous knot contained in a Darboux ball with self–linking number $-d^3(\\xi ,\\xi ^{t(L\\cup \\widetilde{L})})$ .", "Propositions REF and REF imply that $\\Lambda =\\widetilde{L}\\cup K$ satisfies $d^2(\\xi ,\\xi ^{t(L\\cup \\Lambda )})=0$ and $d^3(\\xi ,\\xi ^{t(L\\cup \\Lambda )})=0$ .", "Theorem REF concludes the statement of the Lemma.", "We are now almost ready to state and prove two results, Propositions REF and REF , that will be the two main steps in the proof of Proposition REF .", "Proposition REF will be an adaptation to higher dimensions of Proposition REF .", "We first discuss the smooth model for it.", "Denote by $M_K(\\tau )$ the manifold obtained by surgery along $K\\subset M$ with framing $\\tau $ .", "In case this is a contact surgery along a Legendrian knot, the notation stands for a contact ($-1$ )–surgery.", "The following observation is a strictly differential topological statement.", "Lemma 18 Let $X$ be a smooth 5–manifold and $M$ a codimension–2 submanifold.", "Consider a knot $K$ in $M$ and a framing $\\tau $ of $K$ in $X$ .", "Suppose that $\\tau $ restricts to a framing $\\tau _s$ of $K$ in $M$ .", "Then a surgery on $X$ along $K$ with framing $\\tau $ induces a surgery on $M$ along $K$ with framing $\\tau _s$ .", "The statement can be seen as a consequence of the description of a surgery as a handle attachment.", "The gradient flow used to glue a 6–dimensional 2–handle $H^6\\cong 2\\times 4$ along the attaching sphere $K$ in $X\\times \\lbrace 1\\rbrace \\subset X\\times [0,1]$ restricts to a gradient flow in the submanifold $M\\times \\lbrace 1\\rbrace $ .", "This describes the attachment of a 4–dimensional 2–handle $H^4\\cong 2\\times 2$ along $K$ in $M\\times \\lbrace 1\\rbrace \\subset M\\times [0,1]$ .", "Note that the belt 3–sphere in the handle $H^6$ intersects the surgered submanifold $M_K$ along the belt 1–sphere of the handle $H^4$ .", "Lemma REF provides the smooth model for the symplectic cobordism we shall construct to prove Proposition REF .", "Proposition REF concerns contact 3–manifolds and a 4–dimensional symplectic cobordism.", "In view of Lemma REF we can adapt Proposition REF to the context of a codimension–2 contact submanifold in a contact 5–manifold.", "The result is as follows.", "Proposition 19 Let $(X,\\xi _X)$ be a contact 5–manifold and $(M,\\xi )$ a codimension–2 overtwisted contact submanifold.", "Consider a transverse knot $K$ in $M$ such that $c_1(\\nu _M)=pd([\\lambda _K])$ and denote $\\lambda =\\lambda _K\\cup \\lambda ^\\sigma _K$ .", "Then there exists a framing $\\tau $ of $\\lambda $ in $(X,\\xi _X)$ restricting to the Legendrian framing $\\tau _s$ of $\\lambda $ in $(M,\\xi )$ such that $M_\\lambda (\\tau _s)$ is contactomorphic to $M$ with a Lutz untwist along $K$ , and the symplectic normal bundle of $M_\\lambda (\\tau _s)$ in $X_\\lambda (\\tau )$ is trivial.", "The contact ($-1$ )–surgery that occurs on the contact 3–manifold $(M,\\xi )$ is the procedure described in Proposition REF and Corollary REF .", "It suffices to explain the choice of framing $\\tau $ for the link $\\lambda $ in $X$ .", "The Legendrian framing $\\tau _s$ for $\\lambda $ in $(M,\\xi )$ is extended to a framing $\\tau $ for $\\lambda $ in $X$ .", "This extension is obtained as follows.", "Consider a section $\\mathfrak {s}:M\\longrightarrow \\nu _M$ transverse to the 0–section and such that $\\lambda _K=Z(\\mathfrak {s}),\\mbox{ where }Z(\\mathfrak {s})=\\lbrace p\\in M:\\mathfrak {s}(p)=0\\rbrace .$ This section exists since $c_1(\\nu _M)=pd([\\lambda _K])$ .", "It is used to define the extension of the Legendrian framing $\\tau _s$ to $\\tau $ .", "Let us discuss in detail this and the effect of the surgery.", "It can be considered in two stages.", "First, surgery along the Legendrian link $\\lambda _K$ .", "The required framing along $\\lambda _K$ is defined to be $\\tau =(\\tau _s,\\mathfrak {s}_*\\tau _s)$ .", "Thus $\\tau $ is constructed using the differential $\\mathfrak {s}_*$ of the section $\\mathfrak {s}$ .", "The section $\\mathfrak {s}$ cannot be used since it vanishes along $\\lambda _K$ .", "Consider polar coordinates $(r,w_1,w_2)\\in 4\\subset 2$ with $(w_1,w_2)\\in \\mathbb {S}^3$ .", "The framing $\\tau $ provides a diffeomorphism $f_\\tau :\\mathbb {S}^1\\times 2\\times 2\\longrightarrow {\\mathcal {U}}(\\lambda _K)\\subset X,\\quad (\\theta ;r,w_1,w_2)\\longmapsto f_\\tau (\\theta ;r,w_1,w_2)$ and we can suppose that $f_\\tau (\\mathbb {S}^1\\times 2\\times \\lbrace 0\\rbrace )={\\mathcal {U}}(\\lambda _K)\\cap M$ , for a neighborhood ${\\mathcal {U}}(\\lambda _K)$ of $\\lambda _K\\subset X$ .", "The differential $\\mathfrak {s}_*$ identifies the pull–back $f_\\tau ^*(\\nu _M)$ of the normal bundle with the trivial bundle $\\mathbb {S}^1\\times 2\\times \\lbrace 0\\rbrace $ over a neighborhood of $\\lambda _K\\subset M$ .", "We can also suppose that the section $\\mathfrak {s}$ in these local coordinates is $((f^\\tau )^*\\mathfrak {s})(\\theta ;r,w_1)=rw_1$ .", "The function $rw_1$ is well–defined although the coordinate $w_1$ is not well–defined at $r=0$ .", "The surgery substitutes the core $\\lambda _K\\cong \\mathbb {S}^1\\times \\lbrace 0\\rbrace \\times \\lbrace 0\\rbrace \\subset \\mathbb {S}^1\\times 2\\times 2$ with coordinates $(\\theta ;r,w_1,w_2)$ by $\\lbrace 0\\rbrace \\times \\mathbb {S}^3\\subset 2\\times \\mathbb {S}^3$ with coordinates $(r,\\theta ;w_1,w_2)$ along the common boundary $\\mathbb {S}^1\\times \\mathbb {S}^3=\\lbrace (\\theta ;w_1,w_2)\\rbrace $ .", "The section $((f^\\tau )^*\\mathfrak {s})(\\theta ;r,w_1)=rw_1$ can be substituted by a section of the form $g:2\\times \\mathbb {S}^3\\longrightarrow \\quad (r,\\theta ;w_1,w_2)\\longmapsto g(r,\\theta ;w_1,w_2)=\\rho (r)w_1$ where $\\rho :\\mathbb {R}\\longrightarrow \\mathbb {R}^+$ is a positive smooth function.", "In particular it is non–vanishing and provides a trivialization of the normal bundle of the surgered submanifold $M_{\\lambda _K}(\\tau _s)$ in the surgered manifold $X_{\\lambda _K}(\\tau )$ .", "Second, surgery along the Legendrian link $\\lambda ^\\sigma _K$ .", "The manifold $M_{\\lambda _K}(\\tau _s)$ has trivial normal bundle in $X_{\\lambda _K}(\\tau )$ .", "Thus there exists a global framing $\\tau _{\\nu }$ of this normal bundle.", "Denote the restriction of this global framing $\\tau _{\\nu }$ to the Legendrian knot $\\lambda ^\\sigma _K$ by $\\tau _{\\nu }|_{\\lambda ^\\sigma _K}$ .", "Then the framing $\\lbrace \\tau _s, \\tau _{\\nu }|_{\\lambda ^\\sigma _K}\\rbrace $ is a framing of the normal bundle of $\\lambda ^\\sigma _K$ inside $X_{\\lambda _K}(\\tau )$ .", "Thus, once the surgery along $\\lambda ^\\sigma _K$ is performed with the framing $\\lbrace \\tau _s, \\tau |_{\\lambda ^\\sigma _K} \\rbrace $ , the resulting normal bundle is still trivial.", "Hence the normal bundle of $M_\\lambda (\\tau _s)$ in $X_\\lambda (\\tau )$ is trivial.", "A minor modification of the argument for Proposition REF yields the following result.", "Proposition 20 Let $(X,\\xi _X)$ be a contact 5–manifold and $(M,\\xi )$ a codimension–2 overtwisted contact submanifold with trivial normal bundle.", "Consider a transverse knot $K$ in $M$ and denote $\\lambda =\\lambda _K\\cup \\lambda ^\\sigma _K$ .", "Then there exists a framing $\\tau $ of $\\lambda $ in $(X,\\xi _X)$ restricting to the Legendrian framing $\\tau _s$ of $\\lambda $ in $(M,\\xi )$ such that $M_\\lambda (\\tau _s)$ is contactomorphic to $M$ with a Lutz untwist along $K$ and the symplectic normal bundle of $M_\\lambda (\\tau _s)$ in $X_\\lambda (\\tau )$ is trivial.", "In this case there is no need to use $\\mathfrak {s}_*$ since the section $\\mathfrak {s}$ can be chosen to be non–vanishing.", "Thus we choose the framing described in the second part of the surgery in Proposition REF .", "Id est, the framing induced by $\\mathfrak {s}$ .", "The surgery along $\\lambda $ with this framing preserves the triviality of the normal bundle.", "We are now ready to prove Proposition REF .", "Let $(M,\\xi )$ be an overtwisted contact 3–manifold.", "We want to show that there is a contact embedding with trivial symplectic normal bundle of $(M,\\xi )$ into an exact symplectially fillable contact 5–manifold.", "Fix an overtwisted disc $\\Delta $ in $(M,\\xi )$ and take a Legendrian link $L$ in $(M,\\xi )$ which is disjoint from $\\Delta $ and such that $pd([L]) = c_1(\\xi )$ .", "By Lemma REF we know that there exists a Legendrian link $\\Lambda $ in $(M,\\xi )$ disjoint from $L$ and $\\Delta $ and such that $\\xi $ is isotopic to $\\overline{\\xi } := \\xi ^{t(L \\cup \\Lambda )}$ .", "Consider a contact embedding $(M,\\overline{\\xi }) \\longrightarrow \\mathbb {S}T^{\\ast }M$ defined by some contact form $\\overline{\\alpha }$ for $\\overline{\\xi }$ .", "The symplectic normal bundle of this embedding is isomorphic to $\\overline{\\xi }$ and hence to $\\xi $ .", "Note that $L$ and $\\Lambda $ are still Legendrian in $(M,\\overline{\\xi })$ .", "Consider the transverse push–offs, with respect to $\\overline{\\xi }$ , $K=t(L)$ and $\\kappa = t(\\Lambda )$ .", "First, we apply Proposition REF to $(M,\\overline{\\xi })$ inside $(X,\\xi _X) := \\mathbb {S}T^{\\ast }M$ , and $K=t(L)$ .", "We can apply it because the symplectic normal bundle of $(M,\\overline{\\xi })$ inside $(X,\\xi _X)$ is $\\overline{\\xi }$ and we know that $c_1(\\overline{\\xi }) = c_1(\\xi ) = pd([L]) = pd([K]) = pd([\\lambda _K]).$ The last equality holds by Lemma REF .", "After applying Proposition REF we get contact structures $\\overline{\\xi }^{\\prime }$ on $M$ and $\\xi _X^{\\prime }$ on $X$ such that $(M,\\overline{\\xi }^{\\prime })$ embeds into $(X,\\xi _X^{\\prime })$ with trivial symplectic normal bundle, and $\\overline{\\xi }^{\\prime }$ , $\\xi _X^{\\prime }$ are obtained from $\\overline{\\xi }$ , $\\xi _X$ by performing a Lutz untwist along $K$ .", "Second, consider $\\kappa = t(\\Lambda )$ as a transverse link in $(M,\\overline{\\xi }^{\\prime })$ and apply Proposition REF to $(M,\\overline{\\xi }^{\\prime })$ inside $(X,\\xi _X^{\\prime })$ and $\\kappa = t(\\Lambda )$ .", "We obtain contact structures $\\overline{\\xi }^{\\prime \\prime }$ on $M$ and $\\xi _X^{\\prime \\prime }$ on $X$ such that $(M,\\overline{\\xi }^{\\prime \\prime })$ embeds into $(X,\\xi _X^{\\prime \\prime })$ with trivial symplectic normal bundle and $\\overline{\\xi }^{\\prime \\prime }$ , $\\xi _X^{\\prime \\prime }$ are obtained from $\\overline{\\xi }^{\\prime }$ , $\\xi _X^{\\prime }$ by performing a Lutz untwist along $\\kappa $ .", "Recall that $\\overline{\\xi }$ was obtained from $\\xi $ by performing a Lutz twist along $K \\cup \\kappa $ .", "We have thus that $\\overline{\\xi }^{\\prime \\prime }$ and $\\xi $ are in the same homotopy class.", "Since the overtwisted disk has not been affected by the previous operations, Theorem REF implies that the two contact structures $\\overline{\\xi }^{\\prime \\prime }$ and $\\xi $ are actually isomorphic.", "We have thus obtained an embedding $(M,\\xi ) \\longrightarrow (X,\\xi _X^{\\prime \\prime })$ with trivial symplectic normal bundle.", "Since $(X,\\xi _X)$ is exact symplectically fillable and $\\xi _X^{\\prime \\prime }$ is obtained from $\\xi _X$ by two Lutz untwists, it follows from Corollary REF and the discussion after it that $(X,\\xi _X^{\\prime \\prime })$ is still exact symplectically fillable.", "This finishes the proof of Proposition REF and hence the proof of Theorem REF in the general case." ] ]
1403.0350
[ [ "To the possibility of a mobility edge in a random 1D lattice with\n long-range correlated disorder" ], [ "Abstract Tight-binding 1D random system with long-range correlations is studied numerically using the localisation criterium, which represents the number of sites, covered by the wave function.", "At low degrees of disorder the signs of a mobility edge, predicted in \\cite{Izr}, were found.", "The possibility of exact mobility edge in the system under consideration is discussed." ], [ "Introduction", "Absence of a mobility edge in 1D disordered systems with short-range correlations is well established [1].", "Despite the fact that this statement is proved only for short-range correlated systems, the occurrence of a mobility edge in 1D random system with some particular long-range correlated disorder, reported in [2], seems to be very interesting and unexpected.", "The authors of [2] have managed to construct the correlated random potential for which the inverse localisation length (calculated theoretically for small degrees of disorder) was found to be zero for the central half-band of the energy spectrum and to be a non-zero constant for the remaining half-band.", "From the author's of this note point of view, the infinitiness of localisation length is not an unambiguous criterium of localisation.", "For example, the zero-state in 1D chain with off-diagonal disorder has infinite localisation length [3] but, nevertheless, zero-state was found to be localised in sense of the basic Anderson criterium of localisation [4].", "Therefore the verification of the result obtained in [2] by means of Anderson (or equivalent) criterium of localisation is necessary.", "The main task of this note is to put forward the additional arguments in favor of presence of a mobility edge in correlated 1D random chain, constructed by recipe [2].", "To do this we perform the numerical exploration of eigen vectors of the random matrix of an appropriate disordered Hamiltonian by means of a criterium of localisation, differing from the finiteness of the localisation length." ], [ " 1D long-range correlated random chain: the Hamiltonian.", "The matrix of the Hamiltonian we are going to study has the tight-binding form $H_{rr^{\\prime }}=\\delta _{rr^{\\prime }}\\varepsilon _r+\\delta _{r,r^{\\prime }+1}+\\delta _{r,r^{\\prime }-1},\\hspace{28.45274pt}r,r,=1,2,...,N$ with long-range correlated random potential $\\varepsilon _r$ , generated in accordance with the following algorithm [2]: $\\varepsilon _r=\\sum _{m=-\\infty }^\\infty G(r-m)\\xi _m \\hspace{42.67912pt}G(m)=\\sqrt{2\\over 3}{3\\over 2\\pi m}\\sin \\bigg [{2\\pi m\\over 3}\\bigg ]$ w here $\\xi _m$ are similarly destributed independent random numbers with $\\langle \\xi _m\\rangle =0$ and $\\langle \\xi _m^2\\rangle =d^2/2$ for any integer $m\\in [-\\infty ,+\\infty ]$ .", "The degree of disorder is controlled by $d$ .", "As it was shown in [2], the spectral dependence of the inverse localisation length calculated for the Hamiltonian (REF ) with random potential (REF ) at small degree of disorder $d$ , goes down to zero in the central half-band $E\\in [-1,1]$ and remains constant at $E\\in [\\pm 2,\\pm 1]$ .", "The authors of [2] have interpreted this as the occurrence of a mobility edges at $E=\\pm 1$ .", "Below we verify this statement from the view point of our criterium of localisation, which is equivalent to the Anderson's criterium." ], [ "Number of sites covered by the wave function: the criterium of localisation", "We will start from reminding of a simple criterium of localisation, suggested in [4], which we will use below.", "We will characterise an arbitrary state $\\bf \\Psi $ by a number $N^\\ast $ of sites, covered by an appropriate wave function $\\bf \\Psi $ , calculated as follows.", "As $\\bf \\Psi $ is the eigen vector of a random Hamiltonian matrix in site representation, $\\bf \\Psi $ represents a vector-cloumn with componets $\\Psi _1,..,\\Psi _i,..,\\Psi _N$ where $i$ is the site index.", "If $|\\Psi _i|^2 = 0$ at a certain site $i$ , then this site is not covered by a state $\\bf \\Psi $ and is not taken into account while calculating $N^\\ast $ .", "Vice versa, if $|\\Psi _i|^2$ reaches its maximum $|\\Psi |^2_{max}$ at the site $i$ , then this site is covered completely and its contribution while calculating $N^\\ast $ is equal to unit.", "In the general case, the contribution of an arbitrary site $i$ is equal to $|\\Psi _i|^2/|\\Psi |^2_{max} $ and we obtain the following formula for $ N^\\ast $ : $N^\\ast =N^\\ast \\lbrace {\\bf \\Psi \\rbrace }=\\sum _i^N {|\\Psi _i|^2\\over |\\Psi |^2_{max}}={1\\over |\\Psi |^2_{max}}$ We can now introduce energy depending $N^\\ast (E)$ as follows.", "Let us specify some energy interval $dE<<E_{max}-E_{min}$ where $E_{max}-E_{min}$ is a typical range of eigen energies for the random Hamiltonian under consideration.", "Consider all states with energies within the interval $[E,E+dE]$ and denote by $\\sum _{E,E+dE}$ the summation over all these states.", "Now calculate $N^\\ast (E)$ as the following average value over all states with energies within the interval $[E,E+dE]$ : $\\langle N^\\ast (E)\\rangle =\\bigg \\langle \\sum _{E,E+dE} N^\\ast ({\\bf \\Psi })/\\sum _{E,E+dE} 1\\bigg \\rangle $ with $\\langle \\rangle $ standing for the averaging over realisations of disorder.", "For small enough $dE$ the function (REF ) does not depend on $dE$ .", "For the Hamiltonian represented by a random matrix of size $N$ , the function $\\langle N^\\ast (E)\\rangle $ in the spectral range of localised states should not depend on $N$ (if $N$ is large enough for $\\langle N^\\ast (E)\\rangle <N$ in this spectral region).", "In the spectral range of delocalised states (if it is exist), the function $\\langle N^\\ast (E)\\rangle $ must increase as $\\sim N$ .", "This property of $\\langle N^\\ast (E)\\rangle $ one can use as a criterium for selection of localised and delocalised states in numerical experiments." ], [ "Results ", "In our calculations the auxiliary random values $\\xi _m$ were of gauss type with the distribution function in the form $\\rho (\\xi )={1\\over \\sqrt{\\pi }d}e^{-(\\xi /d)^2}$ The spectral dependences of $\\langle N^\\ast (E)\\rangle $ for various sizes $N=1000,..., 4000$ of a random Hamiltonian matrix (REF ) at variuos degrees of disorder $d=0.1, 0.2, 0.4$ are presented at Fig.REF .", "From Fig.REF a one can see that, at small disorder $d=0.1$ , the well-pronounced signs of delocalised states in the central half-band $E\\in [-1,1]$ are observed.", "In this spectral range the number $\\langle N^\\ast (E)\\rangle $ of sites covered by the wave functions is proportional to the size $N$ of a random matrix (REF ).", "At $E\\in [\\pm 1,\\pm 2]$ the number $\\langle N^\\ast (E)\\rangle $ does not depend on $N$ .", "As it was mentioned above, such a behaviour corresponds to the localised character of states at $E\\in [\\pm 1,\\pm 2]$ .", "So, the signs of a mobility edge at $E=\\pm 1$ , predicted in [2], are observed.", "Despite the fact that the above data have confirmed the statements of [2] there are at least two questions remain to be answered.", "The first question can be formulated as follows.", "In accordance with [2] the localisation length $L_{loc}$ of states with energies in the range $E\\in [\\pm 1,\\pm 2]$ must be constant.", "But our calculations show that the number of sites $\\langle N^\\ast (E)\\rangle $ covered by the localised wave functions is strongly depend on energy in this energy interval (Fig.REF a), varying from 7 – 8 (nearly zero) at $E=\\pm 2$ to $\\sim 200$ at the \"mobility edge” $E=\\pm 1$ .", "It seems strange because both these quantities ($\\langle N^\\ast (E)\\rangle $ and $L_{loc}$ ) must describe the size of localised states and their energy dependence must be similar $L_{loc}\\sim \\langle N^\\ast (E)\\rangle $ .", "In our opinion this inconsistency reveals the ambiguous and disputable character of the localisation length as a measure of localisation of states.", "The second question relates to the behaviour of the above random model at large degrees of disorder.", "The existence of a mobility edge (in the sense of inverse localisation length) was obtained in [2] in the limit of small disorder $d\\rightarrow 0$ .", "Whether it is possible to extend this result for the case of an arbitrary strong disorder?", "To clear up this we have studied the behaviour of $\\langle N^\\ast (E)\\rangle $ at large degrees of disorder.", "The results are presented at Fig.REF (b,c).", "One can see that Fig.REF (b,c) give grounds to conclude that at large disorder all states of the system are localised: the number of covered sites $\\langle N^\\ast (E)\\rangle $ for the states at the center of the band saturate, i.e.", "the nearly linear dependence $\\langle N^\\ast (E)\\rangle $ on $N$ , which takes place for $d=0.1$ (Fig.REF a), violates.", "Having this in mind, one can suppose that if we had an opportunity to make calculations for $d=0.1$ (Fig.REF a) with matrixes of the size greater than 4000, we would also observe the saturation of linear dependence of $\\langle N^\\ast (E)\\rangle $ on $N$ .", "Or there is some critical degree of disorder $d_c$ , such that for $d>d_c$ , the states in the center of the band become localised?", "In our opinion all these questions are still open.", "Even if the further exploration of the correlated random system (REF ), (REF ) will disprove the existence of exact mobility edge in this system, the curious spectral dependence of a number of covered sites $\\langle N^\\ast (E)\\rangle $ with a \"mobility edge” (may be, virtual), looks very interesting." ] ]
1403.0394
[ [ "Quasisymmetries of Sierpi\\'nski carpet Julia sets" ], [ "Abstract We prove that if $\\xi$ is a quasisymmetric homeomorphism between Sierpi\\'nski carpets that are the Julia sets of postcritically-finite rational maps, then $\\xi$ is the restriction of a M\\\"obius transformation to the Julia set.", "This implies that the group of quasisymmetric homeomorphisms of a Sierpi\\'nski carpet Julia set of a postcritically-finite rational map is finite." ], [ "Introduction", "no A Sierpiński carpet is a topological space homeomorphic to the well-known standard Sierpiński carpet fractal.", "A subset $S$ of the Riemann sphere $\\widehat{\\mathbb {C}}$ is a Sierpiński carpet if and only if it has empty interior and can be written as $S=\\widehat{\\mathbb {C}}\\setminus \\bigcup _{k\\in {\\mathbb {N}}} D_k,$ where the sets $D_k\\subseteq \\widehat{\\mathbb {C}}$ , $k\\in {\\mathbb {N}}$ , are pairwise disjoint Jordan regions with $\\partial D_k\\cap \\partial D_l=\\emptyset $ for $k\\ne l$ , and $\\operatorname{diam}(D_k)\\rightarrow 0$ as $k\\rightarrow \\infty $ .", "A Jordan curve $C$ in a Sierpiński carpet $S$ is called a peripheral circle if its removal does not separate $S$ .", "If $S$ is a Sierpiński carpet as in (REF ), then the peripheral circles of $S$ are precisely the boundaries $\\partial D_k$ of the Jordan regions $D_k$ .", "Sierpiński carpets can arise as Julia sets of rational maps.", "For example, in [4] (see also [9]) it is shown that for the family $f_\\lambda (z)=z^2+\\lambda /z^2,$ in each neighborhood of 0 in the parameter plane, there are infinitely many parameters $\\lambda _n$ such that the Julia sets $\\mathcal {J}(f_{\\lambda _n})$ are Sierpiński carpets on which the corresponding maps $f_{\\lambda _n} $ are not topologically conjugate." ], [ "Statement of main results", "Sierpiński carpets have large groups of self-homeomorphisms.", "For example, it follows from results in [27] that if $n\\in {\\mathbb {N}}$ and we are given two $n$ -tuples of distinct peripheral circles of a Sierpiński carpet $S$ , then there exists a homeomorphism on $S$ that takes the peripheral circles of the first $n$ -tuple to the corresponding peripheral circles of the other.", "In contrast, strong rigidity statements are valid if we consider quasisymmetries of Sierpiński carpets.", "By a quasisymmetry of a compact set $K$ we mean a quasisymmetric homeomorphism of $K$ onto itself (see Section ).", "In the present paper, we prove quasisymmetric rigidity results for Sierpiński carpets that are Julia sets of postcritically-finite rational maps (see Section  for the definitions).", "Theorem 1.1 Let $f \\colon \\widehat{\\mathbb {C}}\\rightarrow \\widehat{\\mathbb {C}}$ be a postcritically-finite rational map, and suppose the Julia set $\\mathcal {J}(f)$ of $f$ is a Sierpiński carpet.", "If $\\xi $ is a quasisymmetry of $\\mathcal {J}(f)$ , then $\\xi $ is the restriction to $\\mathcal {J}(f)$ of a Möbius transformation on $\\widehat{\\mathbb {C}}$ .", "Here a Möbius transformation is a fractional linear transformation on $\\widehat{\\mathbb {C}}$ or a complex-conjugate of such a map; so a Möbius transformation can be orientation-reversing.", "An example of a rational map as in the statement of Theorem REF is $f(z)=z^2-1/(16 z^2)$ ; see Figure REF for its Julia set.", "Figure: The Julia set of f(z)=z 2 -1 16z 2 f(z)=z^2-\\frac{1}{16 z^2}.The six critical points of this map are $0, \\infty $ , and $\\@root 4 \\of {-1}/2$ .", "The postcritical set consists of four points: $0, \\infty $ , and $\\sqrt{-1}/2$ .", "The point $\\infty $ is a fixed point and forms the only periodic cycle; so $f$ is a hyperbolic postcritically-finite rational map.", "The group $G$ of Möbius transformations that leave $\\mathcal {J}(f)$ invariant contains the maps $\\xi _1(z)=i z,\\ \\xi _2(z)=\\overline{z}$ , and $\\xi _3(z)=1/(4z)$ .", "It is likely that these maps actually generate $G$ .", "It was shown by Levin [13], [14] (see also [15]) that if $f$ is a hyperbolic rational map that is not equivalent to $z^d,\\ d\\in {\\mathbb {Z}}$ , and whose Julia set $\\mathcal {J}(f)$ is neither the whole sphere, a circle, nor an arc of a circle, then the group of Möbius transformations that keep $\\mathcal {J}(f)$ invariant is finite.", "In our context, we have the following corollary to Theorem REF .", "Corollary 1.2 If $\\mathcal {J}(f)$ is a Sierpiński carpet Julia set of a postcritically-finite rational map $f$ , then the group of quasisymmetries of $\\mathcal {J}(f)$ is finite.", "See the end of Section  for the proof.", "An immediate consequence is the following fact; see [21] for a related result.", "Corollary 1.3 No Sierpiński carpet Julia set of a postcritically-finite rational map is quasisymmetrically equivalent to the limit set of a Kleinian group.", "Indeed, the Kleinian group acts on its limit set, and so limit set has an infinite group of quasisymmetries.", "Theorem REF is a special case of the following statement, which is the main result of this paper.", "Theorem 1.4 Let $f , g\\colon \\widehat{\\mathbb {C}}\\rightarrow \\widehat{\\mathbb {C}}$ be postcritically-finite rational maps, and suppose the corresponding Julia sets $\\mathcal {J}(f)$ and $\\mathcal {J}(g)$ are Sierpiński carpets.", "If $\\xi $ is a quasisymmetric homeomorphism of $\\mathcal {J}(f)$ onto $\\mathcal {J}(g)$ , then $\\xi $ is the restriction to $\\mathcal {J}(f)$ of a Möbius transformation on $\\widehat{\\mathbb {C}}$ .", "Here and in Theorem REF the map $\\xi $ does not respect the dynamics a priori; in particular, we do not assume that $\\xi $ conjugates $f$ and $g$ .", "If $\\xi $ conjugates the two rational maps, then the result is well known.", "It follows from the uniqueness part of Thurston's characterization of postcritically-finite maps that can be extracted from [10].", "The statement of Theorem REF is false if we drop the assumption that the maps are postcritically-finite.", "Indeed, each hyperbolic rational map $f_0$ is quasiconformally structurally stable near its Julia set, i.e., for any rational map $f$ sufficiently close to $f_0$ , there exist (backward invariant) neighborhoods $U_0\\supset \\mathcal {J}(f_0)$ and $U\\supset \\mathcal {J}(f)$ and a quasiconformal homeomorphism $h: U_0\\rightarrow U$ such that $h(f_0 (z)) = f(h(z))$ for each $z\\in f_0^{-1}(U_0)$ (see [17]).", "This conjugacy is quasisymmetric on the Julia set.", "Moreover, $f$ can be selected so that $h$ is not a Möbius transformation (essentially due to the fact that a rational map of degree $d\\ge 2$ depends on $2d+1>3 $ complex parameters, while a Möbius transformation depends only on 3).", "This discussion is applicable, e.g., to the previously mentioned hyperbolic rational map $f_0(z)=z^2-1/(16 z^2)$ whose Julia set is Sierpiński carpet." ], [ "Previous rigidity results for quasisymmetries", "In [6] the authors considered quasisymmetric rigidity questions for Schottky sets, i.e., subsets of the $n$ -dimensional sphere ${\\mathbb {S}}^n$ whose complements are collections of open balls with disjoint closures.", "A Schottky set $S$ is said to be rigid if each quasisymmetric map of $S$ onto any other Schottky set is the restriction to $S$ of a Möbius transformation.", "The main result in dimension $n=2$ is the following theorem.", "Theorem 1.5 [6] A Schottky set in ${\\mathbb {S}}^2$ is rigid if and only if it has spherical measure zero.", "In higher dimensions the “if\" part of this statement is still true, but not the “only if\" part.", "A relative Schottky set in a region $D\\subseteq {\\mathbb {S}}^n$ is a subset of $D$ obtained by removing from $D$ a collection of balls whose closures are contained in $D$ and are pairwise disjoint.", "Theorem 1.6 [6] A quasisymmetric map from a locally porous relative Schottky set $S$ in $D\\subseteq {\\mathbb {S}}^n$ for $n\\ge 3$ onto a relative Schottky set $\\tilde{S}\\subseteq {\\mathbb {S}}^n$ is the restriction to $S$ of a Möbius transformation.", "For the definition of a locally porous relative Schottky set see the end of Section  (where we define it only for $n=2$ , but the definition is essentially the same for $n\\ge 3$ ).", "Note that Möbius transformations preserve the classes of Schottky and relative Schottky sets.", "Theorem REF is no longer true in dimension $n=2$ .", "However, the third author proved the following result.", "Theorem 1.7 [19] Suppose that $S$ is a relative Schottky set of measure zero in a Jordan region $D\\subset .", "Let $ f SS$ be a locally quasisymmetric orientation-preserving homeomorphism from $ S$ onto a relative Schottky set $ S$ in a Jordan region $ D. Then $f$ is conformal in $S$ in the sense that $f^{\\prime }(p) =\\lim _{q\\in S,\\, q\\rightarrow p}\\frac{f(q)-f(p)}{q-p}$ exists for every $p\\in S$ and is not equal to zero.", "Moreover, the map $f$ is locally bi-Lipschitz and the derivative $f^{\\prime }$ is continuous on $S$ .", "We will call conformal maps $f$ as above Schottky maps, see Section .", "Another fractal space for which a quasisymmetric rigidity result has been established is the slit carpet $S_2$ obtained as follows.", "We start with a closed unit square $[0,1]^2$ in the plane and subdivide it into $2\\times 2$ equal subsquares in the obvious way.", "We then create a vertical slit from the middle of the common vertical side of the top two subsquares to the middle of the common vertical side of the bottom two subsquares.", "Finally, we repeat these procedures for all the subsquares and continue indefinitely.", "The metric on $S_2$ is the path metric induced from the plane.", "Theorem 1.8 [18] The group of quasisymmetries of $S_2$ is the group of isometries of $S_2$ , that is, the finite dihedral group $D_2$ consisting of four elements and isomorphic to ${\\mathbb {Z}}/2{\\mathbb {Z}}\\times {\\mathbb {Z}}/2{\\mathbb {Z}}$ .", "Finally, the first and the third authors of this paper proved the following rigidity result for the standard Sierpiński carpet.", "Theorem 1.9 [7] Every quasisymmetry of the standard Sierpiński carpet $S_3$ is a Euclidean isometry.", "For a class of standard square carpets a slightly weaker result is also known, namely that the group of quasisymmetries of such a carpet is finite dihedral [7]." ], [ "Main techniques and an outline of the proof of Theorem ", "The methods used to prove the main result of this paper are different from those employed to establish Theorems REF –REF .", "The first key ingredient in the proof of Theorem REF is well known and basic in complex dynamics, namely the use of conformal elevators.", "Roughly speaking, this means that by using the dynamics of a given subhyperbolic rational map one can “blow up\" a small disk centered in the Julia set to a definite size.", "We will need a careful analysis to control analytic and geometric properties of such blow-ups (see Section ).", "We will apply this to establish uniform geometric properties of the peripheral circles of Sierpiński carpets that arise as Julia sets of subhyperbolic rational maps.", "Theorem 1.10 Let $f\\colon \\widehat{\\mathbb {C}}\\rightarrow \\widehat{\\mathbb {C}}$ be a subhyperbolic rational map whose Julia set $\\mathcal {J}(f)$ is a Sierpiński carpet.", "Then the peripheral circles of the Sierpiński carpet $\\mathcal {J}(f)$ are uniform quasicircles, they are uniformly relatively separated, and they occur on all locations and scales.", "Moreover, $\\mathcal {J}(f)$ is a porous set, and in particular, has measure zero.", "See Sections  and for an explanation of the terminology and for the proof.", "The fact that the Julia set of subhyperbolic rational map has measure zero is well known [16]; moreover, for hyperbolic rational maps the previous theorem is fairly standard.", "A quasisymmetric map $\\xi $ on a Julia set as in the previous theorem can be extended to a quasiconformal map on $\\widehat{\\mathbb {C}}$ by the following fact.", "Theorem 1.11 Suppose that $\\lbrace D_k:k\\in {\\mathbb {N}}\\rbrace $ is a family of Jordan regions in $\\widehat{\\mathbb {C}}$ with pairwise disjoint closures, and let $\\xi \\colon S = \\widehat{\\mathbb {C}}\\setminus \\bigcup _{k\\in {\\mathbb {N}}} D_k \\rightarrow \\widehat{\\mathbb {C}}$ be a quasisymmetric embedding.", "If the Jordan curves $\\partial D_k$ , $k\\in {\\mathbb {N}}$ , form a family of uniform quasicircles, then there exists a quasiconformal homeomorphism $F\\colon \\widehat{\\mathbb {C}}\\rightarrow \\widehat{\\mathbb {C}}$ whose restriction to $S$ is equal to $f$ .", "This follows from [5], where a stronger, quantitative version is formulated.", "Now suppose that, as in Theorem REF , we have a given quasisymmetry $\\xi $ of the Julia set $\\mathcal {J}(f)$ of a postcritically-finite rational map $f$ onto the Julia set of another such map $g$ .", "Since every postcritically-finite map is subhyperbolic, we can apply Theorems REF and REF and extend $\\xi $ (non-uniquely) to a quasiconformal map on $\\widehat{\\mathbb {C}}$ , also denoted by $\\xi $ .", "We then use conformal elevators to produce a sequence $\\lbrace h_k\\rbrace $ of uniformly quasiregular maps defined in a fixed disk centered at a suitably chosen point in $\\mathcal {J}(f)$ .", "The map $h_k$ is a local symmetry of $\\mathcal {J}(f)$ and is given by $h_k=\\xi ^{-1}\\circ g^{m_k}\\circ \\xi \\circ f^{-n_k}$ for appropriate choices of $n_k,m_k\\in {\\mathbb {N}}$ and branches of $f^{-n_k}$ .", "One can also write $h_k$ in the form $h_k=g_\\xi ^{m_k}\\circ f^{-n_k}$ , where $g_\\xi =\\xi ^{-1}\\circ g\\circ \\xi $ .", "This has the advantage that the maps involved are defined on $\\mathcal {J}(f)$ .", "Standard compactness arguments imply that the sequence $\\lbrace h_k\\rbrace $ has a subsequence that converges locally uniformly to a non-constant quasiconformal map $h$ on a disk centered at a point in $\\mathcal {J}(f)$ .", "Now one wants to show that the sequence $\\lbrace h_k\\rbrace $ stabilizes and so $h_k=h=h_{k+1}$ for large $k$ .", "From this one can derive an equation of the form $g^{m^{\\prime }}\\circ \\xi = g^m\\circ \\xi \\circ f^n$ on $\\mathcal {J}(f)$ for some integers $m,m^{\\prime },n\\in {\\mathbb {N}}$ .", "This relates $\\xi $ to the dynamics of $f$ and $g$ .", "When $\\xi $ is a Möbius transformation, this approach goes back to Levin [13] (see also [14] and [15]).", "In this case, the maps involved are analytic, which played a crucial role in establishing that the sequence $\\lbrace h_k\\rbrace $ stabilizes.", "In our case, the maps are not analytic, so we need to invoke a different techniques, namely rigidity results for Schottky maps on relative Schottky sets (see Section ) established by the third author.", "Our situation can be reduced to the Schottky setting due to the following quasisymmetric uniformization result: Theorem 1.12 [5] Suppose that $T\\subseteq \\widehat{\\mathbb {C}}$ is a Sierpiński carpet whose peripheral circles are uniform quasicircles that are uniformly relatively separated .", "Then $T$ can be mapped to a round Sierpiński carpet $S\\subseteq \\widehat{\\mathbb {C}}$ by a quasisymmetric homeomorphism $\\beta \\colon T\\rightarrow S$ .", "Here we say that a Sierpiński carpet $S$ in $\\widehat{\\mathbb {C}}$ is round if all of its peripheral circles are geometric circles; in this case $S$ is a Schottky set.", "According to Theorem REF , we can again extend the map $\\beta $ to a quasiconformal map on the sphere.", "If we apply the previous theorem to $T=\\mathcal {J}(f)$ , then it follows from Theorem REF that each map $h_k$ is conjugate by $\\beta $ to a Schottky map defined in a fixed relatively open $S\\cap V\\subseteq S$ .", "This is a conformal map on $S\\cap V$ in the sense of Theorem REF , with a continuous derivative.", "After conjugation by $\\beta $ we can write this map in the form $h_k=g_\\xi ^{m_k}\\circ f^{-n_k}$ as well, where, by abuse of notation, we do not distinguish between the original maps and their conjugates by $\\beta $ .", "The following result implies that the sequence $\\lbrace h_k\\rbrace $ of these conjugate maps stabilizes.", "Theorem 1.13 [20] Let $S$ be a locally porous relative Schottky set in a region $D\\subseteq , let $ pS$, and let $ U$ be an open neighborhood of $ p$ such that $ SU$ is connected.", "Suppose that there exists a Schottky map $ u SUS$ with $ u(p)=p$ and $ u'(p)1$.$ Let $\\lbrace h_k\\rbrace _{k\\in {\\mathbb {N}}}$ be a sequence of Schottky maps $h_k\\colon S\\cap U\\rightarrow S$ .", "If each map $h_k$ is a homeomorphism onto its image and if the sequence $\\lbrace h_k\\rbrace $ converges locally uniformly to a homeomorphism $h$ , then there exists $N \\in {\\mathbb {N}}$ such that $h_k =h$ in $S\\cap U$ for all $k\\ge N$ .", "After some algebraic manipulations, the relation $h_k=h_{k+1}$ for large enough $k$ gives the equation $g_\\xi ^{m^{\\prime }} = g_\\xi ^m\\circ f^n$ in a relatively open set $S\\cap U$ , for some integers $m,m^{\\prime },n\\in {\\mathbb {N}}$ .", "The following result promotes the validity of this equation to the whole set $S$ .", "Theorem 1.14 [20] Let $S$ be a locally porous relative Schottky set in $D \\subseteq , and let $ U D$ be an open set such that $ S U$ is connected.", "Let $ u,v S U S$ be Schottky maps, and$ E = {p S U u(p) = v(p)}$.", "If $ E$ has an accumulation point in $ U$, then $ E = SU$ and so $ u=v$.$ If we conjugate back by $\\beta ^{-1}$ to the original maps, then we can conclude that (REF ), and hence (REF ), is valid on the whole Julia set $\\mathcal {J}(f)$ .", "In the final part of the argument, we analyze functional equation (REF ) on $\\mathcal {J}(f)$ .", "Lemma REF in combination with auxiliary results established in Section  allow us to conclude that $\\xi $ has a conformal extension into each periodic component of the Fatou set of $f$ .", "Using the description of the dynamics on the Fatou set in the postcritically-finite case (Lemma REF ), we can actually produce a conformal extension of $\\xi $ into every Fatou component of $f$ by a lifting procedure (see Lemma REF ).", "These extensions piece together a global quasiconformal map, again denoted by $\\xi $ , that is conformal on the Fatou set of $f$ .", "Since the Julia set $\\mathcal {J}(f)$ has measure zero, it follows that $\\xi $ is 1-quasiconformal and hence a Möbius transformation (see Lemma REF ).", "Acknowledgments.", "The authors would like to thank A. Eremenko and A. Hinkkanen for providing useful references." ], [ "Quasiconformal maps and related concepts", "no Throughout the paper we assume that the reader is familiar with basic notions and facts from the theory of quasiconformal maps and complex dynamics.", "We will review some relevant definitions and statements related to these topics in this and the following sections.", "We will almost exclusively deal with the complex plane $ equipped with the Euclidean metric (the distance between $ z$ and $ w$ denoted by $ |z-w|$), or the Riemann sphere $ C$equipped with the chordal metric $$.", "We denote by $ {z |z|<1} $ the open unit disk in $ .", "By default a set $M\\subseteq \\widehat{\\mathbb {C}}$ carries (the restriction of) the chordal metric $\\sigma $ , but we will usually specify the relevant metric in a given context.", "With an underlying space $X$ and a metric on $X$ understood, we denote by $B(p,r)$ the open ball of radius $r>0$ centered at $p\\in X$ , by $\\operatorname{diam}(M)$ the diameter, and by $\\operatorname{dist}(M,N)$ the distance of sets $M,N\\subseteq X$ .", "The cardinality of set $M$ is $\\#M\\in {\\mathbb {N}}_0\\cup \\lbrace \\infty \\rbrace $ , and $\\text{id}_M$ the identity map on $M$ .", "If $f\\colon X\\rightarrow Y$ is a map between sets $X$ and $Y$ and $A\\subseteq X$ , then we denote by $f|_A\\colon A\\rightarrow Y$ the restriction of $f$ to $A$ .", "We will now discuss quasiconformal and related maps.", "For general background on this topic we refer to [24], [1], [26], [12].", "A non-constant continuous map $f \\colon U \\rightarrow on a region $ U is called quasiregular if $f$ is in the Sobolev space $W_{\\rm loc}^{1,2}$ and if there exists a constant $K\\ge 1$ such that the (formal) Jacobi matrix $Df$ satisfies $||Df(z)||^2\\le K\\det (Df(z))$ for almost every $z \\in U$ .", "The condition that $f\\in W_{\\rm loc}^{1,2}$ means that the first distributional partial derivatives of $f$ are locally in the Lebesgue space $L^2$ .", "This definition requires only local coordinates and hence the notion of a quasiregularity can be extended to maps $f\\colon U\\rightarrow \\widehat{\\mathbb {C}}$ on regions $U\\subseteq \\widehat{\\mathbb {C}}$ .", "If $f$ is a homeomorphism onto its image in addition, then $f$ is called a quasiconformal map.", "The map $f$ is called locally quasiconformal if for every point $p\\in U$ there exists a region $V$ with $p\\in V\\subseteq U$ such that $f|_V$ is quasiconformal.", "Each quasiregular map $f\\colon U\\rightarrow \\widehat{\\mathbb {C}}$ is a branched covering map.", "This means that $f$ is an open map and the preimage $f^{-1}(q)$ of each point $q\\in \\widehat{\\mathbb {C}}$ is a discrete subset of $U$ .", "A point $p\\in U$ , near which the quasiregular map $f$ is not a local homeomorphism, is called a critical point of $f$ .", "These points are isolated in $U$ , and accordingly, the set $\\operatorname{crit}(f)$ of all critical points of $f$ is a discrete and relatively closed subset of $U$ .", "One can easily derive these statements from the fact that every quasiregular map $f\\colon U\\rightarrow \\widehat{\\mathbb {C}}$ on a region $U\\subseteq \\widehat{\\mathbb {C}}$ can be represented locally in the form $f=g\\circ \\varphi $ , where $g$ is a holomorphic map and $\\varphi $ is quasiconformal [1].", "Let $(X, d_X)$ and $(Y,d_Y)$ be metric spaces and let $f\\colon X\\rightarrow Y$ be a homeomorphism.", "The map $f$ is called quasisymmetric if there exists a homeomorphism $\\eta \\colon [0,\\infty )\\rightarrow [0,\\infty )$ such that $\\frac{d_Y(f(u),f(v))}{d_Y(f(u),f(w))}\\le \\eta \\bigg (\\frac{d_X(u,v)}{d_X(u,w)}\\bigg ),$ for every triple of distinct points $u,v,w\\in X$ .", "Suppose $U$ and $V$ are subregions of $\\widehat{\\mathbb {C}}$ .", "Then every orientation-preserving quasisymmetric homeomorphism $f\\colon U\\rightarrow V$ is quasiconformal.", "Conversely, every quasiconformal homeomorphism $f\\colon U\\rightarrow V$ is locally quasisymmetric, i.e., for every compact set $M\\subseteq U$ , the restriction $f|_M\\colon M\\rightarrow f(M)$ is a quasisymmetry (see [1]).", "Often it is important to keep track of the quantitative information that appears in the definition of quasiregular maps as in (REF ) or quasisymmetric maps as in (REF ).", "Then we speak of a $K$ -quasiregular map, or an $\\eta $ -quasisymmetry, etc.", "A Jordan curve $J\\subseteq \\widehat{\\mathbb {C}}$ is called a quasicircle if there exists a quasisymmetry $f\\colon \\partial J$ .", "This is equivalent to the requirement that there exists a constant $L\\ge 1$ such that $\\operatorname{diam}(\\alpha )\\le L\\sigma (u,v),$ whenever $u,v\\in {\\mathbb {N}}$ , $u\\ne v$ , and $\\alpha $ is the smaller subarc of $J$ with endpoints $u,v$ .", "If $\\lbrace J_k: k\\in {\\mathbb {N}}\\rbrace $ is a family of quasicircles , then we say that it consists of uniform quasicircles if condition (REF ) is true for some constant $L\\ge 1$ independent of $k\\in {\\mathbb {N}}$ .", "The family $\\lbrace J_k: k\\in {\\mathbb {N}}\\rbrace $ is said to be uniformly relatively separated if there exists a constant $c>0$ such that $ \\frac{\\operatorname{dist}(J_k, J_l)}{\\min \\lbrace \\operatorname{diam}(J_k), \\operatorname{diam}(J_l)\\rbrace }\\ge c$ for all $k,l\\in {\\mathbb {N}}$ , $k\\ne l$ .", "We conclude this section with an extension result that is needed in the proof of Theorem REF .", "Lemma 2.1 Let $S\\subseteq \\widehat{\\mathbb {C}}$ be a Sierpiński carpet written in the form $S =\\widehat{\\mathbb {C}}\\setminus \\bigcup _{k\\in {\\mathbb {N}}}D_k$ with pairwise disjoint Jordan regions $D_k\\subseteq \\widehat{\\mathbb {C}}$ , and suppose that the peripheral circles $\\partial D_k$ , $k\\in {\\mathbb {N}}$ , of $S$ are uniform quasicircles.", "Let $\\xi \\colon S\\rightarrow \\widehat{\\mathbb {C}}$ be an orientation-preserving quasisymmetric embedding of $S$ and suppose that each restriction $\\xi |_{\\partial D_k}\\colon \\partial D_k\\rightarrow \\widehat{\\mathbb {C}}$ , $k\\in {\\mathbb {N}}$ , extends to an embedding $\\xi _k\\colon \\overline{D}_k\\rightarrow \\widehat{\\mathbb {C}}$ that is conformal on $D_k$ .", "Then $\\xi $ has a unique quasiconformal extension $\\widetilde{\\xi }\\colon \\widehat{\\mathbb {C}}\\rightarrow \\widehat{\\mathbb {C}}$ that is conformal on $\\widehat{\\mathbb {C}}\\setminus S$ .", "Moreover, if $S$ has measure zero, then $\\widetilde{\\xi }$ is a Möbius transformation.", "Here we say that the embedding $\\xi \\colon S\\rightarrow \\widehat{\\mathbb {C}}$ is orientation-preserving if $\\xi $ has an extension to an orientation-preserving homeomorphism on the whole sphere $\\widehat{\\mathbb {C}}$ .", "This does not depend on the embedding and is equivalent to the following statement: if we orient each peripheral circle $\\partial _k D$ of the Sierpiński carpet as in the theorem so that $S$ lies “to the left\" of $\\partial D_k$ and if we equip $\\xi (\\partial D_k)$ with the induced orientation, then $\\xi (S)$ lies to the left of $\\xi (\\partial D_k)$ .", "The proof relies on the rather subtle, but well-known relation between quasiconformal, quasisymmetric, and quasi-Möbius maps (for the definition of the latter class and related facts see [5]).", "In the proof we will omit some details that can easily be extracted from the considerations in [5].", "Under the given assumption the image $S^{\\prime }=\\xi (S)$ is also a Sierpiński carpet that we can represent in the form $S^{\\prime }=\\widehat{\\mathbb {C}}\\setminus \\bigcup _{k\\in {\\mathbb {N}}}D^{\\prime }_k$ with pairwise disjoint Jordan regions $D^{\\prime }_k$ .", "Since $\\xi $ maps the peripheral circles of $S$ to the peripheral circles of $S^{\\prime }$ , we can choose the labeling so that $\\xi (\\partial D_k)=\\partial D_k^{\\prime }$ for $k\\in {\\mathbb {N}}$ .", "For each embedding $\\xi _k\\colon \\overline{D}_k \\rightarrow \\widehat{\\mathbb {C}}$ as in the statement of the lemma, we necessarily have $\\xi _k(\\overline{D}_k)=\\overline{D}^{\\prime }_k$ , because $\\xi _k(\\partial D_k)=\\xi (\\partial D_k)=\\partial D^{\\prime }_k$ and $\\xi $ is orientation-preserving.", "Moreover, $\\xi _k$ is uniquely determined by $\\xi |_{\\partial D_k}$ ; this follows from the classical fact that a homeomorphic extension of a conformal map between given Jordan regions is uniquely determined by the image of three distinct boundary points.", "Our original map $\\xi $ and the unique maps $\\xi _k$ , $k\\in {\\mathbb {N}}$ , piece together to a homeomorphism $\\tilde{\\xi }\\colon \\widehat{\\mathbb {C}}\\rightarrow \\widehat{\\mathbb {C}}$ that is conformal on $\\widehat{\\mathbb {C}}\\setminus S$ .", "Moreover, $\\tilde{\\xi }$ is the unique homeomorphic extension of $\\xi $ with this conformality property.", "The Jordan curves $\\partial D_k$ , $k\\in {\\mathbb {N}}$ , form a family of uniform quasicircles, and hence also their images $\\partial D^{\\prime }_k=\\xi (\\partial D_k)$ , $k\\in {\\mathbb {N}}$ , under the quasisymmetry $\\xi $ .", "This implies that Jordan regions $D_k$ and $D^{\\prime }_k$ are uniform quasidisks.", "More precisely, there exist $K$ -quasiconformal homeomorphisms $\\alpha _k$ and $\\alpha ^{\\prime }_k$ on $\\widehat{\\mathbb {C}}$ such that $\\alpha _k(\\overline{D}_k)=\\overline{ and \\alpha ^{\\prime }_k(\\overline{D}^{\\prime }_k)=\\overline{ for k\\in {\\mathbb {N}}, where K\\ge 1 is independent of k. Then the maps \\alpha _k and \\alpha ^{\\prime }_kare uniformly quasi-Möbius (see \\cite [Proposition~3.1~(i)]{Bo}).Moreover, the homeomorphisms \\alpha ^{\\prime }_k\\circ \\xi _k\\circ \\alpha _k^{-1}\\colon \\overline{\\rightarrow \\overline{ are uniformly quasiconformal on , and hence also uniformly quasi-Möbiuson \\overline{ (the last implication is essentially well-known; one can reduce to \\cite [Proposition~3.1~(i)]{Bo} by Schwarz reflection in \\partial and use the fact that \\partial is removable for quasiconformal maps \\cite [Section 35]{Va}).", "It follows that the maps\\xi _k, k\\in {\\mathbb {N}}, are uniformly quasi-Möbius.", "}}Now the maps \\xi _k|_{\\partial D_k}=\\xi |_{\\partial D_k}, k\\in {\\mathbb {N}}, are actually uniformly quasisymmetric, because \\xi is a quasisymmetric embedding.", "This implies that the family \\xi _k, k\\in {\\mathbb {N}}, is also uniformly quasisymmetric (this can be seen as in the proof of \\cite [Proposition~5.3~(ii)]{Bo}).", "Since \\xi =\\tilde{\\xi }|_S is quasisymmetricand the maps \\xi _k=\\tilde{\\xi }|_{\\overline{D}^{\\prime }_k} for k\\in {\\mathbb {N}} are uniformlyquasisymmetric, the homeomorphism \\tilde{\\xi } is quasiconformal (this is shown as in the last part of the proof of \\cite [Proposition~5.1]{Bo}).", "}If S has measure zero, then \\tilde{\\xi } is a quasiconformal map that is conformal on the set \\widehat{\\mathbb {C}}\\setminus S of full measure in \\widehat{\\mathbb {C}}.Hence \\tilde{\\xi } is 1-quasiconformal on \\widehat{\\mathbb {C}}, which, as is well-known, implies that \\tilde{\\xi } is aMöbius transformation (one can, for example, derive this from the uniqueness part of Stoilow^{\\prime }s factorization theorem \\cite [p.~179, Theorem 5.5.1]{AIM}).", "}}$ Fatou components of postcritically-finite maps no In this section we record some facts related to complex dynamics.", "For basic definitions and general background we refer to standard sources such as [3], [8], [22], [25].", "Let $f$ be a rational map on the Riemann sphere $\\widehat{\\mathbb {C}}$ of degree $\\deg (f)\\ge 2$ .", "We denote by $f^n$ for $n\\in {\\mathbb {N}}$ the $n$ th-iterate of $f$ , by $\\mathcal {J}(f)$ its Julia set and by $\\mathcal {F}(f)$ its Fatou set.", "Then $\\mathcal {J}(f)=f(\\mathcal {J}(f))=f^{-1}(\\mathcal {J}(f))=\\mathcal {J}(f^n), $ and we have similar relations for the Fatou set.", "We will use these standard facts throughout.", "A continuous map $f\\colon U\\rightarrow V$ between two regions $U,V\\subseteq \\widehat{\\mathbb {C}}$ is called proper if for every compact set $K\\subseteq V$ the set $f^{-1}(K)\\subseteq U$ is also compact.", "If $f$ is a rational map on $\\widehat{\\mathbb {C}}$ , then its restriction $f|_{U}$ to $U$ is a proper map $f|_{U}\\colon U\\rightarrow V$ if and only if $f(U)\\subseteq V$ and $f(\\partial U)\\subseteq \\partial V$ .", "If $f$ is a rational map, it is a proper map between its Fatou components; more precisely, if $U$ is a Fatou component of $f$ , then $V=f(U)$ is also a Fatou component of $f$ , and the restriction is a proper map of $U$ onto $V$ .", "In particular, the boundary of each Fatou component is mapped onto the boundary of another Fatou component.", "Similarly, we can always write $f^{-1}(U)=U_1\\cup \\dots \\cup U_m,$ where $U_1, \\dots , U_m$ are Fatou components of $f$ that are mapped properly onto $U$ .", "If $f\\colon U\\rightarrow V$ is a proper holomorphic map, possibly defined on a larger set than $U$ , then the topological degree $\\deg (f, U)\\in {\\mathbb {N}}$ of $f$ on $U$ is well-defined as the unique number such that $ \\deg (f, U)=\\sum _{p\\in f^{-1}(q)\\cap U}\\deg _f(p)$ for all $q\\in V$ , where $\\deg _f(p)$ is the local degree of $f$ at $p$ .", "Suppose that $U\\subseteq \\widehat{\\mathbb {C}}$ is finitely-connected, i.e., $\\widehat{\\mathbb {C}}\\setminus U$ has only finitely many connected components, and let $k\\in {\\mathbb {N}}_0$ be the number of components of $\\widehat{\\mathbb {C}}\\setminus U$ .", "We call $\\chi (U)=2-k$ the Euler characteristic of $U$ (see [3] for a related discussion).", "The quantity $\\chi (U)$ is invariant under homeomorphisms and can be obtained as a limit of Euler characteristics of polygons (defined in the usual way as for simplicial complexes) forming a suitable exhaustion of $U$ .", "We have $\\chi (U)=2$ if and only if $U=\\widehat{\\mathbb {C}}$ .", "So $\\chi (U)\\le 1$ for finitely-connected proper subregions $U$ of $\\widehat{\\mathbb {C}}$ with $\\chi (U)=1$ if and only if $U$ is simply connected; If $U$ and $V$ are finitely connected regions, and $f\\colon U\\rightarrow V$ is a proper holomorphic map, then a version of the Riemann-Hurwitz relation (see [3] and [25]) says that $\\deg (f, U)\\chi (V)=\\chi (U)+\\sum _{p\\in U} (\\deg _f(p)-1).$ Part of this statement is that the sum on the right-hand side of this identity is defined as it has only finitely many non-vanishing terms.", "The Riemann-Hurwitz formula is valid in a limiting sense for regions that are infinitely connected, i.e., not finitely-connected.", "In this case, the relation simply says that if $U,V\\subseteq \\widehat{\\mathbb {C}}$ are regions and $f\\colon U\\rightarrow V$ is a proper holomorphic map, then $U$ is infinitely connected if and only if $V$ is infinitely connected.", "If $f$ is a rational map on $\\widehat{\\mathbb {C}}$ , then a point is called a postcritical point of $f$ if it is the image of a critical point of $f$ under some iterate of $f$ .", "If we denote the set of these points by $\\operatorname{post}(f)$ , then we have $ \\operatorname{post}(f)=\\bigcup _{n\\in {\\mathbb {N}}}f^n(\\operatorname{crit}(f)).", "$ The map $f$ is called postcritically-finite if every critical point has a finite orbit under iteration of $f$ .", "This is equivalent to the requirement that $\\operatorname{post}(f)$ is a finite set.", "Note that $\\operatorname{post}(f)=\\operatorname{post}(f^n)$ for all $n\\in {\\mathbb {N}}$ .", "We denote by $\\operatorname{post}^{c}(f)\\subseteq \\operatorname{post}(f)$ the set of points that lie in cycles of periodic critical points; so $\\operatorname{post}^c(f)=\\lbrace f^n(c): n\\in {\\mathbb {N}}_0 \\text{ and $c$ is a periodic critical point of $f$} \\rbrace .", "$ If $f$ is postcritically-finite, then $f$ can only have one possible type of periodic Fatou components (for the general classification of periodic Fatou components and their relation to critical points see [3]); namely, every periodic Fatou component $U$ is a Böttcher domain for some iterate of $f$ : there exists an iterate $f^n$ and a superattracting fixed point $p$ of $f^n$ such that $p\\in U$ .", "The following, essentially well-known, lemma describes the dynamics of a postcritically-finite rational map on a fixed Fatou component.", "Here and in the following we will use the notation $P_k$ for the $k$ -th power map given by $P_k(z)=z^k$ for $z\\in , where $ kN$.$ Lemma 3.1 (Dynamics on fixed Fatou components) Suppose $f$ is a postcritically-finite rational map, and $U$ a Fatou component of $f$ with $f(U)=U$ .", "Then $U$ is simply connected, and contains precisely one critical point $p$ of $f$ .", "We have $f(p)=p$ and $ U\\cap \\operatorname{post}(f)=\\lbrace p\\rbrace $ , and there exists a conformal map $\\psi \\colon U\\rightarrow with $ (p)=0$ such that $ f-1=Pk$, where $ k=f(p)2$.$ So $p$ is a superattracting fixed point of $f$ , $U$ is the corresponding Böttcher domain of $p$ , and on $U$ the map $f$ is conjugate to a power map.", "Note that in general the map $\\psi $ is not uniquely determined due to a rotational ambiguity; namely, one can replace $\\psi $ with $a\\psi $ , where $a^{k-1}=1$ .", "As the statement is essentially well-known, we will only give a sketch of the proof.", "By the classification of Fatou components it is clear that $U$ contains a superattracting fixed point $p$ .", "Then $f(p)=p$ and $p$ is a critical point of $f$ .", "Let $k=\\deg _f(p)\\ge 2$ .", "Without loss of generality we may assume that $p=0$ , and $\\infty \\notin U$ .", "Then there exists a holomorphic function $\\varphi $ (the Böttcher function) defined in a neighborhood of 0 with $\\varphi (0)=0=p$ , $\\varphi ^{\\prime }(0)\\ne 0$ , and $f(\\varphi (z))=\\varphi (z^k)$ for $z$ near 0 [25].", "Since the maps $f^n$ , $n\\in {\\mathbb {N}}$ , form a normal family on $U$ and $f^n(z)\\rightarrow p$ for $z$ near $p$ , we have $\\text{$f^n(z)\\rightarrow p=0$ as $n\\rightarrow \\infty $ locally uniformly for $z\\in U$.", "}$ Let $r\\in (0,1]$ be the maximal radius such that $\\varphi $ has a holomorphic extension to the Euclidean disk $B=B(0,r)$ .", "Then (REF ) remains valid on $B$ .", "We claim that $r=1$ , and so $B=;otherwise, $ 0<r<1$, and by using (\\ref {eq:Boett}) and the fact that $ f UU$ is proper,one can show that $(B)U$.", "The equation (\\ref {eq:Boett}) impliesthat every point $ q(B){p}$ has an infinite orbit under iteration of $ f$; by the local uniformity of the convergence in (\\ref {eq:superattraction}) this remains true for $ q(B){p}$.Since $ f$ is postcritically-finite, this implies that no point in $(B){p}$ can be a critical point of $ f$; but then (\\ref {eq:Boett}) allows us to holomorphically extend$$ to a disk $ B(0,r')$ with $ r'>r$.", "This is a contradiction showing that indeed $ r=1$ and $ B=.", "As before by using (REF ), one sees that $\\varphi (\\subseteq U$ .", "Actually, one also observes that for points $q=\\varphi (z)$ with $z\\in closer and closer to $ , the convergence $f^n(q)\\rightarrow p$ is at a slower and slower rate.", "By (REF ) this is only possible if $\\varphi (z)$ is close to $\\partial U$ if $z\\in is close to $ ; in other words, $\\varphi $ is a proper map of $ to $ U$ and in particular $ (=U$.$ It follows from (REF ) that $\\varphi $ cannot have any critical points in $ (to see this, argue by contradiction and consider a critical point $ c of $\\varphi $ with smallest absolute value $|c|$ ).", "The Riemann-Hurwitz formula (REF ) then implies that $\\chi (U)=1$ and $\\deg (\\varphi )=1$ .", "In particular, $U$ is simply connected and $\\varphi $ is a conformal map of $ onto $ U$.", "For the conformal map $ =-1$ from $ U$ onto $ we then have $\\psi (p)=0$ and we get the desired relation $\\psi \\circ f\\circ \\psi ^{-1}=P_k$ .", "This relation (or again (REF )) implies that the fixed point $p$ is the only critical point of $f$ in $U$ and that each point $q\\in U\\setminus \\lbrace p\\rbrace $ has an infinite orbit; so $p$ is the only postcritical point of $f$ in $U$ .", "The following lemma gives us control for the mapping behavior of iterates of a rational map onto regions containing at most one postcritical point.", "Lemma 3.2 Let $f\\colon \\widehat{\\mathbb {C}}\\rightarrow \\widehat{\\mathbb {C}}$ be a rational map, $n\\in {\\mathbb {N}}$ , $U\\subseteq \\widehat{\\mathbb {C}}$ be a simply connected region with $\\#\\widehat{\\mathbb {C}}\\setminus U\\ge 2$ and $\\#(U\\cap \\operatorname{post}(f))\\le 1$ , and $V$ be a component of $f^{-n}(U)$ .", "Let $p\\in U$ be the unique point in $U\\cap \\operatorname{post}(f)$ if $\\#(U\\cap \\operatorname{post}(f))=1$ and $p\\in U$ be arbitrary if $U\\cap \\operatorname{post}(f)=\\emptyset $ , and let $\\psi _U\\colon U\\rightarrow be a conformal map with $ U(p)=0$.$ Then $V$ is simply connected, the map $f^n\\colon V\\rightarrow U$ is proper, and there exists $k\\in {\\mathbb {N}}$ , and a conformal map $\\psi _V\\colon V\\rightarrow with $ Ufn=PkV$.$ Here $k=1$ if $U\\cap \\operatorname{post}(f)=\\emptyset $ .", "Moreover, if $U\\cap \\operatorname{post}^c(f)=\\emptyset $ then $k\\le N$ , where $N=N(f)\\in {\\mathbb {N}}$ is a constant only depending on $f$ .", "In particular, for given $f$ the number $k$ is uniformly bounded by a constant $N$ independently of $n$ and $U$ if $U\\cap \\operatorname{post}^c(f)=\\emptyset $ .", "Under the given assumptions, $V$ is a region, and the map $g:=f^n|_V\\colon V\\rightarrow U$ is proper.", "Since $U\\cap \\operatorname{post}(f)\\subseteq \\lbrace p\\rbrace $ , the point $p$ is the only possible critical value of $g$ .", "It follows from the Riemann-Hurwitz formula (REF ) that $\\chi (V) &=&\\deg (g,V)\\chi (U)-\\sum _{z\\in V}(\\deg _{g}(z)-1)\\\\&=& \\deg (g,V)-(\\deg (g,V)-\\#g^{-1}(p))\\, =\\, \\#g^{-1}(p).$ As $\\chi (V)\\le 1$ , this is only possible if $\\chi (V)=1$ and $\\#g^{-1}(p)=1$ ; so $V$ is simply connected and $p$ has precisely one preimage $q$ in $V$ which is the only possible critical point of $g$ .", "Obviously, $\\#\\widehat{\\mathbb {C}}\\setminus V\\ge 2$ , and so there exists a conformal map $\\psi _V\\colon V\\rightarrow with $ V(q)=0$.", "Then$ (UfnV-1)$ is a proper holomorphic map from $ to itself and hence a finite Blaschke product $B$ .", "Moreover, $B^{-1}(0)=\\lbrace 0\\rbrace $ , and so we can replace $\\psi _V$ by a postcomposition with a suitable rotation around 0 so that $B(z)=z^k$ for $z\\in , where $ k=(g)N$.", "If $ Upost(f)=$, then $ q$ cannot be a critical point of $ g$, and so$ k=1$.$ It remains to produce a uniform upper bound for $k$ if we assume in addition that $U\\cap \\operatorname{post}^c(f)=\\emptyset $ .", "Then in the list $q, f(q), \\dots , f^{n-1}(q)$ each critical point of $f$ can appear at most once; indeed, otherwise the list contains a periodic critical point which implies that $p=f^n(q)\\in U\\cap \\operatorname{post}^c(f)$ , contradicting our additional hypothesis.", "We conclude that $k=\\deg _{f^n}(q)=\\prod _{i=0}^{n-1} \\deg _f(f^i(q))\\le N=N(f):=\\prod _{c\\in \\operatorname{crit}(f)}\\deg _f(c),$ which gives the desired uniform upper bound for $k$ .", "The next lemma describes the dynamics of a postcritically-finite rational map on arbitrary Fatou components.", "Lemma 3.3 (Dynamics on the Fatou components) Let $f\\colon \\widehat{\\mathbb {C}}\\rightarrow \\widehat{\\mathbb {C}}$ be a postcritically-finite rational map, and $\\mathcal {C}$ be the collection of all Fatou components of $f$ .", "Then there exists a family $\\lbrace \\psi _U\\colon U\\rightarrow U\\in \\mathcal {C}\\rbrace $ of conformal maps with the following property: if $U$ and $V$ are Fatou components of $f$ with $f(V)=U$ , then $\\psi _U\\circ f=P_k\\circ \\psi _V$ on $V$ for some $k=k(U,V)\\in {\\mathbb {N}}$ .", "Moreover, for each $U\\in \\mathcal {C}$ the point $p_U:=\\psi _U^{-1}(0)$ is the unique point in $U\\cap \\bigcup _{n\\in {\\mathbb {N}}_0} f^{-n}(\\operatorname{post}(f))$ .", "In contrast to the points $p_U$ the maps $\\psi _U$ are not uniquely determined in general due to a certain rotational freedom.", "As we will see in the proof of the lemma, $p_U$ can also be characterized as the unique point in $U$ with a finite orbit under iteration of $f$ .", "In the following, we will choose $p_U$ as a basepoint in the Fatou component $U$ .", "If we take 0 as a basepoint in $, then in the previous lemmawe get the following commutative diagram of basepoint-preserving maps between{\\em pointed}regions (i.e., regions with a distinguished basepoint):\\begin{equation*}{(V,p_V) [r]^{\\psi _V} [d]_{f} & (0) [d]^{P_k} \\\\(U,p_U) [r]^{\\psi _U} & (0)}\\end{equation*}Note that this implies in particular that $ f-1(pU)={pV}$ and that $ f V{pV}U{pU}$ is a covering map.$ We first construct the desired maps $\\psi _U$ for the periodic Fatou components $U$ of $f$ .", "So fix a periodic Fatou component $U$ of $f$ , and let $n\\in {\\mathbb {N}}$ be the period of $U$ , i.e., if we define $U_0:=U$ and $U_{k+1}=f(U_k)$ for $k=0, \\dots , n-1$ , then the Fatou components $U_0, \\dots , U_{n-1}$ are all distinct, and $U_n=f^n(U)=U$ .", "By Lemma REF applied to the map $f^n$ , for each $k=0, \\dots , n-1$ the Fatou component $U_k$ is simply connected and there exists a unique point $p_k\\in U_k$ that lies in $\\operatorname{post}(f)=\\operatorname{post}(f^n)$ .", "Moreover, there exists a conformal map $\\psi _0\\colon U_0\\rightarrow with $ 0(p0)=0$ such that$ 0fn=Pd0$ for suitable $ dN$.$ Let $\\psi _1\\colon U_1\\rightarrow be a conformal map with $ 1(p1)=0$.", "By the argument in the proof of Lemma~\\ref {lem:deg} we know that $ B=1f 0-1$ is a finite Blaschke product $ B$ with $ B-1(0)={0}$ and so $ B(z)=azd1$ for suitable constants $ d1N$ and $ a with $|a|=1$ .", "By adjusting $\\psi _1$ by a suitable rotation factor if necessary, we may assume that $a=1$ .", "Then $\\psi _1\\circ f=P_{d_1}\\circ \\psi _0$ on $U_0$ .", "If we repeat this argument, then we get conformal maps $\\psi _k\\colon U_k\\rightarrow with $ k(pk)=0$ and\\begin{equation}\\psi _{k}\\circ f=P_{d_k}\\circ \\psi _{k-1}\\end{equation} on $ Uk-1$ with suitable $ dkN$ for$ k=1, ..., n$.", "Note that$$ \\psi _n\\circ f^n = P_{d_n}\\circ \\psi _{n-1}\\circ f^{n-1}=\\dots =P_{d_n}\\circ \\dots \\circ P_{d_1}\\circ \\psi _0=P_{d^{\\prime }}\\circ \\psi _0$$on $ U0$, where $ d'N$.", "On the other hand, $ 0fn=Pd0$ by definition of $ 0$.Hence $ d=fn(p0)=d'$, and so $ nfn= 0fn$ on $ U0$ which implies $ n=0$.", "If we now define $ Uk:=k$ for $ k=0, ..., n-1$, then by (\\ref {eq:perdFatok}) the desired relation (\\ref {eq:desFatcomp}) holds for each suitable pairof Fatou components from the cycle $ U0, ..., Un-1$.", "We also choose $ pUk=pkUk$ as a basepoint in $ Uk$ for $ k=0, ..., n-1$.", "We know that $ pk$ is the unique point in$ Uk$ that lies in $ post(f)$.", "Since $ f$ is postcritically-finite, each point in$ P:=nN0 f-n(post(f))$ has a finite orbit under iteration of $ f$.It follows from Lemma~\\ref {lem:postratFatou} thateach point $ pUk{p}$ has an infinite orbit and therefore cannot lie in$ P$.", "Hence $ pUk$ is the unique point in $ Uk$ that lies in $ P$.$ We repeat this argument for the other finitely many periodic Fatou components $U$ to obtain suitable conformal maps $\\psi _U\\colon U\\rightarrow and unique basepoints $ pU=U-1(0)UP$.$ If $V$ is a non-periodic Fatou component, then it is mapped to a periodic Fatou component by a sufficiently high iterate of $f$ (this is Sullivan's theorem on the non-existence of wandering domains; see [3]).", "We call the smallest number $k\\in {\\mathbb {N}}_0$ such that $f^k(V)$ is a periodic Fatou component the level of $V$ .", "Suppose $V$ is an arbitrary Fatou component of level 1.", "Then $U=f(V)$ is periodic, and so $\\psi _U$ and $p_U$ are already defined and we know that $\\lbrace p_U\\rbrace =\\operatorname{post}(f)\\cap U$ .", "Hence by Lemma REF there exists a conformal map $\\psi _V\\colon V\\rightarrow U$ such that (REF ) is valid.", "If $p_V:=\\psi _V^{-1}(0)$ , then $f(p_V)=p_U\\in \\operatorname{post}(f)$ , and so $p_V\\in V\\cap P$ .", "Moreover, (REF ) shows that $f(V\\setminus \\lbrace p_V\\rbrace )=U\\setminus \\lbrace p_U\\rbrace $ which implies that each point in $V\\setminus \\lbrace p_V\\rbrace $ has an infinite orbit and cannot lie in $P$ .", "It follows that $p_V$ is the unique point in $V$ that lies in $P$ .", "We repeat this argument for Fatou components of higher and higher level.", "Note that if for a Fatou component $U$ a conformal map $\\psi _U\\colon U\\rightarrow has already been constructed and we know that $ pU:=-1(0)$ is the unique point in $ UP$, then$ Upost(f) {pU}$ and we can again apply Lemma~\\ref {lem:deg} for a Fatou component $ V$ with $ f(V)=U$.$ In this way we obtain conformal maps $\\psi _U$ as desired for all Fatou components $U$ .", "The point $p_U=\\psi _U^{-1}(0)$ is the unique point in $U$ that lies in $P$ , because $f^k(p_U)\\in \\operatorname{post}(f)$ for some $k\\in {\\mathbb {N}}_0$ and all other points in $U$ have an infinite orbit.", "We conclude this section with a lemma that is required in the proof of Theorem REF .", "Lemma 3.4 (Lifting lemma) Let $f\\colon \\widehat{\\mathbb {C}}\\rightarrow \\widehat{\\mathbb {C}}$ be a postcritically-finite rational map, $n\\in {\\mathbb {N}}$ , and $(U,p_U)$ and $(V,p_V)$ be pointed Fatou components of $f$ that are Jordan regions with $f^n(U)=V$ .", "Suppose $D\\subseteq \\widehat{\\mathbb {C}}$ is another Jordan region with a basepoint $p_D\\in D$ , and suppose that $\\alpha \\colon \\overline{D}\\rightarrow \\overline{V} $ is a map with the following properties: (i) $\\alpha $ is continuous on $ \\overline{D}$ and holomorphic on $D $ , (ii) $\\alpha ^{-1}(p_V)=\\lbrace p_D\\rbrace $ , (iii) there exists a continuous map $\\beta \\colon \\partial D \\rightarrow \\partial U$ with $f^n\\circ \\beta = \\alpha |_{\\partial D}$ .", "Then there exists a unique continuous map $\\tilde{\\alpha }\\colon \\overline{D}\\rightarrow \\overline{U} $ with $f^n\\circ \\tilde{\\alpha }= \\alpha $ and $\\tilde{\\alpha }|_{\\partial D}=\\beta $ .", "Moreover, $\\tilde{\\alpha }$ is holomorphic on $D$ and satisfies $\\tilde{\\alpha }^{-1}(p_U)=\\lbrace p_D\\rbrace $ .", "If, in addition, $\\beta $ is a homeomorphism of $\\partial D$ onto $\\partial U$ , then $\\tilde{\\alpha }$ is a conformal homeomorphism of $\\overline{D}$ onto $ \\overline{U}$ .", "Here we call a map $\\varphi \\colon \\overline{\\Omega }\\rightarrow \\overline{\\Omega }^{\\prime }$ between the closures of two Jordan regions $\\Omega , \\Omega ^{\\prime }\\subseteq \\widehat{\\mathbb {C}}$ a conformal homeomorphism if $\\varphi $ is a homeomorphism of $\\overline{\\Omega }$ onto $\\overline{\\Omega }^{\\prime }$ and a conformal map of $\\Omega $ onto $\\Omega ^{\\prime }$ .", "Note that in the previous lemma we necessarily have $f^n(\\overline{U})=\\overline{V}$ , $\\alpha (p_D)=p_V=f^n(p_U)$ , and $\\alpha (\\partial D)\\subseteq \\partial V$ by (iii).", "In the conclusion of the lemma we obtain a lift $\\tilde{\\alpha }$ for a given map $\\alpha $ under the branched covering map $f^n$ so that the diagram ${&\\overline{U} [d]^{f^n} \\\\\\overline{D} [ur]^{\\tilde{\\alpha }} [r]^{\\alpha } & \\overline{V}}$ commutes.", "By Lemma REF the map $f^n$ is actually an unbranched covering map from $\\overline{U}\\setminus \\lbrace p_U\\rbrace $ onto $\\overline{V}\\setminus \\lbrace p_V\\rbrace $ .", "The lemma asserts that the existence and uniqueness of a lift $\\tilde{\\alpha }$ is guaranteed if the boundary map $\\alpha |_{\\partial D}$ has a lift (namely $\\beta $ ), and if we have some compatibility condition for branch points (given by condition (ii)).", "The lemma easily follows from some basic theory for covering maps and lifts (see [11] for general background), so we will only sketch the argument and leave some straightforward details to the reader.", "By Lemma REF we can change $\\overline{U}$ and $\\overline{V}$ by conformal homeomorphisms so that we can assume $\\overline{U}=\\overline{V}=\\overline{, p_U=0=p_V, and f^n=P_k for suitable k\\in {\\mathbb {N}} without loss of generality.By classical conformal mapping theory we may also assume that D= and p_D=0.Then condition (ii) translates to \\alpha (0)=0 and \\alpha (z)\\ne 0 for z\\in \\overline{\\setminus \\lbrace 0\\rbrace .", "}We use this to define a homotopy of the boundary map \\alpha |_{\\partial into the base space \\overline{\\setminus \\lbrace 0\\rbrace of the covering map P_k\\colon \\overline{\\setminus \\lbrace 0\\rbrace \\rightarrow \\overline{\\setminus \\lbrace 0\\rbrace .", "Namely, let H\\colon \\partial (0,1]\\rightarrow \\overline{\\setminus \\lbrace 0\\rbrace be defined asH( \\zeta ,t):=\\alpha (t\\zeta ) for \\zeta \\in \\partial and t\\in (0,1].", "It is convenient to think of H as a homotopy running backwards in time t\\in (0,1] starting at t=1.Note that P_k\\circ \\beta =\\alpha |_{\\partial =H(\\cdot , 1).", "So for the initial time t=1 the homotopy has the lift \\beta under the covering map P_k\\colon \\overline{\\setminus \\lbrace 0\\rbrace \\rightarrow \\overline{\\setminus \\lbrace 0\\rbrace .", "By the homotopy lifting theorem \\cite [p.~60, Proposition~1.30]{Ha}, the whole homotopy H has a unique liftstarting at \\beta , i.e., there exists a unique continuous map \\widetilde{H}\\colon \\partial (0,1]\\rightarrow \\overline{\\setminus \\lbrace 0\\rbrace such thatP_k\\circ \\widetilde{H}=H and \\widetilde{H}(\\cdot , 1)=\\beta .", "Now we define \\tilde{\\alpha }(z)=\\widetilde{H}( z/|z|, |z|) for z\\in \\overline{\\setminus \\lbrace 0\\rbrace .", "Then \\tilde{\\alpha } is continuous on\\overline{\\setminus \\lbrace 0\\rbrace , where it satisfies P_k\\circ \\tilde{\\alpha }=\\alpha .", "Since \\alpha (0)=0, this last equation implies that we get a continuous extensionof \\tilde{\\alpha } to \\overline{ by setting \\tilde{\\alpha }(0)=0.", "This extension is a lift \\tilde{\\alpha } of \\alpha .", "Note that \\tilde{\\alpha }^{-1}(0)=0 and\\tilde{\\alpha }|_{\\partial =\\widetilde{H}(\\cdot , 1)=\\beta .", "Moreover, \\tilde{\\alpha } is holomorphic on , because it is a continuous branch of the k-th root of the holomorphic function\\alpha on .", "This shows that \\tilde{\\alpha } has the desired properties.", "The uniqueness of \\tilde{\\alpha } easily follows from the uniqueness of \\widetilde{H}.", "}We have \\beta = \\tilde{\\alpha }|_{\\partial ; so if \\beta is a homeomorphism,then the argument principle implies that \\tilde{\\alpha } is a conformal homeomorphism of\\overline{ onto \\overline{.", "}}}}\\section {The conformal elevator for subhyperbolic maps}\\numero A rational map f is called {\\em subhyperbolic} if each critical point of f in \\mathcal {J}(f) has a finite orbit while each critical point in \\mathcal {F}(f) has an orbit that converges to an attracting or superattracting cycle of f.The map f is called {\\em hyperbolic} if it is subhyperbolic and f does not have critical points in \\mathcal {J}(f).", "Note that every postcritically-finite rational map is subhyperbolic.", "}}}}For the rest of this section, we will assume that f is a subhyperbolic rational map with \\mathcal {J}(f)\\ne \\widehat{\\mathbb {C}}.Moreover, we will make the following additional assumption:\\begin{equation}\\mathcal {J}(f)\\subseteq \\tfrac{1}{2} \\text{and}\\quad f^{-1}( \\subseteq \\end{equation}Here and in what follows, if B is a disk, we denote by \\frac{1}{2} Bthe disk with the same center and whose radius is half the radius of B.", "}The inclusions (\\ref {eq:addinv}) can always be achieved by conjugating fwith an appropriate Möbius transformation so that \\mathcal {J}(f)\\subseteq \\tfrac{1}{2} and \\infty is an attracting or superattractingperiodic point of f. If we then replace f with suitable iterate, we may in additionassume that \\infty becomes an attracting or superattracting fixed point of f withf(\\widehat{\\mathbb {C}}\\setminus \\subseteq \\widehat{\\mathbb {C}}\\setminus .", "The latter inclusion is equivalent tof^{-1}( \\subseteq .", "}}}}Every small disk B centered at a point in \\mathcal {J}(f) can be ``blown up\" by a carefully chosen iterate f^nto a definite size with good control on how sets are distorted under the map f^n.", "We will discuss this in detail as a preparation for the proofs of Theorems~\\ref {thm:circgeom} and \\ref {thm:main2}, and will refer to this procedure as applying the {\\em conformal elevator} to B.", "In the following, all metric notions refer to the Euclidean metric on .", "}}}$ Let $P\\subseteq \\widehat{\\mathbb {C}}$ denote the union of all superattracting or attracting cycles of $f$ .", "This is a non-empty and finite set contained in $ \\mathcal {F}(f)$ .", "Since $f$ is subhyperbolic, every critical point in $\\mathcal {J}(f)$ has a finite orbit, and every critical point in $\\mathcal {J}(f)$ has an orbit that converges to $P$ .", "Hence there exists a neighborhood of $\\mathcal {J}(f)$ that contains only finitely many points in $\\operatorname{post}(f)$ and no points in $\\operatorname{post}^c(f)$ .", "This implies that we can choose $\\epsilon _0>0$ so small that $\\operatorname{diam}(\\mathcal {J}(f))> 2\\epsilon _0$ , and so that every disk $B^{\\prime }=B(q,r^{\\prime })$ centered at a point $q\\in \\mathcal {J}(f)$ with positive radius $r^{\\prime }\\le 8\\epsilon _0$ is contained in $, contains no point in $ postc(f)$ and at most one point in$ post(f)$.$ Let $B=B(p,r)$ be a small disk centered at a point $p\\in \\mathcal {J}(f)$ and of positive radius $r<\\epsilon _0$ .", "Since $B$ is centered at a point in $\\mathcal {J}(f)$ , we have $\\mathcal {J}(f)\\subseteq f^n(B)$ for sufficiently large $n$ (see [3]), and so the images of $B$ under iterates will eventually have diameter $>2\\epsilon _0$ .", "Hence there exists a maximal number $n\\in {\\mathbb {N}}_0$ such that $f^n(B)$ is contained in the disk of radius $\\epsilon _0$ centered at a point $\\tilde{q}\\in \\mathcal {J}(f)$ .", "If $B(\\tilde{q}, 2\\epsilon _0)\\cap \\operatorname{post}(f)=\\emptyset $ , we define $q=\\tilde{q}$ and $B^{\\prime }=B(q, 2\\epsilon _0)$ .", "Otherwise, there exists a unique point $ q\\in B(\\tilde{q}, 2\\epsilon _0)\\cap \\operatorname{post}(f)$ .", "Then we define $B^{\\prime }:=B( q, 8\\epsilon _0)\\supset B(q, 4\\epsilon _0)\\supset B(\\tilde{q}, 2\\epsilon _0)\\supset f^n(B).", "$ In both cases, we have (i) $f^n(B)\\subseteq \\frac{1}{2}B^{\\prime }\\subseteq ,$ (ii) $B^{\\prime }\\cap \\operatorname{post}^c(f)=\\emptyset $ , (iii) $\\#(B^{\\prime }\\cap \\operatorname{post}(f))\\le 1$ with equality only if $B^{\\prime }$ is centered at a point in $\\operatorname{post}(f)$ .", "By definition of $n$ , the set $f^{n+1}(B)$ must have diameter $\\ge \\epsilon _0$ .", "Hence by uniform continuity of $f$ near $\\mathcal {J}(f)$ there exists $\\delta _0>0$ independent of $B$ such that (iv) $\\operatorname{diam}(f^n(B))\\ge \\delta _0$ .", "Let $\\Omega \\subseteq \\widehat{\\mathbb {C}}$ be the unique component of $f^{-n}(B^{\\prime })$ that contains $B$ .", "Then by Lemma REF and by (), (v) $\\Omega $ is simply connected, and $B\\subseteq \\Omega \\subseteq ,$ (vi) the map $f^n|_{\\Omega }\\colon \\Omega \\rightarrow B^{\\prime }$ is proper, (vii) there exists $k\\in {\\mathbb {N}}$ , and conformal maps $\\varphi \\colon B^{\\prime }\\rightarrow and $ such that $(\\varphi \\circ f^n\\circ \\psi ^{-1})(z)= z^k$ for all $z\\in .", "Here $ kN$ is uniformly bounded independent of $ B$.$ If $k\\ge 2$ , then $q=\\varphi ^{-1}(0)\\in \\operatorname{post}(f)\\cap B^{\\prime }$ , and so $q$ is the center of $B^{\\prime }$ .", "If $k=1$ , then we can choose $\\varphi $ so that this is also the case.", "So (viii) $\\varphi $ maps the center $q$ of $B^{\\prime }$ to 0.", "We refer to the choice of $f^n$ and the associated sets $B^{\\prime }$ and $\\Omega $ and the maps $\\varphi $ and $\\psi $ satisfying properties (i)–(viii) as applying the conformal elevator to $B$ .", "Lemma 3.5 There exist constants $\\gamma ,r_1>0$ and $C_1,C_2,C_3\\ge 1$ independent of $B=B(p,r)$ with the following properties: (a) If $A\\subseteq B$ is a connected set, then $\\frac{\\operatorname{diam}(A)}{\\operatorname{diam}(B)}\\le C_1 \\operatorname{diam}(f^n(A))^{\\gamma }.", "$ (b) $B(f^n(p), r_1)\\subseteq f^n(\\frac{1}{2} B)\\subseteq f^n(B)$ .", "(c) If $u,v\\in B$ , then $|f^n(u)- f^n(v)|\\le C_2\\frac{|u-v|}{\\operatorname{diam}(B)}.", "$ (d) If $u,v\\in B$ , $u\\ne v$ , and $f^n(u)=f^n(v)$ , then the center $q$ of $B^{\\prime }$ belongs to $\\operatorname{post}(f)$ and we have $|f^n(u)- q| \\le C_3\\frac{|u-v|}{\\operatorname{diam}(B)}.", "$ So (a) says that a connected set $A\\subseteq B$ comparable in diameter to $B$ is blown up to a definite size under the conformal elevator, and by (b) the image of $B$ contains a disk of a definite size.", "If we consider the maps $f^n|_B$ for different $B$ , then by (c) they are uniformly Lipschitz if we rescale distances in $B$ by $1/\\operatorname{diam}(B)$ .", "In (d) the center $q$ of $B^{\\prime }$ must be a point in $\\operatorname{post}(f)$ for otherwise $f^n$ would be injective; so (d) says that if distinct, but nearby points are mapped to the same image $w$ under $f^n$ , then a postcritical point must be close to this image $w$ .", "In the following we write $a\\lesssim b$ or $a\\gtrsim b$ for two quantities $a,b\\ge 0$ if we can find a constant $C>0$ independent of the disk $B$ such that $a\\le C b$ or $Ca\\ge b$ , respectively.", "We write $a\\approx b$ if we have both $a\\lesssim b$ and $a\\gtrsim b$ , and in this case say that the quantities $a$ and $b$ are comparable.", "We consider the conformal maps $\\varphi \\colon B^{\\prime }\\rightarrow and$ satisfying properties (vii) and (viii) of the conformal elevator as discussed above.", "The exponent $k$ in (vii) is uniformly bounded, say $k\\le N$ , where $N\\in {\\mathbb {N}}$ is independent of $B$ .", "As before, we use the notation $P_k(z)=z^k$ for $z\\in .$ As we will see, the properties (a)–(d) easily follow from distortion properties of the map $P_k$ .", "We discuss the relevant properties of $P_k$ first (the proof is left to the reader).", "The map $P_k$ is Lipschitz with uniformly bounded Lipschitz constant, because $k$ is uniformly bounded.", "If $M\\subseteq is connected, then$$ \\operatorname{diam}(P_k(M))\\gtrsim \\operatorname{diam}(M)^k\\gtrsim \\operatorname{diam}(M)^N.", "$$Moreover, if $ B(z, r0), then $ B(P_k(z), r_1)\\subseteq P_k(B(z, r_0)), $ where $r_1 \\gtrsim r_0^k \\ge r_0^N$ .", "By (vii) the map $\\varphi $ is a Euclidean similarity, and so $\\varphi (\\frac{1}{2} B^{\\prime })=\\frac{1}{2} .Since the radius of $ B'$ is equal to $ 20$ or $ 80$, and hence comparable to $ 1$, wehave\\begin{equation}|\\varphi (u^{\\prime })-\\varphi (v^{\\prime })|\\approx |u^{\\prime }-v^{\\prime }|,\\end{equation}whenever $ u',v'B'$.$ Moreover, for $\\rho :=2^{-1/N}\\in (0,1)$ (which is independent of $B$ ) we have $D:=B(0,\\rho )\\supseteq P_k^{-1}( \\tfrac{1}{2} =P_k^{-1}(\\varphi (\\tfrac{1}{2} B^{\\prime }))=\\psi (f^{-n}(\\tfrac{1}{2} B^{\\prime })\\cap \\Omega ).", "$ Since $f^n(B)\\subseteq \\frac{1}{2} B^{\\prime }$ by (i) and $B\\subseteq \\Omega $ by (v), we then have $\\psi (B) \\subseteq D$ .", "So if $u,v\\in B$ , then $\\psi (u), \\psi (v)\\in D$ .", "Hence by the Koebe distortion theorem we have $|u-v| \\approx |(\\psi ^{-1})^{\\prime }(0)|\\cdot |\\psi (u)-\\psi (v)|$ whenever $u,v\\in B$ .", "In particular, $ \\operatorname{diam}(B)\\approx |(\\psi ^{-1})^{\\prime }(0)|\\cdot \\operatorname{diam}(\\psi (B)) $ On the other hand, by (iv) $1&\\approx \\operatorname{diam}(f^n(B))\\approx \\operatorname{diam}\\big (\\varphi (f^n(B))\\big ) \\\\ &= \\operatorname{diam}\\big (P_k(\\psi (B))\\big ) \\lesssim \\operatorname{diam}(\\psi (B))\\le 2.$ Hence $\\operatorname{diam}(\\psi (B)) \\approx 1$ , and so $\\operatorname{diam}(B)\\approx |(\\psi ^{-1})^{\\prime }(0)|$ .", "This implies that $\\frac{ |u-v|}{\\operatorname{diam}(B)}\\approx |\\psi (u)-\\psi (v)|,$ whenever $u,v\\in B$ .", "Now let $A\\subseteq B$ be connected.", "Then $\\psi (A)$ is connected, which implies $\\frac{ \\operatorname{diam}(A)}{\\operatorname{diam}(B)}&\\approx \\operatorname{diam}(\\psi (A))\\lesssim \\operatorname{diam}\\big (P_k( \\psi (A))\\big )^{1/N} \\\\ & = \\operatorname{diam}\\big (\\varphi (f^n(A))\\big )^{1/N} \\approx \\operatorname{diam}(f^n(A))^{1/N}.$ Inequality (a) follows.", "It follows from (REF ) that there exists $r_0>0$ independent of $B$ such that $ B(\\psi (p), r_0))\\subseteq \\psi (\\frac{1}{2} B)$ .", "By the distortion property of $P_k$ mentioned in the beginning of the proof, $\\varphi (f^n(\\frac{1}{2} B))=P_k(\\psi (\\frac{1}{2} B))$ then contains a disk $B(P_k(\\psi (p)), r_1)$ with $r_1>0$ independent of $B$ .", "Since $\\varphi (f^n(p))=P_k(\\psi (p))$ , and $\\varphi $ distorts distances uniformly, statement (b) follows.", "For (c) note that if $u,v\\in B$ , then $|f^n(u)-f^n(v)|&\\approx |\\varphi (f^n(u))-\\varphi (f^n(v))|= |P_k(\\psi (u))-P_k(\\psi (v))|\\\\&\\lesssim |\\psi (u)-\\psi (v)| \\approx \\frac{ |u-v|}{\\operatorname{diam}(B)}.", "$ We used that $P_k$ is Lipschitz on $ with a uniform Lipschitz constant.$ Finally we prove (d).", "If $u,v\\in B$ , $u\\ne v$ , and $f^n(u)=f^n(v)$ , then $f^n$ is not injective on $B^{\\prime }$ , and so the center $q$ of $B^{\\prime }$ belongs to $\\operatorname{post}(f)$ .", "Moreover, we then have $\\psi (u)^k=\\psi (v)^k$ , but $\\psi (u)\\ne \\psi (v)$ .", "This implies that $ |\\psi (u)-\\psi (v)|\\gtrsim \\frac{1}{k} |\\psi (u)| \\approx |\\psi (u)|.", "$ It follows that $|f^n(u)-q|&\\approx |\\varphi (f^n(u))-\\varphi (q)|= |\\psi (u)^k|\\\\&\\le |\\psi (u)| \\lesssim |\\psi (u)-\\psi (v)| \\approx \\frac{ |u-v|}{\\operatorname{diam}(B)}.", "$ Geometry of the peripheral circles no In this section we will prove Theorem REF .", "We have already defined in Section  what it means for the peripheral circles of a Sierpiński carpet $S$ to be uniform quasicircles and to be uniformly relatively separated.", "We say that the peripheral circles of $S$ occur on all locations and scales if there exists a constant $C\\ge 1$ such that for every $p\\in S$ and every $0<r\\le {\\rm diam}(\\widehat{\\mathbb {C}})=2$ , there exists a peripheral circle $J$ of $S$ with $B(p,r)\\cap J\\ne 0$ and $r/C\\le {\\rm diam}(J)\\le C r.$ Here and below the metric notions refer to the chordal metric $\\sigma $ on $\\widehat{\\mathbb {C}}$ .", "A set $M\\subseteq \\widehat{\\mathbb {C}}$ is called porous if there exists a constant $c>0$ such that for every $p\\in S$ and every $0<r\\le 2$ there exists a point $q\\in B(p,r)$ such that $B(q,cr)\\subseteq \\widehat{\\mathbb {C}}\\setminus M$ .", "Before we turn to the proof of Theorem REF , we require an auxiliary fact.", "Lemma 4.1 Let $f$ be a rational map such that $\\mathcal {J}(f)$ is a Sierpiński carpet, and let $J$ be a peripheral circle of $\\mathcal {J}(f)$ .", "Then $f^n(J)$ is a peripheral circle of $\\mathcal {J}(f)$ , and $f^{-n}(J)$ is a union of finitely many peripheral circles of $\\mathcal {J}(f)$ for each $n\\in {\\mathbb {N}}$ .", "Moreover, $J\\cap \\operatorname{post}(f)=\\emptyset =J\\cap \\operatorname{crit}(f)$ .", "There exists precisely one Fatou component $U$ of $f$ such that $\\partial U=J$ .", "Then $V=f^n(U)$ is also a Fatou component of $f$ .", "Hence $\\partial V$ is a peripheral circle of $\\mathcal {J}(f)$ .", "The map $f^n|_U\\colon U\\rightarrow V$ is proper which implies that $f^n(J)=f^n(\\partial U)=\\partial V$ .", "Similarly, there are finitely many distinct Fatou components $V_1, \\dots , V_k$ of $f$ such that $ f^{-n}(U)=V_1\\cup \\dots \\cup V_k.", "$ Then $f^{-n}(J)=\\partial V_1\\cup \\dots \\cup \\partial V_k, $ and so the preimage of $J$ under $f^n$ consists of the finitely many disjoint Jordan curves $\\partial V_i$ , $i=1, \\dots , k$ , which are peripheral circles of $\\mathcal {J}(f)$ .", "To show $J\\cap \\operatorname{post}(f)=\\emptyset $ , we argue by contradiction, and assume that there exists a point $p\\in \\operatorname{post}(f)\\cap J$ .", "Then there exists $n\\in {\\mathbb {N}}$ , and $c\\in \\operatorname{crit}(f)$ such that $f^n(c)=p$ .", "As we have just seen, the preimage of $J$ under $f^n$ consists of finitely many disjoint Jordan curves, and is hence a topological 1-manifold.", "On the other hand, since $c\\in f^{-n}(p)\\subseteq f^{-n}(J)$ is a critical point of $f$ and hence of $f^n$ , at $c$ the set $f^{-n}(J)$ cannot be a 1-manifold.", "This is a contradiction.", "Finally, suppose that $c\\in J\\cap \\operatorname{crit}(f)$ .", "Then $f(c)\\in \\operatorname{post}(f)\\cap f(J)$ , and $f(J)$ is a peripheral circle of $\\mathcal {J}(f)$ .", "This is impossible by what we have just seen.", "A general idea for the proof is to argue by contradiction, and get locations where the desired statements fail quantitatively in a worse and worse manner.", "One can then use the dynamics to blow up to a global scale and derive a contradiction from topological facts.", "It is fairly easy to implement this idea if we have expanding dynamics given by a group (see, for example, [5]).", "In the present case, one applies the conformal elevator and the estimates as given by Lemma REF .", "We now provide the details.", "We can pass to iterates of the map $f$ , and also conjugate $f$ by a Möbius transformation as properties that we want to establish are Möbius invariant.", "This Möbius invariance is explicitly stated for peripheral circles to be uniform quasicircles and to be uniformly relatively separated in [5].", "The Möbius invariance of the other stated properties immediately follows from the fact that each Möbius transformation is bi-Lipschitz with respect to the chordal metric.", "In this way, we may assume that () is true.", "Then the peripheral circles are subsets of $, where chordal and Euclidean metric are comparable.", "Therefore, we can use the Euclidean metric, and all metric notions will refer to this metric in the following.$ Part I.", "To show that peripheral circles of $ \\mathcal {J}(f)$ are uniform quasicircles, we argue by contradiction.", "Then for each $k\\in {\\mathbb {N}}$ there exists a peripheral circle $J_k$ of $\\mathcal {J}(f)$ , and distinct points $u_k, v_k\\in J_k$ such that if $\\alpha _k,\\beta _k$ are the two subarcs of $J_k$ with endpoints $u_k$ and $v_k$ , then $\\frac{\\min \\lbrace \\operatorname{diam}(\\alpha _k), \\operatorname{diam}(\\beta _k)\\rbrace }{|u_k-v_k|} \\rightarrow \\infty $ as $k\\rightarrow \\infty $ .", "We can pick $r_k>0$ such that $ \\min \\lbrace \\operatorname{diam}(\\alpha _k), \\operatorname{diam}(\\beta _k)\\rbrace /r_k\\rightarrow \\infty $ and $ |u_k-v_k|/r_k\\rightarrow 0$ as $k\\rightarrow \\infty .$ We now apply the conformal elevator to $B_k:=B(u_k, r_k)$ .", "Let $f^{n_k}$ be the corresponding iterate and $B_k^{\\prime }$ be the ball as discussed in Section .", "Define $J^{\\prime }_k=f^{n_k}(J_k)$ , $u^{\\prime }_k=f^{n_k}(u_k)$ , and $v^{\\prime }_k=f^{n_k}(v_k)$ .", "Then Lemma REF  (a) and (REF ) imply that the diameters of the sets $J^{\\prime }_k$ are uniformly bounded away from 0 independently of $k$ .", "Since $J_k^{\\prime }$ is a peripheral circle of the Sierpiński carpet $\\mathcal {J}(f)$ by Lemma REF , there are only finitely many possibilities for the set $J^{\\prime }_k$ .", "By passing to suitable subsequence if necessary, we may assume that $J^{\\prime }=J^{\\prime }_k$ is a fixed peripheral circle of $\\mathcal {J}(f)$ independent of $k$ .", "The points $u^{\\prime }_k,v^{\\prime }_k$ lie in $J^{\\prime }$ and by (REF ) and Lemma REF  (c) we have $|u^{\\prime }_k-v^{\\prime }_k|\\rightarrow 0$ as $k\\rightarrow \\infty $ .", "For large $k$ we want to find a point $w_k\\ne u_k$ in $B_k$ near $u_k$ with $f^{n_k}(u_k)=f^{n_k}(w_k)$ .", "If $u^{\\prime }_k=v^{\\prime }_k$ we can take $w_k=v_k$ .", "Otherwise, if $u^{\\prime }_k\\ne v^{\\prime }_k$ , there are two subarcs of $J^{\\prime }$ with endpoints $u^{\\prime }_k$ and $v^{\\prime }_k$ .", "Let $\\gamma ^{\\prime }_k\\subseteq J^{\\prime }$ be the one with smaller diameter.", "Then by (REF ) we have $\\operatorname{diam}(\\gamma ^{\\prime }_k)\\rightarrow 0$ as $k\\rightarrow \\infty $ (for the moment we only consider such $k$ for which $\\gamma ^{\\prime }_k$ is defined).", "Since $J^{\\prime }\\cap \\operatorname{post}(f)=\\emptyset $ by Lemma REF , the map $f^{n_k}\\colon J_k \\rightarrow J^{\\prime }$ is a covering map.", "So we can lift the arc $\\gamma ^{\\prime }_k$ under $f^{n_k}$ to a subarc $\\gamma _k$ of $J_k$ with initial point $v_k$ and $f^{n_k}(\\gamma _k)=\\gamma ^{\\prime }_k$ .", "By Lemma REF  (b) we have $\\gamma ^{\\prime }_k\\subseteq f^{n_k} (B_k)$ for large $k$ ; then Lemma REF  (a) implies that $\\gamma _k\\subseteq B_k$ for large $k$ , and also $\\operatorname{diam}(\\gamma _k)/r_k\\rightarrow 0$ as $k\\rightarrow \\infty $ .", "Note that if $w_k$ is the other endpoint of $\\gamma _k$ , then $f^{n_k}(w_k)=u^{\\prime }_k$ .", "We have $w_k\\ne u_k$ for large $k$ ; for if $w_k=u_k$ , then $\\gamma _k\\subseteq J_k$ has the endpoints $u_k$ and $v_k$ and so must agree with one of the arcs $\\alpha _k$ or $\\beta _k$ ; but for large $k$ this is impossible by (REF ) and (REF ).", "In addition, we have $|u_k-w_k|/r_k\\le |u_k-v_k|/r_k+\\operatorname{diam}(\\gamma _k)/r_k\\rightarrow 0$ as $k\\rightarrow \\infty $ .", "Note that this is also true if $w_k=v_k$ .", "In summary, for each large $k$ we can find a point $w_k\\in B_k$ with $w_k\\ne u_k$ , $f^{n_k}(u_k)=f^{n_k}(w_k)$ , and $ |u_k-w_k|/r_k\\rightarrow 0$ as $k\\rightarrow \\infty $ .", "Then by Lemma REF  (d) the center $q_k$ of $B^{\\prime }_k$ must belong to the postcritical set of $f$ and $\\operatorname{dist}(J^{\\prime }, \\operatorname{post}(f))\\le |u^{\\prime }_k- q_k|\\rightarrow 0$ as $k\\rightarrow \\infty $ .", "Since $f$ is subhyperbolic, every sufficiently small neighborhood of $\\mathcal {J}(f) \\supseteq J^{\\prime }$ contains only finitely many points in $\\operatorname{post}(f)$ , and so this implies $J^{\\prime }\\cap \\operatorname{post}(f)\\ne \\emptyset $ .", "We know that this is impossible by Lemma REF and so we get a contradiction.", "This shows that the peripheral circles are uniform quasicircles.", "Part II.", "The proof that the peripheral circles of $\\mathcal {J}(f)$ are uniformly relatively separated runs along almost identical lines.", "Again we argue by contradiction.", "Then for $k\\in {\\mathbb {N}}$ we can find distinct peripheral circles $\\alpha _k$ and $\\beta _k$ of $\\mathcal {J}(f)$ , and points $u_k\\in \\alpha _k$ , $v_k\\in \\beta _k$ such that (REF ) is valid.", "We can again pick $r_k>0$ so that the relations (REF ) and (REF ) are true.", "As before we define $B_k=B(u_k, r_k)$ and apply the conformal elevator to $B_k$ which gives us suitable iterate $f^{n_k}$ and a ball $B^{\\prime }_k$ .", "By Lemma REF  (a) the images of $\\alpha _k$ and $\\beta _k$ under $f^{n_k}$ are blown up to a definite size.", "Since there are only finitely many peripheral circles of $\\mathcal {J}(f)$ whose diameter exceeds a given constant, only finitely many such image pairs can arise.", "By passing to a suitable subsequence if necessary, we may assume that $\\alpha =f^{n_k}(\\alpha _k)$ and $\\beta =f^{n_k}(\\alpha _k)$ are peripheral circles independent of $k$ .", "We define $u^{\\prime }_k:=f^{n_k}(u_k)\\in \\alpha $ and $v^{\\prime }_k:=f^{n_k}(v_k)\\in \\beta $ .", "Then again the relation (REF ) holds.", "This is only possible if $\\alpha \\cap \\beta \\ne \\emptyset $ , and so $\\alpha =\\beta $ .", "Again for large $k$ we want to find a point $w_k\\ne u_k$ in $B_k$ near $u_k$ with $f^{n_k}(u_k)=f^{n_k}(w_k)$ .", "If $u^{\\prime }_k=v^{\\prime }_k$ we can take $w_k:=v_k$ .", "Otherwise, if $u^{\\prime }_k\\ne v^{\\prime }_k$ , we let $\\gamma ^{\\prime }_k$ be the subarc of $\\alpha =\\beta $ with endpoints $u^{\\prime }_k $ and $v^{\\prime }_k$ and smaller diameter.", "Then we can lift $\\gamma ^{\\prime }_k$ to a subarc $\\gamma _k\\subseteq \\beta _k$ with initial point $v_k$ such that $f^{n_k}(\\gamma _k)=\\gamma ^{\\prime }_k$ , and we have (REF ).", "If $w_k\\in \\beta _k$ is the other endpoint of $\\gamma _k$ , then $f^{n_k}(w_k)=u^{\\prime }_k=f^{n_k}(u_k)$ , and $w_k\\ne u_k$ , because these points lie in the disjoint sets $\\beta _k$ and $\\alpha _k$ , respectively.", "Again we have (REF ), which implies that the center $q_k$ of $B_k^{\\prime }$ belongs to $\\operatorname{post}(f)$ , and leads to $\\operatorname{dist}(\\alpha , \\operatorname{post}(f))=0$ .", "We know that this is impossible by Lemma REF .", "Part III.", "We will show that peripheral circles of $\\mathcal {J}(f)$ appear on all locations and scales.", "Let $p\\in \\mathcal {J}(f)$ and $r>0$ be arbitrary, and define $B=B(p,r)$ .", "We may assume that $r$ is small, because by a simple compactness argument one can show that disks of definite, but not too large Euclidean size contain peripheral circles of comparable diameter.", "We now apply the conformal elevator to $B$ to obtain an iterate $f^n$ .", "Lemma REF  (b) implies that there exists a fixed constant $r_1>0$ independent of $B$ such that $B(f^n(p), r_1)\\subseteq f^n(\\frac{1}{2} B)$ .", "By part (a) of the same lemma, we can also find a constant $c_1>0$ independent of $B$ with the following property: if $A$ is a connected set with $A\\cap B(p, r/2)\\ne \\emptyset $ and $\\operatorname{diam}(f^n(A))\\le c_1$ , then $A\\subseteq B$ .", "We can now find a peripheral circle $J^{\\prime }$ of $\\mathcal {J}(f)$ such that $J^{\\prime }\\cap f^n(\\frac{1}{2} B)\\ne \\emptyset $ and $0<c_0 < \\operatorname{diam}(J^{\\prime })<c_1$ , where $c_0$ is another positive constant independent of $B$ .", "This easily follows from a compactness argument based on the fact that $f^n(\\frac{1}{2} B)$ contains a disk of a definite size that is centered at a point in $\\mathcal {J}(f)$ .", "The preimage $f^{-n}(J^{\\prime })$ consists of finitely many components that are peripheral circles of $\\mathcal {J}(f)$ .", "One of these peripheral circles $J$ meets $\\frac{1}{2} B$ .", "Since $\\operatorname{diam}(f^n(J))=\\operatorname{diam}(J^{\\prime })<c_1$ , by the choice of $c_1$ we then have $J\\subseteq B$ , and so $\\operatorname{diam}(J)\\le 2r$ .", "Moreover, it follows from Lemma REF  (c) that $\\operatorname{diam}(J)\\ge c_2 \\operatorname{diam}(J^{\\prime })\\operatorname{diam}(B)\\ge c_3 r$ , where again $c_2,c_3>0$ are independent of $B$ .", "The claim follows.", "Part IV.", "Let $p\\in \\mathcal {J}(f)$ be arbitrary and $r\\in (0,1]$ .", "To establish the porosity of $\\mathcal {J}(f)$ , it is enough to show that the Euclidean disk $B(p,r)$ contains a disk of comparable radius that lies in the complement of $\\mathcal {J}(f)$ .", "By what we have just seen, $B(p,r)$ contains a peripheral circle $J$ of diameter comparable to $r$ .", "By possibly allowing a smaller constant of comparability, we may assume that $J$ is distinct from the one peripheral circle $J_0$ that bounds the unbounded Fatou component of $f$ .", "Then $J\\subseteq B(p,r)$ is the boundary of a bounded Fatou component $U$ , and so $U\\subseteq B(p,r)$ .", "Since the peripheral circles of $\\mathcal {J}(f)$ are uniform quasicircles, it follows that $U$ contains a Euclidean disk $D$ of comparable size (for this standard fact see [5]).", "Then $\\operatorname{diam}(D) \\approx \\operatorname{diam}(J)\\approx r$ .", "Since $D\\subseteq U\\subseteq B(p,r)\\cap \\widehat{\\mathbb {C}}\\setminus \\mathcal {J}(f)$ the porosity of $\\mathcal {J}(f)$ follows.", "Finally, the porosity of $\\mathcal {J}(f)$ implies that $\\mathcal {J}(f)$ cannot have Lebesgue density points, and is hence a set of measure zero.", "Relative Schottky sets and Schottky maps no A relative Schottky set $S$ in a region $D\\subseteq \\widehat{\\mathbb {C}}$ is a subset of $D$ whose complement in $D$ is a union of open geometric disks $\\lbrace B_i\\rbrace _{i\\in I}$ with closures $\\overline{B}_i,\\ i\\in I$ , in $D$ , and such that $\\overline{B}_i\\bigcap \\overline{B}_j=\\emptyset ,\\ i\\ne j$ .", "We write $S=D\\setminus \\bigcup _{i\\in I}B_i.$ If $D=\\widehat{\\mathbb {C}}$ or $, we say that $ S$ is a \\emph {Schottky set}.$ Let $A,B\\subseteq \\widehat{\\mathbb {C}}$ and $\\varphi \\colon A\\rightarrow B$ be a continuous map.", "We call $\\varphi $ a local homeomorphism of $A$ to $B$ if for every point $p\\in A$ there exist open sets $U,V\\subseteq with $ pU$, $ f(p)V$ such that $ f|UA$ is a homeomorphism of $ UA$ onto $ VB$.", "Note that this concept depends of course on $ A$, but also crucially on $ B$: if $ B'B$, then we may consider a local homeomorphism $ fAB$ also as a map $ fAB'$, but the second map will not be a local homeomorphism in general.$ Let $D$ and $\\tilde{D}$ be two regions in $\\widehat{\\mathbb {C}}$ , and let $S=D\\setminus \\bigcup _{i\\in I} B_i$ and $\\tilde{S}=\\tilde{D}\\setminus \\bigcup _{j\\in J}\\tilde{B}_j$ be relative Schottky sets in $D$ and $ \\tilde{D}$ , respectively.", "Let $U$ be an open subset of $D$ and let $f\\colon S\\cap U\\rightarrow \\tilde{S}$ be a local homeomorphism.", "According to [20], such a map $f$ is called a Schottky map if it is conformal at every point $p\\in S\\cap U$ , i.e., the derivative $f^{\\prime }(p)=\\lim _{q\\in S,\\, q\\rightarrow p }\\frac{f(q)-f(p)}{q-p}$ exists and does not vanish, and the function $f^{\\prime }$ is continuous on $S\\cap U$ .", "If $p=\\infty $ or $f(p)=\\infty $ , the existence of this limit and the continuity of $f^{\\prime }$ have to be understood after a coordinate change $z\\mapsto 1/z$ near $\\infty $ .", "In all our applications $S\\subseteq and so we can ignore this technicality.$ Theorem REF implies that if $D$ and $\\tilde{D}$ are Jordan regions, the relative Schottky set $S$ has measure zero, and $f\\colon S\\rightarrow \\tilde{S}$ is a locally quasisymmetric homeomorphism that is orientation-preserving (this is defined similarly as for homeomorphisms between Sierpiński carpets; see the discussion after Lemma REF ), then $f$ is a Schottky map.", "We require a more general criterion for maps to be Schottky maps.", "Lemma 5.1 Let $S\\subseteq be a Schottky set of measure zero.Suppose $ UC$ is open and $ UC$ is a locally quasiconformal map with $ -1(S)=US.", "$Then $ USS$ is a Schottky map.$ In particular, if $\\psi \\colon \\widehat{\\mathbb {C}}\\rightarrow \\widehat{\\mathbb {C}}$ is a quasiregular map with $\\psi ^{-1}(S)=S$ , then $\\psi \\colon S\\setminus \\operatorname{crit}(\\psi )\\rightarrow S$ is a Schottky map.", "In the statement the assumption $S\\subseteq (instead of $ SC$) is not really essential, but helps to avoid some technicalities caused by the point $$.$ Our assumption $\\varphi ^{-1}(S)=U\\cap S$ implies that $\\varphi (U\\cap S)\\subseteq S$ .", "So we can consider the restriction of $\\varphi $ to $U\\cap S$ as a map $\\varphi \\colon U\\cap S\\rightarrow S$ (for simplicity we do not use our usual notation $\\varphi |_{U\\cap S}$ for this and other restrictions in the proof).", "This map is a local homeomorphism $\\varphi \\colon U\\cap S\\rightarrow S$ .", "Indeed, let $p\\in U\\cap S$ be arbitrary.", "Since $\\varphi \\colon U\\rightarrow \\widehat{\\mathbb {C}}$ is a local homeomorphism, there exist open sets $V,W\\subseteq \\widehat{\\mathbb {C}}$ with $p\\in V\\subseteq U$ and $f(p)\\in W$ such that $\\varphi $ is a homeomorphism of $V$ onto $W$ .", "Clearly, $\\varphi (V\\cap S)\\subseteq W\\cap S$ .", "Conversely, if $q\\in W\\cap S$ , then there exists a point $q^{\\prime }\\in V$ with $\\varphi (q^{\\prime })=q$ ; since $\\varphi ^{-1}(S)=U\\cap S$ , we have $q^{\\prime }\\in S$ and so $q^{\\prime }\\in V\\cap S$ .", "Hence $\\varphi (V\\cap S)=W\\cap S$ , which implies that $\\varphi $ is a homeomorphism of $V\\cap S$ onto $W\\cap S$ .", "Note that $p\\in U\\cap S$ lies on a peripheral circle of $S$ if and only if $\\varphi (p)$ lies on a peripheral circle of $S$ .", "Indeed, a point $p\\in S$ lies on a peripheral of $S$ if and only if it is accessible by a path in the complement of $S$ , and it is clear this condition is satisfied for a point $p\\in S\\cap U$ if and only if it is true for the image $\\varphi (p)$ (see [20] for a more general related statement).", "We now want to verify the other conditions for $\\varphi $ to be a Schottky map based on Theorem REF .", "It is enough to reduce to this situation locally near each point $p\\in U\\cap S$ .", "We consider two cases depending on whether $p$ belongs to a peripheral circle of $S$ or not.", "So suppose $p$ does not belong to any of the peripheral circles of $S$ .", "Then there exist arbitrarily small Jordan regions $D$ with $p\\in D$ and $\\partial D\\subseteq S$ such that $\\partial D$ does not meet any peripheral circle of $S$ .", "This easily follows from the fact that if we collapse each closure of a complementary component of $S$ in $\\widehat{\\mathbb {C}}$ to a point, then the resulting quotient space is homeomorphic to $\\widehat{\\mathbb {C}}$ by Moore's theorem [23] (for more details on this and the similar argument below, see the proof of [20]).", "In this way we can find a small Jordan region $D$ with the following properties: (i) $p\\in D \\subseteq \\overline{D}\\subseteq U$ , (ii) the boundary $\\partial D$ is contained in $S$ , but does not meet any peripheral circle of $S$ , (iii) $\\varphi $ is a homeomorphism of $\\overline{D}$ onto the closure $\\overline{D}^{\\prime }$ of another Jordan region $D^{\\prime }\\subseteq \\widehat{\\mathbb {C}}$ .", "As in the first part of the proof, we see that $\\varphi $ is a homeomorphism of $D\\cap S$ onto $D^{\\prime }\\cap S$ .", "This homeomorphism is locally quasisymmetric and orientation-preserving as it is the restriction of a locally quasiconformal map.", "Since $\\partial D$ does not meet peripheral circles of $S$ , the same is true of its image $\\partial D^{\\prime }=\\varphi (\\partial D)$ by what we have seen above.", "It follows that the sets $D\\cap S$ and $D^{\\prime }\\cap S$ are relative Schottky sets of measure zero contained in the Jordan regions $D$ and $D^{\\prime }$ , respectively.", "Note that the set $D\\cap S$ is obtained by deleting from $D$ the complementary disks of $S$ that are contained in $D$ , and $D^{\\prime }\\cap S$ is obtained similarly.", "Now Theorem REF implies that $\\varphi \\colon D\\cap S\\rightarrow D^{\\prime }\\cap S$ is a Schottky map which implies that $\\varphi \\colon U\\cap S \\rightarrow S$ is a Schottky map near $p$ .", "For the other case, assume that $p$ lies on a peripheral circle of $S$ , say $p\\in \\partial B$ , where $B$ is one of the disks that form the complement of $S$ .", "The idea is to use a Schwarz reflection procedure to arrive at a situation similar to the previous case.", "This is fairly straightforward, but we will provide the details for sake of completeness.", "Similarly as before (here we collapse all closures of complementary components of $S$ to points except $\\overline{B}$ ), we find a Jordan region $D$ with the following properties: (i) $\\overline{D}\\subseteq U$ and $\\partial D=\\alpha \\cup \\beta $ , where $\\alpha $ and $\\beta $ are two non-overlapping arcs with the same endpoints such that $\\alpha \\subseteq \\partial B$ , $\\beta \\subseteq S$ , $p$ is an interior point of $\\alpha $ , and no interior point of $\\beta $ lies on a peripheral circle of $S$ , (ii) $\\varphi $ is a homeomorphism of $\\overline{D}$ onto the closure $\\overline{D}^{\\prime }$ of another Jordan region $D^{\\prime }\\subseteq \\widehat{\\mathbb {C}}$ .", "Let $\\alpha ^{\\prime }=\\varphi (\\alpha )$ .", "Then $\\alpha $ is contained in a peripheral circle $\\partial B^{\\prime }$ of $S$ , where $B^{\\prime }$ is a suitable complementary disk of $S$ .", "Note that $\\beta ^{\\prime }=\\varphi (\\beta )$ is an arc contained in $S$ , has its endpoints in $\\partial B^{\\prime }$ , and no interior point of $\\beta ^{\\prime }$ lies on a peripheral circle of $S$ .", "Let $R\\colon \\widehat{\\mathbb {C}}\\rightarrow \\widehat{\\mathbb {C}}$ be the reflection in $\\partial B$ , and $R^{\\prime }\\colon \\widehat{\\mathbb {C}}\\rightarrow \\widehat{\\mathbb {C}}$ be the reflection in $\\partial B^{\\prime }$ .", "Define $\\tilde{S}=S\\cup R(S)$ and $\\tilde{S}^{\\prime }=S\\cup R^{\\prime }(S)$ .", "Then $\\tilde{S}$ and $\\tilde{S}^{\\prime }$ are Schottky sets of measure zero, $\\partial B\\subseteq \\tilde{S}$ , $\\partial B^{\\prime }\\subseteq \\tilde{S}^{\\prime }$ , and $\\partial B$ and $\\partial B^{\\prime }$ do not meet any of the peripheral circles of $\\tilde{S}$ and $\\tilde{S}^{\\prime }$ , respectively.", "Let $\\tilde{D}=D\\cup \\operatorname{int}(\\alpha )\\cup R(D)$ and $\\tilde{D}^{\\prime }=D^{\\prime }\\cup \\operatorname{int}(\\alpha ^{\\prime })\\cup R^{\\prime }(D^{\\prime })$ , where $\\operatorname{int}(\\alpha )$ and $ \\operatorname{int}(\\alpha ^{\\prime })$ denote the set of interior points of the arcs $\\alpha $ and $\\alpha ^{\\prime }$ , respectively.", "Then $\\tilde{D}$ and $\\tilde{D}^{\\prime }$ are Jordan regions such that $p\\in \\tilde{D}$ , $\\partial \\tilde{D}\\subseteq \\tilde{S}$ , $\\partial \\tilde{D}^{\\prime }\\subseteq \\tilde{S}^{\\prime }$ , and $\\partial \\tilde{D}$ and $\\partial \\tilde{D}^{\\prime }$ do not meet any of the peripheral circles of $\\tilde{S}$ and $\\tilde{S}^{\\prime }$ , respectively.", "Hence $\\tilde{D}\\cap \\tilde{S}$ and $\\tilde{D}^{\\prime }\\cap \\tilde{S}^{\\prime }$ are relative Schottky sets of measure zero in $\\tilde{D}$ and $\\tilde{D}^{\\prime }$ , respectively.", "We define a map $\\tilde{\\varphi }\\colon \\tilde{D}\\rightarrow \\tilde{D}^{\\prime }$ by $ \\tilde{\\varphi }(z)= \\left\\lbrace \\begin{array} {cl}\\varphi (z)& \\text{for $z\\in D\\cup \\operatorname{int}(\\alpha ) $,}\\\\&\\\\(R^{\\prime }\\circ \\varphi \\circ R)(z)&\\text{for $z\\in R(D)\\cup \\operatorname{int}(\\alpha )$.}", "\\end{array}\\right.$ Note that this definition is consistent on $\\operatorname{int}(\\alpha )$ , because $\\varphi (\\alpha )=\\alpha ^{\\prime }=\\overline{D}^{\\prime }\\cap R^{\\prime }(\\overline{D}^{\\prime })$ .", "It is clear that $\\tilde{\\varphi }\\colon \\tilde{D}\\rightarrow \\tilde{D}^{\\prime }$ is a homeomorphism.", "Moreover, since the circular arc $\\alpha $ (as any set of $\\sigma $ -finite Hausdorff 1-measure) is removable for quasiconformal maps [26], the map $\\tilde{\\varphi }$ is locally quasiconformal, and hence locally quasisymmetric and orientation-preserving.", "It is also straightforward to see from the definitions and the relation $\\varphi ^{-1}(S)=U\\cap S$ that $\\tilde{\\varphi }^{-1}(\\tilde{S}^{\\prime })= \\tilde{D} \\cap \\tilde{S}$ .", "Similarly as in the beginning of the proof this implies that $\\tilde{\\varphi }\\colon \\tilde{D}\\cap \\tilde{S} \\rightarrow \\tilde{D}^{\\prime }\\cap \\tilde{S}^{\\prime }$ is a homeomorphism.", "Since it is also a local quasisymmetry and orientation-preserving, it follows again from Theorem REF that $\\tilde{\\varphi }\\colon \\tilde{D}\\cap \\tilde{S} \\rightarrow \\tilde{D}^{\\prime }\\cap \\tilde{S}^{\\prime }$ is a Schottky map.", "Note that $\\tilde{D}\\cap S= D\\cap S$ , that on this set the maps $\\tilde{\\varphi }$ and $\\varphi $ agree, and that $\\varphi (\\tilde{D}\\cap S)\\subseteq S$ .", "Thus, $\\varphi \\colon \\tilde{D}\\cap S\\rightarrow S$ is a Schottky map, and so $\\varphi \\colon U\\cap S\\rightarrow S$ is a Schottky map near $p$ .", "It follows that $\\varphi \\colon U\\cap S\\rightarrow S$ is a Schottky map as desired.", "The second part of the statement immediately follows from the first; indeed, $\\operatorname{crit}(\\psi )$ is a finite set and so $U=\\widehat{\\mathbb {C}}\\setminus \\operatorname{crit}(\\psi )$ is an open subset $\\widehat{\\mathbb {C}}$ on which $\\varphi =\\psi |_{U}\\colon U\\rightarrow \\widehat{\\mathbb {C}}$ is a locally quasiconformal map.", "Moreover, $\\varphi ^{-1}(S)=\\psi ^{-1}(S)\\cap U=S\\cap U$ .", "By the first part of the proof, $\\varphi $ and hence also $\\psi $ (restricted to $U\\cap S$ ) is a Schottky map of $U\\cap S=S\\setminus \\operatorname{crit}(\\psi )$ into $S$ .", "A relative Schottky set as in (REF ) is called locally porous at $p\\in S$ if there exists a neighborhood $U$ of $p$ , and constants $r_0>0$ and $C\\ge 1$ such that for each $q\\in S\\cap U$ and $r\\in (0, r_0]$ there exists $i\\in I$ with $B_i\\cap B(q,r)\\ne \\emptyset $ and $r/C\\le \\operatorname{diam}(B_i) \\le Cr$ .", "The relative Schottky set $S$ is called locally porous if it is locally porous at every point $p\\in S$ .", "Every locally porous relative Schottky set has measure zero since it cannot have Lebesgue density points.", "For Schottky maps on locally porous Schottky sets very strong rigidity and uniqueness statements are valid such as Theorems REF and REF stated in the introduction.", "We will need another result of a similar flavor.", "Theorem 5.2 (Me3, Theorem 4.1) Let $S$ be a locally porous relative Schottky set in a region $D\\subseteq , let$ U be an open set such that $S\\cap U$ is connected, and $u\\colon S\\cap U\\rightarrow S$ be a Schottky map.", "Suppose that there exists a point $a\\in S\\cap U$ with $u(a)=a$ and $u^{\\prime }(a)=1$ .", "Then $u=\\operatorname{id}|_{S\\cap U}$ .", "A functional equation in the unit disk no As discussed in the introduction, for the proof of Theorem REF we will establish a functional equation of form (REF ) for the maps in question.", "For postcritically-finite maps $f$ and $g$ this leads to strong conclusions based on the following lemma.", "Recall that $P_k(z)=z^k$ for $k\\in {\\mathbb {N}}$ .", "Lemma 6.1 Let $\\phi \\colon \\partial \\partial be an orientation-preserving homeo\\-morphism, and suppose that there exist numbers $ k,l,nN$, $ k2$, such that\\begin{equation} (P_l\\circ \\phi )(z)= (P_n\\circ \\phi \\circ P_k)(z) \\quad \\text{for $z\\in \\partial $.", "}Then l=nk and there exists a\\in with a^{n(k-1)}=1 such that \\phi (z)=az for all z\\in \\partial .\\end{equation}This lemma implies that we can uniquely extend $$ to a conformal homeomorphismfrom $$ onto itself.", "It is also important that this extension preserves the basepoint $ 0$.$ By considering topological degrees, one immediately sees that $l=nk$ .", "So if we introduce the map $\\psi := P_n\\circ \\phi $ , then () can be rewritten as $ P_k\\circ \\psi =\\psi \\circ P_k \\quad \\text{on $\\partial $.", "}Here the map \\psi \\colon \\partial \\partial has degree n. We claim that this in combination with (\\ref {eq:basiceq2}) implies that for a suitable constant b we have \\psi (z)=bz^n for z\\in \\partial .$ Indeed, there exists a continuous function $\\alpha \\colon {\\mathbb {R}}\\rightarrow {\\mathbb {R}}$ with $\\alpha (t+2\\pi )=\\alpha (t)$ such that $ \\psi (e^{ i t}) = \\exp ( i n t+ i\\alpha (t)) \\quad \\text{for $t\\in {\\mathbb {R}}$}.", "$ By (REF ) we have $ \\exp ( i k n t+ i k\\alpha (t)) = (\\psi (e^{ i t}))^k = \\psi (e^{ i k t})=\\exp ( i kn t+ i\\alpha (kt))$ for $t\\in {\\mathbb {R}}$ .", "This implies that there exists a constant $c\\in {\\mathbb {R}}$ such that $ \\alpha (t)=\\frac{1}{k} \\alpha (tk)+c \\quad \\text{for $t\\in {\\mathbb {R}}$}.", "$ Since $\\alpha $ is $2\\pi $ -periodic, the right-hand side of this equation is $2\\pi /k$ -periodic as a function of $t$ .", "Hence $\\alpha $ is $2\\pi /k$ -periodic.", "Repeating this argument, we see that $\\alpha $ is $2\\pi /k^m$ -periodic for all $m\\in {\\mathbb {N}}$ , and so has arbitrarily small periods (note that $k\\ge 2$ ).", "Since $\\alpha $ is continuous, it follows that $\\alpha $ is constant.", "Hence $\\psi (z)=bz^n$ for $z\\in \\partial with a suitable constant $ b.", "It follows that $ \\psi (z)=b z^n=\\phi (z)^n \\quad \\text{for $z\\in \\partial $.$$Therefore, $\\phi (z)=az$ for $z\\in \\partial with a constant $a\\in , $a\\ne 0$.Inserting this expression for $\\phi $ into (\\ref {eq:basiceq}) and using $l=nk$, we conclude that $a^{n(k-1)}=1$ as desired.", "}$ Proof of Theorem  REF The proof will be given in several steps.", "Step I.", "We first fix the setup.", "We can freely pass to iterates of the maps $f$ or $g$ , because this changes neither their Julia sets nor their postcritical sets.", "We can also conjugate the maps by Möbius transformations.", "Therefore, as in Section , we may assume that $\\mathcal {J}(f), \\mathcal {J}(g)\\subseteq \\tfrac{1}{2} { and } f^{-1}( , g^{-1}(\\subseteq $ Moreover, without loss of generality, we may require that $\\xi $ is orientation-preserving, for otherwise we can conjugate $g$ by $z\\mapsto \\overline{z}$ .", "Since the peripheral circles of $\\mathcal {J}(f)$ and $\\mathcal {J}(g)$ are uniform quasicircles, by Theorem REF the map $\\xi $ extends (non-uniquely) to a quasiconformal, and hence quasisymmetric, map of the whole sphere.", "Then $\\xi (\\mathcal {J}(f))=\\mathcal {J}(g)$ and $\\xi (\\mathcal {F}(f))=\\mathcal {F}(g)$ .", "Since $\\infty $ lies in Fatou components of $f$ and $g$ , we may also assume that $\\xi (\\infty )=\\infty $ (this normalization ultimately depends on the fact that for every point $p\\in there exists a quasiconformal homeomorphism $$on $ C$ with $ (0)=p$ that is the identity outside $ ).", "Then $\\xi $ is a quasisymmetry of $ with respect to the Euclidean metric.", "In the following, all metric notions will refer to this metric.Finally, we define $ g=-1 g$.$ Step II.", "We now carefully choose a location for a “blow-down\" by branches of $f^{-n}$ which will be compensated by a “blow-up\" by iterates of $g$ (or rather $g_\\xi $ ).", "Since repelling periodic points of $f$ are dense in $\\mathcal {J}(f)$ (see [3]), we can find such a point $p$ in $\\mathcal {J}(f)$ that does not lie in $\\operatorname{post}(f)$ .", "Let $\\rho >0$ be a small positive number such that the disk $U_0:=B(p, 3\\rho )\\subseteq is disjoint from $ post(f)$.Since $ p$ is periodic, there exists $ dN$ such that $ fd(p)=p$.", "Let $ U1 be the component of $f^{-d}(U_0)$ that contains $p$ .", "Since $U_0\\cap \\operatorname{post}(f)=\\emptyset $ , the set $U_1$ is a simply connected region, and $f^{d}$ is a conformal map from $U_1$ onto $U_0$ as follows from Lemma REF .", "Then there exists a unique inverse branch $f^{-d}$ with $f^{-d}(p)=p$ that is a conformal map of $U_0$ onto $U_1$ .", "Since $p$ is a repelling fixed point for $f$ , it is an attracting fixed point for this branch $f^{-d}$ .", "By possibly choosing a smaller radius $\\rho >0$ in the definition of $U_0=B(p, 3\\rho )$ and by passing to an iterate of $f^d$ , we may assume that $U_1\\subseteq U_0$ and that $\\operatorname{diam}(f^{-n_k}(U_0))\\rightarrow 0$ as $k\\rightarrow \\infty $ .", "Here $n_k=dk$ for $k\\in {\\mathbb {N}}$ , and $f^{-n_k}$ is the branch obtained by iterating the branch $f^{-d}$ $k$ -times.", "Note that $f^{-n_k}(p)=p$ and $f^{-n_k}$ is a conformal map of $U_0$ onto a simply connected region $U_k$ .", "Then $p\\in U_k\\subseteq U_{k-1}$ for $k\\in {\\mathbb {N}}$ , and $\\operatorname{diam}(U_k)\\rightarrow 0$ as $k\\rightarrow \\infty $ .", "The choice of these inverse branches is consistent in the sense that we have $f^{n_{k+1}-n_k}\\circ f^{-n_{k+1}}=f^{-n_k}$ on $B(p, 3\\rho )$ for all $k\\in {\\mathbb {N}}$ .", "Note that this consistency condition remains valid if we replace the original sequence $\\lbrace n_k\\rbrace $ by a subsequence.", "Let $\\tilde{r}_k>0$ be the smallest number such that $f^{-n_k}(B(p, 2\\rho ))\\subseteq \\tilde{B}_k:=B(p,\\tilde{r}_k).$ Since $p\\in f^{-n_k}(B(p, 2\\rho ))$ we have $\\operatorname{diam}\\big (f^{-n_k}(B(p, 2\\rho ))\\big )\\approx \\tilde{r}_k$ .", "Here and below $\\approx $ indicates implicit positive multiplicative constants independent of $k\\in {\\mathbb {N}}$ .", "It follows that $\\tilde{r}_k\\rightarrow 0$ as $k\\rightarrow \\infty $ .", "Moreover, since $f^{-n_k}$ is conformal on the larger disk $B(p,3\\rho )$ , Koebe's distortion theorem implies that $ \\operatorname{diam}\\big (f^{-n_k}(B(p, 2\\rho ))\\big )\\approx \\operatorname{diam}\\big (f^{-n_k}(B(p, \\rho ))\\big ).", "$ Let $r_k>0$ be the smallest number such that $\\xi (\\tilde{B}_k)\\subseteq B_k:=B(\\xi (p), r_k)$ .", "Again $r_k\\rightarrow 0$ as $k\\rightarrow \\infty $ by continuity of $\\xi $ .", "We now want to apply the conformal elevator given by iterates of $g$ to the disks $B_k$ .", "For this we choose $\\epsilon _0>0$ for the map $g$ as in Section .", "By applying the conformal elevator as described in Section , we can find iterates $g^{m_k}$ such that $g^{m_k}(B_k)$ is blown up to a definite, but not too large size, and so $\\operatorname{diam}(g^{m_k}(B_k))\\approx 1$ .", "Step III.", "Now we consider the composition $h_k=g_\\xi ^{m_k}\\circ f^{-n_k}= \\xi ^{-1}\\circ g^{m_k}\\circ \\xi \\circ f^{-n_k}$ defined on $B(p,2\\rho )$ for $k\\in {\\mathbb {N}}$ .", "We want to show that this sequence subconverges locally uniformly on $B(p,2\\rho )$ to a (non-constant) quasiregular map $h\\colon B(p, 2\\rho )\\rightarrow .$ Since $f^{-n_k}$ maps $B(q,2\\rho )$ conformally into $\\tilde{B}_k$ , $\\xi $ is a quasiconformal map with $\\xi (\\tilde{B}_k)\\subseteq B_k$ , and $g^{m_k}$ is holomorphic on $\\tilde{B}_k$ , we conclude that the maps $h_k$ are uniformly quasiregular on $B(q,2\\rho )$ , i.e., $K$ -quasiregular with $K\\ge 1$ independent of $k$ .", "The images $h_k(B(q,2\\rho ))$ are contained in a small Euclidean neighborhood of $\\mathcal {J}(g)$ and hence in a fixed compact subset of $.", "Standardconvergence results for $ K$-quasiregular mappings \\cite [p.~182, Corollary 5.5.7]{AIM} imply that the sequence $ {hk}$subconverges locally uniformly on $ B(q,2)$ to a map $ hB(q,2) that is also quasiregular, but possibly constant.", "By passing to a subsequence if necessary, we may assume that $h_k\\rightarrow h$ locally uniformly on $B(q,2\\rho )$ .", "To rule out that $h$ is constant, it is enough to show that for smaller disk $B(p,\\rho )$ there exists $\\delta >0$ such that $\\operatorname{diam}\\big (h_k(B(p,\\rho ))\\big )\\ge \\delta $ for all $k\\in {\\mathbb {N}}$ .", "We know that $\\operatorname{diam}\\big (f^{-n_k}(B(p,\\rho ))\\big )\\approx \\operatorname{diam}\\big (f^{-n_k}(B(p,2\\rho ))\\big ) \\approx \\operatorname{diam}(\\tilde{B}_k).", "$ Moreover, since $\\xi $ is a quasisymmetry and $f^{-n_k}(B(p,\\rho ))\\subseteq \\tilde{B}_k$ , this implies $\\operatorname{diam}\\big (\\xi (f^{-n_k}(B(p,\\rho )))\\big )\\approx \\operatorname{diam}(\\xi (\\tilde{B}_k))\\approx \\operatorname{diam}(B_k).", "$ So the connected set $\\xi (f^{-n_k}(B(p,\\rho )))\\subseteq B_k$ is comparable in size to $B_k$ .", "By Lemma REF  (a) the conformal elevator blows it up to a definite, but not too large size, i.e., $\\operatorname{diam}\\big ((g^{m_k}\\circ \\xi \\circ f^{-n_k})(B(p,\\rho ))\\big )\\approx 1.$ Since the sets $(g^{m_k}\\circ \\xi \\circ f^{-n_k})(B(p,\\rho ))$ all meet $\\mathcal {J}(g)$ , they stay in a compact part of $, and so we still get a uniform lower bound for the diameter of these sets if we apply the homeomorphism $ -1$; in other words,$$\\operatorname{diam}\\big ((\\xi ^{-1}\\circ g^{m_k}\\circ \\xi \\circ f^{-n_k})(B(p,\\rho ))\\big )=\\operatorname{diam}\\big (h_k(B(p,\\rho ))\\big )\\approx 1$$as claimed.", "We conclude that $ hkh$ locally uniformly on $ B(p, 2r)$, where $ h$ is non-constant and quasiregular.$ The quasiregular map $h$ has at most countably many critical points, and so there exists a point $q\\in B(p, 2\\rho )\\cap \\mathcal {J}(f)$ and a radius $r>0$ such that $B(q, 2r)\\subseteq B(p, 2\\rho )$ and $h$ is injective on $B(q, 2r)$ and hence quasiconformal.", "Standard topological degree arguments imply that at least on the smaller disk $B(q,r)$ the maps $h_k$ are also injective and hence quasiconformal for all $k$ sufficiently large.", "By possibly disregarding finitely of the maps $h_k$ , we may assume that $h_k$ is quasiconformal on $B(q,r)$ for all $k\\in {\\mathbb {N}}$ .", "To summarize, we have found a disk $B(q,r)$ centered at a point $q\\in \\mathcal {J}(f)$ such that the maps $h_k$ are defined and quasiconformal on $B(q,r)$ and converge uniformly on $B(q,r)$ to a quasiconformal map $h$ .", "From the invariance properties of Julia and Fatou sets and the mapping properties of $\\xi $ , it follows that $ h_k(B(q, r)\\cap \\mathcal {J}(f))\\subseteq \\mathcal {J}(f) \\text{ and } h_k(B(q, r)\\cap \\mathcal {F}(f))\\subseteq \\mathcal {F}(f)$ for each $k\\in {\\mathbb {N}}$ .", "Hence $h_k^{-1}(\\mathcal {J}(f))=\\mathcal {J}(f)\\cap B(q,r)$ for each map $h_k\\colon B(q,r)\\rightarrow , $ kN$.$ Since $\\mathcal {J}(f)$ is closed and $h_k\\rightarrow h$ uniformly on $B(q, r)$ , we also have $h(B(q, r)\\cap \\mathcal {J}(f))\\subseteq \\mathcal {J}(f)$ .", "To get a similar inclusion relation also for the Fatou set, we argue by contradiction and assume that there exists a point $z\\in B(q,r)\\cap \\mathcal {F}(f)$ with $h(z)\\notin \\mathcal {F}(f)$ .", "Then $h(z)\\in \\mathcal {J}(f)$ .", "Since $B(q,r)\\cap \\mathcal {F}(f)$ is an open neighborhood of $z$ , it follows again from standard topological degree arguments that for large enough $k\\in {\\mathbb {N}}$ there exists a point $z_k\\in B(q,r)\\cap \\mathcal {F}(f)$ with $h_k(z_k)=h(z)\\in \\mathcal {J}(f)$ .", "This is impossible by (REF ) and so indeed $h(B(q, r)\\cap \\mathcal {F}(f))\\subseteq \\mathcal {F}(f)$ .", "We conclude that $h^{-1}(\\mathcal {J}(f))=\\mathcal {J}(f)\\cap B(q,r).$ Step IV.", "We know by Theorem REF that the peripheral circles of $\\mathcal {J}(f)$ are uniformly relatively separated uniform quasicircles.", "According to Theorems REF and REF , there exists a quasisymmetric map $\\beta $ on $\\widehat{\\mathbb {C}}$ such that $S=\\beta (\\mathcal {J}(f))$ is a round Sierpiński carpet.", "We may assume $S\\subseteq .$ We conjugate the map $f$ by $\\beta $ to define a new map $\\beta \\circ f\\circ \\beta ^{-1}$ .", "By abuse of notation we call this new map also $f$ .", "Note that this map and its iterates are in general not rational anymore, but quasiregular maps on $\\widehat{\\mathbb {C}}$ .", "Similarly, we conjugate $ g_\\xi , h_k, h$ by $\\beta $ to obtain new maps for which we use the same notation for the moment.", "If $V=\\beta (B(q,r))$ , then the new maps $h_k$ and $h$ are quasiconformal on $V$ , and $h_k\\rightarrow h$ uniformly on $V$ .", "Lemma 7.1 There exist $N\\in {\\mathbb {N}}$ and an open set $W\\subseteq V$ such that $S\\cap W$ is non-empty and connected and $h_k\\equiv h$ on $S\\cap W$ for all $k\\ge N$ .", "Since $\\mathcal {J}(f)$ is porous and $S$ is a quasisymmetric image of $\\mathcal {J}(f)$ , the set $S$ is also porous (and in particular locally porous as defined in Section ).", "The maps $h_k$ and $h$ are quasiconformal on $V=\\beta (B(q,r))$ , and $h_k\\rightarrow h$ uniformly on $V$ as $k\\rightarrow \\infty $ .", "The relations (REF ) and (REF ) translate to $h^{-1}(S)=S\\cap V$ and $h_k^{-1}(S)=S\\cap V$ for $k\\in {\\mathbb {N}}$ .", "So Lemma REF implies that the maps $h\\colon S\\cap V\\rightarrow S$ and $h_k\\colon S \\cap V\\rightarrow S$ for $k\\in {\\mathbb {N}}$ are Schottky maps.", "Each of these restrictions is actually a homeomorphism onto its image.", "There are only finitely many peripheral circles of $\\mathcal {J}(f)$ that contain periodic points of our original rational map $f$ ; indeed, if $J$ is such a peripheral circle, then $f^n(J)=J$ for some $n\\in {\\mathbb {N}}$ as follows from Lemma REF ; but then $J$ bounds a periodic Fatou component of $f$ which leaves only finitely many possibilities for $J$ .", "Since the periodic points of $f$ are dense in $\\mathcal {J}(f)$ , we conclude that we can find a periodic point of $f$ in $ \\mathcal {J}(f)\\cap B(q,r)$ that does not lie on a peripheral circle of $\\mathcal {J}(f)$ .", "Translated to the conjugated map $f$ , this yields existence of a point $a\\in S\\cap V$ that does not lie on a peripheral circle of the Sierpiński carpet $S$ such that $f^n(a)=a$ for some $n\\in {\\mathbb {N}}$ .", "The invariance property of the Julia set gives $f^{-n}(S)=S$ , and so Lemma REF implies that $f^n\\colon S\\setminus \\operatorname{crit}(f^n)\\rightarrow S$ is a Schottky map.", "Note that $a\\notin \\operatorname{crit}(f^n)$ as follows from the fact that for our original rational map $f$ , none of its periodic critical points lies in the Julia set.", "Therefore, our Schottky map $f^n\\colon S\\setminus \\operatorname{crit}(f^n)\\rightarrow S$ has a derivative at the point $a\\in S\\setminus \\operatorname{crit}(f^n)$ in the sense of (REF ).", "If $(f^n)^{\\prime }(a)=1$ , then Theorem REF implies that $f^n\\equiv \\operatorname{id}|_{S\\setminus \\operatorname{crit}(f^n)}$ , and hence by continuity $f^n$ is the identity on $S$ .", "This is clearly impossible, and therefore $(f^n)^{\\prime }(a)\\ne 1$ .", "Since $a\\in S\\cap V$ does not lie on a peripheral circle of $S$ , as in the proof of Lemma REF we can find a small Jordan region $W$ with $a\\in W\\subseteq V$ and $W\\cap \\operatorname{crit}(f^n)=\\emptyset $ such that $\\partial W\\subseteq S$ .", "Then $S\\cap W$ is non-empty and connected.", "We now restrict our maps to $W$ .", "Then $h_k\\colon S\\cap W\\rightarrow S$ is a Schottky map and a homeomorphism onto its image for each $k\\in {\\mathbb {N}}$ .", "The same is true for the map $h\\colon S \\cap W\\rightarrow S$ .", "Moreover $h_k\\rightarrow h$ as $k\\rightarrow \\infty $ uniformly on $W\\cap S$ .", "Finally, the map $u=f^n$ is defined on $S\\cap W$ and gives a Schottky map $u\\colon S\\cap W\\rightarrow S$ such that for $a\\in S\\cap W$ we have $u(a)=a$ and $u^{\\prime }(a)\\ne 1$ .", "So we can apply Theorem REF to conclude that there exists $N\\in {\\mathbb {N}}$ such that $h_k\\equiv h$ on $S\\cap W$ for all $k\\ge N$ .", "By the previous lemma we can fix $k\\ge N$ so that $h_k=h_{k+1}$ on $S\\cap W$ .", "If we go back to the definition of the maps $h_k$ and use the consistency of inverse branches (which is also true for the maps conjugated by $\\beta $ ), then we conclude that $ h_{k+1}=g_\\xi ^{m_{k+1}}\\circ f^{-n_{k+1}}= h_k =g_\\xi ^{m_{k}}\\circ f^{-n_{k}}=g_\\xi ^{m_{k}}\\circ f^{n_{k+1}-n_k}\\circ f^{-n_{k+1}}$ on the set $S \\cap W$ .", "Cancellation gives $ g_\\xi ^{m_{k+1}}=g_\\xi ^{m_{k}}\\circ f^{n_{k+1}-n_k}$ on $f^{-n_{k+1}}(S\\cap W)\\subseteq S$ .", "The two maps on both sides of the last equation are quasiregular maps $\\psi \\colon \\widehat{\\mathbb {C}}\\rightarrow \\widehat{\\mathbb {C}}$ with $\\psi ^{-1}(S)=S$ .", "It follows from Lemma REF that they are Schottky maps $S \\cap U\\rightarrow S$ if $U\\subseteq \\widehat{\\mathbb {C}}$ is an open set that does not contain any of the finitely many critical points of the maps; in particular, $g_\\xi ^{m_{k+1}}$ and $g_\\xi ^{m_k}\\circ f^{n_{k+1}-n_k}$ are Schottky maps $S \\cap U\\rightarrow S$ , where $U=\\widehat{\\mathbb {C}}\\setminus (\\operatorname{crit}(g_\\xi ^{m_{k+1}})\\cup \\operatorname{crit}(g_\\xi ^{m_k}\\circ f^{n_{k+1}-n_k}))$ .", "Since $U$ has a finite complement in $\\widehat{\\mathbb {C}}$ , the non-degenerate connected set $f^{-n_{k+1}}(S\\cap W)\\subseteq S$ has an accumulation point in $S \\cap U$ .", "Theorem REF yields $ g_\\xi ^{m_{k+1}}=g_\\xi ^{m_{k}}\\circ f^{n_{k+1}-n_k}$ on $S\\cap U$ , and hence on all of $S$ by continuity.", "If we conjugate back by $\\beta ^{-1}$ , this leads to the relation $ g_\\xi ^{m_{k+1}}=g_\\xi ^{m_{k}}\\circ f^{n_{k+1}-n_k}$ on $\\mathcal {J}(f)$ for the original maps.", "We conclude that there exist integers $m,m^{\\prime },n\\in {\\mathbb {N}}$ such that for the original maps we have $ g^{m^{\\prime }}\\circ \\xi = g^m\\circ \\xi \\circ f^n$ on $\\mathcal {J}(f)$ .", "Step V. Equation (REF ) gives us a crucial relation of $\\xi $ to the dynamics of $f$ and $g$ on their Julia sets.", "We will bring (REF ) into a convenient form by replacing our original maps with iterates.", "Since $\\mathcal {J}(f)$ is backward invariant, counting preimages of generic points in $\\mathcal {J}(f)$ under iterates of $f$ and of points in $\\mathcal {J}(g)$ under iterates of $g$ leads to the relation $\\deg (g)^{m^{\\prime }-m}=\\deg (f)^n,$ and so $m^{\\prime }-m>0$ .", "If we post-compose both sides in (REF ) by a suitable iterate of $g$ , and then replace $f$ by $f^n$ and $g$ by $g^{m^{\\prime }-m}$ , we arrive at a relation of the form $ g^{l+1}\\circ \\xi = g^l\\circ \\xi \\circ f.$ on $\\mathcal {J}(f)$ for some $l\\in {\\mathbb {N}}$ .", "Note that this equation implies that we have $ g^{n+k}\\circ \\xi = g^n\\circ \\xi \\circ f^k \\ \\text{ for all $k,n\\in {\\mathbb {N}}$ with $n\\ge l$}.$ Step VI.", "In this final step of the proof, we disregard the non-canonical extension to $\\widehat{\\mathbb {C}}$ of our original homeomorphism $\\xi \\colon \\mathcal {J}(f)\\rightarrow \\mathcal {J}(g)$ chosen in the beginning.", "Our goal is to apply (REF ) to produce a natural extension of $\\xi $ mapping each Fatou component of $f$ conformally onto a Fatou component of $g$ .", "Note that if $U$ is a Fatou component of $f$ , then $\\partial U$ is a peripheral circle of $\\mathcal {J}(f)$ .", "Since $\\xi $ sends each peripheral circle of $\\mathcal {J}(f)$ to a peripheral circle of $\\mathcal {J}(g)$ , the image $\\xi (\\partial U)$ bounds a unique Fatou component $V$ of $g$ .", "This sets up a natural bijection between the Fatou components of our maps, and our goal is to conformally “fill in the holes\".", "So let $\\mathcal {C}_f$ and $\\mathcal {C}_g$ be the sets of Fatou components of $f$ and $g$ , respectively.", "By Lemma REF we can choose a corresponding family $\\lbrace \\psi _U: U\\in \\mathcal {C}_f\\rbrace $ of conformal maps.", "Since each Fatou component of $f$ is a Jordan region, we can consider $\\psi _U$ as a conformal homeomorphism from $\\overline{U}$ onto $\\overline{.", "Similarly, we obtain a family of conformal homeomorphisms \\tilde{\\psi }_V: \\overline{V}\\rightarrow \\overline{ for V in \\mathcal {C}_g.These Fatou components carry distinguished basepointsp_U=\\psi _U^{-1}(0)\\in U for U\\in \\mathcal {C}_f and \\tilde{p}_V=\\tilde{\\psi }_V^{-1}(0)\\in V for V\\in \\mathcal {C}_g.", "}We will now first extend \\xi to the periodic Fatou components of f, and thenuse the Lifting Lemma~\\ref {lem:lifting} to get extensions to Fatou components of higher and higher level (as defined in the proof of Lemma~\\ref {lem:FatouDyn}).", "In this argument it will be important to ensure that these extensions are basepoint-preserving.", "}First let $ U$ be a periodic Fatou component of $ f$.", "We denote by $ kN$ the period of $ U$, and define $ V$ to be the Fatou component of $ g$ bounded by $ (U)$, and$ W=gl(V)$, where $ lN$ is as in (\\ref {eq:main4}).Then(\\ref {eq:main4}) implies that $ W$, and hence $ W$ itself, is invariant under $ gk$.", "ByLemma~\\ref {lem:FatouDyn} the basepoint-preserving homeomorphisms $ U (U, pU)( , 0)$ and $ W(W,pW)( ,0)$ conjugate $ fk$ and $ gk$, respectively, to power maps.Since $ f$ and $ g$ are postcritically-finite, the periodic Fatou components $ U$ and $ W$ are superattracting, and thus the degrees of these power maps are at least 2.$ Again by Lemma REF the map $\\tilde{\\psi }_W\\circ g^l\\circ \\tilde{\\psi }^{-1}_V$ is a power map.", "Since $U,V,W$ are Jordan regions, the maps $\\psi _U, \\tilde{\\psi }_V, \\tilde{\\psi }_W$ give homeomorphisms between the boundaries of the corresponding Fatou components and $\\partial .", "Since $$ is anorientation-preserving homeomorphism of $ U$ onto $ V$, the map$ =VU-1$gives an orientation-preserving homeomorphism on $ .", "Now (REF ) for $n=l$ implies that on $\\partial we have{\\begin{@align*}{1}{-1} P_{d_3}\\circ \\phi &=\\tilde{\\psi }_W\\circ g^{k+l}\\circ \\tilde{\\psi }_V^{-1} \\circ \\phi = \\tilde{\\psi }_W\\circ g^{k+l}\\circ \\xi \\circ \\psi _U^{-1}\\\\&= \\tilde{\\psi }_W\\circ g^{l}\\circ \\xi \\circ f^k\\circ \\psi ^{-1}_U= \\tilde{\\psi }_W\\circ g^{l}\\circ \\xi \\circ \\psi ^{-1}_U \\circ P_{d_1}\\\\&= \\tilde{\\psi }_W\\circ g^{l}\\circ \\tilde{\\psi }^{-1}_V\\circ \\phi \\circ P_{d_1}= P_{d_2}\\circ \\phi \\circ P_{d_1}\\end{@align*}}for some $ d1, d 2, d3N$ with $ d12$.", "Lemma~\\ref {L:Rot} implies that $$ extends to $$ as a rotation around $ 0$, also denoted by $$.", "In particular, $ (0)=0$, and so $$ preserves the basepoint $ 0$ in $$.", "If we define $ =V-1U$ on $U$, then $$ is a conformal homeomorphismof $ (U,pU)$ onto $ (V, pV)$.$ In this way, we can conformally extend $\\xi $ to every periodic Fatou component of $f$ so that $\\xi $ maps the basepoint of a Fatou component to the basepoint of the image component.", "To get such an extension also for the other Fatou components $V$ of $f$ , we proceed inductively on the level of the Fatou component.", "So suppose that the level of $V$ is $\\ge 1$ and that we have already found an extension for all Fatou components with a level lower than $V$ .", "This applies to the Fatou component $U=f(V)$ of $f$ , and so a conformal extension $(U, p_U)\\rightarrow ( U^{\\prime }, \\tilde{p}_{U^{\\prime }})$ of $\\xi |_{\\partial U}$ exists, where $U^{\\prime }$ is the Fatou component of $g$ bounded by $\\xi (\\partial U)$ .", "Let $V^{\\prime }$ be the Fatou component of $g$ bounded by $\\xi (\\partial V)$ , and $W=g^{l+1}(V^{\\prime })$ .", "Then by using (REF ) on $\\partial V$ we conclude that $g^l(U^{\\prime })=W$ .", "Define $\\alpha =g^l\\circ \\xi |_{\\overline{U}}\\circ f|_{\\overline{V}}$ and $\\beta =\\xi |_{\\partial V}$ .", "Then the assumptions of Lemma REF are satisfied for $D=V$ , $p_D=p_{V}$ , and the iterate $g^{l+1}\\colon V^{\\prime }\\rightarrow W$ of $g$ .", "Indeed, $\\alpha $ is continuous on $\\overline{V}$ and holomorphic on $V$ , we have $\\alpha ^{-1}(p_W)=f^{-1}( \\xi |_{\\overline{U}}^{-1}(\\tilde{p}_{U^{\\prime }}))=f^{-1}(p_U)=\\lbrace p_V\\rbrace ,$ and $g^{l+1}\\circ \\beta = g^{l+1}\\circ \\xi |_{\\partial V}= g^{l}\\circ \\xi |_{\\partial U}\\circ f|_{\\partial V}=\\alpha .$ Since $\\beta $ is a homeomorphism, it follows that there exists a conformal homeomorphism $\\tilde{\\alpha }$ of $(\\overline{V}, p_V)$ onto $(\\overline{V}^{\\prime }, \\tilde{p}_{V^{\\prime }})$ such that $\\tilde{\\alpha }|_{\\partial V}=\\beta =\\xi |_{\\partial V}$ .", "In other words, $\\tilde{\\alpha }$ gives the desired basepoint preserving conformal extension to the Fatou component $V$ .", "This argument shows that $\\xi $ has a (unique) conformal extension to each Fatou component of $f$ .", "We know that the peripheral circles of $\\mathcal {J}(f)$ are uniform quasicircles, and that $\\mathcal {J}(f)$ has measure zero.", "Lemma REF now implies that $\\xi $ extends to a Möbius transformation on $\\widehat{\\mathbb {C}}$ , which completes the proof.", "The techniques discussed also easily lead to a proof of Corollary REF .", "Let $f$ be a postcritically-finite rational map whose Julia set $\\mathcal {J}(f)$ is a Sierpiński carpet.", "Let $G$ be the group of all Möbius transformations $\\xi $ on $\\widehat{\\mathbb {C}}$ with $\\xi (\\mathcal {J}(f))=\\mathcal {J}(f)$ , and $H$ be the subgroup of all elements in $G$ that preserve orientation.", "By Theorem REF it is enough to prove that $G$ is finite.", "Since $G=H$ or $H$ has index 2 in $G$ , this is true if we can show that $H$ is finite.", "Note that the group $H$ is discrete, i.e., there exists $\\delta _0>0$ such that $ \\sup _{p\\in \\widehat{\\mathbb {C}}} \\sigma (\\xi (p), p)\\ge \\delta _0$ for all $\\xi \\in H$ with $\\xi \\ne \\operatorname{id}_{\\widehat{\\mathbb {C}}}$ .", "Indeed, we choose $\\delta _0>0$ so small that there are at least three distinct complementary components $D_1,D_2,D_3$ of $\\mathcal {J}(f)$ that contain disks of radius $\\delta _0$ .", "In order to show (REF ), suppose that $\\xi \\in H$ and $\\sigma (\\xi (p), p)<\\delta _0$ for all $p\\in \\widehat{\\mathbb {C}}$ .", "Then $\\xi (D_i)\\cap D_i\\ne \\emptyset $ , and so $\\xi (\\overline{D}_i)=\\overline{D}_i$ for $i=1, 2,3$ , because $\\xi $ permutes the closures of Fatou components of $f$ .", "This shows that $\\xi $ is a conformal homeomorphism of the closed Jordan region $\\overline{D}_i$ onto itself.", "Hence $\\xi $ has a fixed point in $\\overline{D}_i$ .", "Since $\\mathcal {J}(f)$ is a Sierpiński carpet, the closures $\\overline{D}_1, \\overline{D}_2, \\overline{D}_3$ are pairwise disjoint, and we conclude that $\\xi $ has at least three fixed points.", "Since $\\xi $ is an orientation-preserving Möbius transformation, this implies that $\\xi =\\operatorname{id}_{\\widehat{\\mathbb {C}}}$ , and the discreteness of $H$ follows.", "We will now analyze type of Möbius transformations contained in the group $H$ (for the relevant classification of Möbius transformations up to conjugacy, see [2]).", "So consider an arbitrary $\\xi \\in H$ , $\\xi \\ne \\text{id}_{\\widehat{\\mathbb {C}}}$ .", "Then $\\xi $ cannot be loxodromic; indeed, otherwise $\\xi $ has a repelling fixed point $p$ which necessarily has to lie in $\\mathcal {J}(f)$ .", "We now argue as in the proof of Theorem REF and “blow down\" by iterates $\\xi ^{-n_k}$ near $p$ and “blow up\" by the conformal elevator using iterates $f^{m_k}$ to obtain a sequence of conformal maps of the form $h_k=f^{m_k}\\circ \\xi ^{-n_k}$ that converge uniformly to a (non-constant) conformal limit function $h$ on a disk $B$ centered at a point in $q\\in \\mathcal {J}(f)$ .", "Again this sequence stabilizes, and so $h_{k+1}=h_k$ for large $k$ on a connected non-degenerate subset of $B$ , and hence on $\\widehat{\\mathbb {C}}$ by the uniqueness theorem for analytic functions.", "This leads to a relation of the form $f^k\\circ \\xi ^l=f^m$ , where $k,l,m\\in {\\mathbb {N}}$ .", "Comparing degrees we get $m=k$ , and so we have $f^{k}=f^{k}\\circ \\xi ^{-ln}$ for all $n\\in {\\mathbb {N}}$ .", "This is impossible, because $f^k=f^{k}\\circ \\xi ^{-ln}\\rightarrow f^{k}(p)$ near $p$ as $n\\rightarrow \\infty $ , while $f^k$ is non-constant.", "The Möbius transformation $\\xi $ cannot be parabolic either; otherwise, after conjugation we may assume that $\\xi (z)=z+a$ with $a\\in , $ a0$.", "Then necessarily $ J(f)$.", "On the other hand, we know thatthe peripheral circles of $ J(f)$ are uniform quasicircles that occur on all locations and scales with respect to the chordal metric.", "Translated to the Euclidean metric near $$ this means that $ J(f)$ has complementary components $ D$ with $ D$ that contain Euclidean disks of arbitrarily large radius, and in particular of radius $ >|a|$; but then $$ cannot move $ D$ off itself, and so $ (D)=D$ for the translation $$.This is impossible.$ Finally, $\\xi $ can be elliptic; then after conjugation we have $\\xi (z)=az$ with $a\\in and $ |a|=1$, $ a1$.", "Since $ H$ is discrete,$ a$ must be a root of unity, and so $$is a torsion element of $ H$.$ We conclude that $H$ is a discrete group of Möbius transformations such that each element $\\xi \\in H$ with $\\xi \\ne \\text{id}_{\\widehat{\\mathbb {C}}}$ is a torsion element.", "It is well-known that such a group $H$ is finite (one can derive this from [2] in combination with the considerations in [2]).", "no Throughout the paper we assume that the reader is familiar with basic notions and facts from the theory of quasiconformal maps and complex dynamics.", "We will review some relevant definitions and statements related to these topics in this and the following sections.", "We will almost exclusively deal with the complex plane $ equipped with the Euclidean metric (the distance between $ z$ and $ w$ denoted by $ |z-w|$), or the Riemann sphere $ C$equipped with the chordal metric $$.", "We denote by $ {z |z|<1} $ the open unit disk in $ .", "By default a set $M\\subseteq \\widehat{\\mathbb {C}}$ carries (the restriction of) the chordal metric $\\sigma $ , but we will usually specify the relevant metric in a given context.", "With an underlying space $X$ and a metric on $X$ understood, we denote by $B(p,r)$ the open ball of radius $r>0$ centered at $p\\in X$ , by $\\operatorname{diam}(M)$ the diameter, and by $\\operatorname{dist}(M,N)$ the distance of sets $M,N\\subseteq X$ .", "The cardinality of set $M$ is $\\#M\\in {\\mathbb {N}}_0\\cup \\lbrace \\infty \\rbrace $ , and $\\text{id}_M$ the identity map on $M$ .", "If $f\\colon X\\rightarrow Y$ is a map between sets $X$ and $Y$ and $A\\subseteq X$ , then we denote by $f|_A\\colon A\\rightarrow Y$ the restriction of $f$ to $A$ .", "We will now discuss quasiconformal and related maps.", "For general background on this topic we refer to [24], [1], [26], [12].", "A non-constant continuous map $f \\colon U \\rightarrow on a region $ U is called quasiregular if $f$ is in the Sobolev space $W_{\\rm loc}^{1,2}$ and if there exists a constant $K\\ge 1$ such that the (formal) Jacobi matrix $Df$ satisfies $||Df(z)||^2\\le K\\det (Df(z))$ for almost every $z \\in U$ .", "The condition that $f\\in W_{\\rm loc}^{1,2}$ means that the first distributional partial derivatives of $f$ are locally in the Lebesgue space $L^2$ .", "This definition requires only local coordinates and hence the notion of a quasiregularity can be extended to maps $f\\colon U\\rightarrow \\widehat{\\mathbb {C}}$ on regions $U\\subseteq \\widehat{\\mathbb {C}}$ .", "If $f$ is a homeomorphism onto its image in addition, then $f$ is called a quasiconformal map.", "The map $f$ is called locally quasiconformal if for every point $p\\in U$ there exists a region $V$ with $p\\in V\\subseteq U$ such that $f|_V$ is quasiconformal.", "Each quasiregular map $f\\colon U\\rightarrow \\widehat{\\mathbb {C}}$ is a branched covering map.", "This means that $f$ is an open map and the preimage $f^{-1}(q)$ of each point $q\\in \\widehat{\\mathbb {C}}$ is a discrete subset of $U$ .", "A point $p\\in U$ , near which the quasiregular map $f$ is not a local homeomorphism, is called a critical point of $f$ .", "These points are isolated in $U$ , and accordingly, the set $\\operatorname{crit}(f)$ of all critical points of $f$ is a discrete and relatively closed subset of $U$ .", "One can easily derive these statements from the fact that every quasiregular map $f\\colon U\\rightarrow \\widehat{\\mathbb {C}}$ on a region $U\\subseteq \\widehat{\\mathbb {C}}$ can be represented locally in the form $f=g\\circ \\varphi $ , where $g$ is a holomorphic map and $\\varphi $ is quasiconformal [1].", "Let $(X, d_X)$ and $(Y,d_Y)$ be metric spaces and let $f\\colon X\\rightarrow Y$ be a homeomorphism.", "The map $f$ is called quasisymmetric if there exists a homeomorphism $\\eta \\colon [0,\\infty )\\rightarrow [0,\\infty )$ such that $\\frac{d_Y(f(u),f(v))}{d_Y(f(u),f(w))}\\le \\eta \\bigg (\\frac{d_X(u,v)}{d_X(u,w)}\\bigg ),$ for every triple of distinct points $u,v,w\\in X$ .", "Suppose $U$ and $V$ are subregions of $\\widehat{\\mathbb {C}}$ .", "Then every orientation-preserving quasisymmetric homeomorphism $f\\colon U\\rightarrow V$ is quasiconformal.", "Conversely, every quasiconformal homeomorphism $f\\colon U\\rightarrow V$ is locally quasisymmetric, i.e., for every compact set $M\\subseteq U$ , the restriction $f|_M\\colon M\\rightarrow f(M)$ is a quasisymmetry (see [1]).", "Often it is important to keep track of the quantitative information that appears in the definition of quasiregular maps as in (REF ) or quasisymmetric maps as in (REF ).", "Then we speak of a $K$ -quasiregular map, or an $\\eta $ -quasisymmetry, etc.", "A Jordan curve $J\\subseteq \\widehat{\\mathbb {C}}$ is called a quasicircle if there exists a quasisymmetry $f\\colon \\partial J$ .", "This is equivalent to the requirement that there exists a constant $L\\ge 1$ such that $\\operatorname{diam}(\\alpha )\\le L\\sigma (u,v),$ whenever $u,v\\in {\\mathbb {N}}$ , $u\\ne v$ , and $\\alpha $ is the smaller subarc of $J$ with endpoints $u,v$ .", "If $\\lbrace J_k: k\\in {\\mathbb {N}}\\rbrace $ is a family of quasicircles , then we say that it consists of uniform quasicircles if condition (REF ) is true for some constant $L\\ge 1$ independent of $k\\in {\\mathbb {N}}$ .", "The family $\\lbrace J_k: k\\in {\\mathbb {N}}\\rbrace $ is said to be uniformly relatively separated if there exists a constant $c>0$ such that $ \\frac{\\operatorname{dist}(J_k, J_l)}{\\min \\lbrace \\operatorname{diam}(J_k), \\operatorname{diam}(J_l)\\rbrace }\\ge c$ for all $k,l\\in {\\mathbb {N}}$ , $k\\ne l$ .", "We conclude this section with an extension result that is needed in the proof of Theorem REF .", "Lemma 2.1 Let $S\\subseteq \\widehat{\\mathbb {C}}$ be a Sierpiński carpet written in the form $S =\\widehat{\\mathbb {C}}\\setminus \\bigcup _{k\\in {\\mathbb {N}}}D_k$ with pairwise disjoint Jordan regions $D_k\\subseteq \\widehat{\\mathbb {C}}$ , and suppose that the peripheral circles $\\partial D_k$ , $k\\in {\\mathbb {N}}$ , of $S$ are uniform quasicircles.", "Let $\\xi \\colon S\\rightarrow \\widehat{\\mathbb {C}}$ be an orientation-preserving quasisymmetric embedding of $S$ and suppose that each restriction $\\xi |_{\\partial D_k}\\colon \\partial D_k\\rightarrow \\widehat{\\mathbb {C}}$ , $k\\in {\\mathbb {N}}$ , extends to an embedding $\\xi _k\\colon \\overline{D}_k\\rightarrow \\widehat{\\mathbb {C}}$ that is conformal on $D_k$ .", "Then $\\xi $ has a unique quasiconformal extension $\\widetilde{\\xi }\\colon \\widehat{\\mathbb {C}}\\rightarrow \\widehat{\\mathbb {C}}$ that is conformal on $\\widehat{\\mathbb {C}}\\setminus S$ .", "Moreover, if $S$ has measure zero, then $\\widetilde{\\xi }$ is a Möbius transformation.", "Here we say that the embedding $\\xi \\colon S\\rightarrow \\widehat{\\mathbb {C}}$ is orientation-preserving if $\\xi $ has an extension to an orientation-preserving homeomorphism on the whole sphere $\\widehat{\\mathbb {C}}$ .", "This does not depend on the embedding and is equivalent to the following statement: if we orient each peripheral circle $\\partial _k D$ of the Sierpiński carpet as in the theorem so that $S$ lies “to the left\" of $\\partial D_k$ and if we equip $\\xi (\\partial D_k)$ with the induced orientation, then $\\xi (S)$ lies to the left of $\\xi (\\partial D_k)$ .", "The proof relies on the rather subtle, but well-known relation between quasiconformal, quasisymmetric, and quasi-Möbius maps (for the definition of the latter class and related facts see [5]).", "In the proof we will omit some details that can easily be extracted from the considerations in [5].", "Under the given assumption the image $S^{\\prime }=\\xi (S)$ is also a Sierpiński carpet that we can represent in the form $S^{\\prime }=\\widehat{\\mathbb {C}}\\setminus \\bigcup _{k\\in {\\mathbb {N}}}D^{\\prime }_k$ with pairwise disjoint Jordan regions $D^{\\prime }_k$ .", "Since $\\xi $ maps the peripheral circles of $S$ to the peripheral circles of $S^{\\prime }$ , we can choose the labeling so that $\\xi (\\partial D_k)=\\partial D_k^{\\prime }$ for $k\\in {\\mathbb {N}}$ .", "For each embedding $\\xi _k\\colon \\overline{D}_k \\rightarrow \\widehat{\\mathbb {C}}$ as in the statement of the lemma, we necessarily have $\\xi _k(\\overline{D}_k)=\\overline{D}^{\\prime }_k$ , because $\\xi _k(\\partial D_k)=\\xi (\\partial D_k)=\\partial D^{\\prime }_k$ and $\\xi $ is orientation-preserving.", "Moreover, $\\xi _k$ is uniquely determined by $\\xi |_{\\partial D_k}$ ; this follows from the classical fact that a homeomorphic extension of a conformal map between given Jordan regions is uniquely determined by the image of three distinct boundary points.", "Our original map $\\xi $ and the unique maps $\\xi _k$ , $k\\in {\\mathbb {N}}$ , piece together to a homeomorphism $\\tilde{\\xi }\\colon \\widehat{\\mathbb {C}}\\rightarrow \\widehat{\\mathbb {C}}$ that is conformal on $\\widehat{\\mathbb {C}}\\setminus S$ .", "Moreover, $\\tilde{\\xi }$ is the unique homeomorphic extension of $\\xi $ with this conformality property.", "The Jordan curves $\\partial D_k$ , $k\\in {\\mathbb {N}}$ , form a family of uniform quasicircles, and hence also their images $\\partial D^{\\prime }_k=\\xi (\\partial D_k)$ , $k\\in {\\mathbb {N}}$ , under the quasisymmetry $\\xi $ .", "This implies that Jordan regions $D_k$ and $D^{\\prime }_k$ are uniform quasidisks.", "More precisely, there exist $K$ -quasiconformal homeomorphisms $\\alpha _k$ and $\\alpha ^{\\prime }_k$ on $\\widehat{\\mathbb {C}}$ such that $\\alpha _k(\\overline{D}_k)=\\overline{ and \\alpha ^{\\prime }_k(\\overline{D}^{\\prime }_k)=\\overline{ for k\\in {\\mathbb {N}}, where K\\ge 1 is independent of k. Then the maps \\alpha _k and \\alpha ^{\\prime }_kare uniformly quasi-Möbius (see \\cite [Proposition~3.1~(i)]{Bo}).Moreover, the homeomorphisms \\alpha ^{\\prime }_k\\circ \\xi _k\\circ \\alpha _k^{-1}\\colon \\overline{\\rightarrow \\overline{ are uniformly quasiconformal on , and hence also uniformly quasi-Möbiuson \\overline{ (the last implication is essentially well-known; one can reduce to \\cite [Proposition~3.1~(i)]{Bo} by Schwarz reflection in \\partial and use the fact that \\partial is removable for quasiconformal maps \\cite [Section 35]{Va}).", "It follows that the maps\\xi _k, k\\in {\\mathbb {N}}, are uniformly quasi-Möbius.", "}}Now the maps \\xi _k|_{\\partial D_k}=\\xi |_{\\partial D_k}, k\\in {\\mathbb {N}}, are actually uniformly quasisymmetric, because \\xi is a quasisymmetric embedding.", "This implies that the family \\xi _k, k\\in {\\mathbb {N}}, is also uniformly quasisymmetric (this can be seen as in the proof of \\cite [Proposition~5.3~(ii)]{Bo}).", "Since \\xi =\\tilde{\\xi }|_S is quasisymmetricand the maps \\xi _k=\\tilde{\\xi }|_{\\overline{D}^{\\prime }_k} for k\\in {\\mathbb {N}} are uniformlyquasisymmetric, the homeomorphism \\tilde{\\xi } is quasiconformal (this is shown as in the last part of the proof of \\cite [Proposition~5.1]{Bo}).", "}If S has measure zero, then \\tilde{\\xi } is a quasiconformal map that is conformal on the set \\widehat{\\mathbb {C}}\\setminus S of full measure in \\widehat{\\mathbb {C}}.Hence \\tilde{\\xi } is 1-quasiconformal on \\widehat{\\mathbb {C}}, which, as is well-known, implies that \\tilde{\\xi } is aMöbius transformation (one can, for example, derive this from the uniqueness part of Stoilow^{\\prime }s factorization theorem \\cite [p.~179, Theorem 5.5.1]{AIM}).", "}}$ Fatou components of postcritically-finite maps no In this section we record some facts related to complex dynamics.", "For basic definitions and general background we refer to standard sources such as [3], [8], [22], [25].", "Let $f$ be a rational map on the Riemann sphere $\\widehat{\\mathbb {C}}$ of degree $\\deg (f)\\ge 2$ .", "We denote by $f^n$ for $n\\in {\\mathbb {N}}$ the $n$ th-iterate of $f$ , by $\\mathcal {J}(f)$ its Julia set and by $\\mathcal {F}(f)$ its Fatou set.", "Then $\\mathcal {J}(f)=f(\\mathcal {J}(f))=f^{-1}(\\mathcal {J}(f))=\\mathcal {J}(f^n), $ and we have similar relations for the Fatou set.", "We will use these standard facts throughout.", "A continuous map $f\\colon U\\rightarrow V$ between two regions $U,V\\subseteq \\widehat{\\mathbb {C}}$ is called proper if for every compact set $K\\subseteq V$ the set $f^{-1}(K)\\subseteq U$ is also compact.", "If $f$ is a rational map on $\\widehat{\\mathbb {C}}$ , then its restriction $f|_{U}$ to $U$ is a proper map $f|_{U}\\colon U\\rightarrow V$ if and only if $f(U)\\subseteq V$ and $f(\\partial U)\\subseteq \\partial V$ .", "If $f$ is a rational map, it is a proper map between its Fatou components; more precisely, if $U$ is a Fatou component of $f$ , then $V=f(U)$ is also a Fatou component of $f$ , and the restriction is a proper map of $U$ onto $V$ .", "In particular, the boundary of each Fatou component is mapped onto the boundary of another Fatou component.", "Similarly, we can always write $f^{-1}(U)=U_1\\cup \\dots \\cup U_m,$ where $U_1, \\dots , U_m$ are Fatou components of $f$ that are mapped properly onto $U$ .", "If $f\\colon U\\rightarrow V$ is a proper holomorphic map, possibly defined on a larger set than $U$ , then the topological degree $\\deg (f, U)\\in {\\mathbb {N}}$ of $f$ on $U$ is well-defined as the unique number such that $ \\deg (f, U)=\\sum _{p\\in f^{-1}(q)\\cap U}\\deg _f(p)$ for all $q\\in V$ , where $\\deg _f(p)$ is the local degree of $f$ at $p$ .", "Suppose that $U\\subseteq \\widehat{\\mathbb {C}}$ is finitely-connected, i.e., $\\widehat{\\mathbb {C}}\\setminus U$ has only finitely many connected components, and let $k\\in {\\mathbb {N}}_0$ be the number of components of $\\widehat{\\mathbb {C}}\\setminus U$ .", "We call $\\chi (U)=2-k$ the Euler characteristic of $U$ (see [3] for a related discussion).", "The quantity $\\chi (U)$ is invariant under homeomorphisms and can be obtained as a limit of Euler characteristics of polygons (defined in the usual way as for simplicial complexes) forming a suitable exhaustion of $U$ .", "We have $\\chi (U)=2$ if and only if $U=\\widehat{\\mathbb {C}}$ .", "So $\\chi (U)\\le 1$ for finitely-connected proper subregions $U$ of $\\widehat{\\mathbb {C}}$ with $\\chi (U)=1$ if and only if $U$ is simply connected; If $U$ and $V$ are finitely connected regions, and $f\\colon U\\rightarrow V$ is a proper holomorphic map, then a version of the Riemann-Hurwitz relation (see [3] and [25]) says that $\\deg (f, U)\\chi (V)=\\chi (U)+\\sum _{p\\in U} (\\deg _f(p)-1).$ Part of this statement is that the sum on the right-hand side of this identity is defined as it has only finitely many non-vanishing terms.", "The Riemann-Hurwitz formula is valid in a limiting sense for regions that are infinitely connected, i.e., not finitely-connected.", "In this case, the relation simply says that if $U,V\\subseteq \\widehat{\\mathbb {C}}$ are regions and $f\\colon U\\rightarrow V$ is a proper holomorphic map, then $U$ is infinitely connected if and only if $V$ is infinitely connected.", "If $f$ is a rational map on $\\widehat{\\mathbb {C}}$ , then a point is called a postcritical point of $f$ if it is the image of a critical point of $f$ under some iterate of $f$ .", "If we denote the set of these points by $\\operatorname{post}(f)$ , then we have $ \\operatorname{post}(f)=\\bigcup _{n\\in {\\mathbb {N}}}f^n(\\operatorname{crit}(f)).", "$ The map $f$ is called postcritically-finite if every critical point has a finite orbit under iteration of $f$ .", "This is equivalent to the requirement that $\\operatorname{post}(f)$ is a finite set.", "Note that $\\operatorname{post}(f)=\\operatorname{post}(f^n)$ for all $n\\in {\\mathbb {N}}$ .", "We denote by $\\operatorname{post}^{c}(f)\\subseteq \\operatorname{post}(f)$ the set of points that lie in cycles of periodic critical points; so $\\operatorname{post}^c(f)=\\lbrace f^n(c): n\\in {\\mathbb {N}}_0 \\text{ and $c$ is a periodic critical point of $f$} \\rbrace .", "$ If $f$ is postcritically-finite, then $f$ can only have one possible type of periodic Fatou components (for the general classification of periodic Fatou components and their relation to critical points see [3]); namely, every periodic Fatou component $U$ is a Böttcher domain for some iterate of $f$ : there exists an iterate $f^n$ and a superattracting fixed point $p$ of $f^n$ such that $p\\in U$ .", "The following, essentially well-known, lemma describes the dynamics of a postcritically-finite rational map on a fixed Fatou component.", "Here and in the following we will use the notation $P_k$ for the $k$ -th power map given by $P_k(z)=z^k$ for $z\\in , where $ kN$.$ Lemma 3.1 (Dynamics on fixed Fatou components) Suppose $f$ is a postcritically-finite rational map, and $U$ a Fatou component of $f$ with $f(U)=U$ .", "Then $U$ is simply connected, and contains precisely one critical point $p$ of $f$ .", "We have $f(p)=p$ and $ U\\cap \\operatorname{post}(f)=\\lbrace p\\rbrace $ , and there exists a conformal map $\\psi \\colon U\\rightarrow with $ (p)=0$ such that $ f-1=Pk$, where $ k=f(p)2$.$ So $p$ is a superattracting fixed point of $f$ , $U$ is the corresponding Böttcher domain of $p$ , and on $U$ the map $f$ is conjugate to a power map.", "Note that in general the map $\\psi $ is not uniquely determined due to a rotational ambiguity; namely, one can replace $\\psi $ with $a\\psi $ , where $a^{k-1}=1$ .", "As the statement is essentially well-known, we will only give a sketch of the proof.", "By the classification of Fatou components it is clear that $U$ contains a superattracting fixed point $p$ .", "Then $f(p)=p$ and $p$ is a critical point of $f$ .", "Let $k=\\deg _f(p)\\ge 2$ .", "Without loss of generality we may assume that $p=0$ , and $\\infty \\notin U$ .", "Then there exists a holomorphic function $\\varphi $ (the Böttcher function) defined in a neighborhood of 0 with $\\varphi (0)=0=p$ , $\\varphi ^{\\prime }(0)\\ne 0$ , and $f(\\varphi (z))=\\varphi (z^k)$ for $z$ near 0 [25].", "Since the maps $f^n$ , $n\\in {\\mathbb {N}}$ , form a normal family on $U$ and $f^n(z)\\rightarrow p$ for $z$ near $p$ , we have $\\text{$f^n(z)\\rightarrow p=0$ as $n\\rightarrow \\infty $ locally uniformly for $z\\in U$.", "}$ Let $r\\in (0,1]$ be the maximal radius such that $\\varphi $ has a holomorphic extension to the Euclidean disk $B=B(0,r)$ .", "Then (REF ) remains valid on $B$ .", "We claim that $r=1$ , and so $B=;otherwise, $ 0<r<1$, and by using (\\ref {eq:Boett}) and the fact that $ f UU$ is proper,one can show that $(B)U$.", "The equation (\\ref {eq:Boett}) impliesthat every point $ q(B){p}$ has an infinite orbit under iteration of $ f$; by the local uniformity of the convergence in (\\ref {eq:superattraction}) this remains true for $ q(B){p}$.Since $ f$ is postcritically-finite, this implies that no point in $(B){p}$ can be a critical point of $ f$; but then (\\ref {eq:Boett}) allows us to holomorphically extend$$ to a disk $ B(0,r')$ with $ r'>r$.", "This is a contradiction showing that indeed $ r=1$ and $ B=.", "As before by using (REF ), one sees that $\\varphi (\\subseteq U$ .", "Actually, one also observes that for points $q=\\varphi (z)$ with $z\\in closer and closer to $ , the convergence $f^n(q)\\rightarrow p$ is at a slower and slower rate.", "By (REF ) this is only possible if $\\varphi (z)$ is close to $\\partial U$ if $z\\in is close to $ ; in other words, $\\varphi $ is a proper map of $ to $ U$ and in particular $ (=U$.$ It follows from (REF ) that $\\varphi $ cannot have any critical points in $ (to see this, argue by contradiction and consider a critical point $ c of $\\varphi $ with smallest absolute value $|c|$ ).", "The Riemann-Hurwitz formula (REF ) then implies that $\\chi (U)=1$ and $\\deg (\\varphi )=1$ .", "In particular, $U$ is simply connected and $\\varphi $ is a conformal map of $ onto $ U$.", "For the conformal map $ =-1$ from $ U$ onto $ we then have $\\psi (p)=0$ and we get the desired relation $\\psi \\circ f\\circ \\psi ^{-1}=P_k$ .", "This relation (or again (REF )) implies that the fixed point $p$ is the only critical point of $f$ in $U$ and that each point $q\\in U\\setminus \\lbrace p\\rbrace $ has an infinite orbit; so $p$ is the only postcritical point of $f$ in $U$ .", "The following lemma gives us control for the mapping behavior of iterates of a rational map onto regions containing at most one postcritical point.", "Lemma 3.2 Let $f\\colon \\widehat{\\mathbb {C}}\\rightarrow \\widehat{\\mathbb {C}}$ be a rational map, $n\\in {\\mathbb {N}}$ , $U\\subseteq \\widehat{\\mathbb {C}}$ be a simply connected region with $\\#\\widehat{\\mathbb {C}}\\setminus U\\ge 2$ and $\\#(U\\cap \\operatorname{post}(f))\\le 1$ , and $V$ be a component of $f^{-n}(U)$ .", "Let $p\\in U$ be the unique point in $U\\cap \\operatorname{post}(f)$ if $\\#(U\\cap \\operatorname{post}(f))=1$ and $p\\in U$ be arbitrary if $U\\cap \\operatorname{post}(f)=\\emptyset $ , and let $\\psi _U\\colon U\\rightarrow be a conformal map with $ U(p)=0$.$ Then $V$ is simply connected, the map $f^n\\colon V\\rightarrow U$ is proper, and there exists $k\\in {\\mathbb {N}}$ , and a conformal map $\\psi _V\\colon V\\rightarrow with $ Ufn=PkV$.$ Here $k=1$ if $U\\cap \\operatorname{post}(f)=\\emptyset $ .", "Moreover, if $U\\cap \\operatorname{post}^c(f)=\\emptyset $ then $k\\le N$ , where $N=N(f)\\in {\\mathbb {N}}$ is a constant only depending on $f$ .", "In particular, for given $f$ the number $k$ is uniformly bounded by a constant $N$ independently of $n$ and $U$ if $U\\cap \\operatorname{post}^c(f)=\\emptyset $ .", "Under the given assumptions, $V$ is a region, and the map $g:=f^n|_V\\colon V\\rightarrow U$ is proper.", "Since $U\\cap \\operatorname{post}(f)\\subseteq \\lbrace p\\rbrace $ , the point $p$ is the only possible critical value of $g$ .", "It follows from the Riemann-Hurwitz formula (REF ) that $\\chi (V) &=&\\deg (g,V)\\chi (U)-\\sum _{z\\in V}(\\deg _{g}(z)-1)\\\\&=& \\deg (g,V)-(\\deg (g,V)-\\#g^{-1}(p))\\, =\\, \\#g^{-1}(p).$ As $\\chi (V)\\le 1$ , this is only possible if $\\chi (V)=1$ and $\\#g^{-1}(p)=1$ ; so $V$ is simply connected and $p$ has precisely one preimage $q$ in $V$ which is the only possible critical point of $g$ .", "Obviously, $\\#\\widehat{\\mathbb {C}}\\setminus V\\ge 2$ , and so there exists a conformal map $\\psi _V\\colon V\\rightarrow with $ V(q)=0$.", "Then$ (UfnV-1)$ is a proper holomorphic map from $ to itself and hence a finite Blaschke product $B$ .", "Moreover, $B^{-1}(0)=\\lbrace 0\\rbrace $ , and so we can replace $\\psi _V$ by a postcomposition with a suitable rotation around 0 so that $B(z)=z^k$ for $z\\in , where $ k=(g)N$.", "If $ Upost(f)=$, then $ q$ cannot be a critical point of $ g$, and so$ k=1$.$ It remains to produce a uniform upper bound for $k$ if we assume in addition that $U\\cap \\operatorname{post}^c(f)=\\emptyset $ .", "Then in the list $q, f(q), \\dots , f^{n-1}(q)$ each critical point of $f$ can appear at most once; indeed, otherwise the list contains a periodic critical point which implies that $p=f^n(q)\\in U\\cap \\operatorname{post}^c(f)$ , contradicting our additional hypothesis.", "We conclude that $k=\\deg _{f^n}(q)=\\prod _{i=0}^{n-1} \\deg _f(f^i(q))\\le N=N(f):=\\prod _{c\\in \\operatorname{crit}(f)}\\deg _f(c),$ which gives the desired uniform upper bound for $k$ .", "The next lemma describes the dynamics of a postcritically-finite rational map on arbitrary Fatou components.", "Lemma 3.3 (Dynamics on the Fatou components) Let $f\\colon \\widehat{\\mathbb {C}}\\rightarrow \\widehat{\\mathbb {C}}$ be a postcritically-finite rational map, and $\\mathcal {C}$ be the collection of all Fatou components of $f$ .", "Then there exists a family $\\lbrace \\psi _U\\colon U\\rightarrow U\\in \\mathcal {C}\\rbrace $ of conformal maps with the following property: if $U$ and $V$ are Fatou components of $f$ with $f(V)=U$ , then $\\psi _U\\circ f=P_k\\circ \\psi _V$ on $V$ for some $k=k(U,V)\\in {\\mathbb {N}}$ .", "Moreover, for each $U\\in \\mathcal {C}$ the point $p_U:=\\psi _U^{-1}(0)$ is the unique point in $U\\cap \\bigcup _{n\\in {\\mathbb {N}}_0} f^{-n}(\\operatorname{post}(f))$ .", "In contrast to the points $p_U$ the maps $\\psi _U$ are not uniquely determined in general due to a certain rotational freedom.", "As we will see in the proof of the lemma, $p_U$ can also be characterized as the unique point in $U$ with a finite orbit under iteration of $f$ .", "In the following, we will choose $p_U$ as a basepoint in the Fatou component $U$ .", "If we take 0 as a basepoint in $, then in the previous lemmawe get the following commutative diagram of basepoint-preserving maps between{\\em pointed}regions (i.e., regions with a distinguished basepoint):\\begin{equation*}{(V,p_V) [r]^{\\psi _V} [d]_{f} & (0) [d]^{P_k} \\\\(U,p_U) [r]^{\\psi _U} & (0)}\\end{equation*}Note that this implies in particular that $ f-1(pU)={pV}$ and that $ f V{pV}U{pU}$ is a covering map.$ We first construct the desired maps $\\psi _U$ for the periodic Fatou components $U$ of $f$ .", "So fix a periodic Fatou component $U$ of $f$ , and let $n\\in {\\mathbb {N}}$ be the period of $U$ , i.e., if we define $U_0:=U$ and $U_{k+1}=f(U_k)$ for $k=0, \\dots , n-1$ , then the Fatou components $U_0, \\dots , U_{n-1}$ are all distinct, and $U_n=f^n(U)=U$ .", "By Lemma REF applied to the map $f^n$ , for each $k=0, \\dots , n-1$ the Fatou component $U_k$ is simply connected and there exists a unique point $p_k\\in U_k$ that lies in $\\operatorname{post}(f)=\\operatorname{post}(f^n)$ .", "Moreover, there exists a conformal map $\\psi _0\\colon U_0\\rightarrow with $ 0(p0)=0$ such that$ 0fn=Pd0$ for suitable $ dN$.$ Let $\\psi _1\\colon U_1\\rightarrow be a conformal map with $ 1(p1)=0$.", "By the argument in the proof of Lemma~\\ref {lem:deg} we know that $ B=1f 0-1$ is a finite Blaschke product $ B$ with $ B-1(0)={0}$ and so $ B(z)=azd1$ for suitable constants $ d1N$ and $ a with $|a|=1$ .", "By adjusting $\\psi _1$ by a suitable rotation factor if necessary, we may assume that $a=1$ .", "Then $\\psi _1\\circ f=P_{d_1}\\circ \\psi _0$ on $U_0$ .", "If we repeat this argument, then we get conformal maps $\\psi _k\\colon U_k\\rightarrow with $ k(pk)=0$ and\\begin{equation}\\psi _{k}\\circ f=P_{d_k}\\circ \\psi _{k-1}\\end{equation} on $ Uk-1$ with suitable $ dkN$ for$ k=1, ..., n$.", "Note that$$ \\psi _n\\circ f^n = P_{d_n}\\circ \\psi _{n-1}\\circ f^{n-1}=\\dots =P_{d_n}\\circ \\dots \\circ P_{d_1}\\circ \\psi _0=P_{d^{\\prime }}\\circ \\psi _0$$on $ U0$, where $ d'N$.", "On the other hand, $ 0fn=Pd0$ by definition of $ 0$.Hence $ d=fn(p0)=d'$, and so $ nfn= 0fn$ on $ U0$ which implies $ n=0$.", "If we now define $ Uk:=k$ for $ k=0, ..., n-1$, then by (\\ref {eq:perdFatok}) the desired relation (\\ref {eq:desFatcomp}) holds for each suitable pairof Fatou components from the cycle $ U0, ..., Un-1$.", "We also choose $ pUk=pkUk$ as a basepoint in $ Uk$ for $ k=0, ..., n-1$.", "We know that $ pk$ is the unique point in$ Uk$ that lies in $ post(f)$.", "Since $ f$ is postcritically-finite, each point in$ P:=nN0 f-n(post(f))$ has a finite orbit under iteration of $ f$.It follows from Lemma~\\ref {lem:postratFatou} thateach point $ pUk{p}$ has an infinite orbit and therefore cannot lie in$ P$.", "Hence $ pUk$ is the unique point in $ Uk$ that lies in $ P$.$ We repeat this argument for the other finitely many periodic Fatou components $U$ to obtain suitable conformal maps $\\psi _U\\colon U\\rightarrow and unique basepoints $ pU=U-1(0)UP$.$ If $V$ is a non-periodic Fatou component, then it is mapped to a periodic Fatou component by a sufficiently high iterate of $f$ (this is Sullivan's theorem on the non-existence of wandering domains; see [3]).", "We call the smallest number $k\\in {\\mathbb {N}}_0$ such that $f^k(V)$ is a periodic Fatou component the level of $V$ .", "Suppose $V$ is an arbitrary Fatou component of level 1.", "Then $U=f(V)$ is periodic, and so $\\psi _U$ and $p_U$ are already defined and we know that $\\lbrace p_U\\rbrace =\\operatorname{post}(f)\\cap U$ .", "Hence by Lemma REF there exists a conformal map $\\psi _V\\colon V\\rightarrow U$ such that (REF ) is valid.", "If $p_V:=\\psi _V^{-1}(0)$ , then $f(p_V)=p_U\\in \\operatorname{post}(f)$ , and so $p_V\\in V\\cap P$ .", "Moreover, (REF ) shows that $f(V\\setminus \\lbrace p_V\\rbrace )=U\\setminus \\lbrace p_U\\rbrace $ which implies that each point in $V\\setminus \\lbrace p_V\\rbrace $ has an infinite orbit and cannot lie in $P$ .", "It follows that $p_V$ is the unique point in $V$ that lies in $P$ .", "We repeat this argument for Fatou components of higher and higher level.", "Note that if for a Fatou component $U$ a conformal map $\\psi _U\\colon U\\rightarrow has already been constructed and we know that $ pU:=-1(0)$ is the unique point in $ UP$, then$ Upost(f) {pU}$ and we can again apply Lemma~\\ref {lem:deg} for a Fatou component $ V$ with $ f(V)=U$.$ In this way we obtain conformal maps $\\psi _U$ as desired for all Fatou components $U$ .", "The point $p_U=\\psi _U^{-1}(0)$ is the unique point in $U$ that lies in $P$ , because $f^k(p_U)\\in \\operatorname{post}(f)$ for some $k\\in {\\mathbb {N}}_0$ and all other points in $U$ have an infinite orbit.", "We conclude this section with a lemma that is required in the proof of Theorem REF .", "Lemma 3.4 (Lifting lemma) Let $f\\colon \\widehat{\\mathbb {C}}\\rightarrow \\widehat{\\mathbb {C}}$ be a postcritically-finite rational map, $n\\in {\\mathbb {N}}$ , and $(U,p_U)$ and $(V,p_V)$ be pointed Fatou components of $f$ that are Jordan regions with $f^n(U)=V$ .", "Suppose $D\\subseteq \\widehat{\\mathbb {C}}$ is another Jordan region with a basepoint $p_D\\in D$ , and suppose that $\\alpha \\colon \\overline{D}\\rightarrow \\overline{V} $ is a map with the following properties: (i) $\\alpha $ is continuous on $ \\overline{D}$ and holomorphic on $D $ , (ii) $\\alpha ^{-1}(p_V)=\\lbrace p_D\\rbrace $ , (iii) there exists a continuous map $\\beta \\colon \\partial D \\rightarrow \\partial U$ with $f^n\\circ \\beta = \\alpha |_{\\partial D}$ .", "Then there exists a unique continuous map $\\tilde{\\alpha }\\colon \\overline{D}\\rightarrow \\overline{U} $ with $f^n\\circ \\tilde{\\alpha }= \\alpha $ and $\\tilde{\\alpha }|_{\\partial D}=\\beta $ .", "Moreover, $\\tilde{\\alpha }$ is holomorphic on $D$ and satisfies $\\tilde{\\alpha }^{-1}(p_U)=\\lbrace p_D\\rbrace $ .", "If, in addition, $\\beta $ is a homeomorphism of $\\partial D$ onto $\\partial U$ , then $\\tilde{\\alpha }$ is a conformal homeomorphism of $\\overline{D}$ onto $ \\overline{U}$ .", "Here we call a map $\\varphi \\colon \\overline{\\Omega }\\rightarrow \\overline{\\Omega }^{\\prime }$ between the closures of two Jordan regions $\\Omega , \\Omega ^{\\prime }\\subseteq \\widehat{\\mathbb {C}}$ a conformal homeomorphism if $\\varphi $ is a homeomorphism of $\\overline{\\Omega }$ onto $\\overline{\\Omega }^{\\prime }$ and a conformal map of $\\Omega $ onto $\\Omega ^{\\prime }$ .", "Note that in the previous lemma we necessarily have $f^n(\\overline{U})=\\overline{V}$ , $\\alpha (p_D)=p_V=f^n(p_U)$ , and $\\alpha (\\partial D)\\subseteq \\partial V$ by (iii).", "In the conclusion of the lemma we obtain a lift $\\tilde{\\alpha }$ for a given map $\\alpha $ under the branched covering map $f^n$ so that the diagram ${&\\overline{U} [d]^{f^n} \\\\\\overline{D} [ur]^{\\tilde{\\alpha }} [r]^{\\alpha } & \\overline{V}}$ commutes.", "By Lemma REF the map $f^n$ is actually an unbranched covering map from $\\overline{U}\\setminus \\lbrace p_U\\rbrace $ onto $\\overline{V}\\setminus \\lbrace p_V\\rbrace $ .", "The lemma asserts that the existence and uniqueness of a lift $\\tilde{\\alpha }$ is guaranteed if the boundary map $\\alpha |_{\\partial D}$ has a lift (namely $\\beta $ ), and if we have some compatibility condition for branch points (given by condition (ii)).", "The lemma easily follows from some basic theory for covering maps and lifts (see [11] for general background), so we will only sketch the argument and leave some straightforward details to the reader.", "By Lemma REF we can change $\\overline{U}$ and $\\overline{V}$ by conformal homeomorphisms so that we can assume $\\overline{U}=\\overline{V}=\\overline{, p_U=0=p_V, and f^n=P_k for suitable k\\in {\\mathbb {N}} without loss of generality.By classical conformal mapping theory we may also assume that D= and p_D=0.Then condition (ii) translates to \\alpha (0)=0 and \\alpha (z)\\ne 0 for z\\in \\overline{\\setminus \\lbrace 0\\rbrace .", "}We use this to define a homotopy of the boundary map \\alpha |_{\\partial into the base space \\overline{\\setminus \\lbrace 0\\rbrace of the covering map P_k\\colon \\overline{\\setminus \\lbrace 0\\rbrace \\rightarrow \\overline{\\setminus \\lbrace 0\\rbrace .", "Namely, let H\\colon \\partial (0,1]\\rightarrow \\overline{\\setminus \\lbrace 0\\rbrace be defined asH( \\zeta ,t):=\\alpha (t\\zeta ) for \\zeta \\in \\partial and t\\in (0,1].", "It is convenient to think of H as a homotopy running backwards in time t\\in (0,1] starting at t=1.Note that P_k\\circ \\beta =\\alpha |_{\\partial =H(\\cdot , 1).", "So for the initial time t=1 the homotopy has the lift \\beta under the covering map P_k\\colon \\overline{\\setminus \\lbrace 0\\rbrace \\rightarrow \\overline{\\setminus \\lbrace 0\\rbrace .", "By the homotopy lifting theorem \\cite [p.~60, Proposition~1.30]{Ha}, the whole homotopy H has a unique liftstarting at \\beta , i.e., there exists a unique continuous map \\widetilde{H}\\colon \\partial (0,1]\\rightarrow \\overline{\\setminus \\lbrace 0\\rbrace such thatP_k\\circ \\widetilde{H}=H and \\widetilde{H}(\\cdot , 1)=\\beta .", "Now we define \\tilde{\\alpha }(z)=\\widetilde{H}( z/|z|, |z|) for z\\in \\overline{\\setminus \\lbrace 0\\rbrace .", "Then \\tilde{\\alpha } is continuous on\\overline{\\setminus \\lbrace 0\\rbrace , where it satisfies P_k\\circ \\tilde{\\alpha }=\\alpha .", "Since \\alpha (0)=0, this last equation implies that we get a continuous extensionof \\tilde{\\alpha } to \\overline{ by setting \\tilde{\\alpha }(0)=0.", "This extension is a lift \\tilde{\\alpha } of \\alpha .", "Note that \\tilde{\\alpha }^{-1}(0)=0 and\\tilde{\\alpha }|_{\\partial =\\widetilde{H}(\\cdot , 1)=\\beta .", "Moreover, \\tilde{\\alpha } is holomorphic on , because it is a continuous branch of the k-th root of the holomorphic function\\alpha on .", "This shows that \\tilde{\\alpha } has the desired properties.", "The uniqueness of \\tilde{\\alpha } easily follows from the uniqueness of \\widetilde{H}.", "}We have \\beta = \\tilde{\\alpha }|_{\\partial ; so if \\beta is a homeomorphism,then the argument principle implies that \\tilde{\\alpha } is a conformal homeomorphism of\\overline{ onto \\overline{.", "}}}}\\section {The conformal elevator for subhyperbolic maps}\\numero A rational map f is called {\\em subhyperbolic} if each critical point of f in \\mathcal {J}(f) has a finite orbit while each critical point in \\mathcal {F}(f) has an orbit that converges to an attracting or superattracting cycle of f.The map f is called {\\em hyperbolic} if it is subhyperbolic and f does not have critical points in \\mathcal {J}(f).", "Note that every postcritically-finite rational map is subhyperbolic.", "}}}}For the rest of this section, we will assume that f is a subhyperbolic rational map with \\mathcal {J}(f)\\ne \\widehat{\\mathbb {C}}.Moreover, we will make the following additional assumption:\\begin{equation}\\mathcal {J}(f)\\subseteq \\tfrac{1}{2} \\text{and}\\quad f^{-1}( \\subseteq \\end{equation}Here and in what follows, if B is a disk, we denote by \\frac{1}{2} Bthe disk with the same center and whose radius is half the radius of B.", "}The inclusions (\\ref {eq:addinv}) can always be achieved by conjugating fwith an appropriate Möbius transformation so that \\mathcal {J}(f)\\subseteq \\tfrac{1}{2} and \\infty is an attracting or superattractingperiodic point of f. If we then replace f with suitable iterate, we may in additionassume that \\infty becomes an attracting or superattracting fixed point of f withf(\\widehat{\\mathbb {C}}\\setminus \\subseteq \\widehat{\\mathbb {C}}\\setminus .", "The latter inclusion is equivalent tof^{-1}( \\subseteq .", "}}}}Every small disk B centered at a point in \\mathcal {J}(f) can be ``blown up\" by a carefully chosen iterate f^nto a definite size with good control on how sets are distorted under the map f^n.", "We will discuss this in detail as a preparation for the proofs of Theorems~\\ref {thm:circgeom} and \\ref {thm:main2}, and will refer to this procedure as applying the {\\em conformal elevator} to B.", "In the following, all metric notions refer to the Euclidean metric on .", "}}}$ Let $P\\subseteq \\widehat{\\mathbb {C}}$ denote the union of all superattracting or attracting cycles of $f$ .", "This is a non-empty and finite set contained in $ \\mathcal {F}(f)$ .", "Since $f$ is subhyperbolic, every critical point in $\\mathcal {J}(f)$ has a finite orbit, and every critical point in $\\mathcal {J}(f)$ has an orbit that converges to $P$ .", "Hence there exists a neighborhood of $\\mathcal {J}(f)$ that contains only finitely many points in $\\operatorname{post}(f)$ and no points in $\\operatorname{post}^c(f)$ .", "This implies that we can choose $\\epsilon _0>0$ so small that $\\operatorname{diam}(\\mathcal {J}(f))> 2\\epsilon _0$ , and so that every disk $B^{\\prime }=B(q,r^{\\prime })$ centered at a point $q\\in \\mathcal {J}(f)$ with positive radius $r^{\\prime }\\le 8\\epsilon _0$ is contained in $, contains no point in $ postc(f)$ and at most one point in$ post(f)$.$ Let $B=B(p,r)$ be a small disk centered at a point $p\\in \\mathcal {J}(f)$ and of positive radius $r<\\epsilon _0$ .", "Since $B$ is centered at a point in $\\mathcal {J}(f)$ , we have $\\mathcal {J}(f)\\subseteq f^n(B)$ for sufficiently large $n$ (see [3]), and so the images of $B$ under iterates will eventually have diameter $>2\\epsilon _0$ .", "Hence there exists a maximal number $n\\in {\\mathbb {N}}_0$ such that $f^n(B)$ is contained in the disk of radius $\\epsilon _0$ centered at a point $\\tilde{q}\\in \\mathcal {J}(f)$ .", "If $B(\\tilde{q}, 2\\epsilon _0)\\cap \\operatorname{post}(f)=\\emptyset $ , we define $q=\\tilde{q}$ and $B^{\\prime }=B(q, 2\\epsilon _0)$ .", "Otherwise, there exists a unique point $ q\\in B(\\tilde{q}, 2\\epsilon _0)\\cap \\operatorname{post}(f)$ .", "Then we define $B^{\\prime }:=B( q, 8\\epsilon _0)\\supset B(q, 4\\epsilon _0)\\supset B(\\tilde{q}, 2\\epsilon _0)\\supset f^n(B).", "$ In both cases, we have (i) $f^n(B)\\subseteq \\frac{1}{2}B^{\\prime }\\subseteq ,$ (ii) $B^{\\prime }\\cap \\operatorname{post}^c(f)=\\emptyset $ , (iii) $\\#(B^{\\prime }\\cap \\operatorname{post}(f))\\le 1$ with equality only if $B^{\\prime }$ is centered at a point in $\\operatorname{post}(f)$ .", "By definition of $n$ , the set $f^{n+1}(B)$ must have diameter $\\ge \\epsilon _0$ .", "Hence by uniform continuity of $f$ near $\\mathcal {J}(f)$ there exists $\\delta _0>0$ independent of $B$ such that (iv) $\\operatorname{diam}(f^n(B))\\ge \\delta _0$ .", "Let $\\Omega \\subseteq \\widehat{\\mathbb {C}}$ be the unique component of $f^{-n}(B^{\\prime })$ that contains $B$ .", "Then by Lemma REF and by (), (v) $\\Omega $ is simply connected, and $B\\subseteq \\Omega \\subseteq ,$ (vi) the map $f^n|_{\\Omega }\\colon \\Omega \\rightarrow B^{\\prime }$ is proper, (vii) there exists $k\\in {\\mathbb {N}}$ , and conformal maps $\\varphi \\colon B^{\\prime }\\rightarrow and $ such that $(\\varphi \\circ f^n\\circ \\psi ^{-1})(z)= z^k$ for all $z\\in .", "Here $ kN$ is uniformly bounded independent of $ B$.$ If $k\\ge 2$ , then $q=\\varphi ^{-1}(0)\\in \\operatorname{post}(f)\\cap B^{\\prime }$ , and so $q$ is the center of $B^{\\prime }$ .", "If $k=1$ , then we can choose $\\varphi $ so that this is also the case.", "So (viii) $\\varphi $ maps the center $q$ of $B^{\\prime }$ to 0.", "We refer to the choice of $f^n$ and the associated sets $B^{\\prime }$ and $\\Omega $ and the maps $\\varphi $ and $\\psi $ satisfying properties (i)–(viii) as applying the conformal elevator to $B$ .", "Lemma 3.5 There exist constants $\\gamma ,r_1>0$ and $C_1,C_2,C_3\\ge 1$ independent of $B=B(p,r)$ with the following properties: (a) If $A\\subseteq B$ is a connected set, then $\\frac{\\operatorname{diam}(A)}{\\operatorname{diam}(B)}\\le C_1 \\operatorname{diam}(f^n(A))^{\\gamma }.", "$ (b) $B(f^n(p), r_1)\\subseteq f^n(\\frac{1}{2} B)\\subseteq f^n(B)$ .", "(c) If $u,v\\in B$ , then $|f^n(u)- f^n(v)|\\le C_2\\frac{|u-v|}{\\operatorname{diam}(B)}.", "$ (d) If $u,v\\in B$ , $u\\ne v$ , and $f^n(u)=f^n(v)$ , then the center $q$ of $B^{\\prime }$ belongs to $\\operatorname{post}(f)$ and we have $|f^n(u)- q| \\le C_3\\frac{|u-v|}{\\operatorname{diam}(B)}.", "$ So (a) says that a connected set $A\\subseteq B$ comparable in diameter to $B$ is blown up to a definite size under the conformal elevator, and by (b) the image of $B$ contains a disk of a definite size.", "If we consider the maps $f^n|_B$ for different $B$ , then by (c) they are uniformly Lipschitz if we rescale distances in $B$ by $1/\\operatorname{diam}(B)$ .", "In (d) the center $q$ of $B^{\\prime }$ must be a point in $\\operatorname{post}(f)$ for otherwise $f^n$ would be injective; so (d) says that if distinct, but nearby points are mapped to the same image $w$ under $f^n$ , then a postcritical point must be close to this image $w$ .", "In the following we write $a\\lesssim b$ or $a\\gtrsim b$ for two quantities $a,b\\ge 0$ if we can find a constant $C>0$ independent of the disk $B$ such that $a\\le C b$ or $Ca\\ge b$ , respectively.", "We write $a\\approx b$ if we have both $a\\lesssim b$ and $a\\gtrsim b$ , and in this case say that the quantities $a$ and $b$ are comparable.", "We consider the conformal maps $\\varphi \\colon B^{\\prime }\\rightarrow and$ satisfying properties (vii) and (viii) of the conformal elevator as discussed above.", "The exponent $k$ in (vii) is uniformly bounded, say $k\\le N$ , where $N\\in {\\mathbb {N}}$ is independent of $B$ .", "As before, we use the notation $P_k(z)=z^k$ for $z\\in .$ As we will see, the properties (a)–(d) easily follow from distortion properties of the map $P_k$ .", "We discuss the relevant properties of $P_k$ first (the proof is left to the reader).", "The map $P_k$ is Lipschitz with uniformly bounded Lipschitz constant, because $k$ is uniformly bounded.", "If $M\\subseteq is connected, then$$ \\operatorname{diam}(P_k(M))\\gtrsim \\operatorname{diam}(M)^k\\gtrsim \\operatorname{diam}(M)^N.", "$$Moreover, if $ B(z, r0), then $ B(P_k(z), r_1)\\subseteq P_k(B(z, r_0)), $ where $r_1 \\gtrsim r_0^k \\ge r_0^N$ .", "By (vii) the map $\\varphi $ is a Euclidean similarity, and so $\\varphi (\\frac{1}{2} B^{\\prime })=\\frac{1}{2} .Since the radius of $ B'$ is equal to $ 20$ or $ 80$, and hence comparable to $ 1$, wehave\\begin{equation}|\\varphi (u^{\\prime })-\\varphi (v^{\\prime })|\\approx |u^{\\prime }-v^{\\prime }|,\\end{equation}whenever $ u',v'B'$.$ Moreover, for $\\rho :=2^{-1/N}\\in (0,1)$ (which is independent of $B$ ) we have $D:=B(0,\\rho )\\supseteq P_k^{-1}( \\tfrac{1}{2} =P_k^{-1}(\\varphi (\\tfrac{1}{2} B^{\\prime }))=\\psi (f^{-n}(\\tfrac{1}{2} B^{\\prime })\\cap \\Omega ).", "$ Since $f^n(B)\\subseteq \\frac{1}{2} B^{\\prime }$ by (i) and $B\\subseteq \\Omega $ by (v), we then have $\\psi (B) \\subseteq D$ .", "So if $u,v\\in B$ , then $\\psi (u), \\psi (v)\\in D$ .", "Hence by the Koebe distortion theorem we have $|u-v| \\approx |(\\psi ^{-1})^{\\prime }(0)|\\cdot |\\psi (u)-\\psi (v)|$ whenever $u,v\\in B$ .", "In particular, $ \\operatorname{diam}(B)\\approx |(\\psi ^{-1})^{\\prime }(0)|\\cdot \\operatorname{diam}(\\psi (B)) $ On the other hand, by (iv) $1&\\approx \\operatorname{diam}(f^n(B))\\approx \\operatorname{diam}\\big (\\varphi (f^n(B))\\big ) \\\\ &= \\operatorname{diam}\\big (P_k(\\psi (B))\\big ) \\lesssim \\operatorname{diam}(\\psi (B))\\le 2.$ Hence $\\operatorname{diam}(\\psi (B)) \\approx 1$ , and so $\\operatorname{diam}(B)\\approx |(\\psi ^{-1})^{\\prime }(0)|$ .", "This implies that $\\frac{ |u-v|}{\\operatorname{diam}(B)}\\approx |\\psi (u)-\\psi (v)|,$ whenever $u,v\\in B$ .", "Now let $A\\subseteq B$ be connected.", "Then $\\psi (A)$ is connected, which implies $\\frac{ \\operatorname{diam}(A)}{\\operatorname{diam}(B)}&\\approx \\operatorname{diam}(\\psi (A))\\lesssim \\operatorname{diam}\\big (P_k( \\psi (A))\\big )^{1/N} \\\\ & = \\operatorname{diam}\\big (\\varphi (f^n(A))\\big )^{1/N} \\approx \\operatorname{diam}(f^n(A))^{1/N}.$ Inequality (a) follows.", "It follows from (REF ) that there exists $r_0>0$ independent of $B$ such that $ B(\\psi (p), r_0))\\subseteq \\psi (\\frac{1}{2} B)$ .", "By the distortion property of $P_k$ mentioned in the beginning of the proof, $\\varphi (f^n(\\frac{1}{2} B))=P_k(\\psi (\\frac{1}{2} B))$ then contains a disk $B(P_k(\\psi (p)), r_1)$ with $r_1>0$ independent of $B$ .", "Since $\\varphi (f^n(p))=P_k(\\psi (p))$ , and $\\varphi $ distorts distances uniformly, statement (b) follows.", "For (c) note that if $u,v\\in B$ , then $|f^n(u)-f^n(v)|&\\approx |\\varphi (f^n(u))-\\varphi (f^n(v))|= |P_k(\\psi (u))-P_k(\\psi (v))|\\\\&\\lesssim |\\psi (u)-\\psi (v)| \\approx \\frac{ |u-v|}{\\operatorname{diam}(B)}.", "$ We used that $P_k$ is Lipschitz on $ with a uniform Lipschitz constant.$ Finally we prove (d).", "If $u,v\\in B$ , $u\\ne v$ , and $f^n(u)=f^n(v)$ , then $f^n$ is not injective on $B^{\\prime }$ , and so the center $q$ of $B^{\\prime }$ belongs to $\\operatorname{post}(f)$ .", "Moreover, we then have $\\psi (u)^k=\\psi (v)^k$ , but $\\psi (u)\\ne \\psi (v)$ .", "This implies that $ |\\psi (u)-\\psi (v)|\\gtrsim \\frac{1}{k} |\\psi (u)| \\approx |\\psi (u)|.", "$ It follows that $|f^n(u)-q|&\\approx |\\varphi (f^n(u))-\\varphi (q)|= |\\psi (u)^k|\\\\&\\le |\\psi (u)| \\lesssim |\\psi (u)-\\psi (v)| \\approx \\frac{ |u-v|}{\\operatorname{diam}(B)}.", "$ Geometry of the peripheral circles no In this section we will prove Theorem REF .", "We have already defined in Section  what it means for the peripheral circles of a Sierpiński carpet $S$ to be uniform quasicircles and to be uniformly relatively separated.", "We say that the peripheral circles of $S$ occur on all locations and scales if there exists a constant $C\\ge 1$ such that for every $p\\in S$ and every $0<r\\le {\\rm diam}(\\widehat{\\mathbb {C}})=2$ , there exists a peripheral circle $J$ of $S$ with $B(p,r)\\cap J\\ne 0$ and $r/C\\le {\\rm diam}(J)\\le C r.$ Here and below the metric notions refer to the chordal metric $\\sigma $ on $\\widehat{\\mathbb {C}}$ .", "A set $M\\subseteq \\widehat{\\mathbb {C}}$ is called porous if there exists a constant $c>0$ such that for every $p\\in S$ and every $0<r\\le 2$ there exists a point $q\\in B(p,r)$ such that $B(q,cr)\\subseteq \\widehat{\\mathbb {C}}\\setminus M$ .", "Before we turn to the proof of Theorem REF , we require an auxiliary fact.", "Lemma 4.1 Let $f$ be a rational map such that $\\mathcal {J}(f)$ is a Sierpiński carpet, and let $J$ be a peripheral circle of $\\mathcal {J}(f)$ .", "Then $f^n(J)$ is a peripheral circle of $\\mathcal {J}(f)$ , and $f^{-n}(J)$ is a union of finitely many peripheral circles of $\\mathcal {J}(f)$ for each $n\\in {\\mathbb {N}}$ .", "Moreover, $J\\cap \\operatorname{post}(f)=\\emptyset =J\\cap \\operatorname{crit}(f)$ .", "There exists precisely one Fatou component $U$ of $f$ such that $\\partial U=J$ .", "Then $V=f^n(U)$ is also a Fatou component of $f$ .", "Hence $\\partial V$ is a peripheral circle of $\\mathcal {J}(f)$ .", "The map $f^n|_U\\colon U\\rightarrow V$ is proper which implies that $f^n(J)=f^n(\\partial U)=\\partial V$ .", "Similarly, there are finitely many distinct Fatou components $V_1, \\dots , V_k$ of $f$ such that $ f^{-n}(U)=V_1\\cup \\dots \\cup V_k.", "$ Then $f^{-n}(J)=\\partial V_1\\cup \\dots \\cup \\partial V_k, $ and so the preimage of $J$ under $f^n$ consists of the finitely many disjoint Jordan curves $\\partial V_i$ , $i=1, \\dots , k$ , which are peripheral circles of $\\mathcal {J}(f)$ .", "To show $J\\cap \\operatorname{post}(f)=\\emptyset $ , we argue by contradiction, and assume that there exists a point $p\\in \\operatorname{post}(f)\\cap J$ .", "Then there exists $n\\in {\\mathbb {N}}$ , and $c\\in \\operatorname{crit}(f)$ such that $f^n(c)=p$ .", "As we have just seen, the preimage of $J$ under $f^n$ consists of finitely many disjoint Jordan curves, and is hence a topological 1-manifold.", "On the other hand, since $c\\in f^{-n}(p)\\subseteq f^{-n}(J)$ is a critical point of $f$ and hence of $f^n$ , at $c$ the set $f^{-n}(J)$ cannot be a 1-manifold.", "This is a contradiction.", "Finally, suppose that $c\\in J\\cap \\operatorname{crit}(f)$ .", "Then $f(c)\\in \\operatorname{post}(f)\\cap f(J)$ , and $f(J)$ is a peripheral circle of $\\mathcal {J}(f)$ .", "This is impossible by what we have just seen.", "A general idea for the proof is to argue by contradiction, and get locations where the desired statements fail quantitatively in a worse and worse manner.", "One can then use the dynamics to blow up to a global scale and derive a contradiction from topological facts.", "It is fairly easy to implement this idea if we have expanding dynamics given by a group (see, for example, [5]).", "In the present case, one applies the conformal elevator and the estimates as given by Lemma REF .", "We now provide the details.", "We can pass to iterates of the map $f$ , and also conjugate $f$ by a Möbius transformation as properties that we want to establish are Möbius invariant.", "This Möbius invariance is explicitly stated for peripheral circles to be uniform quasicircles and to be uniformly relatively separated in [5].", "The Möbius invariance of the other stated properties immediately follows from the fact that each Möbius transformation is bi-Lipschitz with respect to the chordal metric.", "In this way, we may assume that () is true.", "Then the peripheral circles are subsets of $, where chordal and Euclidean metric are comparable.", "Therefore, we can use the Euclidean metric, and all metric notions will refer to this metric in the following.$ Part I.", "To show that peripheral circles of $ \\mathcal {J}(f)$ are uniform quasicircles, we argue by contradiction.", "Then for each $k\\in {\\mathbb {N}}$ there exists a peripheral circle $J_k$ of $\\mathcal {J}(f)$ , and distinct points $u_k, v_k\\in J_k$ such that if $\\alpha _k,\\beta _k$ are the two subarcs of $J_k$ with endpoints $u_k$ and $v_k$ , then $\\frac{\\min \\lbrace \\operatorname{diam}(\\alpha _k), \\operatorname{diam}(\\beta _k)\\rbrace }{|u_k-v_k|} \\rightarrow \\infty $ as $k\\rightarrow \\infty $ .", "We can pick $r_k>0$ such that $ \\min \\lbrace \\operatorname{diam}(\\alpha _k), \\operatorname{diam}(\\beta _k)\\rbrace /r_k\\rightarrow \\infty $ and $ |u_k-v_k|/r_k\\rightarrow 0$ as $k\\rightarrow \\infty .$ We now apply the conformal elevator to $B_k:=B(u_k, r_k)$ .", "Let $f^{n_k}$ be the corresponding iterate and $B_k^{\\prime }$ be the ball as discussed in Section .", "Define $J^{\\prime }_k=f^{n_k}(J_k)$ , $u^{\\prime }_k=f^{n_k}(u_k)$ , and $v^{\\prime }_k=f^{n_k}(v_k)$ .", "Then Lemma REF  (a) and (REF ) imply that the diameters of the sets $J^{\\prime }_k$ are uniformly bounded away from 0 independently of $k$ .", "Since $J_k^{\\prime }$ is a peripheral circle of the Sierpiński carpet $\\mathcal {J}(f)$ by Lemma REF , there are only finitely many possibilities for the set $J^{\\prime }_k$ .", "By passing to suitable subsequence if necessary, we may assume that $J^{\\prime }=J^{\\prime }_k$ is a fixed peripheral circle of $\\mathcal {J}(f)$ independent of $k$ .", "The points $u^{\\prime }_k,v^{\\prime }_k$ lie in $J^{\\prime }$ and by (REF ) and Lemma REF  (c) we have $|u^{\\prime }_k-v^{\\prime }_k|\\rightarrow 0$ as $k\\rightarrow \\infty $ .", "For large $k$ we want to find a point $w_k\\ne u_k$ in $B_k$ near $u_k$ with $f^{n_k}(u_k)=f^{n_k}(w_k)$ .", "If $u^{\\prime }_k=v^{\\prime }_k$ we can take $w_k=v_k$ .", "Otherwise, if $u^{\\prime }_k\\ne v^{\\prime }_k$ , there are two subarcs of $J^{\\prime }$ with endpoints $u^{\\prime }_k$ and $v^{\\prime }_k$ .", "Let $\\gamma ^{\\prime }_k\\subseteq J^{\\prime }$ be the one with smaller diameter.", "Then by (REF ) we have $\\operatorname{diam}(\\gamma ^{\\prime }_k)\\rightarrow 0$ as $k\\rightarrow \\infty $ (for the moment we only consider such $k$ for which $\\gamma ^{\\prime }_k$ is defined).", "Since $J^{\\prime }\\cap \\operatorname{post}(f)=\\emptyset $ by Lemma REF , the map $f^{n_k}\\colon J_k \\rightarrow J^{\\prime }$ is a covering map.", "So we can lift the arc $\\gamma ^{\\prime }_k$ under $f^{n_k}$ to a subarc $\\gamma _k$ of $J_k$ with initial point $v_k$ and $f^{n_k}(\\gamma _k)=\\gamma ^{\\prime }_k$ .", "By Lemma REF  (b) we have $\\gamma ^{\\prime }_k\\subseteq f^{n_k} (B_k)$ for large $k$ ; then Lemma REF  (a) implies that $\\gamma _k\\subseteq B_k$ for large $k$ , and also $\\operatorname{diam}(\\gamma _k)/r_k\\rightarrow 0$ as $k\\rightarrow \\infty $ .", "Note that if $w_k$ is the other endpoint of $\\gamma _k$ , then $f^{n_k}(w_k)=u^{\\prime }_k$ .", "We have $w_k\\ne u_k$ for large $k$ ; for if $w_k=u_k$ , then $\\gamma _k\\subseteq J_k$ has the endpoints $u_k$ and $v_k$ and so must agree with one of the arcs $\\alpha _k$ or $\\beta _k$ ; but for large $k$ this is impossible by (REF ) and (REF ).", "In addition, we have $|u_k-w_k|/r_k\\le |u_k-v_k|/r_k+\\operatorname{diam}(\\gamma _k)/r_k\\rightarrow 0$ as $k\\rightarrow \\infty $ .", "Note that this is also true if $w_k=v_k$ .", "In summary, for each large $k$ we can find a point $w_k\\in B_k$ with $w_k\\ne u_k$ , $f^{n_k}(u_k)=f^{n_k}(w_k)$ , and $ |u_k-w_k|/r_k\\rightarrow 0$ as $k\\rightarrow \\infty $ .", "Then by Lemma REF  (d) the center $q_k$ of $B^{\\prime }_k$ must belong to the postcritical set of $f$ and $\\operatorname{dist}(J^{\\prime }, \\operatorname{post}(f))\\le |u^{\\prime }_k- q_k|\\rightarrow 0$ as $k\\rightarrow \\infty $ .", "Since $f$ is subhyperbolic, every sufficiently small neighborhood of $\\mathcal {J}(f) \\supseteq J^{\\prime }$ contains only finitely many points in $\\operatorname{post}(f)$ , and so this implies $J^{\\prime }\\cap \\operatorname{post}(f)\\ne \\emptyset $ .", "We know that this is impossible by Lemma REF and so we get a contradiction.", "This shows that the peripheral circles are uniform quasicircles.", "Part II.", "The proof that the peripheral circles of $\\mathcal {J}(f)$ are uniformly relatively separated runs along almost identical lines.", "Again we argue by contradiction.", "Then for $k\\in {\\mathbb {N}}$ we can find distinct peripheral circles $\\alpha _k$ and $\\beta _k$ of $\\mathcal {J}(f)$ , and points $u_k\\in \\alpha _k$ , $v_k\\in \\beta _k$ such that (REF ) is valid.", "We can again pick $r_k>0$ so that the relations (REF ) and (REF ) are true.", "As before we define $B_k=B(u_k, r_k)$ and apply the conformal elevator to $B_k$ which gives us suitable iterate $f^{n_k}$ and a ball $B^{\\prime }_k$ .", "By Lemma REF  (a) the images of $\\alpha _k$ and $\\beta _k$ under $f^{n_k}$ are blown up to a definite size.", "Since there are only finitely many peripheral circles of $\\mathcal {J}(f)$ whose diameter exceeds a given constant, only finitely many such image pairs can arise.", "By passing to a suitable subsequence if necessary, we may assume that $\\alpha =f^{n_k}(\\alpha _k)$ and $\\beta =f^{n_k}(\\alpha _k)$ are peripheral circles independent of $k$ .", "We define $u^{\\prime }_k:=f^{n_k}(u_k)\\in \\alpha $ and $v^{\\prime }_k:=f^{n_k}(v_k)\\in \\beta $ .", "Then again the relation (REF ) holds.", "This is only possible if $\\alpha \\cap \\beta \\ne \\emptyset $ , and so $\\alpha =\\beta $ .", "Again for large $k$ we want to find a point $w_k\\ne u_k$ in $B_k$ near $u_k$ with $f^{n_k}(u_k)=f^{n_k}(w_k)$ .", "If $u^{\\prime }_k=v^{\\prime }_k$ we can take $w_k:=v_k$ .", "Otherwise, if $u^{\\prime }_k\\ne v^{\\prime }_k$ , we let $\\gamma ^{\\prime }_k$ be the subarc of $\\alpha =\\beta $ with endpoints $u^{\\prime }_k $ and $v^{\\prime }_k$ and smaller diameter.", "Then we can lift $\\gamma ^{\\prime }_k$ to a subarc $\\gamma _k\\subseteq \\beta _k$ with initial point $v_k$ such that $f^{n_k}(\\gamma _k)=\\gamma ^{\\prime }_k$ , and we have (REF ).", "If $w_k\\in \\beta _k$ is the other endpoint of $\\gamma _k$ , then $f^{n_k}(w_k)=u^{\\prime }_k=f^{n_k}(u_k)$ , and $w_k\\ne u_k$ , because these points lie in the disjoint sets $\\beta _k$ and $\\alpha _k$ , respectively.", "Again we have (REF ), which implies that the center $q_k$ of $B_k^{\\prime }$ belongs to $\\operatorname{post}(f)$ , and leads to $\\operatorname{dist}(\\alpha , \\operatorname{post}(f))=0$ .", "We know that this is impossible by Lemma REF .", "Part III.", "We will show that peripheral circles of $\\mathcal {J}(f)$ appear on all locations and scales.", "Let $p\\in \\mathcal {J}(f)$ and $r>0$ be arbitrary, and define $B=B(p,r)$ .", "We may assume that $r$ is small, because by a simple compactness argument one can show that disks of definite, but not too large Euclidean size contain peripheral circles of comparable diameter.", "We now apply the conformal elevator to $B$ to obtain an iterate $f^n$ .", "Lemma REF  (b) implies that there exists a fixed constant $r_1>0$ independent of $B$ such that $B(f^n(p), r_1)\\subseteq f^n(\\frac{1}{2} B)$ .", "By part (a) of the same lemma, we can also find a constant $c_1>0$ independent of $B$ with the following property: if $A$ is a connected set with $A\\cap B(p, r/2)\\ne \\emptyset $ and $\\operatorname{diam}(f^n(A))\\le c_1$ , then $A\\subseteq B$ .", "We can now find a peripheral circle $J^{\\prime }$ of $\\mathcal {J}(f)$ such that $J^{\\prime }\\cap f^n(\\frac{1}{2} B)\\ne \\emptyset $ and $0<c_0 < \\operatorname{diam}(J^{\\prime })<c_1$ , where $c_0$ is another positive constant independent of $B$ .", "This easily follows from a compactness argument based on the fact that $f^n(\\frac{1}{2} B)$ contains a disk of a definite size that is centered at a point in $\\mathcal {J}(f)$ .", "The preimage $f^{-n}(J^{\\prime })$ consists of finitely many components that are peripheral circles of $\\mathcal {J}(f)$ .", "One of these peripheral circles $J$ meets $\\frac{1}{2} B$ .", "Since $\\operatorname{diam}(f^n(J))=\\operatorname{diam}(J^{\\prime })<c_1$ , by the choice of $c_1$ we then have $J\\subseteq B$ , and so $\\operatorname{diam}(J)\\le 2r$ .", "Moreover, it follows from Lemma REF  (c) that $\\operatorname{diam}(J)\\ge c_2 \\operatorname{diam}(J^{\\prime })\\operatorname{diam}(B)\\ge c_3 r$ , where again $c_2,c_3>0$ are independent of $B$ .", "The claim follows.", "Part IV.", "Let $p\\in \\mathcal {J}(f)$ be arbitrary and $r\\in (0,1]$ .", "To establish the porosity of $\\mathcal {J}(f)$ , it is enough to show that the Euclidean disk $B(p,r)$ contains a disk of comparable radius that lies in the complement of $\\mathcal {J}(f)$ .", "By what we have just seen, $B(p,r)$ contains a peripheral circle $J$ of diameter comparable to $r$ .", "By possibly allowing a smaller constant of comparability, we may assume that $J$ is distinct from the one peripheral circle $J_0$ that bounds the unbounded Fatou component of $f$ .", "Then $J\\subseteq B(p,r)$ is the boundary of a bounded Fatou component $U$ , and so $U\\subseteq B(p,r)$ .", "Since the peripheral circles of $\\mathcal {J}(f)$ are uniform quasicircles, it follows that $U$ contains a Euclidean disk $D$ of comparable size (for this standard fact see [5]).", "Then $\\operatorname{diam}(D) \\approx \\operatorname{diam}(J)\\approx r$ .", "Since $D\\subseteq U\\subseteq B(p,r)\\cap \\widehat{\\mathbb {C}}\\setminus \\mathcal {J}(f)$ the porosity of $\\mathcal {J}(f)$ follows.", "Finally, the porosity of $\\mathcal {J}(f)$ implies that $\\mathcal {J}(f)$ cannot have Lebesgue density points, and is hence a set of measure zero.", "Relative Schottky sets and Schottky maps no A relative Schottky set $S$ in a region $D\\subseteq \\widehat{\\mathbb {C}}$ is a subset of $D$ whose complement in $D$ is a union of open geometric disks $\\lbrace B_i\\rbrace _{i\\in I}$ with closures $\\overline{B}_i,\\ i\\in I$ , in $D$ , and such that $\\overline{B}_i\\bigcap \\overline{B}_j=\\emptyset ,\\ i\\ne j$ .", "We write $S=D\\setminus \\bigcup _{i\\in I}B_i.$ If $D=\\widehat{\\mathbb {C}}$ or $, we say that $ S$ is a \\emph {Schottky set}.$ Let $A,B\\subseteq \\widehat{\\mathbb {C}}$ and $\\varphi \\colon A\\rightarrow B$ be a continuous map.", "We call $\\varphi $ a local homeomorphism of $A$ to $B$ if for every point $p\\in A$ there exist open sets $U,V\\subseteq with $ pU$, $ f(p)V$ such that $ f|UA$ is a homeomorphism of $ UA$ onto $ VB$.", "Note that this concept depends of course on $ A$, but also crucially on $ B$: if $ B'B$, then we may consider a local homeomorphism $ fAB$ also as a map $ fAB'$, but the second map will not be a local homeomorphism in general.$ Let $D$ and $\\tilde{D}$ be two regions in $\\widehat{\\mathbb {C}}$ , and let $S=D\\setminus \\bigcup _{i\\in I} B_i$ and $\\tilde{S}=\\tilde{D}\\setminus \\bigcup _{j\\in J}\\tilde{B}_j$ be relative Schottky sets in $D$ and $ \\tilde{D}$ , respectively.", "Let $U$ be an open subset of $D$ and let $f\\colon S\\cap U\\rightarrow \\tilde{S}$ be a local homeomorphism.", "According to [20], such a map $f$ is called a Schottky map if it is conformal at every point $p\\in S\\cap U$ , i.e., the derivative $f^{\\prime }(p)=\\lim _{q\\in S,\\, q\\rightarrow p }\\frac{f(q)-f(p)}{q-p}$ exists and does not vanish, and the function $f^{\\prime }$ is continuous on $S\\cap U$ .", "If $p=\\infty $ or $f(p)=\\infty $ , the existence of this limit and the continuity of $f^{\\prime }$ have to be understood after a coordinate change $z\\mapsto 1/z$ near $\\infty $ .", "In all our applications $S\\subseteq and so we can ignore this technicality.$ Theorem REF implies that if $D$ and $\\tilde{D}$ are Jordan regions, the relative Schottky set $S$ has measure zero, and $f\\colon S\\rightarrow \\tilde{S}$ is a locally quasisymmetric homeomorphism that is orientation-preserving (this is defined similarly as for homeomorphisms between Sierpiński carpets; see the discussion after Lemma REF ), then $f$ is a Schottky map.", "We require a more general criterion for maps to be Schottky maps.", "Lemma 5.1 Let $S\\subseteq be a Schottky set of measure zero.Suppose $ UC$ is open and $ UC$ is a locally quasiconformal map with $ -1(S)=US.", "$Then $ USS$ is a Schottky map.$ In particular, if $\\psi \\colon \\widehat{\\mathbb {C}}\\rightarrow \\widehat{\\mathbb {C}}$ is a quasiregular map with $\\psi ^{-1}(S)=S$ , then $\\psi \\colon S\\setminus \\operatorname{crit}(\\psi )\\rightarrow S$ is a Schottky map.", "In the statement the assumption $S\\subseteq (instead of $ SC$) is not really essential, but helps to avoid some technicalities caused by the point $$.$ Our assumption $\\varphi ^{-1}(S)=U\\cap S$ implies that $\\varphi (U\\cap S)\\subseteq S$ .", "So we can consider the restriction of $\\varphi $ to $U\\cap S$ as a map $\\varphi \\colon U\\cap S\\rightarrow S$ (for simplicity we do not use our usual notation $\\varphi |_{U\\cap S}$ for this and other restrictions in the proof).", "This map is a local homeomorphism $\\varphi \\colon U\\cap S\\rightarrow S$ .", "Indeed, let $p\\in U\\cap S$ be arbitrary.", "Since $\\varphi \\colon U\\rightarrow \\widehat{\\mathbb {C}}$ is a local homeomorphism, there exist open sets $V,W\\subseteq \\widehat{\\mathbb {C}}$ with $p\\in V\\subseteq U$ and $f(p)\\in W$ such that $\\varphi $ is a homeomorphism of $V$ onto $W$ .", "Clearly, $\\varphi (V\\cap S)\\subseteq W\\cap S$ .", "Conversely, if $q\\in W\\cap S$ , then there exists a point $q^{\\prime }\\in V$ with $\\varphi (q^{\\prime })=q$ ; since $\\varphi ^{-1}(S)=U\\cap S$ , we have $q^{\\prime }\\in S$ and so $q^{\\prime }\\in V\\cap S$ .", "Hence $\\varphi (V\\cap S)=W\\cap S$ , which implies that $\\varphi $ is a homeomorphism of $V\\cap S$ onto $W\\cap S$ .", "Note that $p\\in U\\cap S$ lies on a peripheral circle of $S$ if and only if $\\varphi (p)$ lies on a peripheral circle of $S$ .", "Indeed, a point $p\\in S$ lies on a peripheral of $S$ if and only if it is accessible by a path in the complement of $S$ , and it is clear this condition is satisfied for a point $p\\in S\\cap U$ if and only if it is true for the image $\\varphi (p)$ (see [20] for a more general related statement).", "We now want to verify the other conditions for $\\varphi $ to be a Schottky map based on Theorem REF .", "It is enough to reduce to this situation locally near each point $p\\in U\\cap S$ .", "We consider two cases depending on whether $p$ belongs to a peripheral circle of $S$ or not.", "So suppose $p$ does not belong to any of the peripheral circles of $S$ .", "Then there exist arbitrarily small Jordan regions $D$ with $p\\in D$ and $\\partial D\\subseteq S$ such that $\\partial D$ does not meet any peripheral circle of $S$ .", "This easily follows from the fact that if we collapse each closure of a complementary component of $S$ in $\\widehat{\\mathbb {C}}$ to a point, then the resulting quotient space is homeomorphic to $\\widehat{\\mathbb {C}}$ by Moore's theorem [23] (for more details on this and the similar argument below, see the proof of [20]).", "In this way we can find a small Jordan region $D$ with the following properties: (i) $p\\in D \\subseteq \\overline{D}\\subseteq U$ , (ii) the boundary $\\partial D$ is contained in $S$ , but does not meet any peripheral circle of $S$ , (iii) $\\varphi $ is a homeomorphism of $\\overline{D}$ onto the closure $\\overline{D}^{\\prime }$ of another Jordan region $D^{\\prime }\\subseteq \\widehat{\\mathbb {C}}$ .", "As in the first part of the proof, we see that $\\varphi $ is a homeomorphism of $D\\cap S$ onto $D^{\\prime }\\cap S$ .", "This homeomorphism is locally quasisymmetric and orientation-preserving as it is the restriction of a locally quasiconformal map.", "Since $\\partial D$ does not meet peripheral circles of $S$ , the same is true of its image $\\partial D^{\\prime }=\\varphi (\\partial D)$ by what we have seen above.", "It follows that the sets $D\\cap S$ and $D^{\\prime }\\cap S$ are relative Schottky sets of measure zero contained in the Jordan regions $D$ and $D^{\\prime }$ , respectively.", "Note that the set $D\\cap S$ is obtained by deleting from $D$ the complementary disks of $S$ that are contained in $D$ , and $D^{\\prime }\\cap S$ is obtained similarly.", "Now Theorem REF implies that $\\varphi \\colon D\\cap S\\rightarrow D^{\\prime }\\cap S$ is a Schottky map which implies that $\\varphi \\colon U\\cap S \\rightarrow S$ is a Schottky map near $p$ .", "For the other case, assume that $p$ lies on a peripheral circle of $S$ , say $p\\in \\partial B$ , where $B$ is one of the disks that form the complement of $S$ .", "The idea is to use a Schwarz reflection procedure to arrive at a situation similar to the previous case.", "This is fairly straightforward, but we will provide the details for sake of completeness.", "Similarly as before (here we collapse all closures of complementary components of $S$ to points except $\\overline{B}$ ), we find a Jordan region $D$ with the following properties: (i) $\\overline{D}\\subseteq U$ and $\\partial D=\\alpha \\cup \\beta $ , where $\\alpha $ and $\\beta $ are two non-overlapping arcs with the same endpoints such that $\\alpha \\subseteq \\partial B$ , $\\beta \\subseteq S$ , $p$ is an interior point of $\\alpha $ , and no interior point of $\\beta $ lies on a peripheral circle of $S$ , (ii) $\\varphi $ is a homeomorphism of $\\overline{D}$ onto the closure $\\overline{D}^{\\prime }$ of another Jordan region $D^{\\prime }\\subseteq \\widehat{\\mathbb {C}}$ .", "Let $\\alpha ^{\\prime }=\\varphi (\\alpha )$ .", "Then $\\alpha $ is contained in a peripheral circle $\\partial B^{\\prime }$ of $S$ , where $B^{\\prime }$ is a suitable complementary disk of $S$ .", "Note that $\\beta ^{\\prime }=\\varphi (\\beta )$ is an arc contained in $S$ , has its endpoints in $\\partial B^{\\prime }$ , and no interior point of $\\beta ^{\\prime }$ lies on a peripheral circle of $S$ .", "Let $R\\colon \\widehat{\\mathbb {C}}\\rightarrow \\widehat{\\mathbb {C}}$ be the reflection in $\\partial B$ , and $R^{\\prime }\\colon \\widehat{\\mathbb {C}}\\rightarrow \\widehat{\\mathbb {C}}$ be the reflection in $\\partial B^{\\prime }$ .", "Define $\\tilde{S}=S\\cup R(S)$ and $\\tilde{S}^{\\prime }=S\\cup R^{\\prime }(S)$ .", "Then $\\tilde{S}$ and $\\tilde{S}^{\\prime }$ are Schottky sets of measure zero, $\\partial B\\subseteq \\tilde{S}$ , $\\partial B^{\\prime }\\subseteq \\tilde{S}^{\\prime }$ , and $\\partial B$ and $\\partial B^{\\prime }$ do not meet any of the peripheral circles of $\\tilde{S}$ and $\\tilde{S}^{\\prime }$ , respectively.", "Let $\\tilde{D}=D\\cup \\operatorname{int}(\\alpha )\\cup R(D)$ and $\\tilde{D}^{\\prime }=D^{\\prime }\\cup \\operatorname{int}(\\alpha ^{\\prime })\\cup R^{\\prime }(D^{\\prime })$ , where $\\operatorname{int}(\\alpha )$ and $ \\operatorname{int}(\\alpha ^{\\prime })$ denote the set of interior points of the arcs $\\alpha $ and $\\alpha ^{\\prime }$ , respectively.", "Then $\\tilde{D}$ and $\\tilde{D}^{\\prime }$ are Jordan regions such that $p\\in \\tilde{D}$ , $\\partial \\tilde{D}\\subseteq \\tilde{S}$ , $\\partial \\tilde{D}^{\\prime }\\subseteq \\tilde{S}^{\\prime }$ , and $\\partial \\tilde{D}$ and $\\partial \\tilde{D}^{\\prime }$ do not meet any of the peripheral circles of $\\tilde{S}$ and $\\tilde{S}^{\\prime }$ , respectively.", "Hence $\\tilde{D}\\cap \\tilde{S}$ and $\\tilde{D}^{\\prime }\\cap \\tilde{S}^{\\prime }$ are relative Schottky sets of measure zero in $\\tilde{D}$ and $\\tilde{D}^{\\prime }$ , respectively.", "We define a map $\\tilde{\\varphi }\\colon \\tilde{D}\\rightarrow \\tilde{D}^{\\prime }$ by $ \\tilde{\\varphi }(z)= \\left\\lbrace \\begin{array} {cl}\\varphi (z)& \\text{for $z\\in D\\cup \\operatorname{int}(\\alpha ) $,}\\\\&\\\\(R^{\\prime }\\circ \\varphi \\circ R)(z)&\\text{for $z\\in R(D)\\cup \\operatorname{int}(\\alpha )$.}", "\\end{array}\\right.$ Note that this definition is consistent on $\\operatorname{int}(\\alpha )$ , because $\\varphi (\\alpha )=\\alpha ^{\\prime }=\\overline{D}^{\\prime }\\cap R^{\\prime }(\\overline{D}^{\\prime })$ .", "It is clear that $\\tilde{\\varphi }\\colon \\tilde{D}\\rightarrow \\tilde{D}^{\\prime }$ is a homeomorphism.", "Moreover, since the circular arc $\\alpha $ (as any set of $\\sigma $ -finite Hausdorff 1-measure) is removable for quasiconformal maps [26], the map $\\tilde{\\varphi }$ is locally quasiconformal, and hence locally quasisymmetric and orientation-preserving.", "It is also straightforward to see from the definitions and the relation $\\varphi ^{-1}(S)=U\\cap S$ that $\\tilde{\\varphi }^{-1}(\\tilde{S}^{\\prime })= \\tilde{D} \\cap \\tilde{S}$ .", "Similarly as in the beginning of the proof this implies that $\\tilde{\\varphi }\\colon \\tilde{D}\\cap \\tilde{S} \\rightarrow \\tilde{D}^{\\prime }\\cap \\tilde{S}^{\\prime }$ is a homeomorphism.", "Since it is also a local quasisymmetry and orientation-preserving, it follows again from Theorem REF that $\\tilde{\\varphi }\\colon \\tilde{D}\\cap \\tilde{S} \\rightarrow \\tilde{D}^{\\prime }\\cap \\tilde{S}^{\\prime }$ is a Schottky map.", "Note that $\\tilde{D}\\cap S= D\\cap S$ , that on this set the maps $\\tilde{\\varphi }$ and $\\varphi $ agree, and that $\\varphi (\\tilde{D}\\cap S)\\subseteq S$ .", "Thus, $\\varphi \\colon \\tilde{D}\\cap S\\rightarrow S$ is a Schottky map, and so $\\varphi \\colon U\\cap S\\rightarrow S$ is a Schottky map near $p$ .", "It follows that $\\varphi \\colon U\\cap S\\rightarrow S$ is a Schottky map as desired.", "The second part of the statement immediately follows from the first; indeed, $\\operatorname{crit}(\\psi )$ is a finite set and so $U=\\widehat{\\mathbb {C}}\\setminus \\operatorname{crit}(\\psi )$ is an open subset $\\widehat{\\mathbb {C}}$ on which $\\varphi =\\psi |_{U}\\colon U\\rightarrow \\widehat{\\mathbb {C}}$ is a locally quasiconformal map.", "Moreover, $\\varphi ^{-1}(S)=\\psi ^{-1}(S)\\cap U=S\\cap U$ .", "By the first part of the proof, $\\varphi $ and hence also $\\psi $ (restricted to $U\\cap S$ ) is a Schottky map of $U\\cap S=S\\setminus \\operatorname{crit}(\\psi )$ into $S$ .", "A relative Schottky set as in (REF ) is called locally porous at $p\\in S$ if there exists a neighborhood $U$ of $p$ , and constants $r_0>0$ and $C\\ge 1$ such that for each $q\\in S\\cap U$ and $r\\in (0, r_0]$ there exists $i\\in I$ with $B_i\\cap B(q,r)\\ne \\emptyset $ and $r/C\\le \\operatorname{diam}(B_i) \\le Cr$ .", "The relative Schottky set $S$ is called locally porous if it is locally porous at every point $p\\in S$ .", "Every locally porous relative Schottky set has measure zero since it cannot have Lebesgue density points.", "For Schottky maps on locally porous Schottky sets very strong rigidity and uniqueness statements are valid such as Theorems REF and REF stated in the introduction.", "We will need another result of a similar flavor.", "Theorem 5.2 (Me3, Theorem 4.1) Let $S$ be a locally porous relative Schottky set in a region $D\\subseteq , let$ U be an open set such that $S\\cap U$ is connected, and $u\\colon S\\cap U\\rightarrow S$ be a Schottky map.", "Suppose that there exists a point $a\\in S\\cap U$ with $u(a)=a$ and $u^{\\prime }(a)=1$ .", "Then $u=\\operatorname{id}|_{S\\cap U}$ .", "A functional equation in the unit disk no As discussed in the introduction, for the proof of Theorem REF we will establish a functional equation of form (REF ) for the maps in question.", "For postcritically-finite maps $f$ and $g$ this leads to strong conclusions based on the following lemma.", "Recall that $P_k(z)=z^k$ for $k\\in {\\mathbb {N}}$ .", "Lemma 6.1 Let $\\phi \\colon \\partial \\partial be an orientation-preserving homeo\\-morphism, and suppose that there exist numbers $ k,l,nN$, $ k2$, such that\\begin{equation} (P_l\\circ \\phi )(z)= (P_n\\circ \\phi \\circ P_k)(z) \\quad \\text{for $z\\in \\partial $.", "}Then l=nk and there exists a\\in with a^{n(k-1)}=1 such that \\phi (z)=az for all z\\in \\partial .\\end{equation}This lemma implies that we can uniquely extend $$ to a conformal homeomorphismfrom $$ onto itself.", "It is also important that this extension preserves the basepoint $ 0$.$ By considering topological degrees, one immediately sees that $l=nk$ .", "So if we introduce the map $\\psi := P_n\\circ \\phi $ , then () can be rewritten as $ P_k\\circ \\psi =\\psi \\circ P_k \\quad \\text{on $\\partial $.", "}Here the map \\psi \\colon \\partial \\partial has degree n. We claim that this in combination with (\\ref {eq:basiceq2}) implies that for a suitable constant b we have \\psi (z)=bz^n for z\\in \\partial .$ Indeed, there exists a continuous function $\\alpha \\colon {\\mathbb {R}}\\rightarrow {\\mathbb {R}}$ with $\\alpha (t+2\\pi )=\\alpha (t)$ such that $ \\psi (e^{ i t}) = \\exp ( i n t+ i\\alpha (t)) \\quad \\text{for $t\\in {\\mathbb {R}}$}.", "$ By (REF ) we have $ \\exp ( i k n t+ i k\\alpha (t)) = (\\psi (e^{ i t}))^k = \\psi (e^{ i k t})=\\exp ( i kn t+ i\\alpha (kt))$ for $t\\in {\\mathbb {R}}$ .", "This implies that there exists a constant $c\\in {\\mathbb {R}}$ such that $ \\alpha (t)=\\frac{1}{k} \\alpha (tk)+c \\quad \\text{for $t\\in {\\mathbb {R}}$}.", "$ Since $\\alpha $ is $2\\pi $ -periodic, the right-hand side of this equation is $2\\pi /k$ -periodic as a function of $t$ .", "Hence $\\alpha $ is $2\\pi /k$ -periodic.", "Repeating this argument, we see that $\\alpha $ is $2\\pi /k^m$ -periodic for all $m\\in {\\mathbb {N}}$ , and so has arbitrarily small periods (note that $k\\ge 2$ ).", "Since $\\alpha $ is continuous, it follows that $\\alpha $ is constant.", "Hence $\\psi (z)=bz^n$ for $z\\in \\partial with a suitable constant $ b.", "It follows that $ \\psi (z)=b z^n=\\phi (z)^n \\quad \\text{for $z\\in \\partial $.$$Therefore, $\\phi (z)=az$ for $z\\in \\partial with a constant $a\\in , $a\\ne 0$.Inserting this expression for $\\phi $ into (\\ref {eq:basiceq}) and using $l=nk$, we conclude that $a^{n(k-1)}=1$ as desired.", "}$ Proof of Theorem  REF The proof will be given in several steps.", "Step I.", "We first fix the setup.", "We can freely pass to iterates of the maps $f$ or $g$ , because this changes neither their Julia sets nor their postcritical sets.", "We can also conjugate the maps by Möbius transformations.", "Therefore, as in Section , we may assume that $\\mathcal {J}(f), \\mathcal {J}(g)\\subseteq \\tfrac{1}{2} { and } f^{-1}( , g^{-1}(\\subseteq $ Moreover, without loss of generality, we may require that $\\xi $ is orientation-preserving, for otherwise we can conjugate $g$ by $z\\mapsto \\overline{z}$ .", "Since the peripheral circles of $\\mathcal {J}(f)$ and $\\mathcal {J}(g)$ are uniform quasicircles, by Theorem REF the map $\\xi $ extends (non-uniquely) to a quasiconformal, and hence quasisymmetric, map of the whole sphere.", "Then $\\xi (\\mathcal {J}(f))=\\mathcal {J}(g)$ and $\\xi (\\mathcal {F}(f))=\\mathcal {F}(g)$ .", "Since $\\infty $ lies in Fatou components of $f$ and $g$ , we may also assume that $\\xi (\\infty )=\\infty $ (this normalization ultimately depends on the fact that for every point $p\\in there exists a quasiconformal homeomorphism $$on $ C$ with $ (0)=p$ that is the identity outside $ ).", "Then $\\xi $ is a quasisymmetry of $ with respect to the Euclidean metric.", "In the following, all metric notions will refer to this metric.Finally, we define $ g=-1 g$.$ Step II.", "We now carefully choose a location for a “blow-down\" by branches of $f^{-n}$ which will be compensated by a “blow-up\" by iterates of $g$ (or rather $g_\\xi $ ).", "Since repelling periodic points of $f$ are dense in $\\mathcal {J}(f)$ (see [3]), we can find such a point $p$ in $\\mathcal {J}(f)$ that does not lie in $\\operatorname{post}(f)$ .", "Let $\\rho >0$ be a small positive number such that the disk $U_0:=B(p, 3\\rho )\\subseteq is disjoint from $ post(f)$.Since $ p$ is periodic, there exists $ dN$ such that $ fd(p)=p$.", "Let $ U1 be the component of $f^{-d}(U_0)$ that contains $p$ .", "Since $U_0\\cap \\operatorname{post}(f)=\\emptyset $ , the set $U_1$ is a simply connected region, and $f^{d}$ is a conformal map from $U_1$ onto $U_0$ as follows from Lemma REF .", "Then there exists a unique inverse branch $f^{-d}$ with $f^{-d}(p)=p$ that is a conformal map of $U_0$ onto $U_1$ .", "Since $p$ is a repelling fixed point for $f$ , it is an attracting fixed point for this branch $f^{-d}$ .", "By possibly choosing a smaller radius $\\rho >0$ in the definition of $U_0=B(p, 3\\rho )$ and by passing to an iterate of $f^d$ , we may assume that $U_1\\subseteq U_0$ and that $\\operatorname{diam}(f^{-n_k}(U_0))\\rightarrow 0$ as $k\\rightarrow \\infty $ .", "Here $n_k=dk$ for $k\\in {\\mathbb {N}}$ , and $f^{-n_k}$ is the branch obtained by iterating the branch $f^{-d}$ $k$ -times.", "Note that $f^{-n_k}(p)=p$ and $f^{-n_k}$ is a conformal map of $U_0$ onto a simply connected region $U_k$ .", "Then $p\\in U_k\\subseteq U_{k-1}$ for $k\\in {\\mathbb {N}}$ , and $\\operatorname{diam}(U_k)\\rightarrow 0$ as $k\\rightarrow \\infty $ .", "The choice of these inverse branches is consistent in the sense that we have $f^{n_{k+1}-n_k}\\circ f^{-n_{k+1}}=f^{-n_k}$ on $B(p, 3\\rho )$ for all $k\\in {\\mathbb {N}}$ .", "Note that this consistency condition remains valid if we replace the original sequence $\\lbrace n_k\\rbrace $ by a subsequence.", "Let $\\tilde{r}_k>0$ be the smallest number such that $f^{-n_k}(B(p, 2\\rho ))\\subseteq \\tilde{B}_k:=B(p,\\tilde{r}_k).$ Since $p\\in f^{-n_k}(B(p, 2\\rho ))$ we have $\\operatorname{diam}\\big (f^{-n_k}(B(p, 2\\rho ))\\big )\\approx \\tilde{r}_k$ .", "Here and below $\\approx $ indicates implicit positive multiplicative constants independent of $k\\in {\\mathbb {N}}$ .", "It follows that $\\tilde{r}_k\\rightarrow 0$ as $k\\rightarrow \\infty $ .", "Moreover, since $f^{-n_k}$ is conformal on the larger disk $B(p,3\\rho )$ , Koebe's distortion theorem implies that $ \\operatorname{diam}\\big (f^{-n_k}(B(p, 2\\rho ))\\big )\\approx \\operatorname{diam}\\big (f^{-n_k}(B(p, \\rho ))\\big ).", "$ Let $r_k>0$ be the smallest number such that $\\xi (\\tilde{B}_k)\\subseteq B_k:=B(\\xi (p), r_k)$ .", "Again $r_k\\rightarrow 0$ as $k\\rightarrow \\infty $ by continuity of $\\xi $ .", "We now want to apply the conformal elevator given by iterates of $g$ to the disks $B_k$ .", "For this we choose $\\epsilon _0>0$ for the map $g$ as in Section .", "By applying the conformal elevator as described in Section , we can find iterates $g^{m_k}$ such that $g^{m_k}(B_k)$ is blown up to a definite, but not too large size, and so $\\operatorname{diam}(g^{m_k}(B_k))\\approx 1$ .", "Step III.", "Now we consider the composition $h_k=g_\\xi ^{m_k}\\circ f^{-n_k}= \\xi ^{-1}\\circ g^{m_k}\\circ \\xi \\circ f^{-n_k}$ defined on $B(p,2\\rho )$ for $k\\in {\\mathbb {N}}$ .", "We want to show that this sequence subconverges locally uniformly on $B(p,2\\rho )$ to a (non-constant) quasiregular map $h\\colon B(p, 2\\rho )\\rightarrow .$ Since $f^{-n_k}$ maps $B(q,2\\rho )$ conformally into $\\tilde{B}_k$ , $\\xi $ is a quasiconformal map with $\\xi (\\tilde{B}_k)\\subseteq B_k$ , and $g^{m_k}$ is holomorphic on $\\tilde{B}_k$ , we conclude that the maps $h_k$ are uniformly quasiregular on $B(q,2\\rho )$ , i.e., $K$ -quasiregular with $K\\ge 1$ independent of $k$ .", "The images $h_k(B(q,2\\rho ))$ are contained in a small Euclidean neighborhood of $\\mathcal {J}(g)$ and hence in a fixed compact subset of $.", "Standardconvergence results for $ K$-quasiregular mappings \\cite [p.~182, Corollary 5.5.7]{AIM} imply that the sequence $ {hk}$subconverges locally uniformly on $ B(q,2)$ to a map $ hB(q,2) that is also quasiregular, but possibly constant.", "By passing to a subsequence if necessary, we may assume that $h_k\\rightarrow h$ locally uniformly on $B(q,2\\rho )$ .", "To rule out that $h$ is constant, it is enough to show that for smaller disk $B(p,\\rho )$ there exists $\\delta >0$ such that $\\operatorname{diam}\\big (h_k(B(p,\\rho ))\\big )\\ge \\delta $ for all $k\\in {\\mathbb {N}}$ .", "We know that $\\operatorname{diam}\\big (f^{-n_k}(B(p,\\rho ))\\big )\\approx \\operatorname{diam}\\big (f^{-n_k}(B(p,2\\rho ))\\big ) \\approx \\operatorname{diam}(\\tilde{B}_k).", "$ Moreover, since $\\xi $ is a quasisymmetry and $f^{-n_k}(B(p,\\rho ))\\subseteq \\tilde{B}_k$ , this implies $\\operatorname{diam}\\big (\\xi (f^{-n_k}(B(p,\\rho )))\\big )\\approx \\operatorname{diam}(\\xi (\\tilde{B}_k))\\approx \\operatorname{diam}(B_k).", "$ So the connected set $\\xi (f^{-n_k}(B(p,\\rho )))\\subseteq B_k$ is comparable in size to $B_k$ .", "By Lemma REF  (a) the conformal elevator blows it up to a definite, but not too large size, i.e., $\\operatorname{diam}\\big ((g^{m_k}\\circ \\xi \\circ f^{-n_k})(B(p,\\rho ))\\big )\\approx 1.$ Since the sets $(g^{m_k}\\circ \\xi \\circ f^{-n_k})(B(p,\\rho ))$ all meet $\\mathcal {J}(g)$ , they stay in a compact part of $, and so we still get a uniform lower bound for the diameter of these sets if we apply the homeomorphism $ -1$; in other words,$$\\operatorname{diam}\\big ((\\xi ^{-1}\\circ g^{m_k}\\circ \\xi \\circ f^{-n_k})(B(p,\\rho ))\\big )=\\operatorname{diam}\\big (h_k(B(p,\\rho ))\\big )\\approx 1$$as claimed.", "We conclude that $ hkh$ locally uniformly on $ B(p, 2r)$, where $ h$ is non-constant and quasiregular.$ The quasiregular map $h$ has at most countably many critical points, and so there exists a point $q\\in B(p, 2\\rho )\\cap \\mathcal {J}(f)$ and a radius $r>0$ such that $B(q, 2r)\\subseteq B(p, 2\\rho )$ and $h$ is injective on $B(q, 2r)$ and hence quasiconformal.", "Standard topological degree arguments imply that at least on the smaller disk $B(q,r)$ the maps $h_k$ are also injective and hence quasiconformal for all $k$ sufficiently large.", "By possibly disregarding finitely of the maps $h_k$ , we may assume that $h_k$ is quasiconformal on $B(q,r)$ for all $k\\in {\\mathbb {N}}$ .", "To summarize, we have found a disk $B(q,r)$ centered at a point $q\\in \\mathcal {J}(f)$ such that the maps $h_k$ are defined and quasiconformal on $B(q,r)$ and converge uniformly on $B(q,r)$ to a quasiconformal map $h$ .", "From the invariance properties of Julia and Fatou sets and the mapping properties of $\\xi $ , it follows that $ h_k(B(q, r)\\cap \\mathcal {J}(f))\\subseteq \\mathcal {J}(f) \\text{ and } h_k(B(q, r)\\cap \\mathcal {F}(f))\\subseteq \\mathcal {F}(f)$ for each $k\\in {\\mathbb {N}}$ .", "Hence $h_k^{-1}(\\mathcal {J}(f))=\\mathcal {J}(f)\\cap B(q,r)$ for each map $h_k\\colon B(q,r)\\rightarrow , $ kN$.$ Since $\\mathcal {J}(f)$ is closed and $h_k\\rightarrow h$ uniformly on $B(q, r)$ , we also have $h(B(q, r)\\cap \\mathcal {J}(f))\\subseteq \\mathcal {J}(f)$ .", "To get a similar inclusion relation also for the Fatou set, we argue by contradiction and assume that there exists a point $z\\in B(q,r)\\cap \\mathcal {F}(f)$ with $h(z)\\notin \\mathcal {F}(f)$ .", "Then $h(z)\\in \\mathcal {J}(f)$ .", "Since $B(q,r)\\cap \\mathcal {F}(f)$ is an open neighborhood of $z$ , it follows again from standard topological degree arguments that for large enough $k\\in {\\mathbb {N}}$ there exists a point $z_k\\in B(q,r)\\cap \\mathcal {F}(f)$ with $h_k(z_k)=h(z)\\in \\mathcal {J}(f)$ .", "This is impossible by (REF ) and so indeed $h(B(q, r)\\cap \\mathcal {F}(f))\\subseteq \\mathcal {F}(f)$ .", "We conclude that $h^{-1}(\\mathcal {J}(f))=\\mathcal {J}(f)\\cap B(q,r).$ Step IV.", "We know by Theorem REF that the peripheral circles of $\\mathcal {J}(f)$ are uniformly relatively separated uniform quasicircles.", "According to Theorems REF and REF , there exists a quasisymmetric map $\\beta $ on $\\widehat{\\mathbb {C}}$ such that $S=\\beta (\\mathcal {J}(f))$ is a round Sierpiński carpet.", "We may assume $S\\subseteq .$ We conjugate the map $f$ by $\\beta $ to define a new map $\\beta \\circ f\\circ \\beta ^{-1}$ .", "By abuse of notation we call this new map also $f$ .", "Note that this map and its iterates are in general not rational anymore, but quasiregular maps on $\\widehat{\\mathbb {C}}$ .", "Similarly, we conjugate $ g_\\xi , h_k, h$ by $\\beta $ to obtain new maps for which we use the same notation for the moment.", "If $V=\\beta (B(q,r))$ , then the new maps $h_k$ and $h$ are quasiconformal on $V$ , and $h_k\\rightarrow h$ uniformly on $V$ .", "Lemma 7.1 There exist $N\\in {\\mathbb {N}}$ and an open set $W\\subseteq V$ such that $S\\cap W$ is non-empty and connected and $h_k\\equiv h$ on $S\\cap W$ for all $k\\ge N$ .", "Since $\\mathcal {J}(f)$ is porous and $S$ is a quasisymmetric image of $\\mathcal {J}(f)$ , the set $S$ is also porous (and in particular locally porous as defined in Section ).", "The maps $h_k$ and $h$ are quasiconformal on $V=\\beta (B(q,r))$ , and $h_k\\rightarrow h$ uniformly on $V$ as $k\\rightarrow \\infty $ .", "The relations (REF ) and (REF ) translate to $h^{-1}(S)=S\\cap V$ and $h_k^{-1}(S)=S\\cap V$ for $k\\in {\\mathbb {N}}$ .", "So Lemma REF implies that the maps $h\\colon S\\cap V\\rightarrow S$ and $h_k\\colon S \\cap V\\rightarrow S$ for $k\\in {\\mathbb {N}}$ are Schottky maps.", "Each of these restrictions is actually a homeomorphism onto its image.", "There are only finitely many peripheral circles of $\\mathcal {J}(f)$ that contain periodic points of our original rational map $f$ ; indeed, if $J$ is such a peripheral circle, then $f^n(J)=J$ for some $n\\in {\\mathbb {N}}$ as follows from Lemma REF ; but then $J$ bounds a periodic Fatou component of $f$ which leaves only finitely many possibilities for $J$ .", "Since the periodic points of $f$ are dense in $\\mathcal {J}(f)$ , we conclude that we can find a periodic point of $f$ in $ \\mathcal {J}(f)\\cap B(q,r)$ that does not lie on a peripheral circle of $\\mathcal {J}(f)$ .", "Translated to the conjugated map $f$ , this yields existence of a point $a\\in S\\cap V$ that does not lie on a peripheral circle of the Sierpiński carpet $S$ such that $f^n(a)=a$ for some $n\\in {\\mathbb {N}}$ .", "The invariance property of the Julia set gives $f^{-n}(S)=S$ , and so Lemma REF implies that $f^n\\colon S\\setminus \\operatorname{crit}(f^n)\\rightarrow S$ is a Schottky map.", "Note that $a\\notin \\operatorname{crit}(f^n)$ as follows from the fact that for our original rational map $f$ , none of its periodic critical points lies in the Julia set.", "Therefore, our Schottky map $f^n\\colon S\\setminus \\operatorname{crit}(f^n)\\rightarrow S$ has a derivative at the point $a\\in S\\setminus \\operatorname{crit}(f^n)$ in the sense of (REF ).", "If $(f^n)^{\\prime }(a)=1$ , then Theorem REF implies that $f^n\\equiv \\operatorname{id}|_{S\\setminus \\operatorname{crit}(f^n)}$ , and hence by continuity $f^n$ is the identity on $S$ .", "This is clearly impossible, and therefore $(f^n)^{\\prime }(a)\\ne 1$ .", "Since $a\\in S\\cap V$ does not lie on a peripheral circle of $S$ , as in the proof of Lemma REF we can find a small Jordan region $W$ with $a\\in W\\subseteq V$ and $W\\cap \\operatorname{crit}(f^n)=\\emptyset $ such that $\\partial W\\subseteq S$ .", "Then $S\\cap W$ is non-empty and connected.", "We now restrict our maps to $W$ .", "Then $h_k\\colon S\\cap W\\rightarrow S$ is a Schottky map and a homeomorphism onto its image for each $k\\in {\\mathbb {N}}$ .", "The same is true for the map $h\\colon S \\cap W\\rightarrow S$ .", "Moreover $h_k\\rightarrow h$ as $k\\rightarrow \\infty $ uniformly on $W\\cap S$ .", "Finally, the map $u=f^n$ is defined on $S\\cap W$ and gives a Schottky map $u\\colon S\\cap W\\rightarrow S$ such that for $a\\in S\\cap W$ we have $u(a)=a$ and $u^{\\prime }(a)\\ne 1$ .", "So we can apply Theorem REF to conclude that there exists $N\\in {\\mathbb {N}}$ such that $h_k\\equiv h$ on $S\\cap W$ for all $k\\ge N$ .", "By the previous lemma we can fix $k\\ge N$ so that $h_k=h_{k+1}$ on $S\\cap W$ .", "If we go back to the definition of the maps $h_k$ and use the consistency of inverse branches (which is also true for the maps conjugated by $\\beta $ ), then we conclude that $ h_{k+1}=g_\\xi ^{m_{k+1}}\\circ f^{-n_{k+1}}= h_k =g_\\xi ^{m_{k}}\\circ f^{-n_{k}}=g_\\xi ^{m_{k}}\\circ f^{n_{k+1}-n_k}\\circ f^{-n_{k+1}}$ on the set $S \\cap W$ .", "Cancellation gives $ g_\\xi ^{m_{k+1}}=g_\\xi ^{m_{k}}\\circ f^{n_{k+1}-n_k}$ on $f^{-n_{k+1}}(S\\cap W)\\subseteq S$ .", "The two maps on both sides of the last equation are quasiregular maps $\\psi \\colon \\widehat{\\mathbb {C}}\\rightarrow \\widehat{\\mathbb {C}}$ with $\\psi ^{-1}(S)=S$ .", "It follows from Lemma REF that they are Schottky maps $S \\cap U\\rightarrow S$ if $U\\subseteq \\widehat{\\mathbb {C}}$ is an open set that does not contain any of the finitely many critical points of the maps; in particular, $g_\\xi ^{m_{k+1}}$ and $g_\\xi ^{m_k}\\circ f^{n_{k+1}-n_k}$ are Schottky maps $S \\cap U\\rightarrow S$ , where $U=\\widehat{\\mathbb {C}}\\setminus (\\operatorname{crit}(g_\\xi ^{m_{k+1}})\\cup \\operatorname{crit}(g_\\xi ^{m_k}\\circ f^{n_{k+1}-n_k}))$ .", "Since $U$ has a finite complement in $\\widehat{\\mathbb {C}}$ , the non-degenerate connected set $f^{-n_{k+1}}(S\\cap W)\\subseteq S$ has an accumulation point in $S \\cap U$ .", "Theorem REF yields $ g_\\xi ^{m_{k+1}}=g_\\xi ^{m_{k}}\\circ f^{n_{k+1}-n_k}$ on $S\\cap U$ , and hence on all of $S$ by continuity.", "If we conjugate back by $\\beta ^{-1}$ , this leads to the relation $ g_\\xi ^{m_{k+1}}=g_\\xi ^{m_{k}}\\circ f^{n_{k+1}-n_k}$ on $\\mathcal {J}(f)$ for the original maps.", "We conclude that there exist integers $m,m^{\\prime },n\\in {\\mathbb {N}}$ such that for the original maps we have $ g^{m^{\\prime }}\\circ \\xi = g^m\\circ \\xi \\circ f^n$ on $\\mathcal {J}(f)$ .", "Step V. Equation (REF ) gives us a crucial relation of $\\xi $ to the dynamics of $f$ and $g$ on their Julia sets.", "We will bring (REF ) into a convenient form by replacing our original maps with iterates.", "Since $\\mathcal {J}(f)$ is backward invariant, counting preimages of generic points in $\\mathcal {J}(f)$ under iterates of $f$ and of points in $\\mathcal {J}(g)$ under iterates of $g$ leads to the relation $\\deg (g)^{m^{\\prime }-m}=\\deg (f)^n,$ and so $m^{\\prime }-m>0$ .", "If we post-compose both sides in (REF ) by a suitable iterate of $g$ , and then replace $f$ by $f^n$ and $g$ by $g^{m^{\\prime }-m}$ , we arrive at a relation of the form $ g^{l+1}\\circ \\xi = g^l\\circ \\xi \\circ f.$ on $\\mathcal {J}(f)$ for some $l\\in {\\mathbb {N}}$ .", "Note that this equation implies that we have $ g^{n+k}\\circ \\xi = g^n\\circ \\xi \\circ f^k \\ \\text{ for all $k,n\\in {\\mathbb {N}}$ with $n\\ge l$}.$ Step VI.", "In this final step of the proof, we disregard the non-canonical extension to $\\widehat{\\mathbb {C}}$ of our original homeomorphism $\\xi \\colon \\mathcal {J}(f)\\rightarrow \\mathcal {J}(g)$ chosen in the beginning.", "Our goal is to apply (REF ) to produce a natural extension of $\\xi $ mapping each Fatou component of $f$ conformally onto a Fatou component of $g$ .", "Note that if $U$ is a Fatou component of $f$ , then $\\partial U$ is a peripheral circle of $\\mathcal {J}(f)$ .", "Since $\\xi $ sends each peripheral circle of $\\mathcal {J}(f)$ to a peripheral circle of $\\mathcal {J}(g)$ , the image $\\xi (\\partial U)$ bounds a unique Fatou component $V$ of $g$ .", "This sets up a natural bijection between the Fatou components of our maps, and our goal is to conformally “fill in the holes\".", "So let $\\mathcal {C}_f$ and $\\mathcal {C}_g$ be the sets of Fatou components of $f$ and $g$ , respectively.", "By Lemma REF we can choose a corresponding family $\\lbrace \\psi _U: U\\in \\mathcal {C}_f\\rbrace $ of conformal maps.", "Since each Fatou component of $f$ is a Jordan region, we can consider $\\psi _U$ as a conformal homeomorphism from $\\overline{U}$ onto $\\overline{.", "Similarly, we obtain a family of conformal homeomorphisms \\tilde{\\psi }_V: \\overline{V}\\rightarrow \\overline{ for V in \\mathcal {C}_g.These Fatou components carry distinguished basepointsp_U=\\psi _U^{-1}(0)\\in U for U\\in \\mathcal {C}_f and \\tilde{p}_V=\\tilde{\\psi }_V^{-1}(0)\\in V for V\\in \\mathcal {C}_g.", "}We will now first extend \\xi to the periodic Fatou components of f, and thenuse the Lifting Lemma~\\ref {lem:lifting} to get extensions to Fatou components of higher and higher level (as defined in the proof of Lemma~\\ref {lem:FatouDyn}).", "In this argument it will be important to ensure that these extensions are basepoint-preserving.", "}First let $ U$ be a periodic Fatou component of $ f$.", "We denote by $ kN$ the period of $ U$, and define $ V$ to be the Fatou component of $ g$ bounded by $ (U)$, and$ W=gl(V)$, where $ lN$ is as in (\\ref {eq:main4}).Then(\\ref {eq:main4}) implies that $ W$, and hence $ W$ itself, is invariant under $ gk$.", "ByLemma~\\ref {lem:FatouDyn} the basepoint-preserving homeomorphisms $ U (U, pU)( , 0)$ and $ W(W,pW)( ,0)$ conjugate $ fk$ and $ gk$, respectively, to power maps.Since $ f$ and $ g$ are postcritically-finite, the periodic Fatou components $ U$ and $ W$ are superattracting, and thus the degrees of these power maps are at least 2.$ Again by Lemma REF the map $\\tilde{\\psi }_W\\circ g^l\\circ \\tilde{\\psi }^{-1}_V$ is a power map.", "Since $U,V,W$ are Jordan regions, the maps $\\psi _U, \\tilde{\\psi }_V, \\tilde{\\psi }_W$ give homeomorphisms between the boundaries of the corresponding Fatou components and $\\partial .", "Since $$ is anorientation-preserving homeomorphism of $ U$ onto $ V$, the map$ =VU-1$gives an orientation-preserving homeomorphism on $ .", "Now (REF ) for $n=l$ implies that on $\\partial we have{\\begin{@align*}{1}{-1} P_{d_3}\\circ \\phi &=\\tilde{\\psi }_W\\circ g^{k+l}\\circ \\tilde{\\psi }_V^{-1} \\circ \\phi = \\tilde{\\psi }_W\\circ g^{k+l}\\circ \\xi \\circ \\psi _U^{-1}\\\\&= \\tilde{\\psi }_W\\circ g^{l}\\circ \\xi \\circ f^k\\circ \\psi ^{-1}_U= \\tilde{\\psi }_W\\circ g^{l}\\circ \\xi \\circ \\psi ^{-1}_U \\circ P_{d_1}\\\\&= \\tilde{\\psi }_W\\circ g^{l}\\circ \\tilde{\\psi }^{-1}_V\\circ \\phi \\circ P_{d_1}= P_{d_2}\\circ \\phi \\circ P_{d_1}\\end{@align*}}for some $ d1, d 2, d3N$ with $ d12$.", "Lemma~\\ref {L:Rot} implies that $$ extends to $$ as a rotation around $ 0$, also denoted by $$.", "In particular, $ (0)=0$, and so $$ preserves the basepoint $ 0$ in $$.", "If we define $ =V-1U$ on $U$, then $$ is a conformal homeomorphismof $ (U,pU)$ onto $ (V, pV)$.$ In this way, we can conformally extend $\\xi $ to every periodic Fatou component of $f$ so that $\\xi $ maps the basepoint of a Fatou component to the basepoint of the image component.", "To get such an extension also for the other Fatou components $V$ of $f$ , we proceed inductively on the level of the Fatou component.", "So suppose that the level of $V$ is $\\ge 1$ and that we have already found an extension for all Fatou components with a level lower than $V$ .", "This applies to the Fatou component $U=f(V)$ of $f$ , and so a conformal extension $(U, p_U)\\rightarrow ( U^{\\prime }, \\tilde{p}_{U^{\\prime }})$ of $\\xi |_{\\partial U}$ exists, where $U^{\\prime }$ is the Fatou component of $g$ bounded by $\\xi (\\partial U)$ .", "Let $V^{\\prime }$ be the Fatou component of $g$ bounded by $\\xi (\\partial V)$ , and $W=g^{l+1}(V^{\\prime })$ .", "Then by using (REF ) on $\\partial V$ we conclude that $g^l(U^{\\prime })=W$ .", "Define $\\alpha =g^l\\circ \\xi |_{\\overline{U}}\\circ f|_{\\overline{V}}$ and $\\beta =\\xi |_{\\partial V}$ .", "Then the assumptions of Lemma REF are satisfied for $D=V$ , $p_D=p_{V}$ , and the iterate $g^{l+1}\\colon V^{\\prime }\\rightarrow W$ of $g$ .", "Indeed, $\\alpha $ is continuous on $\\overline{V}$ and holomorphic on $V$ , we have $\\alpha ^{-1}(p_W)=f^{-1}( \\xi |_{\\overline{U}}^{-1}(\\tilde{p}_{U^{\\prime }}))=f^{-1}(p_U)=\\lbrace p_V\\rbrace ,$ and $g^{l+1}\\circ \\beta = g^{l+1}\\circ \\xi |_{\\partial V}= g^{l}\\circ \\xi |_{\\partial U}\\circ f|_{\\partial V}=\\alpha .$ Since $\\beta $ is a homeomorphism, it follows that there exists a conformal homeomorphism $\\tilde{\\alpha }$ of $(\\overline{V}, p_V)$ onto $(\\overline{V}^{\\prime }, \\tilde{p}_{V^{\\prime }})$ such that $\\tilde{\\alpha }|_{\\partial V}=\\beta =\\xi |_{\\partial V}$ .", "In other words, $\\tilde{\\alpha }$ gives the desired basepoint preserving conformal extension to the Fatou component $V$ .", "This argument shows that $\\xi $ has a (unique) conformal extension to each Fatou component of $f$ .", "We know that the peripheral circles of $\\mathcal {J}(f)$ are uniform quasicircles, and that $\\mathcal {J}(f)$ has measure zero.", "Lemma REF now implies that $\\xi $ extends to a Möbius transformation on $\\widehat{\\mathbb {C}}$ , which completes the proof.", "The techniques discussed also easily lead to a proof of Corollary REF .", "Let $f$ be a postcritically-finite rational map whose Julia set $\\mathcal {J}(f)$ is a Sierpiński carpet.", "Let $G$ be the group of all Möbius transformations $\\xi $ on $\\widehat{\\mathbb {C}}$ with $\\xi (\\mathcal {J}(f))=\\mathcal {J}(f)$ , and $H$ be the subgroup of all elements in $G$ that preserve orientation.", "By Theorem REF it is enough to prove that $G$ is finite.", "Since $G=H$ or $H$ has index 2 in $G$ , this is true if we can show that $H$ is finite.", "Note that the group $H$ is discrete, i.e., there exists $\\delta _0>0$ such that $ \\sup _{p\\in \\widehat{\\mathbb {C}}} \\sigma (\\xi (p), p)\\ge \\delta _0$ for all $\\xi \\in H$ with $\\xi \\ne \\operatorname{id}_{\\widehat{\\mathbb {C}}}$ .", "Indeed, we choose $\\delta _0>0$ so small that there are at least three distinct complementary components $D_1,D_2,D_3$ of $\\mathcal {J}(f)$ that contain disks of radius $\\delta _0$ .", "In order to show (REF ), suppose that $\\xi \\in H$ and $\\sigma (\\xi (p), p)<\\delta _0$ for all $p\\in \\widehat{\\mathbb {C}}$ .", "Then $\\xi (D_i)\\cap D_i\\ne \\emptyset $ , and so $\\xi (\\overline{D}_i)=\\overline{D}_i$ for $i=1, 2,3$ , because $\\xi $ permutes the closures of Fatou components of $f$ .", "This shows that $\\xi $ is a conformal homeomorphism of the closed Jordan region $\\overline{D}_i$ onto itself.", "Hence $\\xi $ has a fixed point in $\\overline{D}_i$ .", "Since $\\mathcal {J}(f)$ is a Sierpiński carpet, the closures $\\overline{D}_1, \\overline{D}_2, \\overline{D}_3$ are pairwise disjoint, and we conclude that $\\xi $ has at least three fixed points.", "Since $\\xi $ is an orientation-preserving Möbius transformation, this implies that $\\xi =\\operatorname{id}_{\\widehat{\\mathbb {C}}}$ , and the discreteness of $H$ follows.", "We will now analyze type of Möbius transformations contained in the group $H$ (for the relevant classification of Möbius transformations up to conjugacy, see [2]).", "So consider an arbitrary $\\xi \\in H$ , $\\xi \\ne \\text{id}_{\\widehat{\\mathbb {C}}}$ .", "Then $\\xi $ cannot be loxodromic; indeed, otherwise $\\xi $ has a repelling fixed point $p$ which necessarily has to lie in $\\mathcal {J}(f)$ .", "We now argue as in the proof of Theorem REF and “blow down\" by iterates $\\xi ^{-n_k}$ near $p$ and “blow up\" by the conformal elevator using iterates $f^{m_k}$ to obtain a sequence of conformal maps of the form $h_k=f^{m_k}\\circ \\xi ^{-n_k}$ that converge uniformly to a (non-constant) conformal limit function $h$ on a disk $B$ centered at a point in $q\\in \\mathcal {J}(f)$ .", "Again this sequence stabilizes, and so $h_{k+1}=h_k$ for large $k$ on a connected non-degenerate subset of $B$ , and hence on $\\widehat{\\mathbb {C}}$ by the uniqueness theorem for analytic functions.", "This leads to a relation of the form $f^k\\circ \\xi ^l=f^m$ , where $k,l,m\\in {\\mathbb {N}}$ .", "Comparing degrees we get $m=k$ , and so we have $f^{k}=f^{k}\\circ \\xi ^{-ln}$ for all $n\\in {\\mathbb {N}}$ .", "This is impossible, because $f^k=f^{k}\\circ \\xi ^{-ln}\\rightarrow f^{k}(p)$ near $p$ as $n\\rightarrow \\infty $ , while $f^k$ is non-constant.", "The Möbius transformation $\\xi $ cannot be parabolic either; otherwise, after conjugation we may assume that $\\xi (z)=z+a$ with $a\\in , $ a0$.", "Then necessarily $ J(f)$.", "On the other hand, we know thatthe peripheral circles of $ J(f)$ are uniform quasicircles that occur on all locations and scales with respect to the chordal metric.", "Translated to the Euclidean metric near $$ this means that $ J(f)$ has complementary components $ D$ with $ D$ that contain Euclidean disks of arbitrarily large radius, and in particular of radius $ >|a|$; but then $$ cannot move $ D$ off itself, and so $ (D)=D$ for the translation $$.This is impossible.$ Finally, $\\xi $ can be elliptic; then after conjugation we have $\\xi (z)=az$ with $a\\in and $ |a|=1$, $ a1$.", "Since $ H$ is discrete,$ a$ must be a root of unity, and so $$is a torsion element of $ H$.$ We conclude that $H$ is a discrete group of Möbius transformations such that each element $\\xi \\in H$ with $\\xi \\ne \\text{id}_{\\widehat{\\mathbb {C}}}$ is a torsion element.", "It is well-known that such a group $H$ is finite (one can derive this from [2] in combination with the considerations in [2])." ], [ "Fatou components of postcritically-finite maps", "no In this section we record some facts related to complex dynamics.", "For basic definitions and general background we refer to standard sources such as [3], [8], [22], [25].", "Let $f$ be a rational map on the Riemann sphere $\\widehat{\\mathbb {C}}$ of degree $\\deg (f)\\ge 2$ .", "We denote by $f^n$ for $n\\in {\\mathbb {N}}$ the $n$ th-iterate of $f$ , by $\\mathcal {J}(f)$ its Julia set and by $\\mathcal {F}(f)$ its Fatou set.", "Then $\\mathcal {J}(f)=f(\\mathcal {J}(f))=f^{-1}(\\mathcal {J}(f))=\\mathcal {J}(f^n), $ and we have similar relations for the Fatou set.", "We will use these standard facts throughout.", "A continuous map $f\\colon U\\rightarrow V$ between two regions $U,V\\subseteq \\widehat{\\mathbb {C}}$ is called proper if for every compact set $K\\subseteq V$ the set $f^{-1}(K)\\subseteq U$ is also compact.", "If $f$ is a rational map on $\\widehat{\\mathbb {C}}$ , then its restriction $f|_{U}$ to $U$ is a proper map $f|_{U}\\colon U\\rightarrow V$ if and only if $f(U)\\subseteq V$ and $f(\\partial U)\\subseteq \\partial V$ .", "If $f$ is a rational map, it is a proper map between its Fatou components; more precisely, if $U$ is a Fatou component of $f$ , then $V=f(U)$ is also a Fatou component of $f$ , and the restriction is a proper map of $U$ onto $V$ .", "In particular, the boundary of each Fatou component is mapped onto the boundary of another Fatou component.", "Similarly, we can always write $f^{-1}(U)=U_1\\cup \\dots \\cup U_m,$ where $U_1, \\dots , U_m$ are Fatou components of $f$ that are mapped properly onto $U$ .", "If $f\\colon U\\rightarrow V$ is a proper holomorphic map, possibly defined on a larger set than $U$ , then the topological degree $\\deg (f, U)\\in {\\mathbb {N}}$ of $f$ on $U$ is well-defined as the unique number such that $ \\deg (f, U)=\\sum _{p\\in f^{-1}(q)\\cap U}\\deg _f(p)$ for all $q\\in V$ , where $\\deg _f(p)$ is the local degree of $f$ at $p$ .", "Suppose that $U\\subseteq \\widehat{\\mathbb {C}}$ is finitely-connected, i.e., $\\widehat{\\mathbb {C}}\\setminus U$ has only finitely many connected components, and let $k\\in {\\mathbb {N}}_0$ be the number of components of $\\widehat{\\mathbb {C}}\\setminus U$ .", "We call $\\chi (U)=2-k$ the Euler characteristic of $U$ (see [3] for a related discussion).", "The quantity $\\chi (U)$ is invariant under homeomorphisms and can be obtained as a limit of Euler characteristics of polygons (defined in the usual way as for simplicial complexes) forming a suitable exhaustion of $U$ .", "We have $\\chi (U)=2$ if and only if $U=\\widehat{\\mathbb {C}}$ .", "So $\\chi (U)\\le 1$ for finitely-connected proper subregions $U$ of $\\widehat{\\mathbb {C}}$ with $\\chi (U)=1$ if and only if $U$ is simply connected; If $U$ and $V$ are finitely connected regions, and $f\\colon U\\rightarrow V$ is a proper holomorphic map, then a version of the Riemann-Hurwitz relation (see [3] and [25]) says that $\\deg (f, U)\\chi (V)=\\chi (U)+\\sum _{p\\in U} (\\deg _f(p)-1).$ Part of this statement is that the sum on the right-hand side of this identity is defined as it has only finitely many non-vanishing terms.", "The Riemann-Hurwitz formula is valid in a limiting sense for regions that are infinitely connected, i.e., not finitely-connected.", "In this case, the relation simply says that if $U,V\\subseteq \\widehat{\\mathbb {C}}$ are regions and $f\\colon U\\rightarrow V$ is a proper holomorphic map, then $U$ is infinitely connected if and only if $V$ is infinitely connected.", "If $f$ is a rational map on $\\widehat{\\mathbb {C}}$ , then a point is called a postcritical point of $f$ if it is the image of a critical point of $f$ under some iterate of $f$ .", "If we denote the set of these points by $\\operatorname{post}(f)$ , then we have $ \\operatorname{post}(f)=\\bigcup _{n\\in {\\mathbb {N}}}f^n(\\operatorname{crit}(f)).", "$ The map $f$ is called postcritically-finite if every critical point has a finite orbit under iteration of $f$ .", "This is equivalent to the requirement that $\\operatorname{post}(f)$ is a finite set.", "Note that $\\operatorname{post}(f)=\\operatorname{post}(f^n)$ for all $n\\in {\\mathbb {N}}$ .", "We denote by $\\operatorname{post}^{c}(f)\\subseteq \\operatorname{post}(f)$ the set of points that lie in cycles of periodic critical points; so $\\operatorname{post}^c(f)=\\lbrace f^n(c): n\\in {\\mathbb {N}}_0 \\text{ and $c$ is a periodic critical point of $f$} \\rbrace .", "$ If $f$ is postcritically-finite, then $f$ can only have one possible type of periodic Fatou components (for the general classification of periodic Fatou components and their relation to critical points see [3]); namely, every periodic Fatou component $U$ is a Böttcher domain for some iterate of $f$ : there exists an iterate $f^n$ and a superattracting fixed point $p$ of $f^n$ such that $p\\in U$ .", "The following, essentially well-known, lemma describes the dynamics of a postcritically-finite rational map on a fixed Fatou component.", "Here and in the following we will use the notation $P_k$ for the $k$ -th power map given by $P_k(z)=z^k$ for $z\\in , where $ kN$.$ Lemma 3.1 (Dynamics on fixed Fatou components) Suppose $f$ is a postcritically-finite rational map, and $U$ a Fatou component of $f$ with $f(U)=U$ .", "Then $U$ is simply connected, and contains precisely one critical point $p$ of $f$ .", "We have $f(p)=p$ and $ U\\cap \\operatorname{post}(f)=\\lbrace p\\rbrace $ , and there exists a conformal map $\\psi \\colon U\\rightarrow with $ (p)=0$ such that $ f-1=Pk$, where $ k=f(p)2$.$ So $p$ is a superattracting fixed point of $f$ , $U$ is the corresponding Böttcher domain of $p$ , and on $U$ the map $f$ is conjugate to a power map.", "Note that in general the map $\\psi $ is not uniquely determined due to a rotational ambiguity; namely, one can replace $\\psi $ with $a\\psi $ , where $a^{k-1}=1$ .", "As the statement is essentially well-known, we will only give a sketch of the proof.", "By the classification of Fatou components it is clear that $U$ contains a superattracting fixed point $p$ .", "Then $f(p)=p$ and $p$ is a critical point of $f$ .", "Let $k=\\deg _f(p)\\ge 2$ .", "Without loss of generality we may assume that $p=0$ , and $\\infty \\notin U$ .", "Then there exists a holomorphic function $\\varphi $ (the Böttcher function) defined in a neighborhood of 0 with $\\varphi (0)=0=p$ , $\\varphi ^{\\prime }(0)\\ne 0$ , and $f(\\varphi (z))=\\varphi (z^k)$ for $z$ near 0 [25].", "Since the maps $f^n$ , $n\\in {\\mathbb {N}}$ , form a normal family on $U$ and $f^n(z)\\rightarrow p$ for $z$ near $p$ , we have $\\text{$f^n(z)\\rightarrow p=0$ as $n\\rightarrow \\infty $ locally uniformly for $z\\in U$.", "}$ Let $r\\in (0,1]$ be the maximal radius such that $\\varphi $ has a holomorphic extension to the Euclidean disk $B=B(0,r)$ .", "Then (REF ) remains valid on $B$ .", "We claim that $r=1$ , and so $B=;otherwise, $ 0<r<1$, and by using (\\ref {eq:Boett}) and the fact that $ f UU$ is proper,one can show that $(B)U$.", "The equation (\\ref {eq:Boett}) impliesthat every point $ q(B){p}$ has an infinite orbit under iteration of $ f$; by the local uniformity of the convergence in (\\ref {eq:superattraction}) this remains true for $ q(B){p}$.Since $ f$ is postcritically-finite, this implies that no point in $(B){p}$ can be a critical point of $ f$; but then (\\ref {eq:Boett}) allows us to holomorphically extend$$ to a disk $ B(0,r')$ with $ r'>r$.", "This is a contradiction showing that indeed $ r=1$ and $ B=.", "As before by using (REF ), one sees that $\\varphi (\\subseteq U$ .", "Actually, one also observes that for points $q=\\varphi (z)$ with $z\\in closer and closer to $ , the convergence $f^n(q)\\rightarrow p$ is at a slower and slower rate.", "By (REF ) this is only possible if $\\varphi (z)$ is close to $\\partial U$ if $z\\in is close to $ ; in other words, $\\varphi $ is a proper map of $ to $ U$ and in particular $ (=U$.$ It follows from (REF ) that $\\varphi $ cannot have any critical points in $ (to see this, argue by contradiction and consider a critical point $ c of $\\varphi $ with smallest absolute value $|c|$ ).", "The Riemann-Hurwitz formula (REF ) then implies that $\\chi (U)=1$ and $\\deg (\\varphi )=1$ .", "In particular, $U$ is simply connected and $\\varphi $ is a conformal map of $ onto $ U$.", "For the conformal map $ =-1$ from $ U$ onto $ we then have $\\psi (p)=0$ and we get the desired relation $\\psi \\circ f\\circ \\psi ^{-1}=P_k$ .", "This relation (or again (REF )) implies that the fixed point $p$ is the only critical point of $f$ in $U$ and that each point $q\\in U\\setminus \\lbrace p\\rbrace $ has an infinite orbit; so $p$ is the only postcritical point of $f$ in $U$ .", "The following lemma gives us control for the mapping behavior of iterates of a rational map onto regions containing at most one postcritical point.", "Lemma 3.2 Let $f\\colon \\widehat{\\mathbb {C}}\\rightarrow \\widehat{\\mathbb {C}}$ be a rational map, $n\\in {\\mathbb {N}}$ , $U\\subseteq \\widehat{\\mathbb {C}}$ be a simply connected region with $\\#\\widehat{\\mathbb {C}}\\setminus U\\ge 2$ and $\\#(U\\cap \\operatorname{post}(f))\\le 1$ , and $V$ be a component of $f^{-n}(U)$ .", "Let $p\\in U$ be the unique point in $U\\cap \\operatorname{post}(f)$ if $\\#(U\\cap \\operatorname{post}(f))=1$ and $p\\in U$ be arbitrary if $U\\cap \\operatorname{post}(f)=\\emptyset $ , and let $\\psi _U\\colon U\\rightarrow be a conformal map with $ U(p)=0$.$ Then $V$ is simply connected, the map $f^n\\colon V\\rightarrow U$ is proper, and there exists $k\\in {\\mathbb {N}}$ , and a conformal map $\\psi _V\\colon V\\rightarrow with $ Ufn=PkV$.$ Here $k=1$ if $U\\cap \\operatorname{post}(f)=\\emptyset $ .", "Moreover, if $U\\cap \\operatorname{post}^c(f)=\\emptyset $ then $k\\le N$ , where $N=N(f)\\in {\\mathbb {N}}$ is a constant only depending on $f$ .", "In particular, for given $f$ the number $k$ is uniformly bounded by a constant $N$ independently of $n$ and $U$ if $U\\cap \\operatorname{post}^c(f)=\\emptyset $ .", "Under the given assumptions, $V$ is a region, and the map $g:=f^n|_V\\colon V\\rightarrow U$ is proper.", "Since $U\\cap \\operatorname{post}(f)\\subseteq \\lbrace p\\rbrace $ , the point $p$ is the only possible critical value of $g$ .", "It follows from the Riemann-Hurwitz formula (REF ) that $\\chi (V) &=&\\deg (g,V)\\chi (U)-\\sum _{z\\in V}(\\deg _{g}(z)-1)\\\\&=& \\deg (g,V)-(\\deg (g,V)-\\#g^{-1}(p))\\, =\\, \\#g^{-1}(p).$ As $\\chi (V)\\le 1$ , this is only possible if $\\chi (V)=1$ and $\\#g^{-1}(p)=1$ ; so $V$ is simply connected and $p$ has precisely one preimage $q$ in $V$ which is the only possible critical point of $g$ .", "Obviously, $\\#\\widehat{\\mathbb {C}}\\setminus V\\ge 2$ , and so there exists a conformal map $\\psi _V\\colon V\\rightarrow with $ V(q)=0$.", "Then$ (UfnV-1)$ is a proper holomorphic map from $ to itself and hence a finite Blaschke product $B$ .", "Moreover, $B^{-1}(0)=\\lbrace 0\\rbrace $ , and so we can replace $\\psi _V$ by a postcomposition with a suitable rotation around 0 so that $B(z)=z^k$ for $z\\in , where $ k=(g)N$.", "If $ Upost(f)=$, then $ q$ cannot be a critical point of $ g$, and so$ k=1$.$ It remains to produce a uniform upper bound for $k$ if we assume in addition that $U\\cap \\operatorname{post}^c(f)=\\emptyset $ .", "Then in the list $q, f(q), \\dots , f^{n-1}(q)$ each critical point of $f$ can appear at most once; indeed, otherwise the list contains a periodic critical point which implies that $p=f^n(q)\\in U\\cap \\operatorname{post}^c(f)$ , contradicting our additional hypothesis.", "We conclude that $k=\\deg _{f^n}(q)=\\prod _{i=0}^{n-1} \\deg _f(f^i(q))\\le N=N(f):=\\prod _{c\\in \\operatorname{crit}(f)}\\deg _f(c),$ which gives the desired uniform upper bound for $k$ .", "The next lemma describes the dynamics of a postcritically-finite rational map on arbitrary Fatou components.", "Lemma 3.3 (Dynamics on the Fatou components) Let $f\\colon \\widehat{\\mathbb {C}}\\rightarrow \\widehat{\\mathbb {C}}$ be a postcritically-finite rational map, and $\\mathcal {C}$ be the collection of all Fatou components of $f$ .", "Then there exists a family $\\lbrace \\psi _U\\colon U\\rightarrow U\\in \\mathcal {C}\\rbrace $ of conformal maps with the following property: if $U$ and $V$ are Fatou components of $f$ with $f(V)=U$ , then $\\psi _U\\circ f=P_k\\circ \\psi _V$ on $V$ for some $k=k(U,V)\\in {\\mathbb {N}}$ .", "Moreover, for each $U\\in \\mathcal {C}$ the point $p_U:=\\psi _U^{-1}(0)$ is the unique point in $U\\cap \\bigcup _{n\\in {\\mathbb {N}}_0} f^{-n}(\\operatorname{post}(f))$ .", "In contrast to the points $p_U$ the maps $\\psi _U$ are not uniquely determined in general due to a certain rotational freedom.", "As we will see in the proof of the lemma, $p_U$ can also be characterized as the unique point in $U$ with a finite orbit under iteration of $f$ .", "In the following, we will choose $p_U$ as a basepoint in the Fatou component $U$ .", "If we take 0 as a basepoint in $, then in the previous lemmawe get the following commutative diagram of basepoint-preserving maps between{\\em pointed}regions (i.e., regions with a distinguished basepoint):\\begin{equation*}{(V,p_V) [r]^{\\psi _V} [d]_{f} & (0) [d]^{P_k} \\\\(U,p_U) [r]^{\\psi _U} & (0)}\\end{equation*}Note that this implies in particular that $ f-1(pU)={pV}$ and that $ f V{pV}U{pU}$ is a covering map.$ We first construct the desired maps $\\psi _U$ for the periodic Fatou components $U$ of $f$ .", "So fix a periodic Fatou component $U$ of $f$ , and let $n\\in {\\mathbb {N}}$ be the period of $U$ , i.e., if we define $U_0:=U$ and $U_{k+1}=f(U_k)$ for $k=0, \\dots , n-1$ , then the Fatou components $U_0, \\dots , U_{n-1}$ are all distinct, and $U_n=f^n(U)=U$ .", "By Lemma REF applied to the map $f^n$ , for each $k=0, \\dots , n-1$ the Fatou component $U_k$ is simply connected and there exists a unique point $p_k\\in U_k$ that lies in $\\operatorname{post}(f)=\\operatorname{post}(f^n)$ .", "Moreover, there exists a conformal map $\\psi _0\\colon U_0\\rightarrow with $ 0(p0)=0$ such that$ 0fn=Pd0$ for suitable $ dN$.$ Let $\\psi _1\\colon U_1\\rightarrow be a conformal map with $ 1(p1)=0$.", "By the argument in the proof of Lemma~\\ref {lem:deg} we know that $ B=1f 0-1$ is a finite Blaschke product $ B$ with $ B-1(0)={0}$ and so $ B(z)=azd1$ for suitable constants $ d1N$ and $ a with $|a|=1$ .", "By adjusting $\\psi _1$ by a suitable rotation factor if necessary, we may assume that $a=1$ .", "Then $\\psi _1\\circ f=P_{d_1}\\circ \\psi _0$ on $U_0$ .", "If we repeat this argument, then we get conformal maps $\\psi _k\\colon U_k\\rightarrow with $ k(pk)=0$ and\\begin{equation}\\psi _{k}\\circ f=P_{d_k}\\circ \\psi _{k-1}\\end{equation} on $ Uk-1$ with suitable $ dkN$ for$ k=1, ..., n$.", "Note that$$ \\psi _n\\circ f^n = P_{d_n}\\circ \\psi _{n-1}\\circ f^{n-1}=\\dots =P_{d_n}\\circ \\dots \\circ P_{d_1}\\circ \\psi _0=P_{d^{\\prime }}\\circ \\psi _0$$on $ U0$, where $ d'N$.", "On the other hand, $ 0fn=Pd0$ by definition of $ 0$.Hence $ d=fn(p0)=d'$, and so $ nfn= 0fn$ on $ U0$ which implies $ n=0$.", "If we now define $ Uk:=k$ for $ k=0, ..., n-1$, then by (\\ref {eq:perdFatok}) the desired relation (\\ref {eq:desFatcomp}) holds for each suitable pairof Fatou components from the cycle $ U0, ..., Un-1$.", "We also choose $ pUk=pkUk$ as a basepoint in $ Uk$ for $ k=0, ..., n-1$.", "We know that $ pk$ is the unique point in$ Uk$ that lies in $ post(f)$.", "Since $ f$ is postcritically-finite, each point in$ P:=nN0 f-n(post(f))$ has a finite orbit under iteration of $ f$.It follows from Lemma~\\ref {lem:postratFatou} thateach point $ pUk{p}$ has an infinite orbit and therefore cannot lie in$ P$.", "Hence $ pUk$ is the unique point in $ Uk$ that lies in $ P$.$ We repeat this argument for the other finitely many periodic Fatou components $U$ to obtain suitable conformal maps $\\psi _U\\colon U\\rightarrow and unique basepoints $ pU=U-1(0)UP$.$ If $V$ is a non-periodic Fatou component, then it is mapped to a periodic Fatou component by a sufficiently high iterate of $f$ (this is Sullivan's theorem on the non-existence of wandering domains; see [3]).", "We call the smallest number $k\\in {\\mathbb {N}}_0$ such that $f^k(V)$ is a periodic Fatou component the level of $V$ .", "Suppose $V$ is an arbitrary Fatou component of level 1.", "Then $U=f(V)$ is periodic, and so $\\psi _U$ and $p_U$ are already defined and we know that $\\lbrace p_U\\rbrace =\\operatorname{post}(f)\\cap U$ .", "Hence by Lemma REF there exists a conformal map $\\psi _V\\colon V\\rightarrow U$ such that (REF ) is valid.", "If $p_V:=\\psi _V^{-1}(0)$ , then $f(p_V)=p_U\\in \\operatorname{post}(f)$ , and so $p_V\\in V\\cap P$ .", "Moreover, (REF ) shows that $f(V\\setminus \\lbrace p_V\\rbrace )=U\\setminus \\lbrace p_U\\rbrace $ which implies that each point in $V\\setminus \\lbrace p_V\\rbrace $ has an infinite orbit and cannot lie in $P$ .", "It follows that $p_V$ is the unique point in $V$ that lies in $P$ .", "We repeat this argument for Fatou components of higher and higher level.", "Note that if for a Fatou component $U$ a conformal map $\\psi _U\\colon U\\rightarrow has already been constructed and we know that $ pU:=-1(0)$ is the unique point in $ UP$, then$ Upost(f) {pU}$ and we can again apply Lemma~\\ref {lem:deg} for a Fatou component $ V$ with $ f(V)=U$.$ In this way we obtain conformal maps $\\psi _U$ as desired for all Fatou components $U$ .", "The point $p_U=\\psi _U^{-1}(0)$ is the unique point in $U$ that lies in $P$ , because $f^k(p_U)\\in \\operatorname{post}(f)$ for some $k\\in {\\mathbb {N}}_0$ and all other points in $U$ have an infinite orbit.", "We conclude this section with a lemma that is required in the proof of Theorem REF .", "Lemma 3.4 (Lifting lemma) Let $f\\colon \\widehat{\\mathbb {C}}\\rightarrow \\widehat{\\mathbb {C}}$ be a postcritically-finite rational map, $n\\in {\\mathbb {N}}$ , and $(U,p_U)$ and $(V,p_V)$ be pointed Fatou components of $f$ that are Jordan regions with $f^n(U)=V$ .", "Suppose $D\\subseteq \\widehat{\\mathbb {C}}$ is another Jordan region with a basepoint $p_D\\in D$ , and suppose that $\\alpha \\colon \\overline{D}\\rightarrow \\overline{V} $ is a map with the following properties: (i) $\\alpha $ is continuous on $ \\overline{D}$ and holomorphic on $D $ , (ii) $\\alpha ^{-1}(p_V)=\\lbrace p_D\\rbrace $ , (iii) there exists a continuous map $\\beta \\colon \\partial D \\rightarrow \\partial U$ with $f^n\\circ \\beta = \\alpha |_{\\partial D}$ .", "Then there exists a unique continuous map $\\tilde{\\alpha }\\colon \\overline{D}\\rightarrow \\overline{U} $ with $f^n\\circ \\tilde{\\alpha }= \\alpha $ and $\\tilde{\\alpha }|_{\\partial D}=\\beta $ .", "Moreover, $\\tilde{\\alpha }$ is holomorphic on $D$ and satisfies $\\tilde{\\alpha }^{-1}(p_U)=\\lbrace p_D\\rbrace $ .", "If, in addition, $\\beta $ is a homeomorphism of $\\partial D$ onto $\\partial U$ , then $\\tilde{\\alpha }$ is a conformal homeomorphism of $\\overline{D}$ onto $ \\overline{U}$ .", "Here we call a map $\\varphi \\colon \\overline{\\Omega }\\rightarrow \\overline{\\Omega }^{\\prime }$ between the closures of two Jordan regions $\\Omega , \\Omega ^{\\prime }\\subseteq \\widehat{\\mathbb {C}}$ a conformal homeomorphism if $\\varphi $ is a homeomorphism of $\\overline{\\Omega }$ onto $\\overline{\\Omega }^{\\prime }$ and a conformal map of $\\Omega $ onto $\\Omega ^{\\prime }$ .", "Note that in the previous lemma we necessarily have $f^n(\\overline{U})=\\overline{V}$ , $\\alpha (p_D)=p_V=f^n(p_U)$ , and $\\alpha (\\partial D)\\subseteq \\partial V$ by (iii).", "In the conclusion of the lemma we obtain a lift $\\tilde{\\alpha }$ for a given map $\\alpha $ under the branched covering map $f^n$ so that the diagram ${&\\overline{U} [d]^{f^n} \\\\\\overline{D} [ur]^{\\tilde{\\alpha }} [r]^{\\alpha } & \\overline{V}}$ commutes.", "By Lemma REF the map $f^n$ is actually an unbranched covering map from $\\overline{U}\\setminus \\lbrace p_U\\rbrace $ onto $\\overline{V}\\setminus \\lbrace p_V\\rbrace $ .", "The lemma asserts that the existence and uniqueness of a lift $\\tilde{\\alpha }$ is guaranteed if the boundary map $\\alpha |_{\\partial D}$ has a lift (namely $\\beta $ ), and if we have some compatibility condition for branch points (given by condition (ii)).", "The lemma easily follows from some basic theory for covering maps and lifts (see [11] for general background), so we will only sketch the argument and leave some straightforward details to the reader.", "By Lemma REF we can change $\\overline{U}$ and $\\overline{V}$ by conformal homeomorphisms so that we can assume $\\overline{U}=\\overline{V}=\\overline{, p_U=0=p_V, and f^n=P_k for suitable k\\in {\\mathbb {N}} without loss of generality.By classical conformal mapping theory we may also assume that D= and p_D=0.Then condition (ii) translates to \\alpha (0)=0 and \\alpha (z)\\ne 0 for z\\in \\overline{\\setminus \\lbrace 0\\rbrace .", "}We use this to define a homotopy of the boundary map \\alpha |_{\\partial into the base space \\overline{\\setminus \\lbrace 0\\rbrace of the covering map P_k\\colon \\overline{\\setminus \\lbrace 0\\rbrace \\rightarrow \\overline{\\setminus \\lbrace 0\\rbrace .", "Namely, let H\\colon \\partial (0,1]\\rightarrow \\overline{\\setminus \\lbrace 0\\rbrace be defined asH( \\zeta ,t):=\\alpha (t\\zeta ) for \\zeta \\in \\partial and t\\in (0,1].", "It is convenient to think of H as a homotopy running backwards in time t\\in (0,1] starting at t=1.Note that P_k\\circ \\beta =\\alpha |_{\\partial =H(\\cdot , 1).", "So for the initial time t=1 the homotopy has the lift \\beta under the covering map P_k\\colon \\overline{\\setminus \\lbrace 0\\rbrace \\rightarrow \\overline{\\setminus \\lbrace 0\\rbrace .", "By the homotopy lifting theorem \\cite [p.~60, Proposition~1.30]{Ha}, the whole homotopy H has a unique liftstarting at \\beta , i.e., there exists a unique continuous map \\widetilde{H}\\colon \\partial (0,1]\\rightarrow \\overline{\\setminus \\lbrace 0\\rbrace such thatP_k\\circ \\widetilde{H}=H and \\widetilde{H}(\\cdot , 1)=\\beta .", "Now we define \\tilde{\\alpha }(z)=\\widetilde{H}( z/|z|, |z|) for z\\in \\overline{\\setminus \\lbrace 0\\rbrace .", "Then \\tilde{\\alpha } is continuous on\\overline{\\setminus \\lbrace 0\\rbrace , where it satisfies P_k\\circ \\tilde{\\alpha }=\\alpha .", "Since \\alpha (0)=0, this last equation implies that we get a continuous extensionof \\tilde{\\alpha } to \\overline{ by setting \\tilde{\\alpha }(0)=0.", "This extension is a lift \\tilde{\\alpha } of \\alpha .", "Note that \\tilde{\\alpha }^{-1}(0)=0 and\\tilde{\\alpha }|_{\\partial =\\widetilde{H}(\\cdot , 1)=\\beta .", "Moreover, \\tilde{\\alpha } is holomorphic on , because it is a continuous branch of the k-th root of the holomorphic function\\alpha on .", "This shows that \\tilde{\\alpha } has the desired properties.", "The uniqueness of \\tilde{\\alpha } easily follows from the uniqueness of \\widetilde{H}.", "}We have \\beta = \\tilde{\\alpha }|_{\\partial ; so if \\beta is a homeomorphism,then the argument principle implies that \\tilde{\\alpha } is a conformal homeomorphism of\\overline{ onto \\overline{.", "}}}}\\section {The conformal elevator for subhyperbolic maps}\\numero A rational map f is called {\\em subhyperbolic} if each critical point of f in \\mathcal {J}(f) has a finite orbit while each critical point in \\mathcal {F}(f) has an orbit that converges to an attracting or superattracting cycle of f.The map f is called {\\em hyperbolic} if it is subhyperbolic and f does not have critical points in \\mathcal {J}(f).", "Note that every postcritically-finite rational map is subhyperbolic.", "}}}}For the rest of this section, we will assume that f is a subhyperbolic rational map with \\mathcal {J}(f)\\ne \\widehat{\\mathbb {C}}.Moreover, we will make the following additional assumption:\\begin{equation}\\mathcal {J}(f)\\subseteq \\tfrac{1}{2} \\text{and}\\quad f^{-1}( \\subseteq \\end{equation}Here and in what follows, if B is a disk, we denote by \\frac{1}{2} Bthe disk with the same center and whose radius is half the radius of B.", "}The inclusions (\\ref {eq:addinv}) can always be achieved by conjugating fwith an appropriate Möbius transformation so that \\mathcal {J}(f)\\subseteq \\tfrac{1}{2} and \\infty is an attracting or superattractingperiodic point of f. If we then replace f with suitable iterate, we may in additionassume that \\infty becomes an attracting or superattracting fixed point of f withf(\\widehat{\\mathbb {C}}\\setminus \\subseteq \\widehat{\\mathbb {C}}\\setminus .", "The latter inclusion is equivalent tof^{-1}( \\subseteq .", "}}}}Every small disk B centered at a point in \\mathcal {J}(f) can be ``blown up\" by a carefully chosen iterate f^nto a definite size with good control on how sets are distorted under the map f^n.", "We will discuss this in detail as a preparation for the proofs of Theorems~\\ref {thm:circgeom} and \\ref {thm:main2}, and will refer to this procedure as applying the {\\em conformal elevator} to B.", "In the following, all metric notions refer to the Euclidean metric on .", "}}}$ Let $P\\subseteq \\widehat{\\mathbb {C}}$ denote the union of all superattracting or attracting cycles of $f$ .", "This is a non-empty and finite set contained in $ \\mathcal {F}(f)$ .", "Since $f$ is subhyperbolic, every critical point in $\\mathcal {J}(f)$ has a finite orbit, and every critical point in $\\mathcal {J}(f)$ has an orbit that converges to $P$ .", "Hence there exists a neighborhood of $\\mathcal {J}(f)$ that contains only finitely many points in $\\operatorname{post}(f)$ and no points in $\\operatorname{post}^c(f)$ .", "This implies that we can choose $\\epsilon _0>0$ so small that $\\operatorname{diam}(\\mathcal {J}(f))> 2\\epsilon _0$ , and so that every disk $B^{\\prime }=B(q,r^{\\prime })$ centered at a point $q\\in \\mathcal {J}(f)$ with positive radius $r^{\\prime }\\le 8\\epsilon _0$ is contained in $, contains no point in $ postc(f)$ and at most one point in$ post(f)$.$ Let $B=B(p,r)$ be a small disk centered at a point $p\\in \\mathcal {J}(f)$ and of positive radius $r<\\epsilon _0$ .", "Since $B$ is centered at a point in $\\mathcal {J}(f)$ , we have $\\mathcal {J}(f)\\subseteq f^n(B)$ for sufficiently large $n$ (see [3]), and so the images of $B$ under iterates will eventually have diameter $>2\\epsilon _0$ .", "Hence there exists a maximal number $n\\in {\\mathbb {N}}_0$ such that $f^n(B)$ is contained in the disk of radius $\\epsilon _0$ centered at a point $\\tilde{q}\\in \\mathcal {J}(f)$ .", "If $B(\\tilde{q}, 2\\epsilon _0)\\cap \\operatorname{post}(f)=\\emptyset $ , we define $q=\\tilde{q}$ and $B^{\\prime }=B(q, 2\\epsilon _0)$ .", "Otherwise, there exists a unique point $ q\\in B(\\tilde{q}, 2\\epsilon _0)\\cap \\operatorname{post}(f)$ .", "Then we define $B^{\\prime }:=B( q, 8\\epsilon _0)\\supset B(q, 4\\epsilon _0)\\supset B(\\tilde{q}, 2\\epsilon _0)\\supset f^n(B).", "$ In both cases, we have (i) $f^n(B)\\subseteq \\frac{1}{2}B^{\\prime }\\subseteq ,$ (ii) $B^{\\prime }\\cap \\operatorname{post}^c(f)=\\emptyset $ , (iii) $\\#(B^{\\prime }\\cap \\operatorname{post}(f))\\le 1$ with equality only if $B^{\\prime }$ is centered at a point in $\\operatorname{post}(f)$ .", "By definition of $n$ , the set $f^{n+1}(B)$ must have diameter $\\ge \\epsilon _0$ .", "Hence by uniform continuity of $f$ near $\\mathcal {J}(f)$ there exists $\\delta _0>0$ independent of $B$ such that (iv) $\\operatorname{diam}(f^n(B))\\ge \\delta _0$ .", "Let $\\Omega \\subseteq \\widehat{\\mathbb {C}}$ be the unique component of $f^{-n}(B^{\\prime })$ that contains $B$ .", "Then by Lemma REF and by (), (v) $\\Omega $ is simply connected, and $B\\subseteq \\Omega \\subseteq ,$ (vi) the map $f^n|_{\\Omega }\\colon \\Omega \\rightarrow B^{\\prime }$ is proper, (vii) there exists $k\\in {\\mathbb {N}}$ , and conformal maps $\\varphi \\colon B^{\\prime }\\rightarrow and $ such that $(\\varphi \\circ f^n\\circ \\psi ^{-1})(z)= z^k$ for all $z\\in .", "Here $ kN$ is uniformly bounded independent of $ B$.$ If $k\\ge 2$ , then $q=\\varphi ^{-1}(0)\\in \\operatorname{post}(f)\\cap B^{\\prime }$ , and so $q$ is the center of $B^{\\prime }$ .", "If $k=1$ , then we can choose $\\varphi $ so that this is also the case.", "So (viii) $\\varphi $ maps the center $q$ of $B^{\\prime }$ to 0.", "We refer to the choice of $f^n$ and the associated sets $B^{\\prime }$ and $\\Omega $ and the maps $\\varphi $ and $\\psi $ satisfying properties (i)–(viii) as applying the conformal elevator to $B$ .", "Lemma 3.5 There exist constants $\\gamma ,r_1>0$ and $C_1,C_2,C_3\\ge 1$ independent of $B=B(p,r)$ with the following properties: (a) If $A\\subseteq B$ is a connected set, then $\\frac{\\operatorname{diam}(A)}{\\operatorname{diam}(B)}\\le C_1 \\operatorname{diam}(f^n(A))^{\\gamma }.", "$ (b) $B(f^n(p), r_1)\\subseteq f^n(\\frac{1}{2} B)\\subseteq f^n(B)$ .", "(c) If $u,v\\in B$ , then $|f^n(u)- f^n(v)|\\le C_2\\frac{|u-v|}{\\operatorname{diam}(B)}.", "$ (d) If $u,v\\in B$ , $u\\ne v$ , and $f^n(u)=f^n(v)$ , then the center $q$ of $B^{\\prime }$ belongs to $\\operatorname{post}(f)$ and we have $|f^n(u)- q| \\le C_3\\frac{|u-v|}{\\operatorname{diam}(B)}.", "$ So (a) says that a connected set $A\\subseteq B$ comparable in diameter to $B$ is blown up to a definite size under the conformal elevator, and by (b) the image of $B$ contains a disk of a definite size.", "If we consider the maps $f^n|_B$ for different $B$ , then by (c) they are uniformly Lipschitz if we rescale distances in $B$ by $1/\\operatorname{diam}(B)$ .", "In (d) the center $q$ of $B^{\\prime }$ must be a point in $\\operatorname{post}(f)$ for otherwise $f^n$ would be injective; so (d) says that if distinct, but nearby points are mapped to the same image $w$ under $f^n$ , then a postcritical point must be close to this image $w$ .", "In the following we write $a\\lesssim b$ or $a\\gtrsim b$ for two quantities $a,b\\ge 0$ if we can find a constant $C>0$ independent of the disk $B$ such that $a\\le C b$ or $Ca\\ge b$ , respectively.", "We write $a\\approx b$ if we have both $a\\lesssim b$ and $a\\gtrsim b$ , and in this case say that the quantities $a$ and $b$ are comparable.", "We consider the conformal maps $\\varphi \\colon B^{\\prime }\\rightarrow and$ satisfying properties (vii) and (viii) of the conformal elevator as discussed above.", "The exponent $k$ in (vii) is uniformly bounded, say $k\\le N$ , where $N\\in {\\mathbb {N}}$ is independent of $B$ .", "As before, we use the notation $P_k(z)=z^k$ for $z\\in .$ As we will see, the properties (a)–(d) easily follow from distortion properties of the map $P_k$ .", "We discuss the relevant properties of $P_k$ first (the proof is left to the reader).", "The map $P_k$ is Lipschitz with uniformly bounded Lipschitz constant, because $k$ is uniformly bounded.", "If $M\\subseteq is connected, then$$ \\operatorname{diam}(P_k(M))\\gtrsim \\operatorname{diam}(M)^k\\gtrsim \\operatorname{diam}(M)^N.", "$$Moreover, if $ B(z, r0), then $ B(P_k(z), r_1)\\subseteq P_k(B(z, r_0)), $ where $r_1 \\gtrsim r_0^k \\ge r_0^N$ .", "By (vii) the map $\\varphi $ is a Euclidean similarity, and so $\\varphi (\\frac{1}{2} B^{\\prime })=\\frac{1}{2} .Since the radius of $ B'$ is equal to $ 20$ or $ 80$, and hence comparable to $ 1$, wehave\\begin{equation}|\\varphi (u^{\\prime })-\\varphi (v^{\\prime })|\\approx |u^{\\prime }-v^{\\prime }|,\\end{equation}whenever $ u',v'B'$.$ Moreover, for $\\rho :=2^{-1/N}\\in (0,1)$ (which is independent of $B$ ) we have $D:=B(0,\\rho )\\supseteq P_k^{-1}( \\tfrac{1}{2} =P_k^{-1}(\\varphi (\\tfrac{1}{2} B^{\\prime }))=\\psi (f^{-n}(\\tfrac{1}{2} B^{\\prime })\\cap \\Omega ).", "$ Since $f^n(B)\\subseteq \\frac{1}{2} B^{\\prime }$ by (i) and $B\\subseteq \\Omega $ by (v), we then have $\\psi (B) \\subseteq D$ .", "So if $u,v\\in B$ , then $\\psi (u), \\psi (v)\\in D$ .", "Hence by the Koebe distortion theorem we have $|u-v| \\approx |(\\psi ^{-1})^{\\prime }(0)|\\cdot |\\psi (u)-\\psi (v)|$ whenever $u,v\\in B$ .", "In particular, $ \\operatorname{diam}(B)\\approx |(\\psi ^{-1})^{\\prime }(0)|\\cdot \\operatorname{diam}(\\psi (B)) $ On the other hand, by (iv) $1&\\approx \\operatorname{diam}(f^n(B))\\approx \\operatorname{diam}\\big (\\varphi (f^n(B))\\big ) \\\\ &= \\operatorname{diam}\\big (P_k(\\psi (B))\\big ) \\lesssim \\operatorname{diam}(\\psi (B))\\le 2.$ Hence $\\operatorname{diam}(\\psi (B)) \\approx 1$ , and so $\\operatorname{diam}(B)\\approx |(\\psi ^{-1})^{\\prime }(0)|$ .", "This implies that $\\frac{ |u-v|}{\\operatorname{diam}(B)}\\approx |\\psi (u)-\\psi (v)|,$ whenever $u,v\\in B$ .", "Now let $A\\subseteq B$ be connected.", "Then $\\psi (A)$ is connected, which implies $\\frac{ \\operatorname{diam}(A)}{\\operatorname{diam}(B)}&\\approx \\operatorname{diam}(\\psi (A))\\lesssim \\operatorname{diam}\\big (P_k( \\psi (A))\\big )^{1/N} \\\\ & = \\operatorname{diam}\\big (\\varphi (f^n(A))\\big )^{1/N} \\approx \\operatorname{diam}(f^n(A))^{1/N}.$ Inequality (a) follows.", "It follows from (REF ) that there exists $r_0>0$ independent of $B$ such that $ B(\\psi (p), r_0))\\subseteq \\psi (\\frac{1}{2} B)$ .", "By the distortion property of $P_k$ mentioned in the beginning of the proof, $\\varphi (f^n(\\frac{1}{2} B))=P_k(\\psi (\\frac{1}{2} B))$ then contains a disk $B(P_k(\\psi (p)), r_1)$ with $r_1>0$ independent of $B$ .", "Since $\\varphi (f^n(p))=P_k(\\psi (p))$ , and $\\varphi $ distorts distances uniformly, statement (b) follows.", "For (c) note that if $u,v\\in B$ , then $|f^n(u)-f^n(v)|&\\approx |\\varphi (f^n(u))-\\varphi (f^n(v))|= |P_k(\\psi (u))-P_k(\\psi (v))|\\\\&\\lesssim |\\psi (u)-\\psi (v)| \\approx \\frac{ |u-v|}{\\operatorname{diam}(B)}.", "$ We used that $P_k$ is Lipschitz on $ with a uniform Lipschitz constant.$ Finally we prove (d).", "If $u,v\\in B$ , $u\\ne v$ , and $f^n(u)=f^n(v)$ , then $f^n$ is not injective on $B^{\\prime }$ , and so the center $q$ of $B^{\\prime }$ belongs to $\\operatorname{post}(f)$ .", "Moreover, we then have $\\psi (u)^k=\\psi (v)^k$ , but $\\psi (u)\\ne \\psi (v)$ .", "This implies that $ |\\psi (u)-\\psi (v)|\\gtrsim \\frac{1}{k} |\\psi (u)| \\approx |\\psi (u)|.", "$ It follows that $|f^n(u)-q|&\\approx |\\varphi (f^n(u))-\\varphi (q)|= |\\psi (u)^k|\\\\&\\le |\\psi (u)| \\lesssim |\\psi (u)-\\psi (v)| \\approx \\frac{ |u-v|}{\\operatorname{diam}(B)}.", "$ Geometry of the peripheral circles no In this section we will prove Theorem REF .", "We have already defined in Section  what it means for the peripheral circles of a Sierpiński carpet $S$ to be uniform quasicircles and to be uniformly relatively separated.", "We say that the peripheral circles of $S$ occur on all locations and scales if there exists a constant $C\\ge 1$ such that for every $p\\in S$ and every $0<r\\le {\\rm diam}(\\widehat{\\mathbb {C}})=2$ , there exists a peripheral circle $J$ of $S$ with $B(p,r)\\cap J\\ne 0$ and $r/C\\le {\\rm diam}(J)\\le C r.$ Here and below the metric notions refer to the chordal metric $\\sigma $ on $\\widehat{\\mathbb {C}}$ .", "A set $M\\subseteq \\widehat{\\mathbb {C}}$ is called porous if there exists a constant $c>0$ such that for every $p\\in S$ and every $0<r\\le 2$ there exists a point $q\\in B(p,r)$ such that $B(q,cr)\\subseteq \\widehat{\\mathbb {C}}\\setminus M$ .", "Before we turn to the proof of Theorem REF , we require an auxiliary fact.", "Lemma 4.1 Let $f$ be a rational map such that $\\mathcal {J}(f)$ is a Sierpiński carpet, and let $J$ be a peripheral circle of $\\mathcal {J}(f)$ .", "Then $f^n(J)$ is a peripheral circle of $\\mathcal {J}(f)$ , and $f^{-n}(J)$ is a union of finitely many peripheral circles of $\\mathcal {J}(f)$ for each $n\\in {\\mathbb {N}}$ .", "Moreover, $J\\cap \\operatorname{post}(f)=\\emptyset =J\\cap \\operatorname{crit}(f)$ .", "There exists precisely one Fatou component $U$ of $f$ such that $\\partial U=J$ .", "Then $V=f^n(U)$ is also a Fatou component of $f$ .", "Hence $\\partial V$ is a peripheral circle of $\\mathcal {J}(f)$ .", "The map $f^n|_U\\colon U\\rightarrow V$ is proper which implies that $f^n(J)=f^n(\\partial U)=\\partial V$ .", "Similarly, there are finitely many distinct Fatou components $V_1, \\dots , V_k$ of $f$ such that $ f^{-n}(U)=V_1\\cup \\dots \\cup V_k.", "$ Then $f^{-n}(J)=\\partial V_1\\cup \\dots \\cup \\partial V_k, $ and so the preimage of $J$ under $f^n$ consists of the finitely many disjoint Jordan curves $\\partial V_i$ , $i=1, \\dots , k$ , which are peripheral circles of $\\mathcal {J}(f)$ .", "To show $J\\cap \\operatorname{post}(f)=\\emptyset $ , we argue by contradiction, and assume that there exists a point $p\\in \\operatorname{post}(f)\\cap J$ .", "Then there exists $n\\in {\\mathbb {N}}$ , and $c\\in \\operatorname{crit}(f)$ such that $f^n(c)=p$ .", "As we have just seen, the preimage of $J$ under $f^n$ consists of finitely many disjoint Jordan curves, and is hence a topological 1-manifold.", "On the other hand, since $c\\in f^{-n}(p)\\subseteq f^{-n}(J)$ is a critical point of $f$ and hence of $f^n$ , at $c$ the set $f^{-n}(J)$ cannot be a 1-manifold.", "This is a contradiction.", "Finally, suppose that $c\\in J\\cap \\operatorname{crit}(f)$ .", "Then $f(c)\\in \\operatorname{post}(f)\\cap f(J)$ , and $f(J)$ is a peripheral circle of $\\mathcal {J}(f)$ .", "This is impossible by what we have just seen.", "A general idea for the proof is to argue by contradiction, and get locations where the desired statements fail quantitatively in a worse and worse manner.", "One can then use the dynamics to blow up to a global scale and derive a contradiction from topological facts.", "It is fairly easy to implement this idea if we have expanding dynamics given by a group (see, for example, [5]).", "In the present case, one applies the conformal elevator and the estimates as given by Lemma REF .", "We now provide the details.", "We can pass to iterates of the map $f$ , and also conjugate $f$ by a Möbius transformation as properties that we want to establish are Möbius invariant.", "This Möbius invariance is explicitly stated for peripheral circles to be uniform quasicircles and to be uniformly relatively separated in [5].", "The Möbius invariance of the other stated properties immediately follows from the fact that each Möbius transformation is bi-Lipschitz with respect to the chordal metric.", "In this way, we may assume that () is true.", "Then the peripheral circles are subsets of $, where chordal and Euclidean metric are comparable.", "Therefore, we can use the Euclidean metric, and all metric notions will refer to this metric in the following.$ Part I.", "To show that peripheral circles of $ \\mathcal {J}(f)$ are uniform quasicircles, we argue by contradiction.", "Then for each $k\\in {\\mathbb {N}}$ there exists a peripheral circle $J_k$ of $\\mathcal {J}(f)$ , and distinct points $u_k, v_k\\in J_k$ such that if $\\alpha _k,\\beta _k$ are the two subarcs of $J_k$ with endpoints $u_k$ and $v_k$ , then $\\frac{\\min \\lbrace \\operatorname{diam}(\\alpha _k), \\operatorname{diam}(\\beta _k)\\rbrace }{|u_k-v_k|} \\rightarrow \\infty $ as $k\\rightarrow \\infty $ .", "We can pick $r_k>0$ such that $ \\min \\lbrace \\operatorname{diam}(\\alpha _k), \\operatorname{diam}(\\beta _k)\\rbrace /r_k\\rightarrow \\infty $ and $ |u_k-v_k|/r_k\\rightarrow 0$ as $k\\rightarrow \\infty .$ We now apply the conformal elevator to $B_k:=B(u_k, r_k)$ .", "Let $f^{n_k}$ be the corresponding iterate and $B_k^{\\prime }$ be the ball as discussed in Section .", "Define $J^{\\prime }_k=f^{n_k}(J_k)$ , $u^{\\prime }_k=f^{n_k}(u_k)$ , and $v^{\\prime }_k=f^{n_k}(v_k)$ .", "Then Lemma REF  (a) and (REF ) imply that the diameters of the sets $J^{\\prime }_k$ are uniformly bounded away from 0 independently of $k$ .", "Since $J_k^{\\prime }$ is a peripheral circle of the Sierpiński carpet $\\mathcal {J}(f)$ by Lemma REF , there are only finitely many possibilities for the set $J^{\\prime }_k$ .", "By passing to suitable subsequence if necessary, we may assume that $J^{\\prime }=J^{\\prime }_k$ is a fixed peripheral circle of $\\mathcal {J}(f)$ independent of $k$ .", "The points $u^{\\prime }_k,v^{\\prime }_k$ lie in $J^{\\prime }$ and by (REF ) and Lemma REF  (c) we have $|u^{\\prime }_k-v^{\\prime }_k|\\rightarrow 0$ as $k\\rightarrow \\infty $ .", "For large $k$ we want to find a point $w_k\\ne u_k$ in $B_k$ near $u_k$ with $f^{n_k}(u_k)=f^{n_k}(w_k)$ .", "If $u^{\\prime }_k=v^{\\prime }_k$ we can take $w_k=v_k$ .", "Otherwise, if $u^{\\prime }_k\\ne v^{\\prime }_k$ , there are two subarcs of $J^{\\prime }$ with endpoints $u^{\\prime }_k$ and $v^{\\prime }_k$ .", "Let $\\gamma ^{\\prime }_k\\subseteq J^{\\prime }$ be the one with smaller diameter.", "Then by (REF ) we have $\\operatorname{diam}(\\gamma ^{\\prime }_k)\\rightarrow 0$ as $k\\rightarrow \\infty $ (for the moment we only consider such $k$ for which $\\gamma ^{\\prime }_k$ is defined).", "Since $J^{\\prime }\\cap \\operatorname{post}(f)=\\emptyset $ by Lemma REF , the map $f^{n_k}\\colon J_k \\rightarrow J^{\\prime }$ is a covering map.", "So we can lift the arc $\\gamma ^{\\prime }_k$ under $f^{n_k}$ to a subarc $\\gamma _k$ of $J_k$ with initial point $v_k$ and $f^{n_k}(\\gamma _k)=\\gamma ^{\\prime }_k$ .", "By Lemma REF  (b) we have $\\gamma ^{\\prime }_k\\subseteq f^{n_k} (B_k)$ for large $k$ ; then Lemma REF  (a) implies that $\\gamma _k\\subseteq B_k$ for large $k$ , and also $\\operatorname{diam}(\\gamma _k)/r_k\\rightarrow 0$ as $k\\rightarrow \\infty $ .", "Note that if $w_k$ is the other endpoint of $\\gamma _k$ , then $f^{n_k}(w_k)=u^{\\prime }_k$ .", "We have $w_k\\ne u_k$ for large $k$ ; for if $w_k=u_k$ , then $\\gamma _k\\subseteq J_k$ has the endpoints $u_k$ and $v_k$ and so must agree with one of the arcs $\\alpha _k$ or $\\beta _k$ ; but for large $k$ this is impossible by (REF ) and (REF ).", "In addition, we have $|u_k-w_k|/r_k\\le |u_k-v_k|/r_k+\\operatorname{diam}(\\gamma _k)/r_k\\rightarrow 0$ as $k\\rightarrow \\infty $ .", "Note that this is also true if $w_k=v_k$ .", "In summary, for each large $k$ we can find a point $w_k\\in B_k$ with $w_k\\ne u_k$ , $f^{n_k}(u_k)=f^{n_k}(w_k)$ , and $ |u_k-w_k|/r_k\\rightarrow 0$ as $k\\rightarrow \\infty $ .", "Then by Lemma REF  (d) the center $q_k$ of $B^{\\prime }_k$ must belong to the postcritical set of $f$ and $\\operatorname{dist}(J^{\\prime }, \\operatorname{post}(f))\\le |u^{\\prime }_k- q_k|\\rightarrow 0$ as $k\\rightarrow \\infty $ .", "Since $f$ is subhyperbolic, every sufficiently small neighborhood of $\\mathcal {J}(f) \\supseteq J^{\\prime }$ contains only finitely many points in $\\operatorname{post}(f)$ , and so this implies $J^{\\prime }\\cap \\operatorname{post}(f)\\ne \\emptyset $ .", "We know that this is impossible by Lemma REF and so we get a contradiction.", "This shows that the peripheral circles are uniform quasicircles.", "Part II.", "The proof that the peripheral circles of $\\mathcal {J}(f)$ are uniformly relatively separated runs along almost identical lines.", "Again we argue by contradiction.", "Then for $k\\in {\\mathbb {N}}$ we can find distinct peripheral circles $\\alpha _k$ and $\\beta _k$ of $\\mathcal {J}(f)$ , and points $u_k\\in \\alpha _k$ , $v_k\\in \\beta _k$ such that (REF ) is valid.", "We can again pick $r_k>0$ so that the relations (REF ) and (REF ) are true.", "As before we define $B_k=B(u_k, r_k)$ and apply the conformal elevator to $B_k$ which gives us suitable iterate $f^{n_k}$ and a ball $B^{\\prime }_k$ .", "By Lemma REF  (a) the images of $\\alpha _k$ and $\\beta _k$ under $f^{n_k}$ are blown up to a definite size.", "Since there are only finitely many peripheral circles of $\\mathcal {J}(f)$ whose diameter exceeds a given constant, only finitely many such image pairs can arise.", "By passing to a suitable subsequence if necessary, we may assume that $\\alpha =f^{n_k}(\\alpha _k)$ and $\\beta =f^{n_k}(\\alpha _k)$ are peripheral circles independent of $k$ .", "We define $u^{\\prime }_k:=f^{n_k}(u_k)\\in \\alpha $ and $v^{\\prime }_k:=f^{n_k}(v_k)\\in \\beta $ .", "Then again the relation (REF ) holds.", "This is only possible if $\\alpha \\cap \\beta \\ne \\emptyset $ , and so $\\alpha =\\beta $ .", "Again for large $k$ we want to find a point $w_k\\ne u_k$ in $B_k$ near $u_k$ with $f^{n_k}(u_k)=f^{n_k}(w_k)$ .", "If $u^{\\prime }_k=v^{\\prime }_k$ we can take $w_k:=v_k$ .", "Otherwise, if $u^{\\prime }_k\\ne v^{\\prime }_k$ , we let $\\gamma ^{\\prime }_k$ be the subarc of $\\alpha =\\beta $ with endpoints $u^{\\prime }_k $ and $v^{\\prime }_k$ and smaller diameter.", "Then we can lift $\\gamma ^{\\prime }_k$ to a subarc $\\gamma _k\\subseteq \\beta _k$ with initial point $v_k$ such that $f^{n_k}(\\gamma _k)=\\gamma ^{\\prime }_k$ , and we have (REF ).", "If $w_k\\in \\beta _k$ is the other endpoint of $\\gamma _k$ , then $f^{n_k}(w_k)=u^{\\prime }_k=f^{n_k}(u_k)$ , and $w_k\\ne u_k$ , because these points lie in the disjoint sets $\\beta _k$ and $\\alpha _k$ , respectively.", "Again we have (REF ), which implies that the center $q_k$ of $B_k^{\\prime }$ belongs to $\\operatorname{post}(f)$ , and leads to $\\operatorname{dist}(\\alpha , \\operatorname{post}(f))=0$ .", "We know that this is impossible by Lemma REF .", "Part III.", "We will show that peripheral circles of $\\mathcal {J}(f)$ appear on all locations and scales.", "Let $p\\in \\mathcal {J}(f)$ and $r>0$ be arbitrary, and define $B=B(p,r)$ .", "We may assume that $r$ is small, because by a simple compactness argument one can show that disks of definite, but not too large Euclidean size contain peripheral circles of comparable diameter.", "We now apply the conformal elevator to $B$ to obtain an iterate $f^n$ .", "Lemma REF  (b) implies that there exists a fixed constant $r_1>0$ independent of $B$ such that $B(f^n(p), r_1)\\subseteq f^n(\\frac{1}{2} B)$ .", "By part (a) of the same lemma, we can also find a constant $c_1>0$ independent of $B$ with the following property: if $A$ is a connected set with $A\\cap B(p, r/2)\\ne \\emptyset $ and $\\operatorname{diam}(f^n(A))\\le c_1$ , then $A\\subseteq B$ .", "We can now find a peripheral circle $J^{\\prime }$ of $\\mathcal {J}(f)$ such that $J^{\\prime }\\cap f^n(\\frac{1}{2} B)\\ne \\emptyset $ and $0<c_0 < \\operatorname{diam}(J^{\\prime })<c_1$ , where $c_0$ is another positive constant independent of $B$ .", "This easily follows from a compactness argument based on the fact that $f^n(\\frac{1}{2} B)$ contains a disk of a definite size that is centered at a point in $\\mathcal {J}(f)$ .", "The preimage $f^{-n}(J^{\\prime })$ consists of finitely many components that are peripheral circles of $\\mathcal {J}(f)$ .", "One of these peripheral circles $J$ meets $\\frac{1}{2} B$ .", "Since $\\operatorname{diam}(f^n(J))=\\operatorname{diam}(J^{\\prime })<c_1$ , by the choice of $c_1$ we then have $J\\subseteq B$ , and so $\\operatorname{diam}(J)\\le 2r$ .", "Moreover, it follows from Lemma REF  (c) that $\\operatorname{diam}(J)\\ge c_2 \\operatorname{diam}(J^{\\prime })\\operatorname{diam}(B)\\ge c_3 r$ , where again $c_2,c_3>0$ are independent of $B$ .", "The claim follows.", "Part IV.", "Let $p\\in \\mathcal {J}(f)$ be arbitrary and $r\\in (0,1]$ .", "To establish the porosity of $\\mathcal {J}(f)$ , it is enough to show that the Euclidean disk $B(p,r)$ contains a disk of comparable radius that lies in the complement of $\\mathcal {J}(f)$ .", "By what we have just seen, $B(p,r)$ contains a peripheral circle $J$ of diameter comparable to $r$ .", "By possibly allowing a smaller constant of comparability, we may assume that $J$ is distinct from the one peripheral circle $J_0$ that bounds the unbounded Fatou component of $f$ .", "Then $J\\subseteq B(p,r)$ is the boundary of a bounded Fatou component $U$ , and so $U\\subseteq B(p,r)$ .", "Since the peripheral circles of $\\mathcal {J}(f)$ are uniform quasicircles, it follows that $U$ contains a Euclidean disk $D$ of comparable size (for this standard fact see [5]).", "Then $\\operatorname{diam}(D) \\approx \\operatorname{diam}(J)\\approx r$ .", "Since $D\\subseteq U\\subseteq B(p,r)\\cap \\widehat{\\mathbb {C}}\\setminus \\mathcal {J}(f)$ the porosity of $\\mathcal {J}(f)$ follows.", "Finally, the porosity of $\\mathcal {J}(f)$ implies that $\\mathcal {J}(f)$ cannot have Lebesgue density points, and is hence a set of measure zero.", "Relative Schottky sets and Schottky maps no A relative Schottky set $S$ in a region $D\\subseteq \\widehat{\\mathbb {C}}$ is a subset of $D$ whose complement in $D$ is a union of open geometric disks $\\lbrace B_i\\rbrace _{i\\in I}$ with closures $\\overline{B}_i,\\ i\\in I$ , in $D$ , and such that $\\overline{B}_i\\bigcap \\overline{B}_j=\\emptyset ,\\ i\\ne j$ .", "We write $S=D\\setminus \\bigcup _{i\\in I}B_i.$ If $D=\\widehat{\\mathbb {C}}$ or $, we say that $ S$ is a \\emph {Schottky set}.$ Let $A,B\\subseteq \\widehat{\\mathbb {C}}$ and $\\varphi \\colon A\\rightarrow B$ be a continuous map.", "We call $\\varphi $ a local homeomorphism of $A$ to $B$ if for every point $p\\in A$ there exist open sets $U,V\\subseteq with $ pU$, $ f(p)V$ such that $ f|UA$ is a homeomorphism of $ UA$ onto $ VB$.", "Note that this concept depends of course on $ A$, but also crucially on $ B$: if $ B'B$, then we may consider a local homeomorphism $ fAB$ also as a map $ fAB'$, but the second map will not be a local homeomorphism in general.$ Let $D$ and $\\tilde{D}$ be two regions in $\\widehat{\\mathbb {C}}$ , and let $S=D\\setminus \\bigcup _{i\\in I} B_i$ and $\\tilde{S}=\\tilde{D}\\setminus \\bigcup _{j\\in J}\\tilde{B}_j$ be relative Schottky sets in $D$ and $ \\tilde{D}$ , respectively.", "Let $U$ be an open subset of $D$ and let $f\\colon S\\cap U\\rightarrow \\tilde{S}$ be a local homeomorphism.", "According to [20], such a map $f$ is called a Schottky map if it is conformal at every point $p\\in S\\cap U$ , i.e., the derivative $f^{\\prime }(p)=\\lim _{q\\in S,\\, q\\rightarrow p }\\frac{f(q)-f(p)}{q-p}$ exists and does not vanish, and the function $f^{\\prime }$ is continuous on $S\\cap U$ .", "If $p=\\infty $ or $f(p)=\\infty $ , the existence of this limit and the continuity of $f^{\\prime }$ have to be understood after a coordinate change $z\\mapsto 1/z$ near $\\infty $ .", "In all our applications $S\\subseteq and so we can ignore this technicality.$ Theorem REF implies that if $D$ and $\\tilde{D}$ are Jordan regions, the relative Schottky set $S$ has measure zero, and $f\\colon S\\rightarrow \\tilde{S}$ is a locally quasisymmetric homeomorphism that is orientation-preserving (this is defined similarly as for homeomorphisms between Sierpiński carpets; see the discussion after Lemma REF ), then $f$ is a Schottky map.", "We require a more general criterion for maps to be Schottky maps.", "Lemma 5.1 Let $S\\subseteq be a Schottky set of measure zero.Suppose $ UC$ is open and $ UC$ is a locally quasiconformal map with $ -1(S)=US.", "$Then $ USS$ is a Schottky map.$ In particular, if $\\psi \\colon \\widehat{\\mathbb {C}}\\rightarrow \\widehat{\\mathbb {C}}$ is a quasiregular map with $\\psi ^{-1}(S)=S$ , then $\\psi \\colon S\\setminus \\operatorname{crit}(\\psi )\\rightarrow S$ is a Schottky map.", "In the statement the assumption $S\\subseteq (instead of $ SC$) is not really essential, but helps to avoid some technicalities caused by the point $$.$ Our assumption $\\varphi ^{-1}(S)=U\\cap S$ implies that $\\varphi (U\\cap S)\\subseteq S$ .", "So we can consider the restriction of $\\varphi $ to $U\\cap S$ as a map $\\varphi \\colon U\\cap S\\rightarrow S$ (for simplicity we do not use our usual notation $\\varphi |_{U\\cap S}$ for this and other restrictions in the proof).", "This map is a local homeomorphism $\\varphi \\colon U\\cap S\\rightarrow S$ .", "Indeed, let $p\\in U\\cap S$ be arbitrary.", "Since $\\varphi \\colon U\\rightarrow \\widehat{\\mathbb {C}}$ is a local homeomorphism, there exist open sets $V,W\\subseteq \\widehat{\\mathbb {C}}$ with $p\\in V\\subseteq U$ and $f(p)\\in W$ such that $\\varphi $ is a homeomorphism of $V$ onto $W$ .", "Clearly, $\\varphi (V\\cap S)\\subseteq W\\cap S$ .", "Conversely, if $q\\in W\\cap S$ , then there exists a point $q^{\\prime }\\in V$ with $\\varphi (q^{\\prime })=q$ ; since $\\varphi ^{-1}(S)=U\\cap S$ , we have $q^{\\prime }\\in S$ and so $q^{\\prime }\\in V\\cap S$ .", "Hence $\\varphi (V\\cap S)=W\\cap S$ , which implies that $\\varphi $ is a homeomorphism of $V\\cap S$ onto $W\\cap S$ .", "Note that $p\\in U\\cap S$ lies on a peripheral circle of $S$ if and only if $\\varphi (p)$ lies on a peripheral circle of $S$ .", "Indeed, a point $p\\in S$ lies on a peripheral of $S$ if and only if it is accessible by a path in the complement of $S$ , and it is clear this condition is satisfied for a point $p\\in S\\cap U$ if and only if it is true for the image $\\varphi (p)$ (see [20] for a more general related statement).", "We now want to verify the other conditions for $\\varphi $ to be a Schottky map based on Theorem REF .", "It is enough to reduce to this situation locally near each point $p\\in U\\cap S$ .", "We consider two cases depending on whether $p$ belongs to a peripheral circle of $S$ or not.", "So suppose $p$ does not belong to any of the peripheral circles of $S$ .", "Then there exist arbitrarily small Jordan regions $D$ with $p\\in D$ and $\\partial D\\subseteq S$ such that $\\partial D$ does not meet any peripheral circle of $S$ .", "This easily follows from the fact that if we collapse each closure of a complementary component of $S$ in $\\widehat{\\mathbb {C}}$ to a point, then the resulting quotient space is homeomorphic to $\\widehat{\\mathbb {C}}$ by Moore's theorem [23] (for more details on this and the similar argument below, see the proof of [20]).", "In this way we can find a small Jordan region $D$ with the following properties: (i) $p\\in D \\subseteq \\overline{D}\\subseteq U$ , (ii) the boundary $\\partial D$ is contained in $S$ , but does not meet any peripheral circle of $S$ , (iii) $\\varphi $ is a homeomorphism of $\\overline{D}$ onto the closure $\\overline{D}^{\\prime }$ of another Jordan region $D^{\\prime }\\subseteq \\widehat{\\mathbb {C}}$ .", "As in the first part of the proof, we see that $\\varphi $ is a homeomorphism of $D\\cap S$ onto $D^{\\prime }\\cap S$ .", "This homeomorphism is locally quasisymmetric and orientation-preserving as it is the restriction of a locally quasiconformal map.", "Since $\\partial D$ does not meet peripheral circles of $S$ , the same is true of its image $\\partial D^{\\prime }=\\varphi (\\partial D)$ by what we have seen above.", "It follows that the sets $D\\cap S$ and $D^{\\prime }\\cap S$ are relative Schottky sets of measure zero contained in the Jordan regions $D$ and $D^{\\prime }$ , respectively.", "Note that the set $D\\cap S$ is obtained by deleting from $D$ the complementary disks of $S$ that are contained in $D$ , and $D^{\\prime }\\cap S$ is obtained similarly.", "Now Theorem REF implies that $\\varphi \\colon D\\cap S\\rightarrow D^{\\prime }\\cap S$ is a Schottky map which implies that $\\varphi \\colon U\\cap S \\rightarrow S$ is a Schottky map near $p$ .", "For the other case, assume that $p$ lies on a peripheral circle of $S$ , say $p\\in \\partial B$ , where $B$ is one of the disks that form the complement of $S$ .", "The idea is to use a Schwarz reflection procedure to arrive at a situation similar to the previous case.", "This is fairly straightforward, but we will provide the details for sake of completeness.", "Similarly as before (here we collapse all closures of complementary components of $S$ to points except $\\overline{B}$ ), we find a Jordan region $D$ with the following properties: (i) $\\overline{D}\\subseteq U$ and $\\partial D=\\alpha \\cup \\beta $ , where $\\alpha $ and $\\beta $ are two non-overlapping arcs with the same endpoints such that $\\alpha \\subseteq \\partial B$ , $\\beta \\subseteq S$ , $p$ is an interior point of $\\alpha $ , and no interior point of $\\beta $ lies on a peripheral circle of $S$ , (ii) $\\varphi $ is a homeomorphism of $\\overline{D}$ onto the closure $\\overline{D}^{\\prime }$ of another Jordan region $D^{\\prime }\\subseteq \\widehat{\\mathbb {C}}$ .", "Let $\\alpha ^{\\prime }=\\varphi (\\alpha )$ .", "Then $\\alpha $ is contained in a peripheral circle $\\partial B^{\\prime }$ of $S$ , where $B^{\\prime }$ is a suitable complementary disk of $S$ .", "Note that $\\beta ^{\\prime }=\\varphi (\\beta )$ is an arc contained in $S$ , has its endpoints in $\\partial B^{\\prime }$ , and no interior point of $\\beta ^{\\prime }$ lies on a peripheral circle of $S$ .", "Let $R\\colon \\widehat{\\mathbb {C}}\\rightarrow \\widehat{\\mathbb {C}}$ be the reflection in $\\partial B$ , and $R^{\\prime }\\colon \\widehat{\\mathbb {C}}\\rightarrow \\widehat{\\mathbb {C}}$ be the reflection in $\\partial B^{\\prime }$ .", "Define $\\tilde{S}=S\\cup R(S)$ and $\\tilde{S}^{\\prime }=S\\cup R^{\\prime }(S)$ .", "Then $\\tilde{S}$ and $\\tilde{S}^{\\prime }$ are Schottky sets of measure zero, $\\partial B\\subseteq \\tilde{S}$ , $\\partial B^{\\prime }\\subseteq \\tilde{S}^{\\prime }$ , and $\\partial B$ and $\\partial B^{\\prime }$ do not meet any of the peripheral circles of $\\tilde{S}$ and $\\tilde{S}^{\\prime }$ , respectively.", "Let $\\tilde{D}=D\\cup \\operatorname{int}(\\alpha )\\cup R(D)$ and $\\tilde{D}^{\\prime }=D^{\\prime }\\cup \\operatorname{int}(\\alpha ^{\\prime })\\cup R^{\\prime }(D^{\\prime })$ , where $\\operatorname{int}(\\alpha )$ and $ \\operatorname{int}(\\alpha ^{\\prime })$ denote the set of interior points of the arcs $\\alpha $ and $\\alpha ^{\\prime }$ , respectively.", "Then $\\tilde{D}$ and $\\tilde{D}^{\\prime }$ are Jordan regions such that $p\\in \\tilde{D}$ , $\\partial \\tilde{D}\\subseteq \\tilde{S}$ , $\\partial \\tilde{D}^{\\prime }\\subseteq \\tilde{S}^{\\prime }$ , and $\\partial \\tilde{D}$ and $\\partial \\tilde{D}^{\\prime }$ do not meet any of the peripheral circles of $\\tilde{S}$ and $\\tilde{S}^{\\prime }$ , respectively.", "Hence $\\tilde{D}\\cap \\tilde{S}$ and $\\tilde{D}^{\\prime }\\cap \\tilde{S}^{\\prime }$ are relative Schottky sets of measure zero in $\\tilde{D}$ and $\\tilde{D}^{\\prime }$ , respectively.", "We define a map $\\tilde{\\varphi }\\colon \\tilde{D}\\rightarrow \\tilde{D}^{\\prime }$ by $ \\tilde{\\varphi }(z)= \\left\\lbrace \\begin{array} {cl}\\varphi (z)& \\text{for $z\\in D\\cup \\operatorname{int}(\\alpha ) $,}\\\\&\\\\(R^{\\prime }\\circ \\varphi \\circ R)(z)&\\text{for $z\\in R(D)\\cup \\operatorname{int}(\\alpha )$.}", "\\end{array}\\right.$ Note that this definition is consistent on $\\operatorname{int}(\\alpha )$ , because $\\varphi (\\alpha )=\\alpha ^{\\prime }=\\overline{D}^{\\prime }\\cap R^{\\prime }(\\overline{D}^{\\prime })$ .", "It is clear that $\\tilde{\\varphi }\\colon \\tilde{D}\\rightarrow \\tilde{D}^{\\prime }$ is a homeomorphism.", "Moreover, since the circular arc $\\alpha $ (as any set of $\\sigma $ -finite Hausdorff 1-measure) is removable for quasiconformal maps [26], the map $\\tilde{\\varphi }$ is locally quasiconformal, and hence locally quasisymmetric and orientation-preserving.", "It is also straightforward to see from the definitions and the relation $\\varphi ^{-1}(S)=U\\cap S$ that $\\tilde{\\varphi }^{-1}(\\tilde{S}^{\\prime })= \\tilde{D} \\cap \\tilde{S}$ .", "Similarly as in the beginning of the proof this implies that $\\tilde{\\varphi }\\colon \\tilde{D}\\cap \\tilde{S} \\rightarrow \\tilde{D}^{\\prime }\\cap \\tilde{S}^{\\prime }$ is a homeomorphism.", "Since it is also a local quasisymmetry and orientation-preserving, it follows again from Theorem REF that $\\tilde{\\varphi }\\colon \\tilde{D}\\cap \\tilde{S} \\rightarrow \\tilde{D}^{\\prime }\\cap \\tilde{S}^{\\prime }$ is a Schottky map.", "Note that $\\tilde{D}\\cap S= D\\cap S$ , that on this set the maps $\\tilde{\\varphi }$ and $\\varphi $ agree, and that $\\varphi (\\tilde{D}\\cap S)\\subseteq S$ .", "Thus, $\\varphi \\colon \\tilde{D}\\cap S\\rightarrow S$ is a Schottky map, and so $\\varphi \\colon U\\cap S\\rightarrow S$ is a Schottky map near $p$ .", "It follows that $\\varphi \\colon U\\cap S\\rightarrow S$ is a Schottky map as desired.", "The second part of the statement immediately follows from the first; indeed, $\\operatorname{crit}(\\psi )$ is a finite set and so $U=\\widehat{\\mathbb {C}}\\setminus \\operatorname{crit}(\\psi )$ is an open subset $\\widehat{\\mathbb {C}}$ on which $\\varphi =\\psi |_{U}\\colon U\\rightarrow \\widehat{\\mathbb {C}}$ is a locally quasiconformal map.", "Moreover, $\\varphi ^{-1}(S)=\\psi ^{-1}(S)\\cap U=S\\cap U$ .", "By the first part of the proof, $\\varphi $ and hence also $\\psi $ (restricted to $U\\cap S$ ) is a Schottky map of $U\\cap S=S\\setminus \\operatorname{crit}(\\psi )$ into $S$ .", "A relative Schottky set as in (REF ) is called locally porous at $p\\in S$ if there exists a neighborhood $U$ of $p$ , and constants $r_0>0$ and $C\\ge 1$ such that for each $q\\in S\\cap U$ and $r\\in (0, r_0]$ there exists $i\\in I$ with $B_i\\cap B(q,r)\\ne \\emptyset $ and $r/C\\le \\operatorname{diam}(B_i) \\le Cr$ .", "The relative Schottky set $S$ is called locally porous if it is locally porous at every point $p\\in S$ .", "Every locally porous relative Schottky set has measure zero since it cannot have Lebesgue density points.", "For Schottky maps on locally porous Schottky sets very strong rigidity and uniqueness statements are valid such as Theorems REF and REF stated in the introduction.", "We will need another result of a similar flavor.", "Theorem 5.2 (Me3, Theorem 4.1) Let $S$ be a locally porous relative Schottky set in a region $D\\subseteq , let$ U be an open set such that $S\\cap U$ is connected, and $u\\colon S\\cap U\\rightarrow S$ be a Schottky map.", "Suppose that there exists a point $a\\in S\\cap U$ with $u(a)=a$ and $u^{\\prime }(a)=1$ .", "Then $u=\\operatorname{id}|_{S\\cap U}$ .", "A functional equation in the unit disk no As discussed in the introduction, for the proof of Theorem REF we will establish a functional equation of form (REF ) for the maps in question.", "For postcritically-finite maps $f$ and $g$ this leads to strong conclusions based on the following lemma.", "Recall that $P_k(z)=z^k$ for $k\\in {\\mathbb {N}}$ .", "Lemma 6.1 Let $\\phi \\colon \\partial \\partial be an orientation-preserving homeo\\-morphism, and suppose that there exist numbers $ k,l,nN$, $ k2$, such that\\begin{equation} (P_l\\circ \\phi )(z)= (P_n\\circ \\phi \\circ P_k)(z) \\quad \\text{for $z\\in \\partial $.", "}Then l=nk and there exists a\\in with a^{n(k-1)}=1 such that \\phi (z)=az for all z\\in \\partial .\\end{equation}This lemma implies that we can uniquely extend $$ to a conformal homeomorphismfrom $$ onto itself.", "It is also important that this extension preserves the basepoint $ 0$.$ By considering topological degrees, one immediately sees that $l=nk$ .", "So if we introduce the map $\\psi := P_n\\circ \\phi $ , then () can be rewritten as $ P_k\\circ \\psi =\\psi \\circ P_k \\quad \\text{on $\\partial $.", "}Here the map \\psi \\colon \\partial \\partial has degree n. We claim that this in combination with (\\ref {eq:basiceq2}) implies that for a suitable constant b we have \\psi (z)=bz^n for z\\in \\partial .$ Indeed, there exists a continuous function $\\alpha \\colon {\\mathbb {R}}\\rightarrow {\\mathbb {R}}$ with $\\alpha (t+2\\pi )=\\alpha (t)$ such that $ \\psi (e^{ i t}) = \\exp ( i n t+ i\\alpha (t)) \\quad \\text{for $t\\in {\\mathbb {R}}$}.", "$ By (REF ) we have $ \\exp ( i k n t+ i k\\alpha (t)) = (\\psi (e^{ i t}))^k = \\psi (e^{ i k t})=\\exp ( i kn t+ i\\alpha (kt))$ for $t\\in {\\mathbb {R}}$ .", "This implies that there exists a constant $c\\in {\\mathbb {R}}$ such that $ \\alpha (t)=\\frac{1}{k} \\alpha (tk)+c \\quad \\text{for $t\\in {\\mathbb {R}}$}.", "$ Since $\\alpha $ is $2\\pi $ -periodic, the right-hand side of this equation is $2\\pi /k$ -periodic as a function of $t$ .", "Hence $\\alpha $ is $2\\pi /k$ -periodic.", "Repeating this argument, we see that $\\alpha $ is $2\\pi /k^m$ -periodic for all $m\\in {\\mathbb {N}}$ , and so has arbitrarily small periods (note that $k\\ge 2$ ).", "Since $\\alpha $ is continuous, it follows that $\\alpha $ is constant.", "Hence $\\psi (z)=bz^n$ for $z\\in \\partial with a suitable constant $ b.", "It follows that $ \\psi (z)=b z^n=\\phi (z)^n \\quad \\text{for $z\\in \\partial $.$$Therefore, $\\phi (z)=az$ for $z\\in \\partial with a constant $a\\in , $a\\ne 0$.Inserting this expression for $\\phi $ into (\\ref {eq:basiceq}) and using $l=nk$, we conclude that $a^{n(k-1)}=1$ as desired.", "}$ Proof of Theorem  REF The proof will be given in several steps.", "Step I.", "We first fix the setup.", "We can freely pass to iterates of the maps $f$ or $g$ , because this changes neither their Julia sets nor their postcritical sets.", "We can also conjugate the maps by Möbius transformations.", "Therefore, as in Section , we may assume that $\\mathcal {J}(f), \\mathcal {J}(g)\\subseteq \\tfrac{1}{2} { and } f^{-1}( , g^{-1}(\\subseteq $ Moreover, without loss of generality, we may require that $\\xi $ is orientation-preserving, for otherwise we can conjugate $g$ by $z\\mapsto \\overline{z}$ .", "Since the peripheral circles of $\\mathcal {J}(f)$ and $\\mathcal {J}(g)$ are uniform quasicircles, by Theorem REF the map $\\xi $ extends (non-uniquely) to a quasiconformal, and hence quasisymmetric, map of the whole sphere.", "Then $\\xi (\\mathcal {J}(f))=\\mathcal {J}(g)$ and $\\xi (\\mathcal {F}(f))=\\mathcal {F}(g)$ .", "Since $\\infty $ lies in Fatou components of $f$ and $g$ , we may also assume that $\\xi (\\infty )=\\infty $ (this normalization ultimately depends on the fact that for every point $p\\in there exists a quasiconformal homeomorphism $$on $ C$ with $ (0)=p$ that is the identity outside $ ).", "Then $\\xi $ is a quasisymmetry of $ with respect to the Euclidean metric.", "In the following, all metric notions will refer to this metric.Finally, we define $ g=-1 g$.$ Step II.", "We now carefully choose a location for a “blow-down\" by branches of $f^{-n}$ which will be compensated by a “blow-up\" by iterates of $g$ (or rather $g_\\xi $ ).", "Since repelling periodic points of $f$ are dense in $\\mathcal {J}(f)$ (see [3]), we can find such a point $p$ in $\\mathcal {J}(f)$ that does not lie in $\\operatorname{post}(f)$ .", "Let $\\rho >0$ be a small positive number such that the disk $U_0:=B(p, 3\\rho )\\subseteq is disjoint from $ post(f)$.Since $ p$ is periodic, there exists $ dN$ such that $ fd(p)=p$.", "Let $ U1 be the component of $f^{-d}(U_0)$ that contains $p$ .", "Since $U_0\\cap \\operatorname{post}(f)=\\emptyset $ , the set $U_1$ is a simply connected region, and $f^{d}$ is a conformal map from $U_1$ onto $U_0$ as follows from Lemma REF .", "Then there exists a unique inverse branch $f^{-d}$ with $f^{-d}(p)=p$ that is a conformal map of $U_0$ onto $U_1$ .", "Since $p$ is a repelling fixed point for $f$ , it is an attracting fixed point for this branch $f^{-d}$ .", "By possibly choosing a smaller radius $\\rho >0$ in the definition of $U_0=B(p, 3\\rho )$ and by passing to an iterate of $f^d$ , we may assume that $U_1\\subseteq U_0$ and that $\\operatorname{diam}(f^{-n_k}(U_0))\\rightarrow 0$ as $k\\rightarrow \\infty $ .", "Here $n_k=dk$ for $k\\in {\\mathbb {N}}$ , and $f^{-n_k}$ is the branch obtained by iterating the branch $f^{-d}$ $k$ -times.", "Note that $f^{-n_k}(p)=p$ and $f^{-n_k}$ is a conformal map of $U_0$ onto a simply connected region $U_k$ .", "Then $p\\in U_k\\subseteq U_{k-1}$ for $k\\in {\\mathbb {N}}$ , and $\\operatorname{diam}(U_k)\\rightarrow 0$ as $k\\rightarrow \\infty $ .", "The choice of these inverse branches is consistent in the sense that we have $f^{n_{k+1}-n_k}\\circ f^{-n_{k+1}}=f^{-n_k}$ on $B(p, 3\\rho )$ for all $k\\in {\\mathbb {N}}$ .", "Note that this consistency condition remains valid if we replace the original sequence $\\lbrace n_k\\rbrace $ by a subsequence.", "Let $\\tilde{r}_k>0$ be the smallest number such that $f^{-n_k}(B(p, 2\\rho ))\\subseteq \\tilde{B}_k:=B(p,\\tilde{r}_k).$ Since $p\\in f^{-n_k}(B(p, 2\\rho ))$ we have $\\operatorname{diam}\\big (f^{-n_k}(B(p, 2\\rho ))\\big )\\approx \\tilde{r}_k$ .", "Here and below $\\approx $ indicates implicit positive multiplicative constants independent of $k\\in {\\mathbb {N}}$ .", "It follows that $\\tilde{r}_k\\rightarrow 0$ as $k\\rightarrow \\infty $ .", "Moreover, since $f^{-n_k}$ is conformal on the larger disk $B(p,3\\rho )$ , Koebe's distortion theorem implies that $ \\operatorname{diam}\\big (f^{-n_k}(B(p, 2\\rho ))\\big )\\approx \\operatorname{diam}\\big (f^{-n_k}(B(p, \\rho ))\\big ).", "$ Let $r_k>0$ be the smallest number such that $\\xi (\\tilde{B}_k)\\subseteq B_k:=B(\\xi (p), r_k)$ .", "Again $r_k\\rightarrow 0$ as $k\\rightarrow \\infty $ by continuity of $\\xi $ .", "We now want to apply the conformal elevator given by iterates of $g$ to the disks $B_k$ .", "For this we choose $\\epsilon _0>0$ for the map $g$ as in Section .", "By applying the conformal elevator as described in Section , we can find iterates $g^{m_k}$ such that $g^{m_k}(B_k)$ is blown up to a definite, but not too large size, and so $\\operatorname{diam}(g^{m_k}(B_k))\\approx 1$ .", "Step III.", "Now we consider the composition $h_k=g_\\xi ^{m_k}\\circ f^{-n_k}= \\xi ^{-1}\\circ g^{m_k}\\circ \\xi \\circ f^{-n_k}$ defined on $B(p,2\\rho )$ for $k\\in {\\mathbb {N}}$ .", "We want to show that this sequence subconverges locally uniformly on $B(p,2\\rho )$ to a (non-constant) quasiregular map $h\\colon B(p, 2\\rho )\\rightarrow .$ Since $f^{-n_k}$ maps $B(q,2\\rho )$ conformally into $\\tilde{B}_k$ , $\\xi $ is a quasiconformal map with $\\xi (\\tilde{B}_k)\\subseteq B_k$ , and $g^{m_k}$ is holomorphic on $\\tilde{B}_k$ , we conclude that the maps $h_k$ are uniformly quasiregular on $B(q,2\\rho )$ , i.e., $K$ -quasiregular with $K\\ge 1$ independent of $k$ .", "The images $h_k(B(q,2\\rho ))$ are contained in a small Euclidean neighborhood of $\\mathcal {J}(g)$ and hence in a fixed compact subset of $.", "Standardconvergence results for $ K$-quasiregular mappings \\cite [p.~182, Corollary 5.5.7]{AIM} imply that the sequence $ {hk}$subconverges locally uniformly on $ B(q,2)$ to a map $ hB(q,2) that is also quasiregular, but possibly constant.", "By passing to a subsequence if necessary, we may assume that $h_k\\rightarrow h$ locally uniformly on $B(q,2\\rho )$ .", "To rule out that $h$ is constant, it is enough to show that for smaller disk $B(p,\\rho )$ there exists $\\delta >0$ such that $\\operatorname{diam}\\big (h_k(B(p,\\rho ))\\big )\\ge \\delta $ for all $k\\in {\\mathbb {N}}$ .", "We know that $\\operatorname{diam}\\big (f^{-n_k}(B(p,\\rho ))\\big )\\approx \\operatorname{diam}\\big (f^{-n_k}(B(p,2\\rho ))\\big ) \\approx \\operatorname{diam}(\\tilde{B}_k).", "$ Moreover, since $\\xi $ is a quasisymmetry and $f^{-n_k}(B(p,\\rho ))\\subseteq \\tilde{B}_k$ , this implies $\\operatorname{diam}\\big (\\xi (f^{-n_k}(B(p,\\rho )))\\big )\\approx \\operatorname{diam}(\\xi (\\tilde{B}_k))\\approx \\operatorname{diam}(B_k).", "$ So the connected set $\\xi (f^{-n_k}(B(p,\\rho )))\\subseteq B_k$ is comparable in size to $B_k$ .", "By Lemma REF  (a) the conformal elevator blows it up to a definite, but not too large size, i.e., $\\operatorname{diam}\\big ((g^{m_k}\\circ \\xi \\circ f^{-n_k})(B(p,\\rho ))\\big )\\approx 1.$ Since the sets $(g^{m_k}\\circ \\xi \\circ f^{-n_k})(B(p,\\rho ))$ all meet $\\mathcal {J}(g)$ , they stay in a compact part of $, and so we still get a uniform lower bound for the diameter of these sets if we apply the homeomorphism $ -1$; in other words,$$\\operatorname{diam}\\big ((\\xi ^{-1}\\circ g^{m_k}\\circ \\xi \\circ f^{-n_k})(B(p,\\rho ))\\big )=\\operatorname{diam}\\big (h_k(B(p,\\rho ))\\big )\\approx 1$$as claimed.", "We conclude that $ hkh$ locally uniformly on $ B(p, 2r)$, where $ h$ is non-constant and quasiregular.$ The quasiregular map $h$ has at most countably many critical points, and so there exists a point $q\\in B(p, 2\\rho )\\cap \\mathcal {J}(f)$ and a radius $r>0$ such that $B(q, 2r)\\subseteq B(p, 2\\rho )$ and $h$ is injective on $B(q, 2r)$ and hence quasiconformal.", "Standard topological degree arguments imply that at least on the smaller disk $B(q,r)$ the maps $h_k$ are also injective and hence quasiconformal for all $k$ sufficiently large.", "By possibly disregarding finitely of the maps $h_k$ , we may assume that $h_k$ is quasiconformal on $B(q,r)$ for all $k\\in {\\mathbb {N}}$ .", "To summarize, we have found a disk $B(q,r)$ centered at a point $q\\in \\mathcal {J}(f)$ such that the maps $h_k$ are defined and quasiconformal on $B(q,r)$ and converge uniformly on $B(q,r)$ to a quasiconformal map $h$ .", "From the invariance properties of Julia and Fatou sets and the mapping properties of $\\xi $ , it follows that $ h_k(B(q, r)\\cap \\mathcal {J}(f))\\subseteq \\mathcal {J}(f) \\text{ and } h_k(B(q, r)\\cap \\mathcal {F}(f))\\subseteq \\mathcal {F}(f)$ for each $k\\in {\\mathbb {N}}$ .", "Hence $h_k^{-1}(\\mathcal {J}(f))=\\mathcal {J}(f)\\cap B(q,r)$ for each map $h_k\\colon B(q,r)\\rightarrow , $ kN$.$ Since $\\mathcal {J}(f)$ is closed and $h_k\\rightarrow h$ uniformly on $B(q, r)$ , we also have $h(B(q, r)\\cap \\mathcal {J}(f))\\subseteq \\mathcal {J}(f)$ .", "To get a similar inclusion relation also for the Fatou set, we argue by contradiction and assume that there exists a point $z\\in B(q,r)\\cap \\mathcal {F}(f)$ with $h(z)\\notin \\mathcal {F}(f)$ .", "Then $h(z)\\in \\mathcal {J}(f)$ .", "Since $B(q,r)\\cap \\mathcal {F}(f)$ is an open neighborhood of $z$ , it follows again from standard topological degree arguments that for large enough $k\\in {\\mathbb {N}}$ there exists a point $z_k\\in B(q,r)\\cap \\mathcal {F}(f)$ with $h_k(z_k)=h(z)\\in \\mathcal {J}(f)$ .", "This is impossible by (REF ) and so indeed $h(B(q, r)\\cap \\mathcal {F}(f))\\subseteq \\mathcal {F}(f)$ .", "We conclude that $h^{-1}(\\mathcal {J}(f))=\\mathcal {J}(f)\\cap B(q,r).$ Step IV.", "We know by Theorem REF that the peripheral circles of $\\mathcal {J}(f)$ are uniformly relatively separated uniform quasicircles.", "According to Theorems REF and REF , there exists a quasisymmetric map $\\beta $ on $\\widehat{\\mathbb {C}}$ such that $S=\\beta (\\mathcal {J}(f))$ is a round Sierpiński carpet.", "We may assume $S\\subseteq .$ We conjugate the map $f$ by $\\beta $ to define a new map $\\beta \\circ f\\circ \\beta ^{-1}$ .", "By abuse of notation we call this new map also $f$ .", "Note that this map and its iterates are in general not rational anymore, but quasiregular maps on $\\widehat{\\mathbb {C}}$ .", "Similarly, we conjugate $ g_\\xi , h_k, h$ by $\\beta $ to obtain new maps for which we use the same notation for the moment.", "If $V=\\beta (B(q,r))$ , then the new maps $h_k$ and $h$ are quasiconformal on $V$ , and $h_k\\rightarrow h$ uniformly on $V$ .", "Lemma 7.1 There exist $N\\in {\\mathbb {N}}$ and an open set $W\\subseteq V$ such that $S\\cap W$ is non-empty and connected and $h_k\\equiv h$ on $S\\cap W$ for all $k\\ge N$ .", "Since $\\mathcal {J}(f)$ is porous and $S$ is a quasisymmetric image of $\\mathcal {J}(f)$ , the set $S$ is also porous (and in particular locally porous as defined in Section ).", "The maps $h_k$ and $h$ are quasiconformal on $V=\\beta (B(q,r))$ , and $h_k\\rightarrow h$ uniformly on $V$ as $k\\rightarrow \\infty $ .", "The relations (REF ) and (REF ) translate to $h^{-1}(S)=S\\cap V$ and $h_k^{-1}(S)=S\\cap V$ for $k\\in {\\mathbb {N}}$ .", "So Lemma REF implies that the maps $h\\colon S\\cap V\\rightarrow S$ and $h_k\\colon S \\cap V\\rightarrow S$ for $k\\in {\\mathbb {N}}$ are Schottky maps.", "Each of these restrictions is actually a homeomorphism onto its image.", "There are only finitely many peripheral circles of $\\mathcal {J}(f)$ that contain periodic points of our original rational map $f$ ; indeed, if $J$ is such a peripheral circle, then $f^n(J)=J$ for some $n\\in {\\mathbb {N}}$ as follows from Lemma REF ; but then $J$ bounds a periodic Fatou component of $f$ which leaves only finitely many possibilities for $J$ .", "Since the periodic points of $f$ are dense in $\\mathcal {J}(f)$ , we conclude that we can find a periodic point of $f$ in $ \\mathcal {J}(f)\\cap B(q,r)$ that does not lie on a peripheral circle of $\\mathcal {J}(f)$ .", "Translated to the conjugated map $f$ , this yields existence of a point $a\\in S\\cap V$ that does not lie on a peripheral circle of the Sierpiński carpet $S$ such that $f^n(a)=a$ for some $n\\in {\\mathbb {N}}$ .", "The invariance property of the Julia set gives $f^{-n}(S)=S$ , and so Lemma REF implies that $f^n\\colon S\\setminus \\operatorname{crit}(f^n)\\rightarrow S$ is a Schottky map.", "Note that $a\\notin \\operatorname{crit}(f^n)$ as follows from the fact that for our original rational map $f$ , none of its periodic critical points lies in the Julia set.", "Therefore, our Schottky map $f^n\\colon S\\setminus \\operatorname{crit}(f^n)\\rightarrow S$ has a derivative at the point $a\\in S\\setminus \\operatorname{crit}(f^n)$ in the sense of (REF ).", "If $(f^n)^{\\prime }(a)=1$ , then Theorem REF implies that $f^n\\equiv \\operatorname{id}|_{S\\setminus \\operatorname{crit}(f^n)}$ , and hence by continuity $f^n$ is the identity on $S$ .", "This is clearly impossible, and therefore $(f^n)^{\\prime }(a)\\ne 1$ .", "Since $a\\in S\\cap V$ does not lie on a peripheral circle of $S$ , as in the proof of Lemma REF we can find a small Jordan region $W$ with $a\\in W\\subseteq V$ and $W\\cap \\operatorname{crit}(f^n)=\\emptyset $ such that $\\partial W\\subseteq S$ .", "Then $S\\cap W$ is non-empty and connected.", "We now restrict our maps to $W$ .", "Then $h_k\\colon S\\cap W\\rightarrow S$ is a Schottky map and a homeomorphism onto its image for each $k\\in {\\mathbb {N}}$ .", "The same is true for the map $h\\colon S \\cap W\\rightarrow S$ .", "Moreover $h_k\\rightarrow h$ as $k\\rightarrow \\infty $ uniformly on $W\\cap S$ .", "Finally, the map $u=f^n$ is defined on $S\\cap W$ and gives a Schottky map $u\\colon S\\cap W\\rightarrow S$ such that for $a\\in S\\cap W$ we have $u(a)=a$ and $u^{\\prime }(a)\\ne 1$ .", "So we can apply Theorem REF to conclude that there exists $N\\in {\\mathbb {N}}$ such that $h_k\\equiv h$ on $S\\cap W$ for all $k\\ge N$ .", "By the previous lemma we can fix $k\\ge N$ so that $h_k=h_{k+1}$ on $S\\cap W$ .", "If we go back to the definition of the maps $h_k$ and use the consistency of inverse branches (which is also true for the maps conjugated by $\\beta $ ), then we conclude that $ h_{k+1}=g_\\xi ^{m_{k+1}}\\circ f^{-n_{k+1}}= h_k =g_\\xi ^{m_{k}}\\circ f^{-n_{k}}=g_\\xi ^{m_{k}}\\circ f^{n_{k+1}-n_k}\\circ f^{-n_{k+1}}$ on the set $S \\cap W$ .", "Cancellation gives $ g_\\xi ^{m_{k+1}}=g_\\xi ^{m_{k}}\\circ f^{n_{k+1}-n_k}$ on $f^{-n_{k+1}}(S\\cap W)\\subseteq S$ .", "The two maps on both sides of the last equation are quasiregular maps $\\psi \\colon \\widehat{\\mathbb {C}}\\rightarrow \\widehat{\\mathbb {C}}$ with $\\psi ^{-1}(S)=S$ .", "It follows from Lemma REF that they are Schottky maps $S \\cap U\\rightarrow S$ if $U\\subseteq \\widehat{\\mathbb {C}}$ is an open set that does not contain any of the finitely many critical points of the maps; in particular, $g_\\xi ^{m_{k+1}}$ and $g_\\xi ^{m_k}\\circ f^{n_{k+1}-n_k}$ are Schottky maps $S \\cap U\\rightarrow S$ , where $U=\\widehat{\\mathbb {C}}\\setminus (\\operatorname{crit}(g_\\xi ^{m_{k+1}})\\cup \\operatorname{crit}(g_\\xi ^{m_k}\\circ f^{n_{k+1}-n_k}))$ .", "Since $U$ has a finite complement in $\\widehat{\\mathbb {C}}$ , the non-degenerate connected set $f^{-n_{k+1}}(S\\cap W)\\subseteq S$ has an accumulation point in $S \\cap U$ .", "Theorem REF yields $ g_\\xi ^{m_{k+1}}=g_\\xi ^{m_{k}}\\circ f^{n_{k+1}-n_k}$ on $S\\cap U$ , and hence on all of $S$ by continuity.", "If we conjugate back by $\\beta ^{-1}$ , this leads to the relation $ g_\\xi ^{m_{k+1}}=g_\\xi ^{m_{k}}\\circ f^{n_{k+1}-n_k}$ on $\\mathcal {J}(f)$ for the original maps.", "We conclude that there exist integers $m,m^{\\prime },n\\in {\\mathbb {N}}$ such that for the original maps we have $ g^{m^{\\prime }}\\circ \\xi = g^m\\circ \\xi \\circ f^n$ on $\\mathcal {J}(f)$ .", "Step V. Equation (REF ) gives us a crucial relation of $\\xi $ to the dynamics of $f$ and $g$ on their Julia sets.", "We will bring (REF ) into a convenient form by replacing our original maps with iterates.", "Since $\\mathcal {J}(f)$ is backward invariant, counting preimages of generic points in $\\mathcal {J}(f)$ under iterates of $f$ and of points in $\\mathcal {J}(g)$ under iterates of $g$ leads to the relation $\\deg (g)^{m^{\\prime }-m}=\\deg (f)^n,$ and so $m^{\\prime }-m>0$ .", "If we post-compose both sides in (REF ) by a suitable iterate of $g$ , and then replace $f$ by $f^n$ and $g$ by $g^{m^{\\prime }-m}$ , we arrive at a relation of the form $ g^{l+1}\\circ \\xi = g^l\\circ \\xi \\circ f.$ on $\\mathcal {J}(f)$ for some $l\\in {\\mathbb {N}}$ .", "Note that this equation implies that we have $ g^{n+k}\\circ \\xi = g^n\\circ \\xi \\circ f^k \\ \\text{ for all $k,n\\in {\\mathbb {N}}$ with $n\\ge l$}.$ Step VI.", "In this final step of the proof, we disregard the non-canonical extension to $\\widehat{\\mathbb {C}}$ of our original homeomorphism $\\xi \\colon \\mathcal {J}(f)\\rightarrow \\mathcal {J}(g)$ chosen in the beginning.", "Our goal is to apply (REF ) to produce a natural extension of $\\xi $ mapping each Fatou component of $f$ conformally onto a Fatou component of $g$ .", "Note that if $U$ is a Fatou component of $f$ , then $\\partial U$ is a peripheral circle of $\\mathcal {J}(f)$ .", "Since $\\xi $ sends each peripheral circle of $\\mathcal {J}(f)$ to a peripheral circle of $\\mathcal {J}(g)$ , the image $\\xi (\\partial U)$ bounds a unique Fatou component $V$ of $g$ .", "This sets up a natural bijection between the Fatou components of our maps, and our goal is to conformally “fill in the holes\".", "So let $\\mathcal {C}_f$ and $\\mathcal {C}_g$ be the sets of Fatou components of $f$ and $g$ , respectively.", "By Lemma REF we can choose a corresponding family $\\lbrace \\psi _U: U\\in \\mathcal {C}_f\\rbrace $ of conformal maps.", "Since each Fatou component of $f$ is a Jordan region, we can consider $\\psi _U$ as a conformal homeomorphism from $\\overline{U}$ onto $\\overline{.", "Similarly, we obtain a family of conformal homeomorphisms \\tilde{\\psi }_V: \\overline{V}\\rightarrow \\overline{ for V in \\mathcal {C}_g.These Fatou components carry distinguished basepointsp_U=\\psi _U^{-1}(0)\\in U for U\\in \\mathcal {C}_f and \\tilde{p}_V=\\tilde{\\psi }_V^{-1}(0)\\in V for V\\in \\mathcal {C}_g.", "}We will now first extend \\xi to the periodic Fatou components of f, and thenuse the Lifting Lemma~\\ref {lem:lifting} to get extensions to Fatou components of higher and higher level (as defined in the proof of Lemma~\\ref {lem:FatouDyn}).", "In this argument it will be important to ensure that these extensions are basepoint-preserving.", "}First let $ U$ be a periodic Fatou component of $ f$.", "We denote by $ kN$ the period of $ U$, and define $ V$ to be the Fatou component of $ g$ bounded by $ (U)$, and$ W=gl(V)$, where $ lN$ is as in (\\ref {eq:main4}).Then(\\ref {eq:main4}) implies that $ W$, and hence $ W$ itself, is invariant under $ gk$.", "ByLemma~\\ref {lem:FatouDyn} the basepoint-preserving homeomorphisms $ U (U, pU)( , 0)$ and $ W(W,pW)( ,0)$ conjugate $ fk$ and $ gk$, respectively, to power maps.Since $ f$ and $ g$ are postcritically-finite, the periodic Fatou components $ U$ and $ W$ are superattracting, and thus the degrees of these power maps are at least 2.$ Again by Lemma REF the map $\\tilde{\\psi }_W\\circ g^l\\circ \\tilde{\\psi }^{-1}_V$ is a power map.", "Since $U,V,W$ are Jordan regions, the maps $\\psi _U, \\tilde{\\psi }_V, \\tilde{\\psi }_W$ give homeomorphisms between the boundaries of the corresponding Fatou components and $\\partial .", "Since $$ is anorientation-preserving homeomorphism of $ U$ onto $ V$, the map$ =VU-1$gives an orientation-preserving homeomorphism on $ .", "Now (REF ) for $n=l$ implies that on $\\partial we have{\\begin{@align*}{1}{-1} P_{d_3}\\circ \\phi &=\\tilde{\\psi }_W\\circ g^{k+l}\\circ \\tilde{\\psi }_V^{-1} \\circ \\phi = \\tilde{\\psi }_W\\circ g^{k+l}\\circ \\xi \\circ \\psi _U^{-1}\\\\&= \\tilde{\\psi }_W\\circ g^{l}\\circ \\xi \\circ f^k\\circ \\psi ^{-1}_U= \\tilde{\\psi }_W\\circ g^{l}\\circ \\xi \\circ \\psi ^{-1}_U \\circ P_{d_1}\\\\&= \\tilde{\\psi }_W\\circ g^{l}\\circ \\tilde{\\psi }^{-1}_V\\circ \\phi \\circ P_{d_1}= P_{d_2}\\circ \\phi \\circ P_{d_1}\\end{@align*}}for some $ d1, d 2, d3N$ with $ d12$.", "Lemma~\\ref {L:Rot} implies that $$ extends to $$ as a rotation around $ 0$, also denoted by $$.", "In particular, $ (0)=0$, and so $$ preserves the basepoint $ 0$ in $$.", "If we define $ =V-1U$ on $U$, then $$ is a conformal homeomorphismof $ (U,pU)$ onto $ (V, pV)$.$ In this way, we can conformally extend $\\xi $ to every periodic Fatou component of $f$ so that $\\xi $ maps the basepoint of a Fatou component to the basepoint of the image component.", "To get such an extension also for the other Fatou components $V$ of $f$ , we proceed inductively on the level of the Fatou component.", "So suppose that the level of $V$ is $\\ge 1$ and that we have already found an extension for all Fatou components with a level lower than $V$ .", "This applies to the Fatou component $U=f(V)$ of $f$ , and so a conformal extension $(U, p_U)\\rightarrow ( U^{\\prime }, \\tilde{p}_{U^{\\prime }})$ of $\\xi |_{\\partial U}$ exists, where $U^{\\prime }$ is the Fatou component of $g$ bounded by $\\xi (\\partial U)$ .", "Let $V^{\\prime }$ be the Fatou component of $g$ bounded by $\\xi (\\partial V)$ , and $W=g^{l+1}(V^{\\prime })$ .", "Then by using (REF ) on $\\partial V$ we conclude that $g^l(U^{\\prime })=W$ .", "Define $\\alpha =g^l\\circ \\xi |_{\\overline{U}}\\circ f|_{\\overline{V}}$ and $\\beta =\\xi |_{\\partial V}$ .", "Then the assumptions of Lemma REF are satisfied for $D=V$ , $p_D=p_{V}$ , and the iterate $g^{l+1}\\colon V^{\\prime }\\rightarrow W$ of $g$ .", "Indeed, $\\alpha $ is continuous on $\\overline{V}$ and holomorphic on $V$ , we have $\\alpha ^{-1}(p_W)=f^{-1}( \\xi |_{\\overline{U}}^{-1}(\\tilde{p}_{U^{\\prime }}))=f^{-1}(p_U)=\\lbrace p_V\\rbrace ,$ and $g^{l+1}\\circ \\beta = g^{l+1}\\circ \\xi |_{\\partial V}= g^{l}\\circ \\xi |_{\\partial U}\\circ f|_{\\partial V}=\\alpha .$ Since $\\beta $ is a homeomorphism, it follows that there exists a conformal homeomorphism $\\tilde{\\alpha }$ of $(\\overline{V}, p_V)$ onto $(\\overline{V}^{\\prime }, \\tilde{p}_{V^{\\prime }})$ such that $\\tilde{\\alpha }|_{\\partial V}=\\beta =\\xi |_{\\partial V}$ .", "In other words, $\\tilde{\\alpha }$ gives the desired basepoint preserving conformal extension to the Fatou component $V$ .", "This argument shows that $\\xi $ has a (unique) conformal extension to each Fatou component of $f$ .", "We know that the peripheral circles of $\\mathcal {J}(f)$ are uniform quasicircles, and that $\\mathcal {J}(f)$ has measure zero.", "Lemma REF now implies that $\\xi $ extends to a Möbius transformation on $\\widehat{\\mathbb {C}}$ , which completes the proof.", "The techniques discussed also easily lead to a proof of Corollary REF .", "Let $f$ be a postcritically-finite rational map whose Julia set $\\mathcal {J}(f)$ is a Sierpiński carpet.", "Let $G$ be the group of all Möbius transformations $\\xi $ on $\\widehat{\\mathbb {C}}$ with $\\xi (\\mathcal {J}(f))=\\mathcal {J}(f)$ , and $H$ be the subgroup of all elements in $G$ that preserve orientation.", "By Theorem REF it is enough to prove that $G$ is finite.", "Since $G=H$ or $H$ has index 2 in $G$ , this is true if we can show that $H$ is finite.", "Note that the group $H$ is discrete, i.e., there exists $\\delta _0>0$ such that $ \\sup _{p\\in \\widehat{\\mathbb {C}}} \\sigma (\\xi (p), p)\\ge \\delta _0$ for all $\\xi \\in H$ with $\\xi \\ne \\operatorname{id}_{\\widehat{\\mathbb {C}}}$ .", "Indeed, we choose $\\delta _0>0$ so small that there are at least three distinct complementary components $D_1,D_2,D_3$ of $\\mathcal {J}(f)$ that contain disks of radius $\\delta _0$ .", "In order to show (REF ), suppose that $\\xi \\in H$ and $\\sigma (\\xi (p), p)<\\delta _0$ for all $p\\in \\widehat{\\mathbb {C}}$ .", "Then $\\xi (D_i)\\cap D_i\\ne \\emptyset $ , and so $\\xi (\\overline{D}_i)=\\overline{D}_i$ for $i=1, 2,3$ , because $\\xi $ permutes the closures of Fatou components of $f$ .", "This shows that $\\xi $ is a conformal homeomorphism of the closed Jordan region $\\overline{D}_i$ onto itself.", "Hence $\\xi $ has a fixed point in $\\overline{D}_i$ .", "Since $\\mathcal {J}(f)$ is a Sierpiński carpet, the closures $\\overline{D}_1, \\overline{D}_2, \\overline{D}_3$ are pairwise disjoint, and we conclude that $\\xi $ has at least three fixed points.", "Since $\\xi $ is an orientation-preserving Möbius transformation, this implies that $\\xi =\\operatorname{id}_{\\widehat{\\mathbb {C}}}$ , and the discreteness of $H$ follows.", "We will now analyze type of Möbius transformations contained in the group $H$ (for the relevant classification of Möbius transformations up to conjugacy, see [2]).", "So consider an arbitrary $\\xi \\in H$ , $\\xi \\ne \\text{id}_{\\widehat{\\mathbb {C}}}$ .", "Then $\\xi $ cannot be loxodromic; indeed, otherwise $\\xi $ has a repelling fixed point $p$ which necessarily has to lie in $\\mathcal {J}(f)$ .", "We now argue as in the proof of Theorem REF and “blow down\" by iterates $\\xi ^{-n_k}$ near $p$ and “blow up\" by the conformal elevator using iterates $f^{m_k}$ to obtain a sequence of conformal maps of the form $h_k=f^{m_k}\\circ \\xi ^{-n_k}$ that converge uniformly to a (non-constant) conformal limit function $h$ on a disk $B$ centered at a point in $q\\in \\mathcal {J}(f)$ .", "Again this sequence stabilizes, and so $h_{k+1}=h_k$ for large $k$ on a connected non-degenerate subset of $B$ , and hence on $\\widehat{\\mathbb {C}}$ by the uniqueness theorem for analytic functions.", "This leads to a relation of the form $f^k\\circ \\xi ^l=f^m$ , where $k,l,m\\in {\\mathbb {N}}$ .", "Comparing degrees we get $m=k$ , and so we have $f^{k}=f^{k}\\circ \\xi ^{-ln}$ for all $n\\in {\\mathbb {N}}$ .", "This is impossible, because $f^k=f^{k}\\circ \\xi ^{-ln}\\rightarrow f^{k}(p)$ near $p$ as $n\\rightarrow \\infty $ , while $f^k$ is non-constant.", "The Möbius transformation $\\xi $ cannot be parabolic either; otherwise, after conjugation we may assume that $\\xi (z)=z+a$ with $a\\in , $ a0$.", "Then necessarily $ J(f)$.", "On the other hand, we know thatthe peripheral circles of $ J(f)$ are uniform quasicircles that occur on all locations and scales with respect to the chordal metric.", "Translated to the Euclidean metric near $$ this means that $ J(f)$ has complementary components $ D$ with $ D$ that contain Euclidean disks of arbitrarily large radius, and in particular of radius $ >|a|$; but then $$ cannot move $ D$ off itself, and so $ (D)=D$ for the translation $$.This is impossible.$ Finally, $\\xi $ can be elliptic; then after conjugation we have $\\xi (z)=az$ with $a\\in and $ |a|=1$, $ a1$.", "Since $ H$ is discrete,$ a$ must be a root of unity, and so $$is a torsion element of $ H$.$ We conclude that $H$ is a discrete group of Möbius transformations such that each element $\\xi \\in H$ with $\\xi \\ne \\text{id}_{\\widehat{\\mathbb {C}}}$ is a torsion element.", "It is well-known that such a group $H$ is finite (one can derive this from [2] in combination with the considerations in [2])." ], [ "Geometry of the peripheral circles", "no In this section we will prove Theorem REF .", "We have already defined in Section  what it means for the peripheral circles of a Sierpiński carpet $S$ to be uniform quasicircles and to be uniformly relatively separated.", "We say that the peripheral circles of $S$ occur on all locations and scales if there exists a constant $C\\ge 1$ such that for every $p\\in S$ and every $0<r\\le {\\rm diam}(\\widehat{\\mathbb {C}})=2$ , there exists a peripheral circle $J$ of $S$ with $B(p,r)\\cap J\\ne 0$ and $r/C\\le {\\rm diam}(J)\\le C r.$ Here and below the metric notions refer to the chordal metric $\\sigma $ on $\\widehat{\\mathbb {C}}$ .", "A set $M\\subseteq \\widehat{\\mathbb {C}}$ is called porous if there exists a constant $c>0$ such that for every $p\\in S$ and every $0<r\\le 2$ there exists a point $q\\in B(p,r)$ such that $B(q,cr)\\subseteq \\widehat{\\mathbb {C}}\\setminus M$ .", "Before we turn to the proof of Theorem REF , we require an auxiliary fact.", "Lemma 4.1 Let $f$ be a rational map such that $\\mathcal {J}(f)$ is a Sierpiński carpet, and let $J$ be a peripheral circle of $\\mathcal {J}(f)$ .", "Then $f^n(J)$ is a peripheral circle of $\\mathcal {J}(f)$ , and $f^{-n}(J)$ is a union of finitely many peripheral circles of $\\mathcal {J}(f)$ for each $n\\in {\\mathbb {N}}$ .", "Moreover, $J\\cap \\operatorname{post}(f)=\\emptyset =J\\cap \\operatorname{crit}(f)$ .", "There exists precisely one Fatou component $U$ of $f$ such that $\\partial U=J$ .", "Then $V=f^n(U)$ is also a Fatou component of $f$ .", "Hence $\\partial V$ is a peripheral circle of $\\mathcal {J}(f)$ .", "The map $f^n|_U\\colon U\\rightarrow V$ is proper which implies that $f^n(J)=f^n(\\partial U)=\\partial V$ .", "Similarly, there are finitely many distinct Fatou components $V_1, \\dots , V_k$ of $f$ such that $ f^{-n}(U)=V_1\\cup \\dots \\cup V_k.", "$ Then $f^{-n}(J)=\\partial V_1\\cup \\dots \\cup \\partial V_k, $ and so the preimage of $J$ under $f^n$ consists of the finitely many disjoint Jordan curves $\\partial V_i$ , $i=1, \\dots , k$ , which are peripheral circles of $\\mathcal {J}(f)$ .", "To show $J\\cap \\operatorname{post}(f)=\\emptyset $ , we argue by contradiction, and assume that there exists a point $p\\in \\operatorname{post}(f)\\cap J$ .", "Then there exists $n\\in {\\mathbb {N}}$ , and $c\\in \\operatorname{crit}(f)$ such that $f^n(c)=p$ .", "As we have just seen, the preimage of $J$ under $f^n$ consists of finitely many disjoint Jordan curves, and is hence a topological 1-manifold.", "On the other hand, since $c\\in f^{-n}(p)\\subseteq f^{-n}(J)$ is a critical point of $f$ and hence of $f^n$ , at $c$ the set $f^{-n}(J)$ cannot be a 1-manifold.", "This is a contradiction.", "Finally, suppose that $c\\in J\\cap \\operatorname{crit}(f)$ .", "Then $f(c)\\in \\operatorname{post}(f)\\cap f(J)$ , and $f(J)$ is a peripheral circle of $\\mathcal {J}(f)$ .", "This is impossible by what we have just seen.", "A general idea for the proof is to argue by contradiction, and get locations where the desired statements fail quantitatively in a worse and worse manner.", "One can then use the dynamics to blow up to a global scale and derive a contradiction from topological facts.", "It is fairly easy to implement this idea if we have expanding dynamics given by a group (see, for example, [5]).", "In the present case, one applies the conformal elevator and the estimates as given by Lemma REF .", "We now provide the details.", "We can pass to iterates of the map $f$ , and also conjugate $f$ by a Möbius transformation as properties that we want to establish are Möbius invariant.", "This Möbius invariance is explicitly stated for peripheral circles to be uniform quasicircles and to be uniformly relatively separated in [5].", "The Möbius invariance of the other stated properties immediately follows from the fact that each Möbius transformation is bi-Lipschitz with respect to the chordal metric.", "In this way, we may assume that () is true.", "Then the peripheral circles are subsets of $, where chordal and Euclidean metric are comparable.", "Therefore, we can use the Euclidean metric, and all metric notions will refer to this metric in the following.$ Part I.", "To show that peripheral circles of $ \\mathcal {J}(f)$ are uniform quasicircles, we argue by contradiction.", "Then for each $k\\in {\\mathbb {N}}$ there exists a peripheral circle $J_k$ of $\\mathcal {J}(f)$ , and distinct points $u_k, v_k\\in J_k$ such that if $\\alpha _k,\\beta _k$ are the two subarcs of $J_k$ with endpoints $u_k$ and $v_k$ , then $\\frac{\\min \\lbrace \\operatorname{diam}(\\alpha _k), \\operatorname{diam}(\\beta _k)\\rbrace }{|u_k-v_k|} \\rightarrow \\infty $ as $k\\rightarrow \\infty $ .", "We can pick $r_k>0$ such that $ \\min \\lbrace \\operatorname{diam}(\\alpha _k), \\operatorname{diam}(\\beta _k)\\rbrace /r_k\\rightarrow \\infty $ and $ |u_k-v_k|/r_k\\rightarrow 0$ as $k\\rightarrow \\infty .$ We now apply the conformal elevator to $B_k:=B(u_k, r_k)$ .", "Let $f^{n_k}$ be the corresponding iterate and $B_k^{\\prime }$ be the ball as discussed in Section .", "Define $J^{\\prime }_k=f^{n_k}(J_k)$ , $u^{\\prime }_k=f^{n_k}(u_k)$ , and $v^{\\prime }_k=f^{n_k}(v_k)$ .", "Then Lemma REF  (a) and (REF ) imply that the diameters of the sets $J^{\\prime }_k$ are uniformly bounded away from 0 independently of $k$ .", "Since $J_k^{\\prime }$ is a peripheral circle of the Sierpiński carpet $\\mathcal {J}(f)$ by Lemma REF , there are only finitely many possibilities for the set $J^{\\prime }_k$ .", "By passing to suitable subsequence if necessary, we may assume that $J^{\\prime }=J^{\\prime }_k$ is a fixed peripheral circle of $\\mathcal {J}(f)$ independent of $k$ .", "The points $u^{\\prime }_k,v^{\\prime }_k$ lie in $J^{\\prime }$ and by (REF ) and Lemma REF  (c) we have $|u^{\\prime }_k-v^{\\prime }_k|\\rightarrow 0$ as $k\\rightarrow \\infty $ .", "For large $k$ we want to find a point $w_k\\ne u_k$ in $B_k$ near $u_k$ with $f^{n_k}(u_k)=f^{n_k}(w_k)$ .", "If $u^{\\prime }_k=v^{\\prime }_k$ we can take $w_k=v_k$ .", "Otherwise, if $u^{\\prime }_k\\ne v^{\\prime }_k$ , there are two subarcs of $J^{\\prime }$ with endpoints $u^{\\prime }_k$ and $v^{\\prime }_k$ .", "Let $\\gamma ^{\\prime }_k\\subseteq J^{\\prime }$ be the one with smaller diameter.", "Then by (REF ) we have $\\operatorname{diam}(\\gamma ^{\\prime }_k)\\rightarrow 0$ as $k\\rightarrow \\infty $ (for the moment we only consider such $k$ for which $\\gamma ^{\\prime }_k$ is defined).", "Since $J^{\\prime }\\cap \\operatorname{post}(f)=\\emptyset $ by Lemma REF , the map $f^{n_k}\\colon J_k \\rightarrow J^{\\prime }$ is a covering map.", "So we can lift the arc $\\gamma ^{\\prime }_k$ under $f^{n_k}$ to a subarc $\\gamma _k$ of $J_k$ with initial point $v_k$ and $f^{n_k}(\\gamma _k)=\\gamma ^{\\prime }_k$ .", "By Lemma REF  (b) we have $\\gamma ^{\\prime }_k\\subseteq f^{n_k} (B_k)$ for large $k$ ; then Lemma REF  (a) implies that $\\gamma _k\\subseteq B_k$ for large $k$ , and also $\\operatorname{diam}(\\gamma _k)/r_k\\rightarrow 0$ as $k\\rightarrow \\infty $ .", "Note that if $w_k$ is the other endpoint of $\\gamma _k$ , then $f^{n_k}(w_k)=u^{\\prime }_k$ .", "We have $w_k\\ne u_k$ for large $k$ ; for if $w_k=u_k$ , then $\\gamma _k\\subseteq J_k$ has the endpoints $u_k$ and $v_k$ and so must agree with one of the arcs $\\alpha _k$ or $\\beta _k$ ; but for large $k$ this is impossible by (REF ) and (REF ).", "In addition, we have $|u_k-w_k|/r_k\\le |u_k-v_k|/r_k+\\operatorname{diam}(\\gamma _k)/r_k\\rightarrow 0$ as $k\\rightarrow \\infty $ .", "Note that this is also true if $w_k=v_k$ .", "In summary, for each large $k$ we can find a point $w_k\\in B_k$ with $w_k\\ne u_k$ , $f^{n_k}(u_k)=f^{n_k}(w_k)$ , and $ |u_k-w_k|/r_k\\rightarrow 0$ as $k\\rightarrow \\infty $ .", "Then by Lemma REF  (d) the center $q_k$ of $B^{\\prime }_k$ must belong to the postcritical set of $f$ and $\\operatorname{dist}(J^{\\prime }, \\operatorname{post}(f))\\le |u^{\\prime }_k- q_k|\\rightarrow 0$ as $k\\rightarrow \\infty $ .", "Since $f$ is subhyperbolic, every sufficiently small neighborhood of $\\mathcal {J}(f) \\supseteq J^{\\prime }$ contains only finitely many points in $\\operatorname{post}(f)$ , and so this implies $J^{\\prime }\\cap \\operatorname{post}(f)\\ne \\emptyset $ .", "We know that this is impossible by Lemma REF and so we get a contradiction.", "This shows that the peripheral circles are uniform quasicircles.", "Part II.", "The proof that the peripheral circles of $\\mathcal {J}(f)$ are uniformly relatively separated runs along almost identical lines.", "Again we argue by contradiction.", "Then for $k\\in {\\mathbb {N}}$ we can find distinct peripheral circles $\\alpha _k$ and $\\beta _k$ of $\\mathcal {J}(f)$ , and points $u_k\\in \\alpha _k$ , $v_k\\in \\beta _k$ such that (REF ) is valid.", "We can again pick $r_k>0$ so that the relations (REF ) and (REF ) are true.", "As before we define $B_k=B(u_k, r_k)$ and apply the conformal elevator to $B_k$ which gives us suitable iterate $f^{n_k}$ and a ball $B^{\\prime }_k$ .", "By Lemma REF  (a) the images of $\\alpha _k$ and $\\beta _k$ under $f^{n_k}$ are blown up to a definite size.", "Since there are only finitely many peripheral circles of $\\mathcal {J}(f)$ whose diameter exceeds a given constant, only finitely many such image pairs can arise.", "By passing to a suitable subsequence if necessary, we may assume that $\\alpha =f^{n_k}(\\alpha _k)$ and $\\beta =f^{n_k}(\\alpha _k)$ are peripheral circles independent of $k$ .", "We define $u^{\\prime }_k:=f^{n_k}(u_k)\\in \\alpha $ and $v^{\\prime }_k:=f^{n_k}(v_k)\\in \\beta $ .", "Then again the relation (REF ) holds.", "This is only possible if $\\alpha \\cap \\beta \\ne \\emptyset $ , and so $\\alpha =\\beta $ .", "Again for large $k$ we want to find a point $w_k\\ne u_k$ in $B_k$ near $u_k$ with $f^{n_k}(u_k)=f^{n_k}(w_k)$ .", "If $u^{\\prime }_k=v^{\\prime }_k$ we can take $w_k:=v_k$ .", "Otherwise, if $u^{\\prime }_k\\ne v^{\\prime }_k$ , we let $\\gamma ^{\\prime }_k$ be the subarc of $\\alpha =\\beta $ with endpoints $u^{\\prime }_k $ and $v^{\\prime }_k$ and smaller diameter.", "Then we can lift $\\gamma ^{\\prime }_k$ to a subarc $\\gamma _k\\subseteq \\beta _k$ with initial point $v_k$ such that $f^{n_k}(\\gamma _k)=\\gamma ^{\\prime }_k$ , and we have (REF ).", "If $w_k\\in \\beta _k$ is the other endpoint of $\\gamma _k$ , then $f^{n_k}(w_k)=u^{\\prime }_k=f^{n_k}(u_k)$ , and $w_k\\ne u_k$ , because these points lie in the disjoint sets $\\beta _k$ and $\\alpha _k$ , respectively.", "Again we have (REF ), which implies that the center $q_k$ of $B_k^{\\prime }$ belongs to $\\operatorname{post}(f)$ , and leads to $\\operatorname{dist}(\\alpha , \\operatorname{post}(f))=0$ .", "We know that this is impossible by Lemma REF .", "Part III.", "We will show that peripheral circles of $\\mathcal {J}(f)$ appear on all locations and scales.", "Let $p\\in \\mathcal {J}(f)$ and $r>0$ be arbitrary, and define $B=B(p,r)$ .", "We may assume that $r$ is small, because by a simple compactness argument one can show that disks of definite, but not too large Euclidean size contain peripheral circles of comparable diameter.", "We now apply the conformal elevator to $B$ to obtain an iterate $f^n$ .", "Lemma REF  (b) implies that there exists a fixed constant $r_1>0$ independent of $B$ such that $B(f^n(p), r_1)\\subseteq f^n(\\frac{1}{2} B)$ .", "By part (a) of the same lemma, we can also find a constant $c_1>0$ independent of $B$ with the following property: if $A$ is a connected set with $A\\cap B(p, r/2)\\ne \\emptyset $ and $\\operatorname{diam}(f^n(A))\\le c_1$ , then $A\\subseteq B$ .", "We can now find a peripheral circle $J^{\\prime }$ of $\\mathcal {J}(f)$ such that $J^{\\prime }\\cap f^n(\\frac{1}{2} B)\\ne \\emptyset $ and $0<c_0 < \\operatorname{diam}(J^{\\prime })<c_1$ , where $c_0$ is another positive constant independent of $B$ .", "This easily follows from a compactness argument based on the fact that $f^n(\\frac{1}{2} B)$ contains a disk of a definite size that is centered at a point in $\\mathcal {J}(f)$ .", "The preimage $f^{-n}(J^{\\prime })$ consists of finitely many components that are peripheral circles of $\\mathcal {J}(f)$ .", "One of these peripheral circles $J$ meets $\\frac{1}{2} B$ .", "Since $\\operatorname{diam}(f^n(J))=\\operatorname{diam}(J^{\\prime })<c_1$ , by the choice of $c_1$ we then have $J\\subseteq B$ , and so $\\operatorname{diam}(J)\\le 2r$ .", "Moreover, it follows from Lemma REF  (c) that $\\operatorname{diam}(J)\\ge c_2 \\operatorname{diam}(J^{\\prime })\\operatorname{diam}(B)\\ge c_3 r$ , where again $c_2,c_3>0$ are independent of $B$ .", "The claim follows.", "Part IV.", "Let $p\\in \\mathcal {J}(f)$ be arbitrary and $r\\in (0,1]$ .", "To establish the porosity of $\\mathcal {J}(f)$ , it is enough to show that the Euclidean disk $B(p,r)$ contains a disk of comparable radius that lies in the complement of $\\mathcal {J}(f)$ .", "By what we have just seen, $B(p,r)$ contains a peripheral circle $J$ of diameter comparable to $r$ .", "By possibly allowing a smaller constant of comparability, we may assume that $J$ is distinct from the one peripheral circle $J_0$ that bounds the unbounded Fatou component of $f$ .", "Then $J\\subseteq B(p,r)$ is the boundary of a bounded Fatou component $U$ , and so $U\\subseteq B(p,r)$ .", "Since the peripheral circles of $\\mathcal {J}(f)$ are uniform quasicircles, it follows that $U$ contains a Euclidean disk $D$ of comparable size (for this standard fact see [5]).", "Then $\\operatorname{diam}(D) \\approx \\operatorname{diam}(J)\\approx r$ .", "Since $D\\subseteq U\\subseteq B(p,r)\\cap \\widehat{\\mathbb {C}}\\setminus \\mathcal {J}(f)$ the porosity of $\\mathcal {J}(f)$ follows.", "Finally, the porosity of $\\mathcal {J}(f)$ implies that $\\mathcal {J}(f)$ cannot have Lebesgue density points, and is hence a set of measure zero." ], [ "Relative Schottky sets and Schottky maps", "no A relative Schottky set $S$ in a region $D\\subseteq \\widehat{\\mathbb {C}}$ is a subset of $D$ whose complement in $D$ is a union of open geometric disks $\\lbrace B_i\\rbrace _{i\\in I}$ with closures $\\overline{B}_i,\\ i\\in I$ , in $D$ , and such that $\\overline{B}_i\\bigcap \\overline{B}_j=\\emptyset ,\\ i\\ne j$ .", "We write $S=D\\setminus \\bigcup _{i\\in I}B_i.$ If $D=\\widehat{\\mathbb {C}}$ or $, we say that $ S$ is a \\emph {Schottky set}.$ Let $A,B\\subseteq \\widehat{\\mathbb {C}}$ and $\\varphi \\colon A\\rightarrow B$ be a continuous map.", "We call $\\varphi $ a local homeomorphism of $A$ to $B$ if for every point $p\\in A$ there exist open sets $U,V\\subseteq with $ pU$, $ f(p)V$ such that $ f|UA$ is a homeomorphism of $ UA$ onto $ VB$.", "Note that this concept depends of course on $ A$, but also crucially on $ B$: if $ B'B$, then we may consider a local homeomorphism $ fAB$ also as a map $ fAB'$, but the second map will not be a local homeomorphism in general.$ Let $D$ and $\\tilde{D}$ be two regions in $\\widehat{\\mathbb {C}}$ , and let $S=D\\setminus \\bigcup _{i\\in I} B_i$ and $\\tilde{S}=\\tilde{D}\\setminus \\bigcup _{j\\in J}\\tilde{B}_j$ be relative Schottky sets in $D$ and $ \\tilde{D}$ , respectively.", "Let $U$ be an open subset of $D$ and let $f\\colon S\\cap U\\rightarrow \\tilde{S}$ be a local homeomorphism.", "According to [20], such a map $f$ is called a Schottky map if it is conformal at every point $p\\in S\\cap U$ , i.e., the derivative $f^{\\prime }(p)=\\lim _{q\\in S,\\, q\\rightarrow p }\\frac{f(q)-f(p)}{q-p}$ exists and does not vanish, and the function $f^{\\prime }$ is continuous on $S\\cap U$ .", "If $p=\\infty $ or $f(p)=\\infty $ , the existence of this limit and the continuity of $f^{\\prime }$ have to be understood after a coordinate change $z\\mapsto 1/z$ near $\\infty $ .", "In all our applications $S\\subseteq and so we can ignore this technicality.$ Theorem REF implies that if $D$ and $\\tilde{D}$ are Jordan regions, the relative Schottky set $S$ has measure zero, and $f\\colon S\\rightarrow \\tilde{S}$ is a locally quasisymmetric homeomorphism that is orientation-preserving (this is defined similarly as for homeomorphisms between Sierpiński carpets; see the discussion after Lemma REF ), then $f$ is a Schottky map.", "We require a more general criterion for maps to be Schottky maps.", "Lemma 5.1 Let $S\\subseteq be a Schottky set of measure zero.Suppose $ UC$ is open and $ UC$ is a locally quasiconformal map with $ -1(S)=US.", "$Then $ USS$ is a Schottky map.$ In particular, if $\\psi \\colon \\widehat{\\mathbb {C}}\\rightarrow \\widehat{\\mathbb {C}}$ is a quasiregular map with $\\psi ^{-1}(S)=S$ , then $\\psi \\colon S\\setminus \\operatorname{crit}(\\psi )\\rightarrow S$ is a Schottky map.", "In the statement the assumption $S\\subseteq (instead of $ SC$) is not really essential, but helps to avoid some technicalities caused by the point $$.$ Our assumption $\\varphi ^{-1}(S)=U\\cap S$ implies that $\\varphi (U\\cap S)\\subseteq S$ .", "So we can consider the restriction of $\\varphi $ to $U\\cap S$ as a map $\\varphi \\colon U\\cap S\\rightarrow S$ (for simplicity we do not use our usual notation $\\varphi |_{U\\cap S}$ for this and other restrictions in the proof).", "This map is a local homeomorphism $\\varphi \\colon U\\cap S\\rightarrow S$ .", "Indeed, let $p\\in U\\cap S$ be arbitrary.", "Since $\\varphi \\colon U\\rightarrow \\widehat{\\mathbb {C}}$ is a local homeomorphism, there exist open sets $V,W\\subseteq \\widehat{\\mathbb {C}}$ with $p\\in V\\subseteq U$ and $f(p)\\in W$ such that $\\varphi $ is a homeomorphism of $V$ onto $W$ .", "Clearly, $\\varphi (V\\cap S)\\subseteq W\\cap S$ .", "Conversely, if $q\\in W\\cap S$ , then there exists a point $q^{\\prime }\\in V$ with $\\varphi (q^{\\prime })=q$ ; since $\\varphi ^{-1}(S)=U\\cap S$ , we have $q^{\\prime }\\in S$ and so $q^{\\prime }\\in V\\cap S$ .", "Hence $\\varphi (V\\cap S)=W\\cap S$ , which implies that $\\varphi $ is a homeomorphism of $V\\cap S$ onto $W\\cap S$ .", "Note that $p\\in U\\cap S$ lies on a peripheral circle of $S$ if and only if $\\varphi (p)$ lies on a peripheral circle of $S$ .", "Indeed, a point $p\\in S$ lies on a peripheral of $S$ if and only if it is accessible by a path in the complement of $S$ , and it is clear this condition is satisfied for a point $p\\in S\\cap U$ if and only if it is true for the image $\\varphi (p)$ (see [20] for a more general related statement).", "We now want to verify the other conditions for $\\varphi $ to be a Schottky map based on Theorem REF .", "It is enough to reduce to this situation locally near each point $p\\in U\\cap S$ .", "We consider two cases depending on whether $p$ belongs to a peripheral circle of $S$ or not.", "So suppose $p$ does not belong to any of the peripheral circles of $S$ .", "Then there exist arbitrarily small Jordan regions $D$ with $p\\in D$ and $\\partial D\\subseteq S$ such that $\\partial D$ does not meet any peripheral circle of $S$ .", "This easily follows from the fact that if we collapse each closure of a complementary component of $S$ in $\\widehat{\\mathbb {C}}$ to a point, then the resulting quotient space is homeomorphic to $\\widehat{\\mathbb {C}}$ by Moore's theorem [23] (for more details on this and the similar argument below, see the proof of [20]).", "In this way we can find a small Jordan region $D$ with the following properties: (i) $p\\in D \\subseteq \\overline{D}\\subseteq U$ , (ii) the boundary $\\partial D$ is contained in $S$ , but does not meet any peripheral circle of $S$ , (iii) $\\varphi $ is a homeomorphism of $\\overline{D}$ onto the closure $\\overline{D}^{\\prime }$ of another Jordan region $D^{\\prime }\\subseteq \\widehat{\\mathbb {C}}$ .", "As in the first part of the proof, we see that $\\varphi $ is a homeomorphism of $D\\cap S$ onto $D^{\\prime }\\cap S$ .", "This homeomorphism is locally quasisymmetric and orientation-preserving as it is the restriction of a locally quasiconformal map.", "Since $\\partial D$ does not meet peripheral circles of $S$ , the same is true of its image $\\partial D^{\\prime }=\\varphi (\\partial D)$ by what we have seen above.", "It follows that the sets $D\\cap S$ and $D^{\\prime }\\cap S$ are relative Schottky sets of measure zero contained in the Jordan regions $D$ and $D^{\\prime }$ , respectively.", "Note that the set $D\\cap S$ is obtained by deleting from $D$ the complementary disks of $S$ that are contained in $D$ , and $D^{\\prime }\\cap S$ is obtained similarly.", "Now Theorem REF implies that $\\varphi \\colon D\\cap S\\rightarrow D^{\\prime }\\cap S$ is a Schottky map which implies that $\\varphi \\colon U\\cap S \\rightarrow S$ is a Schottky map near $p$ .", "For the other case, assume that $p$ lies on a peripheral circle of $S$ , say $p\\in \\partial B$ , where $B$ is one of the disks that form the complement of $S$ .", "The idea is to use a Schwarz reflection procedure to arrive at a situation similar to the previous case.", "This is fairly straightforward, but we will provide the details for sake of completeness.", "Similarly as before (here we collapse all closures of complementary components of $S$ to points except $\\overline{B}$ ), we find a Jordan region $D$ with the following properties: (i) $\\overline{D}\\subseteq U$ and $\\partial D=\\alpha \\cup \\beta $ , where $\\alpha $ and $\\beta $ are two non-overlapping arcs with the same endpoints such that $\\alpha \\subseteq \\partial B$ , $\\beta \\subseteq S$ , $p$ is an interior point of $\\alpha $ , and no interior point of $\\beta $ lies on a peripheral circle of $S$ , (ii) $\\varphi $ is a homeomorphism of $\\overline{D}$ onto the closure $\\overline{D}^{\\prime }$ of another Jordan region $D^{\\prime }\\subseteq \\widehat{\\mathbb {C}}$ .", "Let $\\alpha ^{\\prime }=\\varphi (\\alpha )$ .", "Then $\\alpha $ is contained in a peripheral circle $\\partial B^{\\prime }$ of $S$ , where $B^{\\prime }$ is a suitable complementary disk of $S$ .", "Note that $\\beta ^{\\prime }=\\varphi (\\beta )$ is an arc contained in $S$ , has its endpoints in $\\partial B^{\\prime }$ , and no interior point of $\\beta ^{\\prime }$ lies on a peripheral circle of $S$ .", "Let $R\\colon \\widehat{\\mathbb {C}}\\rightarrow \\widehat{\\mathbb {C}}$ be the reflection in $\\partial B$ , and $R^{\\prime }\\colon \\widehat{\\mathbb {C}}\\rightarrow \\widehat{\\mathbb {C}}$ be the reflection in $\\partial B^{\\prime }$ .", "Define $\\tilde{S}=S\\cup R(S)$ and $\\tilde{S}^{\\prime }=S\\cup R^{\\prime }(S)$ .", "Then $\\tilde{S}$ and $\\tilde{S}^{\\prime }$ are Schottky sets of measure zero, $\\partial B\\subseteq \\tilde{S}$ , $\\partial B^{\\prime }\\subseteq \\tilde{S}^{\\prime }$ , and $\\partial B$ and $\\partial B^{\\prime }$ do not meet any of the peripheral circles of $\\tilde{S}$ and $\\tilde{S}^{\\prime }$ , respectively.", "Let $\\tilde{D}=D\\cup \\operatorname{int}(\\alpha )\\cup R(D)$ and $\\tilde{D}^{\\prime }=D^{\\prime }\\cup \\operatorname{int}(\\alpha ^{\\prime })\\cup R^{\\prime }(D^{\\prime })$ , where $\\operatorname{int}(\\alpha )$ and $ \\operatorname{int}(\\alpha ^{\\prime })$ denote the set of interior points of the arcs $\\alpha $ and $\\alpha ^{\\prime }$ , respectively.", "Then $\\tilde{D}$ and $\\tilde{D}^{\\prime }$ are Jordan regions such that $p\\in \\tilde{D}$ , $\\partial \\tilde{D}\\subseteq \\tilde{S}$ , $\\partial \\tilde{D}^{\\prime }\\subseteq \\tilde{S}^{\\prime }$ , and $\\partial \\tilde{D}$ and $\\partial \\tilde{D}^{\\prime }$ do not meet any of the peripheral circles of $\\tilde{S}$ and $\\tilde{S}^{\\prime }$ , respectively.", "Hence $\\tilde{D}\\cap \\tilde{S}$ and $\\tilde{D}^{\\prime }\\cap \\tilde{S}^{\\prime }$ are relative Schottky sets of measure zero in $\\tilde{D}$ and $\\tilde{D}^{\\prime }$ , respectively.", "We define a map $\\tilde{\\varphi }\\colon \\tilde{D}\\rightarrow \\tilde{D}^{\\prime }$ by $ \\tilde{\\varphi }(z)= \\left\\lbrace \\begin{array} {cl}\\varphi (z)& \\text{for $z\\in D\\cup \\operatorname{int}(\\alpha ) $,}\\\\&\\\\(R^{\\prime }\\circ \\varphi \\circ R)(z)&\\text{for $z\\in R(D)\\cup \\operatorname{int}(\\alpha )$.}", "\\end{array}\\right.$ Note that this definition is consistent on $\\operatorname{int}(\\alpha )$ , because $\\varphi (\\alpha )=\\alpha ^{\\prime }=\\overline{D}^{\\prime }\\cap R^{\\prime }(\\overline{D}^{\\prime })$ .", "It is clear that $\\tilde{\\varphi }\\colon \\tilde{D}\\rightarrow \\tilde{D}^{\\prime }$ is a homeomorphism.", "Moreover, since the circular arc $\\alpha $ (as any set of $\\sigma $ -finite Hausdorff 1-measure) is removable for quasiconformal maps [26], the map $\\tilde{\\varphi }$ is locally quasiconformal, and hence locally quasisymmetric and orientation-preserving.", "It is also straightforward to see from the definitions and the relation $\\varphi ^{-1}(S)=U\\cap S$ that $\\tilde{\\varphi }^{-1}(\\tilde{S}^{\\prime })= \\tilde{D} \\cap \\tilde{S}$ .", "Similarly as in the beginning of the proof this implies that $\\tilde{\\varphi }\\colon \\tilde{D}\\cap \\tilde{S} \\rightarrow \\tilde{D}^{\\prime }\\cap \\tilde{S}^{\\prime }$ is a homeomorphism.", "Since it is also a local quasisymmetry and orientation-preserving, it follows again from Theorem REF that $\\tilde{\\varphi }\\colon \\tilde{D}\\cap \\tilde{S} \\rightarrow \\tilde{D}^{\\prime }\\cap \\tilde{S}^{\\prime }$ is a Schottky map.", "Note that $\\tilde{D}\\cap S= D\\cap S$ , that on this set the maps $\\tilde{\\varphi }$ and $\\varphi $ agree, and that $\\varphi (\\tilde{D}\\cap S)\\subseteq S$ .", "Thus, $\\varphi \\colon \\tilde{D}\\cap S\\rightarrow S$ is a Schottky map, and so $\\varphi \\colon U\\cap S\\rightarrow S$ is a Schottky map near $p$ .", "It follows that $\\varphi \\colon U\\cap S\\rightarrow S$ is a Schottky map as desired.", "The second part of the statement immediately follows from the first; indeed, $\\operatorname{crit}(\\psi )$ is a finite set and so $U=\\widehat{\\mathbb {C}}\\setminus \\operatorname{crit}(\\psi )$ is an open subset $\\widehat{\\mathbb {C}}$ on which $\\varphi =\\psi |_{U}\\colon U\\rightarrow \\widehat{\\mathbb {C}}$ is a locally quasiconformal map.", "Moreover, $\\varphi ^{-1}(S)=\\psi ^{-1}(S)\\cap U=S\\cap U$ .", "By the first part of the proof, $\\varphi $ and hence also $\\psi $ (restricted to $U\\cap S$ ) is a Schottky map of $U\\cap S=S\\setminus \\operatorname{crit}(\\psi )$ into $S$ .", "A relative Schottky set as in (REF ) is called locally porous at $p\\in S$ if there exists a neighborhood $U$ of $p$ , and constants $r_0>0$ and $C\\ge 1$ such that for each $q\\in S\\cap U$ and $r\\in (0, r_0]$ there exists $i\\in I$ with $B_i\\cap B(q,r)\\ne \\emptyset $ and $r/C\\le \\operatorname{diam}(B_i) \\le Cr$ .", "The relative Schottky set $S$ is called locally porous if it is locally porous at every point $p\\in S$ .", "Every locally porous relative Schottky set has measure zero since it cannot have Lebesgue density points.", "For Schottky maps on locally porous Schottky sets very strong rigidity and uniqueness statements are valid such as Theorems REF and REF stated in the introduction.", "We will need another result of a similar flavor.", "Theorem 5.2 (Me3, Theorem 4.1) Let $S$ be a locally porous relative Schottky set in a region $D\\subseteq , let$ U be an open set such that $S\\cap U$ is connected, and $u\\colon S\\cap U\\rightarrow S$ be a Schottky map.", "Suppose that there exists a point $a\\in S\\cap U$ with $u(a)=a$ and $u^{\\prime }(a)=1$ .", "Then $u=\\operatorname{id}|_{S\\cap U}$ ." ], [ "A functional equation in the unit disk", "no As discussed in the introduction, for the proof of Theorem REF we will establish a functional equation of form (REF ) for the maps in question.", "For postcritically-finite maps $f$ and $g$ this leads to strong conclusions based on the following lemma.", "Recall that $P_k(z)=z^k$ for $k\\in {\\mathbb {N}}$ .", "Lemma 6.1 Let $\\phi \\colon \\partial \\partial be an orientation-preserving homeo\\-morphism, and suppose that there exist numbers $ k,l,nN$, $ k2$, such that\\begin{equation} (P_l\\circ \\phi )(z)= (P_n\\circ \\phi \\circ P_k)(z) \\quad \\text{for $z\\in \\partial $.", "}Then l=nk and there exists a\\in with a^{n(k-1)}=1 such that \\phi (z)=az for all z\\in \\partial .\\end{equation}This lemma implies that we can uniquely extend $$ to a conformal homeomorphismfrom $$ onto itself.", "It is also important that this extension preserves the basepoint $ 0$.$ By considering topological degrees, one immediately sees that $l=nk$ .", "So if we introduce the map $\\psi := P_n\\circ \\phi $ , then () can be rewritten as $ P_k\\circ \\psi =\\psi \\circ P_k \\quad \\text{on $\\partial $.", "}Here the map \\psi \\colon \\partial \\partial has degree n. We claim that this in combination with (\\ref {eq:basiceq2}) implies that for a suitable constant b we have \\psi (z)=bz^n for z\\in \\partial .$ Indeed, there exists a continuous function $\\alpha \\colon {\\mathbb {R}}\\rightarrow {\\mathbb {R}}$ with $\\alpha (t+2\\pi )=\\alpha (t)$ such that $ \\psi (e^{ i t}) = \\exp ( i n t+ i\\alpha (t)) \\quad \\text{for $t\\in {\\mathbb {R}}$}.", "$ By (REF ) we have $ \\exp ( i k n t+ i k\\alpha (t)) = (\\psi (e^{ i t}))^k = \\psi (e^{ i k t})=\\exp ( i kn t+ i\\alpha (kt))$ for $t\\in {\\mathbb {R}}$ .", "This implies that there exists a constant $c\\in {\\mathbb {R}}$ such that $ \\alpha (t)=\\frac{1}{k} \\alpha (tk)+c \\quad \\text{for $t\\in {\\mathbb {R}}$}.", "$ Since $\\alpha $ is $2\\pi $ -periodic, the right-hand side of this equation is $2\\pi /k$ -periodic as a function of $t$ .", "Hence $\\alpha $ is $2\\pi /k$ -periodic.", "Repeating this argument, we see that $\\alpha $ is $2\\pi /k^m$ -periodic for all $m\\in {\\mathbb {N}}$ , and so has arbitrarily small periods (note that $k\\ge 2$ ).", "Since $\\alpha $ is continuous, it follows that $\\alpha $ is constant.", "Hence $\\psi (z)=bz^n$ for $z\\in \\partial with a suitable constant $ b.", "It follows that $ \\psi (z)=b z^n=\\phi (z)^n \\quad \\text{for $z\\in \\partial $.$$Therefore, $\\phi (z)=az$ for $z\\in \\partial with a constant $a\\in , $a\\ne 0$.Inserting this expression for $\\phi $ into (\\ref {eq:basiceq}) and using $l=nk$, we conclude that $a^{n(k-1)}=1$ as desired.", "}$ Proof of Theorem  REF The proof will be given in several steps.", "Step I.", "We first fix the setup.", "We can freely pass to iterates of the maps $f$ or $g$ , because this changes neither their Julia sets nor their postcritical sets.", "We can also conjugate the maps by Möbius transformations.", "Therefore, as in Section , we may assume that $\\mathcal {J}(f), \\mathcal {J}(g)\\subseteq \\tfrac{1}{2} { and } f^{-1}( , g^{-1}(\\subseteq $ Moreover, without loss of generality, we may require that $\\xi $ is orientation-preserving, for otherwise we can conjugate $g$ by $z\\mapsto \\overline{z}$ .", "Since the peripheral circles of $\\mathcal {J}(f)$ and $\\mathcal {J}(g)$ are uniform quasicircles, by Theorem REF the map $\\xi $ extends (non-uniquely) to a quasiconformal, and hence quasisymmetric, map of the whole sphere.", "Then $\\xi (\\mathcal {J}(f))=\\mathcal {J}(g)$ and $\\xi (\\mathcal {F}(f))=\\mathcal {F}(g)$ .", "Since $\\infty $ lies in Fatou components of $f$ and $g$ , we may also assume that $\\xi (\\infty )=\\infty $ (this normalization ultimately depends on the fact that for every point $p\\in there exists a quasiconformal homeomorphism $$on $ C$ with $ (0)=p$ that is the identity outside $ ).", "Then $\\xi $ is a quasisymmetry of $ with respect to the Euclidean metric.", "In the following, all metric notions will refer to this metric.Finally, we define $ g=-1 g$.$ Step II.", "We now carefully choose a location for a “blow-down\" by branches of $f^{-n}$ which will be compensated by a “blow-up\" by iterates of $g$ (or rather $g_\\xi $ ).", "Since repelling periodic points of $f$ are dense in $\\mathcal {J}(f)$ (see [3]), we can find such a point $p$ in $\\mathcal {J}(f)$ that does not lie in $\\operatorname{post}(f)$ .", "Let $\\rho >0$ be a small positive number such that the disk $U_0:=B(p, 3\\rho )\\subseteq is disjoint from $ post(f)$.Since $ p$ is periodic, there exists $ dN$ such that $ fd(p)=p$.", "Let $ U1 be the component of $f^{-d}(U_0)$ that contains $p$ .", "Since $U_0\\cap \\operatorname{post}(f)=\\emptyset $ , the set $U_1$ is a simply connected region, and $f^{d}$ is a conformal map from $U_1$ onto $U_0$ as follows from Lemma REF .", "Then there exists a unique inverse branch $f^{-d}$ with $f^{-d}(p)=p$ that is a conformal map of $U_0$ onto $U_1$ .", "Since $p$ is a repelling fixed point for $f$ , it is an attracting fixed point for this branch $f^{-d}$ .", "By possibly choosing a smaller radius $\\rho >0$ in the definition of $U_0=B(p, 3\\rho )$ and by passing to an iterate of $f^d$ , we may assume that $U_1\\subseteq U_0$ and that $\\operatorname{diam}(f^{-n_k}(U_0))\\rightarrow 0$ as $k\\rightarrow \\infty $ .", "Here $n_k=dk$ for $k\\in {\\mathbb {N}}$ , and $f^{-n_k}$ is the branch obtained by iterating the branch $f^{-d}$ $k$ -times.", "Note that $f^{-n_k}(p)=p$ and $f^{-n_k}$ is a conformal map of $U_0$ onto a simply connected region $U_k$ .", "Then $p\\in U_k\\subseteq U_{k-1}$ for $k\\in {\\mathbb {N}}$ , and $\\operatorname{diam}(U_k)\\rightarrow 0$ as $k\\rightarrow \\infty $ .", "The choice of these inverse branches is consistent in the sense that we have $f^{n_{k+1}-n_k}\\circ f^{-n_{k+1}}=f^{-n_k}$ on $B(p, 3\\rho )$ for all $k\\in {\\mathbb {N}}$ .", "Note that this consistency condition remains valid if we replace the original sequence $\\lbrace n_k\\rbrace $ by a subsequence.", "Let $\\tilde{r}_k>0$ be the smallest number such that $f^{-n_k}(B(p, 2\\rho ))\\subseteq \\tilde{B}_k:=B(p,\\tilde{r}_k).$ Since $p\\in f^{-n_k}(B(p, 2\\rho ))$ we have $\\operatorname{diam}\\big (f^{-n_k}(B(p, 2\\rho ))\\big )\\approx \\tilde{r}_k$ .", "Here and below $\\approx $ indicates implicit positive multiplicative constants independent of $k\\in {\\mathbb {N}}$ .", "It follows that $\\tilde{r}_k\\rightarrow 0$ as $k\\rightarrow \\infty $ .", "Moreover, since $f^{-n_k}$ is conformal on the larger disk $B(p,3\\rho )$ , Koebe's distortion theorem implies that $ \\operatorname{diam}\\big (f^{-n_k}(B(p, 2\\rho ))\\big )\\approx \\operatorname{diam}\\big (f^{-n_k}(B(p, \\rho ))\\big ).", "$ Let $r_k>0$ be the smallest number such that $\\xi (\\tilde{B}_k)\\subseteq B_k:=B(\\xi (p), r_k)$ .", "Again $r_k\\rightarrow 0$ as $k\\rightarrow \\infty $ by continuity of $\\xi $ .", "We now want to apply the conformal elevator given by iterates of $g$ to the disks $B_k$ .", "For this we choose $\\epsilon _0>0$ for the map $g$ as in Section .", "By applying the conformal elevator as described in Section , we can find iterates $g^{m_k}$ such that $g^{m_k}(B_k)$ is blown up to a definite, but not too large size, and so $\\operatorname{diam}(g^{m_k}(B_k))\\approx 1$ .", "Step III.", "Now we consider the composition $h_k=g_\\xi ^{m_k}\\circ f^{-n_k}= \\xi ^{-1}\\circ g^{m_k}\\circ \\xi \\circ f^{-n_k}$ defined on $B(p,2\\rho )$ for $k\\in {\\mathbb {N}}$ .", "We want to show that this sequence subconverges locally uniformly on $B(p,2\\rho )$ to a (non-constant) quasiregular map $h\\colon B(p, 2\\rho )\\rightarrow .$ Since $f^{-n_k}$ maps $B(q,2\\rho )$ conformally into $\\tilde{B}_k$ , $\\xi $ is a quasiconformal map with $\\xi (\\tilde{B}_k)\\subseteq B_k$ , and $g^{m_k}$ is holomorphic on $\\tilde{B}_k$ , we conclude that the maps $h_k$ are uniformly quasiregular on $B(q,2\\rho )$ , i.e., $K$ -quasiregular with $K\\ge 1$ independent of $k$ .", "The images $h_k(B(q,2\\rho ))$ are contained in a small Euclidean neighborhood of $\\mathcal {J}(g)$ and hence in a fixed compact subset of $.", "Standardconvergence results for $ K$-quasiregular mappings \\cite [p.~182, Corollary 5.5.7]{AIM} imply that the sequence $ {hk}$subconverges locally uniformly on $ B(q,2)$ to a map $ hB(q,2) that is also quasiregular, but possibly constant.", "By passing to a subsequence if necessary, we may assume that $h_k\\rightarrow h$ locally uniformly on $B(q,2\\rho )$ .", "To rule out that $h$ is constant, it is enough to show that for smaller disk $B(p,\\rho )$ there exists $\\delta >0$ such that $\\operatorname{diam}\\big (h_k(B(p,\\rho ))\\big )\\ge \\delta $ for all $k\\in {\\mathbb {N}}$ .", "We know that $\\operatorname{diam}\\big (f^{-n_k}(B(p,\\rho ))\\big )\\approx \\operatorname{diam}\\big (f^{-n_k}(B(p,2\\rho ))\\big ) \\approx \\operatorname{diam}(\\tilde{B}_k).", "$ Moreover, since $\\xi $ is a quasisymmetry and $f^{-n_k}(B(p,\\rho ))\\subseteq \\tilde{B}_k$ , this implies $\\operatorname{diam}\\big (\\xi (f^{-n_k}(B(p,\\rho )))\\big )\\approx \\operatorname{diam}(\\xi (\\tilde{B}_k))\\approx \\operatorname{diam}(B_k).", "$ So the connected set $\\xi (f^{-n_k}(B(p,\\rho )))\\subseteq B_k$ is comparable in size to $B_k$ .", "By Lemma REF  (a) the conformal elevator blows it up to a definite, but not too large size, i.e., $\\operatorname{diam}\\big ((g^{m_k}\\circ \\xi \\circ f^{-n_k})(B(p,\\rho ))\\big )\\approx 1.$ Since the sets $(g^{m_k}\\circ \\xi \\circ f^{-n_k})(B(p,\\rho ))$ all meet $\\mathcal {J}(g)$ , they stay in a compact part of $, and so we still get a uniform lower bound for the diameter of these sets if we apply the homeomorphism $ -1$; in other words,$$\\operatorname{diam}\\big ((\\xi ^{-1}\\circ g^{m_k}\\circ \\xi \\circ f^{-n_k})(B(p,\\rho ))\\big )=\\operatorname{diam}\\big (h_k(B(p,\\rho ))\\big )\\approx 1$$as claimed.", "We conclude that $ hkh$ locally uniformly on $ B(p, 2r)$, where $ h$ is non-constant and quasiregular.$ The quasiregular map $h$ has at most countably many critical points, and so there exists a point $q\\in B(p, 2\\rho )\\cap \\mathcal {J}(f)$ and a radius $r>0$ such that $B(q, 2r)\\subseteq B(p, 2\\rho )$ and $h$ is injective on $B(q, 2r)$ and hence quasiconformal.", "Standard topological degree arguments imply that at least on the smaller disk $B(q,r)$ the maps $h_k$ are also injective and hence quasiconformal for all $k$ sufficiently large.", "By possibly disregarding finitely of the maps $h_k$ , we may assume that $h_k$ is quasiconformal on $B(q,r)$ for all $k\\in {\\mathbb {N}}$ .", "To summarize, we have found a disk $B(q,r)$ centered at a point $q\\in \\mathcal {J}(f)$ such that the maps $h_k$ are defined and quasiconformal on $B(q,r)$ and converge uniformly on $B(q,r)$ to a quasiconformal map $h$ .", "From the invariance properties of Julia and Fatou sets and the mapping properties of $\\xi $ , it follows that $ h_k(B(q, r)\\cap \\mathcal {J}(f))\\subseteq \\mathcal {J}(f) \\text{ and } h_k(B(q, r)\\cap \\mathcal {F}(f))\\subseteq \\mathcal {F}(f)$ for each $k\\in {\\mathbb {N}}$ .", "Hence $h_k^{-1}(\\mathcal {J}(f))=\\mathcal {J}(f)\\cap B(q,r)$ for each map $h_k\\colon B(q,r)\\rightarrow , $ kN$.$ Since $\\mathcal {J}(f)$ is closed and $h_k\\rightarrow h$ uniformly on $B(q, r)$ , we also have $h(B(q, r)\\cap \\mathcal {J}(f))\\subseteq \\mathcal {J}(f)$ .", "To get a similar inclusion relation also for the Fatou set, we argue by contradiction and assume that there exists a point $z\\in B(q,r)\\cap \\mathcal {F}(f)$ with $h(z)\\notin \\mathcal {F}(f)$ .", "Then $h(z)\\in \\mathcal {J}(f)$ .", "Since $B(q,r)\\cap \\mathcal {F}(f)$ is an open neighborhood of $z$ , it follows again from standard topological degree arguments that for large enough $k\\in {\\mathbb {N}}$ there exists a point $z_k\\in B(q,r)\\cap \\mathcal {F}(f)$ with $h_k(z_k)=h(z)\\in \\mathcal {J}(f)$ .", "This is impossible by (REF ) and so indeed $h(B(q, r)\\cap \\mathcal {F}(f))\\subseteq \\mathcal {F}(f)$ .", "We conclude that $h^{-1}(\\mathcal {J}(f))=\\mathcal {J}(f)\\cap B(q,r).$ Step IV.", "We know by Theorem REF that the peripheral circles of $\\mathcal {J}(f)$ are uniformly relatively separated uniform quasicircles.", "According to Theorems REF and REF , there exists a quasisymmetric map $\\beta $ on $\\widehat{\\mathbb {C}}$ such that $S=\\beta (\\mathcal {J}(f))$ is a round Sierpiński carpet.", "We may assume $S\\subseteq .$ We conjugate the map $f$ by $\\beta $ to define a new map $\\beta \\circ f\\circ \\beta ^{-1}$ .", "By abuse of notation we call this new map also $f$ .", "Note that this map and its iterates are in general not rational anymore, but quasiregular maps on $\\widehat{\\mathbb {C}}$ .", "Similarly, we conjugate $ g_\\xi , h_k, h$ by $\\beta $ to obtain new maps for which we use the same notation for the moment.", "If $V=\\beta (B(q,r))$ , then the new maps $h_k$ and $h$ are quasiconformal on $V$ , and $h_k\\rightarrow h$ uniformly on $V$ .", "Lemma 7.1 There exist $N\\in {\\mathbb {N}}$ and an open set $W\\subseteq V$ such that $S\\cap W$ is non-empty and connected and $h_k\\equiv h$ on $S\\cap W$ for all $k\\ge N$ .", "Since $\\mathcal {J}(f)$ is porous and $S$ is a quasisymmetric image of $\\mathcal {J}(f)$ , the set $S$ is also porous (and in particular locally porous as defined in Section ).", "The maps $h_k$ and $h$ are quasiconformal on $V=\\beta (B(q,r))$ , and $h_k\\rightarrow h$ uniformly on $V$ as $k\\rightarrow \\infty $ .", "The relations (REF ) and (REF ) translate to $h^{-1}(S)=S\\cap V$ and $h_k^{-1}(S)=S\\cap V$ for $k\\in {\\mathbb {N}}$ .", "So Lemma REF implies that the maps $h\\colon S\\cap V\\rightarrow S$ and $h_k\\colon S \\cap V\\rightarrow S$ for $k\\in {\\mathbb {N}}$ are Schottky maps.", "Each of these restrictions is actually a homeomorphism onto its image.", "There are only finitely many peripheral circles of $\\mathcal {J}(f)$ that contain periodic points of our original rational map $f$ ; indeed, if $J$ is such a peripheral circle, then $f^n(J)=J$ for some $n\\in {\\mathbb {N}}$ as follows from Lemma REF ; but then $J$ bounds a periodic Fatou component of $f$ which leaves only finitely many possibilities for $J$ .", "Since the periodic points of $f$ are dense in $\\mathcal {J}(f)$ , we conclude that we can find a periodic point of $f$ in $ \\mathcal {J}(f)\\cap B(q,r)$ that does not lie on a peripheral circle of $\\mathcal {J}(f)$ .", "Translated to the conjugated map $f$ , this yields existence of a point $a\\in S\\cap V$ that does not lie on a peripheral circle of the Sierpiński carpet $S$ such that $f^n(a)=a$ for some $n\\in {\\mathbb {N}}$ .", "The invariance property of the Julia set gives $f^{-n}(S)=S$ , and so Lemma REF implies that $f^n\\colon S\\setminus \\operatorname{crit}(f^n)\\rightarrow S$ is a Schottky map.", "Note that $a\\notin \\operatorname{crit}(f^n)$ as follows from the fact that for our original rational map $f$ , none of its periodic critical points lies in the Julia set.", "Therefore, our Schottky map $f^n\\colon S\\setminus \\operatorname{crit}(f^n)\\rightarrow S$ has a derivative at the point $a\\in S\\setminus \\operatorname{crit}(f^n)$ in the sense of (REF ).", "If $(f^n)^{\\prime }(a)=1$ , then Theorem REF implies that $f^n\\equiv \\operatorname{id}|_{S\\setminus \\operatorname{crit}(f^n)}$ , and hence by continuity $f^n$ is the identity on $S$ .", "This is clearly impossible, and therefore $(f^n)^{\\prime }(a)\\ne 1$ .", "Since $a\\in S\\cap V$ does not lie on a peripheral circle of $S$ , as in the proof of Lemma REF we can find a small Jordan region $W$ with $a\\in W\\subseteq V$ and $W\\cap \\operatorname{crit}(f^n)=\\emptyset $ such that $\\partial W\\subseteq S$ .", "Then $S\\cap W$ is non-empty and connected.", "We now restrict our maps to $W$ .", "Then $h_k\\colon S\\cap W\\rightarrow S$ is a Schottky map and a homeomorphism onto its image for each $k\\in {\\mathbb {N}}$ .", "The same is true for the map $h\\colon S \\cap W\\rightarrow S$ .", "Moreover $h_k\\rightarrow h$ as $k\\rightarrow \\infty $ uniformly on $W\\cap S$ .", "Finally, the map $u=f^n$ is defined on $S\\cap W$ and gives a Schottky map $u\\colon S\\cap W\\rightarrow S$ such that for $a\\in S\\cap W$ we have $u(a)=a$ and $u^{\\prime }(a)\\ne 1$ .", "So we can apply Theorem REF to conclude that there exists $N\\in {\\mathbb {N}}$ such that $h_k\\equiv h$ on $S\\cap W$ for all $k\\ge N$ .", "By the previous lemma we can fix $k\\ge N$ so that $h_k=h_{k+1}$ on $S\\cap W$ .", "If we go back to the definition of the maps $h_k$ and use the consistency of inverse branches (which is also true for the maps conjugated by $\\beta $ ), then we conclude that $ h_{k+1}=g_\\xi ^{m_{k+1}}\\circ f^{-n_{k+1}}= h_k =g_\\xi ^{m_{k}}\\circ f^{-n_{k}}=g_\\xi ^{m_{k}}\\circ f^{n_{k+1}-n_k}\\circ f^{-n_{k+1}}$ on the set $S \\cap W$ .", "Cancellation gives $ g_\\xi ^{m_{k+1}}=g_\\xi ^{m_{k}}\\circ f^{n_{k+1}-n_k}$ on $f^{-n_{k+1}}(S\\cap W)\\subseteq S$ .", "The two maps on both sides of the last equation are quasiregular maps $\\psi \\colon \\widehat{\\mathbb {C}}\\rightarrow \\widehat{\\mathbb {C}}$ with $\\psi ^{-1}(S)=S$ .", "It follows from Lemma REF that they are Schottky maps $S \\cap U\\rightarrow S$ if $U\\subseteq \\widehat{\\mathbb {C}}$ is an open set that does not contain any of the finitely many critical points of the maps; in particular, $g_\\xi ^{m_{k+1}}$ and $g_\\xi ^{m_k}\\circ f^{n_{k+1}-n_k}$ are Schottky maps $S \\cap U\\rightarrow S$ , where $U=\\widehat{\\mathbb {C}}\\setminus (\\operatorname{crit}(g_\\xi ^{m_{k+1}})\\cup \\operatorname{crit}(g_\\xi ^{m_k}\\circ f^{n_{k+1}-n_k}))$ .", "Since $U$ has a finite complement in $\\widehat{\\mathbb {C}}$ , the non-degenerate connected set $f^{-n_{k+1}}(S\\cap W)\\subseteq S$ has an accumulation point in $S \\cap U$ .", "Theorem REF yields $ g_\\xi ^{m_{k+1}}=g_\\xi ^{m_{k}}\\circ f^{n_{k+1}-n_k}$ on $S\\cap U$ , and hence on all of $S$ by continuity.", "If we conjugate back by $\\beta ^{-1}$ , this leads to the relation $ g_\\xi ^{m_{k+1}}=g_\\xi ^{m_{k}}\\circ f^{n_{k+1}-n_k}$ on $\\mathcal {J}(f)$ for the original maps.", "We conclude that there exist integers $m,m^{\\prime },n\\in {\\mathbb {N}}$ such that for the original maps we have $ g^{m^{\\prime }}\\circ \\xi = g^m\\circ \\xi \\circ f^n$ on $\\mathcal {J}(f)$ .", "Step V. Equation (REF ) gives us a crucial relation of $\\xi $ to the dynamics of $f$ and $g$ on their Julia sets.", "We will bring (REF ) into a convenient form by replacing our original maps with iterates.", "Since $\\mathcal {J}(f)$ is backward invariant, counting preimages of generic points in $\\mathcal {J}(f)$ under iterates of $f$ and of points in $\\mathcal {J}(g)$ under iterates of $g$ leads to the relation $\\deg (g)^{m^{\\prime }-m}=\\deg (f)^n,$ and so $m^{\\prime }-m>0$ .", "If we post-compose both sides in (REF ) by a suitable iterate of $g$ , and then replace $f$ by $f^n$ and $g$ by $g^{m^{\\prime }-m}$ , we arrive at a relation of the form $ g^{l+1}\\circ \\xi = g^l\\circ \\xi \\circ f.$ on $\\mathcal {J}(f)$ for some $l\\in {\\mathbb {N}}$ .", "Note that this equation implies that we have $ g^{n+k}\\circ \\xi = g^n\\circ \\xi \\circ f^k \\ \\text{ for all $k,n\\in {\\mathbb {N}}$ with $n\\ge l$}.$ Step VI.", "In this final step of the proof, we disregard the non-canonical extension to $\\widehat{\\mathbb {C}}$ of our original homeomorphism $\\xi \\colon \\mathcal {J}(f)\\rightarrow \\mathcal {J}(g)$ chosen in the beginning.", "Our goal is to apply (REF ) to produce a natural extension of $\\xi $ mapping each Fatou component of $f$ conformally onto a Fatou component of $g$ .", "Note that if $U$ is a Fatou component of $f$ , then $\\partial U$ is a peripheral circle of $\\mathcal {J}(f)$ .", "Since $\\xi $ sends each peripheral circle of $\\mathcal {J}(f)$ to a peripheral circle of $\\mathcal {J}(g)$ , the image $\\xi (\\partial U)$ bounds a unique Fatou component $V$ of $g$ .", "This sets up a natural bijection between the Fatou components of our maps, and our goal is to conformally “fill in the holes\".", "So let $\\mathcal {C}_f$ and $\\mathcal {C}_g$ be the sets of Fatou components of $f$ and $g$ , respectively.", "By Lemma REF we can choose a corresponding family $\\lbrace \\psi _U: U\\in \\mathcal {C}_f\\rbrace $ of conformal maps.", "Since each Fatou component of $f$ is a Jordan region, we can consider $\\psi _U$ as a conformal homeomorphism from $\\overline{U}$ onto $\\overline{.", "Similarly, we obtain a family of conformal homeomorphisms \\tilde{\\psi }_V: \\overline{V}\\rightarrow \\overline{ for V in \\mathcal {C}_g.These Fatou components carry distinguished basepointsp_U=\\psi _U^{-1}(0)\\in U for U\\in \\mathcal {C}_f and \\tilde{p}_V=\\tilde{\\psi }_V^{-1}(0)\\in V for V\\in \\mathcal {C}_g.", "}We will now first extend \\xi to the periodic Fatou components of f, and thenuse the Lifting Lemma~\\ref {lem:lifting} to get extensions to Fatou components of higher and higher level (as defined in the proof of Lemma~\\ref {lem:FatouDyn}).", "In this argument it will be important to ensure that these extensions are basepoint-preserving.", "}First let $ U$ be a periodic Fatou component of $ f$.", "We denote by $ kN$ the period of $ U$, and define $ V$ to be the Fatou component of $ g$ bounded by $ (U)$, and$ W=gl(V)$, where $ lN$ is as in (\\ref {eq:main4}).Then(\\ref {eq:main4}) implies that $ W$, and hence $ W$ itself, is invariant under $ gk$.", "ByLemma~\\ref {lem:FatouDyn} the basepoint-preserving homeomorphisms $ U (U, pU)( , 0)$ and $ W(W,pW)( ,0)$ conjugate $ fk$ and $ gk$, respectively, to power maps.Since $ f$ and $ g$ are postcritically-finite, the periodic Fatou components $ U$ and $ W$ are superattracting, and thus the degrees of these power maps are at least 2.$ Again by Lemma REF the map $\\tilde{\\psi }_W\\circ g^l\\circ \\tilde{\\psi }^{-1}_V$ is a power map.", "Since $U,V,W$ are Jordan regions, the maps $\\psi _U, \\tilde{\\psi }_V, \\tilde{\\psi }_W$ give homeomorphisms between the boundaries of the corresponding Fatou components and $\\partial .", "Since $$ is anorientation-preserving homeomorphism of $ U$ onto $ V$, the map$ =VU-1$gives an orientation-preserving homeomorphism on $ .", "Now (REF ) for $n=l$ implies that on $\\partial we have{\\begin{@align*}{1}{-1} P_{d_3}\\circ \\phi &=\\tilde{\\psi }_W\\circ g^{k+l}\\circ \\tilde{\\psi }_V^{-1} \\circ \\phi = \\tilde{\\psi }_W\\circ g^{k+l}\\circ \\xi \\circ \\psi _U^{-1}\\\\&= \\tilde{\\psi }_W\\circ g^{l}\\circ \\xi \\circ f^k\\circ \\psi ^{-1}_U= \\tilde{\\psi }_W\\circ g^{l}\\circ \\xi \\circ \\psi ^{-1}_U \\circ P_{d_1}\\\\&= \\tilde{\\psi }_W\\circ g^{l}\\circ \\tilde{\\psi }^{-1}_V\\circ \\phi \\circ P_{d_1}= P_{d_2}\\circ \\phi \\circ P_{d_1}\\end{@align*}}for some $ d1, d 2, d3N$ with $ d12$.", "Lemma~\\ref {L:Rot} implies that $$ extends to $$ as a rotation around $ 0$, also denoted by $$.", "In particular, $ (0)=0$, and so $$ preserves the basepoint $ 0$ in $$.", "If we define $ =V-1U$ on $U$, then $$ is a conformal homeomorphismof $ (U,pU)$ onto $ (V, pV)$.$ In this way, we can conformally extend $\\xi $ to every periodic Fatou component of $f$ so that $\\xi $ maps the basepoint of a Fatou component to the basepoint of the image component.", "To get such an extension also for the other Fatou components $V$ of $f$ , we proceed inductively on the level of the Fatou component.", "So suppose that the level of $V$ is $\\ge 1$ and that we have already found an extension for all Fatou components with a level lower than $V$ .", "This applies to the Fatou component $U=f(V)$ of $f$ , and so a conformal extension $(U, p_U)\\rightarrow ( U^{\\prime }, \\tilde{p}_{U^{\\prime }})$ of $\\xi |_{\\partial U}$ exists, where $U^{\\prime }$ is the Fatou component of $g$ bounded by $\\xi (\\partial U)$ .", "Let $V^{\\prime }$ be the Fatou component of $g$ bounded by $\\xi (\\partial V)$ , and $W=g^{l+1}(V^{\\prime })$ .", "Then by using (REF ) on $\\partial V$ we conclude that $g^l(U^{\\prime })=W$ .", "Define $\\alpha =g^l\\circ \\xi |_{\\overline{U}}\\circ f|_{\\overline{V}}$ and $\\beta =\\xi |_{\\partial V}$ .", "Then the assumptions of Lemma REF are satisfied for $D=V$ , $p_D=p_{V}$ , and the iterate $g^{l+1}\\colon V^{\\prime }\\rightarrow W$ of $g$ .", "Indeed, $\\alpha $ is continuous on $\\overline{V}$ and holomorphic on $V$ , we have $\\alpha ^{-1}(p_W)=f^{-1}( \\xi |_{\\overline{U}}^{-1}(\\tilde{p}_{U^{\\prime }}))=f^{-1}(p_U)=\\lbrace p_V\\rbrace ,$ and $g^{l+1}\\circ \\beta = g^{l+1}\\circ \\xi |_{\\partial V}= g^{l}\\circ \\xi |_{\\partial U}\\circ f|_{\\partial V}=\\alpha .$ Since $\\beta $ is a homeomorphism, it follows that there exists a conformal homeomorphism $\\tilde{\\alpha }$ of $(\\overline{V}, p_V)$ onto $(\\overline{V}^{\\prime }, \\tilde{p}_{V^{\\prime }})$ such that $\\tilde{\\alpha }|_{\\partial V}=\\beta =\\xi |_{\\partial V}$ .", "In other words, $\\tilde{\\alpha }$ gives the desired basepoint preserving conformal extension to the Fatou component $V$ .", "This argument shows that $\\xi $ has a (unique) conformal extension to each Fatou component of $f$ .", "We know that the peripheral circles of $\\mathcal {J}(f)$ are uniform quasicircles, and that $\\mathcal {J}(f)$ has measure zero.", "Lemma REF now implies that $\\xi $ extends to a Möbius transformation on $\\widehat{\\mathbb {C}}$ , which completes the proof.", "The techniques discussed also easily lead to a proof of Corollary REF .", "Let $f$ be a postcritically-finite rational map whose Julia set $\\mathcal {J}(f)$ is a Sierpiński carpet.", "Let $G$ be the group of all Möbius transformations $\\xi $ on $\\widehat{\\mathbb {C}}$ with $\\xi (\\mathcal {J}(f))=\\mathcal {J}(f)$ , and $H$ be the subgroup of all elements in $G$ that preserve orientation.", "By Theorem REF it is enough to prove that $G$ is finite.", "Since $G=H$ or $H$ has index 2 in $G$ , this is true if we can show that $H$ is finite.", "Note that the group $H$ is discrete, i.e., there exists $\\delta _0>0$ such that $ \\sup _{p\\in \\widehat{\\mathbb {C}}} \\sigma (\\xi (p), p)\\ge \\delta _0$ for all $\\xi \\in H$ with $\\xi \\ne \\operatorname{id}_{\\widehat{\\mathbb {C}}}$ .", "Indeed, we choose $\\delta _0>0$ so small that there are at least three distinct complementary components $D_1,D_2,D_3$ of $\\mathcal {J}(f)$ that contain disks of radius $\\delta _0$ .", "In order to show (REF ), suppose that $\\xi \\in H$ and $\\sigma (\\xi (p), p)<\\delta _0$ for all $p\\in \\widehat{\\mathbb {C}}$ .", "Then $\\xi (D_i)\\cap D_i\\ne \\emptyset $ , and so $\\xi (\\overline{D}_i)=\\overline{D}_i$ for $i=1, 2,3$ , because $\\xi $ permutes the closures of Fatou components of $f$ .", "This shows that $\\xi $ is a conformal homeomorphism of the closed Jordan region $\\overline{D}_i$ onto itself.", "Hence $\\xi $ has a fixed point in $\\overline{D}_i$ .", "Since $\\mathcal {J}(f)$ is a Sierpiński carpet, the closures $\\overline{D}_1, \\overline{D}_2, \\overline{D}_3$ are pairwise disjoint, and we conclude that $\\xi $ has at least three fixed points.", "Since $\\xi $ is an orientation-preserving Möbius transformation, this implies that $\\xi =\\operatorname{id}_{\\widehat{\\mathbb {C}}}$ , and the discreteness of $H$ follows.", "We will now analyze type of Möbius transformations contained in the group $H$ (for the relevant classification of Möbius transformations up to conjugacy, see [2]).", "So consider an arbitrary $\\xi \\in H$ , $\\xi \\ne \\text{id}_{\\widehat{\\mathbb {C}}}$ .", "Then $\\xi $ cannot be loxodromic; indeed, otherwise $\\xi $ has a repelling fixed point $p$ which necessarily has to lie in $\\mathcal {J}(f)$ .", "We now argue as in the proof of Theorem REF and “blow down\" by iterates $\\xi ^{-n_k}$ near $p$ and “blow up\" by the conformal elevator using iterates $f^{m_k}$ to obtain a sequence of conformal maps of the form $h_k=f^{m_k}\\circ \\xi ^{-n_k}$ that converge uniformly to a (non-constant) conformal limit function $h$ on a disk $B$ centered at a point in $q\\in \\mathcal {J}(f)$ .", "Again this sequence stabilizes, and so $h_{k+1}=h_k$ for large $k$ on a connected non-degenerate subset of $B$ , and hence on $\\widehat{\\mathbb {C}}$ by the uniqueness theorem for analytic functions.", "This leads to a relation of the form $f^k\\circ \\xi ^l=f^m$ , where $k,l,m\\in {\\mathbb {N}}$ .", "Comparing degrees we get $m=k$ , and so we have $f^{k}=f^{k}\\circ \\xi ^{-ln}$ for all $n\\in {\\mathbb {N}}$ .", "This is impossible, because $f^k=f^{k}\\circ \\xi ^{-ln}\\rightarrow f^{k}(p)$ near $p$ as $n\\rightarrow \\infty $ , while $f^k$ is non-constant.", "The Möbius transformation $\\xi $ cannot be parabolic either; otherwise, after conjugation we may assume that $\\xi (z)=z+a$ with $a\\in , $ a0$.", "Then necessarily $ J(f)$.", "On the other hand, we know thatthe peripheral circles of $ J(f)$ are uniform quasicircles that occur on all locations and scales with respect to the chordal metric.", "Translated to the Euclidean metric near $$ this means that $ J(f)$ has complementary components $ D$ with $ D$ that contain Euclidean disks of arbitrarily large radius, and in particular of radius $ >|a|$; but then $$ cannot move $ D$ off itself, and so $ (D)=D$ for the translation $$.This is impossible.$ Finally, $\\xi $ can be elliptic; then after conjugation we have $\\xi (z)=az$ with $a\\in and $ |a|=1$, $ a1$.", "Since $ H$ is discrete,$ a$ must be a root of unity, and so $$is a torsion element of $ H$.$ We conclude that $H$ is a discrete group of Möbius transformations such that each element $\\xi \\in H$ with $\\xi \\ne \\text{id}_{\\widehat{\\mathbb {C}}}$ is a torsion element.", "It is well-known that such a group $H$ is finite (one can derive this from [2] in combination with the considerations in [2])." ], [ "Proof of Theorem ", "The proof will be given in several steps.", "Step I.", "We first fix the setup.", "We can freely pass to iterates of the maps $f$ or $g$ , because this changes neither their Julia sets nor their postcritical sets.", "We can also conjugate the maps by Möbius transformations.", "Therefore, as in Section , we may assume that $\\mathcal {J}(f), \\mathcal {J}(g)\\subseteq \\tfrac{1}{2} { and } f^{-1}( , g^{-1}(\\subseteq $ Moreover, without loss of generality, we may require that $\\xi $ is orientation-preserving, for otherwise we can conjugate $g$ by $z\\mapsto \\overline{z}$ .", "Since the peripheral circles of $\\mathcal {J}(f)$ and $\\mathcal {J}(g)$ are uniform quasicircles, by Theorem REF the map $\\xi $ extends (non-uniquely) to a quasiconformal, and hence quasisymmetric, map of the whole sphere.", "Then $\\xi (\\mathcal {J}(f))=\\mathcal {J}(g)$ and $\\xi (\\mathcal {F}(f))=\\mathcal {F}(g)$ .", "Since $\\infty $ lies in Fatou components of $f$ and $g$ , we may also assume that $\\xi (\\infty )=\\infty $ (this normalization ultimately depends on the fact that for every point $p\\in there exists a quasiconformal homeomorphism $$on $ C$ with $ (0)=p$ that is the identity outside $ ).", "Then $\\xi $ is a quasisymmetry of $ with respect to the Euclidean metric.", "In the following, all metric notions will refer to this metric.Finally, we define $ g=-1 g$.$ Step II.", "We now carefully choose a location for a “blow-down\" by branches of $f^{-n}$ which will be compensated by a “blow-up\" by iterates of $g$ (or rather $g_\\xi $ ).", "Since repelling periodic points of $f$ are dense in $\\mathcal {J}(f)$ (see [3]), we can find such a point $p$ in $\\mathcal {J}(f)$ that does not lie in $\\operatorname{post}(f)$ .", "Let $\\rho >0$ be a small positive number such that the disk $U_0:=B(p, 3\\rho )\\subseteq is disjoint from $ post(f)$.Since $ p$ is periodic, there exists $ dN$ such that $ fd(p)=p$.", "Let $ U1 be the component of $f^{-d}(U_0)$ that contains $p$ .", "Since $U_0\\cap \\operatorname{post}(f)=\\emptyset $ , the set $U_1$ is a simply connected region, and $f^{d}$ is a conformal map from $U_1$ onto $U_0$ as follows from Lemma REF .", "Then there exists a unique inverse branch $f^{-d}$ with $f^{-d}(p)=p$ that is a conformal map of $U_0$ onto $U_1$ .", "Since $p$ is a repelling fixed point for $f$ , it is an attracting fixed point for this branch $f^{-d}$ .", "By possibly choosing a smaller radius $\\rho >0$ in the definition of $U_0=B(p, 3\\rho )$ and by passing to an iterate of $f^d$ , we may assume that $U_1\\subseteq U_0$ and that $\\operatorname{diam}(f^{-n_k}(U_0))\\rightarrow 0$ as $k\\rightarrow \\infty $ .", "Here $n_k=dk$ for $k\\in {\\mathbb {N}}$ , and $f^{-n_k}$ is the branch obtained by iterating the branch $f^{-d}$ $k$ -times.", "Note that $f^{-n_k}(p)=p$ and $f^{-n_k}$ is a conformal map of $U_0$ onto a simply connected region $U_k$ .", "Then $p\\in U_k\\subseteq U_{k-1}$ for $k\\in {\\mathbb {N}}$ , and $\\operatorname{diam}(U_k)\\rightarrow 0$ as $k\\rightarrow \\infty $ .", "The choice of these inverse branches is consistent in the sense that we have $f^{n_{k+1}-n_k}\\circ f^{-n_{k+1}}=f^{-n_k}$ on $B(p, 3\\rho )$ for all $k\\in {\\mathbb {N}}$ .", "Note that this consistency condition remains valid if we replace the original sequence $\\lbrace n_k\\rbrace $ by a subsequence.", "Let $\\tilde{r}_k>0$ be the smallest number such that $f^{-n_k}(B(p, 2\\rho ))\\subseteq \\tilde{B}_k:=B(p,\\tilde{r}_k).$ Since $p\\in f^{-n_k}(B(p, 2\\rho ))$ we have $\\operatorname{diam}\\big (f^{-n_k}(B(p, 2\\rho ))\\big )\\approx \\tilde{r}_k$ .", "Here and below $\\approx $ indicates implicit positive multiplicative constants independent of $k\\in {\\mathbb {N}}$ .", "It follows that $\\tilde{r}_k\\rightarrow 0$ as $k\\rightarrow \\infty $ .", "Moreover, since $f^{-n_k}$ is conformal on the larger disk $B(p,3\\rho )$ , Koebe's distortion theorem implies that $ \\operatorname{diam}\\big (f^{-n_k}(B(p, 2\\rho ))\\big )\\approx \\operatorname{diam}\\big (f^{-n_k}(B(p, \\rho ))\\big ).", "$ Let $r_k>0$ be the smallest number such that $\\xi (\\tilde{B}_k)\\subseteq B_k:=B(\\xi (p), r_k)$ .", "Again $r_k\\rightarrow 0$ as $k\\rightarrow \\infty $ by continuity of $\\xi $ .", "We now want to apply the conformal elevator given by iterates of $g$ to the disks $B_k$ .", "For this we choose $\\epsilon _0>0$ for the map $g$ as in Section .", "By applying the conformal elevator as described in Section , we can find iterates $g^{m_k}$ such that $g^{m_k}(B_k)$ is blown up to a definite, but not too large size, and so $\\operatorname{diam}(g^{m_k}(B_k))\\approx 1$ .", "Step III.", "Now we consider the composition $h_k=g_\\xi ^{m_k}\\circ f^{-n_k}= \\xi ^{-1}\\circ g^{m_k}\\circ \\xi \\circ f^{-n_k}$ defined on $B(p,2\\rho )$ for $k\\in {\\mathbb {N}}$ .", "We want to show that this sequence subconverges locally uniformly on $B(p,2\\rho )$ to a (non-constant) quasiregular map $h\\colon B(p, 2\\rho )\\rightarrow .$ Since $f^{-n_k}$ maps $B(q,2\\rho )$ conformally into $\\tilde{B}_k$ , $\\xi $ is a quasiconformal map with $\\xi (\\tilde{B}_k)\\subseteq B_k$ , and $g^{m_k}$ is holomorphic on $\\tilde{B}_k$ , we conclude that the maps $h_k$ are uniformly quasiregular on $B(q,2\\rho )$ , i.e., $K$ -quasiregular with $K\\ge 1$ independent of $k$ .", "The images $h_k(B(q,2\\rho ))$ are contained in a small Euclidean neighborhood of $\\mathcal {J}(g)$ and hence in a fixed compact subset of $.", "Standardconvergence results for $ K$-quasiregular mappings \\cite [p.~182, Corollary 5.5.7]{AIM} imply that the sequence $ {hk}$subconverges locally uniformly on $ B(q,2)$ to a map $ hB(q,2) that is also quasiregular, but possibly constant.", "By passing to a subsequence if necessary, we may assume that $h_k\\rightarrow h$ locally uniformly on $B(q,2\\rho )$ .", "To rule out that $h$ is constant, it is enough to show that for smaller disk $B(p,\\rho )$ there exists $\\delta >0$ such that $\\operatorname{diam}\\big (h_k(B(p,\\rho ))\\big )\\ge \\delta $ for all $k\\in {\\mathbb {N}}$ .", "We know that $\\operatorname{diam}\\big (f^{-n_k}(B(p,\\rho ))\\big )\\approx \\operatorname{diam}\\big (f^{-n_k}(B(p,2\\rho ))\\big ) \\approx \\operatorname{diam}(\\tilde{B}_k).", "$ Moreover, since $\\xi $ is a quasisymmetry and $f^{-n_k}(B(p,\\rho ))\\subseteq \\tilde{B}_k$ , this implies $\\operatorname{diam}\\big (\\xi (f^{-n_k}(B(p,\\rho )))\\big )\\approx \\operatorname{diam}(\\xi (\\tilde{B}_k))\\approx \\operatorname{diam}(B_k).", "$ So the connected set $\\xi (f^{-n_k}(B(p,\\rho )))\\subseteq B_k$ is comparable in size to $B_k$ .", "By Lemma REF  (a) the conformal elevator blows it up to a definite, but not too large size, i.e., $\\operatorname{diam}\\big ((g^{m_k}\\circ \\xi \\circ f^{-n_k})(B(p,\\rho ))\\big )\\approx 1.$ Since the sets $(g^{m_k}\\circ \\xi \\circ f^{-n_k})(B(p,\\rho ))$ all meet $\\mathcal {J}(g)$ , they stay in a compact part of $, and so we still get a uniform lower bound for the diameter of these sets if we apply the homeomorphism $ -1$; in other words,$$\\operatorname{diam}\\big ((\\xi ^{-1}\\circ g^{m_k}\\circ \\xi \\circ f^{-n_k})(B(p,\\rho ))\\big )=\\operatorname{diam}\\big (h_k(B(p,\\rho ))\\big )\\approx 1$$as claimed.", "We conclude that $ hkh$ locally uniformly on $ B(p, 2r)$, where $ h$ is non-constant and quasiregular.$ The quasiregular map $h$ has at most countably many critical points, and so there exists a point $q\\in B(p, 2\\rho )\\cap \\mathcal {J}(f)$ and a radius $r>0$ such that $B(q, 2r)\\subseteq B(p, 2\\rho )$ and $h$ is injective on $B(q, 2r)$ and hence quasiconformal.", "Standard topological degree arguments imply that at least on the smaller disk $B(q,r)$ the maps $h_k$ are also injective and hence quasiconformal for all $k$ sufficiently large.", "By possibly disregarding finitely of the maps $h_k$ , we may assume that $h_k$ is quasiconformal on $B(q,r)$ for all $k\\in {\\mathbb {N}}$ .", "To summarize, we have found a disk $B(q,r)$ centered at a point $q\\in \\mathcal {J}(f)$ such that the maps $h_k$ are defined and quasiconformal on $B(q,r)$ and converge uniformly on $B(q,r)$ to a quasiconformal map $h$ .", "From the invariance properties of Julia and Fatou sets and the mapping properties of $\\xi $ , it follows that $ h_k(B(q, r)\\cap \\mathcal {J}(f))\\subseteq \\mathcal {J}(f) \\text{ and } h_k(B(q, r)\\cap \\mathcal {F}(f))\\subseteq \\mathcal {F}(f)$ for each $k\\in {\\mathbb {N}}$ .", "Hence $h_k^{-1}(\\mathcal {J}(f))=\\mathcal {J}(f)\\cap B(q,r)$ for each map $h_k\\colon B(q,r)\\rightarrow , $ kN$.$ Since $\\mathcal {J}(f)$ is closed and $h_k\\rightarrow h$ uniformly on $B(q, r)$ , we also have $h(B(q, r)\\cap \\mathcal {J}(f))\\subseteq \\mathcal {J}(f)$ .", "To get a similar inclusion relation also for the Fatou set, we argue by contradiction and assume that there exists a point $z\\in B(q,r)\\cap \\mathcal {F}(f)$ with $h(z)\\notin \\mathcal {F}(f)$ .", "Then $h(z)\\in \\mathcal {J}(f)$ .", "Since $B(q,r)\\cap \\mathcal {F}(f)$ is an open neighborhood of $z$ , it follows again from standard topological degree arguments that for large enough $k\\in {\\mathbb {N}}$ there exists a point $z_k\\in B(q,r)\\cap \\mathcal {F}(f)$ with $h_k(z_k)=h(z)\\in \\mathcal {J}(f)$ .", "This is impossible by (REF ) and so indeed $h(B(q, r)\\cap \\mathcal {F}(f))\\subseteq \\mathcal {F}(f)$ .", "We conclude that $h^{-1}(\\mathcal {J}(f))=\\mathcal {J}(f)\\cap B(q,r).$ Step IV.", "We know by Theorem REF that the peripheral circles of $\\mathcal {J}(f)$ are uniformly relatively separated uniform quasicircles.", "According to Theorems REF and REF , there exists a quasisymmetric map $\\beta $ on $\\widehat{\\mathbb {C}}$ such that $S=\\beta (\\mathcal {J}(f))$ is a round Sierpiński carpet.", "We may assume $S\\subseteq .$ We conjugate the map $f$ by $\\beta $ to define a new map $\\beta \\circ f\\circ \\beta ^{-1}$ .", "By abuse of notation we call this new map also $f$ .", "Note that this map and its iterates are in general not rational anymore, but quasiregular maps on $\\widehat{\\mathbb {C}}$ .", "Similarly, we conjugate $ g_\\xi , h_k, h$ by $\\beta $ to obtain new maps for which we use the same notation for the moment.", "If $V=\\beta (B(q,r))$ , then the new maps $h_k$ and $h$ are quasiconformal on $V$ , and $h_k\\rightarrow h$ uniformly on $V$ .", "Lemma 7.1 There exist $N\\in {\\mathbb {N}}$ and an open set $W\\subseteq V$ such that $S\\cap W$ is non-empty and connected and $h_k\\equiv h$ on $S\\cap W$ for all $k\\ge N$ .", "Since $\\mathcal {J}(f)$ is porous and $S$ is a quasisymmetric image of $\\mathcal {J}(f)$ , the set $S$ is also porous (and in particular locally porous as defined in Section ).", "The maps $h_k$ and $h$ are quasiconformal on $V=\\beta (B(q,r))$ , and $h_k\\rightarrow h$ uniformly on $V$ as $k\\rightarrow \\infty $ .", "The relations (REF ) and (REF ) translate to $h^{-1}(S)=S\\cap V$ and $h_k^{-1}(S)=S\\cap V$ for $k\\in {\\mathbb {N}}$ .", "So Lemma REF implies that the maps $h\\colon S\\cap V\\rightarrow S$ and $h_k\\colon S \\cap V\\rightarrow S$ for $k\\in {\\mathbb {N}}$ are Schottky maps.", "Each of these restrictions is actually a homeomorphism onto its image.", "There are only finitely many peripheral circles of $\\mathcal {J}(f)$ that contain periodic points of our original rational map $f$ ; indeed, if $J$ is such a peripheral circle, then $f^n(J)=J$ for some $n\\in {\\mathbb {N}}$ as follows from Lemma REF ; but then $J$ bounds a periodic Fatou component of $f$ which leaves only finitely many possibilities for $J$ .", "Since the periodic points of $f$ are dense in $\\mathcal {J}(f)$ , we conclude that we can find a periodic point of $f$ in $ \\mathcal {J}(f)\\cap B(q,r)$ that does not lie on a peripheral circle of $\\mathcal {J}(f)$ .", "Translated to the conjugated map $f$ , this yields existence of a point $a\\in S\\cap V$ that does not lie on a peripheral circle of the Sierpiński carpet $S$ such that $f^n(a)=a$ for some $n\\in {\\mathbb {N}}$ .", "The invariance property of the Julia set gives $f^{-n}(S)=S$ , and so Lemma REF implies that $f^n\\colon S\\setminus \\operatorname{crit}(f^n)\\rightarrow S$ is a Schottky map.", "Note that $a\\notin \\operatorname{crit}(f^n)$ as follows from the fact that for our original rational map $f$ , none of its periodic critical points lies in the Julia set.", "Therefore, our Schottky map $f^n\\colon S\\setminus \\operatorname{crit}(f^n)\\rightarrow S$ has a derivative at the point $a\\in S\\setminus \\operatorname{crit}(f^n)$ in the sense of (REF ).", "If $(f^n)^{\\prime }(a)=1$ , then Theorem REF implies that $f^n\\equiv \\operatorname{id}|_{S\\setminus \\operatorname{crit}(f^n)}$ , and hence by continuity $f^n$ is the identity on $S$ .", "This is clearly impossible, and therefore $(f^n)^{\\prime }(a)\\ne 1$ .", "Since $a\\in S\\cap V$ does not lie on a peripheral circle of $S$ , as in the proof of Lemma REF we can find a small Jordan region $W$ with $a\\in W\\subseteq V$ and $W\\cap \\operatorname{crit}(f^n)=\\emptyset $ such that $\\partial W\\subseteq S$ .", "Then $S\\cap W$ is non-empty and connected.", "We now restrict our maps to $W$ .", "Then $h_k\\colon S\\cap W\\rightarrow S$ is a Schottky map and a homeomorphism onto its image for each $k\\in {\\mathbb {N}}$ .", "The same is true for the map $h\\colon S \\cap W\\rightarrow S$ .", "Moreover $h_k\\rightarrow h$ as $k\\rightarrow \\infty $ uniformly on $W\\cap S$ .", "Finally, the map $u=f^n$ is defined on $S\\cap W$ and gives a Schottky map $u\\colon S\\cap W\\rightarrow S$ such that for $a\\in S\\cap W$ we have $u(a)=a$ and $u^{\\prime }(a)\\ne 1$ .", "So we can apply Theorem REF to conclude that there exists $N\\in {\\mathbb {N}}$ such that $h_k\\equiv h$ on $S\\cap W$ for all $k\\ge N$ .", "By the previous lemma we can fix $k\\ge N$ so that $h_k=h_{k+1}$ on $S\\cap W$ .", "If we go back to the definition of the maps $h_k$ and use the consistency of inverse branches (which is also true for the maps conjugated by $\\beta $ ), then we conclude that $ h_{k+1}=g_\\xi ^{m_{k+1}}\\circ f^{-n_{k+1}}= h_k =g_\\xi ^{m_{k}}\\circ f^{-n_{k}}=g_\\xi ^{m_{k}}\\circ f^{n_{k+1}-n_k}\\circ f^{-n_{k+1}}$ on the set $S \\cap W$ .", "Cancellation gives $ g_\\xi ^{m_{k+1}}=g_\\xi ^{m_{k}}\\circ f^{n_{k+1}-n_k}$ on $f^{-n_{k+1}}(S\\cap W)\\subseteq S$ .", "The two maps on both sides of the last equation are quasiregular maps $\\psi \\colon \\widehat{\\mathbb {C}}\\rightarrow \\widehat{\\mathbb {C}}$ with $\\psi ^{-1}(S)=S$ .", "It follows from Lemma REF that they are Schottky maps $S \\cap U\\rightarrow S$ if $U\\subseteq \\widehat{\\mathbb {C}}$ is an open set that does not contain any of the finitely many critical points of the maps; in particular, $g_\\xi ^{m_{k+1}}$ and $g_\\xi ^{m_k}\\circ f^{n_{k+1}-n_k}$ are Schottky maps $S \\cap U\\rightarrow S$ , where $U=\\widehat{\\mathbb {C}}\\setminus (\\operatorname{crit}(g_\\xi ^{m_{k+1}})\\cup \\operatorname{crit}(g_\\xi ^{m_k}\\circ f^{n_{k+1}-n_k}))$ .", "Since $U$ has a finite complement in $\\widehat{\\mathbb {C}}$ , the non-degenerate connected set $f^{-n_{k+1}}(S\\cap W)\\subseteq S$ has an accumulation point in $S \\cap U$ .", "Theorem REF yields $ g_\\xi ^{m_{k+1}}=g_\\xi ^{m_{k}}\\circ f^{n_{k+1}-n_k}$ on $S\\cap U$ , and hence on all of $S$ by continuity.", "If we conjugate back by $\\beta ^{-1}$ , this leads to the relation $ g_\\xi ^{m_{k+1}}=g_\\xi ^{m_{k}}\\circ f^{n_{k+1}-n_k}$ on $\\mathcal {J}(f)$ for the original maps.", "We conclude that there exist integers $m,m^{\\prime },n\\in {\\mathbb {N}}$ such that for the original maps we have $ g^{m^{\\prime }}\\circ \\xi = g^m\\circ \\xi \\circ f^n$ on $\\mathcal {J}(f)$ .", "Step V. Equation (REF ) gives us a crucial relation of $\\xi $ to the dynamics of $f$ and $g$ on their Julia sets.", "We will bring (REF ) into a convenient form by replacing our original maps with iterates.", "Since $\\mathcal {J}(f)$ is backward invariant, counting preimages of generic points in $\\mathcal {J}(f)$ under iterates of $f$ and of points in $\\mathcal {J}(g)$ under iterates of $g$ leads to the relation $\\deg (g)^{m^{\\prime }-m}=\\deg (f)^n,$ and so $m^{\\prime }-m>0$ .", "If we post-compose both sides in (REF ) by a suitable iterate of $g$ , and then replace $f$ by $f^n$ and $g$ by $g^{m^{\\prime }-m}$ , we arrive at a relation of the form $ g^{l+1}\\circ \\xi = g^l\\circ \\xi \\circ f.$ on $\\mathcal {J}(f)$ for some $l\\in {\\mathbb {N}}$ .", "Note that this equation implies that we have $ g^{n+k}\\circ \\xi = g^n\\circ \\xi \\circ f^k \\ \\text{ for all $k,n\\in {\\mathbb {N}}$ with $n\\ge l$}.$ Step VI.", "In this final step of the proof, we disregard the non-canonical extension to $\\widehat{\\mathbb {C}}$ of our original homeomorphism $\\xi \\colon \\mathcal {J}(f)\\rightarrow \\mathcal {J}(g)$ chosen in the beginning.", "Our goal is to apply (REF ) to produce a natural extension of $\\xi $ mapping each Fatou component of $f$ conformally onto a Fatou component of $g$ .", "Note that if $U$ is a Fatou component of $f$ , then $\\partial U$ is a peripheral circle of $\\mathcal {J}(f)$ .", "Since $\\xi $ sends each peripheral circle of $\\mathcal {J}(f)$ to a peripheral circle of $\\mathcal {J}(g)$ , the image $\\xi (\\partial U)$ bounds a unique Fatou component $V$ of $g$ .", "This sets up a natural bijection between the Fatou components of our maps, and our goal is to conformally “fill in the holes\".", "So let $\\mathcal {C}_f$ and $\\mathcal {C}_g$ be the sets of Fatou components of $f$ and $g$ , respectively.", "By Lemma REF we can choose a corresponding family $\\lbrace \\psi _U: U\\in \\mathcal {C}_f\\rbrace $ of conformal maps.", "Since each Fatou component of $f$ is a Jordan region, we can consider $\\psi _U$ as a conformal homeomorphism from $\\overline{U}$ onto $\\overline{.", "Similarly, we obtain a family of conformal homeomorphisms \\tilde{\\psi }_V: \\overline{V}\\rightarrow \\overline{ for V in \\mathcal {C}_g.These Fatou components carry distinguished basepointsp_U=\\psi _U^{-1}(0)\\in U for U\\in \\mathcal {C}_f and \\tilde{p}_V=\\tilde{\\psi }_V^{-1}(0)\\in V for V\\in \\mathcal {C}_g.", "}We will now first extend \\xi to the periodic Fatou components of f, and thenuse the Lifting Lemma~\\ref {lem:lifting} to get extensions to Fatou components of higher and higher level (as defined in the proof of Lemma~\\ref {lem:FatouDyn}).", "In this argument it will be important to ensure that these extensions are basepoint-preserving.", "}First let $ U$ be a periodic Fatou component of $ f$.", "We denote by $ kN$ the period of $ U$, and define $ V$ to be the Fatou component of $ g$ bounded by $ (U)$, and$ W=gl(V)$, where $ lN$ is as in (\\ref {eq:main4}).Then(\\ref {eq:main4}) implies that $ W$, and hence $ W$ itself, is invariant under $ gk$.", "ByLemma~\\ref {lem:FatouDyn} the basepoint-preserving homeomorphisms $ U (U, pU)( , 0)$ and $ W(W,pW)( ,0)$ conjugate $ fk$ and $ gk$, respectively, to power maps.Since $ f$ and $ g$ are postcritically-finite, the periodic Fatou components $ U$ and $ W$ are superattracting, and thus the degrees of these power maps are at least 2.$ Again by Lemma REF the map $\\tilde{\\psi }_W\\circ g^l\\circ \\tilde{\\psi }^{-1}_V$ is a power map.", "Since $U,V,W$ are Jordan regions, the maps $\\psi _U, \\tilde{\\psi }_V, \\tilde{\\psi }_W$ give homeomorphisms between the boundaries of the corresponding Fatou components and $\\partial .", "Since $$ is anorientation-preserving homeomorphism of $ U$ onto $ V$, the map$ =VU-1$gives an orientation-preserving homeomorphism on $ .", "Now (REF ) for $n=l$ implies that on $\\partial we have{\\begin{@align*}{1}{-1} P_{d_3}\\circ \\phi &=\\tilde{\\psi }_W\\circ g^{k+l}\\circ \\tilde{\\psi }_V^{-1} \\circ \\phi = \\tilde{\\psi }_W\\circ g^{k+l}\\circ \\xi \\circ \\psi _U^{-1}\\\\&= \\tilde{\\psi }_W\\circ g^{l}\\circ \\xi \\circ f^k\\circ \\psi ^{-1}_U= \\tilde{\\psi }_W\\circ g^{l}\\circ \\xi \\circ \\psi ^{-1}_U \\circ P_{d_1}\\\\&= \\tilde{\\psi }_W\\circ g^{l}\\circ \\tilde{\\psi }^{-1}_V\\circ \\phi \\circ P_{d_1}= P_{d_2}\\circ \\phi \\circ P_{d_1}\\end{@align*}}for some $ d1, d 2, d3N$ with $ d12$.", "Lemma~\\ref {L:Rot} implies that $$ extends to $$ as a rotation around $ 0$, also denoted by $$.", "In particular, $ (0)=0$, and so $$ preserves the basepoint $ 0$ in $$.", "If we define $ =V-1U$ on $U$, then $$ is a conformal homeomorphismof $ (U,pU)$ onto $ (V, pV)$.$ In this way, we can conformally extend $\\xi $ to every periodic Fatou component of $f$ so that $\\xi $ maps the basepoint of a Fatou component to the basepoint of the image component.", "To get such an extension also for the other Fatou components $V$ of $f$ , we proceed inductively on the level of the Fatou component.", "So suppose that the level of $V$ is $\\ge 1$ and that we have already found an extension for all Fatou components with a level lower than $V$ .", "This applies to the Fatou component $U=f(V)$ of $f$ , and so a conformal extension $(U, p_U)\\rightarrow ( U^{\\prime }, \\tilde{p}_{U^{\\prime }})$ of $\\xi |_{\\partial U}$ exists, where $U^{\\prime }$ is the Fatou component of $g$ bounded by $\\xi (\\partial U)$ .", "Let $V^{\\prime }$ be the Fatou component of $g$ bounded by $\\xi (\\partial V)$ , and $W=g^{l+1}(V^{\\prime })$ .", "Then by using (REF ) on $\\partial V$ we conclude that $g^l(U^{\\prime })=W$ .", "Define $\\alpha =g^l\\circ \\xi |_{\\overline{U}}\\circ f|_{\\overline{V}}$ and $\\beta =\\xi |_{\\partial V}$ .", "Then the assumptions of Lemma REF are satisfied for $D=V$ , $p_D=p_{V}$ , and the iterate $g^{l+1}\\colon V^{\\prime }\\rightarrow W$ of $g$ .", "Indeed, $\\alpha $ is continuous on $\\overline{V}$ and holomorphic on $V$ , we have $\\alpha ^{-1}(p_W)=f^{-1}( \\xi |_{\\overline{U}}^{-1}(\\tilde{p}_{U^{\\prime }}))=f^{-1}(p_U)=\\lbrace p_V\\rbrace ,$ and $g^{l+1}\\circ \\beta = g^{l+1}\\circ \\xi |_{\\partial V}= g^{l}\\circ \\xi |_{\\partial U}\\circ f|_{\\partial V}=\\alpha .$ Since $\\beta $ is a homeomorphism, it follows that there exists a conformal homeomorphism $\\tilde{\\alpha }$ of $(\\overline{V}, p_V)$ onto $(\\overline{V}^{\\prime }, \\tilde{p}_{V^{\\prime }})$ such that $\\tilde{\\alpha }|_{\\partial V}=\\beta =\\xi |_{\\partial V}$ .", "In other words, $\\tilde{\\alpha }$ gives the desired basepoint preserving conformal extension to the Fatou component $V$ .", "This argument shows that $\\xi $ has a (unique) conformal extension to each Fatou component of $f$ .", "We know that the peripheral circles of $\\mathcal {J}(f)$ are uniform quasicircles, and that $\\mathcal {J}(f)$ has measure zero.", "Lemma REF now implies that $\\xi $ extends to a Möbius transformation on $\\widehat{\\mathbb {C}}$ , which completes the proof.", "The techniques discussed also easily lead to a proof of Corollary REF .", "Let $f$ be a postcritically-finite rational map whose Julia set $\\mathcal {J}(f)$ is a Sierpiński carpet.", "Let $G$ be the group of all Möbius transformations $\\xi $ on $\\widehat{\\mathbb {C}}$ with $\\xi (\\mathcal {J}(f))=\\mathcal {J}(f)$ , and $H$ be the subgroup of all elements in $G$ that preserve orientation.", "By Theorem REF it is enough to prove that $G$ is finite.", "Since $G=H$ or $H$ has index 2 in $G$ , this is true if we can show that $H$ is finite.", "Note that the group $H$ is discrete, i.e., there exists $\\delta _0>0$ such that $ \\sup _{p\\in \\widehat{\\mathbb {C}}} \\sigma (\\xi (p), p)\\ge \\delta _0$ for all $\\xi \\in H$ with $\\xi \\ne \\operatorname{id}_{\\widehat{\\mathbb {C}}}$ .", "Indeed, we choose $\\delta _0>0$ so small that there are at least three distinct complementary components $D_1,D_2,D_3$ of $\\mathcal {J}(f)$ that contain disks of radius $\\delta _0$ .", "In order to show (REF ), suppose that $\\xi \\in H$ and $\\sigma (\\xi (p), p)<\\delta _0$ for all $p\\in \\widehat{\\mathbb {C}}$ .", "Then $\\xi (D_i)\\cap D_i\\ne \\emptyset $ , and so $\\xi (\\overline{D}_i)=\\overline{D}_i$ for $i=1, 2,3$ , because $\\xi $ permutes the closures of Fatou components of $f$ .", "This shows that $\\xi $ is a conformal homeomorphism of the closed Jordan region $\\overline{D}_i$ onto itself.", "Hence $\\xi $ has a fixed point in $\\overline{D}_i$ .", "Since $\\mathcal {J}(f)$ is a Sierpiński carpet, the closures $\\overline{D}_1, \\overline{D}_2, \\overline{D}_3$ are pairwise disjoint, and we conclude that $\\xi $ has at least three fixed points.", "Since $\\xi $ is an orientation-preserving Möbius transformation, this implies that $\\xi =\\operatorname{id}_{\\widehat{\\mathbb {C}}}$ , and the discreteness of $H$ follows.", "We will now analyze type of Möbius transformations contained in the group $H$ (for the relevant classification of Möbius transformations up to conjugacy, see [2]).", "So consider an arbitrary $\\xi \\in H$ , $\\xi \\ne \\text{id}_{\\widehat{\\mathbb {C}}}$ .", "Then $\\xi $ cannot be loxodromic; indeed, otherwise $\\xi $ has a repelling fixed point $p$ which necessarily has to lie in $\\mathcal {J}(f)$ .", "We now argue as in the proof of Theorem REF and “blow down\" by iterates $\\xi ^{-n_k}$ near $p$ and “blow up\" by the conformal elevator using iterates $f^{m_k}$ to obtain a sequence of conformal maps of the form $h_k=f^{m_k}\\circ \\xi ^{-n_k}$ that converge uniformly to a (non-constant) conformal limit function $h$ on a disk $B$ centered at a point in $q\\in \\mathcal {J}(f)$ .", "Again this sequence stabilizes, and so $h_{k+1}=h_k$ for large $k$ on a connected non-degenerate subset of $B$ , and hence on $\\widehat{\\mathbb {C}}$ by the uniqueness theorem for analytic functions.", "This leads to a relation of the form $f^k\\circ \\xi ^l=f^m$ , where $k,l,m\\in {\\mathbb {N}}$ .", "Comparing degrees we get $m=k$ , and so we have $f^{k}=f^{k}\\circ \\xi ^{-ln}$ for all $n\\in {\\mathbb {N}}$ .", "This is impossible, because $f^k=f^{k}\\circ \\xi ^{-ln}\\rightarrow f^{k}(p)$ near $p$ as $n\\rightarrow \\infty $ , while $f^k$ is non-constant.", "The Möbius transformation $\\xi $ cannot be parabolic either; otherwise, after conjugation we may assume that $\\xi (z)=z+a$ with $a\\in , $ a0$.", "Then necessarily $ J(f)$.", "On the other hand, we know thatthe peripheral circles of $ J(f)$ are uniform quasicircles that occur on all locations and scales with respect to the chordal metric.", "Translated to the Euclidean metric near $$ this means that $ J(f)$ has complementary components $ D$ with $ D$ that contain Euclidean disks of arbitrarily large radius, and in particular of radius $ >|a|$; but then $$ cannot move $ D$ off itself, and so $ (D)=D$ for the translation $$.This is impossible.$ Finally, $\\xi $ can be elliptic; then after conjugation we have $\\xi (z)=az$ with $a\\in and $ |a|=1$, $ a1$.", "Since $ H$ is discrete,$ a$ must be a root of unity, and so $$is a torsion element of $ H$.$ We conclude that $H$ is a discrete group of Möbius transformations such that each element $\\xi \\in H$ with $\\xi \\ne \\text{id}_{\\widehat{\\mathbb {C}}}$ is a torsion element.", "It is well-known that such a group $H$ is finite (one can derive this from [2] in combination with the considerations in [2])." ] ]
1403.0392
[ [ "Skew-symmetric matrices and their principal minors" ], [ "Abstract Let $V$ be a nonempty finite set and $A=(a_{ij})_{i,j\\in V}$ be a matrix with entries in a field $\\mathbb{K}$.", "For a subset $X$ of $V$, we denote by $A[X]$ the submatrix of $A$ having row and column indices in $X$.", "We study the following problem.", "Given a positive integer $k$, what is the relationship between two matrices $A=(a_{ij})_{i,j\\in V}$, $B=(b_{ij})_{i,j\\in V}$ with entries in $\\mathbb{K}$ and such that $\\det(A\\left[ X\\right])=\\det(B\\left[ X\\right])$ for any subset $X$ of $V$ of size at most $k$ ?", "The Theorem that we get in this Note is an improvement of a result of R. Loewy [5] for skew-symmetric matrices whose all off-diagonal entries are nonzero." ], [ "Introduction", "Our original motivation comes from the following open problem, called the Principal Minors Assignement Problem ( PMAP for short) (see [9]).", "Problem 1 Find, a necessary and sufficient conditions for a collection of $2^{n}$ numbers to arise as the principal minors of a matrix of order $n$ ?", "The PMAP has attracted some attention in recent years.", "O. Holtz and B. Strumfels [10] approched this problem algebraically and showed that a real vector of length $2^{n}$ , assuming it strictly satisfies the Hadamard-Fischer inequalities, is the list of principal minors of some real symmetric matrix if and only if it satisfies a certain system of polynomial equations.", "L. Oeding [14] later proved a more general conjecture of O. Holtz and B. Strumfels [10], removing the Hadamard-Fischer assumption and set-theoretically characterizing the variety of principal minors of symmetric matrices.", "Griffin and Tsatsomeros gave an algorithmic solution to the PMAP [6], [7].", "Their work gives an algorithm, which, under a certain “genericity” condition, either outputs a solution matrix or determines that none exists.", "Very recently, J.", "Rising, A. Kulesza, B. Taskar [15] gave an algorithm to solve the PMAP for the symmetric case.", "For the general case, the algebraic relations between the principal minors of a generic matrix of order $n$ are somewhat mysterious.", "S. Lin and B. Sturmfels [12] proved that the ideal of all polynomial relations among the principal minors of an arbitrary matrix of order 4 is minimally generated by 65 polynomials of degree 12.", "R. Kenyon and R. Pemantle [11] showed that by adding in certain 'almost' principal minors, the ideal of relations is generated by translations of a single relation.", "A natural problem in connection with the PMAP is the following.", "Problem 2 What is the relationship between two matrices having equal corresponding principal minors of all orders ?", "Given two matrices $A$ , $B$ of order $n$ with entries in a field $\\mathbb {K}$ , we say that $A$ , $B$ are diagonally similar up to transposition if there exist a nonsingular diagonal matrix $D$ such that $B=D^{-1}AD$ or $B^{t}=D^{-1}AD$ (where $B^{t}$ is the transpose of $B$ ).", "Clearly diagonal similarity up to transposition preserve all principal minors.", "D. J. Hartfiel and R. Loewy [8], and then R. Lowey [13] found sufficient conditions under which diagonal similarity up to transposition is precisely the relationship that must exist between two matrices having equal corresponding principal minors.", "To state the main theorem of [13], we need the following notations.", "Let $V$ be a nonempty finite set and $A=(a_{ij})_{i,j\\in V}$ be a matrix with entries in a field $\\mathbb {K}$ and having row and column indices in $V$ .", "For two nonempty subsets $X$ , $Y$ of $V $ , we denote by $A\\left[ X,Y\\right] $ the submatrix of $A$ having row indices in $X$ and column indices in $Y$ .", "The submatrix $A[X,X]$ is denoted simply by $A[X]$ .", "The matrix $A$ is irreducible if for any proper subset $X$ of $V$ , the two matrices $A[X,V\\setminus X]$ and $A[V\\setminus X,X]$ are nonzero.", "The matrix $A$ is HL-indecomposable (HL after D. J. Hartfiel and R. Lowey) if for any subset $X$ of $V$ such that $2\\le |X|\\le |V|-2$ , either $A[X,V\\setminus X]$ or $A[V\\setminus X,X]$ is of rank at least 2.", "Otherwise, it is HL-decomposable.", "The main theorem of R. Loewy [13] can be stated as follows.", "Theorem 3 Let $V$ be a nonempty set of size at least 4 and $A=(a_{ij})_{i,j\\in V}$ , $B=(b_{ij})_{i,j\\in V}$ two matrices with entries in a field $\\mathbb {K}$ .", "Assume that $A$ is irreducible and HL-indecomposable.", "If $\\det (A[X])=\\det (B[X])$ for any subset $X$ of $V$ , then $A$ and $B$ are diagonally similar up to transposition.", "For symmetric matrices, the Problem REF has been solved by G. M. Engel and H. Schneider [5].", "The following theorem is a directed consequence of Theorem 3.5 (see [5]).", "Theorem 4 Let $V$ be a nonempty set of size at least 4 and $A=(a_{ij})_{i,j\\in V}$ , $B=(b_{ij})_{i,j\\in V}$ two complex symmetric matrices.", "If $\\det (A[X])=\\det (B[X])$ for any subset $X$ of $V$ then there exists a diagonal matrix $D$ with diagonal entries $d_{i}\\in \\left\\lbrace -1,1\\right\\rbrace $ such that $B=DAD^{-1}$ .", "J.", "Rising, A. Kulesza, B. Taskar [15] gave a simple combinatorial proof of Theorem REF .", "Moreover, they pointed out that the hypotheses of Theorem REF can be weakened in sevral special cases.", "For symmetric matrices with no zeros off the diagonal, we can improve this theorem by using the following result.", "If $A$ is a complex symmetric matrix with no zeros off the diagonal, then the principal minors of order at most 3 of $A$ determine the rest of the principal minors.", "This result was obtained by L. Oeding for the generic symmetric matrices (see Remark 7.4, [14]).", "Nevertheless, it is not valid for an arbitrary symmetric matrix.", "For this, we consider the following example.", "Example 5 Let $V:=\\lbrace 1,...,n\\rbrace $ with $n\\ge 4$ .", "Consider the matrices $A_{n}:=(a_{ij})_{i,j\\in V}$ , $B_{n}:=(b_{ij})_{i,j\\in V}$ were $a_{i,i+1}=a_{i+1,i}=b_{i,i+1}=b_{i+1,i}=1$ for $i=1,\\ldots ,n-1$ , $a_{n,1}=a_{1,n}=-b_{n,1}=-b_{1,n}=1$ and $a_{ij}=b_{ij}=0$ , otherwise.", "One can check that $\\det (A_{n}\\left[ X\\right] )=\\det (B_{n}\\left[ X\\right] )$ for any proper subset $X$ of $V$ , but $\\det (A_{n})\\ne \\det (B_{n})$ .", "Consider now, the skew-symmetric version of Problem REF .", "We can ask as for the symmetric matrices if the hypotheses of Theorem REF can be weakened in special cases.", "Clearly, two skew-symmetric matrices of orders $n$ have equal corresponding principal minors of order 3 if and only if they differ up to the sign of their off-diagonal entries and they are not are always diagonally similar up to transposition.", "Then, it is necessary to consider principal minors of order 4.", "More precisely, we can suggest the following problem.", "Problem 6 Given a positive integer $k\\ge 4$ , what is the relationship between two skew-symmetric matrices of orders $n$ having equal corresponding principal minors of order at most $k$ ?", "Our goal, in this article, is to study this problem for skew-symmetric matrices whose all off-diagonal entries are nonzero.", "Such matrices are called dense matrices [16].", "A partial answer can be obtained from the work of G. Wesp [16] about principally unimodular matrices.", "A square matrix is principally unimodular if every principal submatrix has determinant 0, 1 or $-1$ .", "([1], [16]).", "G. Wesp [16] showed that a skew-symmetric dense matrix $A=(a_{ij})_{i,j\\in V}$ with entries in $\\left\\lbrace -1,0,1\\right\\rbrace $ is principally unimodular if and only if $\\det (A[X])=1$ for any subset $X$ of $V$ , of size 4.", "It follows that if $A$ , $B$ are two skew-symmetric dense matrices have equal corresponding principal minors of order at most 4, then they are both principally unimodular or not.", "Our main result is the following Theorem which improves Theorem REF for skew-symmetric dense matrices.", "Theorem 7 Let $V$ be a nonempty set of size at least 4 and $A=(a_{ij})_{i,j\\in V}$ , $B=(b_{ij})_{i,j\\in V}$ two skew-symmetric matrices with entries in a field $\\mathbb {K}$ of characteristic not equal to 2.", "Assume that $A$ is dense and HL-indecomposable.", "If $\\det (A[X])=\\det (B[X])$ for any subset $X$ of $V$ of size at most 4 then there exists a diagonal matrix $D$ with diagonal entries $d_{i}\\in \\left\\lbrace -1,1\\right\\rbrace $ such that $B=DAD$ or $B^{t}=DAD$ , in particular, $\\det (A)=\\det (B)$ .", "Note the Theorem REF is not valid for arbitrary HL-indecomposable skew-symmetric matrices.", "For this, we consider the following example.", "Example 8 Let $V:=\\lbrace 1,...,n\\rbrace $ where $n$ is a even integer such that $n\\ge 6$ .", "Consider the matrices $A_{n}:=(a_{ij})_{i,j\\in V}$ , $B_{n}:=(b_{ij})_{i,j\\in V}$ where $a_{i,i+1}=b_{i,i+1}=-a_{i+1,i}=-b_{i+1,i}=1$ for $i=1,\\ldots ,n-1$ , $a_{n,1}=-b_{n,1}=-a_{1,n}=b_{1,n}=1$ and $a_{ij}=b_{ij}=0$ , otherwise.", "One can check that $A_{n}$ and $B_{n}$ are HL-indecomposable and $\\det (A_{n}\\left[ X\\right] )=\\det (B_{n}\\left[ X\\right] )$ for any proper subset $X$ of $V$ , but $\\det (A_{n})=0$ and $\\det (B_{n})=4$ .", "Throughout this paper, we consider only matrices whose entries are in a field $\\mathbb {K}$ of characteristic not equal to 2." ], [ "Decomposability of Matrices", "Let $A=(a_{ij})_{i,j\\in V}$ be a matrix.", "A subset $X$ of $V$ is a HL-clan of $A$ if both of matrices $A\\left[ X,V\\setminus X\\right] $ and $A\\left[ V\\setminus X,X\\right] $ have rank at most 1.", "By definition, the complement of an HL-clan is an HL-clan.", "Moreover, $\\emptyset $ , $V$ , singletons $\\lbrace x\\rbrace $ and $V\\setminus \\lbrace x\\rbrace $ (where $x\\in V$ ) are HL-clans of $A$ called trivial HL-clans.", "It follows that a matrix is HL-indecomposable if all its HL-clans are trivial.", "From clan's definition for 2-structures [4], we can introduce the concept of clan for matrices which is stronger that of HL-clan.", "A subset $I$ of $V$ is a clan of $A$ if for every $i,j\\in I$ and $x\\in V\\setminus I $ , $a_{xi}=a_{xj}$ and $a_{ix}=a_{jx}$ .", "For example, $\\emptyset $ , $\\left\\lbrace x\\right\\rbrace $ where $x\\in V$ and $V$ are clans of $A$ called trivial clans.", "A matrix is indecomposable if all its clans are trivial.", "Otherwise, it is decomposable.", "In the next proposition, we present some basic properties of clans that can be deduced easily from the definition.", "Proposition 9 Let $A=(a_{ij})_{i,j\\in V}$ be a matrix and let $X,Y$ be two subsets of $V$ .", "i) If $X$ is a clan of $A$ and $Y$ is a clan of $A[X]$ then $Y$ is a clan of $A$ .", "ii) If $X$ is a clan of $A$ then $X\\cap Y$ is a clan of $A\\left[ Y\\right] $ .", "iii) If $X$ and $Y$ are clans of $A$ then $X\\cap Y$ is a clan of $A$ .", "iv) If $X$ and $Y$ are clans of $A$ such that $X\\cap Y\\ne \\emptyset $ , then $X\\cup Y$ is a clan of $A$ .", "v) If $X$ and $Y$ are clans of $A$ such that $X\\setminus Y\\ne \\emptyset $ , then $Y\\setminus X$ is a clan of $A$ .", "Let $A=(a_{ij})_{i,j\\in V}$ be a matrix and let $X,Y$ be two nonempty disjoint subsets of $V$ .", "If for some $\\alpha \\in \\mathbb {K}$ , we have $a_{xy}=\\alpha $ for every $x\\in X$ and every $y\\in Y$ then we write $A_{(X,Y)}=\\alpha $ .", "Clearly, if $X,Y$ are two nonempty disjoint clans of $A$ then there is $\\alpha \\in \\mathbb {K}$ such that $A_{(I,J)}=\\alpha $ .", "A clan-partition of $A$ is a partition $\\mathcal {P}$ of $V$ such that $X$ is a clan of $A$ for every $X\\in \\mathcal {P}$ .", "The following lemma gives a relationship between decomposability and HL-decomposability.", "Lemma 10 Let $V$ be a finite set of size at least 4 and $A=(a_{ij})_{i,j\\in V}$ a matrix.", "We assume that there is $v\\in V$ and $\\lambda $ , $\\kappa $ $\\in \\mathbb {K}\\setminus \\lbrace 0\\rbrace $ such that $a_{vj}=\\lambda $ et $a_{jv}=\\kappa $ for $j\\ne v$ .", "Then $A$ is HL-decomposable if and only if $A[V\\setminus \\lbrace v\\rbrace ]$ is decomposable.", "Clearly $V\\setminus \\lbrace v\\rbrace $ is a clan of $A$ , it follows by Proposition REF that a clan of $A[V\\setminus \\lbrace v\\rbrace ]$ is a clan of $A$ and hence it is a HL-clan of $A$ .", "Therefore, if $A[V\\setminus \\lbrace v\\rbrace ]$ is decomposable then $A$ is HL-decomposable.", "Conversely, suppose that $A$ is HL-decomposable and $I$ is a nontrivial HL-clan of $A$ .", "As $V\\setminus I$ is also a nontrivial HL-clan of $A$ then, by interchanging $I\\ $ and $V\\setminus I$ , we can assume that $I\\subseteq V\\setminus \\left\\lbrace v\\right\\rbrace $ .", "We will show that $I$ is a clan of $A[V\\setminus \\lbrace v\\rbrace ]$ .", "For this, let $i,j\\in I$ and $k\\in (V\\setminus \\left\\lbrace v\\right\\rbrace )\\setminus I$ .", "The matrix $A[\\lbrace v,k\\rbrace ,\\lbrace i,j\\rbrace ]$ (resp.", "$A[\\lbrace i,j\\rbrace ,\\lbrace v,k\\rbrace ]$ ) is a submatrix of $A[V\\setminus I,I]$ (resp.", "$A[I,V\\setminus I]$ ).", "As, $I$ and $V\\setminus I$ are HL-clans of $A$ then both of matrices $A[V\\setminus I,I]$ and $A[I,V\\setminus I]$ have rank at most 1.", "It follows that $\\det (A\\left[\\left\\lbrace v,k\\right\\rbrace ,\\left\\lbrace i,j\\right\\rbrace \\right] )=0$ , $\\det (A\\left[ \\left\\lbrace i,j\\right\\rbrace ,\\left\\lbrace v,k\\right\\rbrace \\right] )=0$ and so, $a_{ki}=a_{kj}$ and $a_{ik}=a_{jk}$ .", "We conclude that $I$ is a nontrivial clan of $A[V\\setminus \\lbrace v\\rbrace ]$ .", "Lemma 11 Let $V$ be a finite set, $A=(a_{ij})_{i,j\\in V}$ a matrix and $D=(d_{ij})_{i,j\\in V}$ a nonsingular diagonal matrix.", "The matrices $A$ and $DAD$ have the same HL-clans.", "In particular, $A$ is HL-indecomposable if and only if $DAD$ is HL-indecomposable.", "Let $X$ be a subset of $V$ .", "We have the following equalities : $(DAD)\\left[ V\\setminus X,X\\right] =(D\\left[ V\\setminus X\\right] )(A\\left[V\\setminus X,X\\right] )(D\\left[ X\\right] )$ $(DAD)\\left[ X,V\\setminus X\\right] =(D\\left[ X\\right] )(A\\left[ X,V\\setminus X\\right] )(D\\left[ V\\setminus X\\right] )$ But, the matrices $D\\left[ X\\right] $ and $D\\left[ V\\setminus X\\right] $ are nonsingular, then $(DAD)\\left[ V\\setminus X,X\\right] $ and $A\\left[V\\setminus X,X\\right] $ (resp.", "$(DAD)\\left[ X,V\\setminus X\\right] $ and $(A\\left[ X,V\\setminus X\\right] )$ have the same rank.", "Therfore, the matrices $A $ and $DAD$ have the same HL-clans.", "Separable matrices We introduce here the notion of separable matrix that comes from clan-decomposability of 2-structures (see [3] ).", "A skew-symmetric matrix $A=(a_{ij})_{i,j\\in V}$ is separable if it has a clan whose complement is also a clan.", "Otherwise, it is inseparable.", "Lemma 12 Let $A=(a_{ij})_{i,j\\in V}$ be a separable matrix and $\\left\\lbrace X,Y\\right\\rbrace $ a clan-partition of $A$ .", "If there is $x\\in X$ such that $A\\left[ V\\setminus \\left\\lbrace x\\right\\rbrace \\right] $ is inseparable, then $X=\\left\\lbrace x\\right\\rbrace $ and $Y=V\\setminus \\left\\lbrace x\\right\\rbrace $ .", "We assume, for contradiction, that $X\\ne \\left\\lbrace x\\right\\rbrace $ .", "From Proposition REF , $\\lbrace X\\cap ( V\\setminus \\left\\lbrace x\\right\\rbrace ), Y\\rbrace $ is a clan-partition of $A\\left[ V\\setminus \\left\\lbrace x\\right\\rbrace \\right] $ but this contradicts the fact that $A\\left[V\\setminus \\left\\lbrace x\\right\\rbrace \\right] $ is an inseparable matrix.", "It follows that $X=\\left\\lbrace x\\right\\rbrace $ and $Y=V\\setminus \\left\\lbrace x\\right\\rbrace $ .", "Lemma 13 Let $V$ be a finite set with $\\left|V\\right|\\ge 5$ , $A=(a_{ij})_{i,j\\in V}$ a skew-symmetric matrix and $Y$ a subset of $V$ such that $\\left|Y\\right|\\ge 2$ .", "If $C$ is a clan of $A$ such that $|C\\cap Y|=1$ and $V=C\\cup Y$ , then $A$ is inseparable if and only if $A[Y]$ is inseparable.", "Let $C\\cap Y:=\\lbrace y\\rbrace $ .", "Suppose that $A[Y]$ is separable and let $\\lbrace Z,Z^{\\prime }\\rbrace $ a clan-partition of $A[Y]$ .", "By interchanging $Z$ by $Z^{\\prime }$ , we can assume that $y\\in Z$ .", "It is easy to see that $\\left\\lbrace Z\\cup C,Z^{\\prime }\\right\\rbrace $ is a clan-partition of $A$ and hence $A$ is separable.", "Conversely, suppose that $A[Y]$ is inseparable.", "Let $W$ be a subset be of $V$ containing $Y$ and such that $A[W]$ is inseparable.", "Choose $W$ so that its cardinality is maximal.", "To prove that $A$ is inseparable, we will show that $W=V$ .", "For contradiction, suppose that $W\\ne V$ and let $v\\in V\\setminus W$ .", "By maximality of $\\left|W\\right|$ , the matrix $A[W\\cup \\lbrace v\\rbrace ]$ is separable and by Lemma REF , $\\left\\lbrace \\left\\lbrace v\\right\\rbrace ,W\\right\\rbrace $ is a clan-partition of $A[W\\cup \\lbrace v\\rbrace ]$ .", "Furthermore, by Proposition REF , $C\\cap (W\\cup \\lbrace v\\rbrace )$ is a clan of $A[W\\cup \\lbrace v\\rbrace ]$ .", "Moreover, as $v\\in C$ because $C\\cup Y=V$ , $Y\\subseteq W$ and $v\\in V\\setminus W$ , it follows that $v\\in (C\\cap (W\\cup \\lbrace v\\rbrace ))\\setminus W$ and hence $(C\\cap (W\\cup \\lbrace v\\rbrace ))\\setminus W\\ne \\emptyset $ .", "By applying Proposition REF , the subset $W\\setminus (C\\cap \\left(W\\cup \\left\\lbrace v\\right\\rbrace \\right) )=W\\setminus C$ is a clan of $A\\left[W\\cup \\left\\lbrace v\\right\\rbrace \\right] $ .", "Thus $\\left\\lbrace W\\setminus C,C\\cap W\\right\\rbrace $ is a clan-partition of $A[W]$ but this contradicts the fact that $A[W]$ is inseparable.", "As a consequence of this result, we obtain the following corollary.", "Corollaire 14 Let $V$ be a finite set of size $n$ with $n\\ge 5$ and $A=(a_{ij})_{i,j\\in V}$ an inseparable skew-symmetric matrix.", "Then, there is $x\\in V$ such that the matrix $A[V\\setminus \\lbrace x\\rbrace ]$ is inseparable.", "First, suppose that the matrix $A$ is decomposable.", "Let $I$ be a nontrivial clan of $A$ and $y\\in I$ .", "Put $Y:=(V\\setminus I)\\cup \\lbrace y\\rbrace $ .", "Clearly, $Y\\cap I=\\left\\lbrace y\\right\\rbrace $ and $V=I\\cup Y$ .", "By Lemma REF , the matrix $A[Y]$ is inseparable.", "Let $x\\in I\\setminus \\lbrace y\\rbrace $ .", "As $I$ is a clan of $A$ , then by Proposition REF , $I\\setminus \\lbrace x\\rbrace $ is a clan of $A[V\\setminus \\lbrace x\\rbrace ]$ .", "In addition, $Y\\cap (I\\setminus \\lbrace x\\rbrace )=\\lbrace y\\rbrace $ and $(I\\setminus \\lbrace x\\rbrace )\\cup Y=V\\setminus \\lbrace x\\rbrace $ , then, by applying again Lemma REF , we deduce that the matrix $A\\left[ V\\setminus \\left\\lbrace x\\right\\rbrace \\right] $ is necessarily inseparable.", "Suppose, now, that the matrix $A$ is indecomposable.", "Firstly, by proceeding as in the proof of Theorem 6.1 (pp.", "93 [4]), we will show that there is a subset $I$ of $V$ such that $3\\le |I|\\le n-1 $ and $A[I]$ is inseparable.", "Assume the contrary and let $x\\in V$ .", "The matrix $A[V\\setminus \\lbrace x\\rbrace ]$ is, then, separable.", "Let $\\lbrace X,X^{\\prime }\\rbrace $ be a clan-partition of $A[V\\setminus \\lbrace x\\rbrace ]$ .", "Without loss of generality, we can assume that $|X^{\\prime }|\\ge |X|$ .", "It follows that $|X^{\\prime }|\\ge 2$ because $|V|\\ge 5$ .", "As $3\\le |X^{\\prime }\\cup \\left\\lbrace x\\right\\rbrace |\\le n-1$ , the submatrix $A[X^{\\prime }\\cup \\left\\lbrace x\\right\\rbrace ]$ is separable.", "Let $\\lbrace Y,Y^{\\prime }\\rbrace $ be a clan-partition of $A[X^{\\prime }\\cup \\lbrace x\\rbrace ]$ such that $x\\in Y$ .", "Let $A_{(Y^{\\prime },Y)}=a$ and $A_{(X^{\\prime },X)}=b$ .", "Since $Y^{\\prime }\\subseteq X^{\\prime }$ , we have $A_{(Y^{\\prime },X)}=b$ .", "In addition, $X\\cup Y=V\\setminus $ $Y^{\\prime }$ , then $Y^{\\prime }$ is a clan of $A$ .", "It follows that $Y$ is a singleton because $A$ is indecomposable.", "Let $Y^{\\prime }:=\\lbrace y\\rbrace $ .", "If $a=b$ then $A_{(y,X)}=b=a$ and $A_{(y,Y)}=a$ .", "It follows that $A_{(y,V\\setminus \\left\\lbrace y\\right\\rbrace )}=a$ and this contradicts the fact that $A$ is indecomposable.", "So $a\\ne b$ .", "If$\\ A_{(x,X)}=b$ , then $A_{(V\\setminus X,X)}=b$ because $A_{(X^{\\prime },X)}=b$ and $V\\setminus X=X^{\\prime }\\cup \\left\\lbrace x\\right\\rbrace $ and this contradicts that $A$ is indecomposable.", "It follows that there is $z\\in X$ such that $a_{xz}\\ne b$ .", "Now, by hypothesis, $A[x,y,z]$ is separable, so $a_{zx}=a$ .", "Furthermore, $(X^{\\prime }\\cup \\left\\lbrace x\\right\\rbrace )\\setminus \\lbrace x,y\\rbrace \\ne \\emptyset $ because $|X^{\\prime }\\cup \\left\\lbrace x\\right\\rbrace |\\ge 3$ .", "Let $v\\in (X^{\\prime }\\cup \\left\\lbrace x\\right\\rbrace )\\setminus \\left\\lbrace x,y\\right\\rbrace =X^{\\prime }\\setminus \\left\\lbrace y\\right\\rbrace $ .", "We can choose $v$ such that $a_{vx}\\ne a$ , because otherwise $X$ would be a nontrivial clan of $A$ and this contradicts the fact that $A$ is indecomposable.", "The submatrix $A[x,z,v]$ is also separable, so $a_{vx}=b$ .", "It follows that the submatrix $A\\left[ x,y,z,v\\right] $ is indecomposable, and then it is inseparable, but this contradicts our assumption.", "Thus there is a subset $I$ of $V$ such that $3\\le \\left|I\\right|\\le n-1$ and $A\\left[ I\\right] $ is inseparable.", "To continue the proof, we choose a subset $I$ of $V$ with maximal cardinality such that $3\\le \\left|I\\right|\\le n-1$ and $A\\left[ I\\right] $ is inseparable.", "We will show that $\\left|I\\right|=n-1$ .", "We assume, for contradiction, that $\\left|I\\right|<n-1$ .", "Let $x\\in V\\setminus I$ .", "By maximality of $\\left|I\\right|$ , the matrix $A[I\\cup \\lbrace x\\rbrace ]$ is separable.", "Let $\\lbrace Z,W\\rbrace $ a clan-partition of $A[I\\cup \\lbrace x\\rbrace ]$ such that $x\\in Z$ .", "As $A\\left[ I\\right] $ is inseparable, then by Lemma REF , we have $Z=\\lbrace x\\rbrace $ and $W=I$ .", "It follows that $I$ is a proper clan of $A$ which contradicts the fact that $A$ is indecomposable.", "HL-equivalent matrices Let $V$ be a finite set and $k$ a positive integer.", "Two matrices $A=(a_{ij})_{i,j\\in V}$ , $B=(b_{ij})_{i,j\\in V}$ are $(\\le k)-$HL-equivalents if for any subset $X$ of $V$ , of size at most $k$ , we have $\\det (A\\left[ X\\right] )=\\det (B\\left[ X\\right] )$ .", "In the rest of this paper, we will consider only skew symmetric dense matrices.", "It is clear that if two skew symmetric dense matrices $A=(a_{ij})_{i,j\\in V}$ , $B=(b_{ij})_{i,j\\in V}$ are $(\\le 2)-$ HL-equivalents then, $b_{xy}=a_{xy}$ or $b_{xy}=-a_{xy}$ for all $x\\ne y\\in V$ .", "We define on $V$ two equivalence relations $\\mathcal {E}_{A,B}$ and $\\mathcal {D}_{A,B}$ as follows.", "For $x\\ne y\\in V$ , $x\\mathcal {E}_{A,B}y$ (resp.", "$x\\mathcal {D}_{A,B}y$ ) if there is a sequence $x_{0}=x,x_{1},\\ldots ,x_{n}=y$ such that $a_{x_{i}x_{i+1}}=b_{x_{i}x_{i+1}}$ (resp.", "$a_{x_{i}x_{i+1}}=-b_{x_{i}x_{i+1}} $ ) for $i=0,\\ldots ,n-1$ .", "Let $x\\in V$ , we denote by $\\mathcal {E}_{A,B} \\langle x \\rangle $ (resp.", "$\\mathcal {D}_{A,B}\\langle x \\rangle $ ) the equivalence class of $x$ for $\\mathcal {E}_{A,B}$ ( resp.", "for $\\mathcal {D}_{A,B}$ ) Let $V$ be a finite set and $A=(a_{ij})_{i,j\\in V}$ a dense skew-symmetric matrix.", "Let $V^{\\infty }:=V\\cup \\left\\lbrace \\infty \\right\\rbrace $ and $A^{\\infty }:=(a_{ij})_{i,j\\in V^{\\infty }}$ were $a_{\\infty \\infty }=0$ , $a_{\\infty j}=1$ and $a_{j\\infty }=-1$ for $j\\in V$ .", "Lemma 15 Let $V=\\left\\lbrace i,j,k\\right\\rbrace $ and $A=(a_{ij})_{i,j\\in V}$ , $B=(b_{ij})_{i,j\\in V}$ two dense skew-symmetric matrices such that $a_{ij}=b_{ij}$ , $a_{ki}=b_{ik}$ and $a_{kj}=b_{jk}$ .", "If $\\det \\left(A^{\\infty }\\right) =\\det \\left( B^{\\infty }\\right) $ then $a_{ik}=a_{jk}$ and $b_{ik}=b_{jk}$ .", "We have the following equalities : $\\det (A^{\\infty })=a_{ij}^{2}-2a_{ij}a_{ik}+2a_{ij}a_{jk}+a_{ik}^{2}-2a_{ik}a_{jk}+a_{jk}^{2}$ , $\\det (B^{\\infty })=b_{ij}^{2}-2b_{ij}b_{ik}+2b_{ij}b_{jk}+b_{ik}^{2}-2b_{ik}b_{jk}+b_{jk}^{2}$ .", "But, by hypothesis, $b_{ij}=a_{ij}$ , $b_{ik}=a_{ki}=-a_{ik}$ and $b_{jk}=a_{kj}=-a_{jk}$ , then we have, $\\det (B^{\\infty })=a_{ij}^{2}+2a_{ij}a_{ik}-2a_{ij}a_{jk}+a_{ik}^{2}-2a_{ik}a_{jk}+a_{jk}^{2}$ .", "It follows that, if $\\det (A^{\\infty })=\\det (B^{\\infty })$ then $a_{jk}=a_{ik}$ because $a_{ij}\\ne 0$ .", "Moreover, $b_{ik}=-a_{ik}=-a_{jk}=-b_{kj}=b_{jk}$ .", "Then, we have also $b_{ik}=b_{jk}$ .", "Lemma 16 Let $V$ be a finite set with $|V|\\ge 3$ and $A=(a_{ij})_{i,j\\in V}$ , $B=(b_{ij})_{i,j\\in V}$ two dense skew-symmetric matrices.", "If $A^{\\infty }$ and $B^{\\infty }$ are $(\\le 4)$ -HL-equivalent then the equivalence classes of $\\mathcal {E}_{A,B}$ and those of $\\mathcal {D}_{A,B}$ are clans of $A$ and of $B$ .", "First, note that $\\mathcal {D}_{A,B}=\\mathcal {E}_{A,B^{t}}$ and the matrices $B^{\\infty }$ , $(B^{t})^{\\infty }$ are $(\\le 4)$ -HL -equivalents.", "So, by interchanging $B$ and $B^{t}$ , it suffices to show that an equivalence class $C$ of $\\mathcal {E}_{A,B}$ is a clan of $A$ and of $B$ .", "For this, let $x\\ne y\\in C$ and $z\\in V\\setminus C$ .", "By definition of $\\mathcal {E}_{A,B}$ , there exists a sequence $x_{0}=x,x_{1},\\ldots ,x_{r}=y$ such that $a_{x_{i}x_{i+1}}=b_{x_{i}x_{i+1}}$ for $i=0,\\ldots ,r-1$ .", "As $z\\notin C$ , then for $j=0,\\ldots ,r$ , we have $a_{zx_{j}}=$ $-b_{zx_{j}}=b_{x_{j}z}$ .", "It follows, from Lemma REF , that $a_{zx_{i}}=a_{zx_{i+1}}$ and $b_{zx_{i}}=b_{zx_{i+1}}$ for $i=0,\\ldots ,r-1$ .", "Therefore $a_{zx}=a_{zy}$ , $b_{zx}=b_{zy}$ and hence $C$ is a clan of $A$ and of $B$ .", "Proposition 17 Let $V$ be a finite set with $|V|\\ge 3$ and $A=(a_{ij})_{i,j\\in V}$ , $B=(b_{ij})_{i,j\\in V}$ two dense skew-symmetric matrices such that $A^{\\infty }$ and $B^{\\infty }$ are $(\\le 4)$ -HL-equivalents.", "If $A$ is inseparable, then, $\\mathcal {D}_{A,B}$ or $\\mathcal {E}_{A,B}$ has at least two equivalence classes.", "The result is trivial for $|V|=2$ or $|V|=3$ .", "For $|V|=4$ , let $V:=\\lbrace i,j,k,l\\rbrace $ .", "Suppose that $V$ is an equivalence class for $\\mathcal {D}_{A,B}$ and for $\\mathcal {E}_{A,B}$ .", "We will show that $A$ is separable.", "Without loss of generality, we can assume that $a_{il}=-b_{il}$ , $a_{lk}=-b_{lk}$ , $a_{kj}=-b_{kj}$ , and $a_{ki}=b_{ki}$ , $a_{ij}=b_{ij}$ , $a_{jl}=b_{jl}$ .", "As $A^{\\infty }$ and $B^{\\infty }$ are $(\\le 4)$ -HL-equivalents, then $\\det (A^{\\infty }\\left[ \\left\\lbrace \\infty \\right\\rbrace \\cup X\\right] )=\\det (B^{\\infty }\\left[ \\left\\lbrace \\infty \\right\\rbrace \\cup X\\right] )$ for $X=\\lbrace i,j,k\\rbrace $ , $X=\\lbrace i,j,l\\rbrace $ , $X=\\left\\lbrace i,k,l\\right\\rbrace $ and $X=\\left\\lbrace i,k,l\\right\\rbrace $ .", "It follows that $4a_{ij}a_{jk}-4a_{ik}a_{jk}=0$ , $-4a_{ij}a_{il}-4a_{il}a_{jl}=0$ , $4a_{ik}a_{kl}-4a_{ik}a_{il}=0$ , $-4a_{jk}a_{jl}-4a_{jl}a_{kl}=0$ and then, $a_{ij}=a_{ik}$ , $a_{ij}=-a_{jl}$ , $a_{kl}=a_{il}$ , $a_{jk}=-a_{kl}$ .", "Furthermore, $\\det (A)=\\det (B)$ , so $4a_{kl}a_{ij}\\left( a_{ik}a_{jl}-a_{il}a_{jk}\\right) =0$ and consequently $a_{ik}a_{jl}-a_{il}a_{jk}=0$ .", "We conclude that $a_{ij}^{2}-a_{kl}^{2}=0$ .", "If $a_{ij}=a_{kl}$ then $a_{ij}=a_{ik}=a_{il}=a_{kl}=a_{kj}=a_{lj}$ and hence $\\lbrace i,j\\rbrace $ and $\\lbrace k,l\\rbrace $ are two clans of $A$ .", "If $a_{ij}=-a_{kl}$ then $a_{ij}=a_{ik}=a_{li}=a_{lj}=a_{jk}=a_{lk}$ .", "So $\\lbrace j,k\\rbrace $ and $\\lbrace i,l\\rbrace $ are two clans of $A$ and hence $A$ is separable.", "As in the proof of Proposition 3 (see [3]), we will continue by induction on $|V|$ for $|V|\\ge 5$ .", "Suppose, by contradiction, that $A$ is inseparable and that $V$ is an equivalence class for $\\mathcal {D}_{A,B}$ and for $\\mathcal {E}_{A,B}$ .", "By Corollary REF , there is $v\\in V$ such that the submatrix $A[V\\setminus \\lbrace v\\rbrace ]$ is inseparable.", "Let $\\mathcal {D}^{\\prime }:=\\mathcal {D}_{A\\left[ V\\setminus \\left\\lbrace v\\right\\rbrace \\right] ,B\\left[ V\\setminus \\left\\lbrace v\\right\\rbrace \\right] }$ and $\\mathcal {E}^{\\prime }:=\\mathcal {E}_{A\\left[ V\\setminus \\left\\lbrace v\\right\\rbrace \\right] ,B\\left[V\\setminus \\left\\lbrace v\\right\\rbrace \\right] }$ .", "By induction hypothesis and without loss of generality, we can assume that $\\mathcal {D}^{\\prime }$ has at least two equivalence classes.", "As $\\mathcal {E}\\langle v\\rangle =V$ , then there exist $x\\ne v$ such that $a_{xv}=b_{xv}$ .", "Moreover, the matrix $A[V\\setminus \\lbrace v\\rbrace ]$ is inseparable, then there exist $y_{1}\\in \\mathcal {D}^{\\prime }\\langle x\\rangle $ and $y_{2}\\in (V\\setminus \\left\\lbrace v\\right\\rbrace )\\setminus \\mathcal {D}^{\\prime }\\langle x\\rangle $ such that $a_{xv}\\ne $ $b_{y_{1}y_{2}}$ .", "Since $\\mathcal {D}^{\\prime }\\langle y_{2}\\rangle V$ and $\\mathcal {D}_{A,B}\\langle y_{2}\\rangle =V$ , there exists $z_{2}\\in \\mathcal {D}^{\\prime }\\langle y_{2}\\rangle $ and $w_{1}\\in V\\setminus \\mathcal {D}^{\\prime }\\langle y_{2}\\rangle $ such that $a_{z_{2}w_{1}}\\ne b_{z_{2}w_{1}}$ .", "But, $\\mathcal {D}^{\\prime }\\langle y_{2}\\rangle =\\mathcal {D}^{\\prime }\\langle z_{2}\\rangle $ , so for any $w\\in (V\\setminus \\left\\lbrace v\\right\\rbrace )\\setminus \\mathcal {D}^{\\prime }\\langle y_{2}\\rangle $ we have $a_{z_{2}w}=b_{z_{2}w}$ .", "We have necessarily, $w_{1}=v$ and hence $a_{z_{2}v}\\ne b_{z_{2}v}$ .", "Furthermore, $\\mathcal {D}^{\\prime }\\langle y_{2}\\rangle =\\mathcal {D}^{\\prime }\\langle z_{2}\\rangle $ and $\\mathcal {D}^{\\prime }\\langle y_{2}\\rangle \\cap \\mathcal {D}^{\\prime }\\langle x\\rangle =\\emptyset $ because $y_{2}\\in (V\\setminus \\left\\lbrace v\\right\\rbrace )\\setminus \\mathcal {D}^{\\prime }\\langle x\\rangle $ , then $z_{2}$ $\\notin \\mathcal {D}^{\\prime }\\langle x\\rangle $ and hence $a_{z_{2}x}=b_{z_{2}x}$ .", "To obtain the contradiction, we will apply Lemma REF to $X=\\left\\lbrace v,x,z_{2}\\right\\rbrace $ .", "By the foregoing, we have $a_{z_{2}v}\\ne b_{z_{2}v}$ , $a_{xv}=b_{xv}$ and $a_{z_{2}x}=b_{z_{2}x}$ .", "But, by Lemma REF , $\\mathcal {D}^{\\prime }\\langle x\\rangle $ and $\\mathcal {D}^{\\prime }\\langle y_{2}\\rangle $ are disjoint clans of $A[V\\setminus \\lbrace v\\rbrace ]$ , so $a_{y_{1}y_{2}}=a_{xz_{2}}$ because $x,y_{1}\\in \\mathcal {D}^{\\prime }\\langle x\\rangle $ and $y_{2},z_{2}\\in \\mathcal {D}^{\\prime }\\langle y_{2}\\rangle $ .", "It follows that $a_{xz_{2}}\\ne a_{xv}$ because $a_{xv}\\ne a_{y_{1}y_{2}}$ and this contradicts Lemma REF .", "Proof of the main Theorem By using the proposition REF and Lemma REF , we obtain the following result which is the key of the proof of the main Theorem.", "Proposition 18 Let $V$ be a finite set with $|V|\\ge 2$ and $A=(a_{ij})_{i,j\\in V}$ , $B=(b_{ij})_{i,j\\in V}$ two dense skew-symmetric matrices such that $A^{\\infty }$ and $B^{\\infty }$ are $(\\le 4)$ - HL-equivalent.", "If $A$ is indecomposable then $B=A$ or $B=A^{t}$ .", "Assume that the matrix $A$ is indecomposable, then it is inseparable.", "By using proposition REF and by interchanging $A$ and $A^{t}$ , we can assume that $\\mathcal {D}_{A,B}$ at least two equivalence classes.", "By Lemma REF these classes are clans of $A$ .", "So these are singletons because $A$ is indecomposable.", "Consequently $B=A$ .", "Let $u\\in V$ and $D:=(d_{ij})_{i,j\\in V}$ (resp.", "$D^{\\prime }:=(d_{ij}^{\\prime })_{i,j\\in V}$ ) the diagonal matrix such that $d_{uu}=1$ and $d_{zz}=\\frac{1}{a_{uz}}$ (resp.", "$d_{uu}^{\\prime }=1$ and $d_{zz}^{\\prime }=\\frac{1}{b_{uz}}$ ) if $z\\ne u$ .", "We consider the two matrices $\\widehat{A}:=DAD:=(\\widehat{a}_{ij})_{i,j\\in V}$ and $\\widehat{B}:=D^{\\prime }BD^{\\prime }:=(\\widehat{b}_{ij})_{i,j\\in V}$ .", "As the matrix $A$ is HL-indecomposable, then by lemma REF , the matrix $\\widehat{A}$ is also HL-indecomposable.", "However $\\widehat{a}_{vj}=\\widehat{b}_{vj}=1$ for $j\\ne v$ , then by Lemma REF , $\\widehat{A}\\left[ V\\setminus \\left\\lbrace v\\right\\rbrace \\right] $ is indecomposable.", "We will apply Proposition REF .", "For this, let $X$ be a subset of $V$ such that $\\left|X\\right|\\le 4$ .", "We have $\\det (\\widehat{A}\\left[ X\\right] )=\\left( \\det D\\left[ X\\right]\\right) ^{2}\\det (A\\left[ X\\right] )$ .", "Similarly, $\\det (\\widehat{B}\\ \\left[ X\\right] )=\\left( \\det D^{\\prime }\\left[ X\\right] \\right) ^{2}\\det (B\\left[ X\\right] )$ .", "But for all $z\\ne u$ , $b_{uz}=a_{uz}$ or $b_{uz}=-a_{uz}$ , then $\\left( \\det D\\left[ X\\right] \\right) ^{2}=\\left( \\det D^{\\prime }\\left[ X\\right] \\right) ^{2}$ .", "Moreover, by hypothesis, $\\det (A[X])=\\det (B[X])$ , then $\\det (\\widehat{A}\\left[ X\\right] )=\\det (\\widehat{B}\\ \\left[ X\\right] )$ and hence the matrices $\\widehat{A}$ , $\\widehat{B}\\ $ are $(\\le 4)$ -HL-equivalents.", "Now, by Proposition REF , $\\widehat{A}\\left[V\\setminus \\left\\lbrace v\\right\\rbrace \\right] =\\widehat{B}\\left[ V\\setminus \\left\\lbrace v\\right\\rbrace \\right] $ or $\\widehat{A}\\left[ V\\setminus \\left\\lbrace v\\right\\rbrace \\right]=(\\widehat{B}\\left[ V\\setminus \\left\\lbrace v\\right\\rbrace \\right] )^{t}$ .", "If $\\widehat{A}\\left[ V\\setminus \\left\\lbrace v\\right\\rbrace \\right] =\\widehat{B}\\left[V\\setminus \\left\\lbrace v\\right\\rbrace \\right] $ then $\\widehat{A}=\\widehat{B}$ and hence $B=(D^{\\prime })^{-1}DA(D^{\\prime })^{-1}D$ .", "Suppose that $\\widehat{A}\\left[ V\\setminus \\left\\lbrace v\\right\\rbrace \\right] =(\\widehat{B}\\left[V\\setminus \\left\\lbrace v\\right\\rbrace \\right] )^{t}$ and let $\\Delta :=(\\delta _{ij})_{i,j\\in V}$ the diagonal matrix such that $\\delta _{uu}=1$ and $\\delta _{zz}=-1$ if $z\\ne u$ .", "Clearly, $\\Delta \\widehat{A}\\Delta =(\\widehat{B})^{t}$ .", "It follows that $B^{t}=(D^{\\prime })^{-1}D\\Delta A(D^{\\prime })^{-1}D\\Delta $ .", "However for all $z\\in V$ , $d_{zz}=d_{zz}^{\\prime } $ or $d_{zz}=-d_{zz}^{\\prime }$ .", "then $(D^{\\prime })^{-1}D$ and $(D^{\\prime })^{-1}D\\Delta $ are diagonal matrices with diagonal entries in $\\left\\lbrace -1,1\\right\\rbrace $ .", "Some remarks about Theorem REF Let $A$ be a skew-symmetric matrix.", "Clearly, the principal minors of order 2 determine the off diagonal entries of $A$ up to sign.", "So, generically there are at most finitely many skew-symmetric matrices with equal corresponding principal minors as a given matrix, and one should expect Theorem REF to hold for sufficiently generic skew-symmetric matrices.", "Nevertheless, for dense skew-symmetric matrix, Theorem REF is the best one.", "To see this, let $A=(a_{ij})_{i,j\\in V}$ be an HL-decomposable matrix and $X$ a nontrivial HL-clan of $A$ .", "Consider the matrix $B=(b_{ij})_{i,j\\in V}$ such that $b_{ij}=-a_{ij}$ if $i$ ,$j\\in X$ and $b_{ij}=a_{ij}$ otherwise.", "By adapting the proof of Lemma 5 (see [8] ), we can prove that $A$ and $B$ have the same principal minors, but they are not diagonally similar up to transposition.", "We can ask if Theorem REF can be obtained from Theorem REF via specialization to skew-symmetric matrices.", "For this, we must prove that for a dense skew-symmetric matrix, the principal minors of order at most 4 determine the rest of its principal minors.", "One way to do this is to give an expression of the determinant of a skew-symmetric matrix of order at least 5 from its principal minors.", "More precisely, let $M$ be a generic skew-symmetric matrix of order $n$ where $n\\ge 5$ and the entries $x_{ij}$ with $i$ $<j$ are indeterminates.", "Let $R=\\mathbb {K}[x_{ij}|1\\le i<j\\le n]$ be the polynomial ring generated by these indeteminates.", "The problem is to find an expression of $\\det \\left( M\\right) $ from the collection of the principal minors $\\det \\left( M\\left[ \\alpha \\right]\\right) $ where $\\alpha $ is a subset of $\\left\\lbrace 1,\\ldots ,n\\right\\rbrace $ of size a most 4.", "Because of the example REF , such expression can not be a polynomial.", "Nevertheless, it is reasonable to suggest the following problem Problem 19 Let $\\left\\langle n\\right\\rangle _{4}$ the collection of all subset $\\alpha \\subseteq \\left\\lbrace 1,\\ldots ,n\\right\\rbrace $ with $\\left|\\alpha \\right|$ $=2$ or 4 and consider the polynomial ring $R=\\mathbb {K}[X_{\\alpha },\\alpha $ $\\in \\left\\langle n\\right\\rangle _{4}]$ where $X_{\\alpha }$ $,\\alpha $ $\\in \\left\\langle n\\right\\rangle _{4}$ are indeterminates.", "Is there a polynomial $Q(X_{\\alpha },\\alpha \\in \\left\\langle n\\right\\rangle _{4})\\in R$ such that $\\prod \\limits _{1\\le i<j\\le n} x_{ij}^{2}\\det \\left( M\\right) =Q(\\det \\left( M\\left[ \\alpha \\right] \\right) ,\\alpha \\in \\left\\langle n\\right\\rangle _{4})$ ?", "Note that a positive answer to this problem combined with the Theorem REF allows to obtain our main theorem.", "Finally, as the principal minors are squares of the corresponding Pfaffians, we can strengthen the assumptions of the Problem REF by replacing the principal minors by sub-pfaffians.", "However, two skew-symmetric matrices with the same corresponding principal sub-pfaffians of order 2 are equal.", "So, it is interesting to consider the following problem.", "Problem 20 Given a positive integer $n\\ge 5$ , what is the relationship between two skew-symmetric matrices of orders $n$ which differ up to the sign of their off-diagonal terms and having equal corresponding principal Pfaffian minors of order 4 ?" ], [ "Separable matrices", "We introduce here the notion of separable matrix that comes from clan-decomposability of 2-structures (see [3] ).", "A skew-symmetric matrix $A=(a_{ij})_{i,j\\in V}$ is separable if it has a clan whose complement is also a clan.", "Otherwise, it is inseparable.", "Lemma 12 Let $A=(a_{ij})_{i,j\\in V}$ be a separable matrix and $\\left\\lbrace X,Y\\right\\rbrace $ a clan-partition of $A$ .", "If there is $x\\in X$ such that $A\\left[ V\\setminus \\left\\lbrace x\\right\\rbrace \\right] $ is inseparable, then $X=\\left\\lbrace x\\right\\rbrace $ and $Y=V\\setminus \\left\\lbrace x\\right\\rbrace $ .", "We assume, for contradiction, that $X\\ne \\left\\lbrace x\\right\\rbrace $ .", "From Proposition REF , $\\lbrace X\\cap ( V\\setminus \\left\\lbrace x\\right\\rbrace ), Y\\rbrace $ is a clan-partition of $A\\left[ V\\setminus \\left\\lbrace x\\right\\rbrace \\right] $ but this contradicts the fact that $A\\left[V\\setminus \\left\\lbrace x\\right\\rbrace \\right] $ is an inseparable matrix.", "It follows that $X=\\left\\lbrace x\\right\\rbrace $ and $Y=V\\setminus \\left\\lbrace x\\right\\rbrace $ .", "Lemma 13 Let $V$ be a finite set with $\\left|V\\right|\\ge 5$ , $A=(a_{ij})_{i,j\\in V}$ a skew-symmetric matrix and $Y$ a subset of $V$ such that $\\left|Y\\right|\\ge 2$ .", "If $C$ is a clan of $A$ such that $|C\\cap Y|=1$ and $V=C\\cup Y$ , then $A$ is inseparable if and only if $A[Y]$ is inseparable.", "Let $C\\cap Y:=\\lbrace y\\rbrace $ .", "Suppose that $A[Y]$ is separable and let $\\lbrace Z,Z^{\\prime }\\rbrace $ a clan-partition of $A[Y]$ .", "By interchanging $Z$ by $Z^{\\prime }$ , we can assume that $y\\in Z$ .", "It is easy to see that $\\left\\lbrace Z\\cup C,Z^{\\prime }\\right\\rbrace $ is a clan-partition of $A$ and hence $A$ is separable.", "Conversely, suppose that $A[Y]$ is inseparable.", "Let $W$ be a subset be of $V$ containing $Y$ and such that $A[W]$ is inseparable.", "Choose $W$ so that its cardinality is maximal.", "To prove that $A$ is inseparable, we will show that $W=V$ .", "For contradiction, suppose that $W\\ne V$ and let $v\\in V\\setminus W$ .", "By maximality of $\\left|W\\right|$ , the matrix $A[W\\cup \\lbrace v\\rbrace ]$ is separable and by Lemma REF , $\\left\\lbrace \\left\\lbrace v\\right\\rbrace ,W\\right\\rbrace $ is a clan-partition of $A[W\\cup \\lbrace v\\rbrace ]$ .", "Furthermore, by Proposition REF , $C\\cap (W\\cup \\lbrace v\\rbrace )$ is a clan of $A[W\\cup \\lbrace v\\rbrace ]$ .", "Moreover, as $v\\in C$ because $C\\cup Y=V$ , $Y\\subseteq W$ and $v\\in V\\setminus W$ , it follows that $v\\in (C\\cap (W\\cup \\lbrace v\\rbrace ))\\setminus W$ and hence $(C\\cap (W\\cup \\lbrace v\\rbrace ))\\setminus W\\ne \\emptyset $ .", "By applying Proposition REF , the subset $W\\setminus (C\\cap \\left(W\\cup \\left\\lbrace v\\right\\rbrace \\right) )=W\\setminus C$ is a clan of $A\\left[W\\cup \\left\\lbrace v\\right\\rbrace \\right] $ .", "Thus $\\left\\lbrace W\\setminus C,C\\cap W\\right\\rbrace $ is a clan-partition of $A[W]$ but this contradicts the fact that $A[W]$ is inseparable.", "As a consequence of this result, we obtain the following corollary.", "Corollaire 14 Let $V$ be a finite set of size $n$ with $n\\ge 5$ and $A=(a_{ij})_{i,j\\in V}$ an inseparable skew-symmetric matrix.", "Then, there is $x\\in V$ such that the matrix $A[V\\setminus \\lbrace x\\rbrace ]$ is inseparable.", "First, suppose that the matrix $A$ is decomposable.", "Let $I$ be a nontrivial clan of $A$ and $y\\in I$ .", "Put $Y:=(V\\setminus I)\\cup \\lbrace y\\rbrace $ .", "Clearly, $Y\\cap I=\\left\\lbrace y\\right\\rbrace $ and $V=I\\cup Y$ .", "By Lemma REF , the matrix $A[Y]$ is inseparable.", "Let $x\\in I\\setminus \\lbrace y\\rbrace $ .", "As $I$ is a clan of $A$ , then by Proposition REF , $I\\setminus \\lbrace x\\rbrace $ is a clan of $A[V\\setminus \\lbrace x\\rbrace ]$ .", "In addition, $Y\\cap (I\\setminus \\lbrace x\\rbrace )=\\lbrace y\\rbrace $ and $(I\\setminus \\lbrace x\\rbrace )\\cup Y=V\\setminus \\lbrace x\\rbrace $ , then, by applying again Lemma REF , we deduce that the matrix $A\\left[ V\\setminus \\left\\lbrace x\\right\\rbrace \\right] $ is necessarily inseparable.", "Suppose, now, that the matrix $A$ is indecomposable.", "Firstly, by proceeding as in the proof of Theorem 6.1 (pp.", "93 [4]), we will show that there is a subset $I$ of $V$ such that $3\\le |I|\\le n-1 $ and $A[I]$ is inseparable.", "Assume the contrary and let $x\\in V$ .", "The matrix $A[V\\setminus \\lbrace x\\rbrace ]$ is, then, separable.", "Let $\\lbrace X,X^{\\prime }\\rbrace $ be a clan-partition of $A[V\\setminus \\lbrace x\\rbrace ]$ .", "Without loss of generality, we can assume that $|X^{\\prime }|\\ge |X|$ .", "It follows that $|X^{\\prime }|\\ge 2$ because $|V|\\ge 5$ .", "As $3\\le |X^{\\prime }\\cup \\left\\lbrace x\\right\\rbrace |\\le n-1$ , the submatrix $A[X^{\\prime }\\cup \\left\\lbrace x\\right\\rbrace ]$ is separable.", "Let $\\lbrace Y,Y^{\\prime }\\rbrace $ be a clan-partition of $A[X^{\\prime }\\cup \\lbrace x\\rbrace ]$ such that $x\\in Y$ .", "Let $A_{(Y^{\\prime },Y)}=a$ and $A_{(X^{\\prime },X)}=b$ .", "Since $Y^{\\prime }\\subseteq X^{\\prime }$ , we have $A_{(Y^{\\prime },X)}=b$ .", "In addition, $X\\cup Y=V\\setminus $ $Y^{\\prime }$ , then $Y^{\\prime }$ is a clan of $A$ .", "It follows that $Y$ is a singleton because $A$ is indecomposable.", "Let $Y^{\\prime }:=\\lbrace y\\rbrace $ .", "If $a=b$ then $A_{(y,X)}=b=a$ and $A_{(y,Y)}=a$ .", "It follows that $A_{(y,V\\setminus \\left\\lbrace y\\right\\rbrace )}=a$ and this contradicts the fact that $A$ is indecomposable.", "So $a\\ne b$ .", "If$\\ A_{(x,X)}=b$ , then $A_{(V\\setminus X,X)}=b$ because $A_{(X^{\\prime },X)}=b$ and $V\\setminus X=X^{\\prime }\\cup \\left\\lbrace x\\right\\rbrace $ and this contradicts that $A$ is indecomposable.", "It follows that there is $z\\in X$ such that $a_{xz}\\ne b$ .", "Now, by hypothesis, $A[x,y,z]$ is separable, so $a_{zx}=a$ .", "Furthermore, $(X^{\\prime }\\cup \\left\\lbrace x\\right\\rbrace )\\setminus \\lbrace x,y\\rbrace \\ne \\emptyset $ because $|X^{\\prime }\\cup \\left\\lbrace x\\right\\rbrace |\\ge 3$ .", "Let $v\\in (X^{\\prime }\\cup \\left\\lbrace x\\right\\rbrace )\\setminus \\left\\lbrace x,y\\right\\rbrace =X^{\\prime }\\setminus \\left\\lbrace y\\right\\rbrace $ .", "We can choose $v$ such that $a_{vx}\\ne a$ , because otherwise $X$ would be a nontrivial clan of $A$ and this contradicts the fact that $A$ is indecomposable.", "The submatrix $A[x,z,v]$ is also separable, so $a_{vx}=b$ .", "It follows that the submatrix $A\\left[ x,y,z,v\\right] $ is indecomposable, and then it is inseparable, but this contradicts our assumption.", "Thus there is a subset $I$ of $V$ such that $3\\le \\left|I\\right|\\le n-1$ and $A\\left[ I\\right] $ is inseparable.", "To continue the proof, we choose a subset $I$ of $V$ with maximal cardinality such that $3\\le \\left|I\\right|\\le n-1$ and $A\\left[ I\\right] $ is inseparable.", "We will show that $\\left|I\\right|=n-1$ .", "We assume, for contradiction, that $\\left|I\\right|<n-1$ .", "Let $x\\in V\\setminus I$ .", "By maximality of $\\left|I\\right|$ , the matrix $A[I\\cup \\lbrace x\\rbrace ]$ is separable.", "Let $\\lbrace Z,W\\rbrace $ a clan-partition of $A[I\\cup \\lbrace x\\rbrace ]$ such that $x\\in Z$ .", "As $A\\left[ I\\right] $ is inseparable, then by Lemma REF , we have $Z=\\lbrace x\\rbrace $ and $W=I$ .", "It follows that $I$ is a proper clan of $A$ which contradicts the fact that $A$ is indecomposable." ], [ "HL-equivalent matrices", "Let $V$ be a finite set and $k$ a positive integer.", "Two matrices $A=(a_{ij})_{i,j\\in V}$ , $B=(b_{ij})_{i,j\\in V}$ are $(\\le k)-$HL-equivalents if for any subset $X$ of $V$ , of size at most $k$ , we have $\\det (A\\left[ X\\right] )=\\det (B\\left[ X\\right] )$ .", "In the rest of this paper, we will consider only skew symmetric dense matrices.", "It is clear that if two skew symmetric dense matrices $A=(a_{ij})_{i,j\\in V}$ , $B=(b_{ij})_{i,j\\in V}$ are $(\\le 2)-$ HL-equivalents then, $b_{xy}=a_{xy}$ or $b_{xy}=-a_{xy}$ for all $x\\ne y\\in V$ .", "We define on $V$ two equivalence relations $\\mathcal {E}_{A,B}$ and $\\mathcal {D}_{A,B}$ as follows.", "For $x\\ne y\\in V$ , $x\\mathcal {E}_{A,B}y$ (resp.", "$x\\mathcal {D}_{A,B}y$ ) if there is a sequence $x_{0}=x,x_{1},\\ldots ,x_{n}=y$ such that $a_{x_{i}x_{i+1}}=b_{x_{i}x_{i+1}}$ (resp.", "$a_{x_{i}x_{i+1}}=-b_{x_{i}x_{i+1}} $ ) for $i=0,\\ldots ,n-1$ .", "Let $x\\in V$ , we denote by $\\mathcal {E}_{A,B} \\langle x \\rangle $ (resp.", "$\\mathcal {D}_{A,B}\\langle x \\rangle $ ) the equivalence class of $x$ for $\\mathcal {E}_{A,B}$ ( resp.", "for $\\mathcal {D}_{A,B}$ ) Let $V$ be a finite set and $A=(a_{ij})_{i,j\\in V}$ a dense skew-symmetric matrix.", "Let $V^{\\infty }:=V\\cup \\left\\lbrace \\infty \\right\\rbrace $ and $A^{\\infty }:=(a_{ij})_{i,j\\in V^{\\infty }}$ were $a_{\\infty \\infty }=0$ , $a_{\\infty j}=1$ and $a_{j\\infty }=-1$ for $j\\in V$ .", "Lemma 15 Let $V=\\left\\lbrace i,j,k\\right\\rbrace $ and $A=(a_{ij})_{i,j\\in V}$ , $B=(b_{ij})_{i,j\\in V}$ two dense skew-symmetric matrices such that $a_{ij}=b_{ij}$ , $a_{ki}=b_{ik}$ and $a_{kj}=b_{jk}$ .", "If $\\det \\left(A^{\\infty }\\right) =\\det \\left( B^{\\infty }\\right) $ then $a_{ik}=a_{jk}$ and $b_{ik}=b_{jk}$ .", "We have the following equalities : $\\det (A^{\\infty })=a_{ij}^{2}-2a_{ij}a_{ik}+2a_{ij}a_{jk}+a_{ik}^{2}-2a_{ik}a_{jk}+a_{jk}^{2}$ , $\\det (B^{\\infty })=b_{ij}^{2}-2b_{ij}b_{ik}+2b_{ij}b_{jk}+b_{ik}^{2}-2b_{ik}b_{jk}+b_{jk}^{2}$ .", "But, by hypothesis, $b_{ij}=a_{ij}$ , $b_{ik}=a_{ki}=-a_{ik}$ and $b_{jk}=a_{kj}=-a_{jk}$ , then we have, $\\det (B^{\\infty })=a_{ij}^{2}+2a_{ij}a_{ik}-2a_{ij}a_{jk}+a_{ik}^{2}-2a_{ik}a_{jk}+a_{jk}^{2}$ .", "It follows that, if $\\det (A^{\\infty })=\\det (B^{\\infty })$ then $a_{jk}=a_{ik}$ because $a_{ij}\\ne 0$ .", "Moreover, $b_{ik}=-a_{ik}=-a_{jk}=-b_{kj}=b_{jk}$ .", "Then, we have also $b_{ik}=b_{jk}$ .", "Lemma 16 Let $V$ be a finite set with $|V|\\ge 3$ and $A=(a_{ij})_{i,j\\in V}$ , $B=(b_{ij})_{i,j\\in V}$ two dense skew-symmetric matrices.", "If $A^{\\infty }$ and $B^{\\infty }$ are $(\\le 4)$ -HL-equivalent then the equivalence classes of $\\mathcal {E}_{A,B}$ and those of $\\mathcal {D}_{A,B}$ are clans of $A$ and of $B$ .", "First, note that $\\mathcal {D}_{A,B}=\\mathcal {E}_{A,B^{t}}$ and the matrices $B^{\\infty }$ , $(B^{t})^{\\infty }$ are $(\\le 4)$ -HL -equivalents.", "So, by interchanging $B$ and $B^{t}$ , it suffices to show that an equivalence class $C$ of $\\mathcal {E}_{A,B}$ is a clan of $A$ and of $B$ .", "For this, let $x\\ne y\\in C$ and $z\\in V\\setminus C$ .", "By definition of $\\mathcal {E}_{A,B}$ , there exists a sequence $x_{0}=x,x_{1},\\ldots ,x_{r}=y$ such that $a_{x_{i}x_{i+1}}=b_{x_{i}x_{i+1}}$ for $i=0,\\ldots ,r-1$ .", "As $z\\notin C$ , then for $j=0,\\ldots ,r$ , we have $a_{zx_{j}}=$ $-b_{zx_{j}}=b_{x_{j}z}$ .", "It follows, from Lemma REF , that $a_{zx_{i}}=a_{zx_{i+1}}$ and $b_{zx_{i}}=b_{zx_{i+1}}$ for $i=0,\\ldots ,r-1$ .", "Therefore $a_{zx}=a_{zy}$ , $b_{zx}=b_{zy}$ and hence $C$ is a clan of $A$ and of $B$ .", "Proposition 17 Let $V$ be a finite set with $|V|\\ge 3$ and $A=(a_{ij})_{i,j\\in V}$ , $B=(b_{ij})_{i,j\\in V}$ two dense skew-symmetric matrices such that $A^{\\infty }$ and $B^{\\infty }$ are $(\\le 4)$ -HL-equivalents.", "If $A$ is inseparable, then, $\\mathcal {D}_{A,B}$ or $\\mathcal {E}_{A,B}$ has at least two equivalence classes.", "The result is trivial for $|V|=2$ or $|V|=3$ .", "For $|V|=4$ , let $V:=\\lbrace i,j,k,l\\rbrace $ .", "Suppose that $V$ is an equivalence class for $\\mathcal {D}_{A,B}$ and for $\\mathcal {E}_{A,B}$ .", "We will show that $A$ is separable.", "Without loss of generality, we can assume that $a_{il}=-b_{il}$ , $a_{lk}=-b_{lk}$ , $a_{kj}=-b_{kj}$ , and $a_{ki}=b_{ki}$ , $a_{ij}=b_{ij}$ , $a_{jl}=b_{jl}$ .", "As $A^{\\infty }$ and $B^{\\infty }$ are $(\\le 4)$ -HL-equivalents, then $\\det (A^{\\infty }\\left[ \\left\\lbrace \\infty \\right\\rbrace \\cup X\\right] )=\\det (B^{\\infty }\\left[ \\left\\lbrace \\infty \\right\\rbrace \\cup X\\right] )$ for $X=\\lbrace i,j,k\\rbrace $ , $X=\\lbrace i,j,l\\rbrace $ , $X=\\left\\lbrace i,k,l\\right\\rbrace $ and $X=\\left\\lbrace i,k,l\\right\\rbrace $ .", "It follows that $4a_{ij}a_{jk}-4a_{ik}a_{jk}=0$ , $-4a_{ij}a_{il}-4a_{il}a_{jl}=0$ , $4a_{ik}a_{kl}-4a_{ik}a_{il}=0$ , $-4a_{jk}a_{jl}-4a_{jl}a_{kl}=0$ and then, $a_{ij}=a_{ik}$ , $a_{ij}=-a_{jl}$ , $a_{kl}=a_{il}$ , $a_{jk}=-a_{kl}$ .", "Furthermore, $\\det (A)=\\det (B)$ , so $4a_{kl}a_{ij}\\left( a_{ik}a_{jl}-a_{il}a_{jk}\\right) =0$ and consequently $a_{ik}a_{jl}-a_{il}a_{jk}=0$ .", "We conclude that $a_{ij}^{2}-a_{kl}^{2}=0$ .", "If $a_{ij}=a_{kl}$ then $a_{ij}=a_{ik}=a_{il}=a_{kl}=a_{kj}=a_{lj}$ and hence $\\lbrace i,j\\rbrace $ and $\\lbrace k,l\\rbrace $ are two clans of $A$ .", "If $a_{ij}=-a_{kl}$ then $a_{ij}=a_{ik}=a_{li}=a_{lj}=a_{jk}=a_{lk}$ .", "So $\\lbrace j,k\\rbrace $ and $\\lbrace i,l\\rbrace $ are two clans of $A$ and hence $A$ is separable.", "As in the proof of Proposition 3 (see [3]), we will continue by induction on $|V|$ for $|V|\\ge 5$ .", "Suppose, by contradiction, that $A$ is inseparable and that $V$ is an equivalence class for $\\mathcal {D}_{A,B}$ and for $\\mathcal {E}_{A,B}$ .", "By Corollary REF , there is $v\\in V$ such that the submatrix $A[V\\setminus \\lbrace v\\rbrace ]$ is inseparable.", "Let $\\mathcal {D}^{\\prime }:=\\mathcal {D}_{A\\left[ V\\setminus \\left\\lbrace v\\right\\rbrace \\right] ,B\\left[ V\\setminus \\left\\lbrace v\\right\\rbrace \\right] }$ and $\\mathcal {E}^{\\prime }:=\\mathcal {E}_{A\\left[ V\\setminus \\left\\lbrace v\\right\\rbrace \\right] ,B\\left[V\\setminus \\left\\lbrace v\\right\\rbrace \\right] }$ .", "By induction hypothesis and without loss of generality, we can assume that $\\mathcal {D}^{\\prime }$ has at least two equivalence classes.", "As $\\mathcal {E}\\langle v\\rangle =V$ , then there exist $x\\ne v$ such that $a_{xv}=b_{xv}$ .", "Moreover, the matrix $A[V\\setminus \\lbrace v\\rbrace ]$ is inseparable, then there exist $y_{1}\\in \\mathcal {D}^{\\prime }\\langle x\\rangle $ and $y_{2}\\in (V\\setminus \\left\\lbrace v\\right\\rbrace )\\setminus \\mathcal {D}^{\\prime }\\langle x\\rangle $ such that $a_{xv}\\ne $ $b_{y_{1}y_{2}}$ .", "Since $\\mathcal {D}^{\\prime }\\langle y_{2}\\rangle V$ and $\\mathcal {D}_{A,B}\\langle y_{2}\\rangle =V$ , there exists $z_{2}\\in \\mathcal {D}^{\\prime }\\langle y_{2}\\rangle $ and $w_{1}\\in V\\setminus \\mathcal {D}^{\\prime }\\langle y_{2}\\rangle $ such that $a_{z_{2}w_{1}}\\ne b_{z_{2}w_{1}}$ .", "But, $\\mathcal {D}^{\\prime }\\langle y_{2}\\rangle =\\mathcal {D}^{\\prime }\\langle z_{2}\\rangle $ , so for any $w\\in (V\\setminus \\left\\lbrace v\\right\\rbrace )\\setminus \\mathcal {D}^{\\prime }\\langle y_{2}\\rangle $ we have $a_{z_{2}w}=b_{z_{2}w}$ .", "We have necessarily, $w_{1}=v$ and hence $a_{z_{2}v}\\ne b_{z_{2}v}$ .", "Furthermore, $\\mathcal {D}^{\\prime }\\langle y_{2}\\rangle =\\mathcal {D}^{\\prime }\\langle z_{2}\\rangle $ and $\\mathcal {D}^{\\prime }\\langle y_{2}\\rangle \\cap \\mathcal {D}^{\\prime }\\langle x\\rangle =\\emptyset $ because $y_{2}\\in (V\\setminus \\left\\lbrace v\\right\\rbrace )\\setminus \\mathcal {D}^{\\prime }\\langle x\\rangle $ , then $z_{2}$ $\\notin \\mathcal {D}^{\\prime }\\langle x\\rangle $ and hence $a_{z_{2}x}=b_{z_{2}x}$ .", "To obtain the contradiction, we will apply Lemma REF to $X=\\left\\lbrace v,x,z_{2}\\right\\rbrace $ .", "By the foregoing, we have $a_{z_{2}v}\\ne b_{z_{2}v}$ , $a_{xv}=b_{xv}$ and $a_{z_{2}x}=b_{z_{2}x}$ .", "But, by Lemma REF , $\\mathcal {D}^{\\prime }\\langle x\\rangle $ and $\\mathcal {D}^{\\prime }\\langle y_{2}\\rangle $ are disjoint clans of $A[V\\setminus \\lbrace v\\rbrace ]$ , so $a_{y_{1}y_{2}}=a_{xz_{2}}$ because $x,y_{1}\\in \\mathcal {D}^{\\prime }\\langle x\\rangle $ and $y_{2},z_{2}\\in \\mathcal {D}^{\\prime }\\langle y_{2}\\rangle $ .", "It follows that $a_{xz_{2}}\\ne a_{xv}$ because $a_{xv}\\ne a_{y_{1}y_{2}}$ and this contradicts Lemma REF ." ], [ "Proof of the main Theorem", "By using the proposition REF and Lemma REF , we obtain the following result which is the key of the proof of the main Theorem.", "Proposition 18 Let $V$ be a finite set with $|V|\\ge 2$ and $A=(a_{ij})_{i,j\\in V}$ , $B=(b_{ij})_{i,j\\in V}$ two dense skew-symmetric matrices such that $A^{\\infty }$ and $B^{\\infty }$ are $(\\le 4)$ - HL-equivalent.", "If $A$ is indecomposable then $B=A$ or $B=A^{t}$ .", "Assume that the matrix $A$ is indecomposable, then it is inseparable.", "By using proposition REF and by interchanging $A$ and $A^{t}$ , we can assume that $\\mathcal {D}_{A,B}$ at least two equivalence classes.", "By Lemma REF these classes are clans of $A$ .", "So these are singletons because $A$ is indecomposable.", "Consequently $B=A$ .", "Let $u\\in V$ and $D:=(d_{ij})_{i,j\\in V}$ (resp.", "$D^{\\prime }:=(d_{ij}^{\\prime })_{i,j\\in V}$ ) the diagonal matrix such that $d_{uu}=1$ and $d_{zz}=\\frac{1}{a_{uz}}$ (resp.", "$d_{uu}^{\\prime }=1$ and $d_{zz}^{\\prime }=\\frac{1}{b_{uz}}$ ) if $z\\ne u$ .", "We consider the two matrices $\\widehat{A}:=DAD:=(\\widehat{a}_{ij})_{i,j\\in V}$ and $\\widehat{B}:=D^{\\prime }BD^{\\prime }:=(\\widehat{b}_{ij})_{i,j\\in V}$ .", "As the matrix $A$ is HL-indecomposable, then by lemma REF , the matrix $\\widehat{A}$ is also HL-indecomposable.", "However $\\widehat{a}_{vj}=\\widehat{b}_{vj}=1$ for $j\\ne v$ , then by Lemma REF , $\\widehat{A}\\left[ V\\setminus \\left\\lbrace v\\right\\rbrace \\right] $ is indecomposable.", "We will apply Proposition REF .", "For this, let $X$ be a subset of $V$ such that $\\left|X\\right|\\le 4$ .", "We have $\\det (\\widehat{A}\\left[ X\\right] )=\\left( \\det D\\left[ X\\right]\\right) ^{2}\\det (A\\left[ X\\right] )$ .", "Similarly, $\\det (\\widehat{B}\\ \\left[ X\\right] )=\\left( \\det D^{\\prime }\\left[ X\\right] \\right) ^{2}\\det (B\\left[ X\\right] )$ .", "But for all $z\\ne u$ , $b_{uz}=a_{uz}$ or $b_{uz}=-a_{uz}$ , then $\\left( \\det D\\left[ X\\right] \\right) ^{2}=\\left( \\det D^{\\prime }\\left[ X\\right] \\right) ^{2}$ .", "Moreover, by hypothesis, $\\det (A[X])=\\det (B[X])$ , then $\\det (\\widehat{A}\\left[ X\\right] )=\\det (\\widehat{B}\\ \\left[ X\\right] )$ and hence the matrices $\\widehat{A}$ , $\\widehat{B}\\ $ are $(\\le 4)$ -HL-equivalents.", "Now, by Proposition REF , $\\widehat{A}\\left[V\\setminus \\left\\lbrace v\\right\\rbrace \\right] =\\widehat{B}\\left[ V\\setminus \\left\\lbrace v\\right\\rbrace \\right] $ or $\\widehat{A}\\left[ V\\setminus \\left\\lbrace v\\right\\rbrace \\right]=(\\widehat{B}\\left[ V\\setminus \\left\\lbrace v\\right\\rbrace \\right] )^{t}$ .", "If $\\widehat{A}\\left[ V\\setminus \\left\\lbrace v\\right\\rbrace \\right] =\\widehat{B}\\left[V\\setminus \\left\\lbrace v\\right\\rbrace \\right] $ then $\\widehat{A}=\\widehat{B}$ and hence $B=(D^{\\prime })^{-1}DA(D^{\\prime })^{-1}D$ .", "Suppose that $\\widehat{A}\\left[ V\\setminus \\left\\lbrace v\\right\\rbrace \\right] =(\\widehat{B}\\left[V\\setminus \\left\\lbrace v\\right\\rbrace \\right] )^{t}$ and let $\\Delta :=(\\delta _{ij})_{i,j\\in V}$ the diagonal matrix such that $\\delta _{uu}=1$ and $\\delta _{zz}=-1$ if $z\\ne u$ .", "Clearly, $\\Delta \\widehat{A}\\Delta =(\\widehat{B})^{t}$ .", "It follows that $B^{t}=(D^{\\prime })^{-1}D\\Delta A(D^{\\prime })^{-1}D\\Delta $ .", "However for all $z\\in V$ , $d_{zz}=d_{zz}^{\\prime } $ or $d_{zz}=-d_{zz}^{\\prime }$ .", "then $(D^{\\prime })^{-1}D$ and $(D^{\\prime })^{-1}D\\Delta $ are diagonal matrices with diagonal entries in $\\left\\lbrace -1,1\\right\\rbrace $ ." ], [ "Some remarks about Theorem ", "Let $A$ be a skew-symmetric matrix.", "Clearly, the principal minors of order 2 determine the off diagonal entries of $A$ up to sign.", "So, generically there are at most finitely many skew-symmetric matrices with equal corresponding principal minors as a given matrix, and one should expect Theorem REF to hold for sufficiently generic skew-symmetric matrices.", "Nevertheless, for dense skew-symmetric matrix, Theorem REF is the best one.", "To see this, let $A=(a_{ij})_{i,j\\in V}$ be an HL-decomposable matrix and $X$ a nontrivial HL-clan of $A$ .", "Consider the matrix $B=(b_{ij})_{i,j\\in V}$ such that $b_{ij}=-a_{ij}$ if $i$ ,$j\\in X$ and $b_{ij}=a_{ij}$ otherwise.", "By adapting the proof of Lemma 5 (see [8] ), we can prove that $A$ and $B$ have the same principal minors, but they are not diagonally similar up to transposition.", "We can ask if Theorem REF can be obtained from Theorem REF via specialization to skew-symmetric matrices.", "For this, we must prove that for a dense skew-symmetric matrix, the principal minors of order at most 4 determine the rest of its principal minors.", "One way to do this is to give an expression of the determinant of a skew-symmetric matrix of order at least 5 from its principal minors.", "More precisely, let $M$ be a generic skew-symmetric matrix of order $n$ where $n\\ge 5$ and the entries $x_{ij}$ with $i$ $<j$ are indeterminates.", "Let $R=\\mathbb {K}[x_{ij}|1\\le i<j\\le n]$ be the polynomial ring generated by these indeteminates.", "The problem is to find an expression of $\\det \\left( M\\right) $ from the collection of the principal minors $\\det \\left( M\\left[ \\alpha \\right]\\right) $ where $\\alpha $ is a subset of $\\left\\lbrace 1,\\ldots ,n\\right\\rbrace $ of size a most 4.", "Because of the example REF , such expression can not be a polynomial.", "Nevertheless, it is reasonable to suggest the following problem Problem 19 Let $\\left\\langle n\\right\\rangle _{4}$ the collection of all subset $\\alpha \\subseteq \\left\\lbrace 1,\\ldots ,n\\right\\rbrace $ with $\\left|\\alpha \\right|$ $=2$ or 4 and consider the polynomial ring $R=\\mathbb {K}[X_{\\alpha },\\alpha $ $\\in \\left\\langle n\\right\\rangle _{4}]$ where $X_{\\alpha }$ $,\\alpha $ $\\in \\left\\langle n\\right\\rangle _{4}$ are indeterminates.", "Is there a polynomial $Q(X_{\\alpha },\\alpha \\in \\left\\langle n\\right\\rangle _{4})\\in R$ such that $\\prod \\limits _{1\\le i<j\\le n} x_{ij}^{2}\\det \\left( M\\right) =Q(\\det \\left( M\\left[ \\alpha \\right] \\right) ,\\alpha \\in \\left\\langle n\\right\\rangle _{4})$ ?", "Note that a positive answer to this problem combined with the Theorem REF allows to obtain our main theorem.", "Finally, as the principal minors are squares of the corresponding Pfaffians, we can strengthen the assumptions of the Problem REF by replacing the principal minors by sub-pfaffians.", "However, two skew-symmetric matrices with the same corresponding principal sub-pfaffians of order 2 are equal.", "So, it is interesting to consider the following problem.", "Problem 20 Given a positive integer $n\\ge 5$ , what is the relationship between two skew-symmetric matrices of orders $n$ which differ up to the sign of their off-diagonal terms and having equal corresponding principal Pfaffian minors of order 4 ?" ] ]
1403.0095
[ [ "On the Complexity of Computing Two Nonlinearity Measures" ], [ "Abstract We study the computational complexity of two Boolean nonlinearity measures: the nonlinearity and the multiplicative complexity.", "We show that if one-way functions exist, no algorithm can compute the multiplicative complexity in time $2^{O(n)}$ given the truth table of length $2^n$, in fact under the same assumption it is impossible to approximate the multiplicative complexity within a factor of $(2-\\epsilon)^{n/2}$.", "When given a circuit, the problem of determining the multiplicative complexity is in the second level of the polynomial hierarchy.", "For nonlinearity, we show that it is #P hard to compute given a function represented by a circuit." ], [ "Introduction", "In many cryptographical settings, such as stream ciphers, block ciphers and hashing, functions being used must be deterministic but should somehow “look” random.", "Since these two desires are contradictory in nature, one might settle with functions satisfying certain properties that random Boolean functions possess with high probability.", "One property is to be somehow different from linear functions.", "This can be quantitatively delineated using so called “nonlinearity measures”.", "Two examples of nonlinearity measures are the nonlinearity, i.e.", "the Hamming distance to the closest affine function, and the multiplicative complexity, i.e.", "the smallest number of AND gates in a circuit over the basis $(\\wedge ,\\oplus ,1)$ computing the function.", "For results relating these measures to each other and cryptographic properties we refer to [6], [4], and the references therein.", "The important point for this paper is that there is a fair number of results on the form “if $f$ has low value according to measure $\\mu $ , $f$ is vulnerable to the following attack ...”.", "Because of this, it was a design criteria in the Advanced Encryption Standard to have parts with high nonlinearity [10].", "In a concrete situation, $f$ is an explicit, finite function, so it is natural to ask how hard it is to compute $\\mu $ given (some representation of) $f$ .", "In this paper, the measure $\\mu $ will be either multiplicative complexity or nonlinearity.", "We consider the two cases where $f$ is being represented by its truth table, or by a circuit computing $f$ .", "We should emphasize that multiplicative complexity is an interesting measure for other reasons than alone being a measure of nonlinearity: In many applications it is harder, in some sense, to handle AND gates than XOR gates, so one is interested in a circuit over $(\\wedge ,\\oplus ,1)$ with a small number of AND gates, rather than a circuit with the smallest number of gates.", "Examples of this include protocols for secure multiparty computation (see e.g.", "[8], [15]), non-interactive secure proofs of knowledge [3], and fully homomorphic encryption (see for example [20]).", "It is a main topic in several papers (see e.g.", "[5], [7], [9]Here we mean concrete finite functions, as opposed to giving good (asymptotic) upper bounds for an infinite family of functions) to find circuits with few AND gates for specific functions using either exact or heuristic techniques.", "Despite this and the applications mentioned above, it appears that the computational hardness has not been studied before.", "The two measures have very different complexities, depending on the representation of $f$ .", "In the following section, we introduce the problems and necessary definitions.", "All our hardness results will be based on assumptions stronger than $\\mathbf {P}\\ne \\mathbf {NP}$ , more precisely the existence of pseudorandom function families and the “Strong Exponential Time Hypothesis”.", "In Section  we show that if pseudorandom function families exist, the multiplicative complexity of a function represented by its truth table cannot be computed (or even approximated with a factor $(2-\\epsilon )^{n/2}$ ) in polynomial time.", "This should be contrasted to the well known fact that nonlinearity can be computed in almost linear time using the Fast Walsh Transformation.", "In Section , we consider the problems when the function is represented by a circuit.", "We show that in terms of time complexity, under our assumptions, the situations differ very little from the case where the function is represented by a truth table.", "However, in terms of complexity classes, the picture looks quite different: Computing the nonlinearity is $\\#\\mathbf {P}$ hard, and multiplicative complexity is in the second level of the polynomial hierarchy." ], [ "Preliminaries", "In the following, we let $\\mathbb {F}_2$ be the finite field of size 2 and $\\mathbb {F}_2^n$ the $n$ -dimensional vector space over $\\mathbb {F}_2$ .", "We denote by $B_n$ the set of Boolean functions, mapping from $\\mathbb {F}_2^n$ into $\\mathbb {F}_2$ .", "We say that $f\\in B_n$ is affine if there exist $\\mathbf {a}\\in \\mathbb {F}_2^n,c\\in \\mathbb {F}_2$ such that $f(\\mathbf {x})=\\mathbf {a}\\cdot \\mathbf {x}+c$ and linear if $f$ is affine with $f(\\mathbf {0})=0$ , with arithmetic over $\\mathbb {F}_2$ .", "This gives the symbol “$+$ ” an overloaded meaning, since we also use it for addition over the reals.", "It should be clear from the context, what is meant.", "In the following an XOR-AND circuit is a circuit with fanin 2 over the basis $(\\wedge ,\\oplus ,1)$ (arithmetic over $GF(2)$ ).", "All circuits from now on are assumed to be XOR-AND circuits.", "We adopt standard terminology for circuits (see e.g.", "[21]).", "If nothing else is specified, for a circuit $C$ we let $n$ be the number of inputs and $m$ be the number of gates, which we refer to as the size of $C$ , denoted $|C|$ .", "For a circuit $C$ we let $f_C$ denote the function computed by $C$ , and $c_\\wedge (C)$ denote the number of AND gates in $C$ .", "For a function $f\\in B_n$ , the multiplicative complexity of $f$ , denoted $c_\\wedge (f)$ , is the smallest number of AND gates necessary and sufficient in an XOR-AND circuit computing $f$ .", "The nonlinearity of a function $f$ , denoted $NL(f)$ is the Hamming distance to its closest affine function, more precisely $NL(f) = 2^{n} -\\max _{\\mathbf {a}\\in \\mathbb {F}_2^n,c\\in \\mathbb {F}_2}|\\lbrace \\mathbf {x}\\in \\mathbb {F}_2^{n}|f(\\mathbf {x})=\\mathbf {a}\\cdot \\mathbf {x}+ c \\rbrace |.$ We consider four decision problems in this paper: $NL_C$ , $NL_{TT}$ , $MC_C$ and $MC_{TT}$ .", "For $NL_C$ (resp $MC_C$ ) the input is a circuit and a target $s\\in \\mathbb {N}$ and the goal is to determine whether the nonlinearity (resp.", "multiplicative complexity) of $f_C$ is at most $s$ .", "For $NL_{TT}$ (resp.", "$MC_{TT}$ ) the input is a truth table of length $2^n$ of a function $f\\in B_n$ and a target $s\\in \\mathbb {N}$ , with the goal to determine whether the nonlinearity (resp.", "multiplicative complexity) of $f$ is at most $s$ .", "We let $a \\in _R D$ denote that $a$ is distributed uniformly at random from $D$ .", "We will need the following definition: A family of Boolean functions $f=\\lbrace f_{n}\\rbrace _{n\\in \\mathbb {N}}$ , $f_n\\colon \\lbrace 0,1 \\rbrace ^n \\times \\lbrace 0,1\\rbrace ^n \\rightarrow \\lbrace 0,1 \\rbrace $ , is a pseudorandom function family if $f$ can be computed in polynomial time and for every probabilistic polynomial time oracle Turing machine $A$ , $|\\Pr _{k\\in _R \\lbrace 0,1\\rbrace ^n}[A^{f_n(k,\\cdot )}(1^n)=1] -\\Pr _{g\\in _R B_n}[A^{g(\\cdot )}(1^n)=1]|\\le n^{-\\omega (1)}.$ Here $A^{H}$ denotes that the algorithm $A$ has oracle access to a function $H$ , that might be $f_n(\\mathbf {k},\\cdot )$ for some $\\mathbf {k}\\in \\mathbb {F}_2^n$ or a random $g\\in B_n$ , for more details see [1].", "Some of our hardness results will be based on the following assumption.", "Assumption 1 There exist pseudorandom function families.", "It is known that pseudorandom function families exist if one-way functions exist [11], [12], [1], so we consider Assumption REF to be very plausible.", "We will also use the following assumptions on the exponential complexity of $SAT$ , due to Impagliazzo and Paturi.", "Assumption 2 (Strong Exponential Time Hypothesis [13]) For any fixed $c<1$ , no algorithm runs in time $2^{cn}$ and computes $SAT$ correctly." ], [ "Truth Table as Input", "It is a well known result that given a function $f\\in B_n$ represented by a truth table of length $2^n$ , the nonlinearity can be computed using $O(n2^{n})$ basic arithmetic operations.", "This is done using the “Fast Walsh Transformation” (See [19] or chapter 1 in [16]).", "In this section we show that the situation is different for multiplicative complexity: Under Assumption REF , $MC_{TT}$ cannot be computed in polynomial time.", "In [14], Kabanets and Cai showed that if subexponentially strong pseudorandom function families exist, the Minimum Circuit Size Problem (MCSP) (the problem of determining the size of a smallest circuit of a function given its truth table) cannot be solved in polynomial time.", "The proof goes by showing that if MCSP could be solved in polynomial time this would induce a natural combinatorial property (as defined in [17]) useful against circuits of polynomial size.", "Now by the celebrated result of Razborov and Rudich [17], this implies the nonexistence of subexponential pseudorandom function families.", "Our proof below is similar in that we use results from [2] in a way similar to what is done in [14], [17] (see also the excellent exposition in [1]).", "However instead of showing the existence of a natural and useful combinatorial property and appealing to limitations of natural proofs, we give an explicit polynomial time algorithm for breaking any pseudorandom function family, contradicting Assumption REF .", "Under Assumption REF , on input a truth table of length $2^n$ , $MC_{TT}$ cannot be computed in time $2^{O(n)}$ .", "Let $\\lbrace f_n\\rbrace _{n\\in \\mathbb {N}}$ be a pseudorandom function family.", "Since $f$ is computable in polynomial time it has circuits of polynomial size (see e.g.", "[1]), so we can choose $c\\ge 2$ such that $c_\\wedge (f_n)\\le n^c$ for all $n\\ge 2$ .", "Suppose for the sake of contradiction that some algorithm computes $MC_{TT}$ in time $2^{O(n)}$ .", "We now describe an algorithm that breaks the pseudorandom function family.", "The algorithm has access to an oracle $H\\in B_n$ , along with the promise either $H(\\mathbf {x})=f_n(\\mathbf {k},\\mathbf {x})$ for $\\mathbf {k}\\in _R \\mathbb {F}_2^n$ or $H(\\mathbf {x})=g(\\mathbf {x})$ for $g\\in _R B_n$ .", "The goal of the algorithm is to distinguish between the two cases.", "Specifically our algorithm will return 0 if $H(\\mathbf {x})=f(\\mathbf {k},\\mathbf {x})$ for some $\\mathbf {k}\\in \\mathbb {F}_2^n$ , and if $H(\\mathbf {x})=g(\\mathbf {x})$ it will return 1 with high probability, where the probability is only taken over the choice of $g$ .", "Let $s=10c\\log n$ and define $h\\in B_s$ as $h(\\mathbf {x})=H(\\mathbf {x}0^{n-s})$ .", "Obtain the complete truth table of $h$ by querying $H$ on all the $2^{s}=2^{10c\\log n}=n^{10c}$ points.", "Now compute $c_\\wedge (h)$ .", "By assumption this can be done in time $poly(n^{10c})$ .", "If $c_\\wedge (h)>n^c$ , output 1, otherwise output 0.", "We now want to argue that this algorithm correctly distinguishes between the two cases.", "Suppose first that $H(\\mathbf {x})=f_n(\\mathbf {k},\\cdot )$ for some $\\mathbf {k}\\in \\mathbb {F}_2^n$ .", "One can think of $h$ as $H$ where some of the input bits are fixed.", "But in this case, $H$ can also be thought of as $f_n$ with $n$ of the input bits fixed.", "Now take the circuit for $f_n$ with the minimal number of AND gates.", "Fixing the value of some of the input bits clearly cannot increase the number of AND gates, hence $c_\\wedge (h) \\le c_\\wedge (f_n)\\le n^c$ .", "Now it remains to argue that if $H$ is a random function, we output 1 with high probability.", "We do this by using the following lemma.", "[Boyar, Peralta, Pochuev] For all $s\\ge 0$ , the number of functions in $B_s$ that can be computed with an XOR-AND circuit using at most $k$ AND gates is at most $2^{k^2+2k+2ks+s+1}$ .", "If $g$ is random on $B_n$ , then $h$ is random on $B_{10c\\log n}$ , so the probability that $c_\\wedge (h)\\le n^c$ is at most: $\\frac{2^{(n^c)^2+2(n^c)+2(n^c)(10c\\log n)+10c\\log n+1}}{ 2^{2^{10c\\log n}}}.$ This tends to 0, so if $H$ is a random function the algorithm returns 0 with probability $o(1)$ .", "In total we have $|\\Pr _{k\\in _R \\lbrace 0,1\\rbrace ^n}[A^{f_n(k,\\cdot )}(1^n)=1] -\\Pr _{g\\in _R B_n}[A^{g(\\cdot )}(1^n)=1]|=|0 - (1- o(1))|,$ concluding that if the polynomial time algoritm for deciding $MC_{TT}$ exists, $f$ is not a pseudorandom function family.", "$\\Box $ A common question to ask about a computationally intractable problem is how well it can be approximated by a polynomial time algorithm.", "An algorithm approximates $c_\\wedge (f)$ with approximation factor $\\rho (n)$ if it always outputs some value in the interval $[c_\\wedge (f),\\rho (n)c_\\wedge (f)]$ .", "By refining the proof above, we see that it is hard to compute $c_\\wedge (f)$ within even a modest factor.", "For every constant $\\epsilon >0$ , under Assumption REF , no algorithm takes the $2^n$ bit truth table of a function $f$ and approximates $c_\\wedge (f)$ with $\\rho (n) \\le (2-\\epsilon )^{n/2}$ in time $2^{O(n)}$ .", "Assume for the sake of contradiction that the algorithm $A$ violates the theorem.", "The algorithm breaking any pseudorandom function family works as the one in the previous proof, but instead we return 1 if the value returned by $A$ is at least $T=(n^c+1)\\cdot (2-\\epsilon )^{n/2}$ .", "Now arguments similar to those in the proof above show that if $A$ returns a value larger than $T$ , $H$ must be random, and if $H$ is random, $h$ has multiplicative complexity at most $(n^c+1)\\cdot (2-\\epsilon )^{n/2}$ with probability at most $\\frac{2^{\\left( (n^c+1)\\cdot (2-\\epsilon )^{(10c\\log n)/2}\\right)^2 +2(n^c+1)\\cdot (2-\\epsilon )^{10c\\log n/2} 10c\\log n +10c\\log n +1}}{2^{2^{10c\\log n}}}$ This tends to zero, implying that under the assumption on $A$ , there is no pseudorandom function family.", "$\\Box $" ], [ "Circuit as Input", "From a practical point of view, the theorems and might seem unrealistic.", "We are allowing the algorithm to be polynomial in the length of the truth table, which is exponential in the number of variables.", "However most functions used for practical purposes admit small circuits.", "To look at the entire truth table might (and in some cases should) be infeasible.", "When working with computational problems on circuits, it is somewhat common to consider the running time in two parameters; the number of inputs to the circuit, denoted by $n$ , and the size of the circuit, denoted by $m$ .", "In the following we assume that $m$ is polynomial in $n$ .", "In this section we show that even determining whether a circuit computes an affine function is $\\mathbf {coNP}$ -complete.", "In addition $NL_C$ can be computed in time $poly(m)2^n$ , and is $\\# \\mathbf {P}$ -hard.", "Under Assumption REF , $MC_C$ cannot be computed in time $poly(m)2^{O(n)}$ , and is contained in the second level of the polynomial hierarchy.", "In the following, we denote by $AFFINE$ the set of circuits computing affine functions.", "$AFFINE$ is $\\mathbf {coNP}$ complete.", "First we show that it actually is in $\\mathbf {coNP}$ .", "Suppose $C\\notin AFFINE$ .", "Then if $f_C(\\mathbf {0})=0$ , there exist $\\mathbf {x},\\mathbf {y}\\in \\mathbb {F}_2^n$ such that $f_C(\\mathbf {x}+\\mathbf {y})\\ne f_C(\\mathbf {x})+f_C(\\mathbf {y})$ and if $C(\\mathbf {0})=1$ , there exists $\\mathbf {x},\\mathbf {y}$ such that $C(\\mathbf {x}+\\mathbf {y})+1\\ne C(\\mathbf {x})+C(\\mathbf {y})$ .", "Given $C,\\mathbf {x}$ and $\\mathbf {y}$ this can clearly be computed in polynomial time.", "To show hardness, we reduce from $TAUTOLOGY$ , which is $\\mathbf {coNP}$ -complete.", "Let $F$ be a formula on $n$ variables, $\\mathbf {x}_1,\\ldots ,\\mathbf {x}_n$ .", "Consider the following reduction: First compute $c=F(0^n)$ , then for every $\\mathbf {e^{(i)}}$ (the vector with all coordinates 0 except the $i$ th) compute $F(\\mathbf {e^{(i)}})$ .", "If any of these or $c$ are 0, clearly $F\\notin TAUTOLOGY$ , so we reduce to a circuit trivially not in $AFFINE$ .", "We claim that $F$ computes an affine function if and only if $F\\in TAUTOLOGY$ .", "Suppose $F$ computes an affine function, then $F(\\mathbf {x})=\\mathbf {a}\\cdot \\mathbf {x}+c$ for some $\\mathbf {a}\\in \\mathbb {F}_2^n$ .", "Then for every $\\mathbf {e^{(i)}}$ , we have $F(\\mathbf {e^{(i)}})=\\mathbf {a}_i+1=1=F(\\mathbf {0}),$ so we must have that $\\mathbf {a}=\\mathbf {0}$ , and $F$ is constant.", "Conversely if it is not affine, it is certainly not constant.", "In particular it is not a tautology.$\\Box $ So even determining whether the multiplicative complexity or nonlinearity is 0 is $\\mathbf {coNP}$ complete.", "In the light of the above reduction, any algorithm for $AFFINE$ induces an algorithm for $SAT$ with essentially the same running time, so under Assumption REF , AFFINE needs time essentially $2^n$ .", "This should be contrasted with the fact that the seemingly harder problem of computing $NL_C$ can be done in time $poly(m)2^n$ by first computing the entire truth table and then using the Fast Walsh Transformation.", "Despite the fact that $NL_C$ does not seem to require much more time to compute than $AFFINE$ , it is hard for a much larger complexity class.", "$NL_C$ is $\\#\\mathbf {P}$ -hard.", "We reduce from $\\# SAT$ .", "Let the circuit $C$ on $n$ variables be an instance of $\\# SAT$ .", "Consider the circuit $C^{\\prime }$ on $n+10$ variables, defined by $C^{\\prime }(\\mathbf {x}_1,\\ldots , \\mathbf {x}_{n+10})=C(\\mathbf {x}_1,\\ldots , \\mathbf {x}_n)\\wedge \\mathbf {x}_{n+1} \\wedge \\mathbf {x}_{n+2}\\wedge \\ldots \\wedge \\mathbf {x}_{n+10}.$ First we claim that independently of $C$ , the best affine approximation of $f_{C^{\\prime }}$ is always 0.", "Notice that 0 agrees with $f_{C^{\\prime }}$ whenever at least one of $x_{n+1},\\ldots ,x_{n+10}$ is 0, and when they are all 1 it agrees on $|\\lbrace \\mathbf {x}\\in \\mathbb {F}_2^n | f_{C^{\\prime }}(\\mathbf {x})=0\\rbrace |$ many points.", "In total 0 and $f_{C^{\\prime }}$ agree on $(2^{10}-1)2^{n}+|\\lbrace \\mathbf {x}\\in \\mathbb {F}_2^n | f_{C}(\\mathbf {x})=0\\rbrace |$ inputs.", "To see that any other affine function approximates $f_{C^{\\prime }}$ worse than 0, notice that any nonconstant affine function is balanced and thus has to disagree with $f_{C^{\\prime }}$ very often.", "The nonlinearity of $f_{C^{\\prime }}$ is therefore $NL(f_{C^{\\prime }}) &= 2^{n+10} -\\max _{\\mathbf {a}\\in \\mathbb {F}_2^{n+10},c\\in \\mathbb {F}_2}|\\lbrace \\mathbf {x}\\in \\mathbb {F}_2^{n+10}|f_{C^{\\prime }}(\\mathbf {x})=\\mathbf {a}\\cdot \\mathbf {x}+ c \\rbrace |\\\\&=2^{n+10} -|\\lbrace \\mathbf {x}\\in \\mathbb {F}_2^{n+10}|f_{C^{\\prime }}(\\mathbf {x})=0 \\rbrace |\\\\&=2^{n+10}-\\left((2^{10}-1)2^{n}+|\\lbrace \\mathbf {x}\\in \\mathbb {F}_2^n | f_{C}(\\mathbf {x})=0\\rbrace |\\right)\\\\&=2^{n}-|\\lbrace \\mathbf {x}\\in \\mathbb {F}_2^n | f_{C}(\\mathbf {x})=0\\rbrace |\\\\&=|\\lbrace \\mathbf {x}\\in \\mathbb {F}_2^n | f_{C}(\\mathbf {x})=1\\rbrace |\\\\$ So the nonlinearity of $f_{C^{\\prime }}$ equals the number satisfying assignments for $C$ .", "$\\Box $ So letting the nonlinearity, $s$ , be a part of the input for $NL_C$ changes the problem from being in level 1 of the polynomial hierarchy to be $\\# \\mathbf {P}$ hard, but does not seem to change the time complexity much.", "The situation for $MC_C$ is essentially the opposite, under Assumption REF , the time $MC_C$ needs is strictly more time than $AFFINE$ , but is contained in $\\Sigma _2^p$ .", "By appealing to Theorem  and , the following theorem follows.", "Under Assumption REF , no polynomial time algorithm computes $MC_C$ .", "Furthermore no algorithm with running time $poly(m)2^{O(n)}$ approximates $c_\\wedge (f)$ with a factor of $(2-\\epsilon )^{n/2}$ for any constant $\\epsilon > 0$ .", "We conclude by showing that although $MC_C$ under Assumption REF requires more time, it is nevertheless contained in the second level of the polynomial hierarchy.", "$MC_C \\in \\Sigma _2^p$ .", "First observe that $MC_C$ written as a language has the right form: $MC_C = \\lbrace (C,s) | \\exists C^{\\prime }\\ \\forall \\mathbf {x}\\in \\mathbb {F}_2^n\\ (C(\\mathbf {x})=C^{\\prime }(\\mathbf {x})\\textrm { and } c_\\wedge (C^{\\prime }) \\le s)\\rbrace .$ Now it only remains to show that one can choose the size of $C^{\\prime }$ is polynomial in $n+|C|$ .", "Specifically, for any $f\\in B_n$ , if $C^{\\prime }$ is the circuit with the smallest number of AND gates computing $f$ , for $n\\ge 3$ , we can assume that $|C^{\\prime }|\\le 2(c_\\wedge (f)+n)^2+c_\\wedge $ .", "For notational convenience let $c_\\wedge (f)=M$ .", "$C^{\\prime }$ consists of XOR and AND gates and each of the $M$ AND gates has exactly two inputs and one output.", "Consider some topological ordering of the AND gates, and call the output of the $i$ th AND gate $o_i$ .", "Each of the inputs to an AND gate is a sum (in $\\mathbb {F}_2$ ) of $x_i$ s, $o_i$ s and possibly the constant 1.", "Thus the $2M$ inputs to the AND gates and the output, can be thought of as $2M+1$ sums over $\\mathbb {F}_2$ over $n+M+1$ variables (we can think of the constant 1 as a variable with a hard-wired value).", "This can be computed with at most $(2M+1)(n+M+1)\\le 2(M+n)^2$ XOR gates, where the inequality holds for $n\\ge 3$ .", "Adding $c_\\wedge (f)$ for the AND gates, we get the claim.", "The theorem now follows, since $c_\\wedge (f)\\le |C|$$\\Box $ The relation between circuit size and multiplicative complexity given in the proof above is not tight, and we do not need it to be.", "See [18] for a tight relationship." ], [ "Acknowledgements", "The author wishes to thank Joan Boyar for helpful discussions." ] ]
1403.0417
[ [ "Turing Instability and Pattern Formation in an Activator-Inhibitor\n System with Nonlinear Diffusion" ], [ "Abstract In this work we study the effect of density dependent nonlinear diffusion on pattern formation in the Lengyel--Epstein system.", "Via the linear stability analysis we determine both the Turing and the Hopf instability boundaries and we show how nonlinear diffusion intensifies the tendency to pattern formation; %favors the mechanism of pattern formation with respect to the classical linear diffusion case; in particular, unlike the case of classical linear diffusion, the Turing instability can occur even when diffusion of the inhibitor is significantly slower than activator's one.", "In the Turing pattern region we perform the WNL multiple scales analysis to derive the equations for the amplitude of the stationary pattern, both in the supercritical and in the subcritical case.", "Moreover, we compute the complex Ginzburg-Landau equation in the vicinity of the Hopf bifurcation point as it gives a slow spatio-temporal modulation of the phase and amplitude of the homogeneous oscillatory solution." ], [ "Introduction", "Self-organized patterning in reaction-diffusion system driven by linear (Fickian) diffusion has been extensively studied since the seminal paper of Turing.", "Nevertheless, in many experimental cases, the gradient of the density of one species induces a flux of another species or of the species itself, therefore nonlinear effects should be taken into account.", "Recently, nonlinear diffusion terms have appeared to model different physical phenomena in diverse contexts like population dynamics and ecology [11], [28], [33], [14], [13], [15], [32], [25], and chemical reactions [3], [21].", "The aim of this work is to describe the Turing pattern formation for the following reaction-diffusion system with nonlinear density-dependent diffusion: $\\begin{split}\\displaystyle \\frac{\\partial U}{\\partial \\tau }&=D_u \\displaystyle \\frac{\\partial }{\\partial \\zeta } \\left(\\left(\\frac{U}{u_0}\\right)^m\\frac{\\partial U}{\\partial \\zeta }\\right)+\\Gamma \\left(a-U-\\frac{4UV}{1+U^2}\\right),\\\\\\displaystyle \\frac{\\partial V}{\\partial \\tau }&=c D_v \\displaystyle \\frac{\\partial }{\\partial \\zeta } \\left(\\left(\\frac{V}{v_0}\\right)^n\\frac{\\partial V}{\\partial \\zeta }\\right)+\\Gamma c b\\left(U-\\frac{UV}{1+U^2}\\right).\\end{split}$ In (REF ), $U(\\zeta ,\\tau )$ and $V(\\zeta ,\\tau )$ , with $\\zeta \\in [0,\\Omega ], \\Omega \\in \\mathbb {R}$ , represent the concentrations of two chemical species, the activator and the inhibitor respectively; the reaction mechanism is chosen as in the Lengyel-Epstein system [23], [22] modeling the chlorite-iodide-malonic acid and starch (CIMA) reaction.", "The parameters $a$ and $b$ are positive constants related to the feed rate, $c >1$ is a rescaling parameter which is bound up with starch concentration and $\\Gamma $ describes the relative strength of the reaction terms.", "The nonlinear density-dependent diffusion terms, given by $D_u(U/u_0)^m$ and $D_v(V/v_0)^n$ , show that when $m,n>0$ , the species tend to diffuse faster (when $U>u_0$ and $V>v_0$ ) or slower (when $U<u_0$ and $V<v_0$ ) than predicted by the linear diffusion.", "$D_u,D_v>0$ are the classical diffusion coefficients and $u_0, v_0>0$ are threshold concentrations, measuring the strength of the interactions between the individuals of the same species.", "Nonlinear diffusion terms as in (REF ) could be employed to model autocatalytic chemical reactions occurring on porous media [36], or in networks of electrical circuits [5], or on surfaces [31], like cellular membranes, or in surface electrodeposition [6], [7].", "Various experimental and numerical studies have been conducted on the Lengyel– Epstein system coupled with linear diffusion, see e.g.", "[9], [10].", "Also the analytical properties of the system have been widely studied: the Hopf bifurcation analysis has been performed in [27]; Turing instability and the pattern formation driven by linear diffusion have been investigated in different geometries [12], [34], [26], [8].", "The existence and non-existence for the steady states of the system have been proved in [30].", "To the best of our knowledge, the effect of the nonlinear diffusivity on Turing pattern of the Lengyel–Epstein system has not been examined, as exceptions we mention [35], where the authors determine the conditions for the occurring of Turing instabilities when linear diffusion for one species is coupled with the subdiffusion of the other species, and Ref.", "[24], where the authors perform an extensive numerical exploration of the Lengyel-Epstein model with local concentration-dependent diffusivity.", "In this paper we show that the nonlinear diffusion facilitate the Turing instability and the formation of the Turing structures as compared to the case of linear diffusion: in particular, increasing the value of the parameter $n$ in (REF ), the Turing instability arises even when the diffusion of the inhibitor is significantly slower than that of the activator (which is not the case when the diffusion is linear, i.e.", "when $n=0$ , see [30], [34]).", "Moreover, as the Lengyel-Epstein model also supports the Hopf bifurcation, the formation of the Turing structure depends on the mutual location of the Hopf and Turing instability boundaries.", "Through linear stability analysis we show that increasing the value of $n$ favors the Turing pattern formation.", "The effect of the parameter $m$ is exactly the opposite, as its increase hinders the mechanism of pattern formation.", "In Section , we shall obtain the Turing pattern forming region in terms of three key system parameters.", "This will enlighten the crucial role of nonlinear diffusion to achieve pattern formation even in the case not allowed when the mechanism is driven by linear diffusion.", "In Section we shall perform the weakly nonlinear (WNL) analysis to derive the equation ruling the evolution of the amplitude of the most unstable mode, both in the supercritical (Stuart-Landau equation) and the subcritical case (quintic Stuart-Landau equation).", "In Section 4 we shall address the process of pattern formation in the vicinity of the Hopf bifurcation, when it is the complex Ginzburg-Landau equation to provide a spatio-temporal modulation of the phase and amplitude of the homogeneous oscillatory solution [1]." ], [ "Linear instabilities", "In analogy with [18], [20], we rescale (REF ) as follows: $\\begin{split}\\frac{\\partial u}{\\partial t}=&\\,\\frac{\\partial ^2}{\\partial x^2} u^{m+1}+\\Gamma \\left(a-u-\\frac{4uv}{1+u^2}\\right),\\\\\\frac{\\partial v}{\\partial t}=&\\,c d\\frac{\\partial ^2}{\\partial x^2}v^{n+1}+\\Gamma c b\\left(u-\\frac{uv}{1+u^2}\\right),\\end{split}$ using $U=u,\\ V=v,\\ \\tau =t,\\ \\zeta =x^*x\\ $ , where: $x^*=\\sqrt{\\frac{D_u}{(m+1)u_0^m}},$ and the parameter $d$ has been defined as: $d=\\frac{(m+1)D_vu_0^m}{(n+1)D_uv_0^m}.$ In what follows the system (REF ) is supplemented with initial data and, given that we are interested in self-organizing patterns, we impose Neumann boundary conditions.", "The only homogeneous stationary state admitted by the system (REF ) is $(\\bar{u},\\bar{v})\\equiv (\\alpha , 1+\\alpha ^2)$ , where $\\alpha =a/5$ .", "Carrying out the linear stability analysis, we derive the dispersion relation $\\lambda ^2+g(k^2)\\lambda +h(k^2)=0$ which gives the growth rate $\\lambda $ as a function of the wavenumber $k$ , where: $\\begin{split}g(k^2)=&\\; k^2 \\,{\\rm tr}(D)-\\Gamma \\,{\\rm tr}(K) ,\\\\h(k^2)=&\\;{\\rm det}(D)k^4+\\Gamma q k^2+\\Gamma ^2 {\\rm det}(K),\\end{split}$ with: $K=\\Gamma \\left(\\begin{array}{cc}\\displaystyle \\frac{3\\alpha ^2-5}{\\alpha ^2+1} & -\\displaystyle \\frac{4\\alpha }{\\alpha ^2+1} \\\\\\displaystyle \\frac{2c b\\alpha ^2}{\\alpha ^2+1} & -\\displaystyle \\frac{c b}{\\alpha ^2+1}\\end{array}\\right),\\qquad D=\\left(\\begin{array}{cc}(m+1)\\bar{u}^m & 0\\\\0 & c d(n+1)\\bar{v}^n\\end{array}\\right)\\!,$ and $q=-K_{11}D_{22}-K_{22}D_{11}$ .", "Notice that the system (REF ) is an activator-inhibitor system under the condition: $3\\alpha ^2 - 5>0,$ since $K_{11}>0, K_{22}<0, K_{12}<0$ and $K_{21}>0$ (see discussion of activator-inhibitor systems in [29]).", "The steady state $(\\bar{u},\\bar{v})$ can lose its stability both via Hopf and Turing bifurcation.", "Oscillatory instability occurs when $g(k^2)=0$ and $h(k^2)>0$ .", "The minimum values of $b$ and $k$ for which $g(k^2)=0$ are: $b_H=\\frac{3\\alpha ^2-5}{c\\alpha }\\, \\qquad k=0,$ and for $b<b_H$ a spatially homogeneous oscillatory mode emerges.", "Notice that condition (REF ) assures $b_H>0$ .", "The neutral stability Turing boundary corresponds to $h(k^2)=0$ , which has a single minimum $(k_c^2, b_c)$ attained when: $k_c^2=-\\frac{\\Gamma q}{2\\, \\rm {det} (D)} \\, ,$ which requires $q<0$ .", "Therefore a necessary condition for Turing instability is given by: $b<\\bar{b}=d\\frac{(n+1)}{(m+1)} \\frac{(3\\alpha ^2-5)(1+\\alpha ^2)^n}{\\alpha ^{m+1}}.$ The value $\\bar{b}$ is non-negative under the condition (REF ).", "Moreover it can be straightforwardly proved that $\\bar{b}$ is a decreasing function of $m$ and an increasing function of $n$ , which means that larger values of $n$ facilitates Turing instability occurring for any values of $d$ .", "However, when $n=0$ , once fixed $m\\ge 0$ , in order to satisfy the condition (REF ), the value of $d$ should be sufficiently large, i.e.", "the diffusion of the inhibitor should be greater than that of the activator.", "Substituting the expression in (REF ) for the most unstable mode in $h(k_c^2)=0$ , the Turing bifurcation value $b=b_c$ is obtained by imposing: $q^2 -4\\ {\\rm det}(D){\\rm det}(K)=0,$ under the condition $q<0$ .", "Introducing $b=\\bar{b}-\\xi $ , with $\\xi >0$ , in (REF ) one gets: $\\begin{split}(m+1)\\frac{\\alpha ^{m+1}}{(1+\\alpha ^2)^2}\\,\\xi ^2+20(n+1)d(\\alpha ^2+1)^{n-1}\\xi \\\\-20(n+1)^2d^2(\\alpha ^2+1)^{2n-1}&\\frac{3\\alpha ^2-5}{\\alpha ^{m+1}}=0,\\end{split}$ whose positive root: $\\begin{split}\\xi =\\xi ^+=2d\\frac{(n+1)(\\alpha ^2+1)^n}{(m+1)\\alpha ^{m+1}}\\left(\\sqrt{5(3\\alpha ^4m+8\\alpha ^4-2\\alpha ^2m+8\\alpha ^2-5m)}\\right.\\\\\\left.-5(\\alpha ^2+1)\\right)&\\end{split}$ (this choice guarantees the condition $q<0$ ) gives the critical value of the parameter $b$ : $b_c=\\bar{b}-\\xi ^+=d \\frac{(1+\\alpha ^2)^n}{\\alpha ^{m+1}}\\frac{n+1}{m+1}\\left(13\\alpha ^2+5-4\\alpha \\sqrt{10(1+\\alpha ^2)}\\right) \\;,$ which is nonnegative under the condition (REF ).", "In Fig.REF we show the instability regions in the parameter space $(d,\\alpha )$ : the Turing instability region T, the Hopf instability region H and the region TH where a competition between the two instabilities occurs.", "In TH which one would develop, depends on the locations of the respective instability boundaries: as $b$ decreases, if $b_c > b_H$ , Turing instability occurs prior to the oscillatory instability and the Turing structures form.", "Figure: The instabilities region.", "Here the parameters are chosen as m=n=1m=n=1, c=8c=8 and b=1.2b=1.2.Imposing $b_c \\ge b_H$ leads to the following inequality: $d\\ge \\bar{s}=\\frac{(m+1)\\alpha ^m(3\\alpha ^2-5)}{(n+1)c(1+\\alpha ^2)^n(13\\alpha ^2+5-4\\alpha \\sqrt{10(1+\\alpha ^2)})},$ where the value $\\bar{s}$ is nonnegative under the condition (REF ).", "Moreover, it can be easily proved that $\\bar{s}$ is an increasing function with respect to $m$ and a decreasing function with respect to $n$ , which means that larger values of $n$ favors Turing instability and the formation of the relative pattern, also when the parameter $d$ is small.", "The effect of the parameter $m$ is opposite.", "This is also evident in Fig.REF , where the Turing and the Hopf instabilities boundaries are drawn with respect to the parameter $d$ varying $m$ and $n$ .", "Figure: Turing and Hopf instability boundaries varying mm and nn.", "The instabilities stay below the lines." ], [ "WNL analysis and pattern formation", "We use the method of multiple scales to determine the amplitude equation of the pattern close to the instability threshold.", "Introducing the control parameter $\\varepsilon $ , which represents the dimensionless distance from the critical value and is therefore defined as $\\varepsilon ^2=(b-b_c)/b_c$ , the characteristic slow temporal scale $T=\\varepsilon ^2 t$ can be easily obtained via linear analysis (see [17]).", "Let us recast the original system (REF ) in the following form: $\\partial _t \\textbf {w}= \\mathcal {L}^{b} \\textbf {w}+\\mathcal {NL}^{b} \\textbf {w},\\qquad \\textbf {w}\\equiv \\left(\\begin{array}{c}{u-\\bar{u}}\\\\{v-\\bar{v}}\\end{array}\\right) \\; ,$ where the linear operator $\\mathcal {L}^{b}=\\Gamma \\, K^b + D \\nabla ^2$ and the nonlinear operator $\\mathcal {NL}^{b}$ contains the remaining terms.", "The matrix $K^b$ and $D$ are given in (REF ), we made explicit the dependence on the bifurcation parameter ${b}$ just for notational convenience.", "Passing to the asymptotic analysis, we expand $b$ and $\\textbf {w}$ as: $b &=& b_c+\\varepsilon ^2 b^{(2)}+\\varepsilon ^4 b^{(4)}+\\dots ,\\\\\\textbf {w}&=&\\varepsilon \\, \\textbf {w}_1 +\\varepsilon ^2\\,\\textbf {w}_2+\\varepsilon ^3\\,\\textbf {w}_3+\\dots , \\\\t&=&t+\\varepsilon ^2 T_2+\\varepsilon ^4 T_4+\\dots , $ where the coefficients $b^{(i)}$ are negative.", "Substituting (REF )-–() into the full system (REF ), the following sequence of linear equations for $\\textbf {w}_i$ is obtained: $\\ \\,O(\\varepsilon ):$ $\\mathcal {L}^{b_c} {\\bf w}_1=\\mathbf {0},$ $\\ \\,O(\\varepsilon ^2):$ $\\mathcal {L}^{b_c} {\\bf w}_2=\\mathbf {F},$ $\\ \\,O(\\varepsilon ^3):$ $\\mathcal {L}^{b_c} {\\bf w}_3=\\mathbf {G},$ where: $\\mathbf {F}=\\frac{\\partial {\\bf w}_1}{\\partial T_1}-D^{(1)}\\nabla ^2\\left(\\begin{array}{c} u_1^2\\\\v_1^2\\end{array}\\right)+\\frac{\\alpha ((\\alpha ^2-3)u_1-(\\alpha -1)v_1)}{(\\alpha ^2+1)^2}\\mathfrak {u}_1,$ $\\begin{split}\\mathbf {G}=&\\,\\frac{\\partial {\\bf w}_1}{\\partial T_2}-D^{(2)}\\nabla ^2\\left(\\begin{array}{c}u_1^3\\\\v_1^3\\end{array}\\right)-2D^{(1)}\\nabla ^2\\left(\\begin{array}{c}u_1u_2\\\\v_1v_2\\end{array}\\right)-\\frac{c b^{(2)}\\alpha (2\\alpha u_1-v_1)}{\\alpha ^2+1}\\mathfrak {u}_2\\\\-&\\,\\frac{(\\alpha ^4-6\\alpha ^2+1)u_1^3-\\alpha (\\alpha ^2-3)u_1v_1+(\\alpha ^4-1)(u_1v_2+u_2v_1)}{(\\alpha ^2+1)^3}\\mathfrak {u}_1\\\\-&\\,\\frac{2\\alpha (-\\alpha ^4+2\\alpha ^2+3)u_1}{(\\alpha ^2+1)^3}\\mathfrak {u}_1\\end{split}$ and: $\\nonumber \\mathfrak {u}_1=\\Gamma \\left(\\begin{array}{c} 4\\\\c b_c\\end{array}\\right),\\qquad \\qquad \\mathfrak {u}_2=\\Gamma \\left(\\begin{array}{c} 0\\\\1\\end{array}\\right),$ $\\nonumber D^{(1)}=\\left(\\begin{array}{cc}\\displaystyle \\frac{m(m+1)}{2}\\bar{u}^{m-1} & 0\\\\0 & cd\\displaystyle \\frac{n(n+1)}{2}\\bar{v}^{n-1}\\end{array}\\right),$ $\\nonumber D^{(2)}=\\left(\\begin{array}{cc}\\displaystyle \\frac{m(m^2-1)}{6}\\bar{u}^{m-2} & 0\\\\0 & \\displaystyle \\frac{n(n^2-1)}{6}\\bar{v}^{n-2}\\end{array}\\right).$ At the lowest order in $\\varepsilon $ we recover the linear problem $\\mathcal {L}^{b_c} {\\bf w}_1=\\mathbf {0}$ whose solution, satisfying the Neumann boundary conditions, is given by: $ {\\bf w}_1=A(T) \\mbox{$\\rho $}\\, \\cos (\\bar{k}_c x) \\; , \\qquad \\mbox{with}\\quad \\mbox{$\\rho $}\\in \\mbox{Ker}( K^{b_c}-\\bar{k}_c^2D)\\; .$ In the above equation we have denoted with $\\bar{k}_c$ the first admissible unstable mode, while $A(T)$ is the amplitude of the pattern and it is still arbitrary at this level.", "The vector $\\mbox{$\\rho $}$ is defined up to a constant and we shall make the normalization in the following way: $\\mbox{$\\rho $}= \\left(\\begin{array}{c} 1 \\\\M \\end{array}\\right) \\, ,\\qquad \\mbox{with} \\quad M\\equiv \\frac{-D_{21}k_c^2+\\Gamma K^{b_c}_{21}}{D_{22}k_c^2-\\Gamma K^{b_c}_{22}},$ where $D_{ij}, K^{b_c}_{ij}$ are the $i,j$ -entries of the matrices $D$ and $K^{b_c}$ .", "Once substituted in (REF ) the first order solution ${\\bf w}_1$ , the vector ${\\bf F}$ is orthogonal to the kernel of the adjoint of $ \\mathcal {L}^{b_c}$ and the equation (REF ) can be solved right away.", "This is not the case for Eq.", "(REF ).", "In fact the vector $\\mathbf {G}$ has the following expression: ${\\bf G}=\\left(\\displaystyle \\frac{d A}{dT_2}\\mbox{$\\rho $}+A {\\bf G}_1^{(1)}+A^3 {\\bf G}_1^{(3)}\\right)\\cos (\\bar{k}_c x)+{\\bf G}^* ,$ where ${\\bf G}^{*}$ contains automatically orthogonal terms and ${\\bf G}_1^{(j)}, j = 1,3$ have a cumbersome expression here not reported.", "The solvability condition for the equation (REF ) gives the following Stuart-Landau equation (SLE) for the amplitude $A(T)$ : $\\frac{d A}{d T}= \\sigma A -L A^3,$ where the coefficients $\\sigma $ and $L$ are given as follows: $\\nonumber \\sigma =-\\frac{<{\\bf G}_1^{(1)}, \\mbox{$\\psi $}>}{<\\mbox{$\\rho $},\\mbox{$\\psi $}>},\\qquad L=\\frac{<{\\bf G}_1^{(3)}, \\mbox{$\\psi $}>}{<\\mbox{$\\rho $},\\mbox{$\\psi $}>},\\quad {\\rm and\\ }{\\mbox{$\\psi $}} \\in {\\rm Ker}\\left(K^{b^c} -\\bar{k}_c^2D\\right)^\\dag .$ Since the growth rate coefficient $\\sigma $ is always positive, the dynamics of the SLE (REF ) can be divided into two qualitatively different cases depending on the sign of the Landau constant $L$ : the supercritical case, when $L$ is positive, and the subcritical case, when $L$ is negative.", "In Fig.REF , the curves across which $L$ changes its sign are drawn in the space $(d, \\alpha )$ and the pattern-forming region is divided in one supercritical region (I) and two subcritical regions (II).", "Figure: The Turing region: subcritical and supercritical.", "The parameters are chosen as m=n=1m=n=1, c=8c=8 and b=1.2b=1.2.In the supercritical case $A_\\infty =\\sqrt{{\\sigma }/{L}}$ is the stable equilibrium solution of the amplitude equation (REF ) and it corresponds to the asymptotic value of the amplitude $A$ of the pattern.", "In Fig.", "REF we show the comparison between the solution predicted by the WNL analysis up to the $O(\\varepsilon ^2)$ and the stationary state (reached starting from a random perturbation of the constant state) computed solving numerically the full system (REF ).", "In all the performed tests, we have checked that the distance between the WNL approximated solution and the numerical solution of the system (REF ) is $O(\\varepsilon ^3)$ in $L^1$ norm.", "Figure: Comparison between the WNL solution (solid line) and the numerical solution of () (dotted line) in the supercritical case.", "The parameters are chosen as m=n=1m=n=1, Γ=140\\Gamma =140, α=1.6\\alpha =1.6 c=40c=40, d=1.5d=1.5, b H =0.0419<b=b c (1-0.1 2 )<b c ∼0.1959b_H=0.0419<b=b_c(1-0.1^2)<b_c\\sim 0.1959 and k c =4k_c=4.In the subcritical regions indicated with II in Fig.REF , the Landau coefficient $L$ has a negative value and the equation (REF ) is not able to capture the amplitude of the pattern.", "In this case to predict the amplitude of the pattern, one needs to push the WNL expansion at a higher order (for a general discussion on the relevance of the higher order amplitude expansions in the study of subcritical bifurcations, see the recent [2] and references therein).", "Performing the WNL up to $O(\\varepsilon ^5)$ we obtain the following quintic SLE for the amplitude $A$ : $\\frac{d A}{dT_2}=\\bar{\\sigma }A-\\bar{L}A^3+\\bar{Q}A^5\\, .$ Here we skip the details of the analysis, but we want to stress that the coefficients $\\bar{\\sigma }$ and $\\bar{L}$ are $O(\\varepsilon ^2)$ perturbation of the coefficients $\\sigma $ and $L$ of the SLE (REF ), and the coefficient $\\bar{Q}$ is $O(\\varepsilon ^2)$ .", "The predicted amplitude is $O(\\varepsilon ^{-1})$ , and therefore the corresponding emerging pattern is an $O(1)$ perturbation of the equilibrium, which contradicts the basic assumption of the perturbation scheme ().", "In the subcritical case, when the growth rate coefficient $\\bar{\\sigma }>0$ , the Landau coefficient $\\bar{L}<0$ and $\\bar{Q}<0$ , one should therefore expect quantitative discrepancies between the predicted solution of the WNL analysis and the numerical solution of the full system.", "Nevertheless in our numerical tests we have found a qualitatively good agreement, see for example Fig.REF (a).", "Moreover, the bifurcation diagram in Fig.REF (b) constructed using the amplitude equation (REF ) is able to predict very well phenomena like bistability and hysteresis cycle shown also by the full system, see [16].", "Figure: (a) Comparison between the WNL solution (solid line) and the numerical solution of () (dotted line) in the subcritical case.", "(b) The bifurcation diagram in the subcritical case.", "The parameters are chosen as m=n=1m=n=1, Γ=100\\Gamma =100, α=3.6\\alpha =3.6 c=40c=40, d=0.5d=0.5, b H ∼0.2353<b=b c (1-0.1 2 )<b c ∼1.7991b_H\\sim 0.2353<b=b_c(1-0.1^2)<b_c\\sim 1.7991 and k c =4k_c=4 ." ], [ "Oscillating pattern at the Hopf bifurcation", "In the Hopf instability region, labeled with H in Fig.REF , the solution of the system (REF ) has a pure oscillating dynamical behavior, see Fig.REF (a) where the homogeneous state $(\\bar{u}, \\bar{v})$ destabilizes and a stable periodic solution emerges.", "When the values of the Turing bifurcation point $b_c$ and the Hopf bifurcation point $b_H$ are rather close, our numerical investigations have also shown that, even though the parameter $b$ is chosen into the Hopf instability region, the proximity to the Turing instability region influence the emerging solution: there is a transient in which a Turing structure oscillates and the corresponding solution in the time-space plane is given in Fig.REF (b).", "Figure: Oscillating patterns.", "(a) Stable periodic solution arising from the oscillation of the equilibrium (u ¯,v ¯)(\\bar{u}, \\bar{v}).", "The parameters are chosen as m=n=1m=n=1, Γ=140\\Gamma =140, α=1.6\\alpha =1.6 c=2c=2, d=1.5d=1.5, b c ∼0.1959<b=b H (1-0.1 2 )<b H ∼0.8375b_c\\sim 0.1959<b=b_H(1-0.1^2)<b_H\\sim 0.8375.", "(b) A Turing-type pattern oscillates next to the codimension 2 Turing-Hopf bifurcation point.", "The parameters are chosen as m=n=1m=n=1, Γ=140\\Gamma =140, α=1.58\\alpha =1.58 c=40c=40, d=0.3206d=0.3206, b c ∼0.0372b_c\\sim 0.0372 b H ∼0.0394b_H\\sim 0.0394, b=0.03787b=0.03787In the direct numerical simulations of the full system (REF ), the integrator must use a step size sufficiently small to follow all the oscillations.", "The complex Ginzburg-Landau equation gives a universal description of reaction-diffusion systems in the neighborhood of the Hopf bifurcation.", "Using the same asymptotic expansion as in (REF ), where the bifurcation value is now $b_H$ , and taking into account also the slow spatial modulation $X$ (whose characteristic length scale is $O(\\varepsilon ^{-1}$ )), at the lowest order $\\varepsilon $ we recover the linear problem $\\mathcal {L}^{b_H}{\\bf w}_1=\\mathbf {0}$ , whose solution is: $ {\\bf w}_1=\\mathcal {A}(X,T)\\mbox{$\\theta $}e^{ih_c t}+ \\bar{\\mathcal {A}}(X,T)\\overline{\\mbox{$\\theta $}} e^{-ih_c t}\\; ,$ where $h_c=\\sqrt{{\\rm det}(K^{b_H})}$ and the vectors $\\mbox{$\\theta $}$ and $\\overline{\\mbox{$\\theta $}}$ satisfy $K^{b_H}\\mbox{$\\theta $}=ih_c\\mbox{$\\theta $}$ , and $\\overline{\\mbox{$\\theta $}} K^{b_H}=ih_c\\overline{\\mbox{$\\theta $}}$ .", "Pushing the asymptotic analysis up to $O(\\varepsilon ^3)$ (the details are not reproduced here as they follow the same steps as in Section), we find the following complex Ginzburg-Landau equation (CGLE) for the amplitude $\\mathcal {A}$ : $ \\frac{\\partial \\mathcal {A}}{\\partial T}=\\delta \\frac{\\partial ^2 \\mathcal {A}}{\\partial X^2}+\\chi \\mathcal {A}+\\eta |\\mathcal {A}|^2\\mathcal {A},$ where the coefficients $\\eta $ and $\\delta $ are complex and the coefficient $\\chi $ is real.", "The amplitude $\\mathcal {A}$ describes the modulation of local oscillations having frequency $h_c$ and the fact that the fundamental phase $e^{i h_c t}$ has been scaled out in the CGLE has enormous numerical advantages." ], [ "Conclusions and open problems", "In the present paper we have examined the Turing mechanism induced by a nonlinear density-dependent diffusion in the Lengyel-Epstein system.", "We have shown that the presence of nonlinear diffusion favors the Turing instability (which competes with the Hopf instability) and the formation of Turing structure also when the diffusion coefficient of the activator exceeds that one of the inhibitor.", "Through a WNL analysis, we have derived the equations which rule the amplitude the pattern, both in the supercritical and subcritical bifurcation case, identifying in the parameters space the supercritical and the subcritical regions.", "All the numerical tests we have run are in good agreement with the prediction of the WNL analysis.", "We have also numerically investigated the oscillating pattern arising in the Hopf instability region and we have computed the CGLE as it describes the slow spatio-temporal modulation of the amplitude of the homogeneous oscillatory solution.", "Some other aspects of the problem could be examined.", "In a 2D domain new pattern forming phenomena occur, as degeneracy leads to more complex structures, predictable via the WNL [19].", "Moreover, the analytic solutions of the CGLE can be obtained in some special cases, e.g.", "plane wave solutions.", "Even though they are the simplest propagating structures supported by the CGLE, they have a fundamental role as their criteria of stability are necessary conditions for spiral wave instability.", "Finally, we could explore the spatio-temporal chaos in the starting from the numerical investigation of the spatio-temporal chaos in the CGLE [3], [4]." ] ]
1403.0351
[ [ "Union-intersecting set systems" ], [ "Abstract Three intersection theorems are proved.", "First, we determine the size of the largest set system, where the system of the pairwise unions is l-intersecting.", "Then we investigate set systems where the union of any s sets intersect the union of any t sets.", "The maximal size of such a set system is determined exactly if s+t<5, and asymptotically if s+t>4.", "Finally, we exactly determine the maximal size of a k-uniform set system that has the above described (s,t)-union-intersecting property, for large enough n." ], [ "Introduction", "In the present paper we prove three intersection theorems, inspired by the following question.", "Question (J. Körner, [9]) Let $\\mathcal {F}$ be a set system whose elements are subsets of $[n]=\\lbrace 1,2,\\dots n\\rbrace $ .", "Assume that there are no four different sets $F_1, F_2, G_1, G_2\\in \\mathcal {F}$ such that $(F_1\\cup F_2)\\cap (G_1\\cup G_2)=\\emptyset $ .", "What is the maximal possible size of $\\mathcal {F}~$ ?", "We will solve this problem as a special case of Theorem REF .", "(See Remark REF .)", "The paper is organized as follows.", "In the rest of this section we review some theorems that will be used later.", "In Section 2, the size of the largest set system will be determined, where the system of the pairwise unions is $l$ -intersecting.", "In Section 3, we investigate set systems where the union of any $s$ sets intersect the union of any $t$ sets.", "The maximal size of such a set system is determined exactly if $s+t\\le 4$ , and asymptotically if $s+t\\ge 5$ .", "In Section 4, we exactly determine the maximal size of a $k$ -uniform set system that has the above described $(s,t)$ -union-intersecting property, for large enough $n$ .", "The following intersection theorems will be used in the proof of Theorem REF .", "Definition Let ${[n] \\atopwithdelims ()k}$ denote the set of all $k$ -element subsets of $[n]$ .", "A set system $\\mathcal {F}$ is called $l$ -intersecting, if $|A\\cap B|\\ge l$ holds for all $A,B\\in \\mathcal {F}$ ($l>0$ ).", "The following set systems (containing $k$ -element subsets of $[n]$ ) are obviously $l$ -intersecting systems.", "$\\mathcal {F}_i=\\left\\lbrace F\\in {[n]\\atopwithdelims ()k} ~\\Big |~ |F\\cap [l+2i]|\\ge l+i \\right\\rbrace ~~~~~~~0\\le i\\le \\frac{n-l}{2}$ For $1\\le l\\le k\\le n$ let $ AK(n,k,l)=\\max _{0\\le i\\le \\frac{n-l}{2}} |\\mathcal {F}_i|.$ It was conjectured by Frankl [5] that this is the maximum size of a $k$ -uniform $l$ -intersecting family.", "(See also [6].)", "Theorem 1.1 (Ahlswede-Khachatrian, [1]) Let $\\mathcal {F}$ be a $k$ -uniform $l$ -intersecting set system whose elements are subsets of $[n]$ .", "($1\\le l\\le k\\le n$ ) Then $|\\mathcal {F}|\\le AK(n,k,l).$ Theorem 1.2 (Katona, [7], formula (12)) Let $\\mathcal {F}$ be a $t$ -intersecting system of subsets of $[n]$ .", "Then $|\\mathcal {F}^i|+|\\mathcal {F}^{n+t-1-i}|\\le {n \\atopwithdelims ()n+t-1-i}.", "~~~~~~~\\left(0\\le i < \\frac{n+t-1}{2}\\right)$ The following results are about set systems not containing certain subposets.", "We will use them to prove Theorem REF .", "Definition Let $P$ be a finite poset with the relation $\\prec $ , and $\\mathcal {F}$ be a family of subsets of $[n]$ .", "We say that $P$ is contained in $\\mathcal {F}$ if there is an injective mapping $f:P\\rightarrow \\mathcal {F}$ satisfying $a\\prec b \\Rightarrow f(a)\\subset f(b)$ for all $a,b\\in P$ .", "$\\mathcal {F}$ is called $P$ -free if $P$ is not contained in it.", "Definition Let $K_{xy}$ denote the poset with elements $\\lbrace a_1, a_2, \\dots a_x, b_1, \\dots b_y\\rbrace $ , where $a_i < b_j$ for all $(i,j)$ and there is no other relation.", "Theorem 1.3 (Katona-Tarján, [8]) Assume that $\\mathcal {G}$ is a family of subsets of $[n]$ that is $K_{12}$ -free and $K_{21}$ -free.", "Then $|\\mathcal {G}|\\le 2{n-1 \\atopwithdelims ()\\lfloor \\frac{n-1}{2} \\rfloor }.$ Theorem 1.4 (De Bonis-Katona, [2]) Assume that $\\mathcal {G}$ is a $K_{1y}$ -free family of subsets of $[n]$ .", "Then $|\\mathcal {G}|\\le {n\\atopwithdelims ()\\lfloor n/2 \\rfloor } \\left(1+\\frac{2(y-1)}{n}+O(n^{-2})\\right).$ Theorem 1.5 (De Bonis-Katona, [2]) Assume that $\\mathcal {G}$ is a $K_{xy}$ -free family of subsets of $[n]$ .", "Then $|\\mathcal {G}|\\le {n\\atopwithdelims ()\\lfloor n/2 \\rfloor } \\left(2+\\frac{2(x+y-3)}{n}+O(n^{-2})\\right).$ Union-l-intersecting systems Definition Let $\\mathcal {F}$ be a set system and $k\\in \\mathbb {N}$ .", "Then $\\mathcal {F}^k$ denotes the set of the $k$ -element sets in $\\mathcal {F}$ .", "Definition A set system $\\mathcal {F}$ is called union-$l$ -intersecting, if it satisfies $|(F_1\\cup F_2)\\cap (G_1\\cup G_2)|\\ge l$ for all sets $F_1, F_2, G_1, G_2\\in \\mathcal {F}$ , $F_1\\ne F_2$ , $G_1\\ne G_2$ .", "Theorem 2.1 Let $\\mathcal {F}$ be a union-$l$ -intersecting set system whose elements are subsets of $[n]$ .", "($n\\ge 3$ ) Then we have the the following upper bounds for $|\\mathcal {F}|$ .", "If $n+l$ is even, then $|\\mathcal {F}|\\le \\sum _{i=\\frac{n+l}{2}-1}^{n} {n \\atopwithdelims ()i}.$ If $n+l$ is odd, then $|\\mathcal {F}|\\le AK\\left(n, \\frac{n+l-3}{2}, l\\right)+\\sum _{i=\\frac{n+l-1}{2}}^{n} {n \\atopwithdelims ()i}.$ These are the best possible bounds.", "We can assume that $\\mathcal {F}$ is an upset, that is $A\\in \\mathcal {F}$ , $A\\subset B$ imply $B\\in \\mathcal {F}$ .", "(If there are sets $A\\subset B$ , $A\\in \\mathcal {F}$ , $B\\notin \\mathcal {F}$ then we can replace $\\mathcal {F}$ by $\\mathcal {F}-A+B$ .", "After finitely many steps we arrive at an upset of the same size that is still union-$l$ -intersecting.)", "First, assume that $l\\in \\lbrace 1,2\\rbrace $ .", "Note that if $A,B\\in \\mathcal {F}$ , $A\\cap B=\\emptyset $ , and $|A\\cup B|=n+l-3<n$ , then both $A$ and $B$ can not be in $\\mathcal {F}$ at the same time.", "$A,B \\in \\mathcal {F}$ and $\\mathcal {F}$ being an upset would imply that there are two sets $C,D\\in \\mathcal {F}$ such that $A \\subset C$ , $B\\subset D$ and $|(A\\cup C) \\cap (B \\cup D)| = |C\\cap D|= l-1$ , with contradiction.", "For all $0\\le i < \\frac{n+l-3}{2}$ define the bipartite graph $G_i(S_i, T_i, E_i)$ as follows.", "Let $S_i$ be the set of all the $i$ -element subsets of $[n]$ , let $T_i$ be the set of subsets of size $n+l-3-i$ , and connect two sets $A\\in S_i$ and $B\\in T_i$ if they are disjoint.", "Then both vertex classes contain vertices of the same degree, so it follows by Hall's theorem that there is a matching that covers the smaller vertex class, that is $S_i$ .", "Since at most one of two matched subsets can be in $\\mathcal {F}$ , it follows that $ |\\mathcal {F}^i|+|\\mathcal {F}^{n+l-3-i}|\\le {n \\atopwithdelims ()n+l-3-i} ~~~~~~~\\left(0\\le i < \\frac{n+l-3}{2}\\right).$ Now let $l\\ge 3$ .", "We will show that (REF ) holds for all $n$ .", "Assume that there are $A,B\\in \\mathcal {F}$ , $A,B\\ne [n]$ such that $|A\\cap B|\\le l-3$ .", "Take $x\\notin A$ and $y\\notin B$ .", "Then $A\\cup \\lbrace x\\rbrace ,~B\\cup \\lbrace y\\rbrace \\in \\mathcal {F}$ , since $\\mathcal {F}$ is an upset and $|(A\\cup (A\\cup \\lbrace x\\rbrace ))\\cap (B\\cup (B\\cup \\lbrace y\\rbrace ))|=|(A\\cup \\lbrace x\\rbrace )\\cap (B\\cup \\lbrace y\\rbrace )|\\le |(A\\cap B) \\cup \\lbrace x,y\\rbrace |=l-3+2 < l.$ So $\\mathcal {F}-\\lbrace [n]\\rbrace $ is an $(l-2)$ -intersecting system, so $\\mathcal {F}$ is one too.", "Now use Theorem REF with $t=l-2$ .", "($l-2$ is positive since $l\\ge 3$ .)", "It gives us that (REF ) holds for all $n$ and $l$ .", "Assume that $l$ is a positive integer $n+l$ is odd.", "Then the sets in $\\mathcal {F}^{\\frac{n+l-3}{2}}$ form an $l$ -intersecting family.", "$A,B \\in \\mathcal {F}^{\\frac{n+l-3}{2}}$ , $|A\\cap B|\\le l-1$ and $\\mathcal {F}$ being an upset would imply that there are two sets $C,D\\in \\mathcal {F}$ such that $A \\subset C$ , $B\\subset D$ and $|(A\\cup C) \\cap (B \\cup D)| = |C\\cap D|= l-1$ .", "So Theorem REF provides an upper bound: $ |\\mathcal {F}^{\\frac{n+l-3}{2}}|\\le AK\\left(n, \\frac{n+l-3}{2}, l\\right).$ The upper bounds of the theorem follow after some calculations.", "When $l\\le 2$ , the inequalities of (REF ) imply $|\\mathcal {F}|=\\sum _{i=0}^n |\\mathcal {F}^i|=|\\mathcal {F}^{\\frac{n+l-3}{2}}|+\\sum _{i=0}^{\\lfloor \\frac{n+l}{2}-2\\rfloor } (|\\mathcal {F}^i|+|\\mathcal {F}^{n+l-3-i}|)+\\sum _{i=n+l-2}^n |\\mathcal {F}^i| \\le |\\mathcal {F}^{\\frac{n+l-3}{2}}|+\\sum _{i=\\lceil \\frac{n+l}{2}-1\\rceil }^n {n \\atopwithdelims ()i}.$ Let $l\\ge 3$ .", "Since $\\mathcal {F}$ is $(l-2)$ -intersecting, $|\\mathcal {F}^i|=0$ for all $i<l-2$ .", "Using (REF ), we get $|\\mathcal {F}|=\\sum _{i=l-3}^n |\\mathcal {F}^i|= |\\mathcal {F}^{\\frac{n+l-3}{2}}|+\\sum _{i=l-3}^{\\lfloor \\frac{n+l}{2}-2\\rfloor } (|\\mathcal {F}^i|+|\\mathcal {F}^{n+l-3-i}|)\\le |\\mathcal {F}^{\\frac{n+l-3}{2}}|+\\sum _{i=\\lceil \\frac{n+l}{2}-1\\rceil }^n {n \\atopwithdelims ()i}.$ We got the same inequality in the two cases.", "The upper bounds of the theorem follow immediately, since $\\mathcal {F}^{\\frac{n+l-3}{2}}=\\emptyset $ , if $n+l$ is even, and $|\\mathcal {F}^{\\frac{n+l-3}{2}}|\\le AK\\left(n, \\frac{n+l-3}{2}, l\\right)$ , if $n+l$ is odd (see (REF )).", "To verify that the given bounds are best possible, consider the following union-$l$ -intersecting set systems.", "When $n+l$ is even, take all the subsets of size at least $\\frac{n+l}{2}-1$ .", "When $n+l$ is odd, take all the subsets of size at least $\\frac{n+l-1}{2}$ and an $\\frac{n+l-3}{2}$ -uniform $l$ -intersecting set system of size $AK\\left(n, \\frac{n+l-3}{2}, l\\right)$ .", "Remark 2.2 Let us formulate the special case $l=1$ what was originally asked by Körner.", "The Erdős-Ko-Rado theorem [3] states that the size of the largest $k$ -uniform intersecting system of subsets of $[n]$ is ${n-1\\atopwithdelims ()k-1}$ , when $n\\ge 2k$ .", "It means that $AK(n,\\frac{n}{2}-1,1)={n-1\\atopwithdelims ()\\frac{n}{2}-2}$ , so the best upper bound for $|\\mathcal {F}|$ when $l=1$ is $ |\\mathcal {F}|\\le {\\left\\lbrace \\begin{array}{ll} \\displaystyle \\sum _{i=(n-1)/2}^n {n \\atopwithdelims ()i} & \\mbox{if } n\\mbox{ is odd,} \\\\ {n-1 \\atopwithdelims ()\\frac{n}{2}-2}+\\displaystyle \\sum _{i=\\frac{n}{2}}^n {n \\atopwithdelims ()i} & \\mbox{if } n\\mbox{ is even.}", "\\end{array}\\right.}", "$ Considering the union of more subsets   In this section we investigate a variation of the problem where we take the union of $s$ and $t$ subsets instead of 2 and 2.", "Definition A set system $\\mathcal {F}$ is called $(s,t)$ -union-intersecting if it has the property that for all $s+t$ pairwise different sets $F_1, F_2, \\dots F_s, G_1, \\dots G_t\\in \\mathcal {F}$ $\\left(\\bigcup _{i=1}^s F_i\\right) \\cap \\left(\\bigcup _{j=1}^t G_j\\right) \\ne \\emptyset .$ The size of the largest $(s,t)$ -union-intersecting system whose elements are subsets of $[n]$ is denoted by $f(n,s,t)$ .", "In this section we determine the value of $f(n,s,t)$ exactly when $s+t\\le 4$ and asymptotically in the other cases.", "Since $f(n,s,t)=f(n,t,s)$ , we can assume that $s\\le t$ .", "Theorem 3.1 Let $n\\ge 3$ .", "$ f(n,1,1)=2^{n-1}.", "$ $ f(n,1,2) = {\\left\\lbrace \\begin{array}{ll} \\displaystyle \\sum _{i=n/2}^n {n \\atopwithdelims ()i} & \\mbox{if } n\\mbox{ is even,} \\\\ {n-1 \\atopwithdelims ()\\frac{n-3}{2}}+\\displaystyle \\sum _{i=(n+1)/2}^n {n \\atopwithdelims ()i} & \\mbox{if } n\\mbox{ is odd.}", "\\end{array}\\right.}", "$ $ f(n,2,2) = {\\left\\lbrace \\begin{array}{ll} \\displaystyle \\sum _{i=(n-1)/2}^n {n \\atopwithdelims ()i} & \\mbox{if } n\\mbox{ is odd,} \\\\ {n-1 \\atopwithdelims ()\\frac{n}{2}-2}+\\displaystyle \\sum _{i=n/2}^n {n \\atopwithdelims ()i} & \\mbox{if } n\\mbox{ is even.}", "\\end{array}\\right.}", "$ $ f(n,1,3) = {\\left\\lbrace \\begin{array}{ll} \\displaystyle \\sum _{i=n/2}^n {n \\atopwithdelims ()i} & \\mbox{if } n\\mbox{ is even,} \\\\ {n-1 \\atopwithdelims ()\\frac{n-1}{2}}+\\displaystyle \\sum _{i=(n+1)/2}^n {n \\atopwithdelims ()i} & \\mbox{if } n\\mbox{ is odd.}", "\\end{array}\\right.}", "$ If $t\\ge 4$ , then $2^{n-1}+\\frac{1}{2}{n \\atopwithdelims ()\\lfloor n/2\\rfloor } \\le f(n,1,t) \\le 2^{n-1}+{n \\atopwithdelims ()\\lfloor n/2\\rfloor }\\left(\\frac{1}{2}+\\frac{t-2}{n}+O(n^{-2})\\right).", "$ If $s\\ge 2$ and $t\\ge 3$ , then $2^{n-1}+{n \\atopwithdelims ()\\lfloor n/2\\rfloor }\\frac{n}{n+2} \\le f(n,s,t) \\le 2^{n-1}+{n \\atopwithdelims ()\\lfloor n/2\\rfloor }\\left(1+\\frac{t+s-3}{n}+O(n^{-2})\\right).", "$ (See [3].)", "$f(n,1,1)$ is the size of the largest intersecting system among the subsets of $[n]$ .", "It is at most $2^{n-1}$ , since a subset and its complement cannot be in the intersecting system at the same time.", "By choosing all the subsets containing a fixed element, we get an intersecting system of size $2^{n-1}$ .", "Let $\\mathcal {F}$ be a $(1,2)$ -union-intersecting system.", "We can assume that $\\mathcal {F}$ is an upset.", "Note that if $A,B\\in \\mathcal {F}$ , $A\\cap B=\\emptyset $ , and $|A\\cup B|=n-1$ , then $A$ and $B$ can not be in $\\mathcal {F}$ at the same time.", "To see this, take let $\\lbrace x\\rbrace =[n]-(A\\cup B)$ .", "Then $A\\cap (B\\cup (B\\cup \\lbrace x\\rbrace ))=A\\cap (B\\cup \\lbrace x\\rbrace )=\\emptyset $ .", "For all $0\\le i < \\frac{n-1}{2}$ define the bipartite graph $G_i(S_i, T_i, E_i)$ as follows.", "Let $S_i$ be the set of all the $i$ -element subsets of $[n]$ , let $T_i$ be the set of subsets of size $n-1-i$ , and connect two sets $A\\in S_i$ and $B\\in T_i$ if they are disjoint.", "Then both vertex classes contain vertices of the same degree, so it follows by Hall's theorem that there is a matching that covers the smaller vertex class, that is $S_i$ .", "Since at most one of two matched subsets can be in $\\mathcal {F}$ , it follows that $ |\\mathcal {F}^i|+|\\mathcal {F}^{n-1-i}|\\le {n \\atopwithdelims ()n-1-i} ~~~~~~~\\left(0\\le i < \\frac{n-1}{2}\\right).$ When $n$ is even, these inequalities together imply $ |\\mathcal {F}|\\le \\sum _{i=n/2}^{n} {n \\atopwithdelims ()i}.$ Assume that $n$ is odd.", "Then $\\mathcal {F}^{\\frac{n-1}{2}}$ is an intersecting family.", "$A,B \\in \\mathcal {F}^{\\frac{n-1}{2}}$ , $A\\cap B= \\emptyset $ and $\\mathcal {F}$ being an upset would imply that there is a set $C\\in \\mathcal {F}$ such that $B\\subset C$ and $|A\\cap (B \\cup C)| = |A\\cap C|= \\emptyset $ .", "So the Erdős-Ko-Rado theorem provides an upper bound: $ |\\mathcal {F}^{\\frac{n-1}{2}}|\\le {n-1 \\atopwithdelims ()\\frac{n-3}{2}}.$ This, together with the inequalities of (REF ) implies $ |\\mathcal {F}| \\le {n-1 \\atopwithdelims ()\\frac{n-3}{2}}+\\sum _{i=(n+1)/2}^n {n \\atopwithdelims ()i}.$ To verify that the given bounds are best possible, consider the following (1,2)-union-intersecting set systems.", "When $n$ is even, take all the subsets of size at least $\\frac{n}{2}$ .", "When $n$ is odd, take all the subsets of size at least $\\frac{n+1}{2}$ and the subsets of size $\\frac{n-1}{2}$ containing a fixed element.", "We already solved this problem as the case $l=1$ in Theorem REF .", "(See Remark REF .)", "Let $\\mathcal {F}$ be a $(1,3)$ -union-intersecting system of subsets of $[n]$ .", "Let $\\mathcal {F}^{\\prime }=\\lbrace [n]-F~\\big |~F\\in \\mathcal {F}\\rbrace $ , and let $\\mathcal {G}=\\mathcal {F}\\cap \\mathcal {F}^{\\prime }$ .", "Now we prove that $\\mathcal {G}$ is $K_{1,2}$ -free and $K_{2,1}$ -free.", "(See Section 1 for the definitions.)", "Since $\\mathcal {G}$ is invariant to taking complements, it is enough to show that $\\mathcal {G}$ is $K_{12}$ -free.", "Assume that there are three pairwise different sets $A,B,C\\in \\mathcal {G}$ , such that $A\\subset B$ and $A\\subset C$ .", "Then the sets $A,[n]-A, [n]-B, [n]-C\\in \\mathcal {F}$ would satisfy $A\\cap (([n]-A)\\cup ([n]-B)\\cup ([n]-C))=A\\cap ([n]-A)=\\emptyset .", "$ Theorem REF gives us the following upper bound for a set system that is $K_{12}$ -free and $K_{21}$ -free: $|\\mathcal {G}|\\le 2{n-1 \\atopwithdelims ()\\lfloor \\frac{n-1}{2} \\rfloor }.$ Since $2|\\mathcal {F}|\\le 2^n+|\\mathcal {G}|$ , we have $|\\mathcal {F}|\\le 2^{n-1}+{n-1 \\atopwithdelims ()\\lfloor \\frac{n-1}{2} \\rfloor }={\\left\\lbrace \\begin{array}{ll} \\displaystyle \\sum _{i=n/2}^n {n \\atopwithdelims ()i} & \\mbox{if } n\\mbox{ is even,} \\\\ {n-1 \\atopwithdelims ()\\frac{n-1}{2}}+\\displaystyle \\sum _{i=(n+1)/2}^n {n \\atopwithdelims ()i} & \\mbox{if } n\\mbox{ is odd.}", "\\end{array}\\right.", "}$ To verify that the given bounds are best possible, consider the following (1,3)-union-intersecting set systems.", "When $n$ is even, take all the subsets of size at least $\\frac{n}{2}$ .", "When $n$ is odd, take all the subsets of size at least $\\frac{n+1}{2}$ and the subsets of size $\\frac{n-1}{2}$ not containing a fixed element.", "Let $\\mathcal {F}$ be a $(1,t)$ -union-intersecting system of subsets of $[n]$ .", "Define $\\mathcal {G}$ as above, and note that $\\mathcal {G}$ is $K_{1,t-1}$ -free.", "Theorem REF implies $|\\mathcal {G}|\\le {n\\atopwithdelims ()\\lfloor n/2 \\rfloor }\\left(1+\\frac{2(t-2)}{n}+O(n^{-2})\\right).$ Since $2|\\mathcal {F}|\\le 2^n+|\\mathcal {G}|$ , we have $|\\mathcal {F}|\\le 2^{n-1}+{n \\atopwithdelims ()\\lfloor n/2\\rfloor }\\left(\\frac{1}{2}+\\frac{t-2}{n}+O(n^{-2})\\right).$ The lower bound follows obviously from $f(n,1,t)\\ge f(n,1,3)=2^{n-1}+{n-1 \\atopwithdelims ()\\lfloor \\frac{n-1}{2} \\rfloor } \\ge 2^{n-1}+\\frac{1}{2}{n \\atopwithdelims ()\\lfloor n/2\\rfloor }.$ Let $\\mathcal {F}$ be an $(s,t)$ -union-intersecting system of subsets of $[n]$ .", "Define $\\mathcal {G}$ as above.", "Now we prove that $\\mathcal {G}$ is $K_{s,t}$ -free.", "Assume that $A_1, A_2, \\dots A_s, B_1, \\dots B_t\\in \\mathcal {G}$ are pairwise different subsets and $A_i\\subset B_j$ for all $(i,j)$ .", "Then $\\left(\\displaystyle \\bigcup _{i=1}^s A_i\\right) \\cap \\left(\\displaystyle \\bigcup _{j=1}^t ([n]-B_j)\\right) = \\emptyset $ .", "It is a contradiction, since $[n]-B_j\\in \\mathcal {F}$ for all $j$ .", "Theorem REF implies $|\\mathcal {G}|\\le {n\\atopwithdelims ()\\lfloor n/2 \\rfloor } \\left(2+\\frac{2(s+t-3)}{n}+O(n^{-2})\\right).$ Since $2|\\mathcal {F}|\\le 2^n+|\\mathcal {G}|$ , we have $|\\mathcal {F}|\\le 2^{n-1}+{n \\atopwithdelims ()\\lfloor n/2\\rfloor }\\left(1+\\frac{t+s-3}{n}+O(n^{-2})\\right).$ The lower bound follows from $f(n,s,t)\\ge f(n,2,2) \\ge 2^{n-1}+{n \\atopwithdelims ()\\lfloor n/2\\rfloor }\\frac{n}{n+2}.$ (The second inequaility can be verified by elementary calculations since the exact value of $f(n,2,2)$ is known.", "Equality holds when $n$ is even.)", "The k-uniform case   In this section we determine the size of the largest $k$ -uniform $(s,t)$ -union-intersecting set system of subsets of $[n]$ for all large enough $n$ .", "Theorem 4.1 Assume that $1\\le s \\le t$ and $\\mathcal {F}\\subset {[n] \\atopwithdelims ()k}$ is an $(s,t)$ -union-intersecting set system.", "Then $|\\mathcal {F}|\\le {n-1 \\atopwithdelims ()k-1}+s-1$ holds for all $n>n(k,t)$ .", "Remark 4.2 There is $k$ -uniform $(s,t)$ -union-intersecting set system of size ${n-1 \\atopwithdelims ()k-1}+s-1$ .", "Take all $k$ -element sets containing a fixed element, then take $s-1$ arbitrary sets of size $k$ .", "We need some preparation before we can start the proof of Theorem REF .", "Definition A sunflower (or $\\Delta $ -system) with $r$ petals and center $M$ is a family $\\lbrace S_1, S_2,\\dots S_r\\rbrace $ where $S_i\\cap S_j=M$ for all $1\\le i<j\\le r$ .", "Lemma 4.3 (Erdős-Rado [4]) Assume that $\\mathcal {A}\\subset {[n] \\atopwithdelims ()k}$ and $|\\mathcal {A}|>k!", "(r-1)^k$ .", "Then $\\mathcal {A}$ contains a sunflower with $r$ petals as a subfamily.", "We prove the lemma by induction on $k$ .", "The statement of the lemma is obviously true when $k=0$ .", "Let $\\mathcal {A}\\subset {[n] \\atopwithdelims ()k}$ a set system not containing a sunflower with $r$ petals.", "Let $\\lbrace A_1, A_2, \\dots A_m\\rbrace $ be a maximal family of pairwise disjoint sets in $\\mathcal {A}$ .", "Since pairwise disjoint sets form a sunflower, $m\\le r-1$ .", "For every $x\\in \\displaystyle \\bigcup _{i=1}^m A_i$ , let $\\mathcal {A}_x=\\lbrace S-\\lbrace x\\rbrace ~\\big |~ S\\in \\mathcal {A},~ x\\in S\\rbrace $ .", "Then each $\\mathcal {A}_x$ is a $k-1$ -uniform set system not containing sunflowers with $r$ petals.", "By induction we have $|\\mathcal {A}_x|\\le (k-1)!", "(r-1)^{k-1}$ .", "Then $|\\mathcal {A}|\\le (k-1)!", "(r-1)^{k-1}\\left|\\bigcup _{i=1}^m A_i \\right|\\le (k-1)!", "(r-1)^{k-1}\\cdot k(r-1)=k!", "(r-1)^k.$ Lemma 4.4 Let $c$ be a fixed positive integer.", "If $\\mathcal {A}\\subset {[n] \\atopwithdelims ()k}$ , $K\\subset [n]$ , $|K|\\le c$ and $|A\\cap K|\\ge 2$ holds for every $A\\in \\mathcal {A}$ , then $|\\mathcal {A}|\\le O(n^{k-2})$ .", "Choose a set $K\\subset K^{\\prime }$ with $|K^{\\prime }|=c$ .", "The conditions of the lemma are also satisfied with $K^{\\prime }$ .", "$|\\mathcal {A}|\\le \\sum _{i=2}^c {c \\atopwithdelims ()i}{n-c \\atopwithdelims ()k-i}.$ The right hand side is a polynomial of $n$ with degree $k-2$ , so $|\\mathcal {A}|\\le O(n^{k-2})$ .", "Lemma 4.5 Let $s,t$ be fixed positive integers.", "Let $\\mathcal {A}, \\mathcal {B}\\subset {[n] \\atopwithdelims ()k}$ .", "Assume that $|\\mathcal {B}|\\ge s$ , $\\mathcal {A}\\cup \\mathcal {B}$ is $(s,t)$ -union-intersecting, and there is an element $a$ such that $a\\in A$ holds for every $A\\in \\mathcal {A}$ , and $a\\notin \\mathcal {B}$ holds for every $B\\in \\mathcal {B}$ .", "Then $|\\mathcal {A}|\\le O(n^{k-2})$ .", "Choose $s$ different sets $B_1, B_2, \\dots B_s\\in \\mathcal {B}$ .", "Let $\\mathcal {A}^{\\prime }= \\lbrace A\\in \\mathcal {A} ~\\big |~ A\\cap \\displaystyle \\bigcup _{i=1}^s B_i\\ne \\emptyset \\rbrace $ .", "Since $\\mathcal {A}\\cup \\mathcal {B}$ is $(s,t)$ -union-intersecting, $|\\mathcal {A}-\\mathcal {A}^{\\prime }|\\le t-1$ .", "$ \\left|A\\cap \\left(\\lbrace a\\rbrace \\cup \\bigcup _{i=1}^s B_i\\right)\\right|\\ge 2 $ holds for all $A\\in \\mathcal {A^{\\prime }}$ .", "Use Lemma REF with $K=\\lbrace a\\rbrace \\cup \\displaystyle \\bigcup _{i=1}^s B_i$ and $c=sk+1$ , it implies $|\\mathcal {A}^{\\prime }|\\le O(n^{k-2})$ .", "Then $|\\mathcal {A}|\\le O(n^{k-2})+(t-1)=O(n^{k-2})$ .", "(of Theorem REF ) Use Lemma REF with $r=ks+t$ .", "If $|\\mathcal {F}|>k!", "(ks+t)^k$ , then $\\mathcal {F}$ contains a sunflower $\\lbrace S_1, S_2, \\dots S_{ks+t}\\rbrace $ as a subfamily.", "(Note that ${n-1\\atopwithdelims ()k-1}>k!", "(ks+t)^k$ holds for large enough $n$ .)", "Let $M$ denote the center of the sunflower and introduce the notations $|M|=\\lbrace a_1, a_2, \\dots a_m\\rbrace $ and $C_i=S_i-M~~(1\\le i\\le ks+t)$ .", "Let $\\mathcal {F}_0=\\lbrace F\\in \\mathcal {F}~\\big |~F\\cap M=\\emptyset \\rbrace $ , and $\\mathcal {F}_i=\\lbrace F\\in \\mathcal {F}~\\big |~a_i\\in F\\rbrace $ for $1\\le i\\le m$ .", "Assume that $|\\mathcal {F}_0|\\ge s$ .", "Let $B_1, B_2, \\dots B_s\\in \\mathcal {F}_0$ be different sets.", "Since $\\left|\\displaystyle \\bigcup _{i=1}^s B_i\\right|\\le ks$ , and the sets $\\lbrace C_1, C_2, \\dots C_{ks+t}\\rbrace $ are pairwise disjoint, there some indices $i_1, i_2, \\dots i_t$ such that $\\emptyset =\\left(\\bigcup _{i=1}^s B_i\\right) \\cap \\left(\\bigcup _{j=1}^t C_{i_j}\\right) = \\left(\\bigcup _{i=1}^s B_i\\right) \\cap \\left(\\bigcup _{j=1}^t S_{i_j}\\right).$ It contradicts our assumption that $\\mathcal {F}$ is $(s,t)$ -union-intersecting, so $|\\mathcal {F}_0|\\le s-1$ .", "Note that the obvious inequality $|\\mathcal {F}_i|\\le {n-1\\atopwithdelims ()k-1}$ holds for all $1\\le i\\le m$ , so the statement of the theorem is true if $|\\mathcal {F}-\\mathcal {F}_i|\\le s-1$ holds for any $i$ .", "Finally, assume that $|\\mathcal {F}-\\mathcal {F}_i|\\ge s$ holds for all $1\\le i\\le m$ .", "Using Lemma REF with $\\mathcal {A}=\\mathcal {F}_i$ and $\\mathcal {B}=\\mathcal {F}-\\mathcal {F}_i$ , we get that $|\\mathcal {F}_i|=O(n^{k-2})$ .", "Then $|\\mathcal {F}|\\le \\sum _{i=0}^m |\\mathcal {F}_i|\\le (s-1)+m\\cdot O(n^{k-2})= O(n^{k-2}).$ Since ${n-1 \\atopwithdelims ()k-1}+s-1$ is a polynomial of $n$ with degree $k-1$ , $|\\mathcal {F}|\\le {n-1 \\atopwithdelims ()k-1}+s-1$ holds for large enough $n$ .", "Remark 4.6 Note that Theorem REF generalizes the Erdős-Ko-Rado theorem [3] for large enough $n$ , since letting $s=1$ and $t\\ge 2$ , we get the same upper bound for $|\\mathcal {F}|$ while having weaker conditions on $\\mathcal {F}$ .", "Question Let $\\mathcal {F}$ be a set system satisfying the conditions of Theorem REF .", "What is the best upper bound for $|\\mathcal {F}|$ , when $n$ is small?" ], [ "Union-l-intersecting systems", "Definition Let $\\mathcal {F}$ be a set system and $k\\in \\mathbb {N}$ .", "Then $\\mathcal {F}^k$ denotes the set of the $k$ -element sets in $\\mathcal {F}$ .", "Definition A set system $\\mathcal {F}$ is called union-$l$ -intersecting, if it satisfies $|(F_1\\cup F_2)\\cap (G_1\\cup G_2)|\\ge l$ for all sets $F_1, F_2, G_1, G_2\\in \\mathcal {F}$ , $F_1\\ne F_2$ , $G_1\\ne G_2$ .", "Theorem 2.1 Let $\\mathcal {F}$ be a union-$l$ -intersecting set system whose elements are subsets of $[n]$ .", "($n\\ge 3$ ) Then we have the the following upper bounds for $|\\mathcal {F}|$ .", "If $n+l$ is even, then $|\\mathcal {F}|\\le \\sum _{i=\\frac{n+l}{2}-1}^{n} {n \\atopwithdelims ()i}.$ If $n+l$ is odd, then $|\\mathcal {F}|\\le AK\\left(n, \\frac{n+l-3}{2}, l\\right)+\\sum _{i=\\frac{n+l-1}{2}}^{n} {n \\atopwithdelims ()i}.$ These are the best possible bounds.", "We can assume that $\\mathcal {F}$ is an upset, that is $A\\in \\mathcal {F}$ , $A\\subset B$ imply $B\\in \\mathcal {F}$ .", "(If there are sets $A\\subset B$ , $A\\in \\mathcal {F}$ , $B\\notin \\mathcal {F}$ then we can replace $\\mathcal {F}$ by $\\mathcal {F}-A+B$ .", "After finitely many steps we arrive at an upset of the same size that is still union-$l$ -intersecting.)", "First, assume that $l\\in \\lbrace 1,2\\rbrace $ .", "Note that if $A,B\\in \\mathcal {F}$ , $A\\cap B=\\emptyset $ , and $|A\\cup B|=n+l-3<n$ , then both $A$ and $B$ can not be in $\\mathcal {F}$ at the same time.", "$A,B \\in \\mathcal {F}$ and $\\mathcal {F}$ being an upset would imply that there are two sets $C,D\\in \\mathcal {F}$ such that $A \\subset C$ , $B\\subset D$ and $|(A\\cup C) \\cap (B \\cup D)| = |C\\cap D|= l-1$ , with contradiction.", "For all $0\\le i < \\frac{n+l-3}{2}$ define the bipartite graph $G_i(S_i, T_i, E_i)$ as follows.", "Let $S_i$ be the set of all the $i$ -element subsets of $[n]$ , let $T_i$ be the set of subsets of size $n+l-3-i$ , and connect two sets $A\\in S_i$ and $B\\in T_i$ if they are disjoint.", "Then both vertex classes contain vertices of the same degree, so it follows by Hall's theorem that there is a matching that covers the smaller vertex class, that is $S_i$ .", "Since at most one of two matched subsets can be in $\\mathcal {F}$ , it follows that $ |\\mathcal {F}^i|+|\\mathcal {F}^{n+l-3-i}|\\le {n \\atopwithdelims ()n+l-3-i} ~~~~~~~\\left(0\\le i < \\frac{n+l-3}{2}\\right).$ Now let $l\\ge 3$ .", "We will show that (REF ) holds for all $n$ .", "Assume that there are $A,B\\in \\mathcal {F}$ , $A,B\\ne [n]$ such that $|A\\cap B|\\le l-3$ .", "Take $x\\notin A$ and $y\\notin B$ .", "Then $A\\cup \\lbrace x\\rbrace ,~B\\cup \\lbrace y\\rbrace \\in \\mathcal {F}$ , since $\\mathcal {F}$ is an upset and $|(A\\cup (A\\cup \\lbrace x\\rbrace ))\\cap (B\\cup (B\\cup \\lbrace y\\rbrace ))|=|(A\\cup \\lbrace x\\rbrace )\\cap (B\\cup \\lbrace y\\rbrace )|\\le |(A\\cap B) \\cup \\lbrace x,y\\rbrace |=l-3+2 < l.$ So $\\mathcal {F}-\\lbrace [n]\\rbrace $ is an $(l-2)$ -intersecting system, so $\\mathcal {F}$ is one too.", "Now use Theorem REF with $t=l-2$ .", "($l-2$ is positive since $l\\ge 3$ .)", "It gives us that (REF ) holds for all $n$ and $l$ .", "Assume that $l$ is a positive integer $n+l$ is odd.", "Then the sets in $\\mathcal {F}^{\\frac{n+l-3}{2}}$ form an $l$ -intersecting family.", "$A,B \\in \\mathcal {F}^{\\frac{n+l-3}{2}}$ , $|A\\cap B|\\le l-1$ and $\\mathcal {F}$ being an upset would imply that there are two sets $C,D\\in \\mathcal {F}$ such that $A \\subset C$ , $B\\subset D$ and $|(A\\cup C) \\cap (B \\cup D)| = |C\\cap D|= l-1$ .", "So Theorem REF provides an upper bound: $ |\\mathcal {F}^{\\frac{n+l-3}{2}}|\\le AK\\left(n, \\frac{n+l-3}{2}, l\\right).$ The upper bounds of the theorem follow after some calculations.", "When $l\\le 2$ , the inequalities of (REF ) imply $|\\mathcal {F}|=\\sum _{i=0}^n |\\mathcal {F}^i|=|\\mathcal {F}^{\\frac{n+l-3}{2}}|+\\sum _{i=0}^{\\lfloor \\frac{n+l}{2}-2\\rfloor } (|\\mathcal {F}^i|+|\\mathcal {F}^{n+l-3-i}|)+\\sum _{i=n+l-2}^n |\\mathcal {F}^i| \\le |\\mathcal {F}^{\\frac{n+l-3}{2}}|+\\sum _{i=\\lceil \\frac{n+l}{2}-1\\rceil }^n {n \\atopwithdelims ()i}.$ Let $l\\ge 3$ .", "Since $\\mathcal {F}$ is $(l-2)$ -intersecting, $|\\mathcal {F}^i|=0$ for all $i<l-2$ .", "Using (REF ), we get $|\\mathcal {F}|=\\sum _{i=l-3}^n |\\mathcal {F}^i|= |\\mathcal {F}^{\\frac{n+l-3}{2}}|+\\sum _{i=l-3}^{\\lfloor \\frac{n+l}{2}-2\\rfloor } (|\\mathcal {F}^i|+|\\mathcal {F}^{n+l-3-i}|)\\le |\\mathcal {F}^{\\frac{n+l-3}{2}}|+\\sum _{i=\\lceil \\frac{n+l}{2}-1\\rceil }^n {n \\atopwithdelims ()i}.$ We got the same inequality in the two cases.", "The upper bounds of the theorem follow immediately, since $\\mathcal {F}^{\\frac{n+l-3}{2}}=\\emptyset $ , if $n+l$ is even, and $|\\mathcal {F}^{\\frac{n+l-3}{2}}|\\le AK\\left(n, \\frac{n+l-3}{2}, l\\right)$ , if $n+l$ is odd (see (REF )).", "To verify that the given bounds are best possible, consider the following union-$l$ -intersecting set systems.", "When $n+l$ is even, take all the subsets of size at least $\\frac{n+l}{2}-1$ .", "When $n+l$ is odd, take all the subsets of size at least $\\frac{n+l-1}{2}$ and an $\\frac{n+l-3}{2}$ -uniform $l$ -intersecting set system of size $AK\\left(n, \\frac{n+l-3}{2}, l\\right)$ .", "Remark 2.2 Let us formulate the special case $l=1$ what was originally asked by Körner.", "The Erdős-Ko-Rado theorem [3] states that the size of the largest $k$ -uniform intersecting system of subsets of $[n]$ is ${n-1\\atopwithdelims ()k-1}$ , when $n\\ge 2k$ .", "It means that $AK(n,\\frac{n}{2}-1,1)={n-1\\atopwithdelims ()\\frac{n}{2}-2}$ , so the best upper bound for $|\\mathcal {F}|$ when $l=1$ is $ |\\mathcal {F}|\\le {\\left\\lbrace \\begin{array}{ll} \\displaystyle \\sum _{i=(n-1)/2}^n {n \\atopwithdelims ()i} & \\mbox{if } n\\mbox{ is odd,} \\\\ {n-1 \\atopwithdelims ()\\frac{n}{2}-2}+\\displaystyle \\sum _{i=\\frac{n}{2}}^n {n \\atopwithdelims ()i} & \\mbox{if } n\\mbox{ is even.}", "\\end{array}\\right.}", "$ Considering the union of more subsets   In this section we investigate a variation of the problem where we take the union of $s$ and $t$ subsets instead of 2 and 2.", "Definition A set system $\\mathcal {F}$ is called $(s,t)$ -union-intersecting if it has the property that for all $s+t$ pairwise different sets $F_1, F_2, \\dots F_s, G_1, \\dots G_t\\in \\mathcal {F}$ $\\left(\\bigcup _{i=1}^s F_i\\right) \\cap \\left(\\bigcup _{j=1}^t G_j\\right) \\ne \\emptyset .$ The size of the largest $(s,t)$ -union-intersecting system whose elements are subsets of $[n]$ is denoted by $f(n,s,t)$ .", "In this section we determine the value of $f(n,s,t)$ exactly when $s+t\\le 4$ and asymptotically in the other cases.", "Since $f(n,s,t)=f(n,t,s)$ , we can assume that $s\\le t$ .", "Theorem 3.1 Let $n\\ge 3$ .", "$ f(n,1,1)=2^{n-1}.", "$ $ f(n,1,2) = {\\left\\lbrace \\begin{array}{ll} \\displaystyle \\sum _{i=n/2}^n {n \\atopwithdelims ()i} & \\mbox{if } n\\mbox{ is even,} \\\\ {n-1 \\atopwithdelims ()\\frac{n-3}{2}}+\\displaystyle \\sum _{i=(n+1)/2}^n {n \\atopwithdelims ()i} & \\mbox{if } n\\mbox{ is odd.}", "\\end{array}\\right.}", "$ $ f(n,2,2) = {\\left\\lbrace \\begin{array}{ll} \\displaystyle \\sum _{i=(n-1)/2}^n {n \\atopwithdelims ()i} & \\mbox{if } n\\mbox{ is odd,} \\\\ {n-1 \\atopwithdelims ()\\frac{n}{2}-2}+\\displaystyle \\sum _{i=n/2}^n {n \\atopwithdelims ()i} & \\mbox{if } n\\mbox{ is even.}", "\\end{array}\\right.}", "$ $ f(n,1,3) = {\\left\\lbrace \\begin{array}{ll} \\displaystyle \\sum _{i=n/2}^n {n \\atopwithdelims ()i} & \\mbox{if } n\\mbox{ is even,} \\\\ {n-1 \\atopwithdelims ()\\frac{n-1}{2}}+\\displaystyle \\sum _{i=(n+1)/2}^n {n \\atopwithdelims ()i} & \\mbox{if } n\\mbox{ is odd.}", "\\end{array}\\right.}", "$ If $t\\ge 4$ , then $2^{n-1}+\\frac{1}{2}{n \\atopwithdelims ()\\lfloor n/2\\rfloor } \\le f(n,1,t) \\le 2^{n-1}+{n \\atopwithdelims ()\\lfloor n/2\\rfloor }\\left(\\frac{1}{2}+\\frac{t-2}{n}+O(n^{-2})\\right).", "$ If $s\\ge 2$ and $t\\ge 3$ , then $2^{n-1}+{n \\atopwithdelims ()\\lfloor n/2\\rfloor }\\frac{n}{n+2} \\le f(n,s,t) \\le 2^{n-1}+{n \\atopwithdelims ()\\lfloor n/2\\rfloor }\\left(1+\\frac{t+s-3}{n}+O(n^{-2})\\right).", "$ (See [3].)", "$f(n,1,1)$ is the size of the largest intersecting system among the subsets of $[n]$ .", "It is at most $2^{n-1}$ , since a subset and its complement cannot be in the intersecting system at the same time.", "By choosing all the subsets containing a fixed element, we get an intersecting system of size $2^{n-1}$ .", "Let $\\mathcal {F}$ be a $(1,2)$ -union-intersecting system.", "We can assume that $\\mathcal {F}$ is an upset.", "Note that if $A,B\\in \\mathcal {F}$ , $A\\cap B=\\emptyset $ , and $|A\\cup B|=n-1$ , then $A$ and $B$ can not be in $\\mathcal {F}$ at the same time.", "To see this, take let $\\lbrace x\\rbrace =[n]-(A\\cup B)$ .", "Then $A\\cap (B\\cup (B\\cup \\lbrace x\\rbrace ))=A\\cap (B\\cup \\lbrace x\\rbrace )=\\emptyset $ .", "For all $0\\le i < \\frac{n-1}{2}$ define the bipartite graph $G_i(S_i, T_i, E_i)$ as follows.", "Let $S_i$ be the set of all the $i$ -element subsets of $[n]$ , let $T_i$ be the set of subsets of size $n-1-i$ , and connect two sets $A\\in S_i$ and $B\\in T_i$ if they are disjoint.", "Then both vertex classes contain vertices of the same degree, so it follows by Hall's theorem that there is a matching that covers the smaller vertex class, that is $S_i$ .", "Since at most one of two matched subsets can be in $\\mathcal {F}$ , it follows that $ |\\mathcal {F}^i|+|\\mathcal {F}^{n-1-i}|\\le {n \\atopwithdelims ()n-1-i} ~~~~~~~\\left(0\\le i < \\frac{n-1}{2}\\right).$ When $n$ is even, these inequalities together imply $ |\\mathcal {F}|\\le \\sum _{i=n/2}^{n} {n \\atopwithdelims ()i}.$ Assume that $n$ is odd.", "Then $\\mathcal {F}^{\\frac{n-1}{2}}$ is an intersecting family.", "$A,B \\in \\mathcal {F}^{\\frac{n-1}{2}}$ , $A\\cap B= \\emptyset $ and $\\mathcal {F}$ being an upset would imply that there is a set $C\\in \\mathcal {F}$ such that $B\\subset C$ and $|A\\cap (B \\cup C)| = |A\\cap C|= \\emptyset $ .", "So the Erdős-Ko-Rado theorem provides an upper bound: $ |\\mathcal {F}^{\\frac{n-1}{2}}|\\le {n-1 \\atopwithdelims ()\\frac{n-3}{2}}.$ This, together with the inequalities of (REF ) implies $ |\\mathcal {F}| \\le {n-1 \\atopwithdelims ()\\frac{n-3}{2}}+\\sum _{i=(n+1)/2}^n {n \\atopwithdelims ()i}.$ To verify that the given bounds are best possible, consider the following (1,2)-union-intersecting set systems.", "When $n$ is even, take all the subsets of size at least $\\frac{n}{2}$ .", "When $n$ is odd, take all the subsets of size at least $\\frac{n+1}{2}$ and the subsets of size $\\frac{n-1}{2}$ containing a fixed element.", "We already solved this problem as the case $l=1$ in Theorem REF .", "(See Remark REF .)", "Let $\\mathcal {F}$ be a $(1,3)$ -union-intersecting system of subsets of $[n]$ .", "Let $\\mathcal {F}^{\\prime }=\\lbrace [n]-F~\\big |~F\\in \\mathcal {F}\\rbrace $ , and let $\\mathcal {G}=\\mathcal {F}\\cap \\mathcal {F}^{\\prime }$ .", "Now we prove that $\\mathcal {G}$ is $K_{1,2}$ -free and $K_{2,1}$ -free.", "(See Section 1 for the definitions.)", "Since $\\mathcal {G}$ is invariant to taking complements, it is enough to show that $\\mathcal {G}$ is $K_{12}$ -free.", "Assume that there are three pairwise different sets $A,B,C\\in \\mathcal {G}$ , such that $A\\subset B$ and $A\\subset C$ .", "Then the sets $A,[n]-A, [n]-B, [n]-C\\in \\mathcal {F}$ would satisfy $A\\cap (([n]-A)\\cup ([n]-B)\\cup ([n]-C))=A\\cap ([n]-A)=\\emptyset .", "$ Theorem REF gives us the following upper bound for a set system that is $K_{12}$ -free and $K_{21}$ -free: $|\\mathcal {G}|\\le 2{n-1 \\atopwithdelims ()\\lfloor \\frac{n-1}{2} \\rfloor }.$ Since $2|\\mathcal {F}|\\le 2^n+|\\mathcal {G}|$ , we have $|\\mathcal {F}|\\le 2^{n-1}+{n-1 \\atopwithdelims ()\\lfloor \\frac{n-1}{2} \\rfloor }={\\left\\lbrace \\begin{array}{ll} \\displaystyle \\sum _{i=n/2}^n {n \\atopwithdelims ()i} & \\mbox{if } n\\mbox{ is even,} \\\\ {n-1 \\atopwithdelims ()\\frac{n-1}{2}}+\\displaystyle \\sum _{i=(n+1)/2}^n {n \\atopwithdelims ()i} & \\mbox{if } n\\mbox{ is odd.}", "\\end{array}\\right.", "}$ To verify that the given bounds are best possible, consider the following (1,3)-union-intersecting set systems.", "When $n$ is even, take all the subsets of size at least $\\frac{n}{2}$ .", "When $n$ is odd, take all the subsets of size at least $\\frac{n+1}{2}$ and the subsets of size $\\frac{n-1}{2}$ not containing a fixed element.", "Let $\\mathcal {F}$ be a $(1,t)$ -union-intersecting system of subsets of $[n]$ .", "Define $\\mathcal {G}$ as above, and note that $\\mathcal {G}$ is $K_{1,t-1}$ -free.", "Theorem REF implies $|\\mathcal {G}|\\le {n\\atopwithdelims ()\\lfloor n/2 \\rfloor }\\left(1+\\frac{2(t-2)}{n}+O(n^{-2})\\right).$ Since $2|\\mathcal {F}|\\le 2^n+|\\mathcal {G}|$ , we have $|\\mathcal {F}|\\le 2^{n-1}+{n \\atopwithdelims ()\\lfloor n/2\\rfloor }\\left(\\frac{1}{2}+\\frac{t-2}{n}+O(n^{-2})\\right).$ The lower bound follows obviously from $f(n,1,t)\\ge f(n,1,3)=2^{n-1}+{n-1 \\atopwithdelims ()\\lfloor \\frac{n-1}{2} \\rfloor } \\ge 2^{n-1}+\\frac{1}{2}{n \\atopwithdelims ()\\lfloor n/2\\rfloor }.$ Let $\\mathcal {F}$ be an $(s,t)$ -union-intersecting system of subsets of $[n]$ .", "Define $\\mathcal {G}$ as above.", "Now we prove that $\\mathcal {G}$ is $K_{s,t}$ -free.", "Assume that $A_1, A_2, \\dots A_s, B_1, \\dots B_t\\in \\mathcal {G}$ are pairwise different subsets and $A_i\\subset B_j$ for all $(i,j)$ .", "Then $\\left(\\displaystyle \\bigcup _{i=1}^s A_i\\right) \\cap \\left(\\displaystyle \\bigcup _{j=1}^t ([n]-B_j)\\right) = \\emptyset $ .", "It is a contradiction, since $[n]-B_j\\in \\mathcal {F}$ for all $j$ .", "Theorem REF implies $|\\mathcal {G}|\\le {n\\atopwithdelims ()\\lfloor n/2 \\rfloor } \\left(2+\\frac{2(s+t-3)}{n}+O(n^{-2})\\right).$ Since $2|\\mathcal {F}|\\le 2^n+|\\mathcal {G}|$ , we have $|\\mathcal {F}|\\le 2^{n-1}+{n \\atopwithdelims ()\\lfloor n/2\\rfloor }\\left(1+\\frac{t+s-3}{n}+O(n^{-2})\\right).$ The lower bound follows from $f(n,s,t)\\ge f(n,2,2) \\ge 2^{n-1}+{n \\atopwithdelims ()\\lfloor n/2\\rfloor }\\frac{n}{n+2}.$ (The second inequaility can be verified by elementary calculations since the exact value of $f(n,2,2)$ is known.", "Equality holds when $n$ is even.)", "The k-uniform case   In this section we determine the size of the largest $k$ -uniform $(s,t)$ -union-intersecting set system of subsets of $[n]$ for all large enough $n$ .", "Theorem 4.1 Assume that $1\\le s \\le t$ and $\\mathcal {F}\\subset {[n] \\atopwithdelims ()k}$ is an $(s,t)$ -union-intersecting set system.", "Then $|\\mathcal {F}|\\le {n-1 \\atopwithdelims ()k-1}+s-1$ holds for all $n>n(k,t)$ .", "Remark 4.2 There is $k$ -uniform $(s,t)$ -union-intersecting set system of size ${n-1 \\atopwithdelims ()k-1}+s-1$ .", "Take all $k$ -element sets containing a fixed element, then take $s-1$ arbitrary sets of size $k$ .", "We need some preparation before we can start the proof of Theorem REF .", "Definition A sunflower (or $\\Delta $ -system) with $r$ petals and center $M$ is a family $\\lbrace S_1, S_2,\\dots S_r\\rbrace $ where $S_i\\cap S_j=M$ for all $1\\le i<j\\le r$ .", "Lemma 4.3 (Erdős-Rado [4]) Assume that $\\mathcal {A}\\subset {[n] \\atopwithdelims ()k}$ and $|\\mathcal {A}|>k!", "(r-1)^k$ .", "Then $\\mathcal {A}$ contains a sunflower with $r$ petals as a subfamily.", "We prove the lemma by induction on $k$ .", "The statement of the lemma is obviously true when $k=0$ .", "Let $\\mathcal {A}\\subset {[n] \\atopwithdelims ()k}$ a set system not containing a sunflower with $r$ petals.", "Let $\\lbrace A_1, A_2, \\dots A_m\\rbrace $ be a maximal family of pairwise disjoint sets in $\\mathcal {A}$ .", "Since pairwise disjoint sets form a sunflower, $m\\le r-1$ .", "For every $x\\in \\displaystyle \\bigcup _{i=1}^m A_i$ , let $\\mathcal {A}_x=\\lbrace S-\\lbrace x\\rbrace ~\\big |~ S\\in \\mathcal {A},~ x\\in S\\rbrace $ .", "Then each $\\mathcal {A}_x$ is a $k-1$ -uniform set system not containing sunflowers with $r$ petals.", "By induction we have $|\\mathcal {A}_x|\\le (k-1)!", "(r-1)^{k-1}$ .", "Then $|\\mathcal {A}|\\le (k-1)!", "(r-1)^{k-1}\\left|\\bigcup _{i=1}^m A_i \\right|\\le (k-1)!", "(r-1)^{k-1}\\cdot k(r-1)=k!", "(r-1)^k.$ Lemma 4.4 Let $c$ be a fixed positive integer.", "If $\\mathcal {A}\\subset {[n] \\atopwithdelims ()k}$ , $K\\subset [n]$ , $|K|\\le c$ and $|A\\cap K|\\ge 2$ holds for every $A\\in \\mathcal {A}$ , then $|\\mathcal {A}|\\le O(n^{k-2})$ .", "Choose a set $K\\subset K^{\\prime }$ with $|K^{\\prime }|=c$ .", "The conditions of the lemma are also satisfied with $K^{\\prime }$ .", "$|\\mathcal {A}|\\le \\sum _{i=2}^c {c \\atopwithdelims ()i}{n-c \\atopwithdelims ()k-i}.$ The right hand side is a polynomial of $n$ with degree $k-2$ , so $|\\mathcal {A}|\\le O(n^{k-2})$ .", "Lemma 4.5 Let $s,t$ be fixed positive integers.", "Let $\\mathcal {A}, \\mathcal {B}\\subset {[n] \\atopwithdelims ()k}$ .", "Assume that $|\\mathcal {B}|\\ge s$ , $\\mathcal {A}\\cup \\mathcal {B}$ is $(s,t)$ -union-intersecting, and there is an element $a$ such that $a\\in A$ holds for every $A\\in \\mathcal {A}$ , and $a\\notin \\mathcal {B}$ holds for every $B\\in \\mathcal {B}$ .", "Then $|\\mathcal {A}|\\le O(n^{k-2})$ .", "Choose $s$ different sets $B_1, B_2, \\dots B_s\\in \\mathcal {B}$ .", "Let $\\mathcal {A}^{\\prime }= \\lbrace A\\in \\mathcal {A} ~\\big |~ A\\cap \\displaystyle \\bigcup _{i=1}^s B_i\\ne \\emptyset \\rbrace $ .", "Since $\\mathcal {A}\\cup \\mathcal {B}$ is $(s,t)$ -union-intersecting, $|\\mathcal {A}-\\mathcal {A}^{\\prime }|\\le t-1$ .", "$ \\left|A\\cap \\left(\\lbrace a\\rbrace \\cup \\bigcup _{i=1}^s B_i\\right)\\right|\\ge 2 $ holds for all $A\\in \\mathcal {A^{\\prime }}$ .", "Use Lemma REF with $K=\\lbrace a\\rbrace \\cup \\displaystyle \\bigcup _{i=1}^s B_i$ and $c=sk+1$ , it implies $|\\mathcal {A}^{\\prime }|\\le O(n^{k-2})$ .", "Then $|\\mathcal {A}|\\le O(n^{k-2})+(t-1)=O(n^{k-2})$ .", "(of Theorem REF ) Use Lemma REF with $r=ks+t$ .", "If $|\\mathcal {F}|>k!", "(ks+t)^k$ , then $\\mathcal {F}$ contains a sunflower $\\lbrace S_1, S_2, \\dots S_{ks+t}\\rbrace $ as a subfamily.", "(Note that ${n-1\\atopwithdelims ()k-1}>k!", "(ks+t)^k$ holds for large enough $n$ .)", "Let $M$ denote the center of the sunflower and introduce the notations $|M|=\\lbrace a_1, a_2, \\dots a_m\\rbrace $ and $C_i=S_i-M~~(1\\le i\\le ks+t)$ .", "Let $\\mathcal {F}_0=\\lbrace F\\in \\mathcal {F}~\\big |~F\\cap M=\\emptyset \\rbrace $ , and $\\mathcal {F}_i=\\lbrace F\\in \\mathcal {F}~\\big |~a_i\\in F\\rbrace $ for $1\\le i\\le m$ .", "Assume that $|\\mathcal {F}_0|\\ge s$ .", "Let $B_1, B_2, \\dots B_s\\in \\mathcal {F}_0$ be different sets.", "Since $\\left|\\displaystyle \\bigcup _{i=1}^s B_i\\right|\\le ks$ , and the sets $\\lbrace C_1, C_2, \\dots C_{ks+t}\\rbrace $ are pairwise disjoint, there some indices $i_1, i_2, \\dots i_t$ such that $\\emptyset =\\left(\\bigcup _{i=1}^s B_i\\right) \\cap \\left(\\bigcup _{j=1}^t C_{i_j}\\right) = \\left(\\bigcup _{i=1}^s B_i\\right) \\cap \\left(\\bigcup _{j=1}^t S_{i_j}\\right).$ It contradicts our assumption that $\\mathcal {F}$ is $(s,t)$ -union-intersecting, so $|\\mathcal {F}_0|\\le s-1$ .", "Note that the obvious inequality $|\\mathcal {F}_i|\\le {n-1\\atopwithdelims ()k-1}$ holds for all $1\\le i\\le m$ , so the statement of the theorem is true if $|\\mathcal {F}-\\mathcal {F}_i|\\le s-1$ holds for any $i$ .", "Finally, assume that $|\\mathcal {F}-\\mathcal {F}_i|\\ge s$ holds for all $1\\le i\\le m$ .", "Using Lemma REF with $\\mathcal {A}=\\mathcal {F}_i$ and $\\mathcal {B}=\\mathcal {F}-\\mathcal {F}_i$ , we get that $|\\mathcal {F}_i|=O(n^{k-2})$ .", "Then $|\\mathcal {F}|\\le \\sum _{i=0}^m |\\mathcal {F}_i|\\le (s-1)+m\\cdot O(n^{k-2})= O(n^{k-2}).$ Since ${n-1 \\atopwithdelims ()k-1}+s-1$ is a polynomial of $n$ with degree $k-1$ , $|\\mathcal {F}|\\le {n-1 \\atopwithdelims ()k-1}+s-1$ holds for large enough $n$ .", "Remark 4.6 Note that Theorem REF generalizes the Erdős-Ko-Rado theorem [3] for large enough $n$ , since letting $s=1$ and $t\\ge 2$ , we get the same upper bound for $|\\mathcal {F}|$ while having weaker conditions on $\\mathcal {F}$ .", "Question Let $\\mathcal {F}$ be a set system satisfying the conditions of Theorem REF .", "What is the best upper bound for $|\\mathcal {F}|$ , when $n$ is small?" ], [ "Considering the union of more subsets", "In this section we investigate a variation of the problem where we take the union of $s$ and $t$ subsets instead of 2 and 2.", "Definition A set system $\\mathcal {F}$ is called $(s,t)$ -union-intersecting if it has the property that for all $s+t$ pairwise different sets $F_1, F_2, \\dots F_s, G_1, \\dots G_t\\in \\mathcal {F}$ $\\left(\\bigcup _{i=1}^s F_i\\right) \\cap \\left(\\bigcup _{j=1}^t G_j\\right) \\ne \\emptyset .$ The size of the largest $(s,t)$ -union-intersecting system whose elements are subsets of $[n]$ is denoted by $f(n,s,t)$ .", "In this section we determine the value of $f(n,s,t)$ exactly when $s+t\\le 4$ and asymptotically in the other cases.", "Since $f(n,s,t)=f(n,t,s)$ , we can assume that $s\\le t$ .", "Theorem 3.1 Let $n\\ge 3$ .", "$ f(n,1,1)=2^{n-1}.", "$ $ f(n,1,2) = {\\left\\lbrace \\begin{array}{ll} \\displaystyle \\sum _{i=n/2}^n {n \\atopwithdelims ()i} & \\mbox{if } n\\mbox{ is even,} \\\\ {n-1 \\atopwithdelims ()\\frac{n-3}{2}}+\\displaystyle \\sum _{i=(n+1)/2}^n {n \\atopwithdelims ()i} & \\mbox{if } n\\mbox{ is odd.}", "\\end{array}\\right.}", "$ $ f(n,2,2) = {\\left\\lbrace \\begin{array}{ll} \\displaystyle \\sum _{i=(n-1)/2}^n {n \\atopwithdelims ()i} & \\mbox{if } n\\mbox{ is odd,} \\\\ {n-1 \\atopwithdelims ()\\frac{n}{2}-2}+\\displaystyle \\sum _{i=n/2}^n {n \\atopwithdelims ()i} & \\mbox{if } n\\mbox{ is even.}", "\\end{array}\\right.}", "$ $ f(n,1,3) = {\\left\\lbrace \\begin{array}{ll} \\displaystyle \\sum _{i=n/2}^n {n \\atopwithdelims ()i} & \\mbox{if } n\\mbox{ is even,} \\\\ {n-1 \\atopwithdelims ()\\frac{n-1}{2}}+\\displaystyle \\sum _{i=(n+1)/2}^n {n \\atopwithdelims ()i} & \\mbox{if } n\\mbox{ is odd.}", "\\end{array}\\right.}", "$ If $t\\ge 4$ , then $2^{n-1}+\\frac{1}{2}{n \\atopwithdelims ()\\lfloor n/2\\rfloor } \\le f(n,1,t) \\le 2^{n-1}+{n \\atopwithdelims ()\\lfloor n/2\\rfloor }\\left(\\frac{1}{2}+\\frac{t-2}{n}+O(n^{-2})\\right).", "$ If $s\\ge 2$ and $t\\ge 3$ , then $2^{n-1}+{n \\atopwithdelims ()\\lfloor n/2\\rfloor }\\frac{n}{n+2} \\le f(n,s,t) \\le 2^{n-1}+{n \\atopwithdelims ()\\lfloor n/2\\rfloor }\\left(1+\\frac{t+s-3}{n}+O(n^{-2})\\right).", "$ (See [3].)", "$f(n,1,1)$ is the size of the largest intersecting system among the subsets of $[n]$ .", "It is at most $2^{n-1}$ , since a subset and its complement cannot be in the intersecting system at the same time.", "By choosing all the subsets containing a fixed element, we get an intersecting system of size $2^{n-1}$ .", "Let $\\mathcal {F}$ be a $(1,2)$ -union-intersecting system.", "We can assume that $\\mathcal {F}$ is an upset.", "Note that if $A,B\\in \\mathcal {F}$ , $A\\cap B=\\emptyset $ , and $|A\\cup B|=n-1$ , then $A$ and $B$ can not be in $\\mathcal {F}$ at the same time.", "To see this, take let $\\lbrace x\\rbrace =[n]-(A\\cup B)$ .", "Then $A\\cap (B\\cup (B\\cup \\lbrace x\\rbrace ))=A\\cap (B\\cup \\lbrace x\\rbrace )=\\emptyset $ .", "For all $0\\le i < \\frac{n-1}{2}$ define the bipartite graph $G_i(S_i, T_i, E_i)$ as follows.", "Let $S_i$ be the set of all the $i$ -element subsets of $[n]$ , let $T_i$ be the set of subsets of size $n-1-i$ , and connect two sets $A\\in S_i$ and $B\\in T_i$ if they are disjoint.", "Then both vertex classes contain vertices of the same degree, so it follows by Hall's theorem that there is a matching that covers the smaller vertex class, that is $S_i$ .", "Since at most one of two matched subsets can be in $\\mathcal {F}$ , it follows that $ |\\mathcal {F}^i|+|\\mathcal {F}^{n-1-i}|\\le {n \\atopwithdelims ()n-1-i} ~~~~~~~\\left(0\\le i < \\frac{n-1}{2}\\right).$ When $n$ is even, these inequalities together imply $ |\\mathcal {F}|\\le \\sum _{i=n/2}^{n} {n \\atopwithdelims ()i}.$ Assume that $n$ is odd.", "Then $\\mathcal {F}^{\\frac{n-1}{2}}$ is an intersecting family.", "$A,B \\in \\mathcal {F}^{\\frac{n-1}{2}}$ , $A\\cap B= \\emptyset $ and $\\mathcal {F}$ being an upset would imply that there is a set $C\\in \\mathcal {F}$ such that $B\\subset C$ and $|A\\cap (B \\cup C)| = |A\\cap C|= \\emptyset $ .", "So the Erdős-Ko-Rado theorem provides an upper bound: $ |\\mathcal {F}^{\\frac{n-1}{2}}|\\le {n-1 \\atopwithdelims ()\\frac{n-3}{2}}.$ This, together with the inequalities of (REF ) implies $ |\\mathcal {F}| \\le {n-1 \\atopwithdelims ()\\frac{n-3}{2}}+\\sum _{i=(n+1)/2}^n {n \\atopwithdelims ()i}.$ To verify that the given bounds are best possible, consider the following (1,2)-union-intersecting set systems.", "When $n$ is even, take all the subsets of size at least $\\frac{n}{2}$ .", "When $n$ is odd, take all the subsets of size at least $\\frac{n+1}{2}$ and the subsets of size $\\frac{n-1}{2}$ containing a fixed element.", "We already solved this problem as the case $l=1$ in Theorem REF .", "(See Remark REF .)", "Let $\\mathcal {F}$ be a $(1,3)$ -union-intersecting system of subsets of $[n]$ .", "Let $\\mathcal {F}^{\\prime }=\\lbrace [n]-F~\\big |~F\\in \\mathcal {F}\\rbrace $ , and let $\\mathcal {G}=\\mathcal {F}\\cap \\mathcal {F}^{\\prime }$ .", "Now we prove that $\\mathcal {G}$ is $K_{1,2}$ -free and $K_{2,1}$ -free.", "(See Section 1 for the definitions.)", "Since $\\mathcal {G}$ is invariant to taking complements, it is enough to show that $\\mathcal {G}$ is $K_{12}$ -free.", "Assume that there are three pairwise different sets $A,B,C\\in \\mathcal {G}$ , such that $A\\subset B$ and $A\\subset C$ .", "Then the sets $A,[n]-A, [n]-B, [n]-C\\in \\mathcal {F}$ would satisfy $A\\cap (([n]-A)\\cup ([n]-B)\\cup ([n]-C))=A\\cap ([n]-A)=\\emptyset .", "$ Theorem REF gives us the following upper bound for a set system that is $K_{12}$ -free and $K_{21}$ -free: $|\\mathcal {G}|\\le 2{n-1 \\atopwithdelims ()\\lfloor \\frac{n-1}{2} \\rfloor }.$ Since $2|\\mathcal {F}|\\le 2^n+|\\mathcal {G}|$ , we have $|\\mathcal {F}|\\le 2^{n-1}+{n-1 \\atopwithdelims ()\\lfloor \\frac{n-1}{2} \\rfloor }={\\left\\lbrace \\begin{array}{ll} \\displaystyle \\sum _{i=n/2}^n {n \\atopwithdelims ()i} & \\mbox{if } n\\mbox{ is even,} \\\\ {n-1 \\atopwithdelims ()\\frac{n-1}{2}}+\\displaystyle \\sum _{i=(n+1)/2}^n {n \\atopwithdelims ()i} & \\mbox{if } n\\mbox{ is odd.}", "\\end{array}\\right.", "}$ To verify that the given bounds are best possible, consider the following (1,3)-union-intersecting set systems.", "When $n$ is even, take all the subsets of size at least $\\frac{n}{2}$ .", "When $n$ is odd, take all the subsets of size at least $\\frac{n+1}{2}$ and the subsets of size $\\frac{n-1}{2}$ not containing a fixed element.", "Let $\\mathcal {F}$ be a $(1,t)$ -union-intersecting system of subsets of $[n]$ .", "Define $\\mathcal {G}$ as above, and note that $\\mathcal {G}$ is $K_{1,t-1}$ -free.", "Theorem REF implies $|\\mathcal {G}|\\le {n\\atopwithdelims ()\\lfloor n/2 \\rfloor }\\left(1+\\frac{2(t-2)}{n}+O(n^{-2})\\right).$ Since $2|\\mathcal {F}|\\le 2^n+|\\mathcal {G}|$ , we have $|\\mathcal {F}|\\le 2^{n-1}+{n \\atopwithdelims ()\\lfloor n/2\\rfloor }\\left(\\frac{1}{2}+\\frac{t-2}{n}+O(n^{-2})\\right).$ The lower bound follows obviously from $f(n,1,t)\\ge f(n,1,3)=2^{n-1}+{n-1 \\atopwithdelims ()\\lfloor \\frac{n-1}{2} \\rfloor } \\ge 2^{n-1}+\\frac{1}{2}{n \\atopwithdelims ()\\lfloor n/2\\rfloor }.$ Let $\\mathcal {F}$ be an $(s,t)$ -union-intersecting system of subsets of $[n]$ .", "Define $\\mathcal {G}$ as above.", "Now we prove that $\\mathcal {G}$ is $K_{s,t}$ -free.", "Assume that $A_1, A_2, \\dots A_s, B_1, \\dots B_t\\in \\mathcal {G}$ are pairwise different subsets and $A_i\\subset B_j$ for all $(i,j)$ .", "Then $\\left(\\displaystyle \\bigcup _{i=1}^s A_i\\right) \\cap \\left(\\displaystyle \\bigcup _{j=1}^t ([n]-B_j)\\right) = \\emptyset $ .", "It is a contradiction, since $[n]-B_j\\in \\mathcal {F}$ for all $j$ .", "Theorem REF implies $|\\mathcal {G}|\\le {n\\atopwithdelims ()\\lfloor n/2 \\rfloor } \\left(2+\\frac{2(s+t-3)}{n}+O(n^{-2})\\right).$ Since $2|\\mathcal {F}|\\le 2^n+|\\mathcal {G}|$ , we have $|\\mathcal {F}|\\le 2^{n-1}+{n \\atopwithdelims ()\\lfloor n/2\\rfloor }\\left(1+\\frac{t+s-3}{n}+O(n^{-2})\\right).$ The lower bound follows from $f(n,s,t)\\ge f(n,2,2) \\ge 2^{n-1}+{n \\atopwithdelims ()\\lfloor n/2\\rfloor }\\frac{n}{n+2}.$ (The second inequaility can be verified by elementary calculations since the exact value of $f(n,2,2)$ is known.", "Equality holds when $n$ is even.)", "The k-uniform case   In this section we determine the size of the largest $k$ -uniform $(s,t)$ -union-intersecting set system of subsets of $[n]$ for all large enough $n$ .", "Theorem 4.1 Assume that $1\\le s \\le t$ and $\\mathcal {F}\\subset {[n] \\atopwithdelims ()k}$ is an $(s,t)$ -union-intersecting set system.", "Then $|\\mathcal {F}|\\le {n-1 \\atopwithdelims ()k-1}+s-1$ holds for all $n>n(k,t)$ .", "Remark 4.2 There is $k$ -uniform $(s,t)$ -union-intersecting set system of size ${n-1 \\atopwithdelims ()k-1}+s-1$ .", "Take all $k$ -element sets containing a fixed element, then take $s-1$ arbitrary sets of size $k$ .", "We need some preparation before we can start the proof of Theorem REF .", "Definition A sunflower (or $\\Delta $ -system) with $r$ petals and center $M$ is a family $\\lbrace S_1, S_2,\\dots S_r\\rbrace $ where $S_i\\cap S_j=M$ for all $1\\le i<j\\le r$ .", "Lemma 4.3 (Erdős-Rado [4]) Assume that $\\mathcal {A}\\subset {[n] \\atopwithdelims ()k}$ and $|\\mathcal {A}|>k!", "(r-1)^k$ .", "Then $\\mathcal {A}$ contains a sunflower with $r$ petals as a subfamily.", "We prove the lemma by induction on $k$ .", "The statement of the lemma is obviously true when $k=0$ .", "Let $\\mathcal {A}\\subset {[n] \\atopwithdelims ()k}$ a set system not containing a sunflower with $r$ petals.", "Let $\\lbrace A_1, A_2, \\dots A_m\\rbrace $ be a maximal family of pairwise disjoint sets in $\\mathcal {A}$ .", "Since pairwise disjoint sets form a sunflower, $m\\le r-1$ .", "For every $x\\in \\displaystyle \\bigcup _{i=1}^m A_i$ , let $\\mathcal {A}_x=\\lbrace S-\\lbrace x\\rbrace ~\\big |~ S\\in \\mathcal {A},~ x\\in S\\rbrace $ .", "Then each $\\mathcal {A}_x$ is a $k-1$ -uniform set system not containing sunflowers with $r$ petals.", "By induction we have $|\\mathcal {A}_x|\\le (k-1)!", "(r-1)^{k-1}$ .", "Then $|\\mathcal {A}|\\le (k-1)!", "(r-1)^{k-1}\\left|\\bigcup _{i=1}^m A_i \\right|\\le (k-1)!", "(r-1)^{k-1}\\cdot k(r-1)=k!", "(r-1)^k.$ Lemma 4.4 Let $c$ be a fixed positive integer.", "If $\\mathcal {A}\\subset {[n] \\atopwithdelims ()k}$ , $K\\subset [n]$ , $|K|\\le c$ and $|A\\cap K|\\ge 2$ holds for every $A\\in \\mathcal {A}$ , then $|\\mathcal {A}|\\le O(n^{k-2})$ .", "Choose a set $K\\subset K^{\\prime }$ with $|K^{\\prime }|=c$ .", "The conditions of the lemma are also satisfied with $K^{\\prime }$ .", "$|\\mathcal {A}|\\le \\sum _{i=2}^c {c \\atopwithdelims ()i}{n-c \\atopwithdelims ()k-i}.$ The right hand side is a polynomial of $n$ with degree $k-2$ , so $|\\mathcal {A}|\\le O(n^{k-2})$ .", "Lemma 4.5 Let $s,t$ be fixed positive integers.", "Let $\\mathcal {A}, \\mathcal {B}\\subset {[n] \\atopwithdelims ()k}$ .", "Assume that $|\\mathcal {B}|\\ge s$ , $\\mathcal {A}\\cup \\mathcal {B}$ is $(s,t)$ -union-intersecting, and there is an element $a$ such that $a\\in A$ holds for every $A\\in \\mathcal {A}$ , and $a\\notin \\mathcal {B}$ holds for every $B\\in \\mathcal {B}$ .", "Then $|\\mathcal {A}|\\le O(n^{k-2})$ .", "Choose $s$ different sets $B_1, B_2, \\dots B_s\\in \\mathcal {B}$ .", "Let $\\mathcal {A}^{\\prime }= \\lbrace A\\in \\mathcal {A} ~\\big |~ A\\cap \\displaystyle \\bigcup _{i=1}^s B_i\\ne \\emptyset \\rbrace $ .", "Since $\\mathcal {A}\\cup \\mathcal {B}$ is $(s,t)$ -union-intersecting, $|\\mathcal {A}-\\mathcal {A}^{\\prime }|\\le t-1$ .", "$ \\left|A\\cap \\left(\\lbrace a\\rbrace \\cup \\bigcup _{i=1}^s B_i\\right)\\right|\\ge 2 $ holds for all $A\\in \\mathcal {A^{\\prime }}$ .", "Use Lemma REF with $K=\\lbrace a\\rbrace \\cup \\displaystyle \\bigcup _{i=1}^s B_i$ and $c=sk+1$ , it implies $|\\mathcal {A}^{\\prime }|\\le O(n^{k-2})$ .", "Then $|\\mathcal {A}|\\le O(n^{k-2})+(t-1)=O(n^{k-2})$ .", "(of Theorem REF ) Use Lemma REF with $r=ks+t$ .", "If $|\\mathcal {F}|>k!", "(ks+t)^k$ , then $\\mathcal {F}$ contains a sunflower $\\lbrace S_1, S_2, \\dots S_{ks+t}\\rbrace $ as a subfamily.", "(Note that ${n-1\\atopwithdelims ()k-1}>k!", "(ks+t)^k$ holds for large enough $n$ .)", "Let $M$ denote the center of the sunflower and introduce the notations $|M|=\\lbrace a_1, a_2, \\dots a_m\\rbrace $ and $C_i=S_i-M~~(1\\le i\\le ks+t)$ .", "Let $\\mathcal {F}_0=\\lbrace F\\in \\mathcal {F}~\\big |~F\\cap M=\\emptyset \\rbrace $ , and $\\mathcal {F}_i=\\lbrace F\\in \\mathcal {F}~\\big |~a_i\\in F\\rbrace $ for $1\\le i\\le m$ .", "Assume that $|\\mathcal {F}_0|\\ge s$ .", "Let $B_1, B_2, \\dots B_s\\in \\mathcal {F}_0$ be different sets.", "Since $\\left|\\displaystyle \\bigcup _{i=1}^s B_i\\right|\\le ks$ , and the sets $\\lbrace C_1, C_2, \\dots C_{ks+t}\\rbrace $ are pairwise disjoint, there some indices $i_1, i_2, \\dots i_t$ such that $\\emptyset =\\left(\\bigcup _{i=1}^s B_i\\right) \\cap \\left(\\bigcup _{j=1}^t C_{i_j}\\right) = \\left(\\bigcup _{i=1}^s B_i\\right) \\cap \\left(\\bigcup _{j=1}^t S_{i_j}\\right).$ It contradicts our assumption that $\\mathcal {F}$ is $(s,t)$ -union-intersecting, so $|\\mathcal {F}_0|\\le s-1$ .", "Note that the obvious inequality $|\\mathcal {F}_i|\\le {n-1\\atopwithdelims ()k-1}$ holds for all $1\\le i\\le m$ , so the statement of the theorem is true if $|\\mathcal {F}-\\mathcal {F}_i|\\le s-1$ holds for any $i$ .", "Finally, assume that $|\\mathcal {F}-\\mathcal {F}_i|\\ge s$ holds for all $1\\le i\\le m$ .", "Using Lemma REF with $\\mathcal {A}=\\mathcal {F}_i$ and $\\mathcal {B}=\\mathcal {F}-\\mathcal {F}_i$ , we get that $|\\mathcal {F}_i|=O(n^{k-2})$ .", "Then $|\\mathcal {F}|\\le \\sum _{i=0}^m |\\mathcal {F}_i|\\le (s-1)+m\\cdot O(n^{k-2})= O(n^{k-2}).$ Since ${n-1 \\atopwithdelims ()k-1}+s-1$ is a polynomial of $n$ with degree $k-1$ , $|\\mathcal {F}|\\le {n-1 \\atopwithdelims ()k-1}+s-1$ holds for large enough $n$ .", "Remark 4.6 Note that Theorem REF generalizes the Erdős-Ko-Rado theorem [3] for large enough $n$ , since letting $s=1$ and $t\\ge 2$ , we get the same upper bound for $|\\mathcal {F}|$ while having weaker conditions on $\\mathcal {F}$ .", "Question Let $\\mathcal {F}$ be a set system satisfying the conditions of Theorem REF .", "What is the best upper bound for $|\\mathcal {F}|$ , when $n$ is small?" ], [ "The k-uniform case", "In this section we determine the size of the largest $k$ -uniform $(s,t)$ -union-intersecting set system of subsets of $[n]$ for all large enough $n$ .", "Theorem 4.1 Assume that $1\\le s \\le t$ and $\\mathcal {F}\\subset {[n] \\atopwithdelims ()k}$ is an $(s,t)$ -union-intersecting set system.", "Then $|\\mathcal {F}|\\le {n-1 \\atopwithdelims ()k-1}+s-1$ holds for all $n>n(k,t)$ .", "Remark 4.2 There is $k$ -uniform $(s,t)$ -union-intersecting set system of size ${n-1 \\atopwithdelims ()k-1}+s-1$ .", "Take all $k$ -element sets containing a fixed element, then take $s-1$ arbitrary sets of size $k$ .", "We need some preparation before we can start the proof of Theorem REF .", "Definition A sunflower (or $\\Delta $ -system) with $r$ petals and center $M$ is a family $\\lbrace S_1, S_2,\\dots S_r\\rbrace $ where $S_i\\cap S_j=M$ for all $1\\le i<j\\le r$ .", "Lemma 4.3 (Erdős-Rado [4]) Assume that $\\mathcal {A}\\subset {[n] \\atopwithdelims ()k}$ and $|\\mathcal {A}|>k!", "(r-1)^k$ .", "Then $\\mathcal {A}$ contains a sunflower with $r$ petals as a subfamily.", "We prove the lemma by induction on $k$ .", "The statement of the lemma is obviously true when $k=0$ .", "Let $\\mathcal {A}\\subset {[n] \\atopwithdelims ()k}$ a set system not containing a sunflower with $r$ petals.", "Let $\\lbrace A_1, A_2, \\dots A_m\\rbrace $ be a maximal family of pairwise disjoint sets in $\\mathcal {A}$ .", "Since pairwise disjoint sets form a sunflower, $m\\le r-1$ .", "For every $x\\in \\displaystyle \\bigcup _{i=1}^m A_i$ , let $\\mathcal {A}_x=\\lbrace S-\\lbrace x\\rbrace ~\\big |~ S\\in \\mathcal {A},~ x\\in S\\rbrace $ .", "Then each $\\mathcal {A}_x$ is a $k-1$ -uniform set system not containing sunflowers with $r$ petals.", "By induction we have $|\\mathcal {A}_x|\\le (k-1)!", "(r-1)^{k-1}$ .", "Then $|\\mathcal {A}|\\le (k-1)!", "(r-1)^{k-1}\\left|\\bigcup _{i=1}^m A_i \\right|\\le (k-1)!", "(r-1)^{k-1}\\cdot k(r-1)=k!", "(r-1)^k.$ Lemma 4.4 Let $c$ be a fixed positive integer.", "If $\\mathcal {A}\\subset {[n] \\atopwithdelims ()k}$ , $K\\subset [n]$ , $|K|\\le c$ and $|A\\cap K|\\ge 2$ holds for every $A\\in \\mathcal {A}$ , then $|\\mathcal {A}|\\le O(n^{k-2})$ .", "Choose a set $K\\subset K^{\\prime }$ with $|K^{\\prime }|=c$ .", "The conditions of the lemma are also satisfied with $K^{\\prime }$ .", "$|\\mathcal {A}|\\le \\sum _{i=2}^c {c \\atopwithdelims ()i}{n-c \\atopwithdelims ()k-i}.$ The right hand side is a polynomial of $n$ with degree $k-2$ , so $|\\mathcal {A}|\\le O(n^{k-2})$ .", "Lemma 4.5 Let $s,t$ be fixed positive integers.", "Let $\\mathcal {A}, \\mathcal {B}\\subset {[n] \\atopwithdelims ()k}$ .", "Assume that $|\\mathcal {B}|\\ge s$ , $\\mathcal {A}\\cup \\mathcal {B}$ is $(s,t)$ -union-intersecting, and there is an element $a$ such that $a\\in A$ holds for every $A\\in \\mathcal {A}$ , and $a\\notin \\mathcal {B}$ holds for every $B\\in \\mathcal {B}$ .", "Then $|\\mathcal {A}|\\le O(n^{k-2})$ .", "Choose $s$ different sets $B_1, B_2, \\dots B_s\\in \\mathcal {B}$ .", "Let $\\mathcal {A}^{\\prime }= \\lbrace A\\in \\mathcal {A} ~\\big |~ A\\cap \\displaystyle \\bigcup _{i=1}^s B_i\\ne \\emptyset \\rbrace $ .", "Since $\\mathcal {A}\\cup \\mathcal {B}$ is $(s,t)$ -union-intersecting, $|\\mathcal {A}-\\mathcal {A}^{\\prime }|\\le t-1$ .", "$ \\left|A\\cap \\left(\\lbrace a\\rbrace \\cup \\bigcup _{i=1}^s B_i\\right)\\right|\\ge 2 $ holds for all $A\\in \\mathcal {A^{\\prime }}$ .", "Use Lemma REF with $K=\\lbrace a\\rbrace \\cup \\displaystyle \\bigcup _{i=1}^s B_i$ and $c=sk+1$ , it implies $|\\mathcal {A}^{\\prime }|\\le O(n^{k-2})$ .", "Then $|\\mathcal {A}|\\le O(n^{k-2})+(t-1)=O(n^{k-2})$ .", "(of Theorem REF ) Use Lemma REF with $r=ks+t$ .", "If $|\\mathcal {F}|>k!", "(ks+t)^k$ , then $\\mathcal {F}$ contains a sunflower $\\lbrace S_1, S_2, \\dots S_{ks+t}\\rbrace $ as a subfamily.", "(Note that ${n-1\\atopwithdelims ()k-1}>k!", "(ks+t)^k$ holds for large enough $n$ .)", "Let $M$ denote the center of the sunflower and introduce the notations $|M|=\\lbrace a_1, a_2, \\dots a_m\\rbrace $ and $C_i=S_i-M~~(1\\le i\\le ks+t)$ .", "Let $\\mathcal {F}_0=\\lbrace F\\in \\mathcal {F}~\\big |~F\\cap M=\\emptyset \\rbrace $ , and $\\mathcal {F}_i=\\lbrace F\\in \\mathcal {F}~\\big |~a_i\\in F\\rbrace $ for $1\\le i\\le m$ .", "Assume that $|\\mathcal {F}_0|\\ge s$ .", "Let $B_1, B_2, \\dots B_s\\in \\mathcal {F}_0$ be different sets.", "Since $\\left|\\displaystyle \\bigcup _{i=1}^s B_i\\right|\\le ks$ , and the sets $\\lbrace C_1, C_2, \\dots C_{ks+t}\\rbrace $ are pairwise disjoint, there some indices $i_1, i_2, \\dots i_t$ such that $\\emptyset =\\left(\\bigcup _{i=1}^s B_i\\right) \\cap \\left(\\bigcup _{j=1}^t C_{i_j}\\right) = \\left(\\bigcup _{i=1}^s B_i\\right) \\cap \\left(\\bigcup _{j=1}^t S_{i_j}\\right).$ It contradicts our assumption that $\\mathcal {F}$ is $(s,t)$ -union-intersecting, so $|\\mathcal {F}_0|\\le s-1$ .", "Note that the obvious inequality $|\\mathcal {F}_i|\\le {n-1\\atopwithdelims ()k-1}$ holds for all $1\\le i\\le m$ , so the statement of the theorem is true if $|\\mathcal {F}-\\mathcal {F}_i|\\le s-1$ holds for any $i$ .", "Finally, assume that $|\\mathcal {F}-\\mathcal {F}_i|\\ge s$ holds for all $1\\le i\\le m$ .", "Using Lemma REF with $\\mathcal {A}=\\mathcal {F}_i$ and $\\mathcal {B}=\\mathcal {F}-\\mathcal {F}_i$ , we get that $|\\mathcal {F}_i|=O(n^{k-2})$ .", "Then $|\\mathcal {F}|\\le \\sum _{i=0}^m |\\mathcal {F}_i|\\le (s-1)+m\\cdot O(n^{k-2})= O(n^{k-2}).$ Since ${n-1 \\atopwithdelims ()k-1}+s-1$ is a polynomial of $n$ with degree $k-1$ , $|\\mathcal {F}|\\le {n-1 \\atopwithdelims ()k-1}+s-1$ holds for large enough $n$ .", "Remark 4.6 Note that Theorem REF generalizes the Erdős-Ko-Rado theorem [3] for large enough $n$ , since letting $s=1$ and $t\\ge 2$ , we get the same upper bound for $|\\mathcal {F}|$ while having weaker conditions on $\\mathcal {F}$ .", "Question Let $\\mathcal {F}$ be a set system satisfying the conditions of Theorem REF .", "What is the best upper bound for $|\\mathcal {F}|$ , when $n$ is small?" ] ]
1403.0088
[ [ "Quintessence reconstruction of interacting HDE in a non-flat universe" ], [ "Abstract In this paper we consider quintessence reconstruction of interacting holographic dark energy in a non-flat background.", "As system's IR cutoff we choose the radius of the event horizon measured on the sphere of the horizon, defined as $L=ar(t)$.", "To this end we construct a quintessence model by a real, single scalar field.", "Evolution of the potential, $V(\\phi)$, as well as the dynamics of the scalar field, $\\phi$, are obtained according to the respective holographic dark energy.", "The reconstructed potentials show a cosmological constant behavior for the present time.", "We constrain the model parameters in a flat universe by using the observational data, and applying the Monte Carlo Markov chain simulation.", "We obtain the best fit values of the holographic dark energy model and the interacting parameters as $c=1.0576^{+0.3010+0.3052}_{-0.6632-0.6632}$ and $\\zeta=0.2433^{+0.6373+0.6373}_{-0.2251-0.2251}$, respectively.", "From the data fitting results we also find that the model can cross the phantom line in the present universe where the best fit value of of the dark energy equation of state is $w_D=-1.2429$." ], [ "Introduction", "A wide range of observational evidences support the present acceleration of the Universe expansion.", "The first evidence for the mentioned acceleration is the cosmological observations from Type Ia supernovae (SN Ia) [1].", "Subsequently such acceleration was repeatedly confirmed by Cosmic Microwave Background (CMB) anisotropies measured by the WMAP satellite [2], Large Scale Structure [3], weak lensing [4] and the integrated Sach-Wolfe effect [5].", "Based on the Einstein's theory of gravity, such an acceleration needs an exotic type of matter with negative pressure, usually called dark energy (DE) in the literatures.", "This new component consists more than $70\\%$ of the present energy content of the universe.", "The simplest alternative which can explain the phase of acceleration is the so called cosmological constant which originally was presented by Einstein to build a static solution for the universe in the context of general relativity.", "Although cosmological constant can explain the acceleration of the universe but it suffers the “fine tuning\" and “coincidence\" problems.", "Of interesting models of DE are those which called scalar field models.", "A typical property of these models is their time varying equation of state parameter ($w=\\frac{P}{\\rho }$ ) favored by cosmic observations [6], [7], [8].", "A plenty of these models have been presented in the literature which an incomplete list is quintessence, tachyon, K-essence, agegraphic, ghost and so on (see [9], [10], [11] and references therein).", "Among different candidate to DE, holographic dark energy (HDE) is one which contains interesting features.", "This model is based on the holographic principle which states that the entropy of a system scales not with it's volume, but with it's surface area [12] and it should be constrained by an infrared cutoff [13].", "Applying such a principle to the DE issue and taking the whole universe into account, then the vacuum energy related to this holographic principle is viewed as DE, usually called HDE [13], [14], [15].", "According to these statements the holographic energy density can be written as [13] $\\rho _{D}= \\frac{3c^2M^2_p}{L^2},$ where $c^2$ is a numerical constant, $M_p^{2} =( 8\\pi G)^{-1}$ , and $L$ is an infrared (IR) cutoff radius.", "It is worth mentioning that the holographic principle does not determine the IR cutoff and we have still freedom to choose $L$ .", "Different choices for IR cutoff parameter, $L$ , have been proposed in the literature, among them are, the particle horizon [16], the future event horizon [17], the Hubble horizon [18], [19] and the apparent horizon [20].", "Each of these choices solve some features and lead new problems.", "For instance in the HDE model with Hubble horizon, the fine tuning problem is solved and the coincidence problem is also alleviated, however, the effective equation of state for such vacuum energy is zero and the universe is decelerating [18] unless the interaction is taken into account [19].", "For a complete list of papers concerning HDE one can refer to [21] and references therein.", "Nowadays, every model which can explain the acceleration of the Universe expansion, and is consistent with observational evidences, could be accepted as a DE candidate.", "Due to the lack of observational evidences about DE models, many approaches are presented to answer the puzzle of the unexpected acceleration of the Universe.", "The number and variety of these models are so increasing which we should classify them in any way.", "One main task in this way is to find equivalent theories presented in different frameworks, however, they seems to have distinct origins.", "One valuable approach which recently has attracted a lot of attention is to make scalar field dual of the DE models [22], [23], [24], [25], [26].", "This interest in the scalar field models of DE partly comes from the fact that scalar fields naturally arise in particle physics including supersymmetric field theories and string/M theory.", "Beside with clarifying the status of the models in the literature maybe sometimes we can use the corresponding scalar field dual of DE model predicting new features and setting observational constraints on the free parameters.", "In this paper our aim is to establish a correspondence between the HDE and quintessence model of DE in a non-flat universe.", "As systems's IR cutoff we shall choose the radius of the event horizon measured on the sphere of the horizon, defined as $L=ar(t)$ .", "Quintessence assumes a canonical scalar field $\\phi $ and a self interacting potential $V(\\phi )$ minimally coupled to the other component in the universe.", "Quintessence is described by the Lagrangian of the form ${\\cal L}=-\\frac{1}{2}g^{\\mu \\nu }\\partial _{\\mu }\\phi \\partial _{\\nu }\\phi -V(\\phi ).$ The energy-momentum tensor of quintessence is $T_{\\mu \\nu }=\\partial _{\\mu }\\phi \\partial _{\\nu }\\phi -g_{\\mu \\nu }\\left[\\frac{1}{2}g^{\\alpha \\beta }\\partial _{\\alpha }\\phi \\partial _{\\beta }\\phi +V(\\phi )\\right].$ In quintessence model we choose a convenient potential $V(\\phi )$ to obtain desirable result in agreement with observations.", "Hence, our goal in this paper is to reconstruct the potential $V(\\phi )$ corresponds to the HDE and investigate the evolution of different parameters in the model.", "Our work differs from Ref.", "[22] in that we consider the interacting HDE model in a non-flat universe, while the author of [22] studied the non-interacting case in a flat universe.", "It also differs from Refs.", "[27], [28], in that we take $L=ar(t)$ as system's IR cutoff not the Hubble radius $L=H^{-1}$ proposed in [27], nor the Ricci scalar like cutoff, $L^{-2}=\\alpha H^2+\\beta \\dot{H}$ , introduced in [28].", "This paper is organized as follows.", "In section , we reconstruct the non-interacting holographic quintessence model with $L=ar(t)$ as IR cutoff.", "In section , we extend our study to the case where there is an interaction between DE and dark matter.", "In order to check the viability of the model, in section , we constrain the holographic interacting quintessence model by using the cosmological data.", "We summarize our results in section ." ], [ "Quintessence reconstruction of HDE ", "Consider the non-flat Friedmann-Robertson-Walker (FRW) universe which is described by the line element $ds^2=dt^2-a^2(t)\\left(\\frac{dr^2}{1-kr^2}+r^2d\\Omega ^2\\right),$ where $a(t)$ is the scale factor, and $k$ is the curvature parameter with $k = -1, 0, 1$ corresponding to open, flat, and closed universes, respectively.", "The first Friedmann equation is $H^2+\\frac{k}{a^2}=\\frac{1}{3M_p^2} \\left( \\rho _m+\\rho _D \\right).$ We introduce, as usual, the fractional energy densities such as $\\Omega _m=\\frac{\\rho _m}{3M_p^2H^2}, \\hspace{14.22636pt}\\Omega _D=\\frac{\\rho _D}{3M_p^2H^2},\\hspace{14.22636pt}\\Omega _k=\\frac{k}{H^2 a^2},$ thus, the Friedmann equation can be written $\\Omega _m+\\Omega _D=1+\\Omega _k.$ We shall assume the quintessence scalar field model of DE is the effective underlying theory.", "The energy density and pressure for the quintessence scalar field are given by [9] $\\rho _\\phi =\\frac{1}{2}\\dot{\\phi }^2+V(\\phi ),\\\\p_\\phi =\\frac{1}{2}\\dot{\\phi }^2-V(\\phi ).", "$ Thus the potential and the kinetic energy term can be written as $&&V(\\phi )=\\frac{1-w_D}{2}\\rho _{\\phi },\\\\&&\\dot{\\phi }^2=(1+w_D)\\rho _\\phi .", "$ Next we implement the HDE model with quintessence field.", "The holographic energy density has the form (REF ), where the radius $L$ in a nonflat universe is chosen as $L=ar(t),$ and the function $r(t)$ can be obtained from the following relation $\\int _{0}^{r(t)}{\\frac{dr}{\\sqrt{1-kr^2}}}=\\int _{0}^{\\infty }{\\frac{dt}{a}}=\\frac{R_h}{a}.$ It is important to note that in the non-flat universe the characteristic length which plays the role of the IR-cutoff is the radius $L$ of the event horizon measured on the sphere of the horizon and not the radial size $R_h$ of the horizon.", "Solving the above equation for general case of the non-flat FRW universe, we have $r(t)=\\frac{1}{\\sqrt{k}}\\sin y,$ where $y=\\sqrt{k} R_h/a$ .", "For latter convenience we rewrite the second Eq.", "(REF ) in the form $HL=\\frac{c}{\\sqrt{\\Omega _D}}.", "$ Taking derivative with respect to the cosmic time $t$ from Eq.", "(REF ) and using Eqs.", "(REF ) and (REF ) we obtain $\\dot{L}=HL+a\\dot{r}(t)=\\frac{c}{\\sqrt{\\Omega _D}}-\\cos y.$ Consider the FRW universe filled with DE and dust (dark matter) which evolves according to their conservation laws $&&\\dot{\\rho }_D+3H\\rho _D(1+w_D)=0,\\\\&&\\dot{\\rho }_m+3H\\rho _m=0, $ where $w_D$ is the equation of state parameter of DE.", "Taking the derivative of Eq.", "(REF ) with respect to time and using Eq.", "(REF ) we find $\\dot{\\rho }_D=-2H\\rho _D\\left(1-\\frac{\\sqrt{\\Omega _D}}{c}\\cos y\\right).$ Inserting this equation in conservation law (REF ), we obtain the equation of state parameter $w_D=-\\frac{1}{3}-\\frac{2\\sqrt{\\Omega _D}}{3c}\\cos y.$ Differentiating Eq.", "(REF ) and using relation ${\\dot{\\Omega }_D}={\\Omega ^{\\prime }_D}H$ , we reach ${\\Omega ^{\\prime }_D}=\\Omega _D\\left(-2\\frac{\\dot{H}}{H^2}-2+\\frac{2}{c}\\sqrt{\\Omega _D}\\cos y\\right),$ where the dot and the prime denote the derivative with respect to the cosmic time and $x=\\ln {a}$ , respectively.", "Taking the derivative of both side of the Friedman equation (REF ) with respect to the cosmic time, and using Eqs.", "(REF ), (REF ), (REF ) and (), it is easy to show that $\\frac{2\\dot{H}}{H^2}=-3-\\Omega _k+\\Omega _D+\\frac{2\\Omega ^{3/2}_D}{c}\\cos y.$ Substituting this relation into Eq.", "(REF ), we obtain the equation of motion of HDE ${\\Omega ^{\\prime }_D}&=&\\Omega _D\\left[(1-\\Omega _D)\\left(1+\\frac{2}{c}\\sqrt{\\Omega _D}\\cos y\\right)+\\Omega _k\\right].$ We have plotted in Figs.", "1 and 2 the evolutions of the $w_D$ and $\\Omega _D$ for the HDE with different parameter $c$ .", "One can see from Fig.", "1 that increasing $c$ leads to a faster evolution of $w_D$ toward more negative values, while a reverse behavior is seen for $\\Omega _D$ and increasing $c$ results a slower evolution of $\\Omega _D$ .", "Now we suggest a correspondence between the HDE and quintessence scalar field namely, we identify $\\rho _\\phi $ with $\\rho _D$ .", "Using relation $\\rho _\\phi =\\rho _D={3M_p^2H^2}\\Omega _D$ and Eq.", "(REF ) we can rewrite the scalar potential and kinetic energy term as Figure: The evolution of w D w_D (left) and Ω D \\Omega _{D} (right)for HDE with different parameter cc.", "Here we takeΩ D0 =0.72\\Omega _{D0}=0.72 and Ω k =0.01.\\Omega _{k}=0.01.Figure: The evolution of the scalar-field φ(a)\\phi (a) (left) andthe potential V(φ)V(\\phi ) (right) for HDE with different parametercc where φ\\phi is in unit of m p m_p and V(φ)V(\\phi ) in ρ c0 \\rho _{c0}.Here we have taken Ω D0 =0.72\\Omega _{D0}=0.72 and Ω k =0.01.\\Omega _{k}=0.01.$V(\\phi )&=&M^2_pH^2 \\Omega _D\\left(2+\\frac{\\sqrt{\\Omega _D}}{c}\\cos y\\right),\\\\\\dot{\\phi }&=&M_pH \\left( 2\\Omega _D-\\frac{2}{c}{\\Omega ^{3/2}_D}\\cos y\\right)^{1/2}.$ Finally we obtain the evolutionary form of the field by integrating the above equation $\\phi (a)-\\phi (a_0)=M_p \\int _{a_0}^{a}{\\frac{da}{a}\\sqrt{ 2\\Omega _D-\\frac{2}{c}{\\Omega ^{3/2}_D}\\cos y}},$ where $a_0$ is the present value of the scale factor, and $\\Omega _D$ is given by Eq.", "(REF ).", "Basically, from Eqs.", "(REF ) and (REF ) one can derive $\\phi =\\phi (a)$ and then combining the result with (REF ) one finds $V=V(\\phi )$ .", "Unfortunately, the analytical form of the potential in terms of the scalar field cannot be determined due to the complexity of the equations involved.", "However, we can obtain it numerically.", "For simplicity we take $\\Omega _k\\simeq 0.01$ fixed in the numerical discussion.", "The reconstructed quintessence potential $V(\\phi )$ and the evolutionary form of the field are plotted in Figs.", "2, where we have taken $\\phi (a_0=1)=0$ .", "A notable point in this figure is that the reconstructed potentials for different values of $c$ have a nonzero value at the present time, which can be interpreted as a cosmological constant behavior of the model desirable from the perspective of $\\Lambda $ CDM model." ], [ "Quintessence reconstruction of interacting HDE model ", "In this section, we consider the interaction between dark matter and DE.", "In this case the continuity equations take the form $&&\\dot{\\rho }_m+3H\\rho _m=Q, \\\\&& \\dot{\\rho }_D+3H\\rho _D(1+w_D)=-Q.$ where $Q$ denotes the interaction term and can be taken as $Q =3\\zeta H{\\rho _D}({1+u})$ with $\\zeta $ being a coupling constant and $u=\\rho _m/\\rho _D$ is the energy density ratio.", "Inserting Eq.", "(REF ) in conservation law (), we obtain the equation of state parameter $w_D=-\\frac{1}{3}-\\frac{2\\sqrt{\\Omega _D}}{3c}\\cos y-\\frac{\\zeta }{\\Omega _D}({1+\\Omega _k}).$ Taking the derivative of both side of the Friedman equation (REF ) with respect to the cosmic time, and using Eqs.", "(REF ), (REF ), () and (REF ), it is easy to show that $\\frac{2\\dot{H}}{H^2}=-3-\\Omega _k+\\Omega _D+\\frac{2\\Omega ^{3/2}_D}{c}+{3{\\zeta }({1+\\Omega _k})}.$ Substituting this relation into Eq.", "(REF ), we obtain the equation of motion of HDE ${\\Omega ^{\\prime }_D}&=&\\Omega _D\\left[(1-\\Omega _D)\\left(1+\\frac{2}{c}\\sqrt{\\Omega _D}\\cos y\\right)-{3{\\zeta }({1+\\Omega _k})}+\\Omega _k\\right].$ We plot in Fig.", "3 the evolutions of $w_D$ and $\\Omega _D$ for interacting HDE with different parameter $c$ .", "Now we implement a correspondence between interacting HDE and quintessence scalar field.", "In this case we find $V(\\phi )&=&M^2_pH^2 \\Omega _D\\left(2+{3{\\zeta }({1+\\Omega _k})}+\\frac{\\sqrt{\\Omega _D}}{c}\\cos y\\right),\\\\\\dot{\\phi }&=&M_pH \\left(2\\Omega _D-3{\\zeta }({1+\\Omega _k})-\\frac{2}{c}{\\Omega ^{3/2}_D}\\cos y\\right)^{1/2}.$ Finally, the evolutionary form of the field can be obtained by integrating the above equation.", "We obtain $\\phi (a)-\\phi (a_0)=M_p \\int _{a_0}^{a}{\\frac{da}{a}\\sqrt{ 2\\Omega _D-\\frac{2}{c}{\\Omega ^{3/2}_D}\\cos y-3{\\zeta }({1+\\Omega _k})}},$ where $\\Omega _D$ is given by Eq.", "(REF ).", "Again, the analytical form of the potential in terms of the scalar field cannot be determined due to the complexity of the equations involved and we do a numerical discussion.", "The reconstructed quintessence potential $V(\\phi )$ and the evolutionary form of the field are plotted in Fig.", "4 and 5, where we have taken $\\phi (a_0=1)=0$ .", "For simplicity we take $\\Omega _k\\simeq 0.01$ fixed in the numerical discussion.", "In the interacting case there exist a different manner of evolution for $w_D$ .", "In the pervious section we found that increasing $c$ leads a faster evolution for $w_D$ toward more negative values while in the interacting case increasing $c$ cause $w_D$ to evolve toward less negative values which can predict a slower rate of expansion for the future HDE dominated universe.", "Also One can find from Figs.", "4 and 5 that the reconstructed potentials evolve toward a nonzero minima at the present as mentioned in the noninteracting case." ], [ "Model Fitting", "In this section we will fit the interacting Quintessence HDE model, in a flat universe, by using the cosmological data.", "To obtain the best fit values of the model parameters, we apply the maximum likelihood method.", "In this method the total likelihood function $\\mathcal {L}_{\\rm total}=e^{-\\chi _{\\rm tot}^2/2}$ can be defined as the product of the separate likelihood functions of uncorrelated observational data with $\\chi ^2_{\\rm tot}=\\chi ^2_{\\rm SNIa}+\\chi ^2_{\\rm CMB}+\\chi ^{2}_{\\rm BAO}+\\chi ^2_{\\rm gas}\\;,$ where SNIa stands for type Ia supernovae, CMB for cosmic microwave background radiation, BAO for baryon acoustic oscillation and gas stands for X-ray gas mass fraction data.", "The details of obtaining each $\\chi $ is discussed in [29].", "Best fit values of parameters are obtained by minimizing $\\chi _{\\rm tot}^2$ .", "In the current paper we will use CMB data from seven-year WMAP [30], type Ia supernovae data from 557 Union2 [31], baryon acoustic oscillation data from SDSS DR7 [32], and the cluster X-ray gas mass fraction data from the Chandra X-ray observations [33].", "We apply a Markov chain Monte Carlo (MCMC) simulation on the parameters of the model by using the publicly available CosmoMC code [34] and considering the parameter vector $\\lbrace \\Omega _{\\rm b}h^2,\\Omega _{\\rm DM}h^2, \\zeta \\rbrace $ .", "The MCMC simulation results are summarized in table I and the two dimensional contours are plotted in figure REF .", "From table I it is clear that the main cosmological parameters $\\Omega _{\\rm b}h^2$ , $\\Omega _{\\rm DM}h^2 $ , $\\Omega _{\\rm D}$ are compatible with the results of the $\\Lambda $ CDM model [35] as one can see from the third column in table I.", "We can see that the best fit value for the dark energy equation of state crossed the phantom line where $w_D=-1.249243^{+0.624537+0.636920}_{-0.455913-0.455913}$ .", "In addition the best fit value of the HDE parameter $c=1.0576^{+0.3010+0.3052}_{-0.6632-0.6632}$ is compatible with the previous numerical analysis works such as $c=0.91^{+0.21}_{-0.18}$ in [36], $c=0.84^{+0.14}_{-0.12}$ in [37] and $c=0.68^{+0.03}_{-0.02}$ in [38].", "Here we obtained a positive best fit value for the interacting parameter in 1$\\sigma $ and 2$\\sigma $ confidence levels as $\\zeta =0.2433^{+0.6373+0.6373}_{-0.2251-0.2251}$ in spite of we took negative values in the prior of the parameter $\\zeta $ as well.", "This positive value suggests only conversion of dark matter to dark energy.", "Therefore in this model there is no chance for converting of DE to dark matter.", "The interacting parameter in the HDE model has been constrained by observational data by many authors although with different parametrization of the interaction term $Q$ [39], [40], [41], [42], [43], [44].", "In [44] the authors have considered the same parametrization as ours in this paper but they have chosen the prior on parameter $\\zeta $ as $\\zeta =[0, 0.02]$ and therefore obtained the best fit value of parameter $\\zeta $ as $\\zeta =0.0006\\pm 0.0006$ .", "Table: The best fit values of the cosmological and model parameters in the interacting Quintessence HDE model in a flat universe with 1σ1\\sigma and 2σ2\\sigma regions.Here CMB, SNIa and BAO and X-ray mass gas fraction data together with the BBN constraints have been used.", "For comparison,the results for the Λ\\Lambda CDM model from the Planck data are presented as well .Figure: 2-dimensional constraint of the cosmological and model parameters contoursin the flat interacting Quintessence HDE model with 1σ1\\sigma and 2σ2\\sigma regions.", "To produce these plots,Union2+CMB+BAO+X-ray gas mass fraction data together with the BBN constraints have been used." ], [ "Conclusions", "Adopting the viewpoint that the quintessence scalar field model of DE is the effective underlying theory of DE, we are able to establish a connection between the quintessence scalar field and interacting HDE scenario.", "This connection allows us to reconstruct the quintessence scalar field model according to the evolutionary behavior of interacting holographic energy density.", "We have reconstructed the potential as well as the dynamics of the quintessence scalar field which describe the quintessence cosmology.", "Unfortunately, the analytical form of the potential in terms of the scalar field cannot be determined due to the complexity of the equations involved.", "However, we have plotted their evolution numerically.", "A close look at these figures shows several notable points.", "In the noninteracting case, we found that increasing $c$ leads to a faster evolution for $w_D$ toward more negative values, while in the interacting case, increasing $c$ cause $w_D$ to evolve toward less negative values which can predict a slower rate of expansion for the future HDE dominated universe.", "Also the evolutionary behavior of the potential, $V(\\phi )$ , revealed that in both interacting/noninteracting cases the potential evolves a non zero value at the present time implying a cosmological constant behavior of the model in this epoch of its evolution.", "By constraining the cosmological parameters of the quintessence HDE model in a flat universe, we found that the best fit values of the main cosmological parameters $\\Omega _{\\rm b}h^2$ , $\\Omega _{\\rm DM}h^2 $ , $\\Omega _{\\rm D}$ are in agreement with the $\\Lambda $ CDM model as one can see from table I.", "The best fit values of the HDE parameter $c$ and interacting parameter $\\zeta $ are compatible with the results of the previous constraining works on the HDE in the presence of interaction between DE and dark matter.", "Moreover, according to our data fitting our model can cross the phantom line in $1\\sigma $ confidence level in the present time of the Universe expansion.", "We thank the referee for constructive comments which helped us to improve the paper significantly.", "A. Sheykhi thanks the Research Council of Shiraz University.", "The work of A. Sheykhi has been supported financially by Research Institute for Astronomy & Astrophysics of Maragha (RIAAM), Iran." ] ]
1403.0196
[ [ "Unary Pushdown Automata and Straight-Line Programs" ], [ "Abstract We consider decision problems for deterministic pushdown automata over a unary alphabet (udpda, for short).", "Udpda are a simple computation model that accept exactly the unary regular languages, but can be exponentially more succinct than finite-state automata.", "We complete the complexity landscape for udpda by showing that emptiness (and thus universality) is P-hard, equivalence and compressed membership problems are P-complete, and inclusion is coNP-complete.", "Our upper bounds are based on a translation theorem between udpda and straight-line programs over the binary alphabet (SLPs).", "We show that the characteristic sequence of any udpda can be represented as a pair of SLPs---one for the prefix, one for the lasso---that have size linear in the size of the udpda and can be computed in polynomial time.", "Hence, decision problems on udpda are reduced to decision problems on SLPs.", "Conversely, any SLP can be converted in logarithmic space into a udpda, and this forms the basis for our lower bound proofs.", "We show coNP-hardness of the ordered matching problem for SLPs, from which we derive coNP-hardness for inclusion.", "In addition, we complete the complexity landscape for unary nondeterministic pushdown automata by showing that the universality problem is $\\Pi_2 \\mathrm P$-hard, using a new class of integer expressions.", "Our techniques have applications beyond udpda.", "We show that our results imply $\\Pi_2 \\mathrm P$-completeness for a natural fragment of Presburger arithmetic and coNP lower bounds for compressed matching problems with one-character wildcards." ], [ "Introduction", "Any model of computation comes with a set of fundamental decision questions: emptiness (does a machine accept some input?", "), universality (does it accept all inputs?", "), inclusion (are all inputs accepted by one machine also accepted by another?", "), and equivalence (do two machines accept exactly the same inputs?).", "The theoretical computer science community has a fairly good understanding of the precise complexity of these problems for most “classical” models, such as finite and pushdown automata, with only a few prominent open questions (e. g., the precise complexity of equivalence for deterministic pushdown automata).", "In this paper, we study a simple class of machines: deterministic pushdown automata working on unary alphabets (unary dpda, or udpda for short).", "A classic theorem of Ginsburg and Rice [7] shows that they accept exactly the unary regular languages, albeit with potentially exponential succinctness when compared to finite automata.", "However, the precise complexity of most basic decision problems for udpda has remained open.", "Our first and main contribution is that we close the complexity picture for these devices.", "We show that emptiness is already $\\mathbf {P}$ -hard for udpda (even when the stack is bounded by a linear function of the number of states) and thus $\\mathbf {P}$ -complete.", "By closure under complementation, it follows that universality is $\\mathbf {P}$ -complete as well.", "Our main technical construction shows equivalence is in $\\mathbf {P}$ (and so $\\mathbf {P}$ -complete).", "Somewhat unexpectedly, inclusion is $\\mathbf {coNP}$ -complete.", "In addition, we study the compressed membership problem: given a udpda over the alphabet $\\lbrace a\\rbrace $ and a number $n$ in binary, is $a^n$ in the language?", "We show that this problem is $\\mathbf {P}$ -complete too.", "A natural attempt at a decision procedure for equivalence or compressed membership would go through translations to finite automata (since udpda only accept regular languages, such a translation is possible).", "Unfortunately, these automata can be exponentially larger than the udpda and, as we demonstrate, such algorithms are not optimal.", "Instead, our approach establishes a connection to straight-line programs (SLPs) on binary words (see, e. g., Lohrey [20]).", "An SLP $\\mathcal {P}$ is a context-free grammar generating a single word, denoted $\\mathrm {eval}(\\mathcal {P})$ , over $\\lbrace 0, 1\\rbrace $ .", "Our main construction is a translation theorem: for any udpda, we construct in polynomial time two SLPs $\\mathcal {P}^{\\prime }$ and $\\mathcal {P}^{\\prime \\prime }$ such that the infinite sequence $\\mathrm {eval}(\\mathcal {P}^{\\prime }) \\cdot \\mathrm {eval}(\\mathcal {P}^{\\prime \\prime })^\\omega \\in \\lbrace 0, 1\\rbrace ^\\omega $ is the characteristic sequence of the language of the udpda (for any $i \\ge 0$ , its $i$ th element is 1 iff $a^i$ is in the language).", "With this construction, decision problems on udpda reduce to decision problems on compressed words.", "Conversely, we show that from any pair $(\\mathcal {P}^{\\prime }, \\mathcal {P}^{\\prime \\prime })$ of SLPs, one can compute, in logarithmic space, a udpda accepting the language with characteristic sequence $\\mathrm {eval}(\\mathcal {P}^{\\prime })\\cdot \\mathrm {eval}(\\mathcal {P}^{\\prime \\prime })^\\omega $ .", "Thus, as regards the computational complexity of decision problems, lower bounds for udpda may be obtained from lower bounds for SLPs.", "Indeed, we show $\\mathbf {coNP}$ -hardness of inclusion via $\\mathbf {coNP}$ -hardness of the ordered matching problem for compressed words (i. e., is $\\mathrm {eval}(\\mathcal {P}_1) \\le \\mathrm {eval}(\\mathcal {P}_2)$ letter-by-letter, where the alphabet comes with an ordering $\\le $ ), a problem of independent interest.", "As a second contribution, we complete the complexity picture for unary non-deterministic pushdown automata (unpda, for short).", "For unpda, the precise complexity of most decision problems was already known [14].", "The remaining open question was the precise complexity of the universality problem, and we show that it is $\\mathbf {\\mathrm {\\Pi }_2 P}$ -hard (membership in $\\mathbf {\\mathrm {\\Pi }_2 P}$ was shown earlier by Huynh [14]).", "An equivalent question was left open in Kopczyński and To [18] in 2010, but the question was posed as early as in 1976 by Hunt III, Rosenkrantz, and Szymanski [12], where it was asked whether the problem was in $\\mathbf {NP}$ or $\\mathbf {PSPACE}$ or outside both.", "Huynh's $\\mathbf {\\mathrm {\\Pi }_2 P}$ -completeness result for equivalence [14] showed, in particular, that universality was in $\\mathbf {PSPACE}$ , and our $\\mathbf {\\mathrm {\\Pi }_2 P}$ -hardness result reveals that membership in $\\mathbf {NP}$ is unlikely under usual complexity assumptions.", "As a corollary, we characterize the complexity of the $\\forall _{\\mathrm {bounded}}\\,\\exists ^*$ -fragment of Presburger arithmetic, where the universal quantifier ranges over numbers at most exponential in the size of the formula.", "To show $\\mathbf {\\mathrm {\\Pi }_2 P}$ -hardness, we show hardness of the universality problem for a class of integer expressions.", "Several decision problems of this form, with the set of operations $\\lbrace +, \\cup \\rbrace $ , were studied in the classic paper of Stockmeyer and Meyer [31], and we show that checking universality of expressions over $\\lbrace +, \\cup , \\times 2, {} {\\times } \\mathbb {N}\\rbrace $ is $\\mathbf {\\mathrm {\\Pi }_2 P}$ -complete (the upper bound follows from Huynh [14]).", "Related work.", "Table REF provides the current complexity picture, including the results in this paper.", "Results on general alphabets are mostly classical and included for comparison.", "Note that the complexity landscape for udpda differs from those for unpda, dpda, and finite automata.", "Upper bounds for emptiness and universality are classical, and the lower bounds for emptiness are originally by Jones and Laaser [17] and Goldschlager [9].", "In the nondeterministic unary case, $\\mathbf {NP}$ -completeness of compressed membership is from Huynh [14], rediscovered later by Plandowski and Rytter [25].", "The $\\mathbf {PSPACE}$ -completeness of the compressed membership problem for binary pushdown automata (see definition in Section ) is by Lohrey [22].", "The main remaining open question is the precise complexity of the equivalence problem for dpda.", "It was shown decidable by Sénizergues [29] and primitive recursive by Stirling [30] and Jančar [15], but only $\\mathbf {P}$ -hardness (from emptiness) is currently known.", "Recently, the equivalence question for dpda when the stack alphabet is unary was shown to be $\\mathbf {NL}$ -complete by Böhm, Göller, and Jančar [4].", "From this, it is easy to show that emptiness and universality are also $\\mathbf {NL}$ -complete.", "Compressed membership, however, remains $\\mathbf {PSPACE}$ -complete (see Caussinus et al.", "[5] and Lohrey [21]), and inclusion is, of course, already undecidable.", "When we additionally restrict dpda to both unary input and unary stack alphabet, all five decision problems are $\\mathbf {L}$ -complete.", "We discuss corollaries of our results and other related work in Section .", "While udpda are a simple class of machines, our proofs show that reasoning about these machines can be surprisingly subtle.", "Acknowledgements.", "We thank Joshua Dunfield for discussions.", "Table: Complexity of decision problems for pushdown automata." ], [ "Pushdown automata.", "A unary pushdown automaton (unpda) over the alphabet $\\lbrace a\\rbrace $ is a finite structure $\\mathcal {A}= (Q, \\mathrm {\\Gamma }, \\bot , q_0, F, \\delta )$ , with $Q$ a set of (control) states, $\\mathrm {\\Gamma }$ a stack alphabet, $\\bot \\in \\mathrm {\\Gamma }$ a bottom-of-the-stack symbol, $q_0 \\in Q$ an initial state, $F \\subseteq Q$ a set of final states, and $\\delta \\subseteq (Q \\times (\\lbrace a\\rbrace \\cup \\lbrace \\varepsilon \\rbrace ) \\times \\mathrm {\\Gamma }) \\times (Q \\times \\mathrm {\\Gamma }^*)$ a set of transitions with the property that, for every $(q_1, \\sigma , \\gamma , q_2, s) \\in \\delta $ , either $\\gamma \\ne \\bot $ and $s \\in (\\mathrm {\\Gamma }\\setminus \\lbrace \\bot \\rbrace )^*$ , or $\\gamma = \\bot $ and $s \\in \\lbrace \\varepsilon \\rbrace \\cup (\\mathrm {\\Gamma }\\setminus \\lbrace \\bot \\rbrace )^* \\bot $ .", "Here and everywhere below $\\varepsilon $ denotes the empty word.", "The semantics of unpda is defined in the following standard way.", "The set of configurations of $\\mathcal {A}$ is $Q \\times (\\mathrm {\\Gamma }\\setminus \\lbrace \\bot \\rbrace )^* \\bot $ .", "Suppose $(q_1, s_1)$ and $(q_2, s_2)$ are configurations; we write $(q_1, s_1) \\vdash _\\sigma \\!", "(q_2, s_2)$ and say that a move to $(q_2, s_2)$ is available to $\\mathcal {A}$ at $(q_1, s_1)$ iff there exists a transition $(q_1, \\sigma , \\gamma , q_2, s) \\in \\delta $ such that, if $\\gamma \\ne \\bot $ or $s \\ne \\varepsilon $ , then $s_1 = \\gamma s^{\\prime }$ and $s_2 = s s^{\\prime }$ for some $s^{\\prime } \\in \\mathrm {\\Gamma }^*$ , or, if $\\gamma = \\bot $ and $s = \\varepsilon $ , then $s_1 = s_2 = \\bot $ .", "A unary pushdown automaton is called deterministic, shortened to udpda, if at every configuration at most one move is available.", "A word $w \\in \\lbrace a\\rbrace ^*$ is accepted by $\\mathcal {A}$ if there exists a configuration $(q_k, s_k)$ with $q_k \\in F$ and a sequence of moves $(q_i, s_i) \\vdash _{\\sigma _i} \\!", "(q_{i + 1}, s_{i + 1})$ , $i = 0, \\ldots , k - 1$ , such that $s_0 = \\bot $ and $\\sigma _0 \\ldots \\sigma _{k - 1} = w$ ; that is, the acceptance is by final state.", "The language of $\\mathcal {A}$ , denoted $L(\\mathcal {A})$ , is the set of all words $w \\in \\lbrace a\\rbrace ^*$ accepted by $\\mathcal {A}$ .", "We define the size of a unary pushdown automaton $\\mathcal {A}$ as $|Q| \\cdot |\\mathrm {\\Gamma }|$ , provided that for all transitions $(q_1, \\sigma , \\gamma , q_2, s) \\in \\delta $ the length of the word $s$ is at most 2 (see also [24]).", "While this definition is better suited for deterministic rather than nondeterministic automata, it already suffices for the purposes of Section , where we handle unpda, because it is always the case that $|\\delta | \\le 2 \\, |Q|^2 \\, |\\mathrm {\\Gamma }|^4$ .", "We consider the following decision problems: emptiness ($L(\\mathcal {A}) =^?\\!\\emptyset $ ), universality ($L(\\mathcal {A}) =^?\\!\\lbrace a\\rbrace ^*$ ), equivalence ($L(\\mathcal {A}_1) =^?\\!L(\\mathcal {A}_2)$ ), and inclusion ($L(\\mathcal {A}_1) \\subseteq ^?\\!", "L(\\mathcal {A}_2)$ ).", "The compressed membership problem for unary pushdown automata is associated with the question $a^n \\in ^?\\!L(\\mathcal {A})$ , with $n$ given in binary as part of the input.", "In the following, hardness is with respect to logarithmic-space reductions.", "Our first result is that emptiness is already $\\mathbf {P}$ -hard for udpda.", "UDPDA-Emptiness and UDPDA-Universality are $\\mathbf {P}$ -complete.", "Emptiness is in $\\mathbf {P}$ for non-deterministic pushdown automata on any alphabet, and deterministic automata can be complemented in polynomial time.", "So, we focus on showing $\\mathbf {P}$ -hardness for emptiness.", "We encode the computations of an alternating logspace Turing machine using an udpda.", "We assume without loss of generality that each configuration $c$ of the machine has exactly two successors, denoted $c_l$ (left successor) and $c_r$ (right successor), and that each run of the machine terminates.", "The udpda encodes a successful run over the computation tree of the TM.", "The states of the udpda are configurations of the Turing machine and an additional context, which can be $a$ (“accepting”), $r$ (“rejecting”), or $x$ (“exploring”).", "The stack alphabet consists of pairs $(c,d)$ , where $c$ is a configurations of the machine and the direction $d\\in {\\lbrace l,r \\rbrace }$ , together with an additional end-of-stack symbol.", "The alphabet has just one symbol 0.", "The initial state is $(c_0, x)$ , where $c_0$ is the initial configuration of the machine, and the stack has the end-of-stack symbol.", "Suppose the current state is $(c,x)$ , where $c$ is not an accepting or rejecting configuration.", "The udpda pushes $(c,l)$ on to the stack and updates its state to $(c_l, x)$ , where $c_l$ is the left successor of $c$ .", "The invariant is that in the exploring phase, the stack maintains the current path in the computation tree, and if the top of the stack is $(c,l)$ (resp.", "$(c,r)$ ) then the current state is the left (resp.", "right) successor of $c$ .", "Suppose now the current state is $(c,x)$ where $c$ is an accepting configuration.", "The context is set to $a$ , and the udpda backtracks up the computation tree using the stack.", "If the top of the stack is the end-of-stack symbol, the machine accepts.", "If the top of the stack $(c^{\\prime }, d)$ consists of an existential configuration $c^{\\prime }$ , then the new state is $(c^{\\prime },a)$ and recursively the machine moves up the stack.", "If the top of the stack $(c^{\\prime },d)$ consists of a universal configuration $c^{\\prime }$ , and $d=l$ , then the new state is $(c^{\\prime }_r, x)$ , the right successor of $c^{\\prime }$ , and the top of stack is replaced by $(c^{\\prime },r)$ .", "If the top of the stack $(c^{\\prime },d)$ consists of a universal configuration $c^{\\prime }$ , and $d=r$ , then the new state is $(c^{\\prime }, a)$ , the stack is popped, and the machine continues to move up the stack.", "Suppose now the current state is $(c,x)$ where $c$ is a rejecting configuration.", "The context is set to $r$ , and the udpda backtracks up the computation tree using the stack.", "If the top of the stack is the end-of-stack symbol, the machine rejects.", "If the top of the stack $(c^{\\prime }, d)$ consists of an existential configuration $c^{\\prime }$ , and $d=l$ , then the new state is $(c^{\\prime }_r,x)$ and the top of stack is replaced with $(c^{\\prime },r)$ .", "If the top of the stack $(c^{\\prime }, d)$ consists of an existential configuration $c^{\\prime }$ , and $d=r$ , then the new state is $(c^{\\prime },r)$ , the stack is popped, and the machine continues to move up the stack.", "The top of stack is replaced with $(c^{\\prime },r)$ .", "If the top of the stack $(c^{\\prime },d)$ consists of a universal configuration $c^{\\prime }$ , then the new state is $(c^{\\prime },r)$ , the stack is popped, and the machine continues to move up the stack.", "It is easy to see that the reduction can be performed in logarithmic space.", "If the TM has an accepting computation tree, the udpda has an accepting run following the computation tree.", "On the other hand, any accepting computation of the udpda is a depth-first traversal of an accepting computation tree of the TM.", "Finally, since udpda can be complemented in logarithmic space, we get the corresponding results for universality.", "This completes the proof.", "$\\Box $ A straight-line program [20], or an SLP, over an alphabet $\\mathrm {\\Sigma }$ is a context-free grammar that generates a single word; in other words, it is a tuple $\\mathcal {P}= (S, \\mathrm {\\Sigma }, \\mathrm {\\Delta }, \\pi )$ , where $\\mathrm {\\Sigma }$ and $\\mathrm {\\Delta }$ are disjoint sets of terminal and nonterminal symbols (terminals and nonterminals), $S \\in \\mathrm {\\Delta }$ is the axiom, and the function $\\pi \\colon \\mathrm {\\Delta }\\rightarrow (\\mathrm {\\Sigma }\\cup \\mathrm {\\Delta })^*$ defines a set of productions written as “$N \\rightarrow w$ ”, $w = \\pi (N)$ , and satisfies the property that the relation $\\lbrace (N, D) \\mid N \\rightarrow w \\text{\\ and\\ }D \\text{\\ occurs in\\ } w \\rbrace $ is acyclic.", "An SLP $\\mathcal {P}$ is said to generate a (unique) word $w \\in \\mathrm {\\Sigma }^*$ , denoted $\\mathrm {eval}(\\mathcal {P})$ , which is the result of applying substitutions $\\pi $ to $S$ .", "An SLP is said to be in Chomsky normal form if for all productions $N \\rightarrow w$ it holds that either $w \\in \\mathrm {\\Sigma }$ or $w \\in \\mathrm {\\Delta }^{\\!2}$ .", "The size of an SLP is the number of nonterminals in its Chomsky normal form." ], [ "Indicator pairs and the translation theorem", "We say that a pair of SLPs $(\\mathcal {P}^{\\prime }, \\mathcal {P}^{\\prime \\prime })$ over an alphabet $\\mathrm {\\Sigma }$ generates a sequence $c \\in \\mathrm {\\Sigma }^\\omega $ if $\\mathrm {eval}(\\mathcal {P}^{\\prime })\\cdot (\\mathrm {eval}(\\mathcal {P}^{\\prime \\prime }))^\\omega = c$ .", "We call an infinite sequence $c \\in \\lbrace 0, 1\\rbrace ^\\omega $ , $c = c_0 c_1 c_2 \\ldots $  , the characteristic sequence of a unary language $L \\subseteq \\lbrace a\\rbrace ^*$ if, for all $i \\ge 0$ , it holds that $c_i$ is 1 if $a^i \\in L$ and 0 otherwise.", "One may note that the characteristic sequence is eventually periodic if and only if $L$ is regular.", "A pair of straight-line programs $(\\mathcal {P}^{\\prime }, \\mathcal {P}^{\\prime \\prime })$ over $\\lbrace 0, 1\\rbrace $ is called an indicator pair for a unary language $L \\subseteq \\lbrace a\\rbrace ^*$ if it generates the characteristic sequence of $L$ .", "A unary language can have several different indicator pairs.", "Indicator pairs form a descriptional system for unary languages, with the size of $(\\mathcal {P}^{\\prime }, \\mathcal {P}^{\\prime \\prime })$ defined as the sum of sizes of $\\mathcal {P}^{\\prime }$ and $\\mathcal {P}^{\\prime \\prime }$ .", "The following translation theorem shows that udpda and indicator pairs are polynomially-equivalent representations for unary regular languages.", "We remark that Theorem  does not give a normal form for udpda because of the non-uniqueness of indicator pairs.", "[translation theorem] For a unary language $L \\subseteq \\lbrace a\\rbrace ^*$ : 1 if there exists a udpda $\\mathcal {A}$ of size $m$ with $L(\\mathcal {A}) = L$ , then there exists an indicator pair for $L$ of size $O(m)$ ; 2 if there exists an indicator pair for $L$ of size $m$ , then there exists a udpda $\\mathcal {A}$ of size $O(m)$ with $L(\\mathcal {A}) = L$ .", "Both statements are supported by polynomial-time algorithms, the second of which works in logarithmic space.", "We only discuss part REF , which presents the main technical challenge.", "The starting point is the simple observation that a udpda $\\mathcal {A}$ has a single infinite computation, provided that the input tape supplies $\\mathcal {A}$ with as many input symbols $a$ as it needs to consume.", "Along this computation, events of two types are encountered: $\\mathcal {A}$ can consume a symbol from the input and can enter a final state.", "The crucial technical task is to construct inductively, using dynamic programming, straight-line programs that record these events along finite computational segments.", "These segments are of two types: first, between matching push and pop moves (“procedure calls”) and, second, from some starting point until a move pops the symbol that has been on top of the stack at that point (“exits from current context”).", "Loops are detected, and infinite computations are associated with pairs of SLPs: in such a pair, one SLP records the initial segment, or prefix of the computation, and the other SLP records events within the loop.", "After constructing these SLPs, it remains to transform the computational “history”, or transcript, associated with the initial configuration of $\\mathcal {A}$ into the characteristic sequence.", "This transformation can easily be performed in polynomial time, without expanding SLPs into the words that they generate.", "The result is the indicator pair for $\\mathcal {A}$ .", "$\\Box $ The full proof of Theorem  is given is Section .", "Note that going from indicator pairs to udpda is useful for obtaining lower bounds on the computational complexity of decision problems for udpda (Theorems REF and REF ).", "For this purpose, it suffices to model just a single SLP, but taking into account the whole pair is interesting from the point of view of descriptional complexity (see also Section ).", "Decision problems for udpda Compressed membership and equivalence For an SLP $\\mathcal {P}$ , by $|\\mathcal {P}|$ we denote the length of the word $\\mathrm {eval}(\\mathcal {P})$ , and by $\\mathcal {P}[n]$ the $n$ th symbol of $\\mathrm {eval}(\\mathcal {P})$ , counting from 0 (that is, $0 \\le n \\le |\\mathcal {P}| - 1$ ).", "We write $\\mathcal {P}_1 \\equiv \\mathcal {P}_2$ if and only if $\\mathrm {eval}(\\mathcal {P}_1) = \\mathrm {eval}(\\mathcal {P}_2)$ .", "The following SLP-Query problem is known to be $\\mathbf {P}$ -complete (see Lifshits and Lohrey [19]): given an SLP $\\mathcal {P}$ over $\\lbrace 0, 1\\rbrace $ and a number $n$ in binary, decide whether $\\mathcal {P}[n] = 1$ .", "The problem SLP-Equivalence is only known to be in $\\mathbf {P}$ (see, e. g., Lohrey [20]): given two SLPs $\\mathcal {P}_1$ , $\\mathcal {P}_2$ , decide whether $\\mathcal {P}_1 \\equiv \\mathcal {P}_2$ .", "UDPDA-Compressed-Membership is $\\mathbf {P}$ -complete.", "The upper bound follows from Theorem .", "Indeed, given a udpda $\\mathcal {A}$ and a number $n$ , first construct an indicator pair $(\\mathcal {P}^{\\prime }, \\mathcal {P}^{\\prime \\prime })$ for $L(\\mathcal {A})$ .", "Now compute $|\\mathcal {P}^{\\prime }|$ and $|\\mathcal {P}^{\\prime \\prime }|$ and then decide if $n \\le |\\mathcal {P}^{\\prime }| - 1$ .", "If so, the answer is given by $\\mathcal {P}^{\\prime }[n]$ , otherwise by $\\mathcal {P}^{\\prime \\prime }[r]$ , where $r = (n - |\\mathcal {P}^{\\prime }|) \\bmod |\\mathcal {P}^{\\prime \\prime }|$ and in both cases 1 is interpreted as “yes” and 0 as “no”.", "To prove the lower bound, we reduce from the SLP-Query problem.", "Take an instance with an SLP $\\mathcal {P}$ and a number $n$ in binary.", "By transforming the pair $(\\mathcal {P}, \\mathcal {P}_0)$ , with $\\mathcal {P}_0$ any fixed SLP over $\\lbrace 0, 1\\rbrace $ , into a udpda $\\mathcal {A}$ using part REF of Theorem , this problem is reduced, in logspace, to whether $a^n \\in L(\\mathcal {A})$ .", "$\\Box $ Recall that emptiness and universality of udpda are $\\mathbf {P}$ -complete by Proposition .", "Our next theorem extends this result to the general equivalence problem for udpda.", "UDPDA-Equivalence is $\\mathbf {P}$ -complete.", "Hardness follows from Proposition .", "We show how Theorem  can be used to prove the upper bound: given udpda $\\mathcal {A}_1$ and $\\mathcal {A}_2$ , first construct indicator pairs $(\\mathcal {P}^{\\prime }_1, \\mathcal {P}^{\\prime \\prime }_1)$ and $(\\mathcal {P}^{\\prime }_2, \\mathcal {P}^{\\prime \\prime }_2)$ for $L(\\mathcal {A}_1)$ and $L(\\mathcal {A}_2)$ , respectively.", "Now reduce the problem of whether $L(\\mathcal {A}_1) = L(\\mathcal {A}_2)$ to SLP-Equivalence.", "The key observation is that an eventually periodic sequence that has periods $|\\mathcal {P}^{\\prime \\prime }_1|$ and $|\\mathcal {P}^{\\prime \\prime }_2|$ also has period $t = \\gcd (|\\mathcal {P}^{\\prime \\prime }_1|, |\\mathcal {P}^{\\prime \\prime }_2|)$ .", "Therefore, it suffices to check that, first, the initial segments of the generated sequences match and, second, that $\\mathcal {P}^{\\prime \\prime }_1$ and $\\mathcal {P}^{\\prime \\prime }_2$ generate powers of the same word up to a certain circular shift.", "In more detail, let us first introduce some auxiliary operations for SLP.", "For SLPs $\\mathcal {P}_1$ and $\\mathcal {P}_2$ , by $\\mathcal {P}_1 \\cdot \\mathcal {P}_2$ we denote an SLP that generates $\\mathrm {eval}(\\mathcal {P}_1) \\cdot \\mathrm {eval}(\\mathcal {P}_2)$ , obtained by “concatenating” $\\mathcal {P}_1$ and $\\mathcal {P}_2$ .", "Now suppose that $\\mathcal {P}$ generates $w = w[0] \\ldots w[|\\mathcal {P}|-1]$ .", "Then the SLP $\\mathcal {P}[a \\,\\text{..}\\,b)$ generates the word $w[a \\,\\text{..}\\,b) = w[a] \\ldots w[b - 1]$ , of length $b - a$ (as in $\\mathcal {P}[n]$ , indexing starts from 0).", "The SLP $\\mathcal {P}^\\alpha $ generates $w^\\alpha $ , with the meaning clear for $\\alpha = 0, 1, 2, \\ldots $  , also extended to $\\alpha \\in \\mathbb {Q}$ with $\\alpha \\cdot |\\mathcal {P}| \\in \\mathbb {Z}_{\\ge 0}$ by setting $w^{k + n / |w|} = w^k \\cdot w[0 \\,\\text{..}\\,n)$ , $n < |w|$ .", "Finally, ${\\mathcal {P}} \\curvearrowleft {s}$ denotes cyclic shift and evaluates to $w[s \\,\\text{..}\\,|w|) \\cdot w[0 \\,\\text{..}\\,s)$ .", "One can easily demonstrate that all these operations can be implemented in polynomial time.", "So, assume that $|\\mathcal {P}^{\\prime }_1| \\ge |\\mathcal {P}^{\\prime }_2|$ .", "First, one needs to check whether $\\mathcal {P}^{\\prime }_1 \\equiv \\mathcal {P}^{\\prime }_2 \\cdot (\\mathcal {P}^{\\prime \\prime }_2)^\\alpha $ , where $\\alpha = (|\\mathcal {P}^{\\prime }_1| - |\\mathcal {P}^{\\prime }_2|) / |\\mathcal {P}^{\\prime \\prime }_2|$ .", "Second, note that an eventually periodic sequence that has periods $|\\mathcal {P}^{\\prime \\prime }_1|$ and $|\\mathcal {P}^{\\prime \\prime }_2|$ also has period $t = \\gcd (|\\mathcal {P}^{\\prime \\prime }_1|, |\\mathcal {P}^{\\prime \\prime }_2|)$ .", "Compute $t$ and an auxiliary SLP $\\mathcal {P}^{\\prime \\prime }= \\mathcal {P}^{\\prime \\prime }_1[0 \\,\\text{..}\\,t)$ , and then check whether $\\mathcal {P}^{\\prime \\prime }_1 \\equiv (\\mathcal {P}^{\\prime \\prime })^{|\\mathcal {P}^{\\prime \\prime }_1| / t}$ and ${\\mathcal {P}^{\\prime \\prime }_2} \\curvearrowleft {s} \\equiv (\\mathcal {P}^{\\prime \\prime })^{|\\mathcal {P}^{\\prime \\prime }_2| / t}$ with $s = (|\\mathcal {P}^{\\prime }_1| - |\\mathcal {P}^{\\prime }_2|) \\bmod |\\mathcal {P}^{\\prime \\prime }_2|$ .", "It is an easy exercise to show that $L(\\mathcal {A}_1) = L(\\mathcal {A}_2)$ iff all the checks are successful.", "This completes the proof.", "$\\Box $ Inclusion A natural idea for handling the inclusion problem for udpda would be to extend the result of Theorem REF , that is, to tackle inclusion similarly to equivalence.", "This raises the problem of comparing the words generated by two SLPs in the componentwise sense with respect to the order $0 \\le 1$ .", "To the best of our knowledge, this problem has not been studied previously, so we deal with it separately.", "As it turns out, here one cannot hope for an efficient algorithm unless $\\mathbf {P}= \\mathbf {NP}$ .", "Let us define the following family of problems, parameterized by partial order $R$ on the alphabet of size at least 2, and denoted $\\textsc {SLP-Compo\\-nent\\-wise-$ R$}$ .", "The input is a pair of SLPs $\\mathcal {P}_1$ , $\\mathcal {P}_2$ over an alphabet partially ordered by $R$ , generating words of equal length.", "The output is “yes” iff for all $i$ , $0 \\le i < |\\mathcal {P}_1|$ , the relation $R(\\mathcal {P}_1[i], \\mathcal {P}_2[i])$ holds.", "By SLP-Componentwise-${(0 \\le 1)}$ we mean a special case of this problem where $R$ is the partial order on $\\lbrace 0, 1\\rbrace $ given by $0 \\le 0$ , $0 \\le 1$ , $1 \\le 1$ .", "$\\textsc {SLP-Compo\\-nent\\-wise-$ R$}$ is $\\mathbf {coNP}$ -complete if $R$ is not the equality relation (that is, if $R(a, b)$ holds for some $a \\ne b$ ), and in $\\mathbf {P}$ otherwise.", "We first prove that SLP-Componentwise-${(0 \\le 1)}$ is $\\mathbf {coNP}$ -hard.", "We show a reduction from the complement of Subset-Sum: suppose we start with an instance of Subset-Sum containing a vector of naturals $w = (w_1, \\ldots , w_n)$ and a natural $t$ , and the question is whether there exists a vector $x = (x_1, \\ldots , x_n) \\in \\lbrace 0, 1\\rbrace ^n$ such that $x \\cdot w = t$ , where $x \\cdot w$ is defined as the inner product $\\sum _{i = 1}^{n} x_i w_i$ .", "Let $s = (1, \\ldots , 1) \\cdot w$ be the sum of all components of $w$ .", "We use the construction of so-called Lohrey words.", "Lohrey shows [22] that given such an instance, it is possible to construct in logarithmic space two SLPs that generate words $W_1 = \\prod _{x \\in \\lbrace 0, 1\\rbrace ^n} a^{x \\cdot w} b a^{s - x \\cdot w}$ and $W_2 = (a^{t} b a^{s - t})^{2^n}$ , where the product in $W_1$ enumerates the $x$ es in the lexicographic order.", "Now $W_1$ and $W_2$ share a symbol $b$ in some position iff the original instance of Subset-Sum is a yes-instance.", "Substitute 0 for $a$ and 1 for $b$ in the first SLP, and 0 for $b$ and 1 for $a$ in the second SLP.", "The new SLPs $\\mathcal {P}_1$ and $\\mathcal {P}_2$ obtained in this way form a no-instance of SLP-Componentwise-${(0 \\le 1)}$ iff the original instance of Subset-Sum is a yes-instance, because now the “distinguished” pair of symbols consists of a 1 in $\\mathcal {P}_1$ and 0 in $\\mathcal {P}_2$ .", "Therefore, SLP-Componentwise-${(0 \\le 1)}$ is $\\mathbf {coNP}$ -hard.", "Now observe that, for any $R$ , membership of $\\textsc {SLP-Compo\\-nent\\-wise-$ R$}$ in $\\mathbf {coNP}$ is obvious, and the hardness is by a simple reduction from SLP-Componentwise-${(0 \\le 1)}$: just substitute $a$ for 0 and $b$ for 1 (recall that by the definition of partial order, $R(b, a)$ would entail $a = b$ , which is false).", "In the last special case in the statement, $R$ is just the equality relation, so deciding $\\textsc {SLP-Compo\\-nent\\-wise-$ R$}$ is the same as deciding SLP-Equivalence, which is in $\\mathbf {P}$ (see Section ).", "This concludes the proof.", "$\\Box $ A corollary of Theorem REF on a problem of matching for compressed partial words is demonstrated in Section .", "An alternative reduction showing hardness of SLP-Componentwise-${(0 \\le 1)}$, this time from $\\textsc {Circuit-SAT}$ , but also making use of Subset-Sum and Lohrey words, can be derived from Bertoni, Choffrut, and Radicioni [3].", "They show that for any Boolean circuit with NAND-gates there exists a pair of straight-line programs $\\mathcal {P}_1$ , $\\mathcal {P}_2$ generating words over $\\lbrace 0, 1\\rbrace $ of the same length with the following property: the function computed by the circuit takes on the value 1 on at least one input combination iff both words share a 1 at some position.", "Moreover, these two SLPs can be constructed in polynomial time.", "As a result, after flipping all terminal symbols in the second of these SLPs, the resulting pair is a no-instance of SLP-Componentwise-${(0 \\le 1)}$ iff the original circuit is satisfiable.", "UDPDA-Inclusion is $\\mathbf {coNP}$ -complete.", "First combine Theorem REF with part REF of Theorem  to prove hardness.", "Indeed, Theorem REF shows that SLP-Componentwise-${(0 \\le 1)}$ is $\\mathbf {coNP}$ -hard.", "Take an instance with two SLPs $\\mathcal {P}_1$ , $\\mathcal {P}_2$ and transform indicator pairs $(\\mathcal {P}_1, \\mathcal {P}_0)$ and $(\\mathcal {P}_2, \\mathcal {P}_0)$ , with $\\mathcal {P}_0$ any fixed SLP over $\\lbrace 0, 1\\rbrace $ , into udpda $\\mathcal {A}_1$ , $\\mathcal {A}_2$ with the help of part REF of Theorem .", "Now the characteristic sequence of $L(\\mathcal {A}_i)$ , $i = 1, 2$ , is equal to $\\mathrm {eval}(\\mathcal {P}_i) \\cdot (\\mathrm {eval}(\\mathcal {P}_0))^\\omega $ .", "As a result, it holds that $\\mathrm {eval}(\\mathcal {P}_1) \\le \\mathrm {eval}(\\mathcal {P}_2)$ in the componentwise sense if and only if $L(\\mathcal {A}_1) \\subseteq L(\\mathcal {A}_2)$ .", "This concludes the hardness proof.", "It remains to show that UDPDA-Inclusion is in $\\mathbf {coNP}$ .", "First note that for any udpda $\\mathcal {A}$ there exists a deterministic pushdown automaton (DFA) that accepts $L(\\mathcal {A})$ and has size at most $2^{O(m)}$ , where $m$ is the size of $\\mathcal {A}$ (see discussion in Section  or Pighizzini [24]).", "Therefore, if $L(\\mathcal {A}_1) \\lnot \\subseteq L(\\mathcal {A}_2)$ , then there exists a witness $a^n \\in L(\\mathcal {A}_2) \\setminus L(\\mathcal {A}_1)$ with $n$ at most exponential in the size of $\\mathcal {A}_1$ and $\\mathcal {A}_2$ .", "By Theorem REF , compressed membership is in $\\mathbf {P}$ , so this completes the proof.", "$\\Box $ Proof of Theorem  Let us first recall some standard definitions and fix notation.", "In a udpda $\\mathcal {A}$ , if $(q_1, s_1) \\vdash _\\sigma \\!", "(q_2, s_2)$ for some $\\sigma $ , we also write $(q_1, s_1) \\vdash (q_2, s_2)$ .", "A computation of a udpda $\\mathcal {A}$ starting at a configuration $(q, s)$ is defined as a (finite or infinite) sequence of configurations $(q_i, s_i)$ with $(q_1, s_1) = (q, s)$ and, for all $i$ , $(q_i, s_i) \\vdash _{\\sigma _i} \\!", "(q_{i + 1}, s_{i + 1})$ for some $\\sigma _i$ .", "If the sequence is finite and ends with $(q_k, s_k)$ , we also write $(q_1, s_1) \\vdash ^*_w \\!", "(q_k, s_k)$ , where $w = \\sigma _1 \\ldots \\sigma _{k - 1} \\in \\lbrace a\\rbrace ^*$ .", "We can also omit the word $w$ when it is not important and say that $(q_k, s_k)$ is reachable from $(q_1, s_1)$ ; in other words, the reachability relation $\\vdash ^*$ is the reflexive and transitive closure of the move relation $\\vdash $ .", "From indicator pairs to udpda Going from indicator pairs to udpda is the easier direction in Theorem .", "We start with an auxiliary lemma that enables one to model a single SLP with a udpda.", "This lemma on its own is already sufficient for lower bounds of Theorem REF and Theorem REF in Section .", "There exists an algorithm that works in logarithmic space and transforms an arbitrary SLP $\\mathcal {P}$ of size $m$ over $\\lbrace 0, 1\\rbrace $ into a udpda $\\mathcal {A}$ of size $O(m)$ over $\\lbrace a\\rbrace $ such that the characteristic sequence of $L(\\mathcal {A})$ is $0 \\cdot \\mathrm {eval}(\\mathcal {P})\\cdot 0^\\omega $ .", "In $\\mathcal {A}$ , it holds that $(q_0, \\bot ) \\vdash ^*_w \\!", "(\\bar{q}_0, \\bot )$ for $w = a^{|\\mathrm {eval}(\\mathcal {P})|}$ , $q_0$ the initial state, and $\\bar{q}_0$ a non-final state without outgoing transitions.", "The main part of the algorithm works as follows.", "Assume that $\\mathcal {P}$ is given in Chomsky normal form.", "With each nonterminal $N$ we associate a gadget in the udpda $\\mathcal {A}$ , whose interface is by definition the entry state $q_N$ and the exit state $\\bar{q}_N$ , which will only have outgoing pop transitions.", "With a production of the form $N \\rightarrow \\sigma $ , $\\sigma \\in \\lbrace 0, 1\\rbrace $ , we associate a single internal transition from $q_N$ to $\\bar{q}_N$ reading an $a$ from the input tape.", "The state $q_N$ is always non-final, and the state $\\bar{q}_N$ is final if and only if $\\sigma = 1$ .", "With a production of the form $N \\rightarrow A B$ we associate two stack symbols $\\gamma _N^1$ , $\\gamma _N^2$ and the following gadget.", "At a state $q_N$ , the udpda pushes a symbol $\\gamma _N^1$ onto the stack and goes to the state $q_A$ .", "We add a pop transition from $\\bar{q}_A$ that reads $\\gamma _N^1$ from the stack and leads to an auxiliary state $q^{\\prime }_N$ .", "The only transition from this state pushes $\\gamma _N^2$ and leads to $q_B$ , and another transition from $\\bar{q}_B$ pops $\\gamma _N^2$ and goes to $\\bar{q}_N$ .", "Here all three states $q_N$ , $q^{\\prime }_N$ , and $\\bar{q}_N$ are non-final, and the four introduced incident transitions do not read from the input.", "Finally, if a nonterminal $N$ is the axiom of $\\mathcal {P}$ , make the state $q_N$ initial and non-final and make $\\bar{q}_N$ a non-accepting sink that reads $a$ from the input and pops $\\bot $ .", "The reader can easily check that the characteristic sequence of the udpda $\\mathcal {A}$ constructed in this way is indeed $0 \\cdot \\mathrm {eval}(\\mathcal {P})\\cdot 0^\\omega $ , and the construction can be performed in logarithmic space.", "Now note that while the udpda $\\mathcal {A}$ satisfies $|Q| = O(m)$ , we may have also introduced up to 2 stack symbols per nonterminal.", "Therefore, the size of $\\mathcal {A}$ can be as large as $\\mathrm {\\Omega }(m^2)$ .", "However, we can use a standard trick from circuit complexity to avoid this blowup and make this size linear in $m$ .", "Indeed, first observe that the number of stack symbols, not counting $\\bot $ , in the construction above can be reduced to $k$ , the maximum, over all nonterminals $N$ , of the number of occurrences of $N$ in the right-hand sides of productions of $\\mathcal {P}$ .", "Second, recall that a straight-line program naturally defines a circuit where productions of the form $N \\rightarrow A B$ correspond to gates performing concatenation.", "The value of $k$ is the maximum fan-out of gates in this circuit, and it is well-known how to reduce it to $O(1)$ with just a constant-factor increase in the number of gates (see, e. g., Savage [26]).", "The construction can be easily performed in logarithmic space, and the only building block is the identity gate, which in our case translates to a production of the form $N \\rightarrow A$ .", "Although such productions are not allowed in Chomsky normal form, the construction above can be adjusted accordingly, in a straightforward fashion.", "This completes the proof.", "$\\Box $ Now, to model an entire indicator pair, we apply Lemma REF twice and combine the results.", "There exists an algorithm that works in logarithmic space and, given an indicator pair $(\\mathcal {P}^{\\prime }, \\mathcal {P}^{\\prime \\prime })$ of size $m$ for some unary language $L \\subseteq \\lbrace a\\rbrace ^*$ , outputs a udpda $\\mathcal {A}$ of size $O(m)$ such that $L(\\mathcal {A}) = L$ .", "We shall use the same notation as in Subsection REF of Section .", "First compute the bit $b = \\mathcal {P}^{\\prime }[0]$ and construct an SLP $\\mathcal {P}^{\\prime }_1$ of size $O(m)$ such that $\\mathrm {eval}(\\mathcal {P}^{\\prime })= b \\cdot \\mathrm {eval}(\\mathcal {P}^{\\prime }_1)$ .", "Note that this can be done in logarithmic space, even though the general SLP-Query problem is $\\mathbf {P}$ -complete.", "Now construct, according to Lemma REF , two udpda $\\mathcal {A}^{\\prime }$ and $\\mathcal {A}^{\\prime \\prime }$ for $\\mathcal {P}^{\\prime }_1$ and $\\mathcal {P}^{\\prime \\prime }$ , respectively.", "Assume that their sets of control states are disjoint and connect them in the following way.", "Add internal $\\varepsilon $ -transitions from the “last” states of both to the initial state of $\\mathcal {A}^{\\prime \\prime }$ .", "Now make the initial state of $\\mathcal {A}^{\\prime }$ the initial state of $\\mathcal {A}$ ; make it also a final state if $b = 1$ .", "It is easily checked that the language of the udpda $\\mathcal {A}$ constructed in this way has characteristic sequence $\\mathrm {eval}(\\mathcal {P}^{\\prime })\\cdot (\\mathrm {eval}(\\mathcal {P}^{\\prime \\prime }))^\\omega $ and, hence, is equal to $L$ .", "$\\Box $ Lemma REF proves part REF in Theorem .", "From udpda to indicator pairs Going from udpda to indicator pairs is the main part of Theorem , and in this subsection we describe our construction in detail.", "The proof of the key technical lemma is deferred until the following Subsection REF .", "Assumptions and notation.", "We assume without loss of generality that the given udpda $\\mathcal {A}$ satisfies the following conditions.", "First, its set of control states, $Q$ , is partitioned into three subsets according to the type of available moves.", "More precisely, we assumeHere and further in the text we use the symbol $\\sqcup $ to denote the union of disjoint sets.", "$Q = Q_{0}\\sqcup Q_{+1}\\sqcup Q_{-1}$ with the property that all transitions $(q, \\sigma , \\gamma , q^{\\prime }, s)$ with states $q$ from $Q_d$ , $d \\in \\lbrace 0, -1, +1\\rbrace $ , have $|s| = 1 + d$ ; moreover, we assume that $s = \\gamma $ whenever $d = 0$ , and $s = \\gamma ^{\\prime } \\gamma $ for some $\\gamma ^{\\prime } \\in \\mathrm {\\Gamma }$ whenever $d = 1$ .", "Second, for convenience of notation we assume that there exists a subset $R \\subseteq Q$ such that all transitions departing from states from $R$ read a symbol from the input tape, and transitions departing from $Q \\setminus R$ do not.", "Third, we assume that $\\delta $ is specified by means of total functions $\\delta _{0}\\colon Q_{0}\\rightarrow Q$ , $\\delta _{+1}\\colon Q_{+1}\\rightarrow Q \\times \\mathrm {\\Gamma }$ , and $\\delta _{-1}\\colon Q_{-1}\\times \\mathrm {\\Gamma }\\rightarrow Q$ .", "We write $\\delta _{0}(q)=q^{\\prime }$ , $\\delta _{+1}(q)=(q^{\\prime },\\gamma )$ , and $\\delta _{-1}(q,\\gamma )=q^{\\prime }$ accordingly; associated transitions and states are called internal, push, and pop transitions and states, respectively.", "Note that this assumption implies that only pop transitions can “look” at the top of the stack.", "An arbitrary udpda $\\mathcal {A}^{\\prime } = (Q^{\\prime }, \\mathrm {\\Gamma }, \\bot , q^{\\prime }_0, F^{\\prime }, \\delta ^{\\prime })$ of size $m$ can be transformed into a udpda $\\mathcal {A}= (Q, \\mathrm {\\Gamma }, \\bot , q_0, F, \\delta )$ that accepts $L(\\mathcal {A}^{\\prime })$ , satisfies the assumptions of this subsubsection, and has $|Q| = O(m)$ control states.", "The proof is easy and left to the reader.", "Note that since $\\mathcal {A}$ is deterministic, it holds that for any configuration $(q, s)$ of $\\mathcal {A}$ there exists a unique infinite computation $(q_i, s_i)_{i = 0}^{\\infty }$ starting at $(q, s)$ , referred to as the computation in the text below.", "This computation can be thought of as a run of $\\mathcal {A}$ on an input tape with an infinite sequence $a^\\omega $ .", "The computation of $\\mathcal {A}$ is, naturally, the computation starting from $(q_0, \\bot )$ .", "Note that it is due to the fact that $\\mathcal {A}$ is unary that we are able to feed it a single infinite word instead of countably many finite words.", "In the text below we shall use the following notation and conventions.", "To refer to an SLP $(S, \\mathrm {\\Sigma }, \\mathrm {\\Delta }, \\pi )$ , we sometimes just use its axiom, $S$ .", "The generated word, $w$ , is denoted by $\\mathrm {eval}(S)$ as usual.", "Note that the set of terminals is often understood from the context and the set of nonterminals is always the set of left-hand sides of productions.", "This enables us to use the notation $\\mathrm {eval}(S)$ to refer to the word generated by the implicitly defined SLP, whenever the set of productions is clear from the context.", "Transcripts of computations and overview of the algorithm.", "Recall that our goal is to describe an algorithm that, given a udpda $\\mathcal {A}$ , produces an indicator pair for $L(\\mathcal {A})$ .", "We first assemble some tools that will allow us to handle the computation of $\\mathcal {A}$ per se.", "To this end, we introduce transcripts of computations, which record “events” that determine whether certain input words are accepted or rejected.", "Consider a (finite or infinite) computation that consists of moves $(q_i, s_i) \\vdash _{\\sigma _i} \\!", "(q_{i + 1}, s_{i + 1})$ , for $1 \\le i \\le k$ or for $i \\ge 1$ , respectively.", "We define the transcript of such a computation as a (finite or infinite) sequence $\\mu (q_1) \\, \\sigma _1 \\, \\mu (q_2) \\, \\sigma _2 \\, \\ldots \\, \\mu (q_k) \\, \\sigma _k\\quad \\text{or}\\quad \\mu (q_1) \\, \\sigma _1 \\, \\mu (q_2) \\, \\sigma _2 \\, \\ldots ,\\quad \\text{respectively,}$ where, for any $q_i$ , $\\mu (q_i) = f$ if $q_i \\in F$ and $\\mu (q_i) = \\varepsilon $ if $q_i \\in Q \\setminus F$ .", "Note that in the finite case the transcript does not include $\\mu (q_{k + 1})$ where $q_{k + 1}$ is the control state in the last configuration.", "In particular, if a computation consists of a single configuration, then its transcript is $\\varepsilon $ .", "In general, transcripts are finite words and infinite sequences over the auxiliary alphabet $\\lbrace a, f\\rbrace $ .", "The reader may notice that our definition for the finite case basically treats finite computations as left-closed, right-open intervals and lets us perform their concatenation in a natural way.", "We note, however, that from a technical point of view, a definition treating them as closed intervals would actually do just as well.", "Note that any sequence $s \\in \\lbrace a, f\\rbrace ^\\omega $ containing infinitely many occurrences of $a$ naturally defines a unique characteristic sequence $c \\in \\lbrace 0, 1\\rbrace ^\\omega $ such that if $s$ is the transcript of a udpda computation, then $c$ is the characteristic sequence of this udpda's language.", "The following lemma shows that this correspondence is efficient if the sequences are represented by pairs of SLPs.", "There exists a polynomial-time algorithm that, given a pair of straight-line programs $(\\mathcal {T}^{\\prime }, \\mathcal {T}^{\\prime \\prime })$ of size $m$ that generates a sequence $s \\in \\lbrace a, f\\rbrace ^\\omega $ and such that the symbol $a$ occurs in $\\mathrm {eval}(\\mathcal {T}^{\\prime \\prime })$ , produces a pair of straight-line programs $(\\mathcal {P}^{\\prime }, \\mathcal {P}^{\\prime \\prime })$ of size $O(m)$ that generates the characteristic sequence defined by $s$ .", "Observe that it suffices to apply to the sequence generated by $(\\mathcal {T}^{\\prime }, \\mathcal {T}^{\\prime \\prime })$ the composition of the following substitutions: $h_1 \\colon a f\\mapsto 1$ , $h_2 \\colon a \\mapsto 0$ , and $h_3 \\colon f\\mapsto \\varepsilon $ .", "One can easily see that applying $h_2$ and $h_3$ reduces to applying them to terminal symbols in SLPs, so it suffices to show that the application of $h_1$ can also be done in polynomial time and increases the number of productions in Chomsky normal form by at most a constant factor.", "We first show how to apply $h_1$ to a single SLP.", "Assume Chomsky normal form and process the productions of the SLP inductively in the bottom-up direction.", "Productions with terminal symbols remain unchanged, and productions of the form $N \\rightarrow A B$ are handled as follows: if $\\mathrm {eval}(A)$ ends with an $a$ and $\\mathrm {eval}(B)$ begins with an $f$ , then replace the production with $N \\rightarrow (A a^{-1}) \\cdot 1 \\cdot (f^{-1} B)$ , otherwise leave it unchanged as well.", "Here we use auxiliary nonterminals of the form $N a^{-1}$ and $f^{-1} N$ with the property that $\\mathrm {eval}(N a^{-1}) \\cdot a = \\mathrm {eval}(N)$ and $f\\cdot \\mathrm {eval}(f^{-1} N) = \\mathrm {eval}(N)$ .", "These nonterminals are easily defined inductively in a straightforward manner, just after processing $N$ .", "At the end of this process one obtains an SLP that generates the result of applying $h_1$ to the word generated by the original SLP.", "We now show how to handle the fact that we need to apply $h_1$ to the entire sequence $\\mathrm {eval}(\\mathcal {T}^{\\prime })\\cdot (\\mathrm {eval}(\\mathcal {T}^{\\prime \\prime }))^\\omega $ .", "Process the SLPs $\\mathcal {T}^{\\prime }$ and $\\mathcal {T}^{\\prime \\prime }$ as described above; for convenience, we shall use the same two names for the obtained programs.", "Then deal with the junction points in the sequence $\\mathrm {eval}(\\mathcal {T}^{\\prime })\\cdot (\\mathrm {eval}(\\mathcal {T}^{\\prime \\prime }))^\\omega $ as follows.", "If $\\mathrm {eval}(\\mathcal {T}^{\\prime \\prime })$ does not start with an $f$ , then there is nothing to do.", "Now suppose it does; then there are two options.", "The first option is that $\\mathrm {eval}(\\mathcal {T}^{\\prime \\prime })$ ends with an $a$ .", "In this case replace $\\mathcal {T}^{\\prime \\prime }$ with $(f^{-1} \\mathcal {T}^{\\prime \\prime }) \\cdot 1$ and $\\mathcal {T}^{\\prime }$ with $(\\mathcal {T}^{\\prime }a^{-1}) \\cdot 1$ or with $(\\mathcal {T}^{\\prime }f)$ according to whether it ends with an $a$ or not.", "The second option is that $\\mathcal {T}^{\\prime \\prime }$ does not end with an $a$ .", "In this case, if $\\mathcal {T}^{\\prime }$ ends with an $a$ , replace it with $(\\mathcal {T}^{\\prime }a^{-1}) \\cdot 1 \\cdot (f^{-1} \\mathcal {T}^{\\prime \\prime })$ , otherwise do nothing.", "One can easily see that the pair of SLPs obtained on this step will generate the image of the original sequence $\\mathrm {eval}(\\mathcal {T}^{\\prime })\\cdot (\\mathrm {eval}(\\mathcal {T}^{\\prime \\prime }))^\\omega $ under $h_1$ .", "This completes the proof.", "$\\Box $ Note that we could use a result by Bertoni, Choffrut, and Radicioni [3] and apply a four-state transducer (however, the underlying automaton needs to be $\\varepsilon $ -free, which would make us figure out the last position “manually”).", "Now it remains to show how to efficiently produce, given a udpda $\\mathcal {A}$ , a pair of SLPs $(\\mathcal {T}^{\\prime }, \\mathcal {T}^{\\prime \\prime })$ generating the transcript of the computation of $\\mathcal {A}$ .", "This is the key part of the entire algorithm, captured by the following lemma.", "There exists a polynomial-time algorithm that, given a udpda $\\mathcal {A}$ of size $m$ , produces a pair of straight-line programs $(\\mathcal {T}^{\\prime }, \\mathcal {T}^{\\prime \\prime })$ of size $O(m)$ that generates the transcript of the computation of $\\mathcal {A}$ .", "The proof of Lemma REF is given in the next subsection.", "Put together, Lemmas REF and REF prove the harder direction (that is, part REF ) of Theorem .", "The only caveat is that if $\\mathrm {eval}(\\mathcal {T}^{\\prime \\prime })\\in \\lbrace f\\rbrace ^*$ , then one needs to replace $\\mathcal {T}^{\\prime \\prime }$ with a simple SLP that generates $a$ and possibly adjust $\\mathcal {T}^{\\prime }$ so that $f$ be appended to the generated word.", "This corresponds to the case where $\\mathcal {A}$ does not read the entire input and enters an infinite loop of $\\varepsilon $ -moves (that is, moves that do not consume $a$ from the input).", "Details: proof of Lemma  REF Returning and non-returning states.", "The main difficulty in proving Lemma REF lies in capturing the structure of a unary deterministic computation.", "To reason about such computations in a convenient manner, we introduce the following definitions.", "We say that a state $q$ is returning if it holds that $(q, \\bot ) \\vdash ^* (q^{\\prime }, \\bot )$ for some state $q^{\\prime } \\in Q_{-1}$ (recall that states from $Q_{-1}$ are pop states).", "In such a case the control state $q^{\\prime }$ of the first configuration of the form $(q^{\\prime }, \\bot )$ , $q^{\\prime } \\in Q_{-1}$ , occurring in the infinite computation starting from $(q, \\bot )$ is called the exit point of $q$ , and the computation between $(q, \\bot )$ and this $(q^{\\prime }, \\bot )$ the return segment from $q$ .", "For example, if $q \\in Q_{-1}$ , then $q$ is its own exit point, and the return segment from $q$ contains no moves.", "Intuitively, the exit point is the first control state in the computation where the bottom-of-the-stack symbol in the configuration $(q, \\bot )$ may matter.", "One can formally show that if $q^{\\prime }$ is the exit point of $q$ , then for any configuration $(q, s)$ it holds that $(q, s) \\vdash ^* (q^{\\prime }, s)$ and, moreover, the transcript of the return segment from $q$ is equal, for any $s$ , to the transcript of the shortest computation from $(q, s)$ to $(q^{\\prime }, s)$ .", "If a control state is not returning, it is called non-returning.", "For such a state $q$ , it holds that for every configuration $(q^{\\prime }, s^{\\prime })$ reachable from $(q, \\bot )$ either $s^{\\prime } \\ne \\bot $ or $q^{\\prime } \\notin Q_{-1}$ .", "One can show formally that infinite computations starting from configurations $(q, s)$ with a fixed non-returning state $q$ and arbitrary $s$ have identical transcripts and, therefore, identical characteristic sequences associated with them.", "As a result, we can talk about infinite computations starting at a non-returning control state $q$ , rather than in a specific configuration $(q, s)$ .", "Now consider a state $q \\notin Q_{-1}$ , an arbitrary configuration $(q, s)$ and the infinite computation starting from $(q, s)$ .", "Suppose that this computation enters a configuration of the form $(\\bar{q}, s)$ after at least one move.", "Then the horizontal successor of $q$ is defined as the control state $\\bar{q}$ of the first such configuration, and the computation between these configurations is called the horizontal segment from $q$ .", "In other cases, horizontal successor and horizontal segment are undefined.", "It is easily seen that the horizontal successor, whenever it exists, is well-defined in the sense that it does not depend upon the choice of $s \\in (\\mathrm {\\Gamma }\\setminus \\lbrace \\bot \\rbrace )^* \\bot $ .", "Similarly, the choice of $s$ only determines the “lower” part of the stack in the configurations of the horizontal segment; since we shall only be interested in the transcripts, this abuse of terminology is harmless.", "Equivalently, suppose that $q \\notin Q_{-1}$ and $(q, s) \\vdash (q^{\\prime }, s^{\\prime })$ .", "If $s^{\\prime } = s$ then the horizontal successor of $q$ is $q^{\\prime }$ .", "Otherwise it holds that $\\delta _{+1}(q)=(q^{\\prime },\\gamma )$ for some $\\gamma \\in \\mathrm {\\Gamma }$ , so that $s^{\\prime } = \\gamma s$ .", "Now if $q^{\\prime }$ is returning, $q^{\\prime \\prime }$ is the exit point of $q^{\\prime }$ , and $\\delta _{-1}(q^{\\prime \\prime },\\gamma )=\\bar{q}$ for the same $\\gamma $ , then $\\bar{q}$ is the horizontal successor of $q$ .", "The horizontal segment is in both cases defined as the shortest non-empty computation of the form $(q, s) \\vdash ^* (\\bar{q}, s)$ .", "General approach and data structures.", "Recall that our goal in this subsection is to define an algorithm that constructs a pair of straight-line programs $(\\mathcal {T}^{\\prime }, \\mathcal {T}^{\\prime \\prime })$ generating the transcript of the infinite computation of $\\mathcal {A}$ .", "The approach that we take is dynamic programming.", "We separate out intermediate goals of several kinds and construct, for an arbitrary control state $q \\in Q$ , SLPs and pairs of SLPs that generate transcripts of the infinite computation starting at $q$ (if $q$ is non-returning), of the return segment from $q$ (if $q$ is returning), and of the horizontal segment from $q$ (whenever it is defined).", "Our algorithm will write productions as it runs, always using, on their right-hand side, only terminal symbols from $\\lbrace a, f\\rbrace $ and nonterminals defined by productions written earlier.", "This enables us to use the notation $\\mathrm {eval}(A)$ for nonterminals $A$ without referring to a specific SLP.", "Once written, a production is never modified.", "The main data structures of the algorithm, apart from the productions it writes, are as follows: three partial functions $\\mathcal {E}, \\mathcal {H}, \\mathcal {W}\\colon Q \\rightarrow Q$ and a subset $\\mathrm {NonRet}\\subseteq Q$ .", "Associated with $\\mathcal {E}$ and $\\mathcal {H}$ are nonterminals $E_q$ and $H_q$ , and with $\\mathrm {NonRet}$ nonterminals $N^{\\prime }_q$ and $N^{\\prime \\prime }_q$ .", "Note that the partial functions from $Q$ to $Q$ can be thought of as digraphs on the set of vertices $Q$ .", "In such digraphs the outdegree of every vertex is at most 1.", "The algorithm will subsequently modify these partial functions, that is, add new edges and/or remove existing ones (however, the outdegree of no vertex will ever be increased to above 1).", "We can also promise that $\\mathcal {E}$ will only increase, i. e., its graph will only get new edges, $\\mathcal {W}$ will only decrease, and $\\mathcal {H}$ can go both ways.", "During its run the algorithm will maintain the following invariants: (I1) $Q = \\operatorname{dom}\\mathcal {E}\\sqcup \\operatorname{dom}\\mathcal {H}\\sqcup \\operatorname{dom}\\mathcal {W}\\sqcup \\mathrm {NonRet}$ , where $\\sqcup $ denotes union of disjoint sets.", "(I2) Whenever $\\mathcal {E}(q) = q^{\\prime }$ , it holds that $q$ is returning, $q^{\\prime }$ is the exit point of $q$ , and $\\mathrm {eval}(E_q)$ is the transcript of the return segment from $q$ .", "(I3) Whenever $\\mathcal {H}(q) = q^{\\prime }$ , it holds that $q^{\\prime }$ is the horizontal successor of $q$ and $\\mathrm {eval}(H_q)$ is the transcript of the horizontal segment from $q$ .", "(I4) Whenever $\\mathcal {W}(q) = q^{\\prime }$ , it holds that $\\delta _{+1}(q)=(q^{\\prime },\\gamma )$ for some $\\gamma \\in \\mathrm {\\Gamma }$ .", "(I5) Whenever $q \\in \\mathrm {NonRet}$ , it holds that $q$ is non-returning and the sequence $\\mathrm {eval}(N^{\\prime }_q) \\cdot (\\mathrm {eval}(N^{\\prime \\prime }_q))^\\omega $ is the transcript of the infinite computation starting at $q$ .", "Description of the algorithm: computing transcripts.", "Our algorithm has three stages: the initialization stage, the main stage, and the $\\bot $ -handling stage.", "The initialization stage of the algorithm works as follows: for each $q \\in Q$ , write $V_q \\rightarrow \\mu (q) \\sigma (q)$ , where $\\mu (q)$ is $f$ if $q \\in F$ and $\\varepsilon $ otherwise, and $\\sigma (q)$ is $a$ if $q \\in R$ (that is, if transitions departing from $q$ read a symbol from the input) and $\\varepsilon $ otherwise; for all $q \\in Q_{-1}$ , set $\\mathcal {E}(q) = q$ and write $E_q \\rightarrow \\varepsilon $ ; for all $q \\in Q_{0}$ , set $\\mathcal {H}(q) = q^{\\prime }$ where $\\delta _{0}(q)=q^{\\prime }$ and write $H_q \\rightarrow V_q$ ; for all $q \\in Q_{+1}$ , set $\\mathcal {W}(q) = q^{\\prime }$ where $\\delta _{+1}(q)=(q^{\\prime },\\gamma )$ for some $\\gamma \\in \\mathrm {\\Gamma }$ ; set $\\mathrm {NonRet}= \\emptyset $ .", "It is easy to see that in this way all invariants REF –REF are initially satisfied (recall that the transcript of an empty computation is $\\varepsilon $ ).", "For convenience, we also introduce two auxiliary objects: a partial function $\\mathcal {G}\\colon Q \\rightarrow Q$ and nonterminals $G_q$ , defined as follows.", "The domain of $\\mathcal {G}$ is $\\operatorname{dom}\\mathcal {G}= \\operatorname{dom}\\mathcal {H}\\sqcup \\operatorname{dom}\\mathcal {W}$ ; note that, according to invariant REF , this union is disjoint.", "We assign $\\mathcal {G}(q) = q^{\\prime }$ iff $\\mathcal {H}(q) = q^{\\prime }$ or $\\mathcal {W}(q) = q^{\\prime }$ .", "We shall assume that $\\mathcal {G}$ is recomputed as $\\mathcal {H}$ and $\\mathcal {W}$ change.", "Now for every $q \\in \\operatorname{dom}\\mathcal {G}$ , we let $G_q$ stand for $H_q$ if $q \\in \\operatorname{dom}\\mathcal {H}$ and for $V_q$ if $q \\in \\operatorname{dom}\\mathcal {W}$ .", "At this point we are ready to describe the main stage of the algorithm.", "During this stage, the algorithm applies the following rules until none of them is applicable (if at some point several rules can be applied, the choice is made arbitrarily; the rules are well-defined whenever invariants REF –REF hold): (R1) If $\\mathcal {G}(q) = q^{\\prime }$ where $q^{\\prime } \\in \\mathrm {NonRet}$ and $q \\in Q$ : remove $q$ from either $\\operatorname{dom}\\mathcal {H}$ or $\\operatorname{dom}\\mathcal {W}$ , add $q$ to $\\mathrm {NonRet}$ , write $N^{\\prime }_q \\rightarrow G_q N^{\\prime }_{q^{\\prime }}$ and $N^{\\prime \\prime }_q \\rightarrow N^{\\prime \\prime }_{q^{\\prime }}$ .", "(R2) If $\\mathcal {H}(q) = q^{\\prime }$ where $q^{\\prime } \\in \\operatorname{dom}\\mathcal {E}$ and $q \\in Q$ : remove $q$ from $\\operatorname{dom}\\mathcal {H}$ , define $\\mathcal {E}(q) = \\mathcal {E}(q^{\\prime })$ , write $E_q \\rightarrow H_q E_{q^{\\prime }}$ .", "(R3) If $\\mathcal {W}(q) = q^{\\prime }$ where $q^{\\prime } \\in \\operatorname{dom}\\mathcal {E}$ and $q \\in Q$ : remove $q$ from $\\operatorname{dom}\\mathcal {W}$ , define $\\mathcal {H}(q) = \\bar{q}$ where $\\mathcal {E}(q^{\\prime }) = q^{\\prime \\prime }$ , $\\delta _{-1}(q^{\\prime \\prime },\\gamma )=\\bar{q}$ , and $\\delta _{+1}(q)=(q^{\\prime },\\gamma )$ (that is, $\\gamma $ is the symbol pushed by the transition leaving $q$ , and $\\bar{q}$ is the state reached by popping $\\gamma $ at $q^{\\prime \\prime }$ , the exit point of $q^{\\prime }$ ).", "Finally, write $H_q \\rightarrow V_q E_{q^{\\prime }} V_{q^{\\prime \\prime }}$ .", "(R4) If $\\mathcal {G}$ contains a simple cycle, that is, if $\\mathcal {G}(q_i) = q_{i + 1}$ for $i = 1, \\ldots , k - 1$ and $\\mathcal {G}(q_k) = q_1$ , where $q_i \\ne q_j$ for $i \\ne j$ , then for each vertex $q_i$ of the cycle remove it from either $\\operatorname{dom}\\mathcal {H}$ or $\\operatorname{dom}\\mathcal {W}$ and add it to $\\mathrm {NonRet}$ ; in addition, write $N^{\\prime }_{q_k} \\rightarrow G_{q_k}$ , $N^{\\prime \\prime }_{q_k} \\rightarrow G_{q_1} \\ldots G_{q_k}$ , and, for each $i = 1, \\ldots , k - 1$ , $N^{\\prime }_{q_i} \\rightarrow G_{q_i} N^{\\prime }_{q_{i + 1}}$ and $N^{\\prime \\prime }_{q_i} \\rightarrow N^{\\prime \\prime }_{q_{i + 1}}$ .", "We shall need two basic facts about this stage of the algorithm.", "Application of rules REF –REF does not violate invariants REF –REF .", "The proof of Claim REF is easy and left to the reader.", "If no rule is applicable, then $\\operatorname{dom}\\mathcal {G}= \\emptyset $ .", "Suppose $\\operatorname{dom}\\mathcal {G}\\ne \\emptyset $ .", "Consider the graph associated with $\\mathcal {G}$ and observe that all vertices in $\\operatorname{dom}\\mathcal {G}$ have outdegree 1.", "This implies that $\\mathcal {G}$ has either a cycle within $\\operatorname{dom}\\mathcal {G}$ or an edge from $\\operatorname{dom}\\mathcal {G}$ to $Q \\setminus \\operatorname{dom}\\mathcal {G}$ .", "In the first case, rule REF is applicable.", "In the second case, we conclude with the help of the invariant REF that the edge leads from a vertex in $\\operatorname{dom}\\mathcal {H}\\sqcup \\operatorname{dom}\\mathcal {W}$ to a vertex in $\\mathrm {NonRet}\\sqcup \\operatorname{dom}\\mathcal {E}$ .", "If the destination is in $\\mathrm {NonRet}$ , then rule REF is applicable; otherwise the destination is in $\\operatorname{dom}\\mathcal {E}$ and one can apply rule REF or rule REF according to whether the source is in $\\operatorname{dom}\\mathcal {H}$ or in $\\operatorname{dom}\\mathcal {W}$ .", "$\\Box $ Now we are ready to describe the $\\bot $ -handling stage of the algorithm.", "By the beginning of this stage, the structure of deterministic computation of $\\mathcal {A}$ has already been almost completely captured by the productions written earlier, and it only remains to account for moves involving $\\bot $ .", "So this last stage of the algorithm takes the initial state $q_0$ of $\\mathcal {A}$ and proceeds as follows.", "If $q_0 \\in \\mathrm {NonRet}$ , then take $N^{\\prime }_{q_0}$ as the axiom of $\\mathcal {T}^{\\prime }$ and $N^{\\prime \\prime }_{q_0}$ as the axiom of $\\mathcal {T}^{\\prime \\prime }$ .", "By invariant REF , these nonterminals are defined and generate appropriate words, so the pair $(\\mathcal {T}^{\\prime }, \\mathcal {T}^{\\prime \\prime })$ indeed generates the transcript of the computation of $\\mathcal {A}$ .", "Since at the beginning of the $\\bot $ -handling stage $\\operatorname{dom}\\mathcal {G}= \\emptyset $ , it remains to consider the case $q_0 \\in \\operatorname{dom}\\mathcal {E}$ .", "Define a partial function $\\mathcal {E}^\\bot \\colon Q \\rightarrow Q$ by setting, for each $q \\in \\operatorname{dom}\\mathcal {E}$ , its value according to $\\mathcal {E}^\\bot (q) = \\bar{q}$ if $\\mathcal {E}(q) = q^{\\prime }$ and $\\delta _{-1}(q^{\\prime },\\bot )=\\bar{q}$ .", "Write productions $E^\\bot _q \\rightarrow E_q V_{q^{\\prime }}$ accordingly.", "Now associate $\\mathcal {E}^\\bot $ with a graph, as earlier, and consider the longest simple path within $\\operatorname{dom}\\mathcal {E}^\\bot $ starting at $q_0$ .", "Suppose it ends at a vertex $q_k$ , where $\\mathcal {E}^\\bot (q_i) = q_{i + 1}$ for $i = 0, \\ldots , k$ .", "There are two subcases here according to why the path cannot go any further.", "The first possible reason is that it reaches $Q \\setminus \\operatorname{dom}\\mathcal {E}^\\bot = \\mathrm {NonRet}$ , that is, that $q_{k + 1}$ belongs to $\\mathrm {NonRet}$ .", "In this subcase write $N^{\\prime }_{q_0} \\rightarrow E^\\bot _{q_0} \\ldots E^\\bot _{q_{k}} N^{\\prime }_{q_{k + 1}}$ and $N^{\\prime \\prime }_{q_0} \\rightarrow N^{\\prime \\prime }_{q_{k + 1}}$ .", "The second possible reason is that $q_{k + 1} = q_i$ where $0 \\le i \\le k$ .", "In this subcase write $N^{\\prime }_{q_0} \\rightarrow E^\\bot _{q_0} \\ldots E^\\bot _{q_{i - 1}}$ and $N^{\\prime \\prime }_{q_0} \\rightarrow E^\\bot _{q_i} \\ldots E^\\bot _{q_k}$ .", "In any of the two subcases above, take $N^{\\prime }_{q_0}$ and $N^{\\prime \\prime }_{q_0}$ as axioms of $\\mathcal {T}^{\\prime }$ and $\\mathcal {T}^{\\prime \\prime }$ , respectively.", "The correctness of this step follows easily from the invariants REF and REF .", "This gives a polynomial algorithm that converts a udpda $\\mathcal {A}$ into a pair of SLPs $(\\mathcal {T}^{\\prime }, \\mathcal {T}^{\\prime \\prime })$ that generates the transcript of the infinite computation of $\\mathcal {A}$ , and the only remaining bit is bounding the size of $(\\mathcal {T}^{\\prime }, \\mathcal {T}^{\\prime \\prime })$ .", "The size of $(\\mathcal {T}^{\\prime }, \\mathcal {T}^{\\prime \\prime })$ is $O(|Q|)$ .", "There are three types of nonterminals whose productions may have growing size: $N^{\\prime \\prime }_{q_k}$ in rule REF , and $N^{\\prime }_{q_0}$ and $N^{\\prime \\prime }_{q_0}$ in the $\\bot $ -handling stage.", "For all three types, the size is bounded by the cardinality of the set of states involved in a cycle or a path.", "Since such sets never intersect, all such nonterminals together contribute at most $|Q|$ productions to the Chomsky normal form.", "The contribution of other nonterminals is also $O(|Q|)$ , because they all have fixed size and each state $q$ is associated with a bounded number of them.", "$\\Box $ Combined with Claim REF in Subsection REF , this completes the proof of Lemma REF and Theorem .", "Universality of unpda In this section we settle the complexity status of the universality problem for unary, possibly nondeterministic pushdown automata.", "While $\\mathbf {\\mathrm {\\Pi }_2 P}$ -completeness of equivalence and inclusion is shown by Huynh [14], it has been unknown whether the universality problem is also $\\mathbf {\\mathrm {\\Pi }_2 P}$ -hard.", "For convenience of notation, we use an auxiliary descriptional system.", "Define integer expressions over the set of operations $\\lbrace +, \\cup , \\times 2, {} {\\times } \\mathbb {N}\\rbrace $ inductively: the base case is a non-negative integer $n$ , written in binary, and the inductive step is associated with binary operations $+$ , $\\cup $ , and unary operations $\\times 2$ , ${} {\\times } \\mathbb {N}$ .", "To each expression $E$ we associate a set of non-negative integers $S(E)$ : $S(n) = \\lbrace n \\rbrace $ , $S(E_1 + E_2) = \\lbrace s_1 + s_2 \\colon s_1 \\in S(E_1), s_2 \\in S(E_2) \\rbrace $ , $S(E_1 \\cup E_2) = S(E_1) \\cup S(E_2)$ , $S(E \\times 2) = S(E + E)$ , $S({E} {\\times } \\mathbb {N}) = \\lbrace s k \\colon s \\in S(E), k = 0, 1, 2, \\ldots \\,\\rbrace $ .", "Expressions $E_1$ and $E_2$ are called equivalent iff $S(E_1) = S(E_2)$ ; an expression $E$ is universal iff it is equivalent to ${1} {\\times } \\mathbb {N}$ .", "The problem of deciding universality is denoted by Integer-$\\lbrace +, \\cup , \\times 2, {} {\\times } \\mathbb {N}\\rbrace $ -Expression-Universality.", "Decision problems for integer expressions have been studied for more than 40 years: Stockmeyer and Meyer [31] showed that for expressions over $\\lbrace +,\\cup \\rbrace $ compressed membership is $\\mathbf {NP}$ -complete and equivalence is $\\mathbf {\\mathrm {\\Pi }_2 P}$ -complete (universality is, of course, trivial).", "For recent results on such problems with operations from $\\lbrace +, \\cup , \\cap , \\times , \\overline{\\phantom{x}}\\rbrace $ , see McKenzie and Wagner [23] and Glaßer et al. [8].", "Integer-$\\lbrace +, \\cup , \\times 2, {} {\\times } \\mathbb {N}\\rbrace $ -Expression-Universality is $\\mathbf {\\mathrm {\\Pi }_2 P}$ -hard.", "The reduction is from the Generalized-Subset-Sum problem, which is defined as follows.", "The input consists of two vectors of naturals, $u = (u_1, \\ldots , u_n)$ and $v = (v_1, \\ldots , v_m)$ , and a natural $t$ , and the problem is to decide whether for all $y \\in \\lbrace 0, 1\\rbrace ^m$ there exists an $x \\in \\lbrace 0, 1\\rbrace ^n$ such that $x \\cdot u + y \\cdot v = t$ , where the middle dot $\\cdot $ once again denotes the inner product.", "This problem was shown to be hard by Berman et al. [1].", "Start with an instance of Generalized-Subset-Sum and let $M$ be a big number, $M > \\sum _{i = 1}^{n} u_i + \\sum _{j = 1}^{m} v_j$ .", "Assume without loss of generality that $M > t$ .", "Consider the integer expression $E$ defined by the following equations: $E &= E^{\\prime } \\cup E^{\\prime \\prime }, \\\\E^{\\prime } &= (2^m M + {1} {\\times } \\mathbb {N}) \\cup ({M} {\\times } \\mathbb {N} + ([0, t - 1] \\cup [t + 1, M - 1])),\\\\E^{\\prime \\prime } &= \\sum _{j = 1}^{m} (0 \\cup (2^{j - 1} M + v_j)) +\\sum _{i = 1}^{n} (0 \\cup u_i), \\\\[a, b] &= a + [0, b - a], \\\\[0, t] &= [0, \\lfloor t/2 \\rfloor ] \\times 2+ (0 \\cup (t \\bmod 2)), \\\\[0, 1] &= 0 \\cup 1, \\\\[0, 0] &= 0.$ Note that the size of $E$ is polynomial in the size of the input, and $E$ can be constructed in logarithmic space.", "We show that $E$ is universal iff the input is a yes-instance of Generalized-Subset-Sum.", "It is immediate that $E$ is universal if and only if $S(E)$ contains $2^m$ numbers of the form $k M + t$ , $0 \\le k < 2^m$ .", "We show that every such number is in $S(E)$ if and only if for the binary vector $y = (y_1, \\ldots , y_m) \\in \\lbrace 0, 1\\rbrace ^m$ , defined by $k = \\sum _{j = 1}^{m} y_j \\, 2^{j - 1}$ , there exists a vector $x \\in \\lbrace 0, 1\\rbrace ^n$ such that $x \\cdot u + y \\cdot v = t$ .", "First consider an arbitrary $y \\in \\lbrace 0, 1\\rbrace ^m$ and choose $k$ as above.", "Suppose that for this $y$ there exists an $x \\in \\lbrace 0, 1\\rbrace ^n$ such that $x \\cdot u + y \\cdot v = t$ .", "One can easily see that appropriate choices in $E^{\\prime \\prime }$ give the number $k M + y \\cdot v + x \\cdot u = k M + t$ .", "Conversely, suppose that $k M + t \\in S(E)$ for some $k$ , $0 \\le k < 2^m$ ; then $k M + t \\in S(E^{\\prime \\prime })$ .", "Since $(1, \\ldots , 1) \\cdot u + (1, \\ldots , 1) \\cdot v < M$ , it holds that $t = y \\cdot v + x \\cdot u$ for binary vectors $y \\in \\lbrace 0, 1\\rbrace ^m$ and $x \\in \\lbrace 0, 1\\rbrace ^n$ that correspond to the choices in the addends.", "Moreover, the same inequality also shows that $k M$ is equal to the sum of some powers of two in the first sum in $E^{\\prime \\prime }$ , and so $k = \\sum _{j = 1}^{m} y_j \\, 2^{j - 1}$ .", "This completes the proof.", "$\\Box $ With circuits instead of formulae (see also [23] and [8]) we would not need doubling.", "Furthermore, we only use ${} {\\times } \\mathbb {N}$ on fixed numbers, so instead we could use any feature for expressing an arithmetic progression with fixed common difference.", "Unary-PDA-Universality is $\\mathbf {\\mathrm {\\Pi }_2 P}$ -complete.", "A reduction from Integer-$\\lbrace +, \\cup , \\times 2, {} {\\times } \\mathbb {N}\\rbrace $ -Expression-Universality, which is $\\mathbf {\\mathrm {\\Pi }_2 P}$ -hard by Lemma , shows hardness.", "Indeed, an integer expression over $\\lbrace +, \\cup , \\times 2, {} {\\times } \\mathbb {N}\\rbrace $ can be transformed into a unary CFG in a straightforward way.", "Binary numbers are encoded by poly-size SLPs, summation is modeled by concatenation, and union by alternatives.", "Doubling is a special case of summation, and ${} {\\times } \\mathbb {N}$ gives rise to productions of the form $N^{\\prime } \\rightarrow \\varepsilon $ and $N^{\\prime } \\rightarrow N N^{\\prime }$ .", "The obtained CFG is then transformed into a unary PDA $\\mathcal {A}$ by a standard algorithm (see, e. g., Savage [26]).", "The result is that $L(\\mathcal {A}) = \\lbrace 1^s \\colon s \\in S(E) \\rbrace $ , and $\\mathcal {A}$ is computed from $E$ in logarithmic space.", "This concludes the proof.", "$\\Box $ We give a simple proof of the $\\mathbf {\\mathrm {\\Pi }_2 P}$ upper bound.", "Let $\\varphi _{\\mathcal {A}}(x)$ be an existential Presburger formula of size polynomial in the size of $\\mathcal {A}$ that characterizes the Parikh image of $L(\\mathcal {A})$ (see Verma, Seidl, and Schwentick [32]).", "To show that an udpda $\\mathcal {A}$ is non-universal, we find an $n\\ge 0$ such that $\\lnot \\varphi _{\\mathcal {A}}(n)$ holds.", "Now we note that for any udpda $\\mathcal {A}$ of size $m$ , there is a deterministic finite automaton of size $2^{O(m)}$ accepting $L(\\mathcal {A})$ (see discussion in Section  and Pighizzini [24]).", "Thus, $n$ is bounded by $2^{O(m)}$ .", "Therefore, checking non-universality can be expressed as a predicate: $\\exists n \\le 2^{O(m)}.\\lnot \\varphi _{\\mathcal {A}}(n)$ .", "This is a $\\mathbf {\\mathrm {\\Sigma }_2 P}$ -predicate, because the $\\exists ^*$ -fragment of Presburger arithmetic is $\\mathbf {NP}$ -complete [33].", "Universality, equivalence, and inclusion are $\\mathbf {\\mathrm {\\Pi }_2 P}$ -complete for (possibly nondeterministic) unary pushdown automata, unary context-free grammars, and integer expressions over $\\lbrace +, \\cup , \\times 2, {} {\\times } \\mathbb {N}\\rbrace $ .", "Another consequence of Theorem  is that deciding equality of a (not necessarily unary) context-free language, given as a context-free grammar, to any fixed context-free language $L_0$ that contains an infinite regular subset, is $\\mathbf {\\mathrm {\\Pi }_2 P}$ -hard and, if $L_0 \\subseteq \\lbrace a\\rbrace ^*$ , $\\mathbf {\\mathrm {\\Pi }_2 P}$ -complete.", "The lower bound is by reduction due to Hunt III, Rosenkrantz, and Szymanski [12], who show that deciding equivalence to $\\lbrace a\\rbrace ^*$ reduces to deciding equivalence to any such $L_0$ .", "The reduction is shown to be polynomial-time, but is easily seen to be logarithmic-space as well.", "The upper bound for the unary case is by Huynh [14]; in the general case, the problem can be undecidable.", "Corollaries and discussion Descriptional complexity aspects of udpda.", "Theorem  can be used to obtain several results on descriptional complexity aspects of udpda proved earlier by Pighizzini [24].", "He shows how to transform a udpda of size $m$ into an equivalent deterministic finite automaton (DFA) with at most $2^m$ states [24] and into an equivalent context-free grammar in Chomsky normal form (CNF) with at most $2 m + 1$ nonterminals [24].", "In our construction $m$ gets multiplied by a small constant, but the advantage is that we now see (the slightly weaker variants of) these results as easy corollaries of a single underlying theorem.", "Indeed, using an indicator pair $(\\mathcal {P}^{\\prime }, \\mathcal {P}^{\\prime \\prime })$ for $L$ , it is straightforward to construct a DFA of size $|\\mathrm {eval}(\\mathcal {P}^{\\prime })| + |\\mathrm {eval}(\\mathcal {P}^{\\prime \\prime })|$ accepting $L$ , as well as to transform the pair into a CFG in CNF that generates $L$ and has at most thrice the size of $(\\mathcal {P}^{\\prime }, \\mathcal {P}^{\\prime \\prime })$ .", "Another result which follows, even more directly, from ours is a lower bound on the size of udpda accepting a specific language $L_1$  [24].", "To obtain this lower bound, Pighizzini employs a known lower bound on the SLP-size of the word $W = W\\!", "[0] \\ldots W\\!", "[K - 1] \\in \\lbrace 0, 1\\rbrace ^K$ such that $a^n \\in L_1$ iff $W\\!", "[n \\bmod K] = 1$ .", "To this end, a udpda $\\mathcal {A}$ accepting $L_1$ is intersected (we are glossing over some technicalities here) with a small deterministic finite automaton that “captures” the end of the word $W$ .", "The obtained udpda, which only accepts $a^K$ , is transformed into an equivalent context-free grammar.", "It is then possible to use the structure of the grammar to transform it into an SLP that produces $W$ (note that such a transformation in general is $\\mathbf {NP}$ -hard).", "While the proof produces from a udpda for $L_1$ a related SLP with a polynomial blowup, this construction depends crucially on the structure of the language $L_1$ , so it is difficult to generalize the argument to all udpda and thus obtain Theorem .", "Our proof of Theorem  therefore follows a very different path.", "Relationship to Presburger arithmetic.", "An alternative way to prove the upper bound in Theorem REF is via Presburger arithmetic, using the observation that there is a poly-time computable existential Presburger formula that expresses the membership of a word $a^n$ in $L(\\lnot \\mathcal {A}_1)$ and $L(\\mathcal {A}_2)$ .", "This technique distills the arguments used by Huynh [13], [14] to show that the compressed membership problem for unary pushdown automata is in $\\mathbf {NP}$ .", "It is used in a purified form by Plandowski and Rytter [25], who developed a much shorter proof of the same fact (apparently unaware of the previous proof).", "The same idea was later rediscovered and used in a combination with Presburger arithmetic by Verma, Seidl, and Schwentick [32].", "Another application of this technique provides an alternative proof of the $\\mathbf {\\mathrm {\\Pi }_2 P}$ upper bound for unpda inclusion (Theorem ): to show that $L(\\mathcal {A})$ is universal, we check that $L(\\mathcal {A})$ accepts all words up to length $2^{O(m)}$ (this bound is sufficient because there is a deterministic finite automaton for the language with this size—see the discussion above).", "The proof known to date, due to Huynh [14], involves reproving Parikh's theorem and is more than 10 pages long.", "Reduction to Presburger formulae produces a much simpler proof.", "Also, our $\\mathbf {\\mathrm {\\Pi }_2 P}$ -hardness result for unpda shows that the $\\forall _{\\mathrm {bounded}}\\,\\exists ^*$ -fragment of Presburger arithmetic is $\\mathbf {\\mathrm {\\Pi }_2 P}$ -complete, where the variable bound by the universal quantifier is at most exponential in the size of the formula.", "The upper bound holds because the $\\exists ^*$ -fragment is $\\mathbf {NP}$ -complete [33].", "In comparison, the $\\forall \\, \\exists ^*$ -fragment, without any restrictions on the domain of the universally quantified variable, requires co-nondeterministic $2^{n^{\\mathrm {\\Omega }(1)}}$ time, see Grädel [10].", "Previously known fragments that are complete for the second level of the polynomial hierarchy involve alternation depth 3 and a fixed number of quantifiers, as in Grädel [11] and Schöning [28].", "Also note that the $\\forall ^s \\, \\exists ^t$ -fragment is $\\mathbf {coNP}$ -complete for all fixed $s \\ge 1$ and $t \\ge 2$ , see Grädel [11].", "Problems involving compressed words.", "Recall Theorem REF : given two SLPs, it is $\\mathbf {coNP}$ -complete to compare the generated words componentwise with respect to any partial order different from equality.", "As a corollary, we get precise complexity bounds for SLP equivalence in the presence of wildcards or, equivalently, compressed matching in the model of partial words (see, e. g., Fischer and Paterson [6] and Berstel and Boasson [2]).", "Consider the problem SLP-Partial-Word-Matching: the input is a pair of SLPs $\\mathcal {P}_1$ , $\\mathcal {P}_2$ over the alphabet $\\lbrace a, b, \\text{\\texttt {?", "}}\\rbrace $ , generating words of equal length, and the output is “yes” iff for every $i$ , $0 \\le i < |\\mathcal {P}_1|$ , either $\\mathcal {P}_1[i] = \\mathcal {P}_2[i]$ or at least one of $\\mathcal {P}_1[i]$ and $\\mathcal {P}_2[i]$ is $\\text{\\texttt {?", "}}$ (a hole, or a single-character wildcard).", "Schmidt-Schauß [27] defines a problem equivalent to SLP-Partial-Word-Matching, along with another related problem, where one needs to find occurrences of $\\mathrm {eval}(\\mathcal {P}_1)$ in $\\mathrm {eval}(\\mathcal {P}_2)$ (as in pattern matching), $\\mathcal {P}_2$ is known to contain no holes, and two symbols match iff they are equal or at least one of them is a hole.", "For this related problem, he develops a polynomial-time algorithm that finds (a representation of) all matching occurrences and operates under the assumption that the number of holes in $\\mathrm {eval}(\\mathcal {P}_1)$ is polynomial in the size of the input.", "He also points out that no solution for (the general case of) SLP-Partial-Word-Matching is known—unless a polynomial upper bound on the number of $\\text{\\texttt {?", "}}$ s in $\\mathrm {eval}(\\mathcal {P}_1)$ and $\\mathrm {eval}(\\mathcal {P}_2)$ is given.", "Our next proposition shows that such a solution is not possible unless $\\mathbf {P}= \\mathbf {NP}$ .", "It is an easy consequence of Theorem REF .", "SLP-Partial-Word-Matching is $\\mathbf {coNP}$ -complete.", "Membership in $\\mathbf {coNP}$ is obvious, and the hardness is by a reduction from SLP-Componentwise-${(0 \\le 1)}$.", "Given a pair of SLPs $\\mathcal {P}_1$ , $\\mathcal {P}_2$ over $\\lbrace 0, 1\\rbrace $ , substitute $\\text{\\texttt {?", "}}$ for 0 and $a$ for 1 in $\\mathcal {P}_1$ , and $b$ for 0 and $\\text{\\texttt {?", "}}$ for 1 in $\\mathcal {P}_2$ .", "The resulting pair of SLPs over $\\lbrace a, b, \\text{\\texttt {?", "}}\\rbrace $ is a yes-instance of SLP-Partial-Word-Matching iff the original pair is a yes-instance of SLP-Componentwise-${(0 \\le 1)}$.", "$\\Box $ The wide class of compressed membership problems (deciding $\\mathrm {eval}(\\mathcal {P})\\in L$ ) is studied and discussed in Jeż [16] and Lohrey [20].", "In the case of words over the unary alphabet, $w \\in \\lbrace a\\rbrace ^*$ , expressing $w$ with an SLP is poly-time equivalent to representing it with its length $|w|$ written in binary.", "An easy corollary of Theorem REF is that deciding $w \\in L(\\mathcal {A})$ , where $\\mathcal {A}$ is a (not necessarily unary) deterministic pushdown automaton and $w = a^n$ with $n$ given in binary, is $\\mathbf {P}$ -complete.", "Finally, we note that the precise complexity of SLP equivalence remains open [20].", "We cannot immediately apply lower bounds for udpda equivalence, since we do not know if the translation from udpda to indicator pairs in Theorem  can be implemented in logarithmic (or even polylogarithmic) space." ], [ "Compressed membership and equivalence", "For an SLP $\\mathcal {P}$ , by $|\\mathcal {P}|$ we denote the length of the word $\\mathrm {eval}(\\mathcal {P})$ , and by $\\mathcal {P}[n]$ the $n$ th symbol of $\\mathrm {eval}(\\mathcal {P})$ , counting from 0 (that is, $0 \\le n \\le |\\mathcal {P}| - 1$ ).", "We write $\\mathcal {P}_1 \\equiv \\mathcal {P}_2$ if and only if $\\mathrm {eval}(\\mathcal {P}_1) = \\mathrm {eval}(\\mathcal {P}_2)$ .", "The following SLP-Query problem is known to be $\\mathbf {P}$ -complete (see Lifshits and Lohrey [19]): given an SLP $\\mathcal {P}$ over $\\lbrace 0, 1\\rbrace $ and a number $n$ in binary, decide whether $\\mathcal {P}[n] = 1$ .", "The problem SLP-Equivalence is only known to be in $\\mathbf {P}$ (see, e. g., Lohrey [20]): given two SLPs $\\mathcal {P}_1$ , $\\mathcal {P}_2$ , decide whether $\\mathcal {P}_1 \\equiv \\mathcal {P}_2$ .", "UDPDA-Compressed-Membership is $\\mathbf {P}$ -complete.", "The upper bound follows from Theorem .", "Indeed, given a udpda $\\mathcal {A}$ and a number $n$ , first construct an indicator pair $(\\mathcal {P}^{\\prime }, \\mathcal {P}^{\\prime \\prime })$ for $L(\\mathcal {A})$ .", "Now compute $|\\mathcal {P}^{\\prime }|$ and $|\\mathcal {P}^{\\prime \\prime }|$ and then decide if $n \\le |\\mathcal {P}^{\\prime }| - 1$ .", "If so, the answer is given by $\\mathcal {P}^{\\prime }[n]$ , otherwise by $\\mathcal {P}^{\\prime \\prime }[r]$ , where $r = (n - |\\mathcal {P}^{\\prime }|) \\bmod |\\mathcal {P}^{\\prime \\prime }|$ and in both cases 1 is interpreted as “yes” and 0 as “no”.", "To prove the lower bound, we reduce from the SLP-Query problem.", "Take an instance with an SLP $\\mathcal {P}$ and a number $n$ in binary.", "By transforming the pair $(\\mathcal {P}, \\mathcal {P}_0)$ , with $\\mathcal {P}_0$ any fixed SLP over $\\lbrace 0, 1\\rbrace $ , into a udpda $\\mathcal {A}$ using part REF of Theorem , this problem is reduced, in logspace, to whether $a^n \\in L(\\mathcal {A})$ .", "$\\Box $ Recall that emptiness and universality of udpda are $\\mathbf {P}$ -complete by Proposition .", "Our next theorem extends this result to the general equivalence problem for udpda.", "UDPDA-Equivalence is $\\mathbf {P}$ -complete.", "Hardness follows from Proposition .", "We show how Theorem  can be used to prove the upper bound: given udpda $\\mathcal {A}_1$ and $\\mathcal {A}_2$ , first construct indicator pairs $(\\mathcal {P}^{\\prime }_1, \\mathcal {P}^{\\prime \\prime }_1)$ and $(\\mathcal {P}^{\\prime }_2, \\mathcal {P}^{\\prime \\prime }_2)$ for $L(\\mathcal {A}_1)$ and $L(\\mathcal {A}_2)$ , respectively.", "Now reduce the problem of whether $L(\\mathcal {A}_1) = L(\\mathcal {A}_2)$ to SLP-Equivalence.", "The key observation is that an eventually periodic sequence that has periods $|\\mathcal {P}^{\\prime \\prime }_1|$ and $|\\mathcal {P}^{\\prime \\prime }_2|$ also has period $t = \\gcd (|\\mathcal {P}^{\\prime \\prime }_1|, |\\mathcal {P}^{\\prime \\prime }_2|)$ .", "Therefore, it suffices to check that, first, the initial segments of the generated sequences match and, second, that $\\mathcal {P}^{\\prime \\prime }_1$ and $\\mathcal {P}^{\\prime \\prime }_2$ generate powers of the same word up to a certain circular shift.", "In more detail, let us first introduce some auxiliary operations for SLP.", "For SLPs $\\mathcal {P}_1$ and $\\mathcal {P}_2$ , by $\\mathcal {P}_1 \\cdot \\mathcal {P}_2$ we denote an SLP that generates $\\mathrm {eval}(\\mathcal {P}_1) \\cdot \\mathrm {eval}(\\mathcal {P}_2)$ , obtained by “concatenating” $\\mathcal {P}_1$ and $\\mathcal {P}_2$ .", "Now suppose that $\\mathcal {P}$ generates $w = w[0] \\ldots w[|\\mathcal {P}|-1]$ .", "Then the SLP $\\mathcal {P}[a \\,\\text{..}\\,b)$ generates the word $w[a \\,\\text{..}\\,b) = w[a] \\ldots w[b - 1]$ , of length $b - a$ (as in $\\mathcal {P}[n]$ , indexing starts from 0).", "The SLP $\\mathcal {P}^\\alpha $ generates $w^\\alpha $ , with the meaning clear for $\\alpha = 0, 1, 2, \\ldots $  , also extended to $\\alpha \\in \\mathbb {Q}$ with $\\alpha \\cdot |\\mathcal {P}| \\in \\mathbb {Z}_{\\ge 0}$ by setting $w^{k + n / |w|} = w^k \\cdot w[0 \\,\\text{..}\\,n)$ , $n < |w|$ .", "Finally, ${\\mathcal {P}} \\curvearrowleft {s}$ denotes cyclic shift and evaluates to $w[s \\,\\text{..}\\,|w|) \\cdot w[0 \\,\\text{..}\\,s)$ .", "One can easily demonstrate that all these operations can be implemented in polynomial time.", "So, assume that $|\\mathcal {P}^{\\prime }_1| \\ge |\\mathcal {P}^{\\prime }_2|$ .", "First, one needs to check whether $\\mathcal {P}^{\\prime }_1 \\equiv \\mathcal {P}^{\\prime }_2 \\cdot (\\mathcal {P}^{\\prime \\prime }_2)^\\alpha $ , where $\\alpha = (|\\mathcal {P}^{\\prime }_1| - |\\mathcal {P}^{\\prime }_2|) / |\\mathcal {P}^{\\prime \\prime }_2|$ .", "Second, note that an eventually periodic sequence that has periods $|\\mathcal {P}^{\\prime \\prime }_1|$ and $|\\mathcal {P}^{\\prime \\prime }_2|$ also has period $t = \\gcd (|\\mathcal {P}^{\\prime \\prime }_1|, |\\mathcal {P}^{\\prime \\prime }_2|)$ .", "Compute $t$ and an auxiliary SLP $\\mathcal {P}^{\\prime \\prime }= \\mathcal {P}^{\\prime \\prime }_1[0 \\,\\text{..}\\,t)$ , and then check whether $\\mathcal {P}^{\\prime \\prime }_1 \\equiv (\\mathcal {P}^{\\prime \\prime })^{|\\mathcal {P}^{\\prime \\prime }_1| / t}$ and ${\\mathcal {P}^{\\prime \\prime }_2} \\curvearrowleft {s} \\equiv (\\mathcal {P}^{\\prime \\prime })^{|\\mathcal {P}^{\\prime \\prime }_2| / t}$ with $s = (|\\mathcal {P}^{\\prime }_1| - |\\mathcal {P}^{\\prime }_2|) \\bmod |\\mathcal {P}^{\\prime \\prime }_2|$ .", "It is an easy exercise to show that $L(\\mathcal {A}_1) = L(\\mathcal {A}_2)$ iff all the checks are successful.", "This completes the proof.", "$\\Box $" ], [ "Inclusion", "A natural idea for handling the inclusion problem for udpda would be to extend the result of Theorem REF , that is, to tackle inclusion similarly to equivalence.", "This raises the problem of comparing the words generated by two SLPs in the componentwise sense with respect to the order $0 \\le 1$ .", "To the best of our knowledge, this problem has not been studied previously, so we deal with it separately.", "As it turns out, here one cannot hope for an efficient algorithm unless $\\mathbf {P}= \\mathbf {NP}$ .", "Let us define the following family of problems, parameterized by partial order $R$ on the alphabet of size at least 2, and denoted $\\textsc {SLP-Compo\\-nent\\-wise-$ R$}$ .", "The input is a pair of SLPs $\\mathcal {P}_1$ , $\\mathcal {P}_2$ over an alphabet partially ordered by $R$ , generating words of equal length.", "The output is “yes” iff for all $i$ , $0 \\le i < |\\mathcal {P}_1|$ , the relation $R(\\mathcal {P}_1[i], \\mathcal {P}_2[i])$ holds.", "By SLP-Componentwise-${(0 \\le 1)}$ we mean a special case of this problem where $R$ is the partial order on $\\lbrace 0, 1\\rbrace $ given by $0 \\le 0$ , $0 \\le 1$ , $1 \\le 1$ .", "$\\textsc {SLP-Compo\\-nent\\-wise-$ R$}$ is $\\mathbf {coNP}$ -complete if $R$ is not the equality relation (that is, if $R(a, b)$ holds for some $a \\ne b$ ), and in $\\mathbf {P}$ otherwise.", "We first prove that SLP-Componentwise-${(0 \\le 1)}$ is $\\mathbf {coNP}$ -hard.", "We show a reduction from the complement of Subset-Sum: suppose we start with an instance of Subset-Sum containing a vector of naturals $w = (w_1, \\ldots , w_n)$ and a natural $t$ , and the question is whether there exists a vector $x = (x_1, \\ldots , x_n) \\in \\lbrace 0, 1\\rbrace ^n$ such that $x \\cdot w = t$ , where $x \\cdot w$ is defined as the inner product $\\sum _{i = 1}^{n} x_i w_i$ .", "Let $s = (1, \\ldots , 1) \\cdot w$ be the sum of all components of $w$ .", "We use the construction of so-called Lohrey words.", "Lohrey shows [22] that given such an instance, it is possible to construct in logarithmic space two SLPs that generate words $W_1 = \\prod _{x \\in \\lbrace 0, 1\\rbrace ^n} a^{x \\cdot w} b a^{s - x \\cdot w}$ and $W_2 = (a^{t} b a^{s - t})^{2^n}$ , where the product in $W_1$ enumerates the $x$ es in the lexicographic order.", "Now $W_1$ and $W_2$ share a symbol $b$ in some position iff the original instance of Subset-Sum is a yes-instance.", "Substitute 0 for $a$ and 1 for $b$ in the first SLP, and 0 for $b$ and 1 for $a$ in the second SLP.", "The new SLPs $\\mathcal {P}_1$ and $\\mathcal {P}_2$ obtained in this way form a no-instance of SLP-Componentwise-${(0 \\le 1)}$ iff the original instance of Subset-Sum is a yes-instance, because now the “distinguished” pair of symbols consists of a 1 in $\\mathcal {P}_1$ and 0 in $\\mathcal {P}_2$ .", "Therefore, SLP-Componentwise-${(0 \\le 1)}$ is $\\mathbf {coNP}$ -hard.", "Now observe that, for any $R$ , membership of $\\textsc {SLP-Compo\\-nent\\-wise-$ R$}$ in $\\mathbf {coNP}$ is obvious, and the hardness is by a simple reduction from SLP-Componentwise-${(0 \\le 1)}$: just substitute $a$ for 0 and $b$ for 1 (recall that by the definition of partial order, $R(b, a)$ would entail $a = b$ , which is false).", "In the last special case in the statement, $R$ is just the equality relation, so deciding $\\textsc {SLP-Compo\\-nent\\-wise-$ R$}$ is the same as deciding SLP-Equivalence, which is in $\\mathbf {P}$ (see Section ).", "This concludes the proof.", "$\\Box $ A corollary of Theorem REF on a problem of matching for compressed partial words is demonstrated in Section .", "An alternative reduction showing hardness of SLP-Componentwise-${(0 \\le 1)}$, this time from $\\textsc {Circuit-SAT}$ , but also making use of Subset-Sum and Lohrey words, can be derived from Bertoni, Choffrut, and Radicioni [3].", "They show that for any Boolean circuit with NAND-gates there exists a pair of straight-line programs $\\mathcal {P}_1$ , $\\mathcal {P}_2$ generating words over $\\lbrace 0, 1\\rbrace $ of the same length with the following property: the function computed by the circuit takes on the value 1 on at least one input combination iff both words share a 1 at some position.", "Moreover, these two SLPs can be constructed in polynomial time.", "As a result, after flipping all terminal symbols in the second of these SLPs, the resulting pair is a no-instance of SLP-Componentwise-${(0 \\le 1)}$ iff the original circuit is satisfiable.", "UDPDA-Inclusion is $\\mathbf {coNP}$ -complete.", "First combine Theorem REF with part REF of Theorem  to prove hardness.", "Indeed, Theorem REF shows that SLP-Componentwise-${(0 \\le 1)}$ is $\\mathbf {coNP}$ -hard.", "Take an instance with two SLPs $\\mathcal {P}_1$ , $\\mathcal {P}_2$ and transform indicator pairs $(\\mathcal {P}_1, \\mathcal {P}_0)$ and $(\\mathcal {P}_2, \\mathcal {P}_0)$ , with $\\mathcal {P}_0$ any fixed SLP over $\\lbrace 0, 1\\rbrace $ , into udpda $\\mathcal {A}_1$ , $\\mathcal {A}_2$ with the help of part REF of Theorem .", "Now the characteristic sequence of $L(\\mathcal {A}_i)$ , $i = 1, 2$ , is equal to $\\mathrm {eval}(\\mathcal {P}_i) \\cdot (\\mathrm {eval}(\\mathcal {P}_0))^\\omega $ .", "As a result, it holds that $\\mathrm {eval}(\\mathcal {P}_1) \\le \\mathrm {eval}(\\mathcal {P}_2)$ in the componentwise sense if and only if $L(\\mathcal {A}_1) \\subseteq L(\\mathcal {A}_2)$ .", "This concludes the hardness proof.", "It remains to show that UDPDA-Inclusion is in $\\mathbf {coNP}$ .", "First note that for any udpda $\\mathcal {A}$ there exists a deterministic pushdown automaton (DFA) that accepts $L(\\mathcal {A})$ and has size at most $2^{O(m)}$ , where $m$ is the size of $\\mathcal {A}$ (see discussion in Section  or Pighizzini [24]).", "Therefore, if $L(\\mathcal {A}_1) \\lnot \\subseteq L(\\mathcal {A}_2)$ , then there exists a witness $a^n \\in L(\\mathcal {A}_2) \\setminus L(\\mathcal {A}_1)$ with $n$ at most exponential in the size of $\\mathcal {A}_1$ and $\\mathcal {A}_2$ .", "By Theorem REF , compressed membership is in $\\mathbf {P}$ , so this completes the proof.", "$\\Box $" ], [ "Proof of Theorem ", "Let us first recall some standard definitions and fix notation.", "In a udpda $\\mathcal {A}$ , if $(q_1, s_1) \\vdash _\\sigma \\!", "(q_2, s_2)$ for some $\\sigma $ , we also write $(q_1, s_1) \\vdash (q_2, s_2)$ .", "A computation of a udpda $\\mathcal {A}$ starting at a configuration $(q, s)$ is defined as a (finite or infinite) sequence of configurations $(q_i, s_i)$ with $(q_1, s_1) = (q, s)$ and, for all $i$ , $(q_i, s_i) \\vdash _{\\sigma _i} \\!", "(q_{i + 1}, s_{i + 1})$ for some $\\sigma _i$ .", "If the sequence is finite and ends with $(q_k, s_k)$ , we also write $(q_1, s_1) \\vdash ^*_w \\!", "(q_k, s_k)$ , where $w = \\sigma _1 \\ldots \\sigma _{k - 1} \\in \\lbrace a\\rbrace ^*$ .", "We can also omit the word $w$ when it is not important and say that $(q_k, s_k)$ is reachable from $(q_1, s_1)$ ; in other words, the reachability relation $\\vdash ^*$ is the reflexive and transitive closure of the move relation $\\vdash $ ." ], [ "From indicator pairs to udpda", "Going from indicator pairs to udpda is the easier direction in Theorem .", "We start with an auxiliary lemma that enables one to model a single SLP with a udpda.", "This lemma on its own is already sufficient for lower bounds of Theorem REF and Theorem REF in Section .", "There exists an algorithm that works in logarithmic space and transforms an arbitrary SLP $\\mathcal {P}$ of size $m$ over $\\lbrace 0, 1\\rbrace $ into a udpda $\\mathcal {A}$ of size $O(m)$ over $\\lbrace a\\rbrace $ such that the characteristic sequence of $L(\\mathcal {A})$ is $0 \\cdot \\mathrm {eval}(\\mathcal {P})\\cdot 0^\\omega $ .", "In $\\mathcal {A}$ , it holds that $(q_0, \\bot ) \\vdash ^*_w \\!", "(\\bar{q}_0, \\bot )$ for $w = a^{|\\mathrm {eval}(\\mathcal {P})|}$ , $q_0$ the initial state, and $\\bar{q}_0$ a non-final state without outgoing transitions.", "The main part of the algorithm works as follows.", "Assume that $\\mathcal {P}$ is given in Chomsky normal form.", "With each nonterminal $N$ we associate a gadget in the udpda $\\mathcal {A}$ , whose interface is by definition the entry state $q_N$ and the exit state $\\bar{q}_N$ , which will only have outgoing pop transitions.", "With a production of the form $N \\rightarrow \\sigma $ , $\\sigma \\in \\lbrace 0, 1\\rbrace $ , we associate a single internal transition from $q_N$ to $\\bar{q}_N$ reading an $a$ from the input tape.", "The state $q_N$ is always non-final, and the state $\\bar{q}_N$ is final if and only if $\\sigma = 1$ .", "With a production of the form $N \\rightarrow A B$ we associate two stack symbols $\\gamma _N^1$ , $\\gamma _N^2$ and the following gadget.", "At a state $q_N$ , the udpda pushes a symbol $\\gamma _N^1$ onto the stack and goes to the state $q_A$ .", "We add a pop transition from $\\bar{q}_A$ that reads $\\gamma _N^1$ from the stack and leads to an auxiliary state $q^{\\prime }_N$ .", "The only transition from this state pushes $\\gamma _N^2$ and leads to $q_B$ , and another transition from $\\bar{q}_B$ pops $\\gamma _N^2$ and goes to $\\bar{q}_N$ .", "Here all three states $q_N$ , $q^{\\prime }_N$ , and $\\bar{q}_N$ are non-final, and the four introduced incident transitions do not read from the input.", "Finally, if a nonterminal $N$ is the axiom of $\\mathcal {P}$ , make the state $q_N$ initial and non-final and make $\\bar{q}_N$ a non-accepting sink that reads $a$ from the input and pops $\\bot $ .", "The reader can easily check that the characteristic sequence of the udpda $\\mathcal {A}$ constructed in this way is indeed $0 \\cdot \\mathrm {eval}(\\mathcal {P})\\cdot 0^\\omega $ , and the construction can be performed in logarithmic space.", "Now note that while the udpda $\\mathcal {A}$ satisfies $|Q| = O(m)$ , we may have also introduced up to 2 stack symbols per nonterminal.", "Therefore, the size of $\\mathcal {A}$ can be as large as $\\mathrm {\\Omega }(m^2)$ .", "However, we can use a standard trick from circuit complexity to avoid this blowup and make this size linear in $m$ .", "Indeed, first observe that the number of stack symbols, not counting $\\bot $ , in the construction above can be reduced to $k$ , the maximum, over all nonterminals $N$ , of the number of occurrences of $N$ in the right-hand sides of productions of $\\mathcal {P}$ .", "Second, recall that a straight-line program naturally defines a circuit where productions of the form $N \\rightarrow A B$ correspond to gates performing concatenation.", "The value of $k$ is the maximum fan-out of gates in this circuit, and it is well-known how to reduce it to $O(1)$ with just a constant-factor increase in the number of gates (see, e. g., Savage [26]).", "The construction can be easily performed in logarithmic space, and the only building block is the identity gate, which in our case translates to a production of the form $N \\rightarrow A$ .", "Although such productions are not allowed in Chomsky normal form, the construction above can be adjusted accordingly, in a straightforward fashion.", "This completes the proof.", "$\\Box $ Now, to model an entire indicator pair, we apply Lemma REF twice and combine the results.", "There exists an algorithm that works in logarithmic space and, given an indicator pair $(\\mathcal {P}^{\\prime }, \\mathcal {P}^{\\prime \\prime })$ of size $m$ for some unary language $L \\subseteq \\lbrace a\\rbrace ^*$ , outputs a udpda $\\mathcal {A}$ of size $O(m)$ such that $L(\\mathcal {A}) = L$ .", "We shall use the same notation as in Subsection REF of Section .", "First compute the bit $b = \\mathcal {P}^{\\prime }[0]$ and construct an SLP $\\mathcal {P}^{\\prime }_1$ of size $O(m)$ such that $\\mathrm {eval}(\\mathcal {P}^{\\prime })= b \\cdot \\mathrm {eval}(\\mathcal {P}^{\\prime }_1)$ .", "Note that this can be done in logarithmic space, even though the general SLP-Query problem is $\\mathbf {P}$ -complete.", "Now construct, according to Lemma REF , two udpda $\\mathcal {A}^{\\prime }$ and $\\mathcal {A}^{\\prime \\prime }$ for $\\mathcal {P}^{\\prime }_1$ and $\\mathcal {P}^{\\prime \\prime }$ , respectively.", "Assume that their sets of control states are disjoint and connect them in the following way.", "Add internal $\\varepsilon $ -transitions from the “last” states of both to the initial state of $\\mathcal {A}^{\\prime \\prime }$ .", "Now make the initial state of $\\mathcal {A}^{\\prime }$ the initial state of $\\mathcal {A}$ ; make it also a final state if $b = 1$ .", "It is easily checked that the language of the udpda $\\mathcal {A}$ constructed in this way has characteristic sequence $\\mathrm {eval}(\\mathcal {P}^{\\prime })\\cdot (\\mathrm {eval}(\\mathcal {P}^{\\prime \\prime }))^\\omega $ and, hence, is equal to $L$ .", "$\\Box $ Lemma REF proves part REF in Theorem ." ], [ "From udpda to indicator pairs", "Going from udpda to indicator pairs is the main part of Theorem , and in this subsection we describe our construction in detail.", "The proof of the key technical lemma is deferred until the following Subsection REF ." ], [ "Assumptions and notation.", "We assume without loss of generality that the given udpda $\\mathcal {A}$ satisfies the following conditions.", "First, its set of control states, $Q$ , is partitioned into three subsets according to the type of available moves.", "More precisely, we assumeHere and further in the text we use the symbol $\\sqcup $ to denote the union of disjoint sets.", "$Q = Q_{0}\\sqcup Q_{+1}\\sqcup Q_{-1}$ with the property that all transitions $(q, \\sigma , \\gamma , q^{\\prime }, s)$ with states $q$ from $Q_d$ , $d \\in \\lbrace 0, -1, +1\\rbrace $ , have $|s| = 1 + d$ ; moreover, we assume that $s = \\gamma $ whenever $d = 0$ , and $s = \\gamma ^{\\prime } \\gamma $ for some $\\gamma ^{\\prime } \\in \\mathrm {\\Gamma }$ whenever $d = 1$ .", "Second, for convenience of notation we assume that there exists a subset $R \\subseteq Q$ such that all transitions departing from states from $R$ read a symbol from the input tape, and transitions departing from $Q \\setminus R$ do not.", "Third, we assume that $\\delta $ is specified by means of total functions $\\delta _{0}\\colon Q_{0}\\rightarrow Q$ , $\\delta _{+1}\\colon Q_{+1}\\rightarrow Q \\times \\mathrm {\\Gamma }$ , and $\\delta _{-1}\\colon Q_{-1}\\times \\mathrm {\\Gamma }\\rightarrow Q$ .", "We write $\\delta _{0}(q)=q^{\\prime }$ , $\\delta _{+1}(q)=(q^{\\prime },\\gamma )$ , and $\\delta _{-1}(q,\\gamma )=q^{\\prime }$ accordingly; associated transitions and states are called internal, push, and pop transitions and states, respectively.", "Note that this assumption implies that only pop transitions can “look” at the top of the stack.", "An arbitrary udpda $\\mathcal {A}^{\\prime } = (Q^{\\prime }, \\mathrm {\\Gamma }, \\bot , q^{\\prime }_0, F^{\\prime }, \\delta ^{\\prime })$ of size $m$ can be transformed into a udpda $\\mathcal {A}= (Q, \\mathrm {\\Gamma }, \\bot , q_0, F, \\delta )$ that accepts $L(\\mathcal {A}^{\\prime })$ , satisfies the assumptions of this subsubsection, and has $|Q| = O(m)$ control states.", "The proof is easy and left to the reader.", "Note that since $\\mathcal {A}$ is deterministic, it holds that for any configuration $(q, s)$ of $\\mathcal {A}$ there exists a unique infinite computation $(q_i, s_i)_{i = 0}^{\\infty }$ starting at $(q, s)$ , referred to as the computation in the text below.", "This computation can be thought of as a run of $\\mathcal {A}$ on an input tape with an infinite sequence $a^\\omega $ .", "The computation of $\\mathcal {A}$ is, naturally, the computation starting from $(q_0, \\bot )$ .", "Note that it is due to the fact that $\\mathcal {A}$ is unary that we are able to feed it a single infinite word instead of countably many finite words.", "In the text below we shall use the following notation and conventions.", "To refer to an SLP $(S, \\mathrm {\\Sigma }, \\mathrm {\\Delta }, \\pi )$ , we sometimes just use its axiom, $S$ .", "The generated word, $w$ , is denoted by $\\mathrm {eval}(S)$ as usual.", "Note that the set of terminals is often understood from the context and the set of nonterminals is always the set of left-hand sides of productions.", "This enables us to use the notation $\\mathrm {eval}(S)$ to refer to the word generated by the implicitly defined SLP, whenever the set of productions is clear from the context.", "Recall that our goal is to describe an algorithm that, given a udpda $\\mathcal {A}$ , produces an indicator pair for $L(\\mathcal {A})$ .", "We first assemble some tools that will allow us to handle the computation of $\\mathcal {A}$ per se.", "To this end, we introduce transcripts of computations, which record “events” that determine whether certain input words are accepted or rejected.", "Consider a (finite or infinite) computation that consists of moves $(q_i, s_i) \\vdash _{\\sigma _i} \\!", "(q_{i + 1}, s_{i + 1})$ , for $1 \\le i \\le k$ or for $i \\ge 1$ , respectively.", "We define the transcript of such a computation as a (finite or infinite) sequence $\\mu (q_1) \\, \\sigma _1 \\, \\mu (q_2) \\, \\sigma _2 \\, \\ldots \\, \\mu (q_k) \\, \\sigma _k\\quad \\text{or}\\quad \\mu (q_1) \\, \\sigma _1 \\, \\mu (q_2) \\, \\sigma _2 \\, \\ldots ,\\quad \\text{respectively,}$ where, for any $q_i$ , $\\mu (q_i) = f$ if $q_i \\in F$ and $\\mu (q_i) = \\varepsilon $ if $q_i \\in Q \\setminus F$ .", "Note that in the finite case the transcript does not include $\\mu (q_{k + 1})$ where $q_{k + 1}$ is the control state in the last configuration.", "In particular, if a computation consists of a single configuration, then its transcript is $\\varepsilon $ .", "In general, transcripts are finite words and infinite sequences over the auxiliary alphabet $\\lbrace a, f\\rbrace $ .", "The reader may notice that our definition for the finite case basically treats finite computations as left-closed, right-open intervals and lets us perform their concatenation in a natural way.", "We note, however, that from a technical point of view, a definition treating them as closed intervals would actually do just as well.", "Note that any sequence $s \\in \\lbrace a, f\\rbrace ^\\omega $ containing infinitely many occurrences of $a$ naturally defines a unique characteristic sequence $c \\in \\lbrace 0, 1\\rbrace ^\\omega $ such that if $s$ is the transcript of a udpda computation, then $c$ is the characteristic sequence of this udpda's language.", "The following lemma shows that this correspondence is efficient if the sequences are represented by pairs of SLPs.", "There exists a polynomial-time algorithm that, given a pair of straight-line programs $(\\mathcal {T}^{\\prime }, \\mathcal {T}^{\\prime \\prime })$ of size $m$ that generates a sequence $s \\in \\lbrace a, f\\rbrace ^\\omega $ and such that the symbol $a$ occurs in $\\mathrm {eval}(\\mathcal {T}^{\\prime \\prime })$ , produces a pair of straight-line programs $(\\mathcal {P}^{\\prime }, \\mathcal {P}^{\\prime \\prime })$ of size $O(m)$ that generates the characteristic sequence defined by $s$ .", "Observe that it suffices to apply to the sequence generated by $(\\mathcal {T}^{\\prime }, \\mathcal {T}^{\\prime \\prime })$ the composition of the following substitutions: $h_1 \\colon a f\\mapsto 1$ , $h_2 \\colon a \\mapsto 0$ , and $h_3 \\colon f\\mapsto \\varepsilon $ .", "One can easily see that applying $h_2$ and $h_3$ reduces to applying them to terminal symbols in SLPs, so it suffices to show that the application of $h_1$ can also be done in polynomial time and increases the number of productions in Chomsky normal form by at most a constant factor.", "We first show how to apply $h_1$ to a single SLP.", "Assume Chomsky normal form and process the productions of the SLP inductively in the bottom-up direction.", "Productions with terminal symbols remain unchanged, and productions of the form $N \\rightarrow A B$ are handled as follows: if $\\mathrm {eval}(A)$ ends with an $a$ and $\\mathrm {eval}(B)$ begins with an $f$ , then replace the production with $N \\rightarrow (A a^{-1}) \\cdot 1 \\cdot (f^{-1} B)$ , otherwise leave it unchanged as well.", "Here we use auxiliary nonterminals of the form $N a^{-1}$ and $f^{-1} N$ with the property that $\\mathrm {eval}(N a^{-1}) \\cdot a = \\mathrm {eval}(N)$ and $f\\cdot \\mathrm {eval}(f^{-1} N) = \\mathrm {eval}(N)$ .", "These nonterminals are easily defined inductively in a straightforward manner, just after processing $N$ .", "At the end of this process one obtains an SLP that generates the result of applying $h_1$ to the word generated by the original SLP.", "We now show how to handle the fact that we need to apply $h_1$ to the entire sequence $\\mathrm {eval}(\\mathcal {T}^{\\prime })\\cdot (\\mathrm {eval}(\\mathcal {T}^{\\prime \\prime }))^\\omega $ .", "Process the SLPs $\\mathcal {T}^{\\prime }$ and $\\mathcal {T}^{\\prime \\prime }$ as described above; for convenience, we shall use the same two names for the obtained programs.", "Then deal with the junction points in the sequence $\\mathrm {eval}(\\mathcal {T}^{\\prime })\\cdot (\\mathrm {eval}(\\mathcal {T}^{\\prime \\prime }))^\\omega $ as follows.", "If $\\mathrm {eval}(\\mathcal {T}^{\\prime \\prime })$ does not start with an $f$ , then there is nothing to do.", "Now suppose it does; then there are two options.", "The first option is that $\\mathrm {eval}(\\mathcal {T}^{\\prime \\prime })$ ends with an $a$ .", "In this case replace $\\mathcal {T}^{\\prime \\prime }$ with $(f^{-1} \\mathcal {T}^{\\prime \\prime }) \\cdot 1$ and $\\mathcal {T}^{\\prime }$ with $(\\mathcal {T}^{\\prime }a^{-1}) \\cdot 1$ or with $(\\mathcal {T}^{\\prime }f)$ according to whether it ends with an $a$ or not.", "The second option is that $\\mathcal {T}^{\\prime \\prime }$ does not end with an $a$ .", "In this case, if $\\mathcal {T}^{\\prime }$ ends with an $a$ , replace it with $(\\mathcal {T}^{\\prime }a^{-1}) \\cdot 1 \\cdot (f^{-1} \\mathcal {T}^{\\prime \\prime })$ , otherwise do nothing.", "One can easily see that the pair of SLPs obtained on this step will generate the image of the original sequence $\\mathrm {eval}(\\mathcal {T}^{\\prime })\\cdot (\\mathrm {eval}(\\mathcal {T}^{\\prime \\prime }))^\\omega $ under $h_1$ .", "This completes the proof.", "$\\Box $ Note that we could use a result by Bertoni, Choffrut, and Radicioni [3] and apply a four-state transducer (however, the underlying automaton needs to be $\\varepsilon $ -free, which would make us figure out the last position “manually”).", "Now it remains to show how to efficiently produce, given a udpda $\\mathcal {A}$ , a pair of SLPs $(\\mathcal {T}^{\\prime }, \\mathcal {T}^{\\prime \\prime })$ generating the transcript of the computation of $\\mathcal {A}$ .", "This is the key part of the entire algorithm, captured by the following lemma.", "There exists a polynomial-time algorithm that, given a udpda $\\mathcal {A}$ of size $m$ , produces a pair of straight-line programs $(\\mathcal {T}^{\\prime }, \\mathcal {T}^{\\prime \\prime })$ of size $O(m)$ that generates the transcript of the computation of $\\mathcal {A}$ .", "The proof of Lemma REF is given in the next subsection.", "Put together, Lemmas REF and REF prove the harder direction (that is, part REF ) of Theorem .", "The only caveat is that if $\\mathrm {eval}(\\mathcal {T}^{\\prime \\prime })\\in \\lbrace f\\rbrace ^*$ , then one needs to replace $\\mathcal {T}^{\\prime \\prime }$ with a simple SLP that generates $a$ and possibly adjust $\\mathcal {T}^{\\prime }$ so that $f$ be appended to the generated word.", "This corresponds to the case where $\\mathcal {A}$ does not read the entire input and enters an infinite loop of $\\varepsilon $ -moves (that is, moves that do not consume $a$ from the input).", "The main difficulty in proving Lemma REF lies in capturing the structure of a unary deterministic computation.", "To reason about such computations in a convenient manner, we introduce the following definitions.", "We say that a state $q$ is returning if it holds that $(q, \\bot ) \\vdash ^* (q^{\\prime }, \\bot )$ for some state $q^{\\prime } \\in Q_{-1}$ (recall that states from $Q_{-1}$ are pop states).", "In such a case the control state $q^{\\prime }$ of the first configuration of the form $(q^{\\prime }, \\bot )$ , $q^{\\prime } \\in Q_{-1}$ , occurring in the infinite computation starting from $(q, \\bot )$ is called the exit point of $q$ , and the computation between $(q, \\bot )$ and this $(q^{\\prime }, \\bot )$ the return segment from $q$ .", "For example, if $q \\in Q_{-1}$ , then $q$ is its own exit point, and the return segment from $q$ contains no moves.", "Intuitively, the exit point is the first control state in the computation where the bottom-of-the-stack symbol in the configuration $(q, \\bot )$ may matter.", "One can formally show that if $q^{\\prime }$ is the exit point of $q$ , then for any configuration $(q, s)$ it holds that $(q, s) \\vdash ^* (q^{\\prime }, s)$ and, moreover, the transcript of the return segment from $q$ is equal, for any $s$ , to the transcript of the shortest computation from $(q, s)$ to $(q^{\\prime }, s)$ .", "If a control state is not returning, it is called non-returning.", "For such a state $q$ , it holds that for every configuration $(q^{\\prime }, s^{\\prime })$ reachable from $(q, \\bot )$ either $s^{\\prime } \\ne \\bot $ or $q^{\\prime } \\notin Q_{-1}$ .", "One can show formally that infinite computations starting from configurations $(q, s)$ with a fixed non-returning state $q$ and arbitrary $s$ have identical transcripts and, therefore, identical characteristic sequences associated with them.", "As a result, we can talk about infinite computations starting at a non-returning control state $q$ , rather than in a specific configuration $(q, s)$ .", "Now consider a state $q \\notin Q_{-1}$ , an arbitrary configuration $(q, s)$ and the infinite computation starting from $(q, s)$ .", "Suppose that this computation enters a configuration of the form $(\\bar{q}, s)$ after at least one move.", "Then the horizontal successor of $q$ is defined as the control state $\\bar{q}$ of the first such configuration, and the computation between these configurations is called the horizontal segment from $q$ .", "In other cases, horizontal successor and horizontal segment are undefined.", "It is easily seen that the horizontal successor, whenever it exists, is well-defined in the sense that it does not depend upon the choice of $s \\in (\\mathrm {\\Gamma }\\setminus \\lbrace \\bot \\rbrace )^* \\bot $ .", "Similarly, the choice of $s$ only determines the “lower” part of the stack in the configurations of the horizontal segment; since we shall only be interested in the transcripts, this abuse of terminology is harmless.", "Equivalently, suppose that $q \\notin Q_{-1}$ and $(q, s) \\vdash (q^{\\prime }, s^{\\prime })$ .", "If $s^{\\prime } = s$ then the horizontal successor of $q$ is $q^{\\prime }$ .", "Otherwise it holds that $\\delta _{+1}(q)=(q^{\\prime },\\gamma )$ for some $\\gamma \\in \\mathrm {\\Gamma }$ , so that $s^{\\prime } = \\gamma s$ .", "Now if $q^{\\prime }$ is returning, $q^{\\prime \\prime }$ is the exit point of $q^{\\prime }$ , and $\\delta _{-1}(q^{\\prime \\prime },\\gamma )=\\bar{q}$ for the same $\\gamma $ , then $\\bar{q}$ is the horizontal successor of $q$ .", "The horizontal segment is in both cases defined as the shortest non-empty computation of the form $(q, s) \\vdash ^* (\\bar{q}, s)$ .", "Recall that our goal in this subsection is to define an algorithm that constructs a pair of straight-line programs $(\\mathcal {T}^{\\prime }, \\mathcal {T}^{\\prime \\prime })$ generating the transcript of the infinite computation of $\\mathcal {A}$ .", "The approach that we take is dynamic programming.", "We separate out intermediate goals of several kinds and construct, for an arbitrary control state $q \\in Q$ , SLPs and pairs of SLPs that generate transcripts of the infinite computation starting at $q$ (if $q$ is non-returning), of the return segment from $q$ (if $q$ is returning), and of the horizontal segment from $q$ (whenever it is defined).", "Our algorithm will write productions as it runs, always using, on their right-hand side, only terminal symbols from $\\lbrace a, f\\rbrace $ and nonterminals defined by productions written earlier.", "This enables us to use the notation $\\mathrm {eval}(A)$ for nonterminals $A$ without referring to a specific SLP.", "Once written, a production is never modified.", "The main data structures of the algorithm, apart from the productions it writes, are as follows: three partial functions $\\mathcal {E}, \\mathcal {H}, \\mathcal {W}\\colon Q \\rightarrow Q$ and a subset $\\mathrm {NonRet}\\subseteq Q$ .", "Associated with $\\mathcal {E}$ and $\\mathcal {H}$ are nonterminals $E_q$ and $H_q$ , and with $\\mathrm {NonRet}$ nonterminals $N^{\\prime }_q$ and $N^{\\prime \\prime }_q$ .", "Note that the partial functions from $Q$ to $Q$ can be thought of as digraphs on the set of vertices $Q$ .", "In such digraphs the outdegree of every vertex is at most 1.", "The algorithm will subsequently modify these partial functions, that is, add new edges and/or remove existing ones (however, the outdegree of no vertex will ever be increased to above 1).", "We can also promise that $\\mathcal {E}$ will only increase, i. e., its graph will only get new edges, $\\mathcal {W}$ will only decrease, and $\\mathcal {H}$ can go both ways.", "During its run the algorithm will maintain the following invariants: (I1) $Q = \\operatorname{dom}\\mathcal {E}\\sqcup \\operatorname{dom}\\mathcal {H}\\sqcup \\operatorname{dom}\\mathcal {W}\\sqcup \\mathrm {NonRet}$ , where $\\sqcup $ denotes union of disjoint sets.", "(I2) Whenever $\\mathcal {E}(q) = q^{\\prime }$ , it holds that $q$ is returning, $q^{\\prime }$ is the exit point of $q$ , and $\\mathrm {eval}(E_q)$ is the transcript of the return segment from $q$ .", "(I3) Whenever $\\mathcal {H}(q) = q^{\\prime }$ , it holds that $q^{\\prime }$ is the horizontal successor of $q$ and $\\mathrm {eval}(H_q)$ is the transcript of the horizontal segment from $q$ .", "(I4) Whenever $\\mathcal {W}(q) = q^{\\prime }$ , it holds that $\\delta _{+1}(q)=(q^{\\prime },\\gamma )$ for some $\\gamma \\in \\mathrm {\\Gamma }$ .", "(I5) Whenever $q \\in \\mathrm {NonRet}$ , it holds that $q$ is non-returning and the sequence $\\mathrm {eval}(N^{\\prime }_q) \\cdot (\\mathrm {eval}(N^{\\prime \\prime }_q))^\\omega $ is the transcript of the infinite computation starting at $q$ .", "Description of the algorithm: computing transcripts.", "Our algorithm has three stages: the initialization stage, the main stage, and the $\\bot $ -handling stage.", "The initialization stage of the algorithm works as follows: for each $q \\in Q$ , write $V_q \\rightarrow \\mu (q) \\sigma (q)$ , where $\\mu (q)$ is $f$ if $q \\in F$ and $\\varepsilon $ otherwise, and $\\sigma (q)$ is $a$ if $q \\in R$ (that is, if transitions departing from $q$ read a symbol from the input) and $\\varepsilon $ otherwise; for all $q \\in Q_{-1}$ , set $\\mathcal {E}(q) = q$ and write $E_q \\rightarrow \\varepsilon $ ; for all $q \\in Q_{0}$ , set $\\mathcal {H}(q) = q^{\\prime }$ where $\\delta _{0}(q)=q^{\\prime }$ and write $H_q \\rightarrow V_q$ ; for all $q \\in Q_{+1}$ , set $\\mathcal {W}(q) = q^{\\prime }$ where $\\delta _{+1}(q)=(q^{\\prime },\\gamma )$ for some $\\gamma \\in \\mathrm {\\Gamma }$ ; set $\\mathrm {NonRet}= \\emptyset $ .", "It is easy to see that in this way all invariants REF –REF are initially satisfied (recall that the transcript of an empty computation is $\\varepsilon $ ).", "For convenience, we also introduce two auxiliary objects: a partial function $\\mathcal {G}\\colon Q \\rightarrow Q$ and nonterminals $G_q$ , defined as follows.", "The domain of $\\mathcal {G}$ is $\\operatorname{dom}\\mathcal {G}= \\operatorname{dom}\\mathcal {H}\\sqcup \\operatorname{dom}\\mathcal {W}$ ; note that, according to invariant REF , this union is disjoint.", "We assign $\\mathcal {G}(q) = q^{\\prime }$ iff $\\mathcal {H}(q) = q^{\\prime }$ or $\\mathcal {W}(q) = q^{\\prime }$ .", "We shall assume that $\\mathcal {G}$ is recomputed as $\\mathcal {H}$ and $\\mathcal {W}$ change.", "Now for every $q \\in \\operatorname{dom}\\mathcal {G}$ , we let $G_q$ stand for $H_q$ if $q \\in \\operatorname{dom}\\mathcal {H}$ and for $V_q$ if $q \\in \\operatorname{dom}\\mathcal {W}$ .", "At this point we are ready to describe the main stage of the algorithm.", "During this stage, the algorithm applies the following rules until none of them is applicable (if at some point several rules can be applied, the choice is made arbitrarily; the rules are well-defined whenever invariants REF –REF hold): (R1) If $\\mathcal {G}(q) = q^{\\prime }$ where $q^{\\prime } \\in \\mathrm {NonRet}$ and $q \\in Q$ : remove $q$ from either $\\operatorname{dom}\\mathcal {H}$ or $\\operatorname{dom}\\mathcal {W}$ , add $q$ to $\\mathrm {NonRet}$ , write $N^{\\prime }_q \\rightarrow G_q N^{\\prime }_{q^{\\prime }}$ and $N^{\\prime \\prime }_q \\rightarrow N^{\\prime \\prime }_{q^{\\prime }}$ .", "(R2) If $\\mathcal {H}(q) = q^{\\prime }$ where $q^{\\prime } \\in \\operatorname{dom}\\mathcal {E}$ and $q \\in Q$ : remove $q$ from $\\operatorname{dom}\\mathcal {H}$ , define $\\mathcal {E}(q) = \\mathcal {E}(q^{\\prime })$ , write $E_q \\rightarrow H_q E_{q^{\\prime }}$ .", "(R3) If $\\mathcal {W}(q) = q^{\\prime }$ where $q^{\\prime } \\in \\operatorname{dom}\\mathcal {E}$ and $q \\in Q$ : remove $q$ from $\\operatorname{dom}\\mathcal {W}$ , define $\\mathcal {H}(q) = \\bar{q}$ where $\\mathcal {E}(q^{\\prime }) = q^{\\prime \\prime }$ , $\\delta _{-1}(q^{\\prime \\prime },\\gamma )=\\bar{q}$ , and $\\delta _{+1}(q)=(q^{\\prime },\\gamma )$ (that is, $\\gamma $ is the symbol pushed by the transition leaving $q$ , and $\\bar{q}$ is the state reached by popping $\\gamma $ at $q^{\\prime \\prime }$ , the exit point of $q^{\\prime }$ ).", "Finally, write $H_q \\rightarrow V_q E_{q^{\\prime }} V_{q^{\\prime \\prime }}$ .", "(R4) If $\\mathcal {G}$ contains a simple cycle, that is, if $\\mathcal {G}(q_i) = q_{i + 1}$ for $i = 1, \\ldots , k - 1$ and $\\mathcal {G}(q_k) = q_1$ , where $q_i \\ne q_j$ for $i \\ne j$ , then for each vertex $q_i$ of the cycle remove it from either $\\operatorname{dom}\\mathcal {H}$ or $\\operatorname{dom}\\mathcal {W}$ and add it to $\\mathrm {NonRet}$ ; in addition, write $N^{\\prime }_{q_k} \\rightarrow G_{q_k}$ , $N^{\\prime \\prime }_{q_k} \\rightarrow G_{q_1} \\ldots G_{q_k}$ , and, for each $i = 1, \\ldots , k - 1$ , $N^{\\prime }_{q_i} \\rightarrow G_{q_i} N^{\\prime }_{q_{i + 1}}$ and $N^{\\prime \\prime }_{q_i} \\rightarrow N^{\\prime \\prime }_{q_{i + 1}}$ .", "We shall need two basic facts about this stage of the algorithm.", "Application of rules REF –REF does not violate invariants REF –REF .", "The proof of Claim REF is easy and left to the reader.", "If no rule is applicable, then $\\operatorname{dom}\\mathcal {G}= \\emptyset $ .", "Suppose $\\operatorname{dom}\\mathcal {G}\\ne \\emptyset $ .", "Consider the graph associated with $\\mathcal {G}$ and observe that all vertices in $\\operatorname{dom}\\mathcal {G}$ have outdegree 1.", "This implies that $\\mathcal {G}$ has either a cycle within $\\operatorname{dom}\\mathcal {G}$ or an edge from $\\operatorname{dom}\\mathcal {G}$ to $Q \\setminus \\operatorname{dom}\\mathcal {G}$ .", "In the first case, rule REF is applicable.", "In the second case, we conclude with the help of the invariant REF that the edge leads from a vertex in $\\operatorname{dom}\\mathcal {H}\\sqcup \\operatorname{dom}\\mathcal {W}$ to a vertex in $\\mathrm {NonRet}\\sqcup \\operatorname{dom}\\mathcal {E}$ .", "If the destination is in $\\mathrm {NonRet}$ , then rule REF is applicable; otherwise the destination is in $\\operatorname{dom}\\mathcal {E}$ and one can apply rule REF or rule REF according to whether the source is in $\\operatorname{dom}\\mathcal {H}$ or in $\\operatorname{dom}\\mathcal {W}$ .", "$\\Box $ Now we are ready to describe the $\\bot $ -handling stage of the algorithm.", "By the beginning of this stage, the structure of deterministic computation of $\\mathcal {A}$ has already been almost completely captured by the productions written earlier, and it only remains to account for moves involving $\\bot $ .", "So this last stage of the algorithm takes the initial state $q_0$ of $\\mathcal {A}$ and proceeds as follows.", "If $q_0 \\in \\mathrm {NonRet}$ , then take $N^{\\prime }_{q_0}$ as the axiom of $\\mathcal {T}^{\\prime }$ and $N^{\\prime \\prime }_{q_0}$ as the axiom of $\\mathcal {T}^{\\prime \\prime }$ .", "By invariant REF , these nonterminals are defined and generate appropriate words, so the pair $(\\mathcal {T}^{\\prime }, \\mathcal {T}^{\\prime \\prime })$ indeed generates the transcript of the computation of $\\mathcal {A}$ .", "Since at the beginning of the $\\bot $ -handling stage $\\operatorname{dom}\\mathcal {G}= \\emptyset $ , it remains to consider the case $q_0 \\in \\operatorname{dom}\\mathcal {E}$ .", "Define a partial function $\\mathcal {E}^\\bot \\colon Q \\rightarrow Q$ by setting, for each $q \\in \\operatorname{dom}\\mathcal {E}$ , its value according to $\\mathcal {E}^\\bot (q) = \\bar{q}$ if $\\mathcal {E}(q) = q^{\\prime }$ and $\\delta _{-1}(q^{\\prime },\\bot )=\\bar{q}$ .", "Write productions $E^\\bot _q \\rightarrow E_q V_{q^{\\prime }}$ accordingly.", "Now associate $\\mathcal {E}^\\bot $ with a graph, as earlier, and consider the longest simple path within $\\operatorname{dom}\\mathcal {E}^\\bot $ starting at $q_0$ .", "Suppose it ends at a vertex $q_k$ , where $\\mathcal {E}^\\bot (q_i) = q_{i + 1}$ for $i = 0, \\ldots , k$ .", "There are two subcases here according to why the path cannot go any further.", "The first possible reason is that it reaches $Q \\setminus \\operatorname{dom}\\mathcal {E}^\\bot = \\mathrm {NonRet}$ , that is, that $q_{k + 1}$ belongs to $\\mathrm {NonRet}$ .", "In this subcase write $N^{\\prime }_{q_0} \\rightarrow E^\\bot _{q_0} \\ldots E^\\bot _{q_{k}} N^{\\prime }_{q_{k + 1}}$ and $N^{\\prime \\prime }_{q_0} \\rightarrow N^{\\prime \\prime }_{q_{k + 1}}$ .", "The second possible reason is that $q_{k + 1} = q_i$ where $0 \\le i \\le k$ .", "In this subcase write $N^{\\prime }_{q_0} \\rightarrow E^\\bot _{q_0} \\ldots E^\\bot _{q_{i - 1}}$ and $N^{\\prime \\prime }_{q_0} \\rightarrow E^\\bot _{q_i} \\ldots E^\\bot _{q_k}$ .", "In any of the two subcases above, take $N^{\\prime }_{q_0}$ and $N^{\\prime \\prime }_{q_0}$ as axioms of $\\mathcal {T}^{\\prime }$ and $\\mathcal {T}^{\\prime \\prime }$ , respectively.", "The correctness of this step follows easily from the invariants REF and REF .", "This gives a polynomial algorithm that converts a udpda $\\mathcal {A}$ into a pair of SLPs $(\\mathcal {T}^{\\prime }, \\mathcal {T}^{\\prime \\prime })$ that generates the transcript of the infinite computation of $\\mathcal {A}$ , and the only remaining bit is bounding the size of $(\\mathcal {T}^{\\prime }, \\mathcal {T}^{\\prime \\prime })$ .", "The size of $(\\mathcal {T}^{\\prime }, \\mathcal {T}^{\\prime \\prime })$ is $O(|Q|)$ .", "There are three types of nonterminals whose productions may have growing size: $N^{\\prime \\prime }_{q_k}$ in rule REF , and $N^{\\prime }_{q_0}$ and $N^{\\prime \\prime }_{q_0}$ in the $\\bot $ -handling stage.", "For all three types, the size is bounded by the cardinality of the set of states involved in a cycle or a path.", "Since such sets never intersect, all such nonterminals together contribute at most $|Q|$ productions to the Chomsky normal form.", "The contribution of other nonterminals is also $O(|Q|)$ , because they all have fixed size and each state $q$ is associated with a bounded number of them.", "$\\Box $ Combined with Claim REF in Subsection REF , this completes the proof of Lemma REF and Theorem .", "Universality of unpda In this section we settle the complexity status of the universality problem for unary, possibly nondeterministic pushdown automata.", "While $\\mathbf {\\mathrm {\\Pi }_2 P}$ -completeness of equivalence and inclusion is shown by Huynh [14], it has been unknown whether the universality problem is also $\\mathbf {\\mathrm {\\Pi }_2 P}$ -hard.", "For convenience of notation, we use an auxiliary descriptional system.", "Define integer expressions over the set of operations $\\lbrace +, \\cup , \\times 2, {} {\\times } \\mathbb {N}\\rbrace $ inductively: the base case is a non-negative integer $n$ , written in binary, and the inductive step is associated with binary operations $+$ , $\\cup $ , and unary operations $\\times 2$ , ${} {\\times } \\mathbb {N}$ .", "To each expression $E$ we associate a set of non-negative integers $S(E)$ : $S(n) = \\lbrace n \\rbrace $ , $S(E_1 + E_2) = \\lbrace s_1 + s_2 \\colon s_1 \\in S(E_1), s_2 \\in S(E_2) \\rbrace $ , $S(E_1 \\cup E_2) = S(E_1) \\cup S(E_2)$ , $S(E \\times 2) = S(E + E)$ , $S({E} {\\times } \\mathbb {N}) = \\lbrace s k \\colon s \\in S(E), k = 0, 1, 2, \\ldots \\,\\rbrace $ .", "Expressions $E_1$ and $E_2$ are called equivalent iff $S(E_1) = S(E_2)$ ; an expression $E$ is universal iff it is equivalent to ${1} {\\times } \\mathbb {N}$ .", "The problem of deciding universality is denoted by Integer-$\\lbrace +, \\cup , \\times 2, {} {\\times } \\mathbb {N}\\rbrace $ -Expression-Universality.", "Decision problems for integer expressions have been studied for more than 40 years: Stockmeyer and Meyer [31] showed that for expressions over $\\lbrace +,\\cup \\rbrace $ compressed membership is $\\mathbf {NP}$ -complete and equivalence is $\\mathbf {\\mathrm {\\Pi }_2 P}$ -complete (universality is, of course, trivial).", "For recent results on such problems with operations from $\\lbrace +, \\cup , \\cap , \\times , \\overline{\\phantom{x}}\\rbrace $ , see McKenzie and Wagner [23] and Glaßer et al. [8].", "Integer-$\\lbrace +, \\cup , \\times 2, {} {\\times } \\mathbb {N}\\rbrace $ -Expression-Universality is $\\mathbf {\\mathrm {\\Pi }_2 P}$ -hard.", "The reduction is from the Generalized-Subset-Sum problem, which is defined as follows.", "The input consists of two vectors of naturals, $u = (u_1, \\ldots , u_n)$ and $v = (v_1, \\ldots , v_m)$ , and a natural $t$ , and the problem is to decide whether for all $y \\in \\lbrace 0, 1\\rbrace ^m$ there exists an $x \\in \\lbrace 0, 1\\rbrace ^n$ such that $x \\cdot u + y \\cdot v = t$ , where the middle dot $\\cdot $ once again denotes the inner product.", "This problem was shown to be hard by Berman et al. [1].", "Start with an instance of Generalized-Subset-Sum and let $M$ be a big number, $M > \\sum _{i = 1}^{n} u_i + \\sum _{j = 1}^{m} v_j$ .", "Assume without loss of generality that $M > t$ .", "Consider the integer expression $E$ defined by the following equations: $E &= E^{\\prime } \\cup E^{\\prime \\prime }, \\\\E^{\\prime } &= (2^m M + {1} {\\times } \\mathbb {N}) \\cup ({M} {\\times } \\mathbb {N} + ([0, t - 1] \\cup [t + 1, M - 1])),\\\\E^{\\prime \\prime } &= \\sum _{j = 1}^{m} (0 \\cup (2^{j - 1} M + v_j)) +\\sum _{i = 1}^{n} (0 \\cup u_i), \\\\[a, b] &= a + [0, b - a], \\\\[0, t] &= [0, \\lfloor t/2 \\rfloor ] \\times 2+ (0 \\cup (t \\bmod 2)), \\\\[0, 1] &= 0 \\cup 1, \\\\[0, 0] &= 0.$ Note that the size of $E$ is polynomial in the size of the input, and $E$ can be constructed in logarithmic space.", "We show that $E$ is universal iff the input is a yes-instance of Generalized-Subset-Sum.", "It is immediate that $E$ is universal if and only if $S(E)$ contains $2^m$ numbers of the form $k M + t$ , $0 \\le k < 2^m$ .", "We show that every such number is in $S(E)$ if and only if for the binary vector $y = (y_1, \\ldots , y_m) \\in \\lbrace 0, 1\\rbrace ^m$ , defined by $k = \\sum _{j = 1}^{m} y_j \\, 2^{j - 1}$ , there exists a vector $x \\in \\lbrace 0, 1\\rbrace ^n$ such that $x \\cdot u + y \\cdot v = t$ .", "First consider an arbitrary $y \\in \\lbrace 0, 1\\rbrace ^m$ and choose $k$ as above.", "Suppose that for this $y$ there exists an $x \\in \\lbrace 0, 1\\rbrace ^n$ such that $x \\cdot u + y \\cdot v = t$ .", "One can easily see that appropriate choices in $E^{\\prime \\prime }$ give the number $k M + y \\cdot v + x \\cdot u = k M + t$ .", "Conversely, suppose that $k M + t \\in S(E)$ for some $k$ , $0 \\le k < 2^m$ ; then $k M + t \\in S(E^{\\prime \\prime })$ .", "Since $(1, \\ldots , 1) \\cdot u + (1, \\ldots , 1) \\cdot v < M$ , it holds that $t = y \\cdot v + x \\cdot u$ for binary vectors $y \\in \\lbrace 0, 1\\rbrace ^m$ and $x \\in \\lbrace 0, 1\\rbrace ^n$ that correspond to the choices in the addends.", "Moreover, the same inequality also shows that $k M$ is equal to the sum of some powers of two in the first sum in $E^{\\prime \\prime }$ , and so $k = \\sum _{j = 1}^{m} y_j \\, 2^{j - 1}$ .", "This completes the proof.", "$\\Box $ With circuits instead of formulae (see also [23] and [8]) we would not need doubling.", "Furthermore, we only use ${} {\\times } \\mathbb {N}$ on fixed numbers, so instead we could use any feature for expressing an arithmetic progression with fixed common difference.", "Unary-PDA-Universality is $\\mathbf {\\mathrm {\\Pi }_2 P}$ -complete.", "A reduction from Integer-$\\lbrace +, \\cup , \\times 2, {} {\\times } \\mathbb {N}\\rbrace $ -Expression-Universality, which is $\\mathbf {\\mathrm {\\Pi }_2 P}$ -hard by Lemma , shows hardness.", "Indeed, an integer expression over $\\lbrace +, \\cup , \\times 2, {} {\\times } \\mathbb {N}\\rbrace $ can be transformed into a unary CFG in a straightforward way.", "Binary numbers are encoded by poly-size SLPs, summation is modeled by concatenation, and union by alternatives.", "Doubling is a special case of summation, and ${} {\\times } \\mathbb {N}$ gives rise to productions of the form $N^{\\prime } \\rightarrow \\varepsilon $ and $N^{\\prime } \\rightarrow N N^{\\prime }$ .", "The obtained CFG is then transformed into a unary PDA $\\mathcal {A}$ by a standard algorithm (see, e. g., Savage [26]).", "The result is that $L(\\mathcal {A}) = \\lbrace 1^s \\colon s \\in S(E) \\rbrace $ , and $\\mathcal {A}$ is computed from $E$ in logarithmic space.", "This concludes the proof.", "$\\Box $ We give a simple proof of the $\\mathbf {\\mathrm {\\Pi }_2 P}$ upper bound.", "Let $\\varphi _{\\mathcal {A}}(x)$ be an existential Presburger formula of size polynomial in the size of $\\mathcal {A}$ that characterizes the Parikh image of $L(\\mathcal {A})$ (see Verma, Seidl, and Schwentick [32]).", "To show that an udpda $\\mathcal {A}$ is non-universal, we find an $n\\ge 0$ such that $\\lnot \\varphi _{\\mathcal {A}}(n)$ holds.", "Now we note that for any udpda $\\mathcal {A}$ of size $m$ , there is a deterministic finite automaton of size $2^{O(m)}$ accepting $L(\\mathcal {A})$ (see discussion in Section  and Pighizzini [24]).", "Thus, $n$ is bounded by $2^{O(m)}$ .", "Therefore, checking non-universality can be expressed as a predicate: $\\exists n \\le 2^{O(m)}.\\lnot \\varphi _{\\mathcal {A}}(n)$ .", "This is a $\\mathbf {\\mathrm {\\Sigma }_2 P}$ -predicate, because the $\\exists ^*$ -fragment of Presburger arithmetic is $\\mathbf {NP}$ -complete [33].", "Universality, equivalence, and inclusion are $\\mathbf {\\mathrm {\\Pi }_2 P}$ -complete for (possibly nondeterministic) unary pushdown automata, unary context-free grammars, and integer expressions over $\\lbrace +, \\cup , \\times 2, {} {\\times } \\mathbb {N}\\rbrace $ .", "Another consequence of Theorem  is that deciding equality of a (not necessarily unary) context-free language, given as a context-free grammar, to any fixed context-free language $L_0$ that contains an infinite regular subset, is $\\mathbf {\\mathrm {\\Pi }_2 P}$ -hard and, if $L_0 \\subseteq \\lbrace a\\rbrace ^*$ , $\\mathbf {\\mathrm {\\Pi }_2 P}$ -complete.", "The lower bound is by reduction due to Hunt III, Rosenkrantz, and Szymanski [12], who show that deciding equivalence to $\\lbrace a\\rbrace ^*$ reduces to deciding equivalence to any such $L_0$ .", "The reduction is shown to be polynomial-time, but is easily seen to be logarithmic-space as well.", "The upper bound for the unary case is by Huynh [14]; in the general case, the problem can be undecidable.", "Corollaries and discussion Descriptional complexity aspects of udpda.", "Theorem  can be used to obtain several results on descriptional complexity aspects of udpda proved earlier by Pighizzini [24].", "He shows how to transform a udpda of size $m$ into an equivalent deterministic finite automaton (DFA) with at most $2^m$ states [24] and into an equivalent context-free grammar in Chomsky normal form (CNF) with at most $2 m + 1$ nonterminals [24].", "In our construction $m$ gets multiplied by a small constant, but the advantage is that we now see (the slightly weaker variants of) these results as easy corollaries of a single underlying theorem.", "Indeed, using an indicator pair $(\\mathcal {P}^{\\prime }, \\mathcal {P}^{\\prime \\prime })$ for $L$ , it is straightforward to construct a DFA of size $|\\mathrm {eval}(\\mathcal {P}^{\\prime })| + |\\mathrm {eval}(\\mathcal {P}^{\\prime \\prime })|$ accepting $L$ , as well as to transform the pair into a CFG in CNF that generates $L$ and has at most thrice the size of $(\\mathcal {P}^{\\prime }, \\mathcal {P}^{\\prime \\prime })$ .", "Another result which follows, even more directly, from ours is a lower bound on the size of udpda accepting a specific language $L_1$  [24].", "To obtain this lower bound, Pighizzini employs a known lower bound on the SLP-size of the word $W = W\\!", "[0] \\ldots W\\!", "[K - 1] \\in \\lbrace 0, 1\\rbrace ^K$ such that $a^n \\in L_1$ iff $W\\!", "[n \\bmod K] = 1$ .", "To this end, a udpda $\\mathcal {A}$ accepting $L_1$ is intersected (we are glossing over some technicalities here) with a small deterministic finite automaton that “captures” the end of the word $W$ .", "The obtained udpda, which only accepts $a^K$ , is transformed into an equivalent context-free grammar.", "It is then possible to use the structure of the grammar to transform it into an SLP that produces $W$ (note that such a transformation in general is $\\mathbf {NP}$ -hard).", "While the proof produces from a udpda for $L_1$ a related SLP with a polynomial blowup, this construction depends crucially on the structure of the language $L_1$ , so it is difficult to generalize the argument to all udpda and thus obtain Theorem .", "Our proof of Theorem  therefore follows a very different path.", "Relationship to Presburger arithmetic.", "An alternative way to prove the upper bound in Theorem REF is via Presburger arithmetic, using the observation that there is a poly-time computable existential Presburger formula that expresses the membership of a word $a^n$ in $L(\\lnot \\mathcal {A}_1)$ and $L(\\mathcal {A}_2)$ .", "This technique distills the arguments used by Huynh [13], [14] to show that the compressed membership problem for unary pushdown automata is in $\\mathbf {NP}$ .", "It is used in a purified form by Plandowski and Rytter [25], who developed a much shorter proof of the same fact (apparently unaware of the previous proof).", "The same idea was later rediscovered and used in a combination with Presburger arithmetic by Verma, Seidl, and Schwentick [32].", "Another application of this technique provides an alternative proof of the $\\mathbf {\\mathrm {\\Pi }_2 P}$ upper bound for unpda inclusion (Theorem ): to show that $L(\\mathcal {A})$ is universal, we check that $L(\\mathcal {A})$ accepts all words up to length $2^{O(m)}$ (this bound is sufficient because there is a deterministic finite automaton for the language with this size—see the discussion above).", "The proof known to date, due to Huynh [14], involves reproving Parikh's theorem and is more than 10 pages long.", "Reduction to Presburger formulae produces a much simpler proof.", "Also, our $\\mathbf {\\mathrm {\\Pi }_2 P}$ -hardness result for unpda shows that the $\\forall _{\\mathrm {bounded}}\\,\\exists ^*$ -fragment of Presburger arithmetic is $\\mathbf {\\mathrm {\\Pi }_2 P}$ -complete, where the variable bound by the universal quantifier is at most exponential in the size of the formula.", "The upper bound holds because the $\\exists ^*$ -fragment is $\\mathbf {NP}$ -complete [33].", "In comparison, the $\\forall \\, \\exists ^*$ -fragment, without any restrictions on the domain of the universally quantified variable, requires co-nondeterministic $2^{n^{\\mathrm {\\Omega }(1)}}$ time, see Grädel [10].", "Previously known fragments that are complete for the second level of the polynomial hierarchy involve alternation depth 3 and a fixed number of quantifiers, as in Grädel [11] and Schöning [28].", "Also note that the $\\forall ^s \\, \\exists ^t$ -fragment is $\\mathbf {coNP}$ -complete for all fixed $s \\ge 1$ and $t \\ge 2$ , see Grädel [11].", "Problems involving compressed words.", "Recall Theorem REF : given two SLPs, it is $\\mathbf {coNP}$ -complete to compare the generated words componentwise with respect to any partial order different from equality.", "As a corollary, we get precise complexity bounds for SLP equivalence in the presence of wildcards or, equivalently, compressed matching in the model of partial words (see, e. g., Fischer and Paterson [6] and Berstel and Boasson [2]).", "Consider the problem SLP-Partial-Word-Matching: the input is a pair of SLPs $\\mathcal {P}_1$ , $\\mathcal {P}_2$ over the alphabet $\\lbrace a, b, \\text{\\texttt {?", "}}\\rbrace $ , generating words of equal length, and the output is “yes” iff for every $i$ , $0 \\le i < |\\mathcal {P}_1|$ , either $\\mathcal {P}_1[i] = \\mathcal {P}_2[i]$ or at least one of $\\mathcal {P}_1[i]$ and $\\mathcal {P}_2[i]$ is $\\text{\\texttt {?", "}}$ (a hole, or a single-character wildcard).", "Schmidt-Schauß [27] defines a problem equivalent to SLP-Partial-Word-Matching, along with another related problem, where one needs to find occurrences of $\\mathrm {eval}(\\mathcal {P}_1)$ in $\\mathrm {eval}(\\mathcal {P}_2)$ (as in pattern matching), $\\mathcal {P}_2$ is known to contain no holes, and two symbols match iff they are equal or at least one of them is a hole.", "For this related problem, he develops a polynomial-time algorithm that finds (a representation of) all matching occurrences and operates under the assumption that the number of holes in $\\mathrm {eval}(\\mathcal {P}_1)$ is polynomial in the size of the input.", "He also points out that no solution for (the general case of) SLP-Partial-Word-Matching is known—unless a polynomial upper bound on the number of $\\text{\\texttt {?", "}}$ s in $\\mathrm {eval}(\\mathcal {P}_1)$ and $\\mathrm {eval}(\\mathcal {P}_2)$ is given.", "Our next proposition shows that such a solution is not possible unless $\\mathbf {P}= \\mathbf {NP}$ .", "It is an easy consequence of Theorem REF .", "SLP-Partial-Word-Matching is $\\mathbf {coNP}$ -complete.", "Membership in $\\mathbf {coNP}$ is obvious, and the hardness is by a reduction from SLP-Componentwise-${(0 \\le 1)}$.", "Given a pair of SLPs $\\mathcal {P}_1$ , $\\mathcal {P}_2$ over $\\lbrace 0, 1\\rbrace $ , substitute $\\text{\\texttt {?", "}}$ for 0 and $a$ for 1 in $\\mathcal {P}_1$ , and $b$ for 0 and $\\text{\\texttt {?", "}}$ for 1 in $\\mathcal {P}_2$ .", "The resulting pair of SLPs over $\\lbrace a, b, \\text{\\texttt {?", "}}\\rbrace $ is a yes-instance of SLP-Partial-Word-Matching iff the original pair is a yes-instance of SLP-Componentwise-${(0 \\le 1)}$.", "$\\Box $ The wide class of compressed membership problems (deciding $\\mathrm {eval}(\\mathcal {P})\\in L$ ) is studied and discussed in Jeż [16] and Lohrey [20].", "In the case of words over the unary alphabet, $w \\in \\lbrace a\\rbrace ^*$ , expressing $w$ with an SLP is poly-time equivalent to representing it with its length $|w|$ written in binary.", "An easy corollary of Theorem REF is that deciding $w \\in L(\\mathcal {A})$ , where $\\mathcal {A}$ is a (not necessarily unary) deterministic pushdown automaton and $w = a^n$ with $n$ given in binary, is $\\mathbf {P}$ -complete.", "Finally, we note that the precise complexity of SLP equivalence remains open [20].", "We cannot immediately apply lower bounds for udpda equivalence, since we do not know if the translation from udpda to indicator pairs in Theorem  can be implemented in logarithmic (or even polylogarithmic) space." ], [ "Universality of unpda", "In this section we settle the complexity status of the universality problem for unary, possibly nondeterministic pushdown automata.", "While $\\mathbf {\\mathrm {\\Pi }_2 P}$ -completeness of equivalence and inclusion is shown by Huynh [14], it has been unknown whether the universality problem is also $\\mathbf {\\mathrm {\\Pi }_2 P}$ -hard.", "For convenience of notation, we use an auxiliary descriptional system.", "Define integer expressions over the set of operations $\\lbrace +, \\cup , \\times 2, {} {\\times } \\mathbb {N}\\rbrace $ inductively: the base case is a non-negative integer $n$ , written in binary, and the inductive step is associated with binary operations $+$ , $\\cup $ , and unary operations $\\times 2$ , ${} {\\times } \\mathbb {N}$ .", "To each expression $E$ we associate a set of non-negative integers $S(E)$ : $S(n) = \\lbrace n \\rbrace $ , $S(E_1 + E_2) = \\lbrace s_1 + s_2 \\colon s_1 \\in S(E_1), s_2 \\in S(E_2) \\rbrace $ , $S(E_1 \\cup E_2) = S(E_1) \\cup S(E_2)$ , $S(E \\times 2) = S(E + E)$ , $S({E} {\\times } \\mathbb {N}) = \\lbrace s k \\colon s \\in S(E), k = 0, 1, 2, \\ldots \\,\\rbrace $ .", "Expressions $E_1$ and $E_2$ are called equivalent iff $S(E_1) = S(E_2)$ ; an expression $E$ is universal iff it is equivalent to ${1} {\\times } \\mathbb {N}$ .", "The problem of deciding universality is denoted by Integer-$\\lbrace +, \\cup , \\times 2, {} {\\times } \\mathbb {N}\\rbrace $ -Expression-Universality.", "Decision problems for integer expressions have been studied for more than 40 years: Stockmeyer and Meyer [31] showed that for expressions over $\\lbrace +,\\cup \\rbrace $ compressed membership is $\\mathbf {NP}$ -complete and equivalence is $\\mathbf {\\mathrm {\\Pi }_2 P}$ -complete (universality is, of course, trivial).", "For recent results on such problems with operations from $\\lbrace +, \\cup , \\cap , \\times , \\overline{\\phantom{x}}\\rbrace $ , see McKenzie and Wagner [23] and Glaßer et al. [8].", "Integer-$\\lbrace +, \\cup , \\times 2, {} {\\times } \\mathbb {N}\\rbrace $ -Expression-Universality is $\\mathbf {\\mathrm {\\Pi }_2 P}$ -hard.", "The reduction is from the Generalized-Subset-Sum problem, which is defined as follows.", "The input consists of two vectors of naturals, $u = (u_1, \\ldots , u_n)$ and $v = (v_1, \\ldots , v_m)$ , and a natural $t$ , and the problem is to decide whether for all $y \\in \\lbrace 0, 1\\rbrace ^m$ there exists an $x \\in \\lbrace 0, 1\\rbrace ^n$ such that $x \\cdot u + y \\cdot v = t$ , where the middle dot $\\cdot $ once again denotes the inner product.", "This problem was shown to be hard by Berman et al. [1].", "Start with an instance of Generalized-Subset-Sum and let $M$ be a big number, $M > \\sum _{i = 1}^{n} u_i + \\sum _{j = 1}^{m} v_j$ .", "Assume without loss of generality that $M > t$ .", "Consider the integer expression $E$ defined by the following equations: $E &= E^{\\prime } \\cup E^{\\prime \\prime }, \\\\E^{\\prime } &= (2^m M + {1} {\\times } \\mathbb {N}) \\cup ({M} {\\times } \\mathbb {N} + ([0, t - 1] \\cup [t + 1, M - 1])),\\\\E^{\\prime \\prime } &= \\sum _{j = 1}^{m} (0 \\cup (2^{j - 1} M + v_j)) +\\sum _{i = 1}^{n} (0 \\cup u_i), \\\\[a, b] &= a + [0, b - a], \\\\[0, t] &= [0, \\lfloor t/2 \\rfloor ] \\times 2+ (0 \\cup (t \\bmod 2)), \\\\[0, 1] &= 0 \\cup 1, \\\\[0, 0] &= 0.$ Note that the size of $E$ is polynomial in the size of the input, and $E$ can be constructed in logarithmic space.", "We show that $E$ is universal iff the input is a yes-instance of Generalized-Subset-Sum.", "It is immediate that $E$ is universal if and only if $S(E)$ contains $2^m$ numbers of the form $k M + t$ , $0 \\le k < 2^m$ .", "We show that every such number is in $S(E)$ if and only if for the binary vector $y = (y_1, \\ldots , y_m) \\in \\lbrace 0, 1\\rbrace ^m$ , defined by $k = \\sum _{j = 1}^{m} y_j \\, 2^{j - 1}$ , there exists a vector $x \\in \\lbrace 0, 1\\rbrace ^n$ such that $x \\cdot u + y \\cdot v = t$ .", "First consider an arbitrary $y \\in \\lbrace 0, 1\\rbrace ^m$ and choose $k$ as above.", "Suppose that for this $y$ there exists an $x \\in \\lbrace 0, 1\\rbrace ^n$ such that $x \\cdot u + y \\cdot v = t$ .", "One can easily see that appropriate choices in $E^{\\prime \\prime }$ give the number $k M + y \\cdot v + x \\cdot u = k M + t$ .", "Conversely, suppose that $k M + t \\in S(E)$ for some $k$ , $0 \\le k < 2^m$ ; then $k M + t \\in S(E^{\\prime \\prime })$ .", "Since $(1, \\ldots , 1) \\cdot u + (1, \\ldots , 1) \\cdot v < M$ , it holds that $t = y \\cdot v + x \\cdot u$ for binary vectors $y \\in \\lbrace 0, 1\\rbrace ^m$ and $x \\in \\lbrace 0, 1\\rbrace ^n$ that correspond to the choices in the addends.", "Moreover, the same inequality also shows that $k M$ is equal to the sum of some powers of two in the first sum in $E^{\\prime \\prime }$ , and so $k = \\sum _{j = 1}^{m} y_j \\, 2^{j - 1}$ .", "This completes the proof.", "$\\Box $ With circuits instead of formulae (see also [23] and [8]) we would not need doubling.", "Furthermore, we only use ${} {\\times } \\mathbb {N}$ on fixed numbers, so instead we could use any feature for expressing an arithmetic progression with fixed common difference.", "Unary-PDA-Universality is $\\mathbf {\\mathrm {\\Pi }_2 P}$ -complete.", "A reduction from Integer-$\\lbrace +, \\cup , \\times 2, {} {\\times } \\mathbb {N}\\rbrace $ -Expression-Universality, which is $\\mathbf {\\mathrm {\\Pi }_2 P}$ -hard by Lemma , shows hardness.", "Indeed, an integer expression over $\\lbrace +, \\cup , \\times 2, {} {\\times } \\mathbb {N}\\rbrace $ can be transformed into a unary CFG in a straightforward way.", "Binary numbers are encoded by poly-size SLPs, summation is modeled by concatenation, and union by alternatives.", "Doubling is a special case of summation, and ${} {\\times } \\mathbb {N}$ gives rise to productions of the form $N^{\\prime } \\rightarrow \\varepsilon $ and $N^{\\prime } \\rightarrow N N^{\\prime }$ .", "The obtained CFG is then transformed into a unary PDA $\\mathcal {A}$ by a standard algorithm (see, e. g., Savage [26]).", "The result is that $L(\\mathcal {A}) = \\lbrace 1^s \\colon s \\in S(E) \\rbrace $ , and $\\mathcal {A}$ is computed from $E$ in logarithmic space.", "This concludes the proof.", "$\\Box $ We give a simple proof of the $\\mathbf {\\mathrm {\\Pi }_2 P}$ upper bound.", "Let $\\varphi _{\\mathcal {A}}(x)$ be an existential Presburger formula of size polynomial in the size of $\\mathcal {A}$ that characterizes the Parikh image of $L(\\mathcal {A})$ (see Verma, Seidl, and Schwentick [32]).", "To show that an udpda $\\mathcal {A}$ is non-universal, we find an $n\\ge 0$ such that $\\lnot \\varphi _{\\mathcal {A}}(n)$ holds.", "Now we note that for any udpda $\\mathcal {A}$ of size $m$ , there is a deterministic finite automaton of size $2^{O(m)}$ accepting $L(\\mathcal {A})$ (see discussion in Section  and Pighizzini [24]).", "Thus, $n$ is bounded by $2^{O(m)}$ .", "Therefore, checking non-universality can be expressed as a predicate: $\\exists n \\le 2^{O(m)}.\\lnot \\varphi _{\\mathcal {A}}(n)$ .", "This is a $\\mathbf {\\mathrm {\\Sigma }_2 P}$ -predicate, because the $\\exists ^*$ -fragment of Presburger arithmetic is $\\mathbf {NP}$ -complete [33].", "Universality, equivalence, and inclusion are $\\mathbf {\\mathrm {\\Pi }_2 P}$ -complete for (possibly nondeterministic) unary pushdown automata, unary context-free grammars, and integer expressions over $\\lbrace +, \\cup , \\times 2, {} {\\times } \\mathbb {N}\\rbrace $ .", "Another consequence of Theorem  is that deciding equality of a (not necessarily unary) context-free language, given as a context-free grammar, to any fixed context-free language $L_0$ that contains an infinite regular subset, is $\\mathbf {\\mathrm {\\Pi }_2 P}$ -hard and, if $L_0 \\subseteq \\lbrace a\\rbrace ^*$ , $\\mathbf {\\mathrm {\\Pi }_2 P}$ -complete.", "The lower bound is by reduction due to Hunt III, Rosenkrantz, and Szymanski [12], who show that deciding equivalence to $\\lbrace a\\rbrace ^*$ reduces to deciding equivalence to any such $L_0$ .", "The reduction is shown to be polynomial-time, but is easily seen to be logarithmic-space as well.", "The upper bound for the unary case is by Huynh [14]; in the general case, the problem can be undecidable." ], [ "Descriptional complexity aspects of udpda.", "Theorem  can be used to obtain several results on descriptional complexity aspects of udpda proved earlier by Pighizzini [24].", "He shows how to transform a udpda of size $m$ into an equivalent deterministic finite automaton (DFA) with at most $2^m$ states [24] and into an equivalent context-free grammar in Chomsky normal form (CNF) with at most $2 m + 1$ nonterminals [24].", "In our construction $m$ gets multiplied by a small constant, but the advantage is that we now see (the slightly weaker variants of) these results as easy corollaries of a single underlying theorem.", "Indeed, using an indicator pair $(\\mathcal {P}^{\\prime }, \\mathcal {P}^{\\prime \\prime })$ for $L$ , it is straightforward to construct a DFA of size $|\\mathrm {eval}(\\mathcal {P}^{\\prime })| + |\\mathrm {eval}(\\mathcal {P}^{\\prime \\prime })|$ accepting $L$ , as well as to transform the pair into a CFG in CNF that generates $L$ and has at most thrice the size of $(\\mathcal {P}^{\\prime }, \\mathcal {P}^{\\prime \\prime })$ .", "Another result which follows, even more directly, from ours is a lower bound on the size of udpda accepting a specific language $L_1$  [24].", "To obtain this lower bound, Pighizzini employs a known lower bound on the SLP-size of the word $W = W\\!", "[0] \\ldots W\\!", "[K - 1] \\in \\lbrace 0, 1\\rbrace ^K$ such that $a^n \\in L_1$ iff $W\\!", "[n \\bmod K] = 1$ .", "To this end, a udpda $\\mathcal {A}$ accepting $L_1$ is intersected (we are glossing over some technicalities here) with a small deterministic finite automaton that “captures” the end of the word $W$ .", "The obtained udpda, which only accepts $a^K$ , is transformed into an equivalent context-free grammar.", "It is then possible to use the structure of the grammar to transform it into an SLP that produces $W$ (note that such a transformation in general is $\\mathbf {NP}$ -hard).", "While the proof produces from a udpda for $L_1$ a related SLP with a polynomial blowup, this construction depends crucially on the structure of the language $L_1$ , so it is difficult to generalize the argument to all udpda and thus obtain Theorem .", "Our proof of Theorem  therefore follows a very different path.", "An alternative way to prove the upper bound in Theorem REF is via Presburger arithmetic, using the observation that there is a poly-time computable existential Presburger formula that expresses the membership of a word $a^n$ in $L(\\lnot \\mathcal {A}_1)$ and $L(\\mathcal {A}_2)$ .", "This technique distills the arguments used by Huynh [13], [14] to show that the compressed membership problem for unary pushdown automata is in $\\mathbf {NP}$ .", "It is used in a purified form by Plandowski and Rytter [25], who developed a much shorter proof of the same fact (apparently unaware of the previous proof).", "The same idea was later rediscovered and used in a combination with Presburger arithmetic by Verma, Seidl, and Schwentick [32].", "Another application of this technique provides an alternative proof of the $\\mathbf {\\mathrm {\\Pi }_2 P}$ upper bound for unpda inclusion (Theorem ): to show that $L(\\mathcal {A})$ is universal, we check that $L(\\mathcal {A})$ accepts all words up to length $2^{O(m)}$ (this bound is sufficient because there is a deterministic finite automaton for the language with this size—see the discussion above).", "The proof known to date, due to Huynh [14], involves reproving Parikh's theorem and is more than 10 pages long.", "Reduction to Presburger formulae produces a much simpler proof.", "Also, our $\\mathbf {\\mathrm {\\Pi }_2 P}$ -hardness result for unpda shows that the $\\forall _{\\mathrm {bounded}}\\,\\exists ^*$ -fragment of Presburger arithmetic is $\\mathbf {\\mathrm {\\Pi }_2 P}$ -complete, where the variable bound by the universal quantifier is at most exponential in the size of the formula.", "The upper bound holds because the $\\exists ^*$ -fragment is $\\mathbf {NP}$ -complete [33].", "In comparison, the $\\forall \\, \\exists ^*$ -fragment, without any restrictions on the domain of the universally quantified variable, requires co-nondeterministic $2^{n^{\\mathrm {\\Omega }(1)}}$ time, see Grädel [10].", "Previously known fragments that are complete for the second level of the polynomial hierarchy involve alternation depth 3 and a fixed number of quantifiers, as in Grädel [11] and Schöning [28].", "Also note that the $\\forall ^s \\, \\exists ^t$ -fragment is $\\mathbf {coNP}$ -complete for all fixed $s \\ge 1$ and $t \\ge 2$ , see Grädel [11].", "Recall Theorem REF : given two SLPs, it is $\\mathbf {coNP}$ -complete to compare the generated words componentwise with respect to any partial order different from equality.", "As a corollary, we get precise complexity bounds for SLP equivalence in the presence of wildcards or, equivalently, compressed matching in the model of partial words (see, e. g., Fischer and Paterson [6] and Berstel and Boasson [2]).", "Consider the problem SLP-Partial-Word-Matching: the input is a pair of SLPs $\\mathcal {P}_1$ , $\\mathcal {P}_2$ over the alphabet $\\lbrace a, b, \\text{\\texttt {?", "}}\\rbrace $ , generating words of equal length, and the output is “yes” iff for every $i$ , $0 \\le i < |\\mathcal {P}_1|$ , either $\\mathcal {P}_1[i] = \\mathcal {P}_2[i]$ or at least one of $\\mathcal {P}_1[i]$ and $\\mathcal {P}_2[i]$ is $\\text{\\texttt {?", "}}$ (a hole, or a single-character wildcard).", "Schmidt-Schauß [27] defines a problem equivalent to SLP-Partial-Word-Matching, along with another related problem, where one needs to find occurrences of $\\mathrm {eval}(\\mathcal {P}_1)$ in $\\mathrm {eval}(\\mathcal {P}_2)$ (as in pattern matching), $\\mathcal {P}_2$ is known to contain no holes, and two symbols match iff they are equal or at least one of them is a hole.", "For this related problem, he develops a polynomial-time algorithm that finds (a representation of) all matching occurrences and operates under the assumption that the number of holes in $\\mathrm {eval}(\\mathcal {P}_1)$ is polynomial in the size of the input.", "He also points out that no solution for (the general case of) SLP-Partial-Word-Matching is known—unless a polynomial upper bound on the number of $\\text{\\texttt {?", "}}$ s in $\\mathrm {eval}(\\mathcal {P}_1)$ and $\\mathrm {eval}(\\mathcal {P}_2)$ is given.", "Our next proposition shows that such a solution is not possible unless $\\mathbf {P}= \\mathbf {NP}$ .", "It is an easy consequence of Theorem REF .", "SLP-Partial-Word-Matching is $\\mathbf {coNP}$ -complete.", "Membership in $\\mathbf {coNP}$ is obvious, and the hardness is by a reduction from SLP-Componentwise-${(0 \\le 1)}$.", "Given a pair of SLPs $\\mathcal {P}_1$ , $\\mathcal {P}_2$ over $\\lbrace 0, 1\\rbrace $ , substitute $\\text{\\texttt {?", "}}$ for 0 and $a$ for 1 in $\\mathcal {P}_1$ , and $b$ for 0 and $\\text{\\texttt {?", "}}$ for 1 in $\\mathcal {P}_2$ .", "The resulting pair of SLPs over $\\lbrace a, b, \\text{\\texttt {?", "}}\\rbrace $ is a yes-instance of SLP-Partial-Word-Matching iff the original pair is a yes-instance of SLP-Componentwise-${(0 \\le 1)}$.", "$\\Box $ The wide class of compressed membership problems (deciding $\\mathrm {eval}(\\mathcal {P})\\in L$ ) is studied and discussed in Jeż [16] and Lohrey [20].", "In the case of words over the unary alphabet, $w \\in \\lbrace a\\rbrace ^*$ , expressing $w$ with an SLP is poly-time equivalent to representing it with its length $|w|$ written in binary.", "An easy corollary of Theorem REF is that deciding $w \\in L(\\mathcal {A})$ , where $\\mathcal {A}$ is a (not necessarily unary) deterministic pushdown automaton and $w = a^n$ with $n$ given in binary, is $\\mathbf {P}$ -complete.", "Finally, we note that the precise complexity of SLP equivalence remains open [20].", "We cannot immediately apply lower bounds for udpda equivalence, since we do not know if the translation from udpda to indicator pairs in Theorem  can be implemented in logarithmic (or even polylogarithmic) space." ] ]
1403.0509
[ [ "Existence and linearized stability of solitary waves for a quasilinear\n Benney system" ], [ "Abstract We prove the existence of solitary wave solutions to the quasilinear Benney system $$iu_{t}+u_{xx}=a|u|^pu+uv,\\quad v_t+f(v)_x=(|u|^2)_x$$ where $f(v)=-\\gamma v^3$, $-1<p<+\\infty$ and $a,\\gamma>0$.", "We establish, in particular, the existence of travelling waves with speed arbitrary large if $p<0$ and arbitrary close to $0$ if $p>\\frac 23$.", "We also show the existence of standing waves in the case $-1<p\\leq \\frac 23$, with compact support if $-1<p<0$.\\\\ Finally, we obtain, under certain conditions, the linearized stability of such solutions." ], [ "Introduction", "In the seminal works [9], [10], D.J.", "Benney introduced a number of universal models describing the interaction between short and long waves propagating along a direction $(Ox)$ in a dispersive media.", "One of these models is the system $\\left\\lbrace \\begin{array}{llll}\\displaystyle i\\frac{\\partial u}{\\partial t}+\\frac{\\partial ^2 u}{\\partial x^2}=m_1|u|^2u+m_2uv\\\\\\\\\\displaystyle \\frac{\\partial v}{\\partial t}+m_3\\frac{\\partial v}{\\partial x}=m_4\\frac{\\partial }{\\partial x}(|u|^2),\\quad x\\in \\mathbb {R},\\,t\\ge 0.\\end{array}\\right.$ Here, $m_j$ are real constants, $u=u_{(y)}+iu_{(z)}$ represents, in complex notation, the transverse components $(u_{(y)},u_{(z)})$ of the short wave, and $v$ the density perturbation induced by the long wave.", "This model has been successfully applied to several physical contexts, such as the study of the formation and annihilation of solitons resulting from the interaction between Langmuir and ion sound waves in a magnetized plasma, in the case where the perturbation propagates with a speed close to that of sound ([28], [39]), or the interaction between Alfvén and magneto-acoustic waves in a cold plasma subjected to a strong external magnetic field ([13], [35]).", "In water waves theory, applications of this model include the interaction between gravity-capillary waves in a two-layer fluid, when the group velocity of the surface waves coincides with the phase velocity of the internal waves (see [21],[22],[36].", "See also [34] for an alternative derivation of Benney's equations from the Zakharov formulation of surface gravity waves).", "Other examples, such as long-wave short-wave interaction in bubbly liquids ([1]) or optical-microwave interactions in nonlinear mediums ([14]) can be given.", "The mathematical study of system (REF ), namely the well-posedness of the associated Cauchy Problem or the existence and stability of solitary waves, has been extensively conducted over the years by many authors (see for instance [6], [12], [26], [32], [37],[38] and references therein).", "As pointed out in [9], this system is an adequate model in the case where the amplitude of the long wave is considerably smaller than the amplitude of the short wave.", "When both amplitudes are of the same order, the effect of long waves becomes considerably weaker, and, in this context, (REF ) should be replaced by a system of the form $\\left\\lbrace \\begin{array}{llll}\\displaystyle i\\frac{\\partial u}{\\partial t}+\\frac{\\partial ^2 u}{\\partial x^2}=|u|^2u+uv\\\\\\\\\\displaystyle \\frac{\\partial v}{\\partial t}+\\frac{\\partial }{\\partial x}f(v)=\\frac{\\partial }{\\partial x}(|u|^2),\\end{array}\\right.$ where $f$ is a nonlinear polynomial.", "Contrarely to the linear case $f(v)=mv$ , only recently some attention has been given to the mathematical study of these more general systems.", "In [2], the case of the Schrödinger-Burgers' system ($f(v)=mv^2$ ) was adressed in the half-line.", "The existence and linear stability of shockwave solutions to (REF ) was proved in [4].", "By combining methods from dispersive equations and systems of hyperbolic conservation laws, in [15], [19], the authors studied the existence of global weak solutions and local strong solutions for the corresponding Cauchy problem in the energy space, in the case where $f(v)=av^2-bv^3$ , $a,b>0$ (see also [3], [5], [16], [17], [18] and [20] for related results concerning similar systems).", "Also recently, Bégout and Díaz ([7], [8]) considered nonlinear Schrödinger equations with an “absorbing” singular potential of the form $|u|^p$ , $p<0$ such as the homogenous equation $iu_t+\\Delta u=\\alpha |u|^{p}u,\\qquad -1<p<0.$ Nonlinear Schrödinger equations with singular potentials arise in a large variety of contexts (see e.g.", "[31],[27]).", "The authors proved in particular that under some circumstances such equations admit standing wave solutions of the form $u(x,t)=\\phi (x)e^{i\\beta t}$ with compact support, under the fundamental condition $-1<p<0$ .", "Such localization of solutions is well-known not to exist for ordinary Schrödinger equations and seem to be a special feature of singular potentials of this type.", "With these motivations, in the present work, we are concerned with the existence and behaviour of solitary waves for quasilinear Benney systems of the type $\\left\\lbrace \\begin{array}{llll}\\displaystyle i\\frac{\\partial u}{\\partial t}+\\frac{\\partial ^2 u}{\\partial x^2}=m_1|u|^pu+uv\\\\\\\\\\displaystyle \\frac{\\partial v}{\\partial t}+\\frac{\\partial }{\\partial x}f(v)=\\frac{\\partial }{\\partial x}(|u|^2),\\end{array}\\right.$ where $f(v)=m_2v^3$ and $-1<p<+\\infty $ .", "The rest of this paper is organized as follows: In Sections 2 and 3 we establish the existence of a two-parameter family of solitary-wave solutions to (REF ) of the form $(u(x,t),v(x,t))=(e^{iwt}e^{i\\frac{c}{2}(x-ct)}\\phi (x-ct),{\\psi }(x-ct)),$ where $\\phi $ and $-\\psi $ are non-negative radially decreasing functions vanishing at infinity.", "This result relies on the derivation of sharp estimates for the Lagrange multiplier associated to a variational minimization problem.", "These estimates allow us also to exhibit solitary waves with positive speed $c$ arbitrary large in the case $-1<p<0$ and arbitrary close to 0 for $p>\\frac{2}{3}$ .", "When $0\\le p\\le \\frac{2}{3}$ , we prove, in Section 4, the existence of standing-wave solutions ($c=0$ ) of the form $(u(x,t),v(x,t))=(e^{iwt}\\phi (x),{\\psi }(x))$ by applying a result due to Berestycki and Lions ([11]).", "We also establish the existence of standing waves with compact support.", "The condition for the existence of such localized solutions is $-1<p<0$ , related in particular to the convergence of a singular integral of the type $\\displaystyle \\int _0^{a}\\frac{dx}{x^{1+\\frac{p}{2}}}$ .", "Although we use totally different methods, this is, as mentionned above, the exact same condition used in [7], [8] to derive solutions with compact support.", "Finally, in Section 5, after establishing the global well-posedness of a non-autonomous system consisting of the linearization of (REF ) around a solitary wave, we prove, in the spirit of [23], the linearized stability of solitary wave solutions in the case $p>-\\frac{2}{3}$ , with $c=0$ if $p<0$ (and without restrictions on the speed $c$ if $p>0$ ).", "Our results are synthesized in the following table: Existence and stability of solitary-wave solutions to (REF ) Table: NO_CAPTION" ], [ "Existence of Solitary waves for $-1<p<0$", "We consider the system $\\left\\lbrace \\begin{array}{llll}iu_{t}+u_{xx}=a|u|^pu+uv\\\\v_t+f(v)_x=(|u|^2)_x,\\\\\\end{array}\\right.$ where $f(v)=-\\gamma v^3$ , $-1<p<0$ , $\\gamma >0$ and $a>0$ .", "We look for solutions of the form $(u(x,t),v(x,t))=(e^{iwt}e^{i\\frac{c}{2}(x-ct)}\\phi (x-ct),{\\psi }(x-ct)),$ with $\\phi $ and $\\psi $ real-valued and vanishing at infinity.", "We obtain the system $\\left\\lbrace \\begin{array}{rrrr}-\\phi ^{\\prime \\prime }+c^*\\phi &=&-\\phi \\psi -a|\\phi |^p\\phi \\\\c\\psi &=&-\\phi ^2+f(\\psi ),\\\\\\end{array}\\right.$ where $c^*=w-\\frac{c^2}{4}$ .", "By showing the existence of solutions to (REF ), we will prove the following theorem, describing a two-parameter family of soutions to (REF ): Theorem 2.1 Let $\\displaystyle \\frac{1}{3}<\\alpha < 1$ .", "There exists $\\mu _0=\\mu (\\alpha )>0$ such that for all $\\mu >\\mu _0$ , the system (REF ) has non-trivial solutions of the form $\\left\\lbrace \\begin{array}{llll}u(x,t)=e^{iwt}e^{i\\frac{c}{2}(x-ct)}\\phi _{\\mu ,\\alpha }(x-ct),\\\\\\\\v(x,t)={\\psi }_{\\mu ,\\alpha }(x-ct)\\end{array}\\right.$ where $\\phi _{\\mu ,\\alpha }$ and $-\\psi _{\\mu ,\\alpha }$ are non-negative radially decreasing $H^1$ functions such that $\\Vert \\phi _{\\mu ,\\alpha }\\Vert _{H^1}^2+\\Vert \\psi _{\\mu ,\\alpha }\\Vert _2^2\\ge \\mu ^{\\frac{1}{4}(1-\\alpha )}.$ Furthermore, $c=c(\\mu ,\\alpha )\\approx _{\\mu \\rightarrow +\\infty }\\mu ^{\\frac{1}{2}(3-\\alpha )}.$ The minimization problem For $u\\in H^1(\\mathbb {R})\\cap L^{p+2}(\\mathbb {R})$ , $-1<p<0$ , and $v\\in L^2(\\mathbb {R})\\cap L^4(\\mathbb {R})$ , let $\\tau (u,v)=\\frac{2a}{p+2}\\int |u|^{p+2}+\\int vu^2+\\frac{\\gamma }{4}\\int v^4.$ Also, for $d,\\mu >0$ , let $X_{\\mu ,d}=\\lbrace (u,v)\\in H^1(\\mathbb {R})\\cap L^{p+2}(\\mathbb {R})\\times (L^2(\\mathbb {R})\\times L^4(\\mathbb {R})) \\,:\\\\\\,{N_d}(u,v)=\\Vert u\\Vert _2^2+\\Vert u^{\\prime }\\Vert _2^2+d\\Vert v\\Vert _2^2=\\mu \\rbrace $ and $\\mathcal {I}(\\mu ,d)=inf \\lbrace \\tau (u,v)\\,:\\,(u,v)\\in X_{\\mu ,d}\\rbrace .$ If $(u,v)$ is a minimizer, then there exists a Lagrange multiplier $\\lambda $ such that $\\nabla \\tau =\\lambda \\nabla N_d$ , that is $\\left\\lbrace \\begin{array}{lllll}2a|u|^pu+2vu&=&\\lambda (-2u^{\\prime \\prime }+2u)\\\\u^2+\\gamma v^3&=&2\\lambda dv\\end{array}\\right.$ and $\\left\\lbrace \\begin{array}{lllllllll}\\lambda u^{\\prime \\prime }-\\lambda u&=&-uv-a|u|^pu\\\\-2d\\lambda v&=&-u^2+f(v).\\end{array}\\right.$ If $\\lambda <0$ , the change of variable $x^{\\prime }=x\\sqrt{-\\lambda }$ leads to a solution $(\\phi (x),\\psi (x))=\\left(u\\left(\\sqrt{-\\lambda }x\\right),v\\left(\\sqrt{-\\lambda }x\\right)\\right)$ of system (REF ) for $c*=-\\lambda \\quad \\textrm { and }\\quad c=-2\\lambda d.$ Proposition 2.2 For $\\mu ,d>0$ , $\\mathcal {I}(\\mu ,d)>-\\infty $ .", "Proof: We only have to notice that for $(u,v)\\in X_{\\mu ,d}$ , $\\tau (u,v)\\ge - \\int |v|u^2\\ge -\\Vert v\\Vert _2\\Vert u\\Vert _4^2\\ge -C\\Vert v\\Vert _2\\Vert u^{\\prime }\\Vert _2^{\\frac{1}{2}}\\Vert u\\Vert _2^{\\frac{3}{2}}\\ge -C\\frac{\\mu ^{\\frac{3}{2}}}{d^{\\frac{1}{2}}},$ by the Gagliardo-Nirenberg inequality ($C>0$ ).$\\blacksquare $ Proposition 2.3 For $\\mu ,d>0$ , $\\displaystyle \\mathcal {I}(\\mu ,d)\\le - \\frac{3}{8\\sqrt{\\pi }}\\frac{\\mu ^{\\frac{3}{2}}}{d^{\\frac{1}{2}}}+C\\left(\\mu ^{1+\\frac{p}{2}}+{\\gamma }\\frac{\\mu ^2}{d^2}\\right),$ where $C$ is a positive constant.", "In particular, for $\\displaystyle \\frac{1}{3}<\\alpha <1$ , $d=\\mu ^{\\alpha }$ and $\\mu $ large enough, $\\mathcal {I}(\\mu ,d)<0$ .", "Proof: For $B>0$ , we consider the following functions $u(x)=\\frac{B}{1+x^2}$ and $v(x)=-\\frac{1}{\\sqrt{d}}u(x).$ A simple computation shows that $\\Vert u\\Vert _2^2+\\Vert u^{\\prime }\\Vert _2^2+d\\Vert v\\Vert _2^2=B^2\\pi ,$ hence, by taking $\\displaystyle B=\\sqrt{\\frac{\\mu }{\\pi }}$ , $(u,v)\\in X_{\\mu ,d}$ .", "Furthermore, $\\displaystyle \\int vu^2=-\\frac{B^3}{\\sqrt{d}}\\int \\left(\\frac{1}{1+x^2}\\right)^3=-\\frac{\\frac{3\\pi }{8}}{\\pi ^{\\frac{3}{2}}}\\frac{\\mu ^{\\frac{3}{2}}}{d^{\\frac{1}{2}}}=-\\frac{3}{8\\sqrt{\\pi }}\\frac{\\mu ^{\\frac{3}{2}}}{d^{\\frac{1}{2}}},$ hence $\\tau (u,v)=\\frac{2a}{p+2}\\int |u|^{p+2}+\\int vu^2+\\frac{\\gamma }{4}\\int v^4\\le -\\frac{3}{8\\sqrt{\\pi }}\\frac{\\mu ^{\\frac{3}{2}}}{d^{\\frac{1}{2}}}+C\\left(\\mu ^{1+\\frac{p}{2}}+{\\gamma }\\frac{\\mu ^2}{d^2}\\right),$ where $C\\displaystyle >0$ .$\\blacksquare $ Proposition 2.4 Let $\\mu ,d>0$ and $(u,v)\\in X_{\\mu ,d}$ .", "There exists $\\tilde{u}$ non-negative and $\\tilde{v}$ non-positive, $\\tilde{u}$ and $|\\tilde{v}|$ radially decreasing, such that $\\tau (\\tilde{u},\\tilde{v})\\le \\tau (u,v)$ and $(\\tilde{u},\\tilde{v})\\in X_{\\mu ,d}$ .", "Proof: Let $u_*=|u|^*$ and $v_*=-|v|^*$ , where $f^*$ denotes the Schwarz symmetrization of $f$ .", "On one hand, $\\tau (|u|,-|v|)=\\frac{2a}{p+2}\\int |u|^{p+2}-\\int |v|u^2+\\frac{\\gamma }{4}\\int v^4\\le \\tau (u,v).$ Furthermore, since for $r\\ge 1$ , $\\displaystyle \\int (f^*)^r=\\int f^r$ for every positive function $f$ in $L^r(\\mathbb {R})$ and $\\displaystyle \\int |u|^2|v|\\le \\int (|u|^*)^2|v|^*$ , $\\tau (u_*,v_*)\\le \\tau (u,v).$ By the Polya-Szego inequality, $\\displaystyle \\int ((u_*)^{\\prime })^2\\le \\int (u^{\\prime })^2$ , hence $N_d(u_*,v_*)\\le N_d(u,v)=\\mu .$ If $N_d(u_*,v_*)=\\mu $ , we put $(\\tilde{u},\\tilde{v})=(u_*,v_*)$ .", "If $N_d(u_*,v_*)<\\mu $ we set, for $k>0$ , $\\displaystyle \\tilde{u}(x)=k^{\\frac{1}{4p}}u_*\\left(\\frac{x}{k^{\\frac{p+2}{-4p}}}\\right) \\textrm { and } \\displaystyle \\tilde{v}(x)=k^{\\frac{1}{4}}v_*(kx).$ Since $\\displaystyle \\int |\\tilde{u}|^2=k^{-\\frac{1}{4}}\\int u_*^2$ and $\\displaystyle \\int |\\tilde{v}|^2=k^{-\\frac{1}{2}}\\int v_*^2$ and at least one of these quantities is different from 0, there exists $0<k<1$ such that $N_d(\\tilde{u},\\tilde{v})=\\mu .$ Furthermore, $\\int \\tilde{v}^4=\\int v_*^4,$ $\\int \\tilde{u}^{p+2}=\\int u_*^{p+2}$ and $\\displaystyle \\int \\tilde{u}^2\\tilde{v}=k^{\\frac{1}{2p}+\\frac{1}{4}}\\int u_*^2\\left(\\frac{x}{k^{\\frac{p+2}{-4p}}}\\right)v_*(kx)< k^{\\frac{1}{2p}+\\frac{1}{4}}\\int u_*^2\\left(\\frac{x}{k^{\\frac{p+2}{-4p}}}\\right)v_*\\left(\\frac{x}{k^{\\frac{p+2}{-4p}}}\\right)$ since $\\displaystyle |kx|< \\left|\\frac{x}{k^{\\frac{p+2}{-4p}}}\\right|$ for $x\\ne 0$ and $-v_*$ is non-negative and radially decreasing.", "Finally, $\\displaystyle \\int \\tilde{u}^2\\tilde{v}\\le \\left(k^{\\frac{1}{2p}+\\frac{1}{4}-{\\frac{p+2}{4p}}}\\right)\\int u_*^2v_*=\\int u_*^2v_*$ and $\\tau (\\tilde{u},\\tilde{v})< \\tau (u_*,v_*)\\le \\tau (u,v)$ , which completes the proof.$\\blacksquare $ Proposition 2.5 Let $\\mu ,d>0$ .", "There exists a solution $(u,v)$ for the minimization problem (REF ), with $u$ and $-v$ non-negative and radially decreasing.", "Proof: Let $(u_n,v_n)$ a minimizing sequence in $(H^1_{rd}(\\mathbb {R})\\cap L^{p+2}(\\mathbb {R}))\\times L^2_{rd}(\\mathbb {R})\\cap L^4(\\mathbb {R})$ .", "By the compacity of the injection $H^1_{rd}(\\mathbb {R})\\hookrightarrow L^r(\\mathbb {R})$ , $r>2$ , there exists a subsequence still denoted $u_n$ such that $u_n\\rightarrow u$ in $L^4(\\mathbb {R})$ ; $u_n\\rightharpoonup u$ in $H^1(\\mathbb {R})$ weak; $u_n\\rightarrow u$ almost everywhere (in particular, $u$ is radial decreasing).", "Also, since $\\displaystyle \\Vert v_n\\Vert _2^2\\le \\frac{\\mu }{d}$ is bounded, we can extract a subsequence still denoted $v_n$ such that $v_n\\rightharpoonup v$ in $L^2(\\mathbb {R})$ weak.", "Hence, since $u_n^2\\rightarrow u^2$ in $L^2$ strong and $v_n\\rightharpoonup v$ in $L^2$ weak, $\\int v_nu_n^2\\rightarrow \\int u^2v.$ The sequence $\\frac{\\gamma }{4}\\int v_n^4=\\tau (u_n,v_n)-\\int v_nu_n^2-\\frac{2a}{p+2}\\int |u|^{p+2}$ is thus bounded, and we can extract a subsequence still denoted $v_n$ such that $v_n\\rightharpoonup v$ in $L^4$ weak.", "Since $\\displaystyle \\int v^4\\le \\liminf \\int v_n^4$ and $\\displaystyle \\int |u|^{p+2}\\le \\liminf \\int |u_n|^{p+2}$ , $\\tau (u,v)\\le \\liminf \\tau (u_n,v_n)=\\mathcal {I}(u,v).$ Now, if $\\Vert u\\Vert _2^2+\\Vert u^{\\prime }\\Vert _2^2+d\\Vert v\\Vert _2^2<\\mu $ , the construction made in the proof of Proposition REF shows that there exists $(\\tilde{u},\\tilde{v})\\in X_{\\mu ,d}$ such that $\\tau (\\tilde{u},\\tilde{v})<\\tau (u,v).$ Finally, $\\mathcal {I}(\\mu ,d)\\le \\tau (\\tilde{u},\\tilde{v})<\\tau (u,v)\\le \\liminf \\tau (u_n,v_n)=\\mathcal {I}(\\mu ,d),$ which is absurd, hence $(\\tilde{u},\\tilde{v})\\in X_{\\mu ,d}$ is a minimizer.", "Note that, since $\\mathcal {I}(\\mu ,d)=\\tau (u,v)$ , we have in fact that $\\int |u|^{p+2}=\\liminf \\int |u_n|^{p+2}$ , $\\int v^4=\\liminf \\int v_n^4$ , $\\int v^2=\\liminf \\int v_n^2$ and $\\int u^2+u^{\\prime 2}=\\liminf \\int u_n^2+u_n^{\\prime 2}$ , hence $u_n\\rightarrow u$ in $L^{p+2}(\\mathbb {R})\\cap H^1(\\mathbb {R})$ strong and $v_n\\rightarrow v$ in $L^2(\\mathbb {R})\\cap L^4(\\mathbb {R})$ strong.", "By choosing a new subsequence, $v_n\\rightarrow v$ almost everywhere, hence $-v$ is non-negative and radially decreasing.$\\blacksquare $ If $(u,v)\\in X_{\\mu ,d}$ is a solution to the minimization problem, $u,-v\\ge 0$ , there exists a Lagrange multiplier $\\lambda \\in \\mathbb {R}$ such that $\\left\\lbrace \\begin{array}{rrrr}\\lambda u^{\\prime \\prime }-\\lambda u&=&-uv-a|u|^pu\\\\-2d\\lambda v&=&-u^2+f(v).\\end{array}\\right.$ The next result states the assymptotic behaviour of $\\lambda $ : Proposition 2.6 Let $\\displaystyle \\frac{1}{3}<\\alpha < 1$ and $d=\\mu ^{\\alpha }$ .", "There exists positive constants $M_1,M_2$ such that for $\\mu $ large enough, $M_1 \\left(\\frac{\\mu }{d}\\right)^{\\frac{3}{2}}\\le -\\lambda \\le M_2\\left(\\frac{\\mu }{d}\\right)^{\\frac{3}{2}}.$ Proof: Multiplying the equations in (REF ) respectively by $\\phi $ and $\\psi $ , $2\\lambda d=3\\int u^2v+2a\\int u^{p+2}+\\gamma \\int v^4.$ In particular, $-2\\lambda d=-3\\int u^2v-2a\\int u^{p+2}-\\gamma \\int v^4$ $\\le 3\\left(\\int u^4\\right)^{\\frac{1}{2}}\\left(\\int v^2\\right)^{\\frac{1}{2}}\\le 3C\\mu \\sqrt{\\frac{\\mu }{d}},$ $C>0$ , by the Gagliardo Nirenberg inequality.", "This proves the second inequality by choosing $\\displaystyle M_2=\\frac{3C}{2}$ .", "Now, since $2a\\int u^{p+2}=(p+2)\\tau (u,v)-(p+2)\\int u^2v-(p+2)\\frac{\\gamma }{4}\\int v^4,$ we obtain by (REF ) that $2\\lambda d=(1-p)\\int u^2v+(p+2)\\tau (u,v)+\\frac{\\gamma }{4}\\left(2-p\\right) \\int v^4.$ Since $\\frac{\\gamma }{4}\\int v^4=\\tau (u,v)-\\frac{2a}{p+2}\\int u^{p+2}-\\int u^2v\\le \\tau (u,v)-\\int u^2v,$ $2\\lambda d\\le 4\\tau (u,v)-\\int u^2v\\le 4\\tau (u,v)+\\left(\\int v^2\\right)^{\\frac{1}{2}}\\left(\\int u^4\\right)^{\\frac{1}{2}}$ $\\le 4\\tau (u,v)+C_0^{\\frac{1}{2}}\\frac{\\mu ^{\\frac{3}{2}}}{d^{\\frac{1}{2}}},$ where $C_0$ is the smaller constant for the Gagliardo-Nirenberg inequality $\\Vert u\\Vert ^4_4\\le C_0\\Vert u^{\\prime }\\Vert _2\\Vert u\\Vert _2^3.$ By Proposition $\\ref {estneg}$ , $2\\lambda d\\le \\left(C_0^{\\frac{1}{2}}-\\frac{3}{2\\sqrt{\\pi }}\\right)\\frac{\\mu ^{\\frac{3}{2}}}{d^{\\frac{1}{2}}}+4C\\left(\\mu ^{1+\\frac{p}{2}}+{\\gamma }\\frac{\\mu ^2}{d^2}\\right),$ where $C>0$ .", "Also, one can choose $C_0=\\frac{1}{\\sqrt{3}}$ .", "Indeed, it is known that the sharp constant in the Gagliardo-Nirenberg inequality is given by $C_0=\\frac{4}{\\sqrt{3}\\Vert Q\\Vert _2^2}$ , where $Q(x)=\\sqrt{2}\\operatorname{sech}(x)$ is the positive radial solution of $Q^{\\prime \\prime }+Q^3=Q$ : $\\Vert Q\\Vert _2^2=4$ (see for instance [24], [25]).", "Now, taking $d=\\mu ^{\\alpha }$ and putting $\\displaystyle \\epsilon =\\frac{3}{2\\sqrt{\\pi }}-\\frac{1}{3^{\\frac{1}{4}}}>0$ , $c:=-2\\lambda d\\ge \\epsilon \\frac{\\mu ^{\\frac{3}{2}}}{d^{\\frac{1}{2}}}-C^{\\prime }\\left(\\mu ^{1+\\frac{p}{2}}+\\frac{\\gamma }{2}\\frac{\\mu ^2}{d^2}\\right)=\\epsilon \\mu ^{\\frac{3}{2}-\\frac{\\alpha }{2}}-C^{\\prime }\\mu ^{1+\\frac{p}{2}}-C^{\\prime }\\frac{\\gamma }{2}\\mu ^{2(1-\\alpha )}$ $\\ge \\frac{\\epsilon }{2}\\mu ^{\\frac{3}{2}-\\frac{\\alpha }{2}}$ for $\\mu $ large enough, since for $1\\ge \\alpha >\\frac{1}{3}$ , we have $1+\\frac{p}{2}<\\frac{1}{2}(3-\\alpha )$ and $2(1-\\alpha )<\\frac{1}{2}(3-\\alpha )$ .", "The proof is now complete by taking $\\displaystyle M_1=\\frac{\\epsilon }{4}$ .$\\blacksquare $ End of the proof of Theorem REF : In particular, from Proposition REF , $\\lambda <0$ .", "By the change of variables (REF ), we obtain from a minimizer $(u,v)\\in X_{\\mu ,d}$ a solution $(\\phi _\\mu ,\\psi _\\mu )$ of system (REF ).", "Note that $\\mu =\\Vert u\\Vert _2^2+\\Vert u^{\\prime }\\Vert _2^2+d\\Vert v\\Vert _2^2=\\left\\Vert \\phi _{\\mu ,\\alpha }\\left(\\frac{\\cdot }{\\sqrt{-\\lambda }}\\right)\\right\\Vert _2^2+\\left\\Vert \\phi _{\\mu ,\\alpha }^{\\prime }\\left(\\frac{\\cdot }{\\sqrt{-\\lambda }}\\right)\\right\\Vert _2^2+d\\left\\Vert \\psi _{\\mu ,\\alpha }\\left(\\frac{\\cdot }{\\sqrt{-\\lambda }}\\right)\\right\\Vert _2^2$ and $\\mu =\\sqrt{-\\lambda }\\Vert \\phi _{\\mu ,\\alpha }\\Vert _2^2+\\frac{1}{\\sqrt{-\\lambda }}\\Vert \\phi _{\\mu ,\\alpha }^{\\prime }\\Vert _2^2+d\\sqrt{-\\lambda }\\Vert \\psi _{\\mu ,\\alpha }\\Vert _2^2.$ Hence, $\\mu \\approx \\mu ^{\\frac{3}{4}(1-\\alpha )}\\Vert \\phi _{\\mu ,\\alpha }\\Vert _2^2+ \\mu ^{\\frac{3}{4}(\\alpha -1)}\\Vert \\phi _{\\mu ,\\alpha }^{\\prime }\\Vert _2^2+\\mu ^{\\frac{1}{4}(3+\\alpha )}\\Vert \\psi _{\\mu ,\\alpha }\\Vert _2^2$ $\\le \\mu ^{\\frac{1}{4}(3+\\alpha )}\\left(\\Vert \\phi _\\mu \\Vert _{H^1}^2+\\Vert \\psi _\\mu \\Vert _2^2\\right)$ and $\\Vert \\phi _{\\mu ,\\alpha }\\Vert _{H^1}^2+\\Vert \\psi _{\\mu ,\\alpha }\\Vert _2^2\\ge C\\mu ^{\\frac{1}{4}(1-\\alpha )},\\quad C>0.$ $\\,$$\\blacksquare $" ], [ "Existence of Solitary waves for $p>\\frac{2}{3}$", "In the case of $p>\\frac{2}{3}$ , we prove the following result: Theorem 3.1 Let $\\displaystyle 1-p<\\alpha <\\frac{1}{3}$ .", "There exists $\\mu _0=\\mu (\\alpha )>0$ such that for all $0<\\mu <\\mu _0$ , the system (REF ) has non-trivial solutions of the form $\\left\\lbrace \\begin{array}{llll}u(x,t)=e^{iwt}e^{i\\frac{c}{2}(x-ct)}\\phi _{\\mu ,\\alpha }(x-ct),\\\\\\\\v(x,t)={\\psi }_{\\mu ,\\alpha }(x-ct)\\end{array}\\right.$ where $\\phi _{\\mu ,\\alpha }$ and $-\\psi _{\\mu ,\\alpha }$ are non-negative radially decreasing smooth functions such that, for $0\\displaystyle <\\alpha <\\frac{1}{3}$ , $\\Vert \\phi _{\\mu ,\\alpha }\\Vert _{H^1}^2+\\Vert \\psi _{\\mu ,\\alpha }\\Vert _2^2\\le C\\mu ^{\\frac{1}{4}(1-3\\alpha )},\\quad C>0.$ Furthermore, $c=c(\\mu ,\\alpha )\\approx _{\\mu \\rightarrow 0^+}\\mu ^{\\frac{1}{2}(3-\\alpha )}.$ Proof: We begin by noticing that Propositions REF and REF hold for $\\displaystyle p>\\frac{2}{3}$ .", "Furthermore, estimate (REF ) holds for $p<0$ and for $\\displaystyle p>\\frac{2}{3}$ .", "Hence, the conclusions in Propositions REF and REF can be drawn also in this case.", "Finally, estimate (REF ) $-\\lambda d\\le \\frac{3C}{2}\\frac{\\mu ^{\\frac{3}{2}}}{d^{\\frac{1}{2}}}$ remains valid for all $p$ , and, for $2-p\\ge 0$ , estimate (REF ) $2\\lambda d\\le \\left(C_0^{\\frac{1}{2}}-\\frac{3}{2\\sqrt{\\pi }}\\right)\\frac{\\mu ^{\\frac{3}{2}}}{d^{\\frac{1}{2}}}+4C\\left(\\mu ^{1+\\frac{p}{2}}+{\\gamma }\\frac{\\mu ^2}{d^2}\\right),$ can be derived in the exact same way as in the case $p<0$ .", "On the other hand, if $p> 2$ , we get from (REF ) and (REF ) that $2\\lambda d=4\\tau (u,v)-\\int u^2v+2a\\frac{p-2}{p+2}\\int u^{p+2}.$ By Proposition REF , $2\\lambda d\\le \\left(C_0^{\\frac{1}{2}}-\\frac{3}{2\\sqrt{\\pi }}\\right)\\frac{\\mu ^{\\frac{3}{2}}}{d^{\\frac{1}{2}}}+4C\\left(\\mu ^{1+\\frac{p}{2}}+{\\gamma }\\frac{\\mu ^2}{d^2}\\right)+2a\\frac{p-2}{p+2}\\int u^{p+2}.$ Using the Gagliardo-Nirenberg inequality $\\Vert u\\Vert _{p+2}\\le C\\Vert u\\Vert ^{\\frac{p}{2p+4}}\\Vert u^{\\prime }\\Vert ^{\\frac{p+4}{2p+4}}$ , we obtain $\\displaystyle \\int u^{p+2}\\le C\\mu ^{1+\\frac{p}{2}},$ hence, in all cases, $c=-2\\lambda d\\ge \\epsilon \\mu ^{\\frac{3}{2}-\\frac{\\alpha }{2}}-C_1\\mu ^{1+\\frac{p}{2}}-C_2\\frac{\\gamma }{2}\\mu ^{2(1-\\alpha )},$ where $\\epsilon , C_1,$ and $C_2$ are positive constants and $d=\\mu ^{\\alpha }$ .", "Taking $\\displaystyle 1-p<\\alpha <\\frac{1}{3}$ , $\\displaystyle 1+\\frac{p}{2}>\\frac{3}{2}-\\frac{\\alpha }{2}$ and $\\displaystyle 2(1-\\alpha )>\\frac{3}{2}-\\frac{\\alpha }{2}$ , hence there exists $\\mu _0>0$ such that for all $0<\\mu <\\mu _0$ , $c\\ge \\frac{\\epsilon }{2}\\mu ^{\\frac{3}{2}-\\frac{\\alpha }{2}},$ which, with estimate $(\\ref {estsup})$ , completes the proof of (REF ).", "Finally, estimate (REF ) follows from (REF ).", "Remark 3.2 In whats concerns the regularity of $\\phi $ and $\\psi $ , note that the monotony of $\\phi $ and $\\psi $ garantee, via Lebesgue's Theorem, that $\\phi ^{\\prime }$ and $\\psi ^{\\prime }$ exist almost everywhere.", "Differentiating the second equation in (REF ) then yields $\\psi ^{\\prime }(c+3\\gamma \\psi ^2)=-2\\phi \\phi ^{\\prime }.$ Since $\\phi \\in H^1(\\mathbb {R})$ , $\\int (\\psi ^{\\prime })^2\\le \\frac{4}{c^2} \\int \\phi ^2\\phi ^{\\prime 2}\\le \\frac{4}{c^2} \\Vert \\phi \\Vert _{\\infty }^2\\Vert \\phi ^{\\prime }\\Vert _2^2<+\\infty $ and $\\psi \\in H^1(\\mathbb {R})$ .", "Now, in the case where $p\\ge 0$ , the first equation in (REF ) shows that $\\phi ^{\\prime \\prime }\\in L^2(\\mathbb {R})$ , that is, $\\phi \\in H^2(\\mathbb {R})$ .", "And again, by differentiating the second equation, $\\psi ^{\\prime \\prime }(c+3\\gamma \\psi ^2)=-2(\\phi ^{\\prime })^2-2\\phi \\phi ^{\\prime \\prime }-6\\gamma \\psi (\\psi ^{\\prime })^2,$ and we easily get that in fact $\\psi \\in H^2(\\mathbb {R})$ .", "A bootstrap argument then shows that in this case $\\phi ,\\psi \\in H^{\\infty }(\\mathbb {R})$ .", "Remark 3.3 For $p\\ge 0$ and $c\\ge 0$ , let $ (\\phi ,\\psi )$ be $C^2(\\mathbb {R})\\cap W^{2,\\infty }$ solutions of (REF ) with $\\phi \\ge 0$ and $\\psi \\le 0$ .", "Then $\\phi ^p\\in C^2(\\mathbb {R})\\cap W^{2,\\infty }$ .", "Indeed, in a neighbourhood of a point $x$ such that $\\phi (x)=0$ , from the second equation in (REF ), $\\psi \\sim \\phi ^2$ if $c>0$ ($\\psi \\sim \\phi ^{\\frac{2}{3}}$ if $c=0$ ).", "Hence, we derive from the first equation in (REF ) that $\\phi ^{\\prime \\prime }\\sim \\phi $ .", "Noticing that $\\phi $ is non-negative and non-increasing, $\\phi (x)=0$ implies that $\\phi (y)=0$ if $y>x$ and, in particular, $\\phi ^{\\prime }(x)=0$ .", "Writing $\\phi ^{\\prime }(y)=\\int _x^y\\phi ^{\\prime \\prime }(t)dt$ then shows that $\\phi ^{\\prime }\\sim \\phi $ .", "Finally, $\\phi ^{p-1}\\phi ^{\\prime }$ , $\\phi ^{p-2}(\\phi ^{\\prime })^2$ and $\\phi ^{p-1}\\phi ^{\\prime \\prime }$ vanish at $x$ , which gives the desired result." ], [ "Existence of standing waves for $-1< p\\le \\frac{2}{3}$", "In this section we show the existence of smooth non-trivial standing wave solutions to (REF ).", "More precisely: Proposition 4.1 Let $\\displaystyle -1<p\\le \\frac{2}{3}$ and $\\displaystyle \\gamma ,a,\\omega >0$ , with $\\displaystyle \\gamma ^{-\\frac{1}{3}}>a$ if $p=\\frac{2}{3}$ .", "Then (REF ) admits non trivial solutions of the form $(u(x,t),v(x,t))=(e^{iwt}\\phi (x),\\psi (x)),$ where $\\phi \\in C^2(\\mathbb {R})\\cap W^{2,\\infty }(\\mathbb {R})$ and $-\\psi =\\left(\\frac{\\phi ^2}{\\gamma }\\right)^{\\frac{1}{3}}\\in C^1(\\mathbb {R})\\cap W^{1,\\infty }(\\mathbb {R})$ are non-negative, radially descreasing functions.", "Moreover: if $p>-\\frac{2}{3}$ , $\\phi \\in C^3(\\mathbb {R})\\cap W^{3,\\infty }(\\mathbb {R})$ and $\\psi \\in C^2(\\mathbb {R})\\cap W^{2,\\infty }(\\mathbb {R})$ ; if $-1<p<0$ , $\\phi $ and $\\psi $ are compactly supported.", "Proof: Let us consider the system (REF ) with $c=0$ and $\\phi \\ge 0$ : $\\left\\lbrace \\begin{array}{cccc}-\\phi ^{\\prime \\prime }+w\\phi &=&-\\phi \\psi -a\\phi ^{p+1}\\\\\\phi ^2&=&-\\gamma \\psi ^3.\\\\\\end{array}\\right.$ From the second equation, we obtain $\\psi =-\\left(\\frac{\\phi ^2}{\\gamma }\\right)^{\\frac{1}{3}}$ .", "Replacing in the first equation leads to $\\phi ^{\\prime \\prime }=a\\phi ^{p+1}+\\omega \\phi -\\gamma ^{-\\frac{1}{3}}\\phi ^{\\frac{5}{3}}.$ We first analyse the case $-1<p< 0$ .", "By multiplying (REF ) by $\\phi ^{\\prime }$ and integrating, we deduce, for a solution verifying $\\phi ^{\\prime }(\\xi )=0$ in all points $\\xi $ such that $\\phi (\\xi )=0$ , that $\\phi ^{\\prime 2}=\\frac{2a}{p+2}\\phi ^{p+2}+w\\phi ^2-\\frac{3}{4}\\gamma ^{-\\frac{1}{3}}\\phi ^{\\frac{8}{3}}:=h(\\phi ).$ Now, taking $\\phi _0>0$ such that $h(\\phi _0)=0$ and $h(\\phi )\\ne 0$ for $\\phi \\in ]0,\\phi _0[$ , we can derive from (REF ) the existence of a solution $\\phi \\in C^2(\\mathbb {R})$ to (REF ) with compact support, non-negative, radially decreasing, such that $\\max \\phi =\\phi (0)=\\phi _0$ and $supp(\\phi )=[-x_0,x_0]$ , $x_0=\\int _0^{\\phi _0}(h(\\phi ))^{-\\frac{1}{2}}d\\phi $ (note that this integral is finite for $p<0$ ).", "Moreover, if $\\displaystyle -\\frac{2}{3}<p<0$ , we can easily establish, from (REF ) and (REF ), that $\\phi \\in C^3(\\mathbb {R})$ and $\\psi \\in C^2(\\mathbb {R})$ , with the same support.", "We now turn to the case $\\displaystyle 0\\le p\\le \\frac{2}{3}$ , with $\\displaystyle \\gamma ^{-\\frac{1}{3}}>a$ if $\\displaystyle p=\\frac{2}{3}$ .", "Equation (REF ) can be written as $-\\phi ^{\\prime \\prime }=g(\\phi ):=-a\\phi ^{p+1}-w\\phi +\\gamma ^{-\\frac{1}{3}}\\phi ^{\\frac{5}{3}}.$ We have $g\\in C^1(\\mathbb {R})$ , $g(0)=0$ and $g^{\\prime }(0)=-w<0$ .", "Moreover, putting $\\displaystyle F(\\phi )=\\int _0^{\\phi }g(\\xi )d\\xi $ and $\\phi _0=inf\\lbrace \\xi >0\\,:\\, F(\\xi )=0\\rbrace $ , $\\phi _0>0$ and $g(\\phi _0)=F^{\\prime }(\\phi _0)>0$ .", "By applying Theorem 5 and Remark 6.3 in [11], there exists a unique solution $\\phi \\in C^3(\\mathbb {R})$ of (REF ) such that $\\phi (0)=\\phi _0$ , $\\phi $ positive and radially decreasing, and such that $\\phi (x),|\\phi ^{\\prime }(x)|,|\\phi ^{\\prime \\prime }(x)|\\le Ce^{-\\delta |x|},$ where $C$ and $\\delta $ are positive constants.", "We can easily deduce from (REF ), (REF ) and (REF ) that $\\psi =-\\frac{1}{\\gamma }\\phi ^{\\frac{2}{3}}\\in C^2(\\mathbb {R})$ with $|\\psi (x)|,|\\psi ^{\\prime }(x)|,|\\psi ^{\\prime \\prime }(x)|\\le C^{\\prime }e^{-\\frac{2\\delta }{3} |x|},\\quad C^{\\prime }>0.$" ], [ "Linearized Stability for $p>-\\frac{2}{3}$", "In this section we will consider, for $\\displaystyle p>-\\frac{2}{3}$ , special solutions $(\\tilde{u},\\tilde{v})$ of system (REF ), of the form $\\left\\lbrace \\begin{array}{lllll}\\tilde{u}(x,t)=e^{iwt}e^{i\\frac{c}{2}(x-ct)}\\phi (x-ct)\\\\\\tilde{v}(x,t)=\\psi (x-ct),\\end{array}\\right.$ satisfying the following conditions: $c\\ge 0$ and $c=0$ if $\\displaystyle -\\frac{2}{3}<p<0$ ; $\\phi ,\\psi \\in C^2(\\mathbb {R})\\cap W^{2,\\infty }(\\mathbb {R})$ ; $$ $\\phi ,-\\psi \\ge 0$ , and $\\phi ,-\\psi $ radially decreasing; $\\phi ^p\\in C^2(\\mathbb {R})\\cap W^{2,\\infty }(\\mathbb {R})$ if $p\\ge 0$ (cf.", "Remark REF ).", "By linearizing the system (REF ) around $(\\tilde{u},\\tilde{v})$ (cf.", "[4],[23]), identifying the first order terms and, for sake of simplicity, replacing the solution $(U,V)$ by the new dependent variables $u(x,t)=e^{-iwt}e^{-i\\frac{c^2}{2}t}U(x,t)$ and $v(x,t)=V(x,t)$ , we obtain the system $\\left\\lbrace \\begin{array}{llllllll}iu_t+u_{xx}&=&(w-\\frac{c^2}{2})u+\\frac{a}{2}\\phi ^p[(p+2)u+pe^{icx}\\overline{u}]+e^{i\\frac{c}{2} x}\\phi v+\\psi u\\\\\\\\v_t-3\\gamma (\\psi ^2v)_x&=&2Re(e^{i\\frac{c}{2} x}\\phi u)_x,\\end{array}\\right.$ which we complete with initial data $(u_0,v_0)\\in H^2(\\mathbb {R})\\times H^1(\\mathbb {R}).$ Since, for $p<0$ , $\\phi ^p$ is not, in general, a $C^2\\cap W^{2,\\infty }(\\mathbb {R})$ function, we begin by the study of a regularized system (with the same initial data): $\\left\\lbrace \\begin{array}{llllllll}iu_t+u_{xx}&=&(w-\\frac{c^2}{2})u+\\frac{a}{2}(\\phi +\\epsilon )^p[(p+2)u+pe^{icx}\\overline{u}]+e^{i\\frac{c}{2} x}\\phi v+\\psi u\\\\\\\\v_t-3\\gamma (\\psi ^2v)_x&=&2Re(e^{-i\\frac{c}{2} x}\\phi u)_x,\\end{array}\\right.$ where $\\epsilon >0$ if $p<0$ ($\\epsilon =0$ otherwise).", "We begin by proving the following result concerning this regularized system: Proposition 5.1 For each $p>-\\frac{2}{3}$ there exists a unique solution $(u,v)\\in (C([0,+\\infty [;H^2)\\cap C^1([0,+\\infty [;L^2))\\times (C([0,+\\infty [;H^1)\\cap C^1([0,+\\infty [;L^2))$ of system (REF ) with initial data $(u_0,v_0)\\in H^2(\\mathbb {R})\\times H^1(\\mathbb {R})$ .", "Proof: We follow the technique in [19],[33] and introduce an auxiliary system with non-local source which can be tackled by Kato's theory ([29], [30]).", "This is necessary in order to write the system (REF ) without derivative loss in the nonlinear term (see [19] for details).", "Hence, we consider the system $\\left\\lbrace \\begin{array}{llllllll}iF_t+F_{xx}&=&(w-\\frac{c^2}{2}+\\psi )F+a(\\phi +\\epsilon )^p[\\frac{p}{2}(F+e^{icx}\\overline{F})+F]\\\\\\end{array}&&+e^{i\\frac{c}{2} x}\\phi [3\\gamma (\\psi ^2v)_x+2Re(e^{-i\\frac{c}{2}x}\\phi \\tilde{u})_x]+\\psi _tu\\\\&&+e^{icx}\\phi _tv+ap(\\phi +\\epsilon )^{p-1}\\phi _t[\\frac{p}{2}(u+e^{icx}\\overline{u})]\\\\v_t-3\\gamma (\\psi ^2v)_x&=&2Re(e^{-i\\frac{c}{2} x}\\phi \\tilde{u})_x,\\right.$ where $\\left\\lbrace \\begin{array}{llllllll}u(x,t)&=&u_0(x)+\\int _0^t F(x,s)ds,\\\\\\tilde{u}(x,t)&=&(\\Delta -1)^{-1}([(w-\\frac{c^2}{2})+\\psi +\\frac{a(p+2)}{2}(\\phi +\\epsilon )^p]u\\\\&&+\\frac{ap}{2}\\phi ^pe^{icx}\\overline{u}+e^{i\\frac{c}{2}x}\\phi v-iF),\\end{array}\\right.$ with initial data $F(.,0)=F_0\\in L^2(\\mathbb {R}),\\quad v(.,0)=v_0\\in H^1(\\mathbb {R}).$ Once we have, for a fixed $T>0$ , a solution $F\\in C([0,T]; L^2)\\cap C^1([0,T];H^{-2}),\\quad v\\in C([0,T]; H^1)\\cap C^1([0,T];L^2)$ for the problem (REF )-(REF )-(REF ), we can argue as in [19], Lemma 2.1, and show that $(u,v)$ is the desired solution to system (REF ).", "We only sketch the argument, since it is similar to the one in [4] and [19].", "First, we write (REF ) as a system of three equations, by decomposing $F$ into its real and imaginary parts.", "This allows us to obtain a system with the abstract form $U_t+AU=g(t,U), \\,U(.,0)=U_0,$ with $U=(Re F, Im F, v)$ and $U_0=(Re F_0, Im F_0, v_0)$ , the corresponding initial data.", "Following [4], [19], we decompose the operator $A=\\left[\\begin{array}{cccccccccc}0&\\Delta &0\\\\-\\Delta &0&0\\\\0&0&-3\\gamma [(\\psi ^2)_x+\\psi ^2\\frac{\\partial }{\\partial x}]\\end{array}\\right]$ in the form $SAS^{-1}=A+B$ for some operator $B$ .", "In the present setting, we can choose $S=\\left[\\begin{array}{cccccccccc}1-\\Delta &0&0\\\\0&1-\\Delta &0\\\\0&0&(1-\\Delta )^{\\frac{1}{2}}\\end{array}\\right]$ Note that $S\\,:\\,Y=L^2\\times L^2\\times H^1\\rightarrow X=H^{-2}\\times H^{-2}\\times L^2$ is an isomorphism.", "The relevant properties of $S$ (in particular the ones concerning the entry $(1-\\Delta )^{\\frac{1}{2}}$ ) can be found in [29], Section 8.", "Observe that the right-hand-side of (REF ) is linear in $U$ , hence it is straightforward to derive the necessary estimates for the source term $g$ and we may finally apply Theorem 2 in [30] (or Theorem 7.1 in [29]) and conclude with the existence of a unique pair $(F,v)$ satisfying (REF )-(REF )-(REF ), which achieves the sketch of the proof.$\\blacksquare $ We are now in position to prove the linearized stability result: Proposition 5.2 Let $p>-\\frac{2}{3}$ and consider a special solution $(\\tilde{u},\\tilde{v})$ to (REF ) satisfying (REF )-(REF ).Then $(\\tilde{u},\\tilde{v})$ is linearly stable in the sense that for any $T>0$ and any initial data $(u_0,v_0)\\in H^1\\times L^2$ , the system (REF ) admits a unique weak solution $(u,v)\\in L^{\\infty }(0,T;H^1\\times L^2)$ such that $\\Vert (u,v)\\Vert ^2_{L^{\\infty }(0,T;H^1\\times L^2)}\\le G_T(\\Vert (u_0,v_0)\\Vert ^2_{H^1\\times L^2}),$ where $G_{T}\\,:\\,\\mathbb {R}^+\\rightarrow \\mathbb {R}^+$ is a continuous function vanishing at the origin.", "Moreover, if $(u_0,v_0)\\in H^2\\times H^1$ and $p\\ge 0$ , $(u,v)$ is a strong solution satisfying $(u,v)\\in [C([0,T];H^2)\\cap C^1([0,T];L^2)]\\times [C([0,T];H^1)\\cap C^1([0,T];L^2)]$ and $\\Vert (u,v)\\Vert ^2_{L^{\\infty }(0,T;H^2\\times H^1)}\\le G_T(\\Vert (u_0,v_0)\\Vert ^2_{H^2\\times H^1}).$ Proof: We consider, for fixed $\\epsilon $ , the solution $(u_{\\epsilon },v_{\\epsilon })$ of system (REF ) with initial data $(u_{0\\epsilon },v_{0\\epsilon })\\in H^2\\times H^1,$ with $(u_{0\\epsilon },v_{0\\epsilon })\\rightarrow (u_0,v_0)\\quad \\textrm {in }H^1\\times L^2.$ In what follows, for simplicity, we will drop the subscript $\\epsilon $ .", "By multiplying the first equation in (REF ) by $\\overline{u}$ (respectively by $\\overline{u_t}$ ), taking the imaginary part (respectively the real part) and integrating, we get $\\frac{1}{2}\\frac{d}{dt}\\int |u|^2dx=\\frac{ap}{2} Im\\int (\\phi +\\epsilon )^pe^{i\\frac{c}{2}x}\\overline{u}^2dx+Im\\int e^{i\\frac{c}{2}x}\\phi v\\overline{u}dx$ and $\\frac{d}{dt}\\left\\lbrace \\frac{1}{2}\\int |u_x|^2dx+\\frac{1}{2}\\left(w-\\frac{c^2}{2}\\right)\\int |u|^2dx+\\frac{a(p+2)}{4}\\int (\\phi +\\epsilon )^p|u|^2dx+\\frac{1}{2} \\int \\psi |u|^2dx \\right\\rbrace +$ $\\frac{ap}{2}\\int (\\phi +\\epsilon )^pRe\\left(e^{icx}\\overline{u}\\frac{\\partial \\overline{u}}{\\partial t}\\right)dx+\\int \\phi Re\\left(e^{i\\frac{c}{2}x}v\\frac{\\partial \\overline{u}}{\\partial t}\\right)dx.$ We have $(\\phi +\\epsilon )^pRe\\left(e^{icx}\\overline{u}\\frac{\\partial \\overline{u}}{\\partial t}\\right)=\\frac{1}{2}(\\phi +\\epsilon )^p\\frac{\\partial }{\\partial t}Re\\left(e^{icx}\\overline{u}^2\\right)$ $=\\frac{1}{2}\\frac{d}{dt}\\left\\lbrace (\\phi +\\epsilon )^pRe\\left(e^{icx}\\overline{u}^2\\right)\\right\\rbrace +pc(\\phi +\\epsilon )^{p-1}\\phi ^{\\prime }Re\\left(e^{icx}\\overline{u}^2\\right)$ (recall that $c=0$ if $-\\frac{2}{3}<p<0$ and $\\epsilon =0$ if $p\\ge 0$ ), $\\phi Re\\left(e^{i\\frac{c}{2}x}v\\frac{d\\overline{u}}{dt}\\right)=\\frac{\\partial }{\\partial t}\\left\\lbrace \\phi Re\\left(e^{i\\frac{c}{2}x}v\\overline{u}\\right)\\right\\rbrace +c\\phi ^{\\prime } Re\\left(e^{i\\frac{c}{2}x}v\\overline{u}\\right)- Re\\left(e^{i\\frac{c}{2}x}\\phi \\overline{u}\\frac{\\partial v}{\\partial t}\\right),$ and, by the second equation in (REF ), $ Re\\left(e^{i\\frac{c}{2}x}\\phi \\overline{u}\\frac{\\partial v}{\\partial t}\\right)=Re\\left(3\\gamma e^{i\\frac{c}{2}x}\\phi \\overline{u}(\\psi ^2v)_x\\right)+2Re\\left(e^{i\\frac{c}{2}x}\\phi \\overline{u}\\right)Re\\left(e^{-i\\frac{c}{2}x}\\phi u\\right)_x,$ and so $\\int Re\\left(e^{i\\frac{c}{2}x}\\phi \\overline{u}\\frac{\\partial v}{\\partial t}\\right)dx=-3\\gamma Re\\int \\left(e^{i\\frac{c}{2}x}\\phi \\overline{u}\\right)_x\\psi ^2vdx.$ Now, we also derive, from the second equation in (REF ), $\\frac{1}{2}\\frac{d}{dt}\\int v^2dx-3\\gamma \\int (\\psi ^2v)_xvdx=2\\int Re(e^{-i\\frac{c}{2}x}\\phi u)_xvdx.$ Moreover, $\\int (\\psi ^2v)_xvdx=-\\int \\psi ^2vv_xdx=\\frac{1}{2}\\int (\\psi ^2)_xv^2dx.$ By applying Cauchy-Schwarz and Gronwall inequalities, it is now easy to obtain the following estimate for $t\\in [0,T]$ and where $G_{T}\\,:\\,\\mathbb {R}^+\\rightarrow \\mathbb {R}^+$ is a continuous function vanishing at the origin and indepedent of $\\epsilon $ : $\\Vert u_{\\epsilon }(t)\\Vert _{H^1}^2+\\Vert v_{\\epsilon }(t)\\Vert _{L^2}^2\\le G_T(\\Vert u_0\\Vert _{H^1}^2+\\Vert v_0\\Vert _{H^1}^2),\\quad t\\in [0,T].$ The first part of the Theorem is now an easy consequence of (REF ) and (REF ), since, by (REF ), there exists a subsequence of $\\lbrace (u_{\\epsilon },v_{\\epsilon })\\rbrace $ (still denoted $\\lbrace (u_{\\epsilon },v_{\\epsilon })\\rbrace $ ) and $(u,v)\\in L^{\\infty }(0,T;H^1\\times L^2)$ such that $u_{\\epsilon }\\rightharpoonup u$ in $L^{\\infty }(0,T;H^1)$ weak *; $v_{\\epsilon }\\rightharpoonup v$ in $L^{\\infty }(0,T;L^2)$ weak *; $(u,v)$ satisfies (REF ) and $(u_t,v_t)\\in L^{\\infty }(0,T;H^{-1}\\times H^{-1}).$ Hence, $u\\in C([0,T];L^2)$ , $v\\in C([0,T];H^{-1})$ , $(u(0),v(0))=(u_0,v_0)$ and $(u,v)$ is a weak solution of (REF ).", "The uniqueness follows from (REF ).", "In the case $p\\ge 0$ , we have $\\phi ^p\\in C^2(\\mathbb {R})\\cap W^{2,\\infty }$ (cf.", "Remark REF ), so we do not need to regularize $\\phi $ : we can solve directly (REF ) for initial data $(u_0,v_0)\\in H^2\\times H^1$ .", "In this case we still obtain estimates of $v_x$ , $v_t$ , $u_t$ and $u_{xx}$ in $L^2$ to prove (REF ).", "Differentiating the second equation of the system (REF ), multiplying by $v_x$ , and after a few integrations by parts, we otain $\\frac{1}{2}\\frac{d}{dt}\\int (v_x)^2dx+\\frac{15\\gamma }{2}\\int (\\psi ^2)_x(v_x)^2dx=3\\int Re\\left(e^{-i\\frac{c}{2}x}\\phi u\\right)_{xx}v_xdx.$ From (REF ) and the first equation in (REF ) we deduce, with $G_{T}\\,:\\,\\mathbb {R}^+\\rightarrow \\mathbb {R}^+$ a continuous function vanishing at the origin: $\\Vert v_x\\Vert _2^2\\le G_T(\\Vert v_0\\Vert _{H^1}^2)(\\Vert u_{xx}\\Vert _2^2+\\Vert u_x\\Vert _2^2),\\quad t\\in [0,T].$ Now, the first equation and (REF ) gives $\\Vert u_{xx}\\Vert _2^2\\le \\Vert u_t\\Vert _2^2+G_T(\\Vert (u_0,v_0)\\Vert _{H^1\\times L^2}),\\quad t\\in [0,T].$ Finally, we differentiate with respect to time the first equation of (REF ), multiply by $\\overline{u_t}$ and integrate the imaginary part to obtain $\\frac{d}{dt}\\Vert u_t\\Vert _2^2\\le C(\\Vert u\\Vert _2^2+\\Vert v\\Vert _2^2+\\Vert u_t\\Vert _2^2+\\Vert v_t\\Vert _2^2),\\quad t\\in [0,T].$ From the second equation in (REF ) we also derive $\\Vert v_t\\Vert _2^2\\le C(\\Vert v\\Vert _2^2+\\Vert v_x\\Vert _2^2+\\Vert u\\Vert _2^2+\\Vert u_x\\Vert _2^2),\\quad t\\in [0,T].$ Applying Gronwall's inequality to (REF ) and, by (REF ), (REF ), (REF ) and (REF ), we obtain the estimate (REF ).$\\blacksquare $ Acknowledgements The authors are grateful to Luis Sanchez for many discussions and were partially supported by FCT (Portuguese Foundation for Science and Technology) through the grant PEst-OE/MAT/UI0209/2011." ] ]
1403.0199
[ [ "Penner coordinates for closed surfaces" ], [ "Abstract Penner coordinates are extended to the Teichm\\\"uller spaces of oriented closed surfaces." ], [ "Introduction", "Penner coordinates in decorated Teichmüller spaces of punctured surfaces [8], [9] are distinguished by the following two remarkable properties: the mapping class group action is rational; the Weil–Petersson symplectic form is given explicitly by a simple formula.", "Due to these properties, quantum theory of Teichmüller spaces has been successfully developed in [4], [1] which resulted in construction of a one-parameter family of unitary projective mapping class group representations in infinite dimensional Hilbert spaces.", "For the fundamental groups of punctured surfaces, generalizations of Penner coordinates were constructed for the moduli spaces of faithful $SL(2,$ -representations in [7] and for the moduli spaces of irreducible but not necessarily faithful $PSL(2,\\mathbb {R})$ -representations in [5].", "In this paper, we extend Penner coordinates to the Teichmüller spaces of oriented closed surfaces of genus $g>1$ .", "Let $S$ be a closed oriented surface of genus $g>1$ , and let $R_k\\subset \\operatorname{Hom}(\\pi _1,PSL(2,\\mathbb {R})),\\quad \\pi _1\\equiv \\pi _1( S,x_0),$ be the connected component of representations of Euler number $k\\in \\mathbb {Z}$ with $|k|\\le 2g-2$ .", "According to the result of Goldman [2], the component $R_{2-2g}$ corresponds to discrete faithful representations, so that one has a principal $PSL(2,\\mathbb {R})$ -fibre bundle over the Teichmüller space $\\mathcal {T}\\equiv \\mathcal {T}(S)$ $p\\colon R_{2-2g}\\rightarrow \\mathcal {T}.$ Denoting by $\\Omega $ the space of all horocycles in the hyperbolic plane $\\mathbb {H}^2$ , we consider the associated fibre bundle $\\phi \\colon \\widetilde{\\mathcal {T}}\\rightarrow \\mathcal {T},\\quad \\widetilde{\\mathcal {T}}\\equiv R_{2-2g}\\times _{PSL(2,\\mathbb {R})}\\Omega ,$ as a substitute for Penner's decorated Teichmüller space in the case of closed surfaces.", "We define the $\\lambda $ -distance $\\lambda \\colon \\Omega \\times \\Omega \\rightarrow \\mathbb {R}_{\\ge 0}$ as follows.", "If $h,h^{\\prime }\\in \\Omega $ are based on distinct points of $\\partial \\mathbb {H}^2$ , then $\\lambda (h,h^{\\prime })$ is the hyperbolic length of the horocyclic segment between tangent points of a horocycle tangent simultaneously to both $h$ and $h^{\\prime }$ , and we define $\\lambda (h,h^{\\prime })=0$ if $h$ and $h^{\\prime }$ are based on one and the same point of $\\partial \\mathbb {H}^2$ .", "To any $\\alpha \\in \\pi _1\\setminus \\lbrace 1\\rbrace $ , we associate a function $\\lambda _\\alpha \\colon \\widetilde{\\mathcal {T}}\\rightarrow \\mathbb {R}_{\\ge 0}, \\quad [\\rho ,h]\\mapsto \\lambda (\\rho (\\alpha )h,h).$ It is easily checked that $\\lambda _\\alpha =\\lambda _{\\alpha ^{-1}}.$ The set $\\lambda _\\alpha ^{-1}(0)$ is a sub-bundle of $\\widetilde{\\mathcal {T}}$ with the fibers homemorphic to $\\mathbb {R}\\sqcup \\mathbb {R}$ .", "Moreover, one has $\\alpha \\ne \\beta \\Rightarrow \\lambda _\\alpha ^{-1}(0)\\cap \\lambda _\\beta ^{-1}(0)=\\emptyset .$ For any subset $A\\subset \\pi _1\\setminus \\lbrace 1\\rbrace $ , we associate the subset $\\widetilde{\\mathcal {T}}_A\\equiv \\cap _{\\alpha \\in A}\\lambda ^{-1}_\\alpha (\\mathbb {R}_{>0})$ together with a function $J_A\\colon \\widetilde{\\mathcal {T}}_A\\rightarrow \\mathbb {R}_{>0}^A,\\quad J_A(x)(\\alpha )=\\lambda _\\alpha (x),\\quad \\forall x\\in \\widetilde{\\mathcal {T}}_A,\\ \\forall \\alpha \\in A.$ In what follows, for any cellular complex $X$ , we will denote by $X_i$ the set of its $i$ -dimensional cells.", "We define a triangulation of $(S,x_0)$ as a cellular decomposition with only one vertex at $x_0$ and where all 2-cells are triangles.", "We denote by $\\Delta \\equiv \\Delta ( S,x_0)$ the set of all triangulations of $(S,x_0)$ .", "In principle, the characteristic maps induce orientations on all edges of a triangulation, but we will ignore this part of the information from the cellular structure.", "For any $\\tau \\in \\Delta $ , to any edge $e\\in \\tau _1$ there correspond two mutually inverse elements $\\gamma ^{\\pm 1}\\in \\pi _1$ .", "By abuse of notation, we identify $e$ with any of the functions $\\lambda _{\\gamma ^{\\pm 1}}$ : $e\\equiv \\lambda _{\\gamma ^{\\pm 1}}\\colon \\widetilde{\\mathcal {T}}\\rightarrow \\mathbb {R}_{>0}.$ For any $\\tau \\in \\Delta $ and $e\\in \\tau _1$ , we denote by $\\tau ^e$ the triangulation obtained by the diagonal flip at $e$ , with the flipped edge being denoted as $e_\\tau $ : $\\tau \\ni \\quad \\begin{tikzpicture}[scale=1.5,baseline=-3][color=gray!10] (0:1cm)--(90:1 cm)--(180:1cm)--(-90:1cm)--cycle;[auto] (90:1 cm) to node {e} (-90:1cm) ;(0:1cm) circle (.5pt)--(90:1 cm) circle (.5pt)--(180:1 cm) circle (.5pt)--(-90:1cm) circle (.5pt)--(0:1cm);\\end{tikzpicture}\\quad \\rightsquigarrow \\quad \\begin{tikzpicture}[scale=1.5,baseline=-3][color=gray!10] (0:1cm)--(90:1 cm)--(180:1cm)--(-90:1cm)--cycle;[auto](180:1 cm) to node {e_\\tau } (0:1cm) ;(0:1cm) circle (.5pt)--(90:1 cm) circle (.5pt)--(180:1 cm) circle (.5pt)--(-90:1cm) circle (.5pt)--(0:1cm);\\end{tikzpicture}\\quad \\in \\tau ^e$ It is easily shown that for any $\\tau \\in \\Delta $ , one has a finite covering $\\widetilde{\\mathcal {T}}=\\widetilde{\\mathcal {T}}_{\\tau _1}\\cup (\\cup _{e\\in \\tau _1}\\widetilde{\\mathcal {T}}_{\\tau ^e_1}).$ Our first result gives a realization of $\\widetilde{\\mathcal {T}}_{\\tau _1}$ as an algebraic subset of co-dimension one in $\\mathbb {R}_{>0}^{\\tau _1}$ .", "In more precise terms, the result follows.", "To any pair $(\\tau ,t)$ with $\\tau \\in \\Delta $ and $t\\in \\tau _2$ , we associate a function $\\psi _{\\tau ,t}\\colon \\mathbb {R}_{>0}^{\\tau _1}\\rightarrow \\mathbb {R},\\quad f\\mapsto \\sum _{t^{\\prime }\\in \\tau _2}\\epsilon _t(t^{\\prime })\\frac{a^2+b^2+c^2}{abc},$ where $a,b,c$ are the values of $f$ on three sides of $t^{\\prime }$ , while the function $\\epsilon _t\\colon \\tau _2\\rightarrow \\lbrace -1,1\\rbrace $ takes the value $-1$ on $t$ and the value 1 on all other triangles.", "We remark that $t\\ne t^{\\prime }\\Rightarrow \\psi _{\\tau ,t}^{-1}(0)\\cap \\psi _{\\tau ,t^{\\prime }}^{-1}(0)=\\emptyset .$ We also define $\\psi _\\tau \\equiv \\prod _{t\\in \\tau _2}\\psi _{\\tau ,t}.$ Theorem 1 For any $\\tau \\in \\Delta $ , the map $J_{\\tau _1}\\colon \\widetilde{\\mathcal {T}}_{\\tau _1}\\rightarrow \\mathbb {R}_{>0}^{\\tau _1}$ is an embedding with the image $\\psi _\\tau ^{-1}(0)=\\sqcup _{t\\in \\tau _2}\\psi _{\\tau ,t}^{-1}(0)$ .", "Remark 1 The transition functions $J_{\\tau _1}\\circ J_{\\tau ^e_1}^{-1}$ on the overlaps $\\widetilde{\\mathcal {T}}_{\\tau _1}\\cap \\widetilde{\\mathcal {T}}_{\\tau _1^e}$ are given by the signed Ptolemy transformation of [5] (Proposition 4) with the sign function being given by (REF ).", "This is because the inverse map $J_{\\tau _1}^{-1}$ described in Section  is based on the same combinatorial rules as those of [5].", "Let $\\mathcal {S}\\equiv \\mathcal {S}(S)$ be the set of homotopy classes of essential simple closed curves in $S$ , and $\\Delta ^\\alpha \\subset \\Delta $ the set of triangulations of the form $\\tau ^\\alpha $ with $\\tau $ having an edge representing $\\alpha $ .", "From (REF ), it is easily seen that $\\lambda _\\alpha ^{-1}(0)\\subset \\widetilde{\\mathcal {T}}_{\\tau _1},\\quad \\forall \\tau \\in \\Delta ^\\alpha .$ Our second result gives explicit coordinatization of the sub-bundles $\\lambda _\\alpha ^{-1}(0)$ together with the explicit $\\mathbb {R}_{>0}$ -action along the fibers.", "The result follows.", "For $\\alpha \\in \\mathcal {S}$ , let $\\ell _\\alpha \\colon \\mathcal {T}\\rightarrow \\mathbb {R}_{>0}$ be the hyperbolic length of the geodesic in the homotopy class of $\\alpha $ .", "Any $\\tau \\in \\Delta ^\\alpha $ has a distinguished edge $\\alpha _\\tau $ .", "Let $\\tau _\\alpha $ be the quadrilateral having $\\alpha _\\tau $ as its diagonal.", "Theorem 2 Let $\\alpha \\in \\mathcal {S}$ and $\\tau \\in \\Delta ^\\alpha $ .", "Then (i) one has the inclusion $J_{\\tau _1}(\\lambda _\\alpha ^{-1}(0))\\subset \\cup _{t\\in (\\tau _\\alpha )_2}\\psi _{\\tau ,t}^{-1}(0)$ ; (ii) for any $t\\in (\\tau _\\alpha )_2$ , the map $L_{\\alpha ,\\tau ,t}\\colon \\widetilde{\\mathcal {T}}(\\alpha ,t)\\equiv \\lambda _\\alpha ^{-1}(0)\\cap (\\psi _{\\tau ,t}\\circ J_{\\tau _1})^{-1}(0)\\rightarrow \\mathbb {R}_{>0}\\times \\mathbb {R}^{\\tau _1\\setminus t_1}_{>0}\\\\m\\mapsto (\\ell _\\alpha (\\phi (m)),J_{\\tau _1\\setminus t_1}(m))$ is a homeomorphism; (iii) For any $d\\in \\mathbb {R}_{>0}$ one has the following equivalence $\\phi (m)=\\phi (m^{\\prime })\\Leftrightarrow \\exists \\ c\\in \\mathbb {R}_{>0}\\colon J_{\\tau _1\\setminus t_1}(m)=c\\,J_{\\tau _1\\setminus t_1}(m^{\\prime }),\\\\\\forall m,m^{\\prime }\\in (\\ell _\\alpha \\circ \\phi )^{-1}(d)\\cap \\widetilde{\\mathcal {T}}(\\alpha ,t).$ Remark 2 The space $\\widetilde{\\mathcal {T}}(\\alpha ,t)$ in Theorem REF is a connected component of $\\lambda _\\alpha ^{-1}(0)$ .", "It can also be singled out by fixing an orientation on $\\alpha $ , and considering only the classes $[\\rho ,h]$ , with $h$ based on the attracting fixed point of $\\rho (\\alpha )$ .", "The paper is organized as follows.", "In Section  we collect necessary material on the group $PSL(2,\\mathbb {R})$ and we prove the important Lemmas REF –REF .", "Sections  and contain proofs of Theorems REF and REF respectively." ], [ "Acknowledgements", "This work is supported in part by Swiss National Science Foundation.", "Some of the results were reported at the Oberwolfach workshop “New Trends in Teichmüller Theory and Mapping Class Groups\" in February 2014.", "I would like to thank the participants of this workshop for useful and helpful discussions, especially J. Andersen, M. Burger, L. Chekhov, V. Fock, L. Funar, W. Goldman, N. Kawazumi, F. Luo, G. Masbaum, N. Reshetikhin, R. van der Veen, A. Virelizier, A. Wienhard." ], [ "Factorization in $SL(2)$", "The matrix coefficients of the group $SL(2,\\mathbb {R})$ are the mappings $a,b,c,d\\colon SL(2,\\mathbb {R})\\rightarrow \\mathbb {R}$ such that $g=\\begin{pmatrix}a(g)&b(g)\\\\c(g)&d(g)\\end{pmatrix},\\quad \\forall g\\in SL(2,\\mathbb {R}).$ We fix two group embeddings $u,v\\colon \\mathbb {R}\\rightarrow SL(2,\\mathbb {R})$ defined by $u(x)=\\begin{pmatrix}1&x\\\\0&1\\end{pmatrix},\\quad v(x)=\\begin{pmatrix}1&0\\\\x&1\\end{pmatrix}.$ It is easily verified that an element $g\\in SL(2,\\mathbb {R})$ with nonzero left lower coefficient, i.e.", "$c(g)\\ne 0$ , is uniquely factorized as follows: $g=u(x)v(y)u(z)=\\begin{pmatrix}1+xy&x+z+xyz\\\\y&1+yz\\end{pmatrix}$ where $x=(a(g)-1)c(g)^{-1},\\quad y=c(g),\\quad z=c(g)^{-1}(d(g)-1).$ Moreover, for any $(x,y,z)\\in \\mathbb {R}^3_{>0}$ , there exists a unique triple $(x^{\\prime },y^{\\prime },z^{\\prime })\\in \\mathbb {R}^3_{>0}$ such that $v(x)u(y)v(z)=u(z^{\\prime })v(y^{\\prime })u(x^{\\prime }).$ Explicitly, we have $(x^{\\prime },y^{\\prime },z^{\\prime })=\\left((x+xyz+z)^{-1}xy,x+xyz+z,yz(x+xyz+z)^{-1}\\right)$ Remark 3 The map $R\\colon \\mathbb {R}^3_{>0}\\rightarrow \\mathbb {R}^3_{>0},\\quad (x,y,z)\\mapsto (x^{\\prime },y^{\\prime },z^{\\prime })$ is an involution which solves the set-theoretical tetrahedron equation $R_{123}\\circ R_{145}\\circ R_{246}\\circ R_{356}=R_{356}\\circ R_{246}\\circ R_{145}\\circ R_{123}.$ This solution is related with the star-triangle transformation in electrical networks [3], [6]." ], [ "Universal covering $\\widetilde{SL}(2,\\mathbb {R})$", "Let us define two coordinate charts covering the group manifold of $PSL(2,\\mathbb {R})$ .", "We define two open contractible sets $U_1\\equiv PSL(2,\\mathbb {R})\\setminus |a|^{-1}(0),\\quad U_2\\equiv PSL(2,\\mathbb {R})\\setminus |b|^{-1}(0)$ together with the homeomorphisms $\\varphi _j\\colon U_j\\rightarrow \\mathbb {H}^3,\\ j\\in \\lbrace 1,2\\rbrace ,\\quad \\varphi _1=\\left(\\frac{b}{a},\\frac{c}{a}, |a|\\right),\\ \\varphi _2=\\left(\\frac{a}{b},\\frac{d}{b}, |b|\\right).$ The intersection $U_1\\cap U_2$ consists of two contractible components $U_1\\cap U_2=U_{12}^+\\sqcup U_{12}^-,\\quad U_{12}^\\pm \\equiv \\lbrace \\pm ab>0\\rbrace .$ Let $p\\colon \\widetilde{SL}(2,\\mathbb {R})\\rightarrow PSL(2,\\mathbb {R})$ be the canonical projection from the universal covering space.", "We fix a group isomorphism $\\Phi \\colon \\mathbb {Z}\\rightarrow p^{-1}(\\pm 1)\\simeq \\pi _1(PSL(2,\\mathbb {R}),\\pm 1)$ sending an integer $n$ to the homotopy class of the loop $\\omega _n\\colon [0,1]\\ni t\\mapsto \\pm \\begin{pmatrix}\\cos (n\\pi t)&\\sin (n\\pi t)\\\\-\\sin (n\\pi t)&\\cos (n\\pi t)\\end{pmatrix}.$ We remark that for $n=1$ we have the following inclusions $\\omega _1\\left([0,1]\\setminus \\lbrace 1/2\\rbrace \\right)\\subset U_1,\\quad \\omega _1\\left(]0,1[\\right)\\subset U_2,\\\\\\omega _1\\left(]0,1/2[\\right)\\subset U_{12}^+,\\quad \\omega _1\\left(]1/2,1[\\right)\\subset U_{12}^-.$ For any $x\\in \\mathbb {R}$ , we fix the lifts $\\widetilde{u}(x)$ , $\\widetilde{v}(x)$ to $\\widetilde{SL}(2,\\mathbb {R})$ represented by the paths $\\widetilde{u}(x),\\ \\widetilde{v}(x)\\colon [0,1]\\rightarrow SL(2,\\mathbb {R}),\\quad \\widetilde{u}(x)(t)=u(xt),\\ \\widetilde{v}(x)(t)=v(xt).$ Lemma 1 For any $ x\\in \\mathbb {R}_{>0}$ , let $\\gamma _x$ be the lift of the left hand side of the $SL(2,\\mathbb {R})$ -identity $(v(-x)u(2/x))^2=-1$ obtained by using the lifts (REF ).", "Then the path homotopy class of $\\gamma _x$ is given by the class $\\Phi (1)$ .", "By using (REF ), we have $\\gamma _{x}\\equiv (\\widetilde{v}(-x)\\widetilde{u}(2/x))^2\\colon [0,1]\\rightarrow SL(2,\\mathbb {R}),\\quad t\\mapsto (v(-xt)u(2t/x))^2\\\\=\\begin{pmatrix}1&2t/x\\\\-tx&1-2t^2\\end{pmatrix}^2=\\begin{pmatrix}1-2t^2&4t(1-t^2)/x\\\\-2t(1-t^2)x&1-6t^2+4t^4\\end{pmatrix}$ which has the properties $\\gamma _{x}\\left([0,1]\\setminus \\lbrace 1/\\sqrt{2}\\rbrace \\right)\\subset U_1,\\quad \\gamma _{x}\\left(]0,1[\\right)\\subset U_2,\\\\\\gamma _{x}\\left(]0,1/\\sqrt{2}[\\right)\\subset U_{12}^{+},\\quad \\gamma _{x}\\left(]1/\\sqrt{2},1[\\right)\\subset U_{12}^{-}.$ By comparing (REF ) with (REF ), we conclude that the path homotopy class of $\\gamma _{x}$ coincides with that of $\\omega _1$ .", "Lemma 2 For any $x=(x_1,x_2,x_3)\\in \\mathbb {R}^3_{>0}$ , let $x^{\\prime }\\in \\mathbb {R}_{>0}^3$ be the unique point such that $v(x_1)u(x_2)v(x_3)u(-x_3^{\\prime })v(-x_2^{\\prime })u(-x_1^{\\prime })=1.$ Let $\\alpha _{x}$ be the lift of the left hand side of (REF ) obtained by using the lifts (REF ).", "Then the path homotopy class of $\\alpha _{x}$ is given by the class $\\Phi \\left(0\\right)$ .", "We just remark that the one parameter family of loops $\\lbrace f_t=\\alpha _{tx}\\rbrace _{t\\in [0,1]}$ is a well defined path homotopy between $\\alpha _{x}$ and the constant path.", "Lemma 3 For any $x=(x_1,x_2,x_3)\\in \\mathbb {R}_{>0}^3$ , let $\\bar{x}\\in \\mathbb {R}^3$ be the unique point such that $v(x_1)u(-\\bar{x}_3)v(x_2)u(-\\bar{x}_1)v(x_3)u(-\\bar{x}_2)=\\epsilon \\in \\lbrace -1,1\\rbrace .$ Let $\\beta _{x,\\epsilon }$ be the lift of the left hand side of (REF ) obtained by using the lifts (REF ).", "Then the path homotopy class of $\\beta _{x,\\epsilon }$ is given by the class $\\Phi \\left(-(3+\\epsilon )/2\\right)$ .", "Let $n\\equiv \\Phi ^{-1}([\\beta _{x,\\epsilon }])$ .", "We treat the different values of $\\epsilon $ differently.", "In the case $\\epsilon =1$ , let $i\\in \\lbrace 1,2,3\\rbrace $ be such that $x_i=\\max (x_1,x_2,x_3).$ By making two substitutions $v(x_i)\\rightsquigarrow u(2/x_i)v(-x_i)u(2/x_i),\\quad u(-\\bar{x}_i)\\rightsquigarrow v(-2/\\bar{x}_i)u(\\bar{x}_i)v(-2/\\bar{x}_i)$ in the identity (REF ) and by using cyclic permutations of factors (which do not change the lift) and group homomorphism properties of the maps $u$ and $v$ , we transform (REF ) into a case of identity (REF ).", "By Lemma REF , each substitution in (REF ) increases by one the integer associated to the homotopy class of its lift so that the class of the lift after these two substitutions is given by $\\Phi (n+2)$ .", "On the other hand, by Lemma REF , the same class is given by $\\Phi (0)$ .", "Thus, we arrive at the conclusion that $n+2=0$ .", "The case $\\epsilon =-1$ is treated differently depening on existence or non-existence of Euclidean triangles with side lengths given by the components of $x$ .", "Assume first that there are no such triangles.", "It means that there exists an index $i\\in \\lbrace 1,2,3\\rbrace $ such that $x_i> x_j+x_k$ , where $\\lbrace j,k\\rbrace =\\lbrace 1,2,3\\rbrace \\setminus \\lbrace i\\rbrace $ .", "In that case, the substitution $v(x_i)\\rightsquigarrow u(2/x_i)v(-x_i)u(2/x_i)$ followed by cyclic permutations of factors and subsequent simplifications of products of $u$ -terms transforms identity (REF ) into a case of identity (REF ).", "By a similar reasoning as above for the case with $\\epsilon =1$ , we conclude that $n+1=0$ .", "Assume now that there exists an Euclidean triangle of non-zero area with side lengths given by the components of $x$ .", "In that case, we choose arbitrary index $i\\in \\lbrace 1,2,3\\rbrace $ and make two substitutions $v(x_j)\\rightsquigarrow u(2/x_j)v(-x_j)u(2/x_j),\\quad v(x_k)\\rightsquigarrow u(2/x_k)v(-x_k)u(2/x_k)$ with $\\lbrace j,k\\rbrace =\\lbrace 1,2,3\\rbrace \\setminus \\lbrace i\\rbrace $ .", "Applying necessary cyclic permutations and simplifications of $u$ -terms we arrive at an identity of the same form as (REF ) but with negated components $x_j$ and $x_k$ and with the accordingly modified point $\\bar{x}$ .", "In this new identity, the substitution $u(-\\bar{x}_i)\\rightsquigarrow v(-2/\\bar{x}_i)u(\\bar{x}_i)v(-2/\\bar{x}_i),$ followed by cyclic permutations of factors and subsequent simplifications of products of $v$ -terms, brings it to a case of identity (REF ).", "Under the fist two substitutions (REF ), the integer $n$ is increased by two, while under the last substitution (REF ) it is reduced by one with the final value being $n+2-1=n+1$ .", "Again, by Lemma REF , we conclude that $n+1=0$ .", "Finally, it remains the degenerate case where there exists $i\\in \\lbrace 1,2,3\\rbrace $ such that $x_i=x_j+x_k$ , where $\\lbrace j,k\\rbrace =\\lbrace 1,2,3\\rbrace \\setminus \\lbrace i\\rbrace $ .", "That means that $\\bar{x}_i=0$ , which allows to reduce identity (REF ) to the inverse of (REF ).", "Thus, by Lemma REF , $n=-1$ in this case as well." ], [ "Proof of Theorem ", "Let $\\tau \\in \\Delta $ .", "By cutting out a small open disk $D\\subset S$ centered at $x_0$ , we obtain a cellular decomposition $\\bar{\\tau }$ of $S^{\\prime }\\equiv S\\setminus D$ , where the vertex set is given by the intersection points of edges of $\\tau $ with the boundary of $S^{\\prime }$ , with two types of edges: long edges given by the remnants of the edges of $\\tau $ and short edges given by the boundary segments between the vertices, and hexagonal 2-cells given by truncated triangles of $\\tau $ .", "We assume that the short edges are canonically oriented through the counterclockwise orientation of the boundary of $D$ .", "In what follows, we will abuse the notation by identifying the long edges of $\\bar{\\tau }$ with the edges of $\\tau $ and the hexagonal faces of $\\bar{\\tau }$ with the triangular faces of $\\tau $ , and also we will think of the elements of $\\tau _1$ and $\\tau _2$ as functions on the set $\\mathbb {R}_{>0}^{\\tau _1}\\times \\lbrace -1,1\\rbrace ^{\\tau _2}$ in the sense that $e\\colon \\mathbb {R}_{>0}^{\\tau _1}\\times \\lbrace -1,1\\rbrace ^{\\tau _2}\\rightarrow \\mathbb {R}_{>0}, \\quad (f,\\varepsilon )\\mapsto f(e),\\quad \\forall e\\in \\tau _1,$ and $t\\colon \\mathbb {R}_{>0}^{\\tau _1}\\times \\lbrace -1,1\\rbrace ^{\\tau _2}\\rightarrow \\lbrace -1,1\\rbrace , \\quad (f,\\varepsilon )\\mapsto \\varepsilon (t),\\quad \\forall t\\in \\tau _2.$ Depending on the value of $t$ , the triangle will be called positive or negative.", "Following [5], we start by constructing a map $\\xi _\\tau \\colon \\mathbb {R}_{>0}^{\\tau _1}\\times \\lbrace -1,1\\rbrace ^{\\tau _2}\\rightarrow \\operatorname{Hom}(\\pi _1(S^{\\prime },\\bar{\\tau }_0),PSL(2,\\mathbb {R}))$ where $\\pi _1(S^{\\prime },\\bar{\\tau }_0)$ is the fundamental groupoid of $S^{\\prime }$ with the vertex set $\\bar{\\tau }_0$ .", "The construction is as follows.", "To any long edge $e$ , we associate the element $\\pm w(e)$ where $w(e)\\equiv \\begin{pmatrix}0&-e^{-1}\\\\e&0\\end{pmatrix}.$ As the element $\\pm w(e)$ is of order two, our assignment is valid for both orientations of $e$ .", "For any short edge $e^{\\prime }$ (with the clockwise orientation with respect to the center of the hexagon to which it belongs) we associate the element $\\pm u\\!\\left(t\\frac{a}{bc}\\right)$ , with $t$ being the unique hexagonal face having $e^{\\prime }$ as its side, $a$ is the long side of $t$ opposite to $e^{\\prime }$ , while $b$ and $c$ are two other long sides of $t$ .", "These assignments are illustrated in this picture: $\\begin{tikzpicture}[scale=1.5,baseline=-3,decoration={markings,mark=at position .5 with {{stealth};}}][color=gray!10] (20:1cm)--(100:1 cm)--(140:1 cm)--(-140:1cm)--(-100:1 cm)--(-20:1 cm)--cycle;[auto,color=green,postaction={decorate}] (20:1cm) to node{\\pm u\\!\\left(t\\frac{a}{bc}\\right)} (-20:1cm);[color=green,postaction={decorate}] (140:1 cm)--(100:1 cm);[color=green,postaction={decorate}] (-100:1 cm)--(-140:1 cm);[auto] (20:1cm) circle (.5pt) to node {b} (100:1 cm) circle (.5pt) (140:1 cm) circle (.5pt) to node {a} node[swap]{\\pm w(a)} (-140:1 cm) circle (.5pt)(-100:1cm) circle (.5pt) to node {c} (-20:1 cm) circle (.5pt);\\node at (0,0){t};\\end{tikzpicture}$ where long edges are drawn in black and short edges in green.", "It is straightforward to check that this assignment uniquely extends to a representation of $\\pi _1(S^{\\prime },\\bar{\\tau }_0)$ .", "For $(f,\\varepsilon )\\in \\xi _\\tau ^{-1}(R_{2-2g})$ , by taking the equivalence class of the pair $(\\xi _\\tau (f,\\varepsilon ), h_0)$ , where $h_0$ is the horocycle based at $\\infty $ and passing through $i\\in \\mathbb {H}^2$ , we obtain a map $\\tilde{\\xi }_{\\tau }\\colon \\xi _\\tau ^{-1}(R_{2-2g}) \\rightarrow \\widetilde{\\mathcal {T}}$ which will be shown to be the inverse of $J_{\\tau _1}$ ." ], [ "Calculation of the Euler number", "In the case $g=w(a)$ , the factorization (REF ), (REF ) takes the form $w(a)=u(-1/a)v(a)u(-1/a),\\quad \\forall a\\in \\mathbb {R}_{>0}.$ We implement this factorization combinatorially by transforming $\\bar{\\tau }$ to a new cellular complex $\\tilde{\\tau }$ as follows.", "We insert two new vertices in each short edge of $\\bar{\\tau }$ and connect them inside each hexagonal face by three oriented edges parallel to long edges, the orientations being counterclockwise with respect to the center of the hexagon.", "We associate to these edges the elements of $\\pm v(a)$ where the argument $a$ is the corresponding long edge.", "The following picture summarizes this subdivision: $\\bar{\\tau }\\ni \\quad \\begin{tikzpicture}[scale=1.5,baseline=-3,decoration={markings,mark=at position .5 with {{stealth};}}][color=gray!10] (20:1cm)--(100:1 cm)--(140:1 cm)--(-140:1cm)--(-100:1 cm)--(-20:1 cm)--cycle;[color=green,postaction={decorate}] (140:1 cm)--(100:1 cm);[color=green,postaction={decorate}] (-100:1 cm)--(-140:1 cm);[color=green,postaction={decorate}] (20:1 cm)--(-20:1 cm);(20:1cm) circle (.5pt)--(100:1 cm) circle (.5pt) (140:1 cm) circle (.5pt)(-140:1 cm) circle (.5pt)(-100:1cm) circle (.5pt)--(-20:1 cm) circle (.5pt);[auto] (140:1cm) to node[swap] {a} (-140:1cm);\\end{tikzpicture}\\quad \\rightsquigarrow \\quad \\begin{tikzpicture}[scale=1.5,baseline=-3,decoration={markings,mark=at position .5 with {{stealth};}}][color=gray!10] (20:1cm)--(100:1 cm)--(140:1 cm)--(-140:1cm)--(-100:1 cm)--(-20:1 cm)--cycle;[color=green,postaction={decorate}] (140:1 cm)--(100:1 cm);[color=green,postaction={decorate}] (-100:1 cm)--(-140:1 cm);[color=green,postaction={decorate}] (20:1 cm)--(-20:1 cm);(20:1cm) circle (.5pt)--(100:1 cm) circle (.5pt) (140:1 cm) circle (.5pt)(-140:1 cm) circle (.5pt)(-100:1cm) circle (.5pt)--(-20:1 cm) circle (.5pt);[auto] (140:1cm) to node[swap] {a} (-140:1cm);[color=blue] ((-20:1cm)!", ".2!", "(20:1cm)) circle (.5pt) ((-100:1cm)!", ".2!", "(-140:1cm)) circle (.5pt)((100:1cm)!", ".8!", "(140:1cm)) circle (.5pt) ((-100:1cm)!", ".8!", "(-140:1cm)) circle (.5pt)((-20:1cm)!", ".8!", "(20:1cm)) circle (.5pt)((100:1cm)!", ".2!", "(140:1cm)) circle (.5pt);[color=blue,postaction={decorate}] ((-100:1cm)!", ".2!(-140:1cm))--((-20:1cm)!", ".2!", "(20:1cm)) ;[auto,color=blue,postaction={decorate}] ((100:1cm)!", ".8!", "(140:1cm)) to node {\\pm v(a)} ((-100:1cm)!", ".8!", "(-140:1cm));[color=blue,postaction={decorate}] ((-20:1cm)!", ".8!(20:1cm))--((100:1cm)!", ".2!", "(140:1cm));\\end{tikzpicture}$ where the added vertices and edges are drawn in blue.", "In this subdivided complex, the long edges of $\\bar{\\tau }$ will be called primary long edges while the newly added edges will be called secondary long edges.", "There are now two types of 2-cells: rectangular and hexagonal faces.", "Rectangular faces come naturally in pairs where each pair is associated with a unique primary long edge.", "Within each such pair, let us glue two rectangular faces along their common primary long sides and then erase the primary long edge as is described in this picture: $\\begin{tikzpicture}[scale=1.5,baseline=-3,decoration={markings,mark=at position .5 with {{stealth};}}][color=gray!10] (0:1cm)--(90:1cm)--(180:1cm)--(-90:1cm)--cycle;[color=green] (0:1cm)--(-90:1cm) (180:1cm)--(90:1cm);[color=blue,postaction={decorate}] (-90:1cm)--(180:1cm);[color=blue,postaction={decorate}] (90:1cm)--(0:1cm);[auto] ((180:1cm)!", ".5!", "(90:1cm)) circle (.5pt) to node {a} ((-90:1cm)!", ".5!", "(0:1cm)) circle (.5pt);[color=blue] (0:1cm) circle (.5pt) (90:1cm) circle (.5pt) (180:1cm) circle (.5pt) (-90:1cm) circle (.5pt);\\end{tikzpicture}\\quad \\rightsquigarrow \\quad \\begin{tikzpicture}[scale=1.5,baseline=-3,decoration={markings,mark=at position .5 with {{stealth};}}][color=gray!10] (0:1cm)--(90:1cm)--(180:1cm)--(-90:1cm)--cycle;[color=green] (0:1cm)--(-90:1cm) (180:1cm)--(90:1cm);[color=blue,postaction={decorate}] (-90:1cm)--(180:1cm);[color=blue,postaction={decorate}] (90:1cm)--(0:1cm);\\node at (0,0){a};[color=blue] (0:1cm) circle (.5pt) (90:1cm) circle (.5pt) (180:1cm) circle (.5pt) (-90:1cm) circle (.5pt);\\end{tikzpicture}\\quad \\in \\tilde{\\tau }$ The result is our transformed complex $\\tilde{\\tau }$ which has rectangular faces which are in bijection with the edges of $\\tau $ and and hexagonal faces which are in bijection with the triangles of $\\tau $ .", "By taking into account the group elements associated with the edges, each rectangular face of $\\tilde{\\tau }$ corresponds to a case of the relation (REF ) where variable $x$ is given by the positive number associated with the corresponding primary edge, while each hexagonal face of $\\tilde{\\tau }$ corresponds to a case of the relation (REF ) where three components of the vector $x$ are given by three positive numbers associated with three (primary) long edges around the hexagon and $\\epsilon $ being given by the value of the corresponding function $t\\in \\tau _2$ .", "Under the lifts (REF ), each face contributes an integer to the Euler number of the representation.", "The contributions are controlled by Lemma REF for the rectangular faces and by Lemma REF for the hexagonal faces.", "Let $N_-$ and $N_+$ be the numbers of negative and positive triangles respectively.", "By Lemma REF , the total contribution from all rectangular faces is the number of edges of $\\tau $ , i.e.", "$6g-3$ , while, by Lemma REF , the total contribution of the hexagonal faces is $-N_--2N_+$ .", "Thus, the Euler number of $\\xi _\\tau (f,\\varepsilon )$ is calculated as follows: $e(\\xi _\\tau (f,\\varepsilon ))=6g-3-N_- -2N_+=1+N_--2g,$ where we have taken into account the equality $N_-+N_+=4g-2$ .", "Thus, the subset $\\xi _\\tau ^{-1}(R_{2-2g})$ is completely characterized by the condition $N_-=1$ , i.e.", "that there is only one negative triangle, and the vanishing condition for the boundary holonomy which takes the form $\\sum _{t\\in \\tau _2}t\\frac{p_t}{q_t}=0,\\quad p_t\\equiv \\sum _{e\\in t_1}e^2,\\quad q_t\\equiv \\prod _{e\\in t_1}e.$ These two conditions, in their turn, are equivalent to a single vanishing condition $\\psi _\\tau =0$ in $\\mathbb {R}_{>0}^{\\tau _1}$ with $\\psi _\\tau \\equiv \\prod _{t\\in \\tau _2}\\psi _{\\tau ,t},\\quad \\psi _{\\tau ,t}\\equiv -\\frac{p_t}{q_t}+\\sum _{s\\in \\tau _2\\setminus \\lbrace t\\rbrace }\\frac{p_s}{q_s}.$ Thus, by taking into account the intersection properties (REF ), we conclude that we have a natural identification $\\xi _\\tau ^{-1}(R_{2-2g})\\simeq \\psi _\\tau ^{-1}(0).$" ], [ "Verification of the equality $\\tilde{\\xi }_{\\tau }=J_{\\tau _1}^{-1}$", "The equality $J_{\\tau _1}(\\tilde{\\xi }_{\\tau }(f))=f,\\quad \\forall f\\in \\psi _\\tau ^{-1}(0),$ is checked straightforwardly, while the reverse equality $\\tilde{\\xi }_{\\tau }(J_{\\tau _1}(m))=m,\\quad \\forall m\\in \\widetilde{\\mathcal {T}}_{\\tau _1},$ is checked by choosing a representative $(\\rho ,h_0)$ of $m$ where $h_0$ is the horocycle based at $\\infty $ and passing through $i\\in \\mathbb {H}^2$ .", "For such a representative, the representation $\\rho $ is defined uniquely up to conjugation by elements of the group of upper triangular unipotent matrices (the stabilizer subgroup for $h_0$ ).", "For each triangle $t$ of $\\tau $ , we take three $PSL(2,\\mathbb {R})$ -elements representing its three oriented sides, the orientations being chosen cyclically in the counter-clockwise direction with respect to the center of $t$ , and we choose the $SL(2,\\mathbb {R})$ -representatives of those elements which have positive left lower matrix elements, i.e.", "$c>0$ .", "We associate to $t$ the sign of the cyclic product of those representatives along the boundary of $t$ , thus obtaining a map $\\varepsilon _{m}\\colon \\tau _2\\rightarrow \\lbrace -1,1\\rbrace .$ We also apply the factorization formula (REF ) to each of those representatives thus realizing $\\rho $ as a representation of the form $\\xi _\\tau (f,\\varepsilon _m)$ .", "The Euler number calculation (REF ) implies that $\\varepsilon _{m}=\\epsilon _t$ for some $t\\in \\tau _2$ ." ], [ "Signed Ptolemy transformation", "The transition functions $J_{\\tau _1}\\circ J_{\\tau ^e_1}^{-1}$ on the overlaps $\\widetilde{\\mathcal {T}}_{\\tau _1}\\cap \\widetilde{\\mathcal {T}}_{\\tau _1^e}$ are given by the same signed Ptolemy transformation as in [5] (Proposition 4).", "This is a consequence of the definition of the map $\\xi _\\tau $ based on the same rules (REF ) of assigning group elements on the edges of truncated triangulations.", "Below, following [5], we derive the signed Ptolemy transformation.", "If two pairs with one and the same horocyclic components represent the same point in $\\widetilde{\\mathcal {T}}$ , then the representation components are conjugated by an element of the stabilizer subgroup of the common horocycle which is a parabolic subgroup isomorphic to $\\mathbb {R}$ .", "That means that the parallel transport operators on the short edges of the truncated triangulations, being in the same subgroup, must be the same independently of the triangulation.", "By using the rules (REF ), we can write, for example, the equality for the parallel transport operators along the short edge between the long edges $a$ and $b$ on two sides of this picture $\\tau \\ni \\quad \\begin{tikzpicture}[scale=1.5,baseline=-3,decoration={markings,mark=at position .5 with {{stealth};}}][color=gray!10] (0:1cm)--(45:1 cm)--(90:1cm)--(135:1cm)--(180:1cm)--(-135:1 cm)--(-90:1cm)--(-45:1cm)--cycle;[color=green,postaction={decorate}] (180:1 cm)--(135:1 cm);[color=green,postaction={decorate}] (0:1 cm)--(-45:1 cm);[color=green,postaction={decorate}] (90:1 cm)--((90:1cm)!", ".5!(45:1cm));[color=green,postaction={decorate}]((90:1cm)!", ".5!", "(45:1cm))-- (45:1 cm);[color=green,postaction={decorate}] (-90:1 cm)--((-90:1cm)!", ".5!(-135:1cm));[color=green,postaction={decorate}]((-90:1cm)!", ".5!", "(-135:1cm))-- (-135:1 cm);\\node at (157.5:.5cm){\\alpha };\\node at (-22.5:.5cm){\\beta };\\node (c) at (0,0){e};[auto](0:1 cm)to node[swap] {b}(45:1cm)(90:1cm)to node[swap] {a}(135:1cm)(180:1cm)to node [swap]{d}(-135:1cm)(-90:1cm)to node[swap] {c}(-45:1cm) ;(0:1cm) circle (.5pt)(45:1 cm) circle (.5pt)(90:1 cm) circle (.5pt)(135:1cm) circle (.5pt)(180:1cm) circle (.5pt)(-135:1 cm) circle (.5pt)(-90:1 cm) circle (.5pt)(-45:1cm) circle (.5pt);((-90:1cm)!", ".5!", "(-135:1cm)) circle (.5pt) --(c)-- ((90:1cm)!", ".5!", "(45:1cm)) circle (.5pt);\\end{tikzpicture}\\quad \\leftrightsquigarrow \\quad \\begin{tikzpicture}[scale=1.5,baseline=-3,decoration={markings,mark=at position .5 with {{stealth};}}][color=gray!10] (0:1cm)--(45:1 cm)--(90:1cm)--(135:1cm)--(180:1cm)--(-135:1 cm)--(-90:1cm)--(-45:1cm)--cycle;[color=green,postaction={decorate}] (90:1 cm)--(45:1 cm);[color=green,postaction={decorate}] (-90:1 cm)--(-135:1 cm);[color=green,postaction={decorate}] (0:1 cm)--((0:1cm)!", ".5!(-45:1cm));[color=green,postaction={decorate}]((0:1cm)!", ".5!", "(-45:1cm))-- (-45:1 cm);[color=green,postaction={decorate}] (180:1 cm)--((180:1cm)!", ".5!(135:1cm));[color=green,postaction={decorate}]((180:1cm)!", ".5!", "(135:1cm))-- (135:1 cm);\\node at (67.5:.5cm){\\gamma };\\node at (-112.5:.5cm){\\delta };\\node (c) at (0,0){f};[auto](0:1 cm)to node[swap] {b}(45:1cm)(90:1cm)to node[swap] {a}(135:1cm)(180:1cm)to node [swap]{d}(-135:1cm)(-90:1cm)to node[swap] {c}(-45:1cm) ;(0:1cm) circle (.5pt)(45:1 cm) circle (.5pt)(90:1 cm) circle (.5pt)(135:1cm) circle (.5pt)(180:1cm) circle (.5pt)(-135:1 cm) circle (.5pt)(-90:1 cm) circle (.5pt)(-45:1cm) circle (.5pt);((180:1cm)!", ".5!", "(135:1cm)) circle (.5pt)--(c)--((0:1cm)!", ".5!", "(-45:1cm)) circle (.5pt);\\end{tikzpicture}\\quad \\in \\tau ^e$ The result reads $\\pm u\\left(\\alpha \\frac{d}{ae}\\right)u\\left(\\beta \\frac{c}{eb}\\right)=\\pm u\\left(\\gamma \\frac{f}{ab}\\right)\\Leftrightarrow \\alpha \\frac{d}{ae}+\\beta \\frac{c}{eb}=\\gamma \\frac{f}{ab}\\\\\\Leftrightarrow \\alpha bd+\\beta ac=\\gamma ef$ Doing the same calculation for the short edge between the long edges $c$ and $d$ , we also have $\\pm u\\left(\\beta \\frac{b}{ce}\\right)u\\left(\\alpha \\frac{a}{ed}\\right)=\\pm u\\left(\\delta \\frac{f}{cd}\\right)\\Leftrightarrow \\beta \\frac{b}{ce}+\\alpha \\frac{a}{ed}=\\delta \\frac{f}{cd}\\\\\\Leftrightarrow \\beta bd+\\alpha ac=\\delta ef$ The two equalities can equivalently be rewritten as $\\alpha bd+\\beta ac=\\gamma ef,\\quad \\alpha \\beta =\\gamma \\delta .$ This is exactly the signed Ptolemy relation of [5]." ], [ "Part (i)", "Given $m\\in \\lambda _\\alpha ^{-1}(0)$ .", "Let us choose $\\tau \\in \\Delta ^\\alpha $ .", "The signed Ptolemy transformation formula (REF ) implies that $\\prod _{t\\in (\\tau _\\alpha )_2}\\varepsilon _m(t)=-1,$ see (REF ) for the definition of the function $\\varepsilon _m$ .", "That means that one of the triangles of $\\tau _\\alpha $ is necessarily negative which proves part (i)." ], [ "Part (ii)", "Let $t\\in (\\tau _\\alpha )_2$ be the negative triangle, and let $(\\tau _\\alpha )_1= \\lbrace \\alpha _\\tau , a,b,c,d\\rbrace $ be arranged as in this picture $\\begin{tikzpicture}[scale=1.5,baseline=-3][color=gray!10] (0:1cm)--(90:1 cm)--(180:1cm)--(-90:1cm)--cycle;\\node at (0,.4){t};[auto](180:1 cm)to node[swap] {\\alpha _\\tau }(0:1cm)(0:1cm)to node[swap] {b}(90:1cm)to node [swap]{a}(180:1cm)to node[swap] {d}(-90:1cm)to node[swap] {c}(0:1cm) ;(0:1cm) circle (.5pt)(90:1 cm) circle (.5pt)(180:1 cm) circle (.5pt)(-90:1cm) circle (.5pt);\\end{tikzpicture}$ Apart from equality (REF ), the signed Ptolemy relation also implies that $\\frac{a}{d}=\\frac{b}{c}\\equiv x$ which we can solve for $a$ and $b$ : $a=x d,\\quad b=x c.$ Here we assume that the sides of the quadrilateral $\\tau _\\alpha $ are geometrically pairwise distinctIt is possible that one pair of opposite sides of $\\tau _\\alpha $ are geometrically identical, for example, $b=d$ .", "In that case, instead of (REF ) one will have $a=x^2c$ and $b=d=cx$ .. By substituting (REF ) into $\\psi _{\\tau ,t}$ and writing out explicitly the contributions from the quadrilateral $\\tau _\\alpha $ , we calculate $\\psi _{\\tau ,t}=-\\frac{a}{b\\alpha _\\tau }-\\frac{b}{a\\alpha _\\tau }-\\frac{\\alpha _\\tau }{ab}+\\frac{c}{d\\alpha _\\tau }+\\frac{d}{c\\alpha _\\tau }+\\frac{\\alpha _\\tau }{cd}+\\sum _{s\\in \\tau _2\\setminus (\\tau _\\alpha )_2}\\frac{p_s}{q_s}\\\\=-\\frac{d}{c\\alpha _\\tau }-\\frac{c}{d\\alpha _\\tau }-\\frac{\\alpha _\\tau }{cdx^2}+\\frac{c}{d\\alpha _\\tau }+\\frac{d}{c\\alpha _\\tau }+\\frac{\\alpha _\\tau }{cd}+\\sum _{s\\in \\tau _2\\setminus (\\tau _\\alpha )_2}\\frac{p_s}{q_s}\\\\=-\\frac{\\alpha _\\tau }{cd}\\left(x^{-2}-1\\right)+\\sum _{s\\in \\tau _2\\setminus (\\tau _\\alpha )_2}\\frac{p_s}{q_s}.$ The equality $\\psi _{\\tau ,t}=0$ implies that $x<1$ , and, as the other terms do not contain variable $\\alpha _\\tau $ , we can solve it explicitly for $\\alpha _\\tau $ : $\\alpha _\\tau =\\frac{cd}{x^{-2}-1}\\sum _{s\\in \\tau _2\\setminus (\\tau _\\alpha )_2}\\frac{p_s}{q_s}.$ Finally, by using the rules (REF ), we can calculate the parallel transport operator associated with $\\alpha $ which happens to be represented by an upper triangular matrix with the diagonal elements being given by $\\pm x^{\\pm 1}$ so that we get the relation $x=e^{-\\ell _\\alpha (\\phi (m))/2}.$" ], [ "Part (iii)", "If $\\phi (m)=\\phi (m^{\\prime })$ then, by choosing representatives $(\\rho ,h_0)$ and $(\\rho ^{\\prime },h_0)$ of $m$ and $m^{\\prime }$ respectively, we see that representations $\\rho $ and $\\rho ^{\\prime }$ are conjugated by un upper triangular matrix (it must have $\\infty $ as a fixed point) so that the equivalence (REF ) becomes evident." ] ]
1403.0180
[ [ "Energy eigenfunctions for position-dependent mass particles in a new\n class of molecular hamiltonians" ], [ "Abstract Based on recent results on quasi-exactly solvable Schrodinger equations, we review a new phenomenological potential class lately reported.", "In the present paper we consider the quantum differential equations resulting from position dependent mass (PDM) particles.", "We focus on the PDM version of the hyperbolic potential $V(x) = {a}~\\text{sech}^2x + {b}~\\text{sech}^4x$, which we address analytically with no restrictions on the parameters and the energies.", "This is the celebrated Manning potential, a double-well widely known in molecular physics, until now not investigated for PDM.", "We also evaluate the PDM version of the sixth power hyperbolic potential $V(x) = {a}~{\\text{sech}^6x}+b~{\\text{sech}^4x}$ for which we could find exact expressions under some special settings.", "Finally, we address a triple-well case $V(x) = {a}~{\\text{sech}^6x}+b~{\\text{sech}^4x}+c~\\text{sech}^2x$ of particular interest for its connection to the new trends in atomtronics.", "The PDM Schrodinger equations studied in the present paper yield analytical eigenfunctions in terms of local Heun functions in its confluents forms.", "In all the cases PDM particles are more likely tunneling than ordinary ones.", "In addition, a merging of eigenstates has been observed when the mass becomes nonuniform." ], [ "Introduction", "Over the years, the dynamics of quantum particles in every single substance has been a target for analytical studies in order to have a full understanding of condensed matter systems.", "This is certainly an ambitious task but, although generally frustrating, several phenomenologically relevant models have had its differential equations analytically solved [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17].", "New models involving hyperbolic potentials, typically found in molecular physics, have been reported very recently and in some cases have yield exact wavefunctions for certain relations among the parameters [18], [19], [20].", "In the present paper it is our aim to deal with the family of potentials reported in [18] in connection with the issue of position-dependent mass (PDM) particles.", "The number of solvable potentials in ordinary quantum mechanics is in fact limited, but when we assume the particle mass has a nontrivial space distribution the mathematical difficulties grow even more.", "For some relevant mass distributions some phenomenological potentials have been solved in recent years [21], [22], [23], [24], [25], [26], [27], [28].", "The origin of the PDM approximation can be traced back in the domain of solid state physics [29], [30], [31], [36], [32], [33], [34], [35].", "For instance, the dynamics of electrons in semiconductor heterostructures has been tackled with an effective mass model related to the envelope-function approximation [36], [37], [38].", "Besides this, position dependent mass particles have been used to set to several important issues of low-energy physics related to the understanding of the electronic properties of semiconductors, crystal-growth techniques [39], [40], [41], quantum wells and quantum dots [42], Helium clusters [43], graded crystals [44], quantum liquids [45], and nanowire structures under size variations, impurities, dislocations, and geometrical imperfections [46], among others.", "In this paper, assuming a PDM distribution in the Schrodinger equation, we take up on a new class of potentials [18] recently reported, viz.", "$V(x) = -A\\operatorname{sech}^6x-B\\operatorname{sech}^4x-C\\operatorname{sech}^2x.", "$ For a large variety of constants, this family represents symmetric asymptotically flat double-well potentials, related to problems of solid state and condensed matter physics.", "Double-well potentials are emblematic since they allow studying typical quantal situations involving bound states and particle tunneling through a barrier [47].", "For example, the case $A=0$ of Eq.", "(REF ) results in the renowned Manning potential [6] originally used to address the vibrational normal modes of the NH$_3$ and ND$_3$ molecules and found also appropriate for the understanding of the infrared spectra of organic compounds such as ammonia, formamide and cyanamide.", "Coincidentally, the family given in [19] also includes this potential when the parameter $g>>1$ .", "This class, also phenomenologically rich, was originally related to the double sine-Gordon kink [48] but is known for its interest in different physical subjects [49] such as the study of anti-ferromagnetic chains [50] and experimentally accessible systems like (CH3)$_4$ NMnC1$_3$ (TMMC) [51].", "In both papers [18], [19] the ordinary constant-mass Schrodinger equations have been found some analytical solutions proportional to Heun functions [52].", "These special functions are not very well-known but have been receiving increasing attention, particularly in the last decade [57], [55], [56], [53], [54], [58], [59], [60], [61], [62], [63].", "Interestingly, although not explored in [18], the class given by Eq.", "(REF ) also includes three parameter triple-well potentials particularly interesting in atomtronics, associated with atomic diodes and transistors [64] and laser optics [65].", "Triple-well semiconductor structures have been used in experiments of light transfer in optical waveguides [66], [67] as well as in models of dipolar condensates with phase transitions and metastable states [68].", "In the present work we analyze the new potentials given by Eq.", "(REF ) with a physically significant input, namely a nonuniform mass.", "This phenomenological upgrade of course induces highly nontrivial consequences in the associated differential equations and yields new mathematical and physical results.", "Our goal is to analytically handle the resulting equations and find their general solutions.", "We succeed in some cases which we detail in what follows and find Heun functions in their confluent forms.", "In the case of triple-wells we manage it numerically for its high analytical complexity.", "In every case we compare the PDM results with the equivalent constant mass situations.", "In the next section, , we first address the problem of determining the correct kinetic operator of the PDM Schrodinger equation and then, in Sec.", ", we obtain an effective potential in a convenient space.", "In Sec.", "REF we consider the PDM-Manning potential, a particularly important member of the class (REF ), and find a complete set of eigenstates built in terms of confluent Heun solutions.", "We plot all the six bound-eigenstates of the PDM differential equation together with those of the constant mass problem to show their deviation from the ordinary ones.", "Next, in section REF we find exact expressions for the $E=0$ eigenfunctions of the PDM version of the sixth power potential $V(x) = {a}~{\\operatorname{sech}^6x}+b~{\\operatorname{sech}^4x}$ under some special settings.", "In this case, the eigenstates are proportional to triconfluent forms of the Heun functions.", "Finally, we address the triple-well phase of the potential class, with and without PDM, and discuss our results.", "The final remarks are drawn in Sec.." ], [ "The PDM kinetic operator ", "The effective hamiltonian of a PDM nonrelativistic quantum particle has received much attention along the years for both phenomenological and mathematical reasons [32], [33], [34], [35], [36], [37], [39], [69], [70], [71], [72], [73], [74], [75], [76], [77], [78], [79], [80].", "The full expression for the kinetic-energy hermitian operator for a position-dependent mass $m(x)$ reads $\\hat{T} =\\, \\frac{1}{4} \\, \\hat{T}_0\\,+\\frac{1}{8}\\left\\lbrace \\,\\,\\hat{P}^2\\,m^{-1}(x)+\\, m^{\\alpha }(x)\\,\\hat{P}\\ m^{\\beta }(x)\\,\\hat{P}\\ m^{\\gamma }(x)\\,\\,+\\,m^{\\gamma }(x)\\,\\hat{P}\\ m^{\\beta }(x)\\,\\hat{P}\\ m^{\\alpha }(x)\\, \\right\\rbrace ,$ where, for constant mass, $\\hat{T}_0= \\frac{1}{2}m^{-1}\\, \\hat{P}^2$ is the standard quantum kinetic energy and $\\hat{P}=-i\\hbar \\; d/dx$ is the momentum operator.", "The above parameters have to fulfill the condition $\\alpha +\\beta +\\gamma =-1$ [33].", "Recalling the basic postulate $[\\hat{X},\\hat{P}]=i\\hbar $ , we get $\\hat{T}\\,=\\,\\frac{1}{2\\,m}\\,\\hat{P}^{2}\\,+\\,\\frac{i\\hbar }{2}\\frac{1}{m^{2}}\\frac{dm}{dx}\\,\\,\\hat{P}\\,\\,+\\,U_{\\rm K}\\left( x\\right),$ where $U_{\\rm K}\\left( x\\right) =\\frac{-\\hbar ^{2}}{4m^{3}(x)}\\left[ \\left( \\alpha +\\gamma -1\\right) \\frac{m(x)}{2}\\left(\\frac{d^{2}m}{dx^{2}}\\right)+\\left(1-\\alpha \\gamma -\\alpha -\\gamma \\right) \\left( \\frac{dm}{dx}\\right) ^{2}\\right]$ is an effective potential of kinematic origin.", "This is of course a source of ambiguity in the hamiltonian for it depends on the values of $\\alpha , \\beta , \\gamma $ .", "In order to fix this issue, we can kill the kinematic potential by adding the constraint $\\alpha \\,+\\,\\gamma \\,=\\,1\\,=\\alpha \\,\\gamma \\,+\\,\\alpha \\,+\\,\\gamma .", "$ Its solution is $\\alpha =0$ and $\\gamma =1$ , or $\\alpha =1 $ and $\\gamma =0$ , which corresponds to the Ben-Daniel–Duke $\\hat{T}$ ordering [36].", "Now, the hamiltonian is free of ambiguities but the resulting effective Schrödinger equation still looks weird for it now includes a first order derivative term.", "For an arbitrary external potential $V(x)$ the PDM-Schrödinger equation turns out $\\frac{d^2\\psi (x)}{dx^2} { -\\left(\\frac{1}{m(x)} \\frac{d m(x)}{dx}\\right)} { \\frac{d\\psi (x)}{dx}} +\\frac{2}{\\hbar ^2}\\,{ m(x)}\\, [E-V(x)] \\psi (x) =0.", "$ Noticeably, not only the last term has been strongly modified from the ordinary Schrödinger equation but the differential operator turned out to be dramatically changed.", "This will have of course deep consequences on the physical wave solutions of the system." ], [ "Effective new PDM potential", "Here we will adopt a solitonic smooth effective mass distribution $m(x)=m_0\\, {\\operatorname{sech}}^2(x/d) $ (see e.g.", "[24] and [27]).", "Besides its convenient analytical nature, its shape is familiar in effective models of condensed matter and low energy nuclear physics and depicts a soft symmetric distribution.", "The effective Schrodinger equation (REF ) thus reads $\\psi ^{\\prime \\prime } (x)+2 \\tanh (x) \\psi ^{\\prime }(x)+ \\frac{2m_0}{\\hbar ^2}(E-V(x))\\, \\operatorname{sech}^2(x) \\psi (x)=0, $ but for the ansatz solution $\\psi (x)=\\cosh ^{\\nu }\\!", "(x)\\,\\varphi (x), $ becomes $\\varphi ^{\\prime \\prime }(x)+2 (\\nu +1) \\tanh (x)\\varphi ^{\\prime }(x)+\\left[\\nu (\\nu +2) \\tanh ^2\\!x+ \\left(\\nu +\\frac{2m_0}{\\hbar ^2}(E-V(x)) \\right)\\,{\\operatorname{sech}}^2(x) \\right] \\varphi (x)=0.$ Now, a change of variables $\\operatorname{sech}\\,x = \\cos \\,z, $ maps the domain $(-\\infty ,\\infty ) \\rightarrow (-\\frac{\\pi }{2},\\frac{\\pi }{2})$ and provides $\\varphi ^{\\prime \\prime }(z)+(2\\nu +1)\\tan (z)\\varphi ^{\\prime }(z)+\\left[\\nu +\\nu (\\nu +2)\\tan ^2(z)+\\frac{2m_0}{\\hbar ^2}\\left(E-{V}(z)\\right) \\right]\\varphi (z)=0$ (we call $\\varphi (x(z))=\\varphi (z)$ , etc.).", "The choice $\\nu =-1/2$ allows the removal of the first derivative and grants an ordinary Schrodinger equation $\\left[-\\frac{d^2}{dz^2}+ \\mathcal {V}(z)\\right]\\varphi (z)=\\mathcal {E}\\varphi (z) $ where $\\mathcal {E}=\\frac{2m_0}{\\hbar ^2}E$ .", "The potential class we are dealing with is $V(x) = -A\\operatorname{sech}^6x-B\\operatorname{sech}^4x-C\\operatorname{sech}^2x, $ where the adjustable parameters determine a large variety of possible shapes.", "Thus, in Eq.", "(REF ) we shall employ $\\mathcal {V}(z)=\\frac{1}{2} +\\frac{3}{4}\\tan ^2\\!z -\\mathcal {A}\\cos ^6 z-\\mathcal {B}\\cos ^4 z-\\mathcal {C}\\cos ^2 z, $ where $\\mathcal {A,B,C}$ incorporated the factor $\\frac{2m_0}{\\hbar ^2}$ , e.g.", "$\\mathcal {A}=\\frac{2m_0}{\\hbar ^2}A$ .", "The dynamics of the original PDM particle is therefore described by one of constant mass $m_0$ moving in z-space in a non-ambiguous effective potential with no kinematical contributions.", "Although restricted to within $z =(-\\frac{\\pi }{2},\\frac{\\pi }{2})$ , with $\\varphi (z)=0$ at the borders, we can eventually transform everything back to the original x-variable and $\\psi $ wave-function to obtain the real space solution." ], [ "The ", "The celebrated Manning potential, $V(x) = a\\operatorname{sech}^2x+b\\operatorname{sech}^4x$ , very much used in molecular physics, is here addressed in the very interesting situation of an effective spatially dependent mass.", "This phenomenological potential corresponds to the case $A=0$ of Eq.", "(REF ).", "Note that when $B < 0, C > 0$ and $-C/2B < 1$ , the Manning potential is a double-well potential with two minima at $x =\\pm \\operatorname{arcsech}(\\sqrt{�-C/2B})$ .", "In Fig.REF we show this potential for several values of the free parameters.", "Figure: From top to bottom, plot of the Manning potential V(x)V(x) (left) and𝒱(z)\\mathcal {V}(z) (right),for (ℬ,𝒞)(\\mathcal {B},\\mathcal {C}) = (-500,500)(-500,500), (-1000,1300)(-1000,1300), (-1000,1600)(-1000,1600); and (-1000,1800)(-1000,1800).In $z$ -space we have $\\mathcal {V}(z)=\\frac{1}{2} +\\frac{3}{4}\\tan ^2\\!z - {\\mathcal {B}}\\,{\\cos ^4(z)}- {\\mathcal {C}}\\,{\\cos ^2(z)}, $ to be considered in eq.", "(REF ).", "By means of the ansatz $\\varphi (z)=\\cos ^\\mu \\!z \\,\\phi (z) $ we obtain $\\phi ^{\\prime \\prime }(z)-2\\mu \\tan (z)\\,\\phi ^{\\prime }(z)+\\Big [(\\mu ^2-\\mu -{}^3\\!/{}_{\\!4})\\tan ^2\\!z-\\mu +\\mathcal {E} -{}^1\\!/{}_{\\!2}+ \\mathcal {B}\\cos ^4\\!z+\\mathcal {C}\\cos ^2\\!z\\Big ]\\,\\phi (z) = 0$ which can be simplified by choosing $\\mu ^2-\\mu -{}^3\\!/{}_{\\!4}=0,$ namely $\\mu =3/2$ or $\\mu =-1/2$ .", "If we now transform coordinates by $y=\\sin ^2\\!z$ , the above equation results in $y\\,(1-y)\\,\\phi ^{\\prime \\prime }(y)&+&\\Big [{}^1\\!/{}_{\\!2}-(1+\\mu )\\,y\\Big ]\\,\\phi ^{\\prime }(y)+\\frac{1}{4}\\\\ && \\Big [-\\mu +\\mathcal {E}-\\frac{1}{2}+\\mathcal {B}\\,(1-y)^2+\\mathcal {C}(1-y)\\Big ]\\phi (y) = 0.$ A further transformation $\\phi (y)=e^{\\nu y}H(y),$ puts in evidence its Heun nature $h^{\\prime \\prime }(y) &+& \\left(2\\nu +\\frac{{}^1\\!/{}_{\\!2}}{y}+\\frac{\\mu +{}^1\\!/{}_{\\!2}}{y-1}\\right)\\,h^{\\prime }(y) + \\frac{1}{4y\\,(y-1)}\\Big [\\mu +\\frac{1}{2}-2\\nu -\\mathcal {E-B-C}\\nonumber \\\\&-&4\\Big (\\nu ^2-\\nu -\\mu \\nu -\\frac{\\mathcal {B}}{2}-\\frac{\\mathcal {C}}{4}\\Big )\\,y+(\\mathcal {B}-4\\nu ^2)\\,y^2\\Big ] h(y) = 0.$ Since $\\mu =-1/2$ is misleading and $\\nu $ is arbitrary we choose $\\mu ={}^3\\!/{}_{\\!2}$ and $2\\nu =\\sqrt{\\mathcal {B}}$ .", "This yields $h^{\\prime \\prime }(y)+\\left(\\sqrt{\\mathcal {B}}+\\frac{{}^1\\!/{}_{\\!2}}{y}+\\frac{2}{y-1}\\right) \\,h^{\\prime }(y)+\\nonumber \\\\\\frac{1}{y\\,(y-1)} \\left[\\frac{1}{4}\\Big (\\mathcal {B} +\\mathcal {C} +5\\sqrt{\\mathcal {B}}\\Big )\\,y +\\frac{1}{2}-\\frac{\\sqrt{\\mathcal {B}}+\\mathcal {E}+\\mathcal {B}+\\mathcal {C}}{4}\\right]h(y) = 0, $ which is a canonical non-symmetric confluent Heun equation [53], [60], [61] of the form $Hc^{\\prime \\prime }(y)+ \\left(\\alpha + \\frac{\\beta + 1}{y} + \\frac{\\gamma + 1}{y-1} \\right)Hc^{\\prime }(y)+ \\frac{1}{y(y-1)}\\\\\\left[\\left(\\delta + \\frac{\\alpha }{2}(\\beta + \\gamma +2)\\right)y + \\eta + \\frac{\\beta }{2} + \\frac{1}{2}(\\gamma - \\alpha )(\\beta +1) \\right] Hc(y) = 0, \\nonumber $ with $\\alpha &=& \\sqrt{\\mathcal {B}}\\\\\\beta &=& -{}^1\\!/{}_{\\!2}\\\\\\gamma &=& 1\\\\\\delta &=&\\frac{1}{4}(\\mathcal {B}+\\mathcal {C})\\\\\\eta &=&\\frac{1}{2}-\\frac{\\mathcal {E+\\mathcal {B}+\\mathcal {C}}}{4}.$ Figure: Plot of the normalized symmetric solutionsψ (1) (x)\\psi ^{(1)}(x) [Eq.", "()], when B=-C=-500B=-C=-500(see top (red) curve in Fig.", "),in their three symmetric bound states ℰ 0 =-102.25913969050\\mathcal {E}_0=-102.25913969050 (left),ℰ 2 =-61.4581676270\\mathcal {E}_2=-61.4581676270 (center) and ℰ 4 =-25.9419535530\\mathcal {E}_4=-25.9419535530 (right).Below are shown the corresponding probability densities.Figure: Plot of the normalized antisymmetric solutionsψ (2) (x)\\psi ^{(2)}(x) [Eq.", "()],in their three antisymmetric bound-states when B=-500=-CB=-500=-C(see top (red) curve in Fig.", "), for ℰ 1 =-102.2558018532\\mathcal {E}_1=-102.2558018532 (left),ℰ 3 =-61.3388827970\\mathcal {E}_3=-61.3388827970 (center) and ℰ 5 =-24.20206500\\mathcal {E}_5=-24.20206500 (right).Below are shown the corresponding probability densities.Therefore, the solutions of Eq.", "(REF ) are $h^{(1)}(y) = Hc \\left(\\sqrt{\\mathcal {B}},-\\frac{1}{2}, 1, \\frac{1}{4}(\\mathcal {B}+\\mathcal {C}),\\frac{1}{2} -\\frac{\\mathcal {E+\\mathcal {B}+\\mathcal {C}}}{4};\\, y\\right)\\\\h^{(2)}(y) = \\sqrt{y} Hc \\left(\\sqrt{B},\\frac{1}{2}, 1, \\frac{1}{4}(\\mathcal {B}+\\mathcal {C}),\\frac{1}{2} -\\frac{\\mathcal {E+\\mathcal {B}+\\mathcal {C}}}{4};\\, y\\right),$ which in z-space result $\\varphi ^{(1)}(z) = \\cos ^\\frac{3}{2}\\!z\\, e^{\\frac{\\sqrt{\\mathcal {B}}}{2}\\sin ^2\\!z}Hc\\!", "\\left(\\sqrt{\\mathcal {B}},-\\frac{1}{2}, 1, \\frac{1}{4}(\\mathcal {B}+\\mathcal {C}),\\frac{1}{2}-\\frac{\\mathcal {E+\\mathcal {B}+\\mathcal {C}}}{4};\\, \\sin ^2\\!z \\right)\\\\\\varphi ^{(2)}(z) = \\sin \\!z\\,\\cos ^\\frac{3}{2}\\!z\\, e^{\\frac{\\sqrt{\\mathcal {B}}}{2}\\sin ^2\\!z}Hc\\!", "\\left(\\sqrt{\\mathcal {B}},\\frac{1}{2}, 1, \\frac{1}{4}(\\mathcal {B}+\\mathcal {C}),\\frac{1}{2}-\\frac{\\mathcal {E+\\mathcal {B}+\\mathcal {C}}}{4};\\, \\sin ^2\\!z \\right),$ and in $x$ -space read finally $&&\\psi ^{(1)}(x) = \\operatorname{sech}^2\\!x\\, e^{\\frac{\\sqrt{\\mathcal {B}}}{2}\\tanh ^2\\!x}Hc\\!", "\\left(\\sqrt{\\mathcal {B}},-\\frac{1}{2}, 1, \\frac{1}{4}(\\mathcal {B}+\\mathcal {C}),\\frac{1}{2}-\\frac{\\mathcal {E+\\mathcal {B}+\\mathcal {C}}}{4};\\, \\tanh ^2\\!x \\right)\\\\&&\\psi ^{(2)}(x) = \\tanh x\\,\\operatorname{sech}^2\\!x\\, e^{\\frac{\\sqrt{\\mathcal {B}}}{2}\\tanh ^2\\!x} Hc\\!\\left(\\sqrt{\\mathcal {B}},+\\frac{1}{2}, 1, \\frac{1}{4}(\\mathcal {B}+\\mathcal {C}),\\frac{1}{2}-\\frac{\\mathcal {E+\\mathcal {\\mathcal {B}}+\\mathcal {C}}}{4};\\, \\tanh ^2\\!x \\right).$ In Figs.", "REF and REF we plot the bound-state wavefunctions and probability densities of a PDM particle.", "We have numerically computed the energies of the eigenstates that satisfy vanishing boundary conditions and found just six bound states in this PDM-Manning potential.", "Although the three pairs of probability distributions are very close, the corresponding solutions are certainly different (all the symmetric solutions are nonzero at the origin while the antisymmetric ones are of course null).", "This is particularly apparent for the third pair of eigenfunctions where the tunneling effect is highly manifest in the $\\mathcal {E}_4$ eigenstate.", "As we foreseen, the PDM analytic expressions are quite different from the ordinary constant-mass solutions to the Manning potential found in [18] $&&\\chi ^{(1)}(x) = (\\operatorname{sech}x)^{\\sqrt{-E}}\\,Hc\\!", "\\left(0,-\\frac{1}{2}, \\sqrt{-E}, \\frac{1}{4} {B},\\frac{1}{4}-\\frac{E+ {B}+ {C}}{4};\\, \\tanh ^2\\!x \\right) \\\\&&\\chi ^{(2)}(x) = \\tanh x\\,(\\operatorname{sech}x)^{\\sqrt{-E}}\\,Hc\\!", "\\left(0,\\frac{1}{2}, \\sqrt{-E}, \\frac{1}{4} {B},\\frac{1}{4}-\\frac{E+ {B}+ {C}}{4};\\, \\tanh ^2\\!x \\right).$ In Figs.", "REF and REF we show both pairs of curves for the closest possible eigenenergies found for the two problems.", "We observe similar shapes in both sets with a rapidly increasing deviation from the ordinary constant-mass case for the higher eigensates.", "Note that in all the eigenstates the PDM particle has more probability to be near the origin of coordinates and thus keep on tunneling across the potential barrier.", "Another remarkable point is that while in the constant-mass case there exist fourteen bound-states in the PDM case there are only six.", "This shows a kind of merging of eigensates and a lower number of physical possibilities for growing energies assuming PDM (see Table REF ).", "Table: Complete list of the energy eigenvalues of thePDM and constant-mass Manning hamiltonians for A=0A=0 and B=-C=-500B=-C=-500.The S _S and A _A subindexes at left indicate symmetric and antisymmetric states.Figure: In solid line, the normalized PDM solutionsψ (1) (x)\\psi ^{(1)}(x) [Eq.", "()]for the three symmetric bound states ℰ 0 =-102.25913969050\\mathcal {E}_0=-102.25913969050 (left),ℰ 2 =-61.4581676270\\mathcal {E}_2=-61.4581676270 (center) and ℰ 4 =-25.9419535530\\mathcal {E}_4=-25.9419535530 (right).In dashed line the normalized ordinary solutionsχ (1) (x)\\chi ^{(1)}(x) [Eq.", "()]for the first three symmetric bound states ℰ 0 =-109.94122211881\\mathcal {E}_0=-109.94122211881 (left),ℰ 2 =-81.887958347499\\mathcal {E}_2=-81.887958347499 (center) and ℰ 4 =-57.567702358602\\mathcal {E}_4=-57.567702358602 (right).Here B=-C=-500B=-C=-500; see top (red) Manning potential curve in Fig..Figure: In solid line, the normalized PDM solutionsψ (2) (x)\\psi ^{(2)}(x) [Eq.", "()]for the three antisymmetric bound states ℰ 1 =-102.25580185320\\mathcal {E}_1=-102.25580185320 (left),ℰ 3 =-61.3388827970\\mathcal {E}_3=-61.3388827970 (center) and ℰ 5 =-24.20206500\\mathcal {E}_5=-24.20206500 (right).In dashed line the normalized ordinary solutionsχ (2) (x)\\chi ^{(2)}(x) [Eq.", "()]for the first three antisymmetric bound states ℰ 1 =-109.9940489854443\\mathcal {E}_1=-109.9940489854443 (left),ℰ 3 =-81.87558412881\\mathcal {E}_3=-81.87558412881 (center) and ℰ 5 =-57.474984727067\\mathcal {E}_5=-57.474984727067 (right).Here B=-C=-500B=-C=-500; see top (red) Manning potential curve in Fig.." ], [ "The ", "Regarding the sixth-order members of family (REF ) we have tried to analytically disentangle the full problem but it seems too complex.", "In any case, we have been able to find the exact solution to the PDM-modified differential equation for one free parameter in two specific cases: $B = 0, C=0$ , namely $V(x)= -A \\operatorname{sech}^6(x)$ , and $A = - B, C=0$ , that is $V(x)= -A (\\operatorname{sech}^6(x) - \\operatorname{sech}^4(x))$ , for $E=0$ , both yielding triconfluent Heun eigenfunctions [53] (see also e.g.", "[81]).", "In Fig.", "REF and Fig.", "REF we show these potentials for several values of the free parameter.", "Figure: From top to bottom, plot of well potentials V(x)=Asech 6 (x)V(x)= A \\operatorname{sech}^6(x) (left) andthe corresponding effective potentials 𝒱(z)\\mathcal {V}(z) (right) for A=10,30,50{A} = 10, 30, 50 and 100." ], [ "Case free $A$ and {{formula:89ca7cb6-20e7-4b37-90a1-03ab26e2e2c1}}", "In the first case, the z-space eq.", "(REF ) to solve is $\\varphi ^{\\prime \\prime }(z)- \\left(\\frac{1}{2} + \\frac{3}{4}\\tan ^2(z)-\\mathcal {A}\\cos ^6(z)\\right) \\varphi (z)=0.$ We first factorize $\\varphi (z)=\\cos ^\\sigma (z)\\,\\phi (z)$ , bearing $\\phi ^{\\prime \\prime }(z)-2\\sigma \\tan (z)\\phi ^{\\prime }(z)+\\Big [(\\sigma ^2-\\sigma -{}^3\\!/{}_{\\!4})\\tan (z)^2-\\sigma -{}^1\\!/{}_{\\!2}+\\mathcal {A}\\cos (z)^6\\Big ]\\phi (z) = 0.$ We then cut down this equation by choosing $\\sigma =-{}^1\\!/{}_{\\!2}$ .", "Now we transform the variable by means of $y=\\sin ^2z$ and get $(-y^2+1)\\,\\phi ^{\\prime \\prime }(y)+(A (-y^2+1)^3)\\,\\phi (y) = 0.", "$ It looks as the representative form of the triconfluent Heun equation and therefore we try with an exponential ansatz of the form $\\phi (y)=e^{ay^3+by} h(y)$ (see Proposition E.1.2.1 in [53]), so that Eq.", "(REF ) results $h^{\\prime \\prime }(y)+(6 a y^2+2 b) h^{\\prime }(y)+\\Big [6ay+(3ay^2+b)^2+\\mathcal {A} (-y^2+1)^2\\Big ]\\,h(y) = 0.$ This can be further shorten by means of $a=-{}^1\\!/{}_{\\!3}\\sqrt{-\\mathcal {A}}$ and $b=\\sqrt{-\\mathcal {A}}$ and if we next define the variable $\\bar{y}=\\left(\\frac{2\\sqrt{-\\mathcal {A}}}{3}\\right)^{\\!\\!\\frac{1}{3}}\\,y$ we obtain $h^{\\prime \\prime }(\\bar{y})-\\left[{3}\\,\\bar{y}^2 -(-12 \\mathcal {A})^{\\!\\!", "{}^1\\!/{}_{\\!3}}\\right]h^{\\prime }(\\bar{y}) -3\\,\\bar{y}\\,h(\\bar{y}) = 0.$ Now, this can be readily compared with $H^{\\prime \\prime }(u)- (\\gamma +3u^2)H^{\\prime }(u)+[\\alpha +(\\beta -3) u]H(u)=0$ which is known as the canonical triconfluent form of the Heun equation.", "Its L.I.", "solutions are $H^{(1)}(u)&=&Ht(\\alpha , \\beta , \\gamma ; u)\\\\H^{(2)}(u)&=&e^{u^3+\\gamma u}Ht(\\alpha , -\\beta , \\gamma ; -u).$ The triconfluent Heun equation is obtained from the biconfluent form through a process in which two singularities coalesce by redefining parameters and taking the appropriate limits.", "See [53] for a detailed discussion about the confluence procedure in the case of the Heun equation and its different forms.", "The function $Ht(\\alpha ,\\beta ,\\gamma ;u)$ is a local solution around the origin, which is a regular point.", "Because the single singularity is located at infinity, this series converges in the whole complex plane and consequently its solutions can be related to the Airy functions [82].", "Figure: Plot of symmetric solutions ψ s (x)\\psi _s(x) [up] and thecorresponding probability densities |ψ s (x)| 2 |\\psi _s(x)|^2 [down]given ℬ=𝒞=0\\mathcal {B}=\\mathcal {C}=0 and ℰ=0\\mathcal {E}=0,for 𝒜=3.131784324\\mathcal {A} = 3.131784324 (left);𝒜=41.919051\\mathcal {A} = 41.919051 (center); and 𝒜=125.162981\\mathcal {A} = 125.162981 (right).Figure: Plot of antisymmetric solutions ψ a (x)\\psi _a(x)[up] and the corresponding probability densities |ψ a (x)| 2 |\\psi _a(x)|^2 [down] givenℬ=𝒞=0\\mathcal {B}=\\mathcal {C}=0 and ℰ=0\\mathcal {E}=0, for (left)𝒜=16.962907791\\mathcal {A} = 16.962907791; (center) 𝒜=77.987131\\mathcal {A} = 77.987131;and (right) 𝒜=183.4448166\\mathcal {A} = 183.4448166.Our solutions are thus $h^{(1)}(y)&=&Ht\\left(0, 0, -(-12 \\mathcal {A})^{\\!", "{}^1\\!/{}_{\\!3}};({{}^2\\!/{}_{\\!3}\\sqrt{-\\mathcal {A}}})^{\\!", "{}^1\\!/{}_{\\!3}}\\,y\\right)\\\\h^{(2)}(y)&=&\\exp \\!\\left[{}^2\\!/{}_{\\!3}\\sqrt{-\\mathcal {A}}\\,\\, y\\!\\left(y^2-3\\right)\\right]Ht\\left(0, 0, -(-12 \\mathcal {A})^{\\!", "{}^1\\!/{}_{\\!3}}; - ({{}^2\\!/{}_{\\!3}\\sqrt{-\\mathcal {A}}})^{\\!", "{}^1\\!/{}_{\\!3}}\\,y\\right),$ namely $\\phi ^{(1)}(y)&=&\\exp \\!\\Big [\\!-\\!", "{}^1\\!/{}_{\\!3}\\sqrt{-\\mathcal {A}}\\,y\\, (y^2-3)\\Big ]Ht\\left(0, 0, -(-12 \\mathcal {A})^{\\!", "{}^1\\!/{}_{\\!3}}; ({{}^2\\!/{}_{\\!3}\\sqrt{-\\mathcal {A}}})^{\\!", "{}^1\\!/{}_{\\!3}}\\,y\\right)\\\\\\phi ^{(2)}(y)&=&\\exp \\!\\Big [\\,{}^1\\!/{}_{\\!3}\\sqrt{-\\mathcal {A}}\\,y\\,(y^2-3)\\Big ]Ht\\left(0, 0, -(-12 \\mathcal {A})^{\\!", "{}^1\\!/{}_{\\!3}}; -({{}^2\\!/{}_{\\!3}\\sqrt{-\\mathcal {A}}})^{\\!", "{}^1\\!/{}_{\\!3}}\\,y\\right)\\!\\!.$ In variable $z$ , recalling that $\\varphi (z)=\\cos \\!z\\,\\phi (z)$ , they result $\\varphi ^{(1)}(z)&=&\\cos \\!z\\exp {\\!\\Big [\\!-\\!", "{}^1\\!/{}_{\\!3}\\sqrt{-\\mathcal {A}}\\,\\cos \\!z\\,(\\cos ^2\\!z-3)\\Big ]}Ht\\left(0, 0, -(-12 \\mathcal {A})^{\\!", "{}^1\\!/{}_{\\!3}};({{}^2\\!/{}_{\\!3}\\sqrt{-\\mathcal {A}}})^{\\!", "{}^1\\!/{}_{\\!3}}\\!\\cos z\\!\\right)\\\\\\varphi ^{(2)}(z)&=&\\cos \\!z\\exp \\!\\Big [\\,{}^1\\!/{}_{\\!3}\\sqrt{-\\mathcal {A}}\\,\\cos \\!z\\,(\\cos ^2z-3)\\Big ]Ht\\left(0, 0, -(-12 \\mathcal {A})^{\\!", "{}^1\\!/{}_{\\!3}}; -({{}^2\\!/{}_{\\!3}\\sqrt{-\\mathcal {A}}})^{\\!", "{}^1\\!/{}_{\\!3}}\\!\\cos z\\!\\right)\\!,$ and finally, for $\\psi (x)=\\operatorname{sech}^{\\frac{1}{2}}\\!x\\,\\varphi (x)$ we have $\\psi ^{(1)}(x)&=&\\operatorname{sech}^{\\frac{3}{2}}\\!xe^{\\Big [\\!-\\!", "{}^1\\!/{}_{\\!3}\\sqrt{-\\mathcal {A}}\\,\\operatorname{sech}\\!x\\, (\\operatorname{sech}^2\\!x-3)\\Big ]}Ht\\left(0, 0, -(-12 \\mathcal {A})^{\\!", "{}^1\\!/{}_{\\!3}}; ({{}^2\\!/{}_{\\!3}\\sqrt{-\\mathcal {A}}})^{\\!", "{}^1\\!/{}_{\\!3}}\\operatorname{sech}\\!x\\right)\\\\\\psi ^{(2)}(x)&=&\\operatorname{sech}^{\\frac{3}{2}}\\!xe^{\\Big [\\,{}^1\\!/{}_{\\!3}\\sqrt{-\\mathcal {A}}\\,\\operatorname{sech}\\!x\\, (\\operatorname{sech}^2\\!x-3)\\Big ]}Ht\\left(0, 0, -(-12 \\mathcal {A})^{\\!", "{}^1\\!/{}_{\\!3}}; -({{}^2\\!/{}_{\\!3}\\sqrt{-\\mathcal {A}}})^{\\!", "{}^1\\!/{}_{\\!3}}\\,\\operatorname{sech}\\!x\\right)\\!.$ Note that although these solutions, Eqs.", "(REF ) and (), are both complex functions, their symmetric and antisymmetric combinations are real.", "Furthermore, only the symmetric and antisymmetric solutions satisfy the border conditions and make physical sense.", "There is a discrete number of potential wells with an $E=0$ eigenstate.", "We show these eigenfunctios for the first six values of ${A}$ , see Fig.", "REF and REF .", "It can be seen that the number of nodes of the zero-modes depends directly on the depths of the wells." ], [ "Case free ${B}=-{A}$ , and {{formula:f6392edf-d034-4d60-a0b8-f42aeabec120}} ", "The second case we managed to analytically solve corresponds to the hyperbolic sixth-order double-well plotted in Fig.", "REF .", "In order to show this we have to analyze the following instance of the modified Schrödinger eq.", "(REF ) $\\varphi ^{\\prime \\prime }(z)- \\left(\\frac{1}{2} +\\frac{3}{4}\\tan ^2(z)-\\mathcal {A}\\cos ^6(z)+\\mathcal {A}\\cos ^4(z)\\right) \\varphi (z)=0.$ After transformation $\\varphi (z)=\\cos ^\\mu (z)\\,\\phi (z)$ we get $\\phi ^{\\prime \\prime }(z)-2\\mu \\tan (z)\\phi ^{\\prime }(z)+\\Big [(\\mu ^2-\\mu -3/4)\\tan ^2z-\\mu -1/2+\\mathcal {A}\\cos ^6z-\\mathcal {A}\\cos ^4z\\Big ]\\phi (z) = 0$ which shortens to $\\phi ^{\\prime \\prime }(z)+\\tan (z)\\phi ^{\\prime }(z)+\\Big [\\mathcal {A}\\cos (z)^6 -\\mathcal {A}\\cos (z)^4\\Big ]\\phi (z) = 0,$ provided one chooses $\\mu =-{}^1\\!/{}_{\\!2}$ .", "Now, we change coordinates by $y=\\sin (z)$ yielding $\\phi ^{\\prime \\prime }(y)+\\Big [\\mathcal {A} (1-y^2)^2-\\mathcal {A}(1-y^2)\\Big ]\\phi (y) = 0.$ As in the previous case, we try the ansatz $\\phi (y)=e^{ay^3+by} h(y)$ and get $h^{\\prime \\prime }(y)+(6ay^2+2b)h^{\\prime }(y)+\\Big [6ay+b^2+(\\mathcal {A}+9a^2)\\,y^4+(-\\mathcal {A}+6ab)\\,y^2\\Big ]h(y) = 0,$ which simplifies conveniently when we adopt $3a=-\\sqrt{-\\mathcal {A}}$ and $2b=\\sqrt{-\\mathcal {A}}$ .", "We thus obtain $h^{\\prime \\prime }(y)+\\sqrt{-\\mathcal {A}}\\Big (\\!-2 y^2+1\\Big )\\,h^{\\prime }(y)+\\Big (-2\\sqrt{-\\mathcal {A}}\\,y-\\mathcal {A}/4\\Big )h(y) = 0$ which, by redefining $\\bar{y}=\\left(\\frac{2\\sqrt{-\\mathcal {A}}}{3}\\right)^{\\!\\!\\!", "{}^1\\!/{}_{\\!3}}\\!\\!y$ , gives $h^{\\prime \\prime }(\\bar{y})-\\left[{3}\\,\\bar{y}^2 -\\left(-\\frac{3\\mathcal {A}}{2} \\right)^{\\!\\!", "{}^1\\!/{}_{\\!3}} \\right]h^{\\prime }(\\bar{y})+\\left[-3\\,\\bar{y} +\\left(\\frac{-3\\mathcal {A}}{16}\\right)^{\\!\\!", "{}^2\\!/{}_{\\!3}}\\right]\\,h(\\bar{y}) = 0.$ This is the canonical triconfluent Heun equation $H^{\\prime \\prime }(u)- (3u^2+\\gamma )H^{\\prime }(u)+[(\\beta -3) u+\\alpha ]H(u)=0,$ as soon as we identify $\\alpha &=&\\left(-\\frac{3\\mathcal {A}}{16} \\right)^{\\!\\!", "{}^2\\!/{}_{\\!3}}\\nonumber \\\\\\beta &=&0\\nonumber \\\\\\gamma &=&-\\left(-\\frac{3\\mathcal {A}}{2} \\right)^{\\!\\!", "{}^1\\!/{}_{\\!3}}.\\nonumber $ The solutions of eq.", "(REF ) are then $h^{(1)}(y)&=&Ht\\!\\left(\\left(-\\frac{3\\mathcal {A}}{16}\\right)^{\\!\\!", "{}^2\\!/{}_{\\!3}}\\!\\!\\!\\!\\!,\\,\\,\\,0,-\\!\\left(-\\frac{3\\mathcal {A}}{2} \\right)^{\\!\\!", "{}^1\\!/{}_{\\!3}},\\left(\\frac{2\\sqrt{-\\mathcal {A}}}{3}\\right)^{\\!\\!\\!", "{}^1\\!/{}_{\\!3}}\\!\\!y\\right)\\\\h^{(2)}(y)&=&\\exp \\!\\left[{}^2\\!/{}_{\\!3}\\sqrt{-\\mathcal {A}}\\,\\, y\\!\\left(y^2-{}^3\\!/{}_{\\!2}\\right)\\right]Ht\\!\\left(\\left(-\\frac{3\\mathcal {A}}{16} \\right)^{\\!\\!", "{}^2\\!/{}_{\\!3}}\\!\\!\\!\\!\\!,\\,\\,\\,0, -\\!\\left(-\\frac{3\\mathcal {A}}{2} \\right)^{\\!\\!", "{}^1\\!/{}_{\\!3}},-\\left(\\frac{2\\sqrt{-\\mathcal {A}}}{3}\\right)^{\\!\\!\\!", "{}^1\\!/{}_{\\!3}}\\!\\!y\\!\\!\\right)\\!\\!.$ Finally, by transforming everything back to the original $x$ -space, we have the starting eigenfunctions of the PDM hamiltonian $&&\\psi ^{(1)}\\!", "(x)={\\rm e}^{\\sqrt{-\\mathcal {A}}\\ \\tanh (x)\\!\\left({}^1\\!/{}_{\\!2}-{}^1\\!/{}_{\\!3}\\tanh ^2(x)\\right)}\\ Ht\\!\\left(\\!\\!\\left(-\\frac{3\\mathcal {A}}{16}\\right)^{\\!\\!", "{}^2\\!/{}_{\\!3}}\\!\\!\\!\\!\\!,\\,\\,\\,0, -\\!\\left(-\\frac{3\\mathcal {A}}{2}\\right)^{\\!\\!", "{}^1\\!/{}_{\\!3}}\\!\\!\\!\\!,\\,\\left(\\frac{2\\sqrt{-\\mathcal {A}}}{3}\\right)^{\\!\\!\\!", "{}^1\\!/{}_{\\!3}}\\!\\!\\tanh x\\!\\!\\right)\\\\&&\\psi ^{(2)}\\!", "(x)={\\rm e}^{\\sqrt{-\\mathcal {A}}\\, \\tanh (x)\\!\\left({}^1\\!/{}_{\\!3}\\tanh ^2(x)-{}^1\\!/{}_{\\!2}\\right)}Ht\\!\\left(\\!\\!\\left(-\\frac{3\\mathcal {A}}{16} \\right)^{\\!\\!", "{}^2\\!/{}_{\\!3}}\\!\\!\\!\\!\\!,\\,\\,0,-\\!\\left(-\\frac{3\\mathcal {A}}{2} \\right)^{\\!\\!", "{}^1\\!/{}_{\\!3}}\\!\\!\\!\\!,-\\left(\\frac{2\\sqrt{-\\mathcal {A}}}{3}\\right)^{\\!\\!\\!", "{}^1\\!/{}_{\\!3}}\\!\\!\\!\\tanh x\\!\\!\\right)\\!\\!.$ In this case, we found again that just the symmetric and antisymmetric combinations of Eqs.", "(REF ) and () are real functions and the only that fit the boundary conditions (as expected from the parity of the potential).", "After a numerical survey of the parameter space we found a discrete set of values of potential depths compatible with a zero energy eigenstate.", "In Figs.", "REF and REF we plot the eigenfunctions in the first six cases, three being even and the other antisymmetric, as indicated.", "Figure: Plot of symmetric PDM zero-modes ψ s (x)\\psi _s(x) of V(x)=A(sech 6 (x)-sech 4 (x))V(x)= A (\\operatorname{sech}^6(x) - \\operatorname{sech}^4(x)) [up],and the corresponding probability densities |ψ s (x)| 2 |\\psi _s(x)|^2 [down],for A=-25.125695463186{A} = -25.125695463186 (left);A=-209.2999338840{A} = -209.2999338840 (center); and A=-571.605964500{A}= -571.605964500 (right).Figure: Plot of antisymmetric PDM zero-modes ψ a (x)\\psi _a(x) of V(x)=A(sech 6 (x)-sech 4 (x))V(x)= A (\\operatorname{sech}^6(x) - \\operatorname{sech}^4(x))[up] and the corresponding probability densities |ψ a (x)| 2 |\\psi _a(x)|^2 [down],for A=-56.05506043241{A}=-56.05506043241 (left); A=-284.9369967664{A} = -284.9369967664 (center);A=-691.7230772070{A} = -691.7230772070 (right).For a constant mass, the general solution for this potential is [18] $&&\\chi ^{(1)}(x) = {\\rm e}^{{}^1\\!/{}_{\\!2}\\sqrt{A} \\tanh ^2x}(\\operatorname{sech}x)^{\\sqrt{-E}}\\,Hc\\!", "\\left(\\sqrt{A},-\\frac{1}{2}, \\sqrt{-E},\\ 0,\\frac{1-E}{4};\\, \\tanh ^2\\!x \\right) \\\\&&\\chi ^{(2)}(x) = {\\rm e}^{{}^1\\!/{}_{\\!2}\\sqrt{A} \\tanh ^2x}\\,(\\operatorname{sech}x)^{\\sqrt{-E}}\\tanh x\\,\\,Hc\\!", "\\left(\\sqrt{A},\\frac{1}{2}, \\sqrt{-E},\\ 0,\\frac{1-E}{4};\\, \\tanh ^2\\!x \\right).$ It is noteworthy that we found no zero-energy modes in the ordinary constant mass cases of neither $V(x)= A \\operatorname{sech}^6x$ nor $V(x)= A (\\operatorname{sech}^6x - \\operatorname{sech}^4x)$ potentials." ], [ "Three-term potentials", "The three-term potentials given by Eq.", "(REF ) have three possible phases: hyperbolic single-wells, hyperbolic double-wells and hyperbolic triple-wells.", "Since we have already analyzed in detail the first two situations, among which the PDM Poschl-Teller [21] and PDM Manning potentials respectively, we now focus on the triple-well case which oblige the three terms.", "In Fig.", "REF we show a sequence of triple-wells based on the Manning potential, already represented at the top of Fig.", "REF , now with the addition of a ”$\\sinh ^6x$ ” term.", "It can be seen that in this case the bigger is $A$ the softer is the barrier.", "In Fig.", "REF (up) we show the eigenstates of this three-term PDM-potential $A=60$ (solid) together with the $A=0$ (dashed) Manning potential.", "In Fig.", "REF (down) we show again the $A=60$ and $A=0$ eigenstates but for an ordinary constant mass.", "We put the figures altogether in two lines for a more comprehensive comparison.", "In all the eigenstates we see a higher probability density around the origin in the $V_A(x)$ potential and, remarkably, the PDM particle is always more probably tunneling than the ordinary one.", "We have numerically computed the full spectrum of the $A=60$ potential and found that a for a constant-mass particle there are 14 eigenstates that for PDM merge into eight (see Table REF ).", "Table: Full list of the energy eigenvalues of the PDM and constant-masshamiltonians for B=-C=-500B=-C=-500 and A=60A=60.", "The S _S and A _A subindexes indicatesymmetric and antisymmetric states.Figure: Plot of V(x)=-Asech 6 x-Bsech 4 x-Csech 2 xV(x)= - A \\operatorname{sech}^6x - B \\operatorname{sech}^4x - C\\operatorname{sech}^2xfor A=0 {A}=0 (black), 60 (red), 120 (orange), 500 (blue)and B=-C {B}=- {C} = - 500.", "The first is the Manning case alreadyrepresented at the top of Fig.", "and the following double-wells resultfrom an AA term added to it.Figure: Plot of symmetric eigenstates of V A (x)=V(x) Mann -60sech 6 (x)V_A(x)= V(x)_{Mann}-60 \\operatorname{sech}^6(x) (solid)versus V(x) Mann V(x)_{Mann} (dashed) for PDM [up] and constant mass [down].Likewise, when we add a ”$\\operatorname{sech}^2x$ ” term the zero-modes found in the previous section REF , and the values of $A$ for them to exist, deviate increasingly as $C$ goes bigger.", "In Fig.", "REF we show the first zero-modes for $C=10$ and $C=2$ with respect to $C=0$ .", "These zero-modes take place for $A$ as listed in Table REF .", "Adding a $C=10$ term has a stronger effect and the first two ${A}$ values with a zero-mode are in this case positive.", "Note that as $C$ increases the zero energy particle tends to stay closer to the origin.", "Table REF and the corresponding figures show that, as $A$ grows, the values of $A$ and the curves themselves rapidly converge to a unique one for all the three columns.", "In any case, the number of nodes of these eigenstates grows accordingly.", "Figure: Comparative plot of C=10C=10 (orange solid) vs. C=2C=2 (blue dotted) vs. C=0C=0 (red dashed) forthe first PDM zero-modes of potential V(x)=A(sech 6 (x)-sech 4 (x))+Csech 2 xV(x)= A (\\operatorname{sech}^6(x) - \\operatorname{sech}^4(x)) + C \\operatorname{sech}^2x; see values ofAA in Table .Table: List of AA values for which potentialV(x)=A(sech 6 (x)-sech 4 (x))+Csech 2 xV(x)= A (\\operatorname{sech}^6(x) - \\operatorname{sech}^4(x)) + C \\operatorname{sech}^2x has PDM zero-modes.The S _S and A _A subindexes at left indicate symmetric and antisymmetric states.Now, let us finally deal with the triple-well phase of the family.", "In Fig.", "REF we display a sequence of triple-well potentials constructed by setting specific relations among the parameters.", "In the first (left) figure the potential is about starting the triple-well phase.", "In this case, the first and second derivatives are zero at $x=\\pm \\operatorname{arcsech}\\sqrt{-B/3A}, B/A<0$ , for $B^2=3AC$ .", "We adopt $A>0$ so that $C>0$ and $B<0$ .", "In the second figure we let $B^2>3AC$ and we get a couple of maxima, one at each side of the origin, resulting in three cleanly defined wells.", "In the third figure we let $B^2=4AC$ when both maxima make simultaneously $V^{\\prime }(x)=0$ and $V(x)=0$ .", "For $B^2>4AC$ the potential develops a nonnegative barrier (see fourth figure).", "Figure: Plot of a sequence of potentialsV(x)=-240sech 6 x-Bsech 4 x-160sech 2 xV(x)= - 240\\operatorname{sech}^6x - B \\operatorname{sech}^4x - 160\\operatorname{sech}^2x as described in the text.Figure: Plot of the potentialV T (x)=-800sech 6 x+4ACsech 4 x-449sech 2 xV_T(x)= -800\\operatorname{sech}^6x+\\sqrt{4AC}\\operatorname{sech}^4x-449\\operatorname{sech}^2x.For the triple-well represented in Fig.", "REF , $V_T(x)= -A\\operatorname{sech}^6x+\\sqrt{4AC}\\operatorname{sech}^4x-C\\operatorname{sech}^2x$ ($A=800, C=449$ ), we show the full PDM bound spectrum of eigenfunctions and probability densities in Fig.", "REF .", "As shown in Table REF , there is once again a reduction or merging of eigenstates for PDM in which case the ten (constant-mass) eigenstates result in just four eigenstates when the mass depends on the position.", "Figure: Plot of PDM eigenfunctions and probabilities forV T (x)=-800sech 6 x+4ACsech 4 x-449sech 2 xV_T(x)= -800\\operatorname{sech}^6x+\\sqrt{4AC}\\operatorname{sech}^4x-449\\operatorname{sech}^2x.", "The full sequence of eigenergieshas been numerically computed, see Table .Table: Full list of the energy eigenvalues of thePDM and constant-mass triple-well hamiltonians for A=800A=800, C=449C=449,and B=-4AC≈-1198.67B=-\\sqrt{4AC}\\approx -1198.67.", "The S _S and A _A subindexesat left indicate symmetric and antisymmetric states." ], [ "Conclusion ", "In the present paper we have studied the new hyperbolic potential class recently reported in [18].", "Here, the mass of the particle has gained a position-dependent status in order to enrich the phenomenological possibilities of the models involved and to explore its mathematical consequences.", "After properly setting up the PDM problem we have payed special attention to the celebrated Manning potential, of great interest in molecular physics, here studied in the PDM case for the first time.", "We have analytically obtained the complete set of eigenstates to this hamiltonian and then shown its full bound-state spectrum and eigenfunctions in a case study.", "We have analytically found confluent-Heun expressions in the general case and compared the PDM eigenfunctions to the constant-mass ones.", "Heun functions have recently been receiving increasing attention and have been found in a wide variety of contexts [83], [84], [85], [86], [87], [88].", "PDM particles tend to be more likely tunneling than ordinary ones.", "Next, we have addressed the PDM version of the sixth power hyperbolic potential $V(x) = -{A}~{\\operatorname{sech}^6x}-B~{\\operatorname{sech}^4x}$ and obtained exact expressions for the zero-modes in the $A$ and $A=-B$ cases belonging to a discrete set of parameters.", "All such eigenstates have been found proportional to triconfluent forms of the Heun functions.", "Interestingly, we have met with no zero-modes for any value of the parameter in the ordinary constant-mass counterpart of this potential.", "In both the Manning and sixth-order hyperbolic potentials we have also analyzed the consequences of considering a complementary three-term potential by comparing their eigenfunctions.", "The analysis of these and others three-terms cases has been performed for constant-mass and PDM hamiltonians and has shown interesting differences between their spectra, particularly a reduction of eigenstates in the PDM circumstance.", "Finally, we have discussed the triple-well phase of the potential class and focused especially in a $V_T(x)= -A\\operatorname{sech}^6x+\\sqrt{4AC}\\operatorname{sech}^4x-C\\operatorname{sech}^2x$ case showing the full set of PDM eigenfunctions and probability densities.", "This triple-well also exposes the phenomenon of merging of the ordinary spectrum when the mass turns nonuniform.", "It seems to be a general property of this class of hamiltonians." ] ]
1403.0302
[ [ "Galvanomagnetic effects and manipulation of antiferromagnetic\n interfacial uncompensated magnetic moment in exchange-biased bilayers" ], [ "Abstract In this work, IrMn$_{3}$/insulating-Y$_{3}$Fe$_{5}$O$_{12}$ exchange-biased bilayers are studied.", "The behavior of the net magnetic moment $\\Delta m_{AFM}$ in the antiferromagnet is directly probed by anomalous and planar Hall effects, and anisotropic magnetoresistance.", "The $\\Delta m_{AFM}$ is proved to come from the interfacial uncompensated magnetic moment.", "We demonstrate that the exchange bias and rotational hysteresis are induced by the irreversible switching of the $\\Delta m_{AFM}$.", "In the training effect, the $\\Delta m_{AFM}$ changes continuously.", "This work highlights the fundamental role of the $\\Delta m_{AFM}$ in the exchange bias and facilitates the manipulation of antiferromagnetic spintronic devices." ], [ "Galvanomagnetic effects and manipulation of antiferromagnetic interfacial uncompensated magnetic moment in exchange-biased bilayers X. Zhou Shanghai Key Laboratory of Special Artificial Microstructure and Pohl Institute of Solid State Physics and School of Physics Science and Engineering, Tongji University, Shanghai 200092, China L. Ma Shanghai Key Laboratory of Special Artificial Microstructure and Pohl Institute of Solid State Physics and School of Physics Science and Engineering, Tongji University, Shanghai 200092, China Z. Shi Shanghai Key Laboratory of Special Artificial Microstructure and Pohl Institute of Solid State Physics and School of Physics Science and Engineering, Tongji University, Shanghai 200092, China W. J.", "Fan Shanghai Key Laboratory of Special Artificial Microstructure and Pohl Institute of Solid State Physics and School of Physics Science and Engineering, Tongji University, Shanghai 200092, China R. F. L. Evans Department of Physics, University of York, York YO10 5DD, United Kingdom R. W. Chantrell Department of Physics, University of York, York YO10 5DD, United Kingdom S. Mangin Institut Jean Lamour, UMR CNRS 7198, Universit¨¦ de Lorraine- boulevard des aiguillettes, BP 70239, Vandoeuvre cedex F-54506, France H. W. Zhang State Key Laboratory of Electronic Thin Films and Integrated Devices, University of Electronic Science and Technology of China, Chengdu 610054, China S. M. Zhou$^{\\ddag }$ ${}^{\\ddag }$ Correspondence author.", "Electronic mail: [email protected] Shanghai Key Laboratory of Special Artificial Microstructure and Pohl Institute of Solid State Physics and School of Physics Science and Engineering, Tongji University, Shanghai 200092, China In this work, IrMn$_{3}$ /insulating-Y$_{3}$ Fe$_{5}$ O$_{12}$ exchange-biased bilayers are studied.", "The behavior of the net magnetic moment $\\Delta m_{AFM}$ in the antiferromagnet is directly probed by anomalous and planar Hall effects, and anisotropic magnetoresistance.", "The $\\Delta m_{AFM}$ is proved to come from the interfacial uncompensated magnetic moment.", "We demonstrate that the exchange bias and rotational hysteresis are induced by the irreversible switching of the $\\Delta m_{AFM}$ .", "In the training effect, the $\\Delta m_{AFM}$ changes continuously.", "This work highlights the fundamental role of the $\\Delta m_{AFM}$ in the exchange bias and facilitates the manipulation of antiferromagnetic spintronic devices.", "75.30.Gw; 75.50.Ee; 75.47.-m; 75.70.-i Exchange bias (EB) phenomenon in ferromagnetic (FM)/antiferromagnetic (AFM) systems has attracted lots of attention because of its intriguing physics and technological importance in spin valve based magnetic devices [1], [2], [3], [4].", "After the FM/AFM bilayers are cooled under an external magnetic field from high temperatures to below the Néel temperature of the AFM layers, the hysteresis loops are simultaneously shifted and broadened [5].", "FM/AFM bilayers are now commonly integrated in spintronic devices [6].", "Nevertheless manipulation and characterization of the AFM spins are important to understand and control the exchange bias phenomenon [7].", "Rotatable and frozen AFM spins are generally thought to be responsible for the coercivity enhancement and shift of the FM hysteresis loops [8], [9], [10], [11], [12].", "Ohldag et al found that a nonzero AFM net magnetic moment $\\Delta m_{AFM}$ is necessary to establish the EB [13].", "However, Wu et al thought that the EB can be established without frozen AFM spins [9].", "Therefore, the behavior of AFM spins is still under debate.", "Moreover, the EB training effect is attributed to the relaxation of the $\\Delta m_{AFM}$ towards the equilibrium state during consecutive hysteresis loops [15], [16], [18], [19], [14], [17].", "For FM/AFM bilayers, the rotational hysteresis loss at $H$ larger than the saturation magnetic field is ascribed to the irreversible switching of AFM spins during clock wise (CW) and counter clock wise (CCW) rotations [20], [21], [22], [23].", "Since there is still a lack of direct experimental evidence, it is necessary to elucidate the fundamental mechanism of the AFM spins in FM/AFM bilayers in experiments.", "Figure: (a)Small angle x-ray reflection, (b)large angle x-ray diffraction on IrMn/YIG films, Φ\\Phi and Ψ\\Psi scan with fixed 2θ2\\theta for the (008) reflection of GGG substrate (c) and YIG film (d).", "The room temperature in-plane magnetization hysteresis loop of the YIG layer is shown in the inset of (a).In most studies, the information of AFM spins is indirectly explored from the hysteresis loops of the FM layers with micromagnetic simulations and Monte Carlo calculations [12], [18], [21], [16].", "In sharp contrast, very few methods can be implemented to directly probe the AFM spins due to almost zero net magnetic moment of the AFM layers.", "However, different measurements, combining x-ray magnetic circular dichroism and x-ray magnetic linear dichroism can detect FM and AFM spins due to their element-specific advantage [9], [24], [13].", "Only very recently, tunneling anisotropic magnetoresistance (TAMR) effect, which was initially proposed for tunneling device consisting of a single FM electrode and a nonmagnetic electrode [25], has been used to probe the motion of the AFM spins in AFM spintronic devices [26], [27].", "Since the TAMR arises from the tunneling density of states which depends on the orientation of the AFM spins in a complex way, however, the orientation of the AFM spins cannot be determined directly and in particular the issue whether the $\\Delta m_{AFM}$ does exist or not is still unsolved [28].", "In this Letter, we demonstrate clear evidence of the $\\Delta m_{AFM}$ and reveal its mechanism behind the EB, the training effect, and the rotational hysteresis for IrMn$_{3}$ (=IrMn)/Y$_{3}$ Fe$_{5}$ O$_{12}$ (YIG) bilayers using anomalous Hall effect (AHE), planar Hall effect (PHE), and anisotropic magnetoresistance (AMR) measurements.", "Here, the YIG insulator is used as the FM layer such that all magnetotransport properties are contributed by the metallic IrMn layer.", "Galvanomagnetic measurements allow to probe the entire IrMn layer and not only the interface as in the reported TAMR measurements [26], [27].", "The $\\Delta m_{AFM}$ in metallic IrMn is proved experimentally to arise from the interfacial uncompensated magnetic moment.", "It is clearly demonstrated in experiments that the EB and related phenomena are intrinsically linked to the partial pinning and irreversible motion of the $\\Delta m_{AFM}$ .", "Figure: For IrMn/YIG bilayer, Hall loops at 20 K (a) and 50 K (b) with HH along the film normal direction, and AHC as a function of TT (c).IrMn (5 nm)/YIG (20 nm) bilayers were fabricated by pulsed laser deposition (PLD) and subsequent magnetron sputtering in ultrahigh vacuum on (111)-oriented, single crystalline Gd$_{3}$ Ga$_{5}$ O$_{12}$ (GGG) substrates [29].", "X-ray reflectivity (XRR) measurements show that YIG and IrMn layers are $20\\pm 0.6$ and $5.0\\pm 0.5$ nm, respectively, as shown in Fig.", "REF (a).", "The root mean square surface roughness of the YIG layer is fitted to be 0.6 nm.", "The x-ray diffraction (XRD) spectra in Fig.", "REF (b) show that the GGG substrate and YIG film are of (444) and (888) orientations.", "The pole figures in Figs.", "REF (c) and REF (d) confirm the epitaxial growth of the YIG film.", "As shown in inset of Fig.", "REF (a), the magnetization (134 emu/cm$^{3}$ ) of the YIG film is close to the theoretical value of 131 emu/cm$^{3}$ and the coercivity is small as 6 Oe.", "Before measurements, the films are patterned into normal Hall bar and then cooled from room temperature to 5 K under $H=30$ kOe along the film normal direction.", "The Hall resistivity $\\rho _{xy}$ was measured as a function of the out-of-plane $H$ at various temperatures, as shown in Fig.", "REF .", "The Hall resistivity at spontaneous states $\\rho _{xy}^{+}$ and $\\rho _{xy}^{-}$ were extrapolated from the positive and negative high $H$ and the anomalous Hall resistivity was obtained by the equation $\\rho _{AH}$ =$(\\rho _{xy}^{+}-\\rho _{xy}^{-})/2$ .", "One has the anomalous Hall conductivity (AHC) $\\sigma _{AH}\\approx \\rho _{AH}$ /$\\rho _{xx}^2$ because $\\rho _{AH}$ is two orders of magnitude smaller than the $\\rho _{xx}$  [30].", "Since the $\\rho _{AH}$ decreases sharply and vanishes near $T=50$ K as shown in Figs.", "REF (a) and REF (b), the $\\sigma _{AH}$ is reduced with increasing $T$ and is equal to zero at $T\\ge 50$ K in Fig.", "REF (c).", "More remarkably, since the shifting and asymmetry of the Hall loop both exist at low $T$ and vanish near the same $T=50$ K, the AHC is accompanied by the perpendicular EB [31], [32].", "It is essential to address the physics for the AHC in the IrMn/YIG bilayers.", "With large atomic spin-orbit coupling of heavy Ir atoms and magnetic moment (2.91 $\\mu _B$ ) of Mn atoms, reasonably large AHC is expected in the chemically-ordered $L1_{2}$ IrMn alloy under high $H$  [33], and it should be independent of the film thickness and change slowly with $T$ due to the high Néel temperature of the AFM alloy.", "In sharp contrast, the $\\sigma _{AH}$ in the present IrMn/YIG bilayers changes strongly with the IrMn layer thickness, demonstrating an interfacial nature, as shown in Fig.S1 [29].", "Therefore, the present AHC results should not be caused by the noncollinear spin structure on the kagome lattice [33], which is further confirmed by the vanishing AHC for the 5 nm thick IrMn films on GGG substrates in Fig.S2 [29].", "This is because the present IrMn layers deposited at the ambient temperature are of the chemically disordered face-centered-cubic structure [34].", "With the strong $T$ dependence, the present AHC results cannot be attributed to the spin Hall magnetoresistance either [35].", "As pointed above, however, the AHC is strongly related to the established EB.", "As shown by the AMR results below, any FM layer at the interface can be excluded.", "Therefore, the AHC exclusively hints the existence of the IrMn interfacial uncompensated magnetic moment which is produced by the field cooling procedure.", "Figure: PHE loops (a) and AMR curves (b) of the cycle n=1n=1, 2, and 7 at 5 K. In the schematic picture (c), the film is aligned in the x-y plane, the sensing current i, cooling field H FC H_{FC}, and HH are parallel to the x axis.", "The orientations of the Δm AFM \\Delta m_{AFM} and the FM magnetization are given at stages A(d), B(e), C(f), D(g), and E(h) of the descent branch of the n=1n=1 in (a).", "In (d, e, f, g, h), 0<θ AFM (A)<θ AFM (B)<90 0 0<\\theta _{AFM}(A)<\\theta _{AFM}(B)<90^{0}, and -90 0 <θ AFM (D)<θ AFM (C)<-θ AFM (B)-90^{0}<\\theta _{AFM}(D)<\\theta _{AFM}(C)<-\\theta _{AFM}(B), and 0<θ AFM (A)<θ AFM (E)<90 0 0<\\theta _{AFM}(A)<\\theta _{AFM}(E)<90^{0}.Figures REF (a) and REF (b) show the PHE loops and AMR curves with consecutive cycles after the sample is cooled from room temperature to 5 K with the in-plane $H$ along the cooling field $H_{FC}$ which is parallel to the sensing current $i$ , as shown in Fig.", "REF (c).", "Several distinguished features are demonstrated in the descent branch of the first cycle, $n=1$ .", "Most importantly, for FM metallic films the AMR curves are symmetric, that is to say, the values of the $R_{xx}$ at positive and negative high $H$ are equal to each other [36], [16].", "In striking contrast, however, the AMR curve in Fig.", "REF (b) is asymmetric.", "Therefore, the present AMR results cannot be attributed to any metallic FM layer at the interface but exclusively to the interfacial uncompensated magnetic moment of the IrMn layer.", "Accordingly, the PHE signal and the AMR ratio are proportional to $sin(2\\theta _{AFM})$ and $1-cos^2\\theta _{AFM}$ , respectively, where $\\theta _{AFM}$ refers to the orientation of the $\\Delta m_{AFM}$ with respect to the $x$ axis [36].", "More remarkably, with the monotonic change of the $R_{xx}$ , one has $|\\theta _{AFM}|\\le 90$ degrees.", "In combination with the sign change of the $R_{xy}$ , the $\\Delta m_{AFM}$ should be in either the first or the fourth quadrant [26], as schematically shown in Figs.", "REF (d)-REF (g).", "The IrMn layer is far from the negative saturation within the field of -600 Oe.", "Therefore, the angle between the FM and AFM spins is smaller (larger) than 90 degrees at the positive (negative) high $H$ , and the FM/AFM system is of low (high) interfacial exchange coupling energy, leading to the lateral and vertical shift of the hysteresis loops [13], [37], [38].", "Moreover, when the $H$ changes from stages B to C, the $\\Delta m_{AFM}$ is irreversibly switched from the first quadrant to the fourth one [21], as demonstrated by the variations of $R_{xx}$ and $R_{xy}$ in Figs.", "REF (a) and  REF (b).", "As schematically shown in Figs.", "REF (d)- REF (g), during the sweeping of $H$ , both the magnitude and orientation of the $\\Delta m_{AFM}$ may change, indicating the multidomain process.", "Therefore, the observations of both the $\\Delta m_{AFM}$ and its motion help to elucidate the intriguing physics behind the shifting and broadening of the hysteresis loops in FM/AFM bilayers, and in particular asymmetric magnetization reversal process of the FM magnetization [27], [39].", "For the cycle number $n=1$ , 2, and 7, the descent branch shifts significantly whereas the ascent branch almost does not change as shown in Figs.", "REF (a) and REF (b), in agreement with the first kind of the EB training effect of the (FM) magnetization hysteresis loops in Fig.S3 [29], [14].", "The athermal training effect from $n=1$ to $n=2$ is much larger than those of $n>2$ , which was explained as a result of the switching of AFM spins among easy axes by Hoffmann [15].", "In particular, the PHE signal and AMR ratio at the starting stage A are smaller than those of the ending stage E, indicating that the state of the $\\Delta m_{AFM}$ cannot be recovered after the first cycle, which further confirms the theoretical predictions [15], [18], [19].", "As schematically shown in Figs.", "REF (a) and REF (b), the $\\Delta m_{AFM}$ experiences different trajectories during consecutive cycles, explaining the physics behind the EB training in FM/AFM systems [14], [15], [16], [23], [17], [40].", "Figures REF (a)-REF (d) show the PHE signal as a function of $\\theta _H$ with CW and CCW rotations under different magnitudes of $H$ .", "At $H=50$ Oe, the CW and CCW curves overlap and the FM and AFM spins are expected to rotate reversibly within a small angular region.", "For higher $H$ , the hysteretic behavior begins to occur and becomes strong for $H=300$ and 500 (Oe).", "This effect starts to decrease for $H=1.0$ kOe but still persists at $H=20$ kOe.", "Figures REF (e)- REF (h) show the angular dependence of the PHE signal with CW and CCW rotations under $H=1.0$ kOe at different $T$ .", "At low $T$ , there is a difference between the CW and CCW curves, indicating irreversible rotation of the $\\Delta m_{AFM}$ , and the hysteretic effect becomes weak at enhanced $T$ .", "Near $T_B$ , the measured results can be fitted well with $sin(2\\theta _H)$ due to the ordinary magnetoresistance effect, as demonstrate by the vanishing PHE signal in Fig.S4 [29].", "In a word, the hysteretic behavior of the PHE curves reproduce the rotational hysteresis loss of the FM magnetization in FM/AFM bilayers [1], [5], [22].", "Figure: Angular dependent PHE signal with CW and CCW senses at H=50H=50 (a), 300 (b), 500 (c), and 1000 (d) (Oe) , and at T=5T=5 (e), 15 (f), 25 (g), and 50 (h) (K).", "T=5T=5 K in the left column and H=1.0H=1.0 kOe in the right column.", "In (h), solid cyan line refers to the sin(2θ H )sin(2\\theta _H) fitted results.It is interesting to analyze the magnitude and reversal mechanism of the $\\Delta m_{AFM}$ as a function of $T$ .", "The galvanomagnetic effects in Fig.", "REF and Fig.S4[29] become weak with increasing $T$ , clearly indicating that the $\\Delta m_{AFM}$ is reduced at elevated $T$ and approaches vanishing at $T_B$ .", "Meanwhile, the $\\Delta m_{AFM}$ at low $T$ is reversed irreversibly, leading to the EB establishment.", "At high $T$ , reversible reversal becomes dominant, resulting in the disappearance of the EB.", "The variation of the reversal mode with $T$ confirms the validity of the thermal fluctuation model for polycrystalline AFM systems [41], [8], [23].", "In this model, the reversal possibility is governed by the Arrhenius-Néel law and determined by the competition between the thermal energy and the energy barrier which is equal to the product of the uniaxial anisotropy and the AFM grain volume.", "The low $T_B$ of 50 K is induced by ultrathin thickness and the microstructural deterioration of the IrMn layer which is induced by the lattice mismatch between IrMn and YIG layers [26].", "At $T<T_B$ , the energy barrier is larger than the thermal energy, leading to the irreversible process in most AFM grains.", "Accordingly, the EB is established and accompanied by the sizeable galvanomagnetic effects.", "Since more AFM grains become superparamagnetic for $T$ close to $T_B$ , the $\\Delta m_{AFM}$ , galvanomagnetic effects, and the EB all approach vanishing, as shown in Figs.", "REF and REF , and Fig.S4[29].", "On the other hand, the Meiklejohn-Bean model and the domain state model are not suitable for the present results [5], [42].", "In the former model, AFM spins are fixed during the reversal of the FM magnetization, which is in contradiction with the present results.", "In the latter one, the uncompensated AFM magnetic moment is mainly contributed by the bulk AFM [42] whereas the $\\Delta m_{AFM}$ in the present IrMn/YIG systems mainly stems from the uncompensated magnetic moment at FM/AFM interface.", "In summary, for IrMn/YIG bilayers the interfacial uncompensated magnetic moment $\\Delta m_{AFM}$ is observed by the galvanomagnetic effects.", "The partial pinning and irreversible switching of the $\\Delta m_{AFM}$ are directly proved to be the physical source for the exchange field, coercivity enhancement, and the rotational hysteresis loss.", "The orientation of the $\\Delta m_{AFM}$ is found to continuously change during the EB training effect.", "The present work permits a better understanding of the EB and related phenomena in FM/AFM bilayers.", "It demonstrates that galvanomagnetic measurements allow to probe the behavior of the AFM layer and consequently are a powerful tool to understand FM/AFM systems.", "This technique should be useful in the field of AFM spintronics.", "This work was supported by the State Key Project of Fundamental Research Grant No.", "2015CB921501, the National Science Foundation of China Grant Nos.11374227, 51331004, 51171129, and 51201114, Shanghai Science and Technology Committee Nos.0252nm004, 13XD1403700, and 13520722700." ] ]
1403.0148
[ [ "Rotational properties of two-component Bose gases in the lowest Landau\n level" ], [ "Abstract We study the rotational (yrast) spectra of dilute two-component atomic Bose gases in the low angular momentum regime, assuming equal interspecies and intraspecies interaction.", "Our analysis employs the composite fermion (CF) approach including a pseudospin degree of freedom.", "While the CF approach is not {\\it a priori} expected to work well in this angular momentum regime, we show that composite fermion diagonalization gives remarkably accurate approximations to low energy states in the spectra.", "For angular momenta $0 < L < M$ (where $N$ and $M$ denote the numbers of particles of the two species, and $M \\geq N$), we find that the CF states span the full Hilbert space and provide a convenient set of basis states which, by construction, are eigenstates of the symmetries of the Hamiltonian.", "Within this CF basis, we identify a subset of the basis states with the lowest $\\Lambda$-level kinetic energy.", "Diagonalization within this significally smaller subspace constitutes a major computational simplification and provides very close approximations to ground states and a number of low-lying states within each pseudospin and angular momentum channel." ], [ "Introduction", "In recent years there has been extensive interest in the study of strongly correlated states of cold atoms motivated by analogies with exotic states known from low-dimensional electronic systems.", "Substantial theoretical and experimental effort is being devoted to the possible realisation of quantum Hall-type states in atomic Bose condensates [1].", "Conceptually the simplest way of simulating the magnetic field is by rotation of the atomic cloud [2], although alternative proposals involving synthetic gauge fields [3], [4], [5] are more likely to provide stronger magnetic fields.", "Consequently, a large body of work has focused on the rotational properties of atomic Bose gases.", "In this context, several groups have studied the rotational properties of bosons in the lowest Landau level also at the lowest angular momenta [6], [7], [8], [9], [10], [11] ($L \\le N$ where $N$ is the number of particles).", "Notably, analytically exact ground state wave functions were found in this regime[7], [8].", "Much of these studies have considered small systems.", "Interestingly, a recent experimental paper [12], claiming the first ever realisation of rotating bosons in the quantum Hall regime, precisely involves such small systems (up to ten particles), and includes the lowest angular momenta.", "Even richer physics can be expected in the case of two-species bosons (or fermions, for that sake).", "Two-species Bose gases can be realised as mixtures of two different atoms[13], two isotopes of the same atom[14], or two hyperfine states of the same atom[15].", "As long as all interactions, between atoms of the same species, and between atoms of different species, are the same (we will refer to this as `homogeneous interaction'), the system possesses a pseudospin symmetry, with the two species corresponding to pseudospin “up\" and “down\", respectively.", "Tuning the interaction away from homogeneous can lead to interesting physics, such as a transition from a miscible to an immiscible regime where the interspecies interaction dominates[16].", "It is then obviously of interest to study the rotational properties of such systems.", "Several recent papers have addressed the very interesting topic of possible quantum Hall phases of two-species Bose gases [17], [18], [19], [20], [21], [22].", "In this paper we study the yrast spectra of dilute, rotating two-species Bose gases with homogeneous interaction, at low angular momenta, $L\\le N+M$ , where $N+M$ is the total number of particles.", "Physically, this is the regime where the first vortices (in the two components) enter the system[23].", "A recent study by Papenbrock et al[24] identified a class of analytically exact states in the yrast spectrum of this regime; these include the ground state and some (but not all) excited states.", "Our approach is to study the yrast spectrum in terms of trial wave functions given by the composite fermion (CF) approach[25], with a pseudospin degree of freedom accounting for the two species.", "As has been discussed for the one-species case [26], [10], [11], this is a regime where CF trial wave functions cannot, a priori, be expected to work well, since the physical reasoning behind the CF approach [25] implies that it should apply primarily in the quantum Hall regime where angular momenta are much higher, $L \\sim N^2$ .", "Nevertheless, this approach turned out to work very well for single species gases, and as we will see, the same happens for two-species systems.", "We present results for up to twelve particles, in the disk geometry [27].", "We find that the CF formalism gives very good approximations to the exact low energy states of the system, with overlaps very close to unity.", "In particular, all states of the yrast spectrum, including those of Ref.", "papenbrock, at $0 < L < M$ are reproduced exactly.", "For given $N,M$ and $L$ , the number of CF candidate states of the right quantum numbers is usually larger than one, and one performs a diagonalization within the space spanned by these states.", "This is what we will refer to as CF-diagonalization.", "However, the dimension of this CF subspace is often not significantly smaller than the dimension of the full Hilbert space, and therefore the computational gain over a full numerical diagonalization may not be significant.", "We have identified, within the basis sets of CF candidates for given $(N,M,L)$ , a class of states of particularly simple structure, which in themselves have almost complete overlap with the lowest lying states.", "These states (which we will refer to as `simple states'), are characterised by having at most one composite fermion in each $\\Lambda $ -level, as will be explained below.", "Because of the pseudospin symmetry provided by the homogeneous interaction, from some state at given $L$ and $S_z = (M-N)/2$ , one can obtain, by pseudospin lowering, a state which has the same energy (and overlap) at different $M$ and $N$ .", "Typically, this is an excited state at the new $M, N$ .", "In this way, the simple states will produce excellent trial wave functions for the ground states, as well as a number of low-lying states for the whole pseudospin multiplet at given total number of particles $N+M$ .", "Restriction to the simple CF states constitutes a major computational simplification due to reduction in the dimension of the space.", "Thus, one gets, with relative ease, access to explicit polynomial wave functions that are very close to exact and can be further used to study, e.g., the vortices in the system.", "The paper is organised as follows.", "In section we introduce the theoretical background and formalism used.", "Results for full CF diagonalization as well as simple state calculations are presented in section .", "Finally, in section , we summarise and discuss future perspectives." ], [ "Theory and methods", "System and observables: The system we study consists of two species of bosons in a two dimensional rotating harmonic trap, interacting with each other through a contact potential.", "We will use the pseudospin terminology and refer to one species as \"up\" ($\\uparrow $ ) and the other as \"down\" ($\\downarrow $ ).", "The system of $N$ bosons of type $\\downarrow $ and $M \\ge N$ bosons of type $\\uparrow $ is described by the Hamiltonian $H = \\sum _{i=1}^{N+M} \\left( \\frac{\\mathbf {p}_i^2}{2m}+\\frac{1}{2}m\\omega ^2\\mathbf {r}_i^2 - \\Omega l_i \\right)+ \\sum _{i,j=1}^{N+M}2\\pi g \\delta (\\mathbf {r}_i - \\mathbf {r}_j)$ where $m$ is the mass of the bosons, $\\omega $ is the harmonic trap frequency, $\\Omega $ is the rotational frequency of the trap around the $z$ -axis, $l_i$ are the one-body angular momenta in the $z$ -direction, and $g$ is a parameter specifying the two-body interaction strength.", "As usual [1], in the dilute (weakly interacting) limit, this model may be recast as a two-dimensional lowest Landau level problem in the effective magnetic field $B_{eff}=2m\\omega $ , H = iN+M(-)li + 2g [ i<j=1N (zi - zj) +....             .", "...+ k<l=1M (wk - wl) + i<k=1 (zi - wk) ] with 'flat' Landau levels corresponding to the ideal limit $(\\omega - \\Omega ) \\rightarrow 0$ .", "Here $z = x+iy$ now refers to the positions of the $\\downarrow $ -particles, and $w$ refers to the $\\uparrow $ -particles.", "Notice that we have assumed equal interspecies and intraspecies interaction strength, as well as equal masses for the two species.", "The single-particle states spanning the lowest Landau level (in the symmetric gauge) are $\\psi _{0,l}(z) = N_{l} z^l \\exp {(-z\\bar{z}/4)} \\qquad l\\ge 0 $ where $l$ is the angular momentum of the state and $N_l$ is a normalisation factor.", "The unit of length is $\\sqrt{\\hbar /(2m\\omega )}$ .", "Hereafter we will omit the ubiquitous Gaussians and assume that derivatives do not act on the Gaussian part of the wavefunctions.", "The many-body wavefunctions with total angular momentum $L$ are then homogeneous symmetric (separately in $z_i$ and $w_k$ ) polynomials of total degree $L$ in the coordinates $z_i, w_k$ .", "The Hamiltonian is symmetric under rotation and under change of species of bosons (pseudospin).", "The primary observables of interest (that commute with the Hamiltonian) in the following discussion are the total angular momentum $L_z = \\sum _{i=1}^N z_i\\partial _{z_i} + \\sum _{k=1}^M w_k\\partial _{w_k},$ the center of mass angular momentum $L_c = R\\left( \\sum _{i=1}^N \\partial _{z_i} + \\sum _{k=1}^M \\partial _{w_k} \\right), $ where $R=\\frac{1}{N+M}\\left( \\sum _i^N z_i + \\sum _k^M w_k \\right)$ is the center of mass coordinate, and finally the pseudospin-1/2 operators $\\mathcal {S}^2$ and $\\mathcal {S}_z$ which, in second quantization language, are defined as $\\mathcal {S}_z = \\frac{1}{2}\\sum _{l=0}^\\infty b_{\\uparrow ,l}^\\dagger b_{\\uparrow ,l} - b_{\\downarrow ,l}^\\dagger b_{\\downarrow ,l}$ $\\mathcal {S}^2 = \\mathcal {S}_{-}\\mathcal {S}_{-}^\\dagger + \\mathcal {S}_z(\\mathcal {S}_z+1).$ Here $\\mathcal {S}_{-} = \\sum _{l=0}^\\infty b_{\\downarrow ,l}^\\dagger b_{\\uparrow ,l}$ is the pseudospin lowering operator, taking a state at $(N,M)$ to a state at $(N+1,M-1)$ .", "The $b_{\\downarrow ,l}^\\dagger $ and $b_{\\uparrow ,l}$ are creation and annihilation operators for $\\downarrow $ -type and $\\uparrow $ -type bosons at angular momentum $l$ respectively.", "As stated before, the eigenfunctions of $L_z$ are homogeneous, symmetric polynomials of total degree $L$ .", "The eigenvalues of $L_c$ are $l_c = 0,1,\\ldots ,L$ and the corresponding eigenfunctions are given by $R^{l_c}\\Psi (z,w)$ .", "The function $\\Psi (z,w)$ satisfies $L_c\\Psi (z,w) = 0$ and is therefore a translationally invariant (TI) state (see Appendix ) $\\Psi (z-c,w-c) = \\Psi (z,w).$ We will focus on such states in this paper, realizing that we can create all the states of energy $E$ , angular momentum $L+k$ and center-of-mass (COM) angular momentum $k$ (called COM excitations) by multiplication of $R^k$ with wave functions of energy $E$ , angular momentum $L$ and zero COM angular momentum.", "For a given $N$ and $M$ , the eigenvalue of $\\mathcal {S}_z$ is $(M-N)/2$ , while the eigenvalues of $\\mathcal {S}^2$ are $S(S+1)$ , with $S=S_z,S_z+1,\\ldots ,(N+M)/2$ .", "Since $\\left[ \\mathcal {S}^2,\\mathcal {S}_z \\right] =0$ , we can find the eigenstates in the following way: for a given $A=N+M$ , we find energy eigenstates at $N=0,M=A$ .", "These states all have $S=S_z=(M-N)/2=A/2$ , the only possible value for $S$ .", "Applying the pseudospin lowering operator to these states, we get all possible states at $S=A/2$ for the case $N=1,M=A-1$ .", "The remaining states at $N=1,M=A-1$ must then be the ones with $S=S_z=(A-2)/2$ .", "Since the pseudospin operators commute with the interaction, both energy expectation values and overlaps are unchanged under pseudospin raising and lowering.", "Therefore, for the purpose of comparing the exact energy eigenstates with the CF states of the same quantum number, we consider only the translationally invariant ($L_c=0$ ) highest weight states ($S=S_z$ ), abbreviated as TI-HW states in the following.", "Composite fermion trial wave functions: The composite fermion (CF) approach [25] has been extremely successful in describing the strongly interacting electrons in quantum Hall systems in terms of composite objects comprised of an electron and an even number of vortices (or, loosely speaking, magnetic flux quanta), moving in a reduced effective magnetic field.", "The single particle states available to these emergent particles, form Landau-like levels (often called $\\Lambda $ levels) in this effective magnetic field.", "The single particle wavefunction with an angular momentum $m$ in the $n^{\\rm th}$ $\\Lambda $ -level is $\\psi _{n,m}(z) = N_{n,m} z^m L_n^m\\left( \\frac{z\\bar{z}}{2} \\right),\\quad m \\ge -n ,$ where $L_n^m$ is the associated Laguerre polynomial, and $N_{n,m}$ is a normalization factor.", "As it turns out, composite fermions can be considered as weakly interacting, and in fact, a very accurate description can be obtained by simply approximating them as non-interacting.", "In this approximation, the generic form of a CF trial wave function is $\\Psi _{CF}=\\mathcal {P}_{LLL} \\left(\\Phi \\, J^p \\right)$ where $\\Phi $ is a Slater determinant of CFs, $J$ is a Jastrow factor, $J = \\prod _{i<j}(z_i-z_j)$ , so $J^p$ represents the \"attachment\" of $p$ vortices to each electron ($p$ is an even integer).", "$\\mathcal {P}_{LLL}$ denotes lowest Landau level projection.", "This approach can be straightforwardly modified [1] to describe quantum Hall-type states of bosons, by letting $p$ be an odd number.", "Throughout this paper we will work with the simplest case, $p=1$ .", "The above CF formalism can be generalized [25] to take into account a spin or pseudospin degree of freedom (for example, in a bilayer quantum Hall system).", "In this work we will pursue the analogy to spinful composite fermions, to study the case of two-species rotating bosons in the lowest Landau level, with a pseudospin-$1/2$ degree of freedom accounting for the two species of particles.", "In this case, a CF trial wave function has the general form $\\Psi _{CF}=\\mathcal {P}_{LLL} \\left(\\Phi _\\downarrow \\Phi _\\uparrow J(z,w) \\right)$ where $\\Phi _\\downarrow $ , $\\Phi _\\uparrow $ are Slater determinants for each species of the non-interacting CFs (consisting of the single-particle states (REF )), and the Jastrow factor involves both species, $J(z,w)=\\prod _{i<j=1}^N(z_i-z_j)\\prod _{k<l=1}^M(w_k-w_l)\\prod _{i,k=1}^{N,M}(z_i-w_k).$ Projection to the lowest Landau level is achieved by replacing the conjugate variables $\\overline{z}_i$ , $\\overline{w}_k$ by $\\partial _{z_i}$ , $\\partial _{w_k}$ after moving them all the way to the left in the final polynomial [25].", "A technical comment that is in order here concerns the fact that we are studying low angular momentum states.", "In the quantum Hall regime ($L\\sim \\mathcal {O}(N^2)$ ), where the CF approach is usually applied, many-body ground states involve a small number of filled $\\Lambda $ levels.", "For example, the $\\nu =1/3$ Laughlin state is represented as an integer quantum Hall state of composite fermions with one filled $\\Lambda $ level, the $\\nu =2/5$ quantum Hall state corresponds to two filled $\\Lambda $ levels etc.", "The situation in a low-$L$ state is somewhat different - since the Jastrow factor in (REF ) itself contributes a large angular momentum $L_J=A(A+1)/2$ where $A=N+M$ , the angular momenta of the Slater determinants have to be negative.", "The Slater determinants attain angular momentum by including many derivatives through occupying many $\\Lambda $ levels.", "For this reason, the CF states in the context of lower angular momenta involve many $\\Lambda $ levels, with only a few CFs in each (see e.g.", "the $L=N$ CF candidate discussed in Ref.", "korslund).", "The fact that the interaction is translationally invariant and commutes with the center-of-mass angular momentum allows us to focus our attention on “compact CF states”.", "A CF state is “compact\" if, for every single particle state $\\psi _{n,m}$ in its Slater determinant, the states $\\psi _{n-1,m}$ and $\\psi _{n,m-1}$ are also present.", "That is, every occupied $\\Lambda $ level in the determinant is occupied “compactly”, with all states $m=-n,\\ldots $ up to the largest value of $m$ in that level occupied.", "Fig:REF shows examples of such compact and non-compact states.", "Figure: Example of compact and non-compact states of a single species shown schematically.", "The lines indicate the possible single states that can be occupied by the CFs and the dots indicate occupied statesAll compact states are translationally invariant, and it can be shown that the projected Slater determinant is equivalent to a determinant in which the single particle states have the simple form [25] $\\psi _{n,m}(x_i) \\propto x_i^{n+m}\\partial _i^{n}.$ Because we are interested in the highest weight pseudospin states, the CF states that we consider are required to satisfy Fock's cyclic condition [25].", "For the CF states which are simple products of two Slater determinants times a Jastrow factor, Fock's cyclic condition is satisfied iff for every single CF state that is occupied in the $\\downarrow $ species (the species with fewer particles) the corresponding CF state is occupied in the $\\uparrow $ species (majority species) also.", "When $N=M$ , the two Slater determinants contain the same sets of occupied states $(n,m)$ .", "Note that a CF state which is not a simple product of two determinants and a Jastrow factor may also be a highest weight-state; we do not include such states in our analysis.", "By 'full CF-diagonalization' we thus mean a diagonalization of the interaction in the space spanned by compact CF states with correct angular momentum that satisfy Fock's cyclic condition (in the sense described above).", "Note that the number of linearly independent CF states will generally be less than the number of determinant-pairs that satisfy the relevant restrictions.", "Therefore we reduce the set of projected wave functions to a linearly independent set before diagonalization.", "A class of CF states to which we will devote special attention is the subset of compact CF states which minimize the total $\\Lambda $ -level kinetic energy, for a given $N$ , $M$ and $L$ .", "As mentioned, many $\\Lambda $ -levels need to be involved in order to produce the low angular momenta of the projected states we consider.", "On the other hand, the state vanishes if the $n=A$ level or higher is occupied (due to the high derivatives this implies).", "For $N>0$ and a given angular momentum $0 < L < N+M$ , one observes that the kinetic energy is minimized when the available $\\Lambda $ -levels are either empty or singly occupied.", "This means that all the non-zero single particle wave functions in the Slater determinants are of the form $\\psi _{n,-n}$ , which in the projected Slater determinants simply become $\\psi _{n,-n} \\propto \\partial _i^{n} $ Such states will be called 'simple states' in this paper.", "An example of such a state is the CF candidate for $N = 1, M=3, L=2$ in which the single-particle states $(n,m) = (0,0), (1, -1)$ and $(3, -3)$ are occupied by one species; and $(n,m) = (0,0)$ is occupied by the other (see Fig REF ).", "Figure: Two slater determinants of a simple state shown schematically.", "Left and right sections of the figure show the distribution of the CFs in the two slater determinants involved in the two boson CF state in Eq .", "Each Λ\\Lambda level contains either one or no CFs.The corresponding projected wave function, is $ \\mbox{$$} \\psi (\\lbrace z_i\\rbrace , \\lbrace w_i\\rbrace ) =\\begin{vmatrix}\\partial ^0_{z_1}\\end{vmatrix}\\cdot \\begin{vmatrix}\\partial ^0_{w_1} \\, \\partial ^0_{w_2} \\,\\partial ^0_{w_3}\\\\\\partial ^1_{w_1} \\, \\partial ^1_{w_2} \\,\\partial ^1_{w_3} \\\\\\partial ^3_{w_1} \\, \\partial ^3_{w_2} \\, \\partial ^3_{w_3} \\\\\\end{vmatrix}\\cdot \\, J(z,w)$ Another example are the cases $N=M=L$ , where the CFs occupy every other $\\Lambda $ -level, i.e.", "$(0,0)$ , $(2,-2)$ etc.", "up to $(2L-2,2-2L)$ .", "Incidentally these are exact ground states for the interaction we are considering.", "We find that, in order to capture the lowest energy states, it is sufficient to diagonalize within the restricted subspace of simple CF states instead of diagonalizing within all the compact CF candidates." ], [ "results", "In this section, we present a comparison of spectra from exact diagonalization of the Hamiltonian in the translationally invariant, highest weight sector to the results from the CF diagonalization discussed in Section II.", "We will present results both from full CF diagonalization and specifically from the use of only the 'simple states'.", "Systems of a total of up to 12 particles have been studied.", "For up to 8 particles, the numerics were done in Mathematica, preserving symbolic (i.e.", "infinite) precision up to the point where overlaps are given to machine precision.", "For larger systems a projection algorithm implemented in C was used (Appendix )." ], [ "Full CF diagonalization", "The yrast spectra for $N+M=8$ particles are shown in Fig.", "REF for $0 \\le L \\le N+M$ .", "Each subplot gives the spectrum for a specific $(N,M)$ and thus a fixed pseudospin $\\mathcal {S}_z\\equiv (N-M)/2$ .", "As mentioned before, pseudospin symmetry of the Hamiltonian makes it sufficient to study only the highest weight states $\\mathcal {S}\\equiv \\mathcal {S}_z\\equiv (N-M)/2$ in each spectrum.", "For example, the full spectrum of $(N,M)=(1,7)$ contains states of $\\mathcal {S}_z\\equiv 3$ and $\\mathcal {S}\\equiv 3,4$ .", "Energy and overlap (with CF states) of a state $\\left|n,\\mathcal {S}=4,\\mathcal {S}_z=3\\right\\rangle $ is identical to that of the HW state $S^+\\left|n,\\mathcal {S}=4,\\mathcal {S}_z=3\\right\\rangle $ which is in the HW spectrum of $(N,M)=(0,8)$ .", "Thus the full TI spectrum of (for example) $(N,M)=(2,6)$ should combine the HW spectrum of $(N,M)=(0,8),(1,7)$ and $(2,6)$ .", "For the angular momenta in the range $0\\le L <M$ , the number of linearly independent CF states is equal to the number of basis states in the highest weight sector.", "Therefore, the diagonalization simply reproduces the exact spectrum.", "Even though these CF functions do not extract low energy states, the CF wave functions form a particularly convenient basis (in contrast to the bases of Slater permanents or elementary symmetric polynomials) because the CF states are by construction translationally invariant and highest weight eigenstates of $\\mathcal {S}^2$ .", "For larger angular momenta $M\\le L \\le N+M$ , the overlaps are less than unity, but still very high ($>0.99$ ) for the low lying states in the spectrum, and $>0.9$ for all but a very few cases.", "Since the dimension of the full CF space is only slightly smaller the dimension of the complete Hilbert space, this is not very surprising.", "TABLE REF gives an overview of the dimensions of the spaces in the problem, along with the number of candidate CF wave functions, for $(N,M)=(2,6)$ .", "Again we notice that the CF states span the whole sector for $L<M$ .", "The number of distinct candidate (before projection) pairs of determinants tends to be considerably larger than the number of linearly independent CF basis states.", "We do not know of a systematic way to tell, a priori, which candidate determinants will produce identical (or linearly dependent) CF polynomials, and thus this has to be numerically checked explicitly.", "Figure: (Color online) Exact energy eigenstates (blue dots) and full CF diagonalizationresults (colored rings) in the TI-HW sector, for A=8A=8 particles.", "Numbers below the lowest-lyingstates are overlaps between exact ground state and CF approximation, rounded tofour digits.Table: Dimensionality and number of compact CF candidates for 2+62+6 particles.", "d H d_H is the dimension of the TI-HW eigenspace, d CF d_{CF}is the number of linearly independent compact CF states, and n CF n_{CF} is the number of distinct candidate determinants.As seen above, the space of CF states does not extract any information about the low-lying states in particular.", "In addition, the large number of compact candidate determinants quickly becomes difficult to handle.", "As an example, the dimension of the TI-HW eigenspace at $(N,M)=(3,9)$ and $L=8$ is only 14, while the number of compact candidates is 1799, see TABLE REF .", "We therefore seek to identify a subset of the CF states, to reduce their number without loosing the high overlaps with the lower energy exact states.", "The simple states have been found to accomplish just this.", "Table: Comparing the number of compact candidates to the number of simple candidates." ], [ "Simple CF diagonalization", "In TABLE REF we compare, for different $L$ , the number of candidate determinants for the compact states to the corresponding number of simple state-candidates at $(N,M)=(3,9)$ .", "As we see, the latter is significally lower[28].", "At the two extremes $S_z=0$ and $S_z=A/2$ we have only simple candidates and no simple candidates, respectively.", "Because of the latter fact, and also the fact that systems with $N=0$ are not really two-component systems to begin with, we will omit such cases in the discussion.", "Figure: (Color online) Exact energy eigenstates (blue dots) and simple CF diagonalizationresults (colored rings) in the TI sector, for A=8A=8 particles.", "All values of S 2 S^2 are includedin this plot.", "The numbers denote overlaps between (lowest-lying) simple states and exact ground states.Figure: (Color online) Exact energy eigenstates (blue dots) and simple CF diagonalizationresults (red rings) in the TI-HW sector, for A=12A=12 particles.", "The numbers denote overlaps between (lowest-lying) simple states and exact ground states.The yrast states are shown for 8 and 12 particles in FIGs REF and REF , this time with only the simple CF states included.", "Non-highest-weight states are also included in the spectrum.", "We clearly see that the simple states give very good approximations to the lowest-lying states, while the higher excitations are excluded.", "It is known [24] that the ground states at angular momenta $N \\le L \\le M$ are highest weight states, while this is not necessarily the case for larger $L$ .", "We find that the CF trial states give the correct pseudospin for the ground states.", "Generally, the simple states also cover most of the other low-lying spectrum with high accuracy.", "The cases $(N,M)=(1,A-1)$ are especially interesting.", "As we see from FIGs.", "REF and REF , there is one unique CF wave function for each $L$ , serving as ansatz states for the ground states.", "We also see that a gap to excitations opens up as $L$ approaches $L=A$ from below.", "For $L \\ge A$ however, there is no gap.", "In CF language, this corresponds to a situation where the $\\Lambda $ -level ”kinetic” energy decreases steadily as $L$ grows from 1 to $A-1$ , forming simple states at each $L$ .", "It turns out, however, that there is no way to keep decreasing the $\\Lambda $ -level energy as $L$ increases from $A-1$ to $A$ : at $A-1$ the state is $ \\psi (\\lbrace z_i\\rbrace , \\lbrace w_i\\rbrace ) =\\begin{vmatrix}\\partial ^0_{z_1}\\end{vmatrix}\\cdot \\begin{vmatrix}\\partial ^0_{w_1} & \\partial ^0_{w_2} & \\cdots & \\partial ^0_{w_{A-1}}\\\\\\partial ^1_{w_1} & \\partial ^1_{w_2} & \\cdots & \\partial ^1_{w_{A-1}} \\\\\\vdots & \\vdots & \\vdots & \\vdots \\\\\\partial ^{A-1}_{w_1} & \\partial ^{A-1}_{w_2} & \\cdots & \\partial ^{A-1}_{w_{A-1}} \\\\\\end{vmatrix}\\, \\prod _{k<l} (w_k - w_l) \\prod _{n} (z_1 - w_n)$ where the last part is the Jastrow factor for this system.", "There is no way to create a new simple state by increasing $L$ from the state (REF ).", "Therefore there is no `simple' CF candidate for $L=A$ .", "This matches well with an abrupt vanishing of the gap in exact spectrum, while going from $L=A-1$ to $A$ ." ], [ "Discussion and outlook", "In summary, we have shown that the CF prescription for writing ansatz wave functions for the two-component system of rotating bosons reproduces the exact spectra with very high precision even at the lowest angular momenta (i.e.", "far outside the quantum Hall regime).", "This approach also provides a convenient way of keeping track of the good quantum numbers of the system.", "For the systems studied, we have shown that the compact CF states span the whole TI-HW sector of Hilbert space for $L < M$ , and thus exactly reproduce the many-body spectra in each sector, giving a convenient basis with the desired properties.", "For larger $L$ , we find overlaps remarkably close to unity.", "Secondly, we have identified the lowest-lying CF states for $N>0$ to be composed of states with lowest $\\Lambda $ -level kinetic energy(REF ).", "The restriction to this subset of CF states offers significant computational gains through reduction of dimensionality of the trial-function space while retaining very high overlaps with the low energy part of the exact eigenstates.", "The formalism that we have identified, thus provides very good set of wave functions that can be used to study low energy properties of the system.", "A possible future application is to use these to study details of the structure and formation of vortices in the system.", "A very interesting future direction would be to go beyond the simple limit of homogeneous interactions, i.e.", "break the pseudospin symmetry that we have exploited in this paper.", "In particular, we plan to study the case where the interspecies interaction is tuned away from the intraspecies interaction (which is still assumed to be the same for both species).", "Interesting physics might be expected in this case, such as transitions from cusp states to non-cusps states on the yrast line, or other possibly experimentally verifiable phenomena.", "The particular case of $(N,M)=(1,A-1)$ is especially interesting due to presence of what appears to be a gapped mode that is captured well by the `simple' CF function.", "This case could be used to model the physics of an `impurity' boson on the rotating bose condensate.", "Finally, it is worth commenting on the fact that the number of seemingly distinct CF candidate Slater determinants is much larger than the actual number of linearly independent CF wave functions.", "In particular, in many cases, several different CF Slater determinants turn out to produce the same final polynomial.", "This implies mathematical identities which, at this point, we fail to understand in a systematic way.", "This question links closely to a recent paper[29] which discusses the apparent over-prediction of the number of CF states compared to the number of states seen in exact diagonalization, in excited bands of electronic quantum Hall states.", "Thus, it seems worthwhile to try and understand this issue in the general context of the composite fermion model." ], [ "Acknowledgement", "We would like to thank Jainendra Jain and Thomas Papenbrock for very helpful discussions.", "This work was financially supported by the Research Council of Norway and by NORDITA." ], [ "Translational invariance", "We show that an eigenstate $\\Psi $ of the operator $L_c$ (see Eq.", "REF ) with eigenvalue 0 is a translationally invariant state.", "Note that this translational invariance applies only to the polynomial part of the wavefunction and not the Gaussian part.", "Translation of a boson $z_j$ through a displacement $\\vec{c}$ is achieved by the operator $\\hat{T}_j(\\vec{c}) = e^{-i \\vec{c} \\cdot \\hat{p}_j} = 1 -i \\vec{c} \\cdot \\hat{p}_j - \\frac{1}{2}{(\\vec{c} \\cdot \\hat{p}}_j)^2 + \\cdots $ While considering its action on wavefunctions in the lowest Landau level, we can set $\\partial _{\\bar{z}_j}$ to be 0.", "Thus for any translation $c$ , action of $\\vec{c} \\cdot \\hat{p}_j$ is equivalent to $(c_x+\\imath c_y)\\partial _{z_j} \\equiv c_z\\partial _j$ in the lowest Landau level.", "Since $\\partial _i$ and $\\partial _j$ commute, the translation of all the $N+M$ particles in the system by the same vector is given by $\\begin{split}\\hat{T} & = \\prod _{j=1}^{N+M} \\hat{T}_j(\\vec{c}) \\\\& = \\exp \\left(-ic_z \\sum _{j=1}^{N+M} \\hat{\\partial }_j \\right) \\\\& = 1 - ic_z \\sum _{j=1}^{N+M} \\hat{\\partial }_j -\\frac{c_z^2}{2} \\left( \\sum _{j=1}^{N+M} \\hat{\\partial }_j \\right)^2 + \\cdots \\\\\\end{split}$ For an eigenstate of $L_c$ with eigenvalue 0, $L_c \\Psi (z,w) = R \\left( \\sum _{j=1}^{N+M} \\partial _j \\right) \\Psi (z,w) = 0$ which implies $\\left( \\sum _{j=1}^{N+M} \\partial _j \\right) \\Psi (z,w) = 0$ .", "Therefore, we conclude $\\begin{split}\\hat{T} \\Psi (z,w) & = \\Psi (z,w) -ic_z \\left( \\sum _{j=1}^{N+M} \\partial _j \\right) \\Psi (z,w) - \\frac{c_z^2}{2}\\left( \\sum _{j=1}^{N+M} \\partial _j \\right)^2 \\Psi (z,w) + \\cdots \\\\& = \\Psi (z,w)\\end{split}$ That is, $\\Psi (z,w)$ is translationally invariant." ], [ "Exact Spectrum", "The exact spectrum was obtained by diagonalizing the model Hamiltonian in the subspace of translationally invariant and highest weight pseudospin states (TI-HW states).", "The TI-HW states are obtained by finding the null space for $\\mathcal {O}=\\mathcal {S}_- \\mathcal {S}_+ + L_c$ where $\\mathcal {S}_+$ and $\\mathcal {S}_-$ are the pseudo spin raising and lowering operators respectively, and $L_c$ is the center-of-mass angular momentum operator, as before.", "Since $\\mathcal {S}_-\\mathcal {S}_+$ and $L_c$ are positive definite, $\\left\\langle L_c \\right\\rangle =0$ and $\\left\\langle \\mathcal {S}_- \\mathcal {S}_+\\right\\rangle =0$ is equivalent to $\\left\\langle \\mathcal {O} \\right\\rangle =0$ .", "Exact diagonalization was performed using Lanczos algorithm.", "In this section, we summarize the idea behind the algorithm used for projection calculation.", "The actual implementation uses representation of monomials of $z$ and $w$ in the computer as integer arrays.", "We use the symbol $\\partial ^n_i$ for $n^{\\rm th}$ derivatives with respect to $z_i$ and $D^n_i$ for derivatives with respect to $w_i$ .", "The projection operation, in the case of compact states, involves evaluation of action of complicated derivatives on the Jastrow factor, of the following form $\\Psi =\\det \\left[\\begin{array}{ccc}\\partial _{1}^{n_{1} }z_{1}^{m_{1}} & \\partial _{2}^{n_{1} }z_{2}^{m_{1}} & \\dots \\\\\\partial _{1}^{n_{2}} z_{1}^{m_{2}} & \\partial _{2}^{n_{2}} z_{2}^{m_{2}}& \\dots \\\\\\vdots & \\vdots & \\ddots \\end{array}\\right]\\times \\det \\left[\\begin{array}{ccc}D_{1}^{\\overline{n}_{1}} w_1^{\\overline{m}_{1}} & D_{2}^{\\overline{n}_{1}} w_{2}^{\\overline{m}_{1}} & \\dots \\\\D_{1}^{\\overline{n}_{2}} w_{1}^{\\overline{m}_{2}} & D_{2}^{\\overline{n}_{2}} w_{2}^{\\overline{m}_{2}} & \\dots \\\\\\vdots & \\vdots & \\ddots \\end{array}\\right]J(z,w)$ Note that the derivatives $\\partial _i$ and $D_i$ act also on the Jastrow factor.", "Since $\\Psi $ is symmetric in $\\lbrace z_1,z_2,\\dots z_{N}\\rbrace $ and in $\\lbrace w_1,\\dots w_{M}\\rbrace $ , it has an expansion $\\Psi =\\sum C_{\\lambda ,\\mu } \\mathcal {M}_{\\lambda ,\\mu }$ in the following basis functions $\\mathcal {M}_{\\lambda ,\\mu }=\\mathcal {N}_{\\lambda ,\\mu }\\times \\mathrm {sym}\\left[ z^{\\lambda _1}_1 z^{\\lambda _2}_2 \\dots z^{\\lambda _{N}}_{N}\\right]\\times \\mathrm {sym}\\left[ w^{\\mu _1}_{1} w^{\\mu _2}_{2} \\dots w^{\\mu _{M}}_{M}\\right]$ where $\\mathcal {N}_{\\lambda ,\\mu }$ is the normalization.", "Partitions $\\lambda $ and $\\mu $ of length $N$ and $M$ index the basis states.", "These functions form a poor choice of basis functions for expanding translation invariant pseudospin eigenstates.", "However they are convenient for computational manipulations as they form an orthonormal basis set and can be represented easily on a computer in the form of sorted integer arrays $\\lambda $ and $\\mu $ .", "The coefficients $C_{\\lambda ,\\mu }$ of the expansion can be obtained from computing the coefficients of expansion in a slightly simpler problem as summarized below.", "When the determinants in Eq.", "REF are expanded in permutations of the matrix elements, we get $\\Psi =\\sum _{P\\in s_{N}} \\sum _{Q\\in s_{M}} \\phi _{P,Q}$ where $s_{K}$ is the set of permutations of $\\lbrace 1,2,3..K\\rbrace $ and $\\phi _{P,Q}$ is $\\phi _{P,Q}=\\mathcal {O}_{P,Q}J(z,w)\\\\$ where $\\mathcal {O}_{P,Q}$ is given by $\\mathcal {O}_{P,Q}=\\left[(-1)^P \\prod _{i=1}^{N} \\partial _{P(i)}^{n_i} z^{m_i}_{P(i)} \\right]\\left[(-1)^Q \\prod _{j=1}^{M} \\partial _{Q(j)+N}^{\\bar{n}_j} z^{\\bar{m}_j}_{Q(j)+N}\\right],\\nonumber $ and the Jastrow factor has the expansion: $J(z,w)=\\sum _{R\\in s_{N+M}} (-1)^R \\prod _{k=1}^{N+M} z_k^{R(k)}.\\nonumber $ where $z_k=w_{k-N}$ for $k=N+1,N+2\\dots N+M$ .", "Consider the orthogonal basis functions containing monomials: $\\mathfrak {m}_\\lambda =\\prod _{k=1}^{N+M}z_k^{\\lambda _k}$ .", "Here $\\lambda $ is any non-negative integer sequence of length $N+M$ .", "The Jastrow factor can be easily expanded in this basis as $J(z,w)=\\sum _\\lambda J_\\lambda \\mathfrak {m}_\\lambda $ , where $J_\\lambda =1$ when $\\lambda $ is an even permutation of $(0,1,2,3\\dots N+M-1)$ and $-1$ for odd permutations.", "All we need to calculate is $\\phi _{P=\\mathbb {1},Q=\\mathbb {1}}$ .", "Action of $\\mathcal {O}_{\\mathbb {1},\\mathbb {1}}$ on $\\mathfrak {m}_\\lambda $ gives either 0 or $\\mathcal {O}_{\\mathbb {1},\\mathbb {1}} \\mathfrak {m}_\\lambda &=& \\mathfrak {m}_\\mu \\prod _{i=1}^{N+M}\\frac{(\\lambda _i+m_i) !", "}{(\\lambda _i+m_i-n_i)!", "}\\\\\\mu _i &=& \\lambda _i+m_i-n_i\\nonumber \\text{ where } i=1,\\dots N+M$ $\\mathcal {O}_{\\mathbb {1},\\mathbb {1}} \\mathfrak {m}_\\lambda =0$ if $\\mu _i<0$ for any $i$ .", "Since we know $J_\\lambda $ and the action of $\\mathcal {O}$ on the basis functions, we can evaluate $\\phi _{\\mathbb {1},\\mathbb {1}}$ to get the coefficients $c_\\lambda $ in the expansion of the form: $\\phi _{\\mathbb {1},\\mathbb {1}}=\\sum c_{\\lambda } \\mathfrak {m}_\\lambda $ The above coefficients $c_\\lambda ,\\mu $ are related to the coefficients $C_{\\lambda ,\\mu }$ as shown below.", "From, Eq.REF it can be seen that $\\phi _{P,Q}(z,w)=\\phi _{\\mathbb {1},\\mathbb {1}}(Pz,Qw)$ due to the antisymmetry of $J$ .", "From the expansion of $\\Psi $ in $\\phi _{P,Q}$ , we have $\\Psi =\\sum _\\lambda c_\\lambda \\sum _{P\\in s_N} \\sum _{Q\\in s_M} \\mathfrak {m}_\\lambda (Pz,Qw)$ The array of exponents $\\lambda $ can be split into exponents $\\lambda ^z\\equiv (\\lambda _1\\dots \\lambda _N)$ of $z_k$ and exponents $\\lambda ^w\\equiv (\\lambda _{N+1}\\dots \\lambda _{N+M})$ of $w_k$ ($=z_{k+N}$ ).", "Represent the sorted form of an array $\\lambda ^{z(w)}$ as $\\tilde{\\lambda }^{z(w)}$ .", "In this notation, $\\sum _{P\\in s_N} \\sum _{Q\\in s_M} \\mathfrak {m}_\\lambda (Pz,Qw) = \\frac{1}{\\mathcal {N}_{\\tilde{\\lambda }^z,\\tilde{\\lambda }^w}}\\mathcal {M}_{\\tilde{\\lambda }^z,\\tilde{\\lambda }^w}$ Comparing the expansion Eq.", "REF to the expansion of $\\Psi $ in terms of $\\mathcal {M}$ , we get the relation that can be used to calculate $C_{\\alpha ,\\beta }=\\sum _{\\lambda }^{\\prime } \\frac{c_\\lambda }{\\mathcal {N}_{\\tilde{\\lambda }^z,\\tilde{\\lambda }^w}}$ where the sum is over all $\\lambda $ , such that the parts $\\tilde{\\lambda }^z$ equals $\\alpha $ and $\\tilde{\\lambda }^w$ equals $\\beta $ ." ] ]
1403.0441
[ [ "Heat transport in RbFe2As2 single crystal: evidence for nodal\n superconducting gap" ], [ "Abstract The in-plane thermal conductivity of iron-based superconductor RbFe$_2$As$_2$ single crystal ($T_c \\approx$ 2.1 K) was measured down to 100 mK.", "In zero field, the observation of a significant residual linear term $\\kappa_0/T$ = 0.65 mW K$^{-2}$ cm$^{-1}$ provides clear evidence for nodal superdonducting gap.", "The field dependence of $\\kappa_0/T$ is similar to that of its sister compound CsFe$_2$As$_2$ with comparable residual resistivity $\\rho_0$, and lies between the dirty and clean KFe$_2$As$_2$.", "These results suggest that the (K,Rb,Cs)Fe$_2$As$_2$ serial superconductors have a common nodal gap structure." ], [ "Introduction", "The iron-based superconductors [1], [2] have attracted great attention since Hosono and co-workers reported the discovery of 26 K superconductivity in fluorine doped LaFeAsO in 2008 [1].", "Unfortunately, there is still no consensus on the superconducting mechanism in them, mainly due to their complicated electronic structures [3], [4], [5].", "There are many families of the iron-based superconductors, such as LaO$_{1-x}$ F$_{x}$ FeAs (“1111”), Ba$_{1-x}$ K$_{x}$ Fe$_{2}$ As$_{2}$ (“122”), NaFe$_{1-x}$ Co$_x$ As (“111”), and FeSe$_{x}$ Te$_{1-x}$ (“11”) [6].", "Among them, the “122” family is the most studied one due to the easy growth of sizable high-quality single crystals [7].", "Intriguingly, the members of this family do not share a universal superconducting gap structure.", "While the optimally doped Ba$_{0.6}$ K$_{0.4}$ Fe$_2$ As$_2$ and BaFe$_{1.85}$ Co$_{0.15}$ As$_2$ have nodeless superconducting gaps [4], [8], [9], [10], the extremely hole-doped KFe$_2$ As$_2$ was reported to be a nodal superconductor [11], [12].", "Furthermore, the isovalently doped BaFe$_2$ (As$_{1-x}$ P$_x$ )$_2$ [13], [14], [15] and Ba(Fe$_{1-x}$ Ru$_x$ )$_2$ As$_2$ [16] also manifest nodal superconductivity.", "So far, the origin of these nodal superconducting gaps is still under debate, particularly in KFe$_2$ As$_2$ [11], [12], [17], [18], [19].", "The detailed thermal conductivity study provided compelling evidences for a $d$ -wave gap in KFe$_2$ As$_2$ [17], but the low-temperature angle-resolved photoemission spectroscopy (ARPES) measurements clearly showed nodal $s$ -wave gap [19].", "Recent ARPES and thermal conductivity experiments on highly hole-doped Ba$_{1-x}$ K$_{x}$ Fe$_{2}$ As$_{2}$ also support nodal $s$ -wave gap [20], [21].", "KFe$_2$ As$_2$ has two sister compounds, CsFe$_2$ As$_2$ and RbFe$_2$ As$_2$ , and both of them are superconducting [22], [23].", "While muon-spin spectroscopy measurements on RbFe$_2$ As$_2$ polycrystals suggested that RbFe$_2$ As$_2$ is best described by a two-gap $s$ -wave model [24], [25], recent specific heat and thermal conductivity measurements on CsFe$_2$ As$_2$ single crystals provided clear evidences for nodal superconducting gap in CsFe$_2$ As$_2$ [26], [27].", "To clarify whether the superconducting gap structure of RbFe$_2$ As$_2$ is indeed different from those of KFe$_2$ As$_2$ and CsFe$_2$ As$_2$ , more experiments on RbFe$_2$ As$_2$ single crystals are highly desired.", "In this paper, we present the low-temperature thermal conductivity of RbFe$_2$ As$_2$ single crystal down to 100 mK.", "A significant residual linear term $\\kappa _0/T$ = 0.65 $\\pm $ 0.03 mW K$^{-2}$ cm$^{-1}$ is obtained in zero magnetic field, and the field dependence of $\\kappa _0/T$ mimics that of CsFe$_2$ As$_2$ .", "These results clarify that RbFe$_2$ As$_2$ is also a nodal superconductor.", "The three compounds KFe$_2$ As$_2$ , RbFe$_2$ As$_2$ , and CsFe$_2$ As$_2$ may have a common superconducting gap structure." ], [ "Experiment", "The RbFe$_2$ As$_2$ single crystals were grown by self-flux method for the first time, and the process is the same as the growth of CsFe$_2$ As$_2$ single crystals [26].", "The dc magnetization was measured using a superconducting quantum interference device (MPMS, Quantum design).", "The specific heat measurement above 1.9 K was performed in a physical property measurement system (PPMS, Quantum design) via the relaxation method, and below 1.9 K it was measured in a small dilution refrigerator integrated into the PPMS.", "For transport measurements, the RbFe$_2$ As$_2$ single crystal was cleaved to a rectangular shape of dimensions 2.2 $\\times $ 1.0 mm$^2$ in the $ab$ plane, with 40 $\\mu $ m thickness along the $c$ axis.", "Contacts were made directly on the sample surfaces with silver paint (Dupont 4929N), which were used for both resistivity and thermal conductivity measurements.", "To avoid degradation, the sample was exposed in air for less than 2 hours.", "The contacts are metallic with a typical resistance 100 m$\\Omega $ at 2 K. In-plane thermal conductivity was measured in a dilution refrigerator, using a standard four-wire steady-state method with two RuO$_2$ chip thermometers, calibrated in situ against a reference RuO$_2$ thermometer.", "Magnetic fields were applied along the $c$ axis and perpendicular to the heat current.", "To ensure a homogeneous field distribution in the sample, all fields were applied at a temperature above $T_c$ for transport measurements." ], [ "Results and Discussion", "Figure 1(a) shows the low-temperature dc magnetization of RbFe$_2$ As$_2$ single crystal, measured in $H$ = 20 Oe along $c$ axis, with zero-field cooling process.", "The $T_c \\approx $ 2.10 K is defined at the onset of the diamagnetic transition.", "The magnetization does not saturate down to 1.8 K, where the superconducting volume fraction is already as large as 40%.", "With decreasing temperature, the superconducting volume fraction should further increase to reach the fully shielded state.", "In Fig.", "1(b), we present the low-temperature specific heat of RbFe$_2$ As$_2$ single crystal down to 100 mK in zero field, plotted as $C/T$ vs $T$ .", "A significant jump due to the superconducting transition is observed at $T_c \\approx $ 2.1 K, which indicates the high quality of our sample.", "In order to determine the zero-field normal-state Sommerfeld coefficient $\\gamma _N$ , the specific heat above $T_c$ is fitted to $C_{normal}$ = $\\gamma _NT$ + $\\beta $$T^{3}$ + $\\eta $$T^{5}$ , with $\\beta $ and $\\eta $ as the lattice coefficients.", "The solid line in Fig.", "1(b) is the best fit of $C/T$ from 2.4 to 10 K, which gives $\\gamma _N$ = 127.3 $\\pm $ 0.9 mJ mol$^{-1}$ K$^{-2}$ , $\\beta $ = 0.66 $\\pm $ 0.04 mJ mol$^{-1}$ K$^{-4}$ , and $\\eta $ = 0.0029 $\\pm $ 0.0005 mJ mol$^{-1}$ K$^{-6}$ .", "From the relation $\\theta _D$ = (12$\\pi ^{4}RZ$ / 5$\\beta $ )$^{1/3}$ , where $R$ is the molar gas constant and $Z$ = 5 is the total number of atoms in one unit cell, the Debye temperature $\\theta _D$ = 245 K is estimated.", "This value is comparable to those of KFe$_2$ As$_2$ and CsFe$_2$ As$_2$ [28], [26].", "The in-plane resistivity of RbFe$_2$ A$_2$ single crystal in zero filed is plotted in Fig.", "1(c).", "The $T_c \\approx $ 2.13 K, defined by $\\rho $ = 0, agrees well with the magnetization and specific heat measurements.", "For the polycrystalline sample of RbFe$_2$ A$_2$ , $T_c =$ 2.6 K was defined at the onset of the diamagnetic transition [23], which is 0.5 K higher than our single crystal.", "Similarly, $T_c =$ 2.2 K was defined at the onset of the diamagnetic transition for the CsFe$_2$ As$_2$ polycrystal [22], but $T_c =$ 1.8 K was found in the CsFe$_2$ As$_2$ single crystal [26].", "It is unclear why the $T_c$ shows difference between polycrystalline sample and single crystal for RbFe$_2$ A$_2$ and CsFe$_2$ A$_2$ .", "In case that the single crystals have intrinsic $T_c$ , the $T_c$ of (K, Rb, Cs)Fe$_2$ A$_2$ series (3.8, 2.1, and 1.8 K, respectively) decreases with the increase of the ionic radius of alkali metal.", "In the inset of Fig.", "1(c), the normal-state $\\rho (T)$ below 9 K can be well fitted by $\\rho $ = $\\rho _0$ + $AT^{1.5}$ , with $\\rho _0$ = 1.84 $\\pm $ 0.01 $\\mu $$\\Omega $ cm and $A$ = 0.16 $\\mu $$\\Omega $ cm K$^{-1.5}$ .", "Similar non-Fermi-liquid behavior of $\\rho (T)$ was also observed in KFe$_2$ As$_2$ and CsFe$_2$ As$_2$ [11], [17], [27], which may result from antiferromagnetic spin fluctuations [29].", "The residual resistivity ratio RRR = $\\rho $ (292K)/$\\rho _0 \\approx $ 310 again reflects the high quality of our RbFe$_2$ As$_2$ single crystal.", "Figure 2(a) shows the low-temperature resistivity of RbFe$_2$ As$_2$ single crystal in magnetic fields up to 1.1 T. In order to estimate the zero-temperature upper critical field $H_{c2}(0)$ , the temperature dependence of $H_{c2}(T)$ is plotted in Fig.", "2(b), defined by $\\rho $ = 0 in Fig.", "2(a).", "$H_{c2}(0) \\approx $ 0.97 T is obtained by fitting the data with the Ginzburg-Landau equation $H_{c2}(T) = H_{c2}(0)[1-(T/T_c)^{2}]/[1+(T/T_c)^{2}]$ [30], [31].", "Figure: (Color online).", "Low-temperature in-plane thermalconductivity of RbFe 2 _2As 2 _2 single crystal in zero and magneticfields applied along the cc axis.", "The solid line is a fit of thezero-field data between 0.1 and 0.3 K to κ/T=a+bT\\kappa /T = a + bT, giving a residuallinear term κ 0 /T\\kappa _0/T = 0.65 ±\\pm 0.03 mW K -2 ^{-2} cm -1 ^{-1}.", "The dashedline is the normal-state Wiedemann-Franz law expectationL 0 L_0/ρ 0 \\rho _0, with L 0 L_0 the Lorenz number 2.45 ×\\times 10 -8 ^{-8}W Ω\\Omega K -2 ^{-2} and ρ 0 \\rho _0 = 1.84 μ\\mu Ω\\Omega cm.The low-temperature heat transport measurement is a bulk technique to probe the gap structure of superconductors [32].", "In Fig.", "3, the in-plane thermal conductivity of RbFe$_2$ As$_2$ single crystal in zero and applied field is plotted as $\\kappa /T$ vs $T$ [33], [34].", "The thermal conductivity at very low temperature can be usually fitted to $\\kappa /T$ = $a + bT^{\\alpha -1}$ , in which the two terms $aT$ and $bT^{\\alpha }$ represent contributions from electrons and phonons, respectively.", "The power $\\alpha $ is typically between 2 and 3, due to specular reflections of phonons at the boundary [33], [34].", "Since all the curves in Fig.", "3 are roughly linear, we fix $\\alpha $ to 2.", "The value $\\alpha \\approx $ 2 has been previously observed in dirty KFe$_2$ As$_2$ [11], Ba(Fe$_{1-x}$ Ru$_x$ )$_2$ As$_2$ [16], and CsFe$_2$ As$_2$ single crystals [27].", "Here, we only focus on the electronic term.", "In zero field, the fitting of the data between 0.1 to 0.3 K gives $\\kappa _0/T$ = 0.65 $\\pm $ 0.03 mW K$^{-2}$ cm$^{-1}$ .", "If we slightly change the fitting range, we obtain $\\kappa _0/T$ = 0.62 $\\pm $ 0.03 mW K$^{-2}$ cm$^{-1}$ for the range below 0.27 K and $\\kappa _0/T$ = 0.63 $\\pm $ 0.04 mW K$^{-2}$ cm$^{-1}$ for the range below 0.24 K. Therefore, the value of $\\kappa _0/T$ basically does not depend on the temperature range chosen for the fit.", "Such a significant $\\kappa _0/T$ is usually contributed by nodal quasiparticles, thus it is a strong evidence for nodal superconducting gap [32].", "Previously, $\\kappa _0/T$ = 2.27 $\\pm $ 0.02 and 3.6 $\\pm $ 0.5 mW K$^{-2}$ cm$^{-1}$ were observed for dirty and clean KFe$_2$ As$_2$ single crystals, respectively [17], [11].", "For CsFe$_2$ As$_2$ single crystal with $\\rho _0$ = 1.80 $\\mu $$\\Omega $ cm, $\\kappa _0/T$ = 1.27 $\\pm $ 0.04 mW K$^{-2}$ cm$^{-1}$ was found [27].", "The zero-field value of $\\kappa _0/T$ for RbFe$_2$ As$_2$ is about 5$\\%$ of the normal-state Widemann-Franz law expectation $\\kappa _{N0}/T$ = $L_0/\\rho _0$ = 13.5 mW K$^{-2}$ cm$^{-1}$ , with $L_0$ the Lorenz number 2.45 $\\times $ 10$^{-8}$ W $\\Omega $ K$^{-2}$ and $\\rho _0$ = 1.84 $\\mu $$\\Omega $ cm.", "In $H$ = 0.9 T, the experimental data roughly satisfy the Widemann-Franz law, so we take 0.9 T as the bulk $H_{c2}(0)$ .", "This value is slightly lower than that obtained from resistivity measurements, but it does not affect our discussion of the filed dependence of $\\kappa _0/T$ below.", "Figure: (Color online).", "Normalized residual linear termκ 0 /T\\kappa _0/T of RbFe 2 _2As 2 _2 as a function of H/H c2 H/H_{c2}.", "Forcomparison, similar data are shown for the clean ss-wavesuperconductor Nb , the dd-wave cupratesuperconductor Tl-2201 , the dirty and clean KFe 2 _2As 2 _2 , , and CsFe 2 _2As 2 _2 .The field dependence of $\\kappa _0/T$ may provide more information on the superconducting gap structure [32].", "In Fig.", "4, we plot the normalized $\\kappa _0(H)/T$ of RbFe$_2$ As$_2$ together with the typical $s$ -wave superconductor Nb [35], the $d$ -wave cuprate superconductor Tl$_2$ Ba$_2$ CuO$_{6+\\delta }$ (Tl-2201) [36], the dirty and clean KFe$_2$ As$_2$ [11], [17], and CsFe$_{2}$ As$_{2}$ [27].", "For an $s$ -wave superconductor with isotropic gap, such as Nb, $\\kappa _{0}/T$ grows exponentially with the field [35].", "For the $d$ -wave superconductor Tl-2201, $\\kappa _0/T$ increases roughly proportional to $H^{1/2}$ at low field [36], due to the Volovik effect [37].", "From Fig.", "4, the normalized $\\kappa _0(H)/T$ curve of RbFe$_2$ As$_2$ is very close to that of CsFe$_2$ As$_2$ and lies between the dirty and clean KFe$_2$ As$_2$ .", "Table: The superconducting transiton temperature T c T_c, residual resistivity ρ 0 \\rho _0, zero-field normal-state Sommerfeld coefficient γ N \\gamma _N, upper critical field H c2 (0)H_{c2}(0), and residual linear term κ 0 /T\\kappa _0/T of the clean and dirty KFe 2 _2As 2 _2, CsFe 2 _2As 2 _2, and RbFe 2 _2As 2 _2.", "These values are taken from Refs.", ", , , , , , and this work.As listed in Table I, the $\\rho _0$ of dirty and clean KFe$_2$ As$_2$ differ by 15 times [11], [17], while RbFe$_2$ As$_2$ and CsFe$_2$ As$_2$ have comparable $\\rho _0$ , with values lying between that of dirty and clean KFe$_2$ As$_2$ .", "Therefore, in (K,Rb,Cs)Fe$_2$ As$_2$ serial superconductors, the field dependence of $\\kappa _0/T$ seems to correlate with the impurity level.", "Although Reid $et$ $al.$ argued that the $\\kappa _0(H)/T$ of clean KFe$_2$ As$_2$ is a compelling evidence for $d$ -wave gap [17], recent thermal conductivity measurements on highly hole-doped Ba$_{1-x}$ K$_x$ Fe$_2$ As$_2$ single crystals support nodal $s$ -wave gap [21].", "For such a complex nodal $s$ -wave gap structure, likely with both nodal and nodeless gaps of different magnitudes, it is hard to get a theoretical curve of $\\kappa _0(H)/T$ .", "One needs to carefully consider the effect of impurity on the behavior of $\\kappa _0(H)/T$ .", "Nevertheless, the evolution of the normalized $\\kappa _0(H)/T$ suggests a common nodal gap structure in (K,Rb,Cs)Fe$_2$ As$_2$ serial superconductors.", "In Table I, we also list the $T_c$ , $\\gamma _N$ , $H_{c2}(0)$ , and $\\kappa _0/T$ of the (K,Rb,Cs)Fe$_2$ As$_2$ serial superconductors [11], [17], [27], [26], [38], [39], [40].", "Both $T_c$ and $\\gamma _N$ show a systematic change with increasing the ionic radii of alkali metal.", "The $\\gamma _N$ values of RbFe$_2$ As$_2$ and CsFe$_2$ As$_2$ are very large among all iron-based superconductors, which reflects their abnormally large density of states or effective mass of electrons.", "This may be explained by recent ARPES measurement on CsFe$_2$ As$_2$ single crystals, which suggests that the large separation of FeAs layers along $c$ axis makes the system more two-dimensional and enhances the electronic correlations [41].", "Neither the $H_{c2}(0)$ nor $\\kappa _0/T$ shows a systematic change.", "The $H_{c2}(0)$ of CsFe$_2$ As$_2$ is abnormally high, which may also relate to its much enhanced two dimensionality and electronic correlations.", "As for the $\\kappa _0/T$ , it depends on the very details of the nodal gap, such as the slope of the gap at the node.", "For the accidental nodes appearing in the complex Fermi surfaces of (K,Rb,Cs)Fe$_2$ As$_2$ , the $\\kappa _0/T$ may not necessarily manifest systematic change with the increase of the ionic radii of alkali metal." ], [ "Summary", "In summary, we have measured the magnetization, resistivity, low-temperature specific heat and thermal conductivity of RbFe$_2$ As$_2$ single crystals.", "A nodal superconducting gap in RbFe$_2$ As$_2$ is strongly suggested by the observation of a significant residual linear term $\\kappa _0/T$ = 0.65 mW K$^{-2}$ cm$^{-1}$ in zero magnetic field.", "It is concluded that (K,Rb,Cs)Fe$_2$ As$_2$ serial superconductors may have a common nodal gap structure, and the field dependence of $\\kappa _0/T$ seems to evolve with the impurity level.", "ACKNOWLEDGEMENTS This work is supported by the Natural Science Foundation of China, the Ministry of Science and Technology of China (National Basic Research Program No: 2012CB821402 and 2015CB921401), Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning.", "$^*$ E-mail: shiyan$\\_$ [email protected]" ] ]
1403.0191
[ [ "Diversified homotopic behavior of closed orbits of some R-covered Anosov\n flows" ], [ "Abstract We produce infinitely many examples of Anosov flows in closed 3-manifolds where the set of periodic orbits is partitioned into two infinite subsets.", "In one subset every closed orbit is freely homotopic to infinitely other closed orbits of the flow.", "In the other subset every closed orbit is freely homotopic to only one other closed orbit.", "The examples are obtained by Dehn surgery on geodesic flows.", "The manifolds are toroidal and have Seifert fibered pieces and atoroidal pieces in their torus decompositions." ], [ " Abstract $-$ We produce infinitely many examples of Anosov flows in closed 3-manifolds where the set of periodic orbits is partitioned into two infinite subsets.", "In one subset every closed orbit is freely homotopic to infinitely other closed orbits of the flow.", "In the other subset every closed orbit is freely homotopic to only one other closed orbit.", "The examples are obtained by Dehn surgery on geodesic flows.", "The manifolds are toroidal and have Seifert pieces and atoroidal pieces in their torus decompositions.", "section1-3.5ex plus -1ex minus -.2ex2.3ex plus .2exIntroduction This article deals with the question of free homotopies of closed orbits of Anosov flows [1] in 3-manifolds.", "In particular we deal with the following question: how many closed orbits are freely homotopic to a given closed orbit of the flow?", "Suspension Anosov flows have the property that an arbitrary closed orbit is not freely homotopic to any other closed orbit.", "Geodesic flows have the property that every closed orbit (which corresponds to a geodesic in the surface) is only freely homotopic to one other closed orbit.", "The other orbit corresponds to the same geodesic in the surface, but being traversed in the opposite direction.", "About twenty years ago the author proved that there is an infinite class of Anosov flows in closed hyperbolic 3-manifolds satisfying the property that every closed orbit is freely homotopic to infinitely many other closed orbits [10].", "Obviously this was diametrically opposite to the behavior of the previous two examples and it was also quite unexpected.", "This property in these examples is strongly connected with the large scale properties of Anosov flows when lifted to the universal cover.", "In particular in these examples the property of infinitely orbits which are freely homotopic to each other implies that the flows are not quasigeodesic, that is, orbits are not uniformly efficient in measuring length in the universal cover [10].", "This provided the first examples of Anosov flows in hyperbolic 3-manifolds which are not quasigeodesic.", "The analysis of freely homotopic closed orbits of Anosov flows is also extremely important in other situations, for example: 1) Analysing the interaction between incompressible tori in the manifold and the Anosov flow [4], 2) Studying the structure of the Anosov flow when “restricted\" to a Seifert piece of the torus decomposition of the manifold [4].", "We define the free homotopy class of a closed orbit of a flow to be the collection of closed orbits which are freely homotopic to the original closed orbit.", "For an Anosov flow each free homotopy class is at most infinite countable as there are only countably many closed orbits of the flow [1].", "In this article we are concerned with the cardinality of free homotopy classes.", "Suspensions have all free homotopy classes with cardinality one and geodesic flows have all free homotopy classes with cardinality two.", "In the hyperbolic examples mentioned above every free homotopy class has infinite cardinality.", "In addition if we do finite covers of geodesic flows, where we “unroll the fiber direction\", then we can get examples satisfying the property that every free homotopy class has cardinality $2n$ where $n$ is a positive integer.", "The question we ask is whether we can have mixed behavior for an Anosov flow.", "In other words, can some free homotopy classes be infinite while others have finite cardinality?", "In this article we produce infinitely many examples where this indeed occurs.", "Main theorem $-$ There are infinitely many examples of Anosov flows $\\Phi $ in closed 3-manifolds so that the set of closed orbits is partitioned in two infinite subsets $A$ and $B$ so that the following happens.", "Every closed orbit in $A$ has infinite free homotopy class.", "Every closed orbit in $B$ has free homotopy class of cardinality two.", "The examples are obtained by Dehn surgery on closed orbit of geodesic flows.", "We thank the reviewer, whose suggestions greatly improved the presentation of this article.", "section1-3.5ex plus -1ex minus -.2ex2.3ex plus .2exPrevious results and definitions A three manifold $M$ is irreducible if every sphere bounds a ball [19].", "An incompressible torus is the image of an embedding $f: T^2 \\rightarrow M$ which does not have compressing disks [19].", "A 3-manifold is homotopically atoroidal if every $\\pi _1$ -injective map $f: T^2 \\rightarrow M$ is homotopic into the boundary.", "The manifold is geometrically atoroidal if every incompressible torus is homotopic to the boundary.", "A 3-manifold $M$ is Seifert fibered if it has a 1-dimensional foliation by circles [19], [9].", "The torus decomposition states that every compact, irreducible 3-manifold $M$ can be decomposed by finitely incompressible tori $T_1, ..., T_k$ so that the closure of every component of $(M - \\cup T_k)$ is either Seifert fibered or atoroidal [20], [21].", "A graph manifold is an irreducible 3-manifold whose pieces of the torus decomposition are all Seifert fibered.", "The base space of a Seifert fibered space is the quotient of $M$ by the Seifert fibration.", "It is a 2-dimensional orbifold with finitely many cone points.", "The Seifert space is called small if the base is either the disk with less than 3 cone points or the sphere with less than 4 cone points.", "If $M$ closed admits an Anosov flow then $M$ is irreducible [23].", "But $M$ may be toroidal, for example this happens in the case of geodesic flows or suspensions.", "In the case of geodesic flows the whole manifold is Seifert fibered.", "Let $\\Phi $ be an Anosov flow in $M^3$ and let $\\mbox{$\\Lambda ^s$}, \\mbox{$\\Lambda ^u$}$ be its stable and unstable foliations respectively.", "The leaves of $\\mbox{$\\Lambda ^s$}, \\mbox{$\\Lambda ^u$}$ can only be planes, annuli and Möbius bands [1].", "A leaf $L$ of $\\mbox{$\\Lambda ^s$}$ or $\\mbox{$\\Lambda ^u$}$ is an annulus or Möbius band if and only $L$ contains a closed orbit of $\\Phi $ [1].", "Let $\\mbox{$\\widetilde{\\Lambda }^s$}, \\mbox{$\\widetilde{\\Lambda }^u$}$ be the lifted foliations to the universal cover $\\mbox{$\\widetilde{M}$}$ .", "Let $F$ be a leaf of $\\mbox{$\\widetilde{\\Lambda }^s$}$ or $\\mbox{$\\widetilde{\\Lambda }^u$}$ .", "By the above, the leaf $F$ has non trivial stabilizer if and only if $\\pi (F)$ has a closed orbit of $\\Phi $ .", "Here the map $\\pi : \\mbox{$\\widetilde{M}$}\\rightarrow M$ is the universal covering map.", "The stabilizer of $F$ is $\\lbrace g \\in \\pi _1(M) \\ | \\ g(F) = F \\rbrace $ .", "Theorem 0.1 ([11], [12]) Suppose that $\\Phi $ is an Anosov flow in $M^3$ and suppose that $\\alpha $ and $\\beta $ are closed orbits of $\\Phi $ which are freely homotopic to each other, as oriented curves.", "Then there is $\\gamma $ periodic orbit of $\\Phi $ so that $\\alpha $ is freely homotopic to $\\gamma ^{-1}$ as oriented curves.", "For an arbitrary Anosov flow $\\Phi $ the orbit space $\\mbox{$\\cal O$}$ of the lifted flow $\\mbox{$\\widetilde{\\Phi }$}$ to the universal cover is homeomorphic to the plane $\\mbox{${\\bf R}$}^2$ [10].", "The lifted foliations $\\mbox{$\\widetilde{\\Lambda }^s$}, \\mbox{$\\widetilde{\\Lambda }^u$}$ are invariant by the flow $\\mbox{$\\widetilde{\\Phi }$}$ so they induce one dimensional foliations $\\mbox{${\\cal O}^s$}, \\mbox{${\\cal O}^u$}$ in $\\mbox{$\\cal O$}$ .", "Definition 0.2 A foliation $\\mbox{$\\cal F$}$ in $M$ is $\\mbox{${\\bf R}$}$ -covered if the leaf space of the lifted foliation $\\mbox{$\\widetilde{\\cal F}$}$ to the universal cover $\\mbox{$\\widetilde{M}$}$ is homeomorphic to the real numbers $\\mbox{${\\bf R}$}$ [10].", "Definition 0.3 An Anosov flow is $\\mbox{${\\bf R}$}$ -covered if its stable foliation (or equivalently its unstable foliation [2], [10]) is $\\mbox{${\\bf R}$}$ -covered.", "Examples of $\\mbox{${\\bf R}$}$ -covered Anosov flows are suspensions and geodesic flows [10].", "In [10] the author showed that there is an infinite class of $\\mbox{${\\bf R}$}$ -covered Anosov flows on hyperbolic 3-manifolds.", "Barbot [3] proved that the Handel-Thurston Anosov flows [18] are $\\mbox{${\\bf R}$}$ -covered, as well as an infinite class of Anosov flows in graph manifolds.", "The Anosov flows in the Main theorem are $\\mbox{${\\bf R}$}$ -covered.", "On the other hand the class of non $\\mbox{${\\bf R}$}$ -covered Anosov flows is extremely large.", "For example Barbot [2] proved that $\\mbox{${\\bf R}$}$ -covered Anosov flows are transitive.", "Hence the intransitive Anosov flows constructed by Franks and Williams [14] are not $\\mbox{${\\bf R}$}$ -covered.", "In addition the Bonatti and Langevin examples [7] are transitive Anosov flows which are not covered.", "Barbot [3] constructed many other examples of transitive Anosov flows in graph manifolds and these examples were greatly generalized in [4].", "In all of these examples the underlying manifold is toroidal and consequently not hyperbolic.", "Suppose that $\\Phi $ is an $\\mbox{${\\bf R}$}$ -covered Anosov flow.", "Then there are two possibilities for the topological structure of the lifted stable and unstable foliations $\\mbox{$\\widetilde{\\Lambda }^s$}, \\mbox{$\\widetilde{\\Lambda }^u$}$ to $\\mbox{$\\widetilde{M}$}$ [2], [10] or equivalently for the topological structure of the foliations $\\mbox{${\\cal O}^s$}, \\mbox{${\\cal O}^u$}$ in $\\mbox{$\\cal O$}\\cong \\mbox{${\\bf R}$}^2$ .", "Suppose that every leaf of $\\mbox{$\\widetilde{\\Lambda }^s$}$ intersects every leaf of $\\mbox{$\\widetilde{\\Lambda }^u$}$ .", "In this case $\\Phi $ is said to have the product type.", "Then Barbot [2] showed that $\\Phi $ is topologically equivalent to a suspension Anosov flow.", "This implies that $M$ fibers over the circle with fiber a torus.", "The other possibility is that $\\Phi $ is a skewed $\\mbox{${\\bf R}$}$ -covered Anosov flow.", "This means that $\\mbox{$\\cal O$}$ has a model homeomorphic to an infinite strip $(0,1) \\times \\mbox{${\\bf R}$}$ .", "In addition the model satisfies the following properties.", "The stable foliation $\\mbox{${\\cal O}^s$}$ in $\\mbox{$\\cal O$}$ is a foliation by horizontal segments in $(0,1) \\times \\mbox{${\\bf R}$}$ .", "The unstable foliation is a foliation by parallel segments in $(0,1) \\times \\mbox{${\\bf R}$}$ which make and angle $\\theta $ which is not $\\pi /2$ with the horizontal.", "That is, they are not vertical and hence an unstable leaf does not intersect every stable leaf and vice versa.", "We refer to figure REF .", "Notice that every unstable leaf $u$ of $\\mbox{${\\cal O}^u$}$ intersects an interval $J_u$ of stable leaves of $\\mbox{${\\cal O}^s$}$ .", "This is a strict subset of the leaf space of $\\mbox{${\\cal O}^s$}$ .", "The model implies that different leaves $u$ of $\\mbox{${\\cal O}^u$}$ generate different intervals $J_u$ .", "Suppose that every leaf of $\\mbox{$\\widetilde{\\Lambda }^s$}$ intersects every leaf of $\\mbox{$\\widetilde{\\Lambda }^u$}$ .", "In this case $\\Phi $ is said to have the product type.", "Then Barbot [2] showed that $\\Phi $ is topologically equivalent to a suspension Anosov flow.", "This implies that $M$ fibers over the circle with fiber a torus.", "The other possibility is that $\\Phi $ is a skewed $\\mbox{${\\bf R}$}$ -covered Anosov flow.", "This means that $\\mbox{$\\cal O$}$ has a model homeomorphic to an infinite strip $(0,1) \\times \\mbox{${\\bf R}$}$ .", "In addition the model satisfies the following properties.", "The stable foliation $\\mbox{${\\cal O}^s$}$ in $\\mbox{$\\cal O$}$ is a foliation by horizontal segments in $(0,1) \\times \\mbox{${\\bf R}$}$ .", "The unstable foliation is a foliation by parallel segments in $(0,1) \\times \\mbox{${\\bf R}$}$ which make and angle $\\theta $ which is not $\\pi /2$ with the horizontal.", "That is, they are not vertical and hence an unstable leaf does not intersect every stable leaf and vice versa.", "We refer to figure REF .", "Notice that every unstable leaf $u$ of $\\mbox{${\\cal O}^u$}$ intersects an interval $J_u$ of stable leaves of $\\mbox{${\\cal O}^s$}$ .", "This is a strict subset of the leaf space of $\\mbox{${\\cal O}^s$}$ .", "The model implies that different leaves $u$ of $\\mbox{${\\cal O}^u$}$ generate different intervals $J_u$ .", "In [12] the author proved that if $\\Phi $ is a skewed $\\mbox{${\\bf R}$}$ -covered Anosov flow then the underlying manifold is orientable.", "With this one can also produce infinitely many examples of transitive non $\\mbox{${\\bf R}$}$ -covered Anosov flows where the underlying manifold is hyperbolic.", "Proposition 0.4 Proposition 3.1 of [5] Let $\\gamma $ be a closed orbit of an Anosov flow $\\Phi $ .", "Then any lift $\\widetilde{\\gamma }$ of $\\gamma $ to $\\mbox{$\\widetilde{M}$}$ is unknotted.", "In particular $\\pi _1(\\widetilde{M} - \\widetilde{\\gamma })$ is ${\\bf Z}$ and $\\widetilde{M} - \\widetilde{\\gamma }$ is an open solid torus.", "Dehn filling - Let $N$ be a 3-manifold which is the interior of a compact 3-manifold $\\hat{N}$ so that the boundary of $\\hat{N}$ is a union of tori $P_1, ..., P_k$ .", "Choose simple closed curve generators $a_i, b_i$ for each $\\pi _1(P_i)$ .", "Let $N_{(x_1,y_1),...,(x_k,y_k)}$ be the manifold obtained by Dehn filling $N$ (or more specifically $\\hat{N}$ ) with $k$ solid tori $V_i$ as follows: for each $i$ glue the boundary of the solid torus $V_i$ to $P_i$ by a homeomorphism so that the meridian in $V_i$ is glued to the curve $x_i a_i + y_i b_i$ in $P_i$ .", "We require that the pair of integers $x_i, y_i$ are relatively prime to ensure that $x_i a_i + y_i b_i$ is a simple closed curve in $P_i$ .", "The topological type of the Dehn filled manifold is completely determined by the collection of pairs of integers $(x_i,y_i)$ .", "They are called the Dehn surgery coefficients.", "Dehn surgery - Let $M$ be a closed 3-manifold and $\\gamma $ an orientation preserving simple closed curve in $M$ .", "Let $N(\\gamma )$ be a solid torus neighborhood of $\\gamma $ and $M^{\\prime } = M - \\mathring{N}(\\gamma )$ .", "Choose a pair of generators for $\\partial N(\\gamma ) \\subset \\partial M^{\\prime }$ .", "Then Dehn surgery on $\\gamma $ with coefficients $x,y$ is the Dehn filled manifold $M^{\\prime }_{(x,y)}$ .", "Remarks 1) More generally one can do Dehn surgery on $\\gamma $ if $M$ has boundary, or is not compact.", "2) Dehn surgery is very general: any closed orientable 3-manifold can be obtained from the 3-sphere by iterated Dehn surgery [22].", "Hyperbolic Dehn surgery [24], [25], [6] - Let $M$ be a complete hyperbolic manifold with finite volume and not compact.", "Then $M$ is homeomorphic to the interior of a compact manifold $\\hat{M}$ with boundary components $P_1, ..., P_k$ which we assume are tori.", "Each such torus corresponds to a cusp in $M$ .", "As above one can do Dehn filling to obtain the closed manifold $M_{(x_1,y_1),..., (x_k,y_k)}$ .", "Thurston proved except for finitely many choices of the integers $x_i, y_i$ the manifold $M_{(x_1,y_1),..., (x_k,y_k)}$ admits a hyperbolic structure.", "If one fills only one of the cusps, or a subset of the cusps, then for big enough surgery coefficients, the resulting manifold is hyperbolic but not closed, it still has some cusps.", "The reference [24] has a very extensive and detailed proof of this in the case of the figure eight knot complement in the sphere ${\\bf S}^3$ .", "The book [6] has a proof in the general case.", "Fried's flow Dehn surgery [15] - Suppose that $\\Phi $ is an Anosov flow and that $\\gamma $ is a closed orbit which is orientation preserving in $M$ , and so that the stable leaf $\\mbox{$\\Lambda ^s$}(\\gamma )$ of $\\gamma $ is an annulus (as opposed to a Möbius band).", "Let $N(\\gamma )$ be a solid torus neighborhood of $\\gamma $ .", "Choose generators for $\\pi _1(P)$ where $P = \\partial (N(\\gamma ))$ as follows.", "Let $a$ be a meridian in $N(\\gamma )$ $-$ unique up to inverse in $\\pi _1(P)$ .", "Let $b$ be the intersection of the local stable leaf of $\\gamma $ with $\\partial N(\\gamma )$ .", "Choose the orientation in $b$ to be the one induced by the positive flow direction in $\\gamma $ .", "This is the “longitude\" in $\\partial N(\\gamma )$ in this case.", "Do $(1,n)$ Dehn surgery on $\\gamma $ .", "Fried [15] showed how to do this along the flow: blow up the orbit $\\gamma $ (producing a boundary torus).", "Then blow down this boundary torus to a closed curve according to the surgery coefficients $(1,n)$ .", "The resulting flow is still an Anosov flow and is denoted by $\\Phi _{(1,n)}$ .", "The orbit $\\gamma $ blows up to a torus and then blows down to an orbit of $\\Phi _{(1,n)}$ .", "With this construction notice that there is a bijection between the orbits of $\\Phi $ and the orbits of the surgered flow $\\Phi _{(1,n)}$ .", "Theorem 0.5 (Fe1) Let $\\Phi $ be an $\\mbox{${\\bf R}$}$ -covered Anosov flow in $M^3$ of skewed type.", "Let $\\gamma $ be a closed orbit so that $\\mbox{$\\Lambda ^s$}(\\gamma )$ is an annulus.", "Then under a positivity condition, every Anosov flow $\\Phi _{(1,n)}$ is $\\mbox{${\\bf R}$}$ -covered and of skewed type.", "Under appropriate choices of the basis of $\\pi _1(N(\\gamma ))$ the positivity condition is satisfied for all positive $n$ .", "In particular this is true if $\\Phi $ is the geodesic flow in $T_1 S$ where $S$ is a closed hyperbolic surface.", "In general the positivity condition is satisfied either for all positive $n$ or for all negative $n$ [10].", "The issue is that the meridian is well undefined up to inverse.", "Hence there are two possibilities for the basis, and one of them satisfies the positivity condition for every positive $n$ .", "As it is not needed in this article we do not specify exactly when the positivity condition holds.", "Theorem 0.6 ([10]) Let $M = T_1 S$ where $S$ is a closed, orientable hyperbolic surface and let $\\Phi $ be the geodesic flow in $M$ .", "Let $\\gamma $ be a closed geodesic in $S$ which fills $S$ .", "Let $\\gamma _1$ be a periodic orbit of $\\Phi $ which projects to $\\gamma $ in $S$ .", "Do $(1,n)$ Dehn surgery on $\\gamma _1$ satisfying the positivity condition to generate manifold $M_s$ and Anosov flow $\\Phi _s$ .", "By theorem REF $\\Phi _s$ is $\\mbox{${\\bf R}$}$ -covered.", "Suppose that $n$ is big enough so that $M_s$ is hyperbolic.", "Then every closed orbit of $\\Phi $ is freely homotopic to infinitely many other closed orbits.", "section1-3.5ex plus -1ex minus -.2ex2.3ex plus .2exAtoroidal submanifolds of unit tangent bundles of surfaces Let $S$ be a closed hyperbolic surface and let $M = T_1 S$ be the unit tangent bundle of $S$ .", "Notice that $M$ is orientable, whether $S$ is orientable or not.", "In the next section we will do Dehn surgery on a closed orbit of the geodesic flow to obtain the examples of flows for our Main theorem.", "We use the following notation to denote the projection map $\\tau : \\ M = T_1 S \\ \\rightarrow \\ S$ which is the projection of a unit tangent vector to its basepoint in $S$ .", "Let $\\Phi $ be the geodesic flow in $M$ .", "It is well known that $\\Phi $ is an Anosov flow [1].", "Let $\\alpha $ be a closed geodesic in $S$ .", "This geodesic of $S$ generates two orbits of $\\Phi $ , let $\\alpha _1$ be one such orbit.", "This is equivalent to picking an orientation along $\\alpha $ .", "Let $S_1$ be a subsurface of $S$ that $\\alpha $ fills.", "If $\\alpha $ is simple then $S_1$ is an annulus.", "If $\\alpha $ fills $S$ then $S_1 = S$ .", "Let $S_2$ be the closure in $S$ of $S - S_1$ $-$ which may be empty.", "Let $M_i = T_1 S_i, \\ i = 1,2$ .", "Notice that both $M_1$ and $M_2$ are Seifert fibered (with boundary).", "The purpose of this section is to prove the following result.", "Proposition 0.7 (atoroidal) The submanifold $M_1 - \\alpha _1$ is homotopically atoroidal.", "We will prove that $M_1 - \\alpha _1$ is geometrically atoroidal.", "This statement means the following: notice that $M_1 - \\alpha _1$ is not compact, but $M_1 - \\mathring{N}(\\alpha _1)$ is compact.", "The statement means that $M_1 - \\mathring{N}(\\alpha _1)$ is homotopically atoroidal.", "Notice that $M_1 - \\alpha _1$ is irreducible [19].", "Gabai [16] showed that since $M_1 - \\alpha _1$ is not a small Seifert fibered space then $M_1 - \\alpha _1$ is also homotopically atoroidal.", "Let $T$ be an incompressible torus in $M_1 - \\alpha _1$ .", "We think of $T$ as contained in $M$ .", "There are 2 possibilities: Case 1 $-$ $T$ is $\\pi _1$ -injective in $M$ .", "Here $T$ is contained in $M - M_2$ .", "We use that $M$ is Seifert fibered.", "In addition $T$ is incompressible, so it is an essential lamination in $M$ [17].", "By Brittenham's theorem [8] $T$ is isotopic to either a vertical torus or a horizontal torus in $M$ .", "Vertical torus means it is a union of $S^1$ fibers of the Seifert fibration.", "Horizontal torus means that it is transverse to these fibers.", "Since $S$ is a hyperbolic surface, there is no horizontal torus in $M = T_1 S$ .", "It follows that $T$ is isotopic to a vertical torus $T^{\\prime }$ .", "In addition since $T$ itself is disjoint from $M_2$ and $M_2$ is saturated by the Seifert fibration, we can push the isotopy away from $M_2$ and suppose it is contained in $M_1$ .", "Finally the isotopy forces an isotopy of the orbit $\\alpha _1$ into a curve $\\alpha ^{\\prime }$ disjoint from $T^{\\prime }$ .", "This isotopy projects by $\\tau $ to an homotopy in $S$ from $\\alpha $ to a curve $\\alpha ^*$ and the image of this homotopy is contained in $S_1$ , since the isotopy in $M$ has image in $M_1$ .", "The curve $\\alpha ^*$ is disjoint from the projection $\\tau (T^{\\prime })$ .", "Since $T^{\\prime }$ is vertical this projection is a simple closed curve $\\beta $ in $S_1$ .", "Since $\\alpha $ fills $S_1$ , $\\alpha ^*$ is homotopic to $\\alpha $ in $S$ , and $\\beta $ is disjoint from $\\alpha ^*$ , it now follows that $\\beta $ is a peripheral curve in $S_1$ .", "By another isotopy we can assume that $\\beta $ does not intersect $\\alpha $ or that $T^{\\prime }$ does not intersect $\\alpha _1$ .", "In addition the isotopy from $T$ to $T^{\\prime }$ can be extended to an isotopy from $M_1$ to itself.", "The geometric intersection number of $T, T^{\\prime }$ with $\\alpha _1$ is zero.", "So we can adjust the isotopy so that the images of $T$ under the isotopy never intersect $\\alpha _1$ , and consequently we can further adjust it so that it leaves $\\alpha _1$ fixed pointwise.", "In other words this induces an isotopy in $M_1 - \\alpha _1$ from $T$ to $T^{\\prime }$ .", "This shows that $T$ is peripheral in $M_1 - \\alpha _1$ .", "This finishes the proof in this case.", "Case 2 $-$ $T$ is not $\\pi _1$ -injective in $M$ .", "In particular since $T$ is two sided (as $M$ is orientable), then $T$ is compressible [19].", "This means that there is a closed disk $D$ which compresses $T$ [19], chapter 6.", "Since $T$ is incompressible in $M_1 - \\alpha _1$ , then $D$ intersects $\\alpha _1$ .", "Let $D_1, D_2$ be parallel isotopic copies of $D$ very near $D$ which also are compressing disks for $T$ .", "Then $D_1, D_2$ intersect $T$ in two curves which partition $T$ into two annuli.", "One annulus is very near both $D_1$ and $D_2$ , we call it $A_1$ , let $A$ be the other annulus which is almost all of $T$ .", "Then $A \\cup D_1 \\cup D_2$ is an embedded two dimensional sphere $W$ .", "Since $M$ is irreducible then $W$ bounds a 3-ball $B$ .", "There are two possibilities for the sphere $W$ and ball $B$ .", "In addition $A_2 \\cup D_1 \\cup D_2$ also obviously bounds a ball $B_1$ which is very near the disk $D$ .", "Suppose first that the ball $B$ contains the torus $T$ .", "This means that $A_2$ and consequently also $B_1$ , are both contained in $B$ .", "In addition $B_1$ is a regular tubular neighborhood of a properly embedded arc $\\gamma $ in $B$ .", "The intersection of $\\alpha _1$ with $B_1$ is a collection of arcs which are isotopic to the core $\\gamma $ of $B_1$ .", "Let $\\delta _1$ be one such arc.", "By Proposition REF flow lines of Anosov flows lift to unknotted curves in $\\mbox{$\\widetilde{M}$}$ .", "This implies that $\\gamma $ is unknotted in $B$ and also implies that $\\pi _1(B - \\gamma )$ is ${\\bf Z}$ .", "In particular the torus $T$ is compressible in $B - B_1$ , that is, the closure of $B - B_1$ is a solid torus.", "It follows that the $T$ is compressible in $M_1 - \\alpha _1$ .", "This contradicts the assumption that $T$ is incompressible in $M_1 - \\alpha _1$ .", "The second possibility is that the ball $B$ does not contain $T$ .", "In particular $B$ and $B_1$ have disjoint interiors and the the union $B \\cup B_1$ is a solid torus $V$ with boundary $T$ .", "The union of $B$ and $B_1$ cannot be a solid Klein bottle because $M$ is orientable.", "This solid torus lifts to an infinite solid tube $\\widetilde{V}$ in $\\mbox{$\\widetilde{M}$}$ with boundary $\\widetilde{T}$ which is an infinite cylinder.", "Notice that there is a lift $\\widetilde{\\alpha }_1$ of $\\alpha _1$ contained in $\\widetilde{V}$ so $\\widetilde{T}$ cannot be compact.", "Again by the result of Proposition REF , the infinite curve $\\widetilde{\\alpha }_1$ is unknotted in $\\mbox{$\\widetilde{M}$}$ and hence it is isotopic to the core of $\\widetilde{V}$ .", "Let $\\beta $ be a simple closed curve in $V$ which is isotopic to the core of $V$ .", "If $\\alpha _1$ is not isotopic to $\\beta $ then it is homotopic to a power $\\beta ^n$ where $n > 1$ .", "Projecting $\\beta $ to $\\tau (\\beta )$ in $S$ we obtain a closed curve in $S$ so that $(\\tau (\\beta ))^n$ is freely homotopic to $\\alpha $ .", "But $\\alpha $ is an indivisible closed geodesic and represents an indivisible element of $\\pi _1(S)$ .", "It follows that this cannot happen.", "We conclude that $\\alpha _1$ is isotopic to the core of $V$ .", "It follows that $T$ is isotopic to the boundary of a regular neighborhood of $\\alpha _1$ in $M_1 - \\alpha $ and hence again $T$ is peripheral in $M_1 - \\alpha _1$ .", "This finishes the proof of proposition REF .", "Remark In the case that $\\alpha $ fills $S$ Proposition REF is well known and there is a written proof by Foulon and Hasselblatt in [13].", "Since $M_1 - \\alpha _1$ is atoroidal the geometrization theorem in the Haken case [24], [25] shows that $M_1 - \\alpha _1$ admits a hyperbolic structure.", "The hyperbolic Dehn surgery theorem of Thurston implies that for almost all Dehn fillings along $\\alpha _1$ , the resulting manifold $M_s$ is hyperbolic.", "Notice that since $M_1$ has boundary, the statement $M_s$ is hyperbolic means that the interior of $M_s$ has a complete hyperbolic structure of finite volume, and each (torus) component of $M_1$ generates a cusp in the hyperbolic structure in the interior of $M_s$ .", "section1-3.5ex plus -1ex minus -.2ex2.3ex plus .2exDiversified homotopic behavior of closed orbits First we prove the statements about free homotopy classes of suspension Anosov flows and geodesic flows mentioned in the introduction.", "Suppose first that $\\Phi $ is the geodesic flow in $M = T_1 S$ , where $S$ is a closed, orientable hyperbolic surface.", "Suppose that $\\alpha , \\beta $ are closed orbits of $\\Phi $ which are freely homotopic to each other in $M$ .", "Then the projections $\\tau (\\alpha ), \\tau (\\beta )$ of these orbits to the surface $S$ are freely homotopic in $S$ .", "But $\\tau (\\alpha ), \\tau (\\beta )$ are closed geodesics in a hyperbolic surface, so they are freely homotopic if and only if they are the same geodesic.", "If $\\alpha $ and $\\beta $ are distinct, this can only happen if they represent the same geodesic $\\tau (\\alpha )$ of $S$ which is being traversed in opposite directions.", "Conversely if $\\tau (\\alpha ) = \\tau (\\beta )$ and they are traversed in opposite directions, there is a free homotopy from $\\alpha $ to $\\beta $ .", "This is achieved by considering all unit tangent vectors to $\\tau (\\alpha )$ in the direction of $\\alpha $ and then at time $t$ , $0 \\le t \\le 1$ , rotating all these vectors by an angle of $t \\pi $ .", "At $t = \\pi $ we obtain the tangent vectors to $\\tau (\\alpha )$ pointing in the opposite direction, that is, the direction of $\\beta $ .", "This shows that every free homotopic class of the geodesic flow has exactly two elements.", "The orientability of $S$ is used because if $S$ is not orientable and $\\tau (\\alpha )$ is an orientation reversing closed geodesic, one cannot continuously turn the angle along $\\tau (\\alpha )$ .", "Now consider a suspension Anosov flow $\\Phi $ .", "By Theorem REF , given an arbitrary Anosov flow which admits freely homotopic closed orbits, then the following happens.", "There are closed orbits $\\alpha $ and $\\beta $ so that $\\alpha $ is freely homotopic to $\\beta ^{-1}$ as oriented periodic orbits.", "For suspension Anosov flows this is a problem as follows.", "This is because there is a cross section $W$ which intersects all orbits of $\\Phi $ .", "Suppose that the algebraic intersection number of $\\alpha $ and $W$ is positive.", "Then since $\\alpha $ is freely homotopic to $\\beta ^{-1}$ it follows that the algebraic intersection number of $\\beta $ and $W$ is negative.", "But this is impossible as $W$ is a cross section and transverse to $\\Phi $ .", "This shows that every free homotopy class of a suspension is a singleton.", "Another proof of this fact is the following.", "There is a path metric in $M$ which comes from a Riemannian metric in the universal cover $\\mbox{$\\widetilde{M}$}\\cong \\mbox{${\\bf R}$}^3$ with coordinates $(x,y,t)$ given by the formula $ds^2 \\ = \\ \\lambda ^{2t}_1 dx^2 + \\lambda ^{-2t}_2 dy^2+ dt^2 \\ \\ (1)$ , where $\\lambda _1, \\lambda _2$ are real numbers $> 1$ .", "The lifted flow $\\mbox{$\\widetilde{\\Phi }$}$ has formula $\\mbox{$\\widetilde{\\Phi }$}_t(x,y,t_0) =(x,y, t_0 + t) \\ \\ (2)$ .", "If $\\alpha , \\beta $ are freely homotopic closed orbits of $\\mbox{$\\widetilde{\\Phi }$}$ , then they lift to two distinct orbits of $\\mbox{$\\widetilde{\\Phi }$}$ which are a bounded distance from each other.", "But formulas (1) and (2) show that no two distinct entire orbits of $\\mbox{$\\widetilde{\\Phi }$}$ are a bounded distance from each other.", "This also shows that free homotopy classes are singletons.", "For the property of infinite free homotopy classes for the examples in hyperbolic 3-manifolds see Theorem REF .", "We now proceed with the construction of the examples with diversified homotopic behavior and we prove the Main theorem.", "Let $S$ be a hyperbolic surface and $\\alpha $ a closed geodesic that does not fill $S$ .", "As in the previous section let $S_1$ be a subsurface that $\\alpha $ fills and let $S_2$ be the closure of $S - S_1$ .", "Let $M = T_1 S$ and $\\Phi $ the geodesic flow of $S$ in $M$ .", "Let $\\alpha _1$ be an orbit of $\\Phi $ so that $\\tau (\\alpha _1)= \\alpha $ .", "Let $M_i = T_1 S_i, \\ i = 1, 2$ .", "In the previous section we proved that $M_1 - \\alpha _1$ is atoroidal.", "Now we will do Fried's Dehn surgery on $\\alpha _1$ .", "For simplicity we will assume that the unstable foliation of $\\Phi $ (or equivalently the stable foliation of $\\Phi $ ) is transversely orientable.", "This is equivalent to the surface $S$ being orientable.", "In particular this implies that the stable leaf of $\\alpha _1$ is a annulus.", "Let $Z$ be the boundary of a small tubular solid torus neighborhood $Z_0$ of $\\alpha _1$ contained in $M_1$ .", "Then $Z$ is a two dimensional torus and we will choose a base for $\\pi _1(Z) = H_1(Z)$ .", "We assume that $Z$ is transverse to the local sheet of the stable leaf of $\\alpha _1$ .", "Then this local sheet intersects $Z$ in a pair of simple closed curves.", "Each of these defines a longitude $(0,1)$ in $\\pi _1(Z)$ , choose the direction which is isotopic to the flow forward direction along $\\alpha _1$ .", "The boundary of a meridian disk in $Z_0$ defines the meridian curve $(0,1)$ in $\\pi _1(Z)$ .", "The meridian is well defined up to sign.", "If the stable foliation of $\\Phi $ were not transversely orientable and $\\alpha $ were an orientation reversing curve, then the stable leaf of $\\alpha $ would be a Möbius band and the intersection of the local sheet with $Z$ would be a single closed curve.", "This closed curve would intersect the meridian twice and could not form a basis of $H_1(Z)$ jointly with the meridian.", "We do not want that, hence one of the reasons to restrict to $S$ orientable.", "Now we perform Fried's Dehn surgery on $\\alpha _1$ [15] as described in section REF .", "We do $(1,n)$ surgery on $\\alpha _1$ , so that the following happens.", "The resulting flow is Anosov in the Dehn surgery manifold $M_{\\alpha }$ .", "The meridian is chosen so that for any $n > 0$ the Dehn surgery flow $\\Phi _{\\alpha }$ with new meridian the $(1,n)$ curve is an $\\mbox{${\\bf R}$}$ -covered Anosov flow.", "Recall that there is a bijection between the orbits of the surgered flow $\\Phi _{\\alpha }$ and the orbits of the original flow $\\Phi $ .", "Given an orbit $\\gamma $ of $\\Phi _{\\alpha }$ we let $\\gamma ^{\\prime }$ be the corresponding orbit of $\\Phi $ under this bijection.", "We are now ready to prove the prove the main result of this article, which is restated with more detail below.", "Theorem 0.8 (diversified homotopic behavior) Let $S$ be an orientable, closed hyperbolic surface with a closed geodesic $\\alpha $ which does not fill $S$ .", "Let $S_1$ be a subsurface of $S$ which is filled by $\\alpha $ and let $S_2$ be the closure of $S - S_1$ .", "We assume also that $S_2$ is not a union of annuli.", "Let $M = T_1 S$ with geodesic flow $\\Phi $ and let $M_i = T_1 S_i, \\ i = 1,2$ .", "Let $\\alpha _1$ be a closed orbit of $\\Phi $ which projects to $\\alpha $ in $S$ .", "Do $(1,n)$ Fried's Dehn surgery along $\\alpha _1$ to yield a manifold $M_{\\alpha }$ and an Anosov flow $\\Phi _{\\alpha }$ so that $\\Phi _{\\alpha }$ is $\\mbox{${\\bf R}$}$ -covered.", "Since $M_2$ is disjoint from $\\alpha _1$ it is unaffected by the Dehn surgery and we consider it also as a submanifold of $M_{\\alpha }$ .", "Let $M_3$ be the closure of $M_{\\alpha } - M_2$ .", "We still denote by $\\alpha _1$ the orbit of $\\Phi _{\\alpha }$ corresponding to $\\alpha _1$ orbit of $\\Phi $ .", "Proposition REF implies that $M_3 - \\alpha _1$ is atoroidal and for $n$ big the hyperbolic Dehn surgery theorem [24], [25] implies that $M_3$ is hyperbolic.", "Choose one such $n$ .", "Consider the bijection $\\beta \\rightarrow \\beta ^{\\prime }$ between closed orbits of $\\Phi _{\\alpha }$ and those of $\\Phi $ .", "Then the following happens: i) Let $\\gamma $ be a closed orbit of $\\Phi _{\\alpha }$ so that the corresponding orbit $\\gamma ^{\\prime }$ of $\\Phi $ is homotopic into the submanifold $M_2$ .", "Equivalently $\\gamma ^{\\prime }$ projects to a geodesic in $S$ which is disjoint from $\\alpha $ in $S$ .", "Then $\\gamma $ is freely homotopic in $M_{\\alpha }$ to just one other closed orbit of $\\Phi _{\\alpha }$ .", "ii) Let $\\gamma $ be a closed orbit of $\\Phi _{\\alpha }$ which corresponds to a closed orbit $\\gamma ^{\\prime }$ of $\\Phi $ which is not homotopic into $M_2$ .", "Equivalently $\\gamma ^{\\prime }$ projects to a geodesic in $S$ which transversely intersects $\\alpha $ .", "Then $\\gamma $ is freely homotopic in $M_{\\alpha }$ to infinitely many other closed orbits of $\\Phi _{\\alpha }$ .", "In addition both classes i) and ii) have infinitely many elements.", "i) Let $\\gamma $ be a closed orbit of $\\Phi _{\\alpha }$ so that the corresponding orbit $\\gamma ^{\\prime }$ of $\\Phi $ is homotopic into the submanifold $M_2$ .", "Equivalently $\\gamma ^{\\prime }$ projects to a geodesic in $S$ which is disjoint from $\\alpha $ in $S$ .", "Then $\\gamma $ is freely homotopic in $M_{\\alpha }$ to just one other closed orbit of $\\Phi _{\\alpha }$ .", "ii) Let $\\gamma $ be a closed orbit of $\\Phi _{\\alpha }$ which corresponds to a closed orbit $\\gamma ^{\\prime }$ of $\\Phi $ which is not homotopic into $M_2$ .", "Equivalently $\\gamma ^{\\prime }$ projects to a geodesic in $S$ which transversely intersects $\\alpha $ .", "Then $\\gamma $ is freely homotopic in $M_{\\alpha }$ to infinitely many other closed orbits of $\\Phi _{\\alpha }$ .", "In addition both classes i) and ii) have infinitely many elements.", "First we prove that both classes i) and ii) are infinite.", "Since orbits of $\\Phi _{\\alpha }$ are in one to one correspondence with orbits of $\\Phi $ , one can think of these as statements about closed orbits of $\\Phi $ .", "Any closed geodesic of $S$ which intersects $\\alpha $ is in class ii).", "Clearly there are infinitely many such geodesics so class ii) is infinite.", "On the other hand since $S_2$ is not a union of annuli, there is a component $S^{\\prime }$ which is not an annulus.", "Any geodesic $\\beta $ of $S$ which is homotopic into $S^{\\prime }$ creates an orbit in class i).", "Since $S^{\\prime }$ is not an annulus, there are infinitely many such geodesics $\\beta $ .", "This proves that i) and ii) are infinite subsets.", "An orbit $\\delta $ of $\\Phi $ which projects in $S$ to a geodesic intersecting $\\alpha $ cannot be homotopic into $M_2$ .", "Otherwise the homotopy projects in $S$ to an homotopy from a geodesic intersecting $\\alpha $ to a curve in $S_2$ and hence to a geodesic not intersecting $\\alpha $ .", "This is impossible as closed geodesics in hyperbolic surfaces intersect minimally.", "Conversely if an orbit $\\delta $ projects to a geodesic not intersecting $\\alpha $ , then this geodesic is homotopic to a geodesic contained in $S_2$ .", "This homotopy lifts to a homotopy in $M$ from $\\delta $ to a curve in $M_2$ .", "This proves the equivalence of the first 2 statements in i) and in ii).", "Now we prove that conditions i), ii) imply the respective conclusions about the size of the free homotopy classes.", "Let $\\widetilde{\\Phi }_{\\alpha }$ be the lifted flow to the universal cover $\\mbox{$\\widetilde{M}$}_{\\alpha }$ .", "The flow $\\Phi _{\\alpha }$ is $\\mbox{${\\bf R}$}$ -covered.", "As explained in section REF there are two possibilites for $\\Phi _{\\alpha }$ , either product or skewed.", "If $\\Phi _{\\alpha }$ is product then $M_{\\alpha }$ fibers over the circle with fiber a torus.", "But in our case, $M_{\\alpha }$ has a torus decomposition with one hyperbolic piece $M_3$ and one Seifert piece $M_2$ .", "Therefore it cannot fiber over the circle with fiber a torus.", "We conclude that this case cannot happen.", "Therefore $\\Phi _{\\alpha }$ is skewed.", "Let then $\\beta _0$ be a closed orbit of $\\Phi _{\\alpha }$ .", "since $\\Phi _{\\alpha }$ is an skewed $\\mbox{${\\bf R}$}$ -covered Anosov flow we will produce orbits $\\beta _i, \\ i \\in {\\bf Z}$ which are all freely homotopic to $\\beta _0$ .", "However it is not a priori true that all the orbits $\\beta _i$ are distinct from each other, this will be analysed later.", "Here we identify the fundamental group of the manifold with the set of covering translations of the universal cover.", "Figure: The picture of the orbit space 𝒪≅(0,1)×𝐑\\mbox{$\\cal O$}\\cong (0,1) \\times \\mbox{${\\bf R}$} of a skewed 𝐑\\mbox{${\\bf R}$}-coveredAnosov flow.", "The stable foliation 𝒪 s \\mbox{${\\cal O}^s$} is the foliation by horizontalsegments in (0,1)×𝐑(0,1) \\times \\mbox{${\\bf R}$}.", "The unstable foliation 𝒪 u \\mbox{${\\cal O}^u$} is the foliation bythe parallel slanted segments.This picture also shows how to construct the orbitsβ ˜ i \\widetilde{\\beta }_i starting with the orbit β ˜ 0 \\widetilde{\\beta }_0.Construction of the orbits $\\widetilde{\\beta }_j$ .", "Lift $\\beta _0$ to an orbit $\\widetilde{\\beta }_0$ contained in a stable leaf $l_0$ of $\\mbox{${\\cal O}^s$}$ .", "Let $g$ be the deck transformation of $\\mbox{$\\widetilde{M}$}_{\\alpha }$ which corresponds to $\\beta _0$ in the sense that it generates the stabilizer of $\\widetilde{\\beta }_0$ .", "Then $u = \\mbox{${\\cal O}^u$}(\\widetilde{\\beta }_0)$ intersects an open interval $J_u$ of stable leaves.", "This is a strict subset of the leaf space of $\\mbox{${\\cal O}^s$}$ (equal to leaf space of $\\mbox{$\\widetilde{\\Lambda }^s$}$ ) by the skewed property.", "Let $l_1$ be one of the two stable leaves in the boundary of this interval.", "The fact that there are exactly two boundary leaves in this interval is a direct consequence of the fact that $\\mbox{${\\cal O}^s$}$ (or $\\mbox{$\\widetilde{\\Lambda }^s$}$ ) has leaf space $\\mbox{${\\bf R}$}$ and this fact is not true in general.", "Since $g(\\mbox{${\\cal O}^u$}(\\widetilde{\\beta }_0)) = \\mbox{${\\cal O}^u$}(\\widetilde{\\beta }_0)$ and $g$ preserves the orientation of $\\mbox{${\\cal O}^s$}$ (because $\\mbox{$\\Lambda ^s$}$ is transversely orientable), then $g(l_1) = l_1$ .", "But this implies that there is an orbit $\\widetilde{\\beta }_1$ of $\\mbox{$\\widetilde{\\Phi }$}_{\\alpha }$ in $l_1$ so that $g(\\widetilde{\\beta }_1) = \\widetilde{\\beta }_1$ .", "We refer to fig.", "REF which shows how to obtain leaf $l_1$ and hence the orbit $\\widetilde{\\beta }_1$ .", "This orbit projects to a closed orbit $\\beta _1$ of $\\Phi _{\\alpha }$ in $M_{\\alpha }$ .", "Since both are associated to $g$ , it follows that $\\beta _0, \\beta _1$ are freely homotopic.", "More specifically if we care about orientations then the positively oriented orbit $\\beta _0$ is freely homotopic to the inverse of the positively oriented orbit $\\beta _1$ .", "Remark $-$ Transverse orientability of $\\mbox{$\\Lambda ^s$}$ is necessary for this.", "If for example $\\mbox{$\\Lambda ^s$}$ were not transversely orientable and the unstable leaf of $\\beta _0$ were a Möbius band then the transformation $g$ as constructed above does not preserve the leaf $l_1$ as constructed above.", "Therefore $\\beta _0$ is not freely homotopic to $\\beta _1$ as unoriented curves.", "But $g^2$ preserves $l_1$ and from this it follows that the square $\\beta _0^2$ (as a non simple closed curve) is freely homotopic to $\\beta ^{2}_1$ .", "We proceed with the construction of freely homotopic orbits of $\\Phi _{\\alpha }$ .", "From now on we iterate the procedure above: use $\\mbox{${\\cal O}^u$}(\\widetilde{\\beta }_1)$ to produce a leaf $l_2$ of $\\mbox{${\\cal O}^s$}$ invariant by $g$ , and a closed orbit $\\beta _2$ freely homotopic to $\\beta _1$ $-$ if we consider them just as simple closed curves.", "We again refer to fig.", "REF .", "Now iterate and produce $\\widetilde{\\beta }_i, i \\in {\\bf Z}$ orbits of $\\mbox{$\\widetilde{\\Phi }$}_{\\alpha }$ so that they are all invariant under $g$ and project to closed orbits $\\beta _i$ of $\\Phi _{\\alpha }$ which are all freely homotopic to $\\beta _0$ as unoriented curves.", "Orbits freely homotopic to $\\beta _0$ .", "The covering translation $g$ preserves the leaf $\\mbox{${\\cal O}^u$}(\\widetilde{\\beta }_0)$ .", "Since $\\beta _0$ is the only periodic orbit in $\\mbox{$\\Lambda ^u$}(\\beta _0)$ , it follows that $g$ only preserves the orbit $\\widetilde{\\beta }_0$ in $\\mbox{${\\cal O}^u$}(\\widetilde{\\beta }_0)$ .", "Therefore $g$ does not leave invariant any stable leaf between $\\mbox{${\\cal O}^s$}(\\widetilde{\\beta }_0)$ and $\\mbox{${\\cal O}^s$}(\\widetilde{\\beta }_1)$ and similarly $g$ does not leave invariant any stable leaf between $\\mbox{${\\cal O}^s$}(\\widetilde{\\beta }_i)$ and $\\mbox{${\\cal O}^s$}(\\widetilde{\\beta }_{i+1})$ for any $i \\in {\\bf Z}$ .", "It follows that the collection $\\lbrace \\mbox{${\\cal O}^s$}(\\widetilde{\\beta }_i), i \\in {\\bf Z} \\rbrace $ is exactly the collection of stable leaves left invariant by $g$ .", "Suppose now that $\\delta $ is an orbit of $\\Phi _{\\alpha }$ which is freely homotopic to $\\beta _0$ .", "We can lift the free homotopy so that $\\beta _0$ lifts to $\\widetilde{\\beta }_0$ and $\\delta $ lifts to $\\widetilde{\\delta }$ .", "In particular $g$ leaves invariant $\\widetilde{\\delta }$ and hence leaves invariant $\\mbox{${\\cal O}^s$}(\\widetilde{\\delta })$ .", "It follows that $\\mbox{${\\cal O}^s$}(\\widetilde{\\delta }) = \\mbox{${\\cal O}^s$}(\\widetilde{\\beta }_i)$ for some $i \\in {\\bf Z}$ .", "As a consequence $\\widetilde{\\delta }= \\widetilde{\\beta }_i$ for $\\widetilde{\\beta }_i$ is the only orbit of $\\mbox{$\\widetilde{\\Phi }$}_{\\alpha }$ left invariant by $g$ in $\\mbox{${\\cal O}^s$}(\\widetilde{\\beta }_i)$ .", "It follows that $\\delta $ is one of $\\lbrace \\beta _j, \\ j \\in {\\bf Z} \\rbrace $ .", "Conclusion $-$ The free homotopy class of $\\beta _0$ is finite if and only if the collection $\\lbrace \\beta _i, i \\in {\\bf Z} \\rbrace $ is finite.", "Suppose now that $\\beta _i = \\beta _j$ for some $i, j$ distinct.", "Hence there is $f \\in \\pi _1(M_{\\alpha })$ with $f(\\widetilde{\\beta }_i) = \\widetilde{\\beta }_j$ .", "Then $f$ sends $\\mbox{${\\cal O}^u$}(\\widetilde{\\beta }_i)$ to $\\mbox{${\\cal O}^u$}(\\widetilde{\\beta }_j)$ .", "By the definition of $\\widetilde{\\beta }_{i+1}$ it follows that $f$ sends $\\widetilde{\\beta }_{i+1}$ to $\\widetilde{\\beta }_{j+1}$ .", "Iterating this procedure shows that $f$ preserves the collection $\\lbrace \\widetilde{\\beta }_k, \\ k \\in {\\bf Z} \\rbrace $ .", "In addition it follows easily that $f$ sends $\\widetilde{\\beta }_0$ to $\\widetilde{\\beta }_k$ for $k = j-i$ .", "The free homotopy from $\\beta _0$ to $\\beta _k = \\beta _0$ produces a $\\pi _1$ -injective map of either the torus or the Klein bottle into $M$ .", "We have to consider the Klein bottle because the free homotopy may be from $\\beta _0$ to the inverse of $\\beta _0$ when we account for orientations along orbits.", "Taking the square of this free homotopy if necessary we produce a $\\pi _1$ -injective map of the torus into $M$ .", "The torus theorem [20], [21] shows that the free homotopy is homotopic into a Seifert piece of the torus decomposition of $M_{\\alpha }$ .", "Therefore in our situation the homotopy is freely homotopic into $M_2$ .", "It follows that the orbit $\\beta ^{\\prime }_0$ of $\\Phi $ associated to $\\beta _0$ is freely homotopic into $M_2$ .", "Therefore the geodesic $\\tau (\\beta ^{\\prime }_0)$ of $S$ does not intersect $\\alpha $ .", "This proves part ii) of the theorem: If the geodesic $\\tau (\\beta ^{\\prime }_0)$ intersects $\\alpha $ then the orbit $\\beta _0$ of $\\Phi _{\\alpha }$ is freely homotopic to infinitely many other closed orbits of $\\Phi _{\\alpha }$ .", "Consider now a closed orbit $\\beta _0$ of $\\Phi _{\\alpha }$ so that it corresponds to a geodesic in $S$ which does not intersect $\\alpha $ .", "This geodesic is $\\tau (\\beta ^{\\prime }_0)$ which we denoted by $\\gamma $ .", "There is a non trivial free homotopy in $M = T_1 S$ from $\\beta ^{\\prime }_0$ to itself with the same orientation, obtained by turning the angle along $\\gamma $ by a full turn, from 0 to $2 \\pi $ .", "Notice that this free homotopy at some point is exactly $\\beta ^{\\prime }_0$ and at another point it is exactly the orbit corresponding to the geodesic $\\gamma $ being traversed in the opposite direction.", "This free homotopy is entirely contained in $M_2$ and therefore this free homotopy survives in the Dehn surgered manifold $M_{\\alpha }$ .", "By construction the image of the free homotopy in $M_{\\alpha }$ contains two distinct closed orbits of $\\Phi _{\\alpha }$ , one of which is $\\beta _0$ .", "In particular the free homotopy class of $\\beta _0$ has at least two elements.", "In addition the free homotopy produces a $\\pi _1$ -injective map from $T^2$ into $M$ .", "Choose a basis $g, f$ for $\\pi _1(T^2)$ (seen as covering translations in $\\mbox{$\\widetilde{M}$}$ ) so that $g$ leaves invariant a lift $\\widetilde{\\beta }_0$ of $\\beta _0$ .", "Then $g f(\\widetilde{\\beta }_0) \\ \\ = \\ \\ f g(\\widetilde{\\beta }_0) \\ \\ = \\ \\ f(\\widetilde{\\beta }_0).$ So $g$ also leaves invariant $f(\\widetilde{\\beta }_0)$ .", "As seen in the paragraphs “Orbits freely homotopic to $\\beta _0$ \", it follows that $f(\\widetilde{\\beta }_0) = \\widetilde{\\beta }_j$ for some $j$ in ${\\bf Z}$ .", "This implies that the free homotopy class of $\\beta _0$ is finite.", "Let now $\\delta $ be a closed orbit of $\\Phi _{\\alpha }$ which is freely homotopic to $\\beta _0$ .", "In particular the free homotopy class of $\\delta $ is the same as the free homotopy class of $\\beta _0$ and in the part entitled “Orbits freely homotopic to $\\beta _0$ \" we showed that this free homotopy class is finite in this case.", "In addition from what we already proved in the theorem, it follows that $\\delta $ is isotopic into $M_2$ and choosing $M_2$ appropriately we can assume that $\\beta _0, \\delta $ are contained in $M_2$ .", "Let the free homotopy from $\\beta _0$ to $\\delta $ be realized by a $\\pi _1$ -injective annulus $A$ which is in general position.", "The annulus $A$ is a priori only immersed.", "Let $T = \\partial M_3= \\partial M_2$ an embedded torus in $M_{\\alpha }$ which is $\\pi _1$ -injective.", "Put $A$ in general position with respect to $T$ and analyse the self intersections.", "Any component which is null homotopic in $T$ can be homotoped away because $M_{\\alpha }$ is irreducible [19], [20].", "After this is eliminated each component of $A - T$ is an a priori only immersed annulus.", "But since $M_3$ is a hyperbolic manifold with a single boundary torus $T$ it follows that $M_3$ is acylindrical [24], [25].", "This means that any $\\pi _1$ -injective properly immersed annulus is homotopic rel boundary into the boundary.", "This is because parabolic subgroups of the fundamental group of $M_3$ $-$ as a Kleinian group, have an associated maximal ${\\bf Z}^2$ parabolic subgroup [24], [25].", "In particular this implies that the annulus $A$ can be homotoped away from $M_3$ to be entirely contained in $M_2$ .", "Therefore the free homotopy represented by the annulus $A$ survives if we undo the Dehn surgery on $\\alpha $ .", "This produces a free homotopy between $\\beta ^{\\prime }_0$ and $\\delta ^{\\prime }$ in $M = T_1 S$ .", "But the free homotopy classes of geodesic flows all have exactly two elements.", "Therefore there is only one possibility for $\\delta $ if $\\delta $ is distinct from $\\beta _0$ .", "This shows that the free homotopy class of $\\beta _0$ has exactly two elements.", "This finishes the proof of theorem REF section1-3.5ex plus -1ex minus -.2ex2.3ex plus .2exGeneralizations There are a few ways to generalize the main result of this article.", "Here we mention two of them.", "1) Finite covers and Dehn surgery Let $M = T_1 S$ where $S$ is a closed orientable surface.", "Let $\\Phi $ be the geodesic flow in $M$ .", "First take a finite cover of order $n$ of $M$ unrolling the circle fibers.", "Let this be the manifold $M_1$ with lifted Anosov flow $\\Phi _1$ .", "Then every closed orbit of $\\Phi _1$ is freely homotopic to $2n-1$ other closed orbits of the flow.", "The flow $\\Phi _1$ is a skewed $\\mbox{${\\bf R}$}$ -covered Anosov flow.", "We use the covering map $\\eta : M_1 \\rightarrow M$ and the projection $\\tau : M \\rightarrow S$ .", "We do Dehn surgery on closed orbits of $\\Phi _1$ .", "Essentially the same proof as the Main theorem yields the following result.", "Theorem 0.9 Let $S$ be a closed orientable surface and $M = T_1 S$ with Anosov flow $\\Phi $ .", "Let $M_1$ be a finite cover of $M$ where we unroll the Seifert fibers and let $\\Phi _1$ be the lifted flow to $M_1$ .", "Do $(1,n)$ Fried's flow Dehn surgery on a closed orbit $\\gamma $ of the $\\Phi _1$ so $\\tau \\circ \\eta (\\gamma )$ is a closed geodesic in $S$ which does not fill $S$ and some complementary component of $\\tau \\circ \\eta (\\gamma )$ in $S$ is not an annulus and so that the surgery satisfies the positivity condition.", "Let $\\Phi _s$ be the resulting Anosov flow in the surgery manifold.", "Given $\\beta $ a closed orbit of $\\Phi _s$ , it has an associated unique orbit of $\\Phi _1$ which in turn projects under $\\tau \\circ \\eta $ to a closed geodesic $\\beta ^{\\prime }$ of $S$ .", "Then i) If $\\beta ^{\\prime }$ does not intersect $\\tau \\circ \\eta (\\gamma )$ it follows $\\beta $ is freely homotopic to exactly $2n -1$ other orbits of $\\Phi _s$ .", "In addition ii) If $\\beta ^{\\prime }$ intersects $\\tau \\circ \\eta (\\gamma )$ then $\\beta $ is freely homotopic to infinitely many other closed orbits of $\\Phi _s$ .", "2) Dehn surgery on more than one closed orbit In essentially the same way as in the proof of the Main theorem We obtain a result similar to the Main theorem under Dehn surgery on finitely many closed orbits as follows.", "Theorem 0.10 Let $S$ be a closed orientable surface and $M = T_1 S$ with geodesic flow $\\Phi $ .", "Let $\\lbrace \\alpha _i, \\ \\ 1 \\le i \\le i_0 \\rbrace $ be a finite collection of disjoint closed geodesics in $S$ which are pairwise disjoint and some component of the complement of their union is not an annulus.", "Let $\\gamma _i$ be a closed orbit of $\\Phi $ which projects to $\\alpha _i$ in $S$ .", "For each $i$ do $(1,n_i)$ Fried's flow Dehn surgery on $\\gamma _i$ to yield an Anosov flow $\\Phi _s$ in the Dehn surgery manifold $M_s$ and so that the surgery satisfies the positivity condition.", "Then $\\Phi _1$ is an $\\mbox{${\\bf R}$}$ -covered Anosov flow.", "There is a bijection between orbits of $\\Phi _1$ and orbits of $\\Phi $ .", "Given an orbit $\\beta $ of $\\Phi _s$ consider the orbit of $\\Phi $ associated to it and project it to a closed geodesic $\\beta ^{\\prime }$ in $S$ .", "Then the following happens.", "i) If $\\beta ^{\\prime }$ is disjoint from the union of the $\\lbrace \\alpha _i \\rbrace $ then $\\beta $ is freely homotopic to a single other closed orbit of $\\Phi _s$ .", "ii) If $\\beta ^{\\prime }$ intersects the union of $\\lbrace \\alpha _i \\rbrace $ then $\\beta $ is freely homotopic to infinitely many other closed orbits of $\\Phi _s$ .", "One can also combine the two constructions above.", "Florida State University Tallahassee, FL 32306-4510" ] ]
1403.0310
[ [ "Fast cholesterol flip-flop and lack of swelling in skin lipid\n multilayers" ], [ "Abstract Atomistic simulations were performed on hydrated model lipid multilayers that are representative of the lipid matrix in the outer skin (stratum corneum).", "We find that cholesterol transfers easily between adjacent leaflets belonging to the same bilayer via fast orientational diffusion (tumbling) in the inter-leaflet disordered region, while at the same time there is a large free energy cost against swelling.", "This fast flip-flop may play an important role in accommodating the variety of curvatures that would be required in the three dimensional arrangement of the lipid multilayers in skin, and for enabling mechanical or hydration induced strains without large curvature elastic costs." ], [ "Analysis details and additional results", "In this supplementary material, we provide the methods used in calculating the hydrogen bonds, flip-flop time-scales, in-plane molecular diffusion, and the excess chemical potential of water.", "In our simulation of double bilayer in excess water, only two of the leaflets are in contact with water (Fig.", "REF ).", "We term these two leaflets as the `outer leaflets'.", "The other two `inner' leaflets face each other and are not in contact with water.", "After an equilibration time of 20 ns, we stored 5000 configurations separated by 0.2 ns spanning a total of 1 $\\mu $ s. Unless otherwise stated, the results below are averaged over these 5000 configurations." ], [ "Hydrogen bonds", "We use a geometric criteria [44], [45], [46], [47] to define a hydrogen bond if the distance between the donor and the acceptor atoms is less than $3.5$ Å and simultaneously the absolute angle between the vectors $\\vec{r}_{DH}$ and $\\vec{r}_{AH}$ is less than 30$^{\\circ }$ (Fig.", "REF ).", "Figure: Geometric criteria used to identify hydrogen bonds.Table REF shows the average number of lipid-lipid hydrogen bonds for each lipid species.", "We have separated these into intra-leaflet and inter-leaflet (for inner leaflets that face each other) hydrogen bonds.", "The inner leaflet lipids form a large number of intra-leaflet, as well as inter-leaflet, hydrogen bonds with lipids.", "In addition, lipids in the outer leaflets are also involved in a large number of hydrogen bonds with water molecules.", "Table: Number of hydrogen bonds per lipid molecule in the different leaflets." ], [ "Tail order", "To investigate the alignment of the lipid tails, for any three consecutive CH$_2$ groups (C$_{i-1}$ , C$_{i}$ and C$_{i+1}$ ) in the CER or FFA molecules, we consider the angle $\\theta _z$ of the vector (C$_{i+1}$ - C$_{i-1}$ ) with respect to the z-axis (normal direction to the lipid layers).", "We define an order parameter [48] $S_z (z) = \\left\\langle \\frac{3 \\cos ^2 \\theta _z - 1}{2}\\right\\rangle ,$ where the angular bracket denotes averaging over all CH$_2$ triplets with the central group being at a distance $z$ from the lipid center of mass.", "The usual order parameter is calculated (or measured) as a function of carbon number along the lipid tails [48], while we consider it as a function of $z$ .", "For perfect tail alignment along the $z$ -direction, $S_z=1$ ; random order gives $S_z=0$ ; and perfect aligment perpendicular to the $z$ -direction gives $S_z=-0.5$ .", "Fig.", "REF shows that $S_z$ becomes close to zero in the tail-tail interface region ($z \\simeq \\pm 2.4 \\textrm {nm}$ ) signifying a disordered region.", "The same region shows lower total mass density and a subpopulation of CHOL center of mass." ], [ "Density profile and CHOL flip-flop", "We unfold the molecules in the saved configurations and fix the lipid center of mass at the origin.", "The mass density of the lipids and the number density of the center of masses of the lipids were calculated in this lipid center of mass reference frame.", "Figure: Average density of center of mass of CHOL and identificationof zones (1-6).", "The (red) arrows show the boundaries and local density maximum of zone5, while the (violet) dashed boxes show the criteria for a `flip', which is a transfer betweenadjacent zones.From the distribution of the centers of mass of CHOL, we identify two inter-leaflet liquid like regions centered at $z=\\pm 2.4\\;\\textrm {nm}$ .", "For the peak at $z=2.4 \\;\\textrm {nm}$ , the minima in the distribution of CM are at $z=1.8\\;\\textrm {nm}$ and $z=3.1\\;\\textrm {nm}$ .", "In the first configuration, we assign a zone index to a given CHOL depending on the $z$ -coordinate of its center of mass: (ordered) zone 1: zCM -3.1 (disordered) zone 2: -3.1    zCM < -1.8 (ordered) zone 3: -1.8    zCM < 0 (ordered) zone 4: 0    zCM < 1.8 (disordered) zone 5: 1.8    zCM < 3.1 (ordered) zone 6: 3.1    zCM .", "In subsequent configurations we assign a new provisional zone index to a given CHOL only if has moved into the next zone by at least 10% of width of the next zone.", "This criteria is indicated by dashed boxes in Fig.", "REF .", "Thus, a CHOL initially belonging to zone 5 is considered to have moved to zone 4 only if its $z_{CM} < 1.64$ .", "We define a flip event as a transit between that does not revert to the original zone within two frames (0.4ns).", "There are nearly 3 times more flip events involving the outer leaflets than involving the inner leaflets.", "We define a flip-flop event to be when a CHOL enters a ordered zone after having entered the disordered zone from a different ordered zone.", "In our trajectory we did not find any lipid exchange between the bilayers, so that all flip-flop events involve CHOL exchange between an inner leaflet and an outer leaflet.", "Table REF shows the data used in calculating the flip-flop time scales.", "Table: Statistics for transitions of CHOL molecules between different regions in1μs1\\,\\mu \\textrm {s}, for calculations of flip times and flip-flop times.", "For flip-flop, CHOL begin in the outer (inner) ordered region of a leaflet, and explore the outer (inner) ordered and disordered regions until it enters the inner (outer) ordered regime of the other leaflet of the same bilayer." ], [ "Mean square displacement", "Fig.", "REF shows the $x-y$ component of mean square displacement of the center of mass of the lipids, in a reference frame in which the center of mass of all the lipids is fixed.", "The data is averaged over the lipid molecules and over the time origin.", "CHOL shows higher in-plane mobility (red dashed line) than CER or FFA.", "We separate CHOL into populations that did and did not undergo flip events during the entire trajectory.", "CHOL without any flip events show a mean-square in-plane displacement similar to that of CER or FFA, while CHOL with flip events have a much larger mobility: e. g. $\\left.", "\\frac{\\langle r_{xy}^2\\rangle _{\\textit {flip}}}{\\langle r_{xy}^2\\rangle _{\\textit {no flip}}}\\right|_{(\\tau =500\\,\\textrm {ns})} \\simeq 2.6.$ Figure: Diffusion of center of mass of the leaflets with respect to the center ofmass of all lipid molecules is fixed.In simulations of finite bilayers the leaflets can diffuse with respect to each other [49].", "Since some of the CHOL molecules move between leaflets, we define an approximate leaflet center of mass in terms of the CER and FFA molecules.", "Fig.", "REF shows the diffusion of the leaflets' centers of mass.", "Both the $x$ and $y$ components of the inner two leaflets (red and blue) move together coherently over the entire trajectory.", "Figure: Mean square displacement of lipid center of mass in the reference frame of the leafletcenter of mass.By calculating displacements in the reference frame of a given leaflet, we obtain the mean square displacement of CER and FFA lipids in the reference frame in which the leaflet center of mass is fixed, which is appropriate for a multilayer stack in which leaflet diffusion or sliding is prohibited.", "Fig.", "REF shows that the lipids in the outer leaflets are significantly more mobile than those in the inner leaflets.", "Comparison of Figs.", "REF and REF shows that, even for the outer leaflets, the main contribution to in-plane displacement is from leaflet diffusion, which is pronounced here because of the small membrane size and periodic boundary conditions.", "The fast CHOL exchange and fewer inter-lipid hydrogen bonds render the lipids in the outer leaflets several times more mobile." ], [ "Force fluctuations for constrained water molecules", "We perform simulations in which a single water molecule is constrained to at a given $z$ -separation from the lipid center of mass.", "This constraint induces a rapidly fluctuating force $F_z(z,t)$ in the $z$ direction.", "However, we find a slowly decaying time correlation in $F_z(z,t)$ , particularly in the ordered regions.", "At a fixed $z$ , we fit the autocorrelation of $F_z(z,t)$ to a sum of two generalized exponentials: C(z,t) Fz(z,t) Fz(z,0) =i=12 Ai(z) [- ( ti(z) )i(z) ], where the parameters $A_i, \\tau _i$ , and $\\beta _i$ are fitted with the Levenberg-Marquardt damped least-square method.", "Fig.", "REF shows $C(t)$ in the ordered leaflet region close to bilayer-bilayer interface ($z=0.3\\;\\textrm {nm}$ ) along with the fit with two stretched/compressed exponentials.", "Assigning an average decay time $\\tau _{av,i} = \\frac{\\tau _i}{\\beta _i} \\Gamma \\left(\\frac{1}{\\beta _i} \\right)$ , the fit gives a fast decay time of $0.03\\;\\textrm {ps}$ and a slow decay time of $7.9\\;\\textrm {ns}$ .", "At each $z$ , we ensure that the simulations are longer than the slow decay time.", "Figure: Average decay times of C(z,t)C(z,t) and exponents from fitting ofautocorrelation of F z (z,t)F_z (z,t), at different distances zz from the double bilayercenter.Fig.", "REF shows the variation of the average decay times and the exponents with $z$ .", "There is a fast decay in $C(z,t)$ at all $z$ with $\\tau _{av} \\simeq 0.03\\;\\textrm {ps}$ and exponent $\\beta \\simeq 1.5$ (open circles in Fig.", "REF ).", "Inside the lipid double bilayer, there is an additional slowly decaying component (filled squares in Fig.", "REF ) that follows a stretched exponential with $\\tau _{av} \\sim 20\\;\\textrm {ns}$ in the most ordered part of the leaflets, with a stretching exponent $\\beta \\simeq 0.2$ .", "Inside the ordered lipid leaflets, the $x-y$ diffusion of the constrained water molecule is limited to $40\\;\\textrm {ns}$ timescale (the longest time simulated for the constrained water simulations).", "We use six separate simulations with a different randomly chosen water molecules to calculate $\\langle F_z(z)\\rangle $ at a fixed $z$ .", "Furthermore, we use the antisymmetric property of $\\langle F_z (z)\\rangle $ to get 12 independent estimates of $\\langle F_z (z)\\rangle $ at a given $z$ .", "The error-bars in the excess chemical potential are calculated from error of mean from these 12 estimates of $\\langle F_z (z)\\rangle $ at each $z$ ." ] ]
1403.0030
[ [ "Constraining the spin and the deformation parameters from the black hole\n shadow" ], [ "Abstract Within 5-10 years, very-long baseline interferometry (VLBI) facilities will be able to directly image the accretion flow around SgrA$^*$, the super-massive black hole candidate at the center of the Galaxy, and observe the black hole \"shadow\".", "In 4-dimensional general relativity, the no-hair theorem asserts that uncharged black holes are described by the Kerr solution and are completely specified by their mass $M$ and by their spin parameter $a$.", "In this paper, we explore the possibility of distinguishing Kerr and Bardeen black holes from their shadow.", "In Hioki & Maeda (2009), under the assumption that the background geometry is described by the Kerr solution, the authors proposed an algorithm to estimate the value of $a/M$ by measuring the distortion parameter $\\delta$, an observable quantity that characterizes the shape of the shadow.", "Here, we try to extend their approach.", "Since the Hioki-Maeda distortion parameter is degenerate with respect to the spin and possible deviations from the Kerr solution, one has to measure another quantity to test the Kerr black hole hypothesis.", "We study a few possibilities.", "We find that it is extremely difficult to distinguish Kerr and Bardeen black holes from the sole observation of the shadow, and out of reach for the near future.", "The combination of the measurement of the shadow with possible accurate radio observations of a pulsar in a compact orbit around SgrA$^*$ could be a more promising strategy to verify the Kerr black hole paradigm." ], [ "Introduction", "In 4-dimensional general relativity, the no-hair theorem guarantees that uncharged black holes (BHs) are only described by the Kerr solution, which is completely specified by two parameters; that is, the BH mass $M$ and the BH spin angular momentum $J$  [1], [2], [3].", "A fundamental limit for a Kerr BH is the bound $|a| \\le M$ , where $a = J/M$ is the BH spin parameterThroughout the paper, we use units in which $G_{\\rm N} = c = 1$ , unless stated otherwise..", "This is just the condition for the existence of the event horizon.", "Astrophysical BH candidates are stellar-mass compact objects in X-ray binary systems and super-massive bodies at the center of every normal galaxy [4].", "They are thought to be the Kerr BHs of general relativity simply because they cannot be explained otherwise without introducing new physics, but there is no evidence that the spacetime around them is described by the Kerr metric.", "In the last decade, there have been both significant theoretical work to understand the accretion process onto BH candidates and new observational facilities, so that it might be possible to test the actual nature of these objects in a near future from the properties of the electromagnetic radiation emitted by the accreting material [5], [6].", "A large number of methods have been proposed, including the study of the thermal spectrum of thin accretion disks [7], [8], [9], [10], the analysis of the K$\\alpha $ iron line [11], [12], [13], the observation of the so-called quasi-periodic oscillations (QPOs) [14], [15], [16], [17], the measurement of the radiative efficiency [18], [19], [20], [21], and the estimate of the jet power [22], [23], [24].", "The main difficulty to achieve this goal is that the properties of the electromagnetic radiation from a Kerr BH with dimensionless spin parameter $a/M$ can be very similar, and practically indistinguishable, from the ones of non-Kerr objects with different spin.", "In other words, it is usually impossible to constrain at the same time the value of the spin parameter and possible deviations from the Kerr solution.", "For some metrics, the combination of the analysis of the disk's thermal spectrum and of the K$\\alpha $ iron line cannot break the degeneracy between the spin and the deformation parameters, while in other cases that can be achieved only with very good measurements [25].", "The combination of the analysis of the thermal spectrum of thin disks and the estimate of the jet power can potentially do the job [23], [24], but the latter is not yet a mature technique and therefore it cannot yet be used to test fundamental physics.", "The result is that right now we can only rule out the possibility that BH candidates are some kinds of very exotic objects, like some types of wormholes [26] or some exotic compact objects without event horizon [27], [28].", "The non-observation of electromagnetic radiation emitted by the possible surface of these objects may also be interpreted as an indication for the existence of an event horizon [29], [30] (but see [31], [32]).", "However, more reasonable alternatives, like non-Kerr BHs, are difficult to test.", "Recent observations of SgrA$^*$ at mm wavelength suggest that, hopefully within about 5 years, very-long baseline interferometry (VLBI) facilities at mm/sub-mm wavelength will be able to directly image the accretion flow around the super-massive BH candidate at the center of our Galaxy with a resolution of the order its gravitational radius $r_g = M$  [33], [34].", "These observations will open a completely new window to test gravity in the strong field regime and, in particular, to verify if SgrA$^*$ is a Kerr BH, as expected from general relativity.", "The main goal of these experiments is the observation of the “shadow” of SgrA$^*$ .", "The shadow of a BH is a dark area over a brighter background observed by directly imaging the accretion flow around the compact object [35], [36].", "While the intensity map of the image depends on complicated astrophysical processes related to the accretion properties and the emission mechanisms, the exact shape of the shadow is only determined by the background geometry, being the apparent photon capture sphere as seen by a distant observer.", "A very accurate detection of the boundary of the shadow can thus provide information on the geometry around SgrA$^*$ and test the Kerr BH hypothesis.", "Starting from Refs.", "[37], [38], a number of tests has been proposed in the literature and shadows in different background metrics have been calculated by different groups [39], [40], [41], [42], [43], [44], [45], [46].", "For a recent review, see e.g.", "Ref. [47].", "At first approximation, the shape of the shadow is a circle.", "The radius of the circle corresponds to the apparent photon capture radius, which, for a given metric, is set by the mass of the compact object and its distance from us.", "For SgrA$^*$ and for the other nearby super-massive BH candidates, these two quantities are currently not known with good precision, and therefore the observation of the size of the shadow may not be used to test the spacetime geometry around the compact object (but see Ref.", "[45] and the conclusions of the present work).", "The shape of the shadow is usually thought to be the key-point.", "The first order correction to the circle is due to the spin, as the photon capture radius is different for co-rotating and counter-rotating particles.", "The boundary of the shadow has thus a dent on one side: the deformation is more pronounced for an observer on the equatorial plane (viewing angle $i = 90^\\circ $ ) and decreases as the observer moves towards the spin axis, to completely disappear when $i = 0^\\circ $ or $180^\\circ $ .", "Possible deviations from the Kerr solutions usually introduce smaller corrections and therefore they can be detected only in the case of excellent data.", "In Ref.", "[48], two of us have studied the measurement of the Kerr spin parameter of Kerr BHs and non-Kerr regular BHs; that is, we measured the spin parameter $a/M$ from the shape of the shadow of a BH assuming it was a Kerr BH.", "We used the procedure proposed in Ref.", "[49], which is based on the determination of the distortion parameter $\\delta = D/R$ , where $D$ and $R$ are, respectively, the dent and the radius of the shadow.", "In the case of non-Kerr BHs, this technique provides the correct value of $a/M$ for non-rotating objects, but a quite different spin for near extremal states.", "If we compare this measurement with the frequency of the innermost stable circular orbit (ISCO) that can be potentially obtained by the observations of blobs of plasma orbiting around the BH candidate [50], we find that the nature of the object may be tested in the case of a non-rotating or slow-rotating BHs, while that seems to be impossible for near extremal states, as the two techniques essentially provide the same information on the spacetime geometry.", "In the present paper, we consider a different approach to test the Kerr geometry around SgrA$^*$ .", "We assume to have good observational data of the BH shadow and we try to measure two parameters.", "One parameter of the shadow can indeed only determine one parameter of the background geometry, which is enough in the case of the Kerr metric where there is only the spin.", "If we want to test the Kerr nature of the BH candidate, the spacetime metric will be also characterized by one (or more) deformation parameter(s), measuring possible deviations from the Kerr solution.", "In general, the Hioki-Maeda distortion parameter $\\delta $ is degenerate with respect to the spin and the deformation parameters, in the sense that the same value of $\\delta $ is found for any deformation parameter for a particular value of $a/M$ .", "With the measurement of another parameter of the shadow, it is possible to break such a degeneracy and eventually test the Kerr metric of the spacetime.", "We explore three possibilities.", "We introduce a second distortion parameter, $\\epsilon $ , which characterizes possible deviations from the shape of the shadow of a Kerr BH.", "While a similar approach may sound the most natural extension of Ref.", "[49], it turns out that Kerr BHs and non-Kerr regular BHs have very similar shapes and such a small difference is very difficult to detect in true observational data.", "We thus consider the possibility of measuring the off-set of the center of the shadow with respect to the actual position of the BH.", "Even in this case, the approach can potentially distinguish Kerr BHs and non-Kerr regular BHs, but an accurate measurement of the BH position is very challenging, at least now.", "The third and last case is the measurement of the radius of the shadow, $R$ .", "Here, we need good measurements of the BH mass and distance from us, which are definitively not available today, but they could be possible in the future, for instance from accurate radio observations of a pulsar orbiting SgrA$^*$ with a period shorter than 1 year.", "The same pulsar could also provide a precise estimate of the spin parameter (obtained in the weak field), to be compared with the Hioki-Maeda distortion parameter $\\delta $ of the strong gravity regime.", "It seems therefore that the sole observation of the shadow cannot distinguish Kerr and Bardeen BHs, while the combination of the shadow and pulsar observations is more promising to probe the geometry around SgrA$^*$ .", "The content of the paper is as follows.", "In Section , we review the calculation of the BH shadow, while in Section  we review the procedure proposed in Ref.", "[49] to infer the spin parameter from the determination of the distortion parameter $\\delta $ .", "In Section , we explore the possibility of testing the Kerr metric from the combination of the estimates of the Hioki-Maeda distortion parameter $\\delta $ with, respectively, the distortion parameter $\\epsilon $ , the position of the center of the shadow with respect to the one of the BH, and the shadow radius $R$ .", "Section  is devoted to the discussion.", "Summary and conclusions are in Section ." ], [ "Black hole shadow ", "If a BH is surrounded by an optically thin and geometrically thick accretion flow, a distant observer sees a dark area over a brighter background.", "Such a dark area is the “shadow” of the BH.", "The boundary of the shadow corresponds to the apparent image of the photon capture sphere and therefore it only depends on the geometry of the background [35], [36].", "In this section, we briefly review the calculation of the BH shadow.", "In the case of the Kerr metric, the line element in Boyer-Lindquist coordinates is $ds^2 &=& - \\left(1 - \\frac{2 M r}{\\Sigma }\\right) dt^2- \\frac{4 a M r \\sin ^2 \\theta }{\\Sigma } dt d\\phi + \\frac{\\Sigma }{\\Delta } dr^2 + \\Sigma d\\theta ^2 \\nonumber \\\\&& + \\left(r^2 + a^2 + \\frac{2 a^2 M r \\sin ^2\\theta }{\\Sigma }\\right)\\sin ^2\\theta d\\phi ^2 \\, ,$ where $\\Sigma = r^2 + a^2 \\cos ^2\\theta \\, , \\quad \\Delta = r^2 - 2 M r + a^2 \\, ,$ $M$ is the BH mass, and $a = J/M$ .", "The photon motion is governed by the equations [35] $\\Sigma \\bigg (\\frac{d t}{d \\lambda }\\bigg )&=& \\frac{A E - 2 a M r L_z}{\\Delta } \\, , \\\\\\Sigma ^2\\bigg (\\frac{d r}{d \\lambda }\\bigg )^2&=& \\mathcal {R} \\, , \\\\\\Sigma ^2\\bigg (\\frac{d \\theta }{d \\lambda }\\bigg )^2&=& \\Theta \\, , \\\\\\Sigma \\bigg (\\frac{d \\phi }{d \\lambda }\\bigg )&=& \\frac{2 a M r E + \\left(\\Sigma - 2 M r\\right) L_z \\csc ^2 \\theta }{\\Delta } \\, ,$ where $\\lambda $ is an affine parameter, and $\\mathcal {R} &=& E^2 r^4+\\left(a^2E^2-L_z^2-\\mathcal {Q}\\right)r^2+2 M \\left[(aE-L_z)^2+\\mathcal {Q}\\right]r-a^2\\mathcal {Q}\\, , \\\\\\Theta &=& \\mathcal {Q} \\left(a^2 E^2 - L_z^2 \\csc ^2 \\theta \\right)\\cos ^2 \\theta \\, , \\\\A &=& \\left(r^2 + a^2\\right)^2 - a^2 \\Delta \\sin ^2 \\theta \\, .$ $E$ and $L_z$ are, respectively, the conserved photon energy and the conserved component of the photon angular momentum parallel to the BH spin.", "$\\mathcal {Q}$ is the Carter constant $\\mathcal {Q} &=& p_\\theta ^2+\\cos ^2\\theta \\bigg (\\frac{L_z^2}{\\sin ^2\\theta }-a^2E^2\\bigg ) \\, ,$ and $p_\\theta = \\Sigma \\frac{d\\theta }{d\\lambda }$ is the canonical momentum conjugate to $\\theta $ .", "Motion is only possible when $\\mathcal {R}(r) \\ge 0$ , and therefore the analysis of the position of the roots of $\\mathcal {R}(r)$ can be used to distinguish the capture from the scattered orbits.", "The three kinds of photon orbits are: Capture orbits: $\\mathcal {R}(r)$ has no roots for $r \\ge r_+$ , where $r_+$ is the radial coordinate of the BH event horizon.", "In this case, photons come from infinity and then cross the horizon.", "Scattering orbits: $\\mathcal {R}(r)$ has real roots for $r \\ge r_+$ , which correspond to the photon turning points.", "If the photons come from infinity, they reach a minimum distance from the BH, and then go back to infinity.", "Unstable orbits of constant radius: these orbits separate the capture and the scattering orbits and are determined by $\\mathcal {R}(r_*) = \\frac{\\partial \\mathcal {R}}{\\partial r}(r_*)=0 \\, , \\quad {\\rm and} \\quad \\frac{\\partial ^2 \\mathcal {R}}{\\partial r^2}(r_*) \\ge 0 \\, , $ where $r_*$ is the larger real root of $\\mathcal {R}$ .", "The boundary of the shadow of a BH can be determined by finding the unstable orbits of constant radius.", "Since the photon trajectories are independent of the photon wavelength, it is convenient to introduce the parameters $\\xi = L_z/E$ and $\\eta = \\mathcal {Q}/E^2$ .", "$\\xi $ and $\\eta $ are related to the “celestial coordinates” $\\alpha $ and $\\beta $ of the image plane of the distant observer by $\\alpha = \\frac{\\xi }{\\sin i}\\, , \\quad \\beta = \\pm (\\eta +a^2\\cos ^2 i-\\xi ^2\\cot ^2 i)^{1/2}\\, ,$ where $i$ is the angular coordinate of the observer at infinity.", "Every photon orbit can be characterized by the constants of motion $\\xi $ and $\\eta $ .", "The boundary of the BH shadow is represented by a closed curve determined by the set of unstable circular orbits ($\\xi _c$ , $\\eta _c$ ) on the plane of the distant observer.", "From Eqs.", "(REF ) and (REF ), the equations determining the unstable orbits of constant radius are $\\mathcal {R} &=& r^4+(a^2-\\xi _c^2-\\eta _c)r^2+2M[\\eta _c+(\\xi _c-a)^2]r-a^2\\eta _c = 0 \\, , \\nonumber \\\\\\frac{\\partial \\mathcal {R}}{\\partial r} &=& 4r^3+2(a^2-\\xi _c^2-\\eta _c)r+2M[\\eta _c+(\\xi _c-a)^2] = 0 \\, .", "$ In the Schwarzschild background ($a=0$ ), the BH shadow is a circle of radius $R = 3 \\sqrt{3} \\, M \\approx 5.196 \\, M \\, .$ If $a\\ne 0$ , one finds $\\xi _c &=& \\frac{M(r_*^2-a^2)-r_*(r_*^2-2Mr_*+a^2)}{a(r_*-M)} \\, , \\nonumber \\\\\\eta _c &=& \\frac{r_*^3 [4a^2M-r_*(r_*-3M)^2]}{a^2(r_*-M)^2} \\, ,$ where $r_*$ is the radius of the unstable orbit.", "The shadow of Kerr BHs can be found in many papers in the literature [49].", "In the present work, we want to figure out how we can distinguish Kerr and Bardeen BHs from the observation of the BH shadow.", "In Boyer-Lindquist coordinates, the line element of the rotating Bardeen metric has the same form as the Kerr one, Eqs.", "(REF ) and (REF ), with the mass $M$ replaced by $m$  [51], [52]: $M \\rightarrow m = M \\left(\\frac{r^2}{r^2 + g^2}\\right)^{3/2} \\, , $ without changing $a$ (at least in the simplest form, see Ref.", "[52] for more details).", "Here $g$ can be interpreted as the magnetic charge of a non-linear electromagnetic fieldAs found in Ref.", "[51], the Bardeen metric can be obtained as an exact solution of Einstein's equations coupled to a non-linear electromagnetic field.", "The latter allows to avoid the no-hair theorem.", "or just as a quantity introducing a deviation from the Kerr solution.", "The position of the even horizon is still given by the largest root of $\\Delta = 0$ and there is a bound on the maximum value of the spin parameter, above which there are no BHs.", "The maximum value of $a/M$ is 1 for $g/M = 0$ (Kerr case), and decreases as $g/M$ increases, to reach 0 for $g/M = \\sqrt{16/27} \\approx 0.7698$ .", "There are no BHs for $g/M > \\sqrt{16/27}$ .", "The situation is similar to the Kerr-Newman solution, where the maximum value of $a/M$ is 1 for a vanishing electric charge $Q$ (Kerr case), and decreases as $Q$ increases, to reach 0 for $Q/M = 1$ .", "The Bardeen metric can be seen as the prototype of a large class of metrics, in which the line element in Boyer-Lindquist coordinates has the same form as the Kerr one, with $M$ replaced by a function $m(r)$ that depends only on the radial coordinate.", "Such a family of metrics includes the Kerr-Newman solution, in which $m = M - Q^2/2r$ .", "Since the Bardeen metric (as well as all the other metrics in this family) has the same nice properties as the Kerr one, in particular there exists the Carter constant $\\mathcal {Q}$ and the equations of motion are separable in Boyer-Lindquist coordinates, it is straightforward to generalize the above calculations of the shadow of a Kerr BH to the Bardeen case.", "The system in Eq.", "(REF ) is replaced by $\\mathcal {R} &=& r^4+(a^2-\\xi _c^2-\\eta _c)r^2+2m[\\eta _c+(\\xi _c-a)^2]r-a^2\\eta _c = 0 \\, , \\nonumber \\\\\\frac{\\partial \\mathcal {R}}{\\partial r} &=& 4r^3+2(a^2-\\xi _c^2-\\eta _c)r+2m[\\eta _c+(\\xi _c-a)^2] f = 0 \\, ,$ with $m$ given by Eq.", "(REF ) and $f$ defined by $f = 1 + \\frac{r}{m} \\frac{dm}{dr} = \\frac{r^2 + 4 g^2}{r^2 + g^2} \\, .$ The counterpart of Eq.", "(REF ) is $\\xi _c &=& \\frac{m[(2 - f)r_*^2-fa^2]-r_*(r_*^2-2mr_*+a^2)}{a(r_*-fm)} \\, , \\nonumber \\\\\\eta _c &=& \\frac{r_*^3 \\lbrace 4(2-f)a^2m-r_*[r_*-(4 - f)m]^2\\rbrace }{a^2(r_*- fm)^2} \\, ,$ with $m = m(r_*)$ and $f=f(r_*)$ .", "Example of shadows of Bardeen BHs are reported in Ref. [48].", "Let us note that Eq.", "(REF ) does not hold only for the Bardeen metric, but in the large class of BH solutions in which the line element is given by Eqs.", "(REF ) and (REF ) with $M$ replaced by some $m(r)$ that depend on the radial coordinate only.", "Figure: BH shadow with the three parameters that approximately characterizeits shape: the radius RR, the dent DD, and the distance SS.", "RR is defined as theradius of the circle passing through the three red points, located at the top(β=β max \\beta = \\beta _{\\rm max}), bottom,and most left end of the shadow.", "DD is the difference between the most right pointsof the circle and of the shadow.", "SS is the distance between the center of the circle,CC, and the most right end of the shadow at β=β max /2\\beta = \\beta _{\\rm max}/2.", "TheHioki-Maeda distortion parameter is δ=D/R\\delta = D/R.", "The second distortion parameteris ϵ=S/R\\epsilon = S/R.", "α\\alpha and β\\beta in units M=1M=1.", "See the text for more details." ], [ "Measuring the Kerr spin parameter from the observation of the shadow ", "The boundary of the BH shadow corresponds to the apparent image of the photon capture sphere as seen by the distant observer and it is only determined by the background geometry.", "If we want to infer the value of the parameters of the metric, it is convenient to figure out the features of the shadow that better characterize its shape and that can be measured from the shadow image.", "This strategy was first proposed in [49] to estimate the value of the spin parameter $a/M$ from the observation of the shadow of a Kerr BH.", "In this section, we will briefly review their approach, while in the next section it will be extended to test the Kerr metric of BH candidates.", "At first approximation, the boundary of the shadow is a circle, and it is exactly a circle in the case of a static spherically symmetric solution (like the Schwarzschild metric) or in the one of a stationary axisymmetric solution in which the distant observer is located along the symmetry axis (like the Kerr metric and a viewing angle $i=0^\\circ $ or $180^\\circ $ ).", "We can thus approximate the shadow with a circle passing through the three points located, respectively, at the top position, bottom position, and most left end of its boundary (the three red points in Fig.", "REF ).", "The radius of the shadow, $R$ , is defined as the radius of this circle.", "The first order correction to the circle is due to the spin, because of the spin-orbit coupling between the photon and the BH.", "The gravitational force is indeed stronger if the photon angular momentum is antiparallel to the BH spin (the photon capture radius is thus larger), and weaker in the opposite case (the photon capture radius is smaller).", "The result is that the shadow has a dent on one side, which is larger for a viewing angle $i = 90^\\circ $ , and reduces as the distant observer move to the axis of symmetry, to completely disappear when $i=0^\\circ $ or $180^\\circ $ .", "We define the dent $D$ as the distance between the right endpoints of the circle and of the shadow, see Fig.", "REF .", "We can then introduce the Hioki-Maeda distortion parameter $\\delta = D/R$ , which is a quantity that can be measured from the image of the shadow and can be used to characterize its shape [49].", "Figure: Hioki-Maeda distortion parameter δ\\delta as a function of the spinparameter a/Ma/M for Kerr BHs (red solid line), Bardeen BHs with g/M=0.3g/M=0.3(green dashed line), and Bardeen BHs with g/M=0.6g/M=0.6 (blue dotted line).The inclination angle is i=90 ∘ i = 90^\\circ (left panel) and 45 ∘ 45^\\circ (right panel).The maximum value of the Hioki-Maeda distortion parameter isδ=7/26≈0.269\\delta = 7/26 \\approx 0.269 in the case of an extremal Kerr BH (a/M=1a/M = 1)and a viewing angle i=90 ∘ i = 90^\\circ .If we know that the spacetime geometry is described by the Kerr solution and we have an independent estimate of the viewing angle $i$ , the measurement of the distortion parameter provides an estimate of the BH spin parameter $a/M$ .", "Indeed, for a give $i$ , there is a one-to-one correspondence between $a/M$ and $\\delta $ , and the function $\\delta (a/M)$ can be inverted to obtain $a/M|^{\\rm Kerr}(\\delta )$ .", "If we relax the Kerr BH hypothesis and we want to test the nature of the BH candidate, we have to introduce at least one parameter that quantifies possible deviations from the Kerr geometry.", "If we adopt the Bardeen metric, this role is played by the Bardeen charge $g$ .", "Now the boundary of the shadow, as well as all the other properties of the background metric at small radii, depends on both $a/M$ and $g/M$ .", "The distortion parameter is $\\delta (a/M,g/M)$ and it is not possible to infer $a/M$ without an independent estimate of $g/M$ .", "If we assume that the object is a Kerr BH even if $g/M \\ne 0$ , we can determine the so-called Kerr spin parameter $a/M|^{\\rm Kerr} = a/M|^{\\rm Kerr} [ \\delta (a/M,g/M) ] \\, ,$ which provides a wrong value of the spin for $a/M \\ne 0$  [48].", "Fig.", "REF shows the Hioki-Maeda distortion parameter $\\delta $ as a function of the BH spin parameter $a/M$ for the case of Kerr BHs and Bardeen BHs with $g/M = 0.3$ and $0.6$ .", "It is clear that the same distortion parameter $\\delta $ can characterize the shadow of a Kerr BH with spin $a/M$ or of a Bardeen BH with lower spin.", "Figure: Left panel: shadows of a Kerr BH (blue dashed line) and of a BardeenBH with g/M=0.6g/M = 0.6 (red solid line) with the same mass M=1M = 1, the sameviewing angle i=90 ∘ i = 90^\\circ , and the same Hioki-Maeda distortion parameterδ=0.1500\\delta = 0.1500.", "The Kerr BH has a/M=0.9189a/M = 0.9189, while the Bardeen BH hasa/M=0.5286a/M = 0.5286.", "Right panel: as in the left panel, but for two BHs with the sameshadow radius RR on the sky (M=0.9311M = 0.9311 for the Kerr BH, M=1M=1 for theBardeen one) and the same position of the center of the circle CC (the shadowof the Kerr BH has been shifted by 0.433 along the α\\alpha direction).", "See thetext for more details." ], [ "Measuring the spin and the deformation parameters from the observation of the shadow ", "Since the Hioki-Maeda distortion parameter $\\delta $ is degenerate with respect to the BH spin and possible deviations from the Kerr solution (and it could not be otherwise, because one parameter of the shadow can at most be used to determine one parameter of the background geometry), in this section we look for the best choice of the second parameter to test the Kerr metric.", "If we consider non-rotating BHs, the shadow is a circle for any value of $g/M$ and the sole difference is its radius, which depends on $g/M$ .", "For $a/M \\ne 0$ , we can expect that shadows with the same Hioki-Maeda distortion parameter have different shape and that the difference increases as $g/M$ and $a/M$ increase and $i$ approaches $90^\\circ $ .", "As first step, it is useful to visualize such a difference.", "This is done in Fig.", "REF , where we compare the shadows of a Kerr BH and of a Bardeen BH with $g/M = 0.6$ for $i = 90^\\circ $ .", "We start from imposing that $\\delta = 0.1500$ .", "We find that such a distortion parameter corresponds to a Kerr BH with $a/M = 0.9189$ and to a $g/M = 0.6$ Bardeen BH with $a/M = 0.5286$ .", "The left panel shows the two shadows as computed using the same BH mass $M$ .", "The right panel compares the two shadows after properly rescaling the one of the Kerr BH and shifting it on the celestial plane, so that their radii have the same value and the centers of the shadows coincide.", "We note that the maximum value of the spin for a Bardeen BH with $g/M = 0.6$ is $a/M \\approx 0.5295$ , so we are considering an almost extremal BH.", "Figure: Distortion parameter ϵ\\epsilon as a function of the spinparameter a/Ma/M for Kerr BHs (red solid line), Bardeen BHs with g/M=0.3g/M=0.3(green dashed line), and Bardeen BHs with g/M=0.6g/M=0.6 (blue dotted line).The inclination angle is i=90 ∘ i = 90^\\circ (left panel) and 45 ∘ 45^\\circ (right panel)." ], [ "Distortion parameter $\\epsilon $", "From the right panel in Fig.", "REF , we can realize that the main difference in the shadow shape is in the apparent photon capture radii on the right side, corresponding to the ones associated to corotating orbits.", "We are thus tempted to defined the distortion parameter $\\epsilon $ as follows.", "With reference to Fig.", "REF , we call $S$ the distance between the center of the circle of the shadow, $C$ , and the point on the right side of the boundary with coordinate $\\beta = \\beta _{\\rm max}/2$ , where $\\beta _{\\rm max}$ is the $\\beta $ coordinate of the top end of the shadow used to find $R$ .", "We then define $\\epsilon = S/R$ which, like the Hioki-Maeda distortion parameter $\\delta $ , only depends on the shape of the shadow.", "The distortion parameter $\\epsilon $ as a function of the spin parameter $a/M$ for Kerr BHs, Bardeen BHs with $g/M=0.3$ , and Bardeen BHs with $g/M=0.6$ is shown in Fig.", "REF .", "Figure: Position of the center of the circle on the sky as a function of the spinparameter a/Ma/M for Kerr BHs (red solid line), Bardeen BHs with g/M=0.3g/M=0.3(green dashed line), and Bardeen BHs with g/M=0.6g/M=0.6 (blue dotted line).The inclination angle is i=90 ∘ i = 90^\\circ (left panel) and 45 ∘ 45^\\circ (right panel)." ], [ "Position of the center of the shadow", "The shapes of the shadows of Kerr and Bardeen BHs are clearly very similar and therefore only very accurate image can distinguish the two metrics and provide a meaningful constraint on $g/M$ .", "One can therefore try to follow a different strategy from the exact determination of the shadow shape.", "The left panel in Fig.", "REF may suggest that this could be achieved from the estimate of the exact position on the sky of the center of the circle, $C$ , with respect to the actual center of the system, $\\alpha = \\beta = 0$ .", "In principle, that is surely an available option and Fig.", "REF shows $\\alpha _{\\rm c}/R$ as a function of the spin parameter for the shadows of Kerr and Bardeen BHs.", "We have plotted $\\alpha _{\\rm c}/R$ instead of $\\alpha _{\\rm c}$ or $\\alpha _{\\rm c}/M$ because in this way we do not assume an accurate measurement of the BH mass and distance.", "We just need a very good measurement of the BH position on the sky.", "Figure: Radius of the shadow R/MR/M as a function of the spinparameter a/Ma/M for Kerr BHs (red solid line), Bardeen BHs with g/M=0.3g/M=0.3(green dashed line), and Bardeen BHs with a/M=0.6a/M=0.6 (blue dotted line).The inclination angle is i=90 ∘ i = 90^\\circ (left panel) and 45 ∘ 45^\\circ (right panel)." ], [ "Radius of the shadow $R$", "Lastly, we consider the possibility that we can get very good estimates of the BH mass and distance and we can therefore combine the Hioki-Maeda distortion parameter $\\delta $ with the measurement of the radius of the shadow $R$ .", "$R/M$ as a function of the spin parameter $a/M$ for $g/M = 0.0$ (Kerr), 0.3, and 0.6 is shown in Fig.", "REF .", "It is remarkable that for an observer near the equatorial plane the value of $R/M$ is mainly determined by $g/M$ and it is not very sensitive to the spin.", "The dependence of $R$ on the spin increases as the observer moves towards the axis of symmetry, but it is still weak for $i = 45^\\circ $ , as shown the right panel in Fig.", "REF .", "Figure: The Hioki-Maeda distortion parameter δ\\delta against the distortionparameter ϵ\\epsilon for Kerr BHs (red solid line), Bardeen BHs with g/M=0.3g/M=0.3(green dashed line), and Bardeen BHs with a/M=0.6a/M=0.6 (blue dotted line).The inclination angle is i=90 ∘ i = 90^\\circ (left panel) and 45 ∘ 45^\\circ (right panel).Figure: The Hioki-Maeda distortion parameter δ\\delta against the position ofthe center of the circle on the sky for Kerr BHs (red solid line), Bardeen BHs withg/M=0.3g/M=0.3 (green dashed line), and Bardeen BHs with a/M=0.6a/M=0.6 (blue dotted line).The inclination angle is i=90 ∘ i = 90^\\circ (left panel) and 45 ∘ 45^\\circ (right panel).Figure: The Hioki-Maeda distortion parameter δ\\delta against the radius ofthe shadow R/MR/M for Kerr BHs (red solid line), Bardeen BHs with g/M=0.3g/M=0.3(green dashed line), and Bardeen BHs with a/M=0.6a/M=0.6 (blue dotted line).The inclination angle is i=90 ∘ i = 90^\\circ (left panel) and 45 ∘ 45^\\circ (right panel).Figure: The radius of the shadow R/MR/M as a function of the Bardeen chargeg/Mg/M for different values of the spin parameter and an observer inclinationangle i=90 ∘ i = 90^\\circ (top left panel), 45 ∘ 45^\\circ (top right panel), and10 ∘ 10^\\circ (bottom panel)." ], [ "Discussion ", "At least in principle, the simultaneous measurement of the Hioki-Maeda distortion parameter $\\delta $ and one of the three parameters discussed in the previous section ($\\epsilon $ , $\\alpha _{\\rm c}$ , or $R$ ) breaks the degeneracy between the BH spin and possible deviations from the Kerr geometry and it is therefore a potential approach to test the Kerr metric around SgrA$^*$ from the observation of its shadow.", "This is more evident if we plot the Hioki-Maeda distortion parameter against one of the other three.", "The plots are shown in Fig.", "REF for the distortion parameter $\\epsilon $ , in Fig.", "REF for the shadow center $\\alpha _{\\rm c}/M$ , and in Fig.", "REF for the shadow radius $R/M$ .", "The fact that these curves depend on the value of $g/M$ is enough to conclude that there is no degeneracy.", "However, the above consideration is correct only in principle, in the sense that we have also to check if it is possible to measure $\\epsilon $ , $\\alpha _{\\rm c}$ , or $R$ with sufficient precision to distinguish Kerr and Bardeen BHs.", "In the case of the distortion parameter $\\epsilon $ , the difference between Kerr and Bardeen BHs is clearly tiny, as it was already evident from Fig.", "REF .", "Even in the most optimistic case of an inclination angle $i = 90^\\circ $ , for the same Hioki-Maeda distortion parameter $\\delta $ the difference between the parameter $\\epsilon $ for Kerr BHs and Bardeen BHs with $g/M = 0.3$ is never larger than 0.8%, lower 0.2% for $\\delta < 0.15$ , and lower than 0.08% for $\\delta < 0.10$ .", "If we compare Kerr BHs and Bardeen BHs with $g/M = 0.6$ , we find that shadows with the same $\\delta $ have $\\epsilon $ parameters that differ less than 1.5% (but it is close to 1.5% only when the Bardeen BH is an almost extremal object).", "Such precisions are at least extremely challenging, even in the most favorable conditions for $a/M$ , $g/M$ , and $i$ .", "The uncertainties on $\\delta $ and $\\epsilon $ are $\\frac{\\Delta \\delta }{\\delta } = \\frac{\\Delta R}{R} + \\frac{\\Delta D}{D} \\, , \\quad \\frac{\\Delta \\epsilon }{\\epsilon } = \\frac{\\Delta R}{R} + \\frac{\\Delta S}{S} \\, ,$ where $\\Delta R$ , $\\Delta D$ , and $\\Delta S$ are the uncertainties on $R$ , $D$ , and $S$ .", "For SgrA$^*$ , we expect $R \\approx 30$  $\\mu $ as.", "With an imaging resolution of $\\sim 0.3$  $\\mu $ as, $\\Delta \\epsilon /\\epsilon $ is already 2% or worst, which is not enough even to distinguish a Kerr from a Bardeen BH with $g/M = 0.6$ .", "Here we are also assuming to know the inclination angle $i$ with arbitrary precision, which is surely not the case and its uncertainty introduces an additional error in the estimate of $\\epsilon $ .", "The fact that the shapes of the shadows in Kerr and non-Kerr spacetimes are usually very similar, and therefore difficult to distinguish with observations, was already stressed in Ref.", "[39] from the comparison of Kerr and Tomimatsu-Sato shadows.", "Other non-Kerr metrics might have shadows with more significant deviations, but more often the difference seems to be very small.", "The measurement of the shift between the position of the center of the shadow and the actual center of the system may initially look more promising, because Fig.", "REF shows that the lines for different values of $g/M$ have a large separation.", "Unfortunately, the position of the center of the system on the sky is very difficult to determine and the uncertainty is $\\Delta \\alpha _{\\rm c} \\sim 1$  mas (but see Ref.", "[53], where a position relative to a reference point with an accuracy of order 1 $\\mu $ as might be possible).", "The last parameter of the shadow discussed in the previous section is the apparent size of the radius $R$ (see Figures REF and REF ).", "In this case, we need very good measurement of the BH mass and distance from us.", "At present, these quantities are difficult to measure and the final uncertainty on the expect apparent size on the sky of $R$ is around 15% [45].", "Such an uncertainty is larger than the difference between the radius of the shadow of a Kerr BH and of a Bardeen BH with $g/M = 0.6$ with the same $\\delta $ , which is around 7% for $i = 90^\\circ $ and independently of the spin parameter $a/M$ .", "The measurement of the radius of the shadow may turn out to be the most promising approach in the case of significant improvements of the estimate of the BH mass and of our distance from the galactic center.", "That could be achieved in the case of the discovery of a radio pulsar in a compact orbit (i.e.", "with an orbital periods of a few months) around SgrA$^*$ .", "As discussed in Ref.", "[54], pulsar timing can determine the Keplerian and Post-Keplerian parameters and therefore get a robust estimate of the BH mass, independently of the distance from us from the galactic center, which can instead be inferred with high precision by combining the BH mass with near-infrared astrometric measurements.", "In this case, the uncertainty on $R$ would be determined by the imaging resolution.", "If the resolution is at the level of 0.3 $\\mu $ as, $R$ can be measured with a precision of 1%.", "If SgrA$^*$ is rotating rapidly, after a few years of observations of the radio pulsar, the spin parameter $a/M$ could be determined with a precision of order 0.1% (but the uncertainty would be significantly larger for a mid-rotating or slow-rotating BH) [54].", "Such a measurement could also be combined with the ones of $\\delta $ and $R$ on the spin-Bardeen charge plane.", "Since the pulsar is relatively far from the BH, the pulsar measurement of the frame dragging is really sensitive to the value of $a/M$ , independently of the nature if the SgrA$^*$ , because possible deviations from the Kerr solution are suppressed by powers in $M/r \\ll 1$ , where $r$ is the distance of the pulsarWhile pulsar timing can also measure the BH quadrupole moment and thus test the Kerr nature of SgrA$^*$ (at least in the case of a fast-rotating BH) [54], the combination of the measurement of the shadow and of the pulsar can test if there are deviations of higher order..", "The possible constraints from the simultaneous measurements of the Hioki-Maeda distortion parameter $\\delta $ , the shadow radius $R/M$ , and the spin parameter $a/M$ from a radio pulsar in the case of a Kerr BH with $a/M = 0.7$ are shown in Fig.", "REF .", "The left panel is for an observer's viewing angle $i=90^\\circ $ , while the right panel for $i = 45^\\circ $ .", "For a Kerr BH with $a/M = 0.7$ , the Hioki-Maeda distortion parameter is $\\delta = 0.0668$ ($i=90^\\circ $ ) and $0.0371$ ($i=45^\\circ $ ).", "The red dashed-dotted line in Fig.", "REF indicates the objects with the same Hioki-Maeda distortion parameter, and the two blue dashed lines on the two sides are the boundary of the allowed region assuming an uncertainty on $\\delta $ of 20%In the case $i = 90^\\circ $ , the plot shows just one blue dashed line for $R/M$ because there are no BHs with $R/M$ 1% larger than the one of Kerr BHs..", "In the same way, the shadow radius for a similar BH is $R/M = 5.20$ ($i=90^\\circ $ ) and $R/M=5.12$ ($i=45^\\circ $ ), the red dashed-dotted line is the central value, while the two blue dashed lines are the boundary of the allowed region assuming an uncertainty on $R/M$ of 1%.", "In the case of the spin parameter $a/M$ inferred from a radio pulsar, the uncertainty is assumed to be 1%.", "Figure: Hypothetical constraints from the measurements of the Hioki-Maedadistortion parameter δ\\delta determined with a precision of 20%, of the shadowradius R/MR/M determined with a precision of 1%, and of the spin parameter a/Ma/Minferred from the orbital motion of a pulsar in a compact orbit and determinedwith a precision of 1%, assuming that the object is a Kerr BH with a/M=0.7a/M = 0.7and the inclination angle is i=90 ∘ i = 90^\\circ (left panel, in this case δ=0.0668\\delta = 0.0668and R/M=5.20R/M = 5.20) and 45 ∘ 45^\\circ (right panel, in this case δ=0.0371\\delta = 0.0371and R/M=5.12R/M = 5.12).", "The red dashed-dotted lines are the central values of themeasurements, while the blue dashed curves correspond to their uncertainties.The gray area is the region of objects without event horizon and can be ignored." ], [ "Summary and conclusions ", "Within the next decade, VLBI facilities at mm/sub-mm wavelength will be able to directly image the accretion flow around SgrA$^*$ , the super-massive BH candidate at the center of our Galaxy, and open a new window to test gravity in the strong field regime.", "In particular, it will be possible to obseve the BH “shadow”, whose boundary corresponds to the apparent image of the photon capture sphere and it is therefore determined by the spacetime geometry around the compact object.", "At first approximation, the shadow of a BH is a circle, and its radius depend on the BH mass, distance, and also on the background metric.", "The first order correction to the circle is due to the BH spin, because the photon capture radius is larger for photons with angular momentum antiparallel to the BH spin (the gravitational force is stronger), and smaller in the opposite case (the gravitational force is weaker).", "The final result is that the shadow shows a dent on one side.", "The magnitude of this dent can be measured in terms of the Hioki-Maeda distortion parameter $\\delta $  [49].", "The measurement of $\\delta $ can be used to infer one parameter of the background geometry.", "If the compact object is a Kerr BH and we have an independent estimate of the inclination angle $i$ , $\\delta $ depends only on the BH spin parameter, and therefore its measurement can be used to infer $a/M$ .", "If we want to test the Kerr BH hypothesis, we need to measure another parameter of the shadow in order to break the degeneracy between the spin and possible deviations from the Kerr solution.", "In this work, we have focused the attention on the Bardeen metric, which is characterized by the Bardeen charge $g$ and reduces to the Kerr solution for $g=0$ .", "The Bardeen solution can be seen as the prototype of a large class of non-Kerr BH metrics, in which the line element in Boyer-Lindquist coordinates is the same as the Kerr case with the mass $M$ replaced by a function of the radial coordinate $m$ , which reduces to $M$ at large radii.", "We have investigated how it is possible to constrain the value of $g/M$ from the observation of the BH shadow.", "We have explored three possibilities: the introduction of another distortion parameter of the shadow, $\\epsilon $ , the determination of the center of the shadow with respect to the actual position of the BH, and the estimate of the shadow radius.", "While all the three approach are at least challenging, the third one may be the most promising in the case of significant improvements of the measurement of the mass of SgrA$^*$ and of our distance from the galactic center.", "Since that can be achieved by discovering a radio pulsar with an orbital period of a few months around SgrA$^*$ , we have also briefly discussed the combination of the measurements of the shadow and of the orbital motion of a similar pulsar.", "Such a synergy could turn out a quite interesting tool to test the nature of the super-massive BH candidate at the center of our Galaxy, because the shadow is sensitive to the strong gravitational field very close to the compact object, while the pulsar can accurately probe the weak field at relatively large distances.", "While our calculations have been done in the specific case of the Bardeen background, we stress that it is straightforward to repeat our study for any BH spacetime in which there is the Carter constant and the photon equations of motion are separable.", "The main conclusion of this work are valid even for those BHs.", "We thank Tomohiro Harada for useful comments and suggestions.", "This work was supported by the NSFC grant No.", "11305038, the Shanghai Municipal Education Commission grant for Innovative Programs No.", "14ZZ001, the Thousand Young Talents Program, and Fudan University." ] ]
1403.0371
[ [ "Synchronization in populations of sparsely connected pulse-coupled\n oscillators" ], [ "Abstract We propose a population model for $\\delta$-pulse-coupled oscillators with sparse connectivity.", "The model is given as an evolution equation for the phase density which take the form of a partial differential equation with a non-local term.", "We discuss the existence and stability of stationary solutions and exemplify our approach for integrate-and-fire-like oscillators.", "While for strong couplings, the firing rate of stationary solutions diverges and solutions disappear, small couplings allow for partially synchronous states which emerge at a supercritical Andronov-Hopf bifurcation." ], [ "Synchronization in populations of sparsely connected pulse-coupled oscillators A. Rothkegel A. Rothkegel1,2,3 K. Lehnertz1,2,3 A. Rothkegel & K. Lehnertz 1Department of Epileptology, University of Bonn, Germany 2Helmholtz Institute for Radiation and Nuclear Physics, University of Bonn, Germany 3Interdisciplinary Center for Complex Systems, University of Bonn, Germany 05.45.XtSynchronization; coupled oscillators 89.75.-kComplex systems 84.35.+iNeural networks We propose a population model for $\\delta $ -pulse-coupled oscillators with sparse connectivity.", "The model is given as an evolution equation for the phase density which take the form of a partial differential equation with a non-local term.", "We discuss the existence and stability of stationary solutions and exemplify our approach for integrate-and-fire-like oscillators.", "While for strong couplings, the firing rate of stationary solutions diverges and solutions disappear, small couplings allow for partially synchronous states which emerge at a supercritical Andronov-Hopf bifurcation.", "The collective dynamics of interacting oscillatory systems has been studied in many different contexts in the natural and life sciences [1], [2], [3], [4].", "In the thermodynamic limit, evolution equations for the population density proved to be a useful description [5], [6], [7], in particular to characterize the stability of synchronous and asynchronous states (see, e.g., [8], [9], [10], [11], [12], [13], [14], [15], [16], [17]).", "Usually, dense or all-to-all-coupled networks are considered for these descriptions.", "Motivated by natural systems in which constituents interact with few others only, investigations of complex networks have revealed a large influence of the degree and sparseness of connectivity on network dynamics [18], [19], [20], [21], [22], [23], [24], [25].", "Especially when the knowledge about the connection structure is limited, it suggests itself to assume random connections (as in Erdős-Rényi networks) or random interactions (where excitations are assigned randomly to target oscillators [6], [26], [27], [28]).", "Both approaches often yield comparable dynamics (e.g.", "[29], [30]) whereas random interactions represents a substantial simplification from a mathematical point of view, allowing one to describe the networks in terms of evolution equations for the phase density.", "These equations are usually posed as starting point for the commonly applied mean- or the fluctuation-driven limits.", "However, rarely are they studied in full although it can be expected that sparseness largely influences the collective dynamics as has been discussed for excitable systems [26].", "In this Letter, we propose a population model of $\\delta $ -pulse coupled oscillators with sparse connectivity, derive the governing equations from a general definition of the density flux, and characterize existence and uniqueness of stationary solutions.", "For integrate-and-fire-like oscillators, the latter may either disappear with diverging firing rate or lose stability at a supercritical Andronov-Hopf bifurcation (AHB).", "This is in contrast to the global convergence to complete synchrony for all-to-all coupling that has been shown for finite [8] and for infinite [31] number of oscillators.", "Consider a population of oscillators $n \\in N$ with cyclic phases $\\phi _n(t) \\in [ 0,1 )$ and intrinsic dynamics $\\dot{\\phi }_n(t) = 1$ .", "If for some $t_f$ and some oscillator $n$ the phase reaches 1, the oscillator fires and we introduce a phase jump in all oscillators $n^{\\prime }$ with probability $p= m/N$ [32], [33].", "Here, $m$ is the number of recurrent connections per oscillator.", "The height of the phase jump is defined by the phase response curve $\\Delta (\\phi )$ (PRC) (or equivalently by the phase transition curve $R(\\phi )$ ): $\\phi _{n^{\\prime }}(t_f^+) = \\phi _{n^{\\prime }}(t_f) + \\Delta \\left(\\phi _{n^{\\prime }}(t_f)\\right) = R( \\phi _{n^{\\prime }} ( t_f)).$ The model can be interpreted as an all-to-all coupled network in which connections are not reliable and mediate interactions between oscillators only with a small probability ($p$ ).", "It can also be interpreted as an approximation to an Erdős-Rényi network in which the quenched disorder, imposed by its construction, is replaced by a dynamic coupling structure which takes the form of an ongoing random influence.", "For the limit of large sparse networks ($N \\rightarrow \\infty , m = \\mbox{const.", "}$ ), we represent the network dynamics by a continuity equation for the phase density $\\rho (\\phi ,t)$ $\\partial _t \\rho (\\phi ,t) + \\partial _\\phi J(\\phi ,t) = 0$ with $\\rho (\\phi ,t) \\ge 0$ and $\\int _0^1 \\rho (\\phi ,t) d\\phi = 1$ .", "We assume the probability flux $J(\\phi ,t)$ to be continuous and define both $\\rho $ and $J$ at phases $\\phi \\in [0,1)$ .", "Evaluations at $\\phi = 1$ are meant as left-sided limits towards $\\phi = 1$ .", "$J(1,t)$ is the firing rate.", "Every oscillator is subject to Poisson excitations $\\eta _{\\lambda (t)}$ with inhomogeneous rate $\\lambda (t) = m J(1,t)$ and we can describe its phase variable by the stochastic differential equation $\\partial _t \\phi (t) = 1 + \\eta _{\\lambda (t)}$ .", "To shorten our notation, we will omit in the following the time $t$ as argument of $\\rho $ , $\\lambda $ , and $J$ .", "As we expect $R(\\phi )$ to be non-invertible and to map intervals to a single phase, we have to take care in which way $\\rho $ and $J$ are interpreted at these phases.", "Given some distribution of oscillators phases, we consider $\\rho (\\phi ,t) d\\phi $ as the fraction of oscillators which are contained in a small interval whose left boundary is fixed to $\\phi $ .", "With this definition, $\\rho (\\phi ,t)$ is continuous for right-sided limits and the corresponding $J(\\phi ,t)$ is defined by the oscillators which pass an imaginary boundary which is infinitely close to $\\phi $ and right to $\\phi $ .", "The flux can be formalized in the following way: $J(\\phi ) = \\rho (\\phi ) + \\lambda \\left(\\int \\limits _{I_>(\\phi )} \\rho (\\tilde{\\phi }) d\\tilde{\\phi } - \\int \\limits _{I_\\le (\\phi )} \\rho (\\tilde{\\phi }) d\\tilde{\\phi } \\right),$ where $I_>(\\phi ) := \\lbrace \\tilde{\\phi } < \\phi | R(\\tilde{\\phi }) > \\phi \\rbrace $ is the set of phases smaller than $\\phi $ which is mapped by $R(\\phi )$ to a phase larger than $\\phi $ , and $I_\\le (\\phi ) := \\lbrace \\tilde{\\phi } > \\phi | R(\\tilde{\\phi }) \\le \\phi \\rbrace $ is defined analogously (the order relations in in these formulas are interpreted for unwrapped phases).", "The first term of the r.h.s.", "of (REF ) represents convection due to the intrinsic dynamics of oscillators.", "The integrals represent the fractions of oscillators which are moved across phase $\\phi $ by an excitation, either to smaller or larger values (cf.", "(REF )).", "PRCs which are derived from limit cycle oscillators by phase reduction usually have invertible phase transition curves [34].", "However, (REF ) even holds if $R(\\phi )$ is not invertible and has no or uncountably many inverse images.", "For phases $\\phi $ at which $R(\\phi )$ has at most countably many inverse images, we can represent the sets $I_>(\\phi )$ and $I_\\le (\\phi )$ by a product of two Heaviside functions and derive, differentiating the latter to $\\delta $ -functions, the following expression: $\\partial _\\phi J(\\phi ) = \\partial _\\phi \\rho (\\phi ) + \\lambda \\int _0^1 \\rho (\\tilde{\\phi })\\left( \\delta ( \\phi - \\tilde{\\phi } )- \\delta (\\phi - R(\\tilde{\\phi }) \\right) d\\tilde{\\phi }.$ Denoting with $(R_i^{-1}(\\phi ) | i \\in I)$ an enumeration of the inverse images of $R(\\phi )$ at phase $\\phi $ for an appropriate index set $I$ , the continuity equation (REF ) reads: $\\partial _t \\rho (\\phi ) = - \\partial _\\phi \\rho (\\phi ) - \\lambda \\rho (\\phi ) + \\lambda \\sum _{i \\in I} \\frac{\\rho (R_i^{-1}(\\phi )) }{R^{\\prime }(R^{-1}(\\phi ))}.$ For uncountably many inverse images of some phase $\\varphi $ , they will be contained in $I_>(\\varphi )$ or $I_\\le (\\varphi )$ but not in $I_>(\\varphi ^-)$ and $I_\\le (\\varphi ^-)$ .", "In this case, we obtain a discontinuity between $\\rho (\\varphi ^-)$ and $\\rho (\\varphi )$ which can be expressed by requiring continuity of the flux for left-sided limits at $\\phi = \\varphi $ ($J(\\varphi ^-) = J(\\varphi )$ ).", "Note that the definition in (REF ) automatically ensures continuity for right-sided limits.", "Setting $\\phi = 1$ in (REF ), we obtain the following relationship for the excitation rate $\\lambda = m J(1)$ $\\lambda = m \\rho (1) /\\left( 1 - m \\int \\limits _{I_>(1)} \\rho (\\tilde{\\phi }) d\\tilde{\\phi } + m \\int \\limits _{I_\\le (1)} \\rho (\\tilde{\\phi })d \\tilde{\\phi } \\right).$ Given a PRC and the number of recurrent connections $m$ , the population model for oscillators with sparse connectivity is given by (REF ), (REF ), and by the requirement that $\\rho (\\phi )$ is normalized to allow for an interpretation as probability density function.", "The integral over $I_\\le ( 1)$ in (REF ) corresponds to oscillators which pass the firing threshold in the wrong direction.", "Usually, it is not desirable that such oscillators decrease the firing rate, which can be prevented by requiring the PRC to be bounded by $-\\phi $ from below.", "Note that the excitation rate as defined in (REF ) may diverge or turn negative.", "In these cases every firing oscillator will make, on average, at least one other oscillator fire immediately and a macroscopic amount of oscillators fires in an instant.", "We will refer to this situation as an avalanche.", "Clearly, a numerical integration via some finite difference scheme will break down at this point [35], [36].", "Nevertheless, Monte-Carlo simulations may still be meaningful.", "Let us briefly consider the mean-driven limit, i.e., a sequence of PRCs indexed by $i$ and parameters $m_i$ such that $\\Delta _i(\\phi )$ vanishes as $i \\rightarrow \\infty $ and the product $\\Delta _i(\\phi ) m_i$ converges point-wise to some function $Z(\\phi )$ .", "The flux $J_i(\\phi )$ is then straightforwardly approximated by $J_i(\\phi ) \\rightarrow \\rho (\\phi ) \\left( 1 + J(1) Z(\\phi ) \\right)$ as $i \\rightarrow \\infty $ .", "Setting $\\phi = 1$ in (REF ) gives the expression for the firing rate $\\nu = \\rho (1) / \\left(1 - Z(1) \\rho (1)\\right)$ of the non-linear evolution equation $\\partial _t \\rho (\\phi ) = - \\partial _\\phi \\left[ \\left(1 + \\frac{\\rho (1)}{1 - Z(1)\\rho (1)} Z(\\phi )\\right) \\rho (\\phi ) \\right],$ for which the continuity of the flux in (REF ) leads to the following non-linear boundary condition $\\rho (0) ( 1 + \\lambda Z(0)) = \\rho (1) (1 + \\lambda Z(1)).$ The dynamics of the system defined by (REF ) and (REF ) is easily describable for monotonous PRCs [31].", "For increasing $Z(\\phi )$ the probability density concentrates to a single phase in finite time for arbitrary initial distributions.", "For decreasing $Z(\\phi )$ convergence to the stationary solution $\\rho _0(\\phi ):= c/ ( 1 + c Z(\\phi ))$ can be observed.", "The question of synchronization in the population model with sparse connectivity can be addressed by investigating the existence and the stability of normalized stationary solutions of (REF ) and (REF ).", "Stationary solutions of Eq.", "(REF ) (with $\\partial _t \\rho (\\phi ,t) = 0$ ) can be obtained by segmenting $[0,1]$ into intervals in which oscillators receive either phase advances or retardations.", "For each of these intervals, a solution can then be obtained with some solver for delay differential equations with state dependent delays, stepping towards either larger or smaller phases.", "Phases $\\varphi $ at which $R(\\phi )$ crosses the identity from below serve as suitable starting points for such a stepping approach because the sets $I_>(\\varphi )$ and $I_\\le (\\varphi )$ are empty at such points and we have $\\rho (\\varphi ) = J(\\varphi ) = J(1) = \\lambda /m$ .", "Using this initial value, solutions fulfill (REF ).", "In this way we can obtain stationary solutions $\\rho (\\phi ;\\lambda )$ of (REF ) and (REF ) in sole dependence on $\\lambda $ .", "Let us denote $I(\\lambda ) := \\int _0^1 \\rho (\\phi ;\\lambda ) d \\phi $ .", "In order to allow for a stochastic interpretation of $\\rho (\\phi ;\\lambda )$ , $\\lambda $ must then be chosen with shooting in such a way that $I(\\lambda ) = 1$ .", "However, depending on the PRC such a choice may not be possible.", "We can characterize the condition under which solutions exist by assuming that $R(\\phi )$ is non-decreasing.", "Note that this assumption is valid for commonly considered PRCs including those of integrate-and-fire oscillators [34].", "Under this assumption $I(\\lambda )$ is strictly increasing in $\\lambda $ , which we will show at the end of this letter.", "Stationary solutions are thus unique, and to decide on their existence, it is thus sufficient to investigate the solutions $\\rho (\\phi ;\\lambda )$ for large $\\lambda $ .", "Near $\\varphi $ , the non-local term in (REF ) vanishes, and $\\rho (\\phi )$ decays as $e^{ - \\lambda \\phi } (\\lambda / m)$ which has an integral independent on $\\lambda $ and thus concentrates to $\\delta (\\phi - \\varphi ) / m$ for large $\\lambda $ .", "Analogously, $\\delta $ -peaks are generated at phases $R(\\varphi ), R^2(\\varphi ), ...$ , at which oscillators arrive after having received a certain amount of excitations.", "Integrating over such a sequence of $\\delta $ -peaks, we can express $I(\\infty )$ and thus characterize the existence of asynchronous solutions by the following inequality: $I(\\infty ) = \\max _i \\lbrace i | R^{i} (\\varphi ) \\le 1 + \\varphi \\rbrace / m > 1.$ We now report on findings of a dynamical analysis for the case of integrate-and-fire oscillators [37].", "They are a popular model in many scientific fields ranging from physics and biology to the neurosciences [8].", "Our approach allows us to treat the non-invertible phase transition curves of excitatory and inhibitory oscillators and to study both dynamical regimes from a unified point of view.", "The PRC reads $\\Delta (\\phi ) = \\max \\lbrace \\min \\lbrace a\\phi + b, 1 - \\phi \\rbrace , -\\phi \\rbrace .$ The maximum and minimum bound $\\Delta (\\phi )$ by $-\\phi $ from below and by $1 - \\phi $ from above.", "The bound from above ensures that an excitation of oscillators cannot push them past the firing threshold ($I_>(0) = \\emptyset )$ and leads to uncountably many inverse images $R(\\phi )$ for $\\phi = 1$ .", "This assumption strongly favours synchronization; two oscillators adapt their phases completely after a suprathreshold excitation from one to the other.", "The bound form below prevents oscillators with small phases which receive an inhibitory excitation to attain a negative phase or a phase just below the firing threshold ($I_\\le (1) = \\emptyset $ ).", "We obtain the following boundary condition from (REF ): $\\underbrace{\\rho (0) - \\lambda \\int \\limits _{I_\\le (0)} \\rho (\\tilde{\\phi }) d\\tilde{\\phi }}_{J(0)}= \\underbrace{\\rho (1) + \\lambda \\int \\limits _{I_>(1)} \\rho (\\tilde{\\phi }) d\\tilde{\\phi }}_{J(1)}.$ The parameters $a$ and $b$ control both leakiness and coupling strength.", "For $a, b > 0$ , $\\Delta (\\phi )$ represents excitatory integrate-and-fire oscillators with concave-down charging function.", "For $a,b <0$ , $\\Delta (\\phi )$ represents inhibitory oscillators with concave-down charging function.", "For other parameter combinations, $\\Delta (\\phi )$ represents excitatory and inhibitory oscillators with concave-up charging function and dynamical systems with both positive and negative phase responses.", "Figure: (Colour on-line) Stationary solutions ρ 0 \\rho _0 of () and () for the PRC given in () with different parameter values a,ba, b, and mm obtained by a shooting method with the constraint that ρ 0 \\rho _0 is normalized.", "Top: excitatory oscillators with a=b=0.5/ma = b = 0.5/m, blue: m=500m = 500, red: m=50m=50, black: m=5m = 5.", "The green curve shows the stationary solution in the mean driven limit.", "Bottom: stationary solutions for a=b=0.1a = b = 0.1 and different mean degrees mm.", "Black: m=5.0m = 5.0, red: m=7.0 m = 7.0, blue: m=7.5m = 7.5, green: m=8.0m = 8.0.Figure: (Colour on-line) Top:Mean order parameter r ¯\\bar{r} (r(t):=1/N∑ n∈N e 2πiφ n r(t):= 1 / \\left|N\\right| \\sum _{n \\in N} e^{2 \\pi i \\phi _n} averaged over t∈[100,200]t\\in [100,200]) in dependence on aa and bb for a fixed mean degree m=50m = 50.", "Monte-Carlo simulations with N=10 6 N = 10^6 oscillators starting from uniformly distributed phases φ∈[0,1]\\phi \\in [0,1] .", "The red line indicates the locus of supercritical AHB points obtained by AUTO after a finite difference discretization (2000 dimensions, first order upwind scheme) of () and ().", "The blue line indicates the upper boundary of the parameter regime in which stationary solutions can be obtained by shooting given by ().In Fig.", "REF we show stationary solutions for the PRC given in (REF ).", "Given the PRC, phases in the interval $[0,b]$ are not reachable by excitations.", "The phase density $\\rho (\\phi )$ thus decays exponentially in this interval according to (REF ).", "Phases $\\phi > b$ are reachable, and $\\rho (\\phi )$ exhibits excursions of decreasing amplitudes, in which smoothed versions of the initial exponential segment in $[0,b]$ are repeated (cf. [26]).", "Analogously, inhibitory oscillators with phases near $\\phi =1$ do not receive excitations, which leads to a sharp decrease of $\\rho (\\phi )$ close to the firing threshold (not shown).", "For the mean-driven limit, stationary solutions show oscillations near $\\phi = 0$ with frequencies that diverge with increasing $m$ (cf.", "Fig.", "REF top).", "For large coupling strengths or mean degrees, stationary solutions converge to a series of $\\delta $ -peaks (cf.", "Fig REF bottom) and eventually disappear.", "For small $m$ and depending on oscillator parameters $a$ and $b$ , we can distinguish different dynamics (cf.", "Fig.", "REF ): asynchronous states with oscillator phases distributed according to the stationary solution, and (partially) synchronous states with oscillatory evolutions of the excitation rate $\\lambda $ .", "As for the mean-driven limit, large positive coupling strengths (above the black line in Fig.", "REF ), do not allow for normalized stationary solutions with positive values which can be interpreted as probability density.", "In this regime, oscillators synchronize completely within a few collective oscillations.", "Near this boundary the excitation rate diverges, and we observe no partially synchronous states.", "For smaller coupling strengths, stationary solutions do exist and we now discuss their stability.", "We consider a small, localized perturbation of the stationary solution, which travels periodically around the phase circle.", "An oscillator represented by this perturbation is shifted towards larger phase values due to its intrinsic dynamics and due to excitations.", "Both contributions are reflected by the corresponding terms in (REF ).", "The uncertainty of the oscillator's phase after some time leads to a broadening of the perturbation which increases with the strength of excitations and thus with $|b|$ (and to some degree with $a$ ).", "When the perturbation crosses the firing threshold, positive values of $a$ lead to a larger excitation of oscillators near the perturbation which leads to a sharpening.", "Consequently, the perturbation vanishes for large $|b|$ and small $a$ and increases otherwise.", "The boundary between both behaviors is characterized by a locus of Andronov-Hopf bifurcation (AHB) points.", "The AHB gives rise to oscillatory states with partial synchrony, in which a small perturbation of the stationary solution travels periodically around the phase circle.", "The amplitude of these oscillations increases with the distance to the AHB.", "For negative $b < a$ , oscillators have negative phase responses near $\\phi = 1$ and both integrals in the denominator in (REF ) vanish.", "Consequently, we observe no avalanches and no phase concentrations in $\\rho (\\phi )$ up to some value of $b$ for which complete synchrony is reached.", "For positive $a$ and $b$ , the first integral in (REF ) does not vanish.", "When the amplitude of the oscillations grows so large that the denominator in (REF ) vanishes, an avalanche emerges.", "For larger values of $a$ , subsequent avalanches increase in size leading to complete synchrony after a few oscillations.", "For smaller values of $a$ , these avalanches may, for finite networks, lead to complicated partially synchronous states with recurring avalanches, which, however, lie outside what can be described with the evolution equation.", "We will report on these states elsewhere (Rothkegel and Lehnertz, manuscript in preparation).", "Note that the aforementioned broadening is not present in the mean-driven limit, in which both intrinsic dynamics and excitations are represented by a single convection term.", "If we consider the PRC in (REF ) with parameters $a = \\alpha / m$ and $b = \\beta / m$ , we obtain $Z(\\phi ) = \\alpha \\phi + \\beta $ in the mean-driven limit ($m_i \\rightarrow \\infty $ ).", "The phase density as determined by (REF ) and (REF ) thus converges to the stationary solution $\\rho _0(\\phi )$ for negative $a$ and concentrates for positive $a$ leading to complete synchrony of oscillators.", "In particular, the system does not allow for periodic solutions with partial synchrony of oscillators [31].", "In this case stable stationary solutions cannot be observed for $a > 0$ .", "We have presented a population model of $\\delta $ -pulse-coupled oscillators with sparse connectivity.", "Interactions between oscillators are defined by a phase response curve (PRC).", "We have defined the model in such a way that allowed us to treat non-invertible PRCs which lead to discontinuous distributions of oscillator phases.", "We have demonstrated the uniqueness of asynchronous solutions and characterized their existence.", "Finally, we have shown—using integrate-and-fire-like oscillators—two different mechanism which may lead to loss of asynchronous states.", "Stationary solutions may lose stability, giving rise to oscillations and partially synchronous states, or they may disappear completely, leading to avalanche-like synchronization and a fast convergence to synchrony.", "We are confident that the model may further the understanding of the dynamics of sparsely coupled oscillatory networks.", "Systems that can be modelled as such appear ubiquitously in Nature.", "In this last part of the letter, we will show that $I(\\lambda )$ , the norm of stationary solutions of (5) and (6), is strictly increasing in $\\lambda $ , provided that the phase transition curve $R(\\phi )$ is increasing in $\\phi $ and crosses the identity at one or more points from below.", "For the sake of simplicity, we will shift the phases in such a way, that the crossing occurs at $\\phi = 0$ such that we have $\\rho (0) = J(0) = \\lambda / m$ .", "As first step, we relate $I(\\lambda )$ defined for some PRC to an exit-time problem for the stochastic dynamics of oscillators $\\partial _t \\phi (t) = 1 + \\eta _{\\lambda (t)}$ which is determined by convection with velocity 1 and by Poissonian excitations $\\eta _{\\lambda (t)}$ with inhomogeneous rate $\\lambda (t)$ .", "To this end, we consider the interval $(0,1)$ to be empty at $t = 0$ .", "If we now inject a constant flux $J(0)$ , oscillators will pass the interval and exit at $\\phi = 1$ after some variable time $t_\\mathrm {E}$ .", "We consider the distribution $P(t_\\mathrm {E})$ of these exit times.", "The flux $J(1,t)$ will increase from $J(1,0) = 0$ and will eventually approach the injected amount $J(0)$ , at which time the same amount of oscillators enter and exit the interval.", "If we inject $J(0) = \\lambda /m$ , then the number of oscillators which are in the interval at a large time $t$ , is given by $I(\\lambda )$ and can be expressed by integrating over the difference of incoming and outgoing fluxes: $I(\\lambda ) = \\lim _{\\kappa \\rightarrow \\infty } \\int _0^\\kappa \\frac{\\lambda }{m} - J(1,t) dt.$ Given the distribution of exit times $P(t_\\mathrm {E})$ , we can express the outgoing flux by integrating over the time $t_0$ at which oscillators are injected into the interval: $I(\\lambda ) = \\lim _{\\kappa \\rightarrow \\infty } \\left(\\frac{\\lambda }{m} \\kappa - \\int _0^\\kappa dt \\int _0^t dt_0 P(t - t_0) \\frac{\\lambda }{m} \\right).$ The domain of the integral is the area of the quadrant $(t_0, t) \\in [0,\\kappa ] \\times [0,\\kappa ]$ which lies above the diagonal.", "If we parametrise this domain by $t_\\mathrm {E}:= t - t_0$ and $t_0$ , we obtain, using substitution for multiple variables, $I(\\lambda ) = \\frac{\\lambda }{m} \\lim _{\\kappa \\rightarrow \\infty } \\left( \\kappa - \\int _0^\\kappa dt_\\mathrm {E} \\int _0^{\\kappa - t_\\mathrm {E}} dt_0 P(t_\\mathrm {E}) \\right).$ As every oscillator eventually reaches $\\phi = 1$ , we have $\\int _0^\\infty P(t_\\mathrm {E}) dt_\\mathrm {E} = 1$ , and we obtain a surprisingly simple relationship, which says that the norm of $I(\\lambda )$ is given by the product of the injected flux and the mean exit time: $I(\\lambda ) = \\frac{\\lambda }{m} \\int _0^\\infty { t_\\mathrm {E} P(t_\\mathrm {E}) d t_\\mathrm {E} }.$ Let us represent Eq.", "(REF ) by an integral equation.", "We define $M(\\varphi )$ as the mean time an oscillator with phase $1 - \\varphi $ remains in the unit interval before it reaches $\\phi = 1$ .", "With this definition, the mean exit time for the entire interval is $M(1)$ .", "Oscillators outside of the interval have a vanishing exit time: $M(\\varphi ) = 0$ for $\\varphi < 0$ .", "$M(\\varphi )$ can now be expressed by an average over the time of the next excitation.", "Assuming an exponential distribution $\\iota (t) := \\lambda e^{-\\lambda t}$ for the times between excitations, we can relate these times to probabilities.", "With probability $\\bar{\\iota }(\\varphi ) = 1 - \\int _0^\\varphi \\iota (t) dt$ , the oscillator will leave the interval without receiving another excitation.", "For the case that the oscillator receives an excitation at time $t$ after injection, it has a phase of $1-\\varphi + t + \\Delta (1 - \\varphi + t)$ afterwards.", "The mean time the oscillator needs to pass the remaining phase distance can again be expressed by $M(\\varphi )$ which results in the following integral equation for $M(\\varphi )$ : $M(\\varphi ) = \\bar{\\iota }(\\varphi ) \\varphi + \\int _0^\\varphi \\iota (t) \\left[ t + M(\\varphi - t - \\Delta (1 - \\varphi + t)) \\right] dt .$ Inserting $\\iota (t)$ and multiplying Eq.", "(REF ) by $\\lambda / m$ , we obtain a similar equation for $I(\\lambda ,\\varphi ) : = \\int _0^\\varphi \\rho (\\phi ;\\lambda /m,\\lambda )$ which we define as generalization of $I(\\lambda )$ with $I(\\lambda ,1) = I(\\lambda )$ : $ I (\\lambda , \\varphi ) = \\frac{1 - e^{-\\lambda \\varphi }}{m} + \\int _0^\\varphi \\lambda e^{-\\lambda (\\varphi -t)} I (\\lambda , t- \\Delta (1-t)) dt.$ For convenience, we will use the abbreviations $G(\\lambda ,\\varphi ):= (1- e^{-\\lambda \\varphi }) / m$ and $z(\\varphi , u) := \\varphi + u - \\Delta (1 - \\varphi - u)$ .", "$G(\\lambda ,\\varphi )$ is strictly increasing in both arguments.", "Using our assumption about the PRC, we see that $z(\\varphi ,u)$ is increasing in both arguments but not necessarily strictly increasing.", "Eq.", "(REF ) takes the following form: $ I (\\lambda , \\varphi ) = G(\\lambda , \\varphi ) + \\int _0^\\infty \\lambda e^{-\\lambda u} I (\\lambda , z ( \\varphi , -u)) du.$ We have extended the integral from $\\varphi $ to $\\infty $ using $I(\\lambda ,\\varphi ) = 0$ for $\\varphi < 0$ .", "The equation is a Volterra integral equation of the second kind.", "Note that $z(\\varphi , -u) \\le \\varphi $ such that the equation defines $I(\\lambda , \\varphi )$ in a hierarchical way.", "$I(\\lambda , \\varphi _0)$ is obtained by taking some value $G(\\lambda , \\varphi _0)$ and by adding an weighted average over previous values $I(\\lambda , \\varphi ), \\varphi < \\varphi _0$ .", "We can thus conclude that $I(\\lambda ,\\varphi )$ is positive, if $G(\\lambda ,\\varphi )$ is positive for all $\\varphi $ .", "We demonstrate that $I(\\lambda )$ is strictly increasing in $\\lambda $ for PRCs with $\\partial _\\phi R(\\phi ) \\ge 0$ .", "We first argue that $I(\\lambda ,\\varphi )$ is strictly increasing in its second argument for every $\\lambda > 0$ .", "Differentiating (REF ) by $\\varphi $ , we obtain an integral equation for $\\partial _\\varphi I(\\lambda , \\varphi )$ which is of the same kind as (REF ) and has a non-negative kernel $\\lambda e^{-\\lambda u} \\partial _\\varphi z(\\varphi , -u)$ and a positive function $\\partial _\\varphi G(\\lambda , \\varphi )$ outside of the integral.", "Analogously, we can thus conclude that $\\partial _\\varphi I(\\lambda , \\varphi )$ is positive.", "Finally, we argue that $I(\\lambda , \\varphi ) $ is increasing in $\\lambda $ for every $\\varphi \\le 1$ .", "Taking the derivative of (REF ) with respect to $\\lambda $ , we obtain three terms according to the dependences on $\\lambda $ of $G$ , of the kernel, and of $I$ .", "The first term $\\partial _\\lambda G(\\lambda , \\varphi )$ is positive as G is strictly increasing in its arguments.", "As second term, we obtain $\\int _0^\\infty I(\\lambda , z(\\varphi , -u)) \\partial _\\lambda \\lambda e^{-\\lambda u} du.$ Using $\\partial _\\lambda (\\lambda e^{-\\lambda u}) = \\partial _u ( u e^{-\\lambda u})$ and integrating by parts, we obtain also a positive contribution as $I$ and $z$ are increasing in their second arguments.", "The third term contains a weighted average via a positive kernel of $\\partial _\\lambda I(\\lambda , \\varphi )$ .", "As before, we infer that $\\partial _\\lambda I(\\lambda ,\\varphi )$ is strictly increasing which gives for $\\varphi = 1$ the desired proposition.", "We are grateful to Stefano Cardanobile for fruitful discussions and Gerrit Ansmann for careful revision of an earlier version of the manuscript.", "This work was supported by the Deutsche Forschungsgemeinschaft (LE 660/4-2)." ] ]
1403.0102
[ [ "BCS Instability and Finite Temperature Corrections to Tachyon Mass in\n Intersecting D1-Branes" ], [ "Abstract A holographic description of BCS superconductivity is given in arxiv:1104.2843.", "This model was constructed by insertion of a pair of D8-branes on a D4-background.", "The spectrum of intersecting D8-branes has tachyonic modes indicating an instability which is identified with the BCS instability in superconductors.", "Our aim is to study the stability of the intersecting branes under finite temperature effects.", "Many of the technical aspects of this problem are captured by a simpler problem of two intersecting D1-branes on flat background.", "In the simplified set-up we compute the one-loop finite temperature corrections to the tree-level tachyon mass using the frame-work of SU(2) Yang-Mills theory in (1 + 1)-dimensions.", "We show that the one-loop two-point functions are ultraviolet finite due to cancellation of ultraviolet divergence between the amplitudes containing bosons and fermions in the loop.", "The amplitudes are found to be infrared divergent due to the presence of massless fields in the loops.", "We compute the finite temperature mass correction to all the massless fields and use these temperature dependent masses to compute the tachyonic mass correction.", "We show numerically the existence of a transition temperature at which the effective mass of the tree-level tachyons becomes zero, thereby stabilizing the brane configuration." ], [ "Introduction", "There have been many applications of the AdS/CFT correspondence to understand condensed matter systems.", "When there are gapless modes present these systems are described by conformal field theories at low energies.", "This can happen at second order phase transitions, but also in metals where the excitations above the Fermi surface are gapless.", "For recent developments see [2]-[9].", "At low temperatures metals are unstable towards electron Cooper pair formation and an energy gap develops.", "This is the BCS instability.", "The Cooper pairs are charged and so the condensate breaks the $U(1)$ of electromagnetism and the photon effectively becomes massive as a result of the Higgs phenomenon.", "The energy gap ensures that at low frequencies there is no dissipation of energy when a current flows.", "The mass of the photon results in the exponential fall off of the magnetic field inside a superconductor.", "These are typical characteristics of superconductors.", "Studies of various types of superconductors using holographic techniques have been been the subject of research for the past few years.", "A partial list of references is [10]-[12].", "Inspired by this BCS phenomenon Nambu and Jona-Lasinio gave a description of chiral symmetry breaking in strong interactions [13].", "Their starting point was a non renormalizable model with four Fermi interactions.", "The pairing between quarks and anti-quarks is analogous to Cooper pairing.", "The main point of difference (as summarized in [1]) is that due to the absence of a Fermi surface this instability in QCD happens only for large enough coupling.", "Another point of difference is that the resultant condensate breaks an axial symmetry (rather than a vector symmetry as in BCS)- the $U(1)$ chiral symmetry that is present in QCD (in the absence of bare mass to quarks).", "A holographic dual of 3+1 QCD was constructed in [14] starting from M theory on $AdS_7\\times S^4$ .", "Many interesting calculations have been done with this model including calculation of the glueball mass spectrum - albeit at strong bare coupling [15]-[20].", "An extension of this model to include flavor degrees of freedom was constructed by Sakai and Sugimoto [21].", "Various aspects of this model was further explored in [22]-[27].", "The flavour branes are D8 branes hanging down from the boundary (where they intersect the D4 branes) and are wrapped on $S^4$ .", "It was shown that when there are D8 branes and D8 anti branes, a stable configuration is described by the brane and anti-brane bending towards each other and joining to form a continuous U-shaped brane.", "Since the branes describe left handed quarks and anti-branes describe right handed quarks (or left handed anti quarks) this U configuration breaks chiral symmetry.", "In [1] the Sakai-Sugimoto model was modified to describe BCS superconductivity.", "The Sakai Sugimoto model has unbroken vector like symmetries corresponding to the flavour group.", "Thus for two flavours there is a $U(2)$ .", "In [1] it was shown that in the presence of a finite chemical potential for the $U(1)$ embedded in $SU(2)$ , a D8 brane and an anti D8 brane cross each other.", "Such a configuration is known to be tachyonic and it has been argued that the stable configuration to which this flows has a non zero charged condensate that Higgses the $U(1)$ symmetry [28], [29], [30], [31], [32], [33], [34], [35].", "In [1] analytical solutions were given for such systems by solving the Yang-Mills equations describing intersecting branes in flat space-time.", "Semi analytic and numerical solutions were also given in the curved background of this modified Sakai-Sugimoto model.", "Being a strongly coupled system the expression for the gap in terms of the coupling and other parameters is different from weak coupling BCS.", "The gap and thus the transition temperature are expected to be larger here.", "For weak coupling the relation is $\\Delta \\approx \\epsilon _ce^{-{1\\over g {dn\\over d\\epsilon }}}$ , whereas for strong coupling one expects $\\Delta \\approx \\epsilon _c g {dn\\over d\\epsilon }$ .", "Here $\\epsilon _c$ is some parameter of the metal that fixes the region around the Fermi surface that participates in the pair formation.", "In this paper we attempt to calculate the transition temperature for the model described in [1].", "We do this in flat space-time for simplicity.", "The low energy theory on the brane can be described by the DBI action for the massless fields on the brane.", "This is valid as long as only energies $<<{1\\over \\alpha ^{\\prime }}$ are being probed.", "The DBI action describes the effect of \"integrating out classically\" (i.e.", "via equations of motion) the massive modes of the string.", "However even if the massive modes are integrated out in the quantum theory, the resultant action for the massless modes would look like supersymmetric Yang-Mills corrected by higher dimensional operators down by powers of $\\alpha ^{\\prime }$ - very similar to the DBI action.", "We can study this as a quantum theory with a cutoff $\\Lambda < {1\\over \\sqrt{\\alpha }^{\\prime }}$ and proceed to study the corrections due to the massless mode quantum and thermal fluctuations.", "Since the Yang-Mills action is renormalizable, we know that the effect of the irrelevant higher dimension operators is to make finite renormalizations of the lower dimensional operators.", "This is just a re-phrasing of the decoupling theorem: If the low energy theory is renormalizable the massive modes decouple and further the ambiguities associated with the physics at and above the cutoff scale can be absorbed into a renormalization of the parameters.", "While this is a consistent procedure, this is not good enough for us because if we want to estimate the finite thermal corrections to the action, the finite part should be unambiguous and cannot be the finite part of an infinite term.", "Fortunately, because of supersymmetry, the mass-squared corrections are finite and therefore calculable in principle.", "More precisely if we calculate the corrections as a power series in $\\Lambda $ (with $\\Lambda << {1\\over \\sqrt{\\alpha }^{\\prime }}$ ), one expects that terms that diverge when $\\Lambda \\rightarrow \\infty $ (i.e.", "positive powers and logarithms) are absent.", "Thus all corrections are finite and at most of order ${E\\over \\Lambda } < \\sqrt{\\alpha ^{\\prime }} E$ where $E$ is a typical energy scale.", "Supersymmetry in fact can ensure this even if the theory is not renormalizable as in $Dp$ -branes with $p>3$ .", "The physical quantity we are interested is the temperature correction to the tachyon mass-squared.", "The tachyon mass-squared is $O({\\theta \\over \\alpha ^{\\prime }}) =- q$ , where $\\theta $ is the angle of intersection of the $D$ -branes and $q$ is defined more precisely later.", "Thus we would like to keep this finite as ${1\\over \\alpha ^{\\prime }}\\rightarrow \\infty $ .", "This can be achieved by taking the limit $\\theta \\rightarrow 0, {1\\over \\alpha ^{\\prime }}\\rightarrow \\infty $ such that $q$ is fixed.", "Thus in this limit we can use the supersymmetric Yang-Mills theory on the brane.", "We also simplify further the problem by studying $D1$ branes.", "Classically the solutions we are considering depend only on one coordinate.", "So the solutions are the same for all $Dp$ branes.", "Quantum mechanically the fluctuations will be different and the momentum integrals in Feynman diagrams will be different.", "However many of the techniques used for $D1$ branes should go through since the mass-squared corrections are finite due to supersymmetry.", "Even with these simplifications the calculations are already quite involved.", "The main reason is that the background configuration about which quantum corrections need to be calculated is space dependent.", "The intersecting $D$ -brane configuration is described by one of the adjoint scalars having a value that is linear in $x$ , in the form $\\phi = qx$ .", "Thus in the $x$ -direction one cannot use plane waves as a basis.", "One has to work with eigenfunctions that are essentially harmonic oscillator wave functions.", "One should then calculate the effective potential at finite temperature and then obtain the transition temperature.", "This is a rather difficult calculation.", "In this paper as a first step we adopt the simpler procedure of calculating the corrections to the tachyon mass-squared and finding the temperature at which this turns positive.", "This is not the same thing because positive $(mass)^2$ only ensures local stability.", "In any case with these simplifications the calculation becomes tractable.", "Even so, some of the calculations have to be done numerically.", "There are two technical issues that become complicated because of using Hermite polynomials instead of plane waves.", "One is that of showing UV finiteness.", "As mentioned above, if the theory has divergent mass-squared corrections that need to be renormalized then one cannot calculate the transition temperature from first principles because there is always an arbitrary parameter corresponding to finite mass renormalization.", "Especially when calculations have to be done numerically, one needs to be sure that the series being summed is convergent.", "This demonstration is made difficult, once again because we cannot use a plane wave basis.", "In this paper we show UV finiteness at one loop.", "This check is also useful because it ensures that the degrees of freedom counting has been done correctly.", "The second complication is that there are many massless modes.", "This results in infrared divergences.", "The correct solution to this problem is to use a renormalization group and integrate out high momentum modes first.", "This should typically induce mass-squared corrections to the massless modes (unless they are protected) and the final solution to the RG equations should not have any IR divergences.", "The full RG is difficult to implement.", "However what can be done is to first do a one loop integral where the internal lines have only modes that do not generate IR divergences.", "In this step one can calculate the mass-squared correction to all the massless modes.", "At the next step one includes the remaining unintegrated modes in the tachyon mass-squared correction, with corrected massive propagators and now, because there are no massless modes, there are no IR divergences.", "At this point one is going beyond the one loop approximation and one has in effect summed an infinite number of diagrams.", "In practice one needs to calculate mass-squared corrections only for those modes that are needed for the tachyon mass-squared correction.", "Thus this procedure takes care of the infrared divergences.", "Once these problems are taken care of one can proceed to a calculation of the finite temperature correction to the tachyon mass-squared.", "Both, the temperature independent finite quantum correction to the masses-squared and the temperature dependent finite corrections are calculated.", "However in the final calculation of the tachyon mass-squared, which is done numerically, only the total correction is calculated and plotted.", "Thus we are able to calculate numerically the transition temperature at which the tachyon becomes massive.", "As mentioned above this calculation needs to be generalized to higher branes.", "Also instead of calculating the correction to the mass-squared one should do a more complete calculation and calculate the effective potential.", "Finally one should generalize to curved space time.", "This last may not however be very important because most of the dynamics takes place locally at the intersection point and one should be able to make a simple extrapolation to curved space locally using the equivalence principle.", "This paper is organized as follows.", "Section () is dedicated to the study of mass spectrum of intersecting $D1$ -branes at zero temperature in the Yang-Mills approximation.", "We choose a background given by the vev of one of the scalar field components.", "We compute the normalizable eigenfunctions for all the bosonic fields.", "Those fields which couple to the chosen background become massive at the tree-level and have a discrete mass spectrum.", "Those fields which do not couple to the background remain massless with a continuous spectrum.", "Apart from the massive modes there are also massless modes in the expansion of some of the bosonic fields which are accompanied with eigenfunctions with zero mass eigenvalue.", "The lowest lying modes in the bosonic mass spectrum are the tachyons.", "In section () we present the finite temperature analysis with a single scalar field using background field method.", "In section () we present the finite temperature analysis for the intersecting $D1$ -branes.", "In section (REF ) we present the computations for the one-loop bosonic corrections to the tree-level tachyon mass-squared at finite temperature.", "The fermionic eigenfunctions and their contributions to the one-loop corrections are presented in section (REF ).", "In section () we discuss the problems of ultraviolet and infrared divergences of the amplitudes which arise due to the presence of massless fields in the loop.", "While the amplitudes containing bosons in the loop are both IR and UV divergent the amplitudes containing fermions in the loop are only UV divergent.", "In section (REF ) we present the computations for the cancellation of the UV divergences of the amplitude for the tachyonic mode.", "To tackle the IR problem we first compute the one-loop corrections to all the massless fields.", "Thus we have all massive fields at one-loop level which then allows us to compute the finite two-point functions for the tachyonic modes.", "The one-loop mass-squared corrections to all the massless fields at finite temperature are computed in the sections (REF ) (REF ),(REF ) and (REF ).", "We also discuss their UV finiteness.", "In section (REF ) we present the numerical computation of the transition temperature and an analytical estimate of the behaviour of the masses-squared with varying temperature.", "For large values of temperature, the masses-squared are found to grow linearly with temperature.", "We present relevant details of the computation in the appendices." ], [ "Tree-level spectrum", "In this section we study the classical mass spectrum for an intersecting $D1$ brane configuration at zero temperature in the Yang-Mills approximation.", "The action for two coincident $D1$ branes with gauge group $SU(2)$ has been worked out in appendix .", "The action (REF ) is the dimensional reduction of 10 dimensional ${\\cal N}=1$ Super Yang-Mills.", "The equations of motion in $1+1$ dimensions have a solution $\\Phi ^3_1=qx$ and $A=0$ .", "This corresponds to an intersecting brane configuration with slope $q$ .", "Considering this background solution a fluctuation analysis was done in [28] to analyze the spectrum of the theory.", "To review this analysis we first write down the relevant bosonic part of the action (REF ).", "$S^1_{1+1}&=&\\frac{1}{g^2}\\mbox{tr} \\int d^2x \\left[-\\frac{1}{2}F_{\\mu \\nu }F^{\\mu \\nu }+ D_{\\mu }\\Phi _I D^{\\mu }\\Phi _I+\\frac{1}{2}\\left[\\Phi _I,\\Phi _J\\right]^2\\right]$ The Bosonic Lagrangian up to the quadratic order in fluctuations separates into various decoupled sets.", "In the $A^a_0=0 \\mbox{~} (a=1,2,3)$ gauge we write the decoupled parts separately below.", "$\\begin{split}{\\cal L}(A_x^2,\\Phi _1^1)=&-{1\\over 2}A_x^2 \\partial _0^2 A_x^2-{1\\over 2}\\Phi _1^1 \\partial _0^2 \\Phi _1^1+{1\\over 2}\\Phi _1^1 \\partial _x^2 \\Phi _1^1-q^2 x^2{1\\over 2}(A_x^2)^2\\\\&+ q A_x^2 \\Phi _1^1-qx \\partial _x \\Phi _1^1 A_x^2.\\end{split}$ $\\begin{split}{\\cal L}(A_x^1,\\Phi _1^2)=&-{1\\over 2}A_x^1 \\partial _0^2 A_x^1-{1\\over 2}\\Phi _1^2 \\partial _0^2 \\Phi _1^2+{1\\over 2}\\Phi _1^2 \\partial _x^2 \\Phi _1^2-q^2 x^2 {1\\over 2}(A_x^1)^2\\\\&-q A_x^1 \\Phi _1^2+qx \\partial _x \\Phi _1^2 A_x^1\\end{split}$ $\\begin{split}{\\cal L}(\\Phi _I,A_x^3)=&-{1\\over 2}\\Phi _I^a \\partial _0^2 \\Phi _I^a+{1\\over 2}\\Phi _I^a\\partial _x^2 \\Phi _I^a-{1\\over 2}q^2 x^2 (\\Phi _I^1)^2-{1\\over 2}q^2 x^2 (\\Phi _I^2)^2\\\\&-{1\\over 2}A_x^3 \\partial _0^2 A_x^3 \\mbox{~~~~~(for all $I\\ne 1$)}\\end{split}$ We thus have various decoupled sets of equations at this quadratic order.", "The solutions of these equations give the wave functions corresponding to the normal modes.", "The first two terms ${\\cal L}(A_x^2,\\Phi _1^1)$ and ${\\cal L}(A_x^1,\\Phi _1^2)$ implies that we have a two coupled sets of equations for $(A_x^2,\\Phi _1^1)$ and $(A_x^1,\\Phi _1^2)$ .", "To compute eigenfunctions and the spectrum we first consider the equations for $(A_x^2,\\Phi _1^1)$ fields.", "The eigenvalue equation for the spatial part is, $\\left(\\begin{array}{cc}m^2-q^2x^2&-qx\\partial _x +q\\\\2q+qx\\partial _x&m^2+\\partial _x^2\\end{array}\\right)\\left(\\begin{array}{c}A_x^2\\\\ \\Phi _1^1\\end{array}\\right)=0$ where $m^2$ is the eigenvalue.", "Eliminating, $A_x^2$ from the above equations gives the following equation for $\\Phi _1^1$ , $\\partial _x^2\\Phi _1^1+\\left[P(x)-Q(x)\\right]\\Phi _1^1-x P(x)\\partial _x\\Phi _1^1=0$ where, $P(x)=\\frac{2q^2}{Q(x)}~~~;~~~Q(x)=q^2x^2-m^2$ The asymptotic $( x\\rightarrow \\infty )$ form of the equation (REF ) is, $\\left[\\partial _x^2+m^2-q^2x^2\\right]\\Phi _1^1=0$ which is the Schroedinger's equation for a harmonic oscillator.", "The ground state wave function is $e^{-qx^2/2}$ .", "Thus writing, $A_x^2= e^{-qx^2/2}A ~~~;~~~\\Phi _1^1=e^{-qx^2/2}\\phi $ we get the following equations, $&&(m^2-q^2x^2)A+(q+q^2x^2-qx\\partial _x)\\phi =0\\\\&&(-q^2x^2+qx\\partial _x+2q)A+(m^2-q+q^2x^2-2qx\\partial _x +\\partial _x^2)\\phi =0$ Now assuming series solution of the form, $A=\\sum _k a_k (x\\sqrt{q})^k ~~~;~~~ \\phi =\\sum _k b_k (x\\sqrt{q})^k$ we get the following recursion relations, $\\frac{\\left[(2k-1)-\\frac{m^2}{q}\\right]}{\\left[k-\\frac{m^2}{q}\\right]}b_k-\\frac{(k+1)(k+2)}{\\left[(k+2)-\\frac{m^2}{q}\\right]}b_{k+2}=0$ and $a_k=b_k-\\frac{(k+1)(k+2)}{\\left[(k+2)-\\frac{m^2}{q}\\right]}b_{k+2}$ The quantization condition on $m^2$ is obtained by demanding that the series (REF ) terminates for some value of $k$ that is $n$ .", "This implies that the numerator of the first term of (REF ) vanishes for this value of $k$ .", "We thus have the spectrum given by $m^2=(2n-1)q$ .", "The lowest mode of mass spectrum given by $n=0$ is tachyonic.", "To solve for the eigenfunctions we simply need to compute the various coefficients $(a_k, b_k)$ using the recursion relations.", "For $n=0,2,4 \\cdots $ , $a_k=\\frac{(-1)^{k/2} 2^{k/2}}{(2n-1) k!", "}n (n-2) \\cdots (n-k+2)(k-1)\\\\b_k=\\frac{(-1)^{k/2} 2^{k/2}}{(2n-1) k!", "}n (n-2) \\cdots (n-k+2)(2n-k-1)$ For $n=3,5,7 \\cdots $ , $a_k=\\frac{(-1)^{(k-1)/2} 2^{(k-1)/2}}{(2n-1) k!}", "(n-1)(n-3) \\cdots (n-k+2)(k-1)\\\\b_k=\\frac{(-1)^{(k-1)/2} 2^{(k-1)/2}}{2(n-1) k!}", "(n-1)(n-3) \\cdots (n-k+2)(2n-k-1)$ Putting these back in (REF ), the eigenfunctions are, $A_n=\\sum _{k\\le n}a_k (x\\sqrt{q})^k ~~~;~~~ \\phi _n=\\sum _{k\\le n} b_k (x\\sqrt{q})^k$ The normalized eigenfunctions $\\lbrace A_n(x), \\phi _n(x)\\rbrace $ for both odd and even $n$ can be combined into the following expressions $A_n(x) = {\\cal N}(n)e^{- q x^2/2} \\left(H_n (\\sqrt{q} x) + 2 n H_{n-2} (\\sqrt{q} x) \\right) \\\\\\phi _n(x) = {\\cal N}(n)e^{- q x^2/2} \\left(H_n (\\sqrt{q} x) - 2 n H_{n-2} (\\sqrt{q} x) \\right)$ where $H_n(\\sqrt{q}x)$ are Hermite polynomials and the normalization, ${\\cal N}(n)=1/\\sqrt{\\sqrt{\\pi } 2^n (4 n^2-2n) (n-2)!", "}$ .", "Let us define, $\\zeta _n(x)=\\left(\\begin{array}{c}A_n(x)\\\\\\phi _n(x)\\end{array}\\right).$ $\\zeta _n(x)$ then satisfies the orthogonality condition $\\sqrt{q}\\int dx \\zeta ^{\\dagger }_n(x)\\zeta _{n^{^{\\prime }}}(x)=\\delta _{n,n^{^{\\prime }}}.$ There is also a set of infinitely many degenerate eigenfunctions with $m_n^2=0$ .", "The normalized eigenfunctions for the zero eigenvalues can also be written in terms of Hermite polynomials as $\\tilde{A}_n(x) = \\tilde{{\\cal N}}(n) e^{- q x^2/2} \\left(H_n (\\sqrt{q} x) - 2 (n-1) H_{n-2} (\\sqrt{q} x) \\right) \\\\\\tilde{\\phi }_n(x) = \\tilde{{\\cal N}}(n) e^{- q x^2/2} \\left(H_n (\\sqrt{q} x) + 2 (n-1) H_{n-2} (\\sqrt{q} x) \\right)$ where the normalization, $\\tilde{{\\cal N}}(n)=1/\\sqrt{\\sqrt{\\pi } 2^n (4 n-2)(n-1)!", "}$ .", "We define a different set, $\\tilde{\\zeta }_n(x)=\\left(\\begin{array}{c}\\tilde{A}_n(x)\\\\\\tilde{\\phi }_n(x)\\end{array}\\right).$ $\\tilde{\\zeta }_n(x)$ again satisfies the orthogonality condition as (REF ).", "So that, $\\sqrt{q}\\int dx \\tilde{\\zeta }^{\\dagger }_n(x)\\tilde{\\zeta }_{n^{^{\\prime }}}(x)=\\delta _{n,n^{^{\\prime }}}.$ Along with this we also have, $\\sqrt{q}\\int dx \\zeta ^{\\dagger }_n(x)\\tilde{\\zeta }_{n^{^{\\prime }}}(x)=0 \\mbox{~~ for all~~$n$~and~$n^{^{\\prime }}$~~}.$ Unlike the non-zero eigenvalue sector, in the zero eigenvalue sector we have normalizable eigenfunction for $n=1$ , which is simply $H_1(\\sqrt{q} x)$ .", "There is however no normalizable eigenfunctions for $n=0$ in this sector.", "The spectrum for $m_n^2=0$ is completely degenerate.", "Henceforth in this paper we shall refer the eigenfunctions for $m_n^2=0$ as the “zero-eigenfunctions”.", "From the equations of motion for $(A_x^1,\\Phi _1^2)$ obtained from (REF ), their eigenfunctions are simply $(-A_n(x), \\phi _n(x))$ , and $(-\\tilde{A}_n(x), \\tilde{\\phi }_n(x))$ for $m_n^2=(2n-1)q$ and $m_n^2=0$ respectively.", "There is thus a two fold degeneracy for this spectrum of the theory.", "The term ${\\cal L}(\\Phi _I,A_x^3)$ gives decoupled equations for $\\Phi _I^a$ for each value of $I\\ne 1$ and the gauge index $a$ and another equation for $A_x^3$ alone.The equation of motion for $\\Phi _I^1$ is $\\left(-\\partial _{0}^2+\\partial _x^2-q^2 x^2\\right)\\Phi _I^1=0$ The same equation for $\\Phi _I^2$ .", "The spatial part of the equation is the wave function equation for a Harmonic oscillator.", "The time independent eigenfunctions will thus be given by $\\mathcal {N}^{^{\\prime }}(n)e^{-qx^2/2} H_n(\\sqrt{q} x)$ where $H_n(\\sqrt{q} x)$ are Hermite polynomials.", "The normalization $\\mathcal {N}^{^{\\prime }}(n)=1/\\sqrt{\\sqrt{\\pi }2^n n!", "}$ .The corresponding eigenvalues are $\\gamma _n=(2n+1)q$ .", "The equations of motion for $\\Phi _I^3$ and $A_1^3$ are, $\\left(-\\partial _{0}^2+\\partial _x^2\\right)\\Phi _I^3=0 \\mbox{~~;~~} \\partial _{0}^2 A_x^3=0$ This means that the time independent part of $\\Phi _I^3$ is just a plane wave $e^{ilx}$ .", "Tables REF and REF in appendix summarize the various dimensionfull parameters, normalizations and the eigenfunctions.", "At this point we should note that the only tachyons that arise in the spectrum are those as discussed above.", "There are no other tachyons.", "The presence of the tachyon signals an instability.", "As noted in [1] and the introduction, this instability corresponds to the onset of superconducting phase transition of the baryons in the boundary theory.", "In the brane picture, the tachyon condenses and at the end of the process we are left with a smoothened out brane configuration [28].", "In the following sections we would like to study the quantum theory at finite temperature.", "The main aim is to find the critical temperature at which the tachyonic instability vanishes." ], [ "Finite temperature analysis with one scalar: Warm up exercise", "In this section we first study a simplified model consisting of only one adjoint scalar.", "The purpose of this section is to outline the basic idea involved in the computation of the mass-squared corrections of the tachyon due to finite temperature effects.", "With only one scalar (namely $\\Phi _1^1$ ) we have the equations resulting from (REF ), up to the quadratic order.", "By doing a one-loop integral over the fluctuations we will find an effective action for the tachyonic mode.", "The coefficient of the quadratic part of the effective action gives the mass-squared of the tachyon as function of the temperature.", "We thus start by doing a fluctuation analysis with the doublet of fields ($\\Phi _1^1$ , $A_x^2$ ) fields.", "$A_x^2=A_B+\\delta A ~~~~\\Phi _1^1=\\Phi _B+\\delta \\Phi $ To do a finite temperature analysis, we define the Euclidean coordinate $\\tau =it$ .", "$\\tau $ is periodic with period $\\beta $ , so that the integration limits over $\\tau $ are from 0 to $\\beta $ .", "We now denote the background field and the fluctuations as, $\\zeta (x,\\tau )=\\left(\\begin{array}{c}A_B(x, \\tau )\\\\\\Phi _B(x, \\tau )\\end{array}\\right)~~~,~~~\\delta \\zeta (x,\\tau )=\\left(\\begin{array}{c}\\delta A(x, \\tau )\\\\\\delta \\Phi (x, \\tau )\\end{array}\\right)$ The quadratic background part of the action can be written as, $S_B=\\frac{1}{2g^2}\\int d\\tau dx \\zeta ^{\\dagger }(x,\\tau ){\\cal O}_0(x,\\tau )\\zeta (x,\\tau )$ where ${\\cal O}_0(x,\\tau )=\\left(\\begin{array}{cc}\\partial _{\\tau }^2-q^2x^2&-qx\\partial _x +q\\\\2q+qx\\partial _x&\\partial _{\\tau }^2+\\partial _x^2\\end{array}\\right)$ The mode expansions for the background fields is, $\\begin{split}\\zeta (x,\\tau )&=N^{1/2}\\sum _{w,k}\\left(C_{w,k}\\zeta _k(x) + \\tilde{C}_{w,k}\\tilde{\\zeta }_k(x)\\right)e^{i\\omega _w\\tau }\\\\&=N^{1/2}\\sum _{w,k} \\left(C_{w,k} \\left(\\begin{array}{c}A_k(x)\\\\\\phi _k(x)\\end{array}\\right) + \\tilde{C}_{w,k} \\left(\\begin{array}{c}\\tilde{A}_k(x)\\\\\\tilde{\\phi }_k(x)\\end{array}\\right)\\right)e^{i\\omega _w\\tau }\\end{split}$ Where $\\omega _w=2\\pi w/\\beta $ .", "The normalization constant $N$ is equal to $\\sqrt{q} /\\beta $ .", "$\\zeta _k(x)$ and $\\tilde{\\zeta }_k(x)$ are defined in (REF ) ans (REF ) respectively.", "The corresponding eigenvalues are $-\\lambda _k=-(2k-1)q$ of the first set of eigenfunctions and $\\tilde{\\lambda }_k=0 \\mbox{~}(\\mbox{for~all~} k)$ of the second set.", "The reality of $\\zeta (x,\\tau )$ means that $C_{-w,k}=C_{w,k}^{*}$ and $\\tilde{C}_{-w,k}=\\tilde{C}_{w,k}^{*}$ .", "With these observations, and using the orthogonality properties (REF ), (REF ) and (REF ) the quadratic background part of the action can then be written as, $S_B=-\\frac{1}{2g^2}\\sum _{w,k}\\left(|C_{w,k}|^2 (\\omega _w^2+\\lambda _k) + |\\tilde{C}_{w,k}|^2 \\omega _w^2\\right)$ The mass spectrum now consists of a tower of tachyons of mass-squared $(\\omega _w^2-q)/g^2$ .", "Of these the zero mode, $w=0$ has the lowest value of $mass^2$ .", "We now study the fluctuations about the above background.", "The part of the Lagrangian containing the fluctuations is given by, $S_{\\delta }=\\frac{1}{2g^2}\\int d\\tau dx \\delta \\zeta ^{\\dagger }(x,\\tau ){\\cal O}_{\\delta }(x,\\tau )\\delta \\zeta (x,\\tau )$ where the operator ${\\cal O}_{\\delta }(x,\\tau )$ is, $\\begin{split}{\\cal O}_{\\delta }(x,\\tau )=&{\\cal O}_0(x,\\tau )+{\\cal O}_B(x,\\tau )\\\\=&\\left(\\begin{array}{cc}\\partial _{\\tau }^2-q^2x^2-\\Phi ^2_B(x,\\tau )&-qx\\partial _x +q-2A_B(x,\\tau )\\Phi _B(x,\\tau )\\\\2q+qx\\partial _x-2A_B(x,\\tau )\\Phi _B(x,\\tau )&\\partial _{\\tau }^2+\\partial _x^2-A^2_B(x,\\tau )\\end{array}\\right)\\end{split}$ There will also be terms linear in the fluctuations.", "However since these terms do not contribute to the $1PI$ effective action, we have dropped them here.", "The mode expansions for the fluctuations is, $\\delta \\zeta (x,\\tau ) &=& N^{1/2}\\sum _{m,n}\\left(D_{m,n}\\delta \\zeta _n(x)+\\tilde{D}_{m,n}\\delta \\tilde{\\zeta }_n(x)\\right)e^{i\\omega _m\\tau }\\\\&=& N^{1/2}\\sum _{m,n} \\left(D_{m,n} \\left(\\begin{array}{c}\\delta A_n(x)\\\\\\delta \\phi _n(x)\\end{array}\\right)+\\tilde{D}_{m,n} \\left(\\begin{array}{c}\\delta \\tilde{A}_n(x)\\\\\\delta \\tilde{\\phi }_n(x) \\end{array}\\right)\\right)e^{i\\omega _m\\tau }\\nonumber $ where $\\delta \\zeta _n(x)$ and $\\delta \\tilde{\\zeta }_n(x)$ are now eigenfunctions of the $\\tau $ -independent part of the operator ${\\cal O}_{\\delta }(x,\\tau )$ .", "Let us assume that the corresponding eigenvalues are $-\\Lambda _n$ and $-\\tilde{\\Lambda }_n$ respectively.", "Again since $\\delta \\zeta (x,\\tau )$ is real, $D_{-m,n}=D_{m,n}^{*}$ and $\\tilde{D}_{-m,n}=\\tilde{D}_{m,n}^{*}$ .", "The partition function for the fluctuations is thus, $\\begin{split}Z(\\beta ,q)&=\\int {\\cal D}[D_{m,n}][\\tilde{D}_{m,n}]e^{S_{\\delta }}\\\\&=\\int {\\cal D}[D_{m,n}][\\tilde{D}_{m,n}]e^{-\\frac{1}{2g^2}\\sum _{m,n}\\left[|D_{m,n}|^2 (\\omega _m^2+\\Lambda _n)+|\\tilde{D}_{m,n}|^2 (\\omega _m^2+\\tilde{\\Lambda }_n)\\right]}\\\\&=\\prod _{m,n\\ne 0}\\left[\\frac{1}{(2\\pi g^2)^2}(\\omega _m^2+\\Lambda _n)(\\omega _m^2+\\tilde{\\Lambda }_n)\\right]^{-1/2}\\end{split}$ The eigenvalues $\\Lambda _n$ and $\\tilde{\\Lambda }_n$ are yet to be determined, for which we use perturbation theory by assuming that the background field modes are small.", "We already know the time independent eigenfunctions and the corresponding eigenvalues for the operator ${\\cal O}_0$ .", "We can now treat the background fields in (REF ) as perturbations and find the corrections.", "The background fields can be expanded in terms of the $C_{w,k}$ modes.", "Since we are only interested in the quadratic contribution in $C_{w,k}$ 's, we do not need beyond the leading correction.", "This is because the perturbation matrix ${\\cal O}_B$ contains two powers of background fields giving rise to terms quadratic in the $C_{w,k}$ modes.", "$\\Lambda _n=\\Lambda _n^{(0)}+\\Lambda _n^{(1)}+.... ~~~~~{\\rm with}~~~~~\\Lambda _n^{(0)}=\\lambda _n=(2n-1)q$ $\\tilde{\\Lambda }_n=\\tilde{\\Lambda }_n^{(0)}+\\tilde{\\Lambda }_n^{(1)}+.... ~~~~~{\\rm with}~~~~~\\tilde{\\Lambda }_n^{(0)}=\\tilde{\\lambda }_n=0$ and, $\\delta \\zeta _n(x)=\\delta \\zeta _n^{(0)}(x)+\\delta \\zeta _n^{(1)}(x)+.... ~~~~~{\\rm with}~~~~~\\delta \\zeta _n^{(0)}(x)=\\zeta _n(x)$ $\\delta \\tilde{\\zeta }_n(x)=\\delta \\tilde{\\zeta }_n^{(0)}(x)+\\delta \\tilde{\\zeta }_n^{(1)}(x)+.... ~~~~~{\\rm with}~~~~~\\delta \\tilde{\\zeta }_n^{(0)}(x)=\\tilde{\\zeta }_n(x)$ so that, $\\Lambda _n^{(1)}=-\\int dx d\\tau \\delta \\zeta _n^{(0)\\dagger }(x){\\cal O}_B(x,\\tau )\\delta \\zeta _n^{(0)}(x)$ $\\tilde{\\Lambda }_n^{(1)}=-\\int dx d\\tau \\delta \\tilde{\\zeta }_n^{(0)\\dagger }(x){\\cal O}_B(x,\\tau )\\delta \\tilde{\\zeta }_n^{(0)}(x)$ with this, $\\begin{split}\\log Z(\\beta ,q)&=-{1\\over 2}\\sum _{m,n\\ne 0}\\left[\\log \\left(\\frac{\\omega _m^2+(2n-1)q+\\Lambda _n^{(1)}}{2\\pi g^2}\\right)+\\log \\left(\\frac{\\omega _m^2+\\tilde{\\Lambda }_n^{(1)}}{2 \\pi g^2}\\right)\\right]\\\\&=-{1\\over 2}\\sum _{m,n\\ne 0}\\left[\\log \\left(1+\\frac{\\Lambda _n^{(1)}\\beta ^2}{(2\\pi m)^2+(2n-1)q\\beta ^2}\\right)+\\log \\left(1+\\frac{\\tilde{\\Lambda }_n^{(1)}\\beta ^2}{(2\\pi m)^2}\\right)\\right]\\end{split}$ where in the last line we have omitted the field independent terms.", "For small value of $\\Lambda _n^{(1)}$ the leading term which gives the quadratic correction to the effective action of the background $C_{w,k}$ fields is, $\\log Z(\\beta ,q)=-{1\\over 2}\\sum _{m,n\\ne 0}\\left[\\frac{\\Lambda _n^{(1)}\\beta ^2}{(2\\pi m)^2+(2n-1)q\\beta ^2}+\\frac{\\tilde{\\Lambda }_n^{(1)}\\beta ^2}{(2\\pi m)^2}\\right]$ To find the form of the effective action due to the perturbations we first compute $\\Lambda ^{(1)}_n$ given in equation (REF ).", "The expression for $\\Lambda ^{(1)}_n$ after putting in the mode expansions for the background $\\Phi _B$ and $A_B$ fields reads, $\\Lambda ^{(1)}_n=\\sum _{w,k,k^{^{\\prime }}}\\left[C_{w,k}C^{*}_{w,k^{^{\\prime }}}F_1(k,k^{^{\\prime }},n,n)+\\tilde{C}_{w,k}\\tilde{C}^{*}_{w,k^{^{\\prime }}}F^{^{\\prime }}_1(k,k^{^{\\prime }},n,n)+2 C_{w,k}\\tilde{C}^{*}_{w,k^{^{\\prime }}}F^{^{\\prime \\prime }}_1(k,k^{^{\\prime }},n,n)\\right]\\nonumber \\\\$ $F_1(k,k^{^{\\prime }},n,n)&=&\\sqrt{q}\\int dx \\left[\\phi _k(x)\\phi _{k^{^{\\prime }}}(x)A_n(x)A_n(x)+2A_k\\phi _{k^{^{\\prime }}}(x)\\phi _n(x)A_n(x)\\right.", "\\nonumber \\\\&+& \\left.", "2A_{k^{^{\\prime }}}\\phi _{k}(x)\\phi _n(x)A_n(x)+ A_k(x)A_{k^{^{\\prime }}}(x)\\phi _n(x)\\phi _n(x)\\right]$ $F^{^{\\prime }}_1(k,k^{^{\\prime }},n,n)&=&\\sqrt{q}\\int dx \\left[\\tilde{\\phi }_k(x)\\tilde{\\phi }_{k^{^{\\prime }}}(x)A_n(x)A_n(x)+2\\tilde{A}_k\\tilde{\\phi }_{k^{^{\\prime }}}(x)\\phi _n(x)A_n(x)\\right.", "\\nonumber \\\\&+& \\left.", "2\\tilde{A}_{k^{^{\\prime }}}\\tilde{\\phi }_{k}(x)\\phi _n(x)A_n(x)+ \\tilde{A}_k(x)\\tilde{A}_{k^{^{\\prime }}}(x)\\phi _n(x)\\phi _n(x)\\right]$ $F^{^{\\prime \\prime }}_1(k,k^{^{\\prime }},n,n)&=&\\sqrt{q}\\int dx \\left[{\\phi }_k(x)\\tilde{\\phi }_{k^{^{\\prime }}}(x)A_n(x)A_n(x)+2{A}_k\\tilde{\\phi }_{k^{^{\\prime }}}(x)\\phi _n(x)A_n(x)\\right.", "\\nonumber \\\\&+& \\left.", "2\\tilde{A}_{k^{^{\\prime }}}{\\phi }_{k}(x)\\phi _n(x)A_n(x)+ {A}_k(x)\\tilde{A}_{k^{^{\\prime }}}(x)\\phi _n(x)\\phi _n(x)\\right]$ Similarly expanding $\\tilde{\\Lambda }_n^{(1)}$ given in (REF ) gives, $\\tilde{\\Lambda }^{(1)}_n =\\sum _{w,k,k^{^{\\prime }}}\\left[C_{w,k}C^{*}_{w,k^{^{\\prime }}}\\tilde{F}_1(k,k^{^{\\prime }},n,n)+\\tilde{C}_{w,k}\\tilde{C}^{*}_{w,k^{^{\\prime }}}\\tilde{F}^{^{\\prime }}_1(k,k^{^{\\prime }},n,n)+2 C_{w,k}\\tilde{C}^{*}_{w,k^{^{\\prime }}}\\tilde{F}^{^{\\prime \\prime }}_1(k,k^{^{\\prime }},n,n)\\right]\\nonumber \\\\$ $\\tilde{F}_1(k,k^{^{\\prime }},n,n)&=&\\sqrt{q}\\int dx \\left[\\phi _k(x)\\phi _{k^{^{\\prime }}}(x)\\tilde{A}_n(x)\\tilde{A}_n(x)+2A_k\\phi _{k^{^{\\prime }}}(x)\\tilde{\\phi }_n(x)\\tilde{A}_n(x)\\right.", "\\nonumber \\\\&+& \\left.", "2A_{k^{^{\\prime }}}\\phi _{k}(x)\\tilde{\\phi }_n(x)\\tilde{A}_n(x)+ A_k(x)A_{k^{^{\\prime }}}(x)\\tilde{\\phi }_n(x)\\tilde{\\phi }_n(x)\\right]$ $\\tilde{F}^{^{\\prime }}_1(k,k^{^{\\prime }},n,n)&=&\\sqrt{q}\\int dx \\left[\\tilde{\\phi }_k(x)\\tilde{\\phi }_{k^{^{\\prime }}}(x)\\tilde{A}_n(x)\\tilde{A}_n(x)+2\\tilde{A}_k\\tilde{\\phi }_{k^{^{\\prime }}}(x)\\tilde{\\phi }_n(x)\\tilde{A}_n(x)\\right.", "\\nonumber \\\\&+& \\left.", "2\\tilde{A}_{k^{^{\\prime }}}\\tilde{\\phi }_{k}(x)\\tilde{\\phi }_n(x)\\tilde{A}_n(x)+ \\tilde{A}_k(x)\\tilde{A}_{k^{^{\\prime }}}(x)\\tilde{\\phi }_n(x)\\tilde{\\phi }_n(x)\\right]$ $\\tilde{F}^{^{\\prime \\prime }}_1(k,k^{^{\\prime }},n,n)&=&\\sqrt{q}\\int dx \\left[{\\phi }_k(x)\\tilde{\\phi }_{k^{^{\\prime }}}(x)\\tilde{A}_n(x)\\tilde{A}_n(x)+2{A}_k\\tilde{\\phi }_{k^{^{\\prime }}}(x)\\tilde{\\phi }_n(x)\\tilde{A}_n(x)\\right.", "\\nonumber \\\\&+& \\left.", "2\\tilde{A}_{k^{^{\\prime }}}{\\phi }_{k}(x)\\tilde{\\phi }_n(x)\\tilde{A}_n(x)+ {A}_k(x)\\tilde{A}_{k^{^{\\prime }}}(x)\\tilde{\\phi }_n(x)\\tilde{\\phi }_n(x)\\right]$ Putting these expansions (REF ) and (REF ) in (REF ) we now collect terms containing two $C_{w,k}$ fields, two $\\tilde{C}_{w,k}$ or one $C_{w,k}$ and one $\\tilde{C}_{w,k}$ modes separately.", "A general expression for (REF ) finally is, $\\log Z(\\beta ,q)&=&-\\sum _{w,k,k^{^{\\prime }}}\\left[C_{w,k}C_{w,k^{^{\\prime }}}^*\\Sigma ^{2}(k,k^{^{\\prime }},\\beta ,q)+\\tilde{C}_{w,k}\\tilde{C}_{w,k^{^{\\prime }}}^*\\tilde{\\Sigma }^{2}(k,k^{^{\\prime }},\\beta ,q)\\right.\\nonumber \\\\&+&\\left.2 C_{w,k}\\tilde{C}_{w,k^{^{\\prime }}}^*\\Sigma ^{^{\\prime }2}(k,k^{^{\\prime }},\\beta ,q)\\right]$ The coefficient of the $C_{w,k}C_{w,k^{^{\\prime }}}^*$ term, $\\Sigma ^{2}(k,k^{^{\\prime }},\\beta ,q)$ is given by, $\\Sigma ^{2}(k,k^{^{\\prime }},\\beta ,q)={1\\over 2}\\sqrt{q}\\beta \\sum _{m,n\\ne 0}\\left[\\frac{F_1(k,k^{^{\\prime }},n,n)}{(2\\pi m)^2+(2n-1)q\\beta ^2}+\\frac{\\tilde{F}_1(k,k^{^{\\prime }},n,n)}{(2\\pi m)^2}\\right]$ This is the one-loop correction to the two point amplitude for the $C_{w,k}$ modes.", "Similarly, $\\tilde{\\Sigma }^{2}(k,k^{^{\\prime }},\\beta ,q)={1\\over 2}\\sqrt{q}\\beta \\sum _{m,n\\ne 0}\\left[\\frac{F^{^{\\prime }}_1(k,k^{^{\\prime }},n,n)}{(2\\pi m)^2+(2n-1)q\\beta ^2}+\\frac{\\tilde{F}^{^{\\prime }}_1(k,k^{^{\\prime }},n,n)}{(2\\pi m)^2}\\right]$ and $\\Sigma ^{^{\\prime }2}(k,k^{^{\\prime }},\\beta ,q)={1\\over 2}\\sqrt{q}\\beta \\sum _{m,n\\ne 0}\\left[\\frac{F^{^{\\prime \\prime }}_1(k,k^{^{\\prime }},n,n)}{(2\\pi m)^2+(2n-1)q\\beta ^2}+\\frac{\\tilde{F}^{^{\\prime \\prime }}_1(k,k^{^{\\prime }},n,n)}{(2\\pi m)^2}\\right]$ The fields $C_{w,k}$ and $\\tilde{C}_{w,k}$ for different $k$ and same $w$ are all coupled to each other at the quadratic order.", "This is due to the broken translational along the $x$ direction.", "To compute the mass-squared correction of any of the modes we should compute the mass matrix.", "However this matrix is infinite dimensional.", "We will be interested in the mass-squared corrections of the mode $C_{0,0}$ .", "This we do numerically.", "In this paper for numerical simplicity we just compute the correction term $\\Sigma ^{2}(0,0,\\beta ,q)$ , reserving a more detailed numerical analysis for the future.", "Now, after doing the sum over $m$ in the first term of (REF ), $\\begin{split}\\Sigma ^2(k,k^{^{\\prime }},\\beta ,q)&={1\\over 2}\\sum _{n\\ne 0}\\left[\\frac{F_1(k,k^{^{\\prime }},n,n)}{\\sqrt{(2n-1)}}\\left({1\\over 2}+\\frac{1}{e^{\\sqrt{(2n-1)q}\\beta }-1}\\right)+\\frac{\\tilde{F}_1(k,k^{^{\\prime }},n,n)}{(2\\pi m)^2}\\right]\\\\&= \\Sigma ^2_{vac}+\\Sigma ^2_{\\beta }\\end{split}$ $\\Sigma ^2_{vac}$ is the temperature independent piece.", "This term is potentially ultraviolet divergent.", "In the following sections we shall consider the the theory on the intersecting $D1$ branes.", "This theory is obtained from a finite ${\\cal N}=8$ SYM in two dimensions by giving an expectation value to one of the scalars $\\Phi _1^3=qx$ .", "Supersymmetry in the intersecting $D1$ brane theory is completely broken by the background.", "However the action is supersymmetric and finiteness of the ${\\cal N}=8$ theory implies that the the theory on the intersecting $D1$ branes must also be ultraviolet finite.", "We will see that for the two point functions computed later the ultraviolet divergences cancel between the contributions from the boson and the fermion loops.", "There is also an infrared divergence in (REF ) that comes from the second term for $m=0$ .", "The IR divergence is due to the massless $\\tilde{D}_{m,n}$ modes in the loop.", "To treat these divergences we shall follow the procedure outlined in the introduction.", "A similar computation leading to (REF ), (REF ) and (REF ) can also be performed as follows.", "Expanding both the background and the fluctuation fields using same basis functions that are the eigenfunctions of ${\\cal O}_0$ i.e.", "to the lowest order the fluctuation wave functions, $\\delta A_n(x)=A_n(x)$ , $\\delta \\phi _n(x)=\\phi _n(x)$ , $\\delta \\tilde{A}_n(x)=\\tilde{A}_n(x)$ and $\\delta \\tilde{\\phi }_n(x)=\\tilde{\\phi }_n(x)$ , $\\begin{split}S&=S_B+S_{\\delta }\\\\&=-\\frac{1}{2g^2}\\sum _{w,k}\\left(|C_{w,k}|^2 (\\omega _w^2+\\lambda _k) + |\\tilde{C}_{w,k}|^2 \\omega _w^2 \\right)-\\frac{1}{2g^2}\\sum _{m,n}\\left(|D_{m,n}|^2 (\\omega _m^2+\\lambda _n)+ |\\tilde{D}_{m,n}|^2 \\omega _m^2 \\right)\\\\&+ I + {\\rm background~fields~of~quartic~order}\\end{split}$ where the interaction term $I$ , is $\\begin{split}I=&-\\frac{N}{2g^2}\\sum _{m,m^{^{\\prime }},n,n^{^{\\prime }}}\\sum _{w,w^{^{\\prime }},k,k^{^{\\prime }}}\\left(C_{w,k}C_{w^{^{\\prime }},k^{^{\\prime }}}D_{m,n}D_{m^{^{\\prime }},n^{^{\\prime }}}F_1(k,k^{^{\\prime }},n,n^{^{\\prime }})\\right.\\\\&+\\left.C_{w,k}C_{w^{^{\\prime }},k^{^{\\prime }}}\\tilde{D}_{m,n}\\tilde{D}_{m^{^{\\prime }},n^{^{\\prime }}}\\tilde{F}_1(k,k^{^{\\prime }},n,n^{^{\\prime }})+\\tilde{C}_{w,k}\\tilde{C}_{w^{^{\\prime }},k^{^{\\prime }}}{D}_{m,n}{D}_{m^{^{\\prime }},n^{^{\\prime }}}{F^{^{\\prime }}}_1(k,k^{^{\\prime }},n,n^{^{\\prime }})\\right.\\nonumber \\\\&+\\left.\\tilde{C}_{w,k}\\tilde{C}_{w^{^{\\prime }},k^{^{\\prime }}}\\tilde{D}_{m,n}\\tilde{D}_{m^{^{\\prime }},n^{^{\\prime }}}\\tilde{F^{^{\\prime }}}_1(k,k^{^{\\prime }},n,n^{^{\\prime }})+2 C_{w,k}\\tilde{C}_{w^{^{\\prime }},k^{^{\\prime }}}{D}_{m,n}{D}_{m^{^{\\prime }},n^{^{\\prime }}}{F^{^{\\prime \\prime }}}_1(k,k^{^{\\prime }},n,n^{^{\\prime }})\\right.\\nonumber \\\\&+ \\left.", "2 C_{w,k}\\tilde{C}_{w^{^{\\prime }},k^{^{\\prime }}}\\tilde{D}_{m,n}\\tilde{D}_{m^{^{\\prime }},n^{^{\\prime }}}\\tilde{F^{^{\\prime \\prime }}}_1(k,k^{^{\\prime }},n,n^{^{\\prime }})\\right)\\delta _{m+{m^{^{\\prime }}}+{w}+{w^{^{\\prime }}}, 0}\\end{split}$ where, $N=\\sqrt{q}/\\beta $ .", "Here again we have dropped the terms linear in fluctuations as they do not contribute to the $1PI$ effective action.", "The terms cubic in fluctuations have also not been included as they do not contribute at the one-loop order.", "The tree-level $D_{m,n}$ and $\\tilde{D}_{m,n}$ propagators are then $\\left\\langle D_{m,n}D_{m^{^{\\prime }},n^{^{\\prime }}} \\right\\rangle =g^2\\frac{\\delta _{m,-m^{^{\\prime }}}\\delta _{n,n^{^{\\prime }}}}{\\left[\\omega _m^2+\\lambda _n\\right]}\\mbox{~~;~~}\\left\\langle \\tilde{D}_{m,n}\\tilde{D}_{m^{^{\\prime }},n^{^{\\prime }}} \\right\\rangle =g^2\\frac{\\delta _{m,-m^{^{\\prime }}}\\delta _{n,n^{^{\\prime }}}}{\\omega _m^2}$ With this the one-loop two point amplitudes are same as equations (REF ), (REF ) and (REF ).", "Henceforth in the following sections we will denote both the background and the fluctuation modes as $C_{w,k}$ ." ], [ "Intersecting D1 branes at finite temperature", "We have computed the effective mass-squared of the tachyons as function of temperature in section () for a simplified theory with only one scalar field and no fermions.", "The corrections are infrared divergent due to the presence of the tree-level massless modes in the loops.", "One way of dealing with the problem is by computing the mass-squared corrections to the tree-level massless modes $\\tilde{C}_{w,k}$ , at finite temperature at the one-loop level.", "With the mass-squared corrections, the tree-level massless modes become massive at one-loop level at finite temperature.", "The temperature-dependent effective mass-squared of the tree-level massless modes shift their propagator by $\\omega _m^{-2} \\rightarrow (\\omega ^2_m + m^2(\\beta ))^{-1}$ .", "Now we can use this shifted propagator to calculate the finite effective mass-squared of the tree-level tachyons at finite temperature.", "The first amplitude in (REF ) has a purely temperature independent part and a purely temperature dependent part.", "The temperature dependent part is exponentially damped, hence finite even for large values of the momentum $n$ .", "The temperature independent part however has a non-convergent sum over the momentum $n$ and hence give rise to Ultraviolet divergence.", "The problem of UV divergence can be dealt with by introducing suitable regularization scheme but this in turn renders the finite answer for the mass-squared corrections regularization-dependent.", "Hence there is no unique answer for the effective mass-squared of the tachyon.", "This indicates that the calculation should be done in a framework where cancellation of UV divergences are possible.", "Hence we work in the framework of a originally supersymmetric theory and should consider the contributions from the bosonic as well as fermionic degrees of freedom.", "In this set-up the UV divergences are expected to cancel among the amplitudes containing bosons and fermions in the loop.", "In this section we thus study the finite temperature effects for the full $D1$ brane theory (REF ).", "As seen in section an intersecting brane configuration with only one non-zero angle is given by the background solution $\\Phi _1^3=qx$ and $A=0$ .", "In the following we will study the tachyon mass-squared as a function of the temperature for the theory (REF ) including all the other bosonic fields and the fermions.", "We will compute contribution from Bosons and the Fermions towards the tachyon mass-squared correction separately below." ], [ "Bosons", "In this section we compute the one-loop correction to the tree-level tachyon mass-squared due to the Bosons in the loop.", "To do this we must first write the mode expansions for the individual fields.", "The mode expansion for $(A^2_x,\\Phi _1^1)$ is given in (REF ) in section using the eigenfunctions that have been worked out in section .", "The $(A^1_x,\\Phi _1^2)$ fields satisfy the same mode expansion.", "The only distinction between this and the earlier mode expansion is the sign in front of the eigenfunctions $A_k(x)$ .", "Thus $\\begin{split}\\zeta ^{^{\\prime }}(x,\\tau )&=N^{1/2}\\sum _{w,k}\\left(C^{^{\\prime }}_{w,k}\\zeta _k(x)e^{i\\omega _w\\tau }+ \\tilde{C}^{^{\\prime }}_{w,k}\\tilde{\\zeta }_k(x)e^{i\\omega _w\\tau }\\right)\\\\&=N^{1/2}\\sum _{w,k} \\left(C^{^{\\prime }}_{1w,k} \\left(\\begin{array}{c}-A_k(x)\\\\\\phi _k(x)\\end{array}\\right)e^{i\\omega _w\\tau } + \\tilde{C}^{^{\\prime }}_{w,k} \\left(\\begin{array}{c}-\\tilde{A}_k(x)\\\\\\tilde{\\phi }_k(x)\\end{array}\\right)e^{i\\omega _w\\tau }\\right)\\end{split}$ where $\\zeta ^{^{\\prime }}(x,\\tau )=\\left(\\begin{array}{c}A^1_x(x, \\tau )\\\\\\Phi _1^2(x, \\tau )\\end{array}\\right)$ .", "Similarly since $\\Phi ^1_I$ and $\\Phi ^2_I$ $(I\\ne 1)$ are harmonic oscillators in the $x$ direction and $A^3_x$ and $\\Phi _I^3$ (for all $I$ ) are just plane waves, we have the following mode expansions $\\Phi _I^1(x, \\tau )=N^{1/2} \\mathcal {N}^{^{\\prime }}(n)\\sum _{m,n}\\Phi _I^1(n, m) e^{-qx^2/2}H_n(\\sqrt{q}x)e^{i\\omega _m\\tau },~~ \\lbrace I\\ne 1\\rbrace \\\\\\Phi _I^3(x, \\tau )=\\frac{N^{1/2}}{\\sqrt{q}} \\sum _{m} \\int \\frac{dl}{2\\pi } \\Phi _I^3(l, m)e^{i(\\omega _m\\tau +lx)}, \\lbrace I=1,\\cdots ,8\\rbrace \\\\A^3_x(x,\\tau )= \\frac{N^{1/2}}{\\sqrt{q}} \\int \\frac{dl}{2\\pi }\\sum _{m} A^3_x(m,l)e^{i\\omega _m\\tau +ilx}\\\\$ where $H_{n}(\\sqrt{q}x)$ are the Hermite polynomials and $e^{-qx^2/2}H_n(\\sqrt{q}x)$ are the harmonic oscillator wave functions with eigenvalue $\\gamma _n=(2n+1)q$ .", "The normalization $\\mathcal {N}^{^{\\prime }}(n)=1/\\sqrt{\\sqrt{\\pi }2^n n!", "}$ .", "The propagators and the interaction vertices are listed in the appendix (REF ).", "With these we now write down the contribution to the one loop mass-squared corrections to the background fields.", "The bosonic contributions to the one loop mass-squared corrections at finite temperature can be collected in two groups, namely the ones coming from the four-point vertices and the ones coming from the three-point vertices.", "The bosonic four-point vertices listed in (REF ) together with their corresponding propagators give, $\\Sigma ^1(w,w^{^{\\prime }},k,k^{^{\\prime }},\\beta ,q)&=& {1\\over 2}N \\sum _m \\left[\\sum _n\\left(\\frac{F_1(k,k^{^{\\prime }},n,n)}{\\omega _m^2 + \\lambda _n} + \\frac{\\tilde{F}_1(k,k^{^{\\prime }},n,n)}{\\omega _m^2}+ \\frac{7 F_2(k,k^{^{\\prime }},n,n)}{\\omega _m^2 + \\gamma _n}\\right)\\right.", "\\nonumber \\\\&+& \\left.", "\\int \\frac{dl}{(2\\pi \\sqrt{q})}\\left(\\frac{7 F^{^{\\prime }}_2(k,k^{^{\\prime }},l,-l)}{\\omega _m^2 + l^2}+\\frac{F^{^{\\prime }}_3(k, k^{^{\\prime }},l,-l)}{\\omega _m^2+l^2}\\right)\\right.\\nonumber \\\\&+&\\left.", "\\int \\frac{dl}{2 \\pi \\sqrt{q}} \\frac{F_3(k,k^{^{\\prime }},l,-l)}{\\omega _m^2}\\right] \\delta _{w+w^{^{\\prime }}}$ The Feynman diagrams that constitute the correction (REF ) are given in figure REF .", "In (REF ) the first term represented by the Feynman diagram in figure REF (a) consists of the three-point vertex (REF ) and receives contributions from the $C_{m,n}$ modes with the propagators (REF ) while the second term whose Feynman diagram is given is figure REF (b) comes from the four-point vertex (REF ) and bears contributions from the massless modes $\\tilde{C}_{m,n}$ in the loop with propagator (REF ).", "The third term with the Feynman diagram figure REF (c) comprises of the four-point vertex (REF ) having the seven massive fields $\\Phi ^{1,2}_I(m,n)$ for $I\\ne 1$ with propagators given in (REF ).", "Figure: Feynman diagrams for the amplitudes with four-point vertices.The fourth term in (REF ) is represented by the Feynman diagram figure REF (d) and is the amplitude for the vertex (REF ) which comprise of the seven fields $\\Phi ^3_I$ , for $I\\ne 1$ , with propagator (REF ).", "The fifth term have contributions from the fields $\\Phi ^3_1$ with propagators (REF ) and is depicted in the Feynman diagram figure REF (f) while the sixth term bears the massless gauge field $A^3_x(m,l)$ in the loop with propagator (REF ) and the relevant Feynman diagram given in figure REF (e).", "Similarly, the three-point bosonic vertices listed in (REF ) combined with their respective propagators constitute the one-loop bosonic mass-squared corrections at finite temperature, viz.", "$\\Sigma ^2(w,w^{^{\\prime }},k,k^{^{\\prime }},\\beta ,q)&=&-{1\\over 2}qN \\sum _{m,n}\\left[\\int \\frac{dl}{2 \\pi \\sqrt{q}}\\frac{F_4(k,l,n)F^{*}_4(k^{^{\\prime }},l,n)}{(\\omega _m^2+\\lambda _n)\\omega _{m^{^{\\prime }}}^2}+ \\int \\frac{dl}{2 \\pi \\sqrt{q}}\\frac{\\tilde{F}_4(k,l,n)\\tilde{F}^{*}_4(k^{^{\\prime }},l,n)}{\\omega _m^2 \\omega _{m^{^{\\prime }}}^2}\\right.\\nonumber \\\\&+&\\left.\\int \\frac{dl}{2\\pi \\sqrt{q}}\\left(\\frac{7 F_5(k,l,n)F^{*}_5(k^{^{\\prime }},-l,n)}{(\\omega _m^2+\\gamma _n)(\\omega _{m^{^{\\prime }}}^2+l^2)}+\\frac{F^{^{\\prime }}_5(k,l,n) F^{^{\\prime }*}_5(k^{^{\\prime }},-l,n)}{(\\omega _m^2+\\lambda _n)(\\omega _{m^{^{\\prime }}}^2+l^2)}\\right)\\right.\\nonumber \\\\&+&\\left.\\int \\frac{dl}{2\\pi \\sqrt{q}}\\frac{\\tilde{F}^{^{\\prime }}_5(k,l,n) \\tilde{F}^{^{\\prime }*}_5(k^{^{\\prime }},-l,n)}{(\\omega _m^2)(\\omega _{m^{^{\\prime }}}^2+l^2)}\\right]\\delta _{w+w^{^{\\prime }}}$ where, $w=m+m^{^{\\prime }}$ .", "In (REF ), the first amplitude gets contributions from the fields $C_{m,n}$ with propagator (REF ) and $A^3_x(m^{^{\\prime }},l)$ with propagator (REF ) in the loop with the three-point vertex $F_4(k,l,n)$ given in (REF ).", "The corresponding Feynman diagram is given in figure REF (a).", "In the second amplitude, whose Feynman diagram is given by figure REF (b) comprises of the three-point vertex $\\tilde{F}_4(k,l,n)$ given in (REF ) which gets contributions from the massless modes $\\tilde{C}_{m,n}$ with propagator (REF ) and the massless gauge fields $A^3_x(m,l)$ .", "Figure: Feynman diagrams for the amplitudes with three-point vertices.The third amplitude in (REF ) with the vertices $F_5(k,l,n)$ , has contributions from the pairs of fields $\\Phi ^{(1,2)}_I(m,n)$ with propagator (REF ) and $\\Phi ^3_I(m,l)$ with propagator (REF ) and the Feynman diagram drawn in figure REF (c).", "The fourth amplitude consisting of the three-point vertex $F^{^{\\prime }}_5(k,l,n)$ receives contributions in the loop from $C_{m,n}$ .", "and $\\Phi ^3_1(m,l)$ with propagator (REF ), while the fifth one with vertex $\\tilde{F}^{^{\\prime }}_5(k,l,n)$ comprises of the loop fields $\\tilde{C}_{m,n}$ and $\\Phi ^3_1(m,l)$ .", "The Feynman diagrams for the fourth and fifth terms in (REF ) are given in figure REF (d) and figure REF (e) respectively.", "We are interested in the two point function for the $w=w^{^{\\prime }}=k=k^{^{\\prime }}=0$ mode because this mode is the tachyon with the lowest $mass^2$ value.", "The finite temperature correction involves sum over the Matsubara frequency.", "Upon performing the Matsubara sum (except on the massless modes), the correction (REF ) for $w=w^{^{\\prime }}=0$ can be written as $\\Sigma ^1(0,0,k,k^{^{\\prime }},\\beta ,q)&=&{1\\over 2}\\left[\\sum _n\\frac{F_1(k,k^{^{\\prime }},n,n)}{\\sqrt{(2n-1)}}\\left({1\\over 2}+\\frac{1}{e^{\\beta \\sqrt{(2n-1)q}}-1}\\right)\\right.\\nonumber \\\\&+& N\\left.\\sum _m \\left(\\sum _n\\frac{\\tilde{F}_1(k,k^{^{\\prime }},n,n)}{\\omega ^2_m}+ \\int \\frac{dl}{2 \\pi \\sqrt{q}}\\frac{F_3(k,k^{^{\\prime }},l-l)}{{\\omega ^2_m}}\\right)\\right.\\nonumber \\\\&+& \\left.\\sum _n\\left(\\frac{7 F_2(k,k^{^{\\prime }},n,n)}{\\sqrt{(2n+1)}}\\left({1\\over 2}+\\frac{1}{e^{\\beta \\sqrt{(2n+1)q}}-1}\\right)\\right)\\right.\\nonumber \\\\&+& \\left.\\left(\\int \\frac{dl}{2\\pi \\sqrt{q}}\\frac{(7+1/2)\\delta _{k,k^{^{\\prime }}}}{(l/\\sqrt{q})}\\left({1\\over 2}+\\frac{1}{e^{\\beta l}-1}\\right)\\right)\\right]$ The correction given in (REF ), after the Matsubara sum (leaving out the massless modes) assumes the form $&&\\Sigma ^2(0,0,k,k^{^{\\prime }},\\beta ,q)\\nonumber \\\\&=& -{1\\over 2}\\sum _{n}\\left[\\int \\frac{dl}{2 \\pi \\sqrt{q}} \\frac{F_4(k,l,n)F^{*}_4(k^{^{\\prime }},l,n)}{(2n-1)}\\left[\\left(\\sum _m\\frac{\\sqrt{q}}{\\beta \\omega _m^2} -\\frac{1}{\\sqrt{2n-1}}\\left(\\frac{1}{2}+ \\frac{1}{e^{\\sqrt{(2n-1)q}\\beta }-1}\\right)\\right)\\right]\\right.\\nonumber \\\\&+&\\left.", "qN\\int \\frac{dl}{2 \\pi \\sqrt{q}} \\sum _{m}\\frac{\\tilde{F}_4(k,l,n)\\tilde{F}^{*}_4(k^{^{\\prime }},l,n)}{\\omega _m^4}\\right.\\nonumber \\\\&+&\\left.\\int \\frac{dl}{2 \\pi \\sqrt{q}}\\left[\\frac{7 F_5(k,l,n) F^{*}_5(k^{^{\\prime }},-l,n)}{(l/\\sqrt{q})^2-(2n+1)} \\left(\\frac{1}{\\sqrt{2n+1}}\\left({1\\over 2}+ \\frac{1}{e^{\\sqrt{(2 n+1)q}\\beta }-1}\\right)\\right.\\right.\\right.\\nonumber \\\\&-& \\left.\\left.\\left.\\frac{1}{(l/\\sqrt{q})}\\left({1\\over 2}+ \\frac{1}{e^{l\\beta }-1}\\right)\\right)\\right]\\right.\\nonumber \\\\&+&\\left.\\int \\frac{dl}{2 \\pi \\sqrt{q}}\\left[\\frac{F^{^{\\prime }}_5(k,l,n)F^{^{\\prime }*}_5(k^{^{\\prime }},-l,n)}{(l/\\sqrt{q})^2-(2n-1)} \\left(\\frac{1}{\\sqrt{2n-1}}\\left({1\\over 2}+ \\frac{1}{e^{\\sqrt{(2 n-1)q}\\beta }-1}\\right)\\right.\\right.\\right.\\nonumber \\\\&-& \\left.\\left.\\left.\\frac{1}{(l/\\sqrt{q})}\\left({1\\over 2}+ \\frac{1}{e^{l\\beta }-1}\\right)\\right)\\right]\\right.\\nonumber \\\\&+&\\left.\\int \\frac{dl}{2\\pi \\sqrt{q}}\\frac{\\tilde{F}^{^{\\prime }}_5(k,l,n)\\tilde{F}^{^{\\prime }*}_5(k^{^{\\prime }},-l,n)}{(l/\\sqrt{q})^2}\\left(\\sum _m\\frac{\\sqrt{q}}{\\beta \\omega _{m}^2}-\\frac{1}{(l/\\sqrt{q})}\\left({1\\over 2}+\\frac{1}{e^{l\\beta }-1}\\right)\\right)\\right]\\nonumber \\\\$ In both (REF ) and (REF ), the Matsubara sums give rise to two different kinds of terms, the zero temperature quantum corrections and the temperature dependent part.", "While the zero-temperature parts are independent of $q$ , the temperature dependent parts are functions of both $\\beta $ and $q$ .", "The temperature dependent terms are damped by exponential factors and are hence finite.", "The temperature independent parts however have problems of divergences.", "We shall discuss these problems in the section ()." ], [ "Fermions", "We will now compute the contribution to the tachyon two point amplitude due to fermionic fluctuations.", "The fermions in this contribution only appear in the internal loops.", "Consider first the free and the part of the action (REF ) in appendix that couples to $\\Phi ^3_1$ .", "The corresponding terms are, $\\begin{split}{\\cal L}^{2^{\\prime }}_{1+1}=&\\frac{1}{2}\\left(\\psi ^{aT}_L \\partial _0 \\psi ^{a}_L+\\psi ^{aT}_R \\partial _0 \\psi ^a_R+\\psi ^{aT}_L\\partial _x \\psi ^{a}_L-\\psi ^{aT}_R \\partial _x \\psi ^a_R\\right)\\\\& +\\Phi ^3_1\\left(\\psi ^{1T}_R\\alpha ^T_1\\psi ^{2}_L-\\psi ^{2T}_R\\alpha ^T_1\\psi ^{1}_L\\right)\\end{split}$ We will now call the components of $\\psi ^a_L$ and those of $\\psi ^a_R$ as $L_i^{a}$ and $R_i^{a}$ respectively, where $a$ is the gauge index and $i=1,\\cdots ,8$ is the fermion index.", "In this notation, $\\begin{split}{\\cal L}^{2^{\\prime }}_{1+1}=&\\frac{1}{2}\\left(L^a_i \\partial _0 L_i^a+ R^a_i \\partial _0 R_i^a+ L_i^a \\partial _x L_i^a-R_i^a \\partial _x R_i^a\\right)\\\\& +\\Phi ^3_1\\left(\\psi ^{1T}_R\\alpha ^T_1\\psi ^{2}_L-\\psi ^{2T}_R\\alpha ^T_1\\psi ^{1}_L\\right)\\end{split}$ With the background value of $\\Phi ^3_1=qx$ and putting in the value of $\\alpha _1$ from (REF ), we proceed to diagonalize the action.", "This amounts to solving for the eigenfunctions of $L_i^{a}$ and $R_i^{a}$ .", "We have the following sets of equations from (REF ).", "$(\\partial _0+\\partial _x )L_1^1+qx R_8^2=0\\\\(-\\partial _0+\\partial _x) R_8^2+qx L_1^1 =0$ $(\\partial _0+\\partial _x) L_1^2-qx R_8^1=0\\\\(-\\partial _0+\\partial _x) R_8^1-qx L_1^2 =0$ There are sixteen such sets of decoupled equations.", "The eight sets $(L_1^1,R_8^2), (L_4^1,R_5^2), (L_6^1,R_3^2), (L_7^1,R_2^2),(L_2^2,R_7^1), (L_3^2,R_6^1),(L_5^2,R_4^1), (L_8^2,R_1^1)$ satisfy identical coupled equations like (REF , ).", "The remaining eight sets $(L_1^2,R_8^1), (L_4^2,R_5^1), (L_6^2,R_3^1), (L_7^2,R_2^1),(L_8^1,R_1^2), (L_5^1,R_4^2), (L_3^1,R_6^2), (L_2^1,R_7^2)$ satisfy coupled equations like (REF , ).", "For future convenience we denote the left and right fermions of (REF )as $L(x,t)$ and $R(x,t)$ and those of (REF ) as $\\hat{L}(x,t)$ and $\\hat{R}(x,t)$ .", "Now we solve the differential equations.", "The set of coupled differential equations (REF , ) satisfied by the set of fermionic fields in (REF ) can be promoted to the status of second order differential equations as $(- \\partial ^2_0 + \\partial ^2_x) L(x,t) + q R(x,t) - q^2 x^2 L(x,t) = 0\\\\(- \\partial ^2_0 + \\partial ^2_x) R(x,t) + q L(x,t) - q^2 x^2 R(x,t) = 0$ It is important to note that the set of fermions in (REF ) also satisfy the same set of equations as (REF , ) with $L(x,t)$ and $R(x,t)$ being replaced by $\\hat{L}(x,t)$ and $- \\hat{R}(x,t)$ respectively.", "Let us discuss the solutions to the equations (REF , ).", "Adding (REF ) and () and subtracting () from (REF ) we get the following set of equations $(- \\partial ^2_0 + \\partial ^2_x) F(x,t) + q F(x,t) - q^2 x^2 F(x,t) = 0\\\\(- \\partial ^2_0 + \\partial ^2_x) G(x,t) - q G(x,t) - q^2 x^2 G(x,t) = 0$ where $F(x,t) = L(x,t) + R(x,t)$ and $G(x,t) = L(x,t) - R(x,t)$ .", "In this context let us point out that one can construct similar combinational functions with the fields in (REF ) viz.", "$\\hat{F}(x,t) = \\hat{L}(x,t) + \\hat{R}(x,t)$ and $\\hat{G}(x,t) = \\hat{L}(x,t) - \\hat{R}(x,t)$ , where $\\hat{F} = G$ and $\\hat{G} = F$ .", "As in the case of bosonic fields, the fermionic differential equations can also be analyzed in the asymptotic limit and the fermionic eigenfunctions can be written as $L^a_i(t,x) = e^{- \\frac{q x^2}{2}} \\tilde{L}^a_i (x,t) \\\\R^a_i(t,x) = e^{- \\frac{q x^2}{2}} \\tilde{R}^a_i (x,t)$ Note that although there are two different sets of coupled differential equations; one being (REF , ) satisfied by (REF ) and the other (REF , ) satisfied by (REF ), both sets of equations when recombined give rise to the same differential equations as (REF ) for the left moving fermions and () for the right moving fermions.", "The eigenfunctions from (REF ) and () that also satisfies the first order equations (REF ) are given by $\\psi _n(x) = \\left(\\begin{array}{c}L_n(x)\\\\R_n(x)\\end{array}\\right)$ The corresponding eigenvalue is $=-i\\sqrt{\\lambda ^{^{\\prime }}}=-i\\sqrt{2nq}$ .", "Similarly for the set of fermions given in (REF ) and obeying the equations of motion (REF ), the eigenfunctions can be obtained by repeating the above procedure and we get $\\hat{\\psi }_n(x) = \\left(\\begin{array}{c}\\hat{L}_n(x)\\\\\\hat{R}_n(x)\\end{array}\\right)= \\left(\\begin{array}{c}L_n(x)\\\\-R_n(x)\\end{array}\\right)$ where, $\\begin{split}&L_n(x) = \\hat{L}_n(x) = {\\cal N}_F e^{- \\frac{q x^2}{2}}\\left(- \\frac{i}{\\sqrt{2n}} H_{n}(\\sqrt{q} x)+ H_{n-1} (\\sqrt{q} x)\\right)\\\\&R_n(x) = - \\hat{R}_n(x) ={\\cal N}_F e^{- \\frac{q x^2}{2}}\\left(- \\frac{i}{\\sqrt{2n}} H_{n}(\\sqrt{q} x)- H_{n-1} (\\sqrt{q} x)\\right).\\end{split}$ $H_n(\\sqrt{q}x)$ are the Hermite Polynomials.", "The normalization ${\\cal N}_F=\\sqrt{\\sqrt{\\pi }2^{n+1} (n-1)!", "}$ .", "We now list some important relations satisfied by the eigenfunctions $\\sqrt{q}\\int dx~\\psi ^{\\dagger }_n(x) \\psi _{n^{^{\\prime }}}(x) = \\sqrt{q}\\int dx \\left(L^*_n(x)L_{n^{^{\\prime }}}(x) + R^*_n(x) R_{n^{^{\\prime }}}(x)\\right) = \\delta _{n,n^{^{\\prime }}}.\\\\\\sqrt{q}\\int dx~L^*_n(x) L_{n^{^{\\prime }}}(x) = \\sqrt{q}\\int dx~R^*_n(x) R_{n^{^{\\prime }}}(x) = {1\\over 2}\\delta _{n,n^{^{\\prime }}}\\\\\\sqrt{q}\\int dx~\\psi ^{T}_n(x) \\psi _{n^{^{\\prime }}}(x)=\\sqrt{q}\\int dx \\left(L_n(x) L_{n^{^{\\prime }}}(x)+ R_n(x) R_{n^{^{\\prime }}}(x)\\right) = 0\\\\\\sqrt{q}\\int dx~\\psi ^{\\dagger }_n(x) \\psi ^{*}_{n^{^{\\prime }}}(x)=\\sqrt{q}\\int dx \\left(L^{*}_n(x) L^{*}_{n^{^{\\prime }}}(x)+ R^{*}_n(x) R^{*}_{n^{^{\\prime }}}(x)\\right) = 0$ With the eigenfunctions as defined above, we can now write down the mode expansions for the sixteen pairs defined in (REF ) and (REF ).", "For example we write, $\\left(\\begin{array}{c}L_1^1(x,\\tau )\\\\R_{8}^2(x,\\tau )\\end{array}\\right)= N^{3/4}\\sum ^{\\infty }_{n,m = \\infty }\\left(\\theta _1(m,n)e^{i\\omega _m\\tau } \\left(\\begin{array}{c}L_n(x)\\\\R_n(x)\\end{array}\\right)+ \\theta ^*_1(m,n)e^{-i\\omega _m\\tau }\\left(\\begin{array}{c}L^*_n(x)\\\\R^*_n(x)\\end{array}\\right)\\right)$ where we have used $\\tau = it$ and $N=\\sqrt{q}/{\\beta }$ .", "For each doublet in (REF ) and (REF ) we have a corresponding set of modes $(\\theta _j(m,n),\\theta _j^{*}(m,n))$ .", "So the index $j$ on the $\\theta $ 's run from $1\\cdots 16$ .", "Since $L_i^3$ and $R_i^3$ do not couple to the $\\Phi ^3_1 = qx$ background, they just satisfy the plane wave equations, $(\\partial _0+\\partial _x )L_i^3=0\\mbox{~~~;~~~}(-\\partial _0+\\partial _x) R_i^3=0$ Hence their mode expansions are $\\begin{split}&L_i^3(x,\\tau )=N^{3/4}\\sum _{m}\\frac{1}{\\sqrt{q}}\\int \\frac{dk}{(2\\pi )}L^3_{i}(m,k)e^{i(kx+\\omega _{m}\\tau )} \\\\&R_i^3(x,\\tau )=N^{3/4}\\sum _{m}\\frac{1}{\\sqrt{q}}\\int \\frac{dk}{(2\\pi )}R^3_{i}(m,k)e^{i(kx+\\omega _{m}\\tau )}\\rm {~~~}\\mbox{for all $i=1,\\cdots ,8$}\\end{split}$ where $L^{3*}_i(m,k) = L^{3}_i(m,k)$ .", "Using the orthogonality relations and the mode expansions the quadratic part of the fermionic action can thus be written as, $\\begin{split}S_f=&\\frac{N^{1/2}}{g^2}\\left[\\sum _{m,n,j=1}^{j=16}\\theta _j(m,n)(i\\omega _m-\\sqrt{\\lambda ^{^{\\prime }}_n})\\theta _j^*(m,n)\\right.\\\\&+\\left.", "\\frac{1}{2\\sqrt{q}}\\int \\frac{dk}{2\\pi }\\sum _{m,i=1}^{i=8}L^3_i(m,k)(i \\omega _m+k)L^{3*}_i(m,k)\\right.\\\\&+\\left.", "\\frac{1}{2\\sqrt{q}}\\int \\frac{dk}{2\\pi }\\sum _{m,i=1}^{i=8}R^3_i(m,k)(i\\omega _m-k)R^{3*}_i(m,k)\\right]\\end{split}$ With these we can write down the fermionic propagators as listed in the appendix.", "We now turn to the interaction terms.", "These are the terms in the fermionic action (REF ) that couple to the background fields $\\Phi ^1_1$ and $A^2_x$ that we simply call $\\phi _B$ and $A_B$ respectively as before.", "$\\mathcal {L}_{f} = \\phi _B\\psi ^{2T}_R\\alpha _1^{T}\\psi ^{3}_L-\\phi _B\\psi ^{3T}_R\\alpha _1^{T}\\psi ^{2}_L+A_B\\psi _L^{1T}\\psi _L^3-A_B\\psi _R^{1T}\\psi _R^3$ The corresponding vertices have been worked out in appendix.", "We thus have the following contribution to the tachyon two-point amplitude.", "$\\Sigma ^3(w,w^{^{\\prime }},k,k^{^{\\prime }},\\beta ,q)&=& -(8 N)\\sum _{n,m,m^{^{\\prime }}}\\int \\frac{dl}{2\\pi \\sqrt{q}}\\frac{1}{(i\\omega _m-\\sqrt{\\lambda _n^{^{\\prime }}})}\\nonumber \\\\&\\times &\\left[\\frac{F^R_6(k,n,l)F^{R*}_6(k^{^{\\prime }},n,l)}{(i\\omega _{m^{^{\\prime }}}+l)}+\\frac{F^L_6(k,n,l)F^{L*}_6(k^{^{\\prime }},n,l)}{(i\\omega _{m^{^{\\prime }}}-l)}\\right.\\nonumber \\\\&+&\\left.\\frac{F^L_7(k,n,l)F^{L*}_7(k^{^{\\prime }},n,l)}{(i\\omega _{m^{^{\\prime }}}+l)}+\\frac{F^R_7(k,n,l)F^{R*}_7(k^{^{\\prime }},n,l)}{(i\\omega _{m^{^{\\prime }}}-l)}\\right.\\nonumber \\\\&+&\\left.", "\\frac{F^R_6(k,n,l)F^{L*}_7(k^{^{\\prime }},n,l)+ F^{R*}_6(k,n,l)F^L_7(k^{^{\\prime }},n,l)}{(i\\omega _{m^{^{\\prime }}}+l)}\\right.\\nonumber \\\\&+&\\left.\\frac{F^L_6(k,n,l)F^{R*}_7(k^{^{\\prime }},n,l)+ F^{L*}_6(k,n,l)F^R_7(k^{^{\\prime }},n,l)}{(i\\omega _{m^{^{\\prime }}}-l)}\\right]\\delta _{w+w^{^{\\prime }}}\\nonumber \\\\$ where $w=m+m^{^{\\prime }}$ .", "In (REF ) the Matsubara frequency $\\omega _m =m\\pi /\\beta $ and $m$ is an odd integer due to anti-periodic boundary conditions along the time-cycle for the fermions and $\\omega _{-m}=-\\omega _{m}$ .", "The various diagrams contributing to the amplitude is shown in figures REF and REF .", "Figure REF shows the contractions between $R^2_i$ -$L^1_{9-i}$ and $L^2_i-R^1_{9-i}$ as they are expanded with the same $\\theta $ 's.", "The first term within 3rd bracket in (REF ) is represented by the Feynman diagram in figure REF (a), while the second term by figure REF (b).", "The third and fourth terms are represented by figure REF (c) and figure REF (d) respectively.", "Figure: Feynman diagrams for the amplitudes with three-point fermionic vertices V R/L V^{R/L}and their complex conjugates V *R/L V^{*{R/L}}The fifth and sixth terms in (REF ) results from the “cross”-contraction between the right-moving and left-moving fermions and are depicted in the Feynman diagrams in figures REF (a) REF (b) respectively.", "The functions $F^{R/L}_6$ and $F^{R/L}_7$ are all given in eqns (REF ), (REF ) and (REF ).", "The massive fermions namely $L^{(1,2)}_i$ and $R^{(1,2)}_i$ have their propagators given by (REF ).", "The propagators for the massless fermions which are the 3rd gauge component of the fermionic fields namely $L^3_i$ and $R^3_i$ are given in (REF ).", "Figure: Feynman diagrams showing the cross terms in the amplitude with fermions in the loop.After performing the Matsubara sum in (REF ), the fermionic contribution to the one-loop mass-squared corrections for the tree-level tachyon can be written as $&&\\Sigma ^3(0,0,k,k^{^{\\prime }},\\beta ,q)=\\nonumber \\\\&&(8 N)\\sum _{n}\\left[\\int \\frac{dl}{2\\pi \\sqrt{q}}\\left(\\frac{-\\beta \\tanh \\left(\\frac{\\beta l}{2}\\right)-\\beta \\tanh \\left(\\frac{1}{2} \\beta \\sqrt{2 n q}\\right)}{2 \\left(l+\\sqrt{2 n q}\\right)}\\right)\\right.\\nonumber \\\\&&\\left.\\left[F^R_6(k,n,l)F^{R*}_6(k^{^{\\prime }},n,l) + F^L_7(k,n,l)F^{L*}_7(k^{^{\\prime }},n,l)\\right.\\right.", "\\nonumber \\\\&&\\left.\\left.+ F^R_6(k,n,l)F^{L*}_7(k^{^{\\prime }},n,l) + F^{R*}_6(k,n,l)F^L_7(k^{^{\\prime }},n,l)\\right]\\right.\\nonumber \\\\&+& \\left.", "\\int \\frac{dl}{2 \\pi \\sqrt{q}}\\left(\\frac{-\\beta \\tanh \\left(\\frac{\\beta l}{2}\\right)+\\beta \\tanh \\left(\\frac{1}{2} \\beta \\sqrt{2 n q}\\right)}{2 \\left(l-\\sqrt{2 n q}\\right)}\\right)\\right.\\nonumber \\\\&&+ \\left.\\left[F^L_6(k,n,l)F^{L*}_6(k^{^{\\prime }},n,l) + F^R_7(k,n,l)F^{R*}_7(k^{^{\\prime }},n,l)\\right.\\right.\\nonumber \\\\&&\\left.\\left.+ F^L_6(k,n,l)F^{R*}_7(k^{^{\\prime }},n,l) + F^{L*}_6(k,n,l)F^R_7(k^{^{\\prime }},n,l)\\right]\\right]$ As found in the case of bosonic amplitudes the fermionic counterpart given by (REF ) also can be regrouped into two different parts the zero-temperature quantum corrections and the finite temperature pieces.", "The amplitudes containing fermions in the loop are infrared finite because of anti-periodic boundary conditions imposed on the fermions whereby they pick up temperature dependent mass at tree-level.", "However the fermionic amplitudes (REF ) are ultraviolet divergent.", "The divergence come from the temperature independent pieces in (REF ).", "We shall discuss this problem in details in the following section." ], [ "The Ultraviolet and Infrared Problems", "Each integral as well as the terms bearing the contribution from the massless modes $C_{w,k}$ in (REF ) and (REF ) are infrared divergent for $(\\omega _m=0, l=0)$ .", "Moreover the sums over the momentum $n$ do not converge and integral over the momentum $l$ are log divergent.", "These give rise to ultraviolet divergence in each term in the two-point functions in (REF ), (REF ) and (REF ).", "We deal with the ultraviolet problem first." ], [ "Ultraviolet finiteness of Tachyonic amplitudes", "As mentioned above, every term in the two-point functions (REF ), (REF ) and (REF ) is ultraviolet divergent.", "In the present scenario supersymmetry is completely broken by choice of background, namely, $\\langle \\Phi ^3_1 \\rangle = q x$ , however the equality in the number of bosonic and fermionic degrees of freedom still holds good in the intersecting brane configuration.", "In the ultraviolet limit the degeneracy in the masses of the bosons and fermions is restored and the ultraviolet divergences from the bosonic terms cancel with that from the fermionic terms.", "We now proceed to show this cancellation.", "After performing the Matsubara sum (see Appendix F), the temperature independent part of the bosonic propagators can be written as $\\frac{1}{\\omega _m^2 + \\lambda _n} \\rightarrow \\frac{1}{2 \\sqrt{\\lambda _n}}\\\\\\frac{1}{\\omega _m^2 + \\gamma _n} \\rightarrow \\frac{1}{2 \\sqrt{\\gamma _n}}\\\\\\frac{1}{\\omega _m^2 + l^2} \\rightarrow \\frac{1}{2 l}\\\\\\frac{1}{(\\omega _m^2 + \\lambda _n)(\\omega _m^2 + \\gamma _{n^{^{\\prime }}})} \\rightarrow \\frac{1}{\\gamma _{n^{^{\\prime }}}\\sqrt{\\lambda _n} (\\gamma _{n^{^{\\prime }}}+\\sqrt{\\lambda _n})}\\\\$ Let us now look at the various four-point and three-point vertices.", "We compute the UV limit of the amplitudes for external momentum $k = 0 = k^{^{\\prime }}$ .", "This computation gives rise to Gamma functions summed over their arguments.", "For UV behaviour we take asymptotic expansion of the Gamma functions.", "We first compute the four-pont vertices in the bosonic corrections (REF ) in the limit $n \\rightarrow \\infty $ .", "For one-loop calculation $n=n^{^{\\prime }}$ .", "We use the following properties of $\\Gamma (*)$ functions: $\\lim _{n \\rightarrow \\infty }\\Gamma \\left(n+1\\right) \\sim \\left({\\frac{n}{e}}\\right)^n \\sqrt{2 \\pi n}\\\\\\Gamma \\left(n + {1\\over 2}\\right) = {\\frac{2n!", "}{4^n n!", "}}\\sqrt{\\pi }$ Also the asymptotic expansion of the Hermite Polynomials for $n\\rightarrow \\infty $ gives $e^{-\\frac{x^2}{2}} H_n(x) \\sim \\frac{2^n}{\\sqrt{\\pi }} \\Gamma \\left(\\frac{n+1}{2}\\right)\\cos (\\sqrt{2n}x - n \\frac{\\pi }{2})$ With this asymptotic expansions at our disposal, we can now compute the four-point bosonic vertices.", "The four-point vertex $F_1(0,0,n,n)$ can be written as (using the results of Appendix B,C) $F_1(0,0,n,n) = \\frac{\\mathcal {N}^2(n)}{2 \\sqrt{\\pi }}\\int ^\\infty _\\infty dx~e^{-2 \\sqrt{q}x^2}\\left(6 H_n(\\sqrt{q}x)H_n(\\sqrt{q}x)- 8 n^2 H_{n-2}(\\sqrt{q}x)H_{n-2}(\\sqrt{q}x)\\right)\\nonumber \\\\$ Using the asymptotic expansion of the Hermite polynomials given in (REF ), we get $F_1(0,0,n,n) &=& \\frac{\\mathcal {N}^2(n) \\sqrt{q}}{2 \\sqrt{\\pi }} \\int ^\\infty _{-\\infty } dx~e^{-2 \\sqrt{q} x^2} \\left(6(H_n(\\sqrt{q} x))^2 -8 n^2 (H_{n-2}(\\sqrt{q} x))^2\\right) \\nonumber \\\\&=& \\frac{2^{2n+1}\\mathcal {N}^2(n) \\Gamma ^2\\left(\\frac{n+1}{2}\\right)}{2 \\pi }\\sqrt{q}\\int ^\\infty _{-\\infty } dx~e^{- \\sqrt{q} x^2} \\cos ^2(\\sqrt{2 n q}x - n \\frac{\\pi }{2})\\nonumber \\\\&=& \\left(1+ \\frac{(-1)^n}{e^{2n}}\\right)\\frac{2^{2n} \\left(\\Gamma \\left(\\frac{n+1}{2}\\right)\\right)^2}{\\pi ^2 2^n(4n-2)n(n-2)!", "}.$ In the ultraviolet limit the vertex function $F_1(0,0,n,n)$ reduces to $F_1(0,0,n,n)|_{n\\rightarrow \\infty }=\\left(\\left(1+ \\frac{(-1)^n}{e^{2n}}\\right)\\frac{2^{2n} \\left(\\Gamma \\left(\\frac{n+1}{2}\\right)\\right)^2}{\\pi ^2 2^n (4n-2)n(n-2)!", "}\\right)|_{n\\rightarrow \\infty }= \\frac{1}{2 \\pi \\sqrt{2 n}}.\\nonumber \\\\$ The vertex $\\tilde{F}_1(0,0,n,n)$ has a similar form and in the ultraviolet limit also reduces to $\\tilde{F}_1(0,0,n,n)|_{n \\rightarrow \\infty }=\\left(\\left(1+ \\frac{(-1)^n}{e^{2n}}\\right)\\frac{2^{2n} \\left(\\Gamma \\left(\\frac{n+1}{2}\\right)\\right)^2}{\\pi ^2 2^n (4n-2)(n-1)!", "}\\right)|_{n\\rightarrow \\infty }= \\frac{1}{2 \\pi \\sqrt{2 n}}.$ Putting the external momenta $k= k^{^{\\prime }}=0$ , the four-point vertex function $F_2(0,0,n,n)$ in the limit $n\\rightarrow \\infty $ can be written as $F_2(0,0,n,n) &=&\\frac{1}{2^{n+1} \\sqrt{\\pi } n!", "}\\sqrt{q} \\int ^\\infty _{-\\infty } dx~e^{-2 \\sqrt{q} x^2}(H_n(\\sqrt{q} x))^2 \\\\&=& \\left(1+ \\frac{(-1)^n}{e^{2n}}\\right)\\frac{2^{2n} \\left(\\Gamma \\left(\\frac{n+1}{2}\\right)\\right)^2}{\\pi ^2 2^{n+1} n!", "}$ In the UV limit the vertex function (REF ) reduces to $F_2(0,0,n,n)|_{n\\rightarrow \\infty } =\\left(1+ \\frac{(-1)^n}{e^{2n}}\\right)\\frac{2^{2n} \\left(\\Gamma \\left(\\frac{n+1}{2}\\right)\\right)^2}{\\pi ^2 2^{n+1} n!", "}|_{n\\rightarrow \\infty }=\\frac{1}{\\pi \\sqrt{2n}}.$ The remaining four-point bosonic vertex functions namely $F^{^{\\prime }}_2(0,0,l,-l)$ , $F_3(0,0)$ and $F^{^{\\prime }}_3(0,0,l,-l)$ are independent of $n$ .", "Therefore for a fixed value of the external momenta, namely $k=k^{^{\\prime }}=0$ they can be exactly computed and found to be, $F^{^{\\prime }}_2(0,0,l,-l) =1\\\\F^{^{\\prime }}_3(0,0,l,-l) = {1\\over 2}\\\\F_3(0,0,l,-l) = {1\\over 2}$ The three-point bosonic vertices contain both the continuous momentum $l$ coming from the massless fields as well as the discrete momentum $n$ coming from the fields coupled to the background $ \\left\\langle \\phi ^3_1 \\right\\rangle = q x$ .", "The UV-limit must be taken unambiguously for each term in the amplitude $\\Sigma ^2(0,0,0,0,\\beta ,q)$ .", "Let us first try to compute the amplitude for the three-point vertices $F_4(0,l,n)$ and $\\tilde{F}_4(0,l,n)$ with external momenta $k=0$ .", "The three-point vertex $F_4(0,l,n)$ can be written as $F_4(0,l,n) &=& \\frac{\\mathcal {N}(n)}{\\sqrt{2\\sqrt{\\pi }}} \\sqrt{q} \\int dx~e^{- qx^2} e^{i l x} 4nH_n(\\sqrt{q}x)$ where $\\mathcal {N}(n)$ are the normalization factors for the eigenvectors $\\zeta _n(x)$ and the factor $e^{ilx}$ can be attributed to the presence of the massless field $A^3_x(m,l)$ in the loop.", "Hence the integral in (REF ) is simply the Fourier transform of the Hermite polynomials weighted by Gaussian factor.", "The Fourier transform of a single Hermite Polynomial is given by $\\int ^\\infty _{-\\infty } dx (e^{ilx} e^{- x^2} H_n( x)) = (-1)^n i^n \\sqrt{\\pi } e^{-\\frac{l^2}{4}} l^n$ In the following analysis we will set $q=1$ and will restore factors of $q$ only in the final expressions.", "Using (REF ) in (REF ) the three-point vertex $F_4(0,l,n)$ can be written as $F_4(0,l,n)= \\sqrt{\\pi } (-1)^{n-1} i^{n-1} \\left(\\frac{ 4n e^{-\\frac{l^2}{4}} l^{n-1}}{\\sqrt{2^{n+1}\\pi (4n^2-2n) (n-2)!", "}}\\right)$ We decompose the corresponding propagator into partial fraction (for reference see the first term in (REF ) and the Feynman diagram in fig.", "REF (a)).", "$\\frac{1}{\\omega _m^2(\\omega _m^2 + \\lambda _n)} = \\frac{1}{\\lambda _n} \\left(\\frac{1}{\\omega _m^2} - \\frac{1}{(\\omega _m^2 + \\lambda _n)}\\right)$ The amplitude comprising of the vertex $F_4(0,l,n)$ given in (REF ) is given by $\\sum _{m,n} \\int \\frac{dl}{2\\pi } \\left(\\frac{16 n^2e^{-\\frac{l^2}{2}} l^{2n-2}}{2^{n+1} (4n^2-2n) (n-2)!", "}\\frac{1}{\\lambda _n}\\frac{1}{\\omega ^2_m}-\\frac{16 n^2e^{-\\frac{l^2}{2}}l^{2n-2}}{2^{n+1} (4n^2-2n) (n-2)!", "}\\frac{1}{\\lambda _n(\\omega ^2_m + \\lambda _n)}\\right)\\nonumber \\\\$ In the first term in (REF ) we perform the sum over $n$ and take the UV limit $l \\rightarrow \\infty $ .", "In the second term we first compute the integration over $l$ and then expand the resulting expression asymptotically about $n= \\infty $ .", "The leading order contribution to the UV divergence obtained from this amplitude is thus $\\sum _{m}\\int \\frac{dl}{2\\pi \\sqrt{q}}\\frac{1}{2 \\omega ^2_m} - \\sum _{m,n}\\frac{1}{2 \\pi \\sqrt{2 n}}\\frac{1}{\\omega ^2_m + \\lambda _n}.$ The three-point vertex $\\tilde{F}_4(0,l,n)$ vanishes for all $n$ .", "Hence the corresponding amplitude vanishes.", "The two-point functions corresponding to the three-point vertices $F_5(0,l,n)$ , $F^{^{\\prime }}_5(0,l,n)$ and $\\tilde{F}^{^{\\prime }}_5(0,l,n)$ (obtained from (REF ), (REF ) and (REF )) contain propagators with momentum $l$ as well as those containing the momentum $n$ .", "The amplitude bearing the three-point vertex $F_5(0,l,n)$ contains contributions from the fields $\\Phi ^{1,2}_I(m,n)$ and the massless fields $\\Phi ^3_I(m,l)$ in the loop.", "Therefore the propagators contain both the mass-squared eigenvalues $\\gamma _n = (2 n + 1) q$ of the basis functions of the fields $\\Phi ^{1,2}_I$ and the momentum $l$ of the massless field $\\Phi ^3_I$ .", "Similarly $F^{^{\\prime }}_5(0,l,n)$ has contributions from the fields $C_{m,n}$ and the massless fields $\\Phi ^3_1(m,l)$ .", "Therefore the propagators in the corresponding two-point function has both the mass-squared eigenvalues $\\lambda _n = (2n-1)q$ and the momentum $l$ .", "In both these amplitudes we first perform the integration over $l$ and then asymptotically expand the resulting expressions about $n= \\infty $ .", "As for the two-point function for the three-point vertex $\\tilde{F}^{^{\\prime }}_5(0,l,n)$ the fields participating in the loop are the massless modes $\\tilde{C}_{m,n}$ as well as $\\Phi ^3_1(m,l)$ .", "In this case we first decompose the mixed propagator into two parts.", "We then sum over $n$ and then take the limit $l \\rightarrow \\infty $ .", "Throughout the computations the external momentum is kept fixed at $k=0$ .", "All the three-point vertices $F_5(0,l,n)$ , $F^{^{\\prime }}_5(0,l,n)$ and $\\tilde{F}^{^{\\prime }}_5(0,l,n)$ has the factor $e^{ilx}$ due to the presence of massless fields in the loop.", "As in the cases of $F_4(0,l,n)$ , the integrals over the world-volume coordinate $x$ in these vertex functions amount to evaluating the the Fourier transform of the various Hermite polynomials constituting the vertex functions.", "Using the result from (REF ) in computing the vertex functions (REF ), (REF ) and (REF ) for $k=0$ , we arrive at the following expressions $F_5(0,l,n)= -(-1)^{n+1}i^{(n+1)}e^{-\\frac{l^2}{4}}\\frac{\\left(l^{(n+1)} + 2n l^{(n-1)}\\right)}{\\sqrt{2^{n+1} \\pi n!", "}}$ $F^{^{\\prime }}_5(0,l,n)= (-1)^{n+1}i^{(n+1)}e^{-\\frac{l^2}{4}}\\frac{3 l^{(n+1)} + 4 n (n-2) l^{(n-3)}}{\\sqrt{2^{n+2} \\pi (4 n^2 - 2 n) (n-2)!", "}}$ $\\tilde{F}^{^{\\prime }}_5(0,l,n)= (-1)^{n+1}i^{(n+1)}e^{-\\frac{l^2}{4}}\\frac{3 l^{(n+1)} - 4 n (n-2) l^{(n-3)}}{\\sqrt{2^{n+2} \\pi (4 n - 2) (n-1)!", "}}$ The amplitude for the three-point vertex $F_5(0,l,n)$ can be written as $&&\\sum _{m,n} \\int \\frac{dl}{2\\pi } \\frac{F_5(0,l,n) F^{*}_5(0,l,n)}{(\\omega ^2_m + \\gamma _n)(\\omega ^2_m+ l^2)}\\\\&=& \\sum _{m,n} \\int \\frac{dl}{2\\pi } \\frac{e^{-\\frac{l^2}{2}}}{(\\omega ^2_m + \\gamma _n)(\\omega ^2_m+ l^2)}\\frac{\\left(l^{(n+1)} + 2n l^{(n-1)}\\right)^2}{2^{n+1} \\pi n!", "}\\nonumber \\\\$ where the second line in (REF ) is obtained by plugging in (REF ) in the first line.", "After performing the integral over $l$ , we asymptotically expand the result about $n = \\infty $ This results into $&&\\sum _{m,n} \\int \\frac{dl}{2\\pi } \\frac{F_5(0,l,n) F^{*}_5(0,l,n)}{(\\omega ^2_m + \\gamma _n)(\\omega ^2_m+ l^2)}\\nonumber \\\\&=& {1\\over 2}\\sum _{m,n} \\frac{1}{(\\omega ^2_m + \\gamma _n)}\\left[{\\left(\\frac{2 \\sqrt{2} \\sqrt{\\frac{1}{n}}}{\\pi }-\\frac{\\left(\\left(-7+4\\omega _m^2\\right)\\right) \\left(\\frac{1}{n}\\right)^{3/2}}{2 \\left(\\sqrt{2}\\pi \\right)}+\\mathcal {O}\\left(\\frac{1}{n}\\right)^2\\right)}\\right.\\nonumber \\\\&+&\\left.", "{2^{-n} \\left(\\frac{e}{n}\\right)^n\\left(\\omega _m^2\\right)^n \\sec (n \\pi ) \\left(-\\frac{e^{\\frac{\\omega _m^2}{2}} \\sqrt{\\frac{2}{\\pi }} n^{3/2}}{\\omega _m^3}+\\frac{e^{\\frac{\\omega _m^2}{2}}\\left(24\\omega _m^2+2\\right) \\sqrt{n}}{12 \\sqrt{2 \\pi }\\omega _m^3}+\\mathcal {O}\\left(\\frac{1}{n}\\right)^0\\right)}\\right]\\nonumber \\\\$ In the above expression the terms with odd power of $\\omega _m$ vanishes under summation over $m$ over $\\lbrace -\\infty , \\infty \\rbrace $ .", "The leading order term in the amplitude contributing to UV divergence is $\\sum _{m,n} \\frac{4}{2\\pi \\sqrt{2 n}}\\frac{1}{(\\omega ^2_m + \\gamma _n)}$ Similarly using the result of Fourier Transform in (REF ), the amplitude for the three-point vertex $F^{^{\\prime }}_5$ can be written as $&&\\sum _{m,n} \\int \\frac{dl}{2\\pi } \\frac{F^{^{\\prime }}_5(0,l,n) F^{^{\\prime }*}_5(0,l,n)}{(\\omega ^2_m + \\lambda _n)(l^2+\\omega ^2_m)}\\\\&=& \\sum _{m,n} \\int \\frac{dl}{2\\pi } \\frac{e^{-\\frac{l^2}{2}}}{(\\omega ^2_m + \\lambda _n)(l^2+\\omega ^2_m)}\\frac{\\left(3 l^{(n+1)} + 4 n (n-2) l^{(n-3)}\\right)^2}{2^{(n+2)} \\pi (4 n^2 - 2 n) (n-2)!", "}\\nonumber \\\\$ After integrating the amplitude in (REF ) over $l$ and expanding about $n=\\infty $ we get the expansion $&&\\sum _{m,n} \\int \\frac{dl}{2\\pi } \\frac{F^{^{\\prime }}_5(0,l,n) F^{^{\\prime }*}_5(0,l,n)}{(\\omega ^2_m + \\lambda _n)(l^2+\\omega ^2_m)}\\nonumber \\\\&=&{1\\over 2}\\sum _{m,n}\\frac{1}{(\\omega ^2_m + \\lambda _n)}\\left[\\left(\\frac{2 \\sqrt{2} \\sqrt{\\frac{1}{n}}}{\\pi }+\\mathcal {O}\\left(\\frac{1}{n}\\right)^1\\right)\\right.\\nonumber \\\\&+&\\left.", "2^{-n} \\left(\\frac{e}{n}\\right)^n \\left(\\frac{1}{\\omega _m^2}\\right)^{-n}\\sec (n \\pi ) \\left(-\\frac{e^{\\frac{\\omega _m^2}{2}} \\sqrt{\\frac{2}{\\pi }} n^{7/2}}{\\omega _m^7}+{\\mathcal {O}\\left(\\frac{1}{n}\\right)^3}\\right)\\right]$ The terms with odd powers of $\\omega _m$ vanishes under the sum over $m$ over $\\lbrace -\\infty , \\infty \\rbrace $ .", "The leading order term in the expansion of the amplitude (REF ) that contributes to the UV divergence is $\\sum _{m,n} \\frac{4}{2\\pi \\sqrt{2 n}}\\frac{1}{(\\omega ^2_m + \\lambda _n)}$ As for the amplitude containing the three-point vertex $\\tilde{F}^{^{\\prime }}_5(0,l,n)$ we have $&&\\sum _{m,n} \\int \\frac{dl}{2\\pi } \\frac{\\tilde{F}^{^{\\prime }}_5(0,l,n) \\tilde{F}^{^{\\prime }*}_5(0,l,n)}{\\omega ^2_m (\\omega ^2_m+l^2)}\\\\&=& \\sum _{m,n} \\int \\frac{dl}{2\\pi } \\frac{e^{-\\frac{l^2}{2}}}{\\omega ^2_m (\\omega ^2_m+l^2)}\\frac{\\left(3 l^{(n+1)} - 4 (n-1)(n-2) l^{(n-3)}\\right)^2}{2^{(n+2)} \\pi (4 n - 2) (n-1)!", "}\\nonumber $ We rewrite the propagators as $\\frac{1}{\\omega _m^2(\\omega _m^2 + l^2)} = \\frac{1}{l^2} \\left(\\frac{1}{\\omega _m^2} - \\frac{1}{(\\omega _m^2 + l^2)}\\right)$ we thus have the following expression, $\\sum _{m} \\int \\frac{dl}{2\\pi }~\\frac{e^{-\\frac{l^2}{2}}}{l^2}\\left(\\frac{1}{\\omega ^2_m} - \\frac{1}{\\omega ^2_m+l^2}\\right)\\frac{\\left(3 l^{(n+1)} - 4 (n-1)(n-2) l^{(n-3)}\\right)^2}{2^{(n+2)} \\pi (4 n - 2) (n-1)!", "}$ In the first term of the above expression we perform the integral over $l$ and then take the $n\\rightarrow \\infty $ limit.", "The first term gives $\\sum _{m,n} \\frac{1}{2\\pi \\sqrt{2n}}\\frac{1}{\\omega _m^2}$ In the second term we perform the sum over $n$ , which gives $&-&\\sum _{m} \\int \\frac{dl}{2\\pi }~\\frac{e^{-\\frac{l^2}{2}}}{l^2}\\left(\\frac{1}{\\omega ^2_m+l^2}\\right)\\nonumber \\\\&&\\frac{\\left(-128+96 l^4-18 l^8-9 l^{10}+8 e^{\\frac{l^2}{2}} \\left(16-8 l^2-10 l^4+6 l^6+l^8\\right)\\right)}{16l^6}\\nonumber \\\\$ The divergent piece in the last line is $-{1\\over 2}\\sum _{m} \\int \\frac{dl}{2\\pi \\sqrt{q}}\\left(\\frac{1}{\\omega ^2_m + l^2}\\right)$ Let us now look at the fermionic amplitudes in (REF ).", "We first note that the massless fermionic fields namely $R^3_i$ and $L^3_i$ has propagators $(i\\omega _m \\pm l)^{-1}$ where the “$+$ ”-sign stands for $R^3_i$ and the “$-$ ”-sign for $L^3_i$ .", "However while computing the two-point functions one needs to perform integrations with respect the momentum $l$ over the entire range of $\\lbrace -\\infty , \\infty \\rbrace $ .", "Hence to analyze the UV behaviour it suffices to consider only one one sign for the propagators.", "For our purpose we consider the following propagator and decomposed it into $\\frac{1}{(i\\omega _m - \\sqrt{\\lambda ^{^{\\prime }}_n})(i\\omega _m + l)}= {1\\over 2}\\left(-\\frac{1}{\\omega _m^2 + \\lambda ^{^{\\prime }}_n} - \\frac{1}{\\omega _m^2 + l^2} + \\frac{l^2+ \\lambda ^{^{\\prime }}_n}{(\\omega _m^2 + \\lambda ^{^{\\prime }}_n)(\\omega _m^2 + l^2)}\\right)\\nonumber \\\\$ where we have dropped the terms with odd powers of $\\omega _m$ and $l$ , because they are odd functions of $\\omega _m$ and $l$ and will vanish with respect to sum over $m$ over $\\lbrace -\\infty , \\infty \\rbrace $ as well as integral over $l$ over the same interval.", "The fermionic vertices are all three-point vertices with contributions from the massless fermions $L^3_{i}$ and $R^3_i$ respectively in the loop.", "As found in the bosonic amplitude the integrals in (REF ), (REF ) and (REF ) at $k=0$ also amounts to computing the Fourier transform the Hermite polynomials from the massive fermions which in turn produce the vertex functions in terms of $l$ and $n$ .", "The fermionic vertices an thus be written as $F^L_6(0,n,l)=F^L_7(0,n,l)= (-1)^{n+1}i^{(n+1)}e^{-\\frac{l^2}{4}}\\frac{(\\frac{l^{n}}{\\sqrt{2n}} - l^{(n-1)})}{\\sqrt{2\\sqrt{\\pi }} \\sqrt{2^{n+1}\\sqrt{\\pi }(n-1)!", "}}\\\\F^R_6(0,n,l)= F^R_7(0,n,l)=- (-1)^{n+1} i^{(n+1)}e^{-\\frac{l^2}{4}}\\frac{(\\frac{l^{n}}{\\sqrt{2n}} + l^{(n-1)})}{\\sqrt{2\\sqrt{\\pi }} \\sqrt{2^{n+1}\\sqrt{\\pi }(n-1)!", "}}$ Combining the equations (REF ) with (REF ), (REF ) and (REF ), the total fermion two-point functions for the tree-level tachyon at finite temperature is given by $&&\\Sigma ^3(0,0, 0,0,\\beta ,q)\\nonumber \\\\&&= (8) \\sum _{m,n} \\int \\frac{dl}{2 \\pi }{1\\over 2}\\left(-\\frac{1}{\\omega _m^2 + \\lambda ^{^{\\prime }}_n} - \\frac{1}{\\omega _m^2 + l^2} + \\frac{l^2+ \\lambda ^{^{\\prime }}_n}{(\\omega _m^2 + \\lambda ^{^{\\prime }}_n)(\\omega _m^2 + l^2)}\\right)\\nonumber \\\\&&\\left(e^{-\\frac{l^2}{2}}\\frac{2 (\\frac{l^{n}}{\\sqrt{2n}} - l^{(n-1)})^2 + 2 (\\frac{l^{n}}{\\sqrt{2n}}+ l^{(n-1)})^2 + 4 (\\frac{l^{n}}{\\sqrt{2n}} - l^{(n-1)})(\\frac{l^{n}}{\\sqrt{2n}} + l^{(n-1)})}{2^{n+2}\\pi (n-1)!", "}\\right)\\nonumber \\\\$ Combining all the fermionic vertices, eqn.", "(REF ) can be finally written down as $&&\\Sigma ^3(0,0, 0,0,\\beta ,q) = (8) \\sum _{m,n} \\int \\frac{dl}{2 \\pi }{1\\over 2}\\left[-\\frac{1}{\\omega _m^2 + \\lambda ^{^{\\prime }}_n} \\left(e^{-\\frac{l^2}{2}}\\frac{l^{2 n}}{2^{n+1}\\pi n!", "}\\right)\\right.\\nonumber \\\\&&\\left.- \\frac{1}{\\omega _m^2 + l^2} \\left(e^{-\\frac{l^2}{2}}\\frac{l^{2 n}}{2^{n+1}\\pi n!", "}\\right)+ \\frac{l^2 + \\lambda ^{^{\\prime }}_n}{(\\omega _m^2 + \\lambda ^{^{\\prime }}_n)(\\omega _m^2 + l^2)}\\left(e^{-\\frac{l^2}{2}}\\frac{l^{2 n}}{2^{n+1}\\pi n)!", "}\\right)\\right]\\nonumber \\\\$ The first term in (REF ) upon integration over the momentum $l$ and in the $n \\rightarrow \\infty $ limit yields the leading order fermionic contribution $- {1\\over 2}(16) \\sum _{m,n} \\frac{1}{ 2 \\pi \\sqrt{2n}} \\frac{1}{\\omega ^2_m + \\lambda ^{^{\\prime }}_n}$ In the second term in (REF ) we sum over $n$ and then take the $l \\rightarrow \\infty $ limit and extract the leading order term as $-{1\\over 2}(8) \\int \\frac{dl}{2 \\pi \\sqrt{q}} \\sum _{m} \\frac{1}{\\omega ^2_m + l^2}$ In the 3rd and last term we integrate over $l$ and the leading order term in the large $n$ -expansion is given by ${1\\over 2}(8) \\sum _{m,n} \\frac{4}{ 2 \\pi \\sqrt{2n}} \\frac{1}{\\omega ^2_m + \\lambda ^{^{\\prime }}_n}$ The total leading order contribution to the UV divergence from the bosonic side is $&&\\underbrace{{1\\over 2}\\sum _{m,n} \\frac{1}{ 2 \\pi \\sqrt{2n}} \\frac{1}{\\omega ^2_m+ \\lambda _n} + {1\\over 2}\\sum _{m,n} \\frac{7 \\times 2}{ 2 \\pi \\sqrt{2n}} \\frac{1}{\\omega ^2_m + \\gamma _n}}_{\\text{amplitudes involving $F_1(0,0,n,n)$ and $F_2(0,0,n,n)$}}+ \\underbrace{\\sum _{m,n}\\frac{1}{2\\pi \\sqrt{2n}} \\frac{1}{2 \\omega ^2_m}}_{\\text{ $\\tilde{F}_1(0,0,n,n)$}}\\nonumber \\\\&-&\\underbrace{\\int \\frac{dl}{2 \\pi \\sqrt{q}}\\sum _{m}\\frac{1}{2 \\omega ^2_m}+ {1\\over 2}\\sum _{m,n} \\frac{1}{ 2 \\pi \\sqrt{2n}} \\frac{1}{\\omega ^2_m+ \\lambda _n}}_{\\text{amplitude involving ${F}_4(0,l,n)$}}+ \\underbrace{\\int \\frac{dl}{2\\pi \\sqrt{q}}\\sum _m \\frac{1}{2 \\omega ^2_m}}_{\\text{ $F_3(0,0,l,-l)$}}\\nonumber \\\\&+&\\underbrace{\\left({1\\over 2}(7) \\int \\frac{dl}{2\\pi \\sqrt{q}}\\sum _m \\frac{1}{\\omega ^2_m + l^2}+ {1\\over 2}\\times {1\\over 2}\\int \\frac{dl}{2\\pi \\sqrt{q}}\\sum _m \\frac{1}{\\omega ^2_m + l^2}\\right)}_{\\text{amplitudes involving $F^{^{\\prime }}_2(0,0,n,n)$ and $F^{^{\\prime }}_3(0,0,n,n)$}}\\nonumber \\\\&+&\\underbrace{{1\\over 2}\\times {1\\over 2}\\int \\frac{dl}{2\\pi \\sqrt{q}}\\sum _m \\frac{1}{\\omega ^2_m + l^2}-\\sum _{m,n}\\frac{1}{2\\pi \\sqrt{2n}}\\frac{1}{2 \\omega ^2_m}}_{\\text{amplitude involving $\\tilde{F}^{^{\\prime }}_5(0,l,n)$}}\\nonumber \\\\&-&\\underbrace{\\left({1\\over 2}(7) \\sum _{m,n} \\frac{4}{ 2 \\pi \\sqrt{2n}} \\frac{1}{\\omega ^2_m + \\gamma _n}+ {1\\over 2}\\sum _{m,n} \\frac{4}{2\\pi \\sqrt{2n}} \\frac{1}{\\omega ^2_m + \\lambda _n}\\right)}_{\\text{amplitudes involving $F_5(0,l,n)$ and $F^{^{\\prime }}_5(0,l,n)$}}\\nonumber \\\\$ The total leading order contribution to the UV divergence from the amplitudes containing fermions in the loop is $&&-\\underbrace{{1\\over 2}(16) \\sum _{m,n} \\frac{1}{ 2 \\pi \\sqrt{2n}} \\frac{1}{\\omega ^2_m+ \\lambda ^{^{\\prime }}_n}}_{\\text{1st term in $\\Sigma ^3(0,0,0,0,\\beta , q)$}}- \\underbrace{{1\\over 2}(8) \\int \\frac{dl}{2 \\pi \\sqrt{q}}\\sum _m \\frac{1}{\\omega ^2_m+ l^2}}_{\\text{2nd term in $\\Sigma ^3(0,0,0,0,\\beta , q)$}}\\\\&&+\\underbrace{{1\\over 2}(8) \\sum _{m,n} \\frac{4}{ 2 \\pi \\sqrt{2n}} \\frac{1}{\\omega ^2_m+ \\lambda ^{^{\\prime }}_n}}_{\\text{3rd term in $\\Sigma ^3(0,0,0,0,\\beta , q)$}}$ As $n \\rightarrow \\infty $ , we see that $\\gamma _n=\\lambda _n= \\lambda ^{^{\\prime }}_n = 2 n q$ .", "Comparing (REF ) with (REF ), we see that the leading order terms from bosonic sides cancel with that from the fermionic side.", "In this method of proving UV finiteness of the finite temperature corrections to the tree-level tachyon mass-squared the UV divergence in $l$ or $n$ is thus softened by the fact that the Matsubara sum is left untouched.", "The large $n$ expansion is valid under the assumption that $m<n$ .", "This assumption restricts our proof to a corner in the phase space.", "However the counting of the degrees of freedom on the bosonic and fermionic sides still match which in turns forces the divergences to cancel out.", "We expect the finiteness of the two-point functions to hold for large values of $m$ also because higher order terms in the expansion is heavily suppressed by Gaussian factors." ], [ "Infrared problem", "We now address the problem of infrared divergences.", "The appearance of IR divergence is due to the presence of massless fields namely $\\tilde{C}_{w,k}$ , $A_x^3$ and $\\Phi _I^3$ (for $I=1\\cdots 8$ ) in the loop.", "To compute the IR-finite two-point $C_{w,k}$ amplitude we shall follow a two step procedure as mentioned in the introduction.", "In the first step we compute the temperature corrected masses-squared of the massless fields by integrating over the modes in the internal lines with an IR cutoff.", "The next step is to introduce these masses in the propagators for the massless fields.", "This is equivalent to summing over an infinite set of diagrams which is illustrated in Figure REF .", "In the figure, the $\\chi $ field stands for the modes with tree-level mass zero.", "The bold line is the corrected propagator for the $\\chi $ field due to the sum of an infinite set of diagrams on the right.", "Figure: Diagram showing the correction to the propagator for the χ\\chi field due to mass insertions.At each temperature the sums and the integrals are now IR-finite because all fields in the loops are now massive.", "Thus we can compute the mass-squared corrections and obtain the temperature corrected effective masses-squared of the tree-level tachyon as a function of $q$ and $\\beta $ .", "In the following few sections we compute the two-point functions for the massless modes of the doublet of fields $(\\Phi ^1_1, A^2_x)$ namely the $\\tilde{C}_{w,k}^{\\prime }s$ and the other massless fields namely $\\Phi ^3_1, \\Phi ^3_I, A^3_x$ at finite temperature." ], [ "Two-point functions for the $\\tilde{C}_{w,k}$ modes", "In this section we compute the two point function for the $\\tilde{C}_{w,k}$ modes.", "During these computations we note that there are no normalizable eigenfunctions $\\tilde{A}_n(x)$ and $\\tilde{\\phi }_n(x)$ (see eqn (REF )) for $n=0$ .", "The amplitudes with bosons in the loop and consisting of the four-point vertices for the one-loop masses-squared of the massless modes are given by $\\Sigma ^1_H(w,w^{^{\\prime }},k,k^{^{\\prime }},\\beta ,q)&=& {1\\over 2}N \\sum _m\\left[\\sum _n \\left(\\frac{H_1(k,k^{^{\\prime }},n,n)}{\\omega _m^2 + \\lambda _n} + \\frac{\\tilde{H}_1(k,k^{^{\\prime }},n,n)}{\\omega _m^2}+ \\frac{7 H_2(k,k^{^{\\prime }},n,n)}{\\omega _m^2 + \\gamma _n}\\right)\\right.", "\\nonumber \\\\&+& \\left.", "\\int \\frac{dl}{(2\\pi \\sqrt{q})}\\left(\\frac{7 H^{^{\\prime }}_2(k,k^{^{\\prime }},l,-l)}{\\omega _m^2 + l^2}+\\frac{H^{^{\\prime }}_3(k, k^{^{\\prime }},l,-l)}{\\omega _m^2+l^2}\\right)\\right.\\nonumber \\\\&+&\\left.", "\\int \\frac{dl}{2 \\pi \\sqrt{q}}\\frac{H_3(k,k^{^{\\prime }}l,-l)}{\\omega _m^2}\\right] \\delta _{w+w^{^{\\prime }}},$ where $w=m+m^{^{\\prime }}$ .", "Here `$H$ ' denotes the vertices corresponding to the finite temperature two-point functions for the massless modes $\\tilde{C}_{w,k}$ 's.", "The Feynman diagrams comprising the four-point vertices that contribute to the mass-squared corrections to the massless (at tree-level) modes $\\tilde{C}_{w,k}$ are depicted in figure REF .", "Figure: Feynman diagrams for the two-point C ˜ w,k \\tilde{C}_{w,k} amplitudes with four-point vertices.The amplitudes arising from three-point interaction vertices for the massless modes $\\tilde{C}_{w,k}$ are collected into $\\Sigma ^2_H(w,w^{^{\\prime }},k,k^{^{\\prime }},\\beta ,q)&=&-{1\\over 2}qN \\sum _{m,n}\\left[\\int \\frac{dl}{2 \\pi \\sqrt{q}}\\frac{H_4(k,l,n)H^{*}_4(k^{^{\\prime }},l,n)}{(\\omega _m^2+\\lambda _n)\\omega _{m^{^{\\prime }}}^2}+ \\int \\frac{dl}{2 \\pi \\sqrt{q}}\\frac{\\tilde{H}_4(k,l,n)\\tilde{H}^{*}_4(k^{^{\\prime }},l,n)}{\\omega _m^2 \\omega _{m^{^{\\prime }}}^2}\\right.\\nonumber \\\\&+&\\left.\\int \\frac{dl}{2\\pi \\sqrt{q}}\\left(\\frac{7 H_5(k,l,n)H^{*}_5(k^{^{\\prime }},-l,n)}{(\\omega _m^2+\\gamma _n)(\\omega _{m^{^{\\prime }}}^2+l^2)}+\\frac{H^{^{\\prime }}_5(k,l,n) H^{^{\\prime }*}_5(k^{^{\\prime }},-l,n)}{(\\omega _m^2+\\lambda _n)(\\omega _{m^{^{\\prime }}}^2+l^2)}\\right)\\right.\\nonumber \\\\&+&\\left.\\int \\frac{dl}{2\\pi \\sqrt{q}}\\frac{\\tilde{H}^{^{\\prime }}_5(k,l,n) \\tilde{H}^{^{\\prime }*}_5(k^{^{\\prime }},-l,n)}{(\\omega _m^2)(\\omega _{m^{^{\\prime }}}^2+l^2)}\\right]\\delta _{w+w^{^{\\prime }}}$ where $w=m+m^{^{\\prime }}$ .", "The Feynman diagrams for the three-point interactions are given in figure REF .", "Figure: Feynman diagrams for the two-point C ˜ w,k \\tilde{C}_{w,k} amplitudes with three-point verticesThe various four-point and three-point vertices are given in appendix (REF ).", "Note that the one-loop mass-squared corrections to the massless modes have the same structure as the tachyonic amplitudes.", "This is because they originate from the same set of interactions for the doublet $\\zeta (x,\\tau )$ in the action.", "The only difference between (REF ), (REF ) and (REF ), (REF ) is that the momentum modes in the external legs of the Feynman diagrams for the latter are now the massless modes $\\tilde{C}_{w,k}$ .", "Upon performing the Matsubara sums, the bosonic amplitudes namely (REF ) and (REF ) can be recast as $\\Sigma ^1_H(0,0,k,k^{^{\\prime }},\\beta ,q)&=&{1\\over 2}\\left[\\sum _n\\frac{H_1(k,k^{^{\\prime }},n,n)}{\\sqrt{(2n-1)}} \\left({1\\over 2}+\\frac{1}{e^{\\beta \\sqrt{(2n-1)q}}-1}\\right)\\right.\\nonumber \\\\&+& N\\left.\\sum _m \\left(\\sum _n\\frac{\\tilde{H}_1(k,k^{^{\\prime }},n,n)}{\\omega ^2_m}+ \\int \\frac{dl}{2 \\pi \\sqrt{q}}\\frac{H_3(k,k^{^{\\prime }},l-l)}{{\\omega ^2_m}}\\right)\\right.\\nonumber \\\\&+& \\left.\\sum _n\\left(\\frac{7 H_2(k,k^{^{\\prime }},n,n)}{\\sqrt{(2n+1)}}\\left({1\\over 2}+\\frac{1}{e^{\\beta \\sqrt{(2n+1)q}}-1}\\right)\\right)\\right.\\nonumber \\\\&+& \\left.\\left(\\int \\frac{dl}{2\\pi \\sqrt{q}}\\frac{(7+1/2)\\delta _{k,k^{^{\\prime }}}}{(l/\\sqrt{q})}\\left({1\\over 2}+\\frac{1}{e^{\\beta l}-1}\\right)\\right)\\right]$ $&&\\Sigma ^2_H(k,k^{^{\\prime }},\\beta ,q)\\nonumber \\\\&=& -{1\\over 2}\\sum _{n}\\left[\\int \\frac{dl}{2 \\pi \\sqrt{q}}\\frac{H_4(k,l,n)H^{*}_4(k^{^{\\prime }},l,n)}{(2n-1)}\\left[\\left(\\sum _m\\frac{\\sqrt{q}}{\\beta \\omega _m^2} -\\frac{1}{\\sqrt{2n-1}}\\left(\\frac{1}{2}+ \\frac{1}{e^{\\sqrt{(2n-1)q}\\beta }}-1\\right)\\right)\\right]\\right.\\nonumber \\\\&+&\\left.", "qN\\int \\frac{dl}{2 \\pi \\sqrt{q}}\\sum _{m}\\frac{\\tilde{H}_4(k,l,n)\\tilde{H}^{*}_4(k^{^{\\prime }},l,n)}{\\omega _m^4}\\right.\\nonumber \\\\&+&\\left.\\int \\frac{dl}{2 \\pi \\sqrt{q}}\\left[\\frac{7 H_5(k,l,n) H^{*}_5(k^{^{\\prime }},-l,n)}{(l/\\sqrt{q})^2-(2n+1)} \\left(\\frac{1}{\\sqrt{2n+1}}\\left({1\\over 2}+ \\frac{1}{e^{\\sqrt{(2 n+1)q}\\beta }-1}\\right)\\right.\\right.\\right.\\nonumber \\\\&-& \\left.\\left.\\left.\\frac{1}{(l/\\sqrt{q})}\\left({1\\over 2}+ \\frac{1}{e^{l\\beta }-1}\\right)\\right)\\right]\\right.\\nonumber \\\\&+&\\left.\\int \\frac{dl}{2 \\pi \\sqrt{q}}\\left[\\frac{H^{^{\\prime }}_5(k,l,n) H^{^{\\prime }*}_5(k^{^{\\prime }},-l,n)}{(l/\\sqrt{q})^2-(2n-1)} \\left(\\frac{1}{\\sqrt{2n-1}}\\left({1\\over 2}+ \\frac{1}{e^{\\sqrt{(2 n-1)q}\\beta }-1}\\right)\\right.\\right.\\right.\\nonumber \\\\&-& \\left.\\left.\\left.\\frac{1}{(l/\\sqrt{q})}\\left({1\\over 2}+ \\frac{1}{e^{l\\beta }-1}\\right)\\right)\\right]\\right.\\nonumber \\\\&+&\\left.\\int \\frac{dl}{2\\pi \\sqrt{q}}\\frac{\\tilde{H}^{^{\\prime }}_5(k,l,n)\\tilde{H}^{^{\\prime }*}_5(k^{^{\\prime }},-l,n)}{(l/\\sqrt{q})^2}\\left(\\sum _m\\frac{1}{\\omega _{m}^2}-\\frac{1}{(l/\\sqrt{q})}\\left({1\\over 2}+\\frac{1}{e^{l\\beta }-1}\\right)\\right)\\right]\\nonumber \\\\$ respectively.", "Similarly using the various vertex functions given in appendix (REF ), the finite-temperature contribution due to fermions in the loop to the mass-squared corrections for the massless modes is given by equation (REF ).", "The relevant Feynman diagrams are listed in Figure REF and Figure REF .", "Figure: Feynman diagrams for the amplitudes with three-point fermionic vertices V H R/L V^{R/L}_Hand their complex conjugates V H *R/L V^{*R/L}_HFigure: Feynman diagrams showing the cross terms in the amplitude with fermions in the loop.$\\Sigma ^3_H(w,w^{^{\\prime }},k,k^{^{\\prime }},\\beta ,q)&=&- (8 N)\\sum _{n,m,m^{^{\\prime }}}\\int \\frac{dl}{2\\pi \\sqrt{q}}\\frac{1}{(i\\omega _m-\\sqrt{\\lambda _n^{^{\\prime }}})}\\nonumber \\\\&\\times &\\left[\\frac{H^R_6(k,n,l)H^{R*}_6(k^{^{\\prime }},n,l)}{(i\\omega _{m^{^{\\prime }}}+l)}+\\frac{H^L_6(k,n,l)H^{L*}_6(k^{^{\\prime }},n,l)}{(i\\omega _{m^{^{\\prime }}}-l)}\\right.\\nonumber \\\\&+ &\\left.\\frac{H^L_7(k,n,l)F^{H*}_7(k^{^{\\prime }},n,l)}{(i\\omega _{m^{^{\\prime }}}+l)}+\\frac{H^R_7(k,n,l)H^{R*}_7(k^{^{\\prime }},n,l)}{(i\\omega _{m^{^{\\prime }}}-l)}\\right.\\nonumber \\\\&+&\\left.", "\\frac{H^R_6(k,n,l)H^{L*}_7(k^{^{\\prime }},n,l) + H^{R*}_6(k,n,l)H^L_7(k^{^{\\prime }},n,l)}{(i\\omega _{m^{^{\\prime }}}+l)}\\right.\\nonumber \\\\&+&\\left.\\frac{H^L_6(k,n,l)H^{R*}_7(k^{^{\\prime }},n,l)+ H^{L*}_6(k,n,l)H^R_7(k^{^{\\prime }},n,l)}{(i\\omega _{m^{^{\\prime }}}-l)}\\right]\\delta _{w+w^{^{\\prime }}}\\nonumber \\\\$ where $w=m+m^{^{\\prime }}$ .", "After performing the Matsubara sums the amplitude (REF ) assumes the following form.", "$&&\\Sigma ^3_H(0,0,k,k^{^{\\prime }},\\beta ,q)=\\nonumber \\\\&&(8 N)\\sum _{n}\\left[\\int \\frac{dl}{2\\pi \\sqrt{q}}\\left(\\frac{-\\beta \\tanh \\left(\\frac{\\beta l}{2}\\right)-\\beta \\tanh \\left(\\frac{1}{2} \\beta \\sqrt{2 n q}\\right)}{2\\left(l+\\sqrt{2 n q}\\right)}\\right)\\right.\\nonumber \\\\&&\\left.", "[H^R_6(k,n,l)H^{R*}_6(k^{^{\\prime }},n,l) + H^L_7(k,n,l)H^{L*}_7(k^{^{\\prime }},n,l)\\right.", "\\nonumber \\\\&&\\left.+ H^R_6(k,n,l)H^{L*}_7(k^{^{\\prime }},n,l) + H^{R*}_6(k,n,l)H^L_7(k^{^{\\prime }},n,l)]\\right.\\nonumber \\\\&+& \\left.", "\\int \\frac{dl}{2 \\pi \\sqrt{q}}\\left(\\frac{-\\beta \\tanh \\left(\\frac{\\beta l}{2}\\right)+\\beta \\tanh \\left(\\frac{1}{2} \\beta \\sqrt{2 n q}\\right)}{2 \\left(l-\\sqrt{2 n q}\\right)}\\right)\\right.\\nonumber \\\\&&+ \\left.", "[H^L_6(k,n,l)H^{L*}_6(k^{^{\\prime }},n,l) + H^R_7(k,n,l)H^{R*}_7(k^{^{\\prime }},n,l)\\right.\\nonumber \\\\&&\\left.+ H^L_6(k,n,l)H^{R*}_7(k^{^{\\prime }},n,l) + H^{L*}_6(k,n,l)H^R_7(k^{^{\\prime }},n,l)]\\right]$ At this point we recall that there is no normalizable eigenfunction for the massless modes $\\tilde{C}_{w,0}$ .", "Hence the counting starts from $k=1$ .", "These massless modes appear as the fluctuations $\\tilde{C}_{m,n}$ with $k$ replaced as $n$ in the one-loop diagrams for the tree-level tachyon (see figures REF and REF ) where we need to sum over all $n$ .", "As mentioned before the two point functions for all the $C_{w,k}$ and the $\\tilde{C}_{w,k}$ modes are coupled to each other at the one loop level giving rise to an infinite dimensional mass-matrix.", "To get the corrected spectrum we must re-diagonalize the mass matrix.", "Since our approach is to get the final finite values of the masses-squared numerically, for simplicity we shall work with a finite dimensional matrix for the $\\tilde{C}_{w,k}$ modes.", "Like the two point amplitude for the $C_{w,k}$ , the two-point functions of the massless modes also have contributions from the massless fields $\\Phi _1^3$ , $\\Phi _I^3;I \\ne 1$ and $A^3_x$ in the loop hence has the problem of infrared divergence and will be addressed in the way as mentioned before.", "The UV finiteness of the amplitudes for the two-point functions of the fields $\\tilde{C}_{w,k}$ can be checked using the method used for the tachyonic case." ], [ "Two point function for $\\Phi _1^3$", "Using the vertices computed in appendix REF we first write down the two point amplitude for $\\Phi _1^3$ .", "The Feynman diagrams involving the four-point vertices is depicted in Figure REF Figure: Feynman diagrams for the amplitudes with four-point bosonic verticesV 1 1 ,V 2 1 ,V ˜ 2 1 V^1_{1},~V^1_2,~\\tilde{V}^1_2The one-loop two-point functions involving the four-point vertices given in section (REF ) contributing to the finite-temperature mass-squared corrections to the tree-level massless field $\\Phi ^3_1$ can be collected into $\\Sigma ^1_{\\Phi _1^3-\\Phi _1^3}={1\\over 2}N \\sum _{m,n}\\left[(7\\times 2) \\frac{G_1^1(l,l^{^{\\prime }},n,n)}{(\\omega _m^2+\\gamma _n)}+(2) \\frac{G_2^1(l,l^{^{\\prime }},n,n)}{(\\omega _m^2+\\lambda _n)} +(2) \\frac{\\tilde{G}_2^{1}(l,l^{^{\\prime }},n,n)}{(\\omega _m^2)} \\right]\\delta _{w+w^{^{\\prime }}}$ The first term in (REF ) has contributions from the massive fields $\\Phi ^{(1,2)}_I$ , $I\\ne 1$ in the loop with four-point vertex and propagator given in (REF ) and the corresponding Feynman diagram is given by figure REF (a).", "The second term bears contributions from the fields $C_{m,n}$ with propagator (REF ) and depicted in the Feynman diagram in figure REF (b).", "The third term involves the massless fields $\\tilde{C}_{m,n}$ in the loop with propagator (REF ) and corresponding Feynman diagram in figure REF (c).", "Similarly the three-point bosonic interactions contributing to the finite temperature corrections to the massless field $\\Phi ^3_1(w,l)$ are collected in the mass-squared correction (REF ).", "The corresponding Feynman diagrams are given in Figure REF .", "Figure: Feynman diagrams with three-point bosonic verticesV 1 1 ' ,V 3 1 ,V ˜ 3 1 ,V ˜ 3 1 ' V^{1^{\\prime }}_{1},~V^1_3,~\\tilde{V}^1_3, \\tilde{V}^{1^{\\prime }}_3$\\Sigma ^2_{\\Phi _1^3-\\Phi _1^3}&=&-{1\\over 2}qN \\sum _{m,n,n^{^{\\prime }}}\\left[(7\\times 2)\\frac{G_1^{1^{\\prime }}(l,l^{^{\\prime }},n,n)}{(\\omega _m^2+\\gamma _n)(\\omega _{m^{^{\\prime }}}^2+\\gamma ^{^{\\prime }}_{n^{^{\\prime }}})}+ (2)\\frac{G_3^1(l,n,n^{^{\\prime }})G_3^1(l^{^{\\prime }},n,n^{^{\\prime }})}{(\\omega _m^2+\\lambda _n)(\\omega _{m^{^{\\prime }}}^2+\\lambda _{n^{^{\\prime }}})}\\right.\\nonumber \\\\&+& \\left.", "(2)\\frac{\\tilde{G}_3^{1}(l,n,n^{^{\\prime }})\\tilde{G}_3^{1}(l^{^{\\prime }},n,n^{^{\\prime }})}{(\\omega _m^2)(\\omega _{m^{^{\\prime }}}^2)}+ (2)\\frac{\\tilde{G}_3^{1^{\\prime }}(l,n,n^{^{\\prime }})\\tilde{G}_3^{1^{\\prime }}(l^{^{\\prime }},n,n^{^{\\prime }})}{\\omega _m^2(\\omega _{m^{^{\\prime }}}^2+\\lambda _{n^{^{\\prime }}})}\\right]\\delta _{w+w^{^{\\prime }}}$ where $w=m^{^{\\prime }}+m$ .", "The first term in (REF ) comprising the three-point vertex $G^{1^{\\prime }}_1(l,n,n^{^{\\prime }})$ involves the fields $\\Phi ^{(1,2)}_I$ , $I\\ne 1$ in the loops and is represented by the Feynman diagram in figure REF (a).", "Similarly the second term in (REF ) involving the vertex $G^{1}_3(l,n, n^{^{\\prime }})$ comprises of the fields $C_{m,n}$ s in the loop.", "The corresponding Feynman diagram is given in figure REF (b).", "The third term has contributions from $\\tilde{C}_{m,n}$ s with Feynman diagram in figure REF (c),while the fourth term represented in figure REF (d) has contributions from the fields $C_{m,n}$ and $\\tilde{C}_{m,n}$ .", "Figure: Feynman diagrams involving three-point vertices V f 1 ,V f 2 V^1_{f},~V^2_f.Figure REF shows the amplitude involving the fermions in the loop.", "The corresponding expression is $\\Sigma ^3_{\\Phi _1^3-\\Phi _1^3}&=&{1\\over 2}N \\sum _{m,n,n^{^{\\prime }}}\\left[ (16\\times 2)\\frac{G_f^1(l,n,n^{^{\\prime }})G_f^{1*}(l^{^{\\prime }},n,n^{^{\\prime }})}{(i\\omega _m-\\sqrt{\\lambda _n^{^{\\prime }}})(i\\omega _{m^{^{\\prime }}}-\\sqrt{\\lambda _{n^{^{\\prime }}}^{^{\\prime }}})}\\right.\\nonumber \\\\&-&\\left.", "(16)\\frac{G_f^2(l,n,n^{^{\\prime }})G_f^{2*}(l^{^{\\prime }},n,n^{^{\\prime }})}{(i\\omega _m-\\sqrt{\\lambda _n^{^{\\prime }}})(-i\\omega _{m^{^{\\prime }}}-\\sqrt{\\lambda _{n^{^{\\prime }}}^{^{\\prime }}})}\\right]\\delta _{w+w^{^{\\prime }}}$ with $w=m^{^{\\prime }}+m$ .", "The first and the second term in (REF ) are depicted in the Feynman diagrams given in figures REF (a) and REF (b) respectively.", "The various vertices given in section (REF ) that are involved in the two-point functions realizing the mass-squared corrections to the tree-level massless field $\\Phi ^3_1$ can be exactly computed using the orthogonality relation for Hermite Polynomials.", "Setting the external momenta $l=l^{^{\\prime }}=w=w^{^{\\prime }}=0$ , we can write down the various vertices as $G^1_1[0,0,n,n^{^{\\prime }}] &=& \\delta _{n,n^{^{\\prime }}},~~G^1_2[0,0,n,n^{^{\\prime }}]={1\\over 2}\\delta _{n,n^{^{\\prime }}},~~\\tilde{G}^1_2[0,0,n,n^{^{\\prime }}]={1\\over 2}\\delta _{n,n^{^{\\prime }}},\\\\G^{1^{\\prime }}_1[0,n,n^{^{\\prime }}] &=& \\sqrt{2n} \\delta _{n-1,n^{^{\\prime }}},~~G^1_3[0,n,n^{,}]=2 \\sqrt{\\frac{2n(n-1)(n-2)}{(2n-1)(2n-3)}} \\delta _{n-1,n^{^{\\prime }}},\\\\\\tilde{G}^1_3[0,n,n^{^{\\prime }}]&=& 0,\\\\\\tilde{G}^{1^{\\prime }}_3[0,n,n^{^{\\prime }}]&=& -\\left(\\frac{\\sqrt{2(n-1)}(n+1)}{\\sqrt{(2n-1)(2n+1)}} \\delta _{n+1,n^{^{\\prime }}}- \\frac{\\sqrt{2n}(n-1)}{\\sqrt{(2n-1)(2n-3)}} \\delta _{n-1,n^{^{\\prime }}}\\right)\\\\G^1_f[0,n,n^{^{\\prime }}]&=& -\\frac{i}{4}\\left(2\\delta _{n-1,n^{^{\\prime }}}\\right),~~G^2_f[0,n,n^{^{\\prime }}]=-\\frac{i}{2}\\left(\\delta _{n+1,n^{,}} + \\delta _{n-1,n^{^{\\prime }}}\\right)$ To prove the ultraviolet finiteness of the mass-squared corrections (REF ),(REF ) and (REF ), we set the external momenta $(l,l^{^{\\prime }},w)=0$ and compute the vertices in the large $n$ limit.", "The vertices can be evaluated using the orthogonality condition for Hermite polynomials.", "The various four-point vertices given in (REF ) in the large $n$ limit assume the forms $G_1^1(0,0,n,n)= 1\\mbox{~~;~~} G_2^1(0,0,n,n)=\\tilde{G}_2^1(0,0,n,n)= {1\\over 2}$ where we have used the Kronecker delta's to set $n=n^{^{\\prime }}$ .", "The three-point vertices given in () in the large $n$ limit become $G_1^{1^{\\prime }}(0,n,n^{^{\\prime }})\\sim \\frac{\\sqrt{2n}}{2}[2\\delta _{n^{^{\\prime }},n-1}] \\mbox{~~;~~}G_3^1(0,n,n^{^{\\prime }})\\sim \\frac{\\sqrt{2n}}{8}[8\\delta _{n^{^{\\prime }},n-1}]$ The three-point bosonic vertex $\\tilde{G}^1_3(0,n,n^{^{\\prime }})$ is found to be identically zero for all values $n$ in ().", "Moreover the fermionic three-point vertices are exact for all values $n$ .", "Hence the remaining three-point bosonic vertex () in the limit $n \\rightarrow \\infty $ can be written as $\\tilde{G}_3^{1^{\\prime }}(0,n,n^{^{\\prime }})\\sim \\frac{\\sqrt{2n}}{8}[4\\delta _{n^{^{\\prime }},n-1}-4\\delta _{n^{^{\\prime }},n+1}]$ The ultraviolet contribution to the amplitude can now be written down by putting these asymptotic values of the vertices into (REF ,REF ,REF ).", "We get the following from the bosonic fields in the loop $\\Sigma ^1_{\\Phi _1^3-\\Phi _1^3}\\sim {1\\over 2}N \\sum _{m,n}\\left[(7\\times 2) \\frac{1}{(\\omega _m^2+\\gamma _n)}+(2) \\frac{1/2}{(\\omega _m^2+\\lambda _n)} + (2) \\frac{1/2}{(\\omega _m^2)} \\right]$ $\\Sigma ^2_{\\Phi _1^3-\\Phi _1^3}\\sim -{1\\over 2}qN \\sum _{m,n}\\left[(7\\times 2) \\frac{2n}{(\\omega _m^2+\\gamma _n)^2}+ (2)\\frac{2n}{(\\omega _m^2+\\lambda _n)^2}+ (2)\\frac{(2n)/2}{\\omega _m^2(\\omega _{m}^2+\\lambda _{n})}\\right]\\nonumber \\\\$ Noting that in the large $n$ limit, $\\lambda _n=\\gamma _n\\sim 2nq$ , $\\Sigma ^1_{\\Phi _1^3-\\Phi _1^3}+\\Sigma ^2_{\\Phi _1^3-\\Phi _1^3} &\\sim & N \\sum _{m,n}\\left[(8) \\frac{1}{(\\omega _m^2+2nq)}-(8) \\frac{2nq}{(\\omega _m^2+2nq)^2}\\right]\\\\&\\sim &\\sum _{n}\\frac{2}{\\sqrt{2n}}$ In the last line we have done the sum over the Matsubara frequencies $m$ and omitted all the finite temperature dependent pieces.", "Similarly the asymptotic form of the fermionic contribution is, $\\Sigma ^3_{\\Phi _1^3-\\Phi _1^3}\\sim - N \\sum _{m,n}\\left[ (4)\\frac{1}{(\\omega _m^2+\\lambda ^{^{\\prime }}_n)}-(4)\\frac{1}{(i\\omega _m-\\sqrt{\\lambda ^{^{\\prime }}_n})^2}\\right]\\sim -\\sum _{n}\\frac{2}{\\sqrt{2n}}$ Thus the one-loop $\\Phi _1^3-\\Phi _1^3$ amplitude is ultraviolet finite.", "One can exactly compute the various amplitudes in (REF ),(REF ) and (REF ) using the various vertices presented in (REF -) and their corresponding propagators (see Appendix ()).", "In particular one can write down the effective mass-squared for the field $\\Phi ^3_1$ as a function of $q$ and $\\beta $ as $m^2_{\\Phi ^3_1}(q,\\beta ) = m^2_{10} + m^2_1(q,\\beta )$ where $ m^2_{10}$ denotes the zero temperature quantum corrections which is made dimensionless by dividing the physical $m_{10}^2$ by $g^2$ and $m^2_1(q,\\beta )$ denotes the temperature dependent mass-squared corrections for the massless field $\\Phi ^3_1$ .", "The zero temperature quantum corrections for all $n$ can be written as $m^2_{10}&=& \\left[\\sum ^\\infty _{n=0}\\frac{7}{2 \\sqrt{2n+1}} + \\sum ^\\infty _{n=2}\\frac{1}{4 \\sqrt{2n-1}}-\\frac{7}{2}\\sum ^\\infty _{n=1}\\left(\\frac{n}{\\sqrt{2n-1}}-\\frac{n}{\\sqrt{2n+1}}\\right)\\right.\\nonumber \\\\&+&\\left.\\sum ^\\infty _{n=2} \\left(\\frac{(n-1)}{(2n-1)^{\\frac{5}{2}}}\\left(\\frac{(n+1)^2}{(2n+1)}+ \\frac{n(n-1)}{(2n-3)}\\right)-2\\frac{n(n-1)(n-2)}{(2n-1)(2n-3)}\\left(\\frac{1}{\\sqrt{2n-3}}- \\frac{1}{\\sqrt{2n-1}}\\right)\\right)\\right]\\nonumber \\\\&-&\\sum ^{\\infty }_{n=1}\\frac{4}{\\sqrt{2n} +\\sqrt{2n-2}}$ The zero temperature quantum correction given by (REF ) can be evaluated numerically.", "The convergent value is given by $m^2_{10} = 1.579$ The temperature dependent part in (REF ) can be written as $m^2_1(q,\\beta )&=& \\left[\\sum ^\\infty _{n=0} \\frac{7}{\\sqrt{2n+1}}\\frac{1}{\\left(e^{\\sqrt{(2n+1)q}\\beta }-1\\right)}+ \\sum ^\\infty _{n=2} \\frac{1}{2\\sqrt{2n-1}}\\frac{1}{\\left(e^{\\sqrt{(2n-1)q}\\beta }-1\\right)}\\right.\\nonumber \\\\&-& \\left.", "7 \\sum ^\\infty _{n=1} \\left(\\frac{n}{\\sqrt{2n-1}}\\frac{1}{\\left(e^{\\sqrt{(2n-1)q}\\beta }-1\\right)} -\\frac{n}{\\sqrt{2n+1}}\\frac{1}{\\left(e^{\\sqrt{(2n+1)q}\\beta }-1\\right)}\\right)\\right.\\nonumber \\\\&-&\\left.", "4 \\sum ^\\infty _{n=1} \\frac{n(n-1)(n-2)}{(2n-1)(2n-3)}\\left(\\frac{1}{\\sqrt{2n-3}}\\frac{1}{\\left(e^{\\sqrt{(2n-3)q}\\beta }-1\\right)}-\\frac{1}{\\sqrt{2n-1}}\\frac{1}{\\left(e^{\\sqrt{(2n-1)q}\\beta }-1\\right)}\\right)\\right.\\nonumber \\\\&+&\\left.", "2\\sum ^\\infty _{n=2}\\left(\\frac{(n-1)}{(2n-1)^{\\frac{5}{2}}}\\left(\\frac{(n+1)^2}{(2n+1)}+ \\frac{n(n-1)}{2n-3}\\right) \\frac{1}{\\left(e^{\\sqrt{(2n-1)q}\\beta }-1\\right)}\\right)\\right.\\nonumber \\\\&+&\\left.\\left({1\\over 2}-2\\sum ^\\infty _{n=2} \\left(\\frac{(n-1)}{(2n-1)}\\left(\\frac{(n+1)^2}{(2n+1)^2}+ \\frac{n(n-1)}{(2n-3)^2}\\right)\\right)\\right)\\sum ^{\\infty }_{m=-\\infty } \\frac{\\sqrt{q}\\beta }{4\\pi ^2 m^2}\\right]\\nonumber \\\\&+&\\left[\\sum ^{\\infty }_{n=1}\\left(\\left(4\\frac{1}{(\\sqrt{2n}+ \\sqrt{2(n-1)})}+2\\frac{(\\sqrt{2(n+1)} + \\sqrt{2(n-1)}-2 \\sqrt{2n})}{(\\sqrt{2n}- \\sqrt{2(n-1)})(\\sqrt{2(n+1)}- \\sqrt{2n})}\\right)\\frac{1}{\\left(e^{\\sqrt{2nq}\\beta }+1\\right)}\\right.\\right.\\nonumber \\\\&+&\\left.\\left.", "\\left(4\\frac{1}{(\\sqrt{2n}+ \\sqrt{2(n-1)})} - \\frac{2}{\\sqrt{2n}- \\sqrt{2(n-1)}}\\right)\\frac{1}{\\left(e^{\\sqrt{2(n-1)q}\\beta }+1\\right)}\\right.\\right.\\nonumber \\\\&+&\\left.\\left.\\frac{2}{\\sqrt{2(n+1)} - \\sqrt{2n}}\\frac{1}{\\left(e^{\\sqrt{2(n+1)q}\\beta }+1\\right)}\\right)\\right]$" ], [ "Two point function for $\\Phi _I^3$ {{formula:fdc20c5f-582e-43f6-9655-cbf72888eb93}}", "Let us now write down all the amplitudes that constitute the finite temperature one-loop mass-squared corrections to the tree-level massless fields $\\Phi ^3_I(m,l)$ , where $I\\ne 1$ .", "The computation of the vertices involving the fermions in this section has been done explicitly for $I=2$ .", "However the $SO(7)$ invariance of the theory implies that the two point function is the same for all $I=2\\cdots 8$ .", "Using the vertices listed in appendix REF we write below the expressions for the two point function.", "The Feynman diagrams involving the four-point bosonic interactions are depicted in the figure REF .", "$\\Sigma ^1_{\\Phi _I^3-\\Phi _I^3}={1\\over 2}N \\sum _{m,n}\\left[(6\\times 2) \\frac{G_1^I(l,l^{^{\\prime }},n,n)}{(\\omega _m^2+\\gamma _n)}+(2) \\frac{G_2^I(l,l^{^{\\prime }},n,n)}{(\\omega _m^2+\\lambda _n)}+(2) \\frac{\\tilde{G}_2^I(l,l^{^{\\prime }},n,n)}{\\omega _m^2}\\right]\\delta _{w+w^{^{\\prime }}}$ Figure: Feynman diagrams with four-point vertices V 1 I ,V 2 I ,V ˜ 2 I V^I_{1},~V^I_2,~\\tilde{V}^I_2.The three terms in (REF ) are represented by the Feynman diagrams in figures REF (a), REF (b) and REF (c) in the same order.", "The first term involves the fields $\\Phi ^{(1,2)}_I$ , $I\\ne 1$ , the second term involves the fields $C_{m,n}$ and the third term comprises of the fields $\\tilde{C}_{m,n}$ .", "Similarly the three-point bosonic interactions of $\\Phi ^3_I(m,l)$ are represented in the Feynman diagrams in Figure REF .", "Figure: Feynman diagrams with the three-point vertices V 3 I ,V ˜ 2 I V^I_{3},~\\tilde{V}^I_2The two-point function involving the bosonic three-point vertices and contributing to the one-loop finite temperature mass-corrections for the field $\\Phi ^3_I$ , $I\\ne 1$ is given by $\\Sigma ^2_{\\Phi _I^3-\\Phi _I^3}&=&-{1\\over 2}qN \\sum _{m,n,n^{^{\\prime }}}\\left[(2)\\frac{G_3^I(l,n,n^{^{\\prime }})G_3^I(l^{^{\\prime }},n,n^{^{\\prime }})}{(\\omega _m^2+\\gamma _n)(\\omega _{m^{^{\\prime }}}^2+\\lambda _{n^{^{\\prime }}})}+(2)\\frac{\\tilde{G}_3^I(l,n,n^{^{\\prime }})\\tilde{G}_3^I(l^{^{\\prime }},n,n^{^{\\prime }})}{(\\omega _m^2+\\gamma _{n^{^{\\prime }}})\\omega _{m^{^{\\prime }}}^2}\\right]\\delta _{w+w^{^{\\prime }}}$ where $w=m^{^{\\prime }}+m$ .", "The first term in (REF ) involving the three-point vertex $G_3^I(l,n,n^{^{\\prime }})$ involves the fields $\\Phi ^{(1.2)}_I$ , $I\\ne 1$ and $C_{m,n}$ s. The corresponding Feynman diagram is shown in figure REF (a).", "Similarly the second term in (REF ) involving the three-point vertex $\\tilde{G}_3^I(l,n,n^{^{\\prime }})$ involves the fields $\\Phi ^{(1.2)}_I$ , $I\\ne 1$ and $\\tilde{C}_{m,n}$ s. The relevant Feynman diagram is shown in figure REF (b).", "The Feynman diagram involving the three-point vertex with fermions is drawn in Figure REF .", "Figure: Feynman diagram with three-point vertex V f I V^I_{f} and its complex conjugate.The corresponding amplitude is $\\Sigma ^3_{\\Phi _I^3-\\Phi _I^3}=N \\sum _{m,n,n^{^{\\prime }}}\\left[ (8)\\frac{G_f^I(l,n,n^{^{\\prime }})G_f^{I*}(l^{^{\\prime }},n,n^{^{\\prime }})}{(i \\omega _m - \\sqrt{\\lambda ^{^{\\prime }}_{n}})(i \\omega _{m^{^{\\prime }}}-\\sqrt{\\lambda ^{^{\\prime }}_{n^{^{\\prime }}}})}\\right]\\delta _{w+w^{^{\\prime }}}$ where $w=m^{^{\\prime }}+m$ .", "As in the case of $\\Phi ^3_1$ , we can use the orthogonality relation for Hermite Polynomials to compute exactly, the various vertices given in section (REF ) and constituting the mass-squared corrections (REF ),(REF ) and (REF ).", "We do these computations after setting the external momenta $l=l^{^{\\prime }}= w=w^{^{\\prime }}=0$ .", "$G^I_1[0,0,n,n^{^{\\prime }}] &=& \\delta _{n,n^{^{\\prime }}}\\\\G^I_2[0,0,n,n] &=& \\delta _{n,n^{^{\\prime }}}\\\\\\tilde{G}^I_2[0,0,n,n^{^{\\prime }}] &=& \\delta _{n,n^{^{\\prime }}}\\\\G^I_3[0,n,n^{^{\\prime }}] &=& 0\\\\\\tilde{G}^I_3[0,n,n^{^{\\prime }}] &=&\\sqrt{2n-1} \\delta _{n-1,n^{^{\\prime }}}\\\\G^I_f[0,n,n^{^{\\prime }}]&=& i \\delta _{n,n^{^{\\prime }}}$ We now proceed to establish the UV finiteness of finite temperature mass-squared corrections to $\\Phi ^3_I$ .", "We first analyze the large $n$ -behaviour of the various vertices for the one-loop mass-squared corrections(REF ) (REF ) and (REF ).", "The four-point vertices in the amplitudes constituting the two-point function (REF ) are associated with only one kind of propagator.", "In the large $n$ limit the four-point vertices computed in (REF ), () and () can be written as $G_1^I(0,0,n,n)=G_2^I(0,0,n,n)=\\tilde{G}_2^I(0,0,n,n)= 1$ where we have used the Kronecker deltas to set $n=n^{^{\\prime }}$ .", "Note that the vertex $G^I_3(0,n,n^{^{\\prime }})$ is identically zero for all values of $n$ as shown in ().", "Furthermore the fermionic vertex $G^I_f(0,n,n^{^{\\prime }})$ () can be exactly computed for all $n$ and remains the same in the UV limit.", "The remaining three-point bosonic vertex $\\tilde{G}^I_3(0,n,n^{^{\\prime }})$ () in the UV limit becomes $\\tilde{G}_3^I(0,n,n^{^{\\prime }})\\sim \\sqrt{2n}[\\delta _{n^{^{\\prime }},n-1}]$ This large $n$ behaviour of the vertices in turn gives rise to the following asymptotic forms of the two-point functions for the tree-level massless field $\\Phi ^3_I$ .", "$\\Sigma ^1_{\\Phi _I^3-\\Phi _I^3}\\sim {1\\over 2}N \\sum _{m,n}\\left[(6\\times 2) \\frac{1}{(\\omega _m^2+\\gamma _n)}+(2) \\frac{1}{(\\omega _m^2+\\lambda _n)} + (2) \\frac{1}{(\\omega _m^2)} \\right]$ $\\Sigma ^2_{\\Phi _I^3-\\Phi _I^3}\\sim -{1\\over 2}qN \\sum _{m,n}\\left[ \\frac{(2n)}{\\omega _m^2(\\omega _{m}^2+\\lambda _{n})}\\right]$ Thus the total bosonic contribution in the limit $n \\rightarrow \\infty $ can be written as $\\Sigma ^1_{\\Phi _I^3-\\Phi _I^3}+\\Sigma ^2_{\\Phi _I^3-\\Phi _I^3} \\sim N \\sum _{m,n}\\left[(8) \\frac{1}{(\\omega _m^2+2nq)}\\right]\\sim \\sum _{n}\\frac{4}{\\sqrt{2n}}$ In this large $n$ limit, the contribution from the fermions coming from $\\Sigma ^3_{\\Phi _I^3-\\Phi _I^3}$ is same as the right hand side of eqn(REF ) with opposite sign.", "Combining the vertices in (REF - ) with their respective propagators (see Appendix ()), the effective mass-squared corrections for $\\Phi ^3_I$ , $I\\ne 1$ , can be written down as a function of $q$ and $\\beta $ in the following form; $m^2_{\\Phi ^3_I} (q,\\beta ) = m^2_{I0} + m^2_{I1}(q,\\beta ).$ where $ m^2_{I0}$ and $m^2_{I1}(q,\\beta )$ denote the zero temperature quantum corrections and the finite temperature corrections respectively to the tree-level massless field $\\Phi ^3_I$ , $I\\ne 1$ .", "The zero temperature quantum correction here is made dimensionless in the same way as in the case of $m^2_{10}$ in (REF ) and can be exactly computed as in the case of $\\Phi ^3_1$ and found to be $m^2_{I0} = \\sum ^{\\infty }_{n=0}\\frac{3}{\\sqrt{2n+1}} + \\sum ^{\\infty }_{n=1}\\frac{1}{\\sqrt{2n-1}}- \\sum ^{\\infty }_{n=1}\\frac{4}{\\sqrt{2n}}$ for all $n$ , where the first two terms under summation come from the bosonic contributions and the last term comes from the fermionic contributions.", "The various sums in (REF ) can be reorganized and written in terms of the regularized Riemann Zeta function $\\zeta \\left({1\\over 2}\\right)$ .", "The dimensionless zero temperature quantum correction can be evaluated as $m^2_{I0} = (4(1-\\sqrt{2})\\zeta \\left({1\\over 2}\\right)-1) = 1.495$ Similarly the finite temperature part $ m^2_{I1}(q,\\beta )$ can be written as, $m^2_{I1}(q,\\beta ) &=& \\sum ^{\\infty }_{n=0}\\frac{6}{\\sqrt{2 n +1}}\\frac{1}{e^{\\sqrt{(2n+1)q}\\beta }-1}+\\sum ^{\\infty }_{n=1}\\frac{2}{\\sqrt{2n-1}}\\frac{1}{e^{\\sqrt{(2n-1)q}\\beta }-1}\\nonumber \\\\&+& \\sum ^{\\infty }_{n=1}\\frac{8}{\\sqrt{2n}} \\frac{1}{e^{\\sqrt{2 n q}\\beta }+1}$" ], [ "Two point function for $A^3_x$", "We give below that expression for the two point one loop amplitude for $A^3_x$ .", "The vertices are worked out in appendix REF .", "The Feynman diagrams comprising the four-point bosonic interactions is given in figure REF .", "Figure: Feynman diagrams with four-point vertices V 1 A ,V 2 A ,V ˜ 2 A V^A_{1},~V^A_2,~\\tilde{V}^A_2.The amplitudes that are represented by the Feynman diagrams in Figure REF are collected together into the two-point finite temperature mass-squared corrections to the tree-level massless field $A^3_x$ in the following equation, namely $\\Sigma ^1_{A^3_x-A^3_x}={1\\over 2}N \\sum _{m,n}\\left[(7\\times 2) \\frac{G_1^A(n,n,l,l^{^{\\prime }})}{(\\omega _m^2+\\gamma _n)}+(2) \\frac{G_2^A(n,n,l,l^{^{\\prime }})}{(\\omega _m^2+\\lambda _n)}+ (2) \\frac{\\tilde{G}_2^A(n,n,l,l^{^{\\prime }})}{\\omega _m^2}\\right]\\delta _{w+w^{^{\\prime }}}\\nonumber \\\\$ The first second and third terms in (REF ) are represented by the Feynman diagrams in figures REF (a), REF (b) and REF (c) respectively.", "The fields involved in four-point vertices in the first, second and third terms are $\\Phi ^{(1,2)}_I$ , $I\\ne 1$ , $C_{m,n}$ and $\\tilde{C}_{m,n}$ respectively.", "The Feynman diagrams depicting the various three-point bosonic interactions including $A^3_x$ are presented in figure REF .", "$\\Sigma ^2_{A^3_x-A^3_x}=&-&{1\\over 2}qN \\sum _{m,n,n^{^{\\prime }}}\\left[ \\frac{G_3^A(n,n^{^{\\prime }},l)G_3^A(n,n^{^{\\prime }},l^{^{\\prime }})}{(\\omega _m^2+\\lambda _n)(\\omega _{m^{^{\\prime }}}^2+\\lambda _{n^{^{\\prime }}})}+ \\frac{\\tilde{G}_3^{A}(n,n^{^{\\prime }},l)\\tilde{G}_3^{A}(n,n^{^{\\prime }},l^{^{\\prime }})}{(\\omega _m^2)(\\omega _{m^{^{\\prime }}}^2)}\\right.\\nonumber \\\\&+& \\left.", "(2)\\frac{\\tilde{G}_3^{A^{\\prime }}(n,n^{^{\\prime }},l)\\tilde{G}_3^{A^{\\prime }}(n,n^{^{\\prime }},l^{^{\\prime }})}{(\\omega _m^2+\\lambda _n)(\\omega _{m^{^{\\prime }}}^2)} + (7)\\frac{G_4^A(n,n^{^{\\prime }},l)G_4^A(n,n^{^{\\prime }},l^{^{\\prime }})}{(\\omega _m^2+\\gamma _n)(\\omega _{m^{^{\\prime }}}^2+\\gamma _{n^{^{\\prime }}})}\\right]\\delta _{w+w^{^{\\prime }}}\\nonumber \\\\$ where $w=m^{^{\\prime }}+m$ .", "The first term in (REF ) comprising the three-point vertex $G^{A}_3(l,n,n^{^{\\prime }})$ involves the fields $C_{m,n}$ and $C^{^{\\prime }}_{m,n}$ .", "in the loops and is represented by the Feynman diagram in figure REF (a).", "The second term in (REF ) involving the vertex $\\tilde{G}^{A}_3(l,n, n^{^{\\prime }})$ comprises of the fields $\\tilde{C}_{m,n}$ s and $\\tilde{C}^{^{\\prime }}_{m,n}$ in the loop.", "The corresponding Feynman diagram is given in figure REF (b).", "The third term has contributions from $\\tilde{C}_{m,n}$ s and $\\tilde{C}_{m,n}$ in the loop with Feynman diagram shown in figure REF (c).", "Lastly the fourth term depicted in figure REF (d) has contributions from the fields $\\Phi ^{(1,2)}_I$ , $I\\ne 1$ .", "Figure: Feynman diagrams with three-point verticesV 3 A ,V ˜ 3 A ,V ˜ 3 A ' V^{A}_{3},~\\tilde{V}^A_3,~\\tilde{V}^{A^{\\prime }}_3 and V 3 4 V^4_3.Similarly the amplitude involving fermions in the loop is (REF ).", "The corresponding Feynman diagrams are presented in figure REF .", "$\\Sigma ^3_{A^3_x-A^3_x}=&&{1\\over 2}q N\\sum _{m,n,n^{^{\\prime }}}\\left[(16)\\frac{G_f^{A1}(n,n^{^{\\prime }},l)G_f^{A1*}(n,n^{^{\\prime }},l^{^{\\prime }})}{(i \\omega _m - \\sqrt{\\lambda ^{^{\\prime }}_n})(i \\omega _{m^{^{\\prime }}} - \\sqrt{\\lambda ^{^{\\prime }}_{n^{^{\\prime }}}})})\\right.\\nonumber \\\\&&\\left.-(16)\\frac{G_f^{A2}(n,n^{^{\\prime }},l)G_f^{A2*}(n,n^{^{\\prime }},l^{^{\\prime }})}{(i \\omega _m - \\sqrt{\\lambda ^{^{\\prime }}_n})(-i \\omega _{m^{^{\\prime }}} - \\sqrt{\\lambda ^{^{\\prime }}_{n^{^{\\prime }}}})}) \\right]\\delta _{w+w^{^{\\prime }}}$ where $w=m^{^{\\prime }}+m$ .", "The fermionic three-point vertex $G^{A1}_f(n,n^{^{\\prime }},l)$ constituting the first term in the two-point function (REF ) has contributions from the fermionic fields $\\theta _i(m,n)$ .", "The amplitude is represented in the Feynman diagram presented in figure REF (a).", "The second term in (REF ) on the other hand involves the fields $\\theta _i(m,n)$ and their complex conjugate $\\theta ^{*}_i(m,n)$ in the vertex $G^{A2}_f(n,n^{^{\\prime }},l)$ .", "The corresponding Feynman diagram is given in figure REF (b).", "Figure: Feynman diagrams involving three-point vertices V f A1 ,V f A2 V^{A1}_{f},~V^{A2}_f.The vertices given in section (REF ) and participating in the two-point functions that produce the mass-squared corrections to the tree-level massless field $A^3_x$ can be exactly computed following the same procedure as discussed for $\\Phi ^3_1$ and $\\Phi ^3_I$ , $I\\ne 1$ .", "$G^A_1[n,n^{^{\\prime }},0,0] &=& \\delta _{n,n^{^{\\prime }}},~~G^A_2[n,n^{^{\\prime }},0,0]={1\\over 2}\\delta _{n,n^{^{\\prime }}},~~\\tilde{G}^A_2[n,n^{^{\\prime }},0,0]={1\\over 2}\\delta _{n,n^{^{\\prime }}},\\\\G^{A}_3[n,n^{^{\\prime }},0] &=& 2\\left(\\sqrt{\\frac{2n(n+1)(n-1)}{(2n-1)(2n+1)}} \\delta _{n+1,n^{^{\\prime }}} -\\sqrt{\\frac{2n(n-1)(n-2)}{(2n-1)(2n-3)}} \\delta _{n-1,n^{^{\\prime }}}\\right),\\\\\\tilde{G}^A_3[n,n^{,},0] &=& 0, \\\\\\tilde{G}^{A^{\\prime }}_3[n,n^{^{\\prime }},0]&=&-\\sqrt{2}\\left(\\frac{(n+1)\\sqrt{n-1}}{\\sqrt{(2n-1)(2n+1)}}\\delta _{n+1,n^{^{\\prime }}} + \\frac{\\sqrt{n}(n-1)}{\\sqrt{(2n-1)(2n-3)}}\\delta _{n-1,n^{^{\\prime }}}\\right),\\\\G^A_4[n,n^{^{\\prime }},0]&=& \\left(\\sqrt{2(n+1)}\\delta _{n+1,n^{^{\\prime }}} - \\sqrt{2n}\\delta _{n-1,n^{^{\\prime }}}\\right),\\\\G^{A1}_f[n,n^{^{\\prime }},0]&=& -\\frac{i}{2}\\left(\\delta _{n+1,n^{,}}+\\delta _{n-1,n^{^{\\prime }}}\\right),\\\\G^{A2}_f[n,n^{^{\\prime }},0]&=&-\\frac{i}{2}\\left(2\\delta _{n+1,n^{,}}-2\\delta _{n-1,n^{^{\\prime }}}\\right).$ In the same spirit as for $\\Phi ^3_1$ and $\\Phi ^3_I$ we now proceed to establish the UV finiteness for the one-loop two-point functions for the field $A^3_x$ .", "In the large $n$ limit the various vertices in eqns (-) reduce to $G_1^A(n,n,0)= 1\\mbox{~~;~~} G_2^A(n,n,0)=\\tilde{G}_2^A(n,n,0,0)= {1\\over 2}$ $G_3^{A}(n,n^{^{\\prime }},0)\\sim \\frac{\\sqrt{2n}}{8}[8\\delta _{n^{^{\\prime }},n-1}-8\\delta _{n^{^{\\prime }},n+1}]\\mbox{~~;~~} \\tilde{G}_3^{A^{^{\\prime }}}(n,n^{^{\\prime }},0)\\sim -\\frac{\\sqrt{2n}}{8}[4\\delta _{n^{^{\\prime }},n-1}+4\\delta _{n^{^{\\prime }},n+1}]$ $\\tilde{G}_3^A(n,n^{^{\\prime }},0)= 0 \\mbox{~~;~~}\\tilde{G}_4^{A}(n,n^{^{\\prime }},0)\\sim \\sqrt{2n}[\\delta _{n^{^{\\prime }},n+1}-\\delta _{n^{^{\\prime }},n-1}]$ With these, the amplitudes in the ultraviolet limit is same as the right hand side of the equations (REF ), (REF ) and (REF ), thus showing that the one-loop $A^3_x-A^3_x$ amplitude is ultraviolet finite.", "Once again we write down the effective mass-squared for the field $A^3_x$ as a function of $q$ and $\\beta $ as $m^2_{A^3_x}(q,\\beta ) = m^2_{x0} + m^2_{x1}(q,\\beta )$ where $ m^2_{x0}$ denotes the dimensionless (same as $m^2_{10}$ and $m^2{I0}$ ) zero temperature quantum corrections and $m^2_{x1}(q,\\beta )$ denotes the temperature dependent mass-squared corrections for the tree-level massless field $A^3_x$ .", "The zero temperature quantum corrections can be written as $m^2_{x0}&=& \\left[\\sum ^\\infty _{n=0}\\frac{7}{2 \\sqrt{2n+1}} + \\sum ^\\infty _{n=2}\\frac{1}{4 \\sqrt{2n-1}}-\\frac{7}{4}\\sum ^\\infty _{n=0}\\left(\\frac{1}{\\sqrt{2n+1}}- \\frac{n+1}{\\sqrt{2n+3}}\\right)-\\frac{7}{4}\\sum ^\\infty _{n=1}\\frac{n}{\\sqrt{2n-1}}\\right.\\nonumber \\\\&-&\\left.\\sum ^\\infty _{n=2} \\left(\\frac{n(n-1)}{(2n+1)(2n-3)}\\frac{1}{\\sqrt{2n-1}}- \\frac{n(n+1)(n-1)}{(2n-1)(2n+1)}\\frac{1}{\\sqrt{2n+1}}+ \\frac{n(n-1)(n-2)}{(2n-1)(2n-3)}\\frac{1}{\\sqrt{2n-3}}\\right)\\right.\\nonumber \\\\&+&\\left.\\sum ^{\\infty }_{n=2}\\left(\\frac{(n-1)(n+1)^2}{(2n-1)^{\\frac{5}{2}}(2n+1)}+ \\frac{n(n-1)^2}{(2n-1)^{\\frac{5}{2}}(2n-3)}\\right)\\right]\\nonumber \\\\&-&\\sum ^{\\infty }_{n=1}\\frac{2}{\\sqrt{2n}+\\sqrt{2(n+1)}}+ \\frac{2}{\\sqrt{2n}+\\sqrt{2(n-1)}}$ We compute the dimensionless zero temperature quantum corrections given by (REF ) numerically.", "The convergent value is, $m^2_{x0} = 1.514$ The temperature dependent part in (REF ) can be written as $m^2_{x1}(q,\\beta )&=& \\left[\\sum ^\\infty _{n=0} \\frac{7}{\\sqrt{2n+1}}\\frac{1}{\\left(e^{\\sqrt{(2n+1)q}\\beta }-1\\right)}+ \\sum ^\\infty _{n=2} \\frac{1}{2\\sqrt{2n-1}}\\frac{1}{\\left(e^{\\sqrt{(2n-1)q}\\beta }-1\\right)}\\right.\\nonumber \\\\&-& \\left.", "\\frac{7}{2} \\sum ^\\infty _{n=1} \\left(\\frac{1}{\\sqrt{2n+1}}\\frac{1}{\\left(e^{\\sqrt{(2n+1)q}\\beta }-1\\right)}-\\frac{n+1}{\\sqrt{2n+3}}\\frac{1}{\\left(e^{\\sqrt{(2n+3)q}\\beta }-1\\right)}\\right)\\right.\\nonumber \\\\&-&\\left.", "\\frac{7}{2}\\sum ^\\infty _{n=1}\\frac{n}{\\sqrt{2n-1}}\\frac{1}{\\left(e^{\\sqrt{(2n-1)q}\\beta }-1\\right)}-2 \\sum ^\\infty _{n=2} \\left( \\frac{n(n-1)}{(2n+1)(2n-3)}\\frac{1}{\\sqrt{2n-1}}\\frac{1}{\\left(e^{\\sqrt{(2n-1)q}\\beta }-1\\right)}\\right.\\right.\\nonumber \\\\&-&\\left.\\left.", "\\frac{n(n+1)(n-1)}{(2n-1)(2n+1)^{\\frac{3}{2}}}\\frac{1}{\\left(e^{\\sqrt{(2n+1)q}\\beta }-1\\right)}+ \\frac{n(n-1)(n-2)}{(2n-1)(2n-3)^{\\frac{3}{2}}}\\frac{1}{\\left(e^{\\sqrt{(2n-3)q}\\beta }-1\\right)}\\right)\\right.\\nonumber \\\\&+&\\left.", "2\\sum ^\\infty _{n=2}\\left(\\frac{(n-1)(n+1)^2}{(2n-1)^{\\frac{5}{2}}(2n+1)}+ \\frac{n(n-1)^2}{(2n-1)^{\\frac{5}{2}}(2n-3)}\\right)\\frac{1}{\\left(e^{\\sqrt{(2n-1)q}\\beta }-1\\right)}\\right.\\nonumber \\\\&+&\\left.\\left({1\\over 2}-2 \\sum ^{\\infty }_{n=2}\\left(\\frac{(n-1)(n+1)^2}{(2n-1)(2n+1)^{2}}+ \\frac{n(n-1)^2}{(2n-1)(2n-3)^2}\\right)\\right)\\sum ^{\\infty }_{m=-\\infty } \\frac{\\sqrt{q}\\beta }{4\\pi ^2 m^2}\\right]\\nonumber \\\\&+&\\left[\\sum ^{\\infty }_{n=1}\\left(\\left(2\\frac{(2 \\sqrt{2n} + \\sqrt{2(n-1)}+ \\sqrt{2(n+1)})}{(\\sqrt{2n}+ \\sqrt{2(n-1)})(\\sqrt{2n}+ \\sqrt{2(n+1)})}\\right.\\right.\\right.\\nonumber \\\\&+&\\left.\\left.", "\\left.8\\frac{(\\sqrt{2(n+1)} + \\sqrt{2(n-1)}-2 \\sqrt{2n})}{(\\sqrt{2n}- \\sqrt{2(n-1)})(\\sqrt{2(n+1)}- \\sqrt{2n})}\\right)\\frac{1}{\\left(e^{\\sqrt{2nq}\\beta }+1\\right)}\\right.\\right.\\nonumber \\\\&+&\\left.\\left.", "\\left(\\frac{2}{\\sqrt{2(n-1)} + \\sqrt{2n}} - \\frac{8}{\\sqrt{2n}- \\sqrt{2(n-1)}}\\right)\\frac{1}{\\left(e^{\\sqrt{2(n-1)q}\\beta }+1\\right)}\\right.\\right.\\nonumber \\\\&+&\\left.\\left.\\left(\\frac{2}{\\sqrt{2(n+1)} + \\sqrt{2n}} + \\frac{8}{\\sqrt{2(n+1)}- \\sqrt{2n}}\\right)\\frac{1}{\\left(e^{\\sqrt{2(n+1)q}\\beta }+1\\right)}\\right)\\right]$ In the end the effective masses-squared for the massless fields namely $m^2_{\\Phi ^3_1}$ , $m^2_{\\Phi ^3_I}$ and $m^2_{A^3_x}$ depend only on the parameter $q$ and temperature.", "Later in section (REF ), we present the behaviour of the effective masses-squared with temperature for different values of $q$ ." ], [ "Finite part of effective Tachyon mass", "Having computed the temperature corrected one-loop mass-squared for the various massless fields, we can now proceed to compute the mass-squared corrections for the tree-level tachyons.", "We have already established the UV finiteness of the tachyonic amplitudes by demonstrating the cancellation of leading order divergences from the zero temperature bosonic and fermionic quantum corrections to the tree-level tachyon mass-squared.", "However given the fairly complicated mathematical form of the various corrections given in (REF ),(REF ) and (REF ), extracting the finite part of the amplitudes appears to be very difficult.", "Hence we are unable to give analytical expressions for the finite part of the zero temperature quantum corrections to the tree-level tachyon mass-squared.", "This complications also prevents us from computing the transition temperature analytically.", "Given these handicaps we are compelled to resort to numerical means.", "In the following section we present a numerical computation of the transition temperature.", "Figure: Plots of the mass-squared correction to the massless fieldφ 1 3 \\phi ^3_1 against β=1 T\\beta = \\frac{1}{T} forg 2 =0.01g^2=0.01 and q=0.1q=0.1,0.20.2,0.30.3.Figure: Plot of the mass-squared correction to the massless field φ 1 3 \\phi ^3_1 againstTT forg 2 =0.01g^2=0.01 and q=0.1q=0.1,0.20.2,0.30.3.Figure: Plot of the mass-squared correction to the massless field φ I 3 \\phi ^3_Iagainst β=1 T\\beta =\\frac{1}{T} forg 2 =0.01g^2=0.01 and q=0.1q=0.1,0.20.2,0.30.3.Figure: Plot of the mass-squared correction to the massless field φ I 3 \\phi ^3_I against TTfor g 2 =0.01g^2=0.01 and q=0.1q=0.1,0.20.2,0.30.3.Figure: Plot of the mass-squared correction to the massless field A x 3 A^3_xagainst β=1 T\\beta = \\frac{1}{T} forg 2 =0.01g^2=0.01 and q=0.1q=0.1,0.20.2,0.30.3.Figure: Plot of the mass-squared correction to the massless field A x 3 A^3_x against TT forg 2 =0.01g^2=0.01 and q=0.1q=0.1,0.20.2,0.30.3.Figure: Plot of the mass-squared correction to the tree-level tachyonagainst β=1 T\\beta = \\frac{1}{T} for g 2 =0.01g^2=0.01,q=0.1q=0.1, 0.20.2 and 0.30.3.Figure: Plot of the mass-squared correction to the tree-level tachyonsagainst TT forg 2 =0.01g^2=0.01, q=0.1q=0.1, 0.20.2 and 0.30.3.Figure: Plot of the mass-squared correction to the tree-level tachyons against TT forg 2 =0.01g^2=0.01, q=0.1q=0.1, 0.20.2 and 0.30.3" ], [ "Numerical results", "The computation of the one-loop finite temperature mass-squared for all the tree-level massless degrees of freedom is crucial because these temperature dependent masses modify the corresponding propagators of the massless fields thereby ensuring infra-red finiteness of the one-loop effective masses-squared of the tachyons.", "All the finite temperature corrections to the tree-level masses-squared are now shown to be UV finite.", "The tachyonic instability in the bulk is proposed to give rise to BCS Cooper-pairing instability in the boundary theory [1].", "In this section we demonstrate that the instability is removed by finite temperature effects.", "The tree-level tachyon mass-squared is $-\\frac{q}{g^2}$ , where $g$ is the dimensionfull Yang-Mills coupling in $(1+1)$ -dimensions.", "The finite temperature one-loop correction including the zero temperature quantum corrections is $\\mathcal {O}(1)$ .", "The temperature-dependent mass-squared corrections is always increasing and there exists a critical temperature where the effective mass-squared of the tachyonic fields become zero.", "Beyond the critical temperature the effective mass-squared of the tachyon is found to be positive and increasing.", "This bears hallmark of a phase-transition from the unstable phase to the stable phase.", "In the boundary theory this is proposed in [1] to correspond to a superconducting phase-transition.", "As mentioned in sections and , the critical temperature of phase transition cannot be computed analytically.", "We therefore tread a different path.", "We demonstrate numerically the behaviour of the masses-squared with varying $\\beta $ as well as $T$ due to zero temperature quantum corrections + the finite temperature effects without computing them separately.", "In all the mass-squared corrections, the UV divergent pieces in the zero temperature corrections from the bosonic side cancel with that from the fermionic side.", "At large values of the momenta $n$ and $l$ the finite part of the quantum corrections fall off very sharply and eventually only the finite temperature corrections dominate.", "The parameter $q$ provides a scale for supersymmetry breaking in the present brane-configuration under study and has the dimension of mass-squared.", "It is also related to the angle between the branes as $q= \\frac{1}{\\pi \\alpha ^{^{\\prime }}} \\tan \\left(\\frac{\\theta }{2}\\right)$ The two-point functions for the tachyons have infra-red problem due to the presence of massless fields in the loops.", "As mentioned in section REF we need to modify the propagators of the tree-level massless fields by introducing a mass-squared shift provided by the finite temperature corrections to their tree-level masses-squared.", "The leading order behaviour of the temperature dependent part of the mass-squared corrections in $1+1$ -dimensions is linear with increasing temperature at high temperatures.", "The finite temperature effective mass-squared of the tree-level tachyon in $(1+1)$ -dimensions can thus be estimated within perturbation theory to be (in dimensionless variables) $m^2_{\\text{eff}}(C_{0,0}) = -\\frac{q}{g^2} + \\left[m_0^2+ \\frac{T}{\\sqrt{q}}\\underbrace{\\left(\\sum _n \\frac{1}{\\lambda _n} + \\cdots \\right)}_{=x}\\right]+ \\mathcal {O}(\\frac{g^2}{q}).$ The physical mass is thus $m^2_{eff}g^2$ and $m_0^2$ represents the dimensionless zero temperature quantum corrections independent of $q$ and $g^2$ and collected from all the amplitudes in equations (REF ), (REF ) and (REF ).", "In equation (REF ), $x$ is a dimensionless number (independent of $g^2,$ and $q$ ) specifying the temperature dependent contribution.", "In equation (REF ), $\\mathcal {O}(\\frac{g^2}{q})$ represents the next higher order in quantum corrections given by two-loop Feynman diagrams.", "The dimensionless zero temperature quantum correction to the tree-level tachyon mass-squared and is found numerically to be approximately equal to $m^2_0 = 1.6$ .", "The behaviour of the finite temperature masses-squared of the massless fields as well as the tachyonic fields are depicted pictorially by plotting the masses-squared against temperature $T$ and $\\beta =\\frac{1}{T}$ .", "We proceed to present the plots.", "In all the plots we have displayed the one-loop effective masses-squared as multiplied by $g^2$ .", "The figures (REF ) (REF ) and (REF ) depict the behaviour of the masses-squared namely $m^2_{\\phi ^3_1}$ , $m^2_{\\phi ^3_I}$ and $m^2_{A^3_x}$ with varying $\\beta $ .", "The mass-squared decreases with increasing $\\beta $ as expected.", "In the figures (REF ), (REF ) and (REF ), $m^2_{\\phi ^3_1}$ , $m^2_{\\phi ^3_I}$ and $m^2_{A^3_x}$ are shown to increase almost linearly with increasing temperature.", "This behaviour is expected from finite temperature field theory as finite temperature corrections are always known to be positive and increasing.", "As discussed in section (REF ), we also calculate the mass-matrix for the massless modes $\\tilde{C}_{w,k}$ numerically and diagonalize the matrix.", "Using the temperature dependent masses-squared of the various massless fields discussed above the effective mass-squared for the fields $C_{0,0}$ can be evaluated numerically as a function of $\\beta $ (or $T$ ).", "The effective mass-squared for the tree-level tachyon are plotted against $\\beta $ in the Figure REF .", "for three values of $q$ , namely $q=0.1$ and $q=0.2$ and $q=0.3$ and $g^2=0.01$ .", "The plots of the mass-squared against temperature $T$ for $q=0.1$ and $q=0.2$ and $q=0.3$ are given in Figure REF and Figure REF .", "As expected the finite temperature corrections dominate with increasing temperature which is clear from the $m^2_{eff}(C_{0,0})$ vs $T$ plots in Figure REF where the behaviour of the mass-squared appears to be almost linear at higher temperatures.", "The effective mass-squared $m^2_{eff}(C_{0,0})$ is equal to zero at a value of temperature (the transition temperature $T_c$ ) where the graphs intersect the $\\beta $ and $T$ -axes in the plots given in Figure REF , Figure REF and Figure REF .", "The plots in Figure REF are drawn for smaller range of $T$ in order to show the critical points ($m^2_{\\text{eff}}(C_{0,0})=0$ ) more clearly.", "Putting $m^2_{\\text{eff}}(C_{0,0})=0$ in equation (REF ) we get $T_c = {1\\over x}[q^{\\frac{1}{2}}(\\frac{q}{g^2}- m_0^2)]$ where $T_c$ is the dimensionfull transition temperature (having dimension of mass).", "One can also define a dimensionless transition temperature by $T_c=\\tilde{T}_c q^{\\frac{3}{2}}/g^2$ .", "One should note that the dimensionless tree-level mass-squared of the tachyon for $q=0.1$ , $0.2$ and $0.3$ are $q/g^2= 10$ , 20 and 30 respectively for $g^2=0.01$ , whereas the quantum correction $m_0^2$ is much smaller.", "Now if $m_0^2=0$ , $\\tilde{T}_c = {1\\over x}$ and is thus independent of $q, g^2$ .", "Thus to a good approximation one expects that $\\tilde{T}_c$ will be independent of $q,g^2$ .", "One has to note that this scaling relation for the transition temperature is the dominant term at the level of one-loop.", "At the level of higher loops the dependence of the effective mass-squared on temperature will be much more complicated.", "However at least in weak coupling, the quantum corrections at successive higher loops will be smaller and smaller, and the dominant term in $\\tilde{T}_c$ will still be given by this one loop relation.", "From the plots given in Figure REF and Figure REF , the numerical values of $T_c$ for $g^2=0.01$ and $q=0.1$ , $q=0.2$ and $q=0.3$ are $T_c=3.34$ , $T_c=9.48$ and $T_c=16.73$ respectively.", "This gives $\\tilde{T}_c =1.0562$ , $1.0599$ and $1.0182$ for $q=0.1$ , $0.2$ and $0.3$ and $g^2=0.01$ respectively, confirming our expectation that when $m_0^2$ is small $\\tilde{T}_c$ is approximately independent of $g,q$ ." ], [ "Discussion and Outlook", "We have computed the one-loop finite temperature corrections to the tree-level tachyon mass-squared in intersecting $D1$ -branes in a self consistent manner.", "We have shown the UV finiteness (at one loop).", "We have seen that at high temperatures the tachyonic field becomes massive as expected and we have computed this critical temperature in a one loop approximation - improved by incorporating mass-squared corrections to the massless fields in the spirit of the RG.", "This takes care of the IR divergences.", "Thus our calculation has no UV or IR divergences.", "This model resembles the holographic BCS superconductor model discussed in [1].", "We expect that with the techniques developed in this paper it should be possible to tackle the model in [1] involving higher branes.", "These techniques should also be useful in other contexts where D brane constructions are used and supersymmetry spontaneously broken.", "The entire computation is done in temporal gauge, $A^a_0=0$ .", "This gauge choice helps to avoid ghosts in the theory.", "The original theory describing the world-volume of two $D1$ -branes is a $(1+1)$ -dimensional supersymmetric $SU(2)$ Yang-Mills theory which is a UV finite theory.", "The choice of the background $\\langle \\phi ^3_B \\rangle = qx$ breaks supersymmetry without tampering with the other degrees of freedom.", "Hence we find that the UV behaviour of the amplitudes in a broken supersymmetry scenario remains the same as in the supersymmetric case.", "In order to establish the UV finiteness of the one-loop corrections we rely on asymptotic expansion of the vertices and find that the leading order divergent pieces from the bosonic and fermionic loops cancel among themselves.", "The effective mass-squared of the tachyon is found to grow linearly with temperature in accordance to the expected behaviour in $(1+1)$ -dimensions.", "The kinetic terms for the bosons are scale independent in $(1+1)$ -dimensions.", "Hence the zero-temperature one loop quantum corrections are found to be independent of the supersymmetry breaking scale.", "The crossing of the $m^2_{\\text{eff}}(C_{0,0})$ $\\it {vs}$ $T$ curves from negative to positive values indicates two distinct phases.", "This bears the signature of a phase transition.", "We also find that the dimensionless critical temperature ($T_c g^2/q^{3/2}$ ) for this phase transition has very closely placed values namely $\\tilde{T}_c=1.0562$ , $1.0599$ and $1.0182$ .", "In order to do the complete stability analysis of the intersecting $D1$ -branes at finite temperature one has to compute the full tachyon effective action at finite temperature which rely on higher loop calculations.", "The results demonstrated in this paper can be generalized to higher dimensional branes without much difficulty.", "In particular for two intersecting $Dp$ -branes the effective mass-squared for the tree-level tachyons is expected to grow as $T^{p-1}$ .", "Acknowledgments: The authors would like to acknowledge the Institute of Mathematical Sciences, Chennai for providing a free and vibrant academic atmosphere and the Department of Atomic Energy for providing funds for this project.", "S.P.C.", "would like to thank colleagues and compatriots Swastik Bhattacharya and Shankhadeep Chakrabortty, Saurabh Gupta and Akhilesh Nautiyal and T Geetha for fruitful as well as refreshing lighthearted discussions.", "S.S. would like to thank Institute of Mathematical Sciences for the kind hospitality during various stages of this work.", "The work of S.S. is partially supported by the Research and Development Grant (2013-2014), University of Delhi." ], [ "Dimensional reduction of $D=10$ , {{formula:0349a721-b07d-431c-a6fd-ffd9dc3bd2e7}} , {{formula:793351a4-1b30-4248-8381-71a1e12388d5}} SYM to {{formula:4ad68fd5-522b-48b1-b017-66eb6e826af3}}", "We first write down the action for $D=10$ , ${\\cal N}=1$ SYM We will use the metric $\\mbox{diagonal}(+1, -1, \\cdots , -1)$ ., $S_{9+1}=\\frac{1}{g^2}\\mbox{tr}\\int d^{10}x \\left[-\\frac{1}{2}F_{MN}F^{MN}+i\\bar{\\Psi }\\Gamma ^{M}D_{M}\\Psi \\right]$ $F_{MN}&=&\\partial _{M}A_{N}-\\partial _{N}A_{M}-i\\left[A_{M},A_{N}\\right]\\\\D_{M}\\Psi &=&\\partial _{M}\\Psi -i\\left[A_{M},\\Psi \\right]$ where $M,N=0, \\cdot \\cdot \\cdot 9$ with $A_{M}=\\frac{\\sigma ^a}{2}A^a_{M}$ , $\\Psi =\\frac{\\sigma ^a}{2}\\Psi ^a_{M}$ and, $\\left[\\frac{\\sigma ^a}{2},\\frac{\\sigma ^b}{2}\\right]=i\\epsilon ^{abc}\\frac{\\sigma ^c}{2} \\mbox{~~;~~}\\frac{1}{2} \\mbox{tr} \\left(\\sigma ^a\\sigma ^b\\right)=\\delta ^{ab}$ $\\Gamma ^M$ are $32\\times 32$ imaginary matrices and.", "In $(9+1)$ -dimensions the gamma-matrix which anti commutes with all other gamma matrices is $\\gamma ^{11}= \\left(\\begin{array}{cc}\\mathbb {I}_{16 \\times 16} & 0\\\\0 & -\\mathbb {I}_{16 \\times 16}\\end{array}\\right)$ The chiral projection operator in $(9+1)$ -dimensions giving rise to left and right-moving chiral fermions is given by $\\mathcal {P} = \\frac{1\\pm \\gamma ^{11}}{2}$ Under the chiral projection (REF ), the 32-component Dirac fermions become $\\Psi =\\left(\\begin{array}{c}\\Psi _L \\\\0\\end{array}\\right)$ $\\Psi $ is a 32 component Majorana-Weyl spinor with 16 non-zero components.", "The 32 dimensional $\\Gamma $ matrices satisfy the Dirac algebra $\\lbrace \\Gamma ^M, \\Gamma ^N\\rbrace =2\\eta ^{MN}$ .", "Under the decomposition $SO(9,1)\\rightarrow SO(1,1)\\times SO(8)$ , they have to be written in terms of the 16 dimensional $spin(8)$ matrices.", "For $M$ corresponding to the $SO(8)$ directions (that we label by $I$ ) we call the 16 dimensional $spin(8)$ matrices as $\\gamma ^I$ , $(I=1,...,8)$ .", "The $\\gamma ^I$ 's thus satisfy the $spin(8)$ algebra $\\lbrace \\gamma ^I,\\gamma ^J\\rbrace =2\\delta ^{IJ}$ .", "They are however reducible and can be written in terms of the 8 dimensional representations $\\alpha ^{I}$ as, $\\gamma ^{I}= \\left(\\begin{array}{cc}0 & \\alpha ^I\\\\\\alpha ^{IT} & 0\\end{array}\\right)$ where the $\\alpha ^I$ 's now satisfy $\\lbrace \\alpha ^I,\\alpha ^J\\rbrace =2\\delta ^{IJ}$ .", "A representation of the $\\alpha ^I$ 's can be written follows [36], $\\alpha ^1&=&\\tau \\otimes \\tau \\otimes \\tau \\mbox{~~~~~~~~}\\alpha ^2=1\\otimes \\sigma ^1\\otimes \\tau \\\\\\alpha ^3&=&1\\otimes \\sigma ^3\\otimes \\tau \\mbox{~~~~~~~~}\\alpha ^4=\\sigma ^1\\otimes \\tau \\otimes 1\\nonumber \\\\\\alpha ^5&=&\\sigma ^3\\otimes \\tau \\otimes 1 \\mbox{~~~~~~~~}\\alpha ^6=\\tau \\otimes 1 \\otimes \\sigma ^1\\nonumber \\\\\\alpha ^7&=&\\tau \\otimes 1\\otimes \\sigma ^3 \\mbox{~~~~~~~~}\\alpha ^8= 1\\otimes 1\\otimes 1\\nonumber $ where $\\tau =i\\sigma ^2$ .", "We can construct one more $\\gamma $ matrix that anti commutes with all the other $\\gamma ^I$ 's.", "It is given by $\\gamma ^9=\\gamma ^1\\gamma ^2\\cdot \\cdot \\cdot \\gamma ^8$ .", "In matrix form, $\\gamma ^{9}= \\left(\\begin{array}{cc}1_{8}&0\\\\0&-1_{8}\\end{array}\\right)$ The nine 32 dimensional $\\Gamma ^M$ $(M=1,...9)$ matrices can thus be formed out of the nine $\\gamma $ matrices.", "Since there is no tenth $\\gamma $ matrix, we need to construct a tenth $\\Gamma $ matrix that anti commutes with all the others.", "Ultimately we can write, $\\Gamma ^0&=&\\sigma ^2\\otimes 1_{16}\\\\\\Gamma ^{I}&=&i\\sigma ^1\\otimes \\gamma ^{I}\\nonumber \\\\\\Gamma ^9&=&i\\sigma ^1\\otimes \\gamma ^{9}\\nonumber $ With this, the sixteen dimensional $\\Psi _L$ in equation (REF ) can further be written as, $\\Psi _L=\\left(\\begin{array}{c}\\psi _L \\\\ \\psi _R\\end{array}\\right)$ where $\\psi _L$ and $\\psi _R$ are now 8 component fermions.", "In other words, the sixteen component $\\Psi _L$ decomposes as $16= (1,8)+(\\bar{1},\\bar{8})$ .", "Thus in $1+1$ dimensions we have 8 one-component left-moving plus 8 one-component right-moving fermions.", "We can now write down the dimensionally reduced action, $S_{1+1}&=&S^1_{1+1}+S^2_{1+1}\\\\S^1_{1+1}&=&\\frac{1}{g^2}\\mbox{tr} \\int d^2x \\left[-\\frac{1}{2}F_{\\mu \\nu }F^{\\mu \\nu }+ D_{\\mu }\\Phi _I D^{\\mu }\\Phi _I+\\frac{1}{2}\\left[\\Phi _I,\\Phi _J\\right]^2\\right]\\\\S^2_{1+1}&=&\\frac{i}{g^2}\\mbox{tr} \\int d^2x \\left[\\psi ^T_L D_0 \\psi _L+\\psi ^T_R D_0 \\psi _R+\\psi ^T_L D_1 \\psi _L-\\psi ^T_R D_1 \\psi _R \\right.\\\\&+& \\left.", "2i\\psi ^T_R\\alpha ^T_I\\left[\\Phi _I,\\psi _L\\right]\\right]\\nonumber $ where, $D_{\\mu }=\\partial _{\\mu }-i\\left[A_{\\mu },\\star \\right]$ and all the fermions, $\\psi _L$ and $\\psi _R$ are anti commuting." ], [ "Tables of fields, eigenfunctions and normalizations.", "The various fields together with their eigenfunctions, momenta and momentum modes are tabulated below.", "Table: Eigenfunctions, Tree-level masses and momentum modes." ], [ "Bosons", "Propagator for $C_{m,n}$ $\\left\\langle C_{m,n}C_{m^{^{\\prime }},n^{^{\\prime }}} \\right\\rangle =g^2\\frac{\\delta _{m,-m^{^{\\prime }}}\\delta _{n,n^{^{\\prime }}}}{\\omega _m^2+\\lambda _n}$ Propagator for $\\tilde{C}_{m,n}$ $\\left\\langle \\tilde{C}_{m,n}\\tilde{C}_{m^{^{\\prime }},n^{^{\\prime }}} \\right\\rangle =g^2\\frac{\\delta _{m,-m^{^{\\prime }}}\\delta _{n,n^{^{\\prime }}}}{\\omega _m^2}$ Propagator for the $\\Phi _I^1$ and $\\Phi _I^2$ fluctuations ($I\\ne 1$ )are same and is given by $\\left\\langle \\Phi _I^1(m,n)\\Phi _I^1(m^{^{\\prime }},n^{^{\\prime }}) \\right\\rangle =g^2\\frac{\\delta _{m,-m^{^{\\prime }}}\\delta _{n,n^{^{\\prime }}}}{\\omega _m^2+\\gamma _n}$ Propagator for the $\\Phi _I^3$ fluctuations for all $I$ is given by $\\left\\langle \\Phi _I^3(m,l)\\Phi _I^3(m^{^{\\prime }},l^{^{\\prime }}) \\right\\rangle =g^2 \\frac{\\delta _{m,-m^{^{\\prime }}}2\\pi \\delta (l+l^{^{\\prime }})}{\\omega _m^2+l^2}$ Propagator for the $A^3_x$ fluctuations assumes the form because there is no term in the Lagrangian (REF ) with spatial derivatives on $A^3_x$ .", "$\\left\\langle A^3_x(m,l) A^3_x(m^{^{\\prime }},l^{^{\\prime }}) \\right\\rangle =g^2 \\frac{\\delta _{m,-m^{^{\\prime }}}2\\pi \\delta (l+l^{^{\\prime }})}{\\omega _m^2}$ Figure: V 1 V_1 , V 2 V_2 vertices$V_1=-\\frac{N}{2g^2} F_1(k,k^{^{\\prime }},n,n^{^{\\prime }}) \\delta _{\\omega +\\omega ^{^{\\prime }}+m+m^{^{\\prime }}} \\mbox{~~~~~~~(Figure \\ref {v1v2})}$ $F_1(k,k^{^{\\prime }},n,n^{^{\\prime }})&=&\\sqrt{q}\\int dx \\left[\\phi _k(x)\\phi _{k^{^{\\prime }}}(x)A_n(x) A_{n^{^{\\prime }}}(x) + 2 A_k(x)\\phi _{k^{^{\\prime }}}(x)\\phi _n(x) A_{n^{^{\\prime }}}(x)\\right.\\nonumber \\\\&+& \\left.", "2 \\phi _k(x) A_{k^{^{\\prime }}}(x)\\phi _n(x) A_{n^{^{\\prime }}}(x) + A_k(x) A_{k^{^{\\prime }}}(x) \\phi _n(x) \\phi _{n^{^{\\prime }}}(x)\\right]$ $V_2=-\\frac{N}{2g^2}\\left[F_2(k,k^{^{\\prime }},n,n^{^{\\prime }})\\right]\\delta _{\\omega +\\omega ^{^{\\prime }}+m+m^{^{\\prime }}} \\mbox{~~~~~~~(Figure \\ref {v1v2})}$ $F_2(k,k^{^{\\prime }},n,n^{^{\\prime }})=\\sqrt{q}\\int dx e^{-qx^2}\\left[A_k(x) A_{k^{^{\\prime }}}(x)+\\phi _k(x)\\phi _{k^{^{\\prime }}}(x)\\right] \\left[H_n(x) H_{n^{^{\\prime }}}(x)\\right]$ Figure: V ˜ 1 \\tilde{V}_1 , V 2 ' V^{^{\\prime }}_2 vertices$\\tilde{V}_1=-\\frac{N}{2g^2} \\tilde{F}_1(k,k^{^{\\prime }},n,n^{^{\\prime }}) \\delta _{\\omega +\\omega ^{^{\\prime }}+m+m^{^{\\prime }}}\\mbox{~~~~~~~(Figure \\ref {tv1v2})}$ $\\tilde{F}_1(k,k^{^{\\prime }},n,n^{^{\\prime }})&=&\\sqrt{q}\\int dx \\left[\\phi _k(x)\\phi _{k^{^{\\prime }}}(x)\\tilde{A}_n(x) \\tilde{A}_{n^{^{\\prime }}}(x) +2 A_k\\phi _{k^{^{\\prime }}}(x)\\tilde{\\phi }_n(x) \\tilde{A}_{n^{^{\\prime }}}(x)\\right.\\nonumber \\\\&+& \\left.", "2 \\phi _k(x) A_{k^{^{\\prime }}}(x)\\tilde{\\phi }_n(x) \\tilde{A}_{n^{^{\\prime }}}(x) + A_k(x) A_{k^{^{\\prime }}}(x) \\tilde{\\phi }_n(x) \\tilde{\\phi }_{n^{^{\\prime }}}(x)\\right]$ $V^{^{\\prime }}_2=-\\frac{N}{2g^2}\\left[F^{^{\\prime }}_2(k,k^{^{\\prime }},l,l^{^{\\prime }})\\right]\\delta _{\\omega +\\omega ^{^{\\prime }}+m+m^{^{\\prime }}}\\mbox{~~~~~~~(Figure \\ref {tv1v2})}$ $F^{^{\\prime }}_2(k,k^{^{\\prime }},l,l^{^{\\prime }})=\\sqrt{q}\\int dx \\left[A_k(x) A_{k^{^{\\prime }}}(x)+\\phi _k(x)\\phi _{k^{^{\\prime }}}(x)\\right] \\left[e^{ilx}e^{il^{^{\\prime }}x}\\right]$ Figure: V 3 V_3 and V 3 ' V^{^{\\prime }}_3 vertices$V_3=-\\frac{N}{2g^2}F_3(k,k^{^{\\prime }},l,l^{^{\\prime }})\\delta _{\\omega +\\omega ^{^{\\prime }}+m+m^{^{\\prime }}}\\mbox{~~~~~~~(Figure \\ref {v3v3p})}$ $F_3(k,k^{^{\\prime }},l,l^{^{\\prime }})=\\sqrt{q}\\int dx \\left[\\phi _k(x) \\phi _{k^{^{\\prime }}(x)}\\right]e^{i (l+l^{^{\\prime }}) x}$ $V^{^{\\prime }}_3=-\\frac{N}{2g^2}F^{^{\\prime }}_3(k,k^{^{\\prime }},l,l^{^{\\prime }})\\delta _{\\omega +\\omega ^{^{\\prime }}+m+m^{^{\\prime }}}\\mbox{~~~~~~~(Figure \\ref {v3v3p})}$ $F^{^{\\prime }}_3(k,k^{^{\\prime }},l,l^{^{\\prime }})=\\sqrt{q}\\int dx \\left[A_k(x) A_{k^{^{\\prime }}(x)}e^{ilx}e^{il^{^{\\prime }}x}\\right]$ Figure: V 4 V_4 and V ˜ 4 \\tilde{V}_4 vertices$V_4&=&-\\frac{N^{3/2}}{g^2} F_4(k,l,n) \\beta \\delta _{\\omega +m+m^{^{\\prime }}}\\mbox{~~~~~~~(Figure \\ref {v4v4t})}$ $F_4(k,l,n)&=&\\int dx \\left[-\\phi _n(x)\\partial _x \\phi _k(x)+\\phi _k(x)\\partial _x\\phi _n(x)-\\phi _n(x)A_k(x) (qx)\\right.\\nonumber \\\\&+& \\left.", "A_n(x)\\phi _k(x) (qx) \\right]e^{ilx}$ $\\tilde{V}_4&=&-\\frac{N^{3/2}}{g^2} \\tilde{F}_4(k,l,n) \\beta \\delta _{\\omega +m+m^{^{\\prime }}}\\mbox{~~~~~~~(Figure \\ref {v4v4t})}$ $\\tilde{F}_4(k,l,n)&=&\\int dx \\left[-\\tilde{\\phi }_n(x)\\partial _x\\phi _k(x)+\\phi _k(x)\\partial _x \\tilde{\\phi }_n(x)-\\tilde{\\phi }_n(x)A_k(x)(qx)\\right.\\nonumber \\\\&+& \\left.\\tilde{A}_n(x)\\phi _k(x) (qx)\\right]e^{ilx}$ Figure: V 5 V_5 vertex.$V_5=-\\frac{N^{3/2}}{2g^2}F_5(k,l,n)\\beta \\delta _{\\omega +m+m^{^{\\prime }}} \\mbox{~~~~~~~(Figure \\ref {v5})}$ $F_5(k,l,n)&=&\\int dx e^{-qx^2/2} \\left[e^{ilx}A_k(x)\\partial _x H_n(x)-(qx)e^{ilx}A_k(x) H_n(x)\\right.\\nonumber \\\\&-& \\left.", "(il)e^{ilx}A_k(x) H_n(x)+(qx) e^{ilx}\\phi _k(x) H_n(x)\\right]$ Figure: V 5 ' V^{^{\\prime }}_5 and V ˜ 5 ' \\tilde{V}^{^{\\prime }}_5 vertices$V^{^{\\prime }}_5=-\\frac{N^{3/2}}{g^2}F^{^{\\prime }}_5(k,l,n)\\beta \\delta _{\\omega +m+m^{^{\\prime }}} \\mbox{~~~~~~~(Figure \\ref {v5ptv5p})}$ $F^{^{\\prime }}_5(k,l,n)&=&\\int dx \\left[(il)e^{ilx}\\phi _k(x)A_n(x)+(il)e^{ilx}A_k(x)\\phi _n(x)\\right.\\nonumber \\\\&-&\\left.e^{ilx}\\partial _x\\phi _k(x) A_n(x)-e^{ilx}\\partial _x\\phi _n(x) A_k(x)\\right]$ $\\tilde{V}^{^{\\prime }}_5=-\\frac{N^{3/2}}{g^2}\\tilde{F}^{^{\\prime }}_5(k,l,n)\\beta \\delta _{\\omega +m+m^{^{\\prime }}} \\mbox{~~~~~~~(Figure \\ref {v5ptv5p})}$ $\\tilde{F}^{^{\\prime }}_5(k,l,n)&=&\\int dx \\left[(il)e^{ilx}\\phi _k(x)\\tilde{A}_n(x)+(il)e^{ilx}A_k(x)\\tilde{\\phi }_n(x)\\right.\\nonumber \\\\&&\\left.", "-e^{ilx}\\partial _x\\phi _k(x) \\tilde{A}_n(x)-e^{ilx}\\partial _x\\tilde{\\phi }_n(x) A_k(x)\\right]$" ], [ "Fermions", "Propagator for the $L^a_i$ and $R^a_i$ modes $(a=1,2)$ , $\\left\\langle \\theta _j(m,n)\\theta _k^*(m^{^{\\prime }},n^{^{\\prime }}) \\right\\rangle =\\frac{g^2}{N^{1/2}}\\frac{\\delta _{jk}\\delta _{m,m^{^{\\prime }}}\\delta _{n^{\\prime }n^{^{\\prime }}}}{i\\omega _m+\\sqrt{\\lambda ^{^{\\prime }}_n}}$ $\\lambda ^{^{\\prime }}_n=2nq$ .", "Propagator for the $L^3_i$ and $R^3_i$ modes.", "$\\left\\langle L^3_i(m,l)L^{3}_k(m^{^{\\prime }},l^{^{\\prime }}) \\right\\rangle &=&\\frac{g^2}{N^{1/2}}\\frac{\\delta _{ik}\\delta _{m,-m^{^{\\prime }}}2\\pi \\delta (l+l^{^{\\prime }})}{i\\omega _m+l}\\nonumber \\\\\\left\\langle R^3_i(m,l)R^{3}_k(m^{^{\\prime }},l^{^{\\prime }}) \\right\\rangle &=&\\frac{g^2}{N^{1/2}}\\frac{\\delta _{ik}\\delta _{m,-m^{^{\\prime }}}2\\pi \\delta (l+l^{^{\\prime }})}{i\\omega _m-l}$ Figure: V 6 R V_6^R and V 6 L V^L_6 vertices$V^{R/L}_{6}=i\\frac{N}{g^2}F^{R/L}_6(k,n,l)\\delta _{w+m+m^{^{\\prime }}}\\mbox{~~~~~~~(Figure \\ref {v6rl})}$ Where, $F^R_6(k,n,l)=\\sqrt{q}\\int dx \\phi _k(x)R_n(x)e^{ilx}$ There are eight such vertices involving $R_i^2$ and $L_{(9-i)}^3$ ($i=1 \\cdots 8$ ).", "We also have eight vertices involving $L_i^2$ and $R_{(9-i)}^3$ ($i=1 \\cdots 8$ ).", "The corresponding vertices would be given by replacing $R_n(x)$ with $L_n(x)$ .", "This vertex is thus, $F^L_6(k,n,l)=\\mp \\sqrt{q}\\int dx \\phi _k(x)L_n(x)e^{ilx}$ The $\\mp $ is due to the fact that half of the above vertices come with sign opposite to that of the other half in the Lagrangian.", "There is no $\\mp $ in (REF ) as the minus sign coming from the vertices in the Lagrangian is compensated by the ones from the eigenfunctions.", "Similarly the other eight vertices involving both $R^3_i$ and $R^a_i$ with $(a=1,2)$ (or the $L$ legs) have the same structure.", "Figure: V 7 R V_7^R and V 7 L V^L_7 vertices$V^{R/L}_{7}=i\\frac{N}{g^2}F^{R/L}_7(k,n,l)\\delta _{w+m+m^{^{\\prime }}} \\mbox{~~~~~~~(Figure \\ref {v7rl})}$ $F^R_7(k,n,l)&=&\\mp \\sqrt{q}\\int dx A_k(x)R_n(x)e^{ilx}\\nonumber \\\\F^L_7(k,n,l)&=&\\sqrt{q}\\int dx A_k(x)L_n(x)e^{ilx}$ Here the $\\mp $ in the expression for $F^R_7(k,n,l)$ is due to the fact that half of the eigenfunctions come with a sign opposite to the other half and in the Lagrangian all the terms come with the same sign." ], [ "Vertices for computation of two-point $\\tilde{C}_{w,k}$ amplitudes ", "Since the zero-eigenfunctions are also massless fields we compute their two-point finite temperature amplitudes.", "The amplitudes are given in section (REF ).", "Here we present the various four-point and three-point vertices that occur in the calculation for the amplitudes of the zero-eigenfunctions." ], [ "Fermionic vertices", "The fermionic three-point vertices for the $\\tilde{C}_{w,k}-\\tilde{C}_{w^{^{\\prime }},k^{^{\\prime }}}$ two-point functions at finite temperature participate in the cancellation of UV divergence as in the case of the tachyonic amplitudes.", "The propagators for the fermionic loops are given in appendix (REF ).", "The various three-point vertices are given below.", "Figure: V H 6 R V_{H_6}^R and V H 6 L V^L_{H_6} vertices$\\tilde{V}^{R/L}_{H_6}=i\\frac{N}{g^2}\\tilde{H}^{R/L}_6(k,n,l)\\delta _{w+m+m^{^{\\prime }}}\\mbox{~~~~~~~(Figure \\ref {hv6rl})}$ $\\tilde{H}^R_6(k,n,l)=\\sqrt{q}\\int dx \\tilde{\\phi }_k(x)R_n(x)e^{ilx}$ $\\tilde{H}^L_6(k,n,l)=\\mp \\sqrt{q}\\int dx \\tilde{\\phi }_k(x)L_n(x)e^{ilx}$ Figure: V H 7 R V_{H_7}^R and V H 7 L V^L_{H_7} vertices$\\tilde{V}^{R/L}_{H_7}=i\\frac{N}{g^2}\\tilde{H}^{R/L}_7(k,n,l)\\delta _{w+m+m^{^{\\prime }}} \\mbox{~~~~~~~(Figure \\ref {hv7rl})}$ $\\tilde{H}^R_7(k,n,l)&=&\\mp \\sqrt{q}\\int dx \\tilde{A}_k(x)R_n(x)e^{ilx}\\nonumber \\\\\\tilde{H}^L_7(k,n,l)&=&\\sqrt{q}\\int dx \\tilde{A}_k(x)L_n(x)e^{ilx}$ The origin of the $\\mp $ sign is explained in Appendix REF ." ], [ "$\\Phi _1^3$ vertices", "We list here all the four-point and three-point bosonic and three-point fermionic vertices that constitute the two-point functions for $\\Phi ^3_1-\\Phi ^3_1$ .", "Figure: V 1 1 V^1_1 and V 2 1 V^1_2 vertices$V_1^1=-\\frac{N}{2g^2} G_1^1(l,l^{^{\\prime }},n,n^{^{\\prime }}) \\delta _{w+w^{^{\\prime }}+m+m^{^{\\prime }}} \\mbox{~~~~~~~(Figure \\ref {v11v12})}$ $G_1^1(l,l^{^{\\prime }},n,n^{^{\\prime }})=\\sqrt{q}\\int dx e^{-qx^2}H_n(x)H_{n^{^{\\prime }}}(x)e^{ilx}e^{il^{^{\\prime }}x}$ $V_2^1=-\\frac{N}{2g^2} G_2^1(l,l^{^{\\prime }},n,n^{^{\\prime }}) \\delta _{w+w^{^{\\prime }}+m+m^{^{\\prime }}} \\mbox{~~~~~~~(Figure \\ref {v11v12})}$ $G_2^1(l,l^{^{\\prime }},n,n^{^{\\prime }})=\\sqrt{q}\\int dx A_n(x)A_{n^{^{\\prime }}}(x)e^{ilx}e^{il^{^{\\prime }}x}$ Figure: V ˜ 2 1 \\tilde{V}_2^1 vertex$\\tilde{V}_2^1=-\\frac{N}{2g^2} \\tilde{G}_2^1(l,l^{^{\\prime }},n,n^{^{\\prime }}) \\delta _{w+w^{^{\\prime }}+m+m^{^{\\prime }}}\\mbox{~~~~~~~(Figure \\ref {tv12})}$ $\\tilde{G}_2^1(l,l^{^{\\prime }},n,n^{^{\\prime }})=\\sqrt{q}\\int dx \\tilde{A}_n(x)\\tilde{A}_{n^{^{\\prime }}}(x)e^{ilx}e^{il^{^{\\prime }}x}$ Figure: V 1 1 ' V^{1^{\\prime }}_1 and V 3 1 V^{1}_3 vertices$V_1^{1^{\\prime }}=-\\frac{N}{2g^2} G_1^{1^{\\prime }}(l,n,n^{^{\\prime }}) \\delta _{w+m+m^{^{\\prime }}}\\mbox{~~~~~~~(Figure \\ref {v11pv13})}$ $G_1^{1^{\\prime }}(l,n,n^{^{\\prime }})=\\sqrt{q}\\int dx~\\left[q x e^{-qx^2}H_n(x)H_{n^{^{\\prime }}}(x)e^{ilx}\\right]$ $V_3^1=-\\frac{N^{3/2}}{g^2} G_3^1(l,n,n^{^{\\prime }}) \\beta \\delta _{w+m+m^{^{\\prime }}}\\mbox{~~~~~~~(Figure \\ref {v11pv13})}$ $G_3^1(l,n,n^{^{\\prime }})=\\int dx e^{ilx}\\left[q x A_n(x) A_{n^{^{\\prime }}}(x) + \\partial _x\\phi _{n^{^{\\prime }}}(x)A_n(x)-il A_n(x)\\phi _{n^{^{\\prime }}}(x)\\right]$ Figure: V ˜ 3 1 \\tilde{V}^1_3 and V ˜ 3 1 ' \\tilde{V}^{1^{\\prime }}_3 vertices$\\tilde{V}_3^1=-\\frac{N^{3/2}}{g^2} \\tilde{G}_3^1(l,n,n^{^{\\prime }}) \\beta \\delta _{w+m+m^{^{\\prime }}}\\mbox{~~~~~~~(Figure \\ref {tv13tv13p})}$ $\\tilde{G}_3^1(l,n,n^{^{\\prime }})=\\int dx e^{ilx}\\left[q x \\tilde{A}_n(x) \\tilde{A}_{n^{^{\\prime }}}(x) + \\partial _x\\tilde{\\phi }_{n^{^{\\prime }}}(x)\\tilde{A}_n(x)-il \\tilde{A}_n(x)\\tilde{\\phi }_{n^{^{\\prime }}}(x)\\right]$ $\\tilde{V}_3^{1^{\\prime }}=-\\frac{N^{3/2}}{g^2} \\tilde{G}_3^{1^{\\prime }}(l,n,n^{^{\\prime }}) \\beta \\delta _{w+m+m^{^{\\prime }}}\\mbox{~~~~~~~(Figure \\ref {tv13tv13p})}$ $\\tilde{G}_3^{1^{\\prime }}(l,n,n^{^{\\prime }})&=&\\int dx e^{ilx}\\left[2q x A_n(x) \\tilde{A}_{n^{^{\\prime }}}(x) + \\partial _x\\tilde{\\phi }_{n^{^{\\prime }}}(x)A_n(x)-il A_n(x)\\tilde{\\phi }_{n^{^{\\prime }}}(x)\\right.\\nonumber \\\\&+&\\left.", "\\partial _x \\phi _{n}(x)\\tilde{A}_{n^{^{\\prime }}}(x)e^{ilx}-il \\tilde{A}_{n^{^{\\prime }}}(x)\\phi _{n}(x)\\right]$ Figure: The fermionic vertices V f 1 V^1_f and V f 2 V^2_f for Φ 1 3 -Φ 1 3 \\Phi ^3_1-\\Phi ^3_1 amplitudes$V_f^1=i\\frac{N}{g^2} G_f^1(l,n,n^{^{\\prime }}) \\delta _{w+m+m^{^{\\prime }}}\\mbox{~~~~~~~(Figure \\ref {v1fv2f})}$ $G_f^1(l,n,n^{^{\\prime }})= \\sqrt{q}\\int dx e^{ilx}R_n(x)L_{n^{^{\\prime }}}(x)$ $V_f^2=i\\frac{N}{g^2} G_f^2(l,n,n^{^{\\prime }}) \\delta _{w+m+m^{^{\\prime }}}\\mbox{~~~~~~~(Figure \\ref {v1fv2f})}$ $G_f^{2}(l,n,n^{^{\\prime }})= \\sqrt{q}\\int dx e^{ilx}\\left[R_n(x)L_{n^{^{\\prime }}}^{*}-L_n(x)R_{n^{^{\\prime }}}^{*}(x)\\right]$" ], [ "$\\Phi _I^3$ , {{formula:90c14ef8-7bac-4c00-b464-df292d6a987d}} vertices", "We list here the vertices for the one-loop finite temperature mass-squared corrections for the massless field $\\Phi ^3_I(m,l), I \\ne 1$ .", "Figure: V 1 I V^I_1 and V 2 I V^I_2 vertices$V_1^I=-\\frac{N}{2g^2} G_1^I(l,l^{^{\\prime }},n,n^{^{\\prime }}) \\delta _{w+w^{^{\\prime }}+m+m^{^{\\prime }}}\\mbox{~~~~~~~(Figure \\ref {vi1vi2})}$ $G_1^I(l,l^{^{\\prime }},n,n^{^{\\prime }})=\\sqrt{q}\\int dx e^{-qx^2}H_n(x)H_{n^{^{\\prime }}}(x)e^{ilx}e^{il^{^{\\prime }}x}$ $V_2^I=-\\frac{N}{2g^2} G_2^I(l,l^{^{\\prime }},n,n^{^{\\prime }}) \\delta _{w+w^{^{\\prime }}+m+m^{^{\\prime }}}\\mbox{~~~~~~~(Figure \\ref {vi1vi2})}$ $G_2^I(l,l^{^{\\prime }},n,n^{^{\\prime }})=\\sqrt{q}\\int dx \\left[A_n(x)A_{n^{^{\\prime }}}(x)+\\phi _n(x)\\phi _{n^{^{\\prime }}}(x)\\right]e^{ilx}e^{il^{^{\\prime }}x}$ Figure: V ˜ 2 I \\tilde{V}_2^I vertex$\\tilde{V}_2^I=-\\frac{N}{2g^2} \\tilde{G}_2^I(l,l^{^{\\prime }},n,n^{^{\\prime }}) \\delta _{w+w^{^{\\prime }}+m+m^{^{\\prime }}}\\mbox{~~~~~~~(Figure \\ref {tvi2})}$ $\\tilde{G}_2^I(l,l^{^{\\prime }},n,n^{^{\\prime }})=\\sqrt{q}\\int dx \\left[\\tilde{A}_n(x)\\tilde{A}_{n^{^{\\prime }}}(x)+\\tilde{\\phi }_n(x)\\tilde{\\phi }_{n^{^{\\prime }}}(x)\\right]e^{ilx}e^{il^{^{\\prime }}x}$ Figure: V 3 I V^I_3 and V ˜ 3 I \\tilde{V}^{I}_3 vertices$V_3^I=-\\frac{N^{3/2}}{g^2} G_3^I(l,n,n^{^{\\prime }}) \\beta \\delta _{w+m+m^{^{\\prime }}}\\mbox{~~~~~~~(Figure \\ref {vi3tvi3})}$ $G_3^I(l,n,n^{^{\\prime }})&=&\\int dx e^{ilx}\\left[\\partial _x (e^{-qx^2/2} H_{n^{^{\\prime }}}(x))A_n(x)-il e^{-qx^2/2}A_n(x)H_{n^{^{\\prime }}}(x)\\right.\\nonumber \\\\&&\\left.-qx\\phi _n(x)e^{-qx^2/2}H_{n^{^{\\prime }}}(x)\\right]$ $\\tilde{V}_3^I=-\\frac{N^{3/2}}{g^2} \\tilde{G}_3^I(l,n,n^{^{\\prime }}) \\beta \\delta _{w+m+m^{^{\\prime }}}\\mbox{~~~~~~~(Figure \\ref {vi3tvi3})}$ $\\tilde{G}_3^I(l,n,n^{^{\\prime }})&=&\\int dx e^{ilx}\\left[\\partial _x (e^{-qx^2/2} H_{n^{^{\\prime }}}(x))\\tilde{A}_n(x)-il e^{-qx^2/2}\\tilde{A}_n(x)H_{n^{^{\\prime }}}(x)\\right.\\nonumber \\\\&&\\left.-qx\\tilde{\\phi }_n(x)e^{-qx^2/2}H_{n^{^{\\prime }}}(x)\\right]$ Figure: V f I V^I_f vertex$V_f^I=i\\frac{N}{g^2} G_f^I(l,n,n^{^{\\prime }}) \\delta _{w+m+m^{^{\\prime }}}\\mbox{~~~~~~~(Figure \\ref {vif})}$ $G_f^{I}(l,n,n^{^{\\prime }})= \\sqrt{q}\\int dx e^{ilx}\\left[R_n(x)L_{n^{^{\\prime }}}+L_n(x)R_{n^{^{\\prime }}}(x)\\right]$" ], [ "$A^3_x$ vertices", "In this section we write down the various vertices needed for computing the two point $A^3_x$ amplitude.", "Figure: V 1 A V^A_1 and V 2 A V^A_2 vertices$V_1^A=-\\frac{N}{2g^2} G_1^A(n,n^{^{\\prime }},l,l^{^{\\prime }}) \\delta _{w+w^{^{\\prime }}+m+m^{^{\\prime }}}\\mbox{~~~~~~~(Figure \\ref {va1va2})}$ $G_1^A(n,n^{^{\\prime }}l,l^{^{\\prime }})= \\sqrt{q} \\int dx e^{-q x^2} H_n(\\sqrt{q}x) H_{n^{^{\\prime }}}(\\sqrt{q}x) e^{i(l+ l^{^{\\prime }})x}$ $V_2^A=-\\frac{N}{2g^2} G_2^A(n,n^{^{\\prime }},l,l^{^{\\prime }}) \\delta _{w+w^{^{\\prime }}+m+m^{^{\\prime }}}\\mbox{~~~~~~~(Figure \\ref {va1va2})}$ $G_2^A(n,n^{^{\\prime }},l,l^{^{\\prime }})=\\sqrt{q}\\int dx \\phi _n(x)\\phi _{n^{^{\\prime }}}(x) e^{i(l+ l^{^{\\prime }})x}$ Figure: V ˜ 2 A \\tilde{V}_2^A vertex$\\tilde{V}_2^A=-\\frac{N}{2g^2} \\tilde{G}_2^A(n,n^{^{\\prime }},l,l^{^{\\prime }}) \\delta _{w+w^{^{\\prime }}+m+m^{^{\\prime }}}\\mbox{~~~~~~~(Figure \\ref {tva2})}$ $\\tilde{G}_2^A(n,n^{^{\\prime }},l,l^{^{\\prime }})=\\sqrt{q}\\int dx \\tilde{\\phi }_n(x)\\tilde{\\phi }_{n^{^{\\prime }}}(x) e^{i(l+ l^{^{\\prime }})x}$ Figure: V 3 A V^A_3 and V ˜ 3 A \\tilde{V}^{A}_3 vertices$V_3^A=-\\frac{N^{3/2}}{g^2} G_3^A(n,n^{^{\\prime }},l) \\beta \\delta _{w+m+m^{^{\\prime }}}\\mbox{~~~~~~~(Figure \\ref {va3tva3})}$ $G_3^A(n,n^{^{\\prime }},l)&=& \\int dx \\left[\\partial _x (\\phi _{n^{^{\\prime }}}(x))\\phi _n(x)-\\partial _x (\\phi _n(x))\\phi _{n^{^{\\prime }}}(x)\\right.\\\\\\nonumber &+& \\left.qx A_{n^{^{\\prime }}}(x)\\phi _n(x)-qx A_{n}(x)\\phi _{n^{^{\\prime }}}(x)\\right] e^{ilx}$ $\\tilde{V}_3^A=-\\frac{N^{3/2}}{g^2} \\tilde{G}_3^A(n,n^{^{\\prime }},l) \\beta \\delta _{w+m+m^{^{\\prime }}}\\mbox{~~~~~~~(Figure \\ref {va3tva3})}$ $\\tilde{G}_3^A(n,n^{^{\\prime }},l)&=& \\int dx \\left[\\partial _x (\\tilde{\\phi }_{n^{^{\\prime }}}(x))\\tilde{\\phi }_n(x)-\\partial _x (\\tilde{\\phi }_n(x))\\tilde{\\phi }_{n^{^{\\prime }}}(x)\\right.\\\\\\nonumber &+& \\left.qx \\tilde{A}_{n^{^{\\prime }}}(x)\\tilde{\\phi }_n(x)-qx \\tilde{A}_{n}(x)\\tilde{\\phi }_{n^{^{\\prime }}}(x)\\right] e^{ilx}$ Figure: V ˜ 3 A ' \\tilde{V}^{A^{\\prime }}_3 and V 4 A V^A_4 vertices$\\tilde{V}_3^{A^{\\prime }}=-\\frac{N^{3/2}}{g^2} \\tilde{G}_3^{A^{\\prime }}(n,n^{^{\\prime }},l) \\beta \\delta _{w+m+m^{^{\\prime }}}\\mbox{~~~~~~~(Figure \\ref {tvap3va4})}$ $\\tilde{G}_3^{A^{\\prime }}(n,n^{^{\\prime }},l)&=& \\int dx \\left[\\partial _x (\\tilde{\\phi }_{n^{^{\\prime }}}(x))\\phi _n(x) - \\partial _x (\\phi _n(x))\\tilde{\\phi }_{n^{^{\\prime }}}(x)\\right.\\\\\\nonumber &+& \\left.", "qx \\tilde{A}_{n^{^{\\prime }}}(x)\\phi _n(x)- \\tilde{\\phi }_{n^{^{\\prime }}}(x)A_n(x)\\right] e^{ilx}$ $V_4^A=-\\frac{N^{3/2}}{g^2} G_4^A(n,n^{^{\\prime }},l) \\beta \\delta _{w+m+m^{^{\\prime }}}\\mbox{~~~~~~~(Figure \\ref {tvap3va4})}$ $G_4^A(n,n^{^{\\prime }},l)&=&\\int dx e^{-qx^2/2}\\left[\\partial _x (e^{-qx^2/2}H_{n^{^{\\prime }}}(x))H_n(x)\\right.\\\\\\nonumber &-& \\left.\\partial _x (e^{-qx^2/2}H_n(x))H_{n^{^{\\prime }}}(x)\\right] e^{ilx}$ Figure: V f A1 V^{A1}_f and V f A2 V^{A2}_f vertices$V_f^{A1}=i\\frac{N}{g^2} G_4^{A1}(n,n^{^{\\prime }},l) \\beta \\delta _{w+m+m^{^{\\prime }}}\\mbox{~~~~~~~(Figure \\ref {va1fva2f})}$ $G_f^{A1}(n,n^{^{\\prime }},l)=\\sqrt{q}\\int dx \\left[L_n(x) L_{n^{^{\\prime }}}(x) - R_n(x) R_{n^{^{\\prime }}}(x)\\right] e^{ilx}$ $V_f^{A2}=i\\frac{N}{g^2} G_4^{A2}(n,n^{^{\\prime }},l) \\beta \\delta _{w+m+m^{^{\\prime }}}\\mbox{~~~~~~~(Figure \\ref {va1fva2f})}$ $G_f^{A2}(n,n^{^{\\prime }},l)=\\sqrt{q}\\int dx \\left[L_n(x) L_{n^{^{\\prime }}}^{*}(x) - R_n(x) R_{n^{^{\\prime }}}^{*}(x)\\right]e^{ilx}$" ], [ "Matsubara Sums", "In this appendix we show some sample computations showing sums over Matsubara frequencies.", "Let us evaluate the sum over $m$ in the propagator $\\frac{1}{\\beta }\\sum ^{\\infty }_{n=2,m = -\\infty } \\frac{1}{\\omega ^2_m + \\lambda _n}$ where $\\omega _m=2m\\pi /\\beta $ Following [37] we convert the sum over $m$ into a contour integral as follows.", "By writing $p_0=i\\omega _m$ , define the function $f(p_0) = -\\frac{1}{p_0^2 - \\lambda _n}$ The function $f(p_0)$ does not have poles on the imaginary axis.", "We multiply it by a function with simple poles on the imaginary axis at values $p_0= \\frac{2 im\\pi }{\\beta }$ and analytic and bounded otherwise.", "A function with this property is $\\coth (p_0 \\beta /2)$ .", "The sum over $m$ in (REF ) can now be reproduced from the contour integral $\\frac{1}{2 \\pi i \\beta }\\oint dp_0 \\left(\\frac{\\beta }{2}\\right)\\coth \\left(\\frac{p_0 \\beta }{2}\\right)f(p_0)$ Figure: Contours for doing the Matsubara sums.where the contour is the sum over the contours $C_n$ shown in Figure REF (a).", "The $C_n$ 's can now be deformed into the contour $\\Gamma $ .", "The contour integral can then be written in terms of line integrals as $\\frac{1}{2 \\pi i}\\int ^{-i \\infty - \\epsilon }_{i \\infty -\\epsilon }dp_0 f(p_0)\\left(-{1\\over 2}-\\frac{1}{e^{-p_0 \\beta } -1}\\right)+\\frac{1}{2 \\pi i}\\int ^{i \\infty + \\epsilon }_{-i \\infty +\\epsilon }dp_0 f(p_0)\\left({1\\over 2}+ \\frac{1}{e^{p_0 \\beta } -1}\\right)$ since the function $f(p_0)$ vanishes for $p_0=\\pm i\\infty $ .", "In (REF ) the frequency sum separates into the zero-temperature part and the temperature-dependent part for both the integrals.", "The line integrals above can now be evaluated using the contour integrals over the contours $\\Gamma _1$ and $\\Gamma _2$ in Figure REF (b).", "This is because the line integral over rest of the rectangle vanishes when the length of the sides are taken to infinity.", "These contour integrals now have contributions only from the poles of $f(p_0)$ at $p_0=\\pm \\sqrt{\\lambda _n}$ .", "Thus, $\\frac{1}{\\beta }\\sum ^{\\infty }_{n=2,m = -\\infty } \\frac{1}{\\omega ^2_m + \\lambda _n}= \\sum _{n=2} \\frac{1}{\\sqrt{\\lambda _n}}\\left({1\\over 2}+ \\frac{1}{e^{\\sqrt{\\lambda _n}\\beta }-1}\\right)$ Similarly using the above formula, the several bosonic propagators upon being summed over the Matsubara frequency $\\omega _m$ are, $\\frac{1}{\\beta }\\sum ^{\\infty }_{n=2,m = -\\infty } \\frac{1}{\\omega ^2_m + \\gamma _n}=\\sum ^{\\infty }_{n=2}\\frac{1}{\\sqrt{\\gamma _n}} \\left({1\\over 2}+ \\frac{1}{e^{\\sqrt{\\gamma _n}\\beta } - 1}\\right)$ $\\frac{1}{\\beta }\\sum ^{\\infty }_{m = -\\infty } \\int ^\\infty _{-\\infty } \\frac{dl}{2 \\pi \\sqrt{q}}\\frac{1}{\\omega ^2_m + l^2}=\\int ^\\infty _{-\\infty } \\frac{dl}{2 \\pi \\sqrt{q}}\\frac{1}{l} \\left({1\\over 2}+ \\frac{1}{e^{l\\beta } - 1}\\right)$ $&&\\frac{1}{\\beta }\\sum ^{\\infty }_{n=2,n^{^{\\prime }}=2,m = -\\infty } \\frac{1}{(\\omega ^2_m + \\gamma _n)(\\omega ^2_m + \\lambda _{n^{^{\\prime }}})}=\\nonumber \\\\&&\\sum ^{\\infty }_{n=2, n^{^{\\prime }}=2}\\frac{1}{{\\gamma _n - \\lambda _{n^{^{\\prime }}}}} \\left(\\frac{1}{\\lambda _{n^{^{\\prime }}}}\\left({1\\over 2}+ \\frac{1}{e^{\\sqrt{\\lambda _{n^{^{\\prime }}}}\\beta } - 1}\\right)-\\frac{1}{\\sqrt{\\gamma _n}}\\left({1\\over 2}+ \\frac{1}{e^{\\sqrt{\\gamma _n}\\beta } - 1}\\right)\\right)$ $&&\\frac{1}{\\beta }\\sum ^{\\infty }_{n=2,m = -\\infty }\\int ^\\infty _{-\\infty } \\frac{dl}{2 \\pi \\sqrt{q}} \\frac{1}{(\\omega ^2_m + l^2)(\\omega ^2_m + \\lambda _n)}=\\nonumber \\\\&&\\sum ^{\\infty }_{n=2} \\int ^\\infty _{-\\infty } \\frac{dl}{2 \\pi \\sqrt{q}}\\frac{1}{{l^2 - \\lambda }} \\left(\\frac{1}{\\lambda _n}\\left({1\\over 2}+ \\frac{1}{e^{\\sqrt{\\lambda _n}\\beta } - 1}\\right)-\\frac{1}{l}\\left({1\\over 2}+ \\frac{1}{e^{l \\beta } - 1}\\right)\\right)$ $&&\\frac{1}{\\beta }\\sum ^{\\infty }_{n=2,m = -\\infty }\\int ^\\infty _{-\\infty } \\frac{dl}{2 \\pi \\sqrt{q}} \\frac{1}{(\\omega ^2_m + l^2)(\\omega ^2_m + \\gamma _n)}=\\nonumber \\\\&&\\sum ^{\\infty }_{n=2} \\int ^\\infty _{-\\infty } \\frac{dl}{2 \\pi \\sqrt{q}}\\frac{1}{{l^2 - \\gamma _n}} \\left(\\frac{1}{\\gamma _n}\\left({1\\over 2}+ \\frac{1}{e^{\\sqrt{\\gamma _n}\\beta } - 1}\\right)-\\frac{1}{l}\\left({1\\over 2}+ \\frac{1}{e^{l \\beta } - 1}\\right)\\right)$ The last three propagators in are mixed propagators.", "So we have decomposed them into partial fractions and then and have done the sum separately.", "The fermions due to their anti-periodic boundary conditions along the Euclidean time direction have have their propagators with $\\omega _m = \\frac{(2m+1)\\pi }{\\beta }$ .", "The sum over the odd integers can be performed by converting the sum into a contour integral as above.", "The only change here is that we must introduce $\\tanh (p_0\\beta /2)$ .", "Thus, $\\frac{1}{\\beta }\\sum ^{\\infty }_{m = -\\infty } \\frac{1}{\\omega ^2_m + \\lambda _n^{^{\\prime }}}&=& \\frac{1}{2 \\pi i \\beta }\\oint dp_0 \\left(\\frac{\\beta }{2}\\right)\\tanh \\left(\\frac{p_0 \\beta }{2}\\right)f(p_0)\\nonumber \\\\&=&\\frac{1}{\\sqrt{\\lambda ^{^{\\prime }}_n}}\\left({1\\over 2}- \\frac{1}{e^{\\sqrt{\\lambda ^{^{\\prime }}_n}\\beta }+1}\\right)$ Similarly using this result we can do the sum over the Matsubara frequencies for the following, $&&\\sum ^{\\infty }_{n=0,m=-\\infty }\\int ^{\\infty }_{-\\infty } \\frac{dl}{2 \\pi \\sqrt{q}}\\frac{1}{(i\\omega _m + \\sqrt{\\lambda {^{\\prime }}_n})(i\\omega _m \\pm l)}= \\nonumber \\\\&&-\\sum ^{\\infty }_{n=0,m=-\\infty }\\int ^{\\infty }_{-\\infty } \\frac{dl}{4 \\pi \\sqrt{q}} \\left(\\frac{1}{(\\omega ^2_m + \\lambda ^{^{\\prime }}_n)} + \\frac{1}{(\\omega ^2_m + l^2)}-\\frac{l^2 + \\lambda _n}{(\\omega ^2_m + \\lambda ^{^{\\prime }}_n)(\\omega ^2_m + l^2)}\\right)\\nonumber \\\\$ We can now do the sum over each of the terms separately, which gives $\\sum _n \\int \\frac{dl}{2\\pi \\sqrt{q}}\\left(\\frac{-\\beta \\tanh \\left(\\frac{\\beta l}{2}\\right)+\\beta \\tanh \\left(\\frac{1}{2} \\beta \\sqrt{\\lambda ^{^{\\prime }}_n}\\right)}{2 \\left(l-\\sqrt{\\lambda ^{^{\\prime }}_n}\\right)}\\right)$" ] ]
1403.0389
[ [ "GALEX J194419.33+491257.0: An Unusually Active SU UMa-Type Dwarf Nova\n with a Very Short Orbital Period in the Kepler Data" ], [ "Abstract We studied the background dwarf nova of KIC 11412044 in the Kepler public data and identified it with GALEX J194419.33+491257.0.", "This object turned out to be a very active SU UMa-type dwarf nova having a mean supercycle of about 150 d and frequent normal outbursts having intervals of 4-10 d. The object showed strong persistent signal of the orbital variation with a period of 0.0528164(4) d (76.06 min) and superhumps with a typical period of 0.0548 d during superoutbursts.", "Most of the superoutbursts were accompanied by a precursor outburst.", "All these features are unusual for this very short orbital period.", "We succeeded in detecting the evolving stage of superhumps (stage A superhumps) and obtained a mass ratio of 0.141(2), which is unusually high for this orbital period.", "We suggest that the unusual outburst properties are a result of this high mass ratio.", "We suspect that this object is a member of the recently recognized class of cataclysmic variables (CVs) with a stripped core evolved secondary which are evolving toward AM CVn-type CVs.", "The present determination of the mass ratio using stage A superhumps makes the first case in such systems." ], [ "Introduction", "The Kepler mission ([3]; [14]), which was aimed to detect extrasolar planets, has provided unprecedentedly sampled data on several cataclysmic variables (CVs).", "This satellite also recorded previously unknown CVs as by-products of the main target stars.", "The best documented example has been the background dwarf nova of KIC 4378554 ([2]; [10]).", "In addition to this object, the group Planet Hunters [5] detected several candidate background CVs.$<$ http://talk.planethunters.org/objects/APH51255246/ discussions/DPH101e5xe$>$ .", "We studied one of these background dwarf novae, the one in the field of KIC 11412044 (hereafter J1944).", "This object was discovered by the Planet Hunters group as a background SU UMa-type dwarf nova of KIC 11412044, in which superoutbursts and frequent normal outbursts were recognized.", "$<$ http://keplerlightcurves.blogspot.jp/2012/07/ dwarf-novae-candidates-at-planet.html$>$ .", "Since it was bright enough and it was frequently included in the aperture mask of KIC 11412044, the outburst behavior can be immediately recognized in Kepler SAP_FLUX light curve of KIC 11412044." ], [ "Data Analysis", "We used Kepler public long cadence (LC) data (Q1–Q17) for analysis.", "Since the outbursts were immediately recognizable in each light curve of the Kepler target pixel images, we used a custom aperture consisting of 4–6 pixels showing outbursts as we did in the background dwarf nova of KIC 4378554 [10].", "We used surrounding pixels to subtract the background from KIC 11412044.", "We further corrected small long-term baseline variations by subtracting a locally-weighted polynomial regression (LOWESS: [4]) and spline functions.", "Since the quiescent magnitude is difficult to determine, we artificially set the level to be 22.0 mag." ], [ "Outburst Properties", "The resultant light curve indicates that this object is an SU UMa-type dwarf nova with frequent outbursts (figure REF ).", "There were eight observed superoutbursts, and from the regular pattern, another superoutburst most likely occurred between BJD 2455553 and 2455568 (a data gap in Q8) and we numbered the superoutburst and supercycle assuming that there is a superoutburst in this gap.", "The intervals between successive superoutbursts (supercycles) were in a range of 120–160 d. We determined the mean supercycle of 147(1) d. Most of superoutbursts were associated with a precursor outburst with a various degree of separation from the main superoutburst.", "The typical duration of the superoutburst is $\\sim $ 8 d including the precursor part.", "This duration is shorter than those of many other SU UMa-type dwarf novae.", "The number of normal outbursts in one supercycle ranged from 11 to 21.", "The intervals of normal outbursts were 4–10 d, one of the shortest known except ER UMa stars [9].", "The amplitudes of normal outbursts increased as the supercycle phase progresses.", "Some of the normal outbursts were “failed”, i.e.", "they decayed before reaching the full maximum." ], [ "Frequency Analysis and Source Identification", "As shown in figure REF , a two-dimensional Fourier analysis (using the Hann window function) of the light curve of this object yielded two periods.", "There was a signal of a constant frequency (18.93 cycle d$^{-1}$ ) with the almost constant strength.", "Using all the data segment, we determined the period to be 0.0528164(4) d (18.934 cycle d$^{-1}$ ).", "We refer this signal to “0.0528 d” signal.", "Based on the high stability of the 0.0528 d signal during the entire Kepler observations, we identified this period to be the orbital period ($P_{\\rm orb}$ ) of this object.", "During superoutbursts, there were transient signals of superhumps at frequencies around 18.1–18.5 cycle d$^{-1}$ as expected.", "Let us now examine the source position of the background dwarf nova in figure REF .", "We checked the pixels which showed the dwarf nova-type variation.", "The peak of signal of dwarf nova-type outbursts was found two pixels away from the center of KIC 11412044 (star 1 on the DSS image).", "At this location, there is a GALEX [15] ultraviolet source GALEX J194419.33$+$ 491257.0 [NUV magnitude 21.3(3)] and we identified this source as the UV counterpart of this dwarf nova (figure REF , Q16), confirming the suggestion in the Planet Hunters' page.", "The superhump component and 0.0528 d component were also confirmed at the location of this object (figure REF , Q14), and we consider that the 0.0528 d signal indeed comes from this dwarf nova.", "This has also been confirmed by the non-detection of the 0.0528 d signal in the SAP_FLUX of KIC 11412044 when this dwarf nova was outside the aperture of KIC 11412044." ], [ "Variation of Superhump Period", "Since the superhump period is less than three LC exposures, it is difficult to determine the times of superhump maxima by the conventional method.", "We employed the Markov-chain Monte Carlo (MCMC) modeling used in [10].", "Although we only show the result of SO3 (figure REF ), the pattern is similar in other superoutbursts.", "In the $O-C$ diagram, stages A–C (for an explanation of these stages, see [7]) can be recognized.", "Long-period superhumps (stage A superhumps) with growing superhumps were recorded during the late part of the precursor outburst to the maximum of the superoutburst.", "The overall pattern is very similar to those of other SU UMa-type dwarf novae, including V1504 Cyg and V344 Lyr ([18]; [19]).", "The object makes the fourth case (after V1504 Cyg, V344 Lyr and V516 Lyr) in the Kepler field in which the growing superhumps lead smoothly from the precursor to the main superoutburst and thus gives further support to the thermal-tidal instability (TTI) model [17] as the explanation for the superoutburst.", "Figure: O-CO-C diagram of SO3 of J1944.From top to bottom:(1): Kepler LC light curve.", "(2): O-CO-C diagram.The figure was drawn against a period of 0.05479 d.Stages A–C (cf. )", "can be recognized.Long-period superhump (stage A superhumps)with growing superhumps were recorded duringthe late part of the precursor outburst to the maximumof the superoutburst.", "(3): Amplitudes in electrons s -1 ^{-1}." ], [ "System Properties", "The inferred fractional superhump period excess $\\varepsilon \\equiv P_{\\rm SH}/P_{\\rm orb}-1$ , where $P_{\\rm SH}$ is the superhump period, of $\\sim $ 3.8% is, however, unusually large for this $P_{\\rm orb}$ (cf.", "figure 15 in [7]).", "[11] recently proposed that stage A superhumps can be used to determine the mass ratio ($q=M_2/M_1$ ) and the resultant mass ratios are as accurate as those obtained from eclipse modeling.", "We have succeeded in measuring the period of stage A superhump during the three superoutbursts: 0.0555(2) d (SO3), 0.05546(5) d (SO6), 0.05552(6) d (SO7).", "The corresponding fractional superhump excesses in the frequency unit $\\varepsilon ^* \\equiv 1-P_{\\rm orb}/P_{\\rm SH}$ are 4.8%, 4.77% and 4.88%.", "These values correspond to the $q$ value of 0.14, 0.139 and 0.143, respectively.", "We therefore adopted $q$ =0.141(2).", "This mass ratio implies a massive (approximately two times more massive) secondary for this very short orbital period comparable to most WZ Sge-type dwarf novae (figure REF ).", "This result may alternatively suggest the possibility of an unusually low-mass white dwarf.", "If we assume that the secondary of J1944 has a normal mass for this orbital period, such as 0.066$M_\\odot $ in WZ Sge [11], the mass of the white dwarf must be $\\sim $ 0.47$M_\\odot $ .", "According to [27], the fraction of CVs having white dwarf lighter than 0.5$M_\\odot $ is only 7$\\pm $ 3 %, even including suspicions measurements.", "Furthermore, there is evidence from modern eclipse observations that mass of the white dwarf in short-$P_{\\rm orb}$ CVs is not diverse [23].", "We therefore consider the interpretation of a massive secondary more likely.", "Figure: Location of J1944 on the evolutionary track.The location of J1944 is plotted (star mark) on figure 5 of.", "The filled circles andfilled squares represent qq values determined usingstage A superhumps and quiescent eclipses, respectively.The dashed and solid curves represent the standard and optimalevolutionary tracks in , respectively.The presence of precursor outburst and the high frequency of normal outbursts are usual features of longer-$P_{\\rm orb}$ systems such as V1504 Cyg and V344 Lyr.", "Systems with $P_{\\rm orb}$ like J1944 are usually WZ Sge-type dwarf novae with very rare (super)outbursts (e.g.", "[12]) or ER UMa-type dwarf novae, a rare subgroup with very frequent outbursts and short supercycles (e.g.", "[8]; [22]).", "J1944 does not match the properties of either group.", "This can be understood if the outburst properties are a reflection of the mass ratio rather than the orbital period since the $q$ value of J1944 is closer to those of longer-$P_{\\rm orb}$ SU UMa-type dwarf novae.", "The presence of such a system would pose a problem in terms of the CV evolution since the secondary loses its mass during the CV evolution and $q$ value is expected to be as low as $\\sim $ 0.08 around the orbital period of J1944.", "In recent years, some objects showing hydrogen lines in their spectra (this excludes the possibility of double-degenerate AM CVn-type systems) have been discovered around this period or even in shorter period.", "These objects include EI Psc ([26]; [25]), V485 Cen [1] and GZ Cet [6].", "These objects are considered to be CVs whose secondary had an evolved core at the time of the contact, and are considered to be progenitors of AM CVn-type double white dwarfs ([21]; [16]; [26]; [25]).", "None of these objects have been reported for $q$ determination directly from radial-velocity studies, and $q$ values have only been inferred from the traditional $\\varepsilon $ , which has an unknown uncertainty [11].", "The detection of stage A superhumps in J1944 allowed the first reliable determination of $q$ in such stripped-core ultracompact binaries.", "There is, however, a marked difference of the outburst frequency between J1944 and these known objects since the frequency of outbursts in such systems have been reported to be low [24].", "This suggests that J1944 has an anomalously high mass-transfer rate among these objects.", "The object may be in a phase analogous to ER UMa-type dwarf novae, whose high mass-transfer rates may be a result of a recent classical nova explosion (cf.", "[8]; [20]).", "Since the object can be within the reach of the ground-based telescopes, the exact optical identification and the search for the feature of the secondary star are encouraged to solve the mystery.", "We thank the Kepler Mission team and the data calibration engineers for making Kepler data available to the public.", "We also thank the Planet Hunters group for making their information on the background dwarf novae public which enabled us to study this interesting object.", "This work was supported by the Grant-in-Aid “Initiative for High-Dimensional Data-Driven Science through Deepening of Sparse Modeling” from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan." ] ]
1403.0308
[ [ "Precise comparison of the Gaussian expansion method and the Gamow shell\n model" ], [ "Abstract We perform a detailed comparison of results of the Gamow Shell Model (GSM) and the Gaussian Expansion Method (GEM) supplemented by the complex scaling (CS) method for the same translationally-invariant cluster-orbital shell model (COSM) Hamiltonian.", "As a benchmark test, we calculate the ground state $0^{+}$ and the first excited state $2^{+}$ of mirror nuclei $^{6}$He and $^{6}$Be in the model space consisting of two valence nucleons in $p$-shell outside of a $^{4}$He core.", "We find a good overall agreement of results obtained in these two different approaches, also for many-body resonances." ], [ "Introduction", "In recent years, the playground of nuclear physics has extended towards neutron and proton drip lines [1], [2], [3].", "Huge amount of new experimental data on nuclei far from the valley of stability has been provided by new rare-isotope facilities.", "The knowledge of these nuclei has largely improved also due to the progress in theoretical methods and computing power which allows to calculate light nuclei in ab initio framework taking into account the proximity of the scattering continuum.", "The description of various manifestations of the continuum coupling requires the generalization of existing many-body methods and call for theories which unify structure and reactions.", "Realistic studies of the coupling to continuum in the many-body framework can be made in the open quantum system extension of the Shell Model (SM), the so-called Continuum Shell Model (CSM) [4], [5].", "A recent realization of the CSM is the complex-energy CSM based on the Berggren ensemble [6], the GSM, which finds a mathematical setting in the Rigged Hilbert Space [7].", "This model is a natural generalization of the standard SM for the description of configuration mixing in weakly bound states and resonances.", "Berggren completeness relation can be derived from the Newton completeness relation [8] for the set of real-energy eigenstates by deforming the real momentum axis to include resonant poles which are located in the fourth quadrant of the complex $k$ -plane.", "Thus the Berggren completeness relation which replaces the real-energy scattering states by the resonance contribution and a background of complex-energy continuum states, puts the resonance part of the spectrum on the same footing as the bound and scattering spectrum.", "As the benefit of the explicit inclusion of the non-resonant continuum and resonant poles, the contribution of the unbound states to the one- and two-body matrix elements can be discussed.", "Berggren ensemble has found the application in the GSM [9], time-dependent Green's function approach [10], the no-core GSM [11], the coupled cluster approach [12], the Density Matrix Renormalization Group (DMRG) approach [13], and in the coupled-channel GSM [14], [15] to study various nuclear structure and reaction problems.", "Another approach is the complex scaling (CS) method [16], which has been used to solve many-body resonances in many fields including atomic physics, molecular physics [17], [18] and nuclear physics [19], [20].", "In the CS method, asymptotically-divergent resonant states are described within ${\\cal L}^{2}$ -integrable functions through the rotation of space coordinates and their conjugate momenta in the complex plane.", "As basis functions, the Gaussian Expansion Method (GEM) [21] has been extensively employed for the cluster-orbital shell model (COSM) [22] and coupled rearrangement channel model such as the TV-model [23].", "The CS-COSM has successfully been applied to description of resonant states observed above the many-body decay threshold in $p$ -shell nuclei ($A=5$ -8) using a $^4$ He+$XN$ model, where $X=1$ -4 and $N=p$ , $n$  [24], [25].", "The CS-TV model for the core+2N systems has been shown to reproduce the observed Coulomb breakup cross sections for three-body continuum energy states [26], [27].", "The purpose of these studies is to perform a detailed comparison of the GSM and the GEM+CS results for $^6$ He and $^6$ Be using the same COSM coordinates for valence nucleons [22] and the same Hamiltonian.", "In COSM, all coordinates are taken with respect to the core Center-of-Mass (CoM), so that the translational invariance is strictly preserved.", "COSM combined with CS method has been employed in numerous studies of weakly bound states and resonances in light nuclei [23], [28], [29], [30], [20], [31], [32], [33], [34].", "COSM coordinates have been also used in GSM [35] to investigate isospin mixing in mirror nuclei [36] and charge radii in halo nuclei [37].", "The paper is organized as follows.", "In Section II we present our COSM Hamiltonian and the model space.", "In Section III, the two theoretical approaches, namely the GEM+CS (Section III.A) and the GSM (Section III.B), are briefly introduced.", "GEM+CS and GSM results for $^6$ He and $^6$ Be are presented and discussed in Section IV.", "Finally, Section V gives the main conclusions of these studies." ], [ "The COSM Hamiltonian", "In these studies, we employ the three-body model for $^{4}$ He plus two-nucleon system in the COSM coordinates [22] (see Fig.", "REF ).", "Figure: Coordinate system of the COSM approach.The Hamiltonian is written as follows: $\\hat{H} = \\sum _{i=1}^{2}\\left(\\hat{t}_{i} + \\hat{V}_{i}^{(C)}\\right)+\\left(\\hat{T}_{12}+ \\hat{v}_{12}+ \\hat{V}_{12}^{(C)}\\right)\\mbox{ ,}$ where $\\hat{t}_{i}$ and $\\hat{V}_{i}^{(C)}$ are the kinetic and potential energy operators for the $^{4}$ He core and an $i$ th valence nucleon subsystem.", "In Eq.", "(REF ), the first parenthesis corresponds to the single-particle Hamiltonian for the $i$ th valence nucleon, which is defined as $\\hat{h}_{i} & \\equiv & \\hat{t}_{i} + \\hat{V}_{i}^{(C)}\\mbox{,}\\hspace{8.53581pt}(i = 1, 2)\\mbox{ .", "}$ In the second parenthesis of Eq.", "(REF ), $\\hat{v}_{12}$ is the nucleon-nucleon interaction for valence particles, and: $\\hat{T}_{12} = -\\frac{\\hbar ^{2}}{M^{(C)}} \\nabla _{1} \\cdot \\nabla _{2}\\mbox{ ,}$ is the recoil part which comes from the subtraction of the center of mass (CoM) motion, due to the finite mass $M^{(C)}$ of the core nucleus.", "The last term of Eq.", "(REF ) is the three-body potential of $^{4}$ He and two valence nucleons.", "The interaction $\\hat{V}_{i}^{(C)}$ between the core and the $i$ th valence nucleon contains three terms: $\\hat{V}_{i}^{(C)} = \\hat{V}^{\\alpha n}_{i}+ \\hat{V}_{i}^{\\rm Coul}+ \\lambda \\, \\hat{\\Lambda }_{i}\\mbox{ , }\\hspace{8.53581pt}(i = 1, 2 )\\mbox{ .", "}$ The nuclear interaction part $\\hat{V}^{\\alpha n}_{i}$ is the modified KKNN potential [38], [23], which reproduces the $\\alpha $ -N phase shifts in the low energy region.", "This potential contains a central and an $LS$ parts as $\\hat{ V}^{\\alpha n}_{i} (r_{i}) & = &V^{\\alpha n}_{0}(r_{i})+ 2 V^{\\alpha n}_{LS}(r_{i}) \\, \\mbox{$L$}\\cdot \\mbox{$S$}\\mbox{,}\\hspace{8.53581pt}( i = 1, 2 )\\mbox{ ,}$ where $\\mbox{$r$}_{i}$ is the relative coordinate between $^{4}$ He and the $i$ th valence nucleon.", "The central part of Eq.", "(REF ) is written as: $V^{\\alpha n}_{0}(r_{i})& = & \\sum _{k=1}^{5}\\, [(-1)^{\\ell _{i}}]_{k} \\,V_{k}^{0} \\, \\exp (-\\rho ^{0}_{k} \\,r^{2}_{i})\\mbox{ ,}$ where $[(-1)^{\\ell _{i}}]_{k}$ is given by: $[(-1)^{\\ell _{i}}]_{k} =\\left\\lbrace \\begin{array}{cl}1 & ( k=1, 2) \\\\(-1)^{\\ell _{i}} & (k=3, 4, 5)\\end{array}\\right.\\mbox{ .", "}$ The $LS$ part is: $V^{\\alpha n}_{LS}(r_{i})& = &\\sum _{m=1}^{3} \\, f^{LS}_{m} \\,V_{m}^{LS} \\, \\exp (-\\rho _{m}^{LS} \\,r^{2}_{i})\\mbox{ ,}$ where the factor $f^{LS}_{m}$ is: $f^{LS}_{m} =\\left\\lbrace \\begin{array}{cl}1 & ( m=1) \\\\1 - 0.3\\times (-1)^{\\ell _{i}} & (m=2, 3)\\end{array}\\right.\\mbox{ .", "}$ Parameters of the modified KKNN potential [38], [23] are shown in Table REF .", "Table: Parameters of the modified KKNNpotential , used in this calculation.For the Coulomb part $\\hat{V}_{i}^{\\rm Coul}$ in Eq.", "(REF ), we use a folded-type Coulomb interaction for the $^{4}$ He+$p$ subsystem: $\\hat{V}^{\\rm Coul}_{i}(r_{i}) = \\frac{2e^{2}}{r_{i}} \\,\\mbox{Erf}(\\alpha \\, r_{i})\\mbox{ ,}$ where $ \\mbox{Erf}(r)$ is the error function, and $\\alpha = 0.828$ fm$^{-1}$ .", "To eliminate the spurious states in the relative motion between $^{4}$ He-core and the valence nucleon in CS, we use a projection operator [40]: $\\hat{\\Lambda }_{i} = \\lambda | FS \\rangle \\langle FS |\\mbox{ ,}$ where the forbidden state in the $^{4}$ He+$N$ case; $| FS \\rangle = | 0s_{1/2} \\rangle $ , is given by the harmonic oscillator function with the size parameter $b=1.4$ fm.", "In GSM, the forbidden state is eliminated from the set of the single-particle states, $\\phi _{i}$ , as $\\phi _{i} \\Rightarrow (1 - \\hat{\\Lambda }_{i}) \\phi _{i}\\mbox{ .", "}$ We can confirm that the core-particle potential (REF ) with the parameters given in Table REF , reproduces experimental energies and widths of $3/2_{1}^{-}$ and $1/2_{1}^{-}$ resonances in the $^{5}$ He($^{4}$ He+$n$ ) and $^{5}$ Li( $^{4}$ He+$p$ ) systems.", "For the two-body interaction $\\hat{v_{12}}(\\mbox{$r$}_{12})$ of valence nucleons, where $\\mbox{$r$}_{12} \\equiv \\mbox{$r$}_{1} - \\mbox{$r$}_{2}$ , we use the Minnesota potential [39]: $& & \\hat{v}_{12}(\\mbox{$r$}_{12}) \\nonumber \\\\& & =\\sum _{k=1}^{3} \\, V_{k}^{0} \\,\\left(W^{(u)}_{k} - M^{(u)}_{k} P^{\\sigma } P^{\\tau }+B^{(u)}_{k} P^{\\sigma } - H^{(u)}_{k} P^{\\tau }\\right)\\nonumber \\\\& &\\hspace{22.76219pt} \\times \\exp (-\\rho _{k} \\,\\mbox{$r$}^{2}_{12})\\mbox{ .", "}$ Parameters of this interaction are summarized in Table REF , and the exchange parameter is taken as $u=1.0$ .", "Table: Parameters of the Minnesota potential .The Coulomb interaction between valence protons is taken as an ordinary $1/r$ -type functional form.", "It was shown that the binding energy of $^{6}$ He cannot be reproduced using the reliable one- and two-body potentials for core-particle and particle-particle parts, respectively [23], [29].", "The correct binding energy in a system $^{4}$ He+$N$ +$N$ is recovered by using a simple two-body Gaussian interaction, mimicking a physical three-body effect in the system [29] as: $\\hat{V}_{12}^{\\rm (C)}(r_{1}, r_{2}) = V^{0}_{\\alpha nn} \\,\\exp (-\\rho _{\\alpha nn} (r^{2}_{1}+r^{2}_{2}))$ with the parameters $V^{0}_{\\alpha nn} = -0.41$ MeV and $\\rho _{\\alpha nn} = 5.102 \\times 10^{-3}$ fm$^{-2}$ ." ], [ "The models", "In this section, we discuss two models for solving $^{4}$ He+$2N$ ($N$ is proton or neutron) systems with the COSM Hamiltonian.", "One is the GEM+CS approach, and another one is the GSM approach.", "The essential differences between the GEM+CS and GSM approaches are the choice of the basis functions and the treatment of continuum states.", "The basis function $\\Phi (\\mbox{$r$}_{1} , \\mbox{$r$}_{2} )$ in COSM is defined with a product of the functions with respect to each coordinate from the core to a valence nucleon, $\\Phi (\\mbox{$r$}_{1} , \\mbox{$r$}_{2} )_{JM}& \\equiv &[{\\cal A}\\left\\lbrace \\phi _{\\alpha _{1} }(\\mbox{$r$}_{1})\\otimes \\phi _{\\alpha _{2} }(\\mbox{$r$}_{2})\\right\\rbrace ]_{JM}\\mbox{ .", "}\\nonumber \\\\$ Here, $\\alpha _{i}$ denotes the angular part of the $i$ th particle $\\lbrace j_{i}, \\, \\ell _{i} \\rbrace $ , and its $z$ -components are implicitly included.", "${\\cal A}$ is the antisymmetrizer for particles 1 and 2.", "The basis function for the $i$ th valence nucleon is $\\phi _{\\alpha _{i} }(\\mbox{$r$}_{i})= f(r_{i}) | j_{i} m_{i} \\rangle \\mbox{ .", "}$ The angular momentum part of the basis function is constructed by using the normal $jj$ -coupling scheme as $| J M \\rangle & = &| [j_{1} \\otimes j_{2}]_{JM} \\rangle \\mbox{ .", "}$ The above coupling procedure is the same both for GEM and GSM." ], [ "The Gaussian expansion method with complex scaling", "The radial part of the GEM wave function is not an eigenfunction of the single-particle Hamiltonian $\\hat{h}_{i}$ , but the Gaussian function with the width parameter $a$ as $f_{n_{i}}(r_{i}) & \\equiv & u^{n_{i}}_{\\ell _{1}}(r_{i})\\nonumber \\\\& = & N_{i} r_{i}^{ \\, \\ell _{i}}\\exp (-\\frac{1}{2} a_{n_{i}} r_{i}^{2})\\mbox{ ,}\\hspace{8.53581pt}(i = 1, 2)\\mbox{ ,}$ where $N_{i}$ is the normalization, and $\\ell _{i}$ is angular momentum for the $i$ th nucleon.", "The width parameter $a_{n_{i}}=1/b_{n_{i}}^{2}$ in the GEM basis functions is defined using the geometric progression as: $b_{n_{i}} = b_{0} \\gamma ^{n_{i}-1}$  [21].", "Here, $b_{0}$ and $\\gamma $ are input parameters, and $n_{i}$ is an integer.", "The model space of the system is spanned by basis functions from $n_{i}=1$ to $N_{\\rm max}$ .", "The $k$ th eigenfunction $ \\psi _{k:\\, \\alpha _{i} }$ of the core+$N$ system can be obtained by diagonalizing the single-particle Hamiltonian $\\hat{h}_{i}$ with the Gaussian basis functions, $\\psi _{k:\\, \\alpha _{i}} (\\mbox{$r$}_{i})= \\sum _{m}^{N_{\\rm max}} \\, c_{m}^{(k)} \\phi ^{(m)}_{\\alpha _{i}} (\\mbox{$r$}_{i})\\mbox{ .", "}$ Here, $\\hat{h}_{i} \\psi _{k:\\, \\alpha _{i} } = \\epsilon _{k}\\psi _{\\alpha _{i}}$ , and $c_{m}^{(k)}$ are determined by using the variational principle.", "For solving the core+$2N$ system, the basis function (REF ) is given by the product of basis functions in Eq.", "(REF ) for particle 1 and 2 as follows: $\\Phi ^{(m)}_{JM}& = &{\\cal A}\\left\\lbrace u_{\\ell _{1}}^{(m)}(r_{1}) \\cdot u_{\\ell _{2}}^{(m)}(r_{2})| J M \\rangle ^{(m)}\\right\\rbrace \\mbox{ .", "}\\nonumber \\\\$ Here, the width parameters $a_{i}^{(m)}$ in $ u_{\\ell _{i}}^{(m)}$ are prepared independently for particle 1 and 2.", "$m$ is the index of the one-body basis functions.", "Figure: Complex-scaled eigenstates of the three-bodyHamiltonian for the Borromean system.Solid circles are bound and resonance states,and open circles are continuum states.The calculation of two-body matrix elements (TBME), $ \\langle \\Phi ^{(m)} | \\hat{O}_{12} | \\Phi ^{(n)} \\rangle $ can be performed analytically.", "Even for different Gaussian width parameters, we can obtain the value of TBME without any approximations.", "The solution of the core+$2N$ system can be obtained by diagonalizing the Hamiltonian $\\hat{H} \\Phi _{k: \\, JM} = E_{k} \\, \\Phi _{k: \\, JM}\\mbox{ ,}$ and the corresponding eigenfunction is expressed as a linear combination of the basis functions, $\\Phi _{k: \\, JM} & = &\\sum _{m}^{N_{\\rm Tot}}\\,C_{m}^{(k)} \\, \\Phi ^{(m)}_{JM}\\mbox{ .", "}$ In order to treat the many-body resonant states, we apply the CS method.", "In this method, the coordinate and momentum are transformed using a rotation angle $\\theta $ as: $r \\rightarrow r \\, e^{ i \\theta }\\hspace{8.53581pt}(k \\rightarrow k \\, e^{ -i \\theta })\\mbox{ .", "}$ Resonance wave functions, which diverge in the asymptotic region, can be converged with this transformation for a suitable rotation angle.", "This essential feature is proven by the ABC-Theorem [16], [41].", "After this transformation, all continuum states are aligned along the rotated axis.", "Furthermore, using GEM, the continuum states are automatically discretized through the diagonalization of the Hamiltonian.", "A schematic figure of the bound states and resonances and discretized continuum states are shown for the Borromean system like $^{4}$ He+$N$ +$N$ in Fig.", "REF ." ], [ "Gamow shell model approach", "Another approach to study many-body resonances is the GSM approach [9], [11], [12], [13].", "This generalization of the nuclear SM treats single-particle bound, resonance and continuum states on the same footing using a complete Berggren single-particle basis [6]: $\\mbox{$1$} & = & \\sum _{i \\in b, r} | \\phi _{i} \\rangle \\langle \\phi _{i} |+ \\oint _{\\Gamma _{k}} dk | \\phi (k) \\rangle \\langle \\phi (k) |\\mbox{ ,}$ where $\\Gamma _{k}$ is a deformed momentum contour.", "For each $(\\ell ,j)$ of the resonant single-particle state in the basis, the set $(\\ell ,j)_{\\rm c}$ of continuum states along the discretized contour in $k$ -plane enclosing the resonant state(s) $(\\ell ,j)$ is included in the basis (see Fig.", "REF ): $\\mbox{$1$}& \\simeq & \\sum _{i \\in b, r} | \\phi _{i} \\rangle \\langle \\phi _{i} |+ \\sum _{\\eta \\in {\\rm cont} } | \\phi (k_{\\eta }) \\rangle \\langle \\phi (k_{\\eta }) |\\mbox{ ,}$ where $k_{\\eta }$ are linear momenta discretized on the deformed contour with the parameters of a maximum $k$ and a number of discretized points.", "Different shapes of $(\\ell ,j)$ -contours are equivalent unless the number of resonant states contained in them changes.", "The complete many-body basis is then formed by all Slater determinants where nucleons occupy the single-particle states of a complete Berggren ensemble [9].", "Figure: Deformed contour on the complex momentum plane (a),and discretized continuum states along the deformed contour (b).Solid circles are bound and resonant states, andopen circles are discretized continuum states.In the Berggren basis, the basis function of the core+$2N$ system is: $\\Phi ^{(\\nu )}_{JM}& = &{\\cal A}\\left\\lbrace [\\phi _{1}^{(\\nu )} \\otimes \\phi _{2}^{(\\nu )}]_{J M}\\right\\rbrace \\mbox{ .", "}\\nonumber \\\\$ Here, $\\phi _{i}^{(\\nu )}$ are single-particle bound, resonance, and discretized-continuum states for particles 1 and 2.", "In GSM, the deformed contour for each $(\\ell ,j)$ is varied to obtain the best numerical precision of calculated eigenenergies and eigenvalues for a given discretization of the contour.", "Since the direct calculation of the TBME using the continuum and/or resonant single-particle states is numerically demanding, and even difficult to define from a theoretical point of view for some particular instances, one calculates TBME using the harmonic oscillator (HO) expansion procedure [35].", "For the TBME between GSM basis functions $ \\Phi ^{(i)}_{\\rm GSM} $ and $ \\Phi ^{(j)}_{\\rm GSM}$ , one obtains: $& & \\langle \\Phi ^{(i)}_{\\rm GSM} | \\hat{O}_{12} | \\Phi ^{(j)}_{\\rm GSM}\\rangle \\nonumber \\\\& & \\nonumber \\\\& = &\\sum _{\\alpha , \\beta } \\langle \\Phi ^{(i)}_{\\rm GSM} |\\Phi _{\\rm HO}^{(\\alpha )}\\rangle \\langle \\Phi _{\\rm HO}^{(\\alpha )} \\, |\\hat{O}_{12} |\\Phi _{\\rm HO}^{(\\beta )}\\rangle \\langle \\Phi _{\\rm HO}^{(\\beta )} |\\Phi ^{(j)}_{\\rm GSM}\\rangle \\nonumber \\\\& = &\\sum _{\\alpha , \\beta }d^{*}_{i,\\alpha } \\,d_{j,\\beta } \\langle \\Phi _{\\rm HO}^{(\\alpha )} \\, |\\hat{O}_{12} |\\Phi _{\\rm HO}^{(\\beta )}\\rangle \\mbox{ ,}$ where $ \\Phi _{\\rm HO}^{(\\alpha )} $ are HO basis functions and $d_{i,\\alpha }$ is the overlap between the GSM basis function $\\Phi ^{(i)}_{\\rm GSM}$ and the HO basis function: $d_{i,\\alpha } \\equiv \\langle \\Phi _{\\rm HO}^{(\\beta )} \\, |\\Phi ^{(i)}_{\\rm GSM}\\rangle \\mbox{ .", "}$ The advantage of this procedure is that the TBMEs with the HO expansion can be stored for a fixed $b_{\\rm HO}$ , and one only needs to calculate the overlaps $d_{i,\\alpha }$ , whatever the Berggren states are." ], [ "Results", "For numerical calculations, we define the number of basis states.", "In the GEM+CS approach, the number of radial wave functions for each valence nucleon $N_{\\rm max}$ is $N_{\\rm max} = 22$ .", "The typical value of the Gaussian width parameters are $b_{0} = 0.1$ fm and $\\gamma = 1.3$ .", "Hence, the maximum size of the width parameter becomes $b = b_{0} \\gamma ^{N_{\\rm max}-1} = 0.1 \\times 1.3^{21} \\simeq 25$ fm.", "In GSM, the continuum is discretized with 40 points for each partial wave and the maximum momentum is $k_{\\rm max} = 3.5$ fm$^{-1}$ ." ], [ "$^{6}$ He in the {{formula:7ac56145-778e-425d-bcdf-a0fb2c3ab5e5}} He+{{formula:bb7670e5-20c2-4aed-a1c6-b9da847171c8}} model space", "First, we show results for the ground state $0^{+}_{1}$ and the first excited state $2^{+}_1$ of $^{6}$ He.", "The ground state of $^{6}$ He is bound one with an energy $E=0.97$ MeV from the $^{4}$ He+$2n$ threshold.", "Hence, we can take the rotation angle as $\\theta = 0$ for the calculation of this state in GEM+CS approach.", "Table: Energies of the ground 0 1 + 0_1^{+} andthe first excited 2 1 + 2_1^{+} states of 6 ^{6}Hecalculated using the GEM+CS and GSM approaches.All units except for the angular momentum are in MeV.We calculate energies of $^{6}$ He by changing the maximum angular momentum for the coordinates $\\mbox{$r$}_{1}$ and $\\mbox{$r$}_{2}$ from $\\ell _{\\rm max} =1$ to 5, where $\\ell _{\\rm max}$ is the maximum angular momentum in the basis function for the $^{4}$ He+$n$ subsystem.", "Parameters of the interaction are chosen to reproduce the binding energy of the ground state of $^{6}$ He in a model space with $\\ell _{\\rm max} = 5$ .", "Figure: Convergence of the poles of the ground 0 1 + 0_1^{+} and thefirst excited 2 1 + 2_1^{+} states of 6 ^{6}He, which arecalculated using the GEM+CS approach and the GSM for 1≤ℓ max ≤51\\le \\ell _{\\rm max}\\le 5.Open and solid circles denote GEM+CS and GSM results, respectively.The energies of the $0^{+}_{1}$ state are shown in Table REF for different values of $\\ell _{\\rm max}$ .", "One can see that the calculation for $\\ell _{\\rm max}=1$ , which includes the $s_{1/2}$ -, $p_{3/2}$ - and $p_{1/2}$ -orbits of the $^{4}$ He+$N$ system, is not enough to reproduce the binding energy of $^{6}$ He.", "The inclusion of higher angular momenta ($\\ell _{\\rm max} \\ge 2$ ) improves the calculated energy significantly.", "Nevertheless, even $\\ell _{\\rm max}=5$ is not enough to obtain the converged ground state energy since the T-type Jaccobi configuration of valence neutrons is very important [23].", "However, since the scope of this paper is to compare results of GEM+CS approach and GSM, we restrict the maximum angular momentum for the core+$N$ system to $\\ell _{\\rm max} = 5$ and determine the interaction parameters in this model space.", "We find a good agreement between GEM+CS and GSM for a Borromean $^{6}$ He nucleus.", "The $\\ell _{\\rm max}$ -dependence of the $0^+_1$ and $2^+_1$ energies is shown in Table REF and Fig.", "REF .", "The density of valence neutrons in the $0_1^+$ state of $^6$ He is plotted in Fig.", "REF .", "One can see that the GEM+CS and GSM approaches give indistinguishable results for the density distributions in the $0_1^+$ halo configuration of $^{6}$ He.", "Figure: (Color online)The density of valence neutrons in COSM coordinate systemfor the ground 0 1 + 0_1^{+} state of 6 ^{6}He (color online).The normalization of the density distribution is 1.Results for the $2^{+}_{1}$ narrow resonance are shown in Table REF and Fig.", "REF .", "The difference between GEM+CS and GSM results in this case is at most $\\sim $ 10 keV.", "The trajectories of the $2^{+}_{1}$ state of the GEM+CS and GSM poles are shown in Fig.", "REF .", "Similarly, as for the $0^{+}_{1}$ -state, results of the GEM+CS and GSM approaches agree well." ], [ "$^{6}$ Be in the {{formula:b62f5eaa-4e74-4bd0-9e9a-d8443a6a80ff}} He+{{formula:d88d3bf3-2206-4f1b-9628-30c1e8cb770b}} model space", "The $^{6}$ Be nucleus, the mirror system of $^{6}$ He, is unbound in the ground state.", "In this section, we shall compare results of GEM+CS and GSM for the $0_1^+$ and $2_1^+$ states of $^{6}$ Be described as a $^{4}$ He+$2p$ three-body system.", "Figure: Poles of the ground and first excited statesof 6 ^{6}Be calculated using GEM+CS approachand GSM from ℓ max =1\\ell _{\\rm max}=1 to 5.Open and solid circles correspond to GEM+CS and GSM results, respectively.Calculated energies of the $0^{+}_{1}$ and $2^{+}_{1}$ states for different $\\ell _{\\rm max}$ values are shown in Table REF .", "The difference of GEM+CS and GSM energies is less than $\\sim $ 10 keV for the $0^{+}_{1}$ state and $\\sim $ 20 keV for the $2^{+}_{1}$ state.", "Table: Energies of the ground 0 1 + 0^{+}_{1} and the first excited2 1 + 2^{+}_{1} states of 6 ^{6}Be calculatedusing the GEM+CS and GSM approaches.All units except for the angular momentum are in MeV.The trajectory of the $0_1^+$ and $2_1^+$ poles in the complex energy plane is shown in Fig.", "REF .", "Contrary to the $0^{+}_{1}$ state, one may notice a slight difference between trajectories of $2^{+}_{1}$ -poles in GEM+CS approach and in GSM.", "This difference diminishes with increasing $\\ell _{\\rm max}$ ." ], [ "Discussion", "In the comparison between the GEM+CS and GSM approaches, we obtain a good agreement for the bound state $0^{+}_{1}$ in $^{6}$ He, and narrow resonances; $2^{+}_{1}$ in $^{6}$ He and $0^{+}_{1}$ in $^{6}$ Be.", "A small difference appears only for the $2^{+}_{1}$ broad resonance in $^{6}$ Be.", "Below, we shall discuss a possible origin of such a small difference in the numerical results.", "Both GEM+CS and GSM approaches solve the non-Hermitian problem.", "In the GEM+CS approach, the wave function of a resonance becomes ${\\cal L}^{2}$ -integrable with the help of the complex rotation.", "As a result, the Hamiltonian becomes non-Hermitian.", "The standard procedure to find the optimum values of the parameters is to search for a stationary point of the eigenvalue with respect to the variational parameters.", "The variational parameters are the complex rotation angle $\\theta $ and the parameter $b_{0}$ in a definition of the Gaussian width; $b_{n_{i}} = b_{0} \\gamma ^{n_{i}-1}$  [20] for the Gaussian basis functions.", "The optimization procedure is a simplified version of the generalized variational principle for complex eigenvalues [42].", "The procedure works efficiently and gives very accurate solutions even for broad resonant states [20].", "Figure: Poles of 6 ^{6}He (2 + )(2^{+}) with ℓ max =1\\ell _{\\rm max} = 1.For GEM+CS, we change the rotation angle θ\\theta from 5 ∘ 5^{\\circ } to 25 ∘ 25^{\\circ } in step of 1 ∘ 1^{\\circ }.GSM is formulated in the Berggren set, which includes bound single-particle states, single-particle resonances and scattering states from the discretized contour for each considered $(\\ell ,j)$ .", "Consequently, the Hamiltonian matrix in this basis becomes complex-symmetric.", "The number of scattering states on each discretized contour $(\\ell ,j)$ and the momentum cutoff have to be chosen to assure the completeness of many-body calculations.", "Moreover, in the HO expansion procedure of calculating the TBMEs, the dependence on the oscillator length and the number of oscillator shells should be carefully examined.", "Fig.", "REF presents a trajectory of the $2^{+}_{1}$ narrow resonant pole of $^{6}$ He calculated in the GEM+CS approach by changing the rotation angle $\\theta $ , where the stationary point at the optimum value of the rotation angle is $\\theta _{\\rm opt} = 13^{\\circ }$ , and the optimum point for the GSM calculation, which is obtained with the oscillator length and the number of oscillator shells are $b_{\\rm HO} = 2$ fm and $N = 41$ , respectively.", "In this case, the difference is only $\\sim 1$ keV, and both methods give almost the equivalent result.", "On the other hand, the $2^{+}_{1}$ state of $^{6}$ Be is a broad resonant pole due to the presence of the Coulomb force for all three particles.", "The optimum value of the rotation angle in GEM+CS calculation is $\\theta _{\\rm opt} = 17^{\\circ }$ , and the optimal HO oscillator length in GSM calculations is $b_{\\rm HO} = 3$ fm.", "The difference of complex GEM+CS and GSM eigenenergies becomes in the order of 10 keV.", "To improve the agreement for the eigenvalues obtained by GEM+CS and GSM, it would be necessary to examine the optimization of the variational parameters more precisely.", "However, in the practical point of view, the difference is only less than 1 percent to the total energy.", "The convergence can be tested by introducing an extrapolation procedure, e.g.", "the Richardson extrapolation [43].", "We extrapolate the energy $E$ as a function of $1/\\ell _{\\rm max}$ to $1/\\ell _{\\rm max} = 0$ .", "The energies $E(1/\\ell _{\\rm max})$ of the $2^{+}$ -state of $^{6}$ Be become $2.483-i0.474$ and $2.464 - i0.481$ (MeV) for GSM+CS and GSM, respectively.", "The difference becomes smaller than that of the $\\ell _{\\rm max} = 5$ case.", "Hence, we can conclude that both methods provide a sufficient accuracy even for the calculation of the broad resonant states." ], [ "Summary", "GSM and GEM+CS are two different theoretical approaches which allow to describe unbound resonant states.", "These two approaches differ in the choice of the basis functions and the numerical procedure to obtain the eigenvalues.", "To benchmark GSM and GEM+CS approaches, we have performed a precise comparison for weakly bound and unbound states using the same Hamiltonian in the COSM coordinates preserving the translational invariance.", "For a weakly bound ground-state of $^{6}$ He, GSM and GEM+CS give essentially identical results.", "For the three-body resonance states, GEM+CS and GSM give very close results proving the reliability of both schemes of the calculation for unbound states.", "The slight difference between GSM and GEM+CS results for broad resonances may have different origins.", "The HO expansion procedure in calculating the TBMEs in GSM may lead to rounding errors, especially for broad many-body resonances.", "On the other hand, the stationarity condition in GEM+CS approach could also be a source of small imprecision for broad resonances.", "Based on our results, we conclude that both approaches are essentially equivalent for all quantities studied.", "The other work for a comparison in the $^{6}$ He system has been done and also shows a good agreement between two different approaches [44].", "We would like to thank W. Nazarewicz and members of the nuclear theory group at Hokkaido University for fruitful discussions.", "This work was supported by the Grant-in-Aid for Scientific Research (No.", "21740154) from the Japan Society for the Promotion of Science, and FUSTIPEN (French-U.S.", "Theory Institute for Physics with Exotic Nuclei) under DOE grant number DE-FG02-10ER41700." ] ]
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